Energy storage (ES) and virtual energy storage (VES) are key components to realizing power system decarbonization. Although ES and VES have been proven to deliver various types of grid services, little work has so far provided a systematical framework for quantifying their adequacy contribution and credible capacity value while incorporating human and market behavior. Therefore, this manuscript proposed a novel evaluation framework to evaluate the capacity credit (CC) of ES and VES. To address the system capacity inadequacy and market behavior of storage, a two-stage coordinated dispatch is proposed to achieve the trade-off between day-ahead self-energy management of resources and efficient adjustment to real-time failures. And we further modeled the human behavior with storage operations and incorporate two types of decision-independent uncertainties (DIUs) (operate state and self-consumption) and one type of decision-dependent uncertainty (DDUs) (available capacity) into the proposed dispatch. Furthermore, novel reliability and CC indices (e.g., equivalent physical storage capacity (EPSC)) are introduced to evaluate the practical and theoretical adequacy contribution of ES and VES, as well as the ability to displace generation and physical storage while maintaining equivalent system adequacy. Exhaustive case studies based on the IEEE RTS-79 system and real-world data verify the significant consequence (10%-70% overestimated CC) of overlooking DIUs and DDUs in the previous works, while the proposed method outperforms other and can generate a credible and realistic result. Finally, we investigate key factors affecting the adequacy contribution of ES and VES, and reasonable suggestions are provided for better flexibility utilization of ES and VES in decarbonized power system.

In this paper, we consider a notion of a higher version of the relation between Courant-Dorfman algebras and Poisson vertex algebras. We define a higher Courant-Dorfman algebra, and study the relationship with graded symplectic geometry. In particular, we give graded Poisson algebras of degree $-n$ in the non-degenerate case. For higher Courant-Dorfman algebras coming from finite-dimensional vector bundles, they coincide with the algebras of functions of the associated differential-graded(dg) symplectic manifolds of degree $n$. We define a higher Lie conformal algebra and Poisson vertex algebra, and give a higher (weak) Courant-Dorfman algebraic structure arising from them. Moreover, we prove that the higher Lie conformal algebras and higher Poisson vertex algebras have properties like Lie conformal algebras and Poisson vertex algebras. As an example, we obtain an algebraic description of Batalin-Fradkin-Vilkovisky(BFV) current algebras.

I investigate the two-dimensional Dirac fermion analogue of artificial atoms (Fock-Darwin states, FD) in a circular n-p junction on a topological insulator surface. The FD states in this non-parabolic potential exhibit a unique electron-hole core-shell structure, where the strict Berry-phase switch (BPS) picture breaks down near criticality: the trapped electron-core states evolve into the envelope functions of quantized snake states. This contradicts the sharp BPS seen in experiments. Nevertheless, the BPS scenario remains valid when treating these envelope functions as effective confined states, thereby reconciling theory with experiment. The field-driven evolution from electrostatic to Landau-level confinement is tracked to 14 T experimentally and supported by simulations, establishing topological surface states as a tunable platform for Dirac physics beyond conventional quantum dots.

We consider the 3D compressible isentropic Euler equations describing the motion of a liquid in an unbounded initial domain with a moving boundary and a fixed flat bottom at finite depth. The liquid is under the influence of gravity and surface tension, and it is not assumed to be irrotational. We prove local well-posedness by combining a carefully designed approximate system and a hyperbolic approach, which allows us to avoid using Nash-Moser iteration. The energy estimates yield no regularity loss and are uniform in both Mach number and surface tension coefficient, provided the Rayleigh-Taylor sign condition is satisfied. We thus simultaneously obtain incompressible and zero surface tension limits. Moreover, we can drop the uniform boundedness (with respect to Mach number) on high-order time derivatives by applying the paradifferential calculus to the analysis of the free-surface evolution.

Atomistic simulations are an important tool in materials modeling. Interatomic potentials (IPs) are at the heart of such molecular models, and the accuracy of a model's predictions depends strongly on the choice of IP. Uncertainty quantification (UQ) is an emerging tool for assessing the reliability of atomistic simulations. The Open Knowledgebase of Interatomic Models (OpenKIM) is a cyberinfrastructure project whose goal is to collect and standardize the study of IPs to enable transparent, reproducible research. Part of the OpenKIM framework is the Python package, KIM-based Learning-Integrated Fitting Framework (KLIFF), that provides tools for fitting parameters in an IP to data. This paper introduces a UQ toolbox extension to KLIFF. We focus on two sources of uncertainty: variations in parameters and inadequacy of the functional form of the IP. Our implementation uses parallel-tempered Markov chain Monte Carlo (PTMCMC), adjusting the sampling temperature to estimate the uncertainty due to the functional form of the IP. We demonstrate on a Stillinger--Weber potential that makes predictions for the atomic energies and forces for silicon in a diamond configuration. Finally, we highlight some potential subtleties in applying and using these tools with recommendations for practitioners and IP developers.

Scattering of a tightly focused electron beam by an atom forms one of the bases of modern electron microscopy. A fundamental symmetry breaking occurs when the target atom is displaced from the beam center. This displacement results in a deflection of the beam and an asymmetric angular distribution of the scattered electrons. Here we propose a concept to control the sign and magnitude of the scattering asymmetry by shaping the incident high-energy electron wave packet in momentum space on the atto- to picosecond time scale. The shaping controls the ultrafast real-space dynamics of the wave packet, shifting the balance between two competing contributions of the impact-parameter-dependent quantum interference and the momentum distribution of the wave packet on the target. We find a strong sensitivity of the elastic scattering on the wave packet properties, an effect that will allow wave-packet and target characterization in ultrafast electron microscopy.

In hybrid circuit QED architectures containing both ancilla qubits and bosonic modes, a controlled beam splitter gate is a powerful resource. It can be used to create (up to a controlled-parity operation) an ancilla-controlled SWAP gate acting on two bosonic modes. This is the essential element required to execute the `swap test' for purity, prepare quantum non-Gaussian entanglement and directly measure nonlinear functionals of quantum states. It also constitutes an important gate for hybrid discrete/continuous-variable quantum computation. We propose a new realization of a hybrid cSWAP utilizing `Kerr-cat' qubits -- anharmonic oscillators subject to strong two-photon driving. The Kerr-cat is used to generate a controlled-phase beam splitter (cPBS) operation. When combined with an ordinary beam splitter one obtains a controlled beam-splitter (cBS) and from this a cSWAP. The strongly biased error channel for the Kerr-cat has phase flips which dominate over bit flips. This yields important benefits for the cSWAP gate which becomes non-destructive and transparent to the dominate error. Our proposal is straightforward to implement and, based on currently existing experimental parameters, should achieve controlled beam-splitter gates with high fidelities comparable to current ordinary beam-splitter operations available in circuit QED.

The Mach--Zehnder interferometer is a powerful device for detecting small phase shifts between two light beams. Simple input states -- such as coherent states or single photons -- can reach the standard quantum limit of phase estimation while more complicated states can be used to reach Heisenberg scaling; the latter, however, require complex states at the input of the interferometer which are difficult to prepare. The quest for highly sensitive phase estimation therefore calls for interferometers with nonlinear devices which would make the preparation of these complex states more efficient. Here, we show that the Heisenberg scaling can be recovered with simple input states (including Fock and coherent states) when the linear mirrors in the interferometer are replaced with controlled-swap gates and measurements on ancilla qubits. These swap tests project the input Fock and coherent states onto NOON and entangled coherent states, respectively, leading to improved sensitivity to small phase shifts in one of the interferometer arms. We perform detailed analysis of ancilla errors, showing that biasing the ancilla towards phase flips offers a great advantage, and perform thorough numerical simulations of a possible implementation in circuit quantum electrodynamics. Our results thus present a viable approach to phase estimation approaching Heisenberg-limited sensitivity.

Local actions (actions of a vertex stabiliser on the neighbours of that vertex) have become an important approach to group actions on trees since J. Tits' introduction in 1970 of the independence property (P) and especially since a 2000 paper by M. Burger and Sh. Mozes. This `local-to-global' approach has been critical in the development of the theory of totally disconnected locally compact groups because it allows the construction of nondiscrete group actions on trees while keeping control over the action of a vertex stabiliser, in a way that is not practical under the classical Bass-Serre approach. The majority of constructions of nonlinear nondiscrete locally compact simple groups use (P) and its generalisations. In this article we give a full classification and description of all closed group actions on trees with Tits' independence property (P) using a new coherent theory for local actions that applies to all actions on trees. This theory is a `local action' complement to classical Bass-Serre theory. On the one hand, our theory gives a decomposition of a group acting on a tree into a `local action diagram' (a decorated graph that encodes all `local' information), and on the other hand a construction of a group acting on a tree from a given local action diagram. One can read directly from the local action diagram whether the resulting group has certain properties, like geometric density, compact generation and simplicity.

Using simulated annealing, we find optimal protocols that evolve a simple product state into a three-qubit $W$ state with a Hamiltonian that describes XY coupling and single-qubit gates, and determine the associated quantum speed limit. Applying Pontryagin's minimum principle, we fully characterize the optimal bang-bang protocols. While leakage affects performance, the protocols remain robust to implementation errors and operate well within relaxation and decoherence times. Our findings highlight Pontryagin's principle as a powerful tool for designing pulse shapes that directly link device interactions to specific quantum gates and target states.

We develop a convergence theory for non-monotone approximation schemes for fully nonlinear parabolic partial differential equations. Modern computational methods such as kernel-based collocation, spectral methods, physics-informed neural networks, and deep Galerkin methods are typically non-monotone, since they produce smooth approximate solutions and compute spatial derivatives directly from gradients of the chosen ansatz. Such schemes therefore lie outside the scope of the classical Barles and Souganidis convergence theory. We introduce an abstract framework that replaces strict monotonicity by two pointwise consistency conditions, on the PDE residual and on the terminal mismatch, both directly verifiable for a smooth approximating sequence. The technical key is a max-min representation of the nonlinearity that converts a vanishing classical residual into the viscosity subsolution and supersolution inequalities, and so dispenses with monotonicity in the abstract argument. The framework yields qualitative convergence under standard hypotheses, together with a quantitative error bound for Hamilton-Jacobi-Bellman equations on an unbounded spatial domain, in which the residual on an expanding truncation cylinder is balanced against an exponentially decaying tail term coming from the controlled stochastic differential equation underlying the value function. As a concrete realization, we analyze kernel-based collocation with Wendland radial basis functions, and present numerical experiments on a benchmark Hamilton-Jacobi-Bellman problem in one and two spatial dimensions that confirm the predicted convergence behaviour.

We study $c$-crossing-critical graphs, which are the minimal graphs that require at least $c$ edge-crossings when drawn in the plane. For $c=1$ there are only two such graphs without degree-2 vertices, $K_5$ and $K_{3,3}$, but for any fixed $c>1$ there exist infinitely many $c$-crossing-critical graphs. It has been previously shown that $c$-crossing-critical graphs have bounded path-width and contain only a bounded number of internally disjoint paths between any two vertices. We expand on these results, providing a more detailed description of the structure of crossing-critical graphs. On the way towards this description, we prove a new structural characterisation of plane graphs of bounded path-width. Then we show that every $c$-crossing-critical graph can be obtained from a $c$-crossing-critical graph of bounded size by replicating bounded-size parts that already appear in narrow "bands" or "fans" in the graph. This also gives an algorithm to generate all the $c$-crossing-critical graphs of at most given order $n$ in polynomial time per each generated graph.

Ranking is used for a wide array of problems, most notably information retrieval (search). There are a number of popular approaches to the evaluation of ranking such as Kendall's $τ$, Average Precision, and nDCG. When dealing with problems such as user ranking or recommendation systems, all these measures suffer from various problems, including an inability to deal with elements of the same rank, inconsistent and ambiguous lower bound scores, and an inappropriate cost function. We propose a new measure, rankDCG, that addresses these problems. This is a modification of the popular nDCG algorithm. We provide a number of criteria for any effective ranking algorithm and show that only rankDCG satisfies all of them. Results are presented on constructed and real data sets. We release a publicly available rankDCG evaluation package.

We propose an adaptive proximal gradient method for minimizing the sum of two functions, where one is a simple convex function, and the other belongs to one of the three classes: nonconvex smooth, convex nonsmooth, or convex smooth. The key feature of the method is an adaptive step size that accumulates historical gradient mapping norms in the denominator. Without any modification or knowledge of problem parameters, the method converges across all three problem classes under mild bounded-iterates and bounded-variance assumptions, with rates matching those of the proximal gradient method up to logarithmic factors, in both deterministic and stochastic settings. For the convex setting, we further propose an accelerated variant. It retains a similar near-optimal convergence rate for the nonsmooth case and achieves an improved rate of order $\widetilde{O}\big(1/t^2 + σ/\sqrt{t}\big)$ for the smooth case, which is optimal up to logarithmic factors. Notably, we develop new techniques for controlling the effect of stochastic noise, which are applicable across all three problem classes in the stochastic setting and enable simplified analysis.

The Chow quotient of a projective variety by the action of a complex torus is known to have a very complicated geometry, even in the case of simple varieties, such as rational homogeneous varieties. In this paper we propose an approach in which the geometry of the Chow quotient is encoded in a projective toric variety and a finite subgroup of its birational automorphisms. We then illustrate how to apply our strategy in the case of some particular rational homogeneous varieties.

In piecewise-smooth differential systems, a hyperbolic limit cycle of a subsystem loses its structural stability if it grazes the switching manifold at a tangent point. Such a cycle is called a grazing loop and in this paper we investigate its bifurcations for arbitrary tangent multiplicity. For the low-multiplicity tangency, the recurrences are comprehensively captured by a functional perturbation with two parameters in previous publications, where the parameters characterize the recurrences near the tangent point and the limit cycle respectively. However, for high-multiplicity tangency, these parameters fail to capture the recurrences and thus, Poincare return maps can not be defined as usual. To address these challenges, we construct a functional perturbation with functions to clarify the recurrences and simultaneously, propose a localization method to make these two recurrences equivalent. We finally establish a quantitative relationship between the multiplicity of tangency and the numbers of crossing limit cycles, sliding loops bifurcating from the grazing loop and the number of tangent points on these sliding loops.

Large spectroscopic and astrometric surveys have revealed complex wave-like features in the Milky Way disk, suggesting that its kinematic and chemical structures are shaped by time-dependent perturbations. Recent studies have reported oscillatory patterns in the Rg-Vphi-VR space, hinting at a possible structural transition in the outer disk. We aim to characterise the transition between the inner and outer Galactic thin disk and to investigate whether radial corrugations can provide a plausible physical interpretation of the observed features. We analysed two large stellar samples from LAMOST DR8 and Gaia DR3, combining spatial, kinematic, and chemical diagnostics. A simplified corrugation model consisting of two radial waves propagating in opposite directions was constructed and fitted to the observed VR pattern. We further validated the model using N-body simulations. Both LAMOST and Gaia samples reproduce the previously reported wave-like pattern in the Rg-Vphi-VR plane. We identify a clear transition between the inner and outer disks via the variations in rotational velocity and metallicities. The corrugation model naturally reproduces the periodic variation of VR with galactocentric radius, and the superposition of the inward and outward propagating modes gives rise to a comparable oscillatory pattern in both observations and simulations. Our modelling suggests that radial corrugations can provide a plausible interpretation of the observed kinematic signatures. The results highlight the complex, multi-perturber nature of the Galactic disk and motivate further investigation with upcoming surveys.

Conserved quantities increasingly underpin the inference of physical models. Recently new conserved quantities have been found in this context, that currently lack an interpretation. Here, we show that irreversible reactions in CRNs and Markov Chains lead to emergent conservation laws and broken cycles. Linearly dependent currents - characterized by the "co-production index" - arise due to irreversible reactions. We derive a law relating conserved quantities, broken cycles, and co-production. This resolves a recent conundrum posed by a machine-discovered candidate for a non-integer conservation law. Our findings introduce heretofore overlooked extensions to a widely used index law for CRNs and Markov Chains that undercounts conservation laws. This furnishes new tools and immediate applications for the inference and analysis of models based on conservation laws.

Grid-following inverters have been widely adopted as a grid interface for renewable energy, and ensuring their small-signal and large-signal stability is critical to modern power systems. Their large-signal, or transient, stability is a significant challenge to analyze because of the interaction of the phase-locked loop (PLL), which must maintain synchronism with various outer-loop controllers. Simple analysis in which outer-loop controllers are idealized is insufficient, and the interactions between the nonlinear dynamics of the PLL and the dynamics of the DC-link voltage control (DVC), as well as the AC terminal voltage control (TVC) when present, must be considered. An asymptotic analysis approach, termed the bandwidth separation method, is proposed. This method enables simplification and order reduction of the original differential equations when sufficient bandwidth separation exists. Through this method, the interaction between the DVC and PLL is explicitly characterized, revealing that such interaction degrades system stability and shrinks the stability region. The analysis also indicates that voltage instability, rather than PLL loss of synchronization alone, is often the root cause of transient instability. Optimal bandwidth configurations for the PLL and DVC are identified under various grid fault conditions: a larger PLL bandwidth improves resilience to phase-jump faults, while a larger DVC bandwidth enhances tolerance to power fluctuations. In addition, the influence of the TVC loop is analyzed, showing that a high TVC bandwidth can mitigate the destabilizing effects of PLL-DVC interaction and further improve transient stability. All analytical findings are validated through hardware-in-the-loop (HIL) experiments.

Integrating non-probability samples into finite-population inference typically requires modeling unknown selection probabilities under a missing-at-random (MAR) assumption that is difficult to verify. We propose a design-based alternative in which the non-probability sample is treated as a fully observed certainty stratum and a probability sample is drawn only from the complementary, previously unsampled units. Within this sequential framework, we develop two generalized regression estimators: one fitting the outcome model separately in the complementary stratum, the other pooling both samples; we make two distinct contributions. First, both estimators are design-consistent and admit consistent variance estimators with no assumption whatsoever on the non-probability selection mechanism, including under not-missing-at-random (NMAR) selection. Second, under a working superpopulation model that holds in both strata, the pilot non-probability sample can be used to construct second-stage inclusion probabilities that achieve Isaki-Fuller asymptotic optimality for the separate estimator; this optimality claim relies on assumptions strictly stronger than MAR, but its failure does not invalidate the consistency results above. A diagnostic test for coefficient homogeneity is proposed to guide the choice between the two estimators. Simulations confirm that the sequential estimators remain essentially unbiased under both MAR and NMAR, while propensity-adjusted competitors can be severely biased under NMAR. Two applications from Lithuanian official statistics illustrate that separate regression is preferable when the pilot stratum and its complement are strongly heterogeneous, whereas combined regression offers a modest efficiency gain when the two strata are similar.

A terahertz absorber with controllable and switchable bandwidth and insensitive to polarization is of great interest. Here, we propose and demonstrate a metasurface absorber with switchable bandwidth based on a phase-change material of vanadium dioxide (VO2) and verify its performance by the finite element method simulations. The metasurface absorber is composed of a hybrid cross fractal as a resonator separated from a gold ground-plane by a polyimide spacer. Switching from narrowband to broadband absorber is achieved via connecting VO2 patches to the gold first-order cross fractal converting the resonator to a third-order cross fractal. In the insulator phase of VO2, the main narrowband absorption occurs at the frequency of 6.05 THz with a 0.99 absorption and a full-width half-maximum (FWHM) of 0.35 THz. Upon insulator-to-metal transition of VO2, the metasurface achieves a broadband absorption with the FWHM of 6.17 THz. The simulations indicate that by controlling the partial phase-transition of VO2, we can tune the bandwidth and absorption level of the absorber. Moreover, the designed absorber is insensitive to polarization due to symmetry and works well for a very wide range of incident angles. In the metallic state of VO2, the absorber has an absorption exceeding 0.5 in the 3.57-8.45 THz frequency range with incident angles up to 65°.

Thrombotic vascular diseases contribute to significant global mortality, yet current therapeutic strategies face persistent challenges including bleeding risks, suboptimal efficiency, and procedural complexity. Here, we report a micro-explosive thermochemical thrombolysis (METCT) therapy via injectable liquid alkali metal (LAM) encapsulated in dimethyl silicone (LAM@oil), which enables prompt, efficient and safe vascular recanalization within an ultrafast timeframe (< 90 seconds). This LAM@oil system effectively disrupts thrombus tissue through a synergistic triple-action mechanism: Mechanical micro-explosions forces, alkaline ablation due to highly localized exothermic chemical reactions, and thermal thrombolysis mediated by elevated temperature. Upon thrombolysis completion, the non-toxic reaction byproducts (sodium and potassium ions) exhibit physiologically biocompatible and metabolizable effects. Critically, the LAM@oil demonstrates significantly higher thrombolytic efficacy compared to clinically available thrombolytic drugs (residual thrombus area percent 10.87%+-7.16% for LAM@oil vs. 80.86%+-13.32% for urokinase), with no associated bleeding risks. This strategy opens a byproduct-free, cost-effective, and high-efficiency alternative to conventional thrombolytics, holding big potential for clinical translation in acute thrombosis management.

Increasing evidence suggests that variability in longitudinal biomarkers, in addition to their mean trajectory, carries prognostic information for time-to-event outcomes. However, standard joint models typically capture only the expected value of the biomarker process, assuming constant residual variability across individuals and time. Fully joint extensions that model within-subject variability exist but are computationally demanding and require dedicated software packages. We propose a flexible two-step approach for incorporating biomarker variability into joint models. First, residuals (or their transformations) from a mixed-effects model are used to derive subject- and time-specific measures of variability. Second, these variability measures are included in a standard joint model, allowing their association with survival to be estimated alongside the mean biomarker trajectory. Our approach can also accommodate multiple biomarkers simultaneously and is readily implemented using existing joint modeling software without custom extensions. Through simulations, we show that our method provides reasonable performance for variability effects across a range of scenarios. We further illustrate our approach using longitudinal data of white blood cell counts from a large phase III glioblastoma trial, demonstrating that both mean levels and variability of hematological markers carry prognostic information for overall survival.

Real-world electric vehicle (EV) energy consumption can deviate by 20-40% from rated values, driven by ambient temperature, traffic congestion, and route characteristics. Existing studies treat these factors in isolation or as static loads, leaving the compound effect of co-varying climate and traffic on HVAC energy unquantified and per-route attribution unavailable. We develop a spatiotemporal simulation framework that couples traffic-aware driving speed, time- and location-specific ambient temperature, and physics-based submodels (cabin HVAC, traction, battery thermal management) at the segment level, paired with a regression-based decomposition that attributes HVAC variability to temperature and trip-duration components on a per-route basis. Applied through a factorial design across seven UK cities and eight radial routes from Manchester, the framework shows total energy varying by 14\% across cities while HVAC energy varies by up to 89\%, making cabin thermal management the primary differentiator under winter conditions. Trip duration, set by traffic and road type, is frequently the dominant driver of HVAC variability: in London, 83\% of above-average HVAC energy is attributable to congestion-extended trip time rather than to temperature. The decomposition yields a closed-form HVAC model from three inputs (ambient temperature, average speed, trip distance), with physically interpretable coefficients and straightforward transfer to other vehicles or regions through three coefficient re-fits. EV range variability is substantially shaped by traffic and road-network characteristics, with implications for route planning, infrastructure design, and energy equity.

We numerically demonstrate a reconfigurable extraordinary terahertz transmission based on a phase-change material of vanadium dioxide (VO2). The proposed hybrid metasurface is composed of an array of subwavelength apertures perforated on a gold film. The holes are partially filled with annular VO2 and gold disks to control the effective aperture area and the modes inside the aperture. Switching between the insulator and the metallic phase of VO2 provides a convenient way to shift the transmission window. We present two designs offering redshift or blueshift of the extraordinary terahertz transmission. Upon phase transition from the insulator to the metallic phase, in the first design, the transmission peak redshifts from 1.02 to 0.82 THz while in the second design the transmission peak blueshifts from 0.71 to 0.77 THz. Furthermore, the transmission level and resonance frequency can be modulated by controlling the partial phase transition of the VO2. The potential applications for the proposed structures are terahertz modulators and reconfigurable filters.

Continuous Dynamic Time Warping (CDTW) is a robust similarity measure for polygonal curves that has recently found a variety of applications. Despite its practical use, not much is known about the algorithmic complexity of computing it in 2D, especially when one requires either an exact solution or strong approximation guarantees. We fill this gap by introducing a $5$-approximation algorithm with running time $O(n^5)$ under the 1-norm. This is the first constant-factor approximation for 2D CDTW with polynomial running time. We extend our algorithm to all polygonal norms on $\mathbb{R}^2$, which we subsequently use in order to achieve a $(5+\varepsilon)$-approximation with time complexity $O(n^5 / \varepsilon^{1/2})$ for CDTW in 2D under any fixed norm. The latter result in particular includes the usual Euclidean 2-norm.

We investigate the first two moments of the critical internal energy $E$ in a weakly disordered two-dimensional Baxter eight-vertex model as a function of the system size $L$, evaluated at the pseudo-critical point. Disorder is introduced via an equivalent representation of the pure eight-vertex model in terms of two ferromagnetic Ising models coupled by a four-spin interaction of strength $g_0$, where the Ising couplings consist of a uniform ferromagnetic part $J>0$ supplemented by weak Gaussian spatial disorder. In the critical regime, the model is formulated in terms of interacting Grassmann-Majorana spinor fields with quartic interactions and analyzed, for small positive $g_0$, using a combination of replica and renormalization-group methods. We also run extensive numerical simulations measuring the critical internal energy. Our results show that its relative variance increases with $L$ and approaches a finite constant as $L \to \infty$ for both $\pm g_0$. Hence, fluctuations remain relevant independently of the sign of $g_0$ (and thus of the specific-heat exponent), implying a lack of self-averaging of both the critical internal energy and the free energy. Consequently, reliable estimates of these quantities require averaging over many disorder realizations. In addition, we numerically confirm earlier predictions concerning the absence of self-averaging of the critical internal energy in the disordered Ising model.

Decoding approaches are widely used in neuroscience and machine learning to compare stimulus representations across neural systems, such as different brain regions, organisms, and deep learning models. Popular methods include decoding (perceptual) manifolds and alignment metrics such as Representational Similarity Analysis (RSA) and Dynamic Similarity Analysis (DSA), where similarity in decoding representations is interpreted as evidence for similar computation. This paper demonstrates a fundamental weakness behind this approach: it is misleading to assume that representational geometry is representative of a neuronal population as a whole, when such representations may actually be shaped by a very small subset of neurons. We show that the complementary encoding paradigm addresses this issue directly: it characterizes how neurons are organized globally in terms of their responses to a set of data, providing insight into how the decoding representation is implemented by neurons within a population. We demonstrate across experiments in biological systems and deep learning models that (i) surprisingly, similar decoding behavior and high representational alignment can arise from small, non-representative subpopulations of neurons; and critically, (ii) alignment metrics are insensitive to encoding manifold topology (how function is distributed across neurons), despite this being a key signature of differentiation across biological systems. A controlled MNIST experiment provides causal evidence: decoding metrics remain unchanged even when encoding topology is causally manipulated via the training loss. Overall, similarity in decoding behavior, as measured by classic alignment metrics, does not imply similarity in function or computation, motivating the use of encoding manifolds as a complementary tool for comparing neural systems.

Motivated by the connection between the Kyle equilibrium with static private signal and the Brownian bridge, we study a much broader class of bridges that allow one to consider more general equilibrium models, for example ones including trading costs and default risk. We show that such bridges are solutions to problems of the Schrödinger-type. Leveraging this connection, we obtain that the equilibria in models with trading costs converge to equilibria in the classical Kyle model.

We explore and discuss corrections to the Newton potential from the quantum effects of conformal matter fields. In this special case, one can compare different approaches, including that of effective quantum gravity and another, based on the conformal (trace) anomaly. The comparison of these two methods is the main focus in the present work. Using the anomaly-induced effective action of gravity requires fixing the quantum vacuum state, similar to what is done in the description of black hole evaporation. In the Boulware vacuum state, we compute the anomaly-induced stress tensor and the first-order correction to the classical gravitational law. The quantum correction to the Newton's potential derived in this way, differs from the result calculated in a way analogous to the effective approach to quantum gravity. The only way to reconcile the two approaches for deriving the leading semiclassical corrections to Newtonian potential is to modify the asymptotic behavior of the average of the energy-momentum tensor in the Boulware vacuum state, as has been recently discussed in the literature.