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+Economic Predictive Control with Periodic
+Horizon for Water Distribution Networks
+Mirhan ¨Urkmez ∗ Carsten Kallesøe ∗ Jan Dimon Bendtsen ∗
+John Leth ∗
+∗ Aalborg University, Fredrik Bajers Vej 7c, DK-9220 Aalborg,
+Denmark
+(e-mail: {mu,csk,dimon,jjl}@es.aau.dk)
+Abstract: This paper deals with the control of pumps in large-scale water distribution networks
+with the aim of minimizing economic costs while satisfying operational constraints. Finding a
+control algorithm in combination with a model that can be applied in real-time is a challenging
+problem due to the nonlinearities presented by the pipes and the network sizes. We propose
+a predictive control algorithm with a periodic horizon. The method provides a way for the
+economic operation of large water networks with a small linear model. Economic Predictive
+control with a periodic horizon and a terminal state constraint is constructed to keep the state
+trajectories close to an optimal periodic trajectory. Barrier terms are also included in the cost
+function to prevent constraint violations. The proposed method is tested on the EPANET
+implementation of the water network of a medium size Danish town (Randers) and shown to
+perform as intended under varying conditions.
+Keywords: Water distribution networks, Pump Scheduling, Predictive control, Periodic
+horizon, Economic model predictive control
+1. INTRODUCTION
+Water distribution networks (WDNs) deliver drinkable
+water from water sources to consumers using elements
+such as pumps, pipes, tanks etc. About 7% − 8% of the
+world’s energy is used for water production and distribu-
+tion (Sharif et al., 2019). Water pumps account for a sig-
+nificant part of the energy required for water distribution
+with their percentage ranging from 90% to 95% of the
+total (Abdelsalam and Gabbar, 2021). There have been
+many works to schedule the operation of the pumps in
+WDNs with proper methods so as to reduce energy costs.
+However, pump scheduling is not an easy task because
+of the nonlinearities governing the network elements. The
+problem gets complicated with increasing network size.
+Also, there are constraints to be satisfied such as limits
+on tank levels.
+In the literature, WDNs with both constant and variable
+speed pumps are studied extensively. The control input is
+turning on and off the pump for constant speed pumps. In
+Lindell Ormsbee (2009), a constant-speed pump schedul-
+ing problem is posed as an optimization problem in which
+the decision variables are the operation times of the pumps
+and the objective is the energy cost. After observing that
+the optimal solution would be not running the pumps at
+all without the constraints, the authors try to find the
+solution closest to the origin that also complies with the
+constraints. The proposed way to find such a solution is
+⋆ This work is funded by Independent Research Fund Denmark
+(DFF). We acknowledge Verdo company, Peter Nordahn, and Steffen
+Schmidt for providing us with the EPANET model and the network
+information.
+using a Genetic Algorithm (GA). In Bagirov et al. (2013),
+the Hooke-Jeeves method is used for finding optimal pump
+operating times for a similar problem. Then, network sim-
+ulation algorithms are used to check if the constraints are
+satisfied. In Castro-Gama et al. (2017), binary decision
+variables are used to represent the opening and the closing
+of each pump. The feasibility of the solution found with
+GA is checked with EPANET, a WDN modeling software,
+and a high cost is assigned to the infeasible solutions.
+The number of open pumps is also taken as the input
+to the system in some works, e.g., Wang et al. (2021).
+The problem is then solved using mixed-integer nonlinear
+programming. In Berkel et al. (2018), a network in which
+pressure zones are connected via constant speed pumps
+is considered. Each pressure zone is treated as a subsys-
+tem and distributed model predictive control (DMPC) is
+applied.
+The flow rate of the pumps should be determined for
+networks with variable speed pumps. In Pour et al. (2019),
+Linear Parameter Varying (LPV) system modeling is used
+to replace the nonlinear part of the network, and an
+Economic Model Predictive Control (EMPC) is applied
+on top of the LPV system to find the optimal flow rates.
+In Kallesøe et al. (2017), a network structure with an
+elevated reservoir is considered. Available data is used
+for the identification of a reduced system model. Then,
+EMPC is applied to the model. In the EMPC formulation,
+node pressures are not constrained. It is assumed that the
+pressures would be in the accepted range because there is
+an elevated reservoir. Relaxation of the original problem
+into a simpler one is commonly used because of the large
+network sizes. The relaxation is generally achieved by
+arXiv:2301.13598v1  [eess.SY]  31 Jan 2023
+
+approximating the nonlinear pipe equations with some
+sort of linear equations or inequalities. In Baunsgaard
+et al. (2016), pipe equations are linearized around an
+operating point, and model predictive control (MPC) is
+applied. In Wang et al. (2018), an EMPC is applied to a
+network where the nonlinear pipe equations are relaxed
+into a set of linear inequalities. Before simplifying the
+system model, the network structure is also simplified in
+Fiedler et al. (2020). A hierarchical clustering method is
+used to represent the original network with a smaller one
+which originally had 378 junctions. A system model is
+derived from the simplified structure using a Deep Neural
+Network (DNN) structure. Lagrangian relaxation is used
+to approximate the original problem in Ghaddar et al.
+(2015).
+In this paper, a way for optimal pump scheduling of large-
+scale WDNs is presented. To control the pumps, a linear
+model of the system is derived. Then, a predictive control
+method with a periodic horizon is constructed. Barrier
+functions are utilized to prevent constraint violation due
+to the model-plant mismatch. With the introduction of
+the periodic horizon and the terminal state constraint,
+the chance of finding a feasible solution is increased by
+keeping trajectories close to an optimal periodic trajectory.
+The method is applied to a medium-sized Danish town’s
+network (Randers).
+The outline of the rest of the paper is as follows. The
+network model is given in Section 2. The proposed control
+method is explained in Section 3. The experimental results
+are presented in Section 4. The paper is concluded with
+final remarks in Section 5.
+2. NETWORK MODEL
+A typical water distribution network consists of pipes,
+pumps, tanks, junction nodes and reservoirs. Water in
+the network flows from high hydraulic head to low head.
+Hydraulic head is a measure of the fluid pressure and is
+equal to the height of a fluid in a static column at a point.
+Hydraulic head loss occurring in a pipe can be approxi-
+mated by the Hazen-Williams Equation as
+∆h = h1 − h2 = Kq|q|0.852
+(1)
+where K is the pipe resistance that depends on the physical
+features of a pipe such as diameter and length, q is the flow
+rate, and h1 and h2 are the heads at the two ends of the
+pipe.
+At each node j, the mass conservation law is satisfied. It
+can be expressed as
+�
+i∈Nj
+qij = dj
+(2)
+where qij is the flow entering the node j from node i and
+dj is the demand at node j, which is the water requested
+by the user at node j. The symbol Nj denotes the set of
+neighbor nodes of node j. Note that qij is positive if the
+flow is from node i to the neighbor node j and negative
+vice versa.
+Tanks are storage elements that provide water to the users.
+In the network, tanks are elevated so that water can be
+pressurized enough to be delivered to the consumers. The
+change in the water level of a tank is dependent on the
+flow coming from neighbor nodes and can be written for
+the tank j as
+Aj ˙hj =
+�
+i∈Nj
+qij
+(3)
+where Aj is the cross-sectional area, hj is the level of the
+tank. Tank levels change according to the flow passing
+through the pipes connected to the tanks. Those flows
+are determined by a set of pipe head loss equations (1),
+and mass balance equations (2) throughout the whole
+network. As Equation (1) is nonlinear, flow through pipes
+connected to the tanks are nonlinear functions fi of the
+demand at each node, tank levels, and the amount of water
+coming from the pumps. Explicit forms of those nonlinear
+functions could be derived if the vector d = [d1, d2...]T
+containing the demands of all the nodes is known, which is
+not possible unless demand data for all nodes are available.
+In our work, we assume that the total demand of the zones
+that are supplied by the pumps can be estimated through
+available data with time series analysis methods, but not
+require d vector to be known. Since fi functions can not be
+found without d vector, we approximate them using linear
+models and write tank level change equations as
+˙h(t) = Ah(t) + B1u(t) + B2da(t)
+(4)
+where h(t) ∈ Rn includes tank levels, A ∈ Rn×n, B1 ∈
+Rn×m, B2 ∈ Rn×1 are constant system matrices and da(t)
+is the aggregated demand of controlled zone at time t,
+u(t) ∈ Rm is the input containing pump flows. The reason
+we chose a linear model is to increase the chance of finding
+a feasible solution for the controller which is posed as
+an optimization problem in the next section. Although
+capturing the full dynamics of a large-scale network is not
+possible with a linear model, the proposed control method
+is designed to compensate for model inaccuracies and we
+have observed that it was enough to control the system
+while satisfying the constraints.
+3. PERIODIC HORIZON CONTROL
+In this section, a predictive control algorithm for pump
+scheduling is presented to minimize the economical costs.
+The aim is to run the pumps when the electricity price is
+low and let tanks deliver water when the price is high while
+also satisfying input and output constraints. The problem
+at time t is posed as
+min
+ut
+0,ut
+1···ut
+N(t)−1
+N(t)−1
+�
+j=0
+J(ht
+j, ut
+j)
+(5a)
+ht
+j = Adht
+j−1 + Bd1ut
+j−1 + Bd2da(j − 1)
+(5b)
+ht
+0 = h(t)
+(5c)
+ut
+j ∈ U ⊆ Rm
+(5d)
+ht
+j ∈ H ⊆ Rn
+(5e)
+ht
+N(t) ∈ Htf ⊆ Rn
+(5f)
+where J(ht
+j, ut
+j) is the economic cost function, ht
+=
+[ht
+1 · · · ht
+N(t)] ∈ Rn×N(t) is the predicted future states, ut
+j is
+the input vector, N(t) is the prediction horizon, U ⊆ Rm
+and H ⊆ Rn denotes the input and state constraints
+respectively and Htf ⊆ Rn is the terminal state set. The
+continuous system (4) is discretized and (5b) is obtained.
+
+The optimization problem (5) is solved at every time step
+separated by ∆t and the first term ut
+0 of the optimal input
+sequence ut = [ut
+0 · · · ut
+N(t)−1] ∈ Rm×N(t) is applied to the
+system.
+Input constraints come from the physical limitations and
+working principles of the pumps. A pump can not provide
+water in the opposite direction and it can deliver a maxi-
+mum amount of water per unit of time. These conditions
+are expressed as
+U = {[u1, · · · um] ∈ Rm | 0 ≤ u1 ≤ u1, · · · 0 ≤ um ≤ um}
+(6)
+where u1 · · · um are upper flow limits. Tank levels are also
+constrained so that there is always enough water in the
+tanks in case of an emergency and there is no overflow of
+water. The set H can be defined as
+H = {[h1, · · · h2] ∈ Rn | ˜h1 ≤ h1 ≤ h1, · · · ˜hn ≤ hn ≤ hn}
+(7)
+The cost function includes the electricity costs of the
+pumps. The power provided to the network by the pump
+i is equal to qpi(pout
+i
+− pin
+i ), where qpi is the pump flow,
+pi
+out and pi
+in are the outlet and inlet pressures of the pump
+i. The inlet pressures pin = [pin
+1 , pin
+2 ] are the pressures of
+the related reservoirs and are assumed to be constant. The
+outlet pressures pout = [pout
+1
+, pout
+2
+] are given as the output
+of the linear model
+pout(t) = Aph(t) + Bpu(t)
+(8)
+where Ap and Bp are found using system identification
+on data generated by the EPANET model. Electricity
+cost at time t is then found by multiplying total power
+consumption u(t)T (pout(t) − pin(t)) with the electricity
+price c(t).
+We acknowledge a certain degree of model-plant mismatch
+by using a linear model (4) to represent the whole network.
+This causes actual states h(t) to be different than the
+predicted states ht. We know that the predicted states
+satisfy the state constraints (7) since they are the solution
+to the optimization problem 5, but the actual states
+might violate them. To ensure the satisfaction of the state
+constraints with the model-plant mismatch, we introduce
+new terms to the cost function. First, we rewrite state
+constraints (7) as
+Ci(h) ≤ 0,
+i = 0, 1, · · · 2 × n − 1
+(9)
+where C0(h) = ˜h1 −h1 and the rest of the Ci functions are
+chosen in a similar manner. The cost function terms are
+then defined as
+Jhi(h) = eai(Ci(h)+bi)
+i = 0, 1, · · · 2 × n − 1
+(10)
+where ai, bi ∈ R>0. This can be seen as an exponential
+barrier function. The parameters ai, bi determine a danger-
+ous region close to the boundaries of the state constraints
+where cost function Jhi attains high values. The predicted
+optimal state trajectories ht do not enter the dangerous
+region if possible because of the high cost values in the
+dangerous region. Then, the actual states h(t) do not
+violate the state constraints (7) assuming the difference
+between the predicted state and the actual state is small.
+If the state trajectory enters one of the dangerous regions
+at any step due to the model-plant mismatch, then the
+cost function will try to drive the trajectory out of the
+region.
+∆t
+N(t)∆t
+N(t + ∆t)∆t
+h(t)
+h(t + ∆t)
+ht
+1
+ut
+0
+ut+∆t
+0
+ht+∆t
+ht
+Br(h∗
+Tday/∆t)
+Fig. 1. Predicted state trajectories ht, ht+∆t at times
+t, t + ∆t. Sampling time ∆t, prediction horizons
+N(t), N(t + ∆t) and the applied inputs ut
+0, ut+∆t
+0
+are
+shown. The true state h(t + ∆t) and the predicted
+state ht
+1 are indicated to emphasize the deviation from
+the prediction. The terminal set Br(h∗
+Tday/∆t) is also
+illustrated.
+The overall cost function includes both the electricity
+expense term and the constraint barrier functions and it
+can be expressed as
+J(h(t), u(t)) = c(t)u(t)T (pout(t)−pin(t))+
+2×n−1
+�
+i=0
+Jhi(h(t))
+(11)
+Both electricity price c(t) and total water demand da(t)
+signals can be viewed as consisting of a periodic signal
+with a period of 1 day and a relatively small deviation
+signal. This can be leveraged to find a feasible controller.
+Suppose a sequence of inputs can be found for some
+initial tank levels such that levels after 1 day are equal
+to initial levels. In that case, the problem after 1 day is
+the same as in the beginning assuming deviation signals
+of the electricity price and the demand are zero, hence
+they are periodic. Then, the input sequence from the
+previous day could be applied and produce the same
+path for tank levels. Taking into account the deviation
+signals and supposing that a solution exists such that
+levels after 1 day are close to initial levels, the input
+sequence from the previous day could be a good point of
+start to search for a feasible solution if the map from the
+initial conditions and the demand profile to the optimal
+input sequences is continuous. Therefore, we choose a
+terminal state constraint for the end of each day to
+increase the chance of finding a feasible solution. Now, the
+remaining problem is to decide which tank levels should
+the trajectories turn back to at the end of each day. We
+define the optimal periodic trajectory of the system as the
+solution of
+(u∗, h∗) = arg min
+ui,hi
+(Tday/∆t)−1
+�
+i=0
+J(hi, ui)
+(12a)
+hi = Adhi−1 + Bd1ui−1 + Bd2d∗
+a(i − 1)
+(12b)
+ui ∈ U ⊆ Rm
+(12c)
+hi ∈ H ⊆ Rn
+(12d)
+h0 = hTday/∆t
+(12e)
+where Tday is the duration of a whole day, d∗
+a is the
+average daily demand profile obtained from the past
+measurements. The resulting state trajectory h∗
+=
+[h∗
+0 · · · h∗
+Tday/∆t] ∈ Rn×(Tday/∆t+1) is the optimal periodic
+trajectory because of the constraint (12e). The terminal
+set Htf and the prediction horizon N(t) is chosen to make
+tank levels at the end of each day close to h∗
+Tday/∆t. At
+
+High Zone
+Low Zone
+Fig. 2. Water Distribution Network of Randers. The pump-
+ing stations to be controlled are shown in red. Tanks
+are shown with a ’T’ shaped symbol in yellow.
+any time t, t + N(t)∆t should be equal to the end of the
+day. Htf and N(t) could be written as
+Htf = Br(h∗
+Tday/∆t)
+(13a)
+N(t) = (Tday − t mod Tday)/∆t
+(13b)
+where Br(h∗
+Tday/∆t) is the open ball centered at h∗
+Tday/∆t
+with radius r. Note that N(t) changes so that t + N(t)∆t
+is the end of the day for all t. With these definitions, the
+condition (13a) will translate to tank levels at the end of
+the day being close to the final point in optimal periodic
+trajectory h∗
+Tday/∆t as shown in Figure 1. Therefore, not
+only chance of finding a feasible solution is increased but
+also the solutions are kept around the optimal periodic
+trajectory h∗. If the problem (5) becomes infeasible at
+any time step t, we apply the second term of the input
+sequence from the previous step ut−∆t
+1
+. The reason behind
+this choice is as follows: If we apply the optimal control
+input ut−∆t
+0
+to the network model (4) at time t − ∆t,
+then the optimal sequence in the next time step will be
+ut = [ut−∆t
+1
+· · · ut−∆t
+N(t−∆t)−1] following Bellman’s principle
+of optimality. Then, at time t, ut−∆t
+1
+will be applied to
+the system as calculated at t − ∆t. Assuming the model-
+plant mismatch is small enough, ut−∆t
+1
+is still a good input
+candidate if the problem is infeasible at time t.
+4. APPLICATION
+The presented method is applied to WDN of Randers, a
+Danish city, which is shown in Figure 2. The network con-
+sists of 4549 nodes and 4905 links connecting them. There
+are 8 pumping stations in the network, 6 of which are
+shown in the figure whereas the other 2 are stationed where
+tanks are placed. The goal is to derive the schedules for
+2 of the pumping stations while other pumps are already
+working according to some predetermined strategies. The
+stations to be controlled are shown in red in the figure.
+Their task is to deliver water mostly to the High Zone (HZ)
+and Low Zone (LZ). However, connections exist between
+HZ-LZ and the rest of the city, so we can not think of the
+system as composed of isolated networks entirely. There
+are also 3 tanks in the HZ. While 2 of them are directly
+connected via pipes, the third one stands alone as shown
+in the figure.
+The overall structure of the Randers WDN with tanks and
+pumps to be controlled are given in Figure 3. There are 3
+water tanks in the network, 2 of which have been connected
+Fig. 3. Structure of the WDN.
+with a pipe directly. The tank level changes can be written
+by applying the mass conservation law (3) to the tanks in
+Figure 3 as
+A1 ˙h1 = q1down + q1up + qinter
+(14a)
+= f1(h1, h2, h3, qp1, qp2, d),
+A2 ˙h2 = q2down + q2up − qinter
+(14b)
+= f2(h1, h2, h3, qp1, qp2, d),
+A3 ˙h3 = q3 = f3(h1, h2, h3, qp1, qp2, d),
+(14c)
+where d is the vector containing the demands of all the
+nodes, qp1, qp2 are the pump flows, A1, A2, A3 are the cross
+sectional areas of the tanks and f1, f2, f3 are nonlinear
+flow functions. Water levels at the two connected tanks are
+almost equal h1 ≈ h2 all the time since the pipe connecting
+respective tanks is big enough to oppose the water flows
+coming from neighbor nodes. That enables us to consider
+h1, h2 together as
+(A1 + A2)˙h1,2 ≈ q1down + q2down + qup = f1 + f2.
+(15)
+We have used the EPANET model of the network to
+generate the data required for approximating f1 + f2
+and f3. The model is simulated with various tank level
+initial conditions and flow rates of 2 pumping stations
+to be controlled. The control laws for the remaining
+pumping stations are already defined in the EPANET
+model. Then, the linear model (4) is fitted to simulation
+data using least squares. The state variables for the model
+are h(t) = [h1,2(t), h3(t)] ∈ R2 and the inputs are u(t) =
+[qp1(t), qp2(t)] ∈ R2. The total demand of High and Low
+Zone is used as aggregated demand da in the model since
+mainly those areas are supplied by the controlled pumps.
+4.1 Simulation Results
+The proposed control method is tested on EPANET model
+of Randers water network. Epanet-Matlab toolkit Eliades
+et al. (2016) is used to set the flow of the 2 pumps at
+each time step and simulate the network. The remaining
+pumps are controlled with rule-based control laws that are
+previously defined on EPANET.
+The parameters of exponential barrier functions Jhi are
+chosen as ai = 80, bi = 0.3 for all i. It is assumed
+that the electricity prices are known in advance during
+the test. Tank levels h1, h2 have a maximum value of 3m
+while h3 has 2.8m. Tanks are required to be at least half
+full. Maximum pump flows are set to 100. Sampling time
+∆t is set to 1 hour in the experiments, so the control
+input is recalculated at each hour. We assume that total
+demand da(t) of HZ and LZ can be estimated up to 1 day
+from available data. Although we do not have historical
+
+qup
+qiup
+q2up
+h1
+h2
+h3
+qinter
+q1down
+2down
+q3
+Pump 1
+Pump 2
+9p1
+qp2data on the demand, we imitate this behaviour by using
+a slightly perturbed version of the real demand used
+in EPANET simulation during MPC calculations. The
+perturbations are adapted from a real demand data set of
+a small Danish facility. Normalized difference between the
+average demand and the demand of a random day in data
+set is added to EPANET demand to replicate estimated
+demand. In each experiment a different day from the data
+set is used, so the assumed estimated demand is different
+each time.
+The simulation results when the presented method is
+applied to the EPANET model are given in Figure 4. The
+initial tank levels are equal to h∗
+Tday/∆t in the simulation.
+The top plot shows the evolution of tank levels along
+with the upper and lower thresholds. It is seen that the
+thresholds are not violated and moreover tank levels are
+not getting too close to them, which was the idea behind
+exponential barrier functions. Both the real demand and
+the assumed estimated demand of HZ and LZ are in the
+figure below. Total applied pump flows and electricity
+prices are in the following figures. The expected result is
+pump flows being higher when electricity prices are low,
+and lower when they are high, which seems to be the case
+as can be seen in the plot. Pump flows drop significantly
+when prices are at the peak and they reach their highest
+value at the end of the day when prices are low. A more
+aggressive controller can be obtained by picking a smaller
+bi value for barrier functions at the expense of risking
+constraint violation. In Figure 5, the tank level simulation
+results and control inputs for different initial conditions
+and different assumed estimated demands are given. The
+electricity price profile is the same as before. It is seen that
+the algorithm is able to control the network on various
+cases while satisfying the constraints. Regardless of initial
+tank levels, the pumping profiles have a similar profile:
+high pump flows close to midnight and in the middle of
+the day. The only exception is the bottom plot. In the
+beginning, prices are low but pump flows are not high.
+This can be attributed to water levels h1, h2 being close to
+the upper thresholds and water demand being low in the
+beginning.
+The assumption that the optimal input sequences U(t)
+would not diverge a lot from the one found in previous
+step U(t − ∆t) is the reason we apply ut−∆t
+1
+at time t if
+the problem (5) is infeasible at time t. This assumption is
+tested with initial conditions h1,2,3 = h∗
+Tday/∆t. In figure
+6, total pump flow [1, 1]T ut
+i, i = 0 · · · N(t) − 1 of the
+found optimal input sequences U(t), t = 0, ∆t · · · Tday−∆t,
+except when the problem were infeasible, are given. It can
+be seen that ut−∆t
+1
+is close to the ut
+0 for all t, which shows
+that our assumption is valid at least for this experiment.
+Finally, the ability of the algorithm to decrease economic
+costs is tested with various initial conditions. For each
+case, a demand follower pumping strategy is used as a
+benchmark. The flow of the 2 pumps is adjusted with trial
+and error for each demand follower such that the total flow
+of the 2 pumps is equal to water demand at each time step
+and tank levels satisfy the terminal constraint (13a). The
+demand follower is a natural candidate to be a benchmark
+method since providing as much water as demand is an
+intuitive idea and the constraints in (5) can be satisfied
+(a)
+(b)
+(c)
+(d)
+Fig. 4. Sample simulation. (a) evolution of tank levels
+through 1 day with upper and lower level thresholds;
+(b) real total demand of HZ and LZ used in EPANET
+simulation and the demand used in MPC calculations;
+(c) total flow provided by the 2 pumps; (d) electricity
+price.
+Proposed Method
+Demand Follower
+0.5967
+1
+0.5745
+1
+0.5826
+1
+0.5558
+1
+Table 1. Relative economic costs of the pro-
+posed method and demand follower strategy
+for various demand profiles
+with manual adjustments of pump flows. The economic
+costs are presented relatively in Table 1 As it is seen, the
+proposed algorithm saves between 40% and 45% of the
+cost with different demand profiles.
+5. CONCLUSION
+We have presented a predictive control algorithm with a
+periodic horizon for WDNs. The aim is to minimize the
+
+3.5
+h1
+h2
+upper threshold
+3
+h3
+3 upper threshold
+Tank Levels
+1.5
+1
+0
+5
+10
+15
+20
+25140
+120
+100
+80
+60
+40
+Real Demand
+Known Demand
+20
+0
+5
+10
+15
+20
+25200
+Total Pump Flow
+150
+100
+50
+0
+0
+5
+10
+15
+20
+251.2
+Price
+0.8
+lectricity
+0.6
+0.4
+E
+0.2
+0
+0
+5
+10
+15
+20
+25
+HoursFig. 5. Tank levels and pump flows for different initial
+conditions
+Fig. 6. Evolution of found input sequences U(t) through 1
+day. It can be seen that the solutions remain close to
+the initial optimal sequence U(0).
+economic cost and satisfy the operational constraints. A
+linear model is used to represent Randers WDN to increase
+the chance of finding a solution to the problem (5) at
+expense of a model-plant mismatch. Periodic horizon is
+introduced to the predictive control formulation to keep
+the resulting state trajectories around the optimal periodic
+trajectory. Barrier functions are used to prevent constraint
+violation since there is a model-plant mismatch.
+The presented algorithm is tested on Randers WDN using
+EPANET. It is shown in various situations that the
+method is able to find an economic solution where pump
+flows are adjusted according to electricity prices. Also,
+it is shown that the system trajectories do not enter
+dangerous zones introduced by barrier functions as long
+as the predicted demand and the actual demand are
+somewhat close.
+As future work, we plan to work on theoretical guarantees
+of the existence of solutions to the proposed method. Also,
+the robustness of periodic horizon control of periodical
+systems with barrier functions will be investigated.
+REFERENCES
+Abdelsalam, A.A. and Gabbar, H.A. (2021). Energy saving
+and management of water pumping networks. Heliyon,
+7(8), e07820. doi:https://doi.org/10.1016/j.heliyon.20
+21.e07820.
+Bagirov, A.M., Barton, A., Mala-Jetmarova, H., Nuaimat,
+A.A., Ahmed, S.T., Sultanova, N., and Yearwood, J.
+(2013). An algorithm for minimization of pumping costs
+in water distribution systems using a novel approach to
+pump scheduling. Math. Comput. Model., 57, 873–886.
+Baunsgaard, K.M.H., Ravn, O., Kallesøe, C.S., and
+Poulsen, N.K. (2016).
+Mpc control of water supply
+networks.
+2016 European Control Conference (ECC),
+1770–1775.
+Berkel, F., Caba, S., Bleich, J., and Liu, S. (2018).
+A
+modeling and distributed mpc approach for water dis-
+tribution networks. Control Engineering Practice.
+Castro-Gama, M.E., Pan, Q., Lanfranchi, E.A., Jonoski,
+A., and Solomatine, D.P. (2017). Pump scheduling for a
+large water distribution network. milan, italy. Procedia
+Engineering, 186, 436–443.
+Eliades, D.G., Kyriakou, M., Vrachimis, S., and Polycar-
+pou, M.M. (2016).
+Epanet-matlab toolkit: An open-
+source software for interfacing epanet with matlab. In
+Proc. 14th International Conference on Computing and
+Control for the Water Industry (CCWI), 8. The Nether-
+lands. doi:10.5281/zenodo.831493.
+Fiedler, F., Cominola, A., and Lucia, S. (2020).
+Eco-
+nomic nonlinear predictive control of water distribution
+networks based on surrogate modeling and automatic
+clustering. IFAC-PapersOnLine, 53, 16636–16643.
+Ghaddar, B., Naoum-Sawaya, J., Kishimoto, A., Taheri,
+N., and Eck, B. (2015).
+A lagrangian decomposition
+approach for the pump scheduling problem in water
+networks. Eur. J. Oper. Res., 241, 490–501.
+Kallesøe, C.S., Jensen, T.N., and Bendtsen, J.D. (2017).
+Plug-and-play model predictive control for water supply
+networks with storage. IFAC-PapersOnLine, 50, 6582–
+6587.
+Lindell Ormsbee, Srini Lingireddy, D.C. (2009). Optimal
+pump scheduling for water distribution systems. URL
+http://www.uky.edu/WDST/PDFs/[73.3]%20Ormsbee%
+20Optimal%20Pump%20Scheduling%20Paper.pdf.
+Pour, F.K., Puig, V., and Cembra˜no, G. (2019). Economic
+mpc-lpv control for the operational management of
+water distribution networks. IFAC-PapersOnLine.
+Sharif, N., Haider, H., Farahat, A., Hewage, K., and Sadiq,
+R. (2019). Water energy nexus for water distribution
+systems: A literature review. Environmental Reviews,
+27. doi:10.1139/er-2018-0106.
+Wang, Y., Alamo, T., Puig, V., and Cembra˜no, G. (2018).
+Economic model predictive control with nonlinear con-
+straint relaxation for the operational management of
+water distribution networks. Energies, 11, 991.
+Wang, Y., Yok, K.T., Wu, W., Simpson, A.R., Weyer, E.,
+and Manzie, C. (2021).
+Minimizing pumping energy
+cost in real-time operations of water distribution sys-
+tems using economic model predictive control. ArXiv,
+abs/2010.07477.
+
+200
+Total Pump Flow
+150
+100
+50
+0
+0
+5
+10
+15
+20
+253.5
+h1
+h2
+upperthreshold
+h3
+3 upper threshold
+Tank Levels
+2.5
+1.5
+1
+0
+5
+10
+15
+20
+25250
+Total Pump Flow
+200
+150
+100
+50
+0
+0
+5
+10
+15
+20
+253.5
+h1
+h2
+upper threshold
+3
+h3
+ upper threshold
+Tank Levels
+1.5
+1
+0
+5
+10
+15
+20
+25200
+Total Pump Flow
+150
+100
+50
+0
+0
+5
+10
+15
+20
+25200
+U(0)
+150
+Flow
+100
+U(1)
+50
+0
+0
+5
+10
+15
+20
+25
+Hours3.5
+h1
+h2
+upperthreshold
+3
+h3
+3 upper threshold
+Tank Levels
+2.5
+1.5
+1
+0
+5
+10
+15
+20
+25250
+Total Pump Flow
+200
+150
+100
+50
+0
+0
+5
+10
+15
+20
+253.5
+h1
+h2
+upperthreshold
+3
+h3
+3 upper threshold
+Tank Levels
+1.5
+1
+0
+5
+10
+15
+20
+25

09FRT4oBgHgl3EQflTce/content/tmp_files/load_file.txt

@@ -0,0 +1,420 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf,len=419
+page_content='Economic Predictive Control with Periodic  Horizon for Water Distribution Networks  Mirhan ¨Urkmez ∗ Carsten Kallesøe ∗ Jan Dimon Bendtsen ∗  John Leth ∗  ∗ Aalborg University, Fredrik Bajers Vej 7c, DK-9220 Aalborg,  Denmark  (e-mail: {mu,csk,dimon,jjl}@es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='aau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='dk)  Abstract: This paper deals with the control of pumps in large-scale water distribution networks  with the aim of minimizing economic costs while satisfying operational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Finding a  control algorithm in combination with a model that can be applied in real-time is a challenging  problem due to the nonlinearities presented by the pipes and the network sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' We propose  a predictive control algorithm with a periodic horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The method provides a way for the  economic operation of large water networks with a small linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Economic Predictive  control with a periodic horizon and a terminal state constraint is constructed to keep the state  trajectories close to an optimal periodic trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Barrier terms are also included in the cost  function to prevent constraint violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The proposed method is tested on the EPANET  implementation of the water network of a medium size Danish town (Randers) and shown to  perform as intended under varying conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  Keywords: Water distribution networks, Pump Scheduling, Predictive control, Periodic  horizon, Economic model predictive control  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' INTRODUCTION  Water distribution networks (WDNs) deliver drinkable  water from water sources to consumers using elements  such as pumps, pipes, tanks etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' About 7% − 8% of the  world’s energy is used for water production and distribu-  tion (Sharif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Water pumps account for a sig-  nificant part of the energy required for water distribution  with their percentage ranging from 90% to 95% of the  total (Abdelsalam and Gabbar, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' There have been  many works to schedule the operation of the pumps in  WDNs with proper methods so as to reduce energy costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  However, pump scheduling is not an easy task because  of the nonlinearities governing the network elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The  problem gets complicated with increasing network size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  Also, there are constraints to be satisfied such as limits  on tank levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  In the literature, WDNs with both constant and variable  speed pumps are studied extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The control input is  turning on and off the pump for constant speed pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' In  Lindell Ormsbee (2009), a constant-speed pump schedul-  ing problem is posed as an optimization problem in which  the decision variables are the operation times of the pumps  and the objective is the energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' After observing that  the optimal solution would be not running the pumps at  all without the constraints, the authors try to find the  solution closest to the origin that also complies with the  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The proposed way to find such a solution is  ⋆ This work is funded by Independent Research Fund Denmark  (DFF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' We acknowledge Verdo company, Peter Nordahn, and Steffen  Schmidt for providing us with the EPANET model and the network  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  using a Genetic Algorithm (GA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' In Bagirov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' (2013),  the Hooke-Jeeves method is used for finding optimal pump  operating times for a similar problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Then, network sim-  ulation algorithms are used to check if the constraints are  satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' In Castro-Gama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' (2017), binary decision  variables are used to represent the opening and the closing  of each pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The feasibility of the solution found with  GA is checked with EPANET, a WDN modeling software,  and a high cost is assigned to the infeasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  The number of open pumps is also taken as the input  to the system in some works, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=', Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  The problem is then solved using mixed-integer nonlinear  programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' In Berkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' (2018), a network in which  pressure zones are connected via constant speed pumps  is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Each pressure zone is treated as a subsys-  tem and distributed model predictive control (DMPC) is  applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  The flow rate of the pumps should be determined for  networks with variable speed pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' In Pour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' (2019),  Linear Parameter Varying (LPV) system modeling is used  to replace the nonlinear part of the network, and an  Economic Model Predictive Control (EMPC) is applied  on top of the LPV system to find the optimal flow rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  In Kallesøe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' (2017), a network structure with an  elevated reservoir is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Available data is used  for the identification of a reduced system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Then,  EMPC is applied to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' In the EMPC formulation,  node pressures are not constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' It is assumed that the  pressures would be in the accepted range because there is  an elevated reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Relaxation of the original problem  into a simpler one is commonly used because of the large  network sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The relaxation is generally achieved by  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='13598v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='SY]  31 Jan 2023  approximating the nonlinear pipe equations with some  sort of linear equations or inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' In Baunsgaard  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' (2016), pipe equations are linearized around an  operating point, and model predictive control (MPC) is  applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' In Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' (2018), an EMPC is applied to a  network where the nonlinear pipe equations are relaxed  into a set of linear inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Before simplifying the  system model, the network structure is also simplified in  Fiedler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' A hierarchical clustering method is  used to represent the original network with a smaller one  which originally had 378 junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' A system model is  derived from the simplified structure using a Deep Neural  Network (DNN) structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Lagrangian relaxation is used  to approximate the original problem in Ghaddar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  In this paper, a way for optimal pump scheduling of large-  scale WDNs is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' To control the pumps, a linear  model of the system is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Then, a predictive control  method with a periodic horizon is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Barrier  functions are utilized to prevent constraint violation due  to the model-plant mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' With the introduction of  the periodic horizon and the terminal state constraint,  the chance of finding a feasible solution is increased by  keeping trajectories close to an optimal periodic trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  The method is applied to a medium-sized Danish town’s  network (Randers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  The outline of the rest of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The  network model is given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The proposed control  method is explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The experimental results  are presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The paper is concluded with  final remarks in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' NETWORK MODEL  A typical water distribution network consists of pipes,  pumps, tanks, junction nodes and reservoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Water in  the network flows from high hydraulic head to low head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  Hydraulic head is a measure of the fluid pressure and is  equal to the height of a fluid in a static column at a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  Hydraulic head loss occurring in a pipe can be approxi-  mated by the Hazen-Williams Equation as  ∆h = h1 − h2 = Kq|q|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='852  (1)  where K is the pipe resistance that depends on the physical  features of a pipe such as diameter and length, q is the flow  rate, and h1 and h2 are the heads at the two ends of the  pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  At each node j, the mass conservation law is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' It  can be expressed as  �  i∈Nj  qij = dj  (2)  where qij is the flow entering the node j from node i and  dj is the demand at node j, which is the water requested  by the user at node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The symbol Nj denotes the set of  neighbor nodes of node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Note that qij is positive if the  flow is from node i to the neighbor node j and negative  vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  Tanks are storage elements that provide water to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  In the network, tanks are elevated so that water can be  pressurized enough to be delivered to the consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The  change in the water level of a tank is dependent on the  flow coming from neighbor nodes and can be written for  the tank j as  Aj ˙hj =  �  i∈Nj  qij  (3)  where Aj is the cross-sectional area, hj is the level of the  tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Tank levels change according to the flow passing  through the pipes connected to the tanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Those flows  are determined by a set of pipe head loss equations (1),  and mass balance equations (2) throughout the whole  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' As Equation (1) is nonlinear, flow through pipes  connected to the tanks are nonlinear functions fi of the  demand at each node, tank levels, and the amount of water  coming from the pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Explicit forms of those nonlinear  functions could be derived if the vector d = [d1, d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=']T  containing the demands of all the nodes is known, which is  not possible unless demand data for all nodes are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  In our work, we assume that the total demand of the zones  that are supplied by the pumps can be estimated through  available data with time series analysis methods, but not  require d vector to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Since fi functions can not be  found without d vector, we approximate them using linear  models and write tank level change equations as  ˙h(t) = Ah(t) + B1u(t) + B2da(t)  (4)  where h(t) ∈ Rn includes tank levels, A ∈ Rn×n, B1 ∈  Rn×m, B2 ∈ Rn×1 are constant system matrices and da(t)  is the aggregated demand of controlled zone at time t,  u(t) ∈ Rm is the input containing pump flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The reason  we chose a linear model is to increase the chance of finding  a feasible solution for the controller which is posed as  an optimization problem in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Although  capturing the full dynamics of a large-scale network is not  possible with a linear model, the proposed control method  is designed to compensate for model inaccuracies and we  have observed that it was enough to control the system  while satisfying the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' PERIODIC HORIZON CONTROL  In this section, a predictive control algorithm for pump  scheduling is presented to minimize the economical costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  The aim is to run the pumps when the electricity price is  low and let tanks deliver water when the price is high while  also satisfying input and output constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The problem  at time t is posed as  min  ut  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='ut  1···ut  N(t)−1  N(t)−1  �  j=0  J(ht  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' ut  j)  (5a)  ht  j = Adht  j−1 + Bd1ut  j−1 + Bd2da(j − 1)  (5b)  ht  0 = h(t)  (5c)  ut  j ∈ U ⊆ Rm  (5d)  ht  j ∈ H ⊆ Rn  (5e)  ht  N(t) ∈ Htf ⊆ Rn  (5f)  where J(ht  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' ut  j) is the economic cost function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' ht  =  [ht  1 · · · ht  N(t)] ∈ Rn×N(t) is the predicted future states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' ut  j is  the input vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' N(t) is the prediction horizon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' U ⊆ Rm  and H ⊆ Rn denotes the input and state constraints  respectively and Htf ⊆ Rn is the terminal state set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The  continuous system (4) is discretized and (5b) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  The optimization problem (5) is solved at every time step  separated by ∆t and the first term ut  0 of the optimal input  sequence ut = [ut  0 · · · ut  N(t)−1] ∈ Rm×N(t) is applied to the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  Input constraints come from the physical limitations and  working principles of the pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' A pump can not provide  water in the opposite direction and it can deliver a maxi-  mum amount of water per unit of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' These conditions  are expressed as  U = {[u1, · · · um] ∈ Rm | 0 ≤ u1 ≤ u1, · · · 0 ≤ um ≤ um}  (6)  where u1 · · · um are upper flow limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Tank levels are also  constrained so that there is always enough water in the  tanks in case of an emergency and there is no overflow of  water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The set H can be defined as  H = {[h1, · · · h2] ∈ Rn | ˜h1 ≤ h1 ≤ h1, · · · ˜hn ≤ hn ≤ hn}  (7)  The cost function includes the electricity costs of the  pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The power provided to the network by the pump  i is equal to qpi(pout  i  − pin  i ), where qpi is the pump flow,  pi  out and pi  in are the outlet and inlet pressures of the pump  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The inlet pressures pin = [pin  1 , pin  2 ] are the pressures of  the related reservoirs and are assumed to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The  outlet pressures pout = [pout  1  , pout  2  ] are given as the output  of the linear model  pout(t) = Aph(t) + Bpu(t)  (8)  where Ap and Bp are found using system identification  on data generated by the EPANET model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Electricity  cost at time t is then found by multiplying total power  consumption u(t)T (pout(t) − pin(t)) with the electricity  price c(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  We acknowledge a certain degree of model-plant mismatch  by using a linear model (4) to represent the whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  This causes actual states h(t) to be different than the  predicted states ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' We know that the predicted states  satisfy the state constraints (7) since they are the solution  to the optimization problem 5, but the actual states  might violate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' To ensure the satisfaction of the state  constraints with the model-plant mismatch, we introduce  new terms to the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' First, we rewrite state  constraints (7) as  Ci(h) ≤ 0,  i = 0, 1, · · · 2 × n − 1  (9)  where C0(h) = ˜h1 −h1 and the rest of the Ci functions are  chosen in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The cost function terms are  then defined as  Jhi(h) = eai(Ci(h)+bi)  i = 0, 1, · · · 2 × n − 1  (10)  where ai, bi ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' This can be seen as an exponential  barrier function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The parameters ai, bi determine a danger-  ous region close to the boundaries of the state constraints  where cost function Jhi attains high values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The predicted  optimal state trajectories ht do not enter the dangerous  region if possible because of the high cost values in the  dangerous region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Then, the actual states h(t) do not  violate the state constraints (7) assuming the di��erence  between the predicted state and the actual state is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  If the state trajectory enters one of the dangerous regions  at any step due to the model-plant mismatch, then the  cost function will try to drive the trajectory out of the  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  ∆t  N(t)∆t  N(t + ∆t)∆t  h(t)  h(t + ∆t)  ht  1  ut  0  ut+∆t  0  ht+∆t  ht  Br(h∗  Tday/∆t)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Predicted state trajectories ht, ht+∆t at times  t, t + ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Sampling time ∆t, prediction horizons  N(t), N(t + ∆t) and the applied inputs ut  0, ut+∆t  0  are  shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The true state h(t + ∆t) and the predicted  state ht  1 are indicated to emphasize the deviation from  the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The terminal set Br(h∗  Tday/∆t) is also  illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  The overall cost function includes both the electricity  expense term and the constraint barrier functions and it  can be expressed as  J(h(t), u(t)) = c(t)u(t)T (pout(t)−pin(t))+  2×n−1  �  i=0  Jhi(h(t))  (11)  Both electricity price c(t) and total water demand da(t)  signals can be viewed as consisting of a periodic signal  with a period of 1 day and a relatively small deviation  signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' This can be leveraged to find a feasible controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  Suppose a sequence of inputs can be found for some  initial tank levels such that levels after 1 day are equal  to initial levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' In that case, the problem after 1 day is  the same as in the beginning assuming deviation signals  of the electricity price and the demand are zero, hence  they are periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Then, the input sequence from the  previous day could be applied and produce the same  path for tank levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Taking into account the deviation  signals and supposing that a solution exists such that  levels after 1 day are close to initial levels, the input  sequence from the previous day could be a good point of  start to search for a feasible solution if the map from the  initial conditions and the demand profile to the optimal  input sequences is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Therefore, we choose a  terminal state constraint for the end of each day to  increase the chance of finding a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Now, the  remaining problem is to decide which tank levels should  the trajectories turn back to at the end of each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' We  define the optimal periodic trajectory of the system as the  solution of  (u∗, h∗) = arg min  ui,hi  (Tday/∆t)−1  �  i=0  J(hi, ui)  (12a)  hi = Adhi−1 + Bd1ui−1 + Bd2d∗  a(i − 1)  (12b)  ui ∈ U ⊆ Rm  (12c)  hi ∈ H ⊆ Rn  (12d)  h0 = hTday/∆t  (12e)  where Tday is the duration of a whole day, d∗  a is the  average daily demand profile obtained from the past  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The resulting state trajectory h∗  =  [h∗  0 · · · h∗  Tday/∆t] ∈ Rn×(Tday/∆t+1) is the optimal periodic  trajectory because of the constraint (12e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The terminal  set Htf and the prediction horizon N(t) is chosen to make  tank levels at the end of each day close to h∗  Tday/∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' At  High Zone  Low Zone  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Water Distribution Network of Randers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The pump-  ing stations to be controlled are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Tanks  are shown with a ’T’ shaped symbol in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  any time t, t + N(t)∆t should be equal to the end of the  day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Htf and N(t) could be written as  Htf = Br(h∗  Tday/∆t)  (13a)  N(t) = (Tday − t mod Tday)/∆t  (13b)  where Br(h∗  Tday/∆t) is the open ball centered at h∗  Tday/∆t  with radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Note that N(t) changes so that t + N(t)∆t  is the end of the day for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' With these definitions, the  condition (13a) will translate to tank levels at the end of  the day being close to the final point in optimal periodic  trajectory h∗  Tday/∆t as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Therefore, not  only chance of finding a feasible solution is increased but  also the solutions are kept around the optimal periodic  trajectory h∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' If the problem (5) becomes infeasible at  any time step t, we apply the second term of the input  sequence from the previous step ut−∆t  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The reason behind  this choice is as follows: If we apply the optimal control  input ut−∆t  0  to the network model (4) at time t − ∆t,  then the optimal sequence in the next time step will be  ut = [ut−∆t  1  · · ut−∆t  N(t−∆t)−1] following Bellman’s principle  of optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Then, at time t, ut−∆t  1  will be applied to  the system as calculated at t − ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Assuming the model-  plant mismatch is small enough, ut−∆t  1  is still a good input  candidate if the problem is infeasible at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' APPLICATION  The presented method is applied to WDN of Randers, a  Danish city, which is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The network con-  sists of 4549 nodes and 4905 links connecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' There  are 8 pumping stations in the network, 6 of which are  shown in the figure whereas the other 2 are stationed where  tanks are placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The goal is to derive the schedules for  2 of the pumping stations while other pumps are already  working according to some predetermined strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The  stations to be controlled are shown in red in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  Their task is to deliver water mostly to the High Zone (HZ)  and Low Zone (LZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' However, connections exist between  HZ-LZ and the rest of the city, so we can not think of the  system as composed of isolated networks entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' There  are also 3 tanks in the HZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' While 2 of them are directly  connected via pipes, the third one stands alone as shown  in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  The overall structure of the Randers WDN with tanks and  pumps to be controlled are given in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' There are 3  water tanks in the network, 2 of which have been connected  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Structure of the WDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  with a pipe directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The tank level changes can be written  by applying the mass conservation law (3) to the tanks in  Figure 3 as  A1 ˙h1 = q1down + q1up + qinter  (14a)  = f1(h1, h2, h3, qp1, qp2, d),  A2 ˙h2 = q2down + q2up − qinter  (14b)  = f2(h1, h2, h3, qp1, qp2, d),  A3 ˙h3 = q3 = f3(h1, h2, h3, qp1, qp2, d),  (14c)  where d is the vector containing the demands of all the  nodes, qp1, qp2 are the pump flows, A1, A2, A3 are the cross  sectional areas of the tanks and f1, f2, f3 are nonlinear  flow functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Water levels at the two connected tanks are  almost equal h1 ≈ h2 all the time since the pipe connecting  respective tanks is big enough to oppose the water flows  coming from neighbor nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' That enables us to consider  h1, h2 together as  (A1 + A2)˙h1,2 ≈ q1down + q2down + qup = f1 + f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  (15)  We have used the EPANET model of the network to  generate the data required for approximating f1 + f2  and f3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The model is simulated with various tank level  initial conditions and flow rates of 2 pumping stations  to be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The control laws for the remaining  pumping stations are already defined in the EPANET  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Then, the linear model (4) is fitted to simulation  data using least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The state variables for the model  are h(t) = [h1,2(t), h3(t)] ∈ R2 and the inputs are u(t) =  [qp1(t), qp2(t)] ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The total demand of High and Low  Zone is used as aggregated demand da in the model since  mainly those areas are supplied by the controlled pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='1 Simulation Results  The proposed control method is tested on EPANET model  of Randers water network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Epanet-Matlab toolkit Eliades  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' (2016) is used to set the flow of the 2 pumps at  each time step and simulate the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The remaining  pumps are controlled with rule-based control laws that are  previously defined on EPANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  The parameters of exponential barrier functions Jhi are  chosen as ai = 80, bi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='3 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' It is assumed  that the electricity prices are known in advance during  the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Tank levels h1, h2 have a maximum value of 3m  while h3 has 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='8m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Tanks are required to be at least half  full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Maximum pump flows are set to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Sampling time  ∆t is set to 1 hour in the experiments, so the control  input is recalculated at each hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' We assume that total  demand da(t) of HZ and LZ can be estimated up to 1 day  from available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Although we do not have historical  qup  qiup  q2up  h1  h2  h3  qinter  q1down  2down  q3  Pump 1  Pump 2  9p1  qp2data on the demand, we imitate this behaviour by using  a slightly perturbed version of the real demand used  in EPANET simulation during MPC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The  perturbations are adapted from a real demand data set of  a small Danish facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Normalized difference between the  average demand and the demand of a random day in data  set is added to EPANET demand to replicate estimated  demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' In each experiment a different day from the data  set is used, so the assumed estimated demand is different  each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  The simulation results when the presented method is  applied to the EPANET model are given in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The  initial tank levels are equal to h∗  Tday/∆t in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  The top plot shows the evolution of tank levels along  with the upper and lower thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' It is seen that the  thresholds are not violated and moreover tank levels are  not getting too close to them, which was the idea behind  exponential barrier functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Both the real demand and  the assumed estimated demand of HZ and LZ are in the  figure below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Total applied pump flows and electricity  prices are in the following figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The expected result is  pump flows being higher when electricity prices are low,  and lower when they are high, which seems to be the case  as can be seen in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Pump flows drop significantly  when prices are at the peak and they reach their highest  value at the end of the day when prices are low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' A more  aggressive controller can be obtained by picking a smaller  bi value for barrier functions at the expense of risking  constraint violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' In Figure 5, the tank level simulation  results and control inputs for different initial conditions  and different assumed estimated demands are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The  electricity price profile is the same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' It is seen that  the algorithm is able to control the network on various  cases while satisfying the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Regardless of initial  tank levels, the pumping profiles have a similar profile:  high pump flows close to midnight and in the middle of  the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The only exception is the bottom plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' In the  beginning, prices are low but pump flows are not high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  This can be attributed to water levels h1, h2 being close to  the upper thresholds and water demand being low in the  beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  The assumption that the optimal input sequences U(t)  would not diverge a lot from the one found in previous  step U(t − ∆t) is the reason we apply ut−∆t  1  at time t if  the problem (5) is infeasible at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' This assumption is  tested with initial conditions h1,2,3 = h∗  Tday/∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' In figure  6, total pump flow [1, 1]T ut  i, i = 0 · · · N(t) − 1 of the  found optimal input sequences U(t), t = 0, ∆t · · · Tday−∆t,  except when the problem were infeasible, are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' It can  be seen that ut−∆t  1  is close to the ut  0 for all t, which shows  that our assumption is valid at least for this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  Finally, the ability of the algorithm to decrease economic  costs is tested with various initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' For each  case, a demand follower pumping strategy is used as a  benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The flow of the 2 pumps is adjusted with trial  and error for each demand follower such that the total flow  of the 2 pumps is equal to water demand at each time step  and tank levels satisfy the terminal constraint (13a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The  demand follower is a natural candidate to be a benchmark  method since providing as much water as demand is an  intuitive idea and the constraints in (5) can be satisfied  (a)  (b)  (c)  (d)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Sample simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' (a) evolution of tank levels  through 1 day with upper and lower level thresholds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  (b) real total demand of HZ and LZ used in EPANET  simulation and the demand used in MPC calculations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  (c) total flow provided by the 2 pumps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' (d) electricity  price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  Proposed Method  Demand Follower  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='5967  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='5745  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='5826  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='5558  1  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Relative economic costs of the pro-  posed method and demand follower strategy  for various demand profiles  with manual adjustments of pump flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The economic  costs are presented relatively in Table 1 As it is seen, the  proposed algorithm saves between 40% and 45% of the  cost with different demand profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' CONCLUSION  We have presented a predictive control algorithm with a  periodic horizon for WDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' The aim is to minimize the  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='5  h1  h2  upper threshold  3  h3  3 upper threshold  Tank Levels  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='5  1  0  5  10  15  20  25140  120  100  80  60  40  Real Demand  Known Demand  20  0  5  10  15  20  25200  Total Pump Flow  150  100  50  0  0  5  10  15  20  251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='2  Price  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='8  lectricity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='4  E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='2  0  0  5  10  15  20  25  HoursFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Tank levels and pump flows for different initial  conditions  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Evolution of found input sequences U(t) through 1  day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' It can be seen that the solutions remain close to  the initial optimal sequence U(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  economic cost and satisfy the operational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' A  linear model is used to represent Randers WDN to increase  the chance of finding a solution to the problem (5) at  expense of a model-plant mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Periodic horizon is  introduced to the predictive control formulation to keep  the resulting state trajectories around the optimal periodic  trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Barrier functions are used to prevent constraint  violation since there is a model-plant mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  The presented algorithm is tested on Randers WDN using  EPANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' It is shown in various situations that the  method is able to find an economic solution where pump  flows are adjusted according to electricity prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Also,  it is shown that the system trajectories do not enter  dangerous zones introduced by barrier functions as long  as the predicted demand and the actual demand are  somewhat close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  As future work, we plan to work on theoretical guarantees  of the existence of solutions to the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Also,  the robustness of periodic horizon control of periodical  systems with barrier functions will be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  REFERENCES  Abdelsalam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' and Gabbar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Energy saving  and management of water pumping networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Heliyon,  7(8), e07820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='heliyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='20  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='e07820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  Bagirov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
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+page_content=', Mala-Jetmarova, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=', Nuaimat,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
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+page_content='  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' An algorithm for minimization of pumping costs  in water distribution systems using a novel approach to  pump scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=', 57, 873–886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  Baunsgaard, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=', Ravn, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=', Kallesøe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=', and  Poulsen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
+page_content='  Mpc control of water supply  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'}
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+IMSc/2023/02
+Aspects of the map from Exact RG to Holographic RG in
+AdS and dS
+Pavan Dharanipragada ∗1, 2, Semanti Dutta †3, and B. Sathiapalan ‡1,2
+1Institute of Mathematical Sciences,CIT Campus, Tharamani, Chennai
+600113, India
+2Homi Bhabha National Institute, Training School Complex, Anushakti
+Nagar, Mumbai 400085, India
+3Centre for High Energy Physics, Indian Institute of Science, C.V. Raman
+Avenue, Bangalore 560012, India
+February 1, 2023
+Abstract
+In earlier work the evolution operator for the exact RG equation was mapped to a
+field theory in Euclidean AdS. This gives a simple way of understanding AdS/CFT. We
+explore aspects of this map by studying a simple example of a Schroedinger equation for
+a free particle with time dependent mass. This is an analytic continuation of an ERG
+like equation. We show for instance that it can be mapped to a harmonic oscillator. We
+show that the same techniques can lead to an understanding of dS/CFT too.
+Contents
+1
+Introduction
+3
+2
+Mapping Free Particle with Time Dependent Mass to a Harmonic Oscillator
+3
+2.1
+Mapping Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.1.1
+Lorentzian Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.1.2
+Euclidean Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2.2
+Mapping Schrodinger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.2.1
+Lorentzian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.2.2
+Euclidean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.2.3
+Analytic Continuation
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.3
+Semiclassical Treatment
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+2.3.1
+Using Harmonic Oscillator Formulation . . . . . . . . . . . . . . . . . . .
+8
+2.3.2
+Using ERG formulation
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+∗[email protected]
+†[email protected]
+‡[email protected]
+1
+arXiv:2301.13605v1  [hep-th]  31 Jan 2023
+
+3
+ERG to field theory in dS
+10
+3.1
+Analytic Continuation
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+3.1.1
+Analytic Continuation of the Action
+. . . . . . . . . . . . . . . . . . . .
+10
+3.2
+Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+3.2.1
+Mapping from Quantum Mechanics . . . . . . . . . . . . . . . . . . . . .
+11
+3.2.2
+Mapping from ERG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+3.3
+Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+3.4
+dS-CFT correspondence
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+4
+Obtaining Bulk field from ERG
+16
+5
+Summary and Conclusions
+20
+2
+
+1
+Introduction
+It has been recognized from the early days of the AdS/CFT correspondence [1, 2, 3, 4] that
+the radial coordinate of the AdS space behaves like a scale for the boundary field theory. This
+observation follows directly from the form of the AdS metric in Poincare coordinates:
+ds2 = R2dz2 + dxµdxµ
+z2
+(1.1)
+This leads naturally to the idea of the “Holographic” renormalization group: If the AdS/CFT
+conjecture is correct then radial evolution in the bulk must correspond to RG evolution in the
+boundary theory [[9]-[25]].
+In [5, 6, 7] a mathematically precise connection was made between the exact RG (ERG)
+equation of a boundary theory and holographic RG equations of a bulk theory in Euclidean
+AdS (EAdS) space. It was shown that the ERG evolution operator of the boundary theory
+can be mapped by a field redefinition to a functional integral of a field theory in the bulk
+AdS space. This guarantees the existence of an EAdS bulk dual of a boundary CFT without
+invoking the AdS/CFT conjecture 1
+Given that the crucial ingredient in this connection with ERG is the form of the metric
+(1.1) with the factor z2 in the denominator, one is naturally led to ask if similar mappings can
+be done for the dS metric
+ds2 = L2−dη2 + dxµdxµ
+η2
+(1.2)
+It too has a scaling form. The difference is that the scale is a time like coordinate - so RG
+evolution seems to be related to a real time evolution. In fact this metric is related to the
+EAdS metric by an analytic continuation: iη = z, iL = R. Thus real time evolution should be
+related to RG evolution by analytic continuation. These points have been discussed in many
+of the early papers on de Sitter holography [[30]-[43]], (see also [44] for more recent work and
+further references.)
+This paper is an attempt to address the question of whether the mapping of [5] can be
+generalised to include for instance dS-CFT. One is also led to explore other kinds of mapping
+in an effort to understand the nature of this map better. In [5] the map was first introduced in
+the case of 0-dimensional field theory in the boundary, which gave a one dimensional bulk field
+theory or equivalently a point particle quantum mechanical system. In this paper therefore we
+start by exploring maps for point particle quantum mechanical systems. In Section 2 we show
+that the dynamics of a free particle with a time dependent mass can be mapped to a harmonic
+oscillator. The Euclidean version of this is relevant for the ERG equation. In Section 3 the case
+of mapping a field theory ERG equation to de Sitter space is considered by starting with the
+analytically continued form. This complements the discussion of earlier papers where dS-CFT
+is described as an analytic continuation of EAdS-CFT. In Section 4 we give some examples
+of two point functions obtained using the techniques of [5] being analytically continued to dS
+space. Section 5 contains a summary and conclusions.
+2
+Mapping Free Particle with Time Dependent Mass to
+a Harmonic Oscillator
+In this section we reconsider the construction of [5] where the action for a free field theory
+in D + 1 dimension with a non standard kinetic term was mapped to a free field in AdSD+1.
+1There is still the open question of the locality properties of interaction terms in this bulk field theory. For
+the case of the O(N) model some aspects of this issue were discussed in [7].
+3
+
+When D = 0 this is just a particle: we will map a free particle with time dependent mass to a
+harmonic oscillator.
+2.1
+Mapping Actions
+2.1.1
+Lorentzian Case
+Consider the following action. It defines an evolution operator for free particle (with time
+dependent mass) wave function.
+S = 1
+2
+� tf
+ti
+dt M(t) ˙x2
+(2.3)
+Ψ(x,t) =
+�
+dxi
+�
+x(ti)
+=
+xi
+x(t)
+=
+x
+Dx ei 1
+2
+� t
+ti M(t′) ˙x2dt′Ψ(xi, ti)
+(2.4)
+Let x(t) = f(t)y(t) with f 2(t) =
+1
+M(t). Substitute this in (2.3).
+S = 1
+2
+�
+dt ( ˙y2 + (
+˙f
+f )2y2 + 2
+˙f
+f ˙yy)
+= 1
+2
+�
+dt [ ˙y2 + (d ln f
+dt )2y2 − ( d2
+dt2 ln f)y2] + 1
+2
+�
+dt d
+dt(d ln f
+dt y2)
+Thus, upto the boundary term, the action is
+S = 1
+2
+�
+dt [ ˙y2 + eln f( d2
+dt2e− ln f)y2]
+(2.5)
+Now choose
+eln f( d2
+dt2e− ln f) = −ω2
+0
+(2.6)
+and we get
+¯S = 1
+2
+�
+dt [ ˙y2 − ω2
+0y2]
+(2.7)
+which is the action for a harmonic oscillator. And we define ¯Ψ by absorbing the contribution
+from the boundary term:
+e− 1
+2 i d ln f(t)
+dt
+y2(t)Ψ(f(t)y, t)
+�
+��
+�
+¯Ψ(y,t)
+=
+�
+dyi
+�
+y(ti)
+=
+yi
+y(t)
+=
+y
+Dy ei 1
+2
+� t
+ti[ ˙y2−ω2
+0y2]dt′ e− 1
+2 i d ln f(ti)
+dt
+y2(ti)Ψ(f(ti)yi, ti)
+�
+��
+�
+¯Ψ(yi,ti)
+(2.8)
+¯S thus defines an evolution operator for the harmonic oscillator wave function ¯Ψ. f satisfies
+d2
+dt2
+1
+f = −ω2
+0
+1
+f
+(2.9)
+y obeys the same equation.
+Thus we can take
+1
+f = a cos ω0(t − t0)
+(2.10)
+4
+
+which requires
+M(t) = a2cos2ω0(t − t0)
+Note that one can do more general cases if one is willing to reparametrize time [26, 27].
+Thus let
+dτ =
+dt
+Mf 2
+(2.11)
+Then one gets (2.7), (2.9) and (2.10) with τ replacing t. In terms of t, (2.9) becomes
+d
+dt(M ˙f) =
+ω2
+0
+Mf 3
+(2.12)
+Very interestingly, as pointed out in [26], it is clear from (2.7) that the energy of the
+harmonic oscillator given by
+E = 1
+2( ˙y2 + ω2
+0y2)
+is a conerved quantity. In terms of the original variables this is
+E = 1
+2(( ˙xf − x ˙f
+f 2
+)2 + ω2
+0(x
+f )2)
+These are known as Ermakov-Lewis invariants - see [26] for references to the literature on these
+invariants - and we see a nice interpretation for them.
+2.1.2
+Euclidean Case
+In the Euclidean case the functional integral is
+Ψ(x,τ) =
+�
+dxi
+�
+x(τi)
+=
+xi
+x(τ)
+=
+x
+Dx e− 1
+2
+� τ
+τi M(τ ′) ˙x2dτ ′Ψ(xi, τi)
+(2.13)
+Ψ in this case is not a wave function. It was shown in [5] that the evolution operator for
+a D-dimensional Euclidean field theory is of this form if we take ME(τ) = −
+1
+˙G(τ) and D = 0.
+In this case Ψ can be taken to be e−H[xi,τi] where H is a Hamiltonian or Euclideanized action.
+Alternatively (depending on what ME(τ) is) it can also be eW[J] - a generating functional or
+partition function.
+Setting x = fy with f 2 =
+1
+ME(τ), one goes through the same manipulations but replacing
+(2.6) by
+eln f( d2
+dτ 2e− ln f) = +ω2
+0
+(2.14)
+and (2.7),(2.8) and (2.9) are replaced by
+¯S = 1
+2
+�
+dτ [ ˙y2 + ω2
+0y2]
+(2.15)
+¯Ψ(y, τ) =
+�
+dyi
+�
+y(τi)
+=
+yi
+y(τ)
+=
+y
+Dy e− 1
+2
+� τ
+τi[ ˙y2+ω2
+0y2]dτ ′ ¯Ψ(yi, τi)
+(2.16)
+and
+d2
+dτ 2
+1
+f = ω2
+0
+1
+f
+(2.17)
+5
+
+The solutions are of the form
+f = A sech ω0(τ − τ0)
+(2.18)
+which means ME(τ) =
+1
+A2cosh2ω0(τ − τ0).
+(2.16) has a τ independent action. In this case there are well known physical interpretations
+for the Euclidean theory. The evolution operator, K(y, τ; yi, 0), where
+K(y, τ; yi, 0) =
+�
+y(0)
+=
+yi
+y(τ)
+=
+y
+Dy e− 1
+2
+� τ
+0 [ ˙y2+ω2
+0y2]dτ ′
+(2.19)
+is the density operator of a QM harmonic oscillator in equilibrium at temperature specified by
+β = τ.
+Less well known is that the evolution operator of the Fokker-Planck equation in stochastic
+quantization can be written in the form given in (2.16). ¯Ψ is then related to the probability
+function (see, for instance, [29] for a nice discussion).
+In the next section we discuss the mappings directly for the Schroedinger equation, rather
+than its evolution operator.
+2.2
+Mapping Schrodinger Equations
+2.2.1
+Lorentzian
+Let us consider the same mapping from the point of view of the Schroedinger equation for the
+free particle wave function.
+Schrodinger’s equation for the free particle is
+i∂Ψ(x, t)
+∂t
+= −
+1
+2M(t)
+∂2Ψ(x, t)
+∂x2
+(2.20)
+Ψ given by (2.4) obeys this equation.
+We make a coordinate transformation and a wave function redefinition. Both can be un-
+derstood as canonical transformations [28].
+Let x = f(t)y with f 2 =
+1
+M(t). We take f, M to be dimensionless. We treat this as a 0 + 1
+dimensional field theory where x has the canonical dimension of − 1
+2. So x = L
+1
+2X would define
+a dimensionless X. L is some length scale.
+∂Ψ(x, t)
+∂t
+= ∂Ψ(f(t)y, t)
+∂t
+−
+˙fy
+f
+∂Ψ(f(t)y, t)
+∂y
+Let
+Ψ(f(t)y, t) = e− 1
+2 αy2 ¯Ψ(y, t)
+∂Ψ
+∂t = e− 1
+2 αy2(−1
+2 ˙αy2 + ∂
+∂t)¯Ψ(y, t)
+−i
+˙fy
+f
+∂Ψ(f(t)y, t)
+∂y
+= ie− 1
+2 αy2(α
+˙f
+f y2 −
+˙f
+f y ∂
+∂y)¯Ψ(y, t)
+1
+M
+1
+2
+∂2
+∂x2Ψ = 1
+2
+∂2
+∂y2e− 1
+2 αy2 ¯Ψ = (1
+2e− 1
+2 αy2(α2y2 − 2αy ∂
+∂y − α + ∂2
+∂y2)¯Ψ)
+Collecting all the terms one finds that (2.20) becomes:
+i∂ ¯Ψ
+∂t = (1
+2i ˙α − iα
+˙f
+f − 1
+2α2)y2 ¯Ψ + (i
+˙f
+f y ∂
+∂y + αy ∂
+∂y)¯Ψ + 1
+2αΨ − 1
+2
+∂2
+∂y2 ¯Ψ
+(2.21)
+6
+
+We choose α = −i
+˙f
+f to get rid of the second term on the RHS. We get
+i∂ ¯Ψ
+∂t = [(1
+2
+d2 ln f
+dt2
+− 1
+2(d ln f
+dt )2)y2 + 1
+2α − 1
+2
+∂2
+∂y2]¯Ψ
+As before it can be rewritten as
+i∂ ¯Ψ
+∂t = 1
+2[−eln f( d2
+dt2e− ln f)y2 − ∂2
+∂y2 + α]¯Ψ
+(2.22)
+Set
+d2
+dt2
+1
+f = −ω2
+0
+1
+f
+again as before to get
+i∂ ¯Ψ
+∂t = 1
+2[− ∂2
+∂y2 + ω2
+0y2 + α]¯Ψ
+(2.23)
+The term 1
+2α generates a scale transformation e− 1
+2 ln f(t)
+f(ti) for ¯Ψ.
+2.2.2
+Euclidean
+The Euclidean version is
+∂Ψ(x, τ)
+∂τ
+=
+1
+2ME(τ)
+∂2Ψ(x, τ)
+∂x2
+(2.24)
+As mentioned above, this is of the form of a Polchinski ERG equation (with
+1
+2ME(τ) = − ˙G(τ))
+for H defined by Ψ ≡ e−H. Going through the same steps one finds, with f 2 =
+1
+ME(τ),
+∂ ¯Ψ
+∂τ = (1
+2 ˙α − α
+˙f
+f + 1
+2α2)y2 ¯Ψ + (
+˙f
+f y ∂
+∂y − αy ∂
+∂y)¯Ψ − 1
+2αΨ + 1
+2
+∂2
+∂y2 ¯Ψ
+(2.25)
+the condition α =
+˙f
+f and the equation becomes
+∂ ¯Ψ
+∂t = 1
+2[− eln f( d2
+dt2e− ln f)
+�
+��
+�
+= ω2
+0
+y2 + ∂2
+∂y2 − α]¯Ψ
+(2.26)
+Thus
+∂ ¯Ψ
+∂τ = 1
+2[ ∂2
+∂y2 − ω2
+0y2 − α]¯Ψ
+(2.27)
+And f obeys
+d2
+dt2
+1
+f = ω2
+0
+1
+f
+(2.28)
+This is a Euclidean harmonic oscillator equation.
+Various physical interpretations of this
+equation were given in the last section. The term α in (2.27) provides a multiplicative scaling
+e− 1
+2
+� t
+ti dt′ ∂t′ ln f = ( f(ti)
+f(t) )
+1
+2 of ¯Ψ.
+2.2.3
+Analytic Continuation
+If we set it = τ, (2.20) becomes (2.24) provided M(−iτ) = ME(τ). Similarly (2.23) becomes
+(2.27). Note that in (2.23) α = −i
+˙f
+f . This analytically continues to
+˙f
+f as required.
+7
+
+2.3
+Semiclassical Treatment
+Most of the AdS/CFT calculations invoke large N to do a semiclassical treatment of the bulk
+theory- one can evaluate boundary Green’s function. The analysis in [5, 7] did this for the
+ERG treatment - the evolution of the Wilson action/Generating functional were calculated. In
+[32] a semiclassical treatment was used to obtain the ground state wave function in dS space.
+For completeness we do the same for the simple systems discussed in this paper. This
+illustrates the connection between ERG and dS.
+2.3.1
+Using Harmonic Oscillator Formulation
+Since
+Ψ(x, t) =
+�
+dxi
+�
+x(ti)
+=
+xi
+x(t)
+=
+x
+Dx ei
+� t
+ti L(x(t′), ˙x(t′),t′)dt′Ψ(xi, ti)
+(2.29)
+solves Schroedinger’s equation. For the Harmonic Oscillator
+L = 1
+2( ˙x2 − ω0x2)
+(2.30)
+for the Lorentzian version.
+One can evaluate the path integral semiclassically by plugging in a classical solution with
+some regular boundary condition. We choose x = 0 at t = −∞. The initial state wave function
+is thus a delta function. Classical solution of the EOM is of the form
+x(t) = ae−iω0t + a∗eiω0t
+Since a should annihilate the vacuum state in the far past we would like the solution to look
+like
+x(t) → eiω0t
+in order to ensure that we are in the ground state.
+x(t) = xfe−iω0(tf−t)
+(2.31)
+At t = −∞ we assume that the solution vanishes. This is justified by an infinitesimal rotation
+t → t + iϵt. Evaluated on this solution, the action becomes
+Sclassical = 1
+2x(t) ˙x(t)|
+tf
+−∞
+We get
+Sclassical = 1
+2iω0x2
+f
+(2.32)
+Plugging (2.31) into (2.29) we obtain
+Ψ(xf) ≈ e− 1
+2 ω0x2
+f
+(2.33)
+If we repeat this for the free field in dS space we get the ground state wave functional [32].
+8
+
+2.3.2
+Using ERG formulation
+For the Euclidean version, we set it = τ and we write
+Ψ(x, τ) =
+�
+dxi
+�
+x(τi)
+=
+xi
+x(τ)
+=
+x
+Dx e−
+� τ
+τi LE(x(τ ′), ˙x(τ ′),τ ′)dτ ′Ψ(xi, τi)
+(2.34)
+It is well known that if one does the semiclassical analysis for the Euclidean case with general
+boundary condition one recovers the thermal density matrix. This is for the time independent
+Hamiltonian - such as the harmonic oscillator. We will not do this here. Instead we proceed
+directly to the ERG interpretation of the calculation. Here the Hamiltonian is time dependent.
+In [5] the analysis given below was applied to W[J]. We repeat it here for the Wilson action.
+Our starting action in this case is (Note ˙G < 0):
+S = −1
+2
+� τf
+τi
+˙x2
+˙G
+(2.35)
+EOM is given by,
+∂τ( ˙x
+˙G
+) = 0
+˙x
+˙G
+= b =⇒ x = bG + c
+We choose G so that it vanishes at τ = ∞ .
+For the Euclidean Harmonic oscillator case G has then to be
+G = − 1
+ω0
+(tanh ω(τ − τi) − 1)
+Also x → 0 as τ → ∞. So c = 0.
+x = bG
+(2.36)
+x(τ) = − b
+ω0
+(tanh ω(τ − τi) − 1)
+On shell
+S = −1
+2
+� τf
+τi
+dτ
+d
+dτ (x ˙x
+G )
+= 1
+2(x(τf) − x(τi))b = 1
+2[x(τf)x(τf)
+G(τf)
+− x(τi)x(τi)
+G(τi)
+]
+If we add this change to the initial Wilson action 1
+2
+x(τi)x(τi)
+G(τi)
+we get the final Wilson action
+Hf = 1
+2
+x(τf)x(τf)
+G(τf)
+If, for instance, we are interested in evaluating H semiclassically at τ = τi.
+x(τi) = b
+ω0
+=⇒ b = x(τi)ω0
+x(τ) = −x(0)(tanh ω(τ − τi) − 1)
+˙x(τ) = −x(0)ω0sech2ω0(τ − τi)
+9
+
+The classical action is
+Sclassical = 1
+2ω0x(τi)2
+Thus since G(τi) =
+1
+ω0, H evaluated semiclassically is:
+H[x, τi] ≈ 1
+2ω0x(τi)2
+(2.37)
+Then
+Ψ = e−H[x,τi] = e−ω0x(τi)2
+which coincides with the ground state wave function of the harmonic oscillator. This is essen-
+tially the Hartle Hawking prescription [45]. This also motivates the dS-CFT correspondence
+statement [30, 31, 32] that ΨdS = ZCFT
+This concludes the discussion of the mapping of ERG equation to a Euclidean harmonic
+oscillator. In higher dimensions this gives free field theory in flat space. We now return to the
+case of interest, namely dS space.
+3
+ERG to field theory in dS
+We first map the system to Euclidean AdS. Then analytically continue and obtain dS results.
+Alternatively, one can analytically continue the ERG equation to the Schroedinger equation
+(when D = 0 this is a free particle with a time dependent mass) and then map to de Sitter
+space. This is all exactly as was done for the harmonic oscillator.
+3.1
+Analytic Continuation
+The EAdS metric in Poincare coordinates is
+ds2 = R2[dxidxi + dz2
+z2
+]
+(3.38)
+The dS metric in Poincare coordinates is:
+ds2 = L2[dxidxi − dη2
+η2
+]
+(3.39)
+The metrics are related by analytic continuation:
+iη = z,
+iL = R
+3.1.1
+Analytic Continuation of the Action
+The action generically is
+S = −1
+2
+�
+dD+1x√g[gµν∂µφ∂νφ + m2φ2]
+(3.40)
+10
+
+de Sitter
+In this case we write √−g since g is negative: g = −( L2
+η2 )D+1. Also g00 = − η2
+L2
+and gij = δij η2
+L2.
+Thus
+SdS =
+�
+dDx
+� ∞
+0
+dη (L
+η )D+1[ η2
+L2∂ηφ∂ηφ − η2
+L2∂iφ∂iφ − m2φ2]
+(3.41)
+In momentum space:
+SdS =
+�
+dDp
+(2π)D
+� ∞
+0
+dη (L
+η )D+1[ η2
+L2∂ηφ(p)∂ηφ(−p) − ( η2
+L2p2 + m2)φ(p)φ(−p)]
+(3.42)
+The functional integral description of the quantum mechanical evolution operator for the
+wave functional of the fields in dS space-time is
+¯Ψ[φ(p), t] =
+�
+dφi(p)
+�
+φ(p, ti)
+=
+φi(p)
+φ(p, t)
+=
+φ(p)
+Dφ(p, t) ei 1
+2
+� t
+ti[ ˙φ(p,t′)2−ω2
+0φ(p,t′)2]dt′ ¯Ψ[φi(p), ti]
+(3.43)
+Euclidean Anti de Sitter
+g = ( R2
+z2 )D+1. Also g00 = z2
+R2 and gij = δij z2
+R2.
+SEAdS =
+�
+dDx
+� ∞
+0
+dz (R
+z )D+1[ z2
+R2∂zφ∂zφ + z2
+R2∂iφ∂iφ + m2φ2]
+(3.44)
+In momentum space
+SEAdS =
+�
+dDp
+(2π)D
+� ∞
+0
+dz (R
+z )D+1[ z2
+R2∂zφ(p)∂zφ(−p) + ( z2
+R2p2 + m2)φ(p)φ(−p)]
+(3.45)
+If we set iη = z and iL = R we see that the functional integral (3.43) becomes
+¯Ψ[φ(p), t] =
+�
+dφi(p)
+�
+φ(p, ti)
+=
+φi(p)
+φ(p, t)
+=
+φ(p)
+Dφ(p, t) e− 1
+2
+� t
+ti[ ˙φ(p,t′)2+ω2
+0φ(p,t′)2]dt′ ¯Ψ[φi(p), ti] (3.46)
+In holograhic RG this is interpreted as a Euclidean functional integral giving the evolution in
+the radial direction. ¯Ψ is to be interpreted as e−SI[φ(p),t] where SI is the Wilson action. It was
+shown in [5] (see below) that this can be obtained by mapping an ERG evolution operator.
+The dS functional integral (3.43) above is thus an analytically continued version of this.
+3.2
+Mapping
+3.2.1
+Mapping from Quantum Mechanics
+Let us go back to Section (2.1) and consider the mapping from the Quantum Mechanics of a
+free particle with time dependent mass. We think of it as a 0 + 1 dimensional field theory.
+M(t) is taken to be dimensionless and x has canonical dimensions of − 1
+2.
+S = 1
+2
+�
+dt M(t) ˙x2
+(3.47)
+(In the ERG version M(t) = 1
+˙G)
+The path integral is
+�
+Dx eiS
+(3.48)
+11
+
+As before x(t) = f(t)y(t) with f 2(t) =
+1
+M(t). Substitute this in (3.47) and go through the
+same steps to obtain:
+S = 1
+2
+�
+dt [ ˙y2 + eln f( d2
+dt2e− ln f)y2]
+(3.49)
+Now choose
+eln f( d2
+dt2e− ln f) = −( η2
+L2p2 + m2)
+(3.50)
+where η = Le
+t
+L. to obtain SdS
+SdS = 1
+2
+�
+dt [ ˙y2 − ( η2
+L2p2 + m2)y2]
+= 1
+2
+�
+dη (L
+η )[ η2
+L2∂ηy∂ηy − ( η2
+L2p2 + m2)y2]
+(3.51)
+p, m here are just some parameters. When D > 0 they will stand for momentum and mass
+of the field respectively. So starting from a free particle with time dependent mass we obtain
+the free field action in de Sitter space dSD+1 with D = 0.
+Schroedinger Equation:
+i∂Ψ(x, t)
+∂t
+= −
+1
+2M(t)
+∂2Ψ(x, t)
+∂x2
+(3.52)
+Using the same mapping as in Section (2.2.1), x = fy
+Ψ(f(t)y, t) = e− 1
+2 αy2 ¯Ψ(y, t)
+with α = −i
+˙f
+f one obtains
+i∂ ¯Ψ
+∂t = [(1
+2
+d2 ln f
+dt2
+− 1
+2(d ln f
+dt )2)y2 + 1
+2α − 1
+2
+∂2
+∂y2]¯Ψ
+Using (3.50) this becomes
+i η
+L
+∂ ¯Ψ
+∂η = [−1
+2
+∂2
+∂y2 + 1
+2( η2
+L2p2 + m2)y2 + 1
+2α]¯Ψ
+(3.53)
+If we construct the Schroedinger equation corresponding to the action (3.51) one obtains
+i η
+L
+∂ ¯Ψ
+∂η = [−1
+2
+∂2
+∂y2 + 1
+2( η2
+L2p2 + m2)y2]¯Ψ
+(3.54)
+which barring the field independent term α is exactly the same as (3.53). This term as we
+have seen provides an overall field independent scaling for all wave functions. It is a consequence
+of the ordering ambiguity in going from classical to quantum treatment. (3.54) (or its extension
+to D > 0) describes the quantum mechanical time evolution of the matter field wave functional
+in de Sitter space.
+12
+
+3.2.2
+Mapping from ERG
+Action
+We now consider the Euclidean version of (3.47), which is the Polchinski ERG
+equation. This is what was done in [5]. Thus we replace M(t) by − 1
+˙G.
+S = −1
+2
+�
+dτ ˙x2
+˙G
+(3.55)
+The path integral is ( ˙G < 0)
+�
+Dx e
+1
+2
+�
+dτ
+˙x2
+˙G
+(3.56)
+which can be obtained from (3.52) by setting it = τ. We take z = Re
+τ
+R If we let iη = z, iL =
+R, it = τ then this can be obtained from the corresponding Minkowski case.
+As before x(τ) = f(τ)y(τ) with f 2(τ) = ˙G. Substitute this in (3.55) and go through the
+same steps to obtain:
+S = 1
+2
+�
+dτ [ ˙y2 + eln f( d2
+dτ 2e− ln f)y2]
+(3.57)
+Now choose
+eln f( d2
+dτ 2e− ln f) = ( z2
+R2p2 + m2)
+(3.58)
+where z = Re
+τ
+R. to obtain SEAdS
+SEAdS =
+�
+dz (R
+z )[ z2
+R2∂zy∂zy + ( z2
+R2p2 + m2)y2]
+(3.59)
+ERG Equation
+By analogy with the Schroedinger equation we can see that (3.56) is the
+evolution operator corresponding to the ERG equation
+∂Ψ(x, τ)
+∂τ
+= −1
+2
+˙G∂2Ψ(x, τ)
+∂x2
+(3.60)
+By the same series of transformations as in the de Sitter case, but using (3.58), one obtains
+z
+R
+∂ ¯Ψ
+∂z = [1
+2
+∂2
+∂y2 − ( z2
+R2p2 + m2)y2 − 1
+2α]¯Ψ
+(3.61)
+with α =
+˙f
+f generating an overall scale transformation for ¯Ψ.
+In the ERG context ¯Ψ
+represents eW[J] upto a quadratic term. This equation is the holographic RG equation in the
+AdS/CFT correspondence for an elementary scalar field [5].
+3.3
+Connections
+Let us summarize the various connections obtained above.
+• We start with the quantum mechanics of a free particle having a time dependent mass.
+The Schroedinger equation (SE) for this is (2.20). Analytical continuation of this equation
+(generalized to higher dimensions) gives the Polchinski ERG equation (2.24).
+• The free particle SE (2.20) can be mapped to a SE for a harmonic oscillator (2.23). The
+ERG equation (2.24) can similarly be mapped to a Euclidean harmonic oscillator (2.27)-
+analytically continued version of (2.23).
+13
+
+• The evolution operators for the above equations are defined in terms of path integrals
+over some actions. The same mapping function f maps the corresponding actions to each
+other. Thus the evolution operator for the free particle Schroedinger equation is given by
+the action in (2.3) which is mapped to a harmonic oscillator action (2.7). The analytical
+continuation of these are the Euclidean ERG evolution operator (2.13) mapped to a
+harmonic oscillator Hamiltonian (2.16). These steps are summarized in the flow diagram
+in Figure 1.
+• The mapping function f was originally chosen in [5] to map the free particle ERG action
+(3.55) to an action for free fields in EAdS0+1 given in (3.60). The analytical continuation
+of this problem to real time gives us an action in dS0+1 (3.51).
+• One can also repeat these steps for the corresponding “wave” equations. The Polchinski
+ERG equation for eW[J] gets mapped to an equation in EAdS for eW[J] which is nothing but
+the holographic RG equations. Analytically continuing this, the Schroedinger equation
+for a wave functional is mapped to a Schroedinger equation for wave functionals of fields
+in dS.
+These are summarized in the figure below (Fig.2). The analytic continuation can be done
+before the map with f is applied or after as shown in the figure. It can be done both for the
+actions as well as for the equations.
+ERG
+Equation
+Holographic RG:
+Radial evolution
+in EAdS
+Schroedinger 
+Equation
+Real time QM
+evolution
+In dS
+Map “f”
+Map “f”
+Analytic
+Continuation
+Analytic 
+Continuation
+ERG
+Equation
+Evolution equation
+For Euclidean
+Harmonic Oscillator
+QM Schroedinger 
+Equation
+Real time QM
+Schroedinger 
+Equation for
+ Harmonic Oscillator
+Map “f”
+Map “f”
+Analytic
+Continuation
+Analytic 
+Continuation
+ERG
+Equation
+Holographic RG:
+Radial evolution
+in EAdS
+Schroedinger 
+Equation
+Real time QM
+evolution
+In dS
+Map “f”
+Map “f”
+Analytic
+Continuation
+Analytic 
+Continuation
+Euclidean Action
+For ERG evolution
+By Feynman 
+Path Integral
+Euclidean action for 
+Harmonic Oscillator
+Path Integral
+Lorentzian
+Action for QM 
+Evolution by
+Path Integral 
+QM evolution:
+ Action for 
+Harmonic Oscillator
+Path Integral
+Map “f”
+Map “f”
+Analytic
+Continuation
+Analytic 
+Continuation
+Flow of equations-Harmonic Oscillator
+Flow of actions – Harmonic Oscillator
+Figure 1: Mapping ERG to Harmonic Oscillator
+3.4
+dS-CFT correspondence
+The connections with ERG mentioned above should, if pursued, provide some insights into
+dS-CFT correspondence. We restrict ourselves to some preliminary observations in this paper.
+14
+
+ERG
+Equation
+Holographic RG:
+Radial evolution
+in EAdS
+Schroedinger 
+Equation
+Real time QM
+evolution
+In dS
+Map “f”
+Map “f”
+Analytic
+Continuation
+Analytic 
+Continuation
+ERG
+Equation
+Holographic RG:
+Radial evolution
+equation in EAdS
+QM Schroedinger 
+Equation
+Real time QM
+Schroedinger 
+equation
+in dS
+Map “f”
+Map “f”
+Analytic
+Continuation
+Analytic 
+Continuation
+ERG
+Equation
+Holographic RG:
+Radial evolution
+in EAdS
+Schroedinger 
+Equation
+Real time QM
+evolution
+In dS
+Map “f”
+Map “f”
+Analytic
+Continuation
+Analytic 
+Continuation
+Euclidean Action
+For ERG evolution
+by functional integral
+Holographic RG:
+ Action  in EadS
+for functional 
+integral
+Lorentzian
+Action for QM 
+evolution
+QM evolution by
+Functional integral:
+ Action in dS
+Map “f”
+Map “f”
+Analytic
+Continuation
+Analytic 
+Continuation
+Flow of equations
+Flow of actions
+Figure 2: Mapping ERG to Holographic RG
+The idea of dS-CFT correspondence was suggested in [30, 31, 32]. This has been investigated
+further by many authors, e.g. [33, 34, 38, 39, 35, 37, 36].
+What we see from the above analysis is that considering the relation between the evolution
+equations, one can say that
+Ψ[φ, J]wave−functional in dS = {Z[φ, J]CFT}analytically continued
+(3.62)
+Thus we see that the dS-CFT correspondence suggested by this analysis is one between an
+ERG equation for a CFT generating functional and a real time quantum mechanical evolution
+of a wave functional in dS space time.
+The LHS of (3.62) is a QM wave functional of fields on a D-dimensional spatial slice of
+a D + 1 dimensional dS spacetime. The RHS is the analytically continued partition function
+of a D-dimensional Euclidean CFT - more precisely, either eWΛ[J] or e−SI,Λ[φ]. The precise
+statement has to involve some statement of the boundary conditions. In the next section we
+give a concrete example with boundary conditions specified.
+Note that the LHS is a complex probability amplitude. Expectation values will involve Ψ∗Ψ
+and were calculated first in [30, 31, 32].
+One can proceed to ask whether the expectations on the spatial slice calculated using Ψ∗Ψ
+also correspond to some other Euclidean CFT on the spatial slice. This was explored further
+in [38]. We do not address this question here.
+In the next section we give some examples that explicitly illustrate the connection made by
+(3.62).
+15
+
+4
+Obtaining Bulk field from ERG
+The ERG formulation stated in this paper starts with the boundary fields. The evolution
+operator for this involves bulk fields but with a non standard action. When this action is
+mapped to EAdS action one can interpret the newly mapped field as the EAdS bulk field. This
+analysis for Euclidean AdS is well defined and has been done in [5, 7]. However, this treatment
+does not have a natural interpretation in the context in dS space. We have elaborated that in
+this section.
+Bulk scalar field in Euclidean AdS and dS
+There are conceptual barriers if one tries to do similar analysis to map the ERG evolution
+operator directly to Lorentzian dS. First of all, it is not clear as in EAdS whether the function
+G(t) a.k.a f 2(t) = ˙G(t) is the Green’s function of the dual field theory of dS. It has an oscillatory
+cutoff function. Therefore we analytically continue the ERG action to a Lorentzian action first,
+and then do the mapping.
+The result thus obtained (4.74) matches with the value found in [39] where the authors have
+found the bulk field in semicalssical approximation from dS bulk action. For the Lorentzian dS
+analysis presented here the RG interpretation is not clearly understood - except as an anlytic
+continuation. We have presented it here for sake of completeness.
+Euclidean AdS
+The Euclidean action of the ERG evolution operator in momentum space,
+S = −1
+2
+�
+dτ
+�
+p
+˙φ2
+˙G
+(4.63)
+is mapped to
+SEAdS =
+�
+dDp
+(2π)D
+� ∞
+ϵEAdS
+dz (R
+z )d+1[ z2
+R2∂zyEAdS(p)∂zyEAdS(−p)+( z2
+R2p2+m2)yEAdS(p)yEAdS(−p)]
+(4.64)
+with z = Re
+τ
+R as described in [5]. We have rescaled the field as φ = fyEAdS where f is
+related to the boundary Green’s function G as f 2 = −
+� z
+R
+�−d ˙G.
+The constraint on 1
+f is given by,
+∂
+∂z{
+� z
+R
+�−d+1 ∂
+∂z
+1
+f } =
+� z
+R
+�−d+1 �
+p2 + m2R2
+z2
+� 1
+f
+(4.65)
+The solutions are zd/2Kα(pz) and zd/2Iα(pz) where α2 = m2R2 + d2
+4 .
+So 1
+f can be taken as,
+1
+f(p, z) = (z)d/2 (AKα(pz) + BIα(pz))
+(4.66)
+The Green’s function is
+G(p, z) = CKα(pz) + DIα(pz)
+AKα(pz) + BIα(pz)
+(4.67)
+The large argument asymptotic form of the Modified Bessel function Iα(z) and Kα(z) are
+given by,
+Iα(z) ∼
+ez
+√
+2πz
+�
+1 + O(1
+z)
+�
+for |arg z| < π
+2
+16
+
+Kα(z) ∼
+� π
+2ze−z
+�
+1 + O(1
+z)
+�
+for |arg z| < 3π
+2
+Putting two constraints on G- i)G(pz → ∞) = 0 ii)G(pz → 0) = γEAdS p−2α, we get,
+D = 0; C(p) = γEAdS p−α; B(p) = −
+1
+γEAdS
+pα
+In semiclassical approximation the bulk field yEAdS = bEAdS
+G
+f . If yEAdS satisfies yEAdS
+0
+the
+bulk field is given by,
+yEAdS = yEAdS
+0
+zd/2
+ϵd/2
+Kα(pz)
+Kα(pϵ)
+(4.68)
+Now let’s check by analytic continuation iη = z and iL = R. First of all, α becomes ν. ϵ
+is replaced by iϵ. We get,
+yEAdS|z=iη, R=iL = yEAdS
+0
+|z=iη, R=iL
+(iη)d/2
+(iϵ)d/2
+Kν(ipη)
+Kν(ipϵ)
+(4.69)
+As,
+yEAdS
+0
+= bEAdS ϵd/2
+EAdS
+γEAdS Kα(pϵ)
+pα
+(4.70)
+de Sitter
+We would like to do the same analysis as above for the Lorentzian case.
+The Lorentzian action obtained from (4.63) by analytic continuation, in momentum space,
+S = −
+�
+dt
+�
+dDp
+(2π)D
+1
+2 ˙G(p)
+˙φ(p) ˙φ(−p)
+and needs to be mapped to
+= 1
+2
+� ∞
+ϵdS
+dη
+�
+dDp
+(2π)D
+��L
+η
+�D−1
+{(∂ηydS)2 − p2ydS2 − m2L2
+η2
+ydS2}
+�
+Here η = Le
+t
+L. We do the field redefinition of boundary field
+φ = fydS
+f is a scale dependent quantity which is related to Green’s function G as f 2 = −
+� η
+L
+�−D ˙G.
+Performing the same manipulations as in [5], one can get the constraint on f as,
+� η
+L
+�d−1 �� η
+L
+�−d+1 d
+dη
+�2
+e− ln f =
+� η
+L
+�−d+1 �
+−p2 − m2L2
+η2
+�
+e− ln f
+−d + 1
+η
+∂
+∂η
+1
+f + ∂2
+∂η2
+1
+f =
+�
+−p2 − m2L2
+η2
+� 1
+f
+The solutions are
+� η
+L
+�d/2 H(1)
+ν (pη) and
+� η
+L
+�d/2 H(2)
+ν (pη) with ν2 = d2
+4 − m2L2.
+The 1
+f can be written in general as( note f is dimensionless),
+1
+f(p, η) =
+� η
+L
+�d/2 �
+AH(1)
+ν (pη) + BH(2)
+ν (pη)
+�
+(4.71)
+17
+
+and the Green’s function is 2
+G(pη) = CH(1)
+ν (pη) + DH(2)
+ν (pη)
+AH(1)
+ν (pη) + BH(2)
+ν (pη)
+Physically one can expect G(pη → ∞) = 0 which yields,
+CH(1)
+ν (pη) + DH(2)
+ν (pη) = 0
+(4.72)
+The asymptotic forms of Hankel functions of both kind for large arguments are,
+H(1)
+ν (z) ∼
+�
+2
+πzei(z− νπ
+2 − π
+4 )
+− π < arg z < 2π
+H(2)
+ν (z) ∼
+�
+2
+πze−i(z− νπ
+2 − π
+4 )
+− 2π < arg z < π
+The presence of the oscillatory functions will not let eq.4.72 to be satisfied.
+Hence we
+analytically continue the argument of Green’s function G. The choice of direction of the analytic
+continuation is based on the anticipation that the bulk field will have positive frequency. Hence
+we take
+η = −iz
+(4.73)
+which prompts us to make C = 0. Also, from the constraint AD − BC = 1 we get A = 1
+D.
+Hence the Green’s function now takes the form,
+G(pz) =
+DH(2)
+ν (ipz)
+1
+DH(1)
+ν (ipz) + BH(2)
+ν (ipz)
+Next another constraint will come from the fact that boundary Green’s function is γdS p−2ν.
+So in the limit of z → 0 using the formulae,
+H(1)
+ν (z) = iYν(z); H(2)
+ν (z) = −iYν(z); Yν(z) = −Γ(ν)
+π
+�2
+z
+�ν
+One can get,
+−iD
+i
+D − iB = γdS p−2ν
+On the other side, f should become a p independent constant at boundary x = 0 so that
+it does not modify the boundary Green’s function, also ydS and f should become same field in
+boundary field theory. This gives,
+i
+D − iB = pν
+Finally we get,
+D = iγdS p−ν ; B = i
+�
+1 − 1
+γdS
+�
+pν
+The bulk field ydS is given by,
+2We use the term Green function by analogy with the EAdS case, where G is the propagator of the boundary
+CFT. Also see for instance [39].
+18
+
+ydS = bdS
+G
+f = bdS(iγp−ν) 1
+Ld/2xd/2H(2)
+ν (ipx)
+If we analytically continue back to η we get,
+ydS = bdS(iγp−ν) 1
+Ld/2(−iη)d/2H(2)
+ν (pη)
+If the field ydS satisfies ydS
+0
+at η = ϵdS then,
+ydS = ydS
+0
+ηd/2
+ϵd/2
+dS
+H(2)
+ν (pη)
+H(2)
+ν (pϵdS)
+(4.74)
+ydS satisfies Bunch-Davies condition.
+Relation between bulk fields in EAdS and dS
+The bulk field in EAdS space is given
+by,
+yEAdS = yEAdS
+0
+zd/2
+ϵd/2
+Kα(pz)
+Kα(pϵ)
+(4.75)
+Let’s apply the analytic continuation continuation iη = z and iL = R. First of all, α becomes
+ν. ϵ is replaced by iϵ. We get,
+yEAdS|z=iη, R=iL = yEAdS
+0
+|z=iη, R=iL
+(iη)d/2
+(iϵ)d/2
+Kν(ipη)
+Kν(ipϵ)
+(4.76)
+As,
+yEAdS
+0
+= bEAdS ϵd/2
+EAdS
+γEAdS Kα(pϵ)
+pα
+(4.77)
+Using the relation between Kα(x) and Hα(x),
+Kα(x) = π
+2 iα+1H(1)
+α (ix); − π < arg x ≤ π
+2
+= π
+2 (−i)α+1H(2)
+α (−ix); − π
+2 < arg x ≤ π
+(4.78)
+Here also we want to ensure the bulk field to be of positive frequency, hence choosing
+H(2)(x).
+yEAdS
+0
+|z=iη, R=iL = π
+2 (i)d/2+α+1bEAdSϵd/2γEAdS
+H(2)
+α (pϵ)
+pα
+= bEAdS
+bdS
+γEAdS
+γdS
+π
+2 (i)d/2+α+1ydS
+0
+Hence,
+yEAdS|z=iη, R=iL =bEAdS
+bdS
+γEAdS
+γdS
+π
+2 (i)d/2+α+1ydS
+0
+ηd/2
+ϵd/2
+H(2)
+α (pη)
+H(2)
+α (pϵ)
+= bEAdS
+bdS
+γEAdS
+γdS
+π
+2 (i)d/2+α+1ydS
+(4.79)
+Upto various normalization constants we see that they agree.
+19
+
+5
+Summary and Conclusions
+In [5, 6] an evolution operator for an ERG equation of a perturbed D-dimensional free field
+theory in flat space was mapped to a field theory action in AdSD+1. Similar mappings were done
+subsequently for the interacting O(N) model at both the free fixed point and at the Wilson-
+Fisher fixed point [7]. The main aim of this paper was to understand better the mapping used
+in these papers and to see if there are other examples. A related question was that of analytic
+continuation of these theories. These questions can posed, both for the ERG equation and its
+evolution operator.
+It was shown that a mapping of this type can map the ERG evolution operator of a (zero-
+dimensional) field theory to the action of a Euclidean harmonic oscillator. Furthermore the
+analytic continuation of the ERG evolution operator action gives the path integral for a free
+particle with a time dependent mass. A similar mapping takes this to a harmonic oscillator.
+This method also gives new way of obtaining the Ermakov-Lewis invariants for the original
+theory.
+The analytically continued ERG equation is a Schroedinger like equation for a free field
+theory wave functional. This gets mapped to the Schroedinger equation for a wave functional
+of a free field theory in de Sitter space. These are summarized in Figures 1,2. This is one
+version of the dS-CFT correspondence. From this point of view, the QM evolution of dS field
+theory is also an ERG evolution of a field theory, but accompanied by an analytic continuation.
+An example was worked out to illustrate this correspondence.
+To understand these issues further it would be useful to apply these techniques to the O(N)
+model ERG equation written in [7]. This ERG equation has extra terms and thus the theory
+naturally has interaction terms in the EAdS bulk action.
+Similarly it would be interesting to study the connection between bulk Green functions
+and the QM correlation functions on the space-like time slice of these theories, as considered
+originally in [30, 31, 32].
+Acknowledgements
+SD would like to thank IMSc,Chennai where part of the work was done.
+20
+
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+

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+Nonlinear Dynamics
+ 
+Harmonic-Gaussian double-well potential stochastic resonance with its application to
+enhance weak fault characteristics of machinery
+--Manuscript Draft--
+ 
+Manuscript Number:
+NODY-D-22-01167R2
+Full Title:
+Harmonic-Gaussian double-well potential stochastic resonance with its application to
+enhance weak fault characteristics of machinery
+Article Type:
+Original Research
+Keywords:
+The benefits of noise, weak signature enhancement, fault identification, fault diagnosis
+Corresponding Author:
+Zijian Qiao, Ph.D.
+Ningbo University
+Ningbo, CHINA
+Corresponding Author Secondary
+Information:
+Corresponding Author's Institution:
+Ningbo University
+Corresponding Author's Secondary
+Institution:
+First Author:
+Zijian Qiao, Ph.D.
+First Author Secondary Information:
+Order of Authors:
+Zijian Qiao, Ph.D.
+Shuai Chen
+Zhihui Lai
+Shengtong Zhou
+Miguel A. F. Sanjuán
+Order of Authors Secondary Information:
+Funding Information:
+Foundation of the State Key Laboratory of
+Performance Monitoring and Protecting of
+Rail Transit Infrastructure of East China
+Jiaotong University
+(HJGZ2021114)
+Dr. Zijian Qiao
+Zhejiang Provincial Natural Science
+Foundation of China
+(LQ22E050003)
+Dr. Zijian Qiao
+National Natural Science Foundation of
+China
+(62001210, 51905349)
+Dr. Zhihui Lai
+The Spanish State Research Agency
+(AEI) and the European Regional
+Development Fund (ERDF)
+(PID2019-105554GB-I00)
+Dr. Miguel A. F. Sanjuán
+Abstract:
+Noise would give rise to incorrect filtering frequency-band selection in signal filtering-
+based methods including fast kurtogram, teager energy operators and wavelet packet
+transform filters and meanwhile would result in incorrect selection of useful
+components and even mode mixing, end effects and etc. in signal decomposition-
+based methods including empirical mode decomposition, singular value decomposition
+and local mean decomposition. On the contrary, noise in stochastic resonance (SR) is
+beneficial to enhance weak signals of interest embedded in signals with strong
+background noise. Taking into account that nonlinear systems are crucial ingredients
+to activate the SR, here we investigate the SR in the cases of overdamped and
+underdamped harmonic-Gaussian double-well potential systems subjected to noise
+and a periodic signal. We derive and measure the analytic expression of the output
+signal-to-noise ratio (SNR) and the steady-state probability density (SPD) function
+Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation
+
+under approximate adiabatic conditions. When the harmonic-Gaussian double-well
+potential loses its stability, we can observe the antiresonance phenomenon, whereas
+adding the damped factor into the overdamped system can change the stability of the
+harmonic-Gaussian double-well potential, resulting that the antiresonance behavior
+disappears in the underdamped system. Then, we use the overdamped and
+underdamped harmonic-Gaussian double-well potential SR to enhance weak useful
+characteristics for diagnosing incipient rotating machinery failures. Theoretical and
+experimental results show that adjusting both noise intensity and system parameters
+can activate overdamped and underdamped harmonic-Gaussian double-well potential
+SR in which there is a bell-shaped peak for the SNR. Additionally, the underdamped
+harmonic-Gaussian double-well potential SR is independent of frequency-shifted and
+rescaling transform to process large machine parameter signals and outperforms the
+overdamped one. Finally, comparing the advanced robust local mean decomposition
+(RLMD) method based on signal decomposition and the wavelet transform method
+based on noise cancellation or infogram method based on signal filtering, the
+overdamped or underdamped harmonic-Gaussian double-well potential SR methods
+characterize a better performance to detect a weak signal. Fault characteristics in the
+early stage of failures are successful in improving the incipient fault characteristic
+identification of rolling element bearings.
+Response to Reviewers:
+Please see the attached "response to reviewers".
+Order of Authors (with Contributor Roles): Zijian Qiao, Ph.D. (Funding acquisition: Supporting; Validation: Lead; Writing – original
+draft: Lead)
+Shuai Chen (Data curation: Equal; Visualization: Lead)
+Zhihui Lai (Investigation: Equal; Visualization: Equal; Writing – review & editing:
+Supporting)
+Shengtong Zhou (Data curation: Lead; Formal analysis: Equal; Investigation: Equal;
+Writing – review & editing: Equal)
+Miguel A. F. Sanjuán (Formal analysis: Equal; Funding acquisition: Equal;
+Methodology: Equal; Writing – review & editing: Lead)
+Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation
+
+ 1 / 27 
+ 
+Harmonic-Gaussian double-well potential stochastic resonance with 
+its application to enhance weak fault characteristics of machinery 
+ 
+Zijian Qiao1,2,3,4, , Shuai Chen1, Zhihui Lai5, Shengtong Zhou2, Miguel A. F. Sanjuán6 
+1 School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo 315211, China 
+2. State Key Laboratory of Performance Monitoring and Protecting of Rail Transit Infrastructure, East China 
+Jiaotong University, Nanchang 330013, China 
+3. Laboratory of Yangjiang Offshore Wind Power, Yangjiang 529599, Guangdong, China 
+4. Zhejiang Provincial Key Laboratory of Part Rolling Technology, Ningbo 315211, China 
+5. Guangdong Provincial Key Laboratory of Micro/Nano Optomechatronics Engineering, College of Mechatronics 
+and Control Engineering, Shenzhen University, Shenzhen 518060, China 
+6. Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan 
+Carlos, Tulipán s/n, Móstoles, 28933, Madrid, Spain 
+ 
+Abstract: 
+Noise would give rise to incorrect filtering frequency-band selection in signal 
+filtering-based methods including fast kurtogram, teager energy operators and wavelet 
+packet transform filters and meanwhile would result in incorrect selection of useful 
+components 
+and 
+even 
+mode 
+mixing, 
+end 
+effects 
+and 
+etc. 
+in 
+signal 
+decomposition-based methods including empirical mode decomposition, singular 
+value decomposition and local mean decomposition. On the contrary, noise in 
+stochastic resonance (SR) is beneficial to enhance weak signals of interest embedded 
+in signals with strong background noise. Taking into account that nonlinear systems 
+are crucial ingredients to activate the SR, here we investigate the SR in the cases of 
+overdamped and underdamped harmonic-Gaussian double-well potential systems 
+subjected to noise and a periodic signal. We derive and measure the analytic 
+expression of the output signal-to-noise ratio (SNR) and the steady-state probability 
+                                                             
+Corresponding author. 
+E-mail address: [email protected], [email protected] (Z. Qiao). 
+Manuscript
+Click here to access/download;Manuscript;Manuscript.docx
+Click here to view linked References
+
+ 2 / 27 
+ 
+density (SPD) function under approximate adiabatic conditions. When the 
+harmonic-Gaussian double-well potential loses its stability, we can observe the 
+antiresonance phenomenon, whereas adding the damped factor into the overdamped 
+system can change the stability of the harmonic-Gaussian double-well potential, 
+resulting that the antiresonance behavior disappears in the underdamped system. Then, 
+we use the overdamped and underdamped harmonic-Gaussian double-well potential 
+SR to enhance weak useful characteristics for diagnosing incipient rotating machinery 
+failures. Theoretical and experimental results show that adjusting both noise intensity 
+and 
+system 
+parameters 
+can 
+activate 
+overdamped 
+and 
+underdamped 
+harmonic-Gaussian double-well potential SR in which there is a bell-shaped peak for 
+the SNR. Additionally, the underdamped harmonic-Gaussian double-well potential SR 
+is independent of frequency-shifted and rescaling transform to process large machine 
+parameter signals and outperforms the overdamped one. Finally, comparing the 
+advanced robust local mean decomposition (RLMD) method based on signal 
+decomposition and the wavelet transform method based on noise cancellation or 
+infogram method based on signal filtering, the overdamped or underdamped 
+harmonic-Gaussian double-well potential SR methods characterize a better 
+performance to detect a weak signal. Fault characteristics in the early stage of failures 
+are successful in improving the incipient fault characteristic identification of rolling 
+element bearings. 
+ 
+Key words: The benefits of noise, weak signature enhancement, fault identification, 
+fault diagnosis 
+ 
+1. Introduction 
+Noise is ubiquitous but unwanted in detecting weak signals [1], but noise in 
+biological systems can be used to amplify weak signals embedded by a strong 
+background noise. Such an ingenious phenomenon is observed in a bistable nonlinear 
+system, namely stochastic resonance (SR) [2]. SR is a kind of synchronization 
+mechanism among the nonlinear systems, noise and a weak periodic signal, which 
+
+ 3 / 27 
+ 
+takes place to activate the SR for amplifying weak useful signals [3]. 
+SR has been investigated from theory to engineering application widely [4-6]. 
+Among three ingredients for activating SR including noise, nonlinear systems and 
+weak useful signals, nonlinear systems are crucial ingredients for extracting weak 
+useful signals and moreover can harvest the energy of noise located at the whole 
+frequency band of a noisy signal to enhance or amplify a weak useful signal. For this 
+purpose, most of scholars pay attention to exploring the behaviors of SR in novel 
+nonlinear systems from bistable [7] to multistable ones [8-10], from overdamped [11] 
+and underdamped [12] to fractional-order [13] ones, and even from cascaded [14] and 
+coupled [15, 16] to time-delayed feedback [17] ones and biological systems [18, 19].  
+Because the bistable system is most classical among them, it has been investigated, 
+such as classical bistable potential overdamped systems, noisy confined bistable 
+potential overdamped systems [20], asymmetric bistable potential overdamped 
+systems [21], classical bistable potential underdamped systems, noisy bistable 
+potential fractional-order systems [22] and E-exponential potential underdamped 
+systems [23, 24]. The E-exponential potential named by the references [23, 24] is a 
+narrow version of the harmonic-Gaussian double-well potential. The references above 
+show that overdamped and underdamped harmonic-Gaussian double-well potential 
+SR has not been studied systematically in theory and further applied to enhance 
+incipient fault identification of machinery for providing a tutorial of other readers and 
+researchers on the SR in the overdamped and underdamped systems with novel 
+generalized double-well potentials yet. Even, the comparison between overdamped 
+and underdamped harmonic-Gaussian double-well potential SR has not been made in 
+theory and engineering applications. Therefore, this paper attempts to investigate the 
+SR in the overdamped and underdamped harmonic-Gaussian double-well potential 
+systems theoretically and then apply it to enhance weak fault characteristics and 
+diagnose incipient faults of machinery. Additionally, some comparisons with other 
+advanced signal processing techniques including signal decomposition-based and 
+noise cancellation or signal filtering-based methods for enhancing weak fault 
+characteristics of machinery are given.  
+
+ 4 / 27 
+ 
+The remainder of this paper is organized as follows. Section 2 and Section 3 
+investigate the overdamped and underdamped harmonic-Gaussian double-well 
+potential SR by deriving the analytic expressions of signal-to-noise ratio (SNR) and 
+steady-state probability density (SPD) functions, respectively. In Section 4, we apply 
+the overdamped and underdamped harmonic-Gaussian double-well potential SR to 
+enhance weak fault characteristics and incipient fault identification of rolling element 
+bearings. Finally, conclusions are drawn in Section 5. 
+ 
+2. Overdamped harmonic-Gaussian double-well potential SR 
+The overdamped Langevin equation driven by a harmonic-Gaussian double-well 
+potential under the action of random noise and a periodic signal can be described as 
+[25] 
+d𝑦
+d𝑥 = −
+𝜕𝑈(𝑥)
+𝜕𝑥
++ 𝐴 cos(𝜔0𝑡) + 𝜀(𝑡)                                     (1) 
+where 𝐴 and ω0 are the amplitude and angular frequency of the periodic signal 
+respectively, and 𝜀(𝑡) is the Gaussian white noise with mean zero and variance 𝐷 
+i.e. noise intensity. 
+The harmonic-Gaussian double-well potential which is a variant of a double-well 
+potential can be expressed as [26] 
+𝑈(𝑥) =
+𝑘
+2 𝑥2 + 𝛼exp(−𝛽𝑥2)                                         (2) 
+where two stable states and one unstable state are located at 𝑥± = ±√ln(2𝛼𝛽 𝑘
+⁄ ) 𝛽
+⁄  
+and 
+𝑥𝑢 = 0
+ respectively, 
+and 
+the 
+barrier 
+height 
+is 
+∆𝑈 = α −
+𝑘[1 + ln(2𝛼𝛽 𝑘
+⁄ )] (2𝛽)
+⁄
+. To ensure the stability of the harmonic-Gaussian 
+double-well potential, this condition ln(2𝛼𝛽 𝑘
+⁄ ) > 0 must be satisfied, further 𝑘 <
+2αβ. When 𝑘 = 1 , Fig. 1(a) shows the harmonic-Gaussian double-well potential 
+under different system parameter sets (𝛼, 𝛽), while Fig. 1(b) depicts those with 
+varying 𝑘. It is seen from Fig. 1(a) that adjusting the system parameter 𝛽 controls 
+the potential-well width whereas the potential-barrier height nearly keeps unchanged, 
+but varying 𝛼 changes the potential-barrier height whereas the potential-well width 
+nearly remains unchanged. Such a behavior is helpful to tune the potential-well width 
+
+ 5 / 27 
+ 
+and depth individually to activate the optimal harmonic-Gaussian double-well 
+potential SR. Meanwhile, it is found from Fig. 1(b) that adjusting 𝑘 can also change 
+the slope of the harmonic-Gaussian double-well potential. 
+ 
+Fig. 1 Harmonic-Gaussian double-well potentials under different parameter sets (a) 
+(𝛼, 𝛽) and (b) (𝛼, 𝛽, 𝑘). 
+The Langevin equation in Eq. (1) can be transformed as further [27] 
+∂ρ(𝑥,𝑡)
+∂𝑡
+= −
+𝜕
+𝜕𝑥 [−𝑘𝑥 + 2𝛼𝛽𝑥exp(−𝛽𝑥2) + 𝐴 cos(𝜔0𝑡)]𝜌(𝑥, 𝑡) + 𝐷
+𝜕2
+𝜕𝑥2 𝜌(𝑥, 𝑡) (3) 
+where 𝜌(𝑥, 𝑡) is the probability density function (PDF) of the stochastic process 
+𝑥(𝑡)  which denotes the transition trajectory of Brownian particles in the 
+harmonic-Gaussian double-well potential as time varies. The corresponding SPD 
+function can be denoted as 
+𝜌𝑠(𝑥, 𝑡) =
+𝑁(𝑡)
+√𝐷 exp [−
+∅(𝑥,𝑡)
+𝐷 ]                                           (4) 
+where 𝑁(𝑡) is the normalization constant and 𝑁(𝑡) = √𝐷 ∫
+exp[−∅(𝑥, 𝑡) 𝐷
+⁄ ]d𝑥
+∞
+−∞
+⁄
+, 
+and ∅(𝑥, 𝑡) is the generalized potential 
+∅(𝑥, 𝑡) = 𝑈(𝑥) − 𝑥𝐴 cos(𝜔0𝑡) .                                        (5) 
+Assuming that the periodic signal 𝐴 cos(𝜔0𝑡) can satisfy the requirement of small 
+parameters under approximate adiabatic conditions, i.e., 𝜔0  is larger than the 
+characteristic relaxation time in double potential wells [28]. Then, the transition rates 
+between the two stable states are given by the Kramers-like formulas [29] 
+𝑊±(𝑥, 𝑡) =
+√|𝑈′′(𝑥±,𝑡)𝑈′′(𝑥𝑢,𝑡)|
+2π
+exp [
+∅(𝑥±,𝑡)−∅(𝑥𝑢,𝑡)
+𝐷
+]                        (6) 
+where the notation | ∙ | denotes the absolute value and  
+(a) 
+(b) 
+
+ 6 / 27 
+ 
+𝑈′′(𝑥, 𝑡) = 𝑘 − 2𝛼𝛽exp(−𝛽𝑥2)(1 − 2𝛽𝑥2) 
+𝑈′′(𝑥𝑢, 𝑡) = 𝑘 − 2𝛼𝛽                                                  
+𝑈′′(𝑥±, 𝑡) = 2𝑘ln (
+2𝛼𝛽
+𝑘 )                                               (7) 
+∅(𝑥𝑢, 𝑡) = 𝑈(𝑥𝑢, 𝑡) − 𝑥𝑢𝐴 cos(𝜔0𝑡) = 𝛼 
+∅(𝑥±, 𝑡) = 𝑈(𝑥±, 𝑡)−𝑥±𝐴 cos(𝜔0𝑡) = 𝑘
+2𝛽 (1 + ln 2𝛼𝛽
+𝑘 ) ∓ 𝐴 cos(𝜔0𝑡) √ln(2𝛼𝛽 𝑘
+⁄ )
+𝛽
+ 
+When we introduce Eq. (7) into Eq. (6), we can obtain 
+𝑊±(𝑥, 𝑡) = √𝑘(2𝛼𝛽 − 𝑘)ln(2𝛼𝛽 𝑘
+⁄ )
+√2π
+ 
+× exp [−
+𝛼
+𝐷 +
+𝑘(1+ln(2𝛼𝛽 𝑘
+⁄ ))
+2𝛽𝐷
+∓ 𝐴 cos(𝜔0𝑡) √
+ln(2𝛼𝛽 𝑘
+⁄ )
+𝛽𝐷2
+]                     (8) 
+Furthermore, Eq. (8) can be transformed as 
+𝑊±(𝑥, 𝑡) = 𝑓(𝜇 ± 𝜂0 cos(𝜔0𝑡))                                      (9) 
+where 
+𝜇 =
+𝛼
+𝐷 −
+𝑘
+2𝛽𝐷 (1 + ln
+2𝛼𝛽
+𝑘 )                                           (10) 
+𝜂0 =
+𝐴
+𝐷 √
+ln(2𝛼𝛽 𝑘
+⁄ )
+𝛽
+                                                 (11) 
+Thus, we can simplify Eq. (8) as  
+𝑊±(𝑥, 𝑡) = 𝑓(𝜇 ± 𝜂0 cos(𝜔0𝑡)) =
+√2𝑘(2𝛼𝛽−𝑘)ln2𝛼𝛽
+𝑘
+2π
+exp[−(𝜇 ± 𝜂0 cos(𝜔0𝑡))] (12) 
+The response of the nonlinear system in Eq. (1) can be quantified using a classical 
+measure, i.e., SNR [30]. To derive its analytic expression, the power spectral density 
+of the system response can be described as  
+𝑆(Ω) = [1 −
+𝛼12𝜂02
+2(𝛼02+𝜔02)] (
+4𝑐2𝛼0
+𝛼02+𝜔02) +
+π𝑐2𝜂02𝛼12
+𝛼02+Ω2 [𝛿(Ω − 𝜔0) + 𝛿(Ω + 𝜔0)]        (13) 
+where 
+𝑐 = √
+ln(2𝛼𝛽 𝑘
+⁄ )
+𝛽
+                                                     (14) 
+𝛼1 = 𝛼0 =
+√2𝑘ln(2𝛼𝛽 𝑘
+⁄ )(2𝛼𝛽−𝑘)
+π
+exp(−𝜇)                                 (15) 
+Finally, the output SNR of the response of the overdamped harmonic-Gaussian 
+double-well potential system can be derived as 
+
+ 7 / 27 
+ 
+SNR =
+π𝑐2𝛼12𝜂02
+𝛼02+Ω2 |Ω=𝜔0 ×
+𝛼02+𝜔02
+4𝑐2𝛼0 [1 −
+𝛼12𝜂02
+2(𝛼02+𝜔02)]
+−1
+=
+π𝛼1𝜂02
+4
+[1 −
+𝛼12𝜂02
+2(𝛼02+𝜔02)]
+−1
+    (16) 
+Therefore, we can analyze the function between the output SNR and system 
+parameters using the analytic expression in Eq. (16). Figure 2 shows the output SNR 
+of overdamped harmonic-Gaussian double-well potential SR as system parameters 
+and noise intensity vary. It can be seen from Fig. 2(a) that the output SNR is a 
+nonmonotonic function of noise intensity 𝐷 under different 𝑘 and the peak value of 
+output SNR increases when 𝑘 raises, suggesting that adjusting 𝑘 is able to activate 
+the SR in the overdamped harmonic-Gaussian double-well potential system for 
+improving the output SNR. Similarly, adjusting 𝛼 and 𝛽 can also maximize the 
+output SNR, and the peak value of the output SNR declines as 𝛼 or 𝛽 increases but 
+the resonant noise intensity at the peak value becomes larger, as shown in Fig. 2(b) 
+and Fig. 2(c), respectively. We visualize the two-dimensional function among SNR 
+and two of system parameters (𝛼, 𝛽, 𝑘), as shown in Fig. 2(d), Fig. 2(c) and Fig. 2(d). 
+One can observe from Fig. 2(d) that a moderate parameter set (𝛼, 𝑘) can improve the 
+SNR of a given signal, whereas there exists a negative output SNR because the 
+harmonic-Gaussian double-well potential loses its stability when 𝑘 ≥ 2𝛼𝛽, resulting 
+in an antiresonance phenomenon. Meanwhile, we fix 𝑘 to express the output SNR as 
+a function of (𝛼, 𝛽) in Fig. 2(e), indicating that only an optimal matching between 𝛼 
+and 𝛽 can activate the overdamped harmonic-Gaussian double-well potential SR to 
+enhance the weak periodic signal embedded by a strong background noise. Similarly, 
+Fig. 2(f) also demonstrates that such a parameter matching is necessary to activate the 
+overdamped harmonic-Gaussian double-well potential SR. When 𝑘 ≥ 2𝛼𝛽, one can 
+also see the antiresonance from Fig. 2(e) and 2(f), respectively. The above results 
+demonstrate that the optimal parameter matching among 𝑘, 𝛼 and 𝛽 is able to 
+maximize the SR. 
+
+ 8 / 27 
+ 
+ 
+Fig. 2 SNR of overdamped harmonic-Gaussian double-well potential SR varies with 
+system parameters and noise intensity: SNR as a function of noise intensity under 
+different 𝑘 in (a), 𝛽 in (b) and 𝛼 in (c); SNR as a two-dimensional function of 
+(𝑘, 𝛼) in (d), (𝛽, 𝛼) in (e) and (𝑘, 𝛽) in (f). 
+Figure 3 depicts the SPD function and the corresponding system responses. The 
+SPD indicates the probability of Brownian particles to reside in double potential wells. 
+It is found from Fig. 3(a) that when 𝐷 = 0.3 the particles oscillate at the right 
+potential well located at 𝑥+ = √ln(2𝛼𝛽 𝑘
+⁄ ) 𝛽
+⁄  for activating intra-well SR, which is 
+demonstrated by the system response in Fig. 3(b) further. When we increase the noise 
+intensity 𝐷, the particles can jump across the potential barrier to go back and forth in 
+double wells for activating the inter-well SR marked in red in Fig. 3(a), whose system 
+response characterizes the eye-catching period marked in red in Fig. 3(b). When the 
+noise intensity is fixed as 𝐷 = 3, two peaks of SPD decline and the corresponding 
+(a) 
+(b) 
+(c) 
+(d) 
+(e) 
+(f) 
+
+ 9 / 27 
+ 
+system response marked in green becomes noisy. 
+ 
+Fig. 3 SPD functions and the corresponding system responses of overdamped 
+harmonic-Gaussian double-well potential SR under different noise intensity: (a) the 
+SPD functions and (b) the corresponding system responses. 
+ 
+3. Underdamped harmonic-Gaussian double-well potential SR  
+The underdamped harmonic-Gaussian double-well potential system subjected to a 
+periodic signal and noise can be described as [31] 
+d2𝑥
+d𝑡2 + 𝛾
+d𝑥
+d𝑡 = −
+𝜕𝑈(𝑥)
+𝜕𝑥
++ 𝐴 cos(𝜔0𝑡) + 𝜉(𝑡)                            (17) 
+where 𝛾 is the damped factor and 𝛾 > 0. Equation (17) can be transformed as [32] 
+{
+d𝑥
+d𝑡 = 𝑦
+d𝑦
+d𝑡 = −𝛾𝑦 − 𝑘𝑥 + 2𝛼𝛽𝑥exp(−𝛽𝑥2) + 𝐴 cos(𝜔0𝑡) + 𝜉(𝑡)
+                 (18) 
+Supposing that 𝐴 = 0, 𝐷 = 0, d𝑥 d𝑡
+⁄
+= 0 and d𝑥 d𝑡 = 0
+⁄
+, we can obtain three 
+singular points 
+(𝑥±
+𝑦±) = (±√
+ln(2𝛼𝛽 𝑘
+⁄ )
+𝛽
+0
+) , (𝑥𝑢
+𝑦𝑢) = (0
+0)                                 (19) 
+Let 
+𝜕𝑈(𝑥, 𝑦) 𝜕𝑥
+⁄
+ and 
+𝜕𝑈(𝑥, 𝑦) 𝜕𝑦
+⁄
+ mark 
+as 
+𝑈𝑥(𝑥, 𝑦)  and 
+𝑈𝑦(𝑥, 𝑦) 
+respectively, and then Eq. (18) can be rewritten as 
+{
+𝑈𝑥(𝑥, 𝑦) = 𝑦
+𝑈𝑦(𝑥, 𝑦) = −𝛾𝑦 − 𝑘𝑥 + 2𝛼𝛽𝑥exp(−𝛽𝑥2) + 𝐴 cos(𝜔0𝑡) + 𝜉(𝑡)          (20) 
+The linearization matrix of Eq. (18) can be calculated as 
+(𝑈𝑥𝑥(𝑥, 𝑦)
+𝑈𝑥𝑦(𝑥, 𝑦)
+𝑈𝑦𝑥(𝑥, 𝑦)
+𝑈𝑦𝑦(𝑥, 𝑦)) = (
+0
+1
+−𝑘 + 2𝛼𝛽exp(−𝛽𝑥2)[1 − 2𝛽𝑥2exp(−𝛽𝑥2)]
+−𝛾) 
+(a) 
+(b) 
+
+ 10 / 27 
+ 
+(21) 
+Further, the linearization matrix at the singular points (±√ln(2𝛼𝛽 𝑘
+⁄ ) 𝛽
+⁄
+, 0) can 
+be denoted as 
+(𝑈𝑥𝑥(𝑥, 𝑦)
+𝑈𝑥𝑦(𝑥, 𝑦)
+𝑈𝑦𝑥(𝑥, 𝑦)
+𝑈𝑦𝑦(𝑥, 𝑦)) |
+𝑥=±√ln(2𝛼𝛽 𝑘
+⁄ )
+𝛽
+,𝑦=0 = (
+0
+1
+−
+𝑘2
+𝛼𝛽 ln (
+2𝛼𝛽
+𝑘 )
+−𝛾)         (22) 
+By solving Eq. (22), the corresponding eigenvalues are calculated as 
+𝛽1,2 =
+−𝛾±√𝛾2−4𝑘2
+𝛼𝛽 ln(2𝛼𝛽
+𝑘 )
+2
+                                            (23) 
+Similarly, the linearization matrix at the singular point (0,0) is 
+(𝑈𝑥𝑥(𝑥, 𝑦)
+𝑈𝑥𝑦(𝑥, 𝑦)
+𝑈𝑦𝑥(𝑥, 𝑦)
+𝑈𝑦𝑦(𝑥, 𝑦)) |𝑥=0,𝑦=0 = (
+0
+1
+−𝑘 + 2𝛼𝛽
+−𝛾)                   (24) 
+The corresponding eigenvalues to the linearization matrix in Eq. (24) are 
+𝜆1,2 =
+−𝛾±√𝛾2+4(2𝛼𝛽−𝑘)
+2
+                                            (25) 
+Assuming that 𝜌(𝑥, 𝑦, 𝑡) is the PDF of the stochastic process in Eq. (18), the 
+corresponding the Fokker-Planck equation is [33] 
+𝜕𝜌(𝑥,𝑦,𝑡)
+𝜕𝑡
+= −
+𝜕𝑦
+𝜕𝑥 𝜌(𝑥, 𝑦, 𝑡) −
+𝜕
+𝜕𝑦 [−𝛾𝑦 − 𝑘𝑥 + 2𝛼𝛽𝑥exp(−𝛽𝑥2) +
+𝐴 cos(𝜔0𝑡)]𝜌(𝑥, 𝑦, 𝑡) + 𝛾𝐷
+𝜕2
+𝜕𝑦2 𝜌(𝑥, 𝑦, 𝑡)                                (26) 
+Then, the corresponding SPD function to Eq. (18) can be denoted as 
+𝜌s(𝑥, 𝑦, 𝑡) = 𝑁(𝑡)exp [−
+1
+𝐷 (
+1
+2 𝑦2 +
+𝑘
+2 𝑥2 + 𝛼exp(−𝛽𝑥2) − 𝑥𝐴 cos(𝜔0𝑡))]    (27) 
+where 𝑁(𝑡) stands for the normalization constant [34] 
+𝑁(𝑡) =
+1
+∫
+∫
+exp[−𝑈̂(𝑥,𝑦,𝑡)
+𝐷
+]d𝑥d𝑦
+∞
+−∞
+∞
+−∞
+                                       (28) 
+in which 𝑈̂(𝑥, 𝑦, 𝑡) denotes the generalized potential 
+𝑈̂(𝑥, 𝑦, 𝑡) =
+1
+2 𝑦2 +
+𝑘
+2 𝑥2 + 𝛼exp(−𝛽𝑥2) − 𝑥𝐴 cos(𝜔0𝑡)                  (29) 
+The 
+transition 
+rates 
+of 
+particles 
+at 
+the 
+singular 
+points 
+(𝑥±, 𝑦±) =
+(±√ln(2𝛼𝛽 𝑘
+⁄ ) 𝛽
+⁄
+, 0) can be calculated as [35] 
+𝑊±(𝑡) =
+√𝛽1𝛽2
+2π
+√−
+𝜆1
+𝜆2 exp [
+1
+𝐷 (
+𝑘
+2𝛽 (1 + ln (
+2𝛼𝛽
+𝑘 )) − 𝛼 ∓ √
+ln(2𝛼𝛽 𝑘
+⁄ )
+𝛽
+𝐴 cos(𝜔0𝑡))] 
+(30) 
+
+ 11 / 27 
+ 
+Finally, the analytic expression of the output SNR of the response of the 
+underdamped harmonic-Gaussian double-well potential system in Eq. (17) is derived 
+as 
+SNR =
+π𝑐2𝛼12𝜂02
+𝛼02+Ω2 |Ω=𝜔0 ×
+𝛼02+𝜔02
+4𝑐2𝛼0 [1 −
+𝛼12𝜂02
+2(𝛼02+𝜔02)]
+−1
+=
+π𝛼1𝜂02
+4
+[1 −
+𝛼12𝜂02
+2(𝛼02+𝜔02)]
+−1
+    (31) 
+where 
+𝛼1 = 𝛼0 =
+√𝛽1𝛽2
+π
+√−
+𝜆1
+𝜆2 exp(−𝑢)                                       (32) 
+Figures 4(a)-4(d) show the output SNR as noise intensity 𝐷 varies under different 
+system parameters. It is found from Fig. 4(a) that the output SNR increases and then 
+decreases as noise intensity increases, suggesting that a noise-induced underdamped 
+harmonic-Gaussian double-well potential SR happens. Moreover, increasing 𝑘 can 
+maximize the output SNR. Like this, adjusting 𝛾, 𝛼 and 𝛽 can also improve the 
+output SNR as shown in Fig. 4(b), Fig. 4(c) and Fig. 4(d) respectively, where the peak 
+value of output SNR and the resonant noise intensity are changed. Figures 4(e)-4(h) 
+show the output SNR as the function of system parameters for a given signal. 
+Adjusting the system parameters can activate the underdamped harmonic-Gaussian 
+double-well potential SR, as shown in Fig. 4(e)-4(h). Different from the overdamped 
+harmonic-Gaussian double-well potential SR, it is noticed from Fig. 4(e)-4(h) that the 
+antiresonance disappears in the underdamped one. That is because the damped factor 
+changes the stability of the nonlinear system. 
+
+ 12 / 27 
+ 
+ 
+Fig. 4 SNR of underdamped harmonic-Gaussian double-well potential SR varies with 
+system parameters and noise intensity: SNR as a function of noise intensity under 
+different 𝑘 in (a), 𝛾 in (b) and 𝛼 in (c), 𝛽 in (d); SNR as a two-dimensional 
+function of (𝛽, 𝛼) in (e), (𝛽, 𝑘) in (f), (𝛾, 𝑘) in (g) and (𝛾, 𝛼) in (h). 
+Figure 5 shows the SPD functions and the corresponding system responses. In Fig. 
+5(a), the SPD functions vary from asymmetrical peaks into two symmetrical ones as 
+noise intensity raises, suggesting that the underdamped harmonic-Gaussian 
+double-well potential SR changes from intra-well SR into inter-well one. In Fig. 5(b), 
+a weak period occurs when intra-well SR happens, and then the system response 
+(a) 
+(b) 
+(c) 
+(e) 
+(f) 
+(g) 
+(d) 
+(h) 
+
+ 13 / 27 
+ 
+becomes chaotic when the particles jump randomly between double wells and finally 
+is periodic when the inter-well SR takes place. 
+ 
+Fig. 5 SPD functions and the corresponding system responses of underdamped 
+harmonic-Gaussian double-well potential SR under different noise intensity: (a) the 
+SPD functions and (b) the corresponding system responses. 
+ 
+4. Application of harmonic-Gaussian double-well potential SR to enhance weak 
+fault characteristics of machinery 
+Rotating components of machinery including bearings, gears and rotors are more 
+prone to failures than fixed components due to contact fatigue, uneven lubrication, 
+misalignment and so on [36-38]. Therefore, how to detect weak fault characteristics of 
+rotating components in the early stage becomes a challenge [39]. Lots of scholars 
+have attempted to cancel or suppress the noise embedded in a signal to extract weak 
+fault characteristics further [40, 41]. On the contrary, we would apply 
+harmonic-Gaussian double-well potential SR to enhance weak fault characteristics of 
+machinery by using noise.  
+Four Rexnord ZA-2115 double row bearing run-to-failure experiments under the 
+rotating speed 2000 rpm and radial load 6000 lbs were performed to acquire the 
+bearing failure data by using accelerometers and a data acquisition card. The bearing 
+parameters are listed as below: the ball number 16, the pitch diameter 2.815 inches, 
+the contact angle 15.17 degrees and rolling element diameter 0.331 inches. The 
+bearing experimental rig is shown in Fig. 6(a) and the corresponding sensor 
+placement is illustrated in Fig. 6(b). This experimental rig is composed of four tested 
+bearings, an AC motor and rub belts [42]. In the bearing run-to-failure experiment, the 
+(a) 
+(b) 
+
+ 14 / 27 
+ 
+sampling frequency is 20 kHz and the sampling time is 1.024 seconds. 
+ 
+Fig. 6 Bearing test rigs and sensor placement illustration: (a) bearing test rigs and (b) 
+sensor placement illustration. 
+All failures occurred after exceeding designed life time of the bearing which is 
+more than 100 million revolutions. The data set describes a test-to-failure experiment 
+and consists of individual files that are 1.024-seconds vibration signal snapshots 
+recorded at 10-minutes intervals. The recording duration is from February 12, 2004 
+10:32:39 to February 19, 2004 06:22:39. At the end of the test-to-failure experiment, 
+outer race failure occurred in the tested bearing 1. The root mean square (RMS), an 
+effective health indicator, is often used to reflect the vibration intensity and monitor 
+the health state of bearings further. Therefore, RMS of bearing run-to-failure 
+experimental vibration data is calculated and depicted in Fig. 7 to observe the 
+degradation trend of the tested bearing 1. The degradation trend changes slowly with 
+slight fluctuation in the range of 0~88 hours and then raises into the larger RMS for 
+degradation marked in red dot in the zoomed RMS plot, suggesting that a tiny outer 
+race failure occurs in the early stage of the tested bearing 1. As time went on, it can be 
+seen from Fig. 7 that RMS keeps increasing, indicating that the outer race failure 
+becomes more and more severe. Finally, this test-to-failure experiment was stopped 
+because of strong vibration. In the test-to-failure experiment, the health state of the 
+tested bearing varies from normal to the early failure to severe failure to end of life, 
+which is consistent with the degradation trend reflected by RMS.  
+(a) 
+Accelerometers 
+Bearing 1 
+Bearing 2 
+Bearing 3 
+Bearing 4 
+Motor 
+Rub belts 
+(b) 
+
+ 15 / 27 
+ 
+ 
+Fig. 7 RMS of the bearing run-to-failure vibration signals. 
+The raw vibration signal at the 88.83th hour marked in red dot and its frequency 
+and envelope spectrum are depicted in Fig. 8. We cannot observe the eye-catching 
+spectral peaks at the theoretical outer race/inner race/roller/cage fault characteristic 
+frequency and its harmonics from both the frequency spectrum in Fig. 8(b) and the 
+zoomed envelope spectrum in Fig. 8(c), which are submerged by other spectral peaks 
+from background noise and excited by other normal components. Although we have 
+completed the bearing run-to-failure experiment and observed that a failure occurred 
+at the outer race of the tested bearing 1 by disassembling four tested bearings, we 
+cannot judge what time a tiny failure occurs at the outer race of the tested bearing 1 
+by virtue of the raw vibration signal and its spectrum in Fig. 8, which is very 
+important for early fault diagnosis and remaining useful life prediction. 
+
+ 16 / 27 
+ 
+Fig. 8 The vibration signal and its spectrum of outer race failure bearing at the early 
+stage: (a) the raw signal, (b) its frequency spectrum and (c) zoomed envelope 
+spectrum. 
+We apply the overdamped harmonic-Gaussian double-well potential SR to enhance 
+the weak fault characteristics in the early stage of the tested bearing 1. Figure 9 shows 
+the enhanced results of weak fault characteristics embedded in the raw vibration 
+signal, where the system parameters are given as 𝑘 = 1.1, 𝛼 = 1.2, 𝛽 = 0.24 and 
+the integral step is ℎ=0.035. The overdamped SR cannot be used to process 
+large-parameter signals directly and frequency-shifted and rescaling transform is 
+widely to solve it. Three key parameters of frequency-shifted and rescaling transform 
+in the overdamped harmonic-Gaussian double-well potential SR are given as below 
+by virtue of the theoretical outer race fault characteristic frequency 236.4 Hz that can 
+be calculated according to the structural parameters and rotating speed of the tested 
+bearing 1: the pass-band cut-off frequency 220 Hz, the stop-band cut-off frequency 
+200 Hz and the carrier frequency 200 Hz. These parameters in the frequency-shifted 
+and rescaling transform could be selected according to the reference [43]. One can 
+
+ 17 / 27 
+ 
+observe from Fig. 9 that the enhanced signal characterizes strong impacts and 
+dominant spectral peaks are at the outer race fault characteristic frequency and its 
+second harmonic of the tested bearing 1, suggesting that a tiny failure occurs at the 
+outer race of the tested bearing 1. However, the overdamped harmonic-Gaussian 
+double-well potential SR depends on the high-pass filter to perform the 
+frequency-shifted and rescaling transform, whose parameters are given artificially. 
+In the overdamped harmonic-Gaussian double-well potential SR-based enhanced 
+results, the low-frequency components of the raw vibration signal (<200Hz) have 
+been removed by using the frequency-shifted and rescaling transform. Moreover, the 
+overdamped harmonic-Gaussian double-well potential SR method would suppress the 
+components beyond the nonlinear filtering frequency band of overdamped 
+harmonic-Gaussian 
+double-well 
+potential 
+SR. 
+Although 
+overdamped 
+harmonic-Gaussian double-well potential SR method is able to utilize the noise 
+located in the nonlinear filtering frequency band of overdamped harmonic-Gaussian 
+double-well potential SR for enhancing weak fault characteristics, a part of noise is 
+removed. Therefore, the amplitude of the detected result in Fig. 9 is smaller than that 
+in Fig. 8. 
+ 
+Fig. 9 Overdamped harmonic-Gaussian double-well potential SR-based enhanced 
+results: (a) the enhanced signal and (b) its zoomed frequency spectrum. 
+fouter 
+2fouter 
+
+ 18 / 27 
+ 
+Further, we apply the underdamped harmonic-Gaussian double-well potential SR to 
+enhance weak fault characteristics embedded in the raw vibration signal, as shown in 
+Fig. 10 whose system parameters are given as 𝑘=1.2, 𝛼=1.1, 𝛽=0.24, 𝛾=0.33 and 
+ℎ =0.035. There are obvious repetitive transients in the enhanced signal and 
+eye-catching spectral peaks at the outer race fault characteristic frequency and its 
+second harmonic in the zoomed frequency spectrum as shown in Fig. 10(b). 
+Compared with the overdamped harmonic-Gaussian double-well potential SR-based 
+results, the underdamped one characterizes the higher spectral peaks at the outer race 
+fault characteristic frequency and its second harmonic in the zoomed frequency 
+spectrum. 
+ 
+Fig. 10 Underdamped harmonic-Gaussian double-well potential SR-based enhanced 
+results: (a) the enhanced signal and (b) its zoomed frequency spectrum. 
+For a comparison, we use the advanced robust local mean decomposition (RLMD) 
+[44, 45] to decompose the raw vibration signal of the tested bearing 1 into the product 
+functions (PFs) and a residual component (Res) for extracting weak fault 
+characteristics. The product functions and their zoomed envelope spectrum are shown 
+in Fig. 11(a) and Fig. 11(b), respectively. One cannot observe the obvious spectral 
+peaks at the outer race fault characteristic frequency and its harmonics from the 
+zoomed envelope spectrum.  
+fouter 
+2fouter 
+Rotating frequency 
+and its harmonics 
+
+ 19 / 27 
+ 
+ 
+Fig. 11 RLMD-based results: (a) product functions and (b) their zoomed envelope 
+spectrum. 
+In addition to signal decomposition methods, signal denoising or signal filtering 
+methods also have been widely applied to extract weak fault characteristics of 
+machinery. Among them, wavelet transform [46, 47] is typical to obtain a denoised 
+version of the raw signal by thresholding the wavelet coefficients. Here, the maximal 
+overlap discrete wavelet transform is used to denoise the signal with soft thresholding, 
+level 3 and db4 wavelet. The denoised signal and its zoomed envelope spectrum are 
+shown in Fig. 12 and Fig. 13, respectively. It is found from Fig. 12 that the wavelet 
+transform can cancel strong background noise, but we cannot see any fault 
+characteristics at the first sight from the zoomed envelope spectrum in Fig. 13.  
+(a) 
+(b) 
+PF1 
+PF2 
+PF3 
+PF4 
+PF5 
+Res 
+PF1 
+PF2 
+PF3 
+PF4 
+PF5 
+Res 
+Time [s] 
+Frequency [Hz] 
+Amplitude [g] 
+
+0.2
+h
+0
+0.2
+0
+0.2
+0.4
+0.6
+0.8
+0.1 
+0
+-0.1
+0
+0.2
+0.4
+0.6
+0.8
+1
+0.05
+0.05
+0
+0.2
+0.4
+0.6
+0.8
+0.02
+-0.02
+0
+0.2
+0.4
+0.6
+0.8
+×10-3
+?>>
+-5
+-10
+0
+0.2
+0.4
+0.6
+0.8
+X10-3
+5
+0
+-5
+0
+0.2
+0.4
+0.6
+0.8 20 / 27 
+ 
+ 
+Fig. 12 Undecimated wavelet transform-based denoised signals. 
+ 
+Fig. 13 The zoomed envelope spectrum of undecimated wavelet transform-based 
+denoised signals. 
+A classical symptom of rotating machines failures in vibration signals is the 
+presence of repetitive transients. Antoni [48] proposed an infogram method to capture 
+the signature of repetitive transients in time domain, which is the variant of classical 
+fast kurtogram method. This method is used to process the raw vibration signal for 
+extracting repetitive transients in time domain. The corresponding results are shown 
+in Fig. 14. Although it can see the slight repetitive transients from the filtered signal in 
+Fig. 14(b), it is difficult for us to identify the period of repetitive transients because of 
+strong background noise and other normal vibration components. The above 
+conclusion could be further confirmed by the squared envelope amplitude sepctrum of 
+
+ 21 / 27 
+ 
+the filtered signal in Fig. 14(b), in which we cannot see the eye-catching spectral 
+peaks at the outer race fault characteristic frequency and its harmonics.  
+ 
+Fig. 14 The detected results using infogram: (a) infogram and (b) the filtered signal 
+and its squared envelope amplitude sepctrum. 
+ 
+5. Conclusions 
+The overdamped and underdamped harmonic-Gaussian double-well potential SR 
+are investigated by deriving the output SNR and SPD functions. It is found that both 
+noise-induced SR and parameter-induced SR can be activated in the overdamped and 
+underdamped harmonic-Gaussian double-well potential systems. Moreover, since the 
+harmonic-Gaussian double-well potential in the range of 𝑘 ≥ 2𝛼𝛽 loses the stability, 
+we can observe the antiresonance, whereas adding the damped factor into the 
+overdamped harmonic-Gaussian double-well potential system can change the stability, 
+resulting that the antiresonance disappears. Above conclusion is applicable under all 
+parameters. 
+Finally, 
+we 
+apply 
+both 
+the 
+overdamped 
+and 
+underdamped 
+harmonic-Gaussian double-well potential SR to enhance weak fault characteristics of 
+bearings for incipient fault identification, where the corresponding parameters would 
+be adjusted or optimized instead of all parameters are applicable to activate the 
+optimal SR. The weak fault characteristics are enhanced successfully to identify the 
+early failure of bearings, which somewhat outperforms to the RLMD, wavelet 
+transform and infogram-based results. But the SR-based methods depend on the prior 
+knowledge of the signals to be detected or structural parameters and rotating speeds of 
+bearings, and cannot detect unknown multiple-frequency and multiple-component 
+(a) 
+(b) 
+
+ 22 / 27 
+ 
+coupled signals without any prior knowledge. Therefore, we would study the 
+SR-based signal decomposition method by using noise to decouple and detect 
+unknown 
+multiple-frequency 
+and 
+multiple-component 
+signals, 
+especially 
+time-varying nonstationary signals in the future. 
+ 
+Acknowledgments 
+This research was supported by Foundation of the State Key Laboratory of 
+Performance Monitoring and Protecting of Rail Transit Infrastructure of East China 
+Jiaotong University (HJGZ2021114), Laboratory of Yangjiang Offshore Wind Power 
+(YJOFWD-OF-2022A08), Zhejiang Provincial Natural Science Foundation of China 
+(LQ22E050003), National Natural Science Foundation of China (52205569), Ningbo 
+Science and Technology Major Project (2020Z110, 2022Z057, 2022Z002), National 
+Natural Science Foundation of China (51905349, 62001210, U2013603), Natural 
+Science Foundation of Guangdong Province (2022A1515010126, 2020A1515011509), 
+Ningbo Natural Science Foundation (2022J098) and also sponsored by K.C. Wong 
+Magna Fund in Ningbo University. The Spanish State Research Agency (AEI) and the 
+European 
+Regional 
+Development 
+Fund 
+(ERDF) 
+under 
+Project 
+No. 
+PID2019-105554GB-I00 is also aknowledged. 
+ 
+Conflicct of Interest 
+The authors declare that they have no conflict of interest. 
+ 
+Data availability 
+The datasets generated during and/or analysed during the current study are 
+available from the corresponding author on reasonable request. 
+ 
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+ 
+ 
+ 
+ 
+ 
+ 
+Zhejiang Provincial Key Laboratory of Part Rolling Technology 
+ 
+School of Mechanical Engineering and Mechanics • Ningbo University 
+ 
+Ningbo University 
+ 
+ 
+July 24, 2022 
+ 
+RE: “Harmonic-Gaussian double-well potential stochastic resonance with its 
+application to enhance weak fault characteristics of machinery” by Zijian Qiao, Shuai 
+Chen, Zhihui Lai, Shengtong Zhou and Miguel A. F. Sanjuán (Manuscript Number: 
+NODY-D-22-01167) 
+ 
+ 
+ 
+Dear Editor, 
+ 
+We have carefully revised our paper taking into account your suggestions and the 
+comments of the reviewers. We have uploaded the revised version and the revision 
+notes. Thank you very much for processing our paper. 
+ 
+We appreciate very much the constructive comments and suggestions provided by the 
+reviewers. They have been incorporated in the revised version of this paper. Major 
+changes made in the paper are marked in blue. The following summarizes our 
+response to each point raised by each reviewer. 
+ 
+We would like to thank the three reviewers for their valuable comments and 
+constructive suggestions to improve the quality of this paper. We have fully 
+considered their comments and suggestions and made revisions accordingly. The 
+major revisions are highlighted by BLUE color. The point-to-point explanations and 
+revisions are listed as follow. 
+ 
+We have taken into full consideration all comments of the three referees and made a 
+thorough revision of the paper. 
+ 
+ 
+ 
+Sincerely yours, 
+ 
+Zijian Qiao Ph.D 
+Shuai Chen M.S. 
+Zhihui Lai Ph.D 
+Shengtong Zhou Ph.D 
+Miguel A. F. Sanjuán Ph.D 
+Cover Letter
+Click here to access/download;attachment to
+manuscript;Cover Letter.doc
+Click here to view linked References
+
+波
+大
+漢 
+Page 1 of 1 
+Highlights 
+ 
+RE: “Harmonic-Gaussian double-well potential stochastic resonance with its application to 
+enhance weak fault characteristics of machinery” by Zijian Qiao, Shuai Chen, Zhihui Lai, 
+Shengtong Zhou and Miguel A. F. Sanjuán 
+ 
+ Harmonic-Gaussian double-well potential SR is investigated by deriving and measuring 
+the output SNR. 
+ Steady-state probability density functions are used to evaluate the transition rates of 
+particles in the harmonic-Gaussian double-well potential. 
+ Parameter-induced SR, noise-induced SR and antiresonance are observed by analyzing 
+the output SNR.  
+ Harmonic-Gaussian double-well potential SR is applied to enhance weak fault 
+characteristics of machinery successfully. 
+Highlights
+Click here to access/download;attachment to
+manuscript;Highlights.doc
+Click here to view linked References
+
+Declaration of Interest Statement 
+The authors declare that they have no conflict of interest. 
+Declaration of Interest Statement
+Click here to access/download;attachment to
+manuscript;Declaration of Interest Statement.docx
+Click here to view linked References
+
+Manuscript Number: NODY-D-22-01167R1 
+Title: Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance 
+weak fault characteristics of machinery 
+Response to Editor 
+There are still some minor comments raised by one of the reviewers needed to be addressed. A minor 
+revision is recommended. 
+Response: We appreciate the constructive comments from two reviewers. According to their 
+comments and suggestions, we have made a thorough revision for the manuscript and have addressed 
+all points raised by each reviewer. The major changes made in the manuscript are marked in BLUE 
+color. We also include the major changes of the manuscript into the response point by point. For 
+convenient review, the page numbers or paragraph numbers of the revision in the manuscript are 
+cited below. 
+We hope that this revised submission is satisfactory. The authors thank editors and anonymous 
+reviewers for their valuable and helpful comments to revise and improve our manuscript. 
+Compressed File
+Click here to access/download;Compressed File;Response to
+Reviewers.docx
+
+Manuscript Number: NODY-D-22-01167R1 
+Title: Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance 
+weak fault characteristics of machinery 
+Response to Reviewer #3 
+The authors have correctly taken into consideration the reviewers comments. 
+Response: Thanks for your recommendation.  
+ 
+
+Manuscript Number: NODY-D-22-01167R1 
+Title: Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance 
+weak fault characteristics of machinery 
+Response to Reviewer #5 
+The paper presents that the overdamped or underdamped harmonic Gaussian double-well potential 
+SR methods characterize a better performance to detect a weak signal. The work is organized in a 
+clear form, but the technical content looks not high and there are important aspects that are not 
+discussed. 
+1. Whether the analysis conclusion obtained is the conclusion under these special parameters or 
+whether all parameters are applicable? Please give some explanations. 
+Response: According to the comments of the reviewer #5, we think that two analysis conclusion 
+obtained could be illustrated whether under these special parameters or all parameters. 
+In Sections 2 and 3: Overdamped and underdamped harmonic-Gaussian double-well 
+potential SR 
+We investigate the SR in the cases of overdamped and underdamped harmonic-Gaussian 
+double-well potential systems subjected to noise and a periodic signal. We derive and measure the 
+analytic expression of the output signal-to-noise ratio (SNR) and the steady-state probability density 
+(SPD) function under approximate adiabatic conditions. When the harmonic-Gaussian double-well 
+potential loses its stability, we can observe the antiresonance phenomenon, whereas adding the 
+damped factor into the overdamped system can change the stability of the harmonic-Gaussian 
+double-well potential, resulting that the antiresonance behavior disappears in the underdamped 
+system. Although above analysis conclusion is obtained under these special parameters, other 
+parameters would depict the same findings. As a result, the analysis conclusion obtained in two 
+
+sections is applicable under all parameters. 
+In Section 4: Application of harmonic-Gaussian double-well potential SR to enhance weak 
+fault characteristics of machinery 
+Harmonic-Gaussian double-well potential stochastic resonance is a typical nonlinear filter with the 
+adjusting parameters in which the noise embedded in a signal is able to be utilized to enhance weak 
+useful information by activating the stochastic resonance phenomenon. The stochastic resonance 
+phenomenon could be activated when the optimal matching among the weak useful information, 
+noise and these parameters of stochastic resonance. For a different signal, therefore, these parameters 
+of the harmonic-Gaussian double-well potential stochastic resonance must be tuned to activate the 
+stochastic resonance phenomenon for enhancing weak useful information by using noise. As a result, 
+applying harmonic-Gaussian double-well potential stochastic resonance to enhance weak fault 
+characteristics of machinery, these parameters of harmonic-Gaussian double-well potential stochastic 
+resonance would be adjusted or optimized instead of all parameters are applicable to activate the 
+optimal stochastic resonance phenomenon. (See the conclusion in Section 5 page 21, which is 
+marked in BLUE) 
+ 
+2. "Noise is ubiquitous and unwanted in detecting weak signals", This sentence is repeated and can 
+be deleted. 
+Response: Thanks for your suggestions. We have deleted it in Abstract. (See the abstract in page 1, 
+which is marked in BLUE) 
+ 
+
+3 "Key words" write too long. 
+Response: Thanks for your suggestions. We have reduced the key words as below: The benefits of 
+noise, weak signature enhancement, fault identification, fault diagnosis. (See the key words in page 2, 
+which is marked in BLUE) 
+ 
+4 "The recording duration is from February 12, 2004 10:32:39 to February 19, 2004 06:22:39". 
+Why was it 18 years ago？ 
+Response: That is because Four Rexnord ZA-2115 double row bearing run-to-failure experiments 
+under the rotating speed 2000 rpm and radial load 6000 lbs were performed in 2004 year. In future 
+work, we would perform and conduct new bearing run-to-failure experiments. Now, our team is 
+designing the new experimental platform and project to acquire new bearing and gear vibration data. 
+Thanks for your understanding.  
+ 
+

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+arXiv:2301.04203v1  [math.CV]  10 Jan 2023
+ZERO DISTRIBUTION OF RANDOM BERNOULLI POLYNOMIAL MAPPINGS
+TURGAY BAYRAKTAR & C¸˙I˘GDEM C¸EL˙IK
+ABSTRACT. In this note, we study asymptotic zero distribution of multivariable full sys-
+tem of random polynomials with independent Bernoulli coefficients. We prove that with
+overwhelming probability their simultaneous zeros sets are discrete and the associated
+normalized empirical measure of zeros asymptotic to the Haar measure on the unit torus.
+1. INTRODUCTION
+A random Kac polynomial on the complex plane is of the form
+(1.1)
+fd(z) =
+d
+�
+j=0
+ajzj
+where the coefficients aj are independent copies of the (real or complex) standard Gauss-
+ian. A classical result due to Kac, Hammersley and Shepp & Vanderbei [20, 16, 23] asserts
+that almost surely the normalized empirical measure of zeros δZ(fd) := 1
+d
+�
+fd(ζ)=0 δζ, con-
+verges to normalized arc length measure on S1 := {|z| = 1} as d → ∞. Asymptotic
+zero distribution of Kac polynomials with i.i.d. discrete random coefficients have also
+been studied extensively (see eg. [22, 14]). More recently, Ibragimov and Zaporozhets
+[19] proved that the empirical measure of zeros δZ(fd) almost surely converges to the the
+normalized arc length measure if and only if the moment condition E[log(1 + |ai|)] < ∞
+holds. This property can be considered as a global universality property of the zeros of
+random polynomials (see also [27] for a local version).
+Building upon the work of Shiffman and Zelditch [26], equilibrium distribution of
+random systems of polynomials with Gaussian coefficients was obtained by Bloom &
+Shiffman [9] and Shiffman [24]. More recently, these results were generalized for inde-
+pendent identically distributed (i.i.d.) random coefficients with bounded density [1, 2].
+We refer the reader to the survey [4] and references therein for the state of the art. On
+the other hand, asymptotic zero distribution of random polynomial mappings with dis-
+crete random coefficients remained open (cf. [3, 8, 5]). In this note, we study asymptotic
+zero distribution of multivariable full system of random polynomials with independent
+Bernoulli coefficients.
+1.1. Statement of the results. A random Bernoulli polynomial is of the form
+fd,i(x) =
+�
+|J|≤d
+αi,JxJ ∈ C [x1, . . . , xn]
+T.B. and C¸.C¸. are partially supported by T¨UB˙ITAK grant ARDEB-1001/119F184.
+1
+
+where xJ = xj1
+1 . . . xjn
+n and αi,J are ±1 Bernoulli random variables. Throughout this work,
+we consider systems (fd,1, . . . , fd,n) of random Bernoulli polynomials with independent
+coefficients. We write f d = (fd,1, . . . , fd,n) for short. We denote the collection of all
+systems of polynomials in n variables and of degree d by Polyn,d that is endowed with
+the product probability measure Probd.
+Theorem 1.1. Let f d = (fd,1, . . . , fd,n) be a system of random polynomials with independent
+±1 valued Bernoulli coefficients. Then there exists a dimensional constant K = K(n) >
+0 and an exceptional set En,d ⊂ Polyn,d such that Probd(En,d) ≤ K/d and for all f d ∈
+Polyn,d(A)\En,d the simultaneous solutions of the system f d are isolated with #Z(f d) = dn.
+For a system f d ∈ Polyn,d, if the simultaneous zeros Z(f d) are isolated we denote
+the corresponding normalized empirical measure by δZ(f d). That is δZ(fd) is a probabil-
+ity measure supported on the isolated zeros. We also let νHaar denote the Haar measure
+of (S1)n of total mass 1. As an application of Theorem 1.1 together with a determinis-
+tic equidistribution result [13, Theorem 1.7], we obtain asymptotic zero distribution of
+random Bernoulli polynomial mappings:
+Corollary 1.2. Let f d = (fd,1, . . . , fd,n) be system of random polynomials with independent
+±1 valued Bernoulli coefficients and En,d ⊂ Polyn,d be as in Theorem 1.1. Then for each
+sequence f d ∈ Polyn,d \ En,d we have
+lim
+d→∞ δZ(fd) = νHaar.
+in weak topology. In particular, δZ(f d) → νHaar in probability Probd as d → ∞.
+Finally, we consider the measure valued random variables
+(1.2)
+�Z(f d) =
+��
+ξi∈Z(f d) δ(ξi)
+for f d ∈ Polyn,d \ En,d
+0
+otherwise.
+and define the expected zero measure by
+(1.3)
+�
+E[ �Z(f d)], ϕ
+�
+=
+�
+P olyn,d\En,d
+�
+ξi∈Z(f d)
+ϕ(ξi) dProbd(f d)
+where ϕ is a continuous function with compact support in Cn and En,d denote the excep-
+tional set given by Theorem 1.1.
+Theorem 1.3. Let f d = (fd,1, . . . , fd,n) be a system of random polynomials with independent
+±1 valued Bernoulli coefficients. Then
+lim
+d→∞ d−nE[Z(f d)] = νHaar
+in weak topology.
+The outline of this work as follows. In §2, we introduce some algebraic background
+on resultants. In particular, we recall multi-polynomial resultant and sparse resultant for
+polynomial systems [15, 11] as well as directional resultant [12]. In §3, we prove the
+main result Theorem 1.1. Finally, in §4 we prove Theorem 1.3.
+2
+
+2. PRELIMINARIES
+In this section, we collect some basic facts and algebraic background related to our
+results. More precisely, we discuss the multi-homogenous (classical) resultant and the
+sparse eliminant as well as the relation of these two notions. For a detailed account of the
+subject and proofs we refer the reader to [15, 11]. We also discuss the sparse resultant
+introduced by D’Andrea and Sombra, and corresponding directional sparse resultants
+[13, 12].
+2.1. Lattice points, polytopes. For a nonempty subset P ⊂ Rn, we denote its convex
+hull in Rn by conv(P). For two nonempty convex sets Q1, Q2, their Minkowski sum is
+defined as
+Q1 + Q2 := {q1 + q2 : q1 ∈ Q1, q2 ∈ Q2}
+and for λ ∈ R, the scaled polytope is of the form
+λQ := {λq : q ∈ Q}.
+It is well known that V oln(d1Q1 + . . . + dnQn) is a homogenous polynomial of degree n in
+the variables d1, . . . , dn ∈ Z+ where V oln denotes the normalized volume of the subsets
+in Rn with respect to the Lebesgue measure. The coefficient of the monomial d1 . . . dn
+is called the mixed volume of Q1, . . . , Qn and denoted by MV (Q1, . . . , Qn). One can use
+the polarization formula to compute the mixed volume of the convex sets Q1, . . . , Qn.
+Namely,
+MVn(Q1, . . . , Qn) =
+n
+�
+k=1
+�
+1≤j1≤...≤jk≤n
+(−1)n−kV oln(Qj1 + . . . + Qjk).
+In particular, if Q = Q1 = . . . = Qn then
+MVn(Q) := MVn(Q, . . . , Q) = n!V oln(Q).
+For a convex set Q ⊂ Rn its support function sQ : Rn → R is defined by
+(2.1)
+sQ(v) := inf
+q∈Q ⟨q, v⟩
+where ⟨·, ·⟩ represents the Euclidean inner product of Rn. Then the equation
+⟨q, v⟩ = sQ(v)
+defines supporting hyperplane of Q and v is called an inward pointing normal. The inter-
+section of Q with the supporting hyperplane in the direction v ∈ Rn is denoted by
+(2.2)
+Qv := {q ∈ Q : ⟨q, v⟩ = sQ(v)}.
+Qv is called the face of Q determined by v. If Qv has codimension 1, it is called a facet of
+Q.
+3
+
+2.2. Resultant of polynomial systems.
+2.2.1. Multipolynomial Resultant. We consider homogenous polynomials of degree di
+Fi(t0, . . . , tn) =
+�
+|J|=di
+ui,JtJ
+for i = 0, . . . , n where J is a multi-index (j0, . . . , jn) and tJ indicates the monomial
+tj0
+0 · · · tjn
+n which is of degree |J| = �n
+i=0 ji. One can see that the homogenous polynomi-
+als of degree di form an affine space by identifying �
+|J|=di ui,JtJ with point (ui,J)|J|=di ∈
+CN(di), where N(di) =
+�n+di−1
+n−1
+�
+.
+Letting N := �n
+i=0 N(di), recall that the incidence variety is defined by
+W =
+�
+(a, t) ∈ CN × Pn : F0(a0, t) = · · · = Fn(an, t) = 0
+�
+.
+We let π : CN × Pn → CN be the projection onto first coordinate. By Projective Extension
+Theorem (see eg. [11]) the image π(W) forms a variety in the affine space CN.
+Definition 2.1. The multipolynomial resultant Resd0,...,dn is defined as the irreducible unique
+(up to a sign) polynomial in Z[a0, . . . , an] which is the defining equation of the variety π(W).
+The resultant of the homogeneous polynomials F0, . . . , Fn is the evaluation of Resd0,...,dn at
+the coefficients of F0, . . . , Fn and it is denoted by Resd0,...,dn(F0, . . . , Fn).
+Note that if d0 = . . . = dn = 1, then the evaluation of multipolynomial resultant
+Resd0,...,dn at the coefficients of F0, . . . , Fn is the determinant of the coefficient matrix. For
+more general cases, we have the following result:
+Theorem 2.2 ([15],[11]). Let F0, . . . , Fn ∈ C[x0, . . . , xn] be homogenous polynomials of
+positive total degrees d0, . . . , dn. Then the system F0 = . . . = Fn = 0 has a solution in the
+complex projective space Pn if and only if Resd0...,dn(F0, . . . , Fn) = 0.
+Theorem 2.2 gives a characterization to determine the existence of nontrivial solutions
+for the systems of homogenous polynomials based on the coefficients of the polynomials
+in the system. However, not all the systems of equations are homogenous, and in the
+power series expansions not all the monomial terms appear. Hence, we need to introduce
+a more general version of the multi-homogenous resultant.
+2.2.2. Sparse Eliminant. Following [15], we will recall the definition of sparse resultant.
+Let A0, . . . , An be a non-empty finite subsets of Zn, and let ui = {ui,a}a∈Ai be a group of
+#Ai variables, i = 0, . . . , n and set u = {u0, . . . , un} . For each i, the general Laurent
+polynomial fi with support supp(fi) = Ai given by
+f(x) =
+�
+a∈Ai
+ui,axa ∈ C[u][x±1
+1 , . . . , x±1
+n ].
+We let A = (A0, . . . , An) and consider the incidence variety in this setting defined by
+(2.3)
+WA =
+�
+(u, x) ∈
+n
+�
+i=0
+P(CAi) × (C∗)n : f0(ui, x) = · · · = fn(un, x) = 0
+�
+.
+4
+
+Consider the canonical projection on the first coordinate
+πA :
+n
+�
+i=0
+P(CAi) × (C∗)n →
+n
+�
+i=0
+P(CAi)
+and let πA(WA) denote the Zariski closure of WA under the projection π.
+Definition 2.3. The sparse eliminant, denoted by ResA, is defined as follows: if the variety
+πA(WA) has codimension 1, then the sparse eliminant is the unique (up to sign) irreducible
+polynomial in Z[u] which is the defining equation of πA(WA). If codim(πA(WA)) ≥ 2, then
+ResA is defined to be the constant polynomial 1. The expression
+ResA(f0, . . . , fn)
+is the evaluation of ResA at the coefficients of f0, . . . , fn.
+Example 2.4. For A0 = {0} , A1 = {0, 1} ⊂ Z, we have that ResA0,A1 = ±u00.
+The classical resultant Resd0,...,dn is the special case of the sparse eliminant. Indeed, let
+Ai be the set of all integer points in the di-simplex, i.e., Ai = diΣn ∩ Zn and Σn is the
+standard unit simplex, that is,
+diΣn := {(a0, . . . , an) ∈ Rn+1 : aj ≥ 0,
+�
+j
+aj ≤ di}.
+Following [11] and [15], for simplicity let all the sparse polynomials f0, . . . , fn have the
+same support Ad = dΣn ∩ Zn for some positive integer d and consider the system
+(2.4)
+
+
+
+f0 = u01xα1 + . . . + u0dxαn = 0
+...
+fn = un1xα1 + . . . + undxαn = 0
+We also let t0, . . . , tn be the homogenous coordinates which are related to x1, . . . , xn by
+xi = ti/t0. Then we define the homogenous polynomials
+(2.5)
+Fi(t0, . . . , tn) = td
+0fi(t1/t0, . . . , tn/t0) = td
+0fi(x1, . . . , xn),
+for 0 ≤ i ≤ n. This method gives n+1 homogenous polynomials of total degree d in
+the variables t0, . . . , tn and this procedure is independent of the choice of homogeneous
+coordinates.
+Proposition 2.5 ([11]). Let Ad = dΣn ∩Zn and consider the systems of polynomials F and
+f as above. Then
+ResA(f0, . . . , fn) = ±Resd,...,d(F0, . . . , Fn),
+where A = (Ad, . . . , Ad).
+Using the above proposition, we can give a version of Theorem 2.2 as follows.
+5
+
+Corollary 2.6. Let f = (f1, . . . , fn) be a system of polynomials with supp(fi) = Ad for
+i = 1, . . . , n. Assume that the system F = (F0, . . . , Fn) consists the homogenizations of fi
+according to process in (2.5) and denote the set of simultaneous nonzero solutions of F by
+Z(F ). Suppose that Z(F ) ∩ H∞(t0) = ∅ where H∞(t0) is the hyperplane at infinity for t0 =
+0. Then the system of polynomials f = 0 has no solution if and only if ResAd(f0, . . . , fn) ̸= 0.
+Proof. If ResAd(f0, . . . , fn) ̸= 0, then by definition of the sparse resultant the system
+f0(x) = . . . = fn(x) = 0
+has no solution. Conversely, letting Fi be the homogenization of fi as in (2.5) with the
+corresponding variable t = (t0, . . . , tn), i.e. Fi(t) = td
+0fi(x). If the system of polynomials
+f = 0 has no solution then Fi(t) = 0 for i = 1, . . . , n if and only if t0 = 0 which contradicts
+our assumption. Hence, by Theorem 2.2 we have
+±ResAd(f0, . . . , fn) = Resd0,...,dn(F0, . . . , Fn) ̸= 0.
+□
+2.2.3. Sparse Resultant. In spite of being a generalization of the multipolynomial resul-
+tant and involving considerable large amount of the system of polynomials, the sparse
+eliminant does not satisfy some essential properties which is necessary in many applica-
+tions, such as additivity property and Poisson formula. In 2014, D’Andrea and Sombra
+[12] introduced the following version of the sparse resultant which has the desired fea-
+tures.
+Definition 2.7. The sparse resultant, denoted by ResA, is defined as any primitive poly-
+nomial in Z[u] that is the defining equation of the direct image of WA, (πA)∗(WA) =
+deg(πA|WA)πA(WA) if this variety has codimension one, and otherwise we set ResA = 1.
+The expression
+ResA(f0, . . . , fn)
+is the evaluation of ResA at the coefficients of f0, . . . , fn.
+According to this definition, the sparse resultant is not irreducible but it is a power of
+the irreducible sparse eliminant, i.e.,
+ResA = ±Res
+deg(πA|WA)
+A
+where deg(πA|WA) is the degree of the projection πA. We also remark that ResA ̸= 1
+whenever ResA ̸= 1. For more details we refer the reader to the manuscripts [12] and
+[13].
+Example 2.8. Let A0 = A1 = A2 = {(0, 0), (2, 0), (0, 2)}. Then ResA = det(ui,j) and
+ResA = ±[det(ui,j)]4.
+6
+
+2.2.4. Directional Resultant. For a subset B ⊂ Zn and a polynomial f(x) = �
+b∈B βbxb
+with support B, we write
+Bv := {b ∈ B : ⟨b, v⟩ = sQ(v)}
+and
+f v(x) =
+�
+b∈Bv
+βbxb
+where Q = conv(B) and v ∈ Rn and sconv(B)(v) is defined as equation (2.1).
+Definition 2.9. Let A1, . . . , An ⊂ Zn be a family of n non-empty finite subsets, v ∈ Zn\{0},
+and v⊥ ⊂ Rn the orthogonal subspace. Then there exists bi,v ∈ Zn such that
+Av
+i − bi,v ⊂ Zn ∩ v⊥
+for i = 1, . . . , n. The resultant of A1, . . . , An in the direction of v, denoted ResAv
+1 ,...,Avn is
+defined as the resultant of the family of the finite subsets Av
+i − bi,v.
+Let fi ∈ C[x±1
+1 , . . . , x±1
+n ] be Laurent polynomials with support supp(fi) ⊂ Ai i = 1, . . . , n.
+For each i = 1, . . . , n, we write f v
+i = xbi,vgi,v for a Laurent polynomial gi,v ∈ C[Zn ∩ v⊥] ≃
+C[y±1
+1 , . . . , y±1
+n−1] with supp(gi,v) ⊂ Av
+i − bi,v. The expression
+ResAv
+1 ,...,Avn(f v
+1 , . . . , f v
+n)
+is defined as the evaluation of this resultant at the coefficients of the gi,v.
+We remark that the definition of directional resultant is independent of the choice of
+the vector bi,v (see [12, Proposition 3.3]). Moreover, the directional resultant ResAv
+1 ,...,Avn ̸=
+1 only if the direction vector v is an inward pointing normal to a facet of the Minkowski
+sum �n
+i=1 conv(Ai). Therefore, the nontrivial directional resultants of the family A1, . . . , An
+is finitely many.
+Example 2.10. Let f(x) = a0 + . . . + anxn ∈ C[x] be a polynomial of degree n. Then the
+nontrivial directional resultants are
+ResA(f v) =
+�
+±a0
+if
+v = 1,
+±an
+if
+v = −1
+for the polytope conv(A) = [0, n] ⊂ R.
+3. EQUIDISTRIBUTION OF ZEROS
+3.1. Random Polynomial Systems. First, we recall a theorem of Kozma and Zeitouni
+[21] asserts that overdetermined random Bernoulli polynomial systems have no common
+zeros with overwhelming probability:
+Theorem 3.1. Let f1, . . . , fn+1 ∈ Z[x1, . . . , xn] be n + 1 independent random Bernoulli
+polynomials of degree d and
+P(d, n) := Probd{∃x ∈ Cn : fi(x) = 0 for i = 1 . . . , n + 1}
+denote the probability that the system f1 = . . . = fn+1 = 0 has a common solution. Then
+there exists a dimensional constant K = K(n) < ∞ such that
+P(d, n) ≤ K/d
+7
+
+for all d ∈ Z+.
+Next, we prove our main result:
+Proof of Theorem 1. Let fd,i be a random Bernoulli polynomial of the form
+(3.1)
+fd,i =
+�
+|J|≤d
+αi,JxJ ∈ Z[x1, . . . , xn],
+where {αi,J} is a family of independent Bernoulli random variables for i = 1, . . . , n.
+We investigate the directional resultants of the system f for all nonzero primitive di-
+rection vectors v ∈ Zn. By [12, Proposition 3.3] it is enough to check the inward normals
+to the Minkowski sum of the supports ndΣn which has n + 1 facets with n + 1 inward
+normals given by vm = em for m = 1, . . . , n and vn+1 = − �n
+m=1 em where {em}n
+m=1 is
+the standard basis of Rn.
+For vm = em the intersection of a support A with the supporting hyperplane in the
+direction em is of the form
+(3.2)
+Avm =
+�
+(j1, . . . , jn) ∈ A : jm = 0,
+n
+�
+l=1
+jl ≤ d
+�
+m = 1, . . . , n. Hence, the polynomials f vm
+i
+can be written as
+(3.3)
+f vm
+i
+:=
+�
+J∈Avm
+αi,JxJ
+for i = 1, . . . , n. Note that polynomials f vm
+i
+depend on n − 1 variables. Following the
+Definition 2.9, if we choose the vector bi,vm = 0 such that Avm − bi,vm ⊂ Zn ∩ vm⊥,
+we see that the functions gi,vm := f vm
+i
+satisfies the equation f vm
+i
+= xbi,vmgi,vm for each
+i = 1, . . . , n.
+Recall that for two univariate polynomials h1, h2 ∈ C[x], their resultant Res(h1, h2) is
+zero if and only if h1 and h2 have a common solution in C. Therefore, if n = 2 the
+necessary and sufficient condition for g1,vm and g2,vm have zero resultant is that they
+have a common zero. Theorem 3.1 implies that there exists a constant Km which is
+independent of d so that the aforementioned event has probability at most Km/d.
+On the other hand, when n > 2, we perform the homogenization process to each (n−1)
+variable polynomial gi,vm for i = 1, . . . , n as described in equation (2.5). We obtain the n
+variable homogenous polynomials Gi,vm of the form
+(3.4)
+Gi,vm(t, x) =
+�
+J∈Avm
+αi,Jtd−|J|xJ.
+In order to compare the sparse resultant of the polynomials gi,vm and the multipolynomial
+resultant of the homogeneous polynomials Gi,vm, we check the conditions of Corollary
+2.6. Let Z(G) be the set of nontrivial solutions of the system G = (G1,vm, . . . , Gn,vm)
+and suppose that G has a solution ξ = (t, ξ2, . . . , ξn) in the hyperplane at infinity H∞(t).
+Evaluating these homogeneous polynomials at t = 0, we obtain the top degree homoge-
+neous part of the polynomials gi,vm for i = 1, . . . , n. Since ξ ∈ H∞(t), it has a nonzero
+coordinate ξk for some k ∈ {2, . . . , n}. For simplicity, let us assume k = 2 and define the
+8
+
+new variables zi := ξi+2/ξ2 for i = 1, . . . , n − 2. Applying this change of variables, we
+obtained
+(3.5)
+�Gi,vm(z1, . . . , zn−2) =
+�
+|J|≤d
+αi,Jzϕ(J)
+where ϕ : Rn → Rn−2 with ϕ(j1, . . . , jn) = (j3, . . . , jn). This gives n random Bernoulli
+polynomials of degree d in n − 2 variables. Hence by Theorem 3.1, there exists a pos-
+itive constant Ci, depending only the dimension n such that the probability that the
+overdetermined system of Bernoulli polynomials �Gi,vm(z1, . . . , zn−2) have a common so-
+lution is less than Ci/d. We infer that the system of homogenized polynomials Gi,vm
+has no common zero at hyperplane at infinity H∞(t) except a set that has probability
+at most Ci/d. Then by Corollary 2.6, outside of a set of small probability, the system of
+polynomials consisting gi,vm has a common solution if and only if the directional resul-
+tant ResAvm
+1
+,...,Avn(f v
+1 , . . . , f v
+n ) = 0. Now, since the system of Bernoulli polynomials gi,vm
+contains n polynomials in n − 1 variables, by Theorem 3.1, there is a dimensional con-
+stant ˜Ci so that the probability that this system has common solution is at most ˜Ci/d.
+Hence outside of a set that has probability Ki/d := Ci/d + ˜Ci/d , the directional resultant
+ResAvmf vm
+d
+̸= 0 for all vm for m = 1, . . . , n.
+Next, for the inward normal vector vn+1 = − �n
+m=1 em, we find the minimal weight in
+this direction as Avn+1 = {J ∈ A : |J| = d}. Hence the polynomials in this directions are
+of the form
+(3.6)
+f vn+1
+i
+=
+�
+|J|=d
+αi,JxJ
+In this case Avn+1 is not a subspace of Zn ∩ v⊥
+n+1, hence we need to translate it by sub-
+tracting a suitable vector bi,vn+1. For Laurent polynomial systems, the sparse resultant is
+invariant under translations of supports (see [12], Proposition 3.3). Since the polynomi-
+als fd,i are not Laurent, we need to determine the effects of this translations. Consider
+the system of Bernoulli polynomials f d and set of its simultaneous zeros Z(f d). For a
+solution x = (x1, . . . , xn) ∈ Z(f d) and assume that x1 = 0. In order to examine the
+incidence of this case, we evaluate the system f d at x1 = 0 and we obtain a new system
+of n Bernoulli polynomials with n − 1 variables. By Theorem 3.1, there exists a constant
+C1 which is independent of d such that this system has a common solution with proba-
+bility at most C1/d. Therefore the probability of the event that x1 = 0 is less than C1/d.
+Hence there is no harm of translation of supports outside of a set that has probability at
+most C/d, where C := �n
+i=1 Ci. Now, choosing the vector bi,vn+1 = (d, 0, . . . , 0) so that
+Avn+1 − bi,vn+1 ⊂ Zn ∩ v⊥
+n+1, we obtain the polynomials of the form
+(3.7)
+gi,vn+1 =
+�
+J∈Avn+1−bi,vn+1
+αi,Jxw(J)
+with w : Rn → Rn satisfying (j1, j2, . . . , jn) �→ (−d + j1, j2, . . . , jn). We substitute the new
+variables yi := xi+1/x1 into gi,vn+1, i = 1, . . . , n − 1 and obtain
+9
+
+(3.8)
+gi,vn+1(y) =
+�
+|J|≤d
+αi,Jyσ(J)
+for y ∈ Cn−1 and σ : Rn → Rn with σ(j1, j2, . . . , jn) = (0, j2, . . . , jn). The system con-
+taining the polynomials gi,vn+1(y), i = 1, . . . , n contains n random Bernoulli polynomials
+with n − 1 random variable as in the cases vm = em. By applying the same steps, it can
+be shown that ResAvn+1f vn+1
+d
+̸= 0 outside of a set that has probability at most Ki+1/d.
+Now, we define the exceptional set En,d as a subset of Polyn,d which contains the sys-
+tems f d that has a zero directional resultant for some nonzero primitive vector v or the
+systems f d have a common solution x ∈ Cn with xi = 0 for some i = 1, . . . , n. More
+precisely, letting
+En,d := {f d ∈ Polyn,d : ∃ v ∈ Zn \ {0} ∋ ResAvf v
+d = 0}
+(3.9)
+�
+{f d ∈ Polyn,d : ∃ x ∈ Z(f d) ∋
+�
+xi = 0}.
+we see that there exists a positive constant K which is independent of d such that
+Prob{En,d} ≤ d−1K
+where K := �n+1
+i=1 Ki + C.
+□
+Next, we recall a deterministic equidistribution results for the solutions of systems of
+integer coefficient polynomials [13]. For a polynomial f ∈ C[x1, . . . , xn], the supremum
+norm of f on the unit torus is defined as
+∥f∥sup :=
+sup
+|w1|=...=|wn|=1
+|f(w1, . . . , wn)| .
+Let νHaar be the Haar measure on Cn with support (S1)n and of total mass 1. Assume that
+f ∈ Polyn,d be a polynomial mapping such that the set of simultaneous zeros Z(f) is a
+discrete set. We denote by denote the discrete probability measure on Cn associated to
+the Z(f) by δZ(f). The following result gives the asymptotic distribution of the zeros of
+such a system f if the coefficients are integer:
+Theorem 3.2. [13] Let f = (f1, . . . , fn) be a polynomial mapping with fi ∈ Z[x1, . . . , xn]
+of degree d ≥ 1 for each i = 1, . . . , n. Assume that ResAv
+1 ,...,Avn(f v
+1 , . . . , f v
+n) ̸= 0 for all
+v ∈ Zn \ {0} and log ||fi||sup = o(d). Then
+lim
+d→∞ δZ(f) = νHaar.
+As a corollary of Theorem 1.1 and Theorem 3.2, we have the following equidistribution
+result for random Bernoulli polynomial mappings:
+Proof of Corollary 1.2. Consider the system of Bernoulli polynomials f d = (fd,1, . . . , fd,n).
+Since all the coefficients are 1 or −1, by triangle inequality
+(3.10)
+∥fd,i∥sup =
+sup
+|w1|=...=|wn|=1
+|fd,i(w1, . . . , wn)| ≤
+�n + d
+d
+�
+= O(dn)
+10
+
+which implies that log ∥fd,i∥sup = o(d). Moreover, by Theorem 1.1 for each sequence
+f d ∈ Polyn,d \ En,d we have
+ResAv
+1 ,...,Avn(f v
+1 , . . . , f v
+n ) ̸= 0
+for all v ∈ Zn \ {0}. Hence, by Theorem 3.2
+lim
+d→∞ δZ(f d) = νHaar
+in weak topology. In particular, δZ(f d) → νHaar in probability since Prob{En,d} → 0 as
+d → ∞.
+□
+4. EXPECTED ZERO DISTRIBUTION
+In this section, we introduce radial and angle discrepancies for random Bernoulli poly-
+nomial mappings in order to study asymptotics of expected zero measures. We adapt
+these concepts from [13] and refer the reader to the manuscript [13] and references
+therein for a detailed account of the preliminary results this section.
+Let Z be a 0-dimensional effective cycle in Cn that is there is a non-empty finite col-
+lection of points ξ = (ξ1, . . . , ξn) ∈ Cn and mξ ∈ N, called the multiplicity of ξ, such
+that Z = �
+ξ mξ[ξ]. The degree of Z is defined by deg(Z) = �
+ξ mξ which is a positive
+number.
+Definition 4.1. [13] Let Z be a 0-dimensional effective cycle in Cn. For each α = (α1, . . . , αn)
+and β = (β1, . . . , βn) ∈ Rn such that −π ≤ αj < βj ≤ π, j = 1, . . . , n we consider the cycle
+Zα,β :=
+�
+{ξ∈Z:αj<arg(ξj)≤βj}
+mξ[ξ].
+The angle discrepancy of Z is defined as
+∆ang(Z) := sup
+α,β
+�����
+deg(Zα,β)
+deg(Z)
+−
+n
+�
+j=1
+βj − αj
+2π
+����� .
+For 0 < ε < 1 we consider the cycle
+Zε :=
+�
+{ξ∈Z:1−ε<|ξj|<(1−ε)−1}
+mξ[ξ].
+The radius discrepancy of Z with respect to ε is defined as
+∆rad(Z, ε) := 1 − deg(Zε)
+deg(Z) .
+Note that 0 < ∆ang(Z) ≤ 1 and 0 ≤ ∆rad(Z, ε) ≤ 1. Observe that the angle discrepancy
+and the radial discrepancy are generalizations of their one dimensional versions defined
+in [14, 17].
+Let A1, . . . , An ⊂ Zn be a collection of finite sets and let Qi = conv(Ai) for each
+i = 1, . . . , n. Throughout this section we assume that D := MVRn(Q1, . . . , Qn) ≥ 1.
+For a vector w ∈ Sn−1 in the unit sphere in Rn, let w⊥ be its orthogonal subspace and
+11
+
+πw⊥ : Rn → w⊥ be the corresponding orthogonal projection. We let MVw⊥ denote the
+mixed volume of the convex bodies in w⊥ induced by the Euclidean measure on w⊥. We
+also denote
+Dw,i = MVw⊥ (πw(Q1), . . . , πw(Qi−1), πw(Qi+1), . . . , πw(Qn)) .
+Let f = (f1, . . . , fn) be a mapping such that the coordinates fi are Laurent polynomials
+with supp(fi) = Ai for i = 1, . . . , n. Following [13], we define the Erd¨os-Tur´an size of f
+by
+(4.1)
+η(f) := 1
+D
+sup
+w∈Sn−1 log
+�
+�n
+i=1 ||f||
+Dw,i
+sup
+�
+v |ResAv
+1 ,...,Avn(f v
+1 , . . . , f vn )|
+|⟨v,w⟩|
+2
+�
+,
+where ⟨·, ·⟩ is the standard inner product in Rn and the product in the denominator
+is taken over all primitive vectors v ∈ Zn. We remark that the Erd¨os-Tur´an size of a
+polynomial mapping f coincides with the bound in the Erd¨os-Tur´an Theorem [14] for
+univariate polynomials.
+The next result gives an upper bound for the Erd¨os-Tur´an size of polynomial systems f
+with integer coefficients.
+Proposition 4.2. [13, Proposition 3.15] Let A1, . . . , An be a non-empty finite subsets of
+Zn and set Qi = conv(Ai) with MVRn(Q1, . . . , Qn) ≥ 1. Let di ∈ Z≥1 and bi ∈ Zn so that
+diΣn + bi, i = 1, . . . , n. Suppose that f1, . . . , fn ∈ Z[x±1
+1 , . . . , x±1
+n ] with supp(fi) ⊆ Ai and
+such that ResAv
+1 ,...,Avn(f v
+d,1, . . . , f v
+d,n) ̸= 0 for all v ∈ Zn \ {0}. Then
+η(f) ≤
+1
+MVRn(Q1, . . . , Qn)
+�
+�
+n + √n
+�
+� n
+�
+i=1
+di
+�
+n
+�
+i=1
+log ∥fi∥sup
+di
+�
+.
+The following theorem gives bounds for angle discrepancy and radius discrepancy of
+Z(f) in terms of the Erd¨os-Tur´an size of f. For one dimensional version see for instance
+[14] and [17].
+Theorem 4.3. [13] Let A1, . . . , An be a non-empty finite subsets of Zn such that
+MVRn(Q1, . . . , Qn) ≥ 1
+with Qi = conv(Ai) for n ≥ 2. Let f1, . . . , fn ∈ C[x±1
+1 , . . . , x±1
+n ] with supp(fi) ⊆ Ai and such
+that ResAv
+1 ,...,Avn(f v
+d,1, . . . , f v
+d,n) ̸= 0 for all v ∈ Zn \ {0}. Then
+(4.2)
+∆ang(Z(f)) ≤ 66n2n(18 + log+(η(f)−1))
+2
+3(n−1)η(f)
+1
+3.
+Moreover, for 0 < ε < 1,
+(4.3)
+∆rad(Z(f), ε) ≤ 2n
+ε η(f).
+For a random Bernoulli polynomial mapping f d we let Z(f d) be the set of simultaneous
+zeros of f d. We define the angle discrepancy ∆ang(Z(f)) and the radius discrepancy
+∆rad(Z(f), ε) as above whenever Z(f d) is a discrete set of points. Otherwise, we set
+∆rad(Z(f), ε) = ∆ang(Z(f)) = 1. Note that as our probability space (Polyn,d, Prob) is
+discrete, measurability of these random variables is not an issue in this setting. Next, we
+estimate the asymptotic expected discrepancies:
+12
+
+Proposition 4.4. Let f d = (fd,1, . . . , fd,n) be a random Bernoulli polynomial mapping of
+degree d ≥ 1. Then
+(4.4)
+lim
+d→∞ E[∆ang(Z(f d))] = 0
+and
+lim
+d→∞ E[∆rad(Z(f d))] = 0.
+Proof. We adapt the argument in [[13], Theorem 4.9] to our setting. Consider the ex-
+pected value of the angular discrepancy which is
+(4.5)
+E[Z(f d)] =
+�
+P olyn,d
+∆ang(Z(f d))dProbd(f d).
+Let En,d be the exceptional set which contains all the systems in Polyn,d with zero direc-
+tional resultants for some nonzero primitive vector v ∈ Zn as described in the proof of
+Theorem 1.1. Since 0 < ∆ang(Z(f d)) ≤ 1 there exist constants K1 which is independent
+of d such that
+(4.6)
+0 ≤
+�
+En,d
+∆ang(Z(f d))dProb(fd) ≤ Prob{En,d} ≤ K1d−1.
+Hence,
+�
+En,d
+∆ang(Z(f d))dProbd(f d) → 0
+as d → ∞.
+Let f d ∈ Polyn,d \ En,d, then by Proposition 4.2
+η(f d) ≤ 1
+dn
+�
+dn−1(n + √n)
+n
+�
+i=1
+log ||fd,i||sup
+�
+(4.7)
+≤ 1
+dn
+�
+dn−1(n + √n)
+n
+�
+i=1
+log(d + 1)
+�
+(4.8)
+≤ K2
+log d
+d
+(4.9)
+for a constant K2 which is independent of d. On the other hand, by Theorem 4.3 for
+f d ∈ Polyn,d \ En,d there exists constants K3, K4, K5 and K6 such that
+∆ang(Z(f d)) ≤ K3η(f d)
+1
+3 log
+� K4
+η(f d)
+� 2
+3(n−1)
+(4.10)
+≤ K5
+�log d
+d
+� 1
+3
+log
+�
+d
+log d
+� 2
+3 (n−1)
+�� K6
+log d
+2n
+3 − 1
+3
+d
+1
+3
+.
+(4.11)
+since the function t
+1
+3 log( a
+t )
+n−1
+3
+is increasing for small values of t > 0. Combining the
+equations (4.9) and (4.11), we deduce that lim
+d→∞ E[∆ang(Z(f d))] = 0.
+The proof of the second assertion is analogous and we omit it.
+□
+Proof of Theorem 1.3. We adapt the argument in [13, Theorem 1.8] to our setting. Let us
+denote νd := E[ �Z(f d)]
+dn
+, where E[ �Z(f d)] is the expected zero measure and νHaar be the Haar
+probability on (S1)n. We need to show that for each continuous function ϕ with compact
+13
+
+support in Cn we have
+�
+ϕdνd →
+�
+ϕdνHaar as d → ∞. To this end, it is enough to prove
+the claim for characteristic functions ϕU of the open sets
+(4.12)
+U := {(z1, . . . , zn) ∈ Cn : r1,j < |zj| < r2,j and αj < arg(zj) < βj}
+where 0 ≤ r1,j < r2,j ≤ ∞, ri,j ̸= 1 for i = 1, 2 and −π < αj < βj ≤ π.
+First, we consider the case when U ∩ (S1)n = ∅. Then one can find an 0 < ε < 1 such
+that U is disjoint from the set
+(4.13)
+{(ξ1, . . . , ξn) ∈ Cn : 1 − ε < |ξj| < (1 − ε)−1 for all j}.
+Let En,d be the exceptional set as in the proof of Theorem 1.1. If f d ∈ Polyn,d \ En,d then
+Z(f d) is discrete and
+#{U ∩ Z(f d)} ≤ deg(Z(f d))∆rad(f d, ε) ≤ dn∆rad(f d, ε).
+On the other hand, if f d ∈ En,d then by definition deg( �Z(f d)|U) = 0. Hence,
+νd(U) ≤ E[∆rad( �Z(f d, ε))]
+and by Proposition 4.4,
+lim
+d→∞
+�
+P olyn,d
+ϕUdνd = 0 = νHaar(U).
+If U ∩ (S1)n ̸= ∅ let
+(4.14)
+�U = {z : αj ≤ arg(zj) ≤ βj for all j }.
+Then we have
+νd(U) −
+n
+�
+j=1
+βj − αj
+2π
+=
+�
+νd(�U) −
+n
+�
+j=1
+βj − αj
+2π
+�
+− νd(�U \ U).
+By Theorem 1.1 we have
+�����νd(�U) −
+n
+�
+j=1
+βj − αj
+2π
+����� =
+�
+P olyn,d\En,d
+�����
+deg(Z(fd)α,β)
+dn
+−
+n
+�
+j=1
+βj − αj
+2π
+����� dProbd(f d) + Kn
+d
+≤
+�
+P olyn,d\En,d
+∆ang(Z(f d))dProbd(f d) + Kn
+d .
+(4.15)
+Note that the set �U \U is a union of a finite number of subsets Um of the form (4.12) such
+that Um ∩ (S1)n = ∅ for all m, we have limd→∞ νd(Um) = 0 by previous case and hence
+limd→∞ νd(U \ U) = 0. Therefore, by Proposition 4.4 and (4.15),
+lim
+d→∞ νd(U) = lim
+d→∞(�U) =
+n
+�
+j=1
+βj − αj
+2π
+= νHaar(U)
+which completes the proof.
+□
+14
+
+REFERENCES
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+FACULTY OF ENGINEERING AND NATURAL SCIENCES, SABANCI UNIVERSITY, ˙ISTANBUL, TURKEY
+Email address: [email protected]
+Email address: [email protected]
+16
+

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+Interpreting learning in biological neural networks as
+zero-order optimization method
+Johannes Schmidt-Hieber∗
+Abstract
+Recently, significant progress has been made regarding the statistical understanding
+of artificial neural networks (ANNs). ANNs are motivated by the functioning of the
+brain, but differ in several crucial aspects. In particular, it is biologically implausible
+that the learning of the brain is based on gradient descent. In this work we look at
+the brain as a statistical method for supervised learning. The main contribution is to
+relate the local updating rule of the connection parameters in biological neural networks
+(BNNs) to a zero-order optimization method.
+Keywords:
+Biological neural networks, zero-order optimization, derivative-free methods,
+supervised learning.
+1
+Introduction
+Compared to artificial neural networks (ANNs), the brain learns faster, generalizes better
+to new situations and consumes much less energy. A child only requires a few examples to
+learn to discriminate a dog from a cat. And people only need a few hours to learn how to
+drive a car. AI systems, however, need thousands of training samples for image recognition
+tasks. And the self-driving car is still under development, despite the availability of data
+for millions of kilometers of test drives and billions of kilometers of simulated drives. The
+∗University of Twente, Drienerlolaan 5, 7522 NB Enschede, The Netherlands.
+Email: [email protected]
+This work has tremendously profited from several discussions with Wouter Koolen. The author is moreover
+extremely grateful for helpful suggestions and several interesting remarks that were brought up by Matus
+Telgarsky. The research has been supported by the NWO/STAR grant 613.009.034b and the NWO Vidi
+grant VI.Vidi.192.021.
+1
+arXiv:2301.11777v1  [cs.LG]  27 Jan 2023
+
+Figure 1:
+Artificial neurons receive and output numbers, biological neurons receive and
+output spike trains.
+superhuman performance of AI for some tasks [30, 33, 5] has to be related to the huge
+databases and the enormous computing power required for the training.
+When identifying the causes for the differences in statistical behavior, it is important to
+emphasize that although ANNs are inspired by the functioning of the brain, they are very
+different from biological neural networks (BNNs). Each biological neuron emits a so called
+spike train that can be modelled as a stochastic process or, more precisely, as a point process
+[8, 9] and all computations in BNNs, including the updating of the network parameters, are
+local. The signal in ANNs, however, is passed instantaneously through the whole network
+without a time component such as a spike train structure. In conclusion, ANNs generate
+functions and BNNs point processes.
+Another difference between ANNs and BNNs is the learning. Fitting the network param-
+eters in large ANNs is based on variations of stochastic gradient descent (SGD) using the
+backpropagation algorithm. The parameter update at each network weight is global in the
+sense that every component of the gradient depends, in general, on all the other, possibly
+millions of network weights in the whole network. This means that SGD methods require
+knowledge of the state of the whole network to update one parameter. This is also known
+as the weight transportation problem [14]. As neurons in a biological network do not have
+the capacity to transport all the information about the state of the other weights, learning
+in BNNs cannot be driven by gradient descent [19]. In [7], Francis Crick writes: ”Neverthe-
+less, as far as the learning process is concerned, it is unlikely that the brain actually uses
+backpropagation.”
+In this work, we link the local updating rule for the parameters in a BNN to a derivative-free
+(or more specifically, a zero-order) optimization method that does not require evaluation
+of the gradient. Theorem 1 shows that, in expectation, this scheme does approximately
+gradient descent.
+2
+
+ARTIEICIALNEURON
+BIOLOGICALNEURON
+OUTPUT
+a (W,X, +W,X2 + ... + W.Xd)
+OUTPUT
+W1
+W.
+W1
+Wd
+W2
+W2
+INPUT
+INPUT
+Wd
+W2
+W2
+W1
+W,2
+A brief introduction to biological neural networks (BNNs)
+Using graph theory terminology, a BNN is a directed
+Figure 2:
+Receiving three spike
+trains, the biological neuron su-
+perimposes them and releases
+spikes, whenever the threshold
+value (in red) is exceeded.
+graph, with nodes representing neurons. The nodes/
+neurons can receive spikes via incoming edges and
+emit spikes via outgoing edges. In the directed graph,
+parent nodes are also called presynaptic neurons and
+children nodes are called postsynaptic neurons.
+A
+simple, first model is to think of a spike that is emit-
+ted at time τ as a signal or function t �→ eτ−t1(t ≥ τ)
+with 1(·) the indicator function. If neuron i emits a
+spike at time τ and is connected to neuron j, then
+neuron j receives the signal wijeτ−t1(t ≥ τ), where
+wij is the weight parameter measuring the strength
+of the connection between the neurons i and j. Due
+to the exponential decay, the signal fades out quickly.
+When does neuron j fire/emits a spike?
+Suppose
+neuron j has incoming edges from neurons i1, . . . , im.
+These neurons will occasionally send spikes to j and
+the overall received signal/potential at j is the su-
+perposition of the weighted incoming signals. If the
+combined signal exceeds a threshold S, j fires and
+all children nodes (postsynaptic neurons) of j receive a signal from j. The generation of
+the spike trains is illustrated in Figure 2 for m = 3. The three incoming spike trains were
+chosen for illustrative purposes as triggering a spike in a BNN requires between 20 to 50
+incoming spikes within a short time period [12], p.10. After a spike, neuron j enters a short
+rest phase before it gets back to its normal state. Although this rest phase might play a
+role in the learning, it will be ignored in the analysis.
+The parameters in the BNN are the non-negative weights measuring the strength of the
+connections. Plasticity is the neuroscience term to describe the changes of the network
+parameters. Spike time dependent plasticity (STDP) predicts that the parameter wij mea-
+suring the signal strength between the neurons i and j is decreased if a spike is sent from
+i to j and increased if neuron j emits a spike [21, 2, 42, 34]. The increase becomes bigger
+if the time lag between the arrived spike and the firing of neuron j gets smaller. This
+is known as ”fire together, wire together” and is the main principle underlying Hebbian
+learning [15, 32].
+3
+
+threshold
+received signal
+emitted signalAmong specific forms for the updating formula, as simple but realistic model is to assume
+that if the spike from neuron i to neuron j arrives at time τ, the weight is decreased by
+A−(wij)Ce−c(τ−T−) at time τ and increased by A+(wij)Ce−c(T+−τ) at time T+, where c, C
+are constants and T−, T+ are the last/first spike time of neuron j before/after τ. Regarding
+the amplitude functions, A±(wij), different choices are possible, see Section 19.2.2 in [12]
+From now on we will study the case that A±(x) = x. For C ≤ 1, this choice guarantees that
+the change of the weight is always smaller than the weight itself. Thus positive weights
+remain positive and the network topology does not change during learning. Combining
+both updating steps into one formula, we have
+wij ← wij + wijC(−e−c(τ−T−) + e−c(T+−τ)).
+(2.1)
+As a reward for how well the task has been completed compared to earlier trials and also
+accounting for the total number of trials, a neurotransmitter such as dopamine is released.
+The higher the reward, the more the parameters are changed.
+If the brain performed
+poorly in the past and suddenly manages to solve a task well, much more neurotransmitter
+is released than if the same task has already been completed equally well in the past. To
+take this into account, it has been argued in the neural coding literature that the realized
+reward is the objective reward, how well the task has been completed, minus the expected
+reward measuring how well the brain anticipated to do this task [11]. Denote by R the
+reward and let R be a measure for the anticipated reward. The reward-based synaptic
+plasticity updating rule becomes then
+wij ← wij + (R − R)wijC
+�
+− e−c(τ−T−) + e−c(T+−τ)�
+.
+(2.2)
+The reward is only released after the prediction has been made. In the meantime, several
+spikes could have been sent from neuron i to neuron j. This requires that the system has a
+short-term memory, [28].
+If the brain has to complete a similar task more frequently, it becomes less exciting over
+time, resulting in a smaller reward. This can be incorporated into the dynamics by including
+a learning rate α > 0,
+wij ← wij + α(R − R)wijC
+�
+− e−c(τ−T−) + e−c(T+−τ)�
+.
+(2.3)
+Supervised learning is more commonly formulated in loss functions than rewards. Because a
+high reward corresponds to a small loss and vice versa, L := −R is a loss function, L = −R
+4
+
+is the anticipated loss, and the updating formula becomes
+wij ← wij + α(L − L)wijC
+�
+e−c(τ−T−) − e−c(T+−τ)�
+.
+(2.4)
+A key observation is that these updating formulas are derivative-free in the sense that they
+involve the reward (or loss) but not its gradient.
+Hebbian learning rules, such as (2.4), model the updating of individual weights, but do not
+explain how the brain can learn a task. A brief overview about relevant existing ideas on
+learning in BNNs is given in Section 5
+3
+Zero-order optimization
+Suppose we want to fit a d-dimensional parameter vector θ to the data and write L(θ)
+for the (training) loss incurred by parameter θ. Derivative-free optimization procedures
+do not require computation of the gradient of the loss. A simple iterative derivative-free
+scheme would be to randomly pick in each round a new candidate parameter and update
+the parameter if the loss is decreased. Standard references for derivative-free optimization
+include [36, 6, 10, 16, 20].
+Zero-order methods (sometimes also called zero-th order methods) are specific derivative-
+free optimization procedures. To explain the concept, recall that standard gradient descent
+is an iterative procedure aiming to minimize the loss function θ �→ L(θ) by the iterative
+scheme
+θk+1 = θk − αk+1∇L(θk),
+k = 0, 1, . . .
+where the initial values θ0 are chosen in some way, αk+1 > 0 is the learning rate and ∇L(θk)
+denotes the gradient of the loss function at θk. In contrast, zero-order methods are only
+allowed to access the loss function but not the gradient of the loss. From the loss, one
+can build, however, an estimator for the gradient of the loss. 1-point zero-order methods
+replace −∇L(θk) by
+βL(θk + ξk)ξk
+with ξk a d-dimensional random vector and β a constant. To see how this relates to the
+gradient, consider the specific case that ξk is multivariate normal with zero mean vector
+and covariance matrix σ2Id, where Id denotes the d × d identity matrix. The multivariate
+5
+
+version of Stein’s lemma [38] states that
+E[L(θk + ξk)ξk] = σ2E[∇L(θk + ξk)]
+(3.1)
+under weak regularity conditions ensuring that all expectations are well-defined.
+This
+means that σ−2L(θk + ξk)ξk estimates the gradient at θk + ξk, that is, ∇L(θk + ξk) =
+∇L(θk)+errork. The hope is that over many iterations the noise contributions cancel out
+such that in the long-run, the 1-point zero-order dynamics behaves similarly as gradient
+descent. The argument above can be extended to general symmetric distributions of ξk
+that are not necessarily Gaussian.
+Unfortunately, the variance of the 1-point zero-order gradient estimator (3.1) can be ex-
+tremely large and often scales quadratically in the number of parameters d. As an example,
+suppose that the data are stored in a d-dimensional vector Y = (Y1, . . . , Yd)⊤ and con-
+sider the least squares loss L(θ) = ∥Y − θ∥2
+2. Taking ξk = (ξk1, . . . , ξkd) ∼ N(0, σ2Id) and
+β = σ−2, as above, we have for the j-th component of βL(θk + ξk)ξk that
+σ−2��Y − θk − ξk
+��2
+2ξkj = σ−2�
+Yj − θkj − ξkj
+�2ξkj + σ−2 �
+ℓ:ℓ̸=j
+�
+Yℓ − θkℓ − ξkℓ
+�2ξkj.
+The second term on the right hand side has zero mean. It is pure noise and does not help
+to estimate the gradient. This sum is over d − 1 summands and its variance scales with
+O(d2) in the number of parameters d.
+Due to the large variance, there are many scenarios for which 1-point zero-order dynamics
+quickly diverges to infinity. Indeed if one iterate θk is already far away from the minimum,
+the large loss can result in a parameter update θk+1 which is much further away from the
+minimizer than θk, leading to an even larger loss and an exponential growth of the loss as
+the number of iterations is further increased.
+Regarding theory of zero-order methods, [10] studies a related zero-order methods and
+mirror descent. Assuming that the parameter vector lies in an Euclidean ball, they obtain
+in their Corollary 1 the rate
+�
+d/k with k the number of iterations and also provide a
+corresponding lower bound proving that this rate is optimal (their Proposition 1). The
+large noise causes the factor
+√
+d in the rate, suggesting slow convergence in the high-
+dimensional regime. [25] also finds a suboptimality of order d if zero-order methods are
+compared to gradient descent. Table 1 in [20] shows that the factor
+√
+d or d occurs in all
+known convergence rates unless second-order information is used.
+Due to the large noise, derivative-free methods are in general thought to be inferior com-
+pared to gradient descent.
+This is for instance remarked in [6], Section 1.3: ”Finally,
+6
+
+we want to make a strong statement that often councils against the use of derivative-free
+methods: if you can obtain clean derivatives (even if it requires considerable effort) and the
+functions defining your problem are smooth and free of noise you should not use derivative-
+free methods.”
+Zero-order methods are also not necessarily much faster to compute than gradient descent
+iterates. For the gradient-based backpropagation of ANNs, the number of operations re-
+quired for the forward pass is of the same order as the number of operations required for
+the backwards pass. Evaluation of the loss is therefore not substantially cheaper than com-
+puting the gradient and zero-order methods cannot be computed at a faster order than
+backpropagation.
+Despite these rather discouraging remarks, there is a rapidly increasing interest in derivative-
+free methods and they are successfully applied in practice, for example by Google [13].
+4
+Hebbian learning as zero-order optimization method
+The updating formula (2.4) allows to address supervised learning tasks, where we want to
+learn the functional relationship between inputs and outputs given observations (or training
+data) from input-output pairs (X1, Y1), (X2, Y2), . . . that are all generated from the same,
+unknown distribution as the vector (X, Y ). Well-known examples for this framework are
+classification and regression. For instance to classify cat and dog images, Xi is the i-th
+image containing all the pixel values of the i-th cat image and Yi is the corresponding label
+”cat” or ”dog”, coded as 0 or 1.
+Consider now a feedforward biological neural network (BNN) with m neurons. This means
+that the neurons/nodes form a directed acyclic graph (DAG) with input neurons receiving
+information from the data Xi and possibly several output neurons. For the subsequent
+analysis, we neither have to specify a layered structure as commonly done for ANNs nor
+conversion rules how vector valued inputs are converted into spike trains or output spike
+trains are cast into response variables, such as conversion into labels in a classification
+problem.
+In the k-th instance, we feed the k-th input vector Xk in the BNN, let the BNN run and
+receive then as output the predicted response �Yk. The loss at this round is a measure for the
+difference between the predicted response �Yk and the real response Yk. It will be denoted
+by L(�Yk, Yk) in the following. The anticipated loss that occurs in (2.4) could be modelled
+by a (weighted) average over past iterations. Here we use the loss of the previous iterate
+L(�Yk−1, Yk−1).
+7
+
+During each instance, several spikes can be sent between any two connected neurons. We
+impose the (strong) assumption that for every run, and any connection, exactly one spike
+will be released.
+Number the m nodes, that represent the neurons in the graph, by 1, . . . , m and denote the
+edge set by T . A pair (i, j) is in T if and only if neuron i is a presynaptic neuron for neuron
+j. Equivalently, (i, j) ∈ T iff there is an arrow from i to j in the underlying DAG. We
+consider the case that the BNN topology is static, that is, the edge set T does not change
+during learning.
+If w(k)
+ij
+is the BNN weight after the k-th round, it is then updated in the (k +1)-st iteration
+according to (2.4)
+w(k+1)
+ij
+(4.1)
+= w(k)
+ij + αk+1
+�
+L(�Yk, Yk) − L(�Yk−1, Yk−1)
+�
+w(k)
+ij C
+�
+e−c(τ (k)
+ij −T (k)
+−,j) − e−c(T (k)
++,j−τ (k)
+ij )�
+,
+for all (i, j) ∈ T and αk+1 > 0 the learning rate. Here T (k)
+−,j and T (k)
++,j are the closest spike
+times of the j-th neuron before/after the arrival time τ (k)
+ij
+of the spike that is sent from
+neuron i to neuron j. The constant C can be integrated into the loss function and is from
+now on set to one.
+For the updating, the location of τ (k)
+ij
+is important within the interval [T (k)
+−,j, T (k)
++,j], while the
+interval length seems to play a minor role. Therefore, we assume that the interval length
+is constant and set A := (T (k)
++,j − T (k)
+−,j)/2. We assume moreover that the arrival time of the
+spike from neuron i to neuron j has a negligible influence on the spike times of neuron j,
+that the spike times τ (k)
+ij
+are all independent of each other, and follow a uniform distribution
+on the interval [T (k)
+−,j, T (k)
++,j]. As mentioned before, to trigger a spike, it needs of the order of
+20 − 50 presynaptic neurons to fire in a short time interval. The influence of an individual
+neuron seems therefore rather minor, justifying the previous assumption. The assumptions
+above show that the random variable U (k)
+ij
+:= τ (k)
+ij
+− 1
+2(T (k)
++,j + T (k)
+−,j) are jointly independent
+and uniformly distributed on [−A, A]. Hence, (4.1) becomes
+w(k+1)
+ij
+= w(k)
+ij + αk+1
+�
+L(�Yk, Yk) − L(�Yk−1, Yk−1)
+�
+w(k)
+ij
+�
+e−c(A+U(k)
+i,j ) − e−c(A−U(k)
+i,j )�
+,
+for all (i, j) ∈ T . The factor e−cA can be absorbed into the loss function and the constant c
+can be absorbed into the hyperparameter A. By reparametrization, we obtain the updating
+formula
+w(k+1)
+ij
+= w(k)
+ij + αk+1
+�
+L(�Yk, Yk) − L(�Yk−1, Yk−1)
+�
+w(k)
+ij
+�
+e−U(k)
+i,j − eU(k)
+i,j
+�
+,
+(4.2)
+8
+
+for all (i, j) ∈ T .
+To further analyze this scheme, it is important to understand how the predicted response
+�Yk depends on the parameters. We now argue that, under the same assumptions as before,
+�Yk is a function of the variables w(k)
+ij + eU(k)
+i,j . The high-level rationale is that in this neural
+model, all the information that is further transmitted in the BNN about the parameter
+w(k)
+ij
+sits in the spike times of neuron j and the interarrival spike times only depend on w(k)
+ij
+through w(k)
+ij + eU(k)
+i,j . To see this, fix neuron j. The only information that this node/neuron
+releases to its descendants in the DAG are the spike times of this neuron. This means that
+from all the incoming information that neuron j receives from presynaptic neurons (parent
+nodes) only the part is transmitted that affects the spike times of neuron j. As mentioned in
+Section 2, a spike arriving at neuron j from neuron i at time τ (k)
+ij
+causes the potential t �→
+w(k)
+ij eτ (k)
+ij −t1(t ≥ τ (k)
+ij ) at node j. If every incoming neuron spikes once, the overall potential
+of neuron j is �
+i:(i,j)∈T w(k)
+ij eτ (k)
+ij −t1(t ≥ τ (k)
+ij ). If S denotes the threshold value for the
+potential at which a neuron spikes, then at the spike time T (k)
++,j of the j-th neuron, we have
+by the definition of U (k)
+ij , S = �
+i:(i,j)∈T w(k)
+ij eτ (k)
+ij −T (k)
++,j = �
+i:(i,j)∈T w(k)
+ij eU(k)
+ij − 1
+2 (T (k)
++,j−T (k)
+−,j).
+Rearranging this equation shows that the interarrival spike time T (k)
++,j−T (k)
+−,j can be expressed
+in terms of the variables w(k)
+ij eU(k)
+ij . Introduce wk := (w(k)
+ij )(i,j)∈T , Uk := (U (k)
+ij )(i,j)∈T and
+write wkeUk for (w(k)
+ij eU(k)
+i,j )(i,j)∈T . The previous argument indicates that the predictor �Yk
+is a function of wkeUk and Xk. Thus, the loss L(�Yk, Yk) can be written as a function of the
+form L
+�
+wkeUk, Xk, Yk
+�
+and (4.2) becomes
+w(k+1)
+ij
+(4.3)
+= w(k)
+ij + αk+1
+�
+L
+�
+wkeUk, Xk, Yk
+�
+− L
+�
+wk−1eUk−1, Xk−1, Yk−1
+��
+w(k)
+ij
+�
+e−U(k)
+i,j − eU(k)
+i,j
+�
+.
+In a BNN, the parameters w(k)
+ij are non-negative. We now introduce the real-valued variables
+θ(k)
+ij
+= log(w(k)
+ij ) and θk = (θ(k)
+ij )(i,j)∈T . This means that w(k)
+ij
+= eθ(k)
+ij . A first order Taylor
+expansion shows that for real numbers u, v, ∆ such that e−v∆ is small, eu = ev + ∆ gives
+u = log(ev + ∆) = v + log(1 + e−v∆) ≈ v + e−v∆. Working with this approximation, we
+can rewrite the formula (4.3) in terms of the θ’s as
+θ(k+1)
+ij
+(4.4)
+= θ(k)
+ij + αk+1
+�
+L
+�
+θk + Uk, Xk, Yk
+�
+− L
+�
+θk−1 + Uk−1, Xk−1, Yk−1
+���
+e−U(k)
+i,j − eU(k)
+i,j
+�
+.
+Relating this formula to gradient descent and the weight transportation problem mentioned
+in the introduction, we see that the update of one parameter only depends on all the other
+9
+
+parameters through the value of the loss function.
+In vector notation, the previous equality becomes
+θk+1
+(4.5)
+= θk + αk+1
+�
+L(θk + Uk, Xk, Yk) − L(θk−1 + Uk−1, Xk−1, Yk−1)
+��
+e−Uk − eUk�
+,
+where eUk and e−Uk should be understood as componentwise applying the functions x �→ ex
+and x �→ e−x to the vector Uk. In particular, the loss is always a scalar and eUk, e−Uk are
+d-dimensional vectors.
+So far, we have not specified any initial conditions. From now on, we assume that the
+initial values θ0, θ−1 are given and that all the other parameter updates are determined by
+(4.5) for k = 0, 1, . . . with U−1, U0, U1, U2, . . . drawn i.i.d. from the uniform distribution
+U([−A, A]d).
+As an analogue of (3.1), the next result shows that in average, this dynamic can also be
+understood as a gradient descent method with gradient evaluated not exactly at θk but at
+a random perturbation θk + Uk.
+Theorem 1. Write Uk = (Uk1, . . . , Ukd)⊤ and let (eA −eUk)(eA −e−Uk) be the vector with
+components (eA − eUkj)(eA − e−Ukj). Denoting by ⊙ the Hadamard product (componentwise
+product) of two matrices/vectors of the same dimension(s), we have
+E
+�
+θk+1
+�
+= E
+�
+θk
+�
+− αk+1e−AE
+�
+∇θkL(θk + Uk, Xk, Yk) ⊙
+�
+eA − eUk��
+eA − e−Uk��
+. (4.6)
+Instead of taking the expectation over all randomness, the statement is also true if we only
+take the expectation with respect to Uk, which is the same as the conditional expectation
+E[·|U−1, U0, U1, . . . , Uk−1, (Xℓ, Yℓ)ℓ≥1].
+Note that (eA − eUkj)(eA − e−Ukj) is non-negative. Thus fA(x) = C(A)−1(eA − ex)(eA −
+e−x)1(−A ≤ x ≤ A) defines a probability density function for the positive normalization
+constant C(A) = 2A(e2A + 1) + 2 − 2e2A =
+� A
+−A(eA − ex)(eA − e−x) dx.
+Denoting by
+∂jL(v, Xk, Yk) the partial derivative of L with respect to the j-th component of v, we can
+state the previous result componentwise as
+E
+�
+θk+1,j
+�
+= E
+�
+θkj
+�
+− αk+1e−AC(A)E
+�
+∂jL(θk + U(j)
+k , Xk, Yk)
+�
+,
+(4.7)
+for a random vector U(j)
+k
+= (Uk1, . . . , Uk,j−1, Vkj, Uk,j+1, . . . , Ukd)⊤, with jointly indepen-
+dent random variables Vkj ∼ fA and Ukℓ ∼ U[−A, A], ℓ = 1, . . . , j − 1, j + 1, . . . , d.
+10
+
+Proof of Theorem 1. Throughout the proof, we omit the dependence of the loss function L
+on the data. By conditioning on (U−1, U0, . . . , Uk−1, (Xℓ, Yℓ)ℓ≥1) and the fact that e−Uk
+and eUk have the same distribution, it follows that
+E
+�
+L(θk−1 + Uk−1)
+�
+e−Uk − eUk��
+= E
+�
+L(θk−1 + Uk−1)E
+��
+e−Uk − eUk� ��� U−1, U0, . . . , Uk−1, (Xℓ, Yℓ)ℓ≥1
+��
+= 0.
+(4.8)
+With u = (u1, . . . , ud)⊤, the j-th component of e−AE[∇θkL(θk+Uk)⊙(eA−eUk)(eA−e−Uk)]
+is
+e−A
+(2A)d
+�
+[−A,A]d ∂jL(θk + u)
+�
+eA − euj��
+eA − e−uj�
+du
+= e−A
+(2A)d
+�
+[−A,A]d−1
+� A
+−A
+∂jL(θk + u)
+�
+eA − euj��
+eA − e−uj�
+dujdu1 . . . duj−1duj+1 . . . dud,
+Observe that (eA − euj)(eA − e−uj) vanishes at the boundaries uj ∈ {−A, A} and ∂uj(eA −
+euj)(eA − e−uj) = eA−uj − eA+uj. Thus, applying integration by parts formula to the inner
+integral yields
+� A
+−A
+∂jL(θk + u)
+�
+eA − euj��
+eA − e−uj�
+duj = −eA
+� A
+−A
+L(θk + u)
+�
+e−uj − euj�
+duj
+and therefore
+e−A
+(2A)d
+�
+[−A,A]d ∂jL(θk + u)
+�
+eA − euj��
+eA − e−uj�
+du
+= −
+1
+(2A)d
+�
+[−A,A]d L(θk + u)
+�
+e−uj − euj�
+du
+= −E
+�
+L(θk + Uk)
+�
+e−Ukj − eUkj��
+.
+This holds for all j = 1, . . . , d. The minus on the right hand side cancels out the first minus
+in (4.6). Together with (4.8), the claim follows.
+Equation (4.8) in the proof shows that the theorem still holds if the term L(θk−1 +
+Uk−1, Xk−1, Yk−1) in (4.5) is replaced by zero or any other value that is independent of
+Uk.
+To obtain a proper zero-order method, a crucial assumption is to choose the amplitude
+functions A+, A− in (2.1) to be the same. In the brain, these functions are close, but some
+11
+
+authors argue that there is a slight difference [34]. Such differences would lead to additional,
+small contributions in the iterations that cannot be linked to the gradient.
+A statistical analysis of the zero-order method (4.5) is challenging, even for simple models
+such as data generated from the linear regression model.
+Another open problem is to
+determine whether the convergence rate of (4.5) scales in the number of parameters d in
+the same way as other zero-order methods.
+5
+Literature on learning with BNNs
+This literature survey is aimed to give a quick overview. For a more detailed summary of
+related literature, see [37, 41].
+To train BNNs on data, a natural idea is to ignore Hebbian learning and to fit BNNs via
+gradient descent. Similar as backpropagation efficiently computes the gradient in ANNs,
+SpikeProp [3, 4] is an algorithm to compute the gradient for spiking neural networks.
+The weight transportation problem is caused by the parameter dependence in the backwards
+pass of the backpropagation algorithm. Feedback alignment [18, 26, 17, 1, 19] avoids this
+by using the backpropagation algorithm with random weights. In a network, the feedback
+could be then transmitted via specific feedback neurons.
+If the brain does a version of backpropagation, the difficulty is always the feedback from
+the output backwards to the neurons. Contrastive Hebbian learning [27] assumes that there
+are two different phases. During the first phase the network does prediction and the second
+phase starts after the prediction error is revealed. In one of the phases the learning is
+Hebbian and in the other one, the learning is anti-Hebbian. Anti-Hebbian learning means
+that if two neurons fire together, the connecting weight parameter is decreased instead of
+increased. Equilibrium propagation [29] overcomes the two types of learning in the different
+phases but requires again the computation of a gradient.
+For a biologically more plausible implementation of the weight transportation problem,
+predictive coding [40, 41, 35, 22, 23] uses two types of neurons, named error nodes and value
+nodes. These two nodes are associated to each other and process forward and backward
+information locally.
+[31] proposes the concept of a ”hedonistic synapse” that follows a Hebbian learning rule
+and takes the global reward into account. For the learning, a hedonistic synapse has to be
+able to store information from previous trials in a so-called eligibility trace.
+Closest to our approach is weight perturbation [39]. Weight perturbation adds random
+12
+
+noise to the parameters or the outputs and compares the loss with and without added
+noise to estimate the gradient. Whereas the cause of the noise perturbation is not entirely
+clear in the weight perturbation framework, we have shown in this work, how the spike
+train structure in BNNs implies a random perturbation of the parameters in the loss with
+uniformly distributed noise and how this leads to a specific derivative-free updating formula
+for the weights that also involves the difference of the loss function evaluated for different
+instance of the noisy parameters.
+A more statistical approach is [24] considering unsupervised classification using a small
+BNN. This work identifies a closer link between a Hebbian learning rule and the EM-
+algorithm for mixtures of multinomial distributions.
+Some other ideas on unsupervised
+learning in BNNs are moreover provided in [12], Section 19.3.
+To summarize, there are various theories that are centered around the idea that the learning
+in BNNs should be linked to gradient descent. All of these approaches, however, contain
+still biological implausibilities and lack a theoretical analysis.
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+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022
+1
+Learning from Guided Play:
+Improving Exploration for Adversarial Imitation
+Learning with Simple Auxiliary Tasks
+Trevor Ablett1, Bryan Chan2, and Jonathan Kelly1
+Abstract—Adversarial imitation learning (AIL) has become a
+popular alternative to supervised imitation learning that reduces
+the distribution shift suffered by the latter. However, AIL requires
+effective exploration during an online reinforcement learning
+phase. In this work, we show that the standard, na¨ıve approach
+to exploration can manifest as a suboptimal local maximum
+if a policy learned with AIL sufficiently matches the expert
+distribution without fully learning the desired task. This can
+be particularly catastrophic for manipulation tasks, where the
+difference between an expert and a non-expert state-action pair
+is often subtle. We present Learning from Guided Play (LfGP),
+a framework in which we leverage expert demonstrations of
+multiple exploratory, auxiliary tasks in addition to a main task.
+The addition of these auxiliary tasks forces the agent to explore
+states and actions that standard AIL may learn to ignore.
+Additionally, this particular formulation allows for the reusability
+of expert data between main tasks. Our experimental results in
+a challenging multitask robotic manipulation domain indicate
+that LfGP significantly outperforms both AIL and behaviour
+cloning, while also being more expert sample efficient than these
+baselines. To explain this performance gap, we provide further
+analysis of a toy problem that highlights the coupling between
+a local maximum and poor exploration, and also visualize the
+differences between the learned models from AIL and LfGP.3
+Index Terms—Imitation Learning, Reinforcement Learning,
+Transfer Learning
+I. INTRODUCTION
+E
+XPLORATION is a crucial part of effective reinforce-
+ment learning (RL). A variety of methods have attempted
+to optimize the exploration-exploitation trade-off of RL agents
+[1]–[3], but the development of a technique that generalizes
+across domains remains an open research problem. A simple,
+well-known approach to reduce the need for random explo-
+ration is to provide a dense, or “shaped,” reward to learn from,
+but this can be very challenging to design appropriately [4].
+Furthermore, the environment may not directly provide the
+low-level state information required for such a reward. An
+alternative to providing a dense reward is to learn a reward
+Manuscript received: Nov. 3, 2022; Accepted: Dec. 18, 2022.
+This paper was recommended for publication by Editor Jens Kober upon
+evaluation of the Associate Editor and Reviewers’ comments.
+1Authors are with the Space & Terrestrial Autonomous Robotic Systems
+(STARS) Laboratory at the University of Toronto Institute for Aerospace
+Studies (UTIAS), Toronto, Ontario, Canada, M3H 5T6. Email: <first
+name>.<last name>@robotics.utias.utoronto.ca
+2Author is with the Department of Computing Science at the Uni-
+versity
+of
+Alberta,
+Edmonton,
+Alberta,
+Canada,
+T6G
+2E8.
+Email:
+[email protected]
+Digital Object Identifier (DOI): see top of this page.
+3Code, Blog, Appendix: https://papers.starslab.ca/lfgp
+Fig. 1: Learning from Guided Play (LfGP) finds an effective stacking
+policy by learning to compose multiple simple auxiliary tasks (only
+Reach is shown, for this episode) along with stacking. Discrim-
+inator Actor-Critic (DAC) [7], or off-policy AIL, reaches a local
+maximum action-value function and policy, failing to solve the task.
+Arrow direction indicates mean policy velocity action, red-to-yellow
+(background) indicates low-to-high learned value, while arrow colour
+indicates probability of closing (green) or opening (blue) the gripper.
+function from expert demonstrations of a task, in a process
+known as inverse RL (IRL) [5]. Many modern approaches
+to IRL are part of the adversarial imitation learning (AIL)
+family [6]. In AIL, rather than learning a reward function
+directly, the policy and a learned discriminator form a two-
+player min-max optimization problem, where the policy aims
+to confuse the discriminator by producing expert-like data,
+while the discriminator attempts to classify expert and non-
+expert data.
+Although AIL has been shown to be more expert sample
+efficient than supervised imitation learning (also known as be-
+havioural cloning, or BC) in continuous-control environments
+[6]–[8], its application to long-horizon robotic manipulation
+tasks with a wide distribution of possible initial configurations
+remains challenging [7], [9]. In this work, we investigate the
+use of AIL in a multitask robotic manipulation domain. We
+find that a state-of-the-art AIL method, in which off-policy
+learning is used to maximize environment sample efficiency [7]
+(i.e., reduce the quantity of environment interaction required
+from the online RL portion of AIL), is outperformed by BC
+arXiv:2301.00051v1  [cs.LG]  30 Dec 2022
+
+LfGP
+DAC
+Reach
+Stack
+Pre-Grasp
+Post-Grasp2
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022
+Multitask Environment
+Reach
+Lift
+Bring
+Together
+Insert
+Stack
+Guided Expert Play
+Guide
+Expert
+bring_0
+together
+stack_01
+Multitask Environment
+Reach(    )
+Lift(    )
+Bring(    )
+Insert(    )
+Stack(   )
+Multitask Environment
+Reach
+Lift
+Bring
+Together
+Insert
+Stack
+Guided Expert Play
+Expert
+lift(  )
+Guide
+Guide
+stack(  )
+Guided Agent Play
+Move(    )
+RESET
+NEXT
+Expert
+lift(  )
+Sched
+( )
+stack(  )
+Sched
+(lift(   ))
+Agent
+Multitask AIL 
+Update
+Fig. 2: The main components of our system for learning from guided play. In a multitask environment, a guide prompts an expert for a mix
+of multitask demonstrations, after which we learn a multitask policy through scheduled hierarchical AIL.
+with an equivalent amount of expert data, contradicting previ-
+ous results [6]–[8]. Through a simplified example, simulated
+robotic experiments, and learned model analysis, we show
+that this outcome occurs because a model learned with expert
+data and a discriminator is susceptible to the deceptive reward
+problem [10]. In other words, while AIL, and more generally
+IRL, can provide something akin to a dense reward, this reward
+is not necessarily optimal for teaching, and AIL alone does
+not enforce sufficiently diverse exploration to escape locally
+optimal but globally poor models. A locally-optimal policy has
+converged to match a subset of the expert data, but in doing
+so, avoids crucial states and actions (e.g., in Fig. 1, grasping
+the blue block) required to globally match the full expert set.
+To overcome this limitation of AIL, we present Learning
+from Guided Play (LfGP),4 in which we combine AIL with a
+scheduled approach to hierarchical RL (HRL) [12], allowing
+an agent to ‘play’ in the environment with an expert guide.
+Using expert demonstrations of multiple relevant auxiliary
+tasks (e.g., Reach, Lift, Move-Object), along with a main task
+(e.g., Stack, Bring, Insert), our scheduled hierarchical agent
+is able to learn tasks where AIL alone fails. Crucially, our
+formulation also allows auxiliary expert data to be reused
+between main tasks, further emphasizing the expert sample
+efficiency of our method.
+We use the word play to describe an agent that simulta-
+neously attempts and learns numerous tasks at once, freely
+composing them together, inspired by the playful (as opposed
+to goal-directed) phase of learning experienced by children
+[12]. In our case, guided represents two separate but related
+ideas: first, that the expert guides this play, as opposed to
+requiring hand-crafted sparse rewards as in [12] (right side
+of Fig. 2), and second, that the expert gathering of multitask,
+semi-structured demonstrations is guided by uniform-random
+task selection (middle of Fig. 2), rather than requiring the
+expert to choose transitions between goals, as in [13], [14].
+Our specific contributions are the following:
+1) A novel application of a hierarchical framework [12] to
+AIL that learns a reward and policy for a challenging
+4Originally presented as a non-archival workshop paper [11].
+main task by simultaneously learning rewards and poli-
+cies for auxiliary tasks.
+2) Manipulation experiments in which we demonstrate that
+AIL fails, while LfGP significantly outperforms both
+AIL and BC.
+3) A thorough ablation study to examine the effects of
+various design choices for LfGP and our baselines.
+4) Empirical analysis, including a simplified representative
+example and visualization of the learned models of LfGP
+and AIL, to better understand why AIL fails and how
+LfGP improves upon it.
+II. PROBLEM FORMULATION
+A Markov decision process (MDP) is defined as M =
+⟨S, A, R, P, ρ0, γ⟩, where the sets S and A are respectively
+the state and action space, R : S×A → R is a reward function,
+P is the state-transition environment dynamics distribution, ρ0
+is the initial state distribution, and γ is the discount factor.
+Actions are sampled from a stochastic policy π(a|s). The
+policy π interacts with the environment to yield experience
+(st, at, rt, st+1) for t = 0, . . . , ∞, where s0 ∼ ρ0(·), at ∼
+π(·|st), st+1 ∼ P(·|st, at), rt = R(st, at). When referring to
+finite-horizon tasks, t = T indicates the final timestep of a
+trajectory.
+For notational convenience, we assume infinite-horizon,
+non-terminating environments where t is unbounded, but
+the extension to the finite-horizon case is trivial. We aim
+to learn a policy π that maximizes the expected return
+J(π)
+=
+Eπ [G(τ0:∞)]
+=
+Eπ [�∞
+t=0 γtR(st, at)], where
+τt:∞ = {(st, at), . . . } is the trajectory starting with (st, at),
+and G(τt:∞) is the return of trajectory τ.
+In this work, we focus on imitation learning (IL), where
+R is unknown and instead we are given a finite set of expert
+demonstration (s, a) pairs BE =
+�
+(s, a)E, . . .
+�
+. In AIL, we
+attempt to simultaneously learn π and a discriminator D : S ×
+A → [0, 1] that differentiates between expert samples (s, a)E
+and policy samples (s, a)π and subsequently define R using D
+[6], [7]. To accommodate hierarchical learning, we augment
+M to contain auxiliary tasks, where Taux = {T1, . . . , TK} are
+separate MDPs that share S, A, P, ρ0 and γ with the main
+task Tmain but have their own reward functions, Rk. With this
+
+ABLETT et al.: LEARNING FROM GUIDED PLAY
+3
+Fig. 3: An MDP, analogous to stacking, with an expert demonstration.
+Poor exploration can lead AIL to learn a suboptimal policy.
+modification, we refer to entities in our model that are specific
+to task T ∈ Tall, Tall = Taux ∪ {Tmain}, as (·)T . We assume
+that we have a set of expert data BE
+T for each task.
+III. LOCAL MAXIMUM WITH OFF-POLICY AIL
+In this section, we provide a representative example of how
+AIL can fail by reaching a locally maximum policy due to a
+learned deceptive reward [10] coupled with poor exploration.
+A simple six-state MDP is shown in Fig. 3, with ten state-
+conditional actions. We refer to actions as at = anm and states
+as st = sn where t, n and m refer to the current timestep,
+current state, and next state, respectively. The reward function
+is R(s5, a55) = +1, R(s1, a15) = −5 and 0 for all other state-
+action pairs. The initial state s1 is always s1, the fixed horizon
+length is 5, and no discounting is used.
+The MDP is meant to be roughly analogous to a stacking
+manipulation task: s2, s3, s4 and s6 represent the first block
+being reached, grasped, lifted, and dropped respectively. State
+s5 represents the gripper hovering over the second block
+(whether the first block has been stacked or not), while s1 is
+the reset state, and a15 represents reaching s5 without grasping
+the first block. Taking action a15 results in a total return of
+-1 (because R(s1, a15) = −5), since the first block has not
+actually been grasped. In our case, the agent does not receive
+any reward, and instead an expert demonstration of the optimal
+trajectory is provided. We will assume access to a learned
+(perfect) discriminator, and will use the AIRL [8] reward, so
+state-action pairs in the expert set receive +1 reward and all
+others receive -1.
+We define the action-value Q(st, at) as the expected
+value of taking action at in state st, and initialize it to
+zero for all (s, a) pairs. We define our update rule as the
+standard Q-Learning update [1], Q(st, at) = Q(st, at) +
+α (R(st, at) + maxa Q(st+1, a) − Q(st, at)), with α = 0.1.
+The agent uses ϵ-greedy exploration, storing each (st, at, st+1)
+tuple into a buffer. After each episode, all Q values are updated
+to convergence using the whole buffer.
+After the first complete episode of {a15, a55, a55, a55, a55},
+Q(s1, a15)
+=
+2.7, and Q(s1, a12)
+=
+0. In the second
+({a12, a26, a61, a15, a55}) and third ({a12, a23, a36, a61, a15})
+episodes, the agent initially moves in the correct direction, but
+ultimately still fails. The final Q values in s1 are Q(s1, a15) =
+0.49 and Q(s1, a12) = 0.13.5
+A policy maximizing Q, having simultaneously learned to
+avoid s6 (by avoiding s2 and s3) and exploiting the (s5, a55)
+expert pair, will choose a1 = a15, giving a final return of
+-1 in the real MDP. This behaviour matches what we see in
+Fig. 1: due to the large negative reward from dropping the
+block, AIL learns a policy that avoids stacking altogether and
+merely reaches the second block, just as AIL here learns to
+skip s2 and s3 and exploit a55. In both cases, poor initial
+exploration leads to a deceptive reward, which exacerbates
+poor exploration.
+IV. LEARNING FROM GUIDED PLAY (LFGP)
+We now introduce Learning from Guided Play (LfGP). Our
+primary goal is to learn a policy πTmain that can solve the main
+task Tmain, with a secondary goal of also learning auxiliary task
+policies πT1, . . . , πTK that are used for improved exploration.
+More specifically, we derive a hierarchical learning objective
+that is decomposed into three parts: i) recovering the reward
+function of each task with expert demonstrations, ii) training
+all policies to achieve their respective goals, and iii) using all
+policies for effective exploration in Tmain. For a summary of
+the algorithm, see supplementary material link in Footnote 3.
+A. Learning the Reward Function
+We first describe how to recover the reward functions from
+expert demonstrations. For each task T ∈ Tall, we learn a dis-
+criminator DT (s, a) that is used to define the reward function
+for policy optimization. We construct the joint discriminator
+loss following [7] to train each discriminator in an off-policy
+manner:
+L(D) = −
+�
+T ∈Tall
+EB [log (1 − DT (s, a))]
++EBE
+T [log (DT (s, a))] .
+(1)
+Each resulting discriminator DT attempts to differentiate the
+occupancy measure between the distributions induced by BE
+T
+and B. We can use DT to define various reward functions [7];
+following [8], we define the reward function for each task T
+to be RT (st, at) = log (DT (st, at)) − log (1 − DT (st, at)).
+B. Learning the Hierarchical Agent
+We adapt Scheduled Auxiliary Control (SAC-X) [12] to
+learn the hierarchical agent. The agent includes low-level
+intention policies (equivalently referred to as intentions), a
+high-level scheduler policy, as well as the Q-functions and the
+discriminators. The intentions aim to solve their corresponding
+tasks (i.e., the intention πT aims to maximize the task return
+J(πT )), whereas the scheduler aims to maximize the expected
+return for Tmain by selecting a sequence of intentions to interact
+with the environment. For the remainder of the paper, when
+we refer to a policy, we are referring to an intention policy,
+as opposed to the scheduler, unless otherwise specified.
+5See six_state_mdp.py from open source code to reproduce.
+
+Legend
+2
+S
+-5
+MDP
+C
+2
+3
+S
+S
+S
+6
+S
+a5
+Expert Demo
+a4
+a1
+2
+S
+S
+a2
+a3
+S
+a1
+a2-5
+Suboptimal AIL Policy
+S4
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022
+1) Learning the Intentions: We learn each intention using
+Soft Actor-Critic (SAC) [15], an actor-critic algorithm that
+maximizes the entropy-regularized objective, though any off-
+policy RL algorithm would suffice. The objective is
+J(πT ) = EπT
+� ∞
+�
+t=0
+γt (RT (st, at) + αH(πT (·|st)))
+�
+,
+(2)
+where the learned temperature α determines the importance
+of the entropy term and H(πT (·|st)) is the entropy of the
+intention πT at state st. The soft Q-function is
+QT (st, at) = RT (st, at)
++ EπT
+� ∞
+�
+t=0
+γt(RT (st+1, at+1) + αH(πT (·|st+1)))
+�
+.
+(3)
+The intentions maximize the joint policy objective
+L(πint) =
+�
+T ∈Tall
+Es∼Ball,a∼πT (·|s) [QT (s, a) − α log πT (a|s)] ,
+(4)
+where πint refers to the set of intentions {πTmain, πT1, . . . , πTK}
+and Ball refers to buffer containing every transition from
+interactions and demonstrations, as is done in [16], [17].
+For policy evaluation, the soft Q-functions QT for each πT
+minimize the joint soft Bellman residual
+L(Q) =
+�
+T ∈Tall
+E(s,a,s′)∼Ball,a′∼πT (·|s′)
+�
+(QT (s, a) − δT )2�
+,
+(5)
+δT = RT (s, a) + γ (QT (s′, a′) − α log πT (a′|s′)) .
+(6)
+Crucially, because each task shares the common S, A, P, ρ0,
+and γ, and we are using off-policy learning, all tasks can learn
+from all data, as in [12].
+2) The Scheduler: SAC-X formulates learning the sched-
+uler by maximizing the expected return of the main task
+[12]. In particular, let H be the number of possible intention
+switches within an episode and let each chosen intention
+execute for ξ timesteps. The H intention choices made within
+the episode are defined as T 0:H−1 =
+�
+T (0), . . . , T (H−1)�
+,
+where T (h) ∈ Tall. The return of the main task, given chosen
+intentions, is then defined as
+GTmain(T 0:H−1) =
+H−1
+�
+h=0
+(h+1)ξ−1
+�
+t=hξ
+γtRTmain(st, at),
+(7)
+where at ∼ πT (h)(·|st) is the action taken at timestep t,
+sampled from the chosen intention T (h) in the hth scheduler
+period. The scheduler for the hth period P h
+S aims to maxi-
+mize the expected main task return: E
+�
+GTmain(T h:H−1)|P h
+S
+�
+.
+Although SAC-X describes a method to learn the scheduler
+[12], we find that a combination of two simple task-agnostic
+heuristics performs similarly in practice (see Section V-C2).
+Specifically, we use a weighted random scheduler (WRS)
+combined with handcrafted trajectories (HC). The WRS forms
+a prior categorical distribution over the set of tasks, with a
+higher probability mass pTmain for the main task and
+pTmain
+K
+for
+all other tasks. This approach is comparable to the uniform
+scheduler from [12], with a bias towards the main task. The
+HC component is a small set of handcrafted trajectories of
+tasks that are sampled half of the time, forcing the scheduler
+to explore trajectories that would clearly be beneficial for
+completing the main task. The chosen handcrafted trajectories
+can be found in our code and in our supplementary material.
+C. Breaking Out of Local Maxima with LfGP
+Returning to the discussion in Section III, resolving the
+local maximum problem with LfGP is straightforward. Sup-
+pose we include a go-right auxiliary task with BE
+go-right =
+{(s1, a12), (s2, a23), (s3, a34)}. When the scheduler chooses
+the go-right intention, the agent does not exploit the a55 action
+because the go-right discriminator learns that R(s5, a55) =
+−1. Since the transitions are stored in the shared buffer that
+the main intention also samples from, the agent can quickly
+obtain the correct, optimal value.
+D. Expert Data Collection
+We assume that each T ∈ Tall has, for evaluation purposes
+only, a binary indicator of success. In single-task imitation
+learning where this assumption is valid, expert data is typically
+collected by allowing the expert to control the agent until
+success conditions are met. At that point, the environment is
+reset following ρ0 and collection is repeated for a fixed number
+of episodes or (s, a) pairs. We collect our expert data in this
+way for each T separately.
+V. EXPERIMENTS
+In this work, we are interested in answering the following
+questions about LfGP:
+1) How does the performance of LfGP compare with BC
+and AIL in challenging manipulation tasks, in terms of
+success rate and expert sample efficiency?
+2) What parts of LfGP are necessary for success?
+3) How do the policies and action value functions differ
+between AIL and LfGP?
+A. Experimental Setup
+We complete experiments in a simulation environment con-
+taining a Franka Emika Panda manipulator, one green and
+one blue block in a tray, fixed zones corresponding to the
+green and blue blocks, and one slot in each zone with < 1mm
+Fig. 4: Example successful runs of our four main tasks. Top to
+bottom: Stack, Unstack-Stack, Bring, Insert.
+
+ABLETT et al.: LEARNING FROM GUIDED PLAY
+5
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Stack
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Unstack-Stack
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Bring
+1
+2
+3
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Insert
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Updates/steps (millions)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Success Rate
+LfGP (multi)
+BC (multi)
+DAC (single)
+BC (single)
+Expert
+Fig. 5: Performance results for LfGP, multitask BC, single-task BC, and DAC on all four tasks considered in this work. The x-axis corresponds
+to both gradient updates and environments steps for LfGP and DAC, and gradient updates only for both versions of BC. The shaded area
+corresponds to standard deviation across five seeds. LfGP significantly outperforms the baselines on all tasks, and even in Bring where it is
+matched by single-task BC, it is far more expert sample efficient.
+tolerance for fitting the blocks (see bottom right of Fig. 4).
+The robot is controlled via delta-position commands, and the
+blocks and end-effector can both be reset anywhere above the
+tray. The environment is designed such that several different
+challenging tasks can be completed within a common observa-
+tion and action space. The main tasks that we investigate are
+Stack, Unstack-Stack, Bring, and Insert (see Fig. 4). For more
+details on our environment and definitions of task success, see
+supplementary material link in Footnote 3. We also define a set
+of auxiliary tasks: Open-Gripper, Close-Gripper, Reach, Lift,
+Move-Object, and Bring (Bring is both a main task and an
+auxiliary task for Insert), all of which are reusable between
+main tasks.
+We compare our method to several standard multitask and
+single-task baselines. A multitask algorithm simultaneously
+learns to complete a main task as well as auxiliary tasks,
+while the single-task algorithms only learn to complete the
+main task. In general, we consider a multitask algorithm to be
+more useful than a single-task algorithm, given the potential
+to reuse expert data and trained models for learning new tasks.
+To ensure a fair comparison, we provide single-task algorithms
+with an equivalent amount of total expert data as our multitask
+methods, as shown in Table I.
+In our main experiments, we compare LfGP to a mul-
+titask variant of behavioural cloning (BC), single-task BC,
+and Discriminator-Actor-Critic (DAC) [7], a state-of-the-art
+approach to AIL. We train multitask BC with a multitask mean
+squared error objective,
+L(πint) =
+�
+T ∈Tall
+�
+(s,a)∈BE
+T
+(πT (s) − a)2 ,
+(8)
+while BC is trained with the corresponding single task version.
+Following recent trends in improving BC performance, we
+train our BC baselines with the same number of gradient
+updates as LfGP and DAC, evaluating the policies at the same
+frequency. This adjustment has been shown to dramatically
+increase the performance of BC [18], [19], particularly com-
+pared to the more common practice of using early stopping,
+as is done in [6], [7]. We validate that this change signifi-
+cantly improves BC performance in our ablation study (see
+Section V-C4).
+We gather expert data by first training an expert policy using
+Scheduled Auxiliary Control (SAC-X) [12]. We then run the
+Task
+Dataset Sizes
+Reuse
+Single Total
+Multi
+Stack
+SOCRLM: 1k/task
+5k
+1k
+6k
+task
+U-Stack
+UOCRLM: 1k/task
+5k
+1k
+6k
+Bring
+BOCRLM: 1k/task
+6k
+0
+6k
+Insert
+IBOCRLM: 1k/task
+6k
+1k
+7k
+Single
+Stack
+S: 6k
+0
+6k
+6k
+Task
+U-Stack
+U: 6k
+0
+6k
+6k
+Bring
+B: 6k
+0
+6k
+6k
+Insert
+I: 6k
+0
+7k
+7k
+TABLE I: The number of (s, a) pairs used for each main and auxiliary
+task. The table illustrates the reusability of the expert data used to
+generate the performance results described in Section V-B. Each letter
+under “Dataset Sizes” is the first letter of a single (auxiliary) task,
+and bolded letters indicate that a dataset was reused for more than
+one main task (e.g., Open-Gripper was used for all four main tasks).
+Multitask methods (e.g., LfGP) are able to reuse a large portion of the
+expert data, while single-task methods (e.g., single-task BC) cannot.
+expert policies to collect various amounts of expert data as
+described in Section IV-D and Table I. We also collect an extra
+200 expert (sT , 0) pairs per auxiliary task, where T refers to
+the final timestep of an individual episode and 0 is an action
+of all zeros. This is equivalent to adding example data, as is
+done in example-based RL [20]. This addition improved final
+task performance, likely because it biases the reward towards
+completing the final task. It is worth noting that, in the real
+world, final states are easier to collect than full demonstrations,
+and LfGP does not require any modifications to accommodate
+these extra examples. Finally, even without this addition, LfGP
+still outperforms the baselines (see Section V-C1).
+B. Performance Results
+Performance results for all methods and main tasks are
+shown in Fig. 5. We freeze the policies every 100k steps
+and evaluate those policies for 50 randomized episodes, using
+only the mean action outputs for stochastic policies. For all
+algorithms, we test across five seeds and report the mean and
+standard deviation of all seeds.
+In Stack, Unstack-Stack, and Insert, LfGP achieves expert
+performance, while the baselines all perform significantly
+worse. In Bring, LfGP does not quite achieve expert per-
+formance, and is matched by single-task BC. However, we
+note that LfGP is much more expert data efficient than single-
+task BC because it reuses auxiliary task data (see Table I).
+A more direct comparison is multitask BC, which performs
+
+6
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Stack (no ablations)
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.5|BE
+orig|
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.5|BE
+orig|
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Subsampled BE
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0No Extra Final Examples
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Updates/steps (millions)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Success Rate
+LfGP (multi)
+BC (multi)
+DAC (single)
+BC (single)
+Expert
+Fig. 6: Various dataset ablations for LfGP and all baselines, including dataset size, subsampling of expert dataset, and replacement of extra
+(sT , 0) pairs with an equivalent amount of regular trajectory (s, a) pairs. In all cases, LfGP still significantly outperforms all baselines.
+1
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+LfGP Scheduler
+0.0
+0.5
+1.0
+Updates/steps (millions)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Success Rate
+WRS + HC
+WRS only
+Learned
+No Sched.
+Expert
+1
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Expert Sampling
+0.0
+0.5
+1.0
+Updates/steps (millions)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Success Rate
+LfGP
+LfGP (BE
+for D only)
+LfGP (No
+(sT , 0) bias)
+DAC
+DAC (BE
+for D only)
+DAC (No
+(sT , 0) bias)
+Expert
+1
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+BC/DAC Alternatives
+0.0
+0.5
+1.0
+Updates/steps (millions)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Success Rate
+BC (multi)
+BC (multi,
+early stop)
+DAC
+GAIL
+BC
+BC (early
+stop)
+Expert
+Fig. 7: Left: Scheduler ablations for training LfGP, WRS is weighted random scheduler, HC is handcraft; Middle: Expert sampling ablations
+for training LfGP/DAC; Right: Baseline ablations for training BC/DAC.
+much more poorly than LfGP across all tasks, including Bring.
+Intriguingly, DAC also performs very poorly on all tasks, a
+phenomenon that we further explore in Section VI.
+C. Ablation Study
+While the fundamental idea of LfGP is relatively straight-
+forward, it is worth considering alternatives to some of the
+specific choices made for our experiments. In this section,
+we complete an ablation study where we vary (a) the expert
+dataset, including size, subsampling, and inclusion of extra
+(sT , 0) pairs; (b) the type of scheduler used for LfGP (see
+Section IV-B2); (c) the sampling strategy used for expert data;
+and (d) the method for training our baselines. To reduce the
+computational load of completing these experiments, all of
+these variations were carried out exclusively for our Stack task.
+All ablation results are shown in Fig. 6 and Fig. 7.
+1) Dataset Ablations: We tested the following dataset vari-
+ations: (a) half and one and a half times the original expert
+dataset size; (b) subsampling BE, taking only every 20th
+timestep, as is done in [6], [7]; and (c) replacing the 200 extra
+(sT , 0) pairs in each buffer with 200 regular trajectory (s, a)
+pairs. Notably, even in the challenging regimes of halving
+and subsampling the dataset, LfGP still learns an expert-level
+policy (albeit more slowly).
+2) Scheduler Ablations: We tested the following scheduler
+variations: (a) Weighted Random Scheduler (WRS) only, re-
+moving the Handcrafted (HC) addition; (b) a learned sched-
+uler, as is used in [12]; and (c) no scheduler, in which only the
+main task is attempted, akin to the Intentional Unintentional
+Agent [12], [21]. Both WRS versions learn slightly faster than
+the learned scheduler, but all three methods outperform the No
+Scheduler ablation, replicating results from [12] demonstrating
+the importance of actually exploring all auxiliary tasks. Per-
+haps surprisingly, the HC modification made little difference
+compared with WRS only, but it is possible that for even more
+complex tasks, this could change.
+3) Expert Sampling Ablations: For our main performance
+experiments, we modified standard AIL in two ways: (a) we
+added expert buffer sampling to π and Q updates, in addition
+to the D updates, as is done in [16], [17]; and (b) we biased the
+sampling of BE when training D to be 95% final (sT , 0) pairs.
+We tested both LfGP and DAC without these additions. For
+LfGP, although these modifications improve learning speed,
+they are not required to generate an expert policy. For DAC,
+performance is quite poor regardless of these adjustments.
+4) Baseline Ablations: To verify that we evaluated against
+fair baselines, we tested two alternatives to those used for our
+main performance experiments: (a) an early stopping variation
+of BC, in which each expert buffer is divided into a 70%/30%
+train/validation split, taking the policy after validation error has
+not improved for 100 epochs; and (b) the on-policy variant
+of DAC, also known as Generative Adversarial Imitation
+Learning (GAIL) [6]. Notably, the early stopping variants of
+BC, commonly used as baselines in other AIL work [6], [7],
+[22] perform dramatically more poorly than those used in our
+experiments, verifying recent trends [18], [19].
+VI. LEARNED MODEL ANALYSIS
+In this section, we further examine the learned Stack models
+of LfGP and DAC. We take snapshots of the average per-
+forming models from LfGP and DAC at four points during
+learning: 200k, 400k, 600k, and 800k model updates and
+environment steps. Although the initial gripper and block
+positions are randomized between episodes during learning,
+for each snapshot, we reset the stacking environment to a
+single set of representative initial conditions. We then run the
+
+ABLETT et al.: LEARNING FROM GUIDED PLAY
+7
+LfGP – Open-Gripper
+LfGP – Close-Gripper
+LfGP – Reach
+LfGP – Lift
+LfGP – Move-Object
+LfGP – Stack
+DAC – Stack
+Fig. 8: The policy outputs (arrows) and Q values (background) for each LfGP task and for DAC at 200k environment steps. The arrows show
+velocity direction/magnitude, blue → green indicates open-gripper → close-gripper. For Q values, red → yellow indicates low → high. The
+LfGP policies and Q functions are reasonable for all tasks, while DAC has only learned to reach toward and above the green block.
+snapshot policies for a single exploratory trajectory, using the
+stochastic outputs of each policy as well as, for LfGP, the
+WRS+HC scheduler. Trajectories from these runs are shown
+in Fig. 9.
+DAC is unable to learn to grasp or even reach the blue
+block and ultimately settles on a policy that learns to reach
+and hover near the green block. This is understandable—DAC
+learns a deceptive reward for hovering above the green block
+regardless of the position of the blue block, because it has not
+sufficiently explored the alternative of first grasping the blue
+block. Even if hovering above the green block does not fully
+match the expert data, the DAC policy receives some reward
+for doing so, as evidenced by the learned Q value on the right
+side of Fig. 8.
+In comparison, even after only 200k environment steps,
+LfGP learns to reach and push the blue block, and by 600k
+steps, grasp, move, and nearly stack it. By enforcing explo-
+ration of sub-tasks that are crucial to completing the main task,
+LfGP ensures that the distribution of expert stacking data is
+fully matched.
+VII. RELATED WORK
+Imitation learning is often divided into two main categories:
+behavioural cloning (BC) [23], [24] and inverse reinforcement
+learning (IRL) [5], [25]. BC recovers the expert policy via
+supervised learning, but it suffers from compounding errors
+due to covariate shift [23], [26]. Alternatively, IRL partially
+alleviates the covariate shift problem by estimating the reward
+function and then applying RL using the learned reward.
+A popular approach to IRL is adversarial imitation learning
+(AIL) [6], [7], [27], in which the expert policy is recovered
+by matching the occupancy measure between the generated
+data and the demonstration data. Our proposed method en-
+hances existing AIL algorithms by enabling exploration of
+Fig. 9: LfGP and DAC trajectories of the gripper, blue block, and
+green block for four stack episodes with consistent initial conditions
+throughout the learning process. The LfGP episodes, each including
+auxiliary task sub-trajectories, demonstrate significantly more variety
+than the DAC trajectories.
+key auxiliary tasks via the use of a scheduled multitask model,
+simultaneously resolving the susceptibility of AIL to deceptive
+rewards.
+Agents learned via hierarchical reinforcement learning
+(HRL), which act over multiple levels of temporal abstractions
+in long planning horizon tasks, are shown to provide more
+effective exploration than agents operating over only a single
+level of abstraction [12], [28], [29]. Our approach for learning
+agents most closely resembles hierarchical AIL methods that
+attempt to combine AIL with HRL [27], [30]–[32]. Existing
+work [30]–[32] often formulates the hierarchical agent using
+the Options framework [28] and learns the reward function
+with AIL [6]. Both [30] and [32] leverage task-specific expert
+demonstrations to learn options using mixture-of-experts and
+expectation-maximization strategies, respectively. In contrast,
+our work focuses on expert demonstrations that include multi-
+ple reusable auxiliary tasks, each of which has clear semantic
+meaning.
+In the multitask setting, [27] and [31] leverage unsegmented,
+multitask expert demonstrations to learn low-level policies via
+a latent variable model. Other work has used a large corpus
+of unsegmented but semantically meaningful “play” expert
+data to bootstrap policy learning [13], [14]. We define our
+expert dataset as being derived from guided play, in that the
+expert completes semantically meaningful auxiliary tasks with
+provided transitions, reducing the burden on the expert to
+generate these data arbitrarily and simultaneously providing
+auxiliary task labels. Compared with learning from unseg-
+mented demonstrations, the use of segmented demonstrations,
+as in [33], ensures that we know which auxiliary tasks our
+model will be learning, and opens up the possibility of expert
+data reuse and also transfer learning. Finally, we deviate from
+the Options framework and build upon Scheduled Auxiliary
+Control (SAC-X) to train our hierarchical agent, since SAC-
+X has been shown to work well for challenging manipulation
+tasks [12].
+VIII. LIMITATIONS
+Our approach is not without limitations. While we were
+able to use LfGP in six and seven-task settings, the number
+of tasks for which this method would become intractable is
+unclear. LfGP needs access to segmented expert data as well;
+in many cases, this is reasonable, and is also required to
+be able to reuse auxiliary task data between main tasks, but
+it does necessitate extra care during expert data collection.
+Also, LfGP requires pre-defined auxiliary tasks: while this is
+a common approach to hierarchical RL (see [34], Section 3.1,
+for numerous examples), choosing these tasks may sometimes
+present a challenge. Finally, compared with methods that use
+offline data exclusively (e.g., BC), for our tasks, LfGP requires
+
+200k
+400k
+600k
+800k
+LfGP
+DAC8
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022
+many online environment steps to learn a high-quality policy.
+This data gathering could be costly if human supervision was
+necessary. It is worth noting that, because LfGP is already a
+multitask method, this final point could be partially resolved
+through the use of multitask reset-free RL [35].
+IX. CONCLUSION
+We have shown how adversarial imitation learning can fail
+at challenging manipulation tasks because it learns deceptive
+rewards. We demonstrated that this can be resolved with
+Learning from Guided Play (LfGP), in which we introduce
+auxiliary tasks and the corresponding expert data, guiding the
+agent to playfully explore parts of the state and action space
+that would have been avoided otherwise. We demonstrated that
+our method dramatically outperforms both BC and AIL base-
+lines, particularly in the case of AIL. Furthermore, our method
+can leverage reusable expert data, making it significantly more
+expert sample efficient than the highest-performing baseline,
+and its learned auxiliary task models can be applied to transfer
+learning. In future work, we intend to investigate transfer
+learning to determine if overall policy learning time can be
+reduced.
+ACKNOWLEDGEMENTS
+We gratefully acknowledge the Digital Research Alliance of
+Canada and NVIDIA Inc., who provided the GPUs used in this
+work through their Resources for Research Groups Program
+and their Hardware Grant Program, respectively.
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+(ICRA’21), Apr. 2021.
+
+ABLETT et al.: LEARNING FROM GUIDED PLAY
+9
+APPENDIX A
+LEARNING FROM GUIDED PLAY ALGORITHM
+The complete pseudo-code is given in Algorithm 1. Our
+implementation builds on RL Sandbox [36], an open-source
+PyTorch [37] framework for RL algorithms. For learning
+the discriminators, we follow DAC and apply a gradient
+penalty for regularization [7], [38]. We optimize the intentions
+via the reparameterization trick [40]. As is commonly done
+in deep RL, we use the Clipped Double Q-Learning trick
+[41] to mitigate overestimation bias [42] and use a target
+network to mitigate learning instability [43] when training
+the policies and Q-functions. We also learn the temperature
+parameter αT separately for each task T (see Section 5 of [44]
+for more details on learning α). For Generative Adversarial
+Imitation Learning (GAIL), we use a common open-source
+PyTorch implementation [45]. The hyperparameters chosen for
+all methods are provided in Section G. Please see videos at
+papers.starslab.ca/lfgp for examples of what LfGP looks like
+in practice.
+Algorithm 1 Learning from Guided Play (LfGP)
+Input: Expert replay buffers BE
+main, BE
+1 , . . . , BE
+K, scheduler
+period ξ, sample batch size N
+Parameters: Intentions πT with corresponding Q-functions
+QT and discriminators DT , and scheduler πS (e.g. with Q-
+table QS)
+1: Initialize replay buffer B
+2: for t = 1, . . . , do
+3:
+# Interact with environment
+4:
+For every ξ steps, select intention πT using πS
+5:
+Select action at using πT
+6:
+Execute action at and observe next state s′
+t
+7:
+Store transition ⟨st, at, s′
+t⟩ in B
+8:
+9:
+# Update discriminator DT ′ for each task T ′
+10:
+Sample {(si, ai)}N
+i=1 ∼ B
+11:
+for each task T ′ do
+12:
+Sample {(s′
+i, a′
+i)}B
+i=1 ∼ BE
+k
+13:
+Update DT ′ following Eq. (1) using GAN + Gradient
+Penalty
+14:
+end for
+15:
+16:
+# Update intentions πT ′ and Q-functions QT ′ for each
+task T ′
+17:
+Sample {(si, ai)}N
+i=1 ∼ B
+18:
+Compute reward DT ′(si, ai) for each task T ′
+19:
+Update π and Q following Eq. (4) and Eq. (5)
+20:
+21:
+# Optional Update learned scheduler πS
+22:
+if at the end of effective horizon then
+23:
+Compute main task return GTmain using reward esti-
+mate from Dmain
+24:
+Update πS
+(e.g. update Q-table QS
+following
+Eq. (A.3) and recompute Boltzmann distribution)
+25:
+end if
+26: end for
+A. Scheduler Details
+1) Learning the Scheduler: As stated in our paper, our
+main experiments used a simple weighted random scheduler
+with handcrafted trajectories. In this section, we provide the
+details of our learned scheduler. Following [12], let H be the
+total number of possible intention switches within an episode
+and let each chosen intention execute for ξ timesteps. The
+H intention choices made within the episode are defined as
+T 0:H−1 =
+�
+T (0), . . . , T (H−1)�
+, where T (h) ∈ Tall. The main
+task’s return given chosen intentions is then defined as
+GTmain(T 0:H−1) =
+H−1
+�
+h=0
+(h+1)ξ−1
+�
+t=hξ
+γtRTmain(st, at),
+(A.1)
+where
+at
+∼
+πT (h)(·|st)
+is
+the
+action
+taken
+at
+timestep
+t,
+sampled
+from
+the
+chosen
+intention
+T (h)
+in
+the
+hth
+scheduler
+period.
+We
+further
+define
+the
+Q-function
+for
+the
+scheduler
+as
+QS(T 0:h−1, T (h))
+=
+ET h:H−1∼P h:H−1
+S
+�
+GTmain(T h:H−1)|T 0:h−1�
+and represent the
+scheduler for the hth period as a softmax distribution P h
+S over
+{QS(T 0:h−1, Tmain), QS(T 0:h−1, T1), . . . , QS(T 0:h−1, TK)}.
+The scheduler maximizes the expected return of the main
+task following the scheduler:
+L(S) = ET (0)∼P 0
+S
+�
+QS(∅, T (0))
+�
+.
+(A.2)
+We use Monte Carlo returns to estimate QS, estimating the
+expected return using the exponential moving average:
+QS(T 0:h−1, T (h)) = (1 − φ)QS(T 0:h−1, T (h))
++φ GTmain(T h:H),
+(A.3)
+where φ ∈ [0, 1] represents the amount of discounting on older
+returns and GTmain(T h:H) is the cumulative discounted return
+of the trajectory starting at timestep hξ.
+B. Weighted Random Scheduler Plus Handcrafted Trajectories
+As stated in our paper, the main experiments were com-
+pleted with the described weighted random scheduler (WRS)
+combined with some simple handcrafted trajectories (HC)
+that we expected to be beneficial for learning each of
+the main tasks. In this section, we provide further de-
+tails of these handcrafted scheduler trajectories. Given a
+chosen proportion hyperparameter (0.5 in our experiments),
+we randomly sampled full trajectories from the lists below
+at the beginning of training episodes, and otherwise sam-
+pled from the regular WRS. For all four tasks Main =
+{Stack, Unstack-Stack, Bring, Insert}, we provided the fol-
+lowing set of trajectories:
+1) Reach, Lift, Main, Open-Gripper, Reach, Lift, Main,
+Open-Gripper.
+2) Reach, Lift, Move-Object, Main, Open-Gripper, Reach,
+Lift, Move-Object.
+3) Lift, Main, Open-Gripper, Lift, Main, Open-Gripper,
+Lift, Main.
+4) Main, Open-Gripper, Main, Open-Gripper, Main, Open-
+Gripper, Main, Open-Gripper.
+
+10
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022
+TABLE II: The components used in our environment observations,
+common to all tasks. Grip finger position is a continuous value from
+0 (closed) to 1 (open).
+Component
+Dim
+Unit
+Privileged?
+Extra info
+EE pos.
+3
+m
+No
+rel. to base
+EE velocity
+3
+m/s
+No
+rel. to base
+Grip finger pos.
+6
+[0, 1]
+No
+current, last 2
+Block pos.
+6
+m
+Yes
+both blocks
+Block rot.
+8
+quat
+Yes
+both blocks
+Block trans vel.
+6
+m/s
+Yes
+rel. to base
+Block rot vel.
+6
+rad/s
+Yes
+rel. to base
+Block rel to EE
+6
+m
+Yes
+both blocks
+Block rel to block
+3
+m
+Yes
+in base frame
+Block rel to slot
+6
+m
+Yes
+both blocks
+Force-torque
+6
+N,Nm
+No
+at wrist
+Total
+59
+5) Move-Object, Main, Open-Gripper, Move-Object, Main,
+Open-Gripper, Move-Object, Main.
+For insert, in addition to the trajectories listed above, we added
+two more trajectories to specifically accommodate Bring as an
+auxiliary task:
+1) Bring,
+Insert,
+Open-Gripper,
+Bring,
+Insert,
+Open-
+Gripper, Bring, Insert.
+2) Reach, Lift, Bring, Insert, Open-Gripper, Reach, Lift,
+Bring.
+APPENDIX B
+ENVIRONMENT DETAILS
+Fig. 10: An image of our multitask environment immediately after a
+reset has been carried out.
+A screenshot of our environment, simulated in PyBullet
+[47], is shown in Fig. 10. We chose this environment because
+we desired tasks that a) have a large distribution of possible
+initial states, representative of manipulation in the real world,
+b) have a shared observation/action space with several other
+tasks, allowing the use of auxiliary tasks and transfer learning,
+and c) require a reasonably long horizon and significant use of
+contact to solve. The environment contains a tray with sloped
+edges (to keep the blocks within the reachable workspace of
+the end-effector), as well as a green and a blue block, each
+of which is 4 cm × 4 cm × 4 cm and has a mass of 100 g.
+The dimensions of the lower part of the tray, before reaching
+the sloped edges, are 30 cm × 30 cm. The dimensions of the
+‘bring’ boundaries (shaded blue and green regions) are 8 cm
+× 8 cm, while the dimensions of the insertion slots, which
+are directly in the center of each shaded region, are 4.1 cm ×
+4.1 cm × 1 cm. The boundaries for end-effector movement,
+relative to the tool center point that is directly between the
+gripper fingers, are a 30 cm × 30 cm × 14.5 cm box, where
+the bottom boundary is low enough to allow the gripper to
+interact with objects, but not to collide with the bottom of the
+tray.
+See Table II for a summary of our environment observations.
+In this work, we use privileged state information (e.g., block
+poses), but adapting our method to exclusively use image-
+based data is straightforward since we do not use hand-crafted
+reward functions as in [12].
+The environment movement actions are 3-DOF translational
+position changes, where the position change is relative to the
+current end-effector position. We leverage PyBullet’s built-in
+position-based inverse kinematics function to generate joint
+commands. Our actions also contain a fourth dimension that
+corresponds to actuating the gripper. To allow for the use
+of policy models with exclusively continuous outputs, this
+dimension accepts any real number, with any value greater
+than 0 commanding the gripper to open, and any number less
+than 0 commanding it to close. Actions are supplied at a rate
+of 20 Hz, and each training episode is limited to 18 seconds,
+corresponding to 360 time steps per episode. For play-based
+expert data collection, we also reset the environment manually
+every 360 time steps. Between episodes, block positions are
+randomized to any pose within the tray, and the end-effector
+is randomized to any position between 5 and 14.5 cm above
+the tray, within the earlier stated end-effector bounds, with
+the gripper fully opened. The only exception to these initial
+conditions is during expert data collection and agent training
+of the Unstack-Stack task: in this case, the green block is
+manually set to be on top of the blue block at the start of the
+episode.
+APPENDIX C
+PERFORMANCE RESULTS FOR AUXILIARY TASKS
+The performance results for all multitask methods and
+all auxiliary tasks are shown in Fig. 11. Multitask BC has
+gradually decreasing performance on many of the auxiliary
+tasks as the number of updates increases, which is consistent
+with mild overfitting. Intriguingly, however, multitask BC
+does achieve quite reasonable performance on many of the
+auxiliary tasks (such as Lift) without needing any of the extra
+environment interactions required by an online method such
+as LfGP or DAC. An interesting direction for future work is to
+determine whether pretraining via multitask BC could provide
+
+ABLETT et al.: LEARNING FROM GUIDED PLAY
+11
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Stack
+Stack
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Open
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Close
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Lift
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Reach
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Move
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Unstack-Stack
+Unstack-Stack
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Open
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Close
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Lift
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Reach
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Move
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Bring
+Bring
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Open
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Close
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Lift
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Reach
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+Move
+1
+2
+3
+4
+0.0
+0.5
+1.0
+Insert
+Insert
+1
+2
+3
+4
+0.0
+0.5
+1.0
+Open
+1
+2
+3
+4
+0.0
+0.5
+1.0
+Close
+1
+2
+3
+4
+0.0
+0.5
+1.0
+Bring
+1
+2
+3
+4
+0.0
+0.5
+1.0
+Lift
+1
+2
+3
+4
+0.0
+0.5
+1.0
+Reach
+1
+2
+3
+4
+0.0
+0.5
+1.0
+Move
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Updates/steps (millions)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Success Rate
+LfGP (multi)
+BC (multi)
+DAC (single)
+BC (single)
+Fig. 11: Performance for LfGP and the multitask baselines across all tasks, shaded area corresponds to standard deviation.
+any improvements in environment sample efficiency. We did
+attempt to do this, but found that it resulted in poorer final
+performance than training from scratch.
+APPENDIX D
+PROCEDURE FOR OBTAINING EXPERTS
+As stated, we used SAC-X [12] to train models that we
+used for generating expert data. We used the same hyperpa-
+rameters that we used for LfGP (see Table III), apart from
+the discriminator, which, of course, does not exist in SAC-X.
+See Section E for details on the hand-crafted rewards that we
+used for training these models. For an example of gathering
+play-based expert data, please see our attached video.
+We made two modifications to regular SAC-X to speed up
+learning. First, we pre-trained a Move-Object model before
+transferring this model to each of our main tasks, as we did
+in Section 5.3 of our main paper, since we found that SAC-X
+would plateau when we tried to learn the more challenging
+tasks from scratch. The need for this modification demon-
+strates another noteworthy benefit of LfGP—when training
+LfGP, main tasks could be learned from scratch, and generally
+in fewer time steps, than it took to train our experts. Second,
+during transfer to the main tasks, we used what we called a
+conditional weighted scheduler instead of a Q-Table: we de-
+fined weights for every combination of tasks, so that the sched-
+uler would pick each task with probability P(T (h)|T (h−1)),
+ensuring that ∀T ′ ∈ Tall, �
+T ∈Tall P(T |T ′) = 1. The weights
+that we used were fairly consistent between main tasks, and
+can be found in our packaged code. The conditional weighted
+scheduler ensured that every task was still explored throughout
+the learning process, so that we would have high-quality
+experts for every auxiliary task in addition to the main task.
+This scheduler can be considered as a more complex alter-
+native to the weighted random scheduler or the addition with
+handcrafted trajectories from our main paper, and again shows
+the flexibility of using a semantically-meaningful multitask
+policy with a common observation and action space.
+APPENDIX E
+EVALUATION
+As stated in our paper, we evaluated all algorithms by
+testing the mean output of the main task policy head in
+our environment and determining a success rate based on 50
+randomly selected resets. These evaluation episodes were run
+for 360 time steps to match our training process, and if a
+condition for success was met within that time, they were
+recorded as a success. The rest of this section describes in
+detail how we evaluated ‘success’ for each of our main and
+auxiliary tasks.
+As previously stated, we trained experts using a modified
+SAC-X [12] that required us to define a set of reward functions
+for each task, which we include in this section. The authors
+of [12] focused on sparse rewards but also showed a few
+experiments in which dense rewards reduced the time to learn
+adequate policies, so we chose to use dense rewards. We note
+that many of these reward functions are particularly com-
+plex and required significant manual shaping effort, further
+motivating the use of an imitation learning scheme like the
+one presented in our paper. It is possible that we could have
+made do with sparse rewards, such as those used in [12], but
+our compute resources made this impractical—for example,
+in [12], their agent took 5000 episodes × 36 actors × 360
+time steps = 64.8 M time steps to learn their stacking task,
+which would have taken over a month of wall clock time on
+our fastest machine. To see the specific values used for the
+rewards and success conditions described in these sections,
+please review our code.
+Unless otherwise stated, each of the success conditions in
+this section had to be held for 10 time steps, or 0.5 seconds,
+before being registered as a success. This choice was made
+to prevent registering a success when, for example, the blue
+block slipped off the green block during the Stack task.
+
+12
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022
+A. Common
+For each of these functions, we use the following common
+labels:
+• pb: blue block position,
+• vb: blue block velocity,
+• ab: blue block acceleration,
+• pg: green block position,
+• pe: end-effector tool center point position (TCP),
+• ps: center of a block pushed into one of the slots,
+• g1: (scalar) gripper finger 1 position,
+• g2: (scalar) gripper finger 2 position, and
+• ag: (scalar) gripper open/close action.
+A block is flat on the tray when pb,z = 0 or pg,z = 0. To
+further reduce training time for SAC-X experts, all rewards
+were set to 0 if ∥pb −pe∥ > 0.1 and ∥pg −pe∥ > 0.1 (i.e., the
+TCP must be within 10 cm of either block). During training
+while using the Unstack-Stack variation of our environment,
+a penalty of -0.1 was added to each reward if ∥pg,z∥ > 0.001
+(i.e., there was a penalty to all rewards if the green block was
+not flat on the tray).
+B. Stack/Unstack-Stack
+The evaluation conditions for Stack and Unstack-Stack are
+identical, but in our Unstack-Stack experiments, the environ-
+ment is manually set to have the green block start on top of
+the blue block.
+1) Success: Using internal PyBullet commands, we check
+to see whether the blue block is in contact with the green
+block and is not in contact with either the tray or the gripper.
+2) Reward: We include a term for checking the distance
+between the blue block and the spot above the the green block,
+a term for rewarding increasing distance between the block and
+the TCP once the block is stacked, a term for shaping lifting
+behaviour, a term to reward closing the gripper when the block
+is within a tight reaching tolerance, and a term for rewarding
+the opening the gripper once the block is stacked.
+C. Bring/Insert
+We use the same success and reward calculations for Bring
+and Insert, but for Bring the threshold for success is 3 cm,
+and for insert, it is 2.5 mm.
+1) Success: We check that the distance between pb and
+ps is less than the defined threshold, that the blue block is
+touching the tray, and that the end-effector is not touching the
+block. For Insert, the block can only be within 2.5 mm of the
+insertion target if it is correctly inserted.
+2) Reward: We include a term for checking the distance
+between the pb and ps and a term for rewarding increas-
+ing distance between pb and pe once the blue block is
+brought/inserted.
+D. Open-Gripper/Close-Gripper
+We use the same success and reward calculations for Open-
+Gripper and Close-Gripper, apart from inverting the condition.
+1) Success: For Open-Gripper and Close-Gripper, we check
+to see if ag < 0 or ag > 0 respectively.
+2) Reward: We include a term for checking the action, as
+we do in the success condition, and also include a shaping term
+that discourages high magnitudes of the movement action.
+E. Lift
+1) Success: We check to see if pb,z > 0.06.
+2) Reward: We add a dense reward for checking the height
+of the block, but specifically also check that the gripper
+positions correspond to being closed around the block, so that
+the block does not simply get pushed up the edges of the tray.
+We also include a shaping term for encouraging the gripper
+to close when the block is reached.
+F. Reach
+1) Success: We check to see if ∥pe − pb∥ < 0.015.
+2) Reward: We have a single dense term to check the
+distance between pe and pb.
+G. Move-Object
+For Move-Object, we changed the required holding time for
+success to 1 second, or 20 time steps.
+1) Success: We check to see if the vb > 0.05 and ab < 5.
+The acceleration condition ensures that the arm has learned to
+move the block by following a smooth trajectory, rather than
+vigorously shaking it or continuously picking up and dropping
+it.
+2) Reward: We include a velocity term and an acceleration
+penalty, as in the success condition, but also include a dense
+bonus for lifting the block.
+APPENDIX F
+RETURN PLOTS
+As previously stated, we generated hand-crafted reward
+functions for each of our tasks for the purpose of training
+our SAC-X experts. Given that we have these rewards, we
+can also generate return plots corresponding to our results
+to add extra insight (see Fig. 12 and Fig. 13). The patterns
+displayed in these plots are, for the most part, quite similar
+to the success rate plots. One notable exception is that there
+is an eventual increase in performance when training DAC on
+Insert, indicating that, perhaps for certain tasks, DAC alone
+can eventually make progress. Nevertheless, it is clear that
+LfGP improves learning efficiency, and it is unclear whether
+DAC would plateau even if it was trained for a longer period.
+APPENDIX G
+MODEL ARCHITECTURES AND HYPERPARAMETERS
+All the single-task models share the same network architec-
+tures and all the multitask models share the same network
+architectures. All layers are initialized using the PyTorch
+default methods [37].
+For the single-task variant, the policy is a fully-connected
+network with two hidden layers followed by ReLU activation.
+Each hidden layer consists of 256 hidden units. The output of
+the policy for LfGP and DAC is split into two vectors, mean
+ˆµ and variance ˆσ2. For both variants of BC, only the mean ˆµ
+
+ABLETT et al.: LEARNING FROM GUIDED PLAY
+13
+0.5
+1.0
+1.5
+2.0
+0
+200
+400
+600
+Stack
+0.5
+1.0
+1.5
+2.0
+0
+200
+400
+600
+800
+1000
+Unstack-Stack
+0.5
+1.0
+1.5
+2.0
+0
+100
+200
+300
+400
+500
+Bring
+0
+1
+2
+3
+4
+100
+200
+300
+400
+500
+Insert
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Updates/steps (millions)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Episode Return
+LfGP (multi)
+BC (multi)
+DAC (single)
+BC (single)
+Expert
+Fig. 12: Episode return for LfGP compared with all baselines. Shaded area corresponds to standard deviation.
+0.5
+1.0
+1.5
+2.0
+0
+200
+400
+600
+Stack
+Stack
+0.5
+1.0
+1.5
+2.0
+200
+250
+300
+Open
+0.5
+1.0
+1.5
+2.0
+100
+150
+200
+250
+300
+Close
+0.5
+1.0
+1.5
+2.0
+0
+200
+400
+Lift
+0.5
+1.0
+1.5
+2.0
+100
+150
+200
+250
+300
+Reach
+0.5
+1.0
+1.5
+2.0
+0
+200
+400
+Move
+0.5
+1.0
+1.5
+2.0
+0
+200
+400
+600
+800
+Unstack-Stack
+Unstack-Stack
+0.5
+1.0
+1.5
+2.0
+200
+250
+300
+Open
+0.5
+1.0
+1.5
+2.0
+150
+200
+250
+300
+Close
+0.5
+1.0
+1.5
+2.0
+0
+200
+400
+Lift
+0.5
+1.0
+1.5
+2.0
+0
+100
+200
+Reach
+0.5
+1.0
+1.5
+2.0
+0
+200
+400
+Move
+0.5
+1.0
+1.5
+2.0
+0
+100
+200
+300
+400
+Bring
+Bring
+0.5
+1.0
+1.5
+2.0
+200
+250
+300
+Open
+0.5
+1.0
+1.5
+2.0
+100
+200
+300
+Close
+0.5
+1.0
+1.5
+2.0
+0
+200
+400
+Lift
+0.5
+1.0
+1.5
+2.0
+100
+200
+300
+Reach
+0.5
+1.0
+1.5
+2.0
+0
+200
+400
+Move
+1
+2
+3
+4
+200
+400
+Insert
+Insert
+1
+2
+3
+4
+250
+275
+300
+325
+Open
+1
+2
+3
+4
+100
+200
+300
+Close
+1
+2
+3
+4
+100
+200
+300
+400
+500
+Bring
+1
+2
+3
+4
+0
+200
+400
+Lift
+1
+2
+3
+4
+0
+100
+200
+300
+Reach
+1
+2
+3
+4
+0
+200
+400
+Move
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Updates/steps (millions)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Episode Return
+LfGP (multi)
+BC (multi)
+DAC (single)
+BC (single)
+Fig. 13: Episode return for LfGP compared with multitask baselines on all tasks. Shaded area corresponds to standard deviation.
+output is used. The vectors define a Gaussian distribution (i.e.
+N(ˆµ, ˆσ2I), where I is the identity matrix). When computing
+actions, we squash the samples using the tanh function and
+bound the actions to be in range [−1, 1], as done in SAC
+[44]. The variance ˆσ2 is computed by applying a softplus
+function followed by a sum with an epsilon ϵ = 1e-7 to
+prevent underflow: ˆσi = softplus(ˆxi) + ϵ. The Q-functions
+are fully-connected networks with two hidden layers followed
+by ReLU activations. Each hidden layer consists of 256 units.
+The output of the Q-function is a scalar corresponding to the
+value estimate given the current state-action pair. Finally, the
+discriminator is a fully-connected network with two hidden
+layers followed by tanh activations. Each hidden layer consists
+of 256 units. The output of the discriminator is a scalar logit
+to be used as an input to the sigmoid function. The sigmoid
+function output can be viewed as the probability of the current
+state-action pair coming from the expert distribution.
+For multitask variant, the policies and the Q-functions share
+their initial layers. There are two shared, fully-connected
+layers followed by ReLU activations. Each layer consists of
+256 units. The output of the last shared layer is then fed into
+the policies and Q-functions. Each policy head and Q-function
+head corresponds to one task and has the same architecture:
+a two-layered fully-connected network followed by ReLU
+activations. The output of the policy head corresponds to the
+parameters of a Gaussian distribution, as described previously.
+Similarly, the output of the Q-function head corresponds to the
+value estimate. Finally, the discriminator is a fully-connected
+network with two hidden layers followed by tanh activations.
+Each hidden layer consists of 256 units. The output of the
+discriminator is a vector, where the ith entry corresponds to
+the logit input to the sigmoid function for task Ti. The ith
+sigmoid function output corresponds to the probability of the
+current state-action pair coming from the expert distribution
+in task Ti.
+The hyperparameters for our experiments are listed in
+Table III and Table V. In the early-stopping variant of BC,
+overfit tolerance refers to the number of full dataset training
+epochs without an improvement in validation error before we
+stop training. All models are optimized using Adam Optimizer
+[48] with PyTorch default values, unless specified otherwise.
+APPENDIX H
+OPEN-ACTION AND CLOSE-ACTION
+
+14
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022
+TABLE III: Hyperparameters for AIL algorithms across all tasks.
+Parameters that do not appear in the original version of DAC are
+shown in blue.
+Algorithm
+LfGP
+DAC
+Total Interactions
+2M (4M for Insert)
+Buffer Size
+2M (4M for Insert)
+Buffer Warmup
+25k
+Initial Exploration
+50k
+Evaluations per task
+50
+Evaluation frequency
+100k interactions
+Intention
+γ
+0.99
+Batch Size
+256
+Q Update Freq.
+1
+Target Q Update Freq.
+1
+π Update Freq.
+1
+Polyak Averaging
+1e-4
+Q Learning Rate
+3e-4
+π Learning Rate
+1e-5
+α Learning Rate
+3e-4
+Initial α
+1e-2
+Target Entropy
+−dim(a) = −4
+Max. Gradient Norm
+10
+π Weight Decay
+1e-2
+Q Weight Decay
+1e-2
+BE sampling proportion
+0.1
+BE sampling decay
+0.99999
+Discriminator
+Learning Rate
+3e-4
+Batch Size
+256
+Gradient Penalty λ
+10
+Weight Decay
+1e-2
+(sT , 0) sampling bias
+0.95
+TABLE IV: Hyperparameters for LfGP schedulers.
+Scheduler
+Learned
+WRS
+WRS + HC
+ξ
+45
+N/A
+N/A
+φ
+0.6
+N/A
+N/A
+Initial Temp.
+360
+N/A
+N/A
+Temp. Decay
+0.9995
+N/A
+N/A
+Min. Temp.
+0.1
+N/A
+N/A
+Main Task Rate
+N/A
+0.5
+0.5
+Handcraft Rate
+N/A
+N/A
+0.5
+DISTRIBUTION MATCHING
+There was one exception to the method we used for col-
+lecting our expert data. Specifically, our Open-Gripper and
+Close-Gripper tasks required additional considerations. It is
+worth reminding the reader that our Open-Gripper and Close-
+Gripper tasks were meant to simply open or close the gripper,
+respectively, while remaining reasonably close to either block.
+If we were to use the approach described above verbatim,
+the Open-Gripper and Close-Gripper data would contain no
+(s, a) pairs where the gripper actually released or grasped
+the block, instead immediately opening or closing the gripper
+while simply hovering near the blocks. Perhaps unsurprisingly,
+this was detrimental to our algorithm’s performance: as one
+example, an agent attempting to learn Stack would, if Open-
+Gripper was selected while the blue block was held above
+TABLE V: Hyperparameters for BC algorithms (both single-task and
+multitask) across all tasks.
+Version
+Main Results
+Early Stopping
+Batch Size
+256
+Learning Rate
+1e-5
+Weight Decay
+1e-2
+Total Updates
+2M (4M for Insert)
+N/A
+Overfit Tolerance
+N/A
+100
+the green block, move the grasped blue block away from the
+green block before dropping it on the tray. This behaviour, of
+course, is not what we would want, but it better matches an
+expert distribution when the environment is reset in between
+each task execution.
+To mitigate this, our Open-Gripper data actually contain a
+mix of each of the other sub-tasks called for the first 45 time
+steps, followed by a switch to Open-Gripper, ensuring that
+the expert dataset contains some degree of block-releasing,
+with the trade-off being that 50% of the Open-Gripper expert
+data is specific to whatever the main task happens to be. We
+left this additional detail out of our main paper for clarity,
+since it corresponds to only a small portion of the expert
+data (every other auxiliary task was fully reused). Similarly,
+the Close-Gripper data calls Lift for 15 time steps before
+switching to Close-Gripper, ensuring that the Close-gripper
+dataset will contain a large proportion of data where the block
+is actually grasped. For the Closer-gripper data, however, this
+modification did still allow data to be reused between main
+tasks.
+APPENDIX I
+ATTEMPTED AND FAILED EXPERIMENTS
+In this section, we provide a list of experiments and modi-
+fications that did not improve performance, in addition to the
+alternatives that did.
+1) Pretraining with BC: We attempted to pretrain LfGP
+using multitask BC, and then to transition to online
+learning with LfGP, but we found that this tended to
+produce significantly poorer final performance. Some
+existing work [49], [50] has investigated transitioning
+from BC to online RL, but achieving this consistently,
+especially with off-policy RL, remains an open research
+problem.
+2) Handcrafted Open-Gripper/Close-Gripper policies:
+Given the simplicity of designing a reward function in
+these two cases, a natural question is whether Open-
+Gripper and Close-Gripper could use hand-crafted re-
+ward functions, or even hand-crafted policies, instead of
+these specialized datasets. In our experiments, both of
+these alternatives proved to be quite detrimental to our
+algorithm.
+3) Penalizing Q values: In our early experiments, we
+found that LfGP training progress was harmed by ex-
+ploding Q values. This problem was particularly exac-
+erbated when we added BE sampling to our Q and π
+updates. It appears that this occurs because, at the begin-
+ning of training, the differences between discriminator
+
+ABLETT et al.: LEARNING FROM GUIDED PLAY
+15
+outputs for expert data and non-expert data are so large
+that the bootstrap Q updates quickly jump to unrealistic
+values. We attempted to use various forms of Q penalties
+to resolve this, akin to Conservative Q Learning (CQL)
+[51], but found that all of our modifications ultimately
+harmed final performance. Some of the things we tried,
+in addition to the CQL loss, were reducing γ (.95, .9),
+clipping Q losses to -5, +5, smooth L1 loss, huber loss,
+increased gradient penalty λ for D (50, 100), decreased
+reward scaling (.1), more discriminator updates per π/Q
+update (10), and weight decay in D only (as is done
+in [9]). We ultimately resolved exploding Q values by
+i) decreasing polyak averaging to a significantly lower
+value than is used in much other work (1e-4 as opposed
+to the SAC default of 5e-3), and ii) adding in weight
+decay (with a significantly higher value used than is
+used in other work) to π, Q, and D training (which was
+required to not overfit with the reduced polyak averaging
+value). Without the added weight decay, performance
+started to plateau and eventually to decrease.
+4) Higher Update-to-Data (UTD) Ratio: Recent work in
+RL has started increasing the UTD ratio (i.e., increas-
+ing the number of policy/Q updates per environment
+interaction), with the goal of improving environment
+sample efficiency [53]. We were actually able to increase
+this from 1 to 2 and achieve a marginal improvement
+in environment sample efficiency, but this also nearly
+doubled the running time of our experiments, so we
+opted not to include this modification in our final results.
+Higher values of the UTD ratio also caused our Q values
+to explode.
+APPENDIX J
+EXPERIMENTAL HARDWARE
+For a list of the software we used in this work, see our code
+and instructions. We used a number of different computers and
+GPUs when completing our experiments:
+1) GPU: NVidia Quadro RTX 8000, CPU: AMD - Ryzen
+5950x 3.4 GHz 16-core 32-thread, RAM: 64GB, OS:
+Ubuntu 20.04.
+2) GPU: NVidia V100 SXM2, CPU: Intel Gold 6148
+Skylake @ 2.4 GHz (only used 4 threads), RAM: 32GB,
+OS: CentOS 7.
+3) GPU: Nvidia GeForce RTX 2070, CPU: RYZEN
+Threadripper 2990WX, RAM: 32GB, OS: Ubuntu 20.04.
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+

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+Chapter 1
+Beyond Classroom:
+Making a Difference in
+Diversity in Tech
+Barbora Buhnova
+With all the opportunities and risks that technology holds in connection to our safe and sus-
+tainable future, it is becoming increasingly important to involve a larger portion of our society in
+becoming active co-creators of our digitalized future—moving from the passenger seat to the
+driver seat. Yet, despite extensive efforts around the world, little progress has been made in
+growing the representation of certain communities and groups in software engineering. This
+chapter shares one successful project, called Czechitas, triggering a major social change in
+Czechia, involving 1 000+ volunteers to support 50 000+ women on their way towards software
+engineering education and career.
+arXiv:2301.12000v1  [cs.SE]  27 Jan 2023
+
+CHAPTER 1
+Introduction
+The past decade has witnessed the emergence of hundreds of initiatives around the world
+supporting various underrepresented groups on their pathway towards software engineering,
+whether connected to universities [13], companies [15], or run as independent non-profit or-
+ganizations [14]. Although the initiatives often start with a great vision and high volunteering
+commitment, after a few years into the activities, it becomes challenging to sustain the volun-
+teering energy and commitment in face of the very slow progress towards the better. In those
+moments, the success cases by others can be what helps us keep going.
+The initiative featured in this chapter, called Czechitas [6], started in 2014 in Czechia, with
+a simple idea to bring tech closer to girls and girls closer to tech, in reaction to the strong
+under-representation of women in tech in the country (see Figure 1.1). The prompt snowball
+effect helped us to build a community around the joint vision to empower and encourage girls
+and women to engage in computing education and career transition, and to show them that
+software engineering is an interesting career direction that is not necessarily difficult nor limited
+to one gender. Initially established to provide women in Czechia with an opportunity to put their
+hands on programming, it now contributes to a major social change in the country.
+Over time, Czechitas has become a movement that has attracted a strong community of
+tech-professional volunteers (over 1 000) and companies (over 100), and given rise to a portfo-
+lio of women-tailored courses in various areas of software engineering, such as programming,
+Figure 1.1: Women ICT Professional (Eurostat, 2019 data) [8].
+2
+
+30%
+25%
+20%
+EU = 17.9%
+15%
+10%
+5% 
+0%CHAPTER 1
+web development, mobile app development, data science, cybersecurity or testing (over 1 300
+courses delivered so far). We have influenced over 50 000 women (over 30 000 via live events
+and over 20 000 via online tutorials) who graduated from our courses to use their new tech
+skills to change their education path or advance their careers.
+Czechitas Mission: We inspire, train and guide new talents towards stronger
+diversity and competitiveness in tech.
+Thanks to the success of our education activities with hundreds of events a year (each
+receiving more registrations than its capacity), we have become recognized as the leading
+platform in Czechia actively addressing gender diversity in tech. In this chapter, we share
+the lessons we learned about the low representation of women in tech, effective strategies in
+supporting women on their way to software engineering, discuss the ingredients that helped us
+succeed, the obstacles and challenges we faced, and the progress yet to be made.
+Why are There so Few Women in Tech?
+Across Europe, only 19.1% of tech professionals are women (according to 2021 data) [8], with
+Czechia being the last on the list. The major reasons behind the trend in our region according
+to our recent study (with 70% of participants from Czechia and Germany) [9] are:
+1. Access. The first hole in the leaky pipeline on girls’ pathway towards software engi-
+neering is linked to the missing access to encouragement and support, together with the
+missing access to suitable education that would be able to build on the interests of girls
+that often span across multiple disciplines.
+2. Stereotypes. The ability to see herself as a software engineer is then challenged by the
+perception of the software engineering as a field not leading to a purpose the girl would
+like to dedicate her future to. Often, the close family and friends step-in in this moment
+to direct girls away from software engineering with the intention to protect them from
+a future where they cannot really imagine the girls becoming successful. Interestingly,
+3
+
+CHAPTER 1
+the intentions are meant well, to protect the girls, which shows how crucial it is to help
+parents (and mainly mothers) to understand that software engineering can be a great
+career choice for their daughters.
+3. Confidence. The next hole on the leaky pipeline comes when girls find themselves in
+the classroom, often surrounded by more-experienced learners (typically boys). For the
+little girls who often excel in other subjects, it can be hard to fall in the category of a slow
+novice learner. The girls often mention frustrations of low self-efficacy, inadequacy and
+missing experience of success in presence of a classroom dynamic being monopolized
+by the earlier technology adopters.
+4. Sense of Belonging. The girls who resist through the earlier three challenges and find
+themselves on the education pathway towards software engineering, find themselves in
+classrooms surrounded predominantly by boys. While this is a comfortable environment
+for some, many in the study reported not feeling comfortable to express themselves, fac-
+ing sexism or unwanted attention and missing relatable role models and mentors, which
+led them to reconsider whether this is the environment they would be willing to spend the
+rest of their lives in.
+5. Feeling Valued. The last hole in the leaky pipeline challenges the women who entered
+software engineering careers, as some of them emphasize the struggle of not feeling
+valued at workplace. The reasons are different for the women with stereotypical talent
+spectrum (that matches the talent spectrum typical among their men colleagues, typically
+being very technical) and non-stereotypical talent spectrum (bringing not-that-common
+talents to the table, typically more multidisciplinary and human-oriented). While the first
+group feels ”tired of proving them wrong”, the second group feel frustrated from their
+strengths viewed as second class and from missing appreciation.
+Supporting Women on their Way to Tech
+In Czechitas, we understand that plumbing the leaky pipeline can hardly be done by isolated
+and uncoordinated efforts. This section discusses the interlinked pillars of our activities (see
+Figure 1.2), listing examples of the activities and events we delivered in 2022.
+4
+
+CHAPTER 1
+Czechitas Pillar I – Awareness
+One of the crucial success factors for a change towards improving gender balance in soft-
+ware engineering is the actual understanding that we are in a disbalanced state that further
+reinforces itself due to the factors discussed earlier. The efforts towards encouraging women
+to join software engineering cannot make a difference unless the society, education system
+and corporate environment welcomes and supports the change (understanding it as a push
+towards the real equilibrium, not a push out of it).
+In Czechitas, we are investing substantial effort in awareness around the topic. In 2022
+alone, we participated in over 20 conferences and panel discussions, gave numerous inter-
+views in TV, radio and other media, organized talks to students and teachers at high schools,
+and to tech professionals in our partner companies. We were visible with a booth at 15 festivals
+and family days across Czechia. Over 2022, Czechitas was mentioned in 508 articles, reach-
+ing major part of Czech population. In 2021, we also launched a Czechitas podcast, which in
+2022 reached over 14 676 listens. Furthermore, our website was in 2022 visited by 123 785
+unique visitors, and our newsletter was followed by 25 983 subscribers.
+The next step in raising awareness among the general public is to make it as easy as
+possible to get the first exposure to coding in a fun, enjoyable and community way. To this end,
+we for instance organize an Advent Christmas Coding campaign (following the tradition of an
+advent calendar, in which instead of a sweet treat, each day holds a coding assignment along a
+story of bringing Mr. Gingerbread home for Christmas), which is being followed by hundreds of
+Figure 1.2: The Pillars of Czechitas Activities.
+5
+
+CAREER
+AWARENESS
+TRAINING
+TRANSITION
+COMMUNITYCHAPTER 1
+people. Furthermore, in collaboration with the Ministry of Education, Youth and Sports, we e.g.
+co-organized the #DIGIEDUHACK hackathon. And in collaboration with Czech universities, we
+run the Czechitas Thesis Award to give visibility to exceptional bachelor theses authored by
+girls. All these activities typically repeat every year.
+Czechitas Pillar II – Training
+Since the start of our activities in 2014, we are improving the education design of our courses
+to reflect the needs of our audience—women and girls who are very often later technology
+adopters or career changers—with an emphasis on providing suitable first contact with soft-
+ware engineering, creating safe and supportive environment for novice learners, accommodat-
+ing differences in the learning speed of each student, building self-confidence, and supporting
+sustaining long-term interest, which we also publish [2, 10]. In 2022, we delivered 242 live
+software-engineering courses with 15 316 participants, with the courses around web develop-
+ment and data science scoring as the most popular ones.
+Although most of the training is targeted to women and girls, we are also investing in training
+Figure 1.3: Czechitas-participation (Data as of June 2022).
+6
+
+21,767
+1.482+
+57.683+
+8,686
+educational
+participants
+events
+428
+10
+11,763
+299
+3,752
+8,268
+238
+418
+13,081
+5,821
+3,969
+145
+141
+4,271
+295
+102
+6
+8,011
+84
+2,654
+45
+135
+1,873
+1,266
+4,299
+2015
+2016
+2017
+2018
+2019
+2020
+2021
+2022*
+2015
+2016
+2017
+2018
+2019
+2020
+2021
+2022*
+Number of educational events
+Number of video tutorials
+Educational events participants
+Educational video tutorials participantsCHAPTER 1
+elementary-school and high-school teachers (irrespective of gender). And some mixed-gender
+activities were organized also for children (7 week-long summer camps in the summer of 2022,
+besides others) and high-school kids, although in case of high schools, it is already important
+to offer also girl-only courses (3 week-long summer schools for high-school girls were given
+in 2022). Besides, training courses for mixed audience are also provided on events such as
+Family Days (we were present at over 20 such events in 2022).
+Czechitas Pillar III – Career Transition
+As many women in our community intend to enter software engineering as their future profes-
+sion, some of our activities are intentionally designed to facilitate this journey, whether software
+engineering is to become their first job or they intend to change their career [3].
+In cooperation with our partner companies, we have identified three career pathways that
+appear to be the most suitable entry points to software engineering in Czechia. These are
+(1) web development (including courses on JavaScript, React, HTML/CSS, Bootstrap, Git,
+UX design, and others), (2) data analytics (including courses on Python, databases, SQL,
+statistics, Power BI, and others), (3) testing (including courses on requirements engineering,
+agile processes, manual testing, issue tracking, regression testing, smoke testing, basics of
+automated testing, browsers, API, databases, version control, and others).
+For the three directions, we have developed a complex career-transition support within so-
+called Digital Academies. A Digital Academy is a four-month program for a group of 30 women
+(and involving around 5-15 partner companies), which besides individual courses covering the
+topics outlined above and taking place 3-4 times a week (evenings on working days, full days
+on weekends) includes also pairing of the students with mentors from the companies to support
+them in developing their own projects, a hackathon, career support, and further events offered
+by the partner companies. In 2022, we have run 10 Digital Academies across four major cities
+in Czechia, with over 60% of the graduates receiving a job offer within three months from
+graduating from the academy.
+To facilitate the career transition also for the women who opt to customize their training
+journey (not attending a Digital Academy), our career consultants provide hundreds of career
+consultations each year (327 in 2022), and we twice a year organize a Czechitas Job Fair,
+7
+
+CHAPTER 1
+where our graduates can meet the representatives of our partner companies (each Job Fair
+attended by about 350 graduates and 30 companies).
+Czechitas Foundation – Community
+The foundation that supports all our activities is the community, which involves the participants
+and graduates of our courses, tech professionals who teach with us, mentors, course facili-
+tators, and our partner companies. The fact that many members in our community are men
+helps us not only engage more tech-professional allies in our vision, but also influence a more
+supportive environment in tech companies where our graduates land. To support the blending
+of the community and increasing the sense of belonging of our graduates also in the mixed-
+gender environment, we regularly engage in organization of Tech Meet-ups and Hackathons,
+as well as informal CzechiPubs that regularly take place in 10 different cities across Czechia.
+Making a Difference
+The positive influence of Czechitas activities in Czechia is already visible in the shifted percep-
+tion of software engineering as an education pathway and career choice to be considered by
+any gender. That not only motivates many girls to consider software engineering in their choice
+of a university study field (with the representation of women among ICT students changing from
+12% in 2016 to 17% in 2021 in Czechia [7, 5], moving the country closer to the European av-
+erage, see Figure 1.4) but is likely also having secondary influence on all who so far hesitated
+to join software engineering.
+What Helped us Succeed
+Building Czechitas was only possible thanks to a coordinated effort of hundreds of people (90
+employees and over 1,000 volunteers). Over the past eight years of our existence, we came to
+understand the ingredients without which this would not be possible:
+• Great leadership and love for what we do is giving us the sense of purpose, energy and
+direction, holding us together and keeping us focused. Mentors from partner companies
+8
+
+CHAPTER 1
+Figure 1.4: Women ICT Students (Czech Statistical Office, 2021 data) [5].
+and beyond have been of great help to guide us through the design of our leadership and
+expansion strategy.
+• Visual and playful communication is giving us the fresh flavour of fun and joy that we
+all (students as well as trainers and volunteers) enjoy joining even after a tiring day at
+school or work. The informal and visually attractive communication helps us to share the
+love for our brand.
+• Community and sense of belonging is crucial for connecting those who strive to learn
+with those who strive to share and teach, and those who want to support the connection.
+It helps our student to feel home and make it easier for them to keep going even when
+learning gets hard.
+• Inclusive environment and encouragement makes it safe for our students to make mis-
+takes and experience success, have the opportunity to exchange knowledge, collaborate,
+and get personalized feedback and guidance. Specific strategies and interventions we
+have developed to support novice learners and their self-efficacy have been key in this
+direction [2].
+• Knowledge and understanding is crucial for us to design our activities with insight into
+the frustrations steering women away from software engineering [9] and effective strate-
+gies to support girls and women in tech education [10] and career transition [3]. We
+9
+
+30%
+25%
+EU = 20%
+20%
+15%
+10%
+5%
+0%CHAPTER 1
+invest our time in sharing the lessons we have learned [2, 9, 3], and learning from other
+initiatives from across the world (e.g., within the EUGAIN network, see https://eugain.eu/).
+• Creating and sharing stories helps us to inspire our students, bring them closer to
+relatable role models, and to give them hope and confidence that with some work and
+dedication, a transition into software engineering is possible. The stories (each featuring
+an inspiring woman who changed her career towards tech) are published in our blog,
+communicated via social networks, and used in media articles. These women inspire
+others as speakers and panelists in our events, and as guests in Czechitas Podcast.
+• Sustainable financial model helps us to sustain a team employed to run the organi-
+zation. The model stands on financial participation of the students, partner companies,
+foundations and individual donors, with an intention to reach out also to the government
+level in the future. The most crucial pillar of our financial sustainability is the partner com-
+panies, which are beside their yearly partnership contributions (depending on the level of
+partnership) helping us to cover certain costs (e.g., offering their office spaces for events,
+motivating their employees to volunteer as mentors), and opening doors towards further
+funding opportunities (e.g. with global foundations connected to their company).
+Obstacles and Challenges we Faced
+As any organization that has substantially outgrown its own plans and expectations, Czechitas
+has undergone numerous changes and readjustments over its course of existence. And al-
+though we are trying to publish the effective setup that works for us now [2, 3, 4], our first steps
+were highly organic and experimental, which was key to learning what works for the context
+we were in. With our enthusiasm and ”always yes” spirit, we walked many paths that we failed
+and rolled back, but we also faced numerous obstacles and challenges that we withstood.
+• Scaling the organization. Turning a non-profit start-up into a scale-up is a challenge on
+its own, as the means for achieving stability are different from traditional companies – be-
+sides the discussed financial stability, also in terms of sustained volunteering involvement
+and brand building. We needed to learn to manage the mix of the innovative and largely
+self-sacrificing founding community with the necessary systematic and organized spirit of
+10
+
+CHAPTER 1
+new employees. We needed to learn to prioritize and say no to some activities that the
+team felt strongly for.
+• Being misunderstood. As a large organization, we needed to learn to communicate
+our mission well so that it is not misunderstood, knowing that anything that damages
+the brand may sink the whole boat. Namely, we needed to help our partner companies
+understand what level of expertise is realistic to achieve in our students, help our students
+understand what time investment and commitment it takes to change direction towards
+tech, and help our society understand why our focus on women is key to the success of
+our society as a whole.
+• Quantifying the impact of our activities. One of the important challenges that we are
+still facing is our ability to quantify the impact of our individual interventions and activities,
+as it is difficult to isolate the effects of each one of them. More so that the impact is often
+very subtle and propagates over long periods of time (e.g., a woman making a few steps
+towards tech education inspiring her friend to make a major shift towards tech, who then
+inspires her daughter to study CS at university). So although we have a Data & Impact
+team at Czechitas, with substantial data available, the numbers we have (e.g., the number
+of women who change their career to tech each year) are still only the tip of the iceberg
+of the real impact we strive for, which is the shift in the collective mindset of the entire
+society, leading to a sustained change.
+Progress yet to be Made
+With the increasing number of Czechitas graduates who are joining software engineering in-
+dustry, often as very junior (in terms of their software-engineering expertise) and diverse (in
+terms of their talents and competencies) members, we find it crucial to assist the companies
+to improve the inclusiveness of their environment to integrate and leverage the new diverse
+talent. In 2020, we made the first step towards that goal via designing a Diversity Awareness
+Training, which was since then delivered to over 300 managers (mostly from Central and East-
+ern Europe) across some of our partner companies. The concepts that have shown to be the
+most crucial to discuss and understand during these trainings are outlined below:
+11
+
+CHAPTER 1
+Figure 1.5: Tuckman’s Model of Team Dynamics with an illustration of different dynamics
+observed in homogeneous and heterogeneous teams.
+• Diversity does not come easy, but it pays off. Avoiding diversity is natural to human in-
+dividuals, but dangerous to humankind1. The same is true for corporate environment. We
+need to acknowledge that diverse teams might have a harder time at start (as illustrated
+with the Tuckman’s Model of Team Dynamics in Figure 1.5), but in long-term, diversity is
+firmly correlated with higher performance [11, 12].
+• We too often lose talented people by missing the talent in them. We are all talented,
+in many diverse ways. It is the task of the manager to recognize and direct the talent to-
+wards team success. The fact that a person uses a different talent spectrum (approaches
+problems and situations differently) does not make them more/less suitable for software
+engineering as such. There is no such thing as a second-class citizen when it comes to
+the talents we need in software engineering.
+• Biases evolved to help us navigate complexity, but they are not serving us well
+when making assumptions about the potential in people. The dark side of biases is
+that we tend to judge people’s potential based on how their talent spectrum matches the
+talent of already-successful ones. Without realizing that the successful ones embody the
+skills and conditions that worked when they joined the field (in the past) while we are now
+choosing the software engineers for the future.
+• Connection is built through communication. There are many unhealthy communica-
+1Our quote inspired by the statement ”Diversity is the new Darwinism” by the Great British Diversity Experiment [1].
+12
+
+7
+Forming
+Strorming
+Norming
+Performing
+Adjourning
+个
+Effectiveness
+Homogenous
+team
+Heterogenous
+team
+TimeCHAPTER 1
+tion patterns around diversity, which often go against the purpose of making us all feel
+the sense of belonging. It is important to create safe space, in which we can learn to
+communicate our differences but also ask about the differences of others. Mistakes are
+part of that learning, and forgiveness of the mistakes shall be encouraged if the mistakes
+were done in the process of learning and not repeated blindly. It is important to create a
+safe space to acknowledge our biases and stop shaming one another for them.
+• Avoid the quick fixes, remove the barriers instead. Encourage curiosity about why cer-
+tain communities are under-represented in software engineering. What are the barriers
+they face and what can we do to remove them or make their journey lighter in presence of
+the barriers (e.g. the care-taking on the side of most women)? Avoiding the conversation
+and looking away from the differences in our experiences might lead the community to as-
+sume that the under-representation is the lower-fit problem, which is dangerous because
+it leads to push-back on any diversity support one might try to introduce.
+• Change takes time. Promoting I&D is more complex than it might seem at first. It is
+crucial to know how to start to see the first positive effects soon and be able to use them
+to get more people on board towards promoting I&D further. Choose your first steps well
+and invest in them. The investment will pay off.
+Conclusion
+Making a difference in improving gender balance in software engineering on the scale of the
+whole country is not easy, but is possible. And it is very rewarding to be part of such a move-
+ment. In 2021, the social impact of Czechitas activities was recognized at the European Union
+level via winning the EU Social Economy Award (over 180 organizations nominated) in the
+Digitalisation and Skills category, and in 2022 winning the global Equals in Tech Award (155
+organizations nominated) in the Skills category. We hope our example can inspire others,
+which is also why we are eager to share the lessons learned from our journey.
+13
+
+Bibliography
+[1] Amanda Bennett. Case study: The great British diversity experiment, 2016. FairPlay Ltd.
+[2] Barbora Buhnova and Lucia Happe. Girl-friendly computer science classroom: Czechitas
+experience report. In European Conference on Software Architecture, pages 125–137.
+Springer, 2020.
+[3] Barbora Buhnova, Lucie Jurystova, and Dita Prikrylova.
+Assisting women in career
+change towards software engineering: experience from czechitas ngo. In Proceedings of
+the 13th European Conference on Software Architecture-Volume 2, pages 88–93, 2019.
+[4] Barbora Buhnova and Dita Prikrylova.
+Women want to learn tech: Lessons from the
+czechitas education project. In 2019 IEEE/ACM 2nd International Workshop on Gender
+Equality in Software Engineering (GE), pages 25–28. IEEE, 2019.
+[5] Czech Statistical Office. Human resources in information technology, 2021. Available on-
+line at URL https://www.czso.cz/documents/10180/165376696/063015-21.pdf/c7e96151-
+b285-4388-9384-532e55f4a318?version=1.2.
+[6] Czechitas.
+Czechitas
+annual
+report
+2021,
+2022.
+Available
+online
+at
+URL
+https://is.muni.cz/go/u6ji13.
+[7] Eurostat.
+Female students under-represented in ICT, 2016.
+Available online at URL
+https://ec.europa.eu/eurostat/web/products-eurostat-news/-/edn-20190425-1.
+[8] Eurostat.
+ICT
+specialists
+in
+employment,
+2022.
+Available
+online
+at
+URL
+https://ec.europa.eu/eurostat/statistics-explained/index.php?title=ICT specialists in em-
+ployment.
+
+CHAPTER 1
+[9] Lucia Happe and Barbora Buhnova. Frustrations steering women away from software
+engineering. IEEE Software, 39(4):63–69, 2022.
+[10] Lucia Happe, Barbora Buhnova, Anne Koziolek, and Ingo Wagner. Effective measures to
+foster girls’ interest in secondary computer science education. Education and Information
+Technologies, 26(3):2811–2829, 2021.
+[11] Dame Vivian Hunt, Dennis Layton, and Sara Prince.
+Why diversity matters, 2015.
+McKinsey. Available online at URL https://www.mckinsey.com/capabilities/people-and-
+organizational-performance/our-insights/why-diversity-matters.
+[12] Rocio Lorenzo and Martin Reeves. How and where diversity drives financial performance.
+Business Harward Review, 2018. Available online at URL https://hbr.org/2018/01/how-
+and-where-diversity-drives-financial-performance.
+[13] Minerva Informatics Equality Award. Best practices in supporting women, 2022. Available
+online at URL https://www.informatics-europe.org/society/minerva-informatics-equality-
+award/best-practices-in-supporting-women.html.
+[14] Sarah K. White. 19 organizations advancing women in tech, 2022. Available online at
+URL https://www.cio.com/article/215709/16-organizations-for-women-in-tech.html.
+[15] Hannah Williams.
+Best initiatives for women in tech, 2017.
+Available online at URL
+https://techmonitor.ai/technology/hardware/best-initiatives-women-tech.
+Acknowledgement
+This chapter was made possible thanks to the great dedication and support of the entire
+Czechitas team. Besides, it has been supported by the COST Action CA19122 – European
+Network for Gender Balance in Informatics (EUGAIN).
+15
+

H9FLT4oBgHgl3EQfIi9F/content/tmp_files/load_file.txt

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+page_content='Chapter 1  Beyond Classroom:  Making a Difference in  Diversity in Tech  Barbora Buhnova  With all the opportunities and risks that technology holds in connection to our safe and sus-  tainable future, it is becoming increasingly important to involve a larger portion of our society in  becoming active co-creators of our digitalized future—moving from the passenger seat to the  driver seat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Yet, despite extensive efforts around the world, little progress has been made in  growing the representation of certain communities and groups in software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' This  chapter shares one successful project, called Czechitas, triggering a major social change in  Czechia, involving 1 000+ volunteers to support 50 000+ women on their way towards software  engineering education and career.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='12000v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='SE]  27 Jan 2023  CHAPTER 1  Introduction  The past decade has witnessed the emergence of hundreds of initiatives around the world  supporting various underrepresented groups on their pathway towards software engineering,  whether connected to universities [13], companies [15], or run as independent non-profit or-  ganizations [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Although the initiatives often start with a great vision and high volunteering  commitment, after a few years into the activities, it becomes challenging to sustain the volun-  teering energy and commitment in face of the very slow progress towards the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' In those  moments, the success cases by others can be what helps us keep going.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  The initiative featured in this chapter, called Czechitas [6], started in 2014 in Czechia, with  a simple idea to bring tech closer to girls and girls closer to tech, in reaction to the strong  under-representation of women in tech in the country (see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The prompt snowball  effect helped us to build a community around the joint vision to empower and encourage girls  and women to engage in computing education and career transition, and to show them that  software engineering is an interesting career direction that is not necessarily difficult nor limited  to one gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Initially established to provide women in Czechia with an opportunity to put their  hands on programming, it now contributes to a major social change in the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Over time, Czechitas has become a movement that has attracted a strong community of  tech-professional volunteers (over 1 000) and companies (over 100), and given rise to a portfo-  lio of women-tailored courses in various areas of software engineering, such as programming,  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='1: Women ICT Professional (Eurostat, 2019 data) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  2  30%  25%  20%  EU = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='9%  15%  10%  5%  0%CHAPTER 1  web development, mobile app development, data science, cybersecurity or testing (over 1 300  courses delivered so far).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' We have influenced over 50 000 women (over 30 000 via live events  and over 20 000 via online tutorials) who graduated from our courses to use their new tech  skills to change their education path or advance their careers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Czechitas Mission: We inspire, train and guide new talents towards stronger  diversity and competitiveness in tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Thanks to the success of our education activities with hundreds of events a year (each  receiving more registrations than its capacity), we have become recognized as the leading  platform in Czechia actively addressing gender diversity in tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' In this chapter, we share  the lessons we learned about the low representation of women in tech, effective strategies in  supporting women on their way to software engineering, discuss the ingredients that helped us  succeed, the obstacles and challenges we faced, and the progress yet to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Why are There so Few Women in Tech?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Across Europe, only 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='1% of tech professionals are women (according to 2021 data) [8], with  Czechia being the last on the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The major reasons behind the trend in our region according  to our recent study (with 70% of participants from Czechia and Germany) [9] are:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The first hole in the leaky pipeline on girls’ pathway towards software engi-  neering is linked to the missing access to encouragement and support, together with the  missing access to suitable education that would be able to build on the interests of girls  that often span across multiple disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Stereotypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The ability to see herself as a software engineer is then challenged by the  perception of the software engineering as a field not leading to a purpose the girl would  like to dedicate her future to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Often, the close family and friends step-in in this moment  to direct girls away from software engineering with the intention to protect them from  a future where they cannot really imagine the girls becoming successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Interestingly,  3  CHAPTER 1  the intentions are meant well, to protect the girls, which shows how crucial it is to help  parents (and mainly mothers) to understand that software engineering can be a great  career choice for their daughters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The next hole on the leaky pipeline comes when girls find themselves in  the classroom, often surrounded by more-experienced learners (typically boys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' For the  little girls who often excel in other subjects, it can be hard to fall in the category of a slow  novice learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The girls often mention frustrations of low self-efficacy, inadequacy and  missing experience of success in presence of a classroom dynamic being monopolized  by the earlier technology adopters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Sense of Belonging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The girls who resist through the earlier three challenges and find  themselves on the education pathway towards software engineering, find themselves in  classrooms surrounded predominantly by boys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' While this is a comfortable environment  for some, many in the study reported not feeling comfortable to express themselves, fac-  ing sexism or unwanted attention and missing relatable role models and mentors, which  led them to reconsider whether this is the environment they would be willing to spend the  rest of their lives in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Feeling Valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The last hole in the leaky pipeline challenges the women who entered  software engineering careers, as some of them emphasize the struggle of not feeling  valued at workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The reasons are different for the women with stereotypical talent  spectrum (that matches the talent spectrum typical among their men colleagues, typically  being very technical) and non-stereotypical talent spectrum (bringing not-that-common  talents to the table, typically more multidisciplinary and human-oriented).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' While the first  group feels ”tired of proving them wrong”, the second group feel frustrated from their  strengths viewed as second class and from missing appreciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Supporting Women on their Way to Tech  In Czechitas, we understand that plumbing the leaky pipeline can hardly be done by isolated  and uncoordinated efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' This section discusses the interlinked pillars of our activities (see  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='2), listing examples of the activities and events we delivered in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  4  CHAPTER 1  Czechitas Pillar I – Awareness  One of the crucial success factors for a change towards improving gender balance in soft-  ware engineering is the actual understanding that we are in a disbalanced state that further  reinforces itself due to the factors discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The efforts towards encouraging women  to join software engineering cannot make a difference unless the society, education system  and corporate environment welcomes and supports the change (understanding it as a push  towards the real equilibrium, not a push out of it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  In Czechitas, we are investing substantial effort in awareness around the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' In 2022  alone, we participated in over 20 conferences and panel discussions, gave numerous inter-  views in TV, radio and other media, organized talks to students and teachers at high schools,  and to tech professionals in our partner companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' We were visible with a booth at 15 festivals  and family days across Czechia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Over 2022, Czechitas was mentioned in 508 articles, reach-  ing major part of Czech population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' In 2021, we also launched a Czechitas podcast, which in  2022 reached over 14 676 listens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Furthermore, our website was in 2022 visited by 123 785  unique visitors, and our newsletter was followed by 25 983 subscribers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  The next step in raising awareness among the general public is to make it as easy as  possible to get the first exposure to coding in a fun, enjoyable and community way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' To this end,  we for instance organize an Advent Christmas Coding campaign (following the tradition of an  advent calendar, in which instead of a sweet treat, each day holds a coding assignment along a  story of bringing Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Gingerbread home for Christmas), which is being followed by hundreds of  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='2: The Pillars of Czechitas Activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  5  CAREER  AWARENESS  TRAINING  TRANSITION  COMMUNITYCHAPTER 1  people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Furthermore, in collaboration with the Ministry of Education, Youth and Sports, we e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  co-organized the #DIGIEDUHACK hackathon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' And in collaboration with Czech universities, we  run the Czechitas Thesis Award to give visibility to exceptional bachelor theses authored by  girls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' All these activities typically repeat every year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Czechitas Pillar II – Training  Since the start of our activities in 2014,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' we are improving the education design of our courses  to reflect the needs of our audience—women and girls who are very often later technology  adopters or career changers—with an emphasis on providing suitable first contact with soft-  ware engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' creating safe and supportive environment for novice learners,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' accommodat-  ing differences in the learning speed of each student,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' building self-confidence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' and supporting  sustaining long-term interest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' which we also publish [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' In 2022, we delivered 242 live  software-engineering courses with 15 316 participants, with the courses around web develop-  ment and data science scoring as the most popular ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Although most of the training is targeted to women and girls, we are also investing in training  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='3: Czechitas-participation (Data as of June 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  6  21,767  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='482+  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='683+  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='686  educational  participants  events  428  10  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='763  299  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='752  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='268  238  418  13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='081  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='821  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='969  145  141  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='271  295  102  6  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='011  84  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='654  45  135  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='873  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='266  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='299  2015  2016  2017  2018  2019  2020  2021  2022*  2015  2016  2017  2018  2019  2020  2021  2022*  Number of educational events  Number of video tutorials  Educational events participants  Educational video tutorials participantsCHAPTER 1  elementary-school and high-school teachers (irrespective of gender).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' And some mixed-gender  activities were organized also for children (7 week-long summer camps in the summer of 2022,  besides others) and high-school kids, although in case of high schools, it is already important  to offer also girl-only courses (3 week-long summer schools for high-school girls were given  in 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Besides, training courses for mixed audience are also provided on events such as  Family Days (we were present at over 20 such events in 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Czechitas Pillar III – Career Transition  As many women in our community intend to enter software engineering as their future profes-  sion, some of our activities are intentionally designed to facilitate this journey, whether software  engineering is to become their first job or they intend to change their career [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  In cooperation with our partner companies, we have identified three career pathways that  appear to be the most suitable entry points to software engineering in Czechia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' These are  (1) web development (including courses on JavaScript, React, HTML/CSS, Bootstrap, Git,  UX design, and others), (2) data analytics (including courses on Python, databases, SQL,  statistics, Power BI, and others), (3) testing (including courses on requirements engineering,  agile processes, manual testing, issue tracking, regression testing, smoke testing, basics of  automated testing, browsers, API, databases, version control, and others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  For the three directions, we have developed a complex career-transition support within so-  called Digital Academies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' A Digital Academy is a four-month program for a group of 30 women  (and involving around 5-15 partner companies), which besides individual courses covering the  topics outlined above and taking place 3-4 times a week (evenings on working days, full days  on weekends) includes also pairing of the students with mentors from the companies to support  them in developing their own projects, a hackathon, career support, and further events offered  by the partner companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' In 2022, we have run 10 Digital Academies across four major cities  in Czechia, with over 60% of the graduates receiving a job offer within three months from  graduating from the academy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  To facilitate the career transition also for the women who opt to customize their training  journey (not attending a Digital Academy), our career consultants provide hundreds of career  consultations each year (327 in 2022), and we twice a year organize a Czechitas Job Fair,  7  CHAPTER 1  where our graduates can meet the representatives of our partner companies (each Job Fair  attended by about 350 graduates and 30 companies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Czechitas Foundation – Community  The foundation that supports all our activities is the community, which involves the participants  and graduates of our courses, tech professionals who teach with us, mentors, course facili-  tators, and our partner companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The fact that many members in our community are men  helps us not only engage more tech-professional allies in our vision, but also influence a more  supportive environment in tech companies where our graduates land.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' To support the blending  of the community and increasing the sense of belonging of our graduates also in the mixed-  gender environment, we regularly engage in organization of Tech Meet-ups and Hackathons,  as well as informal CzechiPubs that regularly take place in 10 different cities across Czechia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Making a Difference  The positive influence of Czechitas activities in Czechia is already visible in the shifted percep-  tion of software engineering as an education pathway and career choice to be considered by  any gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' That not only motivates many girls to consider software engineering in their choice  of a university study field (with the representation of women among ICT students changing from  12% in 2016 to 17% in 2021 in Czechia [7, 5], moving the country closer to the European av-  erage, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='4) but is likely also having secondary influence on all who so far hesitated  to join software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  What Helped us Succeed  Building Czechitas was only possible thanks to a coordinated effort of hundreds of people (90  employees and over 1,000 volunteers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Over the past eight years of our existence, we came to  understand the ingredients without which this would not be possible:  Great leadership and love for what we do is giving us the sense of purpose, energy and  direction, holding us together and keeping us focused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Mentors from partner companies  8  CHAPTER 1  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='4: Women ICT Students (Czech Statistical Office, 2021 data) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  and beyond have been of great help to guide us through the design of our leadership and  expansion strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Visual and playful communication is giving us the fresh flavour of fun and joy that we  all (students as well as trainers and volunteers) enjoy joining even after a tiring day at  school or work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The informal and visually attractive communication helps us to share the  love for our brand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Community and sense of belonging is crucial for connecting those who strive to learn  with those who strive to share and teach, and those who want to support the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  It helps our student to feel home and make it easier for them to keep going even when  learning gets hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Inclusive environment and encouragement makes it safe for our students to make mis-  takes and experience success, have the opportunity to exchange knowledge, collaborate,  and get personalized feedback and guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Specific strategies and interventions we  have developed to support novice learners and their self-efficacy have been key in this  direction [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Knowledge and understanding is crucial for us to design our activities with insight into  the frustrations steering women away from software engineering [9] and effective strate-  gies to support girls and women in tech education [10] and career transition [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' We  9  30%  25%  EU = 20%  20%  15%  10%  5%  0%CHAPTER 1  invest our time in sharing the lessons we have learned [2, 9, 3], and learning from other  initiatives from across the world (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=', within the EUGAIN network, see https://eugain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='eu/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Creating and sharing stories helps us to inspire our students, bring them closer to  relatable role models, and to give them hope and confidence that with some work and  dedication, a transition into software engineering is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The stories (each featuring  an inspiring woman who changed her career towards tech) are published in our blog,  communicated via social networks, and used in media articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' These women inspire  others as speakers and panelists in our events, and as guests in Czechitas Podcast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Sustainable financial model helps us to sustain a team employed to run the organi-  zation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The model stands on financial participation of the students, partner companies,  foundations and individual donors, with an intention to reach out also to the government  level in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The most crucial pillar of our financial sustainability is the partner com-  panies, which are beside their yearly partnership contributions (depending on the level of  partnership) helping us to cover certain costs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=', offering their office spaces for events,  motivating their employees to volunteer as mentors), and opening doors towards further  funding opportunities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' with global foundations connected to their company).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Obstacles and Challenges we Faced  As any organization that has substantially outgrown its own plans and expectations, Czechitas  has undergone numerous changes and readjustments over its course of existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' And al-  though we are trying to publish the effective setup that works for us now [2, 3, 4], our first steps  were highly organic and experimental, which was key to learning what works for the context  we were in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' With our enthusiasm and ”always yes” spirit, we walked many paths that we failed  and rolled back, but we also faced numerous obstacles and challenges that we withstood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Scaling the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Turning a non-profit start-up into a scale-up is a challenge on  its own, as the means for achieving stability are different from traditional companies – be-  sides the discussed financial stability, also in terms of sustained volunteering involvement  and brand building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' We needed to learn to manage the mix of the innovative and largely  self-sacrificing founding community with the necessary systematic and organized spirit of  10  CHAPTER 1  new employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' We needed to learn to prioritize and say no to some activities that the  team felt strongly for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Being misunderstood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' As a large organization, we needed to learn to communicate  our mission well so that it is not misunderstood, knowing that anything that damages  the brand may sink the whole boat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Namely, we needed to help our partner companies  understand what level of expertise is realistic to achieve in our students, help our students  understand what time investment and commitment it takes to change direction towards  tech, and help our society understand why our focus on women is key to the success of  our society as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Quantifying the impact of our activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' One of the important challenges that we are  still facing is our ability to quantify the impact of our individual interventions and activities,  as it is difficult to isolate the effects of each one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' More so that the impact is often  very subtle and propagates over long periods of time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=', a woman making a few steps  towards tech education inspiring her friend to make a major shift towards tech, who then  inspires her daughter to study CS at university).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' So although we have a Data & Impact  team at Czechitas, with substantial data available, the numbers we have (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=', the number  of women who change their career to tech each year) are still only the tip of the iceberg  of the real impact we strive for, which is the shift in the collective mindset of the entire  society, leading to a sustained change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Progress yet to be Made  With the increasing number of Czechitas graduates who are joining software engineering in-  dustry, often as very junior (in terms of their software-engineering expertise) and diverse (in  terms of their talents and competencies) members, we find it crucial to assist the companies  to improve the inclusiveness of their environment to integrate and leverage the new diverse  talent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' In 2020, we made the first step towards that goal via designing a Diversity Awareness  Training, which was since then delivered to over 300 managers (mostly from Central and East-  ern Europe) across some of our partner companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The concepts that have shown to be the  most crucial to discuss and understand during these trainings are outlined below:  11  CHAPTER 1  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='5: Tuckman’s Model of Team Dynamics with an illustration of different dynamics  observed in homogeneous and heterogeneous teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Diversity does not come easy, but it pays off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Avoiding diversity is natural to human in-  dividuals, but dangerous to humankind1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The same is true for corporate environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' We  need to acknowledge that diverse teams might have a harder time at start (as illustrated  with the Tuckman’s Model of Team Dynamics in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='5), but in long-term, diversity is  firmly correlated with higher performance [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  We too often lose talented people by missing the talent in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' We are all talented,  in many diverse ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' It is the task of the manager to recognize and direct the talent to-  wards team success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The fact that a person uses a different talent spectrum (approaches  problems and situations differently) does not make them more/less suitable for software  engineering as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' There is no such thing as a second-class citizen when it comes to  the talents we need in software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Biases evolved to help us navigate complexity, but they are not serving us well  when making assumptions about the potential in people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The dark side of biases is  that we tend to judge people’s potential based on how their talent spectrum matches the  talent of already-successful ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Without realizing that the successful ones embody the  skills and conditions that worked when they joined the field (in the past) while we are now  choosing the software engineers for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Connection is built through communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' There are many unhealthy communica-  1Our quote inspired by the statement ”Diversity is the new Darwinism” by the Great British Diversity Experiment [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  12  7  Forming  Strorming  Norming  Performing  Adjourning  个  Effectiveness  Homogenous  team  Heterogenous  team  TimeCHAPTER 1  tion patterns around diversity, which often go against the purpose of making us all feel  the sense of belonging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' It is important to create safe space, in which we can learn to  communicate our differences but also ask about the differences of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Mistakes are  part of that learning, and forgiveness of the mistakes shall be encouraged if the mistakes  were done in the process of learning and not repeated blindly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' It is important to create a  safe space to acknowledge our biases and stop shaming one another for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Avoid the quick fixes, remove the barriers instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Encourage curiosity about why cer-  tain communities are under-represented in software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' What are the barriers  they face and what can we do to remove them or make their journey lighter in presence of  the barriers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' the care-taking on the side of most women)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Avoiding the conversation  and looking away from the differences in our experiences might lead the community to as-  sume that the under-representation is the lower-fit problem, which is dangerous because  it leads to push-back on any diversity support one might try to introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Change takes time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Promoting I&D is more complex than it might seem at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' It is  crucial to know how to start to see the first positive effects soon and be able to use them  to get more people on board towards promoting I&D further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Choose your first steps well  and invest in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' The investment will pay off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Conclusion  Making a difference in improving gender balance in software engineering on the scale of the  whole country is not easy, but is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' And it is very rewarding to be part of such a move-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' In 2021, the social impact of Czechitas activities was recognized at the European Union  level via winning the EU Social Economy Award (over 180 organizations nominated) in the  Digitalisation and Skills category, and in 2022 winning the global Equals in Tech Award (155  organizations nominated) in the Skills category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' We hope our example can inspire others,  which is also why we are eager to share the lessons learned from our journey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  13  Bibliography  [1] Amanda Bennett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Case study: The great British diversity experiment, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' FairPlay Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  [2] Barbora Buhnova and Lucia Happe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Girl-friendly computer science classroom: Czechitas  experience report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' In European Conference on Software Architecture, pages 125–137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  [3] Barbora Buhnova, Lucie Jurystova, and Dita Prikrylova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Assisting women in career  change towards software engineering: experience from czechitas ngo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' In Proceedings of  the 13th European Conference on Software Architecture-Volume 2, pages 88–93, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  [4] Barbora Buhnova and Dita Prikrylova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Women want to learn tech: Lessons from the  czechitas education project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' In 2019 IEEE/ACM 2nd International Workshop on Gender  Equality in Software Engineering (GE), pages 25–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  [5] Czech Statistical Office.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Human resources in information technology, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Available on-  line at URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='czso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='cz/documents/10180/165376696/063015-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='pdf/c7e96151-  b285-4388-9384-532e55f4a318?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='version=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  [6] Czechitas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Czechitas  annual  report  2021,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Available  online  at  URL  https://is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='muni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='cz/go/u6ji13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  [7] Eurostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Female students under-represented in ICT, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Available online at URL  https://ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='eu/eurostat/web/products-eurostat-news/-/edn-20190425-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  [8] Eurostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  ICT  specialists  in  employment,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Available  online  at  URL  https://ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='eu/eurostat/statistics-explained/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='title=ICT specialists in em-  ployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  CHAPTER 1  [9] Lucia Happe and Barbora Buhnova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Frustrations steering women away from software  engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' IEEE Software, 39(4):63–69, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  [10] Lucia Happe, Barbora Buhnova, Anne Koziolek, and Ingo Wagner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Effective measures to  foster girls’ interest in secondary computer science education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Education and Information  Technologies, 26(3):2811–2829, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  [11] Dame Vivian Hunt, Dennis Layton, and Sara Prince.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Why diversity matters, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  McKinsey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Available online at URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='mckinsey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='com/capabilities/people-and-  organizational-performance/our-insights/why-diversity-matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  [12] Rocio Lorenzo and Martin Reeves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' How and where diversity drives financial performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Business Harward Review, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Available online at URL https://hbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='org/2018/01/how-  and-where-diversity-drives-financial-performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  [13] Minerva Informatics Equality Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Best practices in supporting women, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Available  online at URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='informatics-europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='org/society/minerva-informatics-equality-  award/best-practices-in-supporting-women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  [14] Sarah K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' 19 organizations advancing women in tech, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Available online at  URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='cio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='com/article/215709/16-organizations-for-women-in-tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  [15] Hannah Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Best initiatives for women in tech, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Available online at URL  https://techmonitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='ai/technology/hardware/best-initiatives-women-tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  Acknowledgement  This chapter was made possible thanks to the great dedication and support of the entire  Czechitas team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content=' Besides, it has been supported by the COST Action CA19122 – European  Network for Gender Balance in Informatics (EUGAIN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}
+page_content='  15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'}

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+Elastic Cash
+Anup Rao
+University of Washington
+[email protected]
+January 12, 2023
+Abstract
+Elastic Cash is a new decentralized mechanism for regulating the money supply. The mech-
+anism operates by modifying the supply so that an interest rate determined by a public market
+is kept approximately fixed. It can be incorporated into the conventional monetary system to
+improve the elasticity of the US Dollar, and it can be used to design new elastic cryptocurrencies
+that remain decentralized.
+1
+Introduction
+Money is as old as recorded history, and yet it continues to evolve. Even the mighty US Dollar
+has been repeatedly updated over the last 200 years. The recent emergence of Bitcoin and other
+cryptocurrencies is another step in that evolution, and it prompts us to revisit the mechanisms
+that ensure the desirable properties of money. An important property of money is its elasticity:
+money is elastic if the money supply increases in response to demand. In this work, I present a
+new decentralized mechanism to ensure the elasticity of money.
+The US Dollar is elastic, but it uses a convoluted system to achieve its elasticity. The Dollar
+system was not designed from first principles; it was iteratively amended in response to financial
+crises. Elasticity is currently achieved by the combined actions of many institutions: banks and
+non-banks, public and private, domestic and international. No single entity is entirely in charge
+of the money supply, and a relatively small number of investor-owned institutions have undue
+influence. I discuss the mechanics of the US Dollar system and its limitations in Section 2.
+In contrast, Bitcoin was designed to be inelastic. Bitcoin caps the total possible supply at 21M,
+and the available supply, currently about 19.2M, will slowly increase until it hits this limit. Satoshi
+Nakamoto, the creator of Bitcoin, criticized conventional mechanisms for achieving elasticity in an
+early forum post:
+The root problem with conventional currency is all the trust that’s required to make
+it work. The central bank must be trusted not to debase the currency, but the history
+of fiat currencies is full of breaches of that trust. Banks must be trusted to hold our
+money and transfer it electronically, but they lend it out in waves of credit bubbles with
+barely a fraction in reserve.
+Nakamoto eliminated the need for the kind of infrastructure used to ensure Dollar elasticity by
+simply choosing to make Bitcoin inelastic. Additional technological innovations in Bitcoin further
+eliminated the need for any trusted central authority to carry out transactions.
+1
+arXiv:2301.04244v1  [q-fin.GN]  10 Jan 2023
+
+To illustrate the negative consequences of inelasticity, consider the trajectory of home prices
+denominated in Bitcoin over a long period. As the population grows, the demand for houses is
+sure to grow. Even if the supply of houses keeps pace, home prices must fall, because the supply of
+Bitcoin cannot keep pace. Certainly, if the number of concurrent transactions involving home sales
+grows and the prices of houses remain stable, the amount of Bitcoin involved in these transactions
+would have to grow, yet it cannot grow beyond 21M. So, a fixed money supply leads to falling
+prices in a growing economy, even if the underlying supply and demand of items keep pace with
+each other.
+At the other extreme, if the money supply is increased excessively, the currency is debased; the
+excess supply leads to rising prices and inflation. So, a mechanism for correctly setting the money
+supply is essential, because this is the foundation upon which stable prices are built. As much as
+possible, prices should reflect the tension between the supply and demand for items, and nothing
+else. But how much money is too much? It is not just a matter of trust, the problem is that the
+appropriate supply is difficult for anyone to calculate! Is there a principled method to compute
+the supply, and a fair way to create new money? This work gives my answers to these important
+questions.
+Ideally, a mechanism for achieving elasticity should be transparent.
+It should not rely on
+the judgment or integrity of small groups of people, or a few institutions. I will describe a new
+decentralized mechanism called Elastic Cash that enjoys these features. Elastic Cash can replace
+the role played by private institutions in providing elasticity for the US Dollar, and it can be
+combined with the concept of blockchains to make new cryptocurrencies that are elastic, yet do
+not require any trusted central authority.
+Here I will give a high-level description of Elastic Cash; a full description is in Section 3. I will
+refer to the central bank and the algorithm as cash authorities. At the heart of the mechanism is
+a new financial contract issued by the cash authority called a cashbond, and a public market that
+allows anyone to buy and sell cashbonds using public auctions in the market. Each cashbond can be
+redeemed for $1 on a specific date, so it executes a risk-free loan from the holder of the cashbond to
+the cash authority. The loans are risk-free because the cash authority can always create new money
+to repay the loans. Perhaps it is counterintuitive, but the purpose of the market in cashbonds is
+not to give the cash authority a way to borrow money; instead, the function of the market is to use
+the trading activity of participants to compute the risk-free rate of interest, which can be computed
+from the prices of cashbonds in the market. The mechanism requires that cashbonds can only be
+transfered by selling and buying them in the market. For example, no entity should be permitted
+to use cashbonds as collateral to borrow money. This forces participants to liquidate cashbonds
+when they need money, and keeps the mechanism informed about the demands for money.
+It is market participants that determine the money supply in Elastic Cash.
+The supply is
+increased or decreased according to transparent rules ensuring that the risk-free rate of interest
+remains approximately fixed. Trade in cashbonds leads to fluctuations in the rate of interest, and
+the mechanism responds by creating corresponding fluctuations in the money supply. Intuitively,
+the risk-free rate of interest encodes the cost of renting money. By regulating the supply of money to
+keep the cost fixed, the mechanism ensures that the supply stays in equilibrium with the demand for
+liquidity. When the supply is increased, market participants acquire any newly-generated money.
+The supply is decreased by incentivizing market participants to exchange their money for cashbonds.
+Anyone can participate in this market, and money is distributed to or taken from participants
+according to transparent rules, so no single entity can control the flow of money.
+2
+
+Source: Board of Governors of the Federal Reserve System (US)
+fred.stlouisfed.org
+Billions of Dollars
+1960
+1970
+1980
+1990
+2000
+2010
+2020
+0
+4,000
+8,000
+12,000
+16,000
+20,000
+24,000
+M2
+Figure 1: M2, a measure of the supply of US Dollars
+Elastic Cash can be implemented within the framework of conventional currencies like the US
+Dollar by creating regulations that require the central bank to implement the market for cashbonds
+and generate money according to the rules of the mechanism.
+It can be implemented in the
+framework of cryptocurrencies by setting up a distributed algorithm to implement the market for
+cashbonds using the blockchain. Money is generated by the algorithm according to the rules of the
+mechanism. So, one can obtain the positive features of traditional currencies and cryptocurrencies
+in addition to the elasticity of Elastic Cash.
+Outline of this paper
+In Section 2, I give more details about how the US Dollar achieves its
+current elasticity, before turning to describe the new mechanism in Section 3. I discuss how the new
+mechanism can be incorporated into the Dollar system in Section 4 and how it can be implemented
+on the blockchain to give new cryptocurrencies in Section 7.
+2
+US Dollar elasticity: the Fed, banks, and shadow banks
+Figure 1 shows how the supply of US Dollars held as deposits has changed over time. In this section
+I briefly review the history and mechanics of the US Dollar system. I recommend the following
+3
+
+二Dexcellent sources for additional background on the history of the Dollar system [1, 4], and the book
+[2] for a longer history of the technology of money.
+The supply of US Dollars has consistently been deemed too important to be left entirely in
+the control of a government agency. Instead, we have developed a system where money is created
+by investor-owned private entities. These include banks and so-called shadow banks — non-bank
+financial institutions that are able to create Dollars. This privately created money is then back-
+stopped by the central bank, which is the Federal Reserve, or Fed in the US. The Fed is governed
+by laws written by Congress, but these laws are somewhat ambiguous about the Fed’s powers, and
+Congress has repeatedly made adjustments. The Fed has responded to various crises by incentiviz-
+ing or directly making big changes to the money supply, and by sometimes deciding that a new
+category of privately-issued financial instruments can be exchanged for US Dollars.
+At this point, there are at least four kinds of financial institutions that are able to generate
+financial instruments that are de facto US Dollars. It is helpful to view the evolution of the US
+Dollar system according to the events that elevated these instruments to the stature of US Dollars:
+Commercial bank deposits By the early 1900s, deposits of US Dollars were being issued by
+a number of investor-owned private banks.
+These deposits were treated by depositors as
+equivalent to US Dollars, even though the banks were issuing loans by creating deposits that
+did not correspond to cash reserves. This led to a run on the banks in 1907, and the Fed was
+created in 1913 to solve the problem. The Fed was given the power to backstop the money
+created by commercial banks by lending to the banks. This meant that deposits could always
+be exchanged for money created by the Fed. This solved the problem of bank runs, but also
+elevated bank deposits to the stature of cash issued by the Fed, and effectively meant that
+commercial banks were given the authority to create legitimate US Dollars in the process of
+issuing loans. Currently, about $17T is held in commercial bank deposits.
+Reverse purchase agreements (repos) In the 1950s, broker dealers began to enter the banking
+industry using a financial instrument called reverse purchase agreements, or repo. Repos allow
+dealers to borrow money from cash providers using government securities as collateral. Cash
+providers began to treat the repos they had purchased from dealers as equivalent to cash.
+By itself, such an arrangement does not create any new money, because if repos crash in
+value, then they cannot actually be exchanged for cash. However, the properties of repos
+were substantially altered when the Fed decided to backstop dealers by giving them access to
+overnight loans. This meant that the private holders of repos were guaranteed that repos could
+always be exchanged for US Dollars via the Fed’s repo facility, and so repos were elevated to
+the same stature as cash and bank deposits. In 1991, Congress reduced restrictions on the
+Fed to make it even easier to backstop the repo market. Effectively, broker dealers were given
+the power to create new US Dollars. Currently, the size of the repo market is about $4T.
+Eurodollars Foreign companies including both banks and non-banks (e.g. insurance companies)
+have been issuing financial instruments called eurodollars that can be redeemed for US Dollars.
+These eurodollars were not initially backed by actual Dollars. The oil shock of 1973-1974 led
+to problems in the eurodollar market that eventually brought down a domestic US bank. The
+Fed responded by promising to backstop eurodollars by providing actual US Dollars in the
+form of loans to the corresponding foreign central banks. Effectively, the Fed permitted the
+banking systems of other countries to create deposits that could be exchanged for US Dollars.
+The size of the eurodollar market was estimated at about $13T in 2016 [3].
+4
+
+Money market mutual funds These funds emerged in the 1970s as investments whose share
+price was pegged at $1. In reality, these funds held assets whose value could drop below the
+peg, and so it was not possible to guarantee the peg during a financial crisis. In response
+to the great financial crisis of 2008, the Fed began to backstop these funds using its Money
+Market Mutual Fund Liquidity Facility, and so elevated deposits in these funds to the stature
+of US Dollars. Total assets in these funds is about $4.8T.
+In addition to recognizing new forms of the US Dollar, the Fed has resorted to buying assets
+in order to inject liquidity under the Quantitative Easing program. During the Great Financial
+Crisis of 2008, the Fed kicked off this program by buying mortgage backed securities and treasuries,
+which are loans to the US Treasury. Throughout the last decade, the Fed has continued to expand
+its balance sheet, mostly with treasuries. In 2020 the Fed once again bought significant quantities
+of these assets. Currently the Fed holds about $8.5T on its balance sheet.
+The history of the US Dollar is full of ad hoc amendments to maintain stability in the face of
+financial crises. At its heart, the problem is that there is currently no principled way to regulate
+the supply of money. This becomes apparent during times of financial crises, but the imbalance in
+supply is probably always brewing, even in normal times. By now, the pattern of Fed actions is
+familiar, and it is inevitable that it will repeat. During times of crisis, the Fed must act to protect
+money or face a significant crash in the entire system. Because private financial institutions know
+that the Fed will protect their financial instruments from the most negative consequences of their
+choices, they do not have the correct incentives.
+Elastic Cash is meant to provide a clean, transparent, and principled mechanism to achieve
+robust elasticity.
+We do not need to cede control of the money supply to private institutions
+or foreign banks. We do not need to elevate invented forms of money to the stature of the US
+Dollar. Once Elastic Cash is adopted, I believe we can safely bar all private creation of Dollars.
+The mechanism will generate new US Dollars when required, and financial institutions can obtain
+liquidity by participating in the mechanism, just like everyone else.
+Extricating ourselves from the current system and its vested interests is likely to be challenging,
+to say the least. Nevertheless, I describe a path to incorporating the new mechanism in Section 4.
+3
+Elastic Cash: the details
+Elastic Cash uses trade in cashbonds to determine a risk-free rate of interest. The money supply
+is regulated to ensure that this interest rate remains approximately fixed. Cashbonds are issued
+by the central bank (in the case of conventional currencies) or by the distributed algorithm (in the
+case of cryptocurrencies). I refer to these entities as cash authorities.
+The contract cashbond(d) promises that the cash authority will pay the holder of the contract $1
+on the date d. Let us reserve d0 to denote the current date. On date d0, the cash authority pays each
+holder of cashbond(d0) $1, and these contracts expire. Elastic Cash requires that the cash authority
+implement a public market in cashbonds. On the date d0, contracts of the type cashbond(d) for
+d > d0 will be available in the market for cashbonds maintained by the cash authority.
+Cashbonds are a special class of asset, and they should not be treated like other securities.
+Elastic Cash requires that cashbonds can be generated and traded only in the public market that
+is administered by the cash authority. Cashbonds are not transferable, meaning they cannot be
+exchanged outside of the public market, and they cannot be used as collateral for loans. Because of
+5
+
+these restrictions, cashbonds cannot themselves play the same role as money. The purpose of these
+rules is to ensure that holders of cashbonds that desire liquidity will sell their cashbonds in the
+market and so keep the mechanism informed about the demand for liquidity. For the same reasons,
+trade in cashbonds should not be taxed. The transactions of buying and selling cashbonds should
+be viewed as similar to transactions that move money between savings accounts paying varying
+rates of interest, and treated similarly under the law.
+3.1
+Risk-free rate of interest
+The price at which cashbonds trade implies interest rates for risk-free loans of varying durations.
+Let rate(t) denote the interest rate for duration t. Let us write price(d) to denote the price at which
+cashbond(d) last traded in the market. Then, if the current date is d0, the prices of cashbonds can
+be used to compute implied interest rates according to the formula:
+price(t + d0) · (1 + rate(t))t = 1,
+which implies that the interest rate can be expressed as
+rate(t) = price(t + d0)− 1
+t − 1.
+Because the loans executed by cashbonds are risk-free, the values rate(t) capture something about
+the market’s belief about the opportunity cost of making risk-free loans for duration t. Generally,
+one would expect rate(t) to be a monotone function of t, meaning that rate(t) > rate(t′) if t > t′,
+because loans of longer duration usually command higher interest rates. Moreover, if t is much
+larger than t′, then we might expect rate(t) to have higher variance than rate(t′), because predictions
+about the distant future can diverge much more than predictions about the immediate future.
+These rates encode important information about the demand for liquidity. The goal of the
+mechanism is to regulate the money supply so that one of these rates is held approximately fixed.
+It makes the most sense to pick a rate for a relatively short duration, because these rates are likely
+to have the least variance. With that in mind, let τ denote a short time period, say 1 week. The
+goal of the mechanism will be to keep
+rate(τ) ≈ 0.02.
+There is nothing special about 0.02, except that it is convention for central banks around the world
+to use 2% as the target rate of longterm inflation.
+Let us set
+p− = (1 + 0.021)−τ,
+and
+p+ = (1 + 0.019)−τ.
+The goal of the mechanism will be to regulate the money supply so that
+p− ≤ price(τ + d0) ≤ p+,
+where again d0 is the current date. This corresponds to keeping
+0.019 ≤ rate(τ) ≤ 0.021.
+6
+
+3.2
+Using the market to regulate the money supply
+Participants in the cashbond market can put in orders to sell a specific number of cashbonds that
+they hold at a specific price, and can also put in orders to buy a specific number of cashbond(d)
+at a specific price. The cash authority acts as a market maker to match buy orders to sell orders
+and so conduct transactions at a specific price between market participants. Ideally, the market
+for cashbonds will support auctions1 for sellers to sell their cashbonds when needed.
+The cash authority will itself participate in this public market by buying and selling cash bonds
+in prescribed ways. The goal of the mechanism is to maintain rate(τ) approximately fixed, and to
+keep the market in cashbonds sufficiently liquid, so that the money supply can be quickly adjusted
+based on changes to rate(τ). Here is the proposed scheme for buying and selling cashbonds:
+1. The cash authority will buy and sell cashbonds to keep rate(τ) ≈ 0.02. The cash authority
+will place a standing order to buy an infinite number of contracts cashbond(τ + d0) at price
+p−, and a separate standing order to sell an infinite number of cashbond(τ + d0) contracts at
+price p+.
+Because the cash authority is able to generate arbitrary amounts of both money and cash-
+bonds, it will always be able to satisfy any of the resulting transactions. This ensures that
+p− ≤ price(τ + d0) ≤ p+,
+as discussed above.
+2. When cashbonds are redeemed for money, the cash authority will need to sell new cashbonds
+to restore the balance between money and cashbonds.
+It makes sense to pick a particular target distribution on outstanding cashbonds that is
+maintained during normal times. If the current date is d0, we say that cashbond(δ + d0) has
+duration δ. For example, the cash authority might aim to maintain the invariant that at any
+point in time, 1/4 of the outstanding cashbonds have duration between 0 and 1 month, 1/4
+have duration between 1 month and 1 year, 1/4 have duration between 1 year and 4 years,
+and 1/4 have duration between 4 years and 10 years.2.
+Given such a target distribution, the redeemed cashbonds should be replaced by selling new
+cashbonds at auction, picking the dates of the new cashbonds so that the overall distribution
+on duration is maintained as much as possible.
+3. When the demand for money is high, we are likely to reach the point where all of the available
+bonds cashbond(τ + d0) have been purchased by the cash authority.
+In such times, the
+mechanism has run out of the means to inject money into the financial system at a fast enough
+pace according to rule 1. This can be resolved by selling large quantities of cashbond(2τ +d0)
+contracts at auction in the market. Market participants will be incentivized to buy these
+cashbonds and then sell them back after time τ; at that time the cash authority itself will
+be willing to buy the cashbonds at price p−. The net effect will be to inject money into the
+system, while preserving the number of outstanding cashbonds.
+1I will not commit to a specific style of auction here, though any implementation must carefully specifying how
+the cash authority behaves as a market maker and what the rules of the auctions are.
+2There are many considerations for how to choose the target distribution, but here I will not dwell on the choices
+too much.
+7
+
+The number of cashbonds sold in this process is a design choice. The goal should be inject
+significant liquidity, so I would favor an exponentially escalating volume of sales. For exam-
+ple, the cash authority might first sell a quantity that corresponds to 1% of all outstanding
+cashbonds, and a week later escalate it to 2%, then 4%, and so on until the cashbond market
+returns to the state where market participants are no longer willing to sell back the cash-
+bonds of duration τ to the cash authority at p−. These actions may temporarily distort the
+distribution on the durations of outstanding cashbonds, but the distribution will be quickly
+restored when the new cashbonds are redeemed and rule 2 is applied.
+An actual implementation of Elastic Cash would need to resolve many smaller technical details.
+Let me now make a few comments and observations about the Elastic Cash mechanism as I have
+defined it.
+3.3
+Discussion
+Elastic Cash is quite different from a system where the central bank simply allows deposits for
+all with interest rate 2%—such a scheme does not give the central bank a method to inject large
+amounts of money when the liquidity is needed. History has shown that the Fed needs a tool like
+Elastic Cash to inject liquidity into the financial system, since interest rates have proven too weak
+as a tool to inject large quantities of liquidity. As we discussed in Section 2, this has led to the
+Fed buying assets or propping up assets that were liable to crash in value. In doing so, the Fed
+is forced to pick and choose between market participants that get first access to the new liquidity
+that it provides.
+Central bankers should not be attempting to directly reason about the demands for liquidity;
+they do not have enough data to make those decisions. But if they must take such dramatic actions,
+the scheme of Elastic Cash at the very least gives a fair way to do it by trading cashbonds along
+the lines I have suggested above. This removes the ability of the finance sector to control the flow
+of the new money. It is also preferable to having the Fed buy treasuries, because it disentangles the
+actions of the Fed from the needs of the Treasury. There is no need to tie increases in the money
+supply to increases in government spending.
+Cashbonds should not be confused with conventional government securities like US treasuries.
+These instruments are significantly different from each other, and one cannot make inferences about
+the cashbond market, which does not yet exist, based on the behavior of the US treasury market. Let
+me highlight some key differences. The issuance of cashbonds is controlled by strict and transparent
+rules, and is not tied to the spending of the US government. There is no analogue of debt ceilings,
+or any chance that the central bank will default. Cashbonds cannot be used as collateral for loans,
+cannot be transferred outside of the Elastic Cash market, and trade in cashbonds is not taxed.
+It is important for the functioning of Elastic Cash to maintain a large volume of outstanding
+cashbonds of varying durations. Ideally, we would like there to be broad participation in the cash-
+bond market from all kinds of financial entities: banks, companies, pension funds, and individuals.
+Because these participants will be willing to trade at different durations, participation will be in-
+creased if a wide range of durations are available, and the market is liquid at all durations. Even
+though the cash authority only regulates the interest rate for duration τ, this action will affect the
+rates for all durations. One would expect that banks and other sophisticated players will trade
+cashbonds of shorter duration, and perform the arbitrage necessary for information about demands
+for liquidity of all durations to propagate to the shorter durations. I suspect that there is a prin-
+8
+
+cipled way to choose the ideal distribution on durations of outstanding cashbonds, but I have not
+yet been able to convince myself about what it ought to be.
+4
+Adopting Elastic Cash in the US Dollar system
+As discussed in Section 2, the Dollar system involves many different kinds of institutions that are
+currently creating instruments that can be exchanged for US Dollars. Changing the system is not
+going to be straightforward.
+However, I do believe that there is a path to making the change
+somewhat gradually, so that all the parties involved have time to adapt to the new system. Here
+is a proposed sequence of steps to adopting Elastic Cash for the US Dollar:
+1. The Fed begins to populate the cashbond market by gradually selling cashbonds of varying
+duration. Cashbonds are held at accounts maintained by the Fed, which allows the Fed to
+enforce that cashbonds cannot be transferred outside the cashbond market. At this point,
+cashbonds that expire are replaced according to the rules of Elastic Cash, but the risk-free
+rate of interest is allowed to float freely. I would expect this floating rate to converge close to
+the current Fed funds rate.
+2. Once the market for cashbonds is running at significant scale, regulations should be enacted to
+curtail the private creation of US Dollars. This can be done gradually by raising the interest
+rate at which the Fed lends to private entities through its discount window. At the same
+time, the Fed should begin to put bounds on the risk-free rate determined by cashbond, by
+trading in the cashbond market. Eventually, we should end up with a high rate for borrowing
+from the Fed via the discount window, while the risk-free rate in the cashbond market should
+be close to 2%. This will incentivize private entities to participate in the cashbond market
+and raise money there. The current creators of US Dollars can be handled as follows:
+(a) Commercial banks should be barred from creating new deposits that are not backed
+by cash reserves. Banks should fund new lending activity by selling corporate bonds
+instead.
+(b) The Fed’s repo facility and money market fund facility should be closed.
+(c) The eurodollar market is, perhaps, a bigger problem, both because of its size and the fact
+that the institutions cannot be regulated by US law. Still, the Fed can wind down its
+swap lines with foreign central banks gradually, until eurodollars lose their Fed backing.
+Foreign central banks and governments should be allowed and encouraged to participate
+in the cashbond market to obtain liquidity.
+3. The inevitable tantrums in the financial sector should be treated with stoicism.
+It is an understatement that moving from our current system of private money creation to
+Elastic Cash would be a dramatic change. There are likely to be many challenges that need to
+be overcome to implement it, not least the resistance of the finance industry, whose raison d’ˆetre
+is to control the flow of money. Elastic Cash represents a significant loss of control for financial
+firms, and a democratization of the flow of money. For these reasons, it is perhaps more easily
+implemented in a cryptocurrency, as I discuss next.
+9
+
+5
+Elastic Cash in cryptocurrencies
+A major advantage of Elastic Cash over conventional mechanisms for elasticity is that it can be
+implemented in a truly decentralized way, without any trusted central authority. Bitcoin made
+a technological leap when it introduced the concept of a blockchain.
+Since then, a number of
+cryptocurrencies have emerged, with different ways to implement the blockchain. Any of these
+systems can be used to implement Elastic Cash, so here I will keep the discussion at a high level, only
+talking about how the blockchain can be utilized. Because Elastic Cash involves making significant
+changes to the money supply, I do believe that implementing it requires new cryptocurrencies. I
+do not think it can be implemented using a layer built on top of Bitcoin, for example.
+Here is how one can implement Elastic Cash on a blockchain at a high level:
+1. At any point in time, each user of the cryptocurrency is known to hold some amount of money,
+as well as various cashbonds.
+2. Users of the currency can announce transactions of money, as well as orders placed in the
+cashbond market. The orders can be placed with a specific expiry date.
+3. Miners will add both money transactions and orders in the cashbond market to the next block
+of the blockchain. To implement the market in cashbonds:
+(a) Miners will act as market makers to map buy orders to sell orders and so execute the
+trade in cashbonds. There are some subtle issues that need to be addressed here. For
+example, a miner may be incentivized to pick some orders over others to include on
+the blockchain, and choose to ignore some orders when acting as a market maker. In
+particular, miners should themselves be paid the spread between buy and sell orders as
+a transaction fee to carry out their market making function. This removes the incentives
+to manipulate the orders that are added to the most recent block.
+(b) Miners will also execute the algorithm to simulate the activities of the cash authority in
+the cashbond market. New money and cashbonds will be created according to the rules
+of the mechanism, and these will be traded with users based on the orders that have
+been added to the blockchain.
+6
+Conclusions and Questions
+It is an exciting time to think about the technology of money. The US Dollar is experiencing a
+once-in-a-lifetime contraction (see Figure 1), and the demands for a stable global currency have
+never been larger. Elastic Cash is a broad scheme to enable elastic money. I have purposefully
+left the mechanism underspecified, because I believe that more work is required to understand the
+details and trade-offs involved in the particulars of the mechanism.
+Here are some important questions that I feel remain unanswered:
+1. How should the market maker behave in the cashbond market? In the context of conventional
+currencies, can private entities function as market makers? In the context of cryptocurrencies,
+how should the algorithm be set up so that miners do not have an incentive to behave
+dishonestly when they are carrying out the role of market maker?
+10
+
+2. What style of auction would give the best results for the cashbond market?
+3. What is the ideal target distribution on cashbonds? If the cashbonds are concentrated on
+very short durations, this gives the most power for the mechanism to inject large quantities
+of money, but it also means that the market loses information about the demand for liquidity
+over long durations.
+So, there is a trade-off between various choices for distributions on
+durations.
+4. How can we gradually transition the current US Dollar system to such a mechanism? The
+steps I discussed in Section 4 are likely to be difficult to execute. Perhaps there is a way
+to use cashbonds and incentivize the large players in the financial system to adopt Elastic
+Cash without being forced to do it. What is needed is a mechanism to transition to the new
+mechanism!
+5. How should we expect the free floating rate curve rate(t) to behave as a function of t during
+normal times? I would expect this function to be monotone, but I am not sure how to reason
+about it beyond that.
+7
+Acknowledgements
+Thanks to Paul Beame, Siddharth Iyer, Travis Kriplean, James Lee, Noam Nisan, Darcy Rao,
+Eli Ben-Sasson, Oscar Sprumont, Michael Whitmeyer and Amir Yehudayoff for many helpful and
+entertaining conversations about money.
+References
+[1] Fed history overview. https://www.federalreservehistory.org/time-period.
+[2] Christine Desan.
+Making Money: Coin, Currency, and the Coming of Capitalism.
+Oxford
+University Press, 2014.
+[3] Neels Heyneke and Mehul Daya.
+The rise and fall of the eurodollar system.
+https:
+//www.nedbank.co.za/content/dam/nedbank-crp/reports/Strategy/NeelsAndMehul/
+2016/September/TheRiseAndFallOfTheEurodollarSystem_160907.pdf, 2016.
+[4] Lev Menand. The Fed-Unbound: Central Banking in a Time of Crisis. 2022.
+11
+

HdE2T4oBgHgl3EQf_Alo/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf,len=311
+page_content='Elastic Cash  Anup Rao  University of Washington  anuprao@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='edu  January 12, 2023  Abstract  Elastic Cash is a new decentralized mechanism for regulating the money supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The mech-  anism operates by modifying the supply so that an interest rate determined by a public market  is kept approximately fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' It can be incorporated into the conventional monetary system to  improve the elasticity of the US Dollar, and it can be used to design new elastic cryptocurrencies  that remain decentralized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  1  Introduction  Money is as old as recorded history, and yet it continues to evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Even the mighty US Dollar  has been repeatedly updated over the last 200 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The recent emergence of Bitcoin and other  cryptocurrencies is another step in that evolution, and it prompts us to revisit the mechanisms  that ensure the desirable properties of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' An important property of money is its elasticity:  money is elastic if the money supply increases in response to demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' In this work, I present a  new decentralized mechanism to ensure the elasticity of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  The US Dollar is elastic, but it uses a convoluted system to achieve its elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The Dollar  system was not designed from first principles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' it was iteratively amended in response to financial  crises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Elasticity is currently achieved by the combined actions of many institutions: banks and  non-banks, public and private, domestic and international.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' No single entity is entirely in charge  of the money supply, and a relatively small number of investor-owned institutions have undue  influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' I discuss the mechanics of the US Dollar system and its limitations in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  In contrast, Bitcoin was designed to be inelastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Bitcoin caps the total possible supply at 21M,  and the available supply, currently about 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='2M, will slowly increase until it hits this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Satoshi  Nakamoto, the creator of Bitcoin, criticized conventional mechanisms for achieving elasticity in an  early forum post:  The root problem with conventional currency is all the trust that’s required to make  it work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The central bank must be trusted not to debase the currency, but the history  of fiat currencies is full of breaches of that trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Banks must be trusted to hold our  money and transfer it electronically, but they lend it out in waves of credit bubbles with  barely a fraction in reserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Nakamoto eliminated the need for the kind of infrastructure used to ensure Dollar elasticity by  simply choosing to make Bitcoin inelastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Additional technological innovations in Bitcoin further  eliminated the need for any trusted central authority to carry out transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='04244v1  [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='GN]  10 Jan 2023  To illustrate the negative consequences of inelasticity, consider the trajectory of home prices  denominated in Bitcoin over a long period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' As the population grows, the demand for houses is  sure to grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Even if the supply of houses keeps pace, home prices must fall, because the supply of  Bitcoin cannot keep pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Certainly, if the number of concurrent transactions involving home sales  grows and the prices of houses remain stable, the amount of Bitcoin involved in these transactions  would have to grow, yet it cannot grow beyond 21M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' So, a fixed money supply leads to falling  prices in a growing economy, even if the underlying supply and demand of items keep pace with  each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  At the other extreme, if the money supply is increased excessively, the currency is debased;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' the  excess supply leads to rising prices and inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' So, a mechanism for correctly setting the money  supply is essential, because this is the foundation upon which stable prices are built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' As much as  possible, prices should reflect the tension between the supply and demand for items, and nothing  else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' But how much money is too much?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' It is not just a matter of trust, the problem is that the  appropriate supply is difficult for anyone to calculate!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Is there a principled method to compute  the supply, and a fair way to create new money?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This work gives my answers to these important  questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Ideally, a mechanism for achieving elasticity should be transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  It should not rely on  the judgment or integrity of small groups of people, or a few institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' I will describe a new  decentralized mechanism called Elastic Cash that enjoys these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Elastic Cash can replace  the role played by private institutions in providing elasticity for the US Dollar, and it can be  combined with the concept of blockchains to make new cryptocurrencies that are elastic, yet do  not require any trusted central authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Here I will give a high-level description of Elastic Cash;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' a full description is in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' I will  refer to the central bank and the algorithm as cash authorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' At the heart of the mechanism is  a new financial contract issued by the cash authority called a cashbond, and a public market that  allows anyone to buy and sell cashbonds using public auctions in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Each cashbond can be  redeemed for $1 on a specific date, so it executes a risk-free loan from the holder of the cashbond to  the cash authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The loans are risk-free because the cash authority can always create new money  to repay the loans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Perhaps it is counterintuitive, but the purpose of the market in cashbonds is  not to give the cash authority a way to borrow money;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' instead, the function of the market is to use  the trading activity of participants to compute the risk-free rate of interest, which can be computed  from the prices of cashbonds in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The mechanism requires that cashbonds can only be  transfered by selling and buying them in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' For example, no entity should be permitted  to use cashbonds as collateral to borrow money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This forces participants to liquidate cashbonds  when they need money, and keeps the mechanism informed about the demands for money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  It is market participants that determine the money supply in Elastic Cash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  The supply is  increased or decreased according to transparent rules ensuring that the risk-free rate of interest  remains approximately fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Trade in cashbonds leads to fluctuations in the rate of interest, and  the mechanism responds by creating corresponding fluctuations in the money supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Intuitively,  the risk-free rate of interest encodes the cost of renting money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' By regulating the supply of money to  keep the cost fixed, the mechanism ensures that the supply stays in equilibrium with the demand for  liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' When the supply is increased, market participants acquire any newly-generated money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  The supply is decreased by incentivizing market participants to exchange their money for cashbonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Anyone can participate in this market, and money is distributed to or taken from participants  according to transparent rules, so no single entity can control the flow of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  2  Source: Board of Governors of the Federal Reserve System (US)  fred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='stlouisfed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='org  Billions of Dollars  1960  1970  1980  1990  2000  2010  2020  0  4,000  8,000  12,000  16,000  20,000  24,000  M2  Figure 1: M2, a measure of the supply of US Dollars  Elastic Cash can be implemented within the framework of conventional currencies like the US  Dollar by creating regulations that require the central bank to implement the market for cashbonds  and generate money according to the rules of the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  It can be implemented in the  framework of cryptocurrencies by setting up a distributed algorithm to implement the market for  cashbonds using the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Money is generated by the algorithm according to the rules of the  mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' So, one can obtain the positive features of traditional currencies and cryptocurrencies  in addition to the elasticity of Elastic Cash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Outline of this paper  In Section 2, I give more details about how the US Dollar achieves its  current elasticity, before turning to describe the new mechanism in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' I discuss how the new  mechanism can be incorporated into the Dollar system in Section 4 and how it can be implemented  on the blockchain to give new cryptocurrencies in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  2  US Dollar elasticity: the Fed, banks, and shadow banks  Figure 1 shows how the supply of US Dollars held as deposits has changed over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' In this section  I briefly review the history and mechanics of the US Dollar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' I recommend the following  3  二Dexcellent sources for additional background on the history of the Dollar system [1, 4], and the book  [2] for a longer history of the technology of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  The supply of US Dollars has consistently been deemed too important to be left entirely in  the control of a government agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Instead, we have developed a system where money is created  by investor-owned private entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' These include banks and so-called shadow banks — non-bank  financial institutions that are able to create Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This privately created money is then back-  stopped by the central bank, which is the Federal Reserve, or Fed in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The Fed is governed  by laws written by Congress, but these laws are somewhat ambiguous about the Fed’s powers, and  Congress has repeatedly made adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The Fed has responded to various crises by incentiviz-  ing or directly making big changes to the money supply, and by sometimes deciding that a new  category of privately-issued financial instruments can be exchanged for US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  At this point, there are at least four kinds of financial institutions that are able to generate  financial instruments that are de facto US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' It is helpful to view the evolution of the US  Dollar system according to the events that elevated these instruments to the stature of US Dollars:  Commercial bank deposits By the early 1900s, deposits of US Dollars were being issued by  a number of investor-owned private banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  These deposits were treated by depositors as  equivalent to US Dollars, even though the banks were issuing loans by creating deposits that  did not correspond to cash reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This led to a run on the banks in 1907, and the Fed was  created in 1913 to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The Fed was given the power to backstop the money  created by commercial banks by lending to the banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This meant that deposits could always  be exchanged for money created by the Fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This solved the problem of bank runs, but also  elevated bank deposits to the stature of cash issued by the Fed, and effectively meant that  commercial banks were given the authority to create legitimate US Dollars in the process of  issuing loans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Currently, about $17T is held in commercial bank deposits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Reverse purchase agreements (repos) In the 1950s, broker dealers began to enter the banking  industry using a financial instrument called reverse purchase agreements, or repo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Repos allow  dealers to borrow money from cash providers using government securities as collateral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Cash  providers began to treat the repos they had purchased from dealers as equivalent to cash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  By itself, such an arrangement does not create any new money, because if repos crash in  value, then they cannot actually be exchanged for cash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' However, the properties of repos  were substantially altered when the Fed decided to backstop dealers by giving them access to  overnight loans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This meant that the private holders of repos were guaranteed that repos could  always be exchanged for US Dollars via the Fed’s repo facility, and so repos were elevated to  the same stature as cash and bank deposits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' In 1991, Congress reduced restrictions on the  Fed to make it even easier to backstop the repo market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Effectively, broker dealers were given  the power to create new US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Currently, the size of the repo market is about $4T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Eurodollars Foreign companies including both banks and non-banks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' insurance companies)  have been issuing financial instruments called eurodollars that can be redeemed for US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  These eurodollars were not initially backed by actual Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The oil shock of 1973-1974 led  to problems in the eurodollar market that eventually brought down a domestic US bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The  Fed responded by promising to backstop eurodollars by providing actual US Dollars in the  form of loans to the corresponding foreign central banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Effectively, the Fed permitted the  banking systems of other countries to create deposits that could be exchanged for US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  The size of the eurodollar market was estimated at about $13T in 2016 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  4  Money market mutual funds These funds emerged in the 1970s as investments whose share  price was pegged at $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' In reality, these funds held assets whose value could drop below the  peg, and so it was not possible to guarantee the peg during a financial crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' In response  to the great financial crisis of 2008, the Fed began to backstop these funds using its Money  Market Mutual Fund Liquidity Facility, and so elevated deposits in these funds to the stature  of US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Total assets in these funds is about $4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='8T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  In addition to recognizing new forms of the US Dollar, the Fed has resorted to buying assets  in order to inject liquidity under the Quantitative Easing program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' During the Great Financial  Crisis of 2008, the Fed kicked off this program by buying mortgage backed securities and treasuries,  which are loans to the US Treasury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Throughout the last decade, the Fed has continued to expand  its balance sheet, mostly with treasuries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' In 2020 the Fed once again bought significant quantities  of these assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Currently the Fed holds about $8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='5T on its balance sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  The history of the US Dollar is full of ad hoc amendments to maintain stability in the face of  financial crises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' At its heart, the problem is that there is currently no principled way to regulate  the supply of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This becomes apparent during times of financial crises, but the imbalance in  supply is probably always brewing, even in normal times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' By now, the pattern of Fed actions is  familiar, and it is inevitable that it will repeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' During times of crisis, the Fed must act to protect  money or face a significant crash in the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Because private financial institutions know  that the Fed will protect their financial instruments from the most negative consequences of their  choices, they do not have the correct incentives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Elastic Cash is meant to provide a clean, transparent, and principled mechanism to achieve  robust elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  We do not need to cede control of the money supply to private institutions  or foreign banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' We do not need to elevate invented forms of money to the stature of the US  Dollar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Once Elastic Cash is adopted, I believe we can safely bar all private creation of Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  The mechanism will generate new US Dollars when required, and financial institutions can obtain  liquidity by participating in the mechanism, just like everyone else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Extricating ourselves from the current system and its vested interests is likely to be challenging,  to say the least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Nevertheless, I describe a path to incorporating the new mechanism in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  3  Elastic Cash: the details  Elastic Cash uses trade in cashbonds to determine a risk-free rate of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The money supply  is regulated to ensure that this interest rate remains approximately fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Cashbonds are issued  by the central bank (in the case of conventional currencies) or by the distributed algorithm (in the  case of cryptocurrencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' I refer to these entities as cash authorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  The contract cashbond(d) promises that the cash authority will pay the holder of the contract $1  on the date d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Let us reserve d0 to denote the current date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' On date d0, the cash authority pays each  holder of cashbond(d0) $1, and these contracts expire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Elastic Cash requires that the cash authority  implement a public market in cashbonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' On the date d0, contracts of the type cashbond(d) for  d > d0 will be available in the market for cashbonds maintained by the cash authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Cashbonds are a special class of asset, and they should not be treated like other securities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Elastic Cash requires that cashbonds can be generated and traded only in the public market that  is administered by the cash authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Cashbonds are not transferable, meaning they cannot be  exchanged outside of the public market, and they cannot be used as collateral for loans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Because of  5  these restrictions, cashbonds cannot themselves play the same role as money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The purpose of these  rules is to ensure that holders of cashbonds that desire liquidity will sell their cashbonds in the  market and so keep the mechanism informed about the demand for liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' For the same reasons,  trade in cashbonds should not be taxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The transactions of buying and selling cashbonds should  be viewed as similar to transactions that move money between savings accounts paying varying  rates of interest, and treated similarly under the law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='1  Risk-free rate of interest  The price at which cashbonds trade implies interest rates for risk-free loans of varying durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Let rate(t) denote the interest rate for duration t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Let us write price(d) to denote the price at which  cashbond(d) last traded in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Then, if the current date is d0, the prices of cashbonds can  be used to compute implied interest rates according to the formula:  price(t + d0) · (1 + rate(t))t = 1,  which implies that the interest rate can be expressed as  rate(t) = price(t + d0)− 1  t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Because the loans executed by cashbonds are risk-free, the values rate(t) capture something about  the market’s belief about the opportunity cost of making risk-free loans for duration t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Generally,  one would expect rate(t) to be a monotone function of t, meaning that rate(t) > rate(t′) if t > t′,  because loans of longer duration usually command higher interest rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Moreover, if t is much  larger than t′, then we might expect rate(t) to have higher variance than rate(t′), because predictions  about the distant future can diverge much more than predictions about the immediate future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  These rates encode important information about the demand for liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The goal of the  mechanism is to regulate the money supply so that one of these rates is held approximately fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  It makes the most sense to pick a rate for a relatively short duration, because these rates are likely  to have the least variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' With that in mind, let τ denote a short time period, say 1 week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The  goal of the mechanism will be to keep  rate(τ) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  There is nothing special about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='02, except that it is convention for central banks around the world  to use 2% as the target rate of longterm inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Let us set  p− = (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='021)−τ,  and  p+ = (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='019)−τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  The goal of the mechanism will be to regulate the money supply so that  p− ≤ price(τ + d0) ≤ p+,  where again d0 is the current date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This corresponds to keeping  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='019 ≤ rate(τ) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='2  Using the market to regulate the money supply  Participants in the cashbond market can put in orders to sell a specific number of cashbonds that  they hold at a specific price, and can also put in orders to buy a specific number of cashbond(d)  at a specific price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The cash authority acts as a market maker to match buy orders to sell orders  and so conduct transactions at a specific price between market participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Ideally, the market  for cashbonds will support auctions1 for sellers to sell their cashbonds when needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  The cash authority will itself participate in this public market by buying and selling cash bonds  in prescribed ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The goal of the mechanism is to maintain rate(τ) approximately fixed, and to  keep the market in cashbonds sufficiently liquid, so that the money supply can be quickly adjusted  based on changes to rate(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Here is the proposed scheme for buying and selling cashbonds:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The cash authority will buy and sell cashbonds to keep rate(τ) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The cash authority  will place a standing order to buy an infinite number of contracts cashbond(τ + d0) at price  p−, and a separate standing order to sell an infinite number of cashbond(τ + d0) contracts at  price p+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Because the cash authority is able to generate arbitrary amounts of both money and cash-  bonds, it will always be able to satisfy any of the resulting transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This ensures that  p− ≤ price(τ + d0) ≤ p+,  as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' When cashbonds are redeemed for money, the cash authority will need to sell new cashbonds  to restore the balance between money and cashbonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  It makes sense to pick a particular target distribution on outstanding cashbonds that is  maintained during normal times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' If the current date is d0, we say that cashbond(δ + d0) has  duration δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' For example, the cash authority might aim to maintain the invariant that at any  point in time, 1/4 of the outstanding cashbonds have duration between 0 and 1 month, 1/4  have duration between 1 month and 1 year, 1/4 have duration between 1 year and 4 years,  and 1/4 have duration between 4 years and 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Given such a target distribution, the redeemed cashbonds should be replaced by selling new  cashbonds at auction, picking the dates of the new cashbonds so that the overall distribution  on duration is maintained as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' When the demand for money is high, we are likely to reach the point where all of the available  bonds cashbond(τ + d0) have been purchased by the cash authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  In such times, the  mechanism has run out of the means to inject money into the financial system at a fast enough  pace according to rule 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This can be resolved by selling large quantities of cashbond(2τ +d0)  contracts at auction in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Market participants will be incentivized to buy these  cashbonds and then sell them back after time τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' at that time the cash authority itself will  be willing to buy the cashbonds at price p−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The net effect will be to inject money into the  system, while preserving the number of outstanding cashbonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  1I will not commit to a specific style of auction here, though any implementation must carefully specifying how  the cash authority behaves as a market maker and what the rules of the auctions are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  2There are many considerations for how to choose the target distribution, but here I will not dwell on the choices  too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  7  The number of cashbonds sold in this process is a design choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The goal should be inject  significant liquidity, so I would favor an exponentially escalating volume of sales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' For exam-  ple, the cash authority might first sell a quantity that corresponds to 1% of all outstanding  cashbonds, and a week later escalate it to 2%, then 4%, and so on until the cashbond market  returns to the state where market participants are no longer willing to sell back the cash-  bonds of duration τ to the cash authority at p−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' These actions may temporarily distort the  distribution on the durations of outstanding cashbonds, but the distribution will be quickly  restored when the new cashbonds are redeemed and rule 2 is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  An actual implementation of Elastic Cash would need to resolve many smaller technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Let me now make a few comments and observations about the Elastic Cash mechanism as I have  defined it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='3  Discussion  Elastic Cash is quite different from a system where the central bank simply allows deposits for  all with interest rate 2%—such a scheme does not give the central bank a method to inject large  amounts of money when the liquidity is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' History has shown that the Fed needs a tool like  Elastic Cash to inject liquidity into the financial system, since interest rates have proven too weak  as a tool to inject large quantities of liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' As we discussed in Section 2, this has led to the  Fed buying assets or propping up assets that were liable to crash in value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' In doing so, the Fed  is forced to pick and choose between market participants that get first access to the new liquidity  that it provides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Central bankers should not be attempting to directly reason about the demands for liquidity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  they do not have enough data to make those decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' But if they must take such dramatic actions,  the scheme of Elastic Cash at the very least gives a fair way to do it by trading cashbonds along  the lines I have suggested above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This removes the ability of the finance sector to control the flow  of the new money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' It is also preferable to having the Fed buy treasuries, because it disentangles the  actions of the Fed from the needs of the Treasury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' There is no need to tie increases in the money  supply to increases in government spending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Cashbonds should not be confused with conventional government securities like US treasuries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  These instruments are significantly different from each other, and one cannot make inferences about  the cashbond market, which does not yet exist, based on the behavior of the US treasury market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Let  me highlight some key differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The issuance of cashbonds is controlled by strict and transparent  rules, and is not tied to the spending of the US government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' There is no analogue of debt ceilings,  or any chance that the central bank will default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Cashbonds cannot be used as collateral for loans,  cannot be transferred outside of the Elastic Cash market, and trade in cashbonds is not taxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  It is important for the functioning of Elastic Cash to maintain a large volume of outstanding  cashbonds of varying durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Ideally, we would like there to be broad participation in the cash-  bond market from all kinds of financial entities: banks, companies, pension funds, and individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Because these participants will be willing to trade at different durations, participation will be in-  creased if a wide range of durations are available, and the market is liquid at all durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Even  though the cash authority only regulates the interest rate for duration τ, this action will affect the  rates for all durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' One would expect that banks and other sophisticated players will trade  cashbonds of shorter duration, and perform the arbitrage necessary for information about demands  for liquidity of all durations to propagate to the shorter durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' I suspect that there is a prin-  8  cipled way to choose the ideal distribution on durations of outstanding cashbonds, but I have not  yet been able to convince myself about what it ought to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  4  Adopting Elastic Cash in the US Dollar system  As discussed in Section 2, the Dollar system involves many different kinds of institutions that are  currently creating instruments that can be exchanged for US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Changing the system is not  going to be straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  However, I do believe that there is a path to making the change  somewhat gradually, so that all the parties involved have time to adapt to the new system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Here  is a proposed sequence of steps to adopting Elastic Cash for the US Dollar:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The Fed begins to populate the cashbond market by gradually selling cashbonds of varying  duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Cashbonds are held at accounts maintained by the Fed, which allows the Fed to  enforce that cashbonds cannot be transferred outside the cashbond market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' At this point,  cashbonds that expire are replaced according to the rules of Elastic Cash, but the risk-free  rate of interest is allowed to float freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' I would expect this floating rate to converge close to  the current Fed funds rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Once the market for cashbonds is running at significant scale, regulations should be enacted to  curtail the private creation of US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This can be done gradually by raising the interest  rate at which the Fed lends to private entities through its discount window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' At the same  time, the Fed should begin to put bounds on the risk-free rate determined by cashbond, by  trading in the cashbond market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Eventually, we should end up with a high rate for borrowing  from the Fed via the discount window, while the risk-free rate in the cashbond market should  be close to 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This will incentivize private entities to participate in the cashbond market  and raise money there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The current creators of US Dollars can be handled as follows:  (a) Commercial banks should be barred from creating new deposits that are not backed  by cash reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Banks should fund new lending activity by selling corporate bonds  instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  (b) The Fed’s repo facility and money market fund facility should be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  (c) The eurodollar market is, perhaps, a bigger problem, both because of its size and the fact  that the institutions cannot be regulated by US law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Still, the Fed can wind down its  swap lines with foreign central banks gradually, until eurodollars lose their Fed backing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Foreign central banks and governments should be allowed and encouraged to participate  in the cashbond market to obtain liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The inevitable tantrums in the financial sector should be treated with stoicism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  It is an understatement that moving from our current system of private money creation to  Elastic Cash would be a dramatic change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' There are likely to be many challenges that need to  be overcome to implement it, not least the resistance of the finance industry, whose raison d’ˆetre  is to control the flow of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Elastic Cash represents a significant loss of control for financial  firms, and a democratization of the flow of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' For these reasons, it is perhaps more easily  implemented in a cryptocurrency, as I discuss next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  9  5  Elastic Cash in cryptocurrencies  A major advantage of Elastic Cash over conventional mechanisms for elasticity is that it can be  implemented in a truly decentralized way, without any trusted central authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Bitcoin made  a technological leap when it introduced the concept of a blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Since then, a number of  cryptocurrencies have emerged, with different ways to implement the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Any of these  systems can be used to implement Elastic Cash, so here I will keep the discussion at a high level, only  talking about how the blockchain can be utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Because Elastic Cash involves making significant  changes to the money supply, I do believe that implementing it requires new cryptocurrencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' I  do not think it can be implemented using a layer built on top of Bitcoin, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Here is how one can implement Elastic Cash on a blockchain at a high level:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' At any point in time, each user of the cryptocurrency is known to hold some amount of money,  as well as various cashbonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Users of the currency can announce transactions of money, as well as orders placed in the  cashbond market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The orders can be placed with a specific expiry date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Miners will add both money transactions and orders in the cashbond market to the next block  of the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' To implement the market in cashbonds:  (a) Miners will act as market makers to map buy orders to sell orders and so execute the  trade in cashbonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' There are some subtle issues that need to be addressed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' For  example, a miner may be incentivized to pick some orders over others to include on  the blockchain, and choose to ignore some orders when acting as a market maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' In  particular, miners should themselves be paid the spread between buy and sell orders as  a transaction fee to carry out their market making function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' This removes the incentives  to manipulate the orders that are added to the most recent block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  (b) Miners will also execute the algorithm to simulate the activities of the cash authority in  the cashbond market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' New money and cashbonds will be created according to the rules  of the mechanism, and these will be traded with users based on the orders that have  been added to the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  6  Conclusions and Questions  It is an exciting time to think about the technology of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The US Dollar is experiencing a  once-in-a-lifetime contraction (see Figure 1), and the demands for a stable global currency have  never been larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Elastic Cash is a broad scheme to enable elastic money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' I have purposefully  left the mechanism underspecified, because I believe that more work is required to understand the  details and trade-offs involved in the particulars of the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Here are some important questions that I feel remain unanswered:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' How should the market maker behave in the cashbond market?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' In the context of conventional  currencies, can private entities function as market makers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' In the context of cryptocurrencies,  how should the algorithm be set up so that miners do not have an incentive to behave  dishonestly when they are carrying out the role of market maker?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' What style of auction would give the best results for the cashbond market?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' What is the ideal target distribution on cashbonds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' If the cashbonds are concentrated on  very short durations, this gives the most power for the mechanism to inject large quantities  of money, but it also means that the market loses information about the demand for liquidity  over long durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  So, there is a trade-off between various choices for distributions on  durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' How can we gradually transition the current US Dollar system to such a mechanism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The  steps I discussed in Section 4 are likely to be difficult to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' Perhaps there is a way  to use cashbonds and incentivize the large players in the financial system to adopt Elastic  Cash without being forced to do it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' What is needed is a mechanism to transition to the new  mechanism!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' How should we expect the free floating rate curve rate(t) to behave as a function of t during  normal times?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' I would expect this function to be monotone, but I am not sure how to reason  about it beyond that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  7  Acknowledgements  Thanks to Paul Beame, Siddharth Iyer, Travis Kriplean, James Lee, Noam Nisan, Darcy Rao,  Eli Ben-Sasson, Oscar Sprumont, Michael Whitmeyer and Amir Yehudayoff for many helpful and  entertaining conversations about money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  References  [1] Fed history overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='federalreservehistory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='org/time-period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  [2] Christine Desan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Making Money: Coin, Currency, and the Coming of Capitalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  Oxford  University Press, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  [3] Neels Heyneke and Mehul Daya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  The rise and fall of the eurodollar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  https:  //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='nedbank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='za/content/dam/nedbank-crp/reports/Strategy/NeelsAndMehul/  2016/September/TheRiseAndFallOfTheEurodollarSystem_160907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='pdf, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  [4] Lev Menand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' The Fed-Unbound: Central Banking in a Time of Crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}
+page_content='  11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'}

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