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+Theoretical model of membrane protrusions driven by curved active proteins
+Yoav Ravid 1,∗, Samo Peniˇc 2, Yuko Mimori-Kiyosue,3, Shiro Suetsugu,4,5,6, Aleˇs Igliˇc 2, and Nir S. Gov 1,∗
+1Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot, Israel
+2Laboratory of Physics, Faculty of Electrical Engineering, University of Ljubljana, Ljubljana, Slovenia
+3Laboratory for Molecular and Cellular Dynamics,
+RIKEN Center for Biosystems Dynamics Research,
+Minatojima-minaminachi, Chuo-ku, Kobe, Hyogo 650-0047, Japan
+4Division of Biological Science, Graduate School of Science and Technology,
+Nara Institute of Science and Technology 8916-5, Takayama, Ikoma, Nara, 630-0192, Japan
+5 Data Science Center, Nara Institute of Science and Technology, Ikoma 630-0192, Japan
+6 Center for Digital Green-innovation, Nara Institute of Science and Technology, Ikoma 630-0192, Japan
+Eukaryotic cells intrinsically change their shape, by changing the composition of their membrane
+and by restructuring their underlying cytoskeleton. We present here further studies and extensions
+of a minimal physical model, describing a closed vesicle with mobile curved membrane protein
+complexes. The cytoskeletal forces describe the protrusive force due to actin polymerization which is
+recruited to the membrane by the curved protein complexes. We characterize the phase diagrams of
+this model, as function of the magnitude of the active forces, nearest-neighbor protein interactions
+and the proteins’ spontaneous curvature. It was previously shown that this model can explain the
+formation of lamellipodia-like flat protrusions, and here we explore the regimes where the model can
+also give rise to filopodia-like tubular protrusions. We extend the simulation with curved components
+of both convex and concave species, where we find the formation of complex ruffled clusters, as well
+as internalized invaginations that resemble the process of endocytosis and macropinocytosis. We
+alter the force model representing the cytoskeleton to simulate the effects of bundled instead of
+branched structure, resulting in shapes which resemble filopodia.
+Keywords: Cell membrane, Curved inclusions, Monte-Carlo simulations, Closed vesicle shapes, Cell motility, Filopodia
+I.
+INTRODUCTION
+Cells in our body have different shapes depending on their function, and they control their shapes by exerting
+forces on the flexible plasma membrane [1]. These forces are mostly due to the underlying cytoskeleton, dominated
+by the cortical actin network. The actin polymerization near the membrane exerts protrusive forces that can give
+rise to cellular protrusions, such as filopodia and lamellipodia [2]. The control of the actin polymerization in space
+and time is provided by a host of proteins that nucleate actin polymerization where and when it is needed, and
+are in turn controlled by different signalling cascades. One mechanism for controlling the spatial pattern of actin
+polymerization on the membrane, is to couple the actin nucleation to curved membrane components (CMCs), that are
+both bending locally the membrane and are sensitive to the local membrane curvature (such as BAR domain proteins
+[3]). This coupling was shown theoretically to give rise to positive and negative feedbacks [4], that can result in pattern
+formation in both the spatial distribution of the actin nucleators (recruited by the CMCs) and the membrane shape.
+This coupling between curvature and active protrusive forces was explored for a limited regime of parameters in [5].
+Experimental evidence for this coupling between CMC and protrusive forces has been accumulated in the context of
+lamellipodia [6, 7] and filopodia [8–14] formation.
+A summary of the vesicle shapes that we found in [5] are shown in Fig.1, explored as function of temperature and
+CMC density (Fig.1A). The main phases which were identified are [15]:
+• Diffused CMC-gas phase, where CMC are dispersed as entropy dominates over bending and binding energies.
+• Budded phase, where binding and bending leads to CMC forming hemispherical clusters at the CMC spontaneous
+curvature.
+• Flattened ”pancake” phase, where the active forces push the CMC outwards, leading to a large CMC cluster along
+the rim, with two flat bare membrane disc regions. Low temperature is required to prevent lateral membrane
+fluctuations and thermal diffusion of the CMC from breaking up the rim cluster.
+The pancake phase is quite dynamic, and tends to form ”ruffles” along the edges. With insufficient density of CMC,
+there is a ”two-arc” phase with multiple flat edges connected by elongated membrane (Fig.1B). If the CMC density if
+high, the excess CMC form pearled structures along the rim of the pancake (Fig.1C).
+arXiv:2301.13055v1  [cond-mat.soft]  30 Jan 2023
+
+2
+When the active force is weak or zero (passive CMC), at low temperatures the system is phase-separated into
+energy-minimizing ”pearled necklace” of CMC clusters, each at the CMC spontaneous curvature (Fig.1D). When the
+force is strong and the CMC have low spontaneous curvature (flat), there is a phase of highly elongated ”tubular”
+vesicles, where CMC caps apply large forces that pull membrane tethers (tubular protrusions) (fig.1E).
+Here we expand the analysis of the coupling between the spontaneous curvature of the CMC and protrusive forces,
+by exploring the patterns that form as function of the natural parameters that the cell can manipulate, such as the
+strength of the actin-driven force, the binding strength between the CMCs and the spontaneous curvature of the
+CMCs. By gaining a fuller understanding of the space of shapes that this coupling can produce, we are able to explore
+two more complex configurations: a mixture of two CMCs of different intrinsic curvatures, and CMCs that induce
+aligned active forces which model the effects of actin bundling[16]. These more complex systems, can be compared to
+important biological phenomena, such as endocytosis[17] and filopodia.
+II.
+THE MODEL
+We follow the same coarse-grained continuum model used previously [5] and [18], where the physics of the cell shape
+is described by differential geometry and very few energy components [1, 19]. The lipid bilayer membrane is modeled
+as a 2D flexible sheet, with zero spontaneous curvature, except where there are CMCs. Each CMC on the membrane
+surface represents a complex of proteins that have a specific spontaneous curvature. The energy of the surface is
+modeled by the Helfrich hamiltonian
+Hbending =
+��
+κ
+2 (C1 + C2 − C0φ)2
+(1)
+which penalizes deviation of the shape, given by the local curvatures C1 and C2, from a preferred local shape, determined
+by the CMC relative lateral density φ and the CMC’s preferred membrane curvature C0. To simulate, we discretize
+the system as a closed vesicle described by a graph V, E (vertices and edges respectively) with vertices representing
+small area patches of either bare lipid bilayer or CMC. Note that the simulation does not have an intrinsic length
+scale, however the edge length has to represent lengths larger than tens of nanometers for the coarse-grained model to
+be physically valid. We therefore obtain the following discretized energy
+E =
+�
+i∈V
+κ
+2 (2h(i) − ρiC0)2 A(i) +
+�
+⟨i,j⟩∈E
+−wρiρj +
+�
+i∈V
+wad θ (zi − z0)
+(2)
+where ρi = 1 for a CMC vertex and ρi = 0 for a bare vertex, such that the overall density of CMC is given by
+ρ = �
+i∈V ρi/N, where N = 4502 is the total number of vertices in our simulations. The first term is a discretized
+version of the bending energy (Eq.1), κ is the bending modulus, h(i) is the mean curvature calculated at each vertex
+h = (C1 + C2)/2, C0 is the spontaneous curvature of a CMC, and A(i) is the area assigned to the vertex. The second
+term is the CMC-CMC nearest-neighbor binding energy, going over the edges ⟨i, j⟩, where w is the binding energy per
+bond. The third term is adhesion energy of the membrane to a flat rigid surface located at z = z0, which applies to
+all the nodes that are within a distance of ℓmin from this surface. The membrane is prevented from moving below
+z0 − ℓmin.
+This energy model is used in a Monte-Carlo (MC) simulation Trisurf-ng, described in [5], where random movement
+of vertices and bond flips of edges are accepted if they lower the energy or according to a Boltzmann probability:
+P = exp (−∆E − Wi) where Wi represents the work done by the active forces on each node that contains a CMC, as
+follows
+Wi = −f ˆn(i) · δ⃗xi
+(3)
+where ˆn(i) is the local outwards normal unit vector, and δ⃗xi is the vertex displacement.
+The shift in the locations of the vertices are limited such that the length of each edge remains within this range:
+ℓmin < ℓ < ℓmax. The edge length and adhesion surface constraints are enforced by rejecting any MC moves which
+violate them. In a passive system this would lead to thermal equilibrium, but the active work term is unbounded
+from below, so the system is out of equilibrium. The MC simulation does not have time-scale, as it does not include
+the hydrodynamic flows and dissipative processes that determine the relaxation time-scales of the membrane shape
+changes. It does allow us to follow the shape dynamics by evolving the system along decreasing energy gradients, so
+the trajectory in shape space is correctly described.
+The parameters in the model, used in this paper, are given in table I. All the energies in the model are in units of
+kBT (κ, w), while the external force f is in units of kBT/ℓmin.
+
+3
+In addition, we implement optional models of inhibition of the force on the CMC by neighbors, based on [20] which
+shows different protein species can inhibit the activity of polymerization, inhibiting the actin recruitment and thus
+force on the CMCs. We implement a proportional inhibition, where an active (1) and inhibiting (2) CMC species exist
+f prop
+i
+= f
+1
+Nneighbors
+�
+⟨i,j⟩
+�
+1 − ρ(2)
+j
+�
+(4)
+We also implement a disabling inhibition, where any inhibiting CMC species completely disables the force on it’s
+neighbors.
+f dis
+i
+= f
+�
+⟨i,j⟩
+�
+1 − ρ(2)
+j
+�
+(5)
+In biological filopodia, the actin filament are known to bundle by cross-lining proteins [21]. Our model does not
+have a true representation of the cytoskeleton structure, but we can simulate this bundling by adding an alignment to
+the force on the active CMCs, since the shared internal actin bundle would apply a force in the same direction. This is
+added as an Vicsek-like interaction [22]
+ˆf = ˆni + s �
+r ˆnj
+|ˆni + s �
+r ˆnj|
+(6)
+The direction of force on CMC vertex i ˆfi is a weighted average of the normal direction plus a contribution from all the
+vertices j a distance r from the vertex i with a weight of s, normalized. This replaces the ˆn(i) term in the work term
+i.e. the unmediated local normal. This is superficially similar to the normal Vicsek model [22], where self-propelled
+particles similarly align their direction with neighbors, producing flocking behavior, but here the CMCs/particles are
+connected to each other and embedded in a 2D flexible sheet, and we use force in a MC simulation instead of velocity
+in a Langevin simulation.
+III.
+MATERIALS AND METHODS
+A.
+Computational Methods
+The simulations were run using trisurf-ng [5] version fb86a41 (”Modeled trisurf” branch) (see X) with a tape file
+modified from the available default with the different physical parameters (see I), and additional simulation running
+parameters of nshell=30, mcsweeps=50,000-200,000, iterations=100-1,000 (depending on the desired time resolution).
+Each simulation with a set of parameters was ran independently (”embarrassingly parallel”), which took about two
+weeks to finish, with occasional restarts and expansion of the space limits (nxmax). The resulting VTU files were
+viewed and colored in ParaView, but further analysis and graph generation were done by separate python scripts.
+B.
+Experimental Methods
+The cell culture and lattice light sheet microscopic observation U-251 cells were obtained from the Japanese Collection
+of Research Bioresources Cell Bank. The IRSp53 knockout (KO) cells were generated by the CRISPR/Cas9 system, as
+described previously [23]. The guide RNA targeting the first exon of IRSp53 (CCATGGCGATGAAGTTCCGG) was
+designed using the server http://crispr.mit.edu and inserted into the pX330 vector [23]. After transfection, the cells
+were cloned by monitoring the GFP fluorescence from the reporter plasmid pCAG-EGxxFP with the IRSp53 genome
+fragment using a fluorescence-activated cell sorter [FACSAria (BD)] [24]. The expression of GFP or GFP-IRSp53
+in the IRSp53 knockout cells was performed by the retrovirus-mediated gene transfer, as described previously [24].
+All cell lines were cultured in high glucose DMEM (Thermo Fisher Scientific) supplemented with 10% bovine calf
+serum (Thermo Fischer Scientific) and 1% penicillin-streptomycin solution (Thermo Fischer Scientific) and stored in
+an incubator at 37oC in 5% CO2 and humidified conditions. The cells were seeded on coverslips and then imaged with
+the Lattice light-sheet microscope built in the Mimori-Kiyosue laboratory at RIKEN Center for Biosystems Dynamics
+Research following the design of the Betzig laboratory [25] as described previously [26].
+
+4
+IV.
+FORCE-BINDING STRENGTH PHASE DIAGRAM
+In [5] the phases of the vesicle with active CMC, were mostly explored as function of temperature and global density
+of CMC. However, the cell can more easily modify other parameters, such as the strength of the protrusive forces
+produced by actin polymerization and the binding strength between neighboring CMC. The rate of actin polymerization
+recruited to the CMC can be controlled by the cell through various proteins [27–29]. The effective binding strength
+between the neighboring CMC can similarly depend on the lateral concentration and character of the proteins that
+form the CMC [7], as well as on the type of lipids between the CMC [30]. The cell can modify these internal parameters
+spontaneously or in response to external signals.
+We scan over the force f and binding strength w parameters plane (Fig.2A), with the other parameters of the model
+having the following constant values: The bending modulus is taken to be κ = 20KBT, which is a typical value for
+lipid bilayers. The spontaneous curvature of the CMC is taken to be C0 = 1ℓ−1
+min, representing highly curved objects on
+the membrane. The CMC density is ρ = 10%, which is sufficient to form the pancake shapes that require a complete
+circular cluster of CMC along the vesicle rim [5].
+We find that the simulated vesicles can be divided into several distinct phases: gas phase, budded phase, pancake
+phase, and pearling phase. In addition there are more ambiguous, and possibly transient, elongated and mixed phases
+(Fig.2A). In order to distinguish between these phases, we use four measures that characterize the vesicle shape and
+the CMC cluster organization:
+• Mean cluster size ⟨N⟩
+• 1st eigenvalue of the Gyration tensor λ2
+1
+• 2nd eigenvalue of the Gyration tensor λ2
+2
+• Length of CMC-bare membrane boundary ℓp
+The mean cluster size is averaged over all the CMC clusters, each cluster i having a size Ni of vertices
+⟨N⟩ =
+�
+i Ni
+�
+i 1 = Nvertex
+Nclusters
+We plot this measure (Fig.2B), extracted after the simulation reaches its steady-state regime, where the measures do
+not change on average (see SI). we see that it allows to clearly distinguish the gas phase, which has small cluster sizes
+(yellow line in Fig.2A denotes ⟨N⟩ = 1.5). However, it is rather poor at separating the condensed phases, which all
+have large clusters but differ greatly in their morphology and cluster organization. This is due to the dependence
+of this measure on the number of clusters, which gives large weight to small single-vertex clusters. This makes this
+measure too noisy to distinguish between the other phases, except for the gas phase which mostly contains single-vertex
+clusters.
+We therefore use morphological measures in order to clearly distinguish between the different phases where the
+CMCs are condensed in large clusters. The morphology of the vesicle is quantified by the eigenvalues of the gyration
+tensor λ2
+i . The gyration tensor [31] is defined as the average over all the vertices, with respect to the center of mass
+(similar to the moment of inertia tensor for equal-mass vertices)
+RG ij = ⟨rirj⟩ = 1
+N
+�
+vertices
+�
+�
+x2 xy xz
+xy y2 yz
+zx yz z2
+�
+�
+This can be visualized by a unique ellipsoid which has the same gyration tensor
+xT R−1
+G x = (x · e1)2
+λ2
+1
++ (x · e2)2
+λ2
+2
++ (x · e3)2
+λ2
+1
+= 3
+The eigenvectors ei of the gyration tensor are the directions of the semi-axes of the equivalent ellipsoid and the
+eigenvalues are their length squared divided by 3, ordered by their size: λ2
+1 ≤ λ2
+2 ≤ λ2
+3. The first eigenvalue λ1
+essentially gives how thin is the ellipsoid, and is low for both pancake and highly elongated (linear) shapes. The
+second eigenvalue λ2 is large for the pancake shape (as it is roughly equal to the largest eigenvalue λ2 ∼ λ3), but is
+minimized for elongated shapes, where it similar to the value of the smallest eigenvalue, λ2 ∼ λ1. In Fig.2C,d we plot
+the eigenvalues λ2
+1, λ2
+2, respectively. We find that the phase of pancake shapes is distinguished by the lowest λ2
+1 (green
+and dashed green-light blue lines in Fig.2A), indicating its flatness.
+
+5
+We identify a new phase of elongated shapes, which is distinguished by the lowest values of λ2
+2 (between the light
+blue and dashed green-light blue lines in Fig.2A). These elongated phases are somewhat similar to the ”two-arc” phase
+found in [5], which appeared when there are not enough CMCs to form a complete circular cluster along the flat vesicle
+rim. However, here we do have enough CMC to form a complete circular cluster, as shown in the ”flat” regime. The
+origin of the elongated shapes as w increases beyond the ”flat” phase is due to the formation of transient or stable
+pearling clusters. These cluster effectively sequester enough CMC to prevent the formation of the complete circular
+cluster, leading to two curved regions that collect the CMC and stretch the vesicle due to the active forces. The CMC
+clusters have the shape of flat arcs near the boundary with the ”flat” phase, while closer to the ”pearling” phase the
+clusters are pearled and localized near the curved tips of the vesicle.
+While the ”core” of the phases distinguished by λ2
+1,2 is clear, the edges are much less sharp, due to lack of statistics,
+long evolution time, and the fact that intermediate shapes do exist. There is also no obvious normalization: The
+volume changes greatly, and the area is only approximately conserved. For our Nvertex = 4,502 The flat phase is found
+around λ2
+1 < 50, and the elongated phases is found around 80 < λ2
+2 < 150.
+Finally, we wish to distinguish the phases where the CMCs form pearled clusters. The most outstanding property of
+the pearled clusters is that they phase-separate between the CMC and the bare membrane, as also predicted within
+the theory of self-assembly [5]. We therefore measure the average length of the CMC-bare membrane boundary ¯ℓp, per
+CMC, for all clusters larger than 1 (see SI section 1, Fig.S1)
+¯ℓp =
+�ℓpi
+Ni
+�
+Ni>1
+The phase with pearling clusters is distinguished by having very low ¯ℓp < 0.375 (Fig.2E). We find that this measure
+identifies the pearled clusters both in the pearling and in the elongated phases (red dotted line in Fig.2A). In addition,
+a contour of this measure allows us to separate the mixed phase, where the CMC are in both buds and pearled clusters,
+from the phase that contains only buds (red solid line ¯ℓp <= 1.875 in Fig.2A,E).
+Note that we do not know if these phases are necessarily the absolute steady-states of the system in the limit of
+infinite time. The system might be trapped in a local meta-stable configuration due to dynamical barriers that would
+require unreasonably long simulations for them to escape. For example, in the regime of low force f and large binding
+strength w, the global minimum energy configuration should have all the CMC in a single pearled cluster, but during
+the merging of the pearled clusters into a single cluster they have to overcome bending energy barriers that hinder this
+process [32]. In other regimes, such as the elongated phase, we do not know if a stationary steady-state even exists,
+since the presence of active forces may induce a constantly changing configurations. In the SI section 2 we give a
+simple analytic calculation that gives reasonably well the transition line between the pearled and flat phases, which
+are the main stable condensed phases in this phase diagram (Figs.S2,S3).
+The evolution of a handful of chosen simulations are shown in Fig.3, showing flat, elongated-flat, elongated-pearling,
+and pearling phases. All the simulations begin in a disordered uniform distribution of the CMC on the spherical vesicle,
+but in all of them we find that buds form rather quickly (Fig.3B(i)-E(i)). In the budded phase this configuration
+simply remains stable and does not evolve significantly. It takes longer time for the larger clusters of the flat rim, arcs
+and pearls to form. The transition lines separating two different vesicle phases, obtained from our simulations, are not
+precise, and one can obtain either one of the vesicle shapes close to these lines (Fig.3A).
+To conclude, by exploring the f − w phase diagram, we demonstrate the competition between the protein binding
+which drives the formation of pearled clusters, and the active force that drives the formation of arc-like clusters at
+the edge of flat protrusion. This competition is highlighted in the new phases of vesicle morphologies that we found,
+namely the elongated two-arcs and the elongated-pearled phases. The pearling phase appears for large enough values
+of w, as follows also from the theory of self-assembly [5].
+V.
+FORCE-SPONTANEOUS CURVATURE PHASE DIAGRAM
+We now proceed to explore the interplay between the active force and the spontaneous curvature of the CMC in
+determining the morphology of the vesicle. We chose the parameters for a new set of simulations such that we are in
+the flat phase when the CMC are highly curved: ρ = 20%, κ = 28.5, w = 2. The resulting phase diagram is shown in
+Fig4A.
+We find several phases: budded phase, flat phase, elongated (arcs) phase and highly-elongated (tubes) phase. Here
+the boundaries between the different phases were drawn by eye, due to relative sparse scan over the parameters, and the
+self-evident boundaries (Fig.4A). In this parameter regime, we do not find any pearled phase, with the budded phase
+remaining stable due to the bending energy barrier that prevents buds merging (note that the bending modulus is
+larger here), and lower relative w. Similar to the force-binding strength system (Fig.2A), where the budded and pearled
+
+6
+phases exist for low active force, we also find that as the active force is increased the budded phase is destabilized to
+form the flat phase (Fig.4A).
+The flat phase is destabilized as the spontaneous curvature decreases due to the following mechanism: as C0 decreases
+the thickness of the rim cluster increases, which means that there are not enough CMC to complete a circular cluster
+around the edge of the flat shape. The morphology then changes into local arc-like clusters that pull the vesicle into
+elongated shapes. The elongation of these vesicles depends on the magnitude of the active force.
+The main feature of this phase diagram is the appearance of the highly-elongated tubular phase, where the entire
+vesicle is stretch into a several tubes that are pulled by CMC clusters at their tips. We can theoretically estimate
+the location of the phase transition line, above which a vesicle will become highly-elongated, by comparing the force
+exerted by the active CMC cluster and the restoring force of the emerging membrane tube due to bending (Fig.4B).
+A hemispherical CMC cap with radius r = 2/C0 minimizes the bending energy (Eq.1): E ∝
+�
+1
+r1 + 1
+r2 − C0
+�2
+, and
+maximizes the pulling force (since adding any more CMCs to the cluster, beyond the hemisphere, adds force in the
+opposite direction). The total pulling force of this hemispherical cluster is given by
+Fpull = f ·
+1
+2
+����
+geometry
+· 2π(2/C0)2
+s0
+�
+��
+�
+#vertices
+(7)
+where s0 is the area per vertex, and 2πr2/s0 is the number of CMC in the cluster. This hemispherical cap pulls a tube
+with the same radius from the main vesicle body. Note the extra factor of 1/2 due to the hemispherical shape of the
+cup, compared to the calculation done for a flat cluster of active proteins in [5].
+Assuming the restoring force is dominated by the bending energy of the membrane tube, it is given by (Eq.1) [5]
+Frestore = κ
+2
+2π
+(2/C0)
+(8)
+The highly elongated shape is initiated when the pulling force is greater than this restoring force, so the critical value
+is given by equating Eqs.7,8, which gives
+f = AC3
+0
+(9)
+where A is a constant determined by the constant parameters of the simulation (bending modulus and average area
+per vertex). Plotting this simple cubic relation in Fig.4A (blue solid line, where we fit the value of A), shows a good
+agreement with the observed boundary of the regime of the highly-elongated tubular shapes on the phase diagram.
+Note however that the shapes of the vesicles at the transition to the tubular phase are not always simple cylindrical
+tubes with hemispherical clusters at their tips (Fig.4A), as the analytic model assumes (Fig.4B).
+To conclude this section, we have shown that active CMC give rise to flat protrusions when they are highly curved.
+Tubular protrusions can form for weakly curved active CMC, while for highly curved CMC the active force needed
+to produce such slender protrusions increases extremely fast. In the next sections we explore how slender tubular
+protrusions can be produced with highly curved active proteins, by either changing the effective curvature of the CMC
+cluster, or by increasing the effective pulling force of the cluster.
+VI.
+MULTIPLE CURVATURE
+Real cells have many species of membrane protein of both convex and concave intrinsic curvature. While these
+membrane proteins have distinct curvatures, the effective curvature of a cluster of CMC may depend on the composition
+of the cluster, if it contains CMC of different spontaneous curvatures. In order to form clusters of mixed curvatures,
+we explore vesicles that contain CMC of different curvatures (concave and convex), that bind to each other equally. If
+the two CMC types bind only to their own kind, they form separate clusters on the vesicle, and their coupling with
+each other due to curvature alone is rather weak (see SI). The convex CMC maintain their activity, as in the previous
+sections, while the concave CMC is passive.
+In Fig.4C(i) we show snapshots of the steady-state shapes of the vesicles that contain 10% passive concave CMC, i.e.
+a CMC species with C−
+0 < 0 and f − = 0, in addition to convex CMCs (ρ+ = 10%, f = 0.5, and C+
+0 = 0.8). Both
+types of CMC have the same binding strength w = 2, which binds both types equally, leading to strong mixing of the
+two CMC types. For weakly curved concave CMC (C−
+0 = −0.001) the flat phase remains stable (Fig.4C(i6)), driven by
+the convex active CMC. As the concave CMC become more curved (Fig.4C(i) from right to left) the circular cluster at
+the rim of the flat shape breaks up, and highly elongated shapes appear (Fig.4C(i2,i3)).
+
+7
+These shapes can be explained by mapping the vesicles in Fig.4C(i) on the phase diagram (Fig.4A). For each
+simulation, we calculate the average spontaneous curvature of the CMC clusters: C0,eff =
+�
+C+
+0 + C−
+0
+�
+/2, as well as
+the average pulling force per CMC: feff = f/2. In Fig.4D we plot the typical dashed outline of the vesicles from
+Fig.4C(i) on the phase diagram according to these effective parameters C0,eff, feff. Most vesicles match the shape
+of the phase to which they are mapped in this way. The only exception is the vesicle with the most concave CMCs
+(and effective C0,eff = 0), which is not in the shape of highly-elongated tubes, as suggested by the calculated average
+parameters, but fits better the arcs phase. This phenomena is due to the concave CMCs phase-separating into internal
+”sacks” of concave-enriched clusters (Fig. 5Ai), which results in an effective removal of these concave CMC from
+determining the outer shape of the vesicle. To take this into account, we calculate the effective mean curvature of the
+CMCs while removing the concave CMC that are contained in the internalized sacks. This is done by including in the
+calculation of the average curvature only concave CMCs which are connected to at least one convex CMC. Using this
+revised average spontaneous curvature, we plot the locations of the vesicles on the phase diagram (full snapshots),
+and find that except for the most curved concave CMC (A1), the locations of the other vesicles is minimally affected.
+For the case A1, we find that indeed the formations of large sacks of concave CMC, push the vesicle into the arcs
+regime, compatible with its revised location on the phase diagram. The phase separation of the passive concave CMC
+into sacks is driven by the minimization of the total bending energy. The highly elongated tubes cost a high bending
+energy of the bare membrane: in Fig.4C(i2) the average bending energy of the bare membrane is ∼ 25KBT, while in
+the flatter shapes after the phase separation (Fig.4C(i1)) the average bending energy of the bare membrane drops to
+∼ 17KBT.
+In addition to the overall vesicle shape in the system of mixed curvatures, we are interested in the character of the
+CMC clusters. We find that concave and convex CMCs create complex mixed clusters with a ”coral”- or ”sponge”-like
+texture (Fig.4C and close up in Fig. 5Aii). The texture of these clusters seems similar to the membrane ruffles observed
+in [20] behind the leading edge of motile cells. In this work, the ruffles were attributed to the interaction between
+concave and convex membrane proteins, that are also involved in the recruitment of the actin polymerization. It was
+furthermore proposed in [20] that the pattern of ruffles observed in these cells is determined by the interaction between
+a concave membrane protein that inhibits the actin polymerization, which is recruited by the convex CMC. Motivated
+by this proposed mechanism, we explored the resulting shapes of the vesicle and CMC clusters when the concave
+CMCs inhibit the active force exerted by the convex CMCs. We tested two possibilities: inhibition that is proportional
+to the number of concave neighbors (Eq.4, Fig. 4C(ii)), and full inhibition with even one concave neighbor (Eq.5, Fig.
+4C(iii)). In both cases we find that the effective force is reduced, and that the resulting shapes correspond very well to
+their locations on the phase diagram (Fig.4D). The shapes obtained for full inhibition (Fig. 4C(iii)) are very similar
+to those for a vesicle with a mixture of passive CMC (see SI section 3, Fig.S4). Regarding the comparison with the
+experiments [20], we conclude from the model that the ruffle texture of the CMC clusters does not crucially depend
+on the inhibitory interaction between the two CMC types, but rather on their spontaneous curvatures and binding
+interaction.
+Let us now focus on the phase-separated sacks of highly curved concave CMC, which form within the mixed clusters
+(Fig.5). We observed that the neck that connects the sacks to the outer part of the cluster is much narrower when
+the convex CMC exert outwards protrusive forces (compare Fig.5(Aii) and (Cii)). We quantified the area of the
+narrowest part of the neck in Fig.5B,D for the active and passive convex CMC, respectively. The necks are naturally
+narrower for more highly curved concave CMC. The active convex CMC, which push the membrane outwards, exert
+an effective pressure force that squeezes the neck into a narrower radius. Note that for the narrowest necks, we are
+clearly at the limit of the spatial resolution of the simulation. We do not allow membrane fission, and therefore can
+not describe the process of detachment of such sacks as internalized vesicles [33], as occurs in cells during endocytosis
+and macropinocytosis [34].
+In Fig.5E,F we show the dynamics of the cluster formation, whereby a patch of passive concave CMC (blues) increase
+in size, while its rim is populated by active convex CMC (red). In these images the surrounding bare membrane is
+rendered to be invisible. These simulated dynamics resemble those calculated by another model of macropinocytic
+cups [35], which was based on reaction-diffusion dynamics coupled to active forces.
+Finally, when the two CMC types bind exclusively to their own kind, they form separate clusters, with very limited
+coupling between them (see SI section 4, Fig.S5).
+VII.
+FORCE ALIGNMENT
+As we show in Fig.4A, when the highly curved CMC induce a protrusive force that is directed at the outwards
+normal, we require an extremely large force in order for the highly elongated tubes to form. However, cells initiate
+slender, tube-like filopodia protrusions using highly curved membrane proteins, such as IRsp53 [8–10, 12–14], in
+agreement with theoretical calculations [36]. Within the slender filopodia in cells, the actin filaments are organized into
+
+8
+a cross-linked bundle, which efficiently directs the forces of all the polymerizing actin filaments along the protrusion’s
+axis. The actin nucleators at the tip of the filopodia are different from those at the leading edge of the flat lamellipodia
+[14, 21, 37], and initiate the growth of parallel actin filaments that form the bundle at the filopodia core. In our model,
+since we do not explicitly describe the actin filaments organization, we can only describe the effects of the bundling on
+the forces exerted on the membrane. To simulate this kind of bundling, we add an alignment term of a Vicsek-like
+interaction [22], which aligns the forces exerted on the membrane by each CMC that is bound in a cluster
+ˆfi = ˆni + s �
+r ˆnj
+|ˆni + s �
+r ˆnj|
+(10)
+The direction of the active force exerted on each CMC vertex i, ˆfi, is a weighted average of the local outwards normal
+direction (ˆni) and a contribution from all the vertices j within a distance r from the vertex i (and in the same connected
+cluster), with a weight of s.
+In Fig.6A we plot typical steady-state snapshots of the vesicle shape and CMC clusters, as function of the strength
+and range of the alignment interaction of Eq.10. We observe a rather sharp transition from flat shapes for short-range
+alignment (r < 10) to shapes containing thin tube-like protrusions for long-range alignment. As function of the
+parameter s we find only weak dependence: at very small values of s and r = 10, we find that the weak alignment
+is sufficient to increase the net pulling force of the CMC clusters, such that they break the circular rim of the flat
+shape (Fig.4B(iii)). The resulting shape, with ”paddle”-like protrusions, resembles the ”arcs” phase we found in Fig.4
+between the flat and tubes phases. At higher values of s this paddles phase changes to tubes, due to the stronger
+alignment leading to a larger net pulling force.
+At these larger interaction strength the vesicle produces thin, finger-like clusters with a small bulbous ”head” and an
+elongated ”sleeve” (Fig.6B(ii)). This shape allows the CMC to satisfy their spontaneous curvature, with a spherical tip
+that has a radius of rtip = 2/C0, while the sleeve has a thinner radius of rsleeve = 1/C0. Such a cluster configuration
+is stable due to the alignment of the active forces along the tube axis (perpendicular to the membrane along the
+sleeve, Fig.6B(ii)). Once these elongated clusters form, they exert a large pulling force on the remaining membrane,
+thereby pulling elongated bare-membrane tubes behind them. The membrane tube can have a larger radius than the
+radius of the tubular CMC cluster, as it balances the pulling force with the restoring force due to bending energy. The
+alignment of the forces means that the entire CMC cluster pulls along the protrusion axis (Fig.6B(ii)), exerting a
+much larger total force than was possible using purely normal forces at the tip, thereby forming tubes at values of
+the force per protein that are much lower than predicted by Eq.9 and Fig.4A. Smaller clusters that only contain the
+hemispherical tip (such as Fig.6B(i)), do not grow tube-like protrusions, even though their net pulling force is larger
+by up to a factor of 2 compared to normal-force CMC, due to alignment (compare Fig.6B(i) to Fig.4B and Eq.7).
+In Fig.7A we plot the time progression of a vesicle with aligned-force CMC. We observe that initially localized
+hemispherical buds form rapidly. These buds then coalesce to form larger clusters that grow into the typical shape
+of bulbous tip with a thinner tubular part behind it. The size and total force of each of the clusters are plotted as
+function of time, with each point size indicating the cluster size, and its y-axis coordinate giving its total active force,
+respectively. Note that clusters that contain patches of ”trapped” bare membrane undergo large force fluctuations
+(blue and yellow points, largest two clusters shown on the right of Fig.7A). These fluctuations arise from loss of global
+alignment over the entire CMC cluster, due to the bare membrane patch that allows the alignment to change, especially
+between the protrusion tip and the tubular part.
+In Fig.7B we compare the finger-like protrusions that form due to highly curved aligned-force CMC, with the tubular
+shapes that form due to weakly curved normal-force CMC (Fig.4A). The main difference is that the aligned-force
+protrusions are much more stable compared to the tubes formed by the much smaller clusters of normal-force CMC.
+The normal-force CMC undergo frequent fission and coalescence events, that correspond to tubes shrinking and
+regrowing. These differences in dynamics can be seen in the SI movies S1,S2.
+VIII.
+VESICLES WITH BOTH NORMAL AND ALIGNED-FORCE CMC, ADHERED TO A FLAT
+SUBSTRATE
+We simulate a vesicle with a mixture of CMCs (ρ = 5% of each type), both highly curved and convex, one type with
+normal force and the other with strongly aligned force (r = 15, s = 1). Our initial state of the vesicle is obtained by
+letting the vesicle spread over a flat adhesive substrate, while it contains only normal-force CMC. Then, at a time
+where the vesicle is partially spread (time 0 in Fig.8A), we convert randomly half of the CMC to aligned-force behavior.
+We chose an adhesion strength wad = 0.25 (Eq.2), which gives a well-spread vesicle when containing only normal-force
+CMC [18].
+In Fig.8A we show two simulations: one with universal binding between the normal and aligned-force CMCs, and
+the other with exclusive binding, such that normal-normal and aligned-aligned CMC bind to their own type exclusively.
+
+9
+In these examples we see that the rim cluster forms and drives strong spreading of the vesicle, as expected [18]. The
+aligned-force CMC (labeled in yellow) aggregate to form a single filopodia-like protrusion, which is able to recruit into
+it also normal-force CMC (labeled in red). This filopodia is highly dynamic, undergoing periods of attachment to the
+rim cluster, and to the adhesive substrate, as well as detachments from the substrate. The filopodia is observed to
+attach and detach from the rim cluster, leading to meandering motion. When the two types bind exclusively, they
+form segregated clusters along the rim, with the aligned-force clusters protruding slightly more outwards compared to
+the normal-force clusters. The dynamics of this system can be seen in SI movie S3.
+In Fig.8B we show the evolution of the segregation factor in the simulations, which is defined as
+S = 2 · Prob (CMC neighbor is of the same type) − 1
+(11)
+This segregation factor is equal to 0 for well-mixed clusters (where the probability to have a neighbor CMC of the
+same type is equal to 1/2), and it is equal to 1 for complete phase-separation of the types. In the main panels we give
+the segregation factor per cluster for the simulations shown in Fig.8A. The insets show the average of 4 independent
+simulations, which converge to a value of about S = 0.25 for the universal binding and S = 0.9 for the exclusive
+binding. In the universal case, we can see that the segregation is strongest in the filopodia, so the segergation factor
+for the large rim cluster jumps up or down, when the filopodia protrusion cluster attaches or detaches respectively.
+The protrusion cluster is more segregated (S ≈ 0.25), since its tip is enriched with aligned-force CMCs that drive its
+formation, while the rim cluster is nearly perfectly mixed (S ≈ 0). For the exclusive binding, the segregation is high
+both in the filopodia protrusion and in the rim cluster, so it does not change when the filopodia attach or detach from
+the rim.
+Note that along the adhered vesicle rim, the regions of aligned-force CMC protrude slightly more than the normal-
+force regions (Fig.8A, exclusive). This is enhanced when the normal-force CMC are disabled, so that they do not exert
+any active force, as shown in Fig.S6.
+IX.
+COMPARISON WITH EXPERIMENTS
+We can now compare some of our theoretical results to experimental observations, published and new.
+A.
+Membrane shapes driven by branched actin polymerization
+The active protrusive forces in our model are representative of actin polymerization activity near the cell membrane.
+When the actin polymerization is nucleated by proteins that induce branched actin networks (such as WASP, WAVE
+[38–40]), it is more natural to describe the force as a local pressure on the membrane, which therefore acts towards the
+outwards normal.
+The variety of shapes we obtained in our model (Figs.2,3), range from flat lamellipodia-like shapes, to cylindrical
+filopodia, to pearling-like protrusions. Some of these new elongated shapes can be compared with elongated global cell
+shapes, observed in living cells [41].
+B.
+Membrane shapes driven by bundled actin polymerization
+The introduction of alignment in the forces exerted by the CMC represents in our model the case of proteins that
+nucleate parallel actin bundles, such as VASP and Formins [10, 12, 21]. Our model has demonstrated previously that
+curved proteins that apply normal forces, induce the formation of flattened, lamellipodia-like protrusions [5, 18]. Here
+we show that curved proteins that induce polymerization of bundled actin (aligned-force in our model), naturally give
+rise to filopodia-like protrusions (Figs.6,7). This result fits the observation of highly curved convex-shaped proteins
+such as IRSp53 in both the leading edge of lamellipodia [7, 42] and in filopodia [2], where the actin organization is
+very different due to the different type of actin nucleators [38, 43]. Note that the combination of convex curvature, and
+nucleators of bundled actin, can form filopodia even without the explicit presence of I-BAR proteins (such as IRSp53)
+[44, 45].
+Note that protrusions of similar shapes to our aligned-force protrusions, which have a bulbous tip and a slender
+neck (Figs.6,7), were theoretically predicted to form by anisotropic CMC, even without force [46]. Similar thin tubes
+with bulbous tips are observed in cellular nanotubes [47] and in filopodia [48]. Since many curved proteins, such as
+IRSp53 are anisotropic in their intrinsic shape, it will be interesting to extend our work in the future to include such
+anisotropy.
+
+10
+Finally, our simulations of an adhered vesicle (Fig.8) indicate that the filopodia protrusions can undergo attachment
+and detachment from the substrate, resembling such motion observed in experiments [14]. In addition, when we
+mixed the aligned-force and normal-force CMC with exclusive binding between them, we obtained their segregated
+organization along the rim of the adhered vesicle. This is reminiscent of the observations of segregated regions of
+bundled actin and branched actin nucleators along the rim of cellular protrusions extending on adhered substrates
+[37, 49–51]. As in the experiments, the clusters of aligned-force CMC along the rim slightly protrude, as they exert
+a higher local force on the membrane rim, compared to the normal-force CMC. These small protrusions have been
+termed ”spikes” and ”microspikes” along the edge of lamellipodia in cells [45, 50, 52].
+In Fig.9 we show images illustrating the dynamics of filopodia in cells, using lattice light-sheet microscopy, which is
+capable of the high spatial and temporal resolution necessary to view the dynamics of the thin filopodia [53]. The
+curved membrane protein IRSp53 is fluorescently labeled in green (GFP-IRSp53), while the actin filaments are labeled
+in red (mCherry-lifeact). We observe in the experiments several features that are captured by the theoretical model:
+The filopodia are highly dynamic, both at the cell rim and along its dorsal surface (Fig.9A-D), as we also see in the
+model (Fig.8). The filopodia in the experiments migrate on the cell surface, merge with other filopodia, and undergo
+attachments and detachments from the surface (see SI movies 5-8), as we also see in the simulations (SI movies 3 and
+4). Our assumption in the model of uniform adhesion along the membrane, and along the filopodia, agrees with some
+observations [48, 54], and we can add more complex adhesion models in the future if needed. Note that in the cells we
+observe an additional retraction motion that is driven by myosin-II contractile forces, which we do not have in our
+current model.
+The highly curved IRSp53 is observed to aggregate strongly at the tips of the filopodia, while along the lower parts
+of the protrusion its aggregation is more fragmented (Fig.9E,F). This fits with the shapes that we obtained in the
+model (Fig.6B,8A). Furthermore, our simulations of mixtures of aligned-force and normal-force CMC indicate that
+while the aligned-force CMC are essential for forming the filopodia protrusions and occupy its tip region, there can be
+significant amount of normal-force CMC along the lower part of the filpodia. Since the normal-force CMC correspond
+to branched-actin nucleators, this result suggests that along the lower part of filopodia we may expect to find proteins
+such as WAVE, which are usually associated with the leading edge of the lamellipodia. This prediction is supported by
+some experimental observations of WAVE proteins [55], Arp2/3 complexes [56], and small lamellipodia-like protrusions,
+along filopodia shafts [57].
+C.
+Membrane shapes driven by mixtures of passive concave and active convex CMC
+Our mixtures of CMC of opposite curvatures (Figs.4C,5) gives rise to membrane shapes that resemble in their
+texture the ruffles observed in cells [20]. In addition, we find that when the passive concave component is highly
+curved, we observe a phase separation within the CMC clusters, whereby the concave CMC forms an internalized
+spherical invagination. These invaginations are then squeezed at their base by the active forces induced by the convex
+CMC, and the calculated membrane shape dynamics resembles the process of actin-dependent endycytosis [17, 58–60]
+and macropinocytosis [34, 61–63].
+Note that there is some experimental evidence that the internalized membrane, corresponding to our concave CMC
+region, do indeed contain concave membrane components, such as BAR proteins [64]. In addition, there are examples
+where the internalized region contains membrane components that interact with the convex proteins that recruit actin
+and form the squeezing at the narrow neck. In [59] the internalized activated integrins and associated proteins, bind to
+the actin which is nucleated at the neck, recruited there by IRSp53 (convex) proteins. In our model we show that such
+a direct interaction is necessary for robust formation of the internalized sacks with the recruited convex proteins at the
+neck.
+X.
+DISCUSSION
+In this study we greatly extend our theoretical understanding of the space of membrane shapes that are produced
+by curved membrane protein complexes (CMC) that exert active protrusive forces on the membrane [15]. We started
+by mapping the phases as function of the magnitude of the active force and attractive nearest-neighbor interaction
+strength of CMCs (Fig.2A), demonstrating the competition between these two terms: systems dominated by the
+binding interactions tend to have the equilibrium (pearled) shapes of the CMC clusters. The active forces tend to
+break-up the pearled clusters, and induce the formation of either elongated or flat pancake-like membrane shapes.
+Similarly we exposed the phase diagram in terms of the active force and the CMC spontaneous curvature (Fig.4A),
+whereby highly curved CMC induce flattened vesicle shapes, while less curved CMC induce elongated tubular shapes.
+Note that in these studies the protrusive force applied by each CMC is towards the local outwards normal.
+
+11
+Based on these results we further explored systems where highly curved active CMC could induce tubular protrusions.
+We tested two possible scenarios: In the first one, the effective curvature of the CMC cluster is reduced by mixing
+two types of CMC of opposite curvatures, such that a tubular protrusion forms with a rather flat CMC cluster at its
+tip (Fig.4C,D). In the second, the net protrusive force of the CMC cluster is increased by introducing an alignment
+interaction that tends to align the forces exerted by CMC that are bound within the same cluster (Fig.6). This
+alignment is found to stabilize long tubular CMC clusters, since the aligned active forces act along the tube axis and
+do not act to expand the tube, unlike the case of normal protrusive forces.
+We found that that mixtures of CMC of opposite curvatures, specifically passive concave and active convex, lead
+to formation of clusters with complex textures that resemble ruffles on cell membranes (Figs.4C,5). In addition, we
+found in these systems the formation of internalized invaginations, where the convex active CMC form a narrow neck,
+resembling endocytosis and macropinocytosis in cells.
+To conclude, the results presented in this work expand out theoretical understanding of membrane shapes and
+dynamics driven by intrinsic (spontaneous) curvature of membrane components and cytoskeletal active forces. Some
+of these shapes resemble observed membrane dynamics in living cells, suggesting that this coupling between curved
+membrane proteins and cytoskeleton forces gives rise to these biological phenomena. Many of the features that we
+found, such as the ruffles and the internalized invaginations by mixing CMC of different curvatures, remain to be
+further explored in future theoretical studies. In addition, future studies will explore the dynamics of the membranes
+when the CMC have anisotropic spontaneous curvature, and also in the presence of contractile forces.
+Conflict of Interest Statement
+The authors declare that the research was conducted in the absence of any commercial or financial relationships
+that could be construed as a potential conflict of interest.
+Author Contributions
+YR and NG developed the theoretical model; SP and AI developed the software; YR and NG conceived, designed
+and implemented the analysis of the model, and prepared the manuscript. YK and SS cultured and imaged the cells.
+The manuscript was edited by all the authors.
+Funding
+NG is the incumbent of the Lee and William Abramowitz Professorial Chair of Biophysics, and acknowledges support
+by the Ben May Center for Theory and Computation, and the Israel Science Foundation (Grant No. 207/22). AI and
+SM were supported by the Slovenian Research Agency (ARRS) through the Grants No. J3-3066 and J2-4447 and
+Programme No. P2-0232. YK and SS was supported by grants from the JSPS (KAKENHI JP20H03252, JP20KK0341,
+and JP21H05047) and JST CREST (JPMJCR1863) to SS and Takeda Science Foundation, a Grant-in-Aid for
+Challenging Exploratory Research (KAKENHI No. 20K20379), and JST CREST (JPMJCR1863) to YK.
+Acknowledgments
+NG is the incumbent of the Lee and William Abramowitz Professorial Chair of Biophysics. This research is made
+possible in part by the historic generosity of the Harold Perlman Family.
+Supplemental Data
+The SI text, figures, and movies are also available from the Box drive.
+Data Availability Statement
+The code for generating the simulations of this study can be found in the GitHub repository of YR, which is taken
+and modified off the GitBlit repository of SP. Reconstruction of the initial simulation folders are also available from
+
+12
+the Box drive. Further data or code requests will be happily fulfilled by YR.
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+
+13
+parameter
+units
+Fig.1,[5]
+Fig.2
+Fig.4A
+Fig.4C
+Fig.6
+Fig.7
+Fig.8
+f
+KBT/ℓmin
+1
+0 − 1.2 0 − 0.5
+0.5
+0.2
+0.2, 0.5
+0.5
+w
+KBT
+1
+0 − 4.8
+2
+2
+2
+2
+2
+κ
+KBT
+20*
+20
+28.5
+28.5
+28.5
+28.5
+28.5
+ρ
+1
+0%-20%
+10%
+20%
+10%, 10%
+20%
+20%,
+10%, 10%
+C0
+1/ℓmin
+1(0)
+1
+0.8
+-0.75 − 0, 0.8
+0.4
+0.4, 0.1
+0.8
+TABLE I: The values of the model parameters used in the simulations, in the different figures. The energy units are KBT = 1,
+which define the scale of f, w, κ, and the length units are ℓmin = 1, which define the scale of the vertex lattice, the force, and
+spontaneous curvature.
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+
+14
+FIG. 1: Phases of vesicle shapes driven by curved active CMC, as obtained in [5]. (A) Phase diagram in the temperature-density
+plane: mixed (gas), budded, and flattened (pancake). The gas phase is dominated by entropy, hence appears at either high
+temperatures or low densities. The pancake phase is dominated by having favorable binding and bending energy, where the
+active forces are all radial and stabilize the flat shape. This phase requires large stable CMC cluster, and so can only appear at
+low temperatures. The budded phase appears between the two other phases. At a CMC density that is lower than the minimal
+value needed for a closed circular rim, the pancake shape changes to B) a two-arcs phase, while when the CMC concentration is
+very high the pancake forms pearled extensions that contain the surplus CMC (C). There are two other phases in different
+regimes: (D) The pearling phase appears at higher CMC density, where most of the CMC aggregate into long necklace-like
+clusters that minimize the protein-protein binding energy (phase-separation of CMC), and (E) highly-elongated (tubular) phase
+for flat CMCs, where large CMC caps can exert a strong force that pulls out elongated tubes. Pictures taken from [5] Figs.4c,7d,
+and SI.
+
+(A)
+2
+1.5
+Mixed
+I
+Budded
+0.5
+Pancake
+4
+6
+8
+10
+12
+14
+16
+p[%]15
+FIG. 2: Force-binding strength plane. (A) Phase diagram as function of f and w, with: κ = 20, C0 = 1, and ρ = 10%. The
+different phases are indicated by their names, and a typical snapshot of the vesicle after a long simulation is shown. The
+transition lines between the phases were drawn according to the measures shown in the bottom panels. The gas and buds phase
+is separated by mean cluster size ⟨N⟩ = 1.5 (yellow solid line), as obtained from (B). The green line denotes the boundary of the
+flat phase, obtained approximately from a contour of the first (small) gyration eigenvalue λ2
+1, which is minimal for flat shapes
+(C). The light blue line denotes the boundary of the elongated shapes, roughly following a contour of the second (intermediate)
+gyration eigenvalue λ2
+2 (D). The transition line between the buds and mixed phases is given by a contour of CMC perimeter
+length (¯ℓp = 1.875, red solid line), extracted from (E). Finally, the pearling phase transition line (red dotted line) is drawn along
+the contour of small CMC perimeter length (¯ℓp = 0.375), from (E). In panels (B-E) we plot heatmaps of the following quantities:
+(B) Mean cluster size for clusters smaller than 10, ⟨N⟩ > 10 (C) first (small) gyration eigenvalue λ2
+1, (D) second (intermediate)
+gyration eigenvalue λ2
+2, (E) Mean CMC cluster perimeter length (excluding isolated CMC) ¯ℓ.
+
+Elongated
+Mixed16
+FIG. 3: Evolution of the MC simulation at four different points (B-E) denoted on the phase diagram (A) (Fig.2A). (B): f=0.8,
+w=1.6, (C): f=0.8, w=2.88, (D): f=0.8, w=3.20, and (E): f=0.4, w=4.16. The MC time-steps shown in the snapshots are: (i) 10,
+(ii) 50 (ii) and (iii) 200, and the final time-step (299) is shown on the phase diagram (A). At time (i), all simulations are in the
+budded state. At time (ii), arc and pearling clusters begin to form, favoring arcs for large forces and pearling for large binding
+strength. At time (iii), the vesicles are close to their final steady-state shapes. The flat simulation (B) generates several arcs
+in stage (ii), which coalesce to form a circular stable rim. The pearling simulation (E) generates pearling clusters (ii) which
+coalesce into a few larger clusters (coarsening). In contrast, the elongated simulations generate both arcs and pearled clusters at
+the intermediate stage (ii). These arc-like clusters are sufficient stretch the vesicle, even in (D), to give rise to the final elongated
+phase.
+
+Ci
+D i
+11
+Bi
+ili
+Ei
+ili17
+FIG. 4: (A) Phase diagram in the force-spontaneous curvature plane, using the parameters: ρ = 20%, κ = 28.5, w = 2. The
+different phases are denoted by their typical shapes, and the thin colored transition lines were drawn by hand (yellow, red and
+green). With no or weak force, we find a budded phase. As the force is increased, we find for the high spontaneous curvature
+the flat phase. As the spontaneous curvature is reduced, the flat phase is observed to give way to an ”arcs” phase, which is
+finally replaced by a highly-elongated tubular phase. The thick blue line denotes the theoretical calculation for the transition
+line that bounds the tubes phase, which is a cubic equation: f = AC3
+0 (Eq.9), where we use: A ≈ 10.6. This equation is derived
+from the force balance shown schematically in (B). (C) Typical steady-state snapshots of simulations with a mixture of CMC:
+active convex CMC (C0 = 0.8, f = 0.5, ρ = 10%), and passive concave CMCs (ρ = 10%) with different concave curvatures
+C−
+0 (along the x-axis). We show here three cases: i) no inhibition of the active convex CMC, ii) proportional inhibition, where
+the force exerted by a convex CMC is proportional to number of non-concave neighbors, and iii) disabling interaction, where
+the convex CMC do not exert any force if they have a concave neighbor. (D) Mapping of the vesicles shown in (C) to their
+respective locations in the force-spontaneous curvature phase diagram (A), using the average force and spontaneous curvature of
+the mixture (dashed outlines). The snapshots are shown at shifted locations, according to the effective curvature when we take
+into account the phase-separation of the concave CMC, into internalized sacks. These shifts in locations are most dramatic for
+1i,1ii,2ii,1iii,2iii (indicated by arrows), which places the vesicles in a phase which is appropriate for their shapes.
+
+2
+Flat
+SO
+ubes
+Arcs
+Buds
+Flat
+5i
+61
+2ii
+Tubes
+3ii
+4ii
+511
+61l
+1ili
+Arcs
+2ili
+5ili.
+Buds
+6ili
+3ill18
+FIG. 5: Mixed clusters can precipitate internal sacks, which are composed almost entirely of the concave (passive) CMC, when
+the concave CMCs are highly curved C0 = −0.75, −0.6. This is shown in A(i,ii),C(i,ii) for a system with and without active
+force, respectively. This internal sack is connected to the outside by a thin neck, or ”hole”, shown in A(iii) and C(iii). The
+cross-sectional area of the hole was measured by computing the area of the polygon made from the hole edge, which was picked
+by hand (vertices). A histogram of the simulated hole sizes is shown for the system with and without active force respectively
+(B,D). It is clear that the hole size is smaller in systems with force (B), such that it is in the limit of the simulation resolution.
+The holes are also larger as the spontaneous curvature of the passive concave CMC is smaller. The insets of B,D show typical
+examples of sacks (light blue nodes) connected to the outer part of the cluster (blue nodes) through the neck region (grey
+shading). (E) and (F): Snapshots showing the formation of a sack for the system with active force (A), from the initial random
+state. In (E) we show the cluster viewed from outside of the vesicle (where the bare membrane is rendered invisible), looking
+down on the patch that forms the sack, while in (F) we show the same process viewed from within the vesicle, where we see
+clearly the final invagination.
+
+Aii
+A ili
+curvature: -0.75
+curvature: -0.6
+Ai
+B
+10
+5
+20
+25C
+10
+15
+20
+25
+c i
+c ii
+cili
+LO
+15
+20
+25 0
+10
+15
+20
+50
+Hole area
+Hole area
+0219
+FIG. 6: (A) Vesicle steady-state shapes as function of the strength (s) and range (r) of the Vicsek-like alignment interaction
+(Eq.10)(ρ = 20%,κ = 28.5,C0 = 0.4,w = 2,f = 0.2). Interaction radius smaller than 10 leads to a flat phase. Above an interaction
+radius of 10, the system transitions from a flat to a tubes phase. In between the flat and elongated tubes phases, we find a phase
+with ”paddle”-like clusters. The tubular phase is characterized by CMC clusters that are mostly finger-like with a bulbous tip
+and a tubular sleeve, which often stretch a membrane tube behind them. (B) Snapshots of CMC clusters, with the active forces
+indicated by the arrows, and the colormap indicating the dot product of the local force and local outwards normal. In the tubes
+phase (s = 0.75, r = 15) we show in (i) an example of a hemispherical cluster, which is not able to pull an elongated protrusion.
+In (ii) (top) we show an example of a CMC cluster that contains a tubular sleeve, which increases the net pulling force above
+the threshold to pull a membrane tube. Note that at the sleeve base the alignment is weak due to the bare membrane boundary.
+This effect is also shown in (iii) (bottom), where a small patch of bare membrane is trapped between the cluster tip and the
+sleeve, leading to formation of two different alignment domains within the same cluster. Finally, in (iv) we show an example of
+the paddle cluster (s = 0.1, r = 10), where the weak alignment interaction gives rise to shapes similar to the regular arc-like
+clusters (Fig.4A), elongated by the non-normal force.
+
+Tube
+Paddle
+Flat
+iv
+120
+FIG. 7: (A) Dynamics of the formation of the tubular phase, driven by strong alignment interactions (ρ = 20%, κ = 28.5, C0 =
+0.4, w = 2, f = 0.2, s = 0.75, r = 15). Each circle represents a CMC-cluster at different MC time (x axis), the y axis represents
+the total force exerted by the cluster. The circle size represents the size of the CMC cluster (see sidebar). Color gives a persistent
+”identity” to each cluster, which last until fusion or fission. On the top right is a snapshot of the vesicle in the last time step.
+The four largest cluster are highlighted, and also shown on the right of the panel. Below the x-axis, we give snapshots of the
+vesicle. The rapid initial formation of buds is seen followed by slower fusion of clusters to form elongated protrusions. Two
+of the final large clusters, the bud and one of the elongated tube, are relatively stable, while the other two elongated clusters
+have wildly oscillating force. We can see on the right that the fluctuating cluster incorporates a few bare membrane vertices
+(Fig. 6B,iii). (B) The dynamics of tube formation due to aligned force with highly curved CMCs (top, s = 0.5, r = 30, f = 0.2,
+C0 = 0.4) compared to formation due to shallow (weakly curved) CMCs with normal force (bottom, f = 0.5, C0 = 0.1). The
+tubes of the latter are more dynamic and less stable than clusters of the former. This is also seen on the right panel, which
+shows the total force on the largest clusters, which is far less noisy for the former.
+
+50
+Sizes
+40
+2
+30
+4
+8
+20
+16
+32
+10
+64
+128
+256
+20
+40
+60
+80
+100
+120
+140
+timestep
+75
+50
+25
+0
+200
+400
+75
+50
+25
+0
+200
+40021
+FIG. 8: A: Initial progress of simulation with normal-force CMCs (red) and aligned-force CMCs (yellow), in universal binding
+(top) and type-exclusive binding (bottom), from the side and above (ρalign = 10%, ρnormal = 10%, κ = 28.5, C0 = 0.8, w = 2,
+f = 0.5, s = 0, 1, r = 15, wad = 0.25). CMCs in the rim drive the spreading of the vesicle on the surface, while some aligned-force
+CMCs aggregate into a bulb-and-sleeve cluster which drives the formation of a filopodia-like protrusion. This protrusion can
+attach to the rim cluster and then adhere to the substrate, while it can also detach from the substrate, and consequently also from
+the rim cluster. B: Evolution of the segregation factor in the simulations (Eq.11). The colored lines give the segregation factor
+for each cluster, with the cluster size indicated by the line thickness. In the inset we give the average of the total segregation
+factor for 4 independent simulations. In the universal binding simulation we can see the fliopodia-like cluster repeatedly attach
+and detach from the rim cluster. The rim cluster is mostly mixed for this case, while the protrusion is much more segregated, as
+its tip is enriched with aligned-force CMCs.
+
+0.4
+1.00
+1.00
+0.2
+0.75
+0.75
+factor
+0.0
+0.8
+0.50
+time 0
+50
+100　150　200　250
+0.50
+segregation t
+0.6 
+detachments -
+0.25
+0.25
+0.4
+0
+50
+100
+150
+200
+250
+time
+0.00
+0.00
+Rim cluster
+0.25
+-0.25
+-0.50
+-0.50
+0
+50
+100
+150
+200
+250
+0
+50
+100
+150
+200
+250
+time
+time22
+FIG. 9: Movements of IRSp53-localized cellular protrusions. (A, B) The adhesion (A) and apical (B) plane section of the
+three-dimensional images of an IRSp53-knockout U251 glioblastoma cell expressing GFP-IRSp53 (green) and mCherry-lifeact
+(red). In (A) and (B), the region for the ∼ 2 µm thick xz section projection is indicated by the cyan dotted rectangle. (C) The
+xz section of the region of (A). The white lines indicate the plane in (A,B). The yellow line, which was set in the proximity
+of the surface plane of the cell, indicates the line for the kymograph. (D) The kymograph of the cell surface as indicated in
+the yellow line in (C), along with the annotation of the representative motion of the IRSp53. (E-F) The xy and xz sections at
+the regions that are marked in (A,B), from the periphery (E), the middle (F), and the center (G). The plane parallel to the
+plasma membrane was sectioned and the regions that were projected xy and xz sections each others were marked in cyan dotted
+rectangles. Arrows indicate the protrusions. The scale bar, 10 µm (A-D), 2 µm (E-G), and 50 sec (D).
+
+xz section
+apical plane
+moving to
+the center (E)
+moving to
+the periphery (F)
+adhesion plane
+apical plane
+merging (G)Supplementary Material
+S-1.
+CALCULATION OF THE PERIMETER OF CMC CLUSTERS
+The CMC-bare membrane boundary is measured by summing the dual of the edges between the cluster and bare
+membrane. These are the edges in the voronoi lattice, connecting the mid-section of each edge to the circumcenter of
+the adjacent triangles i.e. the center of the inscribing circle (see figure S-1). Partitioning each triangle between its
+vertices is already used in the calculation of the curvature [1].
+S-2.
+ANALYTICAL CALCULATION OF THE FLAT-PEARLING PHASE TRANSITION LINE
+We can make a rough analytical estimation for the flat-pearling transition by equating the active work and energy of
+the flat phase from a mixed phase to the energy of the pearling phase (Fig. S-2). In the flat phase, moving the active
+CMCs outwards from the radius of the sphere rp to the larger radius of the flattened disc rf results in work. The
+pearling phase has binding advantage because all CMC vertices are connected, with −w per edge, while the flat rim
+has large interface (boundary perimeter length) where CMCs vertices neighbor bare membrane vertices, whose edge
+does not contribute. The pearling phase has a bending disadvantage due to the bare membrane body, which is roughly
+spherical with an energy of 8πκ, compare to the flat phase where the bare membrane is in two flat discs with no
+bending energy (both the pearling and rim clusters are curved to fit the CMCs, so they do not have bending energy).
+− (rf − rp) F = −w (χp − χf) + 8πκ
+(S-1)
+The radius difference ∆r = rf − rp (Fig. S-2), and the number of CMC-CMC bonds χp, χf in the pearling and
+flat phases respectively, are dependant on the geometry of the phases, so they should be very weakly dependant on
+the specific model parameters. Therefore ∆r and χp − χf do not depend on w, f, κ, and we end up having a linear
+relation between f and w along the transition line in the f, w phase diagram. In the force-binding strength (f − w)
+system, we take the values for these geometric quantities from simulations and draw the resulting line on the phase
+diagram (Fig. S-3, green line), which qualitatively matches the behavior of the transition observed in the simulations.
+S-3.
+MIXED CURVATURE CMC CLUSTERS
+The concave and convex CMCs generate a wavelike pattern, but analyzing it in terms of wavenumber is difficult,
+since the clusters are part of an irregular, triangulated surface. The undulations of the CMCs in the mixed clusters are
+essentially independent of C0, and f, as shown in Fig.S-4. Note that we are at the limit of the mesh resolution for
+these undulations. We have yet to be able to compare this to the experimental results in [2].
+S-4.
+MIXED CURVATURE WITH EXCLUSIVE BINDING
+The mixed curvature system (Fig. 4c in the main text) was also simulated using exclusive binding, i.e. only
+same-curvature CMCs bind together (Fig.S-5). The result is that the two CMCs types form separated aggregates,
+with the active convex CMCs aggregating along the rim and forming the flat phase. The passive concave CMC form
+separated clusters of different shapes, depending on their spontaneous curvature. Highly concave CMCs (C−
+0 ≤ −0.45)
+aggregate into internal pearling clusters, that do not affect the flat global phase. The shallower concave CMCs
+(C−
+0 ≥ −0.3) aggregate into large, shallow bowl-like patches.
+In some cases, these concave aggregates are able to form with convex CMC along their rim, since their curvatures
+complement each other (see for example at C−
+0 = −0.3). Since the convex active CMC along the rim of the concave
+cluster apply protrusive forces, they end up forming together a ”cup”-like protrusion. When the force is inhibited,
+this aggregation occurs, but it is not elongated as a protrusion (compare ”None” with ”Disable” at C−
+0 = −0.3 in
+Fig.S-5). Other than that, inhibition doesn’t appear to significantly affect the results in Fig.S-5, since there is no
+significant contact between the two CMC types. These shapes, in the form of open bowls, resemble early stages of
+macropinocytosis [3, 4], but do not evolve to induce closure of the ”mouth”, as we observed when the convex and
+concave CMC had direct interactions (Fig.5 in the main text).
+arXiv:2301.13055v1  [cond-mat.soft]  30 Jan 2023
+
+2
+parameter
+units
+Fig.S-3
+Fig.S-4
+Fig.S-5
+Fig.S-6
+movie 1,2 movie 3,4
+f
+KBT/ℓmin 0 − 1.2
+0.5, 0
+0.5
+0.5
+0.2, 0.5
+0.5
+w
+KBT
+0 − 0.48
+2
+2
+2
+2
+2
+κ
+KBT
+20
+28.5
+28.5
+28.5
+28.5
+28.5
+ρ
+1
+10%
+10%, 10%
+10%, 10%
+10%, 10%
+20%
+10%, 10%
+C0
+1/ℓmin
+1
+-0.6 − 0, 0.8 -0.75 − 0, 0.8
+0.8
+0.4,0.1
+0.8
+TABLE I: The values of the model parameters used in the simulations in the different figures. The energy units are KBT = 1,
+which define the scale of f, w, κ, and the length units are ℓmin = 1, which define the scale of the vertex lattice, the force, and
+spontaneous curvature.
+S-5.
+VESICLES WITH BOTH NORMAL AND ALIGNED-FORCE CMC, ADHERED TO A FLAT
+SUBSTRATE
+In Fig.S-6 we show the dynamics of the vesicle that contains the mixture of aligned-force (yellow) and normal-force
+(red) CMC, which have exclusive binding interactions between them (see Fig.8 in the main text). At time t = 250
+we turned off the normal-force CMC, keeping only the aligned-force CMC active. We find that the adhered area
+shape changes, with the rim regions that contain the curved passive (red) CMC retract into the vesicle, while the
+aligned-force regions protrude more prominently along the adhered rim.
+Movies
+• Movie-S1 Aligned-force simulation of the formation of filopodia-like tubular protrusions (corresponding to
+Fig.7B), with parameters κ = 28.5, f = 0.2, w = 2, C0 = 0.4, ρ = 20%, s = 0.5, r = 30
+• Movie-S2 Normal force simulation, in the regime of tubes shapes (corresponding to Fig.7B), with parameters
+κ = 28.5, f = 0.5, w = 2, C0 = 0.1, ρ = 20%
+• Movie-S3 Adhered, universal-binding between normal-force CMCs (red) and aligned-force CMCs (yellow),
+corresponding to Fig.8A. Parameters used: κ = 28.5, f = 0.5, w = 2, wad = 0.25, C0 = 0.8, ρn = 10%, ρa =
+10%, s = 1, r = 15
+• Movie-S4 Adhered, exclusive-binding between normal-force CMCs (red) and aligned-force CMCs (yellow),
+corresponding to Fig.8A. Parameters used: κ = 28.5, f = 0.5, w = 2, wad = 0.25, C0 = 0.8, ρn = 10%, ρa =
+10%, s = 1, r = 15
+• Movie-S5. The 3D movie of the cell in Figure 9A
+• Movie-S6. The movie of the XY and XZ section for Figure 9E
+• Movie-S7. The movie of the XY and XZ section for Figure 9F
+• Movie-S8. The movie of the XY and XZ section for Figure 9G
+[1] G. Gompper and D. M. Kroll, in Statistical Mechanics of Membranes and Surfaces (WORLD SCIENTIFIC, 2004), pp.
+359–426.
+[2] E. Sitarska, S. D. Almeida, M. S. Beckwith, J. Stopp, Y. Schwab, M. Sixt, A. Kreshuk, A. Erzberger, and A. Diz-Mu˜noz,
+bioRxiv p. 2021.03.26.437199 (2021).
+[3] D. M. Veltman, T. D. Williams, G. Bloomfield, B.-C. Chen, E. Betzig, R. H. Insall, and R. R. Kay, Elife 5, e20085 (2016).
+[4] R. R. Kay, Cells & Development 168, 203713 (2021).
+
+3
+FIG. S-1: Sketch of the boundary of connected clusters: for each edge between the cluster and the outside, a line is drawn from
+the middle to the center each of the adjacent triangles. We ignore the single-clusters (dashed line)
+FIG. S-2: Schematic description of the transition between flat and pearling phases, from an initially mixed, spherical phase (at
+the center). Bare membrane is in white, and CMCs in red, and mixed composition in pink. The flat transition result in all
+CMCs moving from the surface of the sphere to the rim of a flat disc, which has a larger radius ∆r. Due to active force f,
+this generates work W = −f∆r. The bending energy of the CMCs on the rim and in the pearling clusters is assumed to be
+approximately 0, but the spherical body of bare membrane in the pearling phase has a bending energy of a closed sphere: 8πκ,
+while it is zero for the flat discs of bare membrane in the flat phase (since they are flat). Finally, the number of CMC-CMC
+bonds in the pearling phase χp is larger than in the flat phase χf, since in the flat phase it is reduced due to the large boundary
+between the rim cluster the the flat bare membrane discs.
+
+f△r
+8πK4
+FIG. S-3: Phase diagram of the force-binding strength system, with an analytically-derived transition line for the pearling-flat
+transition (green line, Eq.S-1).
+FIG. S-4: The undulation of a CMC cluster with (A) highly concave (−0.6) active CMC (B) with shallow concave (−0.001)
+CMC and disabled force. The size and shape of the clusters is very different, but the peaks and troughs patterning due to CMC
+shape is at the limit of the mesh resolution for both.
+
+20 
+12 
+longai
+04
+96
+88
+Buds
+80
+72
+64 
+56
+48
+earlino
+40
+32
+24
+16
+08(A)
+(B)5
+FIG. S-5: Active convex and passive concave system (red and blue, respectively), with binding between same type only. As in
+the universal binding case, the suppressive and disabling inhibition do not have any strong effects, since the types are separated.
+Simulations with C−
+0 ≤ −0.3 are draw semi-transparent. In all cases, the convex CMCs aggregate in a rim, making the vesicle
+flat, and concave CMCs aggregate in pearling for C−
+0 < −0.3, bowl-like patches for C−
+0 > −0.3, and both for C−
+0 = −0.3.
+
+6
+FIG. S-6: Overview of an adhered vesicle with a mixture of aligned-force (yellow) and normal-force (red) CMC, which have
+exclusive binding interactions between them (see Fig.8 in the main text). At time t = 250 the force is disabled for the
+normal-force CMCs, leaving only the aligned-force CMCs active. The original simulation is given on the top (times 0 − 100),
+and the simulation after the normal-force has been disabled is at the bottom.
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+An improved hybrid regularization approach for extreme
+learning machine
+Liangjuan Zhou
+School of Mathematics, Hunan University
+Changsha, China
+Wei Miao∗
+[email protected]
+School of Mathematics, Hunan University
+Changsha, China
+ABSTRACT
+Extreme learning machine (ELM) is a network model that arbitrarily
+initializes the first hidden layer and can be computed speedily.
+In order to improve the classification performance of ELM, a ℓ2
+and ℓ0.5 regularization ELM model (ℓ2-ℓ0.5-ELM) is proposed in
+this paper. An iterative optimization algorithm of the fixed point
+contraction mapping is applied to solve the ℓ2-ℓ0.5-ELM model. The
+convergence and sparsity of the proposed method are discussed
+and analyzed under reasonable assumptions. The performance of
+the proposed ℓ2-ℓ0.5-ELM method is compared with BP, SVM, ELM,
+ℓ0.5-ELM, ℓ1-ELM, ℓ2-ELM and ℓ2-ℓ1ELM, the results show that the
+prediction accuracy, sparsity, and stability of the ℓ2-ℓ0.5-ELM are
+better than the other 7 models.
+CCS CONCEPTS
+• Mathematics of computing → Convex optimization; • Com-
+puting methodologies → Regularization.
+KEYWORDS
+High-dimensional data, Sparsity, Hybird regularization, Dimension-
+ality reduction
+ACM Reference Format:
+Liangjuan Zhou and Wei Miao. 2022. An improved hybrid regularization
+approach for extreme learning machine. In 2022 4th International Conference
+on Advanced Information Science and System (AISS 2022), November 25–27,
+2022, Sanya, China. ACM, New York, NY, USA, 7 pages. https://doi.org/10.
+1145/3573834.3574501
+1
+INTRODUCTION
+Feedforward neural networks(FNNs), as one of the most frequently
+used neural networks which can be defined mathematically as:
+𝐺𝑁 (𝑥𝑖) =
+𝑁
+∑︁
+𝑖=1
+𝛽𝑖𝑔(⟨𝜔𝑖,𝑥𝑖⟩ + 𝑏𝑖),
+where 𝑥𝑖 = (𝑥𝑖1,𝑥𝑖2, . . . ,𝑥𝑖𝑝) ∈ R𝑝 is the input, 𝑏𝑖 is the bias and 𝑔
+is the activation function. ⟨𝜔𝑖,𝑥𝑖⟩ = �𝑝
+𝑗=1 𝜔𝑖𝑗𝑥𝑖𝑗 is the euclidean
+∗Both authors contributed equally to this research.
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for components of this work owned by others than ACM
+must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
+to post on servers or to redistribute to lists, requires prior specific permission and/or a
+fee. Request permissions from [email protected].
+AISS 2022, November 25–27, 2022, Sanya, China
+© 2022 Association for Computing Machinery.
+ACM ISBN 978-1-4503-9793-3/22/11...$15.00
+https://doi.org/10.1145/3573834.3574501
+inner product, 𝜔𝑖 = (𝜔𝑖1,𝜔𝑖2, . . . ,𝜔𝑖𝑝) ∈ R𝑝 are the weights con-
+necting the input and the 𝑖-th hidden node, and 𝛽𝑖 ∈ R are the
+weights connecting the 𝑖-th hidden and output node. In terms of
+the traditional learning algorithm of FNNs, all parameters in the
+network need to be adjusted based on specific tasks. A classical
+learning method is the backpropagation (BP) algorithm, which is
+mainly solved by gradient descent:
+min
+𝜔𝑖,𝛽𝑖,𝑏𝑖
+𝑛
+∑︁
+𝑖=1
+∥𝑡𝑖 − 𝐺𝑁 (𝑥𝑖)∥2
+2,
+where (𝑥𝑖,𝑡𝑖)(𝑖 = 1, 2, . . . ,𝑛) denotes the training samples. How-
+ever, a randomized learner model, different to the traditional learn-
+ing of FNNs, called as Extreme learning machine(ELM) and related
+algorithms were proposed by Huang[10]. In the ELM model, 𝜔𝑖 and
+𝑏𝑖 are randomly assigned without training, so only 𝛽𝑖 needs to be
+trained. Set T = [𝑡1,𝑡2, . . . ,𝑡𝑛] and
+H =
+
+𝑔(⟨𝜔1,𝑥1⟩ + 𝑏1)
+. . .
+𝑔(⟨𝜔𝑁,𝑥1⟩ + 𝑏𝑁 )
+...
+. . .
+...
+𝑔(⟨𝜔1,𝑥𝑛⟩ + 𝑏1)
+. . .
+𝑔(⟨𝜔𝑁,𝑥𝑛⟩ + 𝑏𝑁 )
+
+,
+(1)
+once the input weights and biases are specified randomly with uni-
+form distribution in [−𝑐,𝑐], the hidden output matrix remains un-
+changed during the training phase. Accordingly, the output weights
+could be written by utilizing the least squares method:
+min
+𝛽 ∈R𝑁
+�
+∥H𝛽 − T∥2
+2
+�
+,
+(2)
+the solution to model (2) could be written as 𝛽 = H†T, where H†
+is the Moore–Penrose generalized inverse of hidden output matrix
+H[14].
+The theoretical basis for the general approximation capability of
+ELM networks has been proposed and established by Igelnik[11] ,
+where the range of randomly allocated input weights and biases
+are data related and assigned in a constructive mode. Consequently,
+the scope of parameters in the algorithm implementation should
+be carefully estimated for diverse datasets. On the other hand,
+considering the sparsity of the output parameter 𝛽 for many high-
+dimensional data, Cao et al.[4] proposed a ℓ1 regular ELM model
+based on the sparsity of the ℓ1 regularization term, which takes the
+following form:
+min
+𝛽 ∈R𝑁
+� 1
+2 ∥H𝛽 − T∥2
+2 + 𝜆∥𝛽∥1
+�
+,
+(3)
+where 𝜆 > 0 is a regularization parameter and 𝛽 is the output
+coefficient calculated by iteration. This model is called the Lasso
+model, and has been studied by many scholars in recent years [15].
+arXiv:2301.01458v1  [math.OC]  4 Jan 2023
+
+AISS 2022, November 25–27, 2022, Sanya, China
+Zhou and Miao.
+For the model (2), Fan et al. [8] added a ℓ0.5 regularization term
+to the ELM model, based on the solution generated by ℓ0.5 is sparser
+than the ℓ1 regularization term [16], and the model is defined as
+follows:
+min
+𝛽 ∈R𝑁
+� 1
+2 ∥H𝛽 − T∥2
+2 + 𝜆∥𝛽∥0.5
+�
+,
+(4)
+where 𝜆 > 0 is a regularization parameter, the model can be solved
+by the iterative semi-threshold algorithm [16].
+The other regularization model for model (2) was about the ℓ2
+regularization term (ℓ2-ELM) [5]:
+min
+𝛽 ∈R𝑁
+� 1
+2 ∥H𝛽 − T∥2
+2 + 𝜇∥𝛽∥2
+2
+�
+,
+(5)
+where 𝜇 is a regularization parameter, and when the expression
+H𝑇 H+𝜇I is invertible after choosing the parameter 𝜇, then the solu-
+tion of the model (5) can be written as 𝛽 = (H𝑇 H + 𝜇I)−1I)−1H𝑇 T.
+Hai et al.[9] proposed a ℓ2-ℓ1-ELM hybrid model by integrating
+the sparsity of the ℓ1 regularization term and the stability of the ℓ2
+regularization term as follows:
+min
+𝛽 ∈R𝑁
+� 1
+2 ∥H𝛽 − T∥2
+2 + 𝜆(𝛾∥𝛽∥1 + 𝜀∥𝛽∥2
+2)
+�
+,
+(6)
+where 𝜆 ≥ 0, 𝛾 ≥ 0 and 𝜀 ≥ 0 are regularization parameters. In-
+spired by the ℓ2-ℓ1-ELM model, according to Xu et al.[17], they
+found that the sparsity of the solution of the ℓ𝑝 (𝑝 ∈ (0, 1)) regular-
+ization term: when 0 < 𝑝 < 0.5, there is no significant difference in
+the sparse effect of ℓ𝑝; when 0.5 < 𝑝 < 1, the smaller 𝑝, the better
+the sparse effect, so the ℓ0.5 regularization term can be used as a
+representative element of ℓ𝑝 (𝑝 ∈ (0, 1)); Therefore, we propose the
+ℓ2-ℓ0.5-ELM model by combining the stability of ℓ2 regularization
+term and the sparsity of ℓ0.5 which is sparser than ℓ1, the new
+model is described as:
+min
+𝛽 ∈R𝑁
+� 1
+2 ∥H𝛽 − T∥2
+2 + 𝜆(𝛾∥𝛽∥0.5 + 𝜀∥𝛽∥2
+2)
+�
+,
+(7)
+where the parameters have the same meaning as the expression
+of (6). The thought of adding ℓ0.5 and ℓ2 penalties simultaneously in
+the optimization model could be found in classification [2, 6]. This
+study mainly establishes an iterative algorithm and studies some
+properties of randomized learner model as Hai[9]. In particular, we
+integrate the features of ELM and propose an iterative strategy for
+solving the hybrid model (7). The main contributions of this paper
+can be summarized as follows:
+(i) The whole model is a non-convex, non-smooth and non-
+Lipschitz optimization problem due to the existence of ℓ0.5 norm.
+We propose a new algorithm called as an ℓ2-ℓ0.5-ELM algorithm.
+This algorithm is proved to be effective by analyzing the sum mini-
+mization problem of two convex functions with certain characteris-
+tics.
+(ii) The key theoretical properties such as convergence, sparsity
+are derived to guarantee the feasibility of the proposed method.
+(iii) Numerous experiments were carried out, including some
+UCI datasets collected from experts and intelligent systems fields,
+gene datasets and ORL face image datasets. Experimental results
+show that the better performance of the proposed ℓ2- ℓ0.5-ELM
+algorithm.
+The rest of this paper is organized as follows. Section 2 reviews
+some basic concepts and theories. Section 3 demonstrates the itera-
+tive method by a fixed point equation and proposes a algorithm for
+ℓ2 - ℓ0.5-ELM model. In Section 4, some theoretical results about
+convergence and sparsity are analyzed. In Section 5, experimental
+results on UCI datasets, gene datasets and ORL face image datasets
+are shown. The conclusion is drawn in Section 6.
+2
+PRELIMINARIES
+In this section, we present some fundamental concepts and con-
+vex optimization theorems primarily. Initially, it is about the half-
+thresholding function[16]. 𝒫(𝜆,𝑡) : R → R, 𝜆 > 0, which can be
+written as:
+𝒫(𝜆,𝑡) =
+� 2
+3𝑡
+�
+1 + cos
+� 2(𝜋−𝜙 (𝑡))
+3
+��
+|𝑡| > 3
+4𝜆
+2
+3
+0
+|𝑡| ≤ 3
+4𝜆
+2
+3
+,
+(8)
+where 𝜙(𝑡) = arccos
+�
+𝜆
+8 ( |𝑡 |
+3 )− 3
+2
+�
+, 𝜋 = 3.14, and then the corre-
+sponding half-thresholding operator half(𝜆, 𝛽) : R𝑁 → R𝑁 acts
+component-wise as:
+[half(𝜆, 𝛽)]𝑖 = 𝒫(𝜆, 𝛽𝑖).
+(9)
+Next, we introduce one key characteristic of the half-thresholding
+operator [7, 16]:
+∥half(𝜆,𝑡) − half(𝜆,𝑡 ′)∥ ≤ ∥𝑡 − 𝑡 ′∥.
+(10)
+Another crucial notion of convex optimization is the proximity
+operator [12]:
+prox𝜑𝛽 = arg min
+�
+∥𝑢 − 𝛽∥2
+2 + 𝜑(𝑢)
+�
+,
+where𝜙 is a real-valued convex function on R𝑁 . A primary property
+of the proximity operator is drawn in Proposition 1[7], which will
+be utilized to prove our major result.
+Proposition 1. Let 𝜑 be a real-valued convex function on R𝑁 .
+Suppose 𝜓 (·) = 𝜑 + 𝜌
+2 ∥ · ∥2
+2 + ⟨·,𝑢⟩ + 𝜎, where 𝑢 ∈ R𝑁 , 𝜌 ∈ [0, ∞),
+𝜎 ∈ R, then
+prox𝜓 𝛽 = prox𝜑/(1+𝜌) ((𝛽 − 𝑢)/(1 + 𝜌)).
+(11)
+3
+SOLUTION: FIXED POINT ITERATIVE
+ALGORITHM FOR THE MODEL
+For the ELM, the output matrix H is a bounded linear operator from
+R𝑁 to R𝑚 owing to the activation function 𝑔(·) ∈ (0, 1), which is
+finite. In order to further improve the accuracy and sparsity, we
+employ the regularization model (7) to estimate the output weights
+of the network. We define concisely as:
+𝑝𝛾,𝜀 = 𝛾∥𝛽∥0.5 + 𝜀∥𝛽∥2
+2,
+where 𝜀, 𝛾 ≥ 0, 𝑝𝛾,𝜀 : R𝑁 → [0, ∞). Then the model (7) can be
+redefined as
+min
+𝛽 ∈R𝑁
+� 1
+2 ∥H𝛽 − T∥2
+2 + 𝜆𝑝𝛾,𝜀
+�
+.
+(12)
+Furthermore, we introduce the following Lemma and Theorem
+which will be utilized to solve our model:
+
+An improved hybrid regularization approach for extreme learning machine
+AISS 2022, November 25–27, 2022, Sanya, China
+Lemma 1. For all 𝜆 > 0 and 𝛽 ∈ R𝑁 ,the half-thresholding operator
+(8) can be described as:
+half(𝜆, 𝛽) = arg min
+𝑢
+� 1
+2 ∥𝑢 − 𝛽∥2
+2 + 𝜆∥𝑢∥0.5
+�
+.
+Lemma 2. For all 𝜆
+>
+0,𝛾
+≥
+0,𝜀
+≥
+0 and 𝛽
+∈
+R𝑁 ,
+half(
+𝜆𝛾
+1+2𝜀𝜆,
+𝛽
+1+2𝜀𝜆 ) is the proximity operator of 𝜆𝑝𝛾,𝜀 (𝛽).
+Theorem 1. Let 𝜆 > 0, 𝛾 ≥ 0, 𝜀 ≥ 0 and 𝛿 ∈ (0, ∞). Then 𝛽 is
+a minimizer of function (12) if and only if it meets the fixed point
+equation:
+𝛽 = half
+�
+𝛿𝜆𝛾
+1 + 2𝜀𝜆𝛿 , (I − 𝛿H𝑇 H)𝛽 − 𝛿H𝑇 T
+1 + 2𝜀𝜆𝛿
+�
+,
+(13)
+where the unit operator I : R𝑁 → R𝑁 , the definition of H is shown
+in (1), and H𝑇 represents the adjoint of H.
+Moreover, from the property of the proximity operator, we can
+drive a precise statement for the Lipschitz constant of a contractive
+map and the corresponding theorem as follows.
+Theorem 2. Set 𝜆 > 0,𝛾 ≥ 0,𝜀 ≥ 0 and 𝛿 ∈ (0, ∞). Suppose that
+there exist two positive constants 𝜅0 and 𝜅, such that the norm of the
+output matrix H shown in (1) of the hidden layer is finite by them,
+namely 𝜅0 ≤ ∥H𝑇 H∥2 ≤ 𝜅, Thus 𝛽 is a minimizer of (12) if and
+only if it is a fixed point of the Lipchitz map Γ : R𝑁 → R𝑁 , that is,
+𝛽 = Γ𝛽 where
+Γ𝛽 = half
+�
+𝛿𝜆𝛾
+1 + 2𝜀𝜆𝛿 , (I − 𝛿H𝑇 H)𝛽 + 𝛿H𝑇 T
+1 + 2𝜀𝜆𝛿
+�
+.
+(14)
+Selecting𝛿 =
+2
+𝜅0+𝜅 , the Lipschitz constant is finite by𝑞 = 1− 2𝜅0
+𝜅 + 𝜅0
+≤
+1. In particular, if 𝜅0 > 0, we can get Γ is a contractive map.
+Theorem 1 and Theorem 2 illustrate that the problem of ℓ2-ℓ0.5-
+ELM can be described as a fixed point algorithm. Furthermore, the
+next theorem will introduce the iterative procedure of the ℓ2-ℓ0.5-
+ELM.
+Theorem 3. Suppose that 𝜅0 and 𝜅 are positive constants, such
+that the norm of the output matrix H shown in (1) of the hidden
+layer is finite by them, namely, 𝜅0 ≤ ∥H𝑇 H∥2 ≤ 𝜅, and the sequence
+{𝛽}∞
+𝑙=0 ⊆ R𝑁 is described iteratively as
+𝛽𝑙 = half
+�
+𝛿𝜆𝛾
+1 + 2𝜀𝜆𝛿 , (I − 𝛿H𝑁 H)𝛽𝑙−1 − 𝛿H𝑇 T
+1 + 2𝜀𝜆𝛿
+�
+,
+(15)
+where 𝑙 = 1, 2, 3, . . . , 𝜆 > 0,𝜀 > 0,𝛾 ≥ 0 and 𝛿 =
+2
+𝜅+𝜅0 . Thus {𝛽𝑙 }∞
+𝑙=0
+strongly converges the minimizer of model (10) in spite of the choice
+of 𝛽0.
+Remark 1. It is not difficult to obtain from the proof of Theorem 3.
+∥𝛽𝑙 − 𝛽∗∥2 ≤
+𝜅 + 𝜅0
+𝜅0(𝜅 + 𝜅0 + 4𝜀𝜆)
+�𝜅 − 𝜅0
+𝜅 + 𝜅0
+�𝑙
+∥H𝑇 T∥2.
+Therefore, for each 𝜉 > 0, if
+𝜅 + 𝜅0
+𝜅0(𝜅 + 𝜅0 + 4𝜀𝜆)
+�𝜅 − 𝜅0
+𝜅 + 𝜅0
+�𝑙
+∥𝛽1 − 𝛽0∥2 < 𝜉.
+namely,
+𝑙 >
+log
+� ∥𝛽1−𝛽0 ∥2(𝜅+𝜅0)
+𝜉𝜅0(𝜅+𝜅0+4𝜀𝜆)
+�
+log
+� 𝜅+𝜅0
+𝜅−𝜅0
+�
+,
+thus
+∥𝛽𝑙 − 𝛽∗∥2 < 𝜉.
+As a conclusion, the complete ℓ2-ℓ0.5-ELM algorithm is given in
+Algorithm 1 which integrates the result of Theorem 3 and Remark
+1. Next section, we want give some properties of our proposed
+algorithm.
+Algorithm 1: the algorithm for ℓ2-ℓ0.5-ELM model
+Input:Given
+a
+set
+of
+training
+samples
+𝒻
+=
+�
+(𝑥𝑗,𝑡𝑗) : 𝑥𝑗 ∈ R𝑝,𝑡𝑗 ∈ R𝑚, 𝑗 = 1, 2, . . . ,𝑛
+�,
+activation
+func-
+tion 𝑔, hidden node number 𝑁, the related regularization
+parameters 𝜆 > 0, 𝛾 ≥ 0, 𝜀 ≥ 0, the corresponding parameter 𝛿,
+and an acceptable error 𝜉.
+Step 1: Randomly assign a proper scope for input weight 𝜔𝑖 and
+bias 𝑏𝑖 (𝑖 = 1, 2, . . . , 𝑁)
+Step 2: Compute the hidden layer output matrix H;
+Step
+3:
+Set
+𝛽0
+=
+(0, 0, . . . , 0),
+𝛽1
+=
+half(
+𝛿𝜆𝛾
+1 + 2𝜀𝜆𝛿 , (I − 𝛿H𝑇 H)𝛽0 + 𝛿H𝑇 T
+1 + 2𝜀𝜆𝛿
+), and 𝑙𝑚𝑎𝑥 be a minimal
+positive integer but larger than
+log
+� ∥𝛽1 − 𝛽0∥2(𝜅 + 𝜅0)
+𝜉𝜅0(𝜅 + 𝜅0 + 4𝜀𝜆)
+�
+log
+� 𝜅+𝜅0
+𝜅−𝜅0
+�
+.
+Step 4: For 𝑙 = 1 : 𝑙𝑚𝑎𝑥
+if 𝑙 ≥ 𝑙𝑚𝑎𝑥, stop;
+else 𝑙 := 𝑙 + 1 and update the 𝛽 as follows: 𝛽𝑙+1 =
+half(
+𝛿𝜆𝛾
+1 + 2𝜀𝜆𝛿 , (I − 𝛿H𝑇 H)𝛽𝑙 + 𝛿H𝑇 T
+1 + 2𝜀𝜆𝛿
+).
+repeat Step 4, until that the desired output weight is ^𝛽 = 𝛽𝑚𝑎𝑥.
+Output: Return the output weights ^𝛽;
+4
+SOME CHARACTERISTICS FOR ℓ2-ℓ0.5-ELM
+For the new section, we want to discuss and analyze some key
+characteristics of the estimator regarding ℓ2-ℓ0.5-ELM, such as the
+convergence and sparsity.
+Theorem 4. 𝛽𝑙 strongly converges to the minimum value 𝛽∗ of
+the minimization problem
+min
+𝛽 ∈R𝑁
+� 1
+2 ∥H𝛽 − T∥2
+2 + 𝜆𝑝𝛾𝜀 (𝛽)
+�
+as 𝑙 → ∞.
+𝛽0.5 in the ℓ2-ℓ0.5-ELM is a highly significant part of the sparsity
+of the solution. Thus, we set the Theorem 5 as follows.
+Theorem 5. Suppose 𝜆
+>
+0,𝛾
+>
+0, then the support of
+half(
+𝜆𝛾
+1+2𝜀𝜆,
+𝛽
+1+2𝜀𝜆 ) is finite for any 𝛽 ∈ R𝑁 . Particularly, 𝛽∗ and 𝛽𝑙
+are all finitely supported.
+If the regularization parameters 𝜆 and 𝛾 are fixed as some con-
+stant values, then 𝛽∗ and 𝛽𝑙 have only a few finite nonzero coeffi-
+cients, and hence the solution to (12) is sparse.
+
+AISS 2022, November 25–27, 2022, Sanya, China
+Zhou and Miao.
+Table 1: Details of the 6 datasets
+Dataset
+Type
+Sapmple
+Feature
+Catagory
+Austrian
+UCI
+690
+14
+2
+Ionosphere
+UCI
+151
+34
+2
+Balance
+UCL
+625
+4
+3
+colon
+gene
+62
+2000
+2
+DLBCL
+gene
+77
+7129
+2
+ORL
+image
+400
+10304
+40
+5
+PERFORMANCE EVALUATION
+In the new section, a succession of experiments, containing some
+UCI benchmark datasets[9] and gene data, are carried out to demon-
+strate the performance of the proposed ℓ2-ℓ0.5-ELM method. All
+the experiments are performed in the Mac Pycharm environment
+running on Quad-Core Intel Core i5, CPU (8 GB 2133 MHz LPDDR3)
+processor with the speed of 1.40GHz. The activation function of
+networks used in the experiments is taken as sigmoid function
+𝑔(𝑥) = 1/(1 + 𝑒−𝑥).
+The ℓ2-ℓ0.5-ELM model is compared with seven other models:
+BP, SVM, ELM, ℓ2-ℓ1-ELM, ℓ2-ELM, ℓ1-ELM, ℓ0.5-ELM. BP includes
+only one hidden layer and output layer, and all parameters are
+trained by back-propagation algorithm; ℓ1-ELM and ℓ0.5-ELM are
+the simplified forms of ℓ2-ℓ1-ELM and ℓ2-ℓ0.5-ELM, respectively.
+The activation function is defined as: 𝑔(𝑥) = 1/(1 + 𝑒−𝑥).
+In order to check the algorithm for ℓ2-ℓ0.5-ELM model, three real
+classification datasets from the UCI machine learning repository
+are considered. The basic information of each dataset is shown in
+Table 1. The average of 30 experimental validations was used as
+the final result. For these datasets, the sample size is fixed, but each
+sample is randomly assigned as training or testing data.
+5.1
+Performance for UCI datasets
+To validate the performance of the proposed ℓ2-ℓ0.5-ELM model,
+three types of real classification datasets were used for the experi-
+ments, including UCI[3], gene expression, and ORL face datasets.
+The UCI machine learning repository (2013UCI) contains three
+datasets: Austrian Credit Approval(Austrian), Ionosphere, and Bal-
+ance Scale(Balance). The gene expression datasets contain colon[1]
+and DLBCL[13], both of which are binary datasets. Moreover, the
+ORL face dataset includes 400 images divided into 40 categories.
+Each category contains 10 images with different facial details and
+each image size is 112 × 92. The detail information of all datasets
+are summarized in Table 1. In addition, these data were obtained
+from different application fields, and it is hoped that the ℓ2-ℓ0.5-
+ELM model can be analyzed from multiple perspectives by using
+these data from different backgrounds.
+We repeat 30 trials and take the averages as the final results
+on account of reducing the random error. And the regularization
+parameters are used to control the trade-off between the error and
+the penalty. For Austrian dataset, take the parameters ( ℓ2-ℓ0.5-ELM,
+ℓ2-ℓ1-ELM : 𝜆 = 0.8,𝛾 = 0.1,𝜀 = 0.9) and for Ionosphere dataset,
+take ( ℓ2-ℓ0.5-ELM, ℓ2-ℓ1-ELM : 𝜆 = 0.9,𝛾 = 0.05,𝜀 = 0.9) and
+Balance Scale dataset, ( ℓ2-ℓ0.5-ELM : 𝜆 = 0.8,𝛾 = 1,𝜀 = 1, for ℓ2-ℓ1-
+ELM : 𝜆 = 0.005,𝛾 = 0.5,𝜀 = 0.5), we set the acceptable error 𝜉 =
+Table 2: Performance comparison of 8 models on 3 different
+datasets
+Datasets
+Methods
+Times(s)
+Remaining Nodes
+Accuracy(% ± %)
+Austrain
+BP
+2.1751
+600
+72.58 ± 13.57
+SVM
+0.0448
+—
+79.14 ± 1.98
+ELM
+0.0588
+600
+65.37 ± 3.08
+ℓ0.5-ELM
+5.8542
+48.5
+82.76 ± 0.00
+ℓ1-ELM
+8.1648
+118.5
+81.38 ± 0.00
+ℓ2-ELM
+8.2735
+600
+80.36 ± 0.00
+ℓ2-ℓ1-ELM
+10.041
+492.5
+81.38 ± 0.00
+ℓ2-ℓ0.5-ELM
+7.5875
+118.5
+82.76 ± 0.00
+Ionosphere
+BP
+2.1751
+600
+72.58 ± 13.57
+SVM
+0.0108
+–
+86.51 ± 2.09
+ELM
+0.0003
+600
+91.55 ± 2.78
+ℓ0.5-ELM
+0.0487
+29.5
+96.96 ± 0.00
+ℓ1-ELM
+5.4755
+115.9
+97.24 ± 1.06
+ℓ2-ELM
+0.0520
+600
+96.05 ± 1.57
+ℓ2-ℓ1-ELM
+4.4093
+437.5
+96.84 ± 0.98
+ℓ2-ℓ0.5-ELM
+0.0569
+193
+98.01 ± 0.00
+Balance
+BP
+4.3814
+600
+59.99 ± 25.26
+SVM
+0.0215
+–
+88.63 ± 1.86
+EL,M
+0.0008
+600
+50.72 ± 6.66
+ℓ0.5-ELM
+0.1285
+23.3
+90.55 ± 0.00
+ℓ1-ELM
+6.5074
+42.9
+90.47 ± 1.66
+ℓ2-ELM
+0.1579
+600
+90.55 ± 0.00
+ℓ2-ℓ1-ELM
+6.8678
+246.4
+90.10 ± 1.35
+ℓ2-ℓ0.5-ELM
+0.0974
+52.7
+90.91 ± 0.00
+0.0001, 0.001, 0.0001 respectively. The number of hidden nodes in
+the experiments is 600. Table 2 shows the running time, the number
+of nodes retained, and the accuracy of the test for each dataset for
+the eight models (the standard deviation is kept to 4 significant
+digits, 0.00 in the table indicates a standard deviation of less than
+10−4). These indices are used to measure the sparsity, stability and
+effectiveness of the proposed method. The corresponding figures
+on testing are shown as follows.
+From the results of 1-3, we can see that the accuracy of the ELM
+model is lower than all the regularized ELM models. The standard
+deviation of the ELM model is higher than that of other regularized
+ELM models, which indicates that the stability of the ELM model is
+lower. The accuracy of the ℓ2-ℓ0.5-ELM model at all nodes can be
+compared with other regularized ELM models, and the accuracy at
+most hidden nodes is higher than other comparable regularized ELM
+models. This indicates that the ℓ2-ℓ0.5-ELM model has consistently
+good classification prediction. In terms of the standard deviation
+of different nodes, the ℓ2-ℓ0.5-ELM model is lower than the other
+compared models, indicating that the classification accuracy of this
+method is more stable.
+We can see the performance of ℓ2-ℓ0.5-ELM in detail and draw
+the following conclusions:
+(i) In 3 datasets, the classification accuracy of the regularized
+ELM methods (ℓ2-ℓ0.5-ELM, ℓ0.5-ELM, ℓ2-ℓ1-ELM, ℓ1-ELM, ℓ2-ELM)
+are significantly higher than that of the BP, SVM and ELM methods,
+indicating that the regularized ELM methods have better general-
+ization performance, and the classification accuracy of ℓ2-ℓ0.5-ELM
+methods is higher than that of other compared regularized ELM
+methods.
+(ii) From the perspective of the number of remaining hidden
+nodes, ℓ0.5-ELM has the lowest number of hidden nodes. It is shown
+
+An improved hybrid regularization approach for extreme learning machine
+AISS 2022, November 25–27, 2022, Sanya, China
+200
+300
+400
+500
+600
+700
+800
+900
+1000 1100 1200
+Number of Hidden Nodes
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+Testing accuracy
+ELM
+l0.5
+l1
+l2
+l2l1
+l2l0.5
+200
+300
+400
+500
+600
+700
+800
+900
+1000 1100 1200
+Number of Hidden Nodes
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+0.012
+0.014
+0.016
+0.018
+0.020
+0.022
+0.024
+0.026
+0.028
+0.030
+SDs of Testing
+ELM
+l0.5
+l1
+l2
+l2l1
+l2l0.5
+Figure 1: Performance comparison of 6 models in the Austrian dataset
+200
+300
+400
+500
+600
+700
+800
+900
+1000 1100 1200
+Number of Hidden Nodes
+0.60
+0.62
+0.64
+0.66
+0.68
+0.70
+0.72
+0.74
+0.76
+0.78
+0.80
+0.82
+0.84
+0.86
+0.88
+0.90
+0.92
+0.94
+0.96
+0.98
+1.00
+Testing accuracy
+ELM
+l0.5
+l1
+l2
+l2l1
+l2l0.5
+200
+300
+400
+500
+600
+700
+800
+900
+1000 1100 1200
+Number of Hidden Nodes
+-0.01
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+SDs of Testing
+ELM
+l0.5
+l1
+l2
+l2l1
+l2l0.5
+Figure 2: Performance comparison of 6 models in the Ionosphere dataset
+200
+300
+400
+500
+600
+700
+800
+900
+1000 1100 1200
+Number of Hidden Nodes
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+Testing accuracy
+ELM
+l0.5
+l1
+l2
+l2l1
+l2l0.5
+200
+300
+400
+500
+600
+700
+800
+900
+1000 1100 1200
+Number of Hidden Nodes
+-0.005
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+0.045
+0.050
+0.055
+0.060
+0.065
+0.070
+0.075
+0.080
+0.085
+0.090
+0.095
+0.100
+SDs of Testing
+ELM
+l0.5
+l1
+l2
+l2l1
+l2l0.5
+Figure 3: Performance comparison of 6 models in the Balance dataset
+that the ℓ0.5 or ℓ1-regularization term is beneficial to enhance the
+sparsity of the hidden nodes of the model. Compared with the ℓ2-
+ℓ1-ELM model, the ℓ2-ℓ0.5-ELM model adds the ℓ0.5 regularization
+term to the model, which has a sparser solution and thus a better
+generalization ability.
+(iii) From the perspective of algorithm running time, the ELM
+model runs in the shortest time (the ELM model can obtain the
+analytic solution directly without iterative computation). In com-
+parison, the SVM model runs faster than all ELM methods with
+regularity. Secondly, for the 5 regularized ELM models, the models
+with ℓ0.5 regularization terms (ℓ0.5-ELM, ℓ2-ℓ0.5-ELM) are faster
+than the models with ℓ1 regularization terms (ℓ1-ELM, ℓ2- ℓ1-ELM).
+5.2
+Performance for gene datasets
+In this section, the performance of the ℓ2-ℓ0.5-ELM model is vali-
+dated using the colon and DLBCL data. The training and testing sets
+of each dataset were experimented in the ratio of 1 : 1. The regular-
+ization parameters are set as follows, colon data: (ℓ2-ℓ0.5-ELM and
+ℓ2-ℓ1-ELM : 𝜆 = 0.09,𝛾 = 0.9,𝜀 = 0.9), DLBCL data: (ℓ2-ℓ0.5-ELM
+and ℓ2-ℓ1-ELM : 𝜆 = 0.005,𝛾 = 0.5,𝜀 = 0.5); and 𝜉 = 0.001. Each
+dataset was repeatedly run 30 times, and the average was taken as
+the final result. As shown in Table 3.
+It can be demonstrated that the prediction accuracy of the single-
+layer BP network is very low and does not capture the features of
+
+AISS 2022, November 25–27, 2022, Sanya, China
+Zhou and Miao.
+200
+300
+400
+500
+600
+700
+800
+900
+1000 1100 1200
+Number of Hidden Nodes
+0.72
+0.73
+0.74
+0.75
+0.76
+0.77
+0.78
+0.79
+0.80
+0.81
+0.82
+0.83
+0.84
+0.85
+0.86
+0.87
+0.88
+0.89
+Testing accuracy
+ELM
+l0.5
+l1
+l2
+l2l1
+l2l0.5
+200
+300
+400
+500
+600
+700
+800
+900
+1000 1100 1200
+Number of Hidden Nodes
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+SDs of Testing
+ELM
+l0.5
+l1
+l2
+l2l1
+l2l0.5
+Figure 4: Performance comparison of 6 models in colon dataset
+Table 3: Performance comparison of 8 models in 2 gene
+datasets
+Datasets
+Methods
+Times(s)
+Remaining Nodes
+Accuracy(% ± %)
+colon
+BP
+22.2641
+1000.0
+55.52 ± 9.15
+SVM
+0.0358
+–
+77.5 ± 7.28
+ELM
+0.0056
+1000.0
+83.02 ± 1.92
+ℓ0.5-ELM
+0.0829
+370.5
+75.00 ± 0.00
+ℓ1-ELM
+0.0488
+974.5
+84.79 ± 2.22
+ℓ2-ELM
+0.0815
+1000.0
+84.17 ± 2.20
+ℓ2-ℓ1-ELM
+0.0401
+1000.0
+83.96 ± 2.24
+ℓ2-ℓ0.5-ELM
+0.0879
+877.0
+87.50 ± 0.00
+DLBCL
+BP
+122.3174
+1000.0
+57.24 ± 12.55
+SVM
+0.0968
+–
+87.24 ± 5.98
+ELM
+0.0060
+786.0
+89.90 ± 5.98
+ℓ0.5-ELM
+5.2214
+242.0
+91.43 ± 0.00
+ℓ1-ELM
+18.2957
+188.5
+89.05 ± 5.12
+ℓ2-ELM
+5.2324
+764.0
+89.51 ± 5.48
+ℓ2-ℓ1-ELM
+15.5286
+431.5
+89.62 ± 6.10
+ℓ2-ℓ0.5-ELM
+5.4519
+575.5
+91.43 ± 0.00
+the data very well. It can also be found that the prediction accu-
+racy of the ℓ2-ℓ0.5-ELM model is slightly higher than that of the
+other methods. The standard deviations of the accuracy of the ELM
+methods with ℓ0.5 regularization are much smaller than those of
+BP, SVM, and ELM, indicating that the ELM model variants with
+ℓ0.5 regularization terms can improve the stability of the solutions;
+The number of hidden nodes in the ℓ0.5-ELM and ℓ1-ELM models
+is smaller, that is, the sparsity of these two regularization terms is
+the strongest, indicating that the addition of ℓ0.5 or ℓ1 regularization
+terms in the ELM model enhances the sparsity of the model, while
+the number of hidden nodes in the ℓ2-ELM model is 1000. The
+number of nodes in the ℓ2-ELM model is 1000, indicating that the
+ℓ2-regularization term has no sparse effect on the model. The ℓ2
+norm is used to increase the stability of the model by penalizing
+oversized regularization parameters. This makes the ℓ2-ℓ0.5-ELM
+sparser and model stable, and thus obtains better generalization
+ability.
+From the perspective of algorithm running time, it can be seen
+that the ELM model has the shortest running time (the ELM model
+can obtain the analytical solution directly without iterative solving).
+In contrast, the SVM model runs faster than all ELM methods with
+regularization.
+Further, we use the colon data to verify the effect of different
+number of hidden nodes (200, 400, 600, 800, 1000, 1200) on the sta-
+bility of the ELM correlation model. We perform 30 experiments
+for each hidden node and calculate the ELM, ℓ2-ℓ0.5-ELM, ℓ0.5-ELM,
+ℓ2-ℓ1-ELM, ℓ1-ELM, ℓ2-ELM for the test set accuracy and standard
+deviation as shown in Figure 4. The test accuracy of ℓ2-ℓ0.5-ELM at
+all nodes can be compared with all regularized ELM models, while
+the accuracy at most hidden nodes is higher than other models.
+The standard deviation of ℓ2-ℓ0.5-ELM model is lower than other
+regularized ELM models.
+5.3
+Performance for ORL face dataset
+The ORL face dataset is used for experimental validation. The num-
+ber of hidden nodes for the experiment is 1000. The average of
+30 experiments is used as the final result. Since the original im-
+age has high dimensionality, we preprocess each image by using
+the (2𝐷)2PCA[18] dimensionality reduction technique. And the
+training set and test set are in the ratio of 7 : 3. The values of the
+regular parameters set in the experiment are as follows: ℓ0.5-ELM
+and ℓ1-ELM (𝛾 = 0.05,𝜀 = 0), ℓ2-ELM (𝛾 = 0,𝜀 = 0.5), ℓ2 -ℓ1-ELM,
+ℓ2-ℓ0.5-ELM(𝛾 = 0.05,𝜀 = 0.5); 𝜆 = 0.001 and 𝜀 = 0.0001 are cho-
+sen in all experiments. This experiment validates the performance
+of the model in terms of accuracy and standard deviation. The re-
+sults are shown in Table 4. From the table, it can be seen that the
+Table 4: Performance comparison of 8 models in ORL face
+dataset
+Methods
+Accuracy(%)
+BP
+31.00 ± 4.90
+SVM
+71.53 ± 2.12
+ELM
+70.58 ± 2.95
+ℓ0.5-ELM
+71.00 ± 2.34
+ℓ1-ELM
+70.85 ± 2.86
+ℓ2-ELM
+71.17 ± 2.47
+ℓ2-ℓ1-ELM
+70.58 ± 2.87
+ℓ2-ℓ0.5-ELM
+71.67 ± 2.34
+
+An improved hybrid regularization approach for extreme learning machine
+AISS 2022, November 25–27, 2022, Sanya, China
+Table 5: Performance comparison of 6 models in ORL face dataset
+Nodes
+ELM
+ℓ0.5-ELM
+ℓ1-ELM
+ℓ2-ELM
+ℓ2-ℓ1-ELM
+ℓ2-ℓ0.5-ELM
+500
+52.92±3.04
+66.10±2.55
+60.00 ±1.77
+62.63± 2.38
+59.25 ±2.32
+65.83 ± 2.46
+1500
+76.08±0.73
+77.00±0.93
+76.33 ±0.67
+76.75± 0.75
+76.33 ±0.76
+77.20 ±0.93
+2000
+78.25±2.00
+78.73±2.45
+78.33 ±2.08
+78.63± 2.18
+78.33 ±2.08
+78.83 ±2.45
+2500
+79.58±3.49
+79.74±3.36
+79.67 ±3.44
+79.21±3.29
+79.63 ±3.44
+79.76 ± 3.26
+3000
+81.50±1.98
+81.55±2.69
+81.42 ±2.07
+81.45±2.39
+81.42 ±2.07
+81.58 ± 2.68
+3500
+81.17±1.81
+81.13±2.22
+81.17 ±1.87
+81.17±1.89
+81.17 ±1.87
+81.25 ± 2.12
+4000
+82.00±1.81
+82.00±1.67
+81.92 ±1.74
+81.96±1.64
+81.92 ±1.74
+82.08 ± 1.65
+mean
+75.22±9.12
+77.16±5.33
+76.21 ±7.00
+76.62±6.21
+76.08 ±7.24
+77.26 ± 5.32
+accuracy of the ℓ2-ℓ0.5-ELM model (which is slightly higher than
+the SVM model) is slightly higher than all other models tested.
+Further, we verify the effect of different values of hidden nodes
+on the prediction accuracy. The number of hidden nodes chosen in
+the experiment is 500, 1000, 1500, 2000, 2500, 3000, 3500, 4000.
+The results are shown in Table 5, which show that the test accu-
+racy of ℓ2-ℓ0.5-ELM model is higher than the other comparative
+ELM models. The test accuracy of the ELM model fluctuates the
+most with the changing of the number of hidden nodes, i.e., the
+selection of different nodes has the greatest impact on it, indicating
+that the ELM model is less stable in high-dimensional data. In con-
+trast, the standard deviations of all the regularized ELM methods
+(5.33, 7.00, 6.21, 7.24, 5.32) are lower than those of the ELM meth-
+ods, indicating that the stability of the ELM model is improved by
+adding the regularization term. ELM methods, indicating that the
+stability of the proposed method is better than the other 5 compared
+to ELM methods.
+6
+CONCLUSION
+In order to further improve the stability and generalization of the
+ELM model, this paper proposes a ℓ2-ℓ0.5-ELM model by combin-
+ing the ℓ0.5 and the ℓ2 regularization term. The iterative algorithm
+is applied to solve the model with a fixed points algorithm. The
+convergence and sparsity of this algorithm are proved. Moreover,
+the proposed ℓ2-ℓ0.5-ELM model is compared with BP, SVM, ELM,
+ℓ0.5-ELM, ℓ1-ELM, ℓ2-ELM and ℓ2-ELM. ℓ2-ℓ1-ELM models. Experi-
+mental comparisons on several datasets (UCI dataset, gene dataset,
+ORL face dataset) show that the ℓ2-ℓ0.5-ELM method outperforms
+the other 7 models in terms of prediction accuracy and stability
+on these data. Therefore, the model can be improved as follows:
+the information of previously computed nodes is not used in the
+computation of different hidden nodes, and it can be learned from
+the incremental learning point of view, which can reduce the com-
+putation time to a certain extent.
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+

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+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 10, JANUARY 2023
+1
+FGAHOI: Fine-Grained Anchors for
+Human-Object Interaction Detection
+Shuailei Ma, Yuefeng Wang, Shanze Wang, and Ying Wei
+Abstract—Human-Object Interaction (HOI), as an important problem in computer vision, requires locating the human-object pair and
+identifying the interactive relationships between them. The HOI instance has a greater span in spatial, scale, and task than the
+individual object instance, making its detection more susceptible to noisy backgrounds. To alleviate the disturbance of noisy
+backgrounds on HOI detection, it is necessary to consider the input image information to generate fine-grained anchors which are then
+leveraged to guide the detection of HOI instances. However, it is challenging for the following reasons. 𝑖) how to extract pivotal features
+from the images with complex background information is still an open question. 𝑖𝑖) how to semantically align the extracted features and
+query embeddings is also a difficult issue. In this paper, a novel end-to-end transformer-based framework (FGAHOI) is proposed to
+alleviate the above problems. FGAHOI comprises three dedicated components namely, multi-scale sampling (MSS), hierarchical
+spatial-aware merging (HSAM) and task-aware merging mechanism (TAM). MSS extracts features of humans, objects and
+interaction areas from noisy backgrounds for HOI instances of various scales. HSAM and TAM semantically align and merge the
+extracted features and query embeddings in the hierarchical spatial and task perspectives in turn. In the meanwhile, a novel training
+strategy Stage-wise Training Strategy is designed to reduce the training pressure caused by overly complex tasks done by FGAHOI.
+In addition, we propose two ways to measure the difficulty of HOI detection and a novel dataset, 𝑖.𝑒., HOI-SDC for the two challenges
+(Uneven Distributed Area in Human-Object Pairs and Long Distance Visual Modeling of Human-Object Pairs) of HOI instances
+detection. Experiments are conducted on three benchmarks: HICO-DET, HOI-SDC and V-COCO. Our model outperforms the
+state-of-the-art HOI detection methods, and the extensive ablations reveal the merits of our proposed contribution. The code is
+available at https://github.com/xiaomabufei/FGAHOI.
+Index Terms—Human-Object Interaction, FGAHOI, Fine-Grained Anchors, Noisy Background, Semantically Aligning.
+!
+1
+INTRODUCTION
+H
+UMAN-Object
+interaction
+(HOI)
+detection,
+as
+a
+downstream task of object detection [1], [2], [3], [4],
+[5], has recently received increasing attention due to its
+great application potential. For successful HOI detection, it
+needs to have the ability to understand human activities
+which are abstracted as a set of <human, object, action>
+triplets in this task, requiring a much deeper understanding
+for the semantic information of visual scenes. Without HOI
+detection, machines can only interpret images as collections
+of object bounding boxes, i.e., AI systems can only pick up
+information such as ’A man is on the bike’ or ’A bike is in
+the corner’, but not ’A man rides a bike’.
+Spanning the past and the present, the existing HOI
+detection approaches [6], [7], [8], [9], [10], [11], [12], [13],
+[14], [15], [16], [17], [18], [19], [20], [21] tend to fall into
+two categories, namely two-stage and one-stage methods.
+Conventional two-stage methods [7], [8], [10], [12], [13],
+[14], [18], [20], [22], [23], [24], [25], as an intuitive approach,
+detect human and object instances by leveraging the off-the-
+•
+Shuailei Ma, Yuefeng Wang are with College of Information Science and
+Engineering, Northeastern University, Shenyang, China, 110819.
+E-mail: {xiaomabufei, wangyuefeng0203} @gmail.com
+•
+Shanze Wang is with Changsha Hisense Intelligent System Research
+Institute Co., Ltd. and Information Technology R&D Innovation Center of
+Peking University, Shaoxing, China.
+E-mail: [email protected]
+•
+Ying Wei is the corresponding author, with College of Information Science
+and Engineering, Northeastern University, Shenyang, China, 110819.
+E-mail: [email protected]
+Manuscript received October 26, 2022; revised January 10, 2023.
+FGAHOI
+Low Level
+Middle Level
+High Level
+Fine-Grained 
+Anchors
+Attention Weights
+Fig. 1: FGAHOI leverages the query embeddings and multi-
+scale features to generate fine-grained anchors and the
+corresponding weights for HOI instances of diverse scales.
+Then, they guide the decoder to aid key semantic infor-
+mation of HOI instances to the content embeddings and
+translate the content embeddings to HOI embeddings for
+predicting all elements of the HOI instances.
+shelf object detector [1], [3], [4], utilizing the visual features
+extracted from the located areas to recognize action classes.
+To fully leverage the visual features, several methods [7],
+[10], [14], [20], [22], [23], [24], [25] separately extract vi-
+sual features of human-object pairs and spatial information
+from the located area in a multi-stream architecture, fusing
+them in a post-fusion strategy. In the meanwhile, several
+approaches [8], [10], [20], [23], [24] employ the existing pose
+arXiv:2301.04019v1  [cs.CV]  8 Jan 2023
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 10, JANUARY 2023
+2
+estimation methods, such as [26], [27], [28] to extract pose
+information and fuse it with other features to predict the
+action class. In addition, some works [8], [12], [13], [18],
+[29] leverage the graph neural network to extract complex
+semantic relationship between humans and objects. How-
+ever, the difficulties encountered in the two-stage approach
+lie mainly in the effective fusion of human-object pairs and
+complex semantic information. Besides, owing to the limita-
+tions of the fixed detector and some other components (pose
+estimation etc.), the two-stage method can only achieve a
+sub-optimal solution.
+To achieve high efficiency, one-stage approaches [6], [9],
+[11], [15], [17], [21], [30], [31] which utilize interaction points
+between the human-object pairs to simultaneously predict
+human and object offset vectors and action classes, are
+proposed to detect human-object pairs and recognize inter-
+active relationships in parallel. However, when the human
+and object in the image are far apart from each other, these
+methods are disturbed by ambiguous semantic features.
+The one-stage methods do not achieve much attention until
+the appearance of the Detection Transformer (DETR) [32]
+and QPIC [19] applies it for HOI detection. Then, plenty
+of transformer-based works [6], [9], [16], [17], [33] attempt
+to solve the HOI detection with different encoder-decoder
+structures and backbone models.
+In comparison to object instances, HOI instances have
+a greater span of spatial, scale and task. In most HOI
+instances, there is a certain distance between human and
+objects and their scale varies enormously. Compared with
+simple object classification, it is necessary to consider more
+information between human-object pairs rather than the
+features of humans and objects for interaction classification.
+Therefore, the detection is more susceptible to distractions
+from noisy backgrounds. However, most recent works [19],
+[33] use object detection frameworks [32], [34] directly for
+HOI detection by simply adding the interaction classifica-
+tion head, ignoring these problems. Inspired by [34] which
+leverages the reference points to guide the decoding pro-
+cess, we propose to leverage fine-grained anchors to guide
+the detection of HOI instances and protect it from noisy
+backgrounds. To generate fine-grained anchors for kinds
+of HOI instances, it is obviously necessary to consider the
+input image features. There are, however, two inevitable
+challenges that arise as a result of this. 𝑖) it is difficult to
+extract pivotal features from the images which contain noisy
+background information. 𝑖𝑖) how to semantically align and
+merge the extracted features with query embeddings is also
+an open question.
+In this paper, we propose a novel transformer-based
+model for HOI detection, i.e., FGAHOI: Fine-Grained An-
+chors for Human-Object Interaction Detection (as shown in
+Fig.1). FGAHOI leverages the multi-scale sampling mech-
+anism (MSS) to extract pivotal features from images with
+noisy background information for variable HOI instances.
+Based on the sampling strategy and initial anchor gener-
+ated by the corresponding query embedding, MSS could
+extract hierarchical spatial features of human, object and the
+interaction region for each HOI instance. Besides, the hi-
+erarchical spatial-aware (HSAM) and task-aware merging
+mechanism (TAM) are utilized to semantically align and
+merge the extracted features with the query embeddings.
+HSAM merges the extracted features in the hierarchical
+spatial perspective according to the cross-attention between
+the features and the query embeddings. Meanwhile, the
+extracted features are aligned towards the query embed-
+dings, according to the cross-attention weights of the merg-
+ing process. Thereafter, TAM leverages the switches which
+dynamically switch ON and OFF to merge the input features
+and query embeddings in the task perspective.
+According to experiment results, we investigate that it
+is difficult of the end-to-end training approach to allow the
+transformer-based models to achieve optimal performance
+when more complex task requirements are required. In-
+spired by the stage-wise training [35], [36] for LTR [37], we
+propose a novel stage-wise training strategy for FGAHOI.
+During the training process, we add the important compo-
+nents of the model in turn to clarify the training direction
+of the model at each stage, so as to maximize the savings in
+the training cost of the model.
+To the best of our knowledge, there are no measurements
+for the difficulty of detecting HOI instances. We investigate
+that two difficulties lie in the detection of human-object
+pairs, 𝑖.𝑒., Uneven Distributed Area in Human-Object
+Pairs and Long Distance Visual Modeling of Human-
+Object Pairs. In this paper, we propose two measurements
+and a novel dataset (HOI-SDC) for these two challenges.
+HOI-SDC eliminates the influence of other factors (Too few
+training samples of some HOI categories, too tricky interac-
+tion actions, et.al.) on the model training and focuses on the
+model for these two difficult challenges. Our contributions
+can be summarized fourfold:
+•
+We propose a novel transformer-based human-object
+interaction detector (FGAHOI) which leverages input
+features to generate fine-grained anchors for pro-
+tecting the detection of HOI instances from noisy
+backgrounds.
+•
+We propose a novel training strategy where each
+component of the model is trained in turn to clar-
+ify the training direction at each stage, in order to
+maximize the training cost savings.
+•
+We propose two ways to measure the difficulty of
+HOI detection and a dataset, 𝑖.𝑒., HOI-SDC for the
+two challenges (Uneven Distributed Area in Human-
+Object Pairs and Long Distance Visual Modeling of
+Human-Object Pairs) of detecting HOI instances.
+•
+Our extensive experiments on three benchmarks:
+HICO-DET
+[38],
+HOI-SDC
+and
+V-COCO
+[39],
+demonstrate the effectiveness of the proposed FGA-
+HOI. Specifically, FGAHOI outperforms all existing
+state-of-the-art methods by a large margin.
+2
+RELATED WORKS
+Two-stage HOI Detection Approaches: The two-stage HOI
+detection approaches [7], [8], [10], [12], [13], [14], [18], [20],
+[22], [23], [24], [25], [29] employ the off-the-shelf object de-
+tector [1], [3], [4] to localize humans and objects. Afterwards,
+the features of backbone networks inside the human and
+objects regions are cropped. Part of the two-stage meth-
+ods [8], [12], [13], [18], [29] treat the human and objects
+feature as nodes and employ graph neural networks [40]
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 10, JANUARY 2023
+3
+Encoder
+Content Embeddings
+…
+Initial Anchor
+…
+Positional Encoding
+Human Box
+Object box/Class
+Human Box
+Object box/Class
+Verb Class
+Verb Class
+HOI
+Detection
+Head
+Decoder
+Task-Aware
+Merging 
+Dynamic
+Switch On/Off
+…
+Positional 
+Embeddings
+Multi-Scale 
+Sampling Strategy
+Multi-Scale 
+Features 
+Hierarchical Spatial-Aware 
+Merging 
+Fig. 2: This figure illustrates the overall structure of FGAHOI. FGAHOI utilizes a hierarchical backbone and a deformable
+encoder to extract the semantic features in a multi-scale approach. In the decoding phrase, FGAHOI leverages the multi-
+scale sampling, hierarchical spatial-aware merging and task-aware merging mechanism to align input features with
+query embeddings and assist the generation of fine-grained anchors for the translation of HOI embeddings. At the back
+end of the pipeline, HOI detector leverages the HOI embeddings and initial anchor to predict all elements of the HOI
+instances.
+to predict action classes. The other part of the two-stage
+approach [7], [10], [14], [20], [22], [23], [24], [25] leverages
+multi-stream networks to extract diverse information from
+cropped regions, such as human features, object features,
+spatial information and human pose information. Then,
+the information is fused to predict the action in a post-
+fusion strategy. Two-stage methods mainly concentrate on
+predicting the action class in the second stage. Nevertheless,
+the quality of cropped features from the first stage cannot
+be guaranteed in most cases, so the method cannot achieve
+an optimal solution. More importantly, integrating semantic
+information of human-object pairs requires massive time
+and computing resources.
+One-stage HOI Detection Approaches: The traditional one-
+stage approaches [9], [11], [15], [31] use interaction points
+or union regions to detect human-object pairs and identify
+interactive action classes in parallel. However, these meth-
+ods which and are hampered by distant human-object pairs,
+require a gathering and pairing process. With the creation of
+DETR [32], one-stage approaches have become the current
+mainstream. QPIC [19] converts the object detection head
+of DETR into an interaction detection head to predict HOI
+instance directly. HOITrans [17] combines transformer [41]
+and CNN [42] to straightly predict HOI instances from the
+query embeddings. AS-Net [6] and HOTR [9] each propose a
+two-branch transformer method that consists of an instance
+decoder and an interaction decoder to predict the boxes
+and action classes in parallel. CDN [16] proposes a cascade
+disentangling decoder to decode action classes. QAHOI [33]
+directly combines Swin Transformer [43] and deformable
+DETR [34] to predict HOI instances.
+Anchor-Based Object Detection Transformer: Deformable
+DETR [34] first introduces the reference point concept,
+where the sampling offset is predicted by each reference
+point to perform deformable cross-attention. To facilitate
+extreme region discrimination, Conditional DETR [44] re-
+formulates the attention operation and rebuilt positional
+queries based on reference points. Anchor DETR [45] pro-
+poses to explicitly capitalize on the spatial prior during
+cross-attention and box regression by utilizing a predefined
+2D anchor point [𝑐𝑥, 𝑐𝑦]. DAB-DETR [46] extends such a
+2D concept to a 4D anchor box [𝑐𝑥, 𝑐𝑦, 𝑤, ℎ] and proposed
+to refine it layer-by-layer. SAM-DETR [47] proposes directly
+updating content embeddings by extracting salient points
+from image features. In this paper, we propose a novel
+decoding process for HOI detection. The alignment and fine-
+grained anchor generation is proposed to align the multi-
+scale features with HOI query embeddings and generate
+fine-grained anchors for the diverse HOI instances with
+variable spatial distribution, scales and tasks. Then, the fine-
+grained anchors guide the deformable attention process in
+aiding key information to query embeddings from noisy
+backgrounds.
+3
+PROPOSED METHOD
+In Sec.3.1, we show the overall architecture of FGAHOI.
+Then, we describe the multi-scale feature extractor in Sec.3.2.
+We introduce the multi-scale sampling strategy in Sec.3.3.1.
+The hierarchical spatial-aware, task-aware merging mech-
+anism and the decoding process is proposed in Sec.3.3.2,
+Sec.3.3.3 and Sec.3.3.4, respectively. In Sec.3.4, we present
+the architecture of the HOI detection head. In Sec.3.5, the
+stage-wise training strategy, loss calculation and inference
+process is illustrated.
+3.1
+Overall Architecture
+The overall architecture of our proposed FGAHOI is illus-
+trated in Fig 2. For a given image 𝑥 ∈ R𝐻×𝑊 ×3, FGAHOI
+firstly uses a hierarchical backbone network to extract the
+multi-scale features Z𝑖
+∈ R
+𝐻
+4×2𝑖 × 𝑊
+4×2𝑖 ×2𝑖𝐶𝑠, 𝑖
+= 1, 2, 3. The
+multi-scale features are then projected from dimension C𝑠
+to dimension C𝑑 by using 1×1 convolution. After being
+flattened out, the multi-scale features are concatenated to
+N𝑠 vectors with C𝑑 dimensions. Afterwards, along with
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 10, JANUARY 2023
+4
+supplementary positional encoding 𝑝 ∈ R𝑁𝑠×𝐶𝑑, the multi-
+scale features are sent into the deformable transformer en-
+coder which consists of a set of stacked deformable encoder
+layers to encode semantic features. The encoded semantic
+features 𝑀 ∈ R𝑁𝑠×𝐶𝑑 are then acquired. In the decoding
+process, the content 𝐶 and positional 𝑃 embeddings are both
+a set of learnable vectors {𝑣𝑖 | 𝑣𝑖 ∈ R𝑐𝑑}𝑁𝑞
+𝑖=1. The positional
+embeddings 𝑃 first generate the initial anchor 𝐴 ∈ R𝑁𝑞×2
+according to a linear layer. The positional 𝑃, content 𝐶
+embeddings, inital anchor 𝐴 and encoded features 𝑀 are
+simultaneously sent into the decoder 𝐹𝑑𝑒𝑐𝑜𝑑𝑒𝑟 (·, ·, ·, ·) which
+is a set of stacked decoder layers. In every decoder layer,
+the initial anchor first leverages the multi-scale sampling
+strategy to sample the multi-scale features corresponding
+to the content embeddings. The sampled features assist
+the generation of fine-grained anchors and corresponding
+attention weights through the hierarchical spatial-aware
+and task-aware merging mechanism. The HOI embeddings
+𝐻 = {ℎ𝑖 | ℎ𝑖 ∈ R𝑐𝑑}𝑁𝑞
+𝑖=1 are translated from the query embed-
+dings 𝑄 through the fine-grained anchors, attention weights
+and the deformable attention. The HOI embeddings 𝐻 are
+acquired as 𝐻 = 𝐹𝑑𝑒𝑐𝑜𝑑𝑒𝑟 (𝑀, 𝑃, 𝐶, 𝐴). Eventually, the HOI
+detector leverages the HOI embeddings 𝐻 and initial anchor
+to predict the HOI instances < 𝑏ℎ, 𝑏𝑜, 𝑐𝑜, 𝑐𝑣 >, where 𝑏ℎ, 𝑏𝑜,
+𝑐𝑜 and 𝑐𝑣 stands for the human box coordinate (𝑥, 𝑦, 𝑤, ℎ),
+object box coordinate, object class and verb class, respec-
+tively.
+3.2
+Multi-Scale Features Extractor
+High-quality visual features are a prerequisite for successful
+HOI detection. For extracting the multi-scale features with
+long-range semantic information, FGAHOI leverages the
+multi-scale feature extractor which consists of a hierarchical
+backbone network and a deformable transformer encoder to
+extract features, the folumation is as Equation.1:
+𝑀 = 𝐹𝑒𝑛𝑐𝑜𝑑𝑒𝑟 (𝐹𝑓 𝑙𝑎𝑡𝑡𝑒𝑛(𝜙(𝑥)), 𝑝, 𝑠, 𝑟, 𝑙)
+∈ R𝑁𝑠×𝐶𝑑,
+(1)
+where 𝐹𝑒𝑛𝑐𝑜𝑑𝑒𝑟 (·), 𝐹𝑓 𝑙𝑎𝑡𝑡𝑒𝑛(·) and 𝜙(·) denotes the encoder,
+flatten operation and backbone network, respectively. 𝑝 is
+the position encoding, 𝑠 is the spatial shape of the multi-
+scale features, 𝑟 stands for the valid ratios and 𝑙 represents
+the level index corresponding the multi-scale features. The
+hierarchical backbone network is flexible and can be com-
+posed of any convolutional neural network [42], [48], [49],
+[50] and transformer backbone network [43], [51], [52], [53],
+[54], [55], [56], [57]. However, CNN is poor at capturing
+non-local semantic features like the relationships between
+humans and objects. In this paper, we mainly use Swin
+Transformer tiny and large version [43] to enhance the
+ability of feature extractor for extracting long-range features.
+3.3
+Why FGAHOI Decodes Better?
+During the decoding process, the fine-grained anchors can
+be regarded as a positional prior to let decoder focus on
+the region of interest, directly guiding the decoder to aid
+semantic information to the content embeddings which are
+used to predict all elements of the HOI instances. Therefore,
+fine-grained anchors play the following two crucial roles in
+HOI detection. 𝑖) Fine-grained anchors directly determine
+whether the information gained from input features to
+content embeddings is instance-critical or noisy background
+information. 𝑖𝑖) Fine-grained anchors determine the quality
+of alignment between the query embeddings and multi-
+scale features of input scenarios. Both are crucial factors for
+the quality of decoding results. The existing methods [33],
+[34] directly utilize the query embedding to generate fine-
+grained anchors based on the initial anchor, without consid-
+ering the multi-scale features of the input scenarios and the
+semantic alignment between the query embedding and the
+input features at all. Our FGAHOI proposes a novel fine-
+grained anchors generator which consists of multi-scale
+sampling, hierarchical spatial-aware merging and task-
+aware merging mechanism (as shown in Fig.3). The gen-
+erator adequately leverages the initial anchor, multi-scale
+features and query embeddings for generating suitable fine-
+grained anchors for diverse input scenarios and aligning
+semantic information between different input scenarios and
+query embeddings. The formulation of FGAHOI decoding
+process is as follows:
+𝐻 = Defattn(Task(Hier Spatial({𝑥𝑖
+𝑠}, 𝐶𝑢), 𝐶𝑢), 𝑀, 𝐶𝑢),
+(2)
+where 𝐶𝑢 is the content embeddings updated by the po-
+sitional embeddings, Defattn represents the deformable at-
+tention, 𝑥𝑖
+𝑠 represents the sampled features of the 𝑖-th level
+features. 𝑀 is the encoded input features.
+3.3.1
+Multi-Scale Sampling Mechanism
+The HOI instances contained in the input scenarios usually
+vary in size, where some instances taking up most of the
+area in the input scenarios and others occupying perhaps
+only a few pixels. Our FGAHOI aims at detecting all in-
+stances in the scene, regardless of the size. Therefore, when
+using the initial anchor to sample the multi-scale features,
+for shallow features mainly used to detect instances of small
+size, the sampling strategy only samples a small range of
+features around the initial anchor. In contrast, for deep
+features mainly used to detect instances of large size, the
+sampling strategy samples a large range of features around
+the initial anchor. As shown in Fig.3 (b), in the generator, the
+encoded features are first reshaped to the original shape.
+Based on the initial anchor, generator leverages the sam-
+pling strategy to sample multi-scale features as follows:
+𝑥𝑖
+𝑠 =𝐹𝑠𝑎𝑚𝑝𝑙𝑒( 𝑟𝑒𝑠ℎ𝑎𝑝𝑒(𝑀)𝑖, 𝐴, 𝑠𝑖𝑧𝑒𝑖, 𝑏𝑖𝑙𝑖𝑛𝑒𝑎𝑟 ),
+(3)
+where 𝑠𝑖𝑧𝑒𝑖 (𝑖 = 0, 1, 2) denotes the sampling size of the 𝑖-th
+level features. 𝑀 is the encoded input features. 𝐴 is the ini-
+tial anchor. Inspired by [58], we utilize bilinear interpolation
+in the sampling strategy.
+3.3.2
+Hierarchical Spatial-Aware Merging Mechanism
+In order to better utilize the hierarchical spatial informa-
+tion of sampled features for aligning content embeddings
+with the sampled features, we propose a novel hierarchical
+spatial-aware merging mechanism (HSAM) which utilizes
+the content embeddings to extract hierarchical spatial in-
+formation and merge the sampled features, as shown in
+Fig.3 (c). The content embeddings are first updated by the
+positional embeddings and multi-head self-attention mech-
+anism as follows:
+𝐶𝑢 = 𝐶 + 𝐹MHA
+�
+(𝐶 + 𝑃)𝑊𝑞, (𝐶 + 𝑃)𝑊 𝑘, 𝐶𝑊 𝑣�
+,
+(4)
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 10, JANUARY 2023
+5
+Multi-Scale Sampling Strategy
+Src Shape
+Anchor
+Sampling
+Low
+High 
+Middle
+Positional 
+Embeddings
+Multi-Head
+Self-Attention
+Add & Norm
+Deformable
+Multi-Head
+Cross-Attention
+FFN
+Add & Norm
+Add & Norm
+Encoded Multi-Scale Features
+Alignment &
+Fine-Grained
+Anchor Generation
+Query
+Embeddings
+(������������ , ������������ )
+V
+K
+Q
+V
+Fined-grained 
+Anchors
+Initial Anchor
+Attention
+Weights
+0
+Updated
+Content 
+Embeddings
+Fine-grained 
+Anchors
+Corresponding
+Attention Weights
+Linear
+Linear
+SoftMax
+Generation of Fine-Grained Anchors
+Reshape
+Reshape
+（b）
+（a）
+（d）
+（e）
+Hierarchical Spatial-Aware 
+Merging Mechanism
+Middle
+Multi-Head 
+Attention
+Flatten
+Low
+High 
+Multi-Head Attention
+CAT
+Positional 
+Embeddings
+Content
+Embeddings
+…
+Task-Aware Merging Mechanism
+Dynamic
+Switch On/Off
+Cross-Attn[(
+,
+]
+Linear
+Linear
+RELU
+Normalize
+Updated
+Content 
+Embeddings
+,
+)
+（c）
+Middle
+Low
+High 
+Merge 
+Features
+Fig. 3: The architecture of FGAHOI’s decoder. (a) Illustration of FGAHOI’s decoding process. (b) Illustration of Multi-
+scale sampling mechanism. (c) Illustration of Hierarchical spatial-aware merging mechanism. (d) Illustration of Task-aware
+merging mechanism. (e) Generation process of fine-grained anchors and the corresponding attention weights.
+where 𝑊𝑞, 𝑊 𝑘 and 𝑊 𝑣 denotes the parameter matrices
+for query, key and value in the self-attention mechanism,
+respectively. 𝐹MHA(·) is the multi-head attention mechanism.
+𝐶 and 𝑃 represents the content and position embeddings,
+respectively. Then, the updated content embeddings are
+leveraged to merge the sampled features, the formulation
+is as follows:
+𝑥𝑖
+𝑚 = 𝐹concat
+�head1, . . . , headNH
+� 𝑊𝑂,
+where headn = Softmax
+�
+(𝐶𝑢𝑊𝑞
+n ) · (𝑥𝑖
+𝑠𝑊 𝑘
+n )𝑇
+√𝑑𝑘
+�
+(𝑥𝑖
+𝑠𝑊 𝑣
+n ).
+(5)
+Where 𝑥𝑖
+𝑚 represents the merged features of the 𝑖-th level
+sampled features. 𝐶𝑢 is the content embeddings updated
+by the positional embeddings. 𝑊𝑂 denotes the parameter
+matrices for multi-head concatenation. 𝑊𝑞
+𝑛 , 𝑊 𝑘
+𝑛 and 𝑊 𝑣
+𝑛
+denote the parameter matrices for query, key and value of
+n-th attention head. 𝐹concat is the concatenating operation.
+𝑑𝑘 =
+𝑁ℎ𝑑
+𝑁𝐻 , 𝑁ℎ𝑑 is the hidden dimensions, and 𝑁𝐻 is the
+number of attention head.
+Following the merging of the sampled features at each
+scale based on spatial information, the merged features at
+each scale are first concatenated together as follows:
+𝑋𝑚 = 𝐹concat({𝑥𝑖
+𝑚}𝑖=0,1,2) ∈ R𝐵×𝑁𝑞×𝑁𝐿×𝑁ℎ𝑑,
+(6)
+where 𝑁𝐿 is the number of multi-scale, 𝑥𝑖
+𝑚 represents the
+merged features of the 𝑖-th level sampled features, 𝑋𝑚 is the
+concatenated multi-scale features and merged by the scale-
+aware merging mechanism as follows:
+𝑋𝑢 = 𝐹concat (head1, . . . , headh) 𝑊𝑂,
+where headn = Softmax
+�
+(𝐶𝑢𝑊𝑞
+n ) · (𝑋𝑚𝑊 𝑘
+n )𝑇
+√𝑑𝑘
+�
+(𝑋𝑚𝑊 𝑣
+n ).
+(7)
+Where 𝑋𝑢 is the merged multi-scale features for updating
+the content embeddings.
+3.3.3
+Task-Aware Merging Mechanism
+Considering diverse HOI instances, the task-aware merging
+mechanism is proposed to fuse the merged multi-scale
+features and content embeddings and align the content
+embeddings with the merged feature in the task-aware
+perspective, as shown in Fig.3 (e). It leverages the merged
+multi-scale features and content embeddings to generate
+dynamic switch for selecting suitable channel in the merging
+process. Content embedding and multi-scale information
+after fusion are first stitched together, the formulation is as
+follows:
+𝑋 = 𝐹𝑠𝑡𝑎𝑐𝑘 (𝐶𝑢, 𝑋𝑢) ∈ R𝐵×𝑁𝑞×(2×𝑁ℎ𝑑).
+(8)
+Where 𝐶𝑢 is the content embeddings updated by the posi-
+tional embeddings, 𝑋𝑢 is the merged multi-scale features.
+Thereafter, we use cross-attention mechanism to update
+these as follows:
+𝑋𝑠𝑤𝑖𝑡𝑐ℎ = 𝐹concat (head1, . . . , headh) 𝑊𝑂,
+where headn = Softmax
+�
+(𝐶𝑢𝑊𝑞
+n ) · (𝑋𝑊 𝑘
+n )𝑇
+√𝑑𝑘
+�
+(𝑋𝑊 𝑣
+n ).
+(9)
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 10, JANUARY 2023
+6
+TABLE 1: Instance statistics of two difficulties. We quantify all the instances in the HAKE-HOI [20] dataset according to
+two newly proposed metrics and divide them into ten intervals.
+Dataset
+IMI
+IMI0
+IMI1
+IMI2
+IMI3
+IMI4
+IMI5
+IMI6
+IMI7
+IMI8
+IMI9
+HAKE-HOI
+num𝐴𝑅
+104243
+65499
+44303
+31241
+21982
+11888
+4670
+1818
+598
+168
+num𝐿𝑅
+424
+1243
+1784
+3043
+8668
+70191
+83314
+79427
+34017
+4299
+SDC Train
+num𝐴𝑅
+62526
+30235
+16346
+12013
+10269
+11189
+4223
+1540
+423
+139
+num𝐿𝑅
+177
+515
+874
+1656
+5208
+48798
+38517
+29544
+20265
+3349
+SDC Test
+num𝐴𝑅
+24737
+0
+0
+0
+0
+0
+0
+0
+0
+0
+num𝐿𝑅
+153
+415
+464
+834
+2704
+20167
+0
+0
+0
+0
+Then, the generated information is utilized to gain the
+dynamic switch for merging, the formulation is as follows:
+𝑆𝑤𝑖𝑡𝑐ℎ𝛾 = 𝐹𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒(𝐹𝑚𝑙𝑝(𝑋𝑠𝑤𝑖𝑡𝑐ℎ))𝛾 ∈ R𝐵×𝑁𝑞×2×2,
+(10)
+where 𝑆𝑤𝑖𝑡𝑐ℎ𝛾 is the dynamic switch for 𝛾-th dimension
+of the merged features. 𝐹ℎ𝑠𝑖𝑔𝑚𝑜𝑖𝑑(·) and 𝐹𝑚𝑙𝑝(·) denote the
+hard sigmoid and feed forward network which consists of
+two linear layers and one Relu activation layer, respectively.
+Inspired by [59], the merging mechanism is designed as
+follows:
+𝑈𝛾 = 𝐹𝑀 𝑎𝑥{𝑆𝑤𝑖𝑡𝑐ℎ𝛾
+𝑖,0 ⊙ 𝑋𝛾
+𝑢 + 𝑆𝑤𝑖𝑡𝑐ℎ𝛾
+𝑖,1}𝑖=0,1 + 𝐶𝛾
+𝑢 ,
+(11)
+where 𝑈𝛾 is 𝛾-th features of content embeddings updated by
+the merged multi-scale features. 𝐹𝑀 𝑎𝑥 is the max operation.
+…
+Linear
+Linear
+MLP
+MLP
+Object Class
+Action Class
+Human Box
+Object Box
+HOI Instances
+<Human/Object Boxes, Object 
+Class, Action Class>
+Initial Anchor
+HOI
+Embeddings
+Fig. 4: The prediction process of the HOI detection head. See
+sec 3.4 for more details.
+3.3.4
+Decoding with Fine-Grained Anchor
+As shown in Fig.3 (e), the updated content embeddings
+are used to generate fine-grained anchors and attention
+weights. According to the linear layer, reshape operation
+and softmax function, the formulation is as follows:
+A = 𝐹𝑙𝑖𝑛&𝑟𝑒𝑠(𝑈) ∈ R𝐵×𝑁𝑞×𝑁𝐻 ×𝑁𝐿×𝑁A×2,
+(12)
+W = 𝐹𝑙𝑖𝑛&𝑟𝑒𝑠&𝑠𝑜 𝑓 𝑡 (𝑈) ∈ R𝐵×𝑁𝑞×𝑁𝐻 ×𝑁𝐿×𝑁A,
+(13)
+As shown in Fig.3 (a), the fine-grained anchors and at-
+tention weights are utilized to aid semantic features from
+the encoded features of the input scenarios to the content
+embeddings, the formulation is as follows:
+P𝑞 =
+𝑁𝐻
+∑︁
+𝑛=1
+𝑾𝑛
+� 𝑁𝐿
+∑︁
+𝑙=1
+𝑁A
+∑︁
+𝑘=1
+W𝑙
+𝑛𝑞𝑘 · 𝑾′
+𝑛𝒙𝒍 �
+A𝑙
+𝑛𝑞𝑘
+��
+,
+(14)
+where P𝑞 is the extracted semantic information used for
+translating 𝑞-th content to HOI embeddings. A𝑙
+𝑛𝑞𝑘 and
+W𝑙
+𝑛𝑞𝑘 represent the 𝑘-th fine-grained anchors and corre-
+sponding attention weights of the 𝑛-th attention head for
+the 𝑞-th query embedding. Both 𝑊𝑛 and 𝑊 ′
+𝑛 are parameter
+matrices of the 𝑛-th attention head. 𝑁A is the number of
+fine-grained anchors of each scale in one attention head.
+3.4
+HOI Detection Head
+FGAHOI leverages a simple HOI detection head to predict
+all elements of HOI instances. As shown in Fig.4, the detec-
+tion head utilizes the HOI embeddings and the initial anchor
+to localize the human and object boxes. In this process, each
+initial anchor acts as the base point for the bounding boxes
+of the corresponding pair of a human and an object, the
+formulation is as follows:
+𝑏ℎ = 𝐹𝑚𝑙𝑝(𝐻)[· · · , : 2] + 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑎𝑛𝑐ℎ𝑜𝑟
+∈ R𝑁𝑞×4,
+(15)
+𝑏𝑜 = 𝐹𝑚𝑙𝑝(𝐻)[· · · , : 2] + 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑎𝑛𝑐ℎ𝑜𝑟
+∈ R𝑁𝑞×4,
+(16)
+𝑐𝑜 = 𝐹𝑙𝑖𝑛𝑒𝑎𝑟 (𝐻)
+∈ R𝑁𝑞×𝑛𝑢𝑚𝑜,
+(17)
+𝑐𝑣 = 𝐹𝑙𝑖𝑛𝑒𝑎𝑟 (𝐻)
+∈ R𝑁𝑞×𝑛𝑢𝑚𝑣,
+(18)
+where 𝐹𝑚𝑙𝑝 denotes the feed forward network consists of
+three linear layers and three relu activation layers. 𝐹𝑙𝑖𝑛𝑒𝑎𝑟
+stands for the linear layer. 𝑛𝑢𝑚𝑜 and 𝑛𝑢𝑚𝑣 are the number
+of object and action classes, respectively. 𝐻 denotes the HOI
+embeddings.
+3.5
+Training and Inference
+3.5.1
+Stage-wise Training
+Inspired by the stage-wise training approach [35], [36] which
+decouples feature learning and classifier learning into two
+independent stages for LTR [37], we propose a novel stage-
+wise training strategy for FGAHOI. We start by training
+the base network (FGAHOI without any merging mecha-
+nism) in an end-to-end manner. We then add the merging
+mechanism in turn to the trained base network for another
+short period of training. In this phrase, the parameters
+of the trained base network are leveraged as pretrained
+parameters and no parameters are fixed during the training
+process.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 10, JANUARY 2023
+7
+ ride, fly, sit_on, exit, 
+direct airplane
+ride, straddle, run, 
+hold,  race horse 
+lasso cow
+carry handbag
+ wear  backpack
+hold, stand_under umbrella 
+ ride, race, run
+straddle, hold horse
+fly, pull 
+kite 
+sail, ride, sit_on, 
+stand_on, drive boat
+race, turn, ride, sit on, 
+straddle, hold motorcycle
+wear tie 
+scratch, walk, 
+pet, train  dog      
+ride, sit on,
+drive, board bus
+carry, wear, hold  
+backpack        
+serve, hit sports_ball 
+swing tennis_racket  
+direct, inspect, ride, 
+sit on, fly  airplane
+type on, read, 
+hold laptop
+ sit on  couch
+brush_with, hold 
+toothbrush 
+kick, block, hit, 
+inspect, dribble  
+sports_ball 
+stand_on, ride, jump,  
+hold skateboard
+ ride, fly, sit_on, exit, 
+direct airplane
+ride, straddle, run, 
+hold,  race horse 
+lasso cow
+carry handbag
+ wear  backpack
+hold, stand_under umbrella 
+ ride, race, run
+straddle, hold horse
+fly, pull 
+kite 
+sail, ride, sit_on, 
+stand_on, drive boat
+race, turn, ride, sit on, 
+straddle, hold motorcycle
+wear tie 
+scratch, walk, 
+pet, train  dog      
+ride, sit on,
+drive, board bus
+carry, wear, hold  
+backpack        
+serve, hit sports_ball 
+swing tennis_racket  
+direct, inspect, ride, 
+sit on, fly  airplane
+type on, read, 
+hold laptop
+ sit on  couch
+brush_with, hold 
+toothbrush 
+kick, block, hit, 
+inspect, dribble  
+sports_ball 
+stand_on, ride, jump,  
+hold skateboard
+Fig. 5: Visualization of HOI detection. Humans and objects are represented by pink and blue bounding boxes respectively,
+and interactions are marked by grey lines linking the box centers. Kindly refer to Sec. 5.6.1 for more details.
+(a)
+(b)
+(c)
+catch sport ball
+watch 
+bird
+kick 
+sports ball
+hit sport 
+ball
+fly kite
+hit sport 
+ball
+wear tie
+swing 
+tennis_racket   
+sit_on 
+toilet
+hold 
+surfboard
+ride skateboard
+ride skateboard
+carry surfboard
+ride surfboard
+ride surfboard
+ride surfboard
+ride surfboard
+ride surfboard
+flip 
+skateboard
+catch sport ball
+watch 
+bird
+kick 
+sports ball
+hit sport 
+ball
+fly kite
+hit sport 
+ball
+wear tie
+swing 
+tennis_racket   
+sit_on 
+toilet
+hold 
+surfboard
+ride skateboard
+carry surfboard
+ride surfboard
+ride surfboard
+flip 
+skateboard
+(a)
+(b)
+(c)
+catch sport ball
+watch 
+bird
+kick 
+sports ball
+hit sport 
+ball
+fly kite
+hit sport 
+ball
+wear tie
+swing 
+tennis_racket   
+sit_on 
+toilet
+hold 
+surfboard
+ride skateboard
+carry surfboard
+ride surfboard
+ride surfboard
+flip 
+skateboard
+Fig. 6: (a) illustrates the excellent long-range visual mod-
+elling capabilities. (b) demonstrates remarkable robustness.
+(c) shows the superior capabilities for identifying small HOI
+instances. Kindly refer to Sec. 5.6.1 for more details.
+3.5.2
+Loss Calculation
+Inspired by the set-based training process of HOI-Trans
+[17], QPIC [19], CDN [16] and QAHOI [33], we first use
+the bipartite matching with the Hungarian algorithm to
+match each ground truth with its best-matching prediction.
+For subsequent back-propagation, a loss is then established
+between the matched predictions and the matching ground
+truths. The folumation is as follows:
+𝐿 = 𝜆𝑜𝐿𝑜
+𝑐 + 𝜆𝑣 𝐿𝑣
+𝑐 +
+∑︁
+𝑘 ∈(ℎ,𝑜)
+�
+𝜆𝑏𝐿𝑘
+𝑏 + 𝜆𝐺𝐼𝑜𝑈 𝐿𝑘
+𝐺𝐼𝑜𝑈
+�
+,
+(19)
+where 𝐿𝑜
+𝑐 and 𝐿𝑣
+𝑐 represent the object class and action class
+loss, respectively. We utilize the modified focal loss function
+[60] and sigmoid focal loss function [61] for 𝐿𝑣
+𝑐 and 𝐿𝑜
+𝑐,
+respectively. 𝐿𝑏 is the box regression loss and consists of the
+𝐿1 Loss. 𝐿𝐺𝐼𝑂𝑈 denotes the intersection-over-union loss, the
+same as the function in QPIC [19]. 𝜆𝑜, 𝜆𝑣, 𝜆𝑏 and 𝜆𝐺𝐼𝑜𝑈 are
+the hyper parameters for adjusting the weights of each loss.
+3.5.3
+Inference
+The inference process is to composite the output of the HOI
+detection head to form HOI triplets. Formally, the 𝑖-th out-
+put prediction is generated as < 𝑏ℎ
+𝑖 , 𝑏𝑜
+𝑖 , 𝑎𝑟𝑔𝑚𝑎𝑥𝑘𝑐ℎ𝑜𝑖
+𝑖
+(𝑘) >.
+The HOI triplet score 𝑐ℎ𝑜𝑖
+𝑖
+combined by the scores of action
+𝑐𝑣
+𝑖 and object 𝑐𝑜
+𝑖 classification, formularized as 𝑐ℎ𝑜𝑖
+𝑖
+= 𝑐𝑣
+𝑖 · 𝑐𝑜
+𝑖 .
+4
+PROPOSED DATASET
+There are two main difficulties existing with human-object
+pairs. 𝑖) Uneven size distribution of human and objects in
+human-object pairs. 𝑖𝑖) Excessive distance between person
+and object in human-object pairs. To the best of our knowl-
+edge, there are no relevant metrics to measure these two
+difficulties. In this paper, we propose two metrics 𝐴𝑅 and
+𝐿𝑅 for measuring these two difficulties. Then two novel
+challenges corresponding to these two difficulties are pro-
+posed. In addition, we propose a novel Set for these Double
+Challenges (HOI-SDC). The data is selected from HAKE-
+HOI [20] which is re-split from HAKE [62] and provides
+110K+ images. HAKE-HOI has 117 action classes, 80 object
+classes and 520 HOI categories.
+
+CB福CREERSBK2
+M
+WITDBWEJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 10, JANUARY 2023
+8
+FGAHOI
+QAHOI
+FGAHOI
+QAHOI
+FGAHOI
+QAHOI
+Fine-Grained
+Anchors #1
+Fine-Grained
+Anchors #2
+Fine-Grained
+Anchors #3
+Fine-Grained
+Anchors #4
+Fine-Grained
+Anchors #5
+Fine-Grained
+Anchors #6
+Fine-Grained
+Anchors #7
+Fine-Grained
+Anchors #8
+HOI
+Instance
+Hold 
+Sport Ball
+Ride 
+Motorcycle
+Fly Kite
+Fig. 7: Comparison of fine-grained anchors between FGAHOI and QAHOI. We visualize the fine-grained anchors
+corresponding to all attention heads and the corresponding attention weights, where the shades of colors correspond
+to the magnitude of the weights. Obviously, FGAHOI is more accurate in focusing on humans, objects and interaction
+areas. Kindly refer to Sec. 5.6.2 for more details.
+4.1
+HOI-UDA
+We propose a novel measurement for the challenge of
+Uneven Distributed Area in Human-Object Pairs, the
+formulation is as follow:
+𝐴𝑅 = 𝐴𝑟𝑒𝑎ℎ · 𝐴𝑟𝑒𝑎𝑜
+𝐴𝑟𝑒𝑎2
+ℎ𝑜𝑖
+,
+(20)
+where 𝐴𝑟𝑒𝑎ℎ, 𝐴𝑟𝑒𝑎𝑜 and 𝐴𝑟𝑒𝑎ℎ𝑜𝑖 denote the area of human,
+object and HOI instances, respectively (as shown in Fig.8
+(a)). We quantify all the instances in the HAKE-HOI into ten
+intervals and count the number of instances of each interval
+in the second and fifth row of Table.1. To better evaluate the
+ability of the model to detect HOI for human-object pairs
+with uneven distributed areas, we specially select 24737
+HOI instances of IMIUDA
+0
+in testing set.
+4.2
+HOI-LDVM
+A novel measurement for the challenge of Long Distance
+Visual Modeling of Human-Object Pairs is proposed in
+Eq.21.
+𝐿𝑅 = 𝐿ℎ + 𝐿𝑜
+𝐿ℎ𝑜𝑖
+,
+(21)
+where 𝐿ℎ, 𝐿𝑜 and 𝐿ℎ𝑜𝑖 denote the size we define of human,
+object and HOI instances, respectively (as shown in Fig.8
+(b)). The instances are quantified in the third and sixth row
+of Table.1. To better evaluate the ability of the model to de-
+tect HOI for human-object pairs with with long distance, we
+specially select 24737 HOI instances of IMILDVM
+0
+∼ IMILDVM
+6
+in testing set.
+4.3
+HOI-SDC
+In order to avoid the training process of the model being
+influenced by a portion of HOI classes with a very small
+number of instances, we remove some of the HOI classes
+containing a very small number of instances and HOI
+classes with no interaction from the training Set for the
+Double Challenge. Finally, there are total 321 HOI classes,
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 10, JANUARY 2023
+9
+TABLE 2: Performance comparison with the state-of-the-art methods on the HICO-DET dataset. ’V’, ’S’, ’P’ and ’L’ represent
+the visual feature, spatial feature, human pose feature and language feature respectively. Fine-tuned Detection means the
+parameter of the model is pre-trained on the MS-COCO dataset. Backbone with ’*’ and ’+’ means that they are pre-trained
+on ImageNet-22K with 384×384 input resolution. QAHOI(R) represents that the results are reproduced on the same machine
+with our model. Kindly refer to Sec. 5.4.1 for more details.
+Architecture
+Method
+Backbone
+Fine-tuned
+Feature
+Default (↑)
+Known Object (↑)
+Full
+Rare
+Non-Rare
+Full
+Rare
+Non-Rare
+Two-Stage Methods
+Multi-stream
+No-Frill [23]
+ResNet-152
+
+A+S+P
+17.18
+12.17
+18.08
+-
+-
+-
+PMFNet [24]
+ResNet-50-FPN
+
+A+S
+17.46
+15.65
+18.00
+20.34
+17.47
+21.20
+ACP [25]
+ResNet-101
+
+A+S+L
+21.96
+16.43
+23.62
+-
+-
+-
+PD-Net [10]
+ResNet-152
+
+A+S+P+L
+22.37
+17.61
+23.79
+26.86
+21.70
+28.44
+VCL [7]
+ResNet-50
+
+A+S
+23.63
+17.21
+25.55
+25.98
+19.12
+28.03
+Graph-Based
+RPNN [8]
+ResNet-50
+
+A+P
+17.35
+12.78
+18.71
+-
+-
+-
+VSGNet [13]
+ResNet-152
+
+A+S
+19.80
+16.05
+20.91
+-
+-
+-
+DRG [12]
+ResNet-50-FPN
+
+A+S+L
+24.53
+19.47
+26.04
+27.98
+23.14
+29.43
+SCG [18]
+ResNet-50-FPN
+
+A+S
+31.33
+24.72
+33.31
+34.37
+27.18
+36.50
+One-Stage Methods
+Interaction points
+IP-Net [15]
+ResNet-50-FPN
+
+A
+19.56
+12.79
+21.58
+22.05
+15.77
+23.92
+PPDM [31]
+Hourglass-104
+
+A
+21.73
+13.78
+24.10
+24.58
+16.65
+26.84
+GGNet [11]
+Hourglass-104
+
+A
+23.47
+16.48
+25.60
+27.36
+20.23
+29.48
+Transformer-Based
+HOITrans [17]
+ResNet-101
+
+A
+26.60
+19.15
+28.54
+29.1
+20.98
+31.57
+HOTR [9]
+ResNet-50
+
+A
+23.46
+16.21
+25.65
+-
+-
+-
+ResNet-50
+
+A
+25.10
+17.34
+27.42
+-
+-
+-
+AS-Net [6]
+ResNet-50
+
+A
+24.40
+22.39
+25.01
+27.41
+25.44
+28.00
+ResNet-50
+
+A
+28.87
+24.25
+30.25
+31.74
+27.07
+33.14
+QPIC [19]
+ResNet-50
+
+A
+29.07
+21.85
+31.23
+31.68
+24.14
+33.93
+ResNet-50
+
+A
+24.21
+17.51
+26.21
+-
+-
+-
+QAHOI [33]
+Swin-Tiny
+
+A
+28.47
+22.44
+30.27
+30.99
+24.83
+32.84
+Swin-Large∗
++
+
+A
+35.78
+29.80
+37.56
+37.59
+31.66
+39.36
+QAHOI (R)
+Swin-Tiny
+
+A
+27.67
+20.22
+29.69
+30.06
+22.95
+32.18
+Swin-Large∗
++
+
+A
+35.43
+29.22
+37.29
+37.23
+31.01
+39.09
+FGAHOI
+Swin-Tiny
+
+A
+29.94
+22.24
+32.24
+32.48
+24.16
+34.97
+Swin-Large∗
++
+
+A
+37.18
+30.71
+39.11
+38.93
+31.93
+41.02
+74 object classes and 93 action classes. The training and
+testing set contain 37,155 and 9,666 images, respectively. The
+detailed distribution of HOI instances is shown in Table.1.
+(b)
+(a)
+������������������������������������������������������������
+������������������������������������������������������������
+������������������������������������������������������������������������������������
+������������������������
+������������������������
+������������������������������������������������
+Fig. 8: Proposed metrics for the difficulties existing with HOI
+instances. (a) Metric for uneven size distribution of humans
+and objects. (b) Metric for excessive distance between person
+and object. Kindly refer to Sec. 4.1 and 4.2 for more details.
+5
+EXPERIMENTS
+5.1
+Dataset
+Experiments are conducted on three HOI datasets: HICO-
+DET [38], V-COCO [39] and HOI-SDC dataset
+HICO-DET [38] has 80 object classes, 117 action classes
+and 600 HOI classes. HICO-DET offers 47,776 images with
+TABLE 3: Performance comparison with the state-of-the-art
+methods on the HOI-SDC dataset. Kindly refer to Sec. 5.4.2
+for more details.
+Dataset
+Backbone
+Method
+mAProle (↑)
+HOI-SDC
+Swin-Tiny
+QAHOI
+19.55
+Swin-Tiny
+Baseline
+21.18
+Swin-Tiny
++HSAM
+21.91
+Swin-Tiny
++TAM
+21.84
+Swin-Tiny
+FGAHOI
+22.25
+151,276 HOI instances, including 38,118 images with 117,871
+annotated instances of human-object pairs in the training set
+and 9658 images with 33,405 annotated instances of human-
+object pairs in the testing set. According to the number
+of these HOI classes, the 600 HOI classes in the dataset
+are grouped into three categories: Full (all HOI classes),
+Rare (138 classes with fewer than ten instances) and Non-
+Rare (462 classes with more than ten instances). Following
+HICO [63], we consider two different evaluation settings
+(the results are shown in Table.2: (1) Known object settings:
+For each HOI category (such as ’flying a kite’), the detection
+is only evaluated on the images that contain the target object
+category (such as ’kite’). The difficulty lies in the local-
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 10, JANUARY 2023
+10
+TABLE 4: Performance comparison with the state-of-the-art
+methods on the V-COCO dataset. Kindly refer to Sec. 5.4.3
+for more details.
+Method
+AP𝑆1
+role (↑)
+AP𝑆2
+role (↑)
+Two-stage Method
+VSG-Net
+51.8
+57.0
+PD-Net
+52.0
+-
+ACP
+53.2
+-
+One-stage Method
+HOITrans
+52.9
+-
+AS-Net
+53.9
+-
+HOTR
+55.2
+64.4
+DIRV
+56.1
+-
+QAHOI(R-50)
+58.2
+58.7
+FGAHOI(R-50)
+59.0
+59.3
+FGAHOI(Swin-T)
+60.5
+61.2
+ization of HOI (e.g. human-kite pairs) and distinguishing
+the interaction (e.g. ’flying’). (2) Default setting: For each
+HOI category, the detection is evaluated on the whole test
+set, including images containing and without target object
+categories. This is a more challenging setting because we
+also need to distinguish background images (such as images
+without ’kite’).
+V-COCO [39] contains 80 different object classes and
+29 action categories and is developed from the MS-COCO
+dataset, which includes 4,946 images for the test subset,
+2,533 images for the train subset and 2,867 images for the
+validation subset. The objects are divided into two types:
+“object” and “instrument”.
+5.2
+Metric
+Following the standard evaluation [21], [39], we use role
+mean average precious to evaluate the predicted HOI in-
+stances. A detected bounding box is considered a true
+positive for object detection if it overlaps with a ground
+truth bounding box of the same class with an intersection
+greater than union (𝐼𝑂𝑈) greater than 0.5. In HOI detection,
+we need to predict human-object pairs. The human-object
+pairs whose human overlap 𝐼𝑂𝑈ℎ and object overlap 𝐼𝑂𝑈𝑜
+both exceed 0.5, i.e., min (𝐼𝑂𝑈ℎ, 𝐼𝑂𝑈𝑜) > 0.5 are declared
+a true positive (as shown in Fig 9). Specifically, for HICO-
+DET, besides the full set of 600 HOI classes, the role mAP
+over a rare set of 138 HOI classes that have less than 10
+training instances and a non-rare set of the other 462 HOI
+classes are also reported. Furthermore, we report the role
+mAP of two scenarios for V-COCO: scenario 1 includes the
+cases even without any objects (for the four action categories
+of body motions), while scenario 2 ignores these cases. For
+HOI-SDC, we report the role mean average precision for the
+full set of 321 HOI classes.
+5.3
+Implementation Details
+The Visual Feature Extractor consists of Swin Transformer
+and a deformable transformer encoder. For Swin-Tiny and
+Swin-Large, the dimensions of the feature maps in the first
+stage are set to 𝐶𝑠 = 96 and 𝐶𝑠 = 192, respectively. We pre-
+train Swin-Tiny on the ImageNet-1k dataset. Swin-Large is
+first pre-trained on the ImageNet-22k dataset and finetuned
+Fig. 9: The human-object pairs with human overlap 𝐼𝑂𝑈ℎ
+and object overlap 𝐼𝑂𝑈𝑜 both exceeding 0.5 are declared as
+true positives. Kindly refer to Sec. 5.2 for more details.
+on the ImageNet-1k dataset. Then the weights are used
+to fine-tune the FGAHOI for the HOI detection task. The
+number of both encoder and decoder layers are set to 6
+(𝑁𝐿𝑎𝑦𝑒𝑟 = 6). The number of query embeddings is set to 300
+(𝑁𝑞 = 300), and the hidden dimension of embeddings in the
+transformer is set to 256 (𝐶𝑑 = 256). In the post-processing
+phase, the first 100 HOI instances are selected according
+to object confidence, and we use 𝛿=0.5 to filter the HOI
+instances by the combined 𝐼𝑂𝑈. Following Deformable-
+DETR [34], the AdamW [64] optimizer is used. The learning
+rates of the extractor and the other components are set to
+10−5 and 10−4, respectively. We use 8 RTX 3090 to train the
+model (QAHOI & FGAHOI) with Swin-Tiny. For the model
+with Swin-Large∗
++, we use 16 RTX 3090 to train them. For
+HICO-DET and HOI-SDC, we train the base network for
+150 epochs and carry out the learning rate drop from the
+120th epoch at the first stage of training. For subsequent
+training, we trained the model for 40 epochs, with a learning
+rate drop at the 15th epoch. For V-COCO dataset, we train
+the base network for 90 epochs and drop the learning rate
+from 60th epoch at the first stage of training. For subsequent
+training, we trained the model for 30 epochs, with a learning
+rate drop at the 10th epoch.
+5.4
+Comparison with State-of-the-Arts
+5.4.1
+HICO-DET
+We compare FGAHOI with the state-of-the-art two-stage
+and one-stage methods on the HICO-DET dataset and
+report the results in Table.1. FGAHOI outperforms both
+state-of-the-art methods. In contrast to the state-of-the-art
+two-stage method SCG [18], FGAHOI with Swin-Large*+
+backbone exceeds an especially significant gain of 5.85 mAP
+in default full setting, 5.99 mAP in default rare setting, 5.8
+mAP in default non-rare setting, 4.56 mAP in known object
+full setting, 4.75 mAP in known rare settings and 4.52 mAP
+in known object non-rare setting. For a fair comparison, we
+used the same machine for the reproduction of the QAHOI
+(as shown in Table.2 QAHOI(R)). In comparison to the state-
+of-the-art one-stage method QAHOI, FGAHOI exceeds it
+in all settings for all backbone networks. For Swin-Tiny
+backbone network, FGAHOI exceeds an especially signifi-
+cant gain of 2.27 mAP in default full setting, 2.02 mAP in
+default rare setting, 2.55 mAP in default non-rare setting,
+2.42 mAP in known object full setting, 1.11 mAP in known
+rare settings and 2.79 mAP in known object non-rare setting.
+In addition, FGAHOI with Swin-Large*+ backbone exceeds
+an especially significant gain of 1.75 mAP in default full
+
+OU
+IOU.
+Ground-truth label
+Prediction boxesJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 10, JANUARY 2023
+11
+TABLE 5: Comparison on ten intervals of the two proposed challenges. We divide the HICO-DET dataset into ten intervals
+based on each of the two challenges and compare the performance of QAHOI and FGAHOI on each interval. Kindly refer
+to Sec. 5.5 for more details.
+Challenge
+Method
+Backbone
+mAProle (↑)
+IMI0
+IMI1
+IMI2
+IMI3
+IMI4
+IMI5
+IMI6
+IMI7
+IMI8
+IMI9
+UDA
+QAHOI
+Swin-Tiny
+16.35
+24.72
+29.24
+34.79
+38.70
+46.21
+53.13
+47.60
+58.66
+60.19
+Swin-Large∗
++
+20.53
+33.58
+41.11
+45.41
+45.44
+56.43
+56.25
+63.53
+71.12
+75.08
+FGAHOI
+Swin-Tiny
+19.74
+29.85
+32.20
+39.46
+40.54
+48.55
+51.32
+46.50
+66.44
+78.17
+Swin-Large∗
++
+23.69
+35.85
+42.51
+50.50
+46.89
+56.95
+56.33
+63.04
+75.70
+79.42
+LDVM
+QAHOI
+Swin-Tiny
+1.33
+4.43
+2.57
+5.00
+8.06
+17.87
+22.81
+29.25
+34.03
+42.29
+Swin-Large∗
++
+0.82
+4.08
+2.56
+7.53
+11.42
+22.87
+30.94
+41.38
+45.31
+60.15
+FGAHOI
+Swin-Tiny
+2.50
+4.15
+3.34
+7.58
+9.83
+21.61
+27.64
+33.07
+38.31
+45.07
+Swin-Large∗
++
+1.44
+4.32
+4.57
+7.81
+11.82
+24.92
+32.50
+43.66
+47.26
+60.55
+TABLE 6: We carefully ablate each of the constituent component of FGAHOI. The middle results denote the role mAP. The
+results in the top right corner represent the performance improvement compared to QAHOI. The results in the bottom
+right corner represent the performance improvement compared to the baseline. Kindly refer to Sec. 5.7.1 for more details.
+Method
+Merging Mechanism
+Default
+Known Object
+Hierarchical Spatial-Aware
+Task-Aware
+Full ↑
+Rare ↑
+Non-Rare ↑
+Full ↑
+Rare ↑
+Non-Rare ↑
+QAHOI
+-
+-
+27.67
+20.22
+29.69
+30.06
+22.95
+32.18
+FGAHOI
+
+
+28.45( +0.78 )
+(
+-
+)
+21.07( +0.85 )
+(
+-
+)
+30.66( +0.97 )
+(
+-
+)
+31.08( +1.02 )
+(
+-
+)
+24.02( +1.01 )
+(
+-
+)
+33.19( +1.07 )
+(
+-
+)
+
+
+29.60( +1.93 )
+( +1.15 )
+22.39( +2.17 )
+( +1.32 )
+31.76( +2.07 )
+( +1.10 )
+32.07( +2.01 )
+( +0.99 )
+24.48( +1.53 )
+( +0.46 )
+34.34( +2.16 )
+( +1.15 )
+
+
+29.32( +1.65 )
+( +0.87 )
+22.34( +2.12 )
+( +1.27 )
+31.41( +1.72 )
+( +0.75)
+31.81( +1.75 )
+( +0.73)
+24.30( +1.35 )
+( +0.28)
+34.05( +1.87 )
+( +0.86)
+
+
+29.94( +2.27 )
+( +1.49 )
+22.24( +2.02 )
+( +1.17 )
+32.24( +2.55 )
+( +1.58 )
+32.48( +2.42 )
+( +1.40 )
+24.16( +1.21 )
+( +0.14 )
+34.97( +2.79 )
+( +1.78 )
+setting, 1.49 mAP in default rare setting, 1.82 mAP in default
+non-rare setting, 1.7 mAP in known object full setting, 0.92
+mAP in known rare settings and 1.93 mAP in known object
+non-rare setting.
+5.4.2
+HOI-SDC
+On the dataset we propose, 𝑖.𝑒., HOI-SDC, we compare
+FGAHOI with QAHOI and ablate each component of FGA-
+HOI (As shown in Table.3). The backbone is set to Swin-Tiny.
+The baseline exceeds QAHOI an especially significant gain
+of 1.63 mAP. HSAM and TAM improve a significant gain
+of 0.73 and 0.66 mAP, respectively. Benefit from the MSS,
+HSAM and TAM, FGAHOI achieve 22.25 mAP on HOI-
+SDC.
+5.4.3
+V-COCO
+We compare FGAHOI with the state-of-the-art methods
+on V-COCO dataset and report the results in Table.4. In
+comparison to QAHOI, FGAHOI only exceeds a small
+margin. This phenomenon is mainly caused by too little
+training data in the dataset. We investigate that FGAHOI
+cannot adequately perform when the training data is not
+sufficient due to the complex task requirements. In addition,
+we investigate the transformer backbone is still superior to
+CNN backbone in this case.
+5.5
+Sensitivity Analysis for UDA and LDVM
+According to the two proposed challenges, we divide the
+HICO-DET into ten intervals. At each intervals, we compare
+FGAHOI and QAHOI with Swin-Tiny, Large∗
++ backbone,
+respectively (As shown in Table.5). When compared be-
+tween each interval of UDA and LDVM, we investigate
+that the difficulty of HOI detection decreases as the interval
+level increases. This justifies the original design. Thus, it
+is imperative to consider ability of the model to address
+these two challenges when proposing novel frameworks for
+HOI detection. In the comparison between FGAHOI and
+QAHOI, the results demonstrate that FGAHOI has better
+capability for uneven distributed area and long distance
+visual modeling of human-object pairs.
+5.6
+Qualitative Analysis
+5.6.1
+Visualized Results
+In order to demonstrate our model, several representative
+HOI predictions are visualized. As shown in Fig.5, our
+model can pinpoint HOI instances from noisy backgrounds
+and excels at detecting various complicated HOIs, including
+one object interacting with different humans, one human
+engaging in multiple interactions with various objects, mul-
+tiple interactions within a single pair, and multiple humans
+engaging in various interactions with various objects. In
+addition, our model is good at long-range visual modelling,
+withstanding the impacts of hostile environments and small
+target identification. Fig.6 (a) illustrates that FGAHOI has
+excellent long-range visual modelling capabilities and can
+accurately identify interactions between human-object pairs
+far from each other. As Fig.6 (b) shows, our model has
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 10, JANUARY 2023
+12
+text_on cell_phone
+talk_on cell_phone
+eat orange
+open book
+cut with knife
+repair hair_drier
+hold hotdog
+hop_on elephant
+kick sports_ball
+hold cup
+carry handbag
+jump skateboard
+hold cup
+drink with cup
+text_on cell_phone
+talk_on cell_phone
+eat orange
+open book
+cut with knife
+repair hair_drier
+hold hotdog
+hop_on elephant
+kick sports_ball
+hold cup
+carry handbag
+jump skateboard
+hold cup
+drink with cup
+(a)
+(b)
+(c)
+Level_0
+Level_0
+Level_1
+Level_1
+Level_2
+Level_2
+Low
+High 
+Read
+Laptop
+Read
+Laptop
+Low
+High 
+Read
+Laptop
+Exit
+Airplane
+Sit on
+Airplane
+Hold
+Horse
+Ride
+Horse
+Fly Kite
+(a)
+(b)
+(c)
+Level_0
+Level_1
+Level_2
+Low
+High 
+Read
+Laptop
+Exit
+Airplane
+Sit on
+Airplane
+Hold
+Horse
+Ride
+Horse
+Fly Kite
+Fig. 10: Visualization of fine-grained anchors in the decoding phase, Level 0, Level 1 and Level 2 represent the features
+at different scales respectively, the color of the blue dots from light to dark represents the degrees of attention of the fine-
+grained anchors and red dots represent the positions of interest of fine-grained anchors in current scale features. Kindly
+refer to Sec. 5.6.2 for more details.
+outstanding robustness and can effectively resist disruption
+from harsh environmental factors, including blurring, block-
+ing and glare. Fig.6 (c) demonstrates the superior capabili-
+ties of FGAHOI to identify small HOI instances.
+5.6.2
+What do the fine-grained anchors look at?
+As shown in Fig.7, we compare the fine-grained anchors of
+FGAHOI and QAHOI. First two HOI instances (𝑖.𝑒, hold
+sport ball and ride motorcycles) exhibit that FGAHOI could
+better focus on humans, objects and the interaction areas
+rather than noisy backgrounds. The fourth head of FGAHOI
+still focuses on the HOI instance, while QAHOI focuses
+on the background. When detecting instance with a long
+distance between human and object, FGAHOI could focus
+on the right position, while QAHOI is like a chicken with its
+head cut off (As shown in the last HOI instance).
+To exhibit the effectiveness of the fine-grained anchors
+for identifying HOI instances and demonstrate the working
+mechanism of fine-grained anchors, we visualize the fine-
+grained anchors of the feature maps at different scales in
+the decoding phase. In Fig.10 (a), we visualize the instances
+of two different humans and one object. As shown in Fig.10
+(b), even for exactly the same human-object pair, the areas
+of focus vary from one interaction to another. In Fig.10 (c),
+
+IS人
+D2MVAD2MVAD2MVAD2MVAJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 10, JANUARY 2023
+13
+text_on cell_phone
+talk_on cell_phone
+eat orange
+ride bicycle
+cut with knife
+repair hair_drier
+hold hotdog
+hop_on elephant
+kick sports_ball
+hold cup
+carry handbag
+jump skateboard
+hold cup
+drink with cup
+text_on cell_phone
+talk_on cell_phone
+eat orange
+ride bicycle
+cut with knife
+repair hair_drier
+hold hotdog
+hop_on elephant
+kick sports_ball
+hold cup
+carry handbag
+jump skateboard
+hold cup
+drink with cup
+Fig. 11: Visualization of several representative interactive actions and the corresponding fine-grained anchors. We only
+visualize a single representative interactive action for one human-object pair. Kindly refer to Sec. 5.6.2 for more details.
+we show two instances contain short and long distance
+between humans and objects, respectively. We investigate
+that the fine-grained anchors of low level feature map focus
+on small and fine-grained areas. They play a major role in
+detecting close range and small HOI instances. The fine-
+grained anchors of high level feature maps focus on large
+and coarse-grained areas. It is necessary for detecting long
+distance and large HOI instances.
+In order to explore what the fine-grained anchors focus
+on, we visualize several representative actions in Fig.11.
+Visualization shows that fine-grained anchors could con-
+centrate attention precisely on the location where the in-
+teractive action is generated. For example, the fine-grained
+anchors mainly focus on the hand for ’text on cell phone’,
+the mouth for ’eat orange’ and the ear and the mouth
+for ’talk on cell phone’. For ’kick sports ball’, ’jump skate-
+board’ and ’hop on elephant’, central areas of interest are
+around legs and feet, while fine-grained anchors primarily
+focuses on hands for ’carry handbag’, ’repair hair drier’,
+’hold cup’, ’hold hotdog’ and ’cut with kinfe’.
+5.7
+Ablation Study
+In this subsection, a set of experiments are designed to
+clearly understand the contribution of each of the con-
+stituent components of the proposed methodology: Merg-
+ing mechanism, Multi-Scale Sampling Strategy and Stage-
+wise Training Strategy. We conducted all experiments on
+the HICO-DET dataset.
+5.7.1
+Ablating FGAHOI Components
+To study the contribution of each of the merging mecha-
+nisms in FGAHOI, we design careful ablation experiments
+in Table.6. To ensure a fair comparison, the sampling sizes
+are all set to [1, 3, 5]. For the baseline which does not lever-
+ages the hierarchical spatial-aware and task-aware merging
+mechanism, we use the average and direct summation op-
+eration to merge the sampled features and connect embed-
+dings. For the results in the table, the middle results denote
+the role mAP, the results in the top right corner represent
+the performance improvement compared to QAHOI and the
+results in the bottom right corner represent the performance
+improvement compared to the baseline. In comparison to
+row 1 (QAHOI), row 2 adds the multi-scale sampling
+strategy. The results demonstrate that adding the sampling
+strategy improves the ability of the model to detect HOI
+instances. The row 3 and 4 show that both hierarchical
+spatial-aware and task-aware merging mechanism make
+an essential contribution to the success of FGAHOI. The
+hierarchical spatial-aware merging mechanism, combined
+with the task-aware merging mechanism performs better
+together (row 5) than using either of them separately (row 3
+and 4). Thus, each component in FGAHOI has a critical role
+to play in HOI detection.
+5.7.2
+Sensitivity Analysis On Multi-Scale Sampling Sizes
+Our multi-scale sampling strategy samples multi-scale fea-
+tures according to the pre-determined sampling sizes. We
+vary different sampling sizes to conduct the sensitivity
+
+EJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 10, JANUARY 2023
+14
+analysis for the sampling strategy and report the results
+in Table.7. We find that the sampling strategy is relatively
+stable. Changes in sampling sizes do not have a significant
+impact on the performance of FGAHOI. However, there is
+still a slight degradation in the performance of FGAHOI as
+the sample size increases. We investigate that as the sample
+size increases, too many background features around the
+fine-grained anchors are sampled, resulting in contamina-
+tion of the sampled features and thus the performance of
+the model suffers. Hence, for validation, we set the sampling
+sizes to [1, 3, 5] in all our experiments, which is a sweet spot
+that balances performance.
+TABLE 7: Comparison between different sampling sizes.
+Smpling Size
+Default
+Known Object
+Full
+Rare
+Non-Rare
+Full
+Rare
+Non-Rare
+[ 1, 3, 5 ]
+29.94
+22.24
+32.24
+32.48
+24.16
+34.97
+[ 3, 5, 7 ]
+29.72
+23.03
+31.72
+32.33
+25.67
+34.30
+[ 5, 7, 9 ]
+29.65
+22.64
+31.74
+32.55
+25.64
+34.62
+5.7.3
+Training Strategies
+As shown in Table.8, we leverage the stage-wise and end-
+to-end training strategy to train FGAHOI, respectively. In
+the end-to-end training strategy, we train FGAHOI for 150
+epochs and the learning rate drop is carried out at the
+120th epoch. The stage-wise training strategy promotes 5.96
+mAP for default full setting, 4.61 for default rare, 6.36 for
+default non-rare, 6.04 for known object full, 4.65 for known
+object rare and 6.46 mAP for known object non-rare setting.
+In comparison to the end-to-end training strategy, we in-
+vestigate that the stage-wise training strategy reduces the
+learning difficulty of the FGAHOI and clarify the learning
+direction of the model by emphasizing it to learn what it
+needs at each stage.
+TABLE 8: Comparison between Stage-Wise and End-to-End
+training approach.
+Training Strategy
+Default
+Known Object
+Full
+Rare
+Non-Rare
+Full
+Rare
+Non-Rare
+Stage-Wise
+29.94
+22.24
+32.24
+32.48
+24.16
+34.97
+End-to-End
+23.98
+17.63
+25.88
+26.44
+19.51
+28.51
+6
+CONCLUSION
+In this paper, we propose a novel transformer-based human-
+object interaction detector (FGAHOI) which leverages the
+input features to generate fine-grained anchors for protect-
+ing the detection of HOI instances from noisy backgrounds.
+We propose a novel training strategy where each component
+of the model is trained sequentially to clarify the training
+direction at each stage, for maximizing the savings of the
+training cost. We propose two novel metrics and a novel
+dataset, 𝑖.𝑒., HOI-SDC for the two challenges (Uneven Dis-
+tributed Area in Human-Object Pairs and Long Distance
+Visual Modeling of Human-Object Pairs) of detecting HOI
+instances. Our extensive experiments on three benchmarks:
+HICO-DET, HOI-SDC and V-COCO, demonstrate the effec-
+tiveness of the proposed FGAHOI. Specifically, FGAHOI
+outperforms all existing state-of-the-art methods by a large
+margin.
+ACKNOWLEDGMENTS
+This work is supported by National Natural Science Foun-
+dation of China (grant No.61871106 and No.61370152),
+Key R&D projects of Liaoning Province, China (grant
+No.2020JH2/10100029), and the Open Project Program
+Foundation of the Key Laboratory of Opto-Electronics In-
+formation Processing, Chinese Academy of Sciences (OEIP-
+O-202002).
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+

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@@ -0,0 +1,2097 @@
+Asymptotics in an Asymptotic CFT
+Lucas Schepersa1 and Daniel C. Thompsona,b2
+a Department of Physics, Swansea University,
+Swansea SA2 8PP, United Kingdom
+b Theoretische Natuurkunde, Vrije Universiteit Brussel,
+& The International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium
+Abstract
+In this work we illustrate the resurgent structure of the λ-deformation; a two-
+dimensional integrable quantum field theory that has an RG flow with an SU(N)k
+Wess-Zumino-Witten conformal fixed point in the UV. To do so we use modern
+matched asymptotic techniques applied to the thermodynamic Bethe ansatz formu-
+lation to compute the free energy to 38 perturbative orders in an expansion of large
+applied chemical potential. We find numerical evidence for factorial asymptotic be-
+haviour with both alternating and non-alternating character which we match to
+an analytic expression. A curiosity of the system is that it exhibits the Cheshire
+Cat phenomenon with the leading non-alternating factorial growth vanishing when
+k divides N. The ambiguities associated to Borel resummation of this series are
+suggestive of non-perturbative contributions.
+This is verified with an analytic
+study of the TBA system demonstrating a cancellation between perturbative and
+non-perturbative ambiguities.
+[email protected]
+[email protected]
+arXiv:2301.11803v1  [hep-th]  27 Jan 2023
+
+1
+Introduction
+A complete understanding of the strong coupling dynamics of four dimensional asymp-
+totically free (AF) non-supersymmetric gauge theory, i.e. QCD, remains elusive. To
+gain a foothold we may turn to simplified toy models. One strategy is to reduce the
+dimensionality of the problem considering instead two dimensional quantum field the-
+ories with similar RG behaviour. In the special case of integrable QFTs, an infinite set
+of symmetries completely determine the exact S-matrix [1] providing a powerful toolkit
+that can be used to tackle non-perturbative questions. An early success of this approach
+was the calculation of the exact ratio of the mass gap to cut-off of AF integrable QFTs
+theories [2–5].
+More recently, techniques in integrable models have been used to elucidate even
+deeper questions of the nature of perturbation theory. Typically, perturbation theory is
+asymptotic in nature with perturbative coefficients growing factorially. The programme
+of resurgence asserts that this breakdown of convergence signals the need to include
+non-perturbative physics. Even more strongly, ambiguities inherent in resummations of
+asymptotic perturbative expansions should be cancelled by compensating ambiguities
+in a non-perturbative sector. For a modern overview of resurgence from a physics view
+point see e.g. [6]. Again, integrable two-dimensional models provide an ideal test bed
+for resurgence.
+In semi-classical approaches [7–12], an adiabatic compactification of two-dimension
+models is used to obtain a quantum mechanics which can be probed to large perturbative
+orders. In these cases, two-dimensional finite Euclidean action configurations, known as
+unitons1 are shown to precisely resolve the semi-classical ambiguities of the perturbative
+sector. Whilst intriguing, such approaches intrinsically disregard degrees of freedom in
+compactification restricting to the lowest KK sector. Alongside this features of renorm-
+alisation group are disregarded in the truncation to Quantum Mechanics. Given these
+limitations, one thus prompted to ask if the resurgence paradigm can be established in
+a fully two-dimensional setting.
+A breakthrough was the work of Volin [13, 14], recently refined in [15, 16] (see also the
+recent papers [17–25]), in addressing the Thermodynamic Bethe Ansatz (TBA) system
+that determines free energy in a large chemical potential. By comparing two scaling
+limits, it is possible to reduce the complicated integral TBA equation to a (complicated)
+set of algebraic equations that fix unknown coefficients in an ansatz for a perturbative
+expansion (in the chemical potential or other more refined coupling). This allows access
+to sufficient order in perturbation theory to reveal factorial divergence of perturbative
+coefficients.
+Although we cannot identify an instanton or other semi-classical non-perturbative
+saddle in the TBA approach, we can find a matching ambiguity using a different method.
+[26] showed that it is possible to solve the TBA equations using a transseries. A critical
+step in their solution is an arbitrary choice of branch cut which introduces an ambiguity
+of the transseries. Although this approach, due to its computational difficulty, cannot
+be executed to large orders, it does exhibit a non-perturbative ambiguity that matches
+the ambiguity of the large-order behaviour found in the perturbative sector.
+In this note we will adopt this toolbox to study the resurgent structure of a theory
+that exhibits a different renormalisation group dynamic. We consider a theory in which
+the UV is not Gaussian but instead is described by a non-trivial interacting conformal
+fixed point. The theory we will consider, known as the λ-model [27, 28], is realised as a
+flow away from an SU(N)k Wess-Zumino-Witten (WZW) model driven at leading order
+by a certain current-current bilinear. The IR of the theory is the principal chiral model,
+expressed in non-Abelian T-dual coordinates, and accordingly is gapped. Whilst this
+1Unlike instantons, unitons are not topologically protected.
+1
+
+marginally relevant deformation breaks conformality and the full affine symmetry of the
+WZW current algebra, it does preserve an infinite symmetry associated to integrablity.
+At the quantum level the exact S-matrix is known (based on symmetry grounds pre-
+dating the Lagrangian description) [29, 30] and has been shown to match the λ-model
+Lagrangian using a light cone lattice discretisation and Quantum Inverse Scattering [31].
+The goal of this note is to match the perturbative ambiguity to that of the transseries
+the new context of a λ-model. The outline is as follows: Section 2 provides a more in-
+depth discussion of the λ-model as we consider its RG flow in more detail and present
+its exact S-matrix. In Section 3, we review the recent techniques to perturbatively solve
+TBA equation [13–16] that determine free energy.
+Introducing a special coupling γ
+in Section 3.4 results in a clean (i.e. log-free) series for the λ-model. We analyse its
+asymptotic behaviour in Section 4 and compute the leading ambiguity. This ambiguity
+is matched by a transseries calculation in Section 5. A particularly eye-catching result
+is that the leading UV ambiguity disappears when N divides k. We wrap up with ideas
+for future research in Section 6.
+2
+The λ-Model
+In this section we outline the salient properties of the two-dimensional integrable QFT
+that we are considering: the λ-deformed model. Classically, the λ-model provides a
+Lagrangian interpolation between the conformal Wess-Zumino-Witten (WZW) model
+for a Lie-group G [32] and the principal chiral model (PCM) (written in non-Abelian
+T-dual variables). Remarkably this theory is integrable for all values of the eponymous
+interpolating parameter λ related to the level, k, of WZW and the radius, r, of the PCM
+by
+λ =
+k
+k + r2 .
+(1)
+Whilst the λ-model for the restricted case of G = SU(2) was first proposed long ago
+[29, 33, 34], the pioneering work of Sfetsos [27] in constructing the general theory has
+prompted extensive recent development (for reviews see [35, 36]). The λ-model has been
+extended to Z2 graded symmetric spaces [27, 37] where it constitutes an interpolation
+between a G/H gauged WZW (representing the coset CFT) and the (non-abelian T-dual
+of) the principal chiral model on G/H and even to Z4 graded super-cosets relevant to
+the construction of the AdS5 × S5 superstring [38] underpinned by an elegant quantum
+group at root-of-unity symmetry structure. In this string theory context, λ-deformation
+is in fact marginal, and the world sheet theory can be viewed as a σ-model in some
+target space super-gravity background [39–41]. Here however we will be considering the
+simpler bosonic case for which the λ-deformation does not define a CFT but rather a
+relevant RG flow from a WZW fixed point in the UV to the dualised PCM the IR [28,
+42, 43]. A series of papers [44–48] have shown how the λ-model is actually part of a wide
+tapestry of integrable deformed models linked by (analytically continued) Poisson-Lie
+T-duality transformations.
+2.1
+Lagrangian Construction
+First we sketch the construction of the non-abelian T-dual of the PCM using the Buscher
+procedure [49] as it informs the construction of the λ-model. We start with the action
+2
+
+of the PCM for a group valued field ˜g 2
+SPCM[˜g] = − r2
+4π
+�
+d2σ Tr
+�
+˜g−1∂+˜g˜g−1∂−˜g
+�
+,
+(2)
+and downgrade the left symmetry ˜g → h−1˜g to a gauge symmetry by introducing a
+gauge connection transforming as A → h−1dh + h−1Ah and replacing derivatives to
+covariant derivatives d → D = d + A. This yields the gauged PCM action which we
+denote as SgPCM[˜g, A]. To ensure that the gauged theory is actually equivalent to the
+ungauged theory (at least in trivial topology which we assume throughout) we enforce
+that the connection is flat (i.e. the gauge field is pure gauge). This is implemented by
+introducing a Lagrange-multiplier term, − Tr(νF+−), to the Lagrangian. Integrating
+out the field ν enforces that the field strength F+− vanishes and we recover the original
+PCM after gauge-fixing ˜g = 1. However, if instead we integrate out the gauge fields A,
+after gauge fixing ˜g = 1, we obtain the non-abelian T-dual model in which the field ν
+becomes the fundamental field.
+The construction of the λ-model by Sfetsos [27] is achieved through a modification of
+this Buscher procedure. Instead of adding a Lagrange multiplier term, we add a gauged
+WZW term. Recall that the WZW model is given by
+SWZW,k[g] = − k
+2π
+�
+Σ
+d2σ Tr
+�
+g−1∂µgg−1∂µg
+�
+− ik
+6��
+�
+M3
+Tr
+�
+g−1dg
+�3 ,
+(3)
+in which g is extended to a 3-manifold M3 with boundary ∂(M3) = Σ.
+Standard
+arguments [32] ensure that the path integral is well-defined (independent of choice of
+extension) provided that k is appropriately quantised, and in particular for G = SU(N)
+which we assume henceforth, k ∈ Z. In this sector we gauge the diagonal symmetry
+g → h−1gh leading to a gauged WZW model action SgWZW,k[g, A] [50, 51].
+To construct the λ-model we combine a gauged PCM and a gauged WZW model:
+Sλ,k[g, ˜g, A] = SgPCM[˜g, A] + SgWZW[g, A] .
+(4)
+Notice that the two models are coupled through the fact that they are gauged by the
+same gauge field. The Sfetsos procedure is concluded by gauge fixing ˜g = 1 and integ-
+rating out the gauge field A using its on-shell value
+A+ = λ(1 − λAdg)−1R+ ,
+A− = −λ(1 − λAdg−1)−1L− ,
+(5)
+where we defined R± = ∂±gg−1 and L± = g−1∂±g and the adjoint action AdgX =
+gXg−1. Integrating out the gauge field then yields the action
+Sλ,k[g] = SWZW,k[g] + kλ
+π
+�
+Σ
+d2σ Tr
+�
+R+(1 − λAdg)−1L−
+�
+.
+(6)
+Though not vital for what follows we note that the equation of motion can be understood
+as a zero-curvature condition on the Lax connection [27, 52]
+L±(z) = −
+2
+1 + λ
+A±
+1 ∓ z ,
+(7)
+in which A± are evaluated with the on-shell values eq. (5) and z ∈ C is a spectral
+parameter. This is the starting point of establishing the classical integrability of the
+theory. Further to this one requires strong integrability i.e. that the conserved charges
+built from the monodromy of this Lax are in involution. This is ensured provided that
+the Poisson algebra of the spatial component of the Lax has a particular r-s Maillet form
+as was demonstrated for the λ-model, and its generalisations, in [47, 53, 54].
+2We use light cone coordinates σ± = 1
+2 (t ± x). Derivatives with respect to light cone coordinates
+are denoted by ∂±.
+3
+
+2.2
+Renormalisation
+The parameter λ given by eq. (1) varies from 0 to 1 and we shall now discuss what
+happens in each of those limits. At a quantum level the parameter λ undergoes an RG
+flow [28, 42, 43] given by (to all orders in λ and leading in 1
+k)
+µdλ
+dµ = β(λ) = −2N
+k
+�
+λ
+1 + λ
+�2
+= −β1λ2 − β2λ3 + O(λ4) .
+(8)
+The leading order behaviour, which shall be relevant later, is given by
+β1 = 2N
+k ,
+β2 = −4N
+k .
+(9)
+There is an evident UV fixed point at λ = 0, corresponding to the undeformed WZW
+model. In the vicinity of this λ ≈ 0, or k ≪ r2, we obtain a current-current deformation
+of the WZW model:
+Sλ,k[g] = SWZW,k[g] + λ
+�
+Σ
+d2σ Tr (R+L−) + O(λ2) ,
+(10)
+This, however, is not a marginal deformation (cf. marginal ones [55]), but relevant as it
+moves away from the WZW theory located in the UV.
+To understand the IR regime as λ → 1, we can force k → ∞. If the group element
+is expanded as g = 1 + i
+kνata, the action SgWZW,k reduces to the Lagrange multiplier
+term − Tr(νF+−). Thus in this limit the Sfetsos procedure reduces to the non-Abelian
+T-dualisation Buscher procedure described above. Hence, in this IR limit, we recover the
+non-abelian T-dual of the PCM. Further into the deep IR, one thus anticipates (as with
+the PCM) that the dimensionless parameter λ is transmuted into a mass gap mediated
+through a cut-off Λ.
+2.3
+Quantum Integrability
+Not only is the theory classical integrable, it remains so at the quantum level. The
+existence of higher spin conserved currents ensure that the scattering matrix of the
+theory factorises, and can be fully determined with the 2-to-2 particle scattering matrix
+the fundamental building block. The S-matrix for the SU(N) λ-model was constructed
+many years ago [56] from an algebraic perspective, and was related directly to the
+Lagrangian description for the SU(2) case in [29] by matching the free energy obtained
+by Lagrangian perturbation theory and by S-matrix TBA techniques. Following the
+introduction of the Lagrangian description λ-model by Sfetsos [27] for SU(N) the exact
+S-matrix was conjectured [52] for general ranks. This conjecture was substantiated by
+Appadu et al. [31] in which the form of the S-matrix was ‘derived’ by the Quantum
+Inverse Scattering Method (i.e. a latticed version of the theory that takes the form of a
+spin chain such from the QFT particle states are obtained as excitations over the ground
+state in a continuum limit)3. Rather than present the full details of the S-matrix (for
+which see [52]) we can give a schematic understanding somewhat mirroring the Sfetsos
+procedure.
+3This QISM is in fact rather non-trivial as the δ′(σ) non-ultra-local terms in the fundamental Poisson
+bracket preclude a simple application of QISM. Instead what is proposed is a modification of the
+λ-model, that lies in the same universality class, to which QISM can be applied.
+This provides a
+description as a spin-k XXX spin chain with alternating inhomogeneities.
+This idea was expanded
+to a two-parameter integrable λ-type model [57] realised as a spin-k XXZ spin chain with alternating
+inhomogeneities.
+4
+
+We start with the SU(N) principal chiral model which has in particular an SU(N)L×
+SU(N)R global symmetry. The fundamental particles are massive and transform in fun-
+damental antisymmetric tensor representations of the global symmetry. The scattering
+depends kinematically only on the rapidity difference θ of the particles4. Reflecting this
+global symmetry, the S-matrix of these fundamental excitations has a schematic tensor
+form (suppressing explicit representation labels)
+SPCM(θ) = X(θ)S(θ) ⊗ S(θ) ,
+(11)
+where X(θ) is an overall scalar dressing factor to ensure all S-matrix axioms are obeyed,
+and the S(θ) factors are separately SU(N) invariant (in fact invariant under a larger
+Yangian symmetry). Recalling that in the Sfetsos procedure the left acting SU(N)L
+symmetry was gauged, it is natural that the left hand block of the tensor product of eq.
+(11) is modified in the λ-theory and indeed this is the case with
+Sλ(θ) = Xk(θ)Sk(θ) ⊗ S(θ) .
+(12)
+Here Sk(θ) is a block [56] that furnishes a quantum group symmetry at the q2(k+N) = 1
+root of unity taken in Restricted-Solid-On-Solid (RSOS) picture representing the scat-
+tering of kink degrees of freedom.
+Given knowledge of the exact S-matrix, the Thermodynamic Bethe Ansatz yields
+a set of rather complicated coupled-integral equations can be used to determine the
+free-energy of the theory. Solving these is quite formidable especially as the S-matrix is
+non-diagonal. A powerful simplification is achieved by exposing the system to a chemical
+potential h for a U(1) charge such that only certain particles condense and contribute
+to the ground state. When the charge is chosen appropriately (as the one defined by
+a highest weight of a rank N/2 antisymmetric representation [31]) then only a single
+particle of maximal charge contributes and the TBA system simplifies to a single integral
+equation determined by the identical scattering of this particle.
+In this case, the scattering “matrix” reduces to a simple phase factor S(θ) that
+governs transmission and reflection.
+It shall prove useful in this case to define the
+scattering kernel of this reduced S-matrix by
+K(θ) =
+1
+2πi
+d
+dθ log S(θ) ,
+(13)
+and its Fourier transform
+K(ω) =
+� ∞
+−∞
+dθ eiωθK(θ) .
+(14)
+As a consequence of Hermitian analyticity on the reduced S-matrix, both K(θ) and its
+Fourier transform are symmetric functions. Explicitly we have that the relevant kernel
+is given by [57]
+1 − K(ω) =
+sinh2(πω/2)
+sinh(πω) sinh(πκω) exp(πκω) ,
+(15)
+where κ =
+k
+N . In what follows, it shall prove useful to write the Fourier transform of
+the scattering kernel as a Wiener-Hopf (WH) decomposition
+1 − K(ω) =
+1
+G+(ω)G−(ω) ,
+(16)
+where G−(ω) = G+(−ω), and G+(ω) is analytic in the Upper Half Plane (UHP) and
+normalised such that G+(2is) = 1 + O
+�
+1
+s
+�
+. Explicitly G+(ω) is given by
+G+(ω) =
+√
+4κ
+Γ(1 − iω/2)2
+Γ(1 − iω)Γ(1 − iκω) exp (ibω − iκω log(−iω)) ,
+(17)
+4The mass shell is related to rapidity by p0 = m cosh θ and p1 = m sinh θ.
+5
+
+with
+b = κ(1 − log(κ)) − log(2) .
+(18)
+3
+TBA Techniques
+Polyakov and Wiegmann [58–60] showed in the 80s that it is possible to compute the free
+energy of an integrable system with a chemical potential h turned on using a thermody-
+namic Bethe ansatz (TBA) technique. Using these techniques, Hasenfratz, Niedermayer
+and Maggiore [2, 3] showed in 19905 that it is possible to calculate the mass gap in
+integrable models by comparing the result from TBA with conventional Lagrangian
+pertubation theory. Building from this we will will apply, in section 3.4, the techniques
+pioneered by [13–16] to extract an expansion for the free energy of λ-model in
+1
+h the
+large order behaviour of which we will study extensively in section 4.
+3.1
+Free Energy
+To present the TBA equations we will specialise to the case described above in which we
+introduce a chemical potential h such that only a single particle dominates the ensemble
+at large h.6 With K(θ) the appropriate scattering kernel, the TBA equations determine
+the density distribution of states, χ(θ), via
+m cosh(θ) = χ(θ) −
+� B
+−B
+K(θ − θ′)χ(θ′)dθ′ ,
+θ2 < B2 ,
+(19)
+from which the charge and energy density follow
+e = m
+� −B
+B
+χ(θ) cosh(θ) dθ
+2π ,
+ρ =
+� −B
+B
+χ(θ) dθ
+2π .
+(20)
+A critical complexity of this system is that the occupied states lie within a Fermi surface
+specified by B, which is however a function of h (with large B corresponding to large
+h). Supposing that we have calculated the energy density, thought of as a function of
+the charge density e = e(ρ), then we can reconstruct a free energy density, F(h), from
+a Legendre transform:
+ρ = −F′(h) ,
+F(h) − F(0) = e(ρ) − ρh .
+(21)
+3.2
+Resolvent Approach
+It will prove useful to recast the integral equation that determines χ(θ) in terms of a
+resolvent function defined by
+R(θ) =
+� B
+−B
+χ(θ′)
+θ − θ′ dθ′.
+(22)
+5This computation was intially performed for the O(N) model, but was later also completed for
+Gross-Neveu models [4, 5] and PCM models [61, 62].
+6That we can reduce the TBA system to involve just one species of particle from the fundamental
+representation singled out by the applied chemical potential is of course an assumption that makes the
+problem readily tractable. One anticipates that states of higher mass and higher charge are energetically
+disfavoured, but properly speaking this assumption ought to be proven starting from a complete nested
+TBA system (which we do not attempt here).
+6
+
+The resolvent is analytical everywhere except around the interval [−B, B] where it has
+an ambiguity given by
+χ(θ) = − 1
+2πi
+�
+R+(θ) − R−(θ)
+�
+,
+(23)
+where we use the short hand notation R±(θ) = R(θ ± iϵ). Suppose that the kernel can
+be cast in terms of some operator O as K(θ) =
+1
+2πiO 1
+θ, then the eq. (19) is equivalent
+to a Riemann-Hilbert problem
+R+(θ) − R−(θ) + OR(θ) = −2πim cosh θ .
+(24)
+A determination of R(θ) is then equivalent to solving the TBA system and once known
+the charge density is immediately extracted as
+ρ = − 1
+2π Resθ=∞R(θ) .
+(25)
+We briefly now review the approach of [13–15] which does so by considering ansatz
+solutions for the resolvent in two limits (the edge and bulk) and matching them to fix
+all undetermined coefficients.
+3.2.1
+Edge Ansatz
+We begin first with the edge limit in which the weak coupling limit B → ∞ is taken
+whilst keeping an edge coordinate z = 2(θ −B) fixed and small. This evidently scales to
+large θ and hence probes the properties of χ(θ) around the vicinity of the Fermi energy,
+B. This limit is best studied by considering the Laplace transform of the resolvent (22)
+given by
+R(z) =
+� ∞
+0
+�R(s)e−szds ,
+�R(s) =
+1
+2πi
+� i∞+δ
+−i∞+δ
+eszR(z)dz .
+(26)
+Note at large B the energy density is related to this Laplace transformation by
+e = meB
+4π
+�R(1/2) .
+(27)
+The key result of [15, 16] is that in the edge limit the Laplace transformed resolvent
+has the following form
+�R(s) = meBΦ(s)Φ
+� 1
+2
+�
+2
+�
+1
+s + 1/2 + Q(s)
+�
+,
+Φ(s) = G+(2is) ,
+(28)
+where G+(s) is the WH decomposition (16) of the (Fourier transformed) scattering
+kernel and Q(s) is a series in large s and a perturbative expansion in
+1
+B of the form
+Q(s) = 1
+Bs
+∞
+�
+m,n=0
+Qn,m
+Bm+nsn .
+(29)
+It should be noted that the coefficients Qn,m may still depend on log B.
+3.2.2
+Bulk Ansatz
+In the bulk limit we let B → ∞ and θ → ∞ but we keep u = θ/B fixed, we are hence
+studying the regime where θ is in the bulk, between 0 and B. The precise form of the
+7
+
+Bulk ansatz depends on the model. For the λ-model, we shall take the same bulk ansatz
+for the Gross-Neveu model [15], which is given by
+R(u) =
+∞
+�
+n=1
+∞
+�
+m=0
+n+m
+�
+k=0
+cn,m,k
+ue(k+1)
+Bm+n(u2 − 1)n
+�
+log u − 1
+1 + u
+�k
+,
+(30)
+where e(k) is 0 if k is even and 1 if k is odd.The bulk ansatz can be motivated by
+constructing it using functions that are analytic outside the interval [−B, B], where
+they have a logarithmic branch cut.7 This is precisely the analytic structure demanded
+by eqs. (22) and (23).
+3.3
+Matching
+If we re-expand the bulk ansatz (30) in an edge regime where z = 2(θ − B) is fixed, we
+should recover the expansion in the edge regime given by (28). Here a miraculous feature
+occurs: upon comparing expansions order by order in large B, then order by order in
+large z (which is small s) and then in log(z), we can solve for all the coefficients cn,m,k
+and Qn,m. One peculiarity of the procedure is that we perform this matching only for
+the regular terms of the expansion z−n (n ≥ 0), while we disregard all divergent terms
+zn (n > 0). Using a desktop PC, over the course of a week, we solved the system up
+to 38 orders. Once this calculation is completed, we compute e and ρ. Using equations
+(28) and (25) we can express ρ and e in terms of the coefficients by
+e = m2e2BΦ(1/2)2
+8π
+�
+1 +
+∞
+�
+m=1
+1
+Bm
+m−1
+�
+n=0
+2n+1Qn,m−1−n
+�
+,
+ρ = 2π
+∞
+�
+m=0
+c1,m,0
+Bm .
+(31)
+Explicitly the first few coefficients required to determine up to order B−2 are given by
+c1,0,0 = 4√κ ,
+c1,1,0 = −2κ3/2 ,
+c1,2,0 = 1
+2κ3/2(2 − κ − 4 log 2 + 4κ log(2B/κ)) ,
+Q0,0 = 0 ,
+Q1,0 = 0 ,
+Q0,1 = κ
+4 .
+(32)
+The last step is to calculate the quantity
+e
+ρ2 as an expansion in B the first terms of
+which are
+8κ
+π
+e
+ρ2 = 1 + κ
+B + κ
+B2
+�
+1 − log(2) + κ
+2 + κ log(2B/κ)
+�
++ O(B−3) .
+(33)
+As this result depends on log(B), it is convenient to define a new effective coupling γ in
+terms of which the perturbative expansion is free from logarithms as we shall do in the
+next section.
+3.4
+Perturbative result
+Before introducing the log-free coupling, we show our results are consistent with those
+of [31], which determines the mass gap of this theory. Using standard TBA techniques,
+7This is different from the PCM bulk ansatz which also has a square root branch cut along the
+interval [−B, B].
+8
+
+they find an expansion for the free energy given by
+F(h) − F(0) = −2h2κ
+π
+�
+1 − 2κα + 2κα2�
+2 + κ + log 4 + +2κ log κ + 2κ log α
+�
+− 8κ2α3 log(α)
+�
+(−2 + 2κ + log 4 + 2κ log(κ) + κ log(α)
+�
++ O(α3)
+�
+.
+(34)
+The coupling α is here defined by
+1
+α = 2 log
+�
+2h
+m
+�
+8κ
+π
+�
+.
+(35)
+By using the Legendre transformation (21) we can compute the total energy e from eq.
+(34). Doing so, we obtain the expression
+8κ
+π
+e
+ρ2 =1 + 2ακ − 2κα2(2κ log(ακ) − κ − 2 + log(4))+
+8κ2α3�
+κ log2(α) + (log(α) − 1)(−2 log(4) + 2κ log(κ))
+�
++ O
+�
+α4�
+.
+(36)
+From eq. (33), it follows that
+e
+ρ2 = χ0 + O(α) where χ0 =
+π
+8κ. Therefore to leading
+order we have h = ∂e
+∂ρ = 2χ0ρ, which leads to ρ = 4hκ
+π . Looking at eq. (35), we should
+thus define a coupling by
+1
+α = 2 log
+�
+ρ
+m
+�
+2π
+κ
+�
+.
+(37)
+This defines α in terms of B. Inverting the relation and substituting into the series (33)
+recovers precisely the expansion (36), providing an important consistency check for our
+programme.
+We now take inspiration from the Gross-Neveu treatment of [15] to create a series
+expansions for
+e
+ρ2 that is log-free. This is appropriate because we have that to leading
+order ∆F ∼ −h2 + O(α), which leads to a coupling defined by8
+1
+γ + ξ log γ = log 2πρ
+m/c ,
+ξ = β2
+β2
+1
+= − k
+N = −κ .
+(38)
+One could demand that the right hand side be log 2πρ
+ΛMS , where ΛMS is the cut-off in
+the minimal subtraction scheme. To achieve this one has to tune the constant c = cMS
+such that cMSΛMS = m. A key outcome of [31] determines that cMS = e3/2N −1/2.
+However, we shall exercise the freedom to pick a c of our own choosing,
+c = 2−κΓ(κ)
+π
+,
+(39)
+such that resulting expressions appear considerably simplified. This leads to an expan-
+sion that is log-free in the coupling, given by
+8κ
+π
+e
+ρ2 =
+∞
+�
+n=0
+anγn = 1 + κγ + κ
+2 [2 − κ]γ2+
+κ
+2
+�
+3 − 5κ + 2κ2�
+γ3 + κ
+8
+�
+3(8 − ζ(3)) − 61κ + 52κ2 − 15κ3�
+γ4+
+κ
+12
+�
+90 − 18ζ(3) + κ(33ζ(3) − 288) + 355κ2 − 203κ3 + 46κ4�
+γ5+
+κ
+32
+�
+45(16 − 4ζ(3) − ζ(5)) + 2κ(259ζ(3) − 1338) + 1
+3κ2(12274 − 1329ζ(3))
+− 3285κ3 + 1412κ4 − 787κ5
+3
+�
+γ6 + O(γ7) .
+(40)
+8This is in contrast to the PCM calculation where the free energy has a structure ∆F ∼ − h2
+α +O(α0),
+which leads to a coupling 1
+γ + (ξ − 1) log γ ∝ log ρ.
+9
+
+Figure 1: Left to right, for κ = 0.98, 1 and 1.02, the Borel-Pad´e-poles in the ζ-plane.
+Evident are singularities at ζ = ±2, with the positive pole removed for κ = 1.
+In the next Section we shall explore this perturbative expansion further.
+4
+Asymptotic Analysis
+In this Section, we will quantitatively analyse the 38 orders of the perturbative series
+obtained in the previous Section. The goal shall be to compute an asymptotic formula
+for the growth of the coefficients as a function of κ. After obtaining such a formula, we
+can compute its Borel ambiguity, which can later be compared against an ambiguity of
+a transseries.
+As the perturbative series can readily be seen to exhibit factorial growth, as a first
+step to resummation we introduce the Borel transform
+B
+�8κ
+π
+e
+ρ
+2�
+≡
+∞
+�
+n=0
+an
+n! ζn .
+(41)
+This series has a finite radius of convergence but typically has either, or both, poles
+and branch cuts. The pole/branch point closest to the origin in the ζ plane is governed
+by the leading asymptotic behaviour.
+Of course, numerically one does not have all
+orders with which to establish this Borel transformation, rather only a finite number
+of coefficients an for n < N say. Here the Borel-Pad´e method can be employed: we
+compute BN[ 8κ
+π
+e
+ρ
+2] = �N
+n=0
+an
+n! ζn = P (ζ)
+Q(ζ) + O(ζ)N+1 in which P and Q are polynomials
+in ζ of degree N/2. This results in a picture in which an accumulation of poles (i.e.
+zeros of Q) is indicative of a branch point. We perform this numerically for various
+values of κ and generically we find evidence of branch points at ζ = ±2 whose location
+is independent of κ except that for κ ∈ Z>0 the pole in the positive axis is removed -
+see Figure 1. Pole/ branch points in the negative real axis of the Borel plane indicate
+contributions to an of alternating sign whereas the contributions to an that result in
+poles on the positive axis would have the same sign. Here the analysis indicates that we
+have both. With 38 perturbative coefficients this analysis should only be regarded as
+indicative but is sufficient to inform an educated guess as to the asymptotic behaviour
+of the an which we will robustly verify below.
+Motivated by the Borel-Pad´e analysis we assume the coefficients grow, to leading
+approximation, as
+an ≈ A+Γ(n + 1)/Sn + A−Γ(n + 1)/(−S)n + O(n−1) .
+(42)
+A first verification is to establish the factor S which can be done noting that
+g+,n :=
+a2n
+2n(2n − 1)a2n−2
+≈ 1
+S2 ,
+g−,n :=
+a2n+1
+2n(2n − 1)a2n−1
+≈ 1
+S2 .
+(43)
+10
+
+4
+2
+-2
+2
+4
+.4
+-24
+2
+-2
+2
+4
+4
+.24
+2
+-2
+2
+4
+.4
+-2Figure 2: The series g+,n (left) and g−,n (right) given by eq. (43) displayed for κ = 0.6.
+Circle markers indicate the raw data, square markers the second Richardson transform-
+ation with accelerated convergence. The final values of the second Richardson transform
+differ by 0.11% and 0.05% respectively from the expected value 1
+4.
+We find, see Figure 2, that the series g±,n converge to 1
+4, independent of κ thus estab-
+lishing S = 2 in accordance with the expectation from the Borel-Pad´e analysis.
+Having established the factorially growing character of the perturbative series, we
+now propose a more refined ansatz for the an. Our central claim can be summarised
+by stating that the perturbative series has coefficients that have a leading large order
+behaviour as
+an ≈ A+
+2n
+∞
+�
+l=0
+β+
+l Γ(n + a+ − l) +
+A−
+(−2)n
+∞
+�
+l=0
+β−
+l Γ(n + a− − l) ,
+(44)
+where we normalise β±
+0 = 1 and the first few coefficients are
+a± = ∓2κ ,
+A± = 8±1
+π
+sin(∓κ)Γ(±κ)
+Γ(∓κ) = −
+8±1
+Γ(∓κ)Γ(1 ∓ κ) ,
+β−
+1 = −β−
+2 = −4κ .
+(45)
+To support these claims, we shall define the auxiliary series
+cn =
+2n
+Γ(n + 1)an ,
+(46)
+to take care of the leading factorial and geometric growth. We project to the alternating
+and non-alternating parts of the series by considering
+f ±
+k = c2k ± c2k−1 ,
+(47)
+which have asymptotics
+f ±
+n = 2A±(2n)a±−1�
+1 + O
+� 1
+n
+��
+,
+(48)
+such that the sequences
+σ±
+n = 1 + n log f ±
+n+1
+f ±
+n
+,
+(49)
+converge to a±. With a± determined one can then directly consider the asymptotics of
+f ± to establish A±. Figure 3 illustrates the convergence of this procedure for a fixed
+11
+
++.n
+0.30r
+0.28
+O
+0.26
+boo
+0.24
+0.22
+0.20
+5
+10
+15
+0
+20g-,n
+0.30r
+0.28
+0.26
+0000000
+口
+0.24
+口
+0.22
+口
+0.20
+5
+10
+15
+0Figure 3: The series σ−
+n (left) converges to a− using (49). Using eq. (48) we display
+(right) the sequence that converges to A−. Circle markers indicate the raw data, square
+markers the second Richardson transformation. For both, we display results for κ = 0.9.
+The second Richardson transform converge to the expected results given by eq (45) up
+to errors of 0.011% and 0.00068% respectively.
+Figure 4: The second Richardson transformation of the sequences (49) (left) and (48)
+(right) to determine a± and A± as functions of κ. a+, A+ are indicated by red crosses
+and a−, A+ by blue points with solid lines showing the analytic formula of eq. (45).
+value of κ, and Figure 4 establishes the functional form of these coefficients for various
+values of κ.
+A methodological subtlety is that, from empirical observation, the contributions from
+the alternating sector, i.e.
+A− (and associated subleading terms), are dominant for
+κ > 0 over those of the non-alternating A+ sector. Thus to extract the non-alternating
+contributions we first establish the leading alternating contribution as described above
+and then repeat the process working instead with a new series in which the leading
+alternating contribution has been subtracted. However, when κ becomes sufficiently
+large, the sub-leading alternating contribution becomes comparable to that of the leading
+non-alternating contribution. This limits the reliability of determination numerically of
+the A+, a+ coefficients to small values of κ. However, these coefficients can be more
+readily verified by continuing to the κ < 0 regime where they are more dominant.
+Having determined in this fashion the leading contributions to an, these can then be
+subtracted from the data, the analysis repeated mutatis mutandis, to determine the sub-
+leading βk coefficients (and again for similar reasons to the above the β−
+k coefficients are
+more readily extracted). Figure 5 gives the numerical form of β−
+1 and β−
+2 as a function
+of κ indicating a linear relationship.
+It becomes somewhat challenging to extract further subleading contributions from
+the data available. However, one can consider defining a new series, ˜an, comprised by
+taking the data set and subtracting the already established asymptotic form of eq. (45).
+12
+
+n
+2.00
+1.95
+1.90
+O
+1.85
+1.80
+666660
+口
+1.75
+口
+口
+1.70
+1.65
+1.60
+0
+5
+10
+15A.
+-0.08r
+-0.10
+-0.12
+-0.14
+-0.16
+5
+10
+15
+20
+0a at
+4
+2
+K
+.3
+-2
+2
+3
+-264A- A+
+-2
+2
+-6
+8Figure 5: The sub-leading coefficient β−
+1 (left) and β−
+2 (right) for various values of κ.
+Shown is the terminal value of the second Richardson Transformation of the sequence
+that gives β−
+n constructed from fn after subtraction of leading alternating and non-
+alternating asymptotics.
+Grey lines correspond to β−
+1
+= −4κ and β−
+2
+= +4κ.
+A
+noticeable drift in β−
+2 for larger values of κ suggests pollution from further sub-dominant
+terms contributing at this order of perturbation theory.
+Figure 6: After subtracting the leading alternating and non-alternating contributions,
+we again perform a Borel-Pad´e computation for κ = −0.75 (left) and κ = 0.4 (right).
+This seems to suggest that there is no longer a Borel singularity at ζ = 2, but instead
+finding one at ζ = 4.
+Using the Borel-Pad´e again to this subtracted series produces some evidence, see Figure
+6, of a compelling feature. Instead of poles at ζ = ±2, as would be anticipated should
+the ansatz (44), one finds that leading positive pole appears to be at ζ = +4. The
+interpretation here is that the subtraction has removed the entire non-alternating terms
+with behaviour 2−n, suggesting that all fluctuations β+
+n>0 = 0 and the next contribution
+comes with twice the “action” 4−n.
+This behaviour is in accordance with the Parisi-’t Hooft conjecture [63–65]; the
+leading poles in the Borel plane at ζ = ±2 lie at integer values and the values of
+a± = ∓2κ = ±2ξ are as expected (see [15]).9
+The pole at ζ = +2 is accordingly
+interpreted as an IR renormalon.
+A similar procedure of subtraction (removing the
+IR renormalon) used above (in Figure 6) was performed in [26] to expose new Borel
+renormalon poles that were not in accordance with Parisi-’t Hooft in cases including e.g.
+the Gross-Neveu model. Here however, Figure 6 indicates that the next most proximate
+IR renormalon pole is found in a location that are consistent with Parisi-’t Hooft.
+9We thank M Mari˜no and T Reis for illuminating us on this point.
+13
+
+β1-
+K
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+-2
+-4
+-6
+-8
+-10
+.12β2
+12
+10
+8
+6
+4
+2
+0.5
+1.0
+1.5
+2.0
+2.5
+3.02
+.2
+2
+6
+22
+2
+65
+Transseries and Ambiguity Cancellation
+In this section we compute the leading ambiguity of
+e
+ρ2 in two different ways. First,
+we calculate the Borel ambiguity of the large order behaviour of the perturbative sector
+established in 4. This is compared against an approach which solves the TBA system
+in terms of a transseries.
+5.1
+Borel resummation and Large Order Perturbative Ambigu-
+ity
+Naively, one could try to resum the original asymptotic series by performing a Laplace
+transform on the Borel transform (41)
+1
+γ
+� ∞
+0
+B
+�8κ
+π
+e
+ρ2
+�
+e−ζ/γdζ = 1
+γ
+� ∞
+0
+∞
+�
+n=0
+an
+n! ζne−ζ/γ ≃
+∞
+�
+n=0
+anγn = 8κ
+π
+e
+ρ2 .
+(50)
+However, as we have seen, the Borel transform B
+�
+8κ
+π
+e
+ρ2
+�
+generically has singularities
+along the positive real axis obstructing the contour of this integral. Therefore, we shall
+introduce a directional Borel resummation given by
+Sθ
+�8κ
+π
+e
+ρ2
+�
+= 1
+γ
+� eiθ∞
+0
+B
+�8κ
+π
+e
+ρ2
+�
+e−ζ/γdζ .
+(51)
+This procedure results, when integrating along a line without singularities, in a finite
+answer, which however, depends on the sign of θ, i.e.
+there is an ambiguity in the
+resummation of the perturbative series. This ambiguity, which is a Stokes phenomenon,
+is studied by considering S+ϵ − S−ϵ. This can be done analytically by using, instead of
+the numerically obtained results, a series whose coefficients are exactly the asymptotic
+form an given by eq. (44) for all values of n:
+(S+ϵ − S−ϵ)
+�8κ
+π
+e
+ρ2
+�
+(γ) = 2πiA+
+� 2
+γ
+�a+
+e−2/γ
+∞
+�
+k=0
+β+
+k
+�γ
+2
+�k
+= −
+16πi
+Γ(−κ)Γ(1 − κ)
+�γ
+2
+�2κ
+e−2/γ[1 + O(γ)] .
+(52)
+Similarly, across the negative real axis we find a leading ambiguity given by
+(Sπ+ϵ − Sπ−ϵ)
+�8κ
+π
+e
+ρ2
+�
+(γ) = 2πiA−
+�
+− 2
+γ
+�a−
+e2/γ
+∞
+�
+k=0
+β−
+k
+�
+− z
+2
+�k
+= −
+πi
+4Γ(κ)Γ(1 + κ)
+�
+−γ
+2
+�−2κ
+e2/γ[1 + O(γ)] .
+(53)
+In these expressions we note the presence of an exponentially small parameter, √qγ =
+�
+2
+γ
+�2κ
+e−2/κ (the square root is for convenience later) characteristic of non-perturbative
+physics. The main thrust of the modern resurgence paradigm is that physical quantities,
+here e/ρ2, should be understood as a transseries, i.e. an expansion in √qγ whose terms
+are each formal (asymptotic) series in γ. It is critical that whilst resummation may
+be ambiguous when applied to any individual term in this (here the perturbative √qγ0
+sector), taken altogether the final result is non-ambiguous. In particular, and this goes
+back to the pioneering work of Bogomol’nyi and Zinn-Justin [66–69], the ambiguity
+of this perturbative sector should be compensated by a leading order ambiguity in an
+appropriate non-perturbative sector. In the next section we shall verify that such an
+ambiguity cancellation does take place.
+14
+
+5.2
+Transseries and Leading Non-Perturbative Ambiguity
+In this series we shall apply a different type of analysis to the TBA equations which
+results in a transseries solution. The starting point shall be a reformulation of the TBA
+system as an integral equation for an auxiliary function u(ω),
+u(ω) = i
+ω +
+1
+2πi
+� ∞
+−∞
+dω′ e2iBω′ϱ(ω′)u(ω′)
+ω′ + ω + iδ
+,
+(54)
+where
+ϱ(ω) = 1 − iω
+1 + iω
+G−(ω)
+G+(ω) ,
+(55)
+together with the boundary condition
+u(i) = m
+2heB G+(i)
+G+(0) .
+(56)
+Having established the function u, the free energy is given by
+∆F(h) = − 1
+2π h2u(i)G−(0)2
+�
+1 −
+1
+2πi
+� ∞
+−∞
+dω e2iωBu(ω)ϱ(ω)
+ω − i
+�
+.
+(57)
+We will now apply to the λ-model the techniques pioneered by [26] to solve this re-
+cursively order by order in a perturbative parameter and a non-perturbative parameter.
+The idea is to move the integration contour of the integral equation (54) into the UHP
+so that it envelops all the branch cuts and poles in the UHP. The Sine-Gordon model is
+special as it only has poles but no branch cut. This was studied in [70] and gives rise to
+a convergent rather than asymptotic expansion. However, in the case of the λ-deformed
+model, we are dealing with both poles and a branch cut along the imagine axis of ρ(ω).
+To separate it from the poles, we slightly move the cut away from the imaginary axis
+to the ray C± = {ξeiθ|θ = π
+2 ± δ}. By deforming the integration contour we isolate the
+contributions coming from the discontinuity over the cut and the residues at the poles
+(see Figure 7). As explained in [26] the choice of moving the branch cut to C+ or C− is
+arbitrary and and gives rise to a leading non-perturbative ambiguity. Letting ϱn,± be
+the residues at x = xn with the cut moved to C±, after this contour pulling eq. (54)
+becomes
+u(ix) = 1
+x +
+1
+2πi
+� ∞e±iϵ
+0
+dx′ e−2Bx′u(ix′)δϱ(ix′)
+x′ + x
++
+�
+n
+e−2Bxnunϱn,±
+xn + x
+,
+(58)
+where un ≡ u(ixn) and δϱ is the discontinuity over the cut10.
+From the WH-decomposition (17), we evaluate ϱ(ω) using (55) as
+ϱ(ω) = −ω + i
+ω − i
+Γ
+� iω
+2 + 1
+�2 Γ(1 − iω)Γ(1 − iκω)
+Γ
+�
+1 − iω
+2
+�2 Γ(iω + 1)Γ(iκω + 1)
+e−2ibωeiκω(log(iω)+log(−iω)) .
+(59)
+For generic values of κ, this has poles on the positive real axis at ω = ixn = iµn with
+µ = 2 with residues given by
+ϱn,± =
+Res
+x=xn±iϵ ϱ(ix) = −2ie2n(2b±iπκ−2κ log(2n))n2n + 1
+2n − 1
+((2n)!)2
+(n!)4
+Γ(1 + 2nκ)
+Γ(1 − 2nκ) .
+(60)
+However, when κ is rational some of these poles are removed.
+Suppose we express
+κ ≡
+k
+N = p/q as a reduced fraction with p, q coprime integers (i.e. q = N/gcd(N, k)),
+10For the discontinuity function, we use the convention δρ(ω) = ρ(ω(1 − iϵ) − ρ(ω(1 + iϵ)).
+15
+
+ω
+C−
+C
+ϱ−
+1
+ϱ−
+2
+ϱ−
+3
+ω
+C+
+C
+ϱ+
+1
+ϱ+
+2
+ϱ+
+3
+Figure 7: The contour C = (−∞, ∞) is deformed into either of two ways. The branch
+cut, represented by the curvy line is moved to either the ray C+ or C−. In those cases
+respectively, the contour is deformed into C+ or C−. In both cases we pick up residues
+ϱ±
+n , but their values differ by the branch cut discontinuity.
+then the set of poles are located at x ∈ 2N\qN (rather than x ∈ 2N). Hence, the residue
+ϱn,± evaluates to zero if 2n ∈ 2N ∩ qN, i.e. 2n is a multiple of q. In particular, when
+k is an integer multiple of a half, i.e. q = 1 or q = 2, all poles are removed entirely.
+If ϱ1 = 0, then ϱn = 0 for all n; in what follows we shall consider only the case where
+ϱ1 ̸= 0 which is most relevant to our discussion.
+The discontinuity function is given by
+δϱ(ix) = 2ix + 1
+x − 1e2bxe−2κx log x sin(κπx)Γ(1 − x/2)2Γ(1 + x)Γ(1 + κx)
+Γ(1 + x/2)2Γ(1 − x)Γ(1 − κx) .
+(61)
+Notice this has simple poles at x = 2n, which have residues that vanish for κ half-integer.
+Lastly we shall need11
+ϱ(i ± 0) = 8e2b∓iπκ Γ(1 + κ)
+Γ(1 − κ) = 8
+πκe2b∓iπκΓ(1 + κ)2 sin(πκ) .
+(62)
+Following again [26], the integral equation (58) is simplified by the introduction of
+P(η, v) given by
+e−2Bxδϱ(ix) = −2ive−ηP(η, v) ,
+(63)
+with a change of variables (x, B) → (η, v):
+1
+v + a log v = 2B ,
+x = vη .
+(64)
+Here, a is a constant determined by demanding that P(η, v) is regular in v with, in
+particular, no log(v) terms. From eq. (61), we have that δϱ(ix) ∝ e˜ax log x � dnxn,
+where ˜a = −2κ, therefore this determines ˜a = a. This yields an expansion of P(η, v)
+given by
+P(η, v) = d1,0η + vη2(d2,0 + d2,1 log(η)) + O(v2) ,
+d1,0 = πκ ,
+d2,0 = 2πκ(1 + (1 − γE − log(κ))κ − log(2)) ,
+d2,1 = −2πκ2 .
+(65)
+With the introduction of an integral operator
+I[f](η) = − v
+π
+� ∞
+0
+dη′ e−η′P(η′, v)f(η′)
+η + η′
+,
+(66)
+11Because we are assuming that κ is not integer, ϱ(i±0) is non-zero. If κ < 0, then ϱ(i±0) generically
+has a finite ambiguity.
+16
+
+after this change of variables, eq. (58) can be written as
+u(η) = u(η) + I[u](η) ,
+(67)
+in which the ‘seed’ solution is given as
+u(η) = 1
+vη + 1
+v
+�
+n
+e−2Bvηnunϱn,±
+ηn + η
+.
+(68)
+The formal solution obtained by iteration is thus presented as
+u(η) =
+∞
+�
+l=0
+Il[u](η) ≡ J [u](η) .
+(69)
+To determine the unknown coefficients un = u(ηn) we evaluate eq.(67) at η = ηn =
+µn/v and define In[f] ≡ I[f](η = ηn) to obtain
+un = 1
+µn + In[u] + 1
+µ
+�
+m
+e−2Bvηnumϱm,±
+m + n
+.
+(70)
+Here we have made a slight adaptation compared to [26] to suit the locations of the poles
+at xn = µn (with µ = 2) (cf. the Gross-Neveu model for which xn = 2n+1
+Υ
+for some
+constant Υ). To treat the exponentially small contributions coming from the residue
+term we introduce the small parameter
+q = e−2Bµ = e−µ/vv−µa .
+(71)
+Both the seed and formal solution, and the unkown values un, admit expansion in q
+u(η) =
+�
+s=1
+u(s)(η)qs ,
+u(η) =
+�
+u(s)(η)qs ,
+un =
+�
+s=0
+u(s)
+n qs .
+(72)
+As the operator J does not introduce factors of q we can construct the full solution
+order by order in q noting u(s)(η) = J [u(s)](η). Using (68) one finds that the first few
+terms12 of the seed solution are given by
+u(0) = 1
+vη ,
+u(1) = ϱ1,±u(0)
+1
+vη + µ ,
+u(2) = ϱ1,±u(1)
+1
+vη + µ + ϱ2,±u(0)
+2
+vη + 2µ .
+(74)
+Applying the q-expansion to eq. (70) we have that
+u(0)
+n
+= J [u(0)](ηn) = 1
+µn + In[J [ 1
+vη ]] ,
+u(1)
+n
+= In[J [u(1)]] + 1
+µ
+ϱ1,±u(0)
+1
+1 + n
+.
+(75)
+Let us assume that ϱ1,± ̸= 0 (i.e. κ is not half-integer), such that these two expressions
+are governing the leading behaviour. Suppose now we work formally13 to leading order
+12For n ≥ 1, we have in general
+u(n)(η) =
+n
+�
+m=1
+ϱm,±u(n−m)
+m
+vη + µm
+.
+(73)
+13i.e. ignoring that q is exponentially smaller than higher order polynomial terms in v.
+17
+
+in v and leading order in q . Because each application of I carries a factor v, to leading
+order it is sufficient to consider only the identity operator J = 1+. . . which results in14
+u(0)
+n
+= 1
+µn −
+v
+nπµd1,0 + O(v2) ,
+u(1)
+n
+=
+ϱ1,±
+µ2(n + 1) −
+d1,0ϱ1,±
+µ2π(n + 1)v + O(v2) .
+(76)
+The leading orders of u(η) are obtained by
+u(η) =
+�
+u(0) + I[u(0)] + O(v)
+�
++ q
+�
+u(1) + I[u(1)] + O(v2)
+�
++ O(q2) .
+(77)
+To implement the boundary condition that relates the chemical potential h to q, v,
+we will need
+u(i) = u
+�
+η = 1
+v
+�
+=
+�
+1 − d1,0
+π v + O(v2)
+�
++
+qϱ1,±
+µ(1 + µ)
+�
+1 − d1,0v
+π
++ O(v2)
+�
++ O(q2) .
+(78)
+The next step is to do the Legendre transform and calculate
+e
+ρ2 from ∆F. This can
+then be used to compare against the perturbative calculation. The same procedure of
+resolving the cut away from the poles of ρ and deforming the contour appropriately
+yields
+∆F(h) = − h2
+2π u(i)G+(0)2
+�
+1 + v2
+π
+� e−ηP(η, v)u(η)
+ηv − 1
+dη
+− e−2Bϱ(i ± ϵ)u(i) −
+�
+n≥1
+qnϱn,±un
+µn − 1
+�
+.
+(79)
+The leading orders of eq. (79) are given by
+∆F(h) = − G+(0)2h
+2π
+�
+1 − 2d10
+π v + O(v2)
+�
+×
+�
+1 − ρ(i ± ϵ)q1/µ +
+2ρ1,±
+µ(1 − µ2)q − 2ρ1,±ρ(i ± ϵ)
+µ(1 + µ)
+q1+1/µ + O(q2)
+�
+.
+(80)
+The first step of the Legendre transform is to relate h to the parameters q and v. This
+is done by substituting the expansion (78) for u(i) into the boundary condition (56)
+which, for µ = 2, gives
+h =
+mG+(i)
+12πG+(0)q−1/4�
+π + d1,0v + O(v2)
+��
+6 − ρ1,±q + O(q2)
+�
+.
+(81)
+As a consequence ρ = − d∆F
+dh
+is given by
+ρ = G+(i)G+(0)m
+12π2
+�
+π − d1,0v + O(v2)
+��
+6q−1/4 + ρ1,±q3/4 + O(q7/4)
+�
+,
+(82)
+from which we obtain
+e
+ρ2 as a series in v and q:
+e
+ρ2 =
+1
+6G+(0)2
+�
+π + 2d1,0v + O(v2)
+��
+3 + 3ρ(i + ±ϵ)q1/2 + ρ1,±q + O(q3/2)
+�
+.
+(83)
+14The small v limit can be taken also in the integral:
+I
+� 1
+vη
+�
+(ηn) = − v
+π
+� ∞
+0
+eη′d1,0η
+vη′ + nµ = − v
+π
+� ∞
+0
+�
+eη′d1,0
+nµ
++ O(v)
+�
+= − vd1,0
+nπµ + O(v2) .
+18
+
+We will now write this expansion in terms of the coupling (38) used in the previous
+Sections. Let us introduce a parameter exponentially small in γ, analogous to q being
+exponentially small in v, given by qγ = e−4/γ(γ/2)4κ. We use (38) to write v as a series
+in γ and qγ. Substituting this series for v (and q = q(v)) into (83), we arrive at
+8κ
+π
+e
+ρ2 =
+�
+1 + κγ + O(γ2)
+�
+− 8e∓iπκ Γ(κ)
+Γ(−κ)q1/2
+γ
+(1 + O(γ))
++ 23−4κe∓2iπκ Γ(2κ)
+Γ(−2κ)qγ(1 + O(γ)) .
+(84)
+We see that the first two coefficient of the perturbative series match precisely with eq.
+(40). The presence of transseries parameters qγ = e−4/γ(γ/2)4κ provides concrete pre-
+dictions of the resurgent structure of the perturbative series. In particular, we compute
+the ambiguity of the transseries (84) due to the difference in result if the branch cut is
+left or right of the poles. To leading order in qγ and γ, it is given by
+8κ
+π
+�� e
+ρ2
+�
+−
+−
+� e
+ρ2
+�
++
+�
+=
+16πi
+Γ(−κ)Γ(1 − κ)
+�γ
+2
+�2κ
+e−2/γ .
+(85)
+This is exactly the same ambiguity as obtained through the asymptotic analysis of our
+perturbative calculation - see eq. (52). We thus observe that the Borel-ambiguity of
+the perturbative series can be cancelled precisely by an ambiguity of a transmonomial.
+Therefore, the large order non-perturbative behaviour is unambiguous up to the order
+considered. This mirrors the fabled BZJJ ambiguity cancellation [66–69] in a field theory
+context.
+The analysis above only finds a source of the leading ambiguity on the positive
+real axis of the Borel plane. However, we can do a similar analysis to recover the Borel
+branch singularity on the negative real axis. The critical modification of the programme,
+as realised by [15], is to deform the contour of the integral equation (54) into the lower
+half plane, instead of the upper half plan. The critical analytic data is then given by the
+branch cut and residues at the negative imaginary axis. In the lower half plane, ϱ(−ix)
+has residues at xn = 2n + 1 and at ˜xn := n
+κ. However, as the latter set of residues is
+unambiguous with respect to the branch cut, they do not contribute15. One subtlety
+when using this approach arises when computing u(i). Deforming the contour of eq.
+(54) to an envelopment of the negative imaginary axis picks up a residue at ω = −i,
+which introduces a contribution of u(−i)ρ(−i ± ϵ)q not present in the analysis above.
+We will not present a detailed derivation as it is similar to the one above. Rather, we
+can report that the final result is a transseries with a leading ambiguity given by
+8κ
+π
+�� e
+ρ2
+�
+−
+−
+� e
+ρ2
+�
++
+�
+= −
+πi
+4Γ(κ)Γ(1 + κ)
+�
+−γ
+2
+�−2κ
+e2/γ .
+(86)
+This precisely matches the ambiguity of the perturbative sector around the negative real
+axis found in eq. (53).
+6
+Discussion
+In this note, we have studied the λ-model and brought it into the fold of resurgent
+analysis of [13–15, 26]. The model is particularly interesting, because, distinct from
+previously considered models, it has a interacting CFT fixed point in the UV.
+15They would be part of a transseries solution, but as they are unambiguous, they are not of interest
+to us currently. As a further side remark, when choosing κ < 0, along the positive imaginary axis ϱ(ix)
+also has such unambiguous residues at x = n
+κ .
+19
+
+We have found a perturbative series for the energy density at finite chemical poten-
+tial of the λ-model in Section 3.4 and identified with numerical techniques its asymp-
+totic form in Section 4.
+A key feature is that the Borel resummation of the large
+order behaviour is ambiguous when taken along either the positive or negative real axis.
+These ambiguities are exactly compensated/cancelled by a further ambiguity in a non-
+perturbative sector of a transseries solution in Section 5. These cancellations provide the
+λ-model with a robustly defined foundation which may serve as a paradigmatic example
+for other theories with asymptotic CFT behaviour.
+Of particular note is that the leading ambiguity on the positive axis (and associated
+features in the Borel plane) vanishes for κ ∈ Z>0, i.e. when the WZW level k divides
+the rank N of the gauge group SU(N). This is reminiscent of Cheshire-cat resurgence
+[71–74] in which the full glory of resurgence only becomes apparent as you deform away
+from certain special points at which it truncates.
+Let us finish with some broader questions to ponder following the analysis of the
+λ-model that we hope might stimulate further investigations on the topic:
+• An interesting feature of the WZW CFT that defines the UV of the λ model is
+that it exhibits level-rank duality [75]. It would be valuable to understand the
+extent to which this property constrains, or is encapsulated, in the form of the
+transseries that defines the λ-model.
+• In a QFT it is sometimes possible to directly link poles/branch points in the
+Borel plane to finite action non-pertubative saddle configurations. Remarkably,
+this can be done even in theories without instantons. In a series of paper [9, 11,
+12] finite action ‘uniton’ configurations of 1+1d integrable QFTs were matched
+to Borel poles of a quantum mechanics that followed by dimensional reduction
+with twisted boundary conditions (akin to a chemical potential as deployed here).
+This poses a natural question: can the features of the Borel plane we have found
+here via TBA methods be related to some finite action saddles? Conversely, given
+the knowledge of such uniton configurations, what do they imply for the TBA
+method? Achieving this would serve to put the semi-classical approaches of [9, 11,
+12] on a surer-footing in quantum field theory.
+• On the other hand there are a class of ambiguities which don’t (yet at least) have an
+interpretation as semi-classical saddles. Instead they are renormalon ambiguities
+associated to certain classes of Feynman diagrams. In [18, 76] it was shown how
+to construct such a series of diagrams which source the renormalon ambiguities in
+1/N expansion of the O(N) vector model, the Gross-Neveu and the SU(N) PCM.
+It would be interesting to investigate if there are diagrams that are responsible for
+the ambiguities in the λ-models.
+• The landscape of integrable models in two dimensions has been vastly expanded
+in recent years through variants of this λ-model, and the related Yang-Baxter σ-
+models. It could be rewarding to deploy similar technique across this landscape
+included e.g. to models with multiple deformation parameters or theories based
+on cosets rather than group manifolds.
+• In [57] the Quantum Inverse Scattering Method was applied to give a direct quant-
+isation of the λ-models as a continuum limit of a spin k Heisenberg spin-chain with
+inhomogeneities. The parameter that governs the in-homogeneity becomes a mass.
+Although the ground state of the system is quite a complicated Fermi sea, one can
+identify holes as certain particle excitations. After taking the continuum limit,
+one can obtain a TBA system for these excitations matching that of the QFT.
+An exciting question is if the above resurgent structure can be given a similar ab
+initio derivation within the QISM framework.
+20
+
+Acknowledgements
+DCT is supported by The Royal Society through a University Research FellowshipGen-
+eralised Dualities in String Theory and Holography URF 150185 and in part by STFC
+grant ST/P00055X/1 as well as by the FWO-Vlaanderen through the project G006119N
+and Vrije Universiteit Brussel through the Strategic Research Program “High-Energy
+Physics”. LS is supported by a PhD studentship from The Royal Society and the grant
+RF\ERE\210269. For the purpose of open access, the authors have applied a Creative
+Commons Attribution (CC BY) licence to any Author Accepted Manuscript version
+arising. We thank M Mari˜no and T Reis for helpful comments on a draft and I Aniceto
+for comments relating to this project.
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+arXiv:2301.02205v1  [math.LO]  5 Jan 2023
+The logic with unsharp implication and negation
+Ivan Chajda and Helmut L¨anger
+Abstract
+It is well-known that intuitionistic logics can be formalized by means of Brouwe-
+rian semilattices, i.e. relatively pseudocomplemented semilattices. Then the logical
+connective implication is considered to be the relative pseudocomplement and con-
+junction is the semilattice operation meet.
+If the Brouwerian semilattice has a
+bottom element 0 then the relative pseudocomplement with respect to 0 is called
+the pseudocomplement and it is considered as the connective negation in this logic.
+Our idea is to consider an arbitrary meet-semilattice with 0 satisfying only the
+Ascending Chain Condition, which is trivially satisfied in finite semilattices, and
+introduce the connective negation x0 as the set of all maximal elements z satis-
+fying x ∧ z = 0 and the connective implication x → y as the set of all maximal
+elements z satisfying x∧z ≤ y. Such a negation and implication is “unsharp” since
+it assigns to one entry x or to two entries x and y belonging to the semilattice,
+respectively, a subset instead of an element of the semilattice. Surprisingly, this
+kind of negation and implication, respectively, still shares a number of properties
+of these connectives in intuitionistic logic, in particular the derivation rule Modus
+Ponens. Moreover, unsharp negation and unsharp implication can be characterized
+by means of five, respectively seven simple axioms. Several examples are presented.
+The concepts of a deductive system and of a filter are introduced as well as the
+congruence determined by such a filter. We finally describe certain relationships
+between these concepts.
+AMS Subject Classification: 03G10, 03G25, 03B60, 06A12, 06D20
+Keywords: Semilattice, Brouwerian semilattice, Heyting algebra, intuitionistic logic,
+unsharp negation, unsharp implication, deductive system, filter, congruence
+1
+Introduction
+Intuitionistic logic is usually algebraically formalized by means of Brouwerian semilat-
+tices, i.e. semilattices (S, ∧, ∗) where ∗ denotes relative pseudocomplementation which is
+considered as the connective implication, see [1], [2], [12], [15] and [16]. If (S, ∧, ∗) has a 0
+then x∗0 is the pseudocomplement of x usually denoted by x∗ and considered as negation
+of x in this logic. If (S, ∧, ∗, 0) is even a lattice then it is called a Heyting algebra, see [14]
+and [17]. For posets the concept of pseudocomplementation was extended and studied
+by the authors in [4] and [5].
+It is well-known that every Brouwerian lattice (or Heyting algebra) is distributive.The
+concept of relative pseudocomplementation was extended by the first author to non-
+distributive lattices under the name sectional pseudocomplementation, see [3] and [9].
+1
+
+Hence a kind of non-distributive intuitionistic logic can be created on sectionally pseu-
+docomplemented lattices.
+In their previous papers [6] and [8] the authors showed that some important logics can
+be based also on posets that need not be lattices. An example of such a logic is the
+logic of quantum mechanics based on orthomodular posets, see e.g. [6], [10], [11] and
+[18]. It is evident that in this case some logical connectives such that disjunction or
+conjunction may be only partial operations or, as pointed out by the authors in [8] and
+[7], they may be be considered in an “unsharp version”, i.e. their result need not be a
+single element but may be a subset of the poset in question. Thus also the connective
+implication is created in this way as “unsharp”. For “unsharpness” see also [13]. This
+motivated us to study a variant of intuitionistic logic based on lattices that need neither
+be relatively pseudocomplemented nor even sectionally pseudocomplemented where the
+connective implication is unsharp.
+2
+Preliminaries
+In the following we identify singletons with their unique element, i.e. we will write x
+instead of {x}. Moreover, all posets considered in the sequel are assumed to satisfy the
+Ascending Chain Condition which we will abbreviate by ACC. This implies that every
+element lies under a maximal one. Of course, every finite poset satisfies the ACC. Let
+(P, ≤) be a poset, b ∈ P and A, B ⊆ P. By Max A we will denote the set of all maximal
+elements of A. We define
+A ≤ B if a ≤ b for all a ∈ A and all b ∈ B,
+A ≤1 B if for every a ∈ A there exists some b ∈ B with a ≤ b,
+A ≈1 B if A ≤1 B and B ≤1 A.
+The relation ≤1 is a quasiorder relation on 2P and ≈1 an equivalence relation on 2P. It
+is easy to see that A ≤1 Max B provided A ⊆ B and that A ≤1 b is equivalent to A ≤ b.
+Let S = (S, ∧) be an arbitrary meet-semilattice and A, B ⊆ S. We define
+A ∧ B := {a ∧ b | a ∈ A, b ∈ B}.
+3
+Unsharp negation
+Let (S, ∧, 0) be a meet-semilattice with 0 satisfying the ACC, a ∈ S and A ⊆ S. We
+define
+a0 := Max{x ∈ S|a ∧ x = 0}.
+Hence 0 is a unary operator on the meet-semilattice (S, ∧, 0) with 0 satisfying the ACC
+which assigns to every element x ∈ S the non-void subset x0 ⊆ S. The element a is called
+sharp if a00 = a. Moreover, we define
+A0 := Max{x ∈ S|A ∧ x = 0}.
+We are going to prove the following properties of the operator 0 for every meet-semilattice
+with 0 satisfying the ACC.
+2
+
+Theorem 3.1. Let S = (S, ∧, 0) be a meet-semilattice with 0 satisfying the ACC and
+a, b ∈ S. Then the following holds:
+(i) a0 is an antichain,
+(ii) a ≤1 a00,
+(iii) a ≤ b implies b0 ≤1 a0,
+(iv) 00 = Max S,
+(v) a ∧ a0 = 0,
+(vi) if S is bounded then 00 = 1 and 10 = 0,
+(vii) a ∧ 00 ≈1 a,
+(viii) a ∧ (a ∧ b)0 ≈1 a ∧ b0.
+Proof.
+(i) This is clear.
+(ii) We have a ∈ {x ∈ S | a0 ∧ x = 0}.
+(iii) If a ≤ b then {x ∈ S | b ∧ x = 0} ⊆ {x ∈ S | a ∧ x = 0}.
+(iv) and (v) follow directly from the definition of a0.
+(vi) If S is bounded then according to (iv)
+00 = Max S = 1,
+10 = Max{x ∈ S | 1 ∧ x = 0} = Max{0} = 0.
+(vii) According to (iv) we have a ≤1 Max S = 00 and hence a ≤1 a ∧ 00 ≤ a.
+(viii) Everyone of the following statements implies the next one:
+(a ∧ b) ∧ (a ∧ b)0 = 0,
+b ∧
+�
+a ∧ (a ∧ b)0�
+= 0,
+a ∧ (a ∧ b)0 ≤1 b0,
+a ∧ (a ∧ b)0 ≤1 a ∧ b0.
+From a ∧ b ≤ b we conclude b0 ≤1 (a ∧ b)0 according to (iii) and hence a ∧ b0 ≤1
+a ∧ (a ∧ b)0.
+From (iii) of Theorem 3.1 there follows immediately x0 ∧ y0 ≤ (x ∧ y)0.
+Example 3.2. Consider the meet-semilattice visualized in Fig. 1:
+3
+
+0
+a
+b
+c
+Fig. 1
+Meet-semilattice
+We have
+a = {b, c}0 = a00,
+00 = {a, b, c},
+a ∧ (a ∧ b)0 = a ∧ 00 = a ∧ {a, b, c} = {0, a} = a ∧ {a, c} = a ∧ b0
+in accordance with (ii), (iv) and (viii) of Theorem 3.1, respectively.
+Example 3.3. Consider the modular lattice L depicted in Fig. 2:
+0
+a
+b
+c
+d
+e
+f
+g
+h
+1
+Fig. 2
+Modular lattice
+We have
+a00 = {g, h}0 = a,
+f 00 = {b, c}0 = f,
+a0 ∧ e0 = {g, h} ∧ d = d ̸= {g, h} = a0 = (a ∧ e)0.
+Hence a and f are sharp and the equality x0 ∧ y0 = (x ∧ y)0 does not hold in general. In
+L from Figure 2 we have
+e0 = d and d0 = e.
+Since e ∧ d = 0 and e ∨ d = 1, {0, d, e, 1} is a complemented lattice.
+If a0 is a singleton, it need not be a complement of a, even if the semilattice is a lattice.
+E.g., consider the four-element lattice with atoms a and b and with an additional greatest
+element 1. Then a0 = b, but a ∨ b ̸= 1, i.e., a0 is not a complement of a.
+For every cardinal number n let Mn = (Mn, ∨, ∧) denote the bounded modular lattice of
+length 2 having n atoms.
+The situation from Figure 2 can be generalized as follows.
+4
+
+Remark 3.4. Every element of a direct product of a Boolean algebra and an arbitrary
+number of lattices Mn (possibly different n) is sharp.
+This follows immediately from the fact that every element of a Boolean algebra and every
+element of the lattice Mn is sharp.
+However, if the lattice L is not a direct product of two-element lattices and various Mn
+then the assertion of Remark 3.4 need not hold, see the following example.
+Example 3.5. Consider the lattice visualized in Fig. 3:
+0
+a
+b
+c
+d
+e
+f
+g
+1
+Fig. 3
+Lattice
+We have
+a00 = {b, c}0 = f ̸= a,
+a000 = f 0 = {b, c} = a0,
+b00 = {c, f}0 = b,
+(a0 ∧ b0)00 = ({b, c} ∧ {c, f})00 = {0, c}00 = {b, f}0 = c ̸= {0, c} = a0 ∧ b0,
+(c0 ∧ f 0)00 = ({b, f} ∧ {b, c})00 = {0, b}00 = {c, f}0 = b ̸= {0, b} = e0 ∧ f 0.
+Hence a is not sharp, b is sharp and the equality (x0 ∧ y0)00 = x0 ∧ y0 does not hold in
+general.
+We are going to show that the operator 0 can be characterized by means of four simple
+conditions.
+Theorem 3.6. Let (S, ∧, 0) be a meet-semilattice with 0 satisfying the ACC and 0 a
+unary operator on S. Then the following are equivalent:
+(i) x0 = Max{y ∈ S | x ∧ y = 0} for all x ∈ S,
+(ii) the operator 0 satisfies the following conditions:
+(P1) x0 is an antichain,
+5
+
+(P2) x ∧ 00 ≈1 x,
+(P3) x ∧ x0 ≈ 0,
+(P4) x ∧ (x ∧ y)0 ≈1 x ∧ y0.
+Proof.
+(i) ⇒ (ii):
+This follows from Theorem 3.1.
+(ii) ⇒ (i):
+If x ∧ y = 0 then according to (P2) and (P4) we have
+y ≈1 y ∧ 00 = y ∧ (x ∧ y)0 = y ∧ (y ∧ x)0 ≈1 y ∧ x0 ≤1 x0
+and hence y ≤1 x0. Conversely, if y ≤1 x0 then according to (P3) we have
+x ∧ y ≤1 x ∧ x0 = 0
+and hence x ∧ y = 0. This shows that x ∧ y = 0 is equivalent to y ≤1 x0. We conclude
+Max{y ∈ S | x ∧ y = 0} = Max{y ∈ S | y ≤1 x0} = x0.
+The last equality can be seen as follows. Let z ∈ Max{y ∈ S | y ≤1 x0}. Then z ≤1 x0,
+i.e. there exists some u ∈ x0 with z ≤ u. We have u ≤1 x0. Now z < u would imply
+z /∈ Max{y ∈ S | y ≤1 x0}, a contradiction. This shows z = u ∈ x0. Conversely, assume
+z ∈ x0. Then z ≤1 x0. If z /∈ Max{y ∈ S | y ≤1 x0} then there would exist some u ∈ S
+with z < u ≤1 x0 and hence there would exist some w ∈ x0 with z < u ≤ w contradicting
+(P1). This shows z ∈ Max{y ∈ S | y ≤1 x0}.
+4
+Unsharp implication
+Now we extend the operation of relative pseudocomplementation to arbitrary meet-
+semilattices with 0 satisfying the ACC as follows: Let S = (S, ∧, 0) be a meet-semilattice
+with 0 satisfying the ACC, a, b ∈ S and A, B ⊆ S. We define
+a → b := Max{x ∈ S | a ∧ x ≤ b}.
+Thus → is a binary operator on S assigning to every pair (x, y) ∈ S2 the non-void subset
+x → y ⊆ S. It is evident that
+x0 = x → 0 for each x ∈ S.
+Moreover, we define
+A → B := Max{x ∈ S | A ∧ x ≤ B}.
+Example 4.1. The “operation table” of the operator → in the meet-semilattice of Figure 1
+looks as follows (we write abc instead of {a, b, c} and so on):
+→
+0
+a
+b
+c
+0
+abc
+abc
+abc
+abc
+a
+bc
+abc
+bc
+ab
+b
+ac
+ac
+abc
+ac
+c
+ab
+ab
+ab
+abc
+6
+
+Example 4.2. The “operation table” of the operator → in the meet-semilattice of Figure 3
+looks as follows (we write bc instead of {b, c} and so on):
+→
+0
+a
+b
+c
+d
+e
+f
+g
+1
+0
+1
+1
+1
+1
+1
+1
+1
+1
+1
+a
+bc
+1
+bc
+bc
+1
+1
+1
+1
+1
+b
+cf
+cf
+1
+cf
+cf
+1
+cf
+1
+1
+c
+bf
+bf
+bf
+1
+bf
+1
+bf
+1
+1
+d
+bc
+g
+bc
+bc
+1
+g
+1
+g
+1
+e
+0
+f
+b
+c
+f
+1
+f
+1
+1
+f
+bc
+g
+bc
+bc
+dg
+g
+1
+g
+1
+g
+0
+f
+b
+c
+f
+ef
+f
+1
+1
+1
+0
+a
+b
+c
+d
+e
+f
+g
+1
+The following properties of the binary operator → can be proved.
+Theorem 4.3. Let S = (S, ∧, 0) be a meet-semilattice with 0 satisfying the ACC and
+a, b, c ∈ S. Then the following holds:
+(i) a → b is an antichain,
+(ii) a ≤ b implies a → b = Max S,
+(iii) b ∈ Max S implies b ∈ a → b,
+(iv) b ≤1 a → b,
+(v) a ≤1 (a → b) → b,
+(vi) a ≤ b implies c → a ≤1 c → b and b → c ≤1 a → c.
+(vii) a ∧ (a → b) ≈1 a ∧ b,
+(viii) a → (b ∧ c) ≈1 (a → b) ∧ (a → c),
+(ix) (a → b) ∧ b ≈1 b,
+(x) if S is bounded then 1 → b = b,
+(xi) a ∧ (b → b) ≈1 a,
+(xii) if S is bounded then a → b = 1 if and only if a ≤ b,
+(xiii) b ≤1 a → (a ∧ b).
+Proof.
+(i) This is clear.
+(ii), (iv), (x), (xii) and (xiii) follow immediately from the definition of →.
+(iii) If b ∈ Max S then because of a∧b ≤ b we have b ∈ Max{x ∈ S | a∧x ≤ b} = a → b.
+(v) Since a ∧ x ≤ b for all x ∈ a → b we have a ∧ (a → b) ≤ b, i.e. (a → b) ∧ a ≤ b.
+7
+
+(vi) If a ≤ b then
+{x ∈ S | c ∧ x ≤ a} ⊆ {x ∈ S | c ∧ x ≤ b},
+{x ∈ S | b ∧ x ≤ c} ⊆ {x ∈ S | a ∧ x ≤ c}.
+(vii) We have a ∧ x ≤ b and hence a ∧ x ≤ a ∧ b for all x ∈ a → b and hence a ∧ b ≤1
+a ∧ (a → b) ≤ a ∧ b according to (iv).
+(viii) According to (vii) we have a → (b ∧ c) ≤1 (a → b) ∧ (a → c). Conversely, assume
+d ∈ a → b and e ∈ a → c. Then a∧d ≤ b and a∧e ≤ c and hence a∧(d∧e) ≤ b∧c
+which implies d ∧ e ≤1 a → (b ∧ c). This shows (a → b) ∧ (a → c) ≤1 a → (b ∧ c).
+(ix) We have b ≤1 a → b according to (iv) and hence b ≤1 (a → b) ∧ b ≤ b.
+(xi) According to (ii) we have a ≤1 Max S = b → b and hence a ≤1 a ∧ (b → b) ≤ a.
+From Theorem 4.3 it is evident that the binary operator → shares properties of the
+logical connective implication in intuitionistic logic despite the fact that it is unsharp,
+i.e. for x, y ∈ S the result of x → y need not be a singleton.
+Hence it extends the
+intuitionistic logic based on a Heyting algebra (L, ∨, ∧, ∗, 0) where again ∨ formalizes
+disjunction, ∧ formalizes conjunction, but now → formalizes unsharp implication and 0
+formalizes unsharp negation.
+Example 4.4. Consider the modular lattice visualized in Fig. 4:
+0
+a
+b
+c
+d
+e
+f
+1
+Fig. 4
+Modular lattice
+Then
+e ≤1 {d, e} = {e, f} → e = (d → e) → e,
+d ∧ (d → e) = d ∧ {e, f} = c = d ∧ e,
+(d → e) ∧ e = {e, f} ∧ e = {c, e} ≈1 e
+in accordance with (v), (vii) and (ix) of Theorem 4.3, respectively.
+8
+
+Remark 4.5. It is easy to see that the operation ∧ and the operator → are related by
+so-called unsharp adjointness, i.e.
+a ∧ b ≤ c if and only if a ≤1 b → c.
+Since the operation ∧ is associative, commutative and monotone, it can be considered as
+a t-norm. Thus the semilattice (S, ∧, →) endowed with the operator → is an unsharply
+residuated semilattice. Moreover, by (viii) of Theorem 4.3 we have
+a ∧ (a → b) ≈1 a ∧ b
+showing that (S, ∧, →) satisfies divisibility.
+Similarly as for the unary operator 0 we can characterize the binary operator → on a
+meet-semilattice with 0 satisfying the ACC as follows.
+Theorem 4.6. Let (S, ∧, 0) be a meet-semilattice with 0 satisfying the ACC and → a
+binary operator on S. Then the following are equivalent:
+(i) x → y = Max{z ∈ S | x ∧ z ≤ y} for all x, y ∈ S,
+(ii) The operator → satisfies the following conditions:
+(R1) x → y is an antichain,
+(R2) x ∧ (x → y) ≈1 x ∧ y,
+(R3) (x → y) ∧ y ≈1 y,
+(R4) x → (y ∧ z) ≈1 (x → y) ∧ (x → z),
+(R5) x ∧ (y → y) ≈1 x,
+(R6) y ≤ z implies x → y ≤1 x → z.
+Proof.
+(i) ⇒ (ii):
+This follows from Theorem 4.3.
+(ii) ⇒ (i):
+If x ∧ z ≤ y then according to (R3), (R5), (R4) and (R6) we have
+z ≈1 (x → z) ∧ z ≤1 x ��� z ≈1 (x → x) ∧ (x → z) ≈1 x → (x ∧ z) ≤1 x → y
+and hence z ≤1 x → y. Conversely, if z ≤1 x → y then according to (R2) we have
+x ∧ z ≤1 x ∧ (x → y) ≈1 x ∧ y ≤ y
+and hence x∧z ≤ y. This shows that x∧z ≤ y is equivalent to z ≤1 x → y. We conclude
+Max{z ∈ S | x ∧ z ≤ y} = Max{z ∈ S | z ≤1 x → y} = x → y.
+The last equality can be seen as follows. Let u ∈ Max{z ∈ S | z ≤1 x → y}. Then
+u ≤1 x → y, i.e. there exists some v ∈ x → y with u ≤ v. We have v ≤1 x → y.
+Now u < v would imply u /∈ Max{z ∈ S | z ≤1 x → y}, a contradiction.
+This
+shows u = v ∈ x → y.
+Conversely, assume u ∈ x → y.
+Then u ≤1 x → y.
+If
+u /∈ Max{z ∈ S | z ≤1 x → y} then there would exist some v ∈ S with u < v ≤1 x → y
+and hence there would exist some w ∈ x → y with u < v ≤ w contradicting (R1). This
+shows u ∈ Max{z ∈ S | z ≤1 x → y}.
+9
+
+Example 4.7. Consider the lattice from Figure 4. Then
+d ∧ (d → e) = d ∧ {e, f} = c = d ∧ e,
+(d → e) ∧ e = e ∧ e = e,
+d → (e ∧ f) = d → c = {e, f} ≈1 {c, e, f} = {e, f} ∧ {e, f} = (d → e) ∧ (d → f)
+in accordance with (R2), (R3) and (R4), respectively.
+5
+Deductive systems
+It is well-known that the connective implication in intuitionistic logic is closely related
+to the so-called deductive systems in the corresponding Brouwerian semilattice. In what
+follows we show that a certain modification of the concept of a deductive system plays a
+similar role for logics with unsharp implication. We define
+Definition 5.1. A deductive system of a meet-semilattice S = (S, ∧, 0) with 0 satisfying
+the ACC is a subset D of S satisfying the following conditions for x, y ∈ S:
+(D1) (Max S) ∩ D ̸= ∅,
+(D2) x ∈ D and (x → y) ∩ D ̸= ∅ imply y ∈ D.
+Recall that a filter of a meet-semilattice S = (S, ∧) is a non-empty subset F of S satisfying
+the following conditions for x, y ∈ S:
+(F1) x, y ∈ F implies x ∧ y ∈ F,
+(F2) x ∈ F and x ≤ y imply y ∈ F.
+It is clear that if S is finite then all filters of S are given by the sets [x) := {y ∈ S | x ≤ y},
+x ∈ S, and hence the poset of all filters of S is dually isomorphic to S and therefore a
+join-semilattice where [x) ∨ [y) = [x ∧ y) for all x, y ∈ S.
+For every non-empty subset A of the universe of a meet-semilattice (S, ∧) we define a
+binary relation Θ(A) on S as follows:
+(x, y) ∈ Θ(A) if there exists some a ∈ A with x ∧ a = y ∧ a.
+Although the following result is known, for the reader’s convenience we present the proof.
+Lemma 5.2. Let S = (S, ∧, 1) be a meet-semilattice with 1 and Φ ∈ Con S. Then the
+following holds:
+(i) [1]Φ is an filter of S,
+(ii) Θ([1]Φ) ⊆ Φ.
+Proof.
+(i) (F1) If a, b ∈ [1]Φ then a ∧ b ∈ [1 ∧ 1]Φ = [1]Φ.
+10
+
+(F2) If a ∈ [1]Φ, b ∈ S and a ≤ b then b = 1 ∧ b ∈ [a ∧ b]Φ = [a]Φ = [1]Φ.
+This shows that [1]Φ is a filter of S.
+(ii) If (a, b) ∈ Θ([1]Φ) then there exists some c ∈ [1]Φ with a ∧ c = b ∧ c whence
+a = a ∧ 1 Φ a ∧ c = b ∧ c Φ b ∧ 1 = b
+which shows (a, b) ∈ Φ.
+Although our definition of a deductive system differs from that known for relatively
+pseudocomplemented semilattices, we are still able to prove the following relationships
+between the concepts mentioned before.
+Theorem 5.3. Let S = (S, ∧, 0, 1) be a bounded meet-semilattice satisfying the ACC and
+D a non-empty subset of S. Then the following are equivalent:
+(i) D a deductive system of S,
+(ii) D is an filter of S,
+(iii) Θ(D) ∈ Con S and D = [1]
+�
+Θ(D)
+�
+.
+Proof.
+(i) ⇒ (ii):
+(F2) Assume a ∈ D, b ∈ S and a ≤ b.
+Then a → b = Max S because of (ii) of
+Theorem 4.3. According to (D1) we have (a → b) ∩ D = Max S ∩ D ̸= ∅ and hence
+b ∈ D by (D2).
+(F1) Let a, b ∈ D. Then by (xiii) of Theorem 4.3 we have b ≤1 a → (a ∧ b). Hence there
+exists some c ∈ a → (a ∧ b) with b ≤ c. Now (F2) implies c ∈ D and therefore
+�
+a → (a ∧ b)
+�
+∩ D ̸= ∅ from which we conclude a ∧ b ∈ D by (D2).
+(ii) ⇒ (iii):
+Evidently, Θ(D) is reflexive and symmetric. Let (a, b), (b, c) ∈ Θ(D). Then there exist
+d, e ∈ D with a ∧ d = b ∧ d and b ∧ e = c ∧ e. Because of (F1) we conclude d ∧ e ∈ D.
+Now
+a ∧ (d ∧ e) = (a ∧ d) ∧ e = (b ∧ d) ∧ e = (b ∧ e) ∧ d = (c ∧ e) ∧ d = c ∧ (d ∧ e)
+which yields (a, c) ∈ Θ(D), i.e. Θ(D) is transitive. Further, if f ∈ S then
+(a ∧ f) ∧ d = (a ∧ d) ∧ f = (b ∧ d) ∧ f = (b ∧ f) ∧ d
+showing (a ∧ f, b ∧ f) ∈ Θ(D).
+Hence Θ(D) ∈ Con S.
+If a ∈ D then because of
+a ∧ a = a = 1 ∧ a we have a ∈ [1]
+�
+Θ(D)
+�
+showing D ⊆ [1]
+�
+Θ(D)
+�
+. Conversely, assume
+a ∈ [1]
+�
+Θ(D)
+�
+. Then there exists some b ∈ D with a ∧ b = 1 ∧ b. This implies b ≤ a
+wherefrom we conclude a ∈ D by (F2) showing [1]
+�
+Θ(D)
+�
+⊆ D.
+(iii) ⇒ (i):
+11
+
+(D1) If a ∈ D then, since S satisfies the ACC, there exists some b ∈ Max S with a ≤ b
+and hence
+b = 1 ∧ b ∈ [a ∧ b]
+�
+Θ(D)
+�
+= [a]
+�
+Θ(D)
+�
+= [1]
+�
+Θ(D)
+�
+= D.
+(D2) If a ∈ D, b ∈ S and (a → b) ∩ D ̸= ∅ then there exists some c ∈ D with c ∈ a → b
+and hence a ∧ c ≤ b whence
+b = 1∧1∧b ∈ [a∧c∧b]
+�
+Θ(D)
+�
+= [a∧c]
+�
+Θ(D)
+�
+= [1∧1]
+�
+Θ(D)
+�
+= [1]
+�
+Θ(D)
+�
+= D.
+It is well known that for a filter F of a relatively pseudocomplemented semilattice we
+have (a, b) ∈ Θ(F) if and only if a → b ∈ F and b → a ∈ F. However, we can modify
+this result also for an arbitrary meet-semilattice with 0 satisfying the ACC provided our
+unsharp implication is considered.
+Proposition 5.4. Let S = (S, ∧, 0) be a meet-semilattice with 0 satisfying the ACC, F
+a filter of S and a, b ∈ S. Then the following are equivalent:
+(i) (a, b) ∈ Θ(F),
+(ii) (a → b) ∩ F ̸= ∅ and (b → a) ∩ F ̸= ∅.
+Proof.
+(i) ⇒ (ii):
+There exists some c ∈ F with a ∧ c = b ∧ c. Hence a ∧ c ≤ b and b ∧ c ≤ a and therefore
+there exists some d ∈ a → b with c ≤ d and some e ∈ b → a with c ≤ e. Because of (F2)
+we conclude d, e ∈ F showing (ii).
+(ii) ⇒ (i):
+Let c ∈ (a → b)∩F and d ∈ (b → a)∩F. Then c∧d ∈ F by (F1), a∧c ≤ b and b∧d ≤ a.
+Hence
+a ∧ (c ∧ d) = (a ∧ c) ∧ (c ∧ d) ≤ b ∧ (c ∧ d) = (b ∧ d) ∧ (c ∧ d) ≤ a ∧ (c ∧ d),
+i.e. a ∧ (c ∧ d) = b ∧ (c ∧ d) showing (i).
+Conclusion
+Although the implication within the logic based on the structure (S, ∧, 0, →) is unsharp,
+i.e. x → y may be a subset I of S which need not be a singleton, it has its logical meaning.
+Namely, we ask that x → y is the maximal element c of S satisfying x ∧ c ≤ y (where ∧
+denotes conjunction). And for each c ∈ I this is satisfied. Moreover, the elements of I
+are mutually incomparable. Thus we have no need to prefer one of them with respect to
+others. However, the expression
+x ∧ (x → y) ≤ y
+is nothing else than the derivation rule Modus Ponens (both in classical as well as in
+non-classical logic) since it properly says that the truth value of y cannot be less than
+the truth value of the conjunction x ∧ (x → y) of x and the implication x → y. Hence,
+despite of the fact of unsharpness, such a logic is sound although it is derived from an
+arbitrary meet-semilattice with 0 satisfying the ACC.
+12
+
+References
+[1] L. E. J. Brouwer, De onbetrouwbaarheid der logische principes. Tijdschrift Wijs-
+begeerte 2 (1908), 152–158.
+[2] L. E. J. Brouwer, Intuitionism and formalism. Bull. Amer. Math. Soc. 20 (1913),
+81–96.
+[3] I. Chajda, An extension of relative pseudocomplementation to non-distributive lat-
+tices. Acta Sci. Math. (Szeged) 69 (2003), 491–496.
+[4] I. Chajda, Pseudocomplemented and Stone posets. Acta Univ. Palack. Olomuc. Fac.
+Rerum Natur. Math. 51 (2012), 29–34.
+[5] I. Chajda and H. L¨anger, Algebras describing pseudocomplemented, relatively pseu-
+docomplemented and sectionally pseudocomplemented posets. Symmetry 13 (2021),
+753 (17 pp.)
+[6] I. Chajda and H. L¨anger, Implication in finite posets with pseudocomplemented
+sections. Soft Computing 26 (2022), 5945–5953.
+[7] I. Chajda and H. L¨anger, The logic of orthomodular posets of finite height. Log. J.
+IGPL 30 (2022), 143–154.
+[8] I. Chajda and H. L¨anger, Operator residuation in orthomodular posets of finite
+height. Fuzzy Sets Systems (submitted).
+[9] I. Chajda, H. L¨anger and J. Paseka, Sectionally pseudocomplemented posets. Order
+38 (2021), 527–546.
+[10] D. Fazio, A. Ledda and F. Paoli, On Finch’s conditions for the completion of ortho-
+modular posets. Found. Sci. (2020), https://doi.org/10.1007/s10699-020-09702-z.
+[11] P. D. Finch, On orthomodular posets. J. Austral. Math. Soc. 11 (1970), 57–62.
+[12] O. Frink, Pseudo-complements in semi-lattices. Duke Math. J. 29 (1962), 505–514.
+[13] R. Giuntini and H. Greuling, Toward a formal language for unsharp properties.
+Found. Phys. 19 (1989), 931–945.
+[14] A. Heyting, Die formalen Regeln der intuitionistischen Logik. Sitzungsber. Akad.
+Berlin 1930, 42–56.
+[15] P. K¨ohler, Brouwerian semilattices: the lattice of total subalgebras. Banach Center
+Publ. 9 (1982), 47–56.
+[16] A. Monteiro, Axiomes ind´ependants pour les alg`ebres de Brouwer. Rev. Un. Mat.
+Argentina 17 (1955), 149–160.
+[17] L. Monteiro, Les alg`ebres de Heyting et de Lukasiewicz trivalentes. Notre Dame J.
+Formal Logic 11 (1970), 453–466.
+[18] P. Pt´ak and S. Pulmannov´a, Orthomodular Structures as Quantum Logics. Kluwer,
+Dordrecht 1991. ISBN 0-7923-1207-4.
+13
+
+Authors’ addresses:
+Ivan Chajda
+Palack´y University Olomouc
+Faculty of Science
+Department of Algebra and Geometry
+17. listopadu 12
+771 46 Olomouc
+Czech Republic
+[email protected]
+Helmut L¨anger
+TU Wien
+Faculty of Mathematics and Geoinformation
+Institute of Discrete Mathematics and Geometry
+Wiedner Hauptstraße 8-10
+1040 Vienna
+Austria, and
+Palack´y University Olomouc
+Faculty of Science
+Department of Algebra and Geometry
+17. listopadu 12
+771 46 Olomouc
+Czech Republic
+[email protected]
+14
+

GtE0T4oBgHgl3EQfRQCY/content/tmp_files/load_file.txt

@@ -0,0 +1,389 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf,len=388
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='02205v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='LO]  5 Jan 2023  The logic with unsharp implication and negation  Ivan Chajda and Helmut L¨anger  Abstract  It is well-known that intuitionistic logics can be formalized by means of Brouwe-  rian semilattices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' relatively pseudocomplemented semilattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then the logical  connective implication is considered to be the relative pseudocomplement and con-  junction is the semilattice operation meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  If the Brouwerian semilattice has a  bottom element 0 then the relative pseudocomplement with respect to 0 is called  the pseudocomplement and it is considered as the connective negation in this logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Our idea is to consider an arbitrary meet-semilattice with 0 satisfying only the  Ascending Chain Condition, which is trivially satisfied in finite semilattices, and  introduce the connective negation x0 as the set of all maximal elements z satis-  fying x ∧ z = 0 and the connective implication x → y as the set of all maximal  elements z satisfying x∧z ≤ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Such a negation and implication is “unsharp” since  it assigns to one entry x or to two entries x and y belonging to the semilattice,  respectively, a subset instead of an element of the semilattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Surprisingly, this  kind of negation and implication, respectively, still shares a number of properties  of these connectives in intuitionistic logic, in particular the derivation rule Modus  Ponens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Moreover, unsharp negation and unsharp implication can be characterized  by means of five, respectively seven simple axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Several examples are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  The concepts of a deductive system and of a filter are introduced as well as the  congruence determined by such a filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' We finally describe certain relationships  between these concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  AMS Subject Classification: 03G10, 03G25, 03B60, 06A12, 06D20  Keywords: Semilattice, Brouwerian semilattice, Heyting algebra, intuitionistic logic,  unsharp negation, unsharp implication, deductive system, filter, congruence  1  Introduction  Intuitionistic logic is usually algebraically formalized by means of Brouwerian semilat-  tices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' semilattices (S, ∧, ∗) where ∗ denotes relative pseudocomplementation which is  considered as the connective implication, see [1], [2], [12], [15] and [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' If (S, ∧, ∗) has a 0  then x∗0 is the pseudocomplement of x usually denoted by x∗ and considered as negation  of x in this logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' If (S, ∧, ∗, 0) is even a lattice then it is called a Heyting algebra, see [14]  and [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' For posets the concept of pseudocomplementation was extended and studied  by the authors in [4] and [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  It is well-known that every Brouwerian lattice (or Heyting algebra) is distributive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='The  concept of relative pseudocomplementation was extended by the first author to non-  distributive lattices under the name sectional pseudocomplementation, see [3] and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  1  Hence a kind of non-distributive intuitionistic logic can be created on sectionally pseu-  docomplemented lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  In their previous papers [6] and [8] the authors showed that some important logics can  be based also on posets that need not be lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' An example of such a logic is the  logic of quantum mechanics based on orthomodular posets, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' [6], [10], [11] and  [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' It is evident that in this case some logical connectives such that disjunction or  conjunction may be only partial operations or, as pointed out by the authors in [8] and  [7], they may be be considered in an “unsharp version”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' their result need not be a  single element but may be a subset of the poset in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Thus also the connective  implication is created in this way as “unsharp”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' For “unsharpness” see also [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' This  motivated us to study a variant of intuitionistic logic based on lattices that need neither  be relatively pseudocomplemented nor even sectionally pseudocomplemented where the  connective implication is unsharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  2  Preliminaries  In the following we identify singletons with their unique element, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' we will write x  instead of {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Moreover, all posets considered in the sequel are assumed to satisfy the  Ascending Chain Condition which we will abbreviate by ACC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' This implies that every  element lies under a maximal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Of course, every finite poset satisfies the ACC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Let  (P, ≤) be a poset, b ∈ P and A, B ⊆ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' By Max A we will denote the set of all maximal  elements of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' We define  A ≤ B if a ≤ b for all a ∈ A and all b ∈ B,  A ≤1 B if for every a ∈ A there exists some b ∈ B with a ≤ b,  A ≈1 B if A ≤1 B and B ≤1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  The relation ≤1 is a quasiorder relation on 2P and ≈1 an equivalence relation on 2P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' It  is easy to see that A ≤1 Max B provided A ⊆ B and that A ≤1 b is equivalent to A ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Let S = (S, ∧) be an arbitrary meet-semilattice and A, B ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' We define  A ∧ B := {a ∧ b | a ∈ A, b ∈ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  3  Unsharp negation  Let (S, ∧, 0) be a meet-semilattice with 0 satisfying the ACC, a ∈ S and A ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' We  define  a0 := Max{x ∈ S|a ∧ x = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Hence 0 is a unary operator on the meet-semilattice (S, ∧, 0) with 0 satisfying the ACC  which assigns to every element x ∈ S the non-void subset x0 ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' The element a is called  sharp if a00 = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Moreover, we define  A0 := Max{x ∈ S|A ∧ x = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  We are going to prove the following properties of the operator 0 for every meet-semilattice  with 0 satisfying the ACC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  2  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Let S = (S, ∧, 0) be a meet-semilattice with 0 satisfying the ACC and  a, b ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then the following holds:  (i) a0 is an antichain,  (ii) a ≤1 a00,  (iii) a ≤ b implies b0 ≤1 a0,  (iv) 00 = Max S,  (v) a ∧ a0 = 0,  (vi) if S is bounded then 00 = 1 and 10 = 0,  (vii) a ∧ 00 ≈1 a,  (viii) a ∧ (a ∧ b)0 ≈1 a ∧ b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (i) This is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (ii) We have a ∈ {x ∈ S | a0 ∧ x = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (iii) If a ≤ b then {x ∈ S | b ∧ x = 0} ⊆ {x ∈ S | a ∧ x = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (iv) and (v) follow directly from the definition of a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (vi) If S is bounded then according to (iv)  00 = Max S = 1,  10 = Max{x ∈ S | 1 ∧ x = 0} = Max{0} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (vii) According to (iv) we have a ≤1 Max S = 00 and hence a ≤1 a ∧ 00 ≤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (viii) Everyone of the following statements implies the next one:  (a ∧ b) ∧ (a ∧ b)0 = 0,  b ∧  �  a ∧ (a ∧ b)0�  = 0,  a ∧ (a ∧ b)0 ≤1 b0,  a ∧ (a ∧ b)0 ≤1 a ∧ b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  From a ∧ b ≤ b we conclude b0 ≤1 (a ∧ b)0 according to (iii) and hence a ∧ b0 ≤1  a ∧ (a ∧ b)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  From (iii) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='1 there follows immediately x0 ∧ y0 ≤ (x ∧ y)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Consider the meet-semilattice visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' 1:  3  0  a  b  c  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' 1  Meet-semilattice  We have  a = {b, c}0 = a00,  00 = {a, b, c},  a ∧ (a ∧ b)0 = a ∧ 00 = a ∧ {a, b, c} = {0, a} = a ∧ {a, c} = a ∧ b0  in accordance with (ii), (iv) and (viii) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Consider the modular lattice L depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' 2:  0  a  b  c  d  e  f  g  h  1  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' 2  Modular lattice  We have  a00 = {g, h}0 = a,  f 00 = {b, c}0 = f,  a0 ∧ e0 = {g, h} ∧ d = d ̸= {g, h} = a0 = (a ∧ e)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Hence a and f are sharp and the equality x0 ∧ y0 = (x ∧ y)0 does not hold in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' In  L from Figure 2 we have  e0 = d and d0 = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Since e ∧ d = 0 and e ∨ d = 1, {0, d, e, 1} is a complemented lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  If a0 is a singleton, it need not be a complement of a, even if the semilattice is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=', consider the four-element lattice with atoms a and b and with an additional greatest  element 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then a0 = b, but a ∨ b ̸= 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=', a0 is not a complement of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  For every cardinal number n let Mn = (Mn, ∨, ∧) denote the bounded modular lattice of  length 2 having n atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  The situation from Figure 2 can be generalized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  4  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Every element of a direct product of a Boolean algebra and an arbitrary  number of lattices Mn (possibly different n) is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  This follows immediately from the fact that every element of a Boolean algebra and every  element of the lattice Mn is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  However, if the lattice L is not a direct product of two-element lattices and various Mn  then the assertion of Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='4 need not hold, see the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Consider the lattice visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' 3:  0  a  b  c  d  e  f  g  1  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' 3  Lattice  We have  a00 = {b, c}0 = f ̸= a,  a000 = f 0 = {b, c} = a0,  b00 = {c, f}0 = b,  (a0 ∧ b0)00 = ({b, c} ∧ {c, f})00 = {0, c}00 = {b, f}0 = c ̸= {0, c} = a0 ∧ b0,  (c0 ∧ f 0)00 = ({b, f} ∧ {b, c})00 = {0, b}00 = {c, f}0 = b ̸= {0, b} = e0 ∧ f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Hence a is not sharp, b is sharp and the equality (x0 ∧ y0)00 = x0 ∧ y0 does not hold in  general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  We are going to show that the operator 0 can be characterized by means of four simple  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Let (S, ∧, 0) be a meet-semilattice with 0 satisfying the ACC and 0 a  unary operator on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then the following are equivalent:  (i) x0 = Max{y ∈ S | x ∧ y = 0} for all x ∈ S,  (ii) the operator 0 satisfies the following conditions:  (P1) x0 is an antichain,  5  (P2) x ∧ 00 ≈1 x,  (P3) x ∧ x0 ≈ 0,  (P4) x ∧ (x ∧ y)0 ≈1 x ∧ y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (i) ⇒ (ii):  This follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (ii) ⇒ (i):  If x ∧ y = 0 then according to (P2) and (P4) we have  y ≈1 y ∧ 00 = y ∧ (x ∧ y)0 = y ∧ (y ∧ x)0 ≈1 y ∧ x0 ≤1 x0  and hence y ≤1 x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Conversely, if y ≤1 x0 then according to (P3) we have  x ∧ y ≤1 x ∧ x0 = 0  and hence x ∧ y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' This shows that x ∧ y = 0 is equivalent to y ≤1 x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' We conclude  Max{y ∈ S | x ∧ y = 0} = Max{y ∈ S | y ≤1 x0} = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  The last equality can be seen as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Let z ∈ Max{y ∈ S | y ≤1 x0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then z ≤1 x0,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' there exists some u ∈ x0 with z ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' We have u ≤1 x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Now z < u would imply  z /∈ Max{y ∈ S | y ≤1 x0}, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' This shows z = u ∈ x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Conversely, assume  z ∈ x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then z ≤1 x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' If z /∈ Max{y ∈ S | y ≤1 x0} then there would exist some u ∈ S  with z < u ≤1 x0 and hence there would exist some w ∈ x0 with z < u ≤ w contradicting  (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' This shows z ∈ Max{y ∈ S | y ≤1 x0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  4  Unsharp implication  Now we extend the operation of relative pseudocomplementation to arbitrary meet-  semilattices with 0 satisfying the ACC as follows: Let S = (S, ∧, 0) be a meet-semilattice  with 0 satisfying the ACC, a, b ∈ S and A, B ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' We define  a → b := Max{x ∈ S | a ∧ x ≤ b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Thus → is a binary operator on S assigning to every pair (x, y) ∈ S2 the non-void subset  x → y ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' It is evident that  x0 = x → 0 for each x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Moreover, we define  A → B := Max{x ∈ S | A ∧ x ≤ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' The “operation table” of the operator → in the meet-semilattice of Figure 1  looks as follows (we write abc instead of {a, b, c} and so on):  →  0  a  b  c  0  abc  abc  abc  abc  a  bc  abc  bc  ab  b  ac  ac  abc  ac  c  ab  ab  ab  abc  6  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' The “operation table” of the operator → in the meet-semilattice of Figure 3  looks as follows (we write bc instead of {b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' c} and so on):  →  0  a  b  c  d  e  f  g  1  0  1  1  1  1  1  1  1  1  1  a  bc  1  bc  bc  1  1  1  1  1  b  cf  cf  1  cf  cf  1  cf  1  1  c  bf  bf  bf  1  bf  1  bf  1  1  d  bc  g  bc  bc  1  g  1  g  1  e  0  f  b  c  f  1  f  1  1  f  bc  g  bc  bc  dg  g  1  g  1  g  0  f  b  c  f  ef  f  1  1  1  0  a  b  c  d  e  f  g  1  The following properties of the binary operator → can be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Let S = (S, ∧, 0) be a meet-semilattice with 0 satisfying the ACC and  a, b, c ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then the following holds:  (i) a → b is an antichain,  (ii) a ≤ b implies a → b = Max S,  (iii) b ∈ Max S implies b ∈ a → b,  (iv) b ≤1 a → b,  (v) a ≤1 (a → b) → b,  (vi) a ≤ b implies c → a ≤1 c → b and b → c ≤1 a → c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (vii) a ∧ (a → b) ≈1 a ∧ b,  (viii) a → (b ∧ c) ≈1 (a → b) ∧ (a → c),  (ix) (a → b) ∧ b ≈1 b,  (x) if S is bounded then 1 → b = b,  (xi) a ∧ (b → b) ≈1 a,  (xii) if S is bounded then a → b = 1 if and only if a ≤ b,  (xiii) b ≤1 a → (a ∧ b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (i) This is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (ii), (iv), (x), (xii) and (xiii) follow immediately from the definition of →.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (iii) If b ∈ Max S then because of a∧b ≤ b we have b ∈ Max{x ∈ S | a∧x ≤ b} = a → b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (v) Since a ∧ x ≤ b for all x ∈ a → b we have a ∧ (a → b) ≤ b, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' (a → b) ∧ a ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  7  (vi) If a ≤ b then  {x ∈ S | c ∧ x ≤ a} ⊆ {x ∈ S | c ∧ x ≤ b},  {x ∈ S | b ∧ x ≤ c} ⊆ {x ∈ S | a ∧ x ≤ c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (vii) We have a ∧ x ≤ b and hence a ∧ x ≤ a ∧ b for all x ∈ a → b and hence a ∧ b ≤1  a ∧ (a → b) ≤ a ∧ b according to (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (viii) According to (vii) we have a → (b ∧ c) ≤1 (a → b) ∧ (a → c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Conversely, assume  d ∈ a → b and e ∈ a → c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then a∧d ≤ b and a∧e ≤ c and hence a∧(d∧e) ≤ b∧c  which implies d ∧ e ≤1 a → (b ∧ c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' This shows (a → b) ∧ (a → c) ≤1 a → (b ∧ c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (ix) We have b ≤1 a → b according to (iv) and hence b ≤1 (a → b) ∧ b ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (xi) According to (ii) we have a ≤1 Max S = b → b and hence a ≤1 a ∧ (b → b) ≤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='3 it is evident that the binary operator → shares properties of the  logical connective implication in intuitionistic logic despite the fact that it is unsharp,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' for x, y ∈ S the result of x → y need not be a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Hence it extends the  intuitionistic logic based on a Heyting algebra (L, ∨, ∧, ∗, 0) where again ∨ formalizes  disjunction, ∧ formalizes conjunction, but now → formalizes unsharp implication and 0  formalizes unsharp negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Consider the modular lattice visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' 4:  0  a  b  c  d  e  f  1  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' 4  Modular lattice  Then  e ≤1 {d, e} = {e, f} → e = (d → e) → e,  d ∧ (d → e) = d ∧ {e, f} = c = d ∧ e,  (d → e) ∧ e = {e, f} ∧ e = {c, e} ≈1 e  in accordance with (v), (vii) and (ix) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  8  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' It is easy to see that the operation ∧ and the operator → are related by  so-called unsharp adjointness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  a ∧ b ≤ c if and only if a ≤1 b → c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Since the operation ∧ is associative, commutative and monotone, it can be considered as  a t-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Thus the semilattice (S, ∧, →) endowed with the operator → is an unsharply  residuated semilattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Moreover, by (viii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='3 we have  a ∧ (a → b) ≈1 a ∧ b  showing that (S, ∧, →) satisfies divisibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Similarly as for the unary operator 0 we can characterize the binary operator → on a  meet-semilattice with 0 satisfying the ACC as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Let (S, ∧, 0) be a meet-semilattice with 0 satisfying the ACC and → a  binary operator on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then the following are equivalent:  (i) x → y = Max{z ∈ S | x ∧ z ≤ y} for all x, y ∈ S,  (ii) The operator → satisfies the following conditions:  (R1) x → y is an antichain,  (R2) x ∧ (x → y) ≈1 x ∧ y,  (R3) (x → y) ∧ y ≈1 y,  (R4) x → (y ∧ z) ≈1 (x → y) ∧ (x → z),  (R5) x ∧ (y → y) ≈1 x,  (R6) y ≤ z implies x → y ≤1 x → z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (i) ⇒ (ii):  This follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (ii) ⇒ (i):  If x ∧ z ≤ y then according to (R3), (R5), (R4) and (R6) we have  z ≈1 (x → z) ∧ z ≤1 x → z ≈1 (x → x) ∧ (x → z) ≈1 x → (x ∧ z) ≤1 x → y  and hence z ≤1 x → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Conversely, if z ≤1 x → y then according to (R2) we have  x ∧ z ≤1 x ∧ (x → y) ≈1 x ∧ y ≤ y  and hence x∧z ≤ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' This shows that x∧z ≤ y is equivalent to z ≤1 x → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' We conclude  Max{z ∈ S | x ∧ z ≤ y} = Max{z ∈ S | z ≤1 x → y} = x → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  The last equality can be seen as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Let u ∈ Max{z ∈ S | z ≤1 x → y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then  u ≤1 x → y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' there exists some v ∈ x → y with u ≤ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' We have v ≤1 x → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Now u < v would imply u /∈ Max{z ∈ S | z ≤1 x → y}, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  This  shows u = v ∈ x → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Conversely, assume u ∈ x → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Then u ≤1 x → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  If  u /∈ Max{z ∈ S | z ≤1 x → y} then there would exist some v ∈ S with u < v ≤1 x → y  and hence there would exist some w ∈ x → y with u < v ≤ w contradicting (R1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' This  shows u ∈ Max{z ∈ S | z ≤1 x → y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  9  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Consider the lattice from Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then  d ∧ (d → e) = d ∧ {e, f} = c = d ∧ e,  (d → e) ∧ e = e ∧ e = e,  d → (e ∧ f) = d → c = {e, f} ≈1 {c, e, f} = {e, f} ∧ {e, f} = (d → e) ∧ (d → f)  in accordance with (R2), (R3) and (R4), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  5  Deductive systems  It is well-known that the connective implication in intuitionistic logic is closely related  to the so-called deductive systems in the corresponding Brouwerian semilattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' In what  follows we show that a certain modification of the concept of a deductive system plays a  similar role for logics with unsharp implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' We define  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' A deductive system of a meet-semilattice S = (S, ∧, 0) with 0 satisfying  the ACC is a subset D of S satisfying the following conditions for x, y ∈ S:  (D1) (Max S) ∩ D ̸= ∅,  (D2) x ∈ D and (x → y) ∩ D ̸= ∅ imply y ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Recall that a filter of a meet-semilattice S = (S, ∧) is a non-empty subset F of S satisfying  the following conditions for x, y ∈ S:  (F1) x, y ∈ F implies x ∧ y ∈ F,  (F2) x ∈ F and x ≤ y imply y ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  It is clear that if S is finite then all filters of S are given by the sets [x) := {y ∈ S | x ≤ y},  x ∈ S, and hence the poset of all filters of S is dually isomorphic to S and therefore a  join-semilattice where [x) ∨ [y) = [x ∧ y) for all x, y ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  For every non-empty subset A of the universe of a meet-semilattice (S, ∧) we define a  binary relation Θ(A) on S as follows:  (x, y) ∈ Θ(A) if there exists some a ∈ A with x ∧ a = y ∧ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Although the following result is known, for the reader’s convenience we present the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Let S = (S, ∧, 1) be a meet-semilattice with 1 and Φ ∈ Con S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then the  following holds:  (i) [1]Φ is an filter of S,  (ii) Θ([1]Φ) ⊆ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (i) (F1) If a, b ∈ [1]Φ then a ∧ b ∈ [1 ∧ 1]Φ = [1]Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  10  (F2) If a ∈ [1]Φ, b ∈ S and a ≤ b then b = 1 ∧ b ∈ [a ∧ b]Φ = [a]Φ = [1]Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  This shows that [1]Φ is a filter of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (ii) If (a, b) ∈ Θ([1]Φ) then there exists some c ∈ [1]Φ with a ∧ c = b ∧ c whence  a = a ∧ 1 Φ a ∧ c = b ∧ c Φ b ∧ 1 = b  which shows (a, b) ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Although our definition of a deductive system differs from that known for relatively  pseudocomplemented semilattices, we are still able to prove the following relationships  between the concepts mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Let S = (S, ∧, 0, 1) be a bounded meet-semilattice satisfying the ACC and  D a non-empty subset of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then the following are equivalent:  (i) D a deductive system of S,  (ii) D is an filter of S,  (iii) Θ(D) ∈ Con S and D = [1]  �  Θ(D)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (i) ⇒ (ii):  (F2) Assume a ∈ D, b ∈ S and a ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Then a → b = Max S because of (ii) of  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' According to (D1) we have (a → b) ∩ D = Max S ∩ D ̸= ∅ and hence  b ∈ D by (D2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (F1) Let a, b ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then by (xiii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='3 we have b ≤1 a → (a ∧ b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Hence there  exists some c ∈ a → (a ∧ b) with b ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Now (F2) implies c ∈ D and therefore  �  a → (a ∧ b)  �  ∩ D ̸= ∅ from which we conclude a ∧ b ∈ D by (D2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (ii) ⇒ (iii):  Evidently, Θ(D) is reflexive and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Let (a, b), (b, c) ∈ Θ(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then there exist  d, e ∈ D with a ∧ d = b ∧ d and b ∧ e = c ∧ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Because of (F1) we conclude d ∧ e ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Now  a ∧ (d ∧ e) = (a ∧ d) ∧ e = (b ∧ d) ∧ e = (b ∧ e) ∧ d = (c ∧ e) ∧ d = c ∧ (d ∧ e)  which yields (a, c) ∈ Θ(D), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Θ(D) is transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Further, if f ∈ S then  (a ∧ f) ∧ d = (a ∧ d) ∧ f = (b ∧ d) ∧ f = (b ∧ f) ∧ d  showing (a ∧ f, b ∧ f) ∈ Θ(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Hence Θ(D) ∈ Con S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  If a ∈ D then because of  a ∧ a = a = 1 ∧ a we have a ∈ [1]  �  Θ(D)  �  showing D ⊆ [1]  �  Θ(D)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Conversely, assume  a ∈ [1]  �  Θ(D)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then there exists some b ∈ D with a ∧ b = 1 ∧ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' This implies b ≤ a  wherefrom we conclude a ∈ D by (F2) showing [1]  �  Θ(D)  �  ⊆ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (iii) ⇒ (i):  11  (D1) If a ∈ D then, since S satisfies the ACC, there exists some b ∈ Max S with a ≤ b  and hence  b = 1 ∧ b ∈ [a ∧ b]  �  Θ(D)  �  = [a]  �  Θ(D)  �  = [1]  �  Θ(D)  �  = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (D2) If a ∈ D, b ∈ S and (a → b) ∩ D ̸= ∅ then there exists some c ∈ D with c ∈ a → b  and hence a ∧ c ≤ b whence  b = 1∧1∧b ∈ [a∧c∧b]  �  Θ(D)  �  = [a∧c]  �  Θ(D)  �  = [1∧1]  �  Θ(D)  �  = [1]  �  Θ(D)  �  = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  It is well known that for a filter F of a relatively pseudocomplemented semilattice we  have (a, b) ∈ Θ(F) if and only if a → b ∈ F and b → a ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' However, we can modify  this result also for an arbitrary meet-semilattice with 0 satisfying the ACC provided our  unsharp implication is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Let S = (S, ∧, 0) be a meet-semilattice with 0 satisfying the ACC, F  a filter of S and a, b ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then the following are equivalent:  (i) (a, b) ∈ Θ(F),  (ii) (a → b) ∩ F ̸= ∅ and (b → a) ∩ F ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (i) ⇒ (ii):  There exists some c ∈ F with a ∧ c = b ∧ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Hence a ∧ c ≤ b and b ∧ c ≤ a and therefore  there exists some d ∈ a → b with c ≤ d and some e ∈ b → a with c ≤ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Because of (F2)  we conclude d, e ∈ F showing (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  (ii) ⇒ (i):  Let c ∈ (a → b)∩F and d ∈ (b → a)∩F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Then c∧d ∈ F by (F1), a∧c ≤ b and b∧d ≤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Hence  a ∧ (c ∧ d) = (a ∧ c) ∧ (c ∧ d) ≤ b ∧ (c ∧ d) = (b ∧ d) ∧ (c ∧ d) ≤ a ∧ (c ∧ d),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' a ∧ (c ∧ d) = b ∧ (c ∧ d) showing (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Conclusion  Although the implication within the logic based on the structure (S, ∧, 0, →) is unsharp,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' x → y may be a subset I of S which need not be a singleton, it has its logical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Namely, we ask that x → y is the maximal element c of S satisfying x ∧ c ≤ y (where ∧  denotes conjunction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' And for each c ∈ I this is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Moreover, the elements of I  are mutually incomparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Thus we have no need to prefer one of them with respect to  others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' However, the expression  x ∧ (x → y) ≤ y  is nothing else than the derivation rule Modus Ponens (both in classical as well as in  non-classical logic) since it properly says that the truth value of y cannot be less than  the truth value of the conjunction x ∧ (x → y) of x and the implication x → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Hence,  despite of the fact of unsharpness, such a logic is sound although it is derived from an  arbitrary meet-semilattice with 0 satisfying the ACC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  12  References  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Brouwer, De onbetrouwbaarheid der logische principes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Tijdschrift Wijs-  begeerte 2 (1908), 152–158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Brouwer, Intuitionism and formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' 20 (1913),  81–96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  [3] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Chajda, An extension of relative pseudocomplementation to non-distributive lat-  tices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Acta Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' (Szeged) 69 (2003), 491–496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  [4] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Chajda, Pseudocomplemented and Stone posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Acta Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Palack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Olomuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Fac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  Rerum Natur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' 51 (2012), 29–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  [5] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Chajda and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' L¨anger, Algebras describing pseudocomplemented, relatively pseu-  docomplemented and sectionally pseudocomplemented posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Symmetry 13 (2021),  753 (17 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=')  [6] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Chajda and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' L¨anger, Implication in finite posets with pseudocomplemented  sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Soft Computing 26 (2022), 5945–5953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  [7] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Chajda and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' L¨anger, The logic of orthomodular posets of finite height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  IGPL 30 (2022), 143–154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  [8] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Chajda and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' L¨anger, Operator residuation in orthomodular posets of finite  height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Fuzzy Sets Systems (submitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  [9] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Chajda, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' L¨anger and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Paseka, Sectionally pseudocomplemented posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Order  38 (2021), 527–546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Fazio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Ledda and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Paoli, On Finch’s conditions for the completion of ortho-  modular posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' (2020), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='1007/s10699-020-09702-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  [11] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Finch, On orthomodular posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Austral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
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+page_content=' Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
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+page_content=' Monteiro, Les alg`ebres de Heyting et de Lukasiewicz trivalentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
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+page_content=' Pt´ak and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Pulmannov´a, Orthomodular Structures as Quantum Logics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' Kluwer,  Dordrecht 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' ISBN 0-7923-1207-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='  13  Authors’ addresses:  Ivan Chajda  Palack´y University Olomouc  Faculty of Science  Department of Algebra and Geometry  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' listopadu 12  771 46 Olomouc  Czech Republic  ivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='chajda@upol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='cz  Helmut L¨anger  TU Wien  Faculty of Mathematics and Geoinformation  Institute of Discrete Mathematics and Geometry  Wiedner Hauptstraße 8-10  1040 Vienna  Austria, and  Palack´y University Olomouc  Faculty of Science  Department of Algebra and Geometry  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content=' listopadu 12  771 46 Olomouc  Czech Republic  helmut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='laenger@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}
+page_content='at  14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfRQCY/content/2301.02205v1.pdf'}

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+Do Users Want Platform Moderation or Individual Control? 
+Examining the Role of Third-Person Effects and Free Speech 
+Support in Shaping Moderation Preferences 
+Shagun	Jhaver,	Amy	Zhang	
+Online	platforms	employ	commercial	content	moderators	and	use	automated	systems	to	identify	and	remove	
+the	most	blatantly	inappropriate	content	for	all	users.	They	also	provide	moderation	settings	that	let	users	
+personalize	their	preferences	for	which	posts	they	want	to	avoid	seeing.	This	study	presents	the	results	of	a	
+nationally	representative	survey	of	984	US	adults.	We	examine	how	users	would	prefer	for	three	categories	of	
+norm-violating	 content	 (hate	 speech,	 sexually	 explicit	 content,	 and	 violent	 content)	 to	 be	 regulated.	
+Specifically,	we	analyze	whether	users	prefer	platforms	to	remove	such	content	for	all	users	or	leave	it	up	to	
+each	user	to	decide	if	and	how	much	they	want	to	moderate	it.	We	explore	the	influence	of	presumed	effects	
+on	others	(PME3)	and	support	for	freedom	of	expression	on	user	attitudes,	the	two	critical	factors	identified	
+as	 relevant	 for	 social	 media	 censorship	 attitudes	 by	 prior	 literature,	 about	 this	 choice.	 We	 find	 perceived	
+negative	effects	on	others	and	free	speech	support	as	significant	predictors	of	preference	for	having	personal	
+moderation	 settings	 over	 platform-directed	 moderation	 for	 regulating	 each	 speech	 category.	 Our	 findings	
+show	that	platform	governance	initiatives	need	to	account	for	both	the	actual	and	perceived	media	effects	of	
+norm-violating	speech	categories	to	increase	user	satisfaction.	Our	analysis	also	suggests	that	people	do	not	
+see	personal	moderation	tools	as	an	infringement	on	others’	free	speech	but	as	a	means	to	assert	greater	
+agency	to	shape	their	social	media	feeds.	
+And	 then	 there	 were	 what	 I'll	 call	 the	 technolibertarians.	 For	 them,	 MUD	 rapists	 were	 of	
+course	assholes,	but	the	presence	of	assholes	on	the	system	was	a	technical	inevitability,	like	
+noise	 on	 a	 phone	 line,	 and	 best	 dealt	 with	 not	 through	 repressive	 social	 disciplinary	
+mechanisms	 but	 through	 the	 timely	 deployment	 of	 defensive	 software	 tools.	 Some	 asshole	
+blasting	violent,	graphic	language	at	you?	Don't	whine	to	the	authorities	about	it	–	hit	the	
+@gag	 command	 and	 the	 asshole's	 statements	 will	 be	 blocked	 from	 your	 screen	 (and	 only	
+yours).	It's	simple,	it's	effective,	and	it	censors	no	one.	–	Excerpt	from	“A	Rape	in	Cyberspace”	
+by	Julian	Dibbell	[7]	
+Introduction 
+With	 the	 emergence	 of	 social	 media	 sites	 and	 their	 widespread	 use	 by	 people	 to	
+communicate	 with	 one	 another,	 companies	 like	 Facebook,	 Twitter,	 and	 YouTube	 have	
+become	 the	 new	 governors	 of	 digital	 expression.	 At	 the	 same	 time,	 individuals	 who	 use	
+these	sites	can	also	actively	shape	governance	in	various	ways.	For	example,	they	may	flag	
+posts	 that	 violate	 community	 policy,	 downvote	 inappropriate	 posts,	 serve	 as	 volunteer	
+moderators,	 engage	 in	 counter-speech,	 or	 configure	 moderation	 settings	 to	 automate	
+inappropriate	 post	 removals.	 We	 are	 therefore	 moving	 towards	 a	 “pluralist	 model	 of	
+speech	regulation	[1],”	in	which	speech	must	be	regulated	in	a	multi-stakeholder	fashion	–	
+legislative	entities	enforce	online	speech	laws,	and	platform	operators	set	up	governance	
+regimes	 of	 acceptable	 content.	 However,	 users	 themselves	 can	 also	 intervene	 against	
+content	perceived	as	problematic.		
+
+This	move	to	a	pluralist	model	is	occurring	against	recent	controversies	over	platforms’	
+moderation	decisions	[32,	47]	and	growing	media,	policymakers,	and	public	calls	to	better	
+regulate	their	content	[51].	In	response,	platforms	have	begun	investing	more	resources	
+into	 improving	 how	 inappropriate	 posts	 are	 detected	 and	 removed	 from	 their	 sites.	 We	
+focus	in	this	article	on	platforms’	offering	of	personal	moderation	tools	that	let	end-users	
+configure	content	moderation	of	the	posts	they	see	to	align	with	their	content	preferences.	
+We	are	primarily	concerned	with	tools	offered	by	platforms	such	as	Instagram	and	Twitch	
+that	let	users	specify	their	sensitivity	to	specific	topical	categories	such	as	sexually	explicit	
+content	 and	 hate	 speech.	 Configuring	 such	 tools	 lets	 users	 have	 the	 moderation	 system	
+operate	in	alignment	with	their	tastes	and	thresholds.		From	the	perspective	of	platforms,	a	
+tactical	consequence	of	offering	personal	moderation	tools	is	that	the	obligation	of	making	
+hard	moderation	decisions,	the	concomitant	responsibility	of	making	mistakes	with	them,	
+and	the	cognitive	labor	of	making	the	correct	configurations	are	passed	over	to	the	users.	
+Therefore,	it	is	vital	that	we	understand	how	users	consider	the	choice	between	platform	
+versus	personal	moderation.			
+Our	 research	 responds	 to	 calls	 by	 governance	 scholars	 to	 conduct	 more	 survey-based	
+research	 to	 understand	 users’	 perspectives	 on	 moderation	 interventions	 by	 different	
+regulatory	actors	[8,	41].	So	far,	we	have	little	knowledge	of	how	end-users	perceive	being	
+given	 self-regulating	 authority	 through	 personal	 moderation	 tools.	 We	 do	 not	 know	 the	
+situations	 in	 which	 users	 prefer	 to	 have	 a	 choice	 in	 shaping	 moderation	 and	 when	 they	
+would	instead	prefer	the	platforms	manage	it	for	every	user	–	and	the	different	factors	that	
+shape	these	preferences.		
+Informed	by	the	third-person	effects	(TPE)	hypothesis,	we	fill	this	gap	by	examining	users’	
+preferences	in	the	context	of	three	norm-violating	speech	categories	that	have	been	studied	
+in	prior	literature:	(1)	Hate	speech;	(2)	Sexually	explicit	content;	and	(3)	Violent	content.	
+Previous	research	has	shown	that	perceptions	of	the	effects	of	media	messages	on	others	
+predict	censorship	attitudes	[42,	44].	We	examine	the	role	that	third-person	effects	[6]	play	
+in	 shaping	 user	 attitudes	 about	 deploying	 platform-enacted	 versus	 personal	 moderation	
+tools.		
+We	also	connect	our	findings	to	the	scholarship	on	understanding	public	attitudes	toward	
+freedom	 of	 expression	 and	 its	 consequences.	 Free	 speech	 is	 a	 core	 constitutional	 right	
+highly	valued	by	many	Americans.	However,	the	introduction	of	personal	moderation	tools	
+offers	 affordances	 that	 complicate	 the	 question	 of	 preserving	 free	 speech	 –	 users	 may	
+configure	tools	to	avoid	specific	content	categories.	Still,	others	may	continue	to	see	the	
+same	 content,	 thereby	 preventing	 infringement	 of	 online	 expressions.	 However,	 if	 most	
+users	 choose	 to	 deploy	 these	 tools,	 specific	 content	 categories	 would	 have	 significantly	
+reduced	 visibility.	 Therefore,	 we	 analyze	 how	 users’	 support	 for	 freedom	 of	 expression	
+shapes	their	notion	of	different	moderation	approaches.	
+Understanding	 the	 public	 views	 on	 the	 platform	 and	 user-enacted	 interventions	 can	
+stimulate	debates	about	the	roles	and	strategies	of	various	regulatory	actors.	It	can	also	
+speak	 to	 calls	 for	 evidence-based	 policymaking	 [30,	 40]	 by	 clarifying	 how	 the	 public	
+understands	 moderation	 practices	 and	 identifying	 the	 gaps	 between	 policy	 and	 public	
+demands	[36,	37,	41].	Given	the	rapid	introduction	of	new	moderation	strategies	by	the	
+
+platforms,	 especially	 personal	 moderation	 tools,	 independent	 academic	 assessments	 of	
+users’	 attitudes	 on	 their	 deployment	 are	 vital.	 Platforms	 can	 also	 benefit	 from	 such	
+research	 by	 understanding	 end-users'	 acceptance	 or	 rejection	 of	 various	 regulatory	
+practices	and	the	factors	that	shape	those	perspectives.	Further,	examinations	of	the	public	
+perception	of	free	speech	within	the	context	of	online	activity	may	also	shape	attempts	to	
+protect	free	speech	in	the	long	run.	
+Literature Review and Hypotheses 
+The Perceptual Component of the TPE 
+Since	Davison	[6]	first	argued	that	individuals	perceive	media’s	impact	on	the	attitudes	and	
+behaviors	of	others	to	be	greater	than	it	is	on	themselves,	many	studies	have	shown	this	
+discrepancy	to	be	consistent	across	a	range	of	contexts	[49]	such	as	political	ads	[16,	38],	
+news	 stories	 [39]	 and	 social	 media	 use	 [46].	 This	 hypothesis,	 termed	 by	 Davison	 as	 the	
+third-person	 effect	 (TPE)	 [6],	 has	 become	 a	 widely	 applied	 perspective	 to	 explain	 public	
+opinion	 on	 the	 media	 censorship	 [19-21,	 33].	 The	 TPE	 hypothesis	 has	 two	 major	
+components	–	the	perceptual	and	behavioral	components	[18].	The	perceptual	component	
+predicts	that	presumed	media	effects	on	others	(PME3)	tend	to	be	greater	than	perceived	
+media	 effects	 on	 self	 (PME1).	 In	 the	 context	 of	 social	 media	 messages’	 influence,	 the	
+perceptual	component	of	TPE	predicts	that	participants	will	consider	others	to	be	more	
+negatively	 influenced	 by	 each	 category	 of	 norm-violating	 speech	 than	 themselves.	 We,	
+therefore,	raise	the	following	hypothesis:	
+H1:	For	each	norm-violating	speech	category,	participants	will	perceive	a	greater	effect	of	
+that	speech	on	others	than	on	themselves.	
+The Behavioral Component of the TPE 
+The	behavioral	component	of	the	TPE	argues	that	when	individuals	perceive	the	greater	
+impact	of	media	messages	on	others	than	on	themselves,	they	will	take	remedial	actions	to	
+mitigate	 the	 perceived	 harmful	 effects	 [20].	 Davison	 [6]	 described	 the	 phenomenon	 of	
+censorship	 as	 one	 of	 the	 most	 interesting	 behavioral	 consequences	 of	 third-person	
+perception.	Prior	research	on	TPE	consequences	shows	that	it	leads	to	censorship	support	
+[15,	19,	44],	and	the	effects	are	particularly	salient	when	persuasive	attempts	may	include	
+socially	 undesirable	 effects	 [28].	 In	 studying	 TPE	 effects	 on	 censorship	 attitudes,	 some	
+researchers	 have	 examined	 the	 consequences	 of	 the	 other-self	 perceptual	 disparity	 in	
+media	effects	(DME	=	PME3	–	PME1).	In	contrast,	others	have	focused	on	the	perceived	
+media	impact	on	others	(PME3)	[42].	Examining	past	research	data	on	TPE	consequences,	
+Chung	 and	 Moon	 [5]	 concluded	 that	 the	 media’s	 presumed	 effect	 on	 others	 (PME3)	 is	 a	
+stronger	 predictor	 of	 censorship	 attitudes	 than	 the	 other-self	 disparity	 in	 the	 perceived	
+media	effects.	We,	therefore,	choose	PME3	as	our	primary	predictor	variable	and	raise	the	
+following	hypothesis:	
+H2:	 For	 each	 speech	 category,	 the	 perceived	 effects	 of	 that	 speech	 on	 others	 will	 be	
+positively	related	to	support	for	the	platform’s	banning	of	that	category.	
+
+Researchers	of	TPE	have	long	been	curious	about	the	potential	behaviors	that	could	result	
+from	 the	 perceived	 media	 impact	 on	 others.	 In	 addition	 to	 censorship	 support,	 prior	
+studies	have	interrogated	behavioral	outcomes	such	as	engaging	in	political	action		[10,	43,	
+50],	disseminating	opposing	information	[3,	17],	and	exposing	apparent	biases	[2,	29].	In	
+the	context	of	online	content	moderation,	Jang	and	Kim	found	that	people	with	a	greater	
+level	of	third-person	perception	were	more	likely	to	support	media	literacy	interventions	
+to	 address	 fake	 news	 [22].	 We	 add	 to	 previous	 efforts	 to	 surface	 the	 different	 types	 of	
+behavioral	 consequences	 of	 TPE	 by	 examining	 its	 impact	 on	 users’	 support	 for	 having	
+personal	 moderation	 settings	 to	 moderate	 norm-violating	 content.	 Since	 such	
+configurations	are	also	a	form	of	regulation,	prior	TPE	research	suggests	that	PME3	would	
+predict	support	for	them	[19,	42].	The	following	hypothesis	thus	can	be	raised:	
+H3:	 For	 each	 speech	 category,	 the	 perceived	 effects	 of	 that	 speech	 on	 others	 will	 be	
+positively	related	to	support	for	having	personal	moderation	tools	to	regulate	the	speech	of	
+that	category.	
+Platform-enacted	 moderation	 and	 personal	 configurations	 to	 self-moderate	 content	 are	
+different	 ways	 to	 regulate	 norm-violating	 speech.	 However,	 while	 platform-enacted	
+moderation	 censors	 content	 platform-wide,	 personal	 moderation	 tools	 allow	 users	 to	
+adjust	 whether	 and	 how	 much	 norm-violating	 speech	 they	 are	 willing	 to	 encounter	
+personally.	 Prior	 literature	 generally	 predicts	 that	 TPE	 would	 lead	 to	 support	 for	
+censorship	attitudes	[5,	19].	However,	it	does	not	guide	how	people	will	react	to	a	choice	
+between	letting	platforms	handle	a	specific	content	category	and	allowing	users	to	specify	
+their	moderation	preferences	for	that	category.	We,	therefore,	ask	the	following	research	
+question:	
+RQ1:	For	each	speech	category,	how	do	its	perceived	effects	on	others	relate	to	support	for	
+the	 platform’s	 banning	 of	 that	 category	 versus	 support	 for	 having	 personal	 moderation	
+tools	regulate	it?	
+Support for Freedom of Speech  
+Discussions	about	the	benefits	of	moderation	measures	are	always	intertwined	with	the	
+issue	of	freedom	of	speech.	In	the	United	States,	the	constitution	protects	the	right	to	free	
+expression	as	a	fundamental	human	right.	However,	platforms	are	private	parties.	Section	
+230	of	the	Communications	Decency	Act	of	1996	provides	them	the	legislative	freedom	to	
+police	their	users	as	they	see	fit	while	not	being	held	accountable	for	errors	or	oversights	
+[13,	34].	Additionally,	experts	have	shown	that	despite	Americans’	support	for	freedom	of	
+expression	 generally,	 their	 tolerance	 for	 hate	 speech	 is	 low	 	 [12,	 48,	 52].	 Thus,	 people’s	
+acceptance	of	free	speech	in	the	abstract	may	not	automatically	imply	their	tolerance	for	
+opposing	expressions	[20].	Examining	how	much	people's	support	for	free	speech	affects	
+their	 opinions	 about	 varied	 moderation	 strategies	 may	 help	 us	 better	 understand	 this	
+discrepancy	in	the	context	of	social	media	platforms.	
+Support	 for	 free	 speech	 and	 attitudes	 toward	 content	 moderation	 have	 been	 linked	 in	
+previous	studies.	For	instance,	Naab	et	al.	showed	that	people	who	commit	to	freedom	of	
+expression	are	less	likely	to	support	restrictive	actions	by	Facebook	moderators	[36].	Guo	
+
+and	 Johnson	 showed	 that	 a	 lack	 of	 support	 for	 freedom	 of	 speech	 predicts	 support	 for	
+government	 regulation	 of	 sexist	 hate	 speech	 [20].	 However,	 they	 did	 not	 find	 that	 the	
+former	 could	 predict	 supportive	 attitudes	 toward	 platform	 censorship.	 Jang	 and	 Kim	
+suggested	 that	 support	 for	 free	 expression	 decreases	 support	 for	 regulating	 fake	 news	
+despite	the	existence	of	third-person	effects	[22].	Overall,	this	body	of	research	indicates	
+that	 participants’	 support	 for	 platform	 regulation	 will	 decline	 as	 free	 speech	 support	
+increases.	We,	therefore,	raise	the	following	hypotheses:	
+H4:	 For	 each	 speech	 category,	 participants’	 support	 for	 freedom	 of	 expression	 will	 be	
+negatively	related	to	their	support	for	the	platform’s	banning	of	that	category.	
+While	prior	research	provides	guidance	on	the	expected	relationship	between	free	speech	
+support	 and	 support	 for	 platform-enacted	 moderation,	 the	 literature	 on	 personal	
+moderation	 tools	 is	 scarce.	 It	 does	 not	 offer	 any	 direct	 guidance	 on	 the	 relationship	
+between	 support	 for	 free	 speech	 and	 support	 for	 having	 personal	 moderation	 tools.	
+However,	there	is	some	prior	work	that	speaks	to	related	issues.	For	example,	Naab	et	al.	
+did	not	find	a	relationship	between	users’	commitment	to	free	speech	and	their	intention	to	
+engage	 in	 corrective	 actions	 such	 as	 rebuking	 the	 comment	 author	 or	 reporting	 the	
+comment	 [36].	 It	 is	 unclear	 whether	 that	 finding	 would	 apply	 to	 our	 context	 since	
+deploying	 personal	 moderation	 tools	 is	 a	 restrictive	 action,	 not	 a	 corrective	 action.	 Its	
+expected	 costs	 (e.g.,	 setting	 up	 once	 for	 personal	 moderation	 v/s	 reporting	 every	
+inappropriate	post;	experiencing	retaliation	when	engaging	in	counter-speech	v/s	private	
+post	removals	for	personal	moderation)	are	also	lower.	Riedl	et	al.	conducted	a	survey	that	
+showed	 that	 individuals’	 support	 for	 free	 speech	 does	 not	 increase	 their	 assumed	 self-
+responsibility	 to	 intervene	 against	 problematic	 comments	 [41].	 However,	 they	 did	 not	
+specify	to	the	survey	takers	what	type	of	responsibility	they	should	carry	out.	Besides,	in	
+our	study	context,	users	may	not	consider	personal	moderation	tools	an	obligation.		
+On	 the	 one	 hand,	 support	 for	 free	 speech	 values	 should	 reduce	 support	 for	 restrictive	
+actions	of	any	kind.	But	on	the	other	hand,	people	may	perceive	personal	moderation	tools	
+as	simply	having	a	greater	agency	to	shape	what	they	see	and	not	an	infringement	on	the	
+free	speech	of	others.	In	this	way,	personal	moderation	tools	allow	a	distinction	between	
+freedom	 of	 speech	 and	 the	 obligation	 to	 be	 heard.	 To	 clarify	 the	 direction	 of	 this	
+relationship,	we	ask	the	following	research	question:	
+RQ2:	 Does	 participants’	 support	 for	 freedom	 of	 expression	 relate	 to	 their	 support	 for	
+having	personal	moderation	tools?	
+When	faced	with	a	choice	between	letting	platforms	handle	a	certain	content	category	and	
+allowing	users	to	specify	their	moderation	preferences,	we	expect	the	latter	to	be	perceived	
+as	 more	 free	 speech	 preserving.	 It	 lets	 users	 decide	 whether	 and	 how	 many	 content	
+removals	 should	 occur	 instead	 of	 allowing	 platforms	 to	 handle	 those	 same	 decisions	
+unilaterally.	We,	therefore,	raise	the	following	hypothesis:	
+
+H5:	 For	 each	 speech	 category,	 participants’	 support	 for	 freedom	 of	 expression	 will	 be	
+related	to	greater	support	for	having	personal	moderation	tools	to	moderate	that	category	
+than	their	support	for	the	platform’s	banning	of	that	category.	
+Methods 
+Our	study	was	considered	exempt	from	review	by	the	University	of	Washington	IRB.	We	
+recruited	participants	through	Lucid,1	a	survey	company	that	provides	researchers	access	
+to	 nationally	 representative	 samples.	 Our	 inclusion	 criterion	 for	 the	 survey	 participants	
+was	all	adult	internet	users	in	the	US.	We	paid	participants	through	the	Lucid	system.		
+We	designed	our	survey	questions	to	test	the	hypotheses	identified	in	the	previous	section	
+and	adapted	survey	instruments	from	relevant	prior	literature	to	test	some	measures.	We	
+describe	these	measures	in	more	detail	below.	To	increase	the	validity	of	the	survey,	we	
+sought	feedback	on	an	early	draft	of	the	survey	questionnaire	from	other	students	at	the	
+authors’	institutions.	Nine	students	who	were	not	involved	with	the	project	responded	to	
+our	request	and	provided	feedback	on	the	wording	of	the	questions	and	survey	flow,	which	
+we	incorporated	into	the	final	survey	design.	We	also	piloted	the	survey	with	a	small	subset	
+of	 the	 sample	 (27	 participants).	 During	 this	 field	 test,	 we	 included	 this	 open-ended	
+question	at	three	different	points	in	the	survey:	“Do	you	have	any	feedback	on	any	of	the	
+questions	 so	 far?	 For	 example,	 is	 any	 question	 unclear	 or	 ambiguous?	 Please	 list	 the	
+question	and	describe	your	challenge	with	answering	it.”	At	the	end	of	the	survey,	we	also	
+asked,	“Overall,	how	can	we	improve	this	survey	from	the	perspective	of	survey	takers?		Do	
+you	have	any	other	thoughts	or	feedback	for	us?	Please	describe.”	Results	from	this	survey	
+pretest	resulted	in	another	round	of	iteration	before	our	questionnaire	reached	the	desired	
+quality.	
+Our	survey	questionnaire	consisted	of	three	blocks	containing	similar	questions	about	hate	
+speech,	sexually	explicit	content,	and	violent	posts.	To	counter	the	effects	of	the	order	of	
+presentation	 on	 survey	 results,	 we	 counterbalanced	 the	 order	 in	 which	 question	 blocks	
+related	to	the	three	content	categories	were	shown	to	the	participants.	At	the	beginning	of	
+each	block,	we	specified	the	norm-violating	category	that	the	following	questions	related	to	
+and	defined	that	category.	We	used	the	following	definitions:	
+• Hate	speech:	“Hate	speech	includes	speech	that	is	dehumanizing,	stereotyping,	or	
+insulting,	 on	 the	 basis	 of	 identity	 markers	 such	 as	 race/ethnicity,	 gender,	 sexual	
+orientation,	religion,	etc.”	
+• Violent	 content:	 “Violent	 content	 includes	 threats	 to	 commit	 violence,	 glorifying	
+violence	or	celebrating	suffering,	depictions	of	violence	that	are	gratuitous	or	gory,	
+and	animal	abuse.”		
+	
+1	https://lucidtheorem.com		
+
+• Sexually	explicit	content:	“Sexually	explicit	content	includes	content	showing	sexual	
+activity,	offering	or	requesting	sexual	activity,	female	nipples	(except	breastfeeding,	
+health,	and	acts	of	protest),	nudity	showing	genitals,	and	sexually	explicit	language.”	
+The	above	definitions	were	inspired	by	the	language	provided	by	the	Facebook	site	when	
+reporting	any	post	under	the	category	of	hate	speech,	violence,	and	nudity,	respectively.	
+We	 administered	 the	 survey	 online	 using	 the	 survey	 software	 package	 Qualtrics.	 We	
+launched	the	survey	on	November	29,	2022.	Table	1	presents	the	demographic	details	of	
+our	final	sample	after	data	cleaning	and	compares	it	with	the	demographics	of	the	adult	US	
+internet	population	[45].			
+Table	1:	Demographic	Profile	of	the	US	Survey		
+	
+Authors’	study,	US	survey	
+Nov	2022	(%)	
+American	Community	Survey,	
+US	sample	2021	(%)	
+Age:	
+	
+	
+				18-29	
+18.7	
+17.4	
+				30-49	
+40.1	
+29.5	
+				50-64	
+23.7	
+25.6	
+				65+	
+17.4	
+27.3	
+Gender:	
+	
+	
+				Male	
+48.2	
+48.6	
+				Female	
+51.8	
+51.4	
+Race/Ethnicity:	
+	
+	
+				White	
+73.5	
+68.3	
+				Black	
+12.2	
+9.3	
+				Other	
+14.3	
+22.4	
+Hispanic:	
+	
+	
+				Yes	
+4.5	
+13.7	
+Education:	
+	
+	
+				High	school	or	less	
+31.4	
+33.5	
+				Some	college	
+24.4	
+33.3	
+				College+	
+43.3	
+33.1	
+Measures 
+Perceived Influence on Self and Others 
+For	 each	 norm-violating	 speech	 category,	 we	 asked	 the	 participants	 to	 estimate	 the	
+influence	 of	 that	 category	 on	 the	 self	 and	 others.	 In	 each	 case,	 we	 asked	two	 questions:	
+“Seeing	<speech	category>	posts	on	social	media	has	a	powerful	influence	on	my	attitudes.”	
+and	 “Seeing	 <speech	 category>	 posts	 on	 social	 media	 has	 a	 powerful	 influence	 on	 my	
+behaviors.”	The	response	categories	ranged	on	a	7-point	Likert-type	scale	from	1	(strongly	
+disagree)	to	7	(strongly	agree).	These	two	items	were	averaged	to	create	a	measure	of	the	
+perceived	 influence	 of	 each	 speech	 category	 on	 the	 self	 (Hate	 speech:	 M=3.89,	 SD=1.91,	
+a=.84;	Violent	speech:	M=3.78,	SD=1.87,	a=.81;	Sexually	explicit	speech:	M=3.55,	SD=1.89,	
+
+a=.87).	 We	 asked	 another	 two	 questions	 replacing	 only	 the	 word	 “my”	 with	 “other	
+people’s”	and	averaged	the	responses	to	create	an	index	of	the	perceived	influence	of	each	
+speech	category	on	others	(Hate	speech:	M=5.28,	SD=1.46,	a=.92;	Violent	speech:	M=5.19,	
+SD=1.40,	a=.91;	Sexually	explicit	speech:	M=4.99,	SD=1.46,	a=.93).	
+Support for Freedom of Speech 
+We	used	items	developed	by	Guo	and	Johnson	to	measure	support	for	freedom	of	speech	
+[20].	 Participants	 rated	 these	 four	 statements	 on	 a	 Likert-type	 scale	 ranging	 from	 1	
+(strongly	disagree)	to	7	(strongly	agree):	(1)	In	general,	I	support	the	First	Amendment,	(2)	
+Freedom	of	expression	is	essential	to	democracy,	(3)	Democracy	works	best	when	citizens	
+communicate	 in	 an	 unregulated	 marketplace	 of	 ideas,	 and	 (4)	 Even	 extreme	 viewpoints	
+deserve	to	be	voiced	in	society.	We	included	the	First	Amendment	statement2	in	the	first	
+question	 to	 clarify	 its	 meaning.	 We	 formed	 an	 index	 for	 free	 speech	 support	 using	 the	
+means	of	these	four	items	(M=5.43,	SD=1.15,	a=.80).	
+Dependent Variables Related to Moderation 
+Support	for	platform-enacted	moderation	of	each	speech	category	was	operationalized	by	
+asking	 participants	 to	 rate	 the	 following	 statement:	 “I	 support	 social	 media	 platforms	
+taking	 down	 any	 posts	 they	 consider	 to	 be	 <speech	 category>	 so	 that	 no	 users	 can	 see	
+them.”	 The	 responses	 for	 this	 statement	 ranged	 on	 a	 7-point	 Likert-type	 scale,	 where	
+1=“strongly	disagree”	and	7=“strongly	agree.”		
+For	 operationalizing	 support	 for	 having	 personal	 moderation	 tools	 for	 each	 category,	 we	
+showed	participants	an	example	of	a	personal	moderation	feature	where	every	user	can	
+decide	the	extent	to	which	they	want	the	<speech	category>	content	filtered	out	(see	Fig.	
+1).	We	asked	participants	to	rate	their	support	for	providing	this	kind	of	setting	to	all	users	
+on	a	Likert-type	scale	ranging	from	1	(strongly	disagree)	to	7	(strongly	agree).	
+	
+2	The	First	Amendment	to	the	United	States	Constitution	states:	"Congress	shall	make	no	
+law	 respecting	 an	 establishment	 of	 religion,	 or	 prohibiting	 the	 free	 exercise	 thereof;	 or	
+abridging	the	freedom	of	speech,	or	of	the	press;	or	the	right	of	the	people	peaceably	to	
+assemble,	and	to	petition	the	Government	for	a	redress	of	grievances.”	
+
+	
+Figure	1:	Survey	question	asking	participants	to	rate	their	support	for	platforms	providing	
+personal	moderation	feature	
+In	 addition	 to	 these	 two	 measures,	 we	 also	 operationalized	 choosing	 platform-enacted	
+moderation	 v/s	 personal	 moderation	 for	 each	 speech	 category	 by	 asking	 respondents	 a	
+binary	question:	‘Given	a	choice	between	platform-wide	moderation	and	a	"Choose	your	
+moderation	settings"	feature	to	handle	<speech	category>	posts,	which	would	you	prefer	to	
+have?’	The	response	categories	included:	(1)	Platform-wide	moderation:	Platforms	should	
+have	the	power	to	remove	all	posts	they	identify	as	<speech	category>	across	the	platform	
+or	(2)	"Choose	your	moderation	settings"	feature:	Each	user	should	be	allowed	to	configure	
+the	extent	to	which	<speech	category>	posts	should	be	removed	for	them.	We	note	that	the	
+“Choose	 you	 moderation	 settings”	 feature	 enables	 a	 range	 of	 choices	 –	 from	 “no	
+moderation”	to	“a	range	of	moderation.”	Thus,	users	who	desired	neither	platform-wide	
+nor	 personal	 moderation	 to	 remove	 any	 posts	 for	 them	 could	 select	 this	 feature	 and	
+configure	it	at	the	“no	moderation”	level.		
+
+The below example shows a feature where every user can decide for themselves
+the extentto whichtheywantsexuallyexplicit contentfiltered out.
+Chooseyourmoderationsettings:
+Filteroutsexuallyexplicitpostsbasedonthelevelyouselect:
+Alittle
+Some
+More
+Alotof
+No moderation
+moderation
+moderation
+moderation
+moderation
+Onlythemostsexuallyexplicitpostswillberemovedatthislevel.
+I support platforms providing this kind of setting to all users
+O Strongly disagree
+O Disagree
+OSomewhatdisagree
+ONeitheragree nordisagree
+O Somewhatagree
+O Agree
+O Strongly agree	
+Figure	2:	Frequency	of	participants’	responses	to	survey	questions	about	support	for	platform-
+wide	and	personal	moderation,	measured	in	percentage.	
+Control Variables 
+Prior	 research	 has	 shown	 that	 socio-demographic	 variables	 are	 related	 to	 TPE	 and	 free	
+speech	and	attitude	towards	media	regulation	[19,	26,	27,	31].	Further,	social	media	use	
+has	 also	 been	 associated	 with	 individuals’	 perceptions	 of	 harmful	 content	 and	 their	
+interventions	 against	 such	 [24,	 25,	 35,	 53].	 Therefore,	 we	 controlled	 for	 age,	 education,	
+gender,	 race,	 political	 affiliation	 (1	 =	 “very	 liberal”,	 7=	 “very	 conservative”),	 and	 social	
+media	use	of	each	respondent.	We	operationalized	the	frequency	of	social	media	use	by	
+following	recommendations	by	Ernala	et	al.	[9]	and	prompting	participants	to	respond	to	
+the	question,	‘In	the	past	week,	on	average,	approximately	how	much	time	PER	DAY	have	
+you	spent	actively	using	any	social	media?	
+Results 
+Our	results	show	that	72.8%,	73.1%,	and	66.2%	of	participants	at	least	somewhat	agreed	
+that	 platforms	 should	 ban	 hate	 speech,	 violent	 content,	 and	 sexually	 explicit	 content,	
+respectively.	Further,	69.9%,	72.6%,	and	72.1%	of	participants	at	least	somewhat	agreed	
+that	 platforms	 should	 offer	 personal	 moderation	 tools	 to	 let	 end-users	 regulate	 hate	
+speech,	violent	content,	and	sexually	explicit	content,	respectively.	
+
+Statistics
+Strongly disagree
+Disagree
+Somewhatdisagree
+Personal ModerationofSexuallyExplicitContent
+I Neither agree nor disagree
+Somewhatagree
+lAgree
+IStronglyagree
+PlatformBanofSexualyExplicitContent
+Personal ModerationofViolentContent
+PlatformBanofViolentContent
+PersonalModerationofHateSpeech
+PlatformBanofHateSpeech
+0
+20
+40
+60
+80
+100
+Percentage of ParticipantsIn	line	with	research	on	the	third-person	effects,	H1	predicted	that	for	each	norm-violating	
+content	category,	participants	would	perceive	the	effects	of	that	category	on	others	to	be	
+stronger	 than	 on	 themselves.	 We	 ran	 a	 paired	 t-test	 and	 found	 the	 perceived	 effects	 on	
+others	 significantly	 stronger	 than	 on	 oneself	 for	 each	 category	 (see	 Table	 2).	 Thus,	 our	
+results	support	H1.	
+Table	2:	Mean,	standard	deviations,	standard	errors	of	participants’	perceived	effects	of	hate	
+speech,	 violent	 content,	 and	 sexually	 explicit	 content	 on	 others	 and	 self,	 and	 t-test	 results	
+comparing	perceived	effects	on	others	and	self	for	each	speech	category	(N	=	993).	***	denotes	
+p	<	.001	
+	
+	
+M	
+SD	
+SE	
+t	
+Cohen’s	d	
+Hate	speech	
+Effects	on	others	
+5.28	
+1.46	
+.05	
+25.37***	
+.805	
+Effects	on	self	
+3.89	
+1.91	
+.06	
+Violent	
+content	
+Effects	on	others	
+5.19	
+1.40	
+.05	
+25.44***	
+.807	
+Effects	on	self	
+3.77	
+1.87	
+.06	
+Sexually	
+explicit	
+content	
+Effects	on	others	
+4.99	
+1.46	
+.05	
+25.81***	
+.819	
+Effects	on	self	
+3.55	
+1.88	
+.06	
+	
+Support for Platform Ban 
+We	computed	hierarchical	linear	regression	to	test	our	hypotheses	2	and	4.	For	each	of	the	
+three	norm-violating	categories,	we	created	a	model	where	the	participant’s	support	for	
+banning	that	category	served	as	the	dependent	variable.	In	Step	1	of	the	three	regression	
+models,	we	included	the	control	variables	age,	gender,	education,	race,	political	affiliation,	
+and	social	media	use.	In	Step	2,	we	introduced	the	independent	variables	PME3	(perceived	
+effects	on	others)	for	that	category	and	support	for	free	speech	(Table	3).		
+Table	3:	Hierarchical	multiple	regression	analyses	predicting	support	for	platforms'	banning	
+of	hate	speech,	violent	content,	and	sexually	explicit	content	(N	=	983)	
+Independent	Variable	
+Support	
+for	
+platform	ban	of	
+hate	speech	(β)	
+Support	
+for	
+platform	 ban	 of	
+violent	 content	
+(β)	
+Support	
+for	
+platform	 ban	 of	
+sexually	 explicit	
+content	(β)	
+
+Step	1	
+	
+	
+	
+				Age	
+.119***	
+.055	
+.102**	
+				Gender	(Female)	
+.127***	
+.167***	
+.153***	
+				Race	(White)	
+.018	
+.013	
+-.059	
+				Educationa	
+-.008	
+.039	
+-.022	
+				Political	affiliationb	
+-.156***	
+-.09**	
+.019	
+				Social	media	usec	
+.022	
+.013	
+.019	
+				R2	
+.093***	
+.065***	
+.05***	
+Step	2	
+	
+	
+	
+				Support	for	free	speech	
+-.101***	
+-.039	
+-.006	
+				Perceived	effects	of	hate	speech	on	others	
+.476***	
+-	
+-	
+				Perceived	effects	of	violent	content	on	others	
+-	
+.397***	
+-	
+				Perceived	 effects	 of	 sexually	 explicit	 content	
+on	others	
+-	
+-	
+.391***	
+				R2	change	
+.215	
+.15***	
+.149***	
+Total	R2	
+.307***	
+.215***	
+.199***	
+**p	<	.01,	***p	<	.001	(t	test	for	β,	two-tailed;	F	test	for	R2,	two-tailed).	
+a0=	Less	than	secondary	education;	1=	Secondary	education	or	more.	
+b1=	Strong	Democrat,	7=	Strong	Republican.	
+c1=	Less	than	10	minutes	per	day,	6=	More	than	3	hours	per	day.	
+β	=	standardized	beta	from	the	full	model	(final	beta	controlling	for	all	variables	in	the	model).		
+For	 each	 norm-violating	 speech	 category,	 the	 regression	 models	 show	 significant	
+influences	of	the	participants'	perceived	effects	of	that	category	on	others	(PME3)	on	their	
+support	for	platform	ban	of	that	category	(Model	1:	hate	speech	–	β	=	.476,	p	<	.001;	Model	
+
+2:	violent	content	–	β	=	.397,	p	<	.001;	Model	3:	sexually	explicit	content	–	β	=	.391,	p	<	
+.001),	supporting	H2.		
+Greater	support	for	free	speech	significantly	negatively	influences	support	for	platform	ban	
+of	hate	speech	(β	=	-.101,	p	<	.001).	It	does	not,	however,	influence	support	for	platform	ban	
+of	violent	content	(β	=	-.039,	p	>	.05)	or	sexually	explicit	content	(β	=	-.006,	p	>	.05).	Thus,	
+H4	is	only	partially	supported.		
+Support for Personal Moderation 
+We	computed	hierarchical	linear	regression	to	test	our	hypothesis	3	and	answer	RQ	2.	For	
+each	of	the	three	norm-violating	categories,	we	created	a	model	where	the	participant’s	
+support	 for	 having	 personal	 moderation	 tools	 to	 regulate	 that	 category	 served	 as	 the	
+dependent	 variable.	 In	 Step	 1	 of	 the	 three	 regression	 models,	 we	 included	 the	 control	
+variables	age,	gender,	education,	race,	political	affiliation,	and	social	media	use.	In	Step	2,	
+we	 introduced	 the	 independent	 variables	 PME3	 (perceived	 effects	 on	 others)	 for	 that	
+category	and	support	for	free	speech	(Table	4).	
+Table	 4:	 Hierarchical	 multiple	 regression	 analyses	 predicting	 support	 for	 using	 personal	
+moderation	tools	(PMT)	to	regulate	hate	speech,	violent	content,	and	sexually	explicit	content	
+(N	=	983)	
+Independent	Variable	
+Support	
+for	
+PMT	
+to	
+regulate	
+hate	
+speech	(β)	
+Support	for	PMT	
+to	
+regulate	
+violent	 content	
+(β)	
+Support	for	PMT	
+to	
+regulate	
+sexually	 explicit	
+content	(β)	
+Step	1	
+	
+	
+	
+				Age	
+.014	
+-.017	
+-.010	
+				Gender	(Female)	
+-.016	
+.019	
+-.016	
+				Race	(White)	
+-.019	
+-.007	
+.08*	
+				Educationa	
+-.018	
+.022	
+.003	
+				Political	affiliationb	
+-.019	
+-.049	
+-.074*	
+				Social	media	usec	
+.051	
+.057	
+.086**	
+				R2	
+.009	
+.013	
+.027***	
+Step	2	
+	
+	
+	
+
+				Support	for	free	speech	
+.173***	
+.195***	
+.224***	
+				Perceived	effects	of	hate	speech	on	others	
+.224***	
+-	
+-	
+				Perceived	effects	of	violent	content	on	others	
+-	
+.187***	
+-	
+				Perceived	 effects	 of	 sexually	 explicit	 content	
+on	others	
+-	
+-	
+.225***	
+				R2	change	
+.087	
+.081	
+.111	
+Total	R2	
+.096***	
+.095***	
+.138***	
+*p	<	.05,	**p	<	.01,	***p	<	.001	(t	test	for	β,	two-tailed;	F	test	for	R2,	two-tailed).	
+a0=	Less	than	secondary	education;	1=	Secondary	education	or	more.	
+b1=	Strong	Democrat,	7=	Strong	Republican.	
+c1=	Less	than	10	minutes	per	day,	6=	More	than	3	hours	per	day.	
+β	=	standardized	beta	from	the	full	model	(final	beta	controlling	for	all	variables	in	the	model).	
+For	 each	 norm-violating	 speech	 category,	 the	 regression	 models	 show	 significant	
+influences	of	the	participants'	perceived	effects	of	that	category	on	others	(PME3)	on	their	
+support	 for	 using	 personal	 moderation	 tools	 to	 regulate	 that	 category	 (Model	 4:	 hate	
+speech	–	β	=	.224,	p	<	.001;	Model	5:	violent	content	–	β	=	.187,	p	<	.001;	Model	6:	sexually	
+explicit	content	–	β	=	.225,	p	<	.001),	supporting	H4.		
+Greater	support	for	free	speech	has	a	significant	positive	influence	on	participants’	support	
+for	 using	 personal	 moderation	 tools	 to	 regulate	 each	 norm-violating	 category	 (Model	 4:	
+hate	speech	–	β	=	.173,	p	<	.001;	Model	5:	violent	content	–	β	=	.195,	p	<	.001;	Model	6:	
+sexually	explicit	content	–	β	=	.224,	p	<	.001).	This	answers	our	RQ	2.	
+	
+
+Choosing Between Platform-wide Moderation and Personal 
+Moderation 
+	
+Figure	3:	Percentage	of	participants	preferring	platform-wide	ban	or	personal	moderation	to	
+regulate	hate	speech,	violent	content,	and	sexually	explicit	content	
+Given	 a	 choice	 between	 platform-wide	 moderation	 and	 a	 personal	 moderation	 tool	 to	
+regulate	hate	speech,	violent	content,	and	sexually	explicit	content,	52.4%,	52%,	and	55.3%	
+of	participants,	respectively,	chose	the	personal	moderation	tool.	This	finding	suggests	that	
+more	participants	prefer	autonomy	over	moderation	than	delegating	it	to	platforms	as	they	
+see	fit.	
+We	created	binomial	logistic	regression	models	to	test	our	hypothesis	5	and	answer	RQ	1.	
+For	each	of	the	three	norm-violating	categories,	we	created	a	model	where	the	participants’	
+binary	choice	between	platform-wide	moderation	and	personal	moderation	to	handle	that	
+category	served	as	the	dependent	variable.	In	Step	1	of	the	three	regression	models,	we	
+included	the	control	variables	age,	gender,	education,	race,	political	affiliation,	and	social	
+media	use.	In	Step	2,	we	introduced	the	independent	variables	PME3	(perceived	effects	on	
+others)	for	that	category	and	support	for	free	speech.	For	each	model,	we	used	the	Box-
+Tidwell	procedure	[4,	11]	to	check	the	assumption	of	linearity	in	the	logit	and	found	in	each	
+case	that	our	continuous	variable	support	for	free	speech	was	not	linearly	related	to	the	
+logit	 of	 the	 dependent	 variable.	 To	 address	 this,	 we	 split	 this	 variable	 into	 two	 ordinal	
+categories	 –	 high	 and	 low,	 recoding	 each	 entry	 for	 this	 variable	 based	 on	 whether	 it	
+exceeded	the	median	value.	Rerunning	the	Box-Tidwell	procedure	with	this	transformed	
+support	 for	 the	 free	 speech	 categorical	 variable,	 we	 found	 all	 remaining	 continuous	
+
+Statistics
+Prefer platform-wide ban
+Preferpersonalmoderation
+Sexually explicit content
+Violent content
+Hate speech
+0
+10
+20
+30
+40
+50
+60
+Percentage of Participantsindependent	variables	in	each	model	to	be	linearly	related	to	the	logit	of	the	dependent	
+variable.	We	next	present	these	models'	binomial	logistic	regression	results	(Table	5).		
+Table	 5:	 Binomial	 logistic	 regression	 analyses	 predicting	 support	 for	 using	 personal	
+moderation	tools	(PMT)	over	platform-wide	ban	to	regulate	hate	speech,	violent	content,	and	
+sexually	explicit	content	(N	=	984)	
+Independent	Variable	
+Support	
+for	
+PMT	
+over	
+platform	ban	to	
+regulate	
+hate	
+speech,	
+Odds	
+Ratio	
+Support	for	PMT	
+over	
+platform	
+ban	 to	 regulate	
+violent	 content,	
+Odds	Ratio	
+Support	for	PMT	
+over	
+platform	
+ban	 to	 regulate	
+violent	 content,	
+Odds	Ratio	
+Step	1	
+	
+	
+	
+				Age	
+.986**	
+.992	
+.994	
+				Gender	(Female)	
+.790	
+.655**	
+.674**	
+				Race	(White)	
+1.107	
+1.313	
+1.203	
+				Educationa	
+.934	
+.870	
+.981	
+				Political	affiliationb	
+1.163***	
+1.118***	
+1.074*	
+				Social	media	usec	
+.990	
+1.071	
+1.091*	
+				Nagelkerke	R2	
+.066***	
+.066***	
+.045***	
+Step	2	
+	
+	
+	
+				Support	for	free	speechd	
+2.703***	
+2.239****	
+1.969***	
+				Perceived	effects	of	hate	speech	on	others	
+.735***	
+-	
+-	
+				Perceived	effects	of	violent	content	on	others	
+-	
+.752***	
+-	
+				Perceived	 effects	 of	 sexually	 explicit	 content	
+on	others	
+-	
+-	
+.857**	
+Total	Nagelkerke	R2	
+.165***	
+.138***	
+.087***	
+*p	<	.05,	**p	<	.01,	***p	<	.001	(t	test	for	β,	two-tailed;	Omnibus	Tests	of	Model	Coefficients	for	R2).	
+
+a0=	Less	than	secondary	education;	1=	Secondary	education	or	more.	
+b1=	Strong	Democrat,	7=	Strong	Republican.	
+c1=	Less	than	10	minutes	per	day,	6=	More	than	3	hours	per	day.	
+d0=	low,	1=high.	
+odds	ratio	=	exp(β)	from	full	model.	
+For	 each	 norm-violating	 speech	 category,	 the	 regression	 models	 show	 significant	
+influences	of	the	participants'	perceived	effects	of	that	category	on	others	(PME3)	on	their	
+choice	 of	 using	 personal	 moderation	 tools	 over	 platform-enacted	 bans	 to	 regulate	 that	
+category.	Increasing	PME3	was	associated	with	a	decreased	likelihood	of	choosing	personal	
+moderation	tools	over	platform	bans	(Model	7:	hate	speech	–	exp(β)	=	.735,	p	<	.001;	Model	
+8:	violent	content	–	exp(β)	=	.752,	p	<	.001;	Model	9:	sexually	explicit	content	–	exp(β)	=	
+.857,	p	<	.01).	This	answers	our	RQ	1.		
+Higher	support	for	free	speech	has	a	significant	positive	influence	on	participants’	support	
+for	 using	 personal	 moderation	 tools	 over	 platform-enacted	 bans	 to	 regulate	 each	 norm-
+violating	 category.	 Participants	 who	 showed	 high	 support	 for	 free	 speech	 have	 2.703,	
+2.239,	and	1.969	times	higher	odds	of	choosing	personal	moderation	tools	over	platform-
+enacted	 bans	 to	 regulate	 hate	 speech,	 violent	 content,	 and	 sexually	 explicit	 content,	
+respectively.	Thus,	H5	is	supported.	
+Discussion 
+In	recent	years,	the	debates	surrounding	the	censorship	of	inappropriate	content	on	social	
+media	have	surfaced	more	and	more	due	to	the	increasing	essentiality	of	social	media	in	
+political	discourse	and	growing	controversies	over	how	platforms	regulate	[13,	14].	The	
+purpose	of	this	study	was	to	measure	public	attitudes	regarding	two	important	forms	of	
+social	media	regulation:	platform-wide	moderation,	which	lets	platforms	unilaterally	make	
+moderation	decisions	for	every	user,	and	personal	moderation,	which	empowers	users	to	
+decide	 for	 themselves	 how	 they	 would	 like	 to	 regulate	 different	 content	 categories	 by	
+adjusting	 their	 moderation	 settings.	 Third-person	 effects	 (TPE)	 are	 commonly	 used	 in	
+examining	 people’s	 attitudes	 towards	 censorship	 and	 related	 behaviors	 or	 behavioral	
+intentions	[6,	18].	The	present	research	extends	the	TPE	research	to	managing	hate	speech,	
+violent	content,	and	sexually	explicit	content	on	social	media.		
+We	 explored	 the	 presumed	 negative	 effects	 of	 each	 type	 of	 content	 on	 self	 (PME1)	 and	
+others	(PME3).	The	results	produced	strong	support	for	the	TPE	hypothesis.	As	expected,	
+participants	perceived	the	social	media	hate	speech,	violent	content,	and	sexually	explicit	
+content	to	have	a	greater	influence	on	others	than	on	themselves.	We	also	examined	how	
+PME3	and	support	for	free	speech	affected	participants’	consequent	censorial	behavior.	In	
+each	 case,	 we	 found	 that	 the	 perceived	 effects	 on	 others	 (PME3)	 predicted	 participants’	
+support	 for	 both	 platform-wide	 moderation	 and	 personal	 moderation.	 This	 is	 a	
+theoretically	 significant	 finding	 of	 this	 study	 since	 it	 helps	 advance	 TPE	 research	 by	
+
+showing	 that	 perceived	 effects	 on	 others	 play	 an	 essential	 role	 in	 triggering	 censorial	
+behavior.	Given	a	choice	between	the	platform	and	personal	moderation,	PME3	predicted	
+support	 for	 platform	 moderation	 in	 each	 case.	 This	 finding	 indicates	 that	 when	 users	
+perceive	the	adverse	effects	of	a	content	category	on	the	public,	they	desire	platforms	to	
+take	site-wide	actions	on	that	content	rather	than	regulate	it	for	themselves.		
+Further,	the	relationship	between	support	for	free	speech	and	support	for	platform-wide	
+moderation	received	only	partial	support.	While	the	connection	is	significant	and	negative	
+for	hate	speech,	it	is	not	significant	for	violent	and	sexually	explicit	content.	This	result	is	in	
+line	with	the	mixed	findings	for	free	speech	support	as	a	predictor	of	supportive	attitudes	
+towards	platform	censorship	observed	in	prior	literature	[20].	On	the	other	hand,	we	found	
+that	 support	 for	 free	 speech	 predicted	 support	 for	 the	 use	 of	 personal	 moderation	 for	
+regulating	 each	 inappropriate	 speech	 category.	 This	 suggests	 that	 people	 may	 perceive	
+personal	moderation	tools	as	not	an	infringement	on	the	free	speech	of	others	but	simply	
+having	a	greater	agency	to	shape	what	they	see.	This	is	further	bolstered	by	our	finding	that	
+given	 a	 choice	 between	 the	 platform	 and	 personal	 moderation,	 support	 for	 free	 speech	
+predicts	support	for	personal	moderation	in	each	case.		
+Other	significant	effects,	less	central	to	the	hypotheses	being	tested,	also	were	found.	We	
+found	that	age	was	positively	related	to	support	for	platform	moderation	of	hate	speech	
+and	 sexually	 explicit	 content,	 but	 not	 violent	 content.	 Females	 supported	 platform	
+moderation	of	each	speech	category	more	than	males.	Democrats	were	more	likely	than	
+Republicans	to	support	platform	moderation	of	hate	speech	and	violent	content,	but	not	
+sexually	 explicit	 content.	 Regarding	 support	 for	 personal	 moderation,	 race,	 political	
+affiliation,	and	social	media	use	were	significant	predictors	for	the	sexually	explicit	content	
+category;	however,	no	control	variables	significantly	predicted	personal	moderation	of	hate	
+speech	or	violent	content.		
+The	evidence	presented	here	has	important	implications	for	how	platforms	govern	their	
+sites.	We	show	that	part	of	the	reason	the	public	supports	moderation	of	norm-violating	
+categories	such	as	pornography	and	war	violence	is	that	it	overestimates	these	categories’	
+effects	 on	 others.	 Therefore,	 company-wide	 moderation	 decisions	 and	 public	 debates	
+concerning	 free	 speech	 and	 its	 limitations,	 must	 recognize	 and	 account	 for	 third-person	
+effects.	It	also	points	to	an	urgent	need	to	measure	the	actual	media	effects	as	opposed	to	
+the	perceived	media	effects	of	different	content	types.	We	have	noticed	that	even	during	
+our	 interview	 studies	 on	 online	 harms,	 participants	 tend	 to	 advocate	 for	 specific	
+moderation	initiatives	based	on	their	perceptions	of	what	others	might	need.	To	arrive	at	
+an	 accurate	 needs-gathering,	 scholars	 must	 focus	 on	 understanding	 the	 perspectives	 of	
+online	content	on	users	themselves	–	a	topic	on	which	they	are	an	expert	–	rather	than	the	
+abstract	others	whose	actual	needs	may	considerably	differ.		
+Several	 limitations	 of	 this	 study	 should	 be	 recognized.	 The	 survey	 design	 prohibited	 us	
+from	exploring	the	motivations	for	specific	perceptions	in	depth.	Furthermore,	we	cannot	
+make	 conclusive	 statements	 about	 causal	 relations	 as	 a	 cross-sectional	 study.	 We	 asked	
+participants	 to	 respond	 to	 questions	 about	 speech	 categories	 that	 could	 be	 broadly	
+interpreted.	We	chose	this	instead	of	presenting	a	specific	instance	of	each	speech	category	
+to	 increase	 the	 generalizability	 of	 our	 findings.	 Still,	 different	 users	 may	 have	 different	
+
+perceptions	 of	 what	 counts	 as	 hate	 speech,	 violent	 content,	 or	 sexually	 explicit	 content.	
+Prior	moderation	research	has	recognized	this	as	a	complex	challenge	in	the	social	media	
+regulation	[23].	We	provided	definitions	of	each	speech	category	in	the	survey	to	clarify	the	
+scope	 of	 each	 category	 to	 our	 participants.	 Nevertheless,	 further	 research	 on	 user	
+perceptions	 of	 stimulus-based	 designs	 that	 present	 preselected	 instances	 of	 each	 norm-
+violating	 speech	 category	 to	 participants	 would	 provide	 valuable	 insights.	 We	 did	 not	
+ground	our	survey	questions	in	a	specific	platform	to	increase	the	generalizability	of	our	
+results.	 Studies	 focused	 on	 particular	 social	 media	 sites	 can	 uncover	 whether	 attitudes	
+towards	specific	platforms	influence	users’	perceptions	of	moderation	actions.		
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+

HtE5T4oBgHgl3EQfWg9Q/content/tmp_files/load_file.txt

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='05559v1  [quant-ph]  11 Jan 2023  Supercurrent and Electromotive force generations  by the Berry connection from many-body wave  functions  Hiroyasu Koizumi  Division of Quantum Condensed Matter Physics, Center for Computational Sciences,  University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan  E-mail: koizumi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='hiroyasu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='fn@u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='tsukuba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='jp  January 2023  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  The velocity field composed of the electromagnetic field vector potential  and the Berry connection from many-body wave functions explains supercurrent  generation, Faraday’s law for the electromotive force (EMF) generation, and other  EMF generations whose origins are not electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' An example calculation  for the EMF from the Berry connection is performed using a model for the cuprate  superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' Introduction  The Berry phase first discovered in the context of the adiabatic approximation now  prevails in various fields of physics [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  In particular, it is now an indispensable  mathematical tool to detect topological defects in quantum wave functions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' Recently,  the Berry connection from many-body wave functions was defined and its usefulness to  calculate supercurrent is demonstrated [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' A salient feature of such a formalism is that it  provides a vector potential directly related to the velocity field for electric current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' In the  present work, we consider the supercurrent and electromotive force (EMF) generations  based on the same formalism [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  The EMF is expressed using a non-irrotational ‘electric field’, Eirrot, whose origin  may not be a real electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' It is defined as  E =  �  C  Enon−irrot · dr  (1)  where C is a closed electric circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' This EMF appears due to various causes, such as  chemical reactions in batters or temperature differences in metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  One of the important EMF generation mechanisms is the Faraday’s law of magnetic  induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' It is expressed as a total time-derivative of a magnetic flux of the magnetic  field B  E = − d  dt  �  S  B · dS  (2)  Supercurrent and EMF by Berry connection  2  where S is a surface whose circumference is C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' This EMF formula is often called the  “flux rule”, since  �  S B · dS is the magnetic flux through the surface S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' it has been  claimed curious since it is composed of two different fundamental equations in classical  theory [6], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=', the Faraday’s law of induction and the Lorentz force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The curiosity  is increased by the fact that one of them is an equation for fields only, and the other  includes particles and is an equation for a force on a particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  This peculiarity disappears in quantum theory using the vector potential A that  is more fundamental than the magnetic field B [7, 8, 9], and the wave function makes  the velocity of a particle a velocity field [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' Then, the two contributions in the “flux  rule” are connected by the duality that a U(1) phase factor added on a wave function  describes a whole system motion, and also plays the role of the vector potential when  it is transferred into the Hamiltonian [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  In the present work, we extend the above vector potential and velocity field  approach for the electric current generation to cases where the vector potential of the  Berry connection from many-body wave functions appears [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' We show that the EMF  generation other than the electromagnetic field origin, such as those due to chemical  reactions or temperature gradients can be expressed by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  The organization of the present work is as follows: we explain the velocity field  appearing from the Berry connection from many-body wave functions in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  We reexamine the Faraday’s EMF generation formula using the velocity field from the  electromagnetic vector potential in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' We examine the EMF generation by the  Berry connection in Section 4, and an example calculation is performed for the Nernst  effect in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' Lastly, we conclude the present work by mentioning implications of  the present new theory in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The velocity field from the Berry connection form many-body wave  functions and supercurrent generation  The key ingredient in the present work is the Berry connection from many-body wave  functions for electrons given by  AMB  Ψ (r)=  1  ℏρ(r)Re  ��  dσ1dx2 · · ·dxNΨ∗(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' xN)(−iℏ∇)Ψ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' xN)  �  (3)  where N is the total number of electrons in the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' ‘Re’ denotes the real part,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' Ψ  is the total wave function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' xi collectively stands for the coordinate ri and the spin σi  of the ith electron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' −iℏ∇ is the Schr¨odinger’s momentum operator for the coordinate  vector r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' and ρ(r) is the number density calculated from Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' This Berry connection is  obtained by regarding r as the “adiabatic parameter”[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  Let us consider the electron system whose kinetic energy operator in the Schr¨odinger  Supercurrent and EMF by Berry connection  3  representation is given by  ˆT = −  N  �  j=1  ℏ2  2me  ∇2  j  (4)  where me is the electron mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  For convenience, we also use the following χ defined as  χ(r) = −2  � r  0  AMB  Ψ (r′) · dr′  (5)  and express the many-electron wave function Ψ as  Ψ(x1, · · · , xN) = exp  �  − i  2  N  �  j=1  χ(rj)  �  Ψ0(x1, · · · , xN)  (6)  Then, Ψ0 = Ψ exp  �  i  2  �N  j=1 χ(rj)  �  is a currentless wave function for the current  operator associated with ˆT in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (4) since the contribution from Ψ and that from  exp  �  i  2  �N  j=1 χ(rj)  �  cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' In other words, a wave function is given as a product  of a currentless one, Ψ0, and the factor for the current exp  �  − i  2  �N  j=1 χ(rj)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The total  wave function Ψ must be a single-valued function of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' This makes χ as an  angular variable that satisfies some periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' This periodicity gives rise to non-trivial  topological integer as will be explained, shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  When electromagnetic field is included, the kinetic energy operator becomes  ˆT ′ =  N  �  j=1  1  2me  (−iℏ∇j − qA)2  (7)  where q = −e is the electron charge, and A is the electromagnetic field vector potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  The magnetic field is given by B = ∇ × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  In the following, we will use the same expression, Ψ, for the total wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  Then, the current density for Ψ is given by  j = −eρv  (8)  with the velocity field v given by  v = e  me  �  A − ℏ  2e∇χ  �  = e  me  A + ℏ  me  AMB  Ψ  (9)  The current density in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (8) is known to give rise to the Meissner effect if it is a  stable one due to the fact that it explicitly depends on A [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' For the stable current  case, ∇χ compensates the gauge ambiguity in A and makes v in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (9) gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  If the Meissner effect is realize, the magnetic filed is expelled from the bulk of a  superconductor [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' Then, the flux quantization is observed for magnetic flux through  Supercurrent and EMF by Berry connection  4  a loop C that goes through the bulk of a ring-shaped superconductor  �  S  B · dS =  �  C  A · dr  = ℏ  2e  �  C  ∇χ · dr  = h  2ewC[χ]  (10)  where wC[χ] is the topological integer ‘winding number’ defined by  wC[χ] = 1  2π  �  C  ∇χ · dr  (11)  According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (9), the presence of non-zero wC[χ] means the existence of the  stable velocity field that satisfies  �  C  v · dr =  h  2me  wC[χ]  (12)  In superconductors, the quantized flux persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' This means that the condition  d  dtwC[χ] = 0  (13)  is realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  In normal metals, the time-derivative of the velocity field is often expressed as  dv  dt = −1  τ v  (14)  using a relaxation time approximation, where τ is the relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  Combination of this with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (12) yields  τ d  dtwC[χ] = −wC[χ]  (15)  If the condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (13) with nonzero wC[χ] is realized, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (15) means that τ  must be ∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=', an infinite conductivity, or zero resistivity is realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The vorticity field from the vector potential A and Faraday’s flux rule  In this section, we consider the case where non-trivial AMB  Ψ  is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' When AMB  Ψ  is  trivial, it satisfies  ∇ × AMB  Ψ  = 0  (16)  Thus, by applying ∇× on the both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (9)  ∇ × v = e  me  B  (17)  is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  Taking the total time-derivative of the above yields  ∇ × dv  dt = e  me  ∂tB + e  me  (v · ∇)B  (18)  Supercurrent and EMF by Berry connection  5  where the total time-derivative of the field B is the Eulerian time-derivative given by  dB  dt = ∂tB + (v · ∇)B  (19)  Integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (18) over the surface S, we have  �  C  dv  dt · dr = e  me  �  S  ∂tB · dS + e  me  �  S  (v · ∇)B · dS  (20)  where the Stokes theorem is used to convert the surface integral to the line integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  Noting that the electromotive force for an electron is given by  E = 1  −e  �  C  d(mev)  dt  dr  (21)  where −e is the electron charge and me is the electron mass, the following relation is  obtained  E = −  �  S  ∂tB · dS −  �  S  (v · ∇)B · dS  (22)  This is equal to the Faraday’s formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  In the situation where the circuit C moves with a constant velocity v0, we have the  following relation  (v0 · ∇)B = ∇ × (B × v0) + v0(∇ · B)  = ∇ × (B × v0)  (23)  due to the fact that B satisfies ∇ · B = 0 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  As a consequence, the well-known EMF formula  E = −  �  S  ∂tB · dS +  �  C  (v0 × B) · dr  (24)  is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The first term in it is attributed to the Faraday’s law of induction, and  the second to the Lorentz force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' This formula is composed of two different fundamental  equations in classical theory [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' However, in the quantum mechanical formalism, two  contributions stem from a single relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The EMF generation by the Berry connection  The velocity field in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (9) contains the vector potential AMB  Ψ  in addition to the  electromagnetic vector potential A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' Just like A, AMB  Ψ  will also give rise to the EMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  We now consider a general case where the Berry connection arises from a set of  states {Ψj} and given by  AMB =  �  j  pjAMB  Ψj  (25)  where pj’s are probabilities satisfy  �  j  pj = 1  (26)  and AMB  Ψj is obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (3) by replacing Ψ with Ψj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  Supercurrent and EMF by Berry connection  6  We express AMB using the following density matrix  ˆd =  �  j  pj|Ψj⟩⟨Ψj|  (27)  where the operator ˆAMB is defined through the relation  ⟨Ψj| ˆAMB|Ψj⟩ = AMB  Ψj  (28)  From now on, we allow the time-dependence in Ψj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' When Ψj is time-dependent, AMB  Ψj  is also time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The distribution probability pj can be also time and coordinate  dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  Using the density operator ˆd and the operator ˆAMB, the vector potential from the  Berry connection is given by  AMB = tr  �  ˆd ˆAMB�  (29)  We define BMB by  BMB = ∇ × AMB  (30)  Then, the EMF from the Berry connection is given by  EMB = −ℏ  e  �  S  ∂tBMB · dS − ℏ  e  �  S  (v · ∇)BMB · dS  (31)  The first term in the right hand side can arise from the time-dependence of pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' This  means that if pj varies with time due to chemical reactions, photo excitations, or etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  it will give rise to the EMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The second term will arise if the temperature depends on  the coordinate, T(r), and pj contains the Boltzmann factor exp(−  Ej  kBT(r)), where Ej is  the energy for the state Ψj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' It also arises when pj depends on the coordinate due, for  example, to the concentration gradient of chemical spices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  Now we consider the case where the circuit moves with a constant vector v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The  circuit in this case should be regarded as a region of the system which flows due to the  flow existing in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' Such a motion may arise from a temperature gradient or  concentration gradient in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' In this case, we have the following relation,  (v · ∇)BMB = −∇ × (v0 × BMB)  (32)  due to the fact that ∇ · BMB = ∇ · (∇ × AMB) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  The equation (31) can be cast into the following form  EMB = −ℏ  e  �  C  �  ∂tAMB − v0 × (∇ × AMB)  �  dr  (33)  that only contains AMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' However, the above formula may not be convenient to use due  to the fact that AMB contains topological singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' A convenient one may be the  following  EMB = −ℏ  e  d  dt  �  S  BMB · dS  (34)  where B in the Faraday’s law in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (2) is replaced by BMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  Supercurrent and EMF by Berry connection  7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' Nernst effect  In this section, we examine the Nernst effect observed in cuprate superconductors  [13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' We examine this phenomenon using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' A theory of superconductivity  in the cuprate predicts the appearance of spin-vortices in the CuO2 plane around doped  holes that become small polarons [16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The spin-vortices generate the vector  potential  AMB = −1  2∇χ  (35)  where χ is an angular variable with period 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' This angular variable appears due to  the requirement that the wave function to be a single-valued function of coordinates  in the situation where itinerant motion of electrons around the small polaron hole is a  spin-twisting one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  We can decompose χ as a sum over spin-vortices  χ =  Nh  �  j=1  χj  (36)  where χj is a contribution form the jth small polaron hole, and Nh is the total number  of holes that become small polarons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  Each χj is characterized by its winding number  wj = 1  2π  �  Cj  ∇χj · dr  (37)  where Cj is a loop that only encircles the center of the jth spin-vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' We can assume  wj to be +1 or −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' only odd integers are allowed due to the spin-twisting motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The  numbers ±1 are favorable from the energetic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  C(t)  v0  x  y  Ly  C(t+Δt)  v0  x0+  Δt  x0  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' A schematic picture for the EMF appearing from the Berry connection  generated by spin-vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  The Berry connection creates the vector potential  proportional to ∇χ, which creates vortices (loop currents) denoted by circles with  arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' We consider two loops C(t) and C(t + ∆t), where t and t + ∆t denote two  times with interval ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The loop moves with velocity v0 in the x-direction due to the  temperature gradient in that direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' A constant magnetic field is applied in the z-  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' A voltage is generated across the y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The sample exists 0 ≤ y ≤ Ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  The left edge of the loop at time t is x0 and that at time t + ∆t is x0 + v0∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  Supercurrent and EMF by Berry connection  8  Let us consider the situation depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' We neglect the contribution from  A assuming that it is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The EMF generated across the sample in the y-direction  is given by  EMB = − ℏ  e  1  ∆t  ��  S(t+∆t)  BMB · dS −  �  S(t)  BMB · dS  �  = − ℏ  e  1  ∆t  ��  C(t+∆t)  AMB · dr −  �  C(t)  AMB · dr  �  = ℏ  e  1  ∆t  �  ∆C  AMB · dr  (38)  where S(t+∆t) and S(t) are surfaces in the xy-plane with circumferences C(t+∆t) and  C(t), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' ∆C is the loop encircling the area x0 ≤ x ≤ x0 + v0∆t, 0 ≤ y ≤ Ly,  with the counterclockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  We approximate  �  ∆C AMB · dr by  �  ∆C  AMB · dr = − 1  2  �  ∆C  ∇χ · dr  ≈ − 1  22π(nm − na)Lyv0∆t  (39)  where nm and na are average densities of wj = 1 (‘meron’) and wj = −1 (‘antimeron’)  vortices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' Thus, nmLyv0∆t and naLyv0∆t are expected numbers of wj = 1  and wj = −1 vortices within the loop ∆C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' (38) and (38), the approximate EMB is given by  EMB ≈ hv0  2e (na − nm)Ly  (40)  Thus, the electric field generated by EMB in the y-direction is given by  Ey ≈ hv0  2e (na − nm)  (41)  In our previous work, na is denoted as nd indicting that it yields a diamagnetic  current, and nm as np indicting that it yields a paramagnetic current [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' Using nd  and np, the Nernst signal is obtained as  eN =  Ey  |∂xT| = hv0(nd − np)  2e|∂xT|  (42)  The same formula was obtained previously for the situation where spin-vortices move  by the temperature gradient [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' Here, the situation is different;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' the spin-vortices  do not move, but the electron system affected by ∇χ moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  Considering that the  small polaron movement is negligible at low temperature, the present situation is more  realistic than the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The temperature dependence is the same as the one  that qualitatively explains the experimental result [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  Note that experiments indicating the presence of loop currents different from  ordinary Abrikosov vortices [19] in the cuprate [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The present result indicates  that the observed Nernst can be explained by the presence of spin-vortex-induced loop  currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  Supercurrent and EMF by Berry connection  9  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' Concluding remarks  Since the EMF by the Berry connection is not the electromagnetic field origin, it may  be more appropriate to call it the Berry-connection motive force (BCMF) given by  F BMF = −eEMB = ℏ d  dt  �  S  BMB · dS  (43)  The BCMF will arise from quantum mechanical dynamics of particles other than  electrons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' for example, from proton dynamics, through chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' The non-  trivial Berry phase effect has been predicted [22], and observed in the hydrogen transfer  reactions [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' Quantum mechanical effects are important in such reactions due to the  relatively light mass of protons [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' It is known that the EMF generated by the  proton pumps is a very important chemical process in biological systems, and the Berry-  connection motive force may play some roles in the working of the proton pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content=' It  may be also useful to invent high performance batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
+page_content='  References  [1] Berry M V 1984 Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE5T4oBgHgl3EQfWg9Q/content/2301.05559v1.pdf'}
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