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<|MaskedSetence|> For instance, Seitz et al. <|MaskedSetence|> Other studies, such as those by Nakatani and Chan [55], Murg et al. [54], Szalay et al. <|MaskedSetence|> [23], explore different heuristics for tree construction, leveraging factors like entanglement or interaction strength..
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**A**: Note that the generation of the embedding tree only depends on the tensor network graph structure, rather than the actual tensor data.
The relationship between the embedding tree and the orderings of the edges is further explained in Section 4.3.
While our method introduces a novel perspective on the construction of the embedding tree, it is worth noting that alternative approaches have been proposed for determining tree structures based on various other heuristics.
**B**: [69] propose a method specifically for determining a tree structure based on a quantum circuit.
**C**: [75], and Ferrari et al.
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<|MaskedSetence|> <|MaskedSetence|> This results in higher frequency oscillations for perturbing fields around the black hole. Furthermore, the increase in the damping rate with Q𝑄Qitalic_Q indicates that the perturbations decay more rapidly, meaning that the charged black hole tends to settle down faster after being perturbed. <|MaskedSetence|> Although most observed black holes are expected to have negligible charge due to charge-neutralizing effects in astrophysical environments, highly charged black holes are theoretically possible in certain exotic scenarios. For example, primordial black holes formed in the early universe could retain some charge if they formed in environments where neutralizing particles were absent. Understanding the impact of charge on the QNM spectrum is therefore important for detecting or constraining the properties of such exotic black holes through gravitational wave observations.
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**A**: Physically, the increase in oscillation frequency with higher Q𝑄Qitalic_Q can be understood in the context of the black hole’s enhanced electric field.
**B**: The faster decay is likely due to the increased energy stored in the electromagnetic field, which enhances the dissipation of perturbing waves.
The physical significance of these findings is particularly relevant in the context of astrophysical black holes that may possess charge.
**C**: As the black hole’s charge increases, the strength of its electromagnetic field grows, which leads to more tightly bound perturbations.
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The number of resulting equations can be large at higher order PSA-CC and spin adaptations. This could slow down the computations. Though for a given basis set, the finite dimensional vector space for the computations from the spatial orbitals is smaller than from the spin orbitals. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|>
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**A**: For example, in the present PSA-CC, there are many contractions involving active indices
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**B**: One may then expect a further optimized PSA-CC will be more efficient than the spin-orbital approaches.
**C**: Nevertheless, it may be necessary to advance further improvement including factorizations kucharski1991recursive ; kallay2001higher ; hirata2003tensor ; parkhill2010sparse ; lai2012effective ; pfeifer2014faster ; manzer2017general ; engels2011fully .
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Figure 4:
Formation of the pseudogap in the hole- and electron-doped cuprates and cluster dynamical-mean-field-theory (CDMFT) calculation for the electron-doped cuprates . a,b, Schematic band dispersions along the AFM BZ boundary [cuts #1 and #2 in panel c] for the hole-doped and electron-doped cuprates with the pseudogap without a node, respectively. <|MaskedSetence|> Note that the LHB in the single-band Hubbard model is equivalent to the Zhang-Rice singlet band in the three-band Hubbard model after appropriate parameter conversion Sheshadri et al. (2023). <|MaskedSetence|> <|MaskedSetence|> c, Momentum cuts for d, e, and f. d-f, Spectral function calculated by CDMFT along cuts #1-#3 indicated in panel c. See text for details of the calculation.
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**A**: IGB, coh-LHB, and coh-UHB denote the in-gap band, the coherent part of the lower Hubbard band and that of the upper Hubbard band, respectively, in the electron-fractionalization picture.
**B**: In the hole-doped (electron-doped) cuprates, the LHB (UHB) is fractionalized into the coh-LHB (coh-UHB) and the IGB.
**C**: CDMFT calculations yield smaller pseudogap around (π/2𝜋2\pi/2italic_π / 2, π/2𝜋2\pi/2italic_π / 2) [around (π𝜋\piitalic_π, 0)] for hole (electron)-doped cuprates.
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<|MaskedSetence|> <|MaskedSetence|> The advantage of the M=N−1𝑀𝑁1M=N-1italic_M = italic_N - 1 assumption is that in this case the group UM,Nsubscript𝑈𝑀𝑁U_{M,N}italic_U start_POSTSUBSCRIPT italic_M , italic_N end_POSTSUBSCRIPT is
trivial, so GL(M,𝐎)⋉UM,N(𝐅)GLleft-normal-factor-semidirect-product𝑀𝐎subscript𝑈𝑀𝑁𝐅{\mathop{\operatorname{\rm GL}}}(M,{\mathbf{O}})\ltimes U_{M,N}({\mathbf{F}})roman_GL ( italic_M , bold_O ) ⋉ italic_U start_POSTSUBSCRIPT italic_M , italic_N end_POSTSUBSCRIPT ( bold_F ) is just equal to GL(N−1,𝐎)GL𝑁1𝐎{\mathop{\operatorname{\rm GL}}}(N-1,{\mathbf{O}})roman_GL ( italic_N - 1 , bold_O ) (and the character χ𝜒\chiitalic_χ is trivial as well).
The current paper should be thought of as a sequel to [BFGT]. <|MaskedSetence|> As was noted above, one has to be careful about specializing to non-generic q𝑞qitalic_q. It turns out that for q=1𝑞1q=1italic_q = 1 the correct statement is as follows..
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**A**: What is done in this paper?
In this paper we deal with the case M=N−1𝑀𝑁1M=N-1italic_M = italic_N - 1 for generic q𝑞qitalic_q.
**B**: 1.3.
**C**: There we consider (among other things) the case q=1𝑞1q=1italic_q = 1.
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<|MaskedSetence|> <|MaskedSetence|> For the overall performance of CM2Mc-LPJmL, realistically simulated precipitation fields are therefore crucial. This motivates the work presented below, where we use a specific kind of GAN to transform the AM2 precipitation fields toward fields that are indistinguishable from ERA5 precipitation fields (see below).
The model experiments of this paper are consistent with [Drüke, von Bloh\BCBL \BOthers. (\APACyear2021)]. <|MaskedSetence|> In this way we ensure that the model starts from a consistent equilibrium between the long-term soil carbon pool, vegetation, ocean, and climate.
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**A**: As a stand-alone model LPJmL has been mainly calibrated with respect to reanalysis, and a similarly accurate precipitation output within CM2Mc-LPJmL would hence be favorable to maintain consistency and to obtain realistic surface fluxes from LPJmL.
**B**: In CM2Mc-LPJmL, the fluxes simulated by LPJmL depend, of course, on the precipitation modelled by AM2.
**C**: After a 5000-year stand-alone LPJmL spin-up, a second fully coupled spin-up under pre-industrial conditions without land use was performed for 1250 model years.
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For many important applications entanglement has been proven to be a powerful resource. <|MaskedSetence|> <|MaskedSetence|> (2017, 2018); Contreras and Goyeneche (2022).
Still, the analysis of AME states is important for understanding quantum error correction and regarded as one of the central problems in the field Horodecki et al. (2022); Rather et al. <|MaskedSetence|>
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**A**: An example of resourceful states are the absolutely maximally entangled (AME) states which maximized the entanglement in the bipartitions, but are notoriously difficult to characterize Scott (2004); Facchi et al.
**B**: (2022).
However, multiparticle entanglement offers a complex and rich structure resulting in the impossibility of quantification by means of a single number..
**C**: (2008); Reuvers (2018); Gour and Wallach (2010); Huber et al.
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<|MaskedSetence|> We assume, in particular, that there exist no massless extended objects interpolated by non-local operators. The notion of particles implies that the low energy effective theory flows to an infrared free fixed point at zero energy. <|MaskedSetence|> <|MaskedSetence|> (1997b) are also commonly used in the literature. Even though such a low energy effective theory per se does not define a phase in the Landau sense (due to the lack of center symmetry), it is still a valid description when the coupling between quarks and gluons becomes very strong.
We will study whether such a confining description is compatible with unbroken chiral symmetry in QCD-like theories.
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**A**: In this paper, by ‘confinement’ we mean a low energy effective description of the QCD-like theory, where particles (hadrons) are interpolated by color-singlet local operators.
**B**: A similar confining description is realized in supersymmetric QCD Seiberg (1994) and other supersymmetric gauge theories Csaki et al.
**C**: (1997a, b), and is usually dubbed as s-confinement Intriligator and Seiberg (1996). 333Other terms such as ‘color confinement’ Greensite (2011), ‘quark confinement’ Shifman (2012); Schwartz (2014), ‘fermion trapping’ Weinberg (2013) and ‘screening confinement’ Csaki et al.
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In magnetic systems, coupling between surface acoustic waves and single magnetic skyrmion has been examined [61, 62, 63]. <|MaskedSetence|> It is theoretically suggested that longitudinal acoustic wave induces translational motion of magnetic bubbles [61]. <|MaskedSetence|> <|MaskedSetence|> While experiments on skyrmion collective translation by surface acoustic waves are lacking, it has been known that the transverse and longitudinal acoustic wave couples with magnon in a different manner [87, 59]. Therefore, there may be a possibility to control collective skyrmion translation by the longitudinal and transverse acoustic waves differently.
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**A**: Experimentally, however, shear-vertical component in Rayleigh waves may disturb translation of Néel-type skyrmion [62, 63].
Rather, shear-horizontal wave induces their translation [63].
**B**: Here, surface acoustic waves include Rayleigh wave (shear-vertical and longitudinal strains are excited) and Love wave (shear-horizontal strain is excited) [60, 63].
**C**: Thus, it should be important to separate the role of shear-vertical and longitudinal strains.
For skyrmion condensed states, collective skyrmion translation corresponds to magnon excitation with a particular wave number [86].
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<|MaskedSetence|> To obtain a full description of the interaction we will suggest a new complementary equation inspired by graph theory. <|MaskedSetence|> We will show that the interaction between dark matter and dark energy
is chaotic. On the other hand, we aim to propose a new law of physics, inspired by the possibility that the obtained results may be universal for all thermodynamic systems. <|MaskedSetence|>
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Based on this new theoretical framework, we will show that interaction equations can be derived from the first law of thermodynamics.
**B**: Additionally, we will introduce and explore several new conceptual definitions.
We believe that our results might give a critical contribution not only to cosmology but also to statistical thermodynamics and the understanding of nature and living systems..
**C**: Finally, based on these theorems, we will carry out new interaction equations that describe the interaction between dark matter and dark energy.
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The additional crucial aspect of the present report is the suggested methodology. Each integrator has been implemented on top of the well-maintained open-source molecular dynamics package ESPResSo [22, 23, 24]. Each method passed automated Python tests scripted by independent researchers providing objectivity, reusability and future maintenance of its implementation. <|MaskedSetence|> <|MaskedSetence|> Clean and repeatable run time environment is organized using Docker technology [25] and GitHub tags of the source code including the simulation scripts [26]. The present analysis is straightforward and appears to minimize code errors in different programming languages and environments. <|MaskedSetence|>
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**A**: The code for testing is available in GitHub and referred here (see Table 1).
**B**: The results reported in the following are based on modification of some of these tests and, importantly, the same Python scripts are used to validate each integrator without ad hoc adaptation.
**C**: Hence, we hereby encourage other researchers to follow such open-source approach to avoid publishing algorithms as pseudo-codes or based on proprietary or hidden frameworks, unmaintained personal software codes, and/or out of reach to other investigators.
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All the signal and background processes are passed through all these event selection criteria up to C4 before passing events to MVA. We create two separate signal and background classes. The combined background is the weighted combination of all the different background processes. Each signal and background class is randomly divided into 50%percent5050\%50 % for training and the rest 50%percent5050\%50 % for testing. We use boosted decision tree (BDT) algorithm and choose a set of kinematic variables from a wider collection of variables for MVA. <|MaskedSetence|> Table 5 lists the relative importance of the various kinematic variables involved in the MVA. <|MaskedSetence|> <|MaskedSetence|>
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**A**: The variables with high relative importance distinguishing the signal class from the background class are preferable.
**B**: The optimized BDT hyperparameters used in our analysis are outlined in Table 6.
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**C**: The left (signal) and right (background) tables of Figure 9 show the linear correlation coefficients among the variables employed in MVA for BP1.
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<|MaskedSetence|> Experiments aimed at elucidating the nature of the possible triangular lattice QSL in YbZn2GaO5 are of great importance. Thus, we grew a large and high-quality single crystal of YbZn2GaO5 (see Fig. S1) using the optical floating-zone technique to facilitate such experiments. Our specific heat measurements indicate that at ultra-low temperatures, the specific heat displays a ∼similar-to\sim∼ T2 dependence, indicating a U(1) Dirac QSL behavior [27, 28]. Additionally, we show that the field-induced T-linear component of the specific heat is proportional to the applied magnetic field, further confirming that the ground state of YbZn2GaO5 is best described with U(1) Dirac QSL [27]. <|MaskedSetence|> <|MaskedSetence|> Hence the specific heat scaling and INS spectra independently are best explained by low-energy spinon excitations with Dirac-like spectrum.
II Results & Discussion.
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**A**: To that end, we present a new Yb-based compound, YbZn2GaO5, that features an ideal triangular lattice of effective spin-1/2 moments without detectable intrinsic chemical disorder.
**B**: In addition to the specific heat measurements, we conducted INS investigation on YbZn2GaO5 which reveals gapless, continuum-like spectra at the high-symmetry M and K points, but not throughout the Brillouin zone, and in particular not at the ΓΓ\Gammaroman_Γ point.
**C**: This particular pattern of low-energy spinon excitations is expected for the U(1) Dirac QSL phase [29, 30] and not for a spinon Fermi surface state.
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<|MaskedSetence|> The effective action implies the existence of two distinct types of scalar KK modes that couple with the massive vector modes. <|MaskedSetence|> However, the solvable 6D branes are usually constructed within the non-conformal metrics. By comparing the EOM for the KK modes (deriving from two ways), we revealed that to preserve the gauge invariance of the effective action in these 6D branes, certain constraints on the brane’s geometry must be introduced.
Nevertheless, the gauge invariance can be easily compromised. <|MaskedSetence|> Secondly, even when the brane solutions match the constraints, gauge invariance is only preserved if both types of massive bound scalar KK modes are present within the brane..
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This study begun with an examination of the methodology for deriving the effective action of a massless bulk U(1)𝑈1U(1)italic_U ( 1 ) gauge field through a general KK decomposition within branes of codimension 2.
**B**: While we have established that the effective action maintains gauge invariance in brane models with a conformal metric.
**C**: Firstly, if the brane solutions do not conform to the imposed constraint conditions, the formulation of a gauge-invariant effective action becomes unfeasible.
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<|MaskedSetence|> By thoroughly investigating our proposed model, we explore and identify a parameter space that can successfully account for the observed GW signals while addressing the small-scale structure problems. <|MaskedSetence|> <|MaskedSetence|> In Section IV, we present our numerical results, showcasing the viable parameter space, and engage in further discussions. Finally, we conclude in Section V..
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**A**: Section III focuses on the calculation of the FOPT and the corresponding GW signals.
**B**: This model holds promise for experimental testing in the near future.
The article is organized as follows: Section II provides an introduction to our model and explores the physics associated with SIDM.
**C**:
In this work, we present a concise and comprehensive model within the framework of SIDM to explain the recently reported PTA data, specifically focusing on the NANOGrav and CPTA observations.
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To gain deeper insights into the fabricated THz emitters, we analyzed the correlation between the amplitude of the emitted THz field at 1 THz and the varying thicknesses of the CoFeB layer in stacks with specified NM layers, see Fig. <|MaskedSetence|> <|MaskedSetence|> Specifically, the stack with a 2 nm Pt layer exhibits a THz signal amplitude that is twice as high as the stack with a 2 nm PtBi layer. However, the emitter with PtBi, as shown in Figs. 3 (c) and (d), exhibits a wider bandwidth of approximately 0.35 THz compared to other stacks, along with a higher central frequency of the THz signal. Interestingly, it demonstrates a significant central frequency shift of approximately 0.3 THz when compared to the emitter with 2 nm Pt. These findings highlight the crucial role of the NM layer material in influencing THz characteristics and suggest the potential advantages of the PtBi stack for applications requiring broader bandwidth and higher frequencies, despite a lower THz amplitude. When evaluating the trade-off between THz signal strength and bandwidth, it is crucial to consider the specific requirements of the intended application, as different applications may prioritize either a higher THz signal strength or a broader bandwidth based on their unique needs and constraints. This observation suggests potential advantages in applications that benefit from higher frequency THz signals, including high-resolution imaging, spectroscopy, communications, medical diagnostics, and etc. <|MaskedSetence|>
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**A**: The THz emitters composed of pure Pt layers demonstrate the highest THz amplitude among the emitters.
**B**: 3 (a) and (b).
**C**: The values of the saturated THz amplitudes and THz peak position measured for different emitters are presented in Table 1.
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<|MaskedSetence|> They can efficiently handle a large number of events and is unaffected by a small number of bad events if any. Furthermore, the Bayesian credible intervals have the clear and unambiguous meaning of being regions of the parameter space that contain the true value of said parameters with a certain posterior probability given data.
Among the two approximate Bayesian techniques described in this paper, the narrow Gaussian method has the following issue. The process to find σ𝜎\sigmaitalic_σ is cumbersome and somewhat subjective as it involves partitioning the set of events into subsets and estimating σ𝜎\sigmaitalic_σ from plots, or developing an algorithm that needs to be compared to an independent method. <|MaskedSetence|> the bandwidth, quantitatively from properties of the posterior samples themselves, with its variation having a much more restrictive effect on the estimated density. <|MaskedSetence|>
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The Bayesian methods are free of all aforementioned pathologies that plague the other methods.
**B**: On the other hand, the KDE method has less user-controlled fine-tuning than the narrow Gaussian method as it estimates its control parameter, i.e.
**C**: Hence, we claim that the Bayesian analysis implemented by the KDE method produces the most trustworthy measurements of the SME coefficients..
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Figure S3:
Atomistic disorder in our Si/SiGe fins. (a-c) Plot of the cross section of the triangular fin including three different disorder configurations at the Si/SiGe interfaces. The blue dots depict the discrete lattice points used for the numerical diagonalization of the Hamiltonian in the main text and the red lines mark the interface between the inner SiGe fin and the outer Si shell including atomic size steps at random positions. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|>
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**A**: This is not surprising since the electron wave function is located far away from the interface steps.
**B**: (d) Valley splitting ΔΔ\Deltaroman_Δ plotted against the Ge concentration 1−x1𝑥1-x1 - italic_x and electric field E𝐸Eitalic_E (pointing along y𝑦yitalic_y direction) in the triangular fin.
**C**: The result is for all three configurations exactly the same as for the triangular fin without interface steps.
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Example applications in 18F-FDG PET imaging demonstrated the capabilities of LM TOF SSS modeling and LM-TOF reconstruction using the OSEM and DIPRecon algorithms. <|MaskedSetence|> <|MaskedSetence|> Extensive research on the validity of the DIPRecon method may seek to explore, for example, the effects of different neural network architectures, hyperparameters, and misalignment between the PET/MRI images. <|MaskedSetence|> Users who wish to test different network/optimization configurations are encouraged to view the DIPRecon tutorial available on the documentation website.
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**A**: As apparent in Figure 4, reconstruction of the low count data using DIPRecon was able to reduce noise in the grey matter regions while preserving cortical structures.
**B**: Since PyTomography directly interfaces with PyTorch, the process of designing and testing neural networks and optimization procedures for the DIPRecon method is straightforward.
**C**: While use of the DIPRecon on the low count data resulted in a slight increase in bias in the grey matter region, the corresponding noise bias curve demonstrates less noise than the high count reconstruction using OSEM.
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<|MaskedSetence|> In 2D systems, the extrinsic electric field, 𝓔𝓔\bm{\mathcal{E}}bold_caligraphic_E, can originate from external factors such as gate voltages. <|MaskedSetence|> However, local fields can arise from charged impurities and structural defects, giving rise to extrinsic RSOI, which leads to skew scattering and side-jumping of electrons [25, 26, 27, 28]. These mechanisms have since been experimentally validated and are now recognized as significant contributors to phenomena such as the anomalous Hall and spin Hall effects. <|MaskedSetence|>
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where α𝛼\alphaitalic_α is a material-dependent Rashba constant.
**B**: Notably, these effects have been observed in a wide range of materials, including those of current interest, such as topological insulators and 2D materials [29, 30].
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**C**: In contrast, in 3D metals, uniform electric fields are typically screened.
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In a real universe, ideal Kerr black holes, Reissner-Nordstrom black holes, and Kerr-Newman black holes cannot exist because black holes cannot exist in a world without an external matter distribution.
Numerous astronomical observations have indicated the existence of an exceptionally intricate distribution of matter in proximity to black holes, and the reality of black holes is contingent upon the consideration of external matter distributionShakura and Sunyaev (1973); Antonucci (1993). <|MaskedSetence|> <|MaskedSetence|> (2006); Li et al. <|MaskedSetence|> The solutions to Einstein’s gravitational field equations become exceedingly complex when accounting for the coupling between dark matter and black holes.
In recent years, several approximate solutions have been derivedXu et al. (2018, 2020); Konoplya (2019). By utilizing these black hole metrics, one can investigate the impact of dark matter on testing the weak cosmic censorship conjecture. Recently, Meng,Xu and Tang discovered that by considering the interaction between an ideal fluid dark matter and a black holeMeng et al. (2023), the weak cosmic censorship conjecture can be violated.
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**A**: (2011).
**B**: Of particular interest are cases involving dark matter particles and dark energy surrounding black holesBertone et al.
**C**: (2005); Bertone and Hooper (2018); Copeland et al.
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<|MaskedSetence|> <|MaskedSetence|> An agent may select from four possible actions, each of which is only possible if the ensuing movement does not lead outside the 3×3333\times 33 × 3 gridworld. <|MaskedSetence|> Action 0 indicates a step to the north, action 1 a step to the south, action 2 a step to the west, and action 3 a step to the east. To prevent the VQC from choosing illegal actions, the expected values are normalized to the interval [0,1]01[0,1][ 0 , 1 ] and masked with environmental regulations.
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Each cell of the 3×3333\times 33 × 3 gridworld can contain either agent 1, agent 2, a red or blue coin.
**B**: Empty grids need not be included in the observation, as movement can occur on them without consequence at any time.
**C**: The numerical actions range from 0 to 3.
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The disadvantages in using a coordinate based approach led to models that used the so called turn based encoding approach [6]. The turn based encoding approach maps the turns taken while moving from one molecule to the next to quantum states. This is analogous to a self avoiding walk (SAW) setup. In a 3D dimensional lattice, we could have 6 turns (above, below, right, left, front, back) from any given given point in the lattice. <|MaskedSetence|> For an N𝑁Nitalic_N bead (molecule) system, we will have N−1𝑁1N-1italic_N - 1 turns, and the total number of qubits required will be 3(N−1)3𝑁13(N-1)3 ( italic_N - 1 ). <|MaskedSetence|> This is an improvement as far as qubit requirement is concerned. However, these methods are unable to capture diagonal movements and they also need ancilla qubits for encoding the slack variables required for enforcing the overlap constraints. In [6], the authors also discuss turn-circuit constructions that do not require ancilla qubits. These methods require very involved circuit constructions with higher order many-body terms. <|MaskedSetence|> [6] also discusses several ways to solve protein folding problems by mapping them to quadratic unconstrained binary optimization (QUBO) problem, heuristic satisfiability problems, etc. QUBO problems are essentially 2-local problems (all terms of the optimization problem involve two or less number of variables) and have been widely used in many applications involving quantum computing machines. QUBO formulations are also solvable by classical algorithms like simulated annealing and tools like IBM CPLEX.
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**A**: A 3-qubit encoding will suffice to represent all the movements.
**B**: A further assumption on taking the first turn as given will result in 3(N−2)3𝑁23(N-2)3 ( italic_N - 2 ) qubit requirements.
**C**: These kind of approaches have been tried out in quantum devices using annealing technologies.
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Although the concept of the fractional derivatives and dispersion has a long history, their realization in physical systems, such as nonlinear fiber optics, is a relatively new and emerging topic. <|MaskedSetence|> The current results focus primarily on positive fractional dispersion lengths, corresponding to the “repulsion” case in the force model, which explains the observed spectral valleys. However, equally intriguing results are expected in the “attraction” case, which would illustrate spectral squeezing. Recently two works have demonstrated an “attraction” effect (narrowed spectrum) in similar contexts [65, 66]. <|MaskedSetence|> <|MaskedSetence|> This approach has significant potential for applications to optical encoding and spatiotemporal mode-locked laser architectures [67, 68]. 3) Finally, due to the mathematical similarity between the FNLSE in nonlinear optics and its quantum counterpart, this regime may serve as an effective model for emulating fractional quantum mechanics [1, 69].
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**A**: 2) The second perspective is the realization of the spatial-temporal light synthesis by incorporating both fractional dispersion and fractional diffraction.
**B**: By involving additional parameters, such as higher-order dispersion or stronger nonlinearity, even greater diversity in the pulse dynamics could be uncovered.
**C**: This inspires several promising perspectives: 1) The immediate goal is to explore additional solutions of the FNLSE.
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It is worth mentioning that another way to present one more peak within EIT Liu2018 ; Yin2013 which is similar to our EIT. However, It is quite different from our CIT. They produce one more peak within EIT by employing symmetry-breaking of coupling between the bright mode and dark mode. Our CIT comes from the symmetry-breaking of dark modes to cause the interference of dark modes and then due to bright mode coupling with interference of dark modes, the transmission spectrum emerges the CIT phenomenon.
The previous paper Yahiaoui2018 proposed the EIT from bright-bright modes coupling, which is similar to Fig. 2 (b). However, our CIT is demonstrated in Fig. 2 (c). <|MaskedSetence|> 2 (b) and (c) have different polarization. Therefore, in our paper, two dark modes can not be excited by external field without bright mode (CW). Thus, the physics behind CIT and EIT from bright-bright modes coupling are entirely different.
For better understanding, We plot the electric field and current density field at CIT frequency, as shown in fig. 6 with corresponding to around 0.72 THz for the red line in Fig. <|MaskedSetence|> As we can obtain from the results, the dark modes present two opposite modes which provide destructive interference between the dark modes at CIT frequency. <|MaskedSetence|> Thus, the bright mode can be re-excited by external THz wave and it causes the CIT dip in the transmission spectrum..
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**A**: 2 (c).
**B**: Therefore, the bright mode can not couple to dark modes at this frequency due to the suppression of dark modes, which remains some current density and energies.
**C**: Note that Fig.
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<|MaskedSetence|> Confining or immersed boundaries introduce preferential orientations of the director field with a given strength (for instance, a tangential anchoring condition), these generally lie in competition with the preferred uniformity of the orientation field [83]. If there are multiple immersed bodies or boundaries, the elastic energy may be reduced by altering their relative positions and orientations. <|MaskedSetence|> When many colloids are introduced to a LC they can self-assemble into linear chains [64, 44, 76, 20]. When the bodies are sufficiently well separated, their long-range interactions conjure a related problem, the interaction of topological defects themselves [86, 36].
Near-field interactions, meanwhile, can be strongly nonlinear due to the interaction and positional rearrangement of topological defects [87, 4, 13, 36]. <|MaskedSetence|> Rather than colloid translations and rotations to reduce the system energy, a separate path towards relaxation is available if the immersed particles are deformable [50, 56, 103, 59, 75].
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**A**: The self-assembly of colloids in LCs has seen wide use in the engineering of smart materials, with applications ranging from biosensors to dynamic porous membranes [7].
**B**: Dipolar and quadrupolar far-field interactions between colloids (depending on normal or tangential anchoring conditions) have been investigated in three-dimensions [66, 70, 27, 2], and similarly between a colloid and a confining boundary [26].
**C**: Bodies immersed in a LC (that are much larger than the LC constituents) disturb the orientational order of the bulk liquid crystal.
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<|MaskedSetence|> <|MaskedSetence|> However, the less intuitive forms of these equations that arise from the correspondences derived above may yield important insights. For example, the perspective of the virial theorem as a special case of the equipartition theorem [34] may be fruitful in evolutionary biology [31]. Translation between biology and physics via the virial theorem and the Price equation may also accelerate discovery of generalizations. <|MaskedSetence|> Moreover, the virial theorem has been applied in a variety of fields (for example economics [2]), meaning that understanding its relationship to the Price equation could be relevant beyond physics and biology.
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**A**: While the stochastic Price equation in evolution [41] and the stochastic virial theorem in astronomy [7] were discovered independently, their similarity suggests other generalizations could similarly parallel each other.
**B**:
In genetics, the natural time increment to consider is discrete (generation), whereas in physics continuous-time is more natural.
**C**: Thus, the discrete Price equation pertains to change in a trait after a single generation, whereas the virial theorem is formulated with continuous-time, and is additionally time averaged.
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<|MaskedSetence|> The resulting KS eigenstates are also not particularly localized functions, and DFT is invariant under unitary rotations of the occupied electronic states. However, several electronic-structure methods, aiming at improving or complementing the capabilities of DFT, fundamentally require to be formulated in terms of localized orbitals (see Sec. III.8). In addition, several of these beyond-DFT methods are not deployed directly on the crystal structure, but operate more as corrections to starting DFT calculations. <|MaskedSetence|> <|MaskedSetence|>
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**A**: In this context, WFs provide a robust way to bridge DFT with advanced electronic-structure methods by allowing to systematically construct orthogonal localized states that represent the manifold of interest: WFs are first constructed on the KS DFT solution and then fed into beyond-DFT methods; a technical overview of how this is carried out in practice is the subject of Sec. III.8.
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**B**: Also, beyond-DFT methods can be computational rather intensive, and it is common practice to apply them only on a subset of bands extracted from the entire manifold.
**C**: Advanced electronic-structure methods
DFT simulations of periodic solids can be conveniently (but definitely not necessarily) performed by adopting a plane-wave basis set, in conjunction with smooth pseudopotentials that reproduce the interaction between valence electrons and nuclei plus core electrons [269].
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Part of the work has been done when CD was a visiting reseacher at the Institut des Hautes Études and she would like to thank Laure Saint-Raymond for her support and hospitality. We are grateful to Rupert L. <|MaskedSetence|> We also thank Hajer Bahouri, Morris Brooks, Albert Cohen, Patrick Gérard, Kihyun Kim, Mathieu Lewin, Hoai-Minh Nguyen, Julien Sabin, Nikita Simonov, Thomas Sørensen, Jakob Stern, Hanne van den Bosch and Jean van Schaftingen for interesting and inspiring discussions.
We acknowledge the support from the Deutsche Forschungsgemeinschaft through the DFG project Nr. <|MaskedSetence|> <|MaskedSetence|>
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**A**: CD also acknowledges the support from the Jean-Paul Gimon Fund and from the Erasmus+ programme.
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**B**: 426365943.
**C**: Frank for helpful suggestions, leading to our consideration of Lorentz norms in Theorem 2.
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Figure 1: (a) Intracavity structure consisting of GaAs emitter and absorber layers, AlGaAs claddings, and outer mirrors consisting of Ag. For the results shown in Fig. 4(b), there is also a 50 nm thick vacuum nanogap in the middle of the structure. The centre points of the emitter and absorber layers are located halfway between the mirrors closest to them and the center point of the full cavity (also other configurations have been experimented with, but this choice has provided the largest overall emission so far). <|MaskedSetence|> 2 for the intracavity structure of Fig. 1(a) with a single example. Here, the thicknesses of the GaAs emitter and absorber layers are 20 and 60 nm, respectively. As in Ref. 12, the emitted and absorbed optical powers depend on the cavity length. <|MaskedSetence|> The optical power emitted by a similarly thick GaAs layer in the reference structure of Fig. 1(b) is also plotted as a constant dotted line in Fig. 2. <|MaskedSetence|>
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**A**: (b) Reference structure, with a GaAs emitter layer emitting light into semi-infinite AlGaAs claddings.
The dependence of the optical power on the cavity length is first illustrated in Fig.
**B**: Here, the maximum emission and absorption are reached at a total cavity length of approximately 250 nm.
**C**: It can be seen that at the maximum, the emission in the cavity is roughly 35 % stronger than in the reference structure..
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<|MaskedSetence|> The investigation of isospectral drums, where the Laplacian operator within closed domains with Dirichlet boundary conditions yields identical spectra for distinct regions sharing the same area, has produced significant results for specific cases and subsets of the problem [5]. These isospectral problems in bounded domains are closely linked to inverse problems in open space [6], forming a rich area of research in the past. <|MaskedSetence|> <|MaskedSetence|> However, another strategy proposed in the work by Li et al. [14], the so-called carpet cloak, requires non-singular transformations, making this route more suited for the practical implementation of the equivalent properties. Twinning closed cavities through TA has been recently developed by Lenz et al. in [15], where the discrete spectrum of a closed domain with Dirichlet boundary conditions is successfully matched. Here we consider unbounded open domains; this is not straightforward as spectral problems for open cavities allow for the leakage of energy into the unbounded medium, have complex eigenspectra and further complications for both theoretical and numerical aspects; we are unaware of attempts to achieve isospectral domains in open systems in wave physics and this opens the way to, for instance, twinning optical waveguides.
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**A**: Furthermore, there has been a recent interest in the study of isospectral or quasi-isospectral potentials inspired by supersymmetric transformations as applied to electromagnetism [7].
**B**: We proceed to utilise Transformation Acoustics (TA), which has been extensively applied for manipulating wave propagation in diverse physical fields sharing the same analytical structure, like electromagnetism [8], acoustics [9], elasticity [10, 11, 12], and many others fields [13].
Perhaps the most striking and well-studied effect enabled by TA is cloaking, allowing perfect concealment of an arbitrary object using a singular transformation.
**C**:
The history of isospectral problems can be traced back to the question famously posed by Mark Kac regarding whether one can hear the shape of a drum [4].
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Acknowledgements
R. G, M. K., V. J, G. V. M. W, C. P., B. D. H. B, A. P. and P. <|MaskedSetence|> acknowledge the French National Research Agency (ANR) under the THz-MUFINS project (Grant No. ANR-21-CE42-0030). We wish to warmly thank Frédéric Picca (SOLEIL Synchrotron) for adapting BINoculars to the analysis of diffraction data from CRISTAL beamline. We acknowledge SOLEIL for provision of synchrotron radiation at CRISTAL beamline (proposal number 99190273). H. B. is grateful to C. <|MaskedSetence|> C. <|MaskedSetence|> and L. B. would like to acknowledge the U.S. Department of Defense under the DEPSCoR program (Award No. FA9550-23-1-0500) and the Vannevar Bush.
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**A**: Lichtensteiger for help on the XRD simulation.
**B**: R.
**C**: P.
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<|MaskedSetence|> (2023); Grushin and Repellin (2023); Keskiner et al. (2023). <|MaskedSetence|> This leads generically to chiral spin liquid phases associated with a spontaneously broken time-reversal symmetry, while the spin-liquid tied to the quasicrystal in this work includes only plaquettes with an even number of edges and no tendency towards spontaneous breaking of this symmetry. <|MaskedSetence|>
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**A**: However, unlike the example considered here, the above constructions inevitably host plaquettes with an odd number of edges.
**B**: Moreover, our quasicrystalline graph possesses a forbidden rotational symmetry, not appreciated in previous constructions, which leads to an interesting interplay of localization-delocalization behavior as a function of energy on the low-energy excitation spectrum, which has not been appreciated in some of these previous works.
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**C**:
Construction of a lattice with a fixed coordination number has been recognized as a viable route toward a solvable spin liquid model in various non-crystalline systems Cassella et al.
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The paper is structured as follows: Section II provides an overview of the R-Symmetric Higgs inflation within a generic GUT model, focusing on its formulation within a no-scale-like supergravity framework. <|MaskedSetence|> <|MaskedSetence|> We highlight the parametric range where observable inflationary primordial gravitational waves are expected, along with a brief discussion on successful reheating and leptogenesis scenarios. Additionally, we touch upon the potential of PBHs as dark matter and their relevance in explaining NanoGrav observations. <|MaskedSetence|>
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**A**: Finally, Section V presents our conclusions.
II R-Symmetric Higgs Inflation.
**B**: Section IV extends the analysis to include the multifield treatment, exploring the presence of PBHs and gravitational waves, with particular attention given to the role of the leading-order nonrenormalizable term in the superpotential.
**C**: In Section III, we delve into the effective single-field treatment of the model.
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