In this work, we address the need for efficient and formally stable Recurrent Neural Networks (RNNs) in environments with limited computational resources by analyzing the stability of the Minimal Gated Unit (MGU) network, a lightweight alternative to common gated RNNs used in system identification. We derive sufficient parametric conditions for the MGU network's input-to-state stability and incremental input-to-state stability properties. These conditions enable a-posteriori validation of model stability and form the basis for novel stability-promoting training methodologies, including a warm-start of the network's parameters and a projected gradient-based optimization scheme, both of which are presented in this work. Comparative evaluation, including robustness analysis and validation on synthetic and real-world data (i.e., the Silverbox benchmark), demonstrates that the minimal gated unit network successfully combines formal stability guarantees with superior parameter efficiency and faster inference times compared to other state-of-the-art recurrent neural networks, while maintaining comparable and satisfactory accuracy. Notably, the results attained on the Silverbox benchmark illustrate that the stable MGU network effectively captures the system dynamics, whereas other stable RNNs fail to converge to a reliable model.

In this work, we propose a cohomological approach to studying perturbative anomalies in quantum mechanics. The Hamiltonian $\hat{H}$ together with the symmetry generator $\hat{S}$ forms a unitary representation of the two-dimensional Abelian Lie algebra $\mathfrak{g}\cong \mathbb{R}^{2}$ on the Hilbert space $V$. We show that perturbations of such a system are related to the first Chevalley-Eilenberg cohomology group $H^{1}_{CE}(\mathbb{R}^{2},\mathfrak{u}(V))$. In turn, the perturbative anomalies of the symmetry $\hat{S}$ are related to the second cohomology group $H^{2}_{CE}(\mathbb{R}^{2},\mathfrak{u}(V))$.

Classical first-order optimization methods for imaging inverse problems scale poorly with image resolution. Wavelet based multilevel strategies can accelerate convergence under strong blur, but their fixed coarse-to-fine schedules lose effectiveness in moderate-blur or noise-dominated regimes. In this work, we propose an adaptive multiresolution block coordinate Forward-Backward algorithm for image restoration. Multiresolution block selection is driven by the local magnitude of the proximal update via a stochastic non-smooth Gauss-Southwell rule applied to the wavelet decomposition of the image. This adaptive selection strategy dynamically balances updates across scales, emphasizing coarse or fine blocks according to the degradation regime. As a result, the proposed method automatically adapts to varying blur and noise levels without relying on a predefined hierarchical update scheme.

We study copositive and completely positive cones over symmetric cones of rank at least $5$, with particular emphasis on whether these cones are spectrahedral shadows and on the behavior of a sum-of-squares inner-approximation hierarchy. We examine to what extent known results for nonnegative orthants of dimension at least $5$ carry over to general symmetric cones of rank at least $5$. We first prove that neither the copositive nor the completely positive cone over such a symmetric cone is a spectrahedral shadow. We then generalize the Horn matrix to this setting by introducing Horn transformations and analyzing their geometric and algebraic properties. We show that Horn transformations generate exposed rays of copositive cones over symmetric cones and that they evade the zeroth level of the sum-of-squares inner-approximation hierarchy. Finally, we examine the asymptotic exactness of this hierarchy over positive semidefinite cones. In contrast to the $5$-dimensional nonnegative orthant, where the hierarchy is known to recover the entire copositive cone in the limit, we construct instances over positive semidefinite cones of order at least $5$ certifying that the union of all levels remains strictly included in the copositive cone.

We present measurements of the total and differential cross sections for near-threshold J/$ψ$ photoproduction obtained with the CLAS12 detector at the Thomas Jefferson National Accelerator Facility. The results are based on data collected during the Fall 2018 and Spring 2019 running periods, using electron beams with energies of 10.6 and 10.2 GeV, respectively, scattered off a liquid-hydrogen target. Near-threshold J$/ψ$ photoproduction offers a unique sensitivity to the strong interaction in the non-perturbative regime of Quantum Chromodynamics (QCD). The energy dependence of the cross section constrains the underlying J$/ψ$ production mechanisms, including multi-gluon exchange and potential baryonic excitations. Additionally, the $t$-dependence of the differential cross section can be related to the transverse spatial distribution of gluons in the proton, providing critical input for theoretical descriptions of the gluonic structure of the proton. An interpretation of the results in terms of the gluon content of the proton is presented, providing new experimental constraints on QCD-inspired models of the proton structure and the role of gluonic degrees of freedom in hadronic mass generation.

Topological insulator nanowires provide a tunable platform for studying the interplay between disorder, quantum interference, and symmetry-protected transport. Here we investigate quantum transport in disordered topological insulator nanowires threaded by an axial magnetic flux. By computing the conductance as a function of wire length, magnetic flux, chemical potential, and disorder strength, we extract the localization length to characterize the flux-driven delocalization transition near half-integer flux quanta. We find that the localization length diverges with a robust critical exponent $ν=2$, independent of the chemical potential and disorder strength considered here. This exponent differs from that of the integer quantum Hall transition, pointing to distinct scaling behavior. Near integer flux quanta, we further find that the conductance evolves from a weak-localization dip at low chemical potential to a weak anti-localization peak at higher chemical potential, which splits and is eventually suppressed as the system crosses over to the strongly localized regime.

This paper continues the study initiated in \cite{VogXia25} on nonscattering phenomena for inhomogeneous media. We investigate star-shaped domains in $\mathbb{R}^2$ and establish finiteness results for nonscattering wavenumbers associated with Herglotz incident waves of fixed density. First, for ellipses we establish finiteness for all constant contrasts $q\neq 1$, removing the geometric restrictions required in previous work. Second, for admissible star-shaped domains with $q\in(0,1)$, we introduce a flexible interval-wise geometric framework that unifies and generalizes earlier finiteness results. Our results reveal that infinite sequences of nonscattering wavenumbers are tied to exact radial symmetry and cannot persist under admissible geometric perturbations. %extend those of \cite{VogXia25} to the regime $0<q<1$ and reveal a rigidity phenomenon for nonscattering behavior beyond the radially symmetric setting.

Property-based testing (PBT) is a popular technique for establishing confidence in software, where users write properties -- i.e., executable specifications -- that can be checked many times in a loop by a testing framework. In modern PBT frameworks, properties are usually written in shallowly embedded domain-specific languages, and their definition is tightly coupled to the way they are tested. Such frameworks often provide convenient configuration options to customize aspects of the testing process, but users are limited to precisely what library authors had the prescience to allow for when developing the framework; if they want more flexibility, they may need to write a new framework from scratch. We propose a new, deeper language for properties based on a mixed embedding that we call deferred binding abstract syntax, which reifies properties as a data structure and decouples them from the property runners that execute them. We implement this language in Rocq and Racket, leveraging the power of dependent and dynamic types, respectively. Finally, we showcase the flexibility of this new approach by rapidly prototyping a variety of property runners, highlighting domain-specific testing improvements that can be unlocked by more programmable testing.

We investigate periodic integrable Markov models, constructed from set-theoretical solutions of the Yang-Baxter equation. We first focus on the simplest class of solutions, called Lyubashenko solutions. We show that the resulting models are equivalent to some twisted Symmetric Simple Exclusion Process (SSEP), which are usual periodic SSEP models where a twist is added on a bond of the ring. We also provide various possible interpretations for these Markov models. Then, we study the long time dynamics of the twisted SSEP, characterising its different stationary states and counting them. Allowing the twist to vary, we examine the possible transitions between the different stationary states. Finally, we extend our construction of Markov models to set-theoretical solutions that are more general than Lyubashenko solutions and show that such models are not equivalent to a twisted SSEP in general.

The "high-entropy" paradigm has been applied to a central challenge in materials science, the design of new functional materials with enhanced performance for targeted applications, with some notable successes over the last twenty years. However, the immensity of the high-entropy design space remains a major impediment to discovering optimal compositions with tailored microstructures. Suites of high-throughput computational tools have been developed to address this problem, but there is a compelling need to inform these models with fast, economical, non-destructive, and versatile experimental guidance. In this work, we demonstrate mechanical resonance measurements can fulfill this need. Mechanical resonance measurements enable the rapid, non-destructive assessment of materials created by novel syntheses and/or processes and provide high-accuracy determinations of elastic constants to directly benchmark models. We exemplify these capabilities on W-Ta-Cr-V-Hf and Mo-Nb-Ti-V-Zr refractory high-entropy alloys and suggest methodologies for the wider adoption and application of mechanical resonance measurements.

Critical quantum field theories occupy a central position in modern theoretical physics for their inherent universality stemming from long-range correlations. As an example, the Tomonaga-Luttinger liquid (TLL) describes a wealth of one-dimensional quantum systems at low temperatures. Its behavior is deeply rooted in the emergence of an effective Fermi sea, leading to power-law correlations and Friedel oscillations. A promising direction to realize systems exhibiting novel universal behavior beyond TLL is through the generalization of the underlying Fermi sea. In this Letter, we show that fractional Fermi seas with reduced occupancy arise in an integrable Bose gas driven out of equilibrium by cyclic changes in interactions from repulsive to attractive. The correlation functions feature signatures of criticality incompatible with a conventional TLL, suggesting a novel critical phase. Our predictions, based on Generalized Hydrodynamics, are directly relevant to cold atoms.

A mixed quantum state can be taken as capturing an unspecified form of ignorance; or as describing the lack of knowledge about the true pure state of the system ("proper mixture"); or as arising from entanglement with another system that has been disregarded ("improper mixture"). These different views yield identical density matrices and therefore identical predictions for future measurements. But when used as prior beliefs for inferring the past state from later observations ("retrodiction"), they lead to different updated beliefs. This is a purely quantum feature of Bayesian agency. Based on this observation, we establish a framework for retrodicting on any quantum belief and we prove a necessary and sufficient condition for the equivalence of beliefs. We also illustrate how these differences have operational consequences in quantum state recovery.

Population-adjusted indirect comparisons (PAICs) are widely used to synthesize evidence when randomized controlled trials enroll different patient populations and head-to-head comparisons are unavailable. Although PAICs adjust for observed population differences across trials, adjustment alone does not ensure transportability of estimated effects to decision-relevant populations for health technology assessment (HTA). We examine and formalize transportability in PAICs from an estimand-based perspective. We distinguish conditional and marginal treatment effect estimands and show how transportability depends on effect modification, collapsibility, and alignment between the scale of effect modification and the effect measure. Using illustrative examples, we demonstrate that even when effect modifiers are shared across treatments, marginal effects are generally population-dependent for commonly used non-collapsible measures, including hazard ratios and odds ratios. Conversely, collapsible and conditional effects defined on the linear predictor scale exhibit more favorable transportability properties. We further show that pairwise PAIC approaches typically identify effects defined in the comparator population and that applying these estimates to other populations entails an additional, often implicit, transport step requiring further assumptions. This has direct implications for HTA, where PAIC-derived effects are routinely applied within cost-effectiveness and decision models defined for different target populations. Our results clarify when applying PAIC-derived treatment effects to desired target populations is justified, when doing so requires additional assumptions, and when results should instead be interpreted as population-specific rather than decision-relevant, supporting more transparent and principled use of indirect evidence in HTA and related decision-making contexts.

A two-parameter class of higher-dimensional, strongly coupled, confining field theories in the presence of magnetic fluxes for two Abelian gauge groups admits a top-down, holographic dual description. The corresponding two-parameter family of regular background solutions of the classical equations of maximal supergravity in seven dimensions descends from maximal supergravity in eleven dimensions. We study the global and local stability properties of these solutions. We identify lines of zero-temperature first-order phase transitions, describing a polygon (a square) in the space of parameters, identified with the two fluxes. The transition separates the family of gravity solutions dual to confining theories, inside the polygon, from those outside, in which the field theory is realised in a conformal phase. In the spectrum of fluctuations of the supergravity equations, interpreted as bound states of the dual, confining field theories, we find no evidence of local instabilities (tachyons). Over a significant portion of parameter space, that extends far away from the proximity to the transition, we identify an approximate dilaton, the mass of which is one order of magnitude smaller than the scale set by confinement. Our findings complement those emerging in other holographic models discussed in the literature, in which either the dilaton mass is only mildly lower than the confinement scale (when approaching a first-order transitions), or parametrically suppressed (when reaching the proximity to a second-order one).

In our previous paper with Tudor Pădurariu, we introduced the notion of limit categories for moduli stacks of Higgs bundles and formulated the Dolbeault geometric Langlands correspondence. These limit categories are expected to provide an effective ``classical limit'' of the categories of D-modules on the moduli stack of bundles, and our formulation links categorical Donaldson-Thomas theory with the geometric Langlands correspondence. In this paper, we prove the above Dolbeault geometric Langlands correspondence for $\mathrm{GL}_2$ over the locus in the Hitchin base where the spectral curves are reduced. This is the first non-trivial case in which the relevant moduli stacks are not quasi-compact, and the use of limit categories is essential to the formulation and proof of the correspondence. Our approach also outlines a strategy for proving the correspondence in greater generality and explains the current obstructions to such an extension.

We present Doberman (Detector OBsERving and Monitoring ApplicatioN), a lightweight, modular, and open-source slow control system designed for small-to medium-scale physics experiments. Doberman addresses the gap between heavyweight industrial SCADA frameworks and ad hoc laboratory solutions by providing a flexible software architecture that supports heterogeneous instrumentation, distributed deployment, automated control, and robust alarm handling. The web-based graphical user interface Doberview provides live and continuously updated visualization, configuration, and control of the entire experiment, supporting both routine operation and rapid response to exceptional conditions. Doberman has been deployed and validated in multiple experimental setups, ranging from a remotely operated underground gamma-ray spectrometer to a large, highly instrumented liquid xenon test facility with several hundred monitored quantities. Doberman and Doberview are released under permissive open-source licenses, and the software, documentation, and example device integrations are publicly available.

Using a new approach based on Galois theory, we study subvarieties of complex representations of reductive groups which satisfy restriction properties on their invariant rings and function fields, along the lines of the Chevalley restriction theorem. For a certain well-behaved class of representations, we explicitly parametrize candidates for these restriction properties and explain a technique to understand their deformations in complex families. We also give algebraic and geometric characterizations of the Chevalley restriction property which clarify how this perspective connects back to previous orbit-theoretic approaches. Finally, we utilize these restriction properties to prove explicit formulas for period integrals of some Calabi-Yau families. The key insight is that the restriction property on function fields can be leveraged to locally interpolate between the algebraic and analytic settings. Using this technique, we lift hypergeometric period formulas from subfamilies to obtain novel explicit formulas for periods of Calabi-Yau double covers of projective spaces and elliptic curves in $\mathbb{P}^2$, expressed in terms of invariant functions on their parameter spaces.

Quantum Electrodynamics predicts that the vacuum must behave as a nonlinear optical medium: the vacuum optical index should increase when it is stressed by intense electromagnetic fields. The DeLLight (Deflection of Light by Light) project aims to measure it by using intense and ultra-short laser pulses. The experiment uses a Sagnac interferometer to amplify the tiny deflection signal of a low-intensity probe pulse crossing the vacuum refractive-index gradient produced by an external high-intensity pump pulse. The measurement of the amplified signal by a CCD camera requires a high spatial resolution, which is limited by the ultimate quantum noise of the CCD. However, interferometric phase noise induced by the mechanical vibrations of the interferometer is also amplified and degrades spatial resolution. To overcome this, we propose a new method named High-Frequency Phase Noise Suppression (HFPNS), based on the addition of a delayed replica (5 ns) of the probe pulse. The delayed pulse, which is not affected by the pump but is subject to the same vibration noise, enables offline subtraction of correlated phase noise. In this work, we present an experimental proof-of-concept on a prototype interferometer operating with a limited amplification factor ($\mathcal{A}\simeq25$), about 10 times smaller than the required value of the final experiment. We have succeeded in reducing phase noise by a factor of 40, resulting in a residual noise level 2.3 times higher than the expected quantum noise. The residual noise is linked to delay-line instabilities and incident beam pointing fluctuations present during these tests. This result validates HFPNS as a robust method for future quantum-noise-limited interferometric measurements of vacuum optical nonlinearity, though additional stabilization and higher interferometric amplification are still needed.

We introduce and study a class of Hopf algebras $H(G, χ, η, b, c, β)$ which are two-step Ore extensions of a group algebra $\mathbb{K}[G]$. This construction unifies and generalizes some known families of Hopf algebras such as generalized Taft algebras and Hopf algebras related to $\mathfrak{sl}_2$ constructed by Wang, Wu, and Tan. We analyze the ring theoretical properties of these algebras and classify all finite dimensional simple modules over them. We also consider the tensor products of simple modules in the zero derivation case.

Let $Ω$ be a bounded open set of class $C^{1,1}$ in $\mathbb{R}^N$ and $s\in(\frac{1}{2}, 1)$. We study a family of fractional Hardy-type inequalities \begin{equation} \frac{c_{N,s}}{2}\displaystyle\iint_{Ω\timesΩ}\frac{(u(x)-u(y))^2}{|x-y|^{N+2s}}\ dxdy-\displaystyleλ\int_Ωu^2\ dx\geq C\displaystyle\int_Ω\frac{u^2}{δ^{2s}}\ dx,~~~\quad\forallλ\in\mathbb{R},~~~~~~~(0.1) \end{equation} with $u\in C_c^\infty(Ω)$ and $C=C(Ω,s,N,λ)>0$. We show that the best constant in $(0.1)$ is achieved if and only if $λ>λ^*(s,Ω)$, for some $λ^*(s,Ω)\in\mathbb{R}$. As a by-product, we derive in particular that the best constant in Hardy inequality $μ_{N,s}(Ω)$ is achieved if and only if $μ_{N,s}(Ω)<\mathfrak{h}_{N,s}$, with $\mathfrak{h}_{N,s}$ being the best constant for the fractional Hardy inequality in the half space. Moreover, if $Ω$ is a convex open set, we obtain a lower bound for $λ^*(s,Ω)$ in terms of the volume of $Ω$. Specifically, we prove that $λ^*(s,Ω)\geq a(N,s)|Ω|^{-\frac{2s}{N}}$ with an explicit constant $a(N,s)>0$. Finally, for bounded $C^{1,1}$ domains, we prove that, for $s$ sufficiently close to $\frac{1}{2}$, the optimal Hardy constant is independent of both the geometry and the topology of $Ω$. More precisely, we establish that $μ_{N,s}(Ω)=\mathfrak{h}_{N,s}$. This behavior is in sharp contrast with the local case, where the topology/geometry of the domain strongly influences the value of the optimal constant, and reveals a new rigidity phenomenon in the nonlocal setting.

Let C_1, C_2, ... be a sequence of separable unital C*-algebras, equipped with faithful tracial states and satisfying a mild condition. Let A be a unital direct limit of one dimensional NCCW complexes, also equipped with a faithful tracial state. Suppose there is a unital trace preserving embedding of A in the Jiang-Su algebra which is an isomorphism on K-theory. (For example, A could be C([0,1]) with Lebesgue measure, or the Jiang-Su algebra itself.) Let D be the infinite reduced free product of the algebras C_n. Then the reduced free product A*D is isomorphic to D. If D is exact and the factors satisfy a blockwise real rank zero condition, then in place of A we can use C(X) for any contractible compact metric space X and any faithful tracial state on C(X). An example consequence is that the reduced free product of infinitely many copies of C([0,1]), with Lebesgue measure, is isomorphic to the reduced free product of infinitely many copies of the Jiang-Su algebra.

It is known that when minimizing smooth Polyak-Łojasiewicz (PL) functions, momentum algorithms cannot significantly improve the convergence bound of gradient descent, contrasting with the acceleration phenomenon occurring in the strongly convex case. To bridge this gap, the literature has proposed strongly quasar-convex functions as an intermediate non-convex class, for which accelerated bounds have been suggested to persist. We show that this is not true in general: the additional structure of strong quasar-convexity does not suffice to guaranty better worst-case bounds for momentum compared to gradient descent. As an alternative, we study PL functions under an aiming condition that measures how well the descent direction points toward a minimizer. This perspective clarifies the geometric ingredient enabling provable acceleration by momentum when minimizing PL functions.

The Landau-Lifshitz-Gilbert (LLG) equation, regarded as a gradient flow with manifold constraint, is the fundamental model describing magnetization dynamics in ferromagnetic materials. It is well known that the normalized tangent plane method is able to simultaneously achieve the non-convex manifold constraint and original energy dissipation. However, the associated computational cost of this numerical approach is exceedingly high. By contrast, the projection method is more straightforward to implement, while it often compromises the inherent energy dissipative property of the continuous model, and the error analysis turns out to be even more challenging. In this work, we first construct a linear and fully discrete finite difference numerical scheme, based on the projection method for the LLG equation, which is capable of simultaneously preserving the non-convex manifold constraint \(|\mathbf{m}| = 1\) and an unconditional original energy dissipation. In the error analysis, the classical theoretical technique becomes ineffective, due to the presence of the nonlinear Laplacian term, which in turn poses a significant challenge. To overcome this subtle difficulty, we carefully rewrite the numerical method in an equivalent weak form, in which a point-wise length preserving feature of the numerical solution plays an essential role. As a result of these estimates in the reformulated weak form, an optimal convergence rate could be theoretically established. In our knowledge, this numerical method is the first linear algorithm that preserves the following combined theoretical properties: (i) point-wise length preservation, (ii) unconditional original energy dissipation, (iii) a theoretical justification of convergence analysis and optimal rate error estimate.

We extend Maxwell electrodynamics with a Chern--Simons coupling to a dark sector top form sourced by domain walls. Cosmic birefringence can arise from a distinct mechanism in which photon polarization is rotated when crossing vacuum interfaces, rather than through adiabatic propagation in a background field. Ultrathin walls induce a finite, frequency-independent polarization rotation generated by an electromagnetic Chern-Simons term localized at the interface. The effect persists even in the absence of ultralight axions or other propagating scalar degrees of freedom. For phenomenologically viable parameters, such walls can generate cosmic microwave background polarization rotation at the level $Δ\vartheta \sim 10^{-3}$ rad, providing a signature of the topological structure of the dark-sector vacuum.

Self-interacting dark matter (SIDM) with a sufficiently large cross section has been shown to naturally produce constant dark matter (DM) cores, as well as core-collapse, at the centers of dwarf halos on cosmic timescales, potentially reducing tensions with observation. Here, we present halos from a new dark matter only (DMO) cosmological (SIDM) simulation: Ms.Marvel DMO with a velocity-dependent self-interaction cross section with $σ/m_\text{max} = 50$ cm$^2$/g at $v_\text{max} = 35$ km/s. We compare these to the CDM suite of Storm simulations including both DMO and dark matter + hydrodynamics runs, in order to test core-formation (and core-collapse) across different dark matter models. We show that Ms.Marvel DMO can reproduce core slopes consistent with observations of isolated dwarf galaxies and more massive ($\text{M}_{vir} \gtrsim 10^{10} M_{\odot}$) CDM dwarf halos that include stellar feedback from the matched CDM run (Storm CDM+baryons). We identify nine Ms.Marvel SIDM DMO halos in the core-collapse phase of gravothermal evolution with halo masses below $2\times 10^9 M_{\odot}$. We find that using core slope to measure the core-collapse timescales of Ms.Marvel DMO halos agrees well with predicted collapse times estimated with the parametric model for SIDM halos introduced by Yang et al.(2023). Additionally, compared to central density, core slope is less sensitive to both the radius of measurement and halo merger history. These results indicate that the slope of the inner DM density profile more cleanly differentiates core-collapsed versus core-forming halos than central density amplitude.

Blazars are a class of active galactic nuclei, supermassive black holes located at the centres of distant galaxies characterised by strong emission across the entire electromagnetic spectrum, from radio waves to gamma rays. Their relativistic jets, closely aligned to the line of sight from Earth, are a rich and complex environment, characterised by the presence of strong magnetic fields over parsec-scale lengths. Owing to their cosmological distance from Earth, these sources serve as ideal targets to probe non-standard gamma-ray propagation. In particular, axion-like particles (ALPs) could be detected through their coupling to photons, which enables ALP-photon conversions in external magnetic fields, leading to distinct signatures in the blazars' gamma-ray spectra. In this work, we estimate the potential of the Cherenkov Telescope Array Observatory (CTAO) to constrain the ALP parameter space by simulating observations of two bright blazars, Mrk 501 and PKS 2155$-$304. We obtain projected $2σ$ exclusion regions, demonstrating that CTAO will be able to consistently improve present limits thanks to its greater energy resolution and point-source sensitivity with respect to present ground-based gamma-ray telescopes. In addition to the standard statistical technique based on the likelihood ratio test, we further demonstrate the application of a new method based on machine learning classifiers, which may help in reducing the effect of systematic model-dependent uncertainties in future ALP searches.

Importance sampling is a Monte Carlo method which designs estimators of expectations under a target distribution using weighted samples from a proposal distribution. When the target distribution is complex, such as multimodal distributions in highdimensional spaces, the efficiency of importance sampling critically depends on the choice of the proposal distribution. In this paper, we propose a novel adaptive scheme for the construction of efficient proposal distributions. Our algorithm promotes efficient exploration of the target distribution by combining global sampling mechanisms with a delayed weighting procedure. The proposed weighting mechanism plays a key role by enabling rapid resampling in regions where the proposal distribution is poorly adapted to the target. Our sampling algorithm is shown to be geometrically convergent under mild assumptions and is illustrated through various numerical experiments.

Inverse design has enabled the systematic design of ultra-compact and high-performance nanophotonic components. Yet enforcing foundry design rules during inverse design remains a major challenge, as optimized devices frequently violate constraints on minimum feature size and spacing. Existing fabrication-constrained approaches typically rely on penalty terms, projection filters, or heuristic binarization schedules, which restrict the accessible design space, require extensive hyperparameter tuning, and often fail to guarantee compliance throughout the optimization trajectory. Here, we introduce a framework for nanophotonic inverse design with intrinsic enforcement of design rules through a generative reparameterization of the design space, restricting optimization to a learned manifold of DRC-compliant geometries. We validate this paradigm by designing representative silicon photonic components including broadband power splitters, spectral duplexers, and mode converters operating across the 1,500-1,600 nm band for both electron-beam lithography and photolithography platforms. Across all devices, the manifold-based formulation reaches state-of-the-art performance metrics with over a 5-fold reduction in computational cost compared to pixel-based representations, while ensuring fabrication-compatible geometries throughout the entire design process. By treating fabrication constraints as a fundamental property of the design representation rather than an external penalty, this work establishes a direct pathway toward broadly applicable, platform-agnostic, and intrinsically DRC-compliant nanophotonics.

We investigate nonequilibrium phase coexistence associated with a first-order phase transition by numerically studying a one-dimensional Hamiltonian-Potts model with fractional spatial derivatives. The fractional derivative is introduced so as to reproduce the low-wave-number density of states of the standard two-dimensional model, allowing phase coexistence to occur in a minimal one-dimensional setting under steady heat conduction. By imposing a constant heat flux through boundary heat baths, we observe the stable coexistence of ordered and disordered phases separated by a stationary interface. We find that the temperature at the interface systematically deviates from the equilibrium transition temperature, demonstrating a clear violation of the local equilibrium description. This deviation indicates that equilibrium metastable states can be stabilized and controlled by a steady heat current. Furthermore, the interface temperature obtained in our simulations is in quantitative agreement with the prediction of global thermodynamics for nonequilibrium steady states. These results confirm that the breakdown of local equilibrium and the stabilization of metastable states are intrinsic features of nonequilibrium first-order phase transitions, independent of spatial dimensionality. Our study thus provides a minimal and controlled numerical model for exploring the fundamental limits of thermodynamic descriptions in nonequilibrium steady states.

We present a systematic framework for the maximally-transcendental part of planar QCD scattering amplitudes and perform the first bootstrap computation of six-gluon MHV amplitudes in massless QCD at the symbol level. By analyzing the maximal weight projection of amplitudes at the integrand level, we relate their maximally-transcendental parts to prescriptive unitarity integrals. This reveals a novel analytic structure: the prefactors multiplying the functions of maximal transcendentality are identified with the four-dimensional leading singularities of the theory. As a consequence, these prefactors admit a complete classification and can be computed using on-shell diagrams, a formalism originally developed in $\mathcal{N}{=}4$ super Yang-Mills theory. As a concrete application, we determine the two-loop prefactors for planar MHV gluon amplitudes at arbitrary multiplicity. Combining these prefactors with recent advances in the planar two-loop six-point function space and explicit six-point prescriptive-unitarity input, we construct a complete symbol ansatz and uniquely fix the maximally-transcendental part of the two-loop six-gluon MHV QCD amplitudes by imposing physical constraints. The resulting symbols are expressible in a reduced 137-letter alphabet, suggesting that this alphabet is complete for two-loop six-point massless MHV scattering. We also discuss the implications for multi-collinear splitting and multi-soft functions.