In this paper, we develop the groundwork for a graph theoretic toy model of supersymmetric quantum mechanics. Using discrete Witten-Morse theory, we demonstrate that finite graphs have a natural supersymmetric structure and use this structure to incorporate supersymmetry into an existing model of graph quantum mechanics. We prove that although key characteristics of continuum supersymmetric systems are preserved on finite unweighted graphs, supersymmetry cannot be spontaneously broken. Finally, we prove new results about the behavior of supersymmetric graph quantum systems under edge rewiring.

We present a general, constructive procedure to find the basis for tensors of arbitrary order subject to linear constraints by transforming the problem to that of finding the nullspace of a linear operator. The proposed method utilizes standard numerical linear algebra techniques that are highly optimized and well-behaved. Our primary applications are in mechanics where modulus tensors and so-called structure tensors can be used to characterize anisotropy of functional dependencies on other inputs such as strain. Like modulus tensors, structure tensors are defined by their invariance to transformations by symmetry group generators but have more general applicability. The fully automated method is an alternative to classical, more intuition-reliant methods such as the Pipkin-Rivlin polynomial integrity basis construction. We demonstrate the utility of the procedure by: (a) enumerating elastic modulus tensors for common symmetries, and (b) finding the lowest-order structure tensors that can represent all common point groups/crystal classes. Furthermore, we employ these results in two calibration problems using neural network models following classical function representation theory: (a) learning the symmetry class and orientation of a hyperelastic material given stress-strain data, and (b) representing strain-dependent anisotropy of the stress response of a soft matrix-stiff fiber composite in a sequence of uniaxial loadings. These two examples demonstrate the utility of the method in model selection and calibration by: (a) determining structural tensors of a selected order across multiple symmetry groups, and (b) determining a basis for a given group that allows the characterization of all subgroups. Using a common order in both cases allows sparse regression to operate on a common function representation to select the best-fit symmetry group for the data.

In this paper we consider three-dimensional Schrödinger operators with a simple threshold eigenvalue. We show, under certain assumptions, that when a small magnetic field is introduced, this eigenvalue turns into a resonance in the time-dependent sense. We find the leading term in the asymptotic expansion of the imaginary part of the resonance and discuss the principal differences with respect to resonances induced by weak electric fields obtained previously in the literature.

In this paper, we provide a rigorous proof of the convergence, in the Cauchy principal value sense, of Feynman graph integrals of holomorphic quantum field theories on closed real analytic Kähler manifolds. In the special case of Calabi-Yau manifolds, these graph integrals give rise to new invariants of Calabi-Yau metrics.

We develop a systematic framework for the model reduction of multivariate geometric Brownian motions, a fundamental class of stochastic processes with broad applications in mathematical finance, population biology, and statistical physics. Our approach highlights the interplay between the method of invariant manifolds and the adiabatic elimination procedure in deriving reduced equations in closed form for the deterministic part of the dynamics. An extended formulation of the fluctuation-dissipation theorem is also used to characterize the stochastic component of the reduced description. As a concrete application, we apply our reduction scheme to a geometric Brownian motion arising from a two-state quantum system, showing that the reduced dynamics accurately captures the localization properties of the original model while significantly simplifying the analysis.

In this paper, we are concerned with the following noncommutative Painlevé II equation \begin{equation*} \mathbf{D}^2 \beta_1 = 4\mathbf{s} \beta_1 +4 \beta_1 \mathbf{s} +8 \beta_1^3, \end{equation*} where $\beta_1=\beta_1(\vec{s})$ is an $n \times n$ matrix-valued function of $\vec{s}=(s_1,\ldots,s_n)$, $\mathbf{s}=\diag(s_1,\ldots,s_n)$ and $\mathbf{D}=\sum_{j=1}^n\frac{\partial}{\partial s_j}$. If $n=1$, it reduces to the classical Painlevé II equation up to a scaling. Given an arbitrary $n \times n$ constant matrix $C=\left(c_{j k}\right)_{j, k=1}^n$, a remarkable result due to Bertola and Cafasso asserts that there exists a unique solution $\beta_1(\vec{s})=\beta_1(\vec{s};C)$ of the noncommutative PII equation such that its $(k,l)$-th entry behaves like $-c_{kl} \Ai (s_k+s_l)$ as $S= \frac{1}{n}\sum_{i=1}^n s_j\to+\infty$, where $\Ai$ stands for the standard Airy function. For a class of structured matrices $C$, we establish asymptotics of the associated solutions as $S \to -\infty$, which particularly include the so-called connection formulas. In the present setting, it comes out that the solution exhibits a hybrid behavior in the sense that each entry corresponds to either an extension of the Hastings-McLeod solution or an extension of the Ablowitz-Segur solution for the PII equation. It is worthwhile to emphasize the asymptotics of the $(k,l)$-th entry as $S \to -\infty$ cannot be deduced solely from its behavior as $S \to +\infty$ in general, which actually also depends on the positive infinity asymptotics of the $(l,k)$-th entry. This new and intriguing phenomenon disappears in the scalar case.

A family of discrete Schrödinger equations with imaginary potentials $V(x)$ is studied. Inside the domain ${\cal D}$ of unitarity-compatible values of $V(x)$, the reality of all of the bound-state energies survives up to the ``exceptional-point'' (EP) maximally non-Hermitian spectral-degeneracy boundaries $\partial {\cal D}$. The computer-assisted localization of the EP limits is performed showing that the complexity of the task grows quickly with the number $N$ of grid points $x$.

This paper presents a novel and systematic formalism for deriving classical field equations within the framework ofcausal fermion systems, explicitly accounting for higher-order corrections such as quantum effects and those arising from spacetime discreteness. Our method, which also generalizes to non-abelian gauge fields and gravitation, gives a systematic procedure for evaluating the linearized field equations of causal fermion systems. By probing these equations with specific wave functions and employing Taylor expansions, we reformulate them as a family of tensorial equations of increasing rank. We show that, for rank one, this approach recovers the established classical dynamics corresponding to Maxwell's equations. In addition, the approach gives rise to higher-rank tensorial equations, where the second-rank equations are expected to encode the Einstein equations, and higher-rank tensors potentially reveal new physics and systematic corrections.

We present a decomposition of classical potentials into a conservative (gradient) component and a non-conservative component. The latter generalizes the curl component of the force in the three-dimensional case. The force is transformed into a differential $1$-form, known as the work form. This work form is decomposed into an exact (gradient) component and an antiexact component, which in turn generalizes the curl part of the force. The antiexact component is subsequently decomposed using the Frobenius theorem. This local decomposition is a useful tool for identifying the specific components of classical potentials.

We compute the joint distribution of two consecutive eigenphase spacings and their ratio for Haar-distributed $\mathrm{U}(N)$ matrices (the circular unitary ensemble) using our framework for Jánossy densities in random matrix theory, formulated via the Tracy-Widom system of nonlinear PDEs. Our result shows that the leading finite-$N$ correction in the gap-ratio distribution relative to the universal sine-kernel limit is of $\mathcal{O}(N^{-4})$, reflecting a nontrivial cancellation of the $\mathcal{O}(N^{-2})$ part present in the joint distributions of consecutive spacings. This finding suggests the potential to extract subtle finite-size corrections from the energy spectra of quantum-chaotic systems and explains why the deviation of the gap-ratio distribution of the Riemann zeta zeros $\{1/2+i\gamma_n\}, \gamma_n\approx T\gg1$ from the sine-kernel prediction scales as $\left(\log(T/2\pi)\right)^{-3}$.

We generalize $N \leftrightarrow -N$ duality of dimension formulae of $SU(N)$ representations on a (class of) representations with $N$-dependent Young diagrams (which include the adjoint representation), and on eigenvalues of the Casimir operator for those representations. We discuss the consequences for the hypothesis of universal decomposition of powers of the adjoint representation into Casimir subspaces.

It is known that perturbative expansions in powers of the coupling in quantum mechanics (QM) and quantum field theory (QFT) are asymptotic series. This can be useful at weak coupling but fails at strong coupling. In this work, we present two types of series expansions valid at strong coupling. We apply the series to a basic integral as well as a QM path integral containing a quadratic and quartic term with coupling constant $\lambda$. The first series is the usual asymptotic one, where the quartic interaction is expanded in powers of $\lambda$. The second series is an expansion of the quadratic part where the interaction is left alone. This yields an absolutely convergent series in inverse powers of $\lambda$ valid at strong coupling. For the basic integral, we revisit the first series and identify what makes it diverge even though the original integral is finite. We fix the problem and obtain, remarkably, a series in powers of the coupling which is absolutely convergent and valid at strong coupling. We explain how this series avoids Dyson's argument on convergence. We then consider the QM path integral (discretized with time interval divided into $N$ equal segments). As before, the second series is absolutely convergent and we obtain analytical expressions in inverse powers of $\lambda$ for the $n$th order terms by taking functional derivatives of generalized hypergeometric functions. The expressions are functions of $N$ and we work them out explicitly up to third order. The general procedure has been implemented in a Mathematica program that generates the expressions at any order $n$. We present numerical results at strong coupling for different values of $N$ starting at $N=2$. The series matches the exact numerical value for a given $N$ (up to a certain accuracy). The continuum is formally reached when $N\to \infty$ but in practice this can be reached at small $N$.

The Radiative Vlasov-Maxwell equations model the radiative kinetics of collisionless relativistic plasma. In them the Lorentz force is modified by the addition of radiation reaction forces. The radiation forces produce damping of particle energy but these forces are not divergence-free in momentum space, which has an effect of concentration near zero momentum. We prove unconditional global regularity of solutions for a class of Radiative Vlasov-Maxwell equations with large initial data.

We establish the first finite-time blow-up results for generalized 3D stochastic fractional Navier-Stokes equations \[ \Caputo \mathbf{u} = -(\mathbf{u} \cdot \nabla)\mathbf{u} - \nabla p + \nu \fLaplacian \mathbf{u} + I^{1-\beta}[\sigma(\mathbf{u}) \dot{W}], \quad \nabla \cdot \mathbf{u} = 0, \] with dissipation $(-\Delta)^{\alpha/2}$ for $\alpha \in (1, 3/2)$, Caputo time-memory $\partial_t^\beta$, and superlinear noise $|\mathbf{u}|^{1+\gamma}$, proving that for a critical window of memory, $\beta \in (\frac{\alpha}{\alpha+3}, \beta_c(\alpha,\gamma))$, the second moment of the vorticity supremum explodes due to a vortex-stretching-driven renewal inequality. This work reveals that when a fluid's temporal memory, governed by $\partial_t^\beta$, is short enough to permit instability but long enough for that instability to mature, the relentless self-amplification from vortex stretching, when coupled with explosive stochastic kicks from the $|\mathbf{u}|^{1+\gamma}$ noise, guarantees the vorticity will spin up to infinity in finite time.

Anisotropic anomalous dust diffusion in microgravity dusty plasma is investigated using experimental data from the Plasmakristall-4 (PK-4) facility on board the International Space Station. The PK-4 experiment uses video cameras to track individual dust particles, which allows for the collection of large amounts of statistical information on the dust particle positions and velocities. In Part One of this paper, these statistics were used to quantify anomalous dust diffusion caused by anisotropies in the plasma-mediated dust-dust interactions in PK-4. Here we use scaling relations to convert statistical parameters extracted from data into input parameters for a Hamiltonian spectral model. The kinetic energy term of the Hamiltonian (modeling anomalous diffusion) is informed from the dust displacement distribution functions, while the potential energy term (modeling stochasticity) is informed from fluctuations in the dust positions. The spectrum of energy states for each Hamiltonian is studied to assess probability for extended states (i.e., a continuous portion of the spectrum). The spectral model shows that the combination of nonlocality and stochasticity leads to high probability for transport at certain scales in Hilbert space, which coincide with the characteristic spatial scales of dust particle jumps observed in the experiments. Lastly, we discuss how this spectral approach is generalizable to many complex systems, such as electron transport in 2D materials where statistical models are not feasible.

We give the graded anti-pre-Lie algebraic structures on the Witt algebra $\mathcal W$ by the classification of certain indecomposable weight representations of $\mathcal W$. Their classification in the sense of isomorphism is also given. Furthermore, there does not exist a graded anti-pre-Lie algebraic structure on the Virasoro algebra $\mathcal V$ satisfying some natural conditions.

Electromagnetic materials with a uniaxial effective permittivity tensor, characterized by its transverse ($\epsilon_\perp$) and axial ($\epsilon_\parallel$) components, play a central role in the design of advanced photonic and electromagnetic materials including hyperbolic metamaterials, and biological imaging platforms. Tight bounds on the complex effective permittivity of such metamaterials are critical for predicting and optimizing their macroscopic electromagnetic response. While rigorous tight bounds exist for isotropic two-phase composites, corresponding results for uniaxial composites remain relatively unexplored. In this work, we systematically investigate the attainable range of $\epsilon_\perp$ and $\epsilon_\parallel$ in the quasistatic regime for two-phase metamaterials with isotropic homogeneous phases. By analyzing known microgeometries and constructing hierarchical laminates (HLs), we demonstrate that the classical bounds on $\epsilon_\perp$ are not optimal. We conjecture improved bounds based on numerically fitted circular arcs derived from convex hulls of $\epsilon_\perp$ values obtained from HLs, and we identify optimal rank-4 HL structures that achieve all points on the conjectured bounds. Additionally, we quantify the correlation between $\epsilon_\perp$ and $\epsilon_\parallel$ for fixed volume fractions, and propose a design algorithm to construct HL microstructures achieving prescribed values of $\epsilon_\perp$. Leveraging the Cherkaev-Gibiansky transformation and the translation method, we extend recent techniques developed for isotropic composites by Kern-Miller-Milton to derive translation bounds on the uniaxial complex effective permittivity tensor. Finally, bounds on the sensitivity of the effective permittivity tensor of low-loss composites are obtained and their optimality is shown in two-dimensions.

A Hermitian $\Phi^4$ matrix model with a Kontsevich-type kinetic term is studied. It was recently discovered that the partition function of this matrix model satisfies the Schrödinger equation of the $N$-body harmonic oscillator, and that eigenstates of the Virasoro operators can be derived from this partition function. We extend these results and obtain an explicit formula for such eigenstates in terms of the free energy. Furthermore, the Schrödinger equation for the $N$-body harmonic oscillator can also be reformulated in terms of connected correlation functions. The $U(1)^N$-symmetry allows us to derive loop equations.

Let $\mu$ be a measure that samples a subset of a finite ground set, and let $\mathcal{A}_e$ be the event that element $e$ is sampled. The measure $\mu$ is negatively correlated if for any pair of elements $e, f$ one has $\mu(\mathcal{A}_e \cap \mathcal{A}_f) - \mu(\mathcal{A}_e) \mu(\mathcal{A}_f) \leq 0$. A measure is positively correlated if the direction of the inequality is reversed. For the random cluster model on graphs positive correlation between edges is known for $q \geq 1$ due to the FKG inequality, while the negative correlation is only conjectured for $0 \leq q \leq 1$. The main result of this paper is to give a combinatorial formula for the difference in question at $q=1$. Previously, such a formula was known in the uniform spanning tree case, which is a limit of the random cluster model at $q=0$.

In this paper, we are concerned with integrable semi- and fully discrete analogues of the massive Thirring model in light core coordinates. By using the Hirota's bilinear approach and the KP reduction method, we propose both the semi- and fully discrete massive Thirring models and construct their multi-bright soliton solutions.

In this paper we develop a mathematical framework for indiscernibility of quantum states, arguing that, given a set of observables, the ``distinguishable objects'' are the equivalence classes modulo indiscernibility relative to the observables. The structure of the set of distinguishable objects - called the Holevo space - is investigated in detail, and it is shown that the observables admit a natural lift to continuous functions on the Holevo space. The theory is illustrated by several examples where the ``distinguishable objects'' can be described explicitly. Among other things, the Holevo spaces and the lifted functions are described for position measurements on a free particle and for spin measurements in the EPR and Bell experiments.

The maximum energy of the EPR model on a weighted graph is known to be upper-bounded by the sum of the total weight and the value of maximum-weight fractional matching~(MWFM). Recently, Apte, Parekh and Sud~(APS) conjecture that the bound could be strengthened by replacing MWFM with maximum weight matching~(MWM). Here we test this conjecture on a special class of regular graphs that Henning and Yeo constructed many years ago. On this class of regular graphs, MWMs achieve tight lower bounds. As for the maximum energy of the EPR model, we have recently devised a new algorithm called Fractional Entanglement Distribution~(FED) based on quasi-homogeneous fractional matchings, which could achieve rather high accuracy. Applying the FED algorithm to the EPR model on Henning-Yeo graphs, we could thus obtain energy as high as possible and matching value as low as possible, and then make high-precision tests of the APS conjecture. Nevertheless, our numerical results do not show any evidence that the APS conjecture could be violated.

We consider a gas whose each particle is characterised by a pair $(x,v_x)$ with the position $x\in \mathbb R^d$ and the velocity $v_x\in \mathbb R^d_0= \mathbb R^d\setminus \{0\}$. We define Gibbs measures on the cone of vector-valued measures and aim to prove their existence. We introduce the family of probability measures $\mu_\lambda$ on the cone $\mathbb K(\mathbb R^d)$. We define local Hamiltonian and partition functions for a positive, symmetric, bounded and measurable pair potential. Using those above, we define Gibbs's measure as a solution to the Dobrushin-Lanford-Ruelle equation. In particular, we focus on the subset of tempered Gibbs measures. To prove the existence of the Gibbs measure, we show that the subset of tempered Gibbs measures is non-empty and relatively compact.

In this paper we give a detailed proof of the classification of extremal (=massless) unitary highest weight representations in the Neveu Schwarz and Ramond sectors of the big $N=4$ superconformal algebra which can be found in [5]. Our results agree with the general conjectures about classification of unitary highest weight representation of minimal $W$-algebras attached to basic Lie superalgebras formulated in [10], [11], and complete their proof for the big $N=4$ superconformal algebra.

We present a new theory for orbital angular momentum (OAM) generation by chaotic phase surfaces with discrete integer bias distributions. We derive fundamental selection rules that determine which OAM modes can be coherently generated. Our analysis shows that ensemble-averaged OAM exists only when the bias parameter takes integer values that match the discrete OAM eigenspace, creating "allowed" and "forbidden" OAM levels. We derive analytical expressions for the OAM power spectrum and demonstrate universal caling behavior within the allowed manifold. These theoretical predictions are validated by comprehensive Monte Carlo simulations, which confirm the selection rules with a forbidden-level suppression factor exceeding 10^4 and demonstrate the universal scaling with exceptional accuracy.

Understanding how a quantum many-body state is maintained stably as a nonequilibrium steady state is of fundamental and practical importance for exploration and exploitation of open quantum systems. We establish a general equivalent condition for an open quantum many-body system governed by the Gorini-Kossakowski-Sudarshan-Lindblad dynamics under local drive and/or dissipation to have a quantum independent and identically distributed (i.i.d.) steady state. We present a sufficient condition for a system to have a quantum i.i.d. steady state by identifying a set of operators that commute with arbitrary quantum i.i.d. states. In particular, a set of quantum i.i.d. states is found to be an invariant subset of time evolution superoperators for systems that satisfy the sufficient condition. These findings not only identify a class of models with exactly solvable steady states but also lead to a no-go theorem that precludes quantum entanglement and spatial correlations in a broad class of quantum many-body steady states in a dissipative environment.

We argue that a field theory with a $\mathbb Z_N$ $p$-form symmetry generically admits, in addition to a Higgs phase and a 'confining' phase, a Coulomb phase in which the infrared theory contains Abelian $p$-form electrodynamics, similar to the behaviour of Yang-Mills theory coupled to adjoint or fundamental matter. We illustrate our claim with continuum and lattice examples.

The Hubbard model at $U\to\infty$ has recently been shown to have resonant valence bond (RVB) ground states on the corner-sharing sawtooth and pyrochlore lattices in the dilute doping limit of a single vacancy. In an effort to further generalize those results, I study how the ground state is modified when not all corners are shared between two tetrahedra as in the quasi-1D lattices of a pyrochlore stripe, and how to approach the problem in the case of finite doping. Using a non-Abelian version of the flux inequality, the tetrahedron chain is shown to have degenerate RVB-like ground states. The Bethe ansatz (BA) is adapted to solve the sawtooth chain with spinless or spin-polarized fermions and multiple holons, which is the first example of applying BA to a quasi-1D lattice.

This work provides a rigorous and self-contained introduction to numerical methods for Hamiltonian simulation in quantum computing, with a focus on high-order product formulas for efficiently approximating the time evolution of quantum systems. Aimed at students and researchers seeking a clear mathematical treatment, the study begins with the foundational principles of quantum mechanics and quantum computation before presenting the Lie-Trotter product formula and its higher-order generalizations. In particular, Suzuki's recursive method is explored to achieve improved error scaling. Through theoretical analysis and illustrative examples, the advantages and limitations of these techniques are discussed, with an emphasis on their application to $k$-local Hamiltonians and their role in overcoming classical computational bottlenecks. The work concludes with a brief overview of current advances and open challenges in Hamiltonian simulation.

An Ising model with random couplings on a graph is a model of a spin glass. While the mean field case of the Sherrington-Kirkpatrick model is very well studied, the more realistic lattice setting, known as the Edwards-Anderson (EA) model, has witnessed rather limited progress. In (Chatterjee,'23) chaotic properties of the ground state in the EA model were established via the study of the Fourier spectrum of the two-point spin correlation. A natural direction of research concerns fractal properties of the Fourier spectrum in analogy with critical percolation. In particular, numerical findings (Bray, Moore,'87) seem to support the belief that the fractal dimension of the associated spectral sample drawn according to the Fourier spectrum is strictly bigger than one. Towards this, in this note we introduce a percolation-type argument, relying on the construction of ``barriers'', to obtain new probabilistic lower bounds on the size of the spectral sample.

Ferromagnetic exponential random graph models (ERGMs) are nonlinear exponential tilts of Erdős-Rényi models, under which the presence of certain subgraphs such as triangles may be emphasized. These models are mixtures of metastable wells which each behave macroscopically like new Erdős-Rényi models themselves, exhibiting the same laws of large numbers for the overall edge count as well as all subgraph counts. However, the microscopic fluctuations of these quantities remained elusive for some time. Building on a recent breakthrough by Fang, Liu, Shao and Zhao [FLSZ24] driven by Stein's method, we prove quantitative central limit theorems (CLTs) for these quantities and more in metastable wells under ferromagnetic ERGMs. One main novelty of our results is that they apply also in the supercritical (low temperature) regime of parameters, which has previously been relatively unexplored. To accomplish this, we develop a novel probabilistic technique based on the careful analysis of the evolution of relevant quantities under the ERGM Glauber dynamics. Our technique allows us to deliver the main input to the method developed by [FLSZ24], which is the fact that the fluctuations of subgraph counts are driven by those of the overall edge count. This was first shown for the triangle count by Sambale and Sinulis [SS20] in the Dobrushin (very high temperature) regime via functional-analytic methods. We feel our technique clarifies the underlying mechanisms at play, and it also supplies improved bounds on the Wasserstein and Kolmogorov distances between the observables at hand and the limiting Gaussians, as compared to the results of [FLSZ24] in the subcritical (high temperature) regime beyond the Dobrushin regime. Moreover, our technique is flexible enough to also yield quantitative CLTs for vertex degrees and local subgraph counts, which have not appeared before in any parameter regime.

We study a class of nonlinear Ginzburg-Landau SPDEs in infinite-volume. We show that under a weakly asymmetric scaling, their solutions converge to that of the KPZ equation. The key technical innovation is the analysis of a stochastic heat kernel for the SPDE of interest, which allows for a multi-scale "localization" to the compact setting.

In this paper, we describe all Nijenhuis operators on 2-dimensional complex pre-Lie algebras and 3-dimensional complex associative algebras. As an application, using these operators, we obtain solutions of the classical Yang-Baxter equation on the corresponding sub-adjacent Lie algebras.

We consider series over Young diagrams of products of Schur functions $s_{\lambda\cup\lambda}$, marked with ``fat partitions'' $\lambda\cup\lambda$, which appear in matrix models associated with ensembles of symplectic and orthogonal matrices and quaternion Ginibre ensembles. We consider mixed matrix models that also contain complex Ginibre ensembles labeled by graphs and the three ensembles mentioned above. Cases are identified when a series of perturbations in coupling constants turn out to be tau functions of the DKP hierarchy introduced by the Kyoto school. This topic relates matrix models to random partitions - discrete symplectic ensemble and its modifications.

The standard Eliashberg theory of superconductivity is studied, in which the effective electron-electron interactions are mediated by generally dispersive phonons, with Eliashberg spectral function $\alpha^2 F(\omega)\geq 0$ that is $\propto\omega^2$ for small $\omega>0$ and vanishes for large $\omega$. The Eliashberg function also defines the electron-phonon coupling strength $\lambda:= 2 \int_0^\infty\frac{\alpha^2 F(\omega)}{\omega}d\omega$. Setting $\frac{2\alpha^2 F(\omega)}{\omega}d\omega =: \lambda P(d\omega)$, formally defining a probability measure $P(d\omega)$ with compact support, and assuming as usual that the phase transition between normal and superconductivity coincides with the linear stability boundary $\mathscr{S}_{\!c}$ of the normal region against perturbations toward the superconducting region, it is shown that $\mathscr{S}_{\!c}$ is a graph of a function $\Lambda(P,T)$ that is determined by a variational principle: if $(\lambda,P,T)\in\mathscr{S}_{\!c}$, then $\lambda = 1/\mathfrak{k}(P,T)$, where $\mathfrak{k}(P,T)>0$ is the largest eigenvalue of a compact self-adjoint operator $\mathfrak{K}(P,T)$ on $\ell^2$ sequences constructed in the paper. Given $P$, sufficient conditions on $T$ are stated under which the map $T\mapsto \lambda = \Lambda(P,T)$ is invertible. For sufficiently large $\lambda$ this yields: (i) the existence of a critical temperature $T_c$ as function of $\lambda$ and $P$; (ii) a sequence of lower bounds on $T_c(\lambda,P)$ that converges to $T_c(\lambda,P)$. Also obtained is an upper bound on $T_c(\lambda,P)$. It agrees with the asymptotic form $T_c(\lambda,P) \sim C \sqrt{\langle \omega^2\rangle} \sqrt{\lambda}$ valid for $\lambda\sim\infty$, given $P$, though with a constant $C$ that is a factor $\approx 2.034$ larger than the sharp constant. Here, $\langle\omega^2\rangle := \int_0^\infty \omega^2 P(d\omega)$.

We present two involutivity theorems in the context of Poisson quasi-Nijenhuis %(PqN) manifolds. The second one stems from recursion relations that generalize the so called Lenard-Magri relations on a bi-Hamiltonian manifold. We apply these results to the closed (or periodic) Toda lattices of type $A_n^{(1)}$, $C_n^{(1)}$, $A_{2n}^{(2)}$ and, for the ones of type $A^{(1)}_n$, we show how this geometrical setting relates to their bi-Hamiltonian representation and to their recursion relations.

The study of undecidability in problems arising from physics has experienced a renewed interest, mainly in connection with quantum information problems. The goal of this review is to survey this recent development. After a historical introduction, we first explain the necessary results about undecidability in mathematics and computer science. Then we briefly review the first results about undecidability in physics which emerged mostly in the 80s and early 90s. Finally we focus on the most recent contributions, which we divide in two main categories: many body systems and quantum information problems.

A fourth-order Schrödinger equation for the description of charge transport in semiconductors in the ballistic regime is proposed with the inclusion of non-parabolic effects in the dispersion relation in order to go beyond the simple effective mass approximation. Similarly to the standard (second order) Schrödinger equation, the problem is reduced to a finite spatial domain with appropriate transparent boundary conditions to simulate charge transport in a quantum coupler (Lent and Kirkner in J Appl Phys 67:6353, 1990; Ben Abdallah et al. in ZAMP 48:135-155, 1997; Ben Abdallah in J. Math. Phys. 41:4241-4261, 2000), where an active region representing an electron device is coupled to leads which take the role of reservoirs. Some analytical properties are investigated, and a generalized formula for the current is obtained. Numerical results show the main features of the solutions of the new model. In particular, an effect of interference appears due to a richer wave structure than that arising for the second-order Schrödinger equation in the effective mass approximation.

We present a geometric construction of irreversible dynamics on Poisson manifolds that satisfies the axioms of metriplectic mechanics and the GENERIC framework. Our approach relies solely on the underlying Poisson structure and its deformation theory, without requiring any additional metric structure. Specifically, we show that if the second Lichnerowicz-Poisson cohomology group of a Poisson manifold is nontrivial, one can construct a symmetric bracket that generates irreversible dynamics compatible with energy conservation and entropy production. This bracket is derived from a 2-cocycle that deforms the original Poisson structure, thereby modifying the associated Casimir foliation. We illustrate the construction with two finite-dimensional examples, the duals of the Lie algebras of the special Euclidean group SE(2) and the Galilei group SGal(3). These examples demonstrate the applicability of the method in classical mechanics, control theory, and mathematical physics. Our framework naturally extends to infinite-dimensional settings, which are discussed as directions for future work.

We study the continuous model of the localized wave propagation corresponding to the one-dimensional diatomic crystal lattice. From the mathematical point of view the problem can be described in terms of the Cauchy problem with localized initial data for a system of two pseudo-differential equations. We assume two small parameters in this formulation -- the lattice step and the size if the initial perturbation. We construct the asymptotic solution of the continuous Cauchy problem with respect to the size of perturbation. The ratio of the small parameters drastically affects the form of the solution. We consider two situations -- when the size of the perturbation is sufficiently large and when it is comparable with the lattice step. In each situations we provide analytical formulae for the asymptotic solution via Airy function.

This work presents a geometric formulation for transforming nonconservative mechanical Hamiltonian systems and introduces a new method for regularizing and linearizing central force dynamics -- in particular, Kepler and Manev dynamics -- through a projective transformation. The transformation is formulated as a configuration space diffeomorphism (rather than a submersion) that is lifted to a cotangent bundle (phase space) symplectomorphism and used to pullback the original mechanical Hamiltonian system, Riemannian kinetic energy metric, and other key geometric objects. Full linearization of both Kepler and Manev dynamics (in any finite dimension) is achieved by a subsequent conformal scaling of the projectively-transformed Hamiltonian vector field. Two such conformal scalings are given, both achieving linearization. Arbitrary conservative and nonconservative perturbations are included, with closed-form solutions readily obtained in the unperturbed Kepler or Manev cases.

Distinguishability is fundamental to information theory and extends naturally to quantum systems. While quantum state discrimination is well understood, quantum channel discrimination remains challenging due to the dynamic nature of channels and the variety of discrimination strategies. This work advances the understanding of quantum channel discrimination and its fundamental limits. We develop new tools for quantum divergences, including sharper bounds on the quantum hypothesis testing relative entropy and additivity results for channel divergences. We establish a quantum Stein's lemma for memoryless channel discrimination, and link the strong converse property to the asymptotic equipartition property and continuity of divergences. Notably, we prove the equivalence of exponentially strong converse properties under coherent and sequential strategies. We further explore the interplay among operational regimes, discrimination strategies, and channel divergences, deriving exponents in various settings and contributing to a unified framework for channel discrimination. Finally, we recast quantum communication tasks as discrimination problems, uncovering deep connections between channel capacities, channel discrimination, and the mathematical structure of channel divergences. These results bridge two core areas of quantum information theory and offer new insights for future exploration.

With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality and intermittency of the family of generalized Riemann's non-differentiable functions \begin{equation} R_{x_0}(t) = \sum_{n \neq 0} \frac{e^{2\pi i ( n^2 t + n x_0 ) } }{n^2}, \qquad x_0 \in [0,1]. \end{equation} These functions represent, in a certain limit, the trajectory of regular polygonal vortex filaments that evolve according to the binormal flow. When $x_0$ is rational, we show that $R_{x_0}$ is multifractal and intermittent by completely determining the spectrum of singularities of $R_{x_0}$ and computing the $L^p$ norms of its Fourier high-pass filters, which are analogues of structure functions. We prove that $R_{x_0}$ has a multifractal behavior also when $x_0$ is irrational. The proofs rely on a careful design of Diophantine sets that depend on $x_0$, which we study by crucially using the Duffin-Schaeffer theorem and the Mass Transference Principle.

We prove ``effective'' linear response for certain classes of non-uniformly expanding random dynamical systems which are not necessarily composed in an i.i.d manner. In applications, the results are obtained for base maps with a sufficient amount of mixing. The fact that the rates are effective is then applied to obtain the differentiability of the variance in the CLT as a function of the parameter, as well as the annealed linear response. These two applications are beyond the reach of the linear response obtained in the general case, when all the random variables appearing in the bounds are only tempered. We also provide several wide examples of one-dimensional maps satisfying our conditions, as well as some higher-dimensional examples.

We discuss anyon condensation in mixed-state topological order. The phases were recently conjectured to be classified by pre-modular fusion categories. Just like anyon condensation in pure-state topological order, a bootstrap analysis shows condensable anyons are given by connected étale algebras. We explain how to perform generic anyon condensation including non-invertible anyons and successive condensations. Interestingly, some condensations lead to pure-state topological orders. We clarify when this happens. We also compute topological invariants of equivalence classes.

This paper concerns the large time asymptotic behavior of solutions to the free boundary problem of the compressible primitive equations in atmospheric dynamics with physical vacuum. Up to second order of the perturbations of an equilibrium, we have introduced a model of the compressible primitive equations with a specific viscosity and shown that the physical vacuum free boundary problem for this model system has a global-in-time solution converging to an equilibrium exponentially, provided that the initial data is a small perturbation of the equilibrium. More precisely, we introduce a new coordinate system by choosing the enthalpy (the square of sound speed) as the vertical coordinate, and thanks to the hydrostatic balance, the degenerate density at the free boundary admits a representation with separation of variables in the new coordinates. Such a property allows us to establish horizontal derivative estimates without involving the singular vertical derivative of the density profile, which plays a key role in our analysis.

Quantum state exclusion is an operational task that has significance in studying foundational questions related to interpreting quantum theory. In such a task, one is given a system whose state is randomly selected from a finite set, and the goal is to identify a state from the set that is not the true state of the system. An error, i.e., an unsuccessful exclusion, occurs if and only if the state identified is the true state. In this paper, we study the optimal error probability of quantum state exclusion and its error exponent -- the rate at which the error probability decays asymptotically -- from an information-theoretic perspective. Our main finding is a single-letter upper bound on the error exponent of state exclusion given by the multivariate log-Euclidean Chernoff divergence, and we prove that this improves upon the best previously known upper bound. We also extend our analysis to the more complicated task of quantum channel exclusion, and we establish a single-letter and efficiently computable upper bound on its error exponent, even assuming the use of adaptive strategies. We derive both upper bounds, for state and channel exclusion, based on one-shot analysis and formulate them as a type of multivariate divergence measure called a barycentric Chernoff divergence. Moreover, our result on channel exclusion has implications in two important special cases. First, for the special case of two hypotheses, our upper bound provides the first known efficiently computable upper bound on the error exponent of symmetric binary channel discrimination. Second, for the special case of classical channels, we show that our upper bound is achievable by a parallel strategy, thus solving the exact error exponent of classical channel exclusion and generalising a similar result on symmetric binary classical channel discrimination.

We construct a new algebraic linearization of the discrete periodic Toda flow by using Mumford's algebraic description of the Jacobian of a hyperelliptic curve. In particular, the discrete periodic Toda flow can be expressed in terms of the famous Gauß composition law for quadratic forms adapted to the framework of hyperelliptic curves by Cantor. One surprising consequence of our approach is a new integrality property for the discrete periodic Toda flow which leads to a $p$-adic description of the closely related periodic box-ball flow, which has very surprising connections to number theory.

We say of an isolated macroscopic quantum system in a pure state $\psi$ that it is in macroscopic thermal equilibrium (MATE) if $\psi$ lies in or close to a suitable subspace $\mathcal{H}_{eq}$ of Hilbert space. It is known that every initial state $\psi_0$ will eventually reach and stay there most of the time (``thermalize'') if the Hamiltonian is non-degenerate and satisfies the appropriate version of the eigenstate thermalization hypothesis (ETH), i.e., that every eigenvector is in MATE. Tasaki recently proved the ETH for a certain perturbation $H_\theta^{fF}$ of the Hamiltonian $H_0^{fF}$ of $N\gg 1$ free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of $H_0^{fF}$. Here, we first point out that also for degenerate Hamiltonians all $\psi_0$ thermalize if the ETH holds, i.e., if every eigenbasis lies in MATE, and we prove that this is the case for $H_0^{fF}$. Inspired by the fact that there is one eigenbasis of $H_0^{fF}$ for which MATE can be proved more easily than for the others, with smaller error bounds, and also in higher spatial dimensions, we show for any given $H_0$ that the existence of one eigenbasis in MATE implies quite generally that most eigenbases of $H_0$ lie in MATE. We also show that, as a consequence, after adding a small generic perturbation, $H=H_0+\lambda V$ with $\lambda\ll 1$, for most perturbations $V$ the perturbed Hamiltonian $H$ satisfies ETH and all states thermalize.

In order to preserve the leading role of the action principle in formulating all field theories one needs quantum field theory, with the associated BRST symmetry, and Feynman-DeWitt-Faddeev-Popov ghost fields. Such fields result from the fibre-bundle structure of the space of histories, but the physics-oriented literature used them formally because a rigorous theory of measure and integration was lacking. Motivated by this framework, this paper exploits previous work of Gill and Zachary, where the use of Banach spaces for the Feynman integral was proposed. The Henstock-Kurzweil integral is first introduced, because it makes it possible to integrate functions like the exponential of (i times x squared). The Lebesgue measure on R(infinity) is then built and used to define the measure on every separable Hilbert space. The subsequent step is the construction of a new Hilbert space KS2, which contains square-integrable functions on R**n as a continuous dense embedding, and contains both the test functions and their dual, the Schwartz space of distributions, as continuous embeddings. This space allows us to construct the Feynman path integral in a manner that maintains its intuitive and computational advantages. We also extend this space to KS2[H], where H is any separable Banach space. Last, the existence of a unique universal definition of time, tau(h), that we call historical time, is proved. We use tau(h) as the order parameter for our construction of Feynman's time ordered operator calculus, which in turn is used to extend the path integral in order to include all time dependent groups and semigroups with a kernel.

A new ``Percolation with Clustering'' (PWC) model is introduced, where (the probabilities of) site percolation configurations on the leaf set of a binary tree are rewarded exponentially according to a generic function, which measures the degree of clustering in the configuration. Conditions on such ``clustering function'' are given for the existence of a limiting free energy and a wetting transition, namely the existence of a non-trivial percolation parameter threshold above and only above which the set of ``dry'' (open) sites have an asymptotic density. Several examples of clustering functions are given and studied using the general theory. The results here will be used in a sequel paper to study the wetting transition for the discrete Gaussian free field on the tree subject to a hard wall constraint.

Analyzing Heisenberg--Robertson (HR) and Schrödinger uncertainty relations we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of non--commuting observables, $A$ and $B$, is zero, and which differ from those described in the literature. These states are not eigenstates of either the observable $A$ or $B$. The correlation function for these observables in such states is equal to zero. We have also shown that the so--called "sum uncertainty relations" also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: one is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the non--commuting observables in the state under consideration.

We study the ensemble of states generated by performing projective measurements on the output of a random matchgate (or free-fermionic) quantum circuit. We rigorously show that this `projected ensemble' exhibits deep thermalization: For large system sizes, it converges towards a universal ensemble that is uniform over the manifold of Gaussian fermionic states. As well as proving moment-wise convergence of these ensembles, we demonstrate that the full distribution of any physical observable in the projected ensemble is close to its universal form in Wasserstein-1 distance, which we argue is an appropriate and efficiently computable measure of convergence when studying deep thermalization. Using this metric, we also numerically find that local matchgate circuits deeply thermalize after a timescale $t \sim L^2$ set by the diffusive spreading of quantum information. Our work opens up new avenues to experimentally accessible protocols to probe the emergence of quantum statistical mechanics and benchmark quantum simulators.

We considered the vacuum polarization of a quantized charged scalar matter field in the background of a topological defect modeled by a finite-thickness tube with magnetic flux inside. The tube is impenetrable for quantum matter, and a generalized boundary condition of the Robin type is imposed at its surface. We have shown that in the flat space-time, the total induced vacuum energy does not depend on the coupling $(\xi)$ of the scalar field's interaction with the space-time curvature, only for the partial cases of the Dirichlet and Neumann boundary conditions on the tube's edge. However, for generalized Robin boundary conditions, the total induced energy depends on the coupling $\xi$ in flat space-time, at least for negative values of the boundary condition parameter $-\pi/2<\theta<0$.

We discuss a large class of non-relativistic continuum field theories where the Euclidean spatial symmetry of the classical theory is violated in the quantum theory by an Adler-Bell-Jackiw-like anomaly. In particular, the continuous translation symmetry of the classical theory is broken in the quantum theory to a discrete symmetry. Furthermore, that discrete symmetry is extended by an internal symmetry, making it non-Abelian. This presentation streamlines and extends the discussion in [1]. In an Appendix, we present an elementary introduction to 't Hooft and Adler-Bell-Jackiw anomalies using a well-known system.

Kramers escape from a metastable state in the presence of both thermal and quantum fluctuations under strong damping is treated as a thermally activated process in a quantum modified semiclassical potential. Dirac's time-dependent variational method together with the Jackiw-Kerman function is employed to derive the semiclassical potential. Quantum correction is incorporated in the drift potential, and is determined by quasi-stationary conditions and minimal uncertainty relation. The semiclassical rate obtained here is consistent in form with those from the quantum Smoluchowski equations deduced heuristically by modifying the diffusion coefficient using the path-integral method. Unlike approaches using the path-integral, which involves continuation into imaginary time, the approach here is simpler and more easily understood in terms of classical picture.

The Dirichlet lambda function $\lambda(s)$ is defined for $\mathrm{Re}(s) > 1$ by \[ \lambda(s) = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^s}. \] This function was initially studied by Euler on the real line, where he denoted it by $N(s)$. In this paper, by applying the partial fraction decomposition of $\pi \tan(\pi x)$ and explicit evaluations of the integrals \[ \int_0^{\frac{1}{2}} x^{2m-1} \cos(2l\pi x) dx \quad \text{and} \quad \int_0^{\frac{1}{2}} x^{m-1} \log \cos(\pi x) dx, \] for positive integers $l$ and $m$, we derive closed-form expressions for several classes of infinite series involving $\lambda(s)$. We also demonstrate that the values $\lambda(k)$ for even integers $k \geq 2$ arise as constant terms in the Fourier expansions of Eisenstein series associated with the congruence subgroup \[ \Gamma_0(2) := \left\{ \begin{pmatrix} a & b c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) : c \equiv 0 \pmod{2} \right\}. \]

We derive full asymptotic expansions for the Humbert function $\Phi_1$ in different limiting regimes of its variables. Our derivation employs various asymptotic methods and relies on key transformation formulae established by Erdélyi (1940), and Tuan and Kalla (1987). The efficiency of our asymptotic results are also illustrated through two applications: (1) analytic continuations of Saran's function $F_M$, and (2) two limits arising in the study of the $1D$ Glauber-Ising model. Finally, some promising directions for future research are highlighted.

This paper presents an operator-theoretic framework Linear Operator Causality Analysis (LOCA), for analysing causality in linearised dynamical systems, focusing here on fluid flows. We demonstrate that the matrix exponential of the linearised differential equations fundamentally encodes the causal relationships between system modes at any future time. We further develop a global measure of causality that quantifies the presence and extent of global causality across all time horizons. This approach provides a physics-based alternative to statistical and information-theoretic causality measures such as Granger causality and transfer entropy. Unlike these data-driven techniques that infer causality from time-series data, LOCA leverages the linearised governing equations, yielding a more rigorous and interpretable measure of causal interactions. We show that LOCA gives equivalent results to data-driven methods under certain assumptions, and further discuss connections to key system properties such as controllability, observability, and graph-theoretic transitive closure. To complement this operator-based approach, we introduce a data-driven methodology akin to Dynamic Mode Decomposition (DMD) that estimates causal connections directly from time series data by approximating the matrix exponential. LOCA also mitigates common issues in data-driven causality analyses, such as misleading inferences due to correlated variables or state truncation. We demonstrate our method on linearised Couette flow, demonstrating how our framework captures both direct and indirect causal interactions among flow structures. Through this example, we highlight the advantages of our approach, including its robustness to correlation-induced biases and its ability to identify causally significant modes.

We present an effective set of necessary and sufficient criteria for block-positivity of matrices of order $4$ over $\mathbb{C}$. The approach is based on Sturm sequences and quartic polynomial positivity conditions presented in recent literature. The procedure allows us to test whether a given $4\times 4$ complex matrix corresponds to an entanglement witness, and it is exact when the matrix coefficients belong to the rationals, extended by $\mathrm{i}$. The method can be generalized to $\mathcal{H}_2\otimes\mathcal{H}_d$ systems for $d>2$ to provide necessary but not sufficient criterion for block-positivity. We also outline an alternative approach to the problem relying on Gröbner bases.

We present an approach to construct a class of generalized Frobenius manifold structures on the orbit spaces of affine Weyl groups, and prove that their monodromy groups are proper subgroups of the associated affine Weyl groups.

We consider the notions of $L_{\infty}$-, $P_{\infty}$-, and $S_{\infty}$-algebras (including "shifted" versions) in the $\mathbb{Z}_2 \times \mathbb{Z}$-graded setting. We also consider thick (microformal) morphisms and show how they work in such graded context. In particular, we show that a "shifted $S_{\infty}$-thick morphism" (which we introduce here) induces an $L_{\infty}$-morphism of shifted $S_{\infty}$-structures. The same holds for "shifted $P_{\infty}$-thick morphisms" and shifted $P_{\infty}$-structures, respectively.

A significant advancement in nonlinear projection-based model order reduction (PMOR) is presented through a highly effective methodology. This methodology employs Gaussian process regression (GPR) and radial basis function (RBF) interpolation for closure error modeling in the latent space, offering notable gains in efficiency and expanding the scope of PMOR. Moving beyond the limitations of deep artificial neural networks (ANNs), previously used for this task, this approach provides crucial advantages in terms of interpretability and a reduced demand for extensive training data. The capabilities of GPR and RBFs are showcased in two demanding applications: a two-dimensional parametric inviscid Burgers problem, featuring propagating shocks across the entire computational domain, and a complex three-dimensional turbulent flow simulation around an Ahmed body. The results demonstrate that this innovative approach preserves accuracy and achieves substantial improvements in efficiency and interpretability when contrasted with traditional PMOR and ANN-based closure modeling.

In this review and methodological article we discuss the main ideas of the integral invariants theory. This theory was originated by Poincare and Cartan. We show how ideas of this theory connect such a different fields of mathematical physics as Hamiltonian dynamics, optics and hydrodynamics. We focus our attention on the results that are rarely expounded in the literature.

We discuss the concept of transformations among reference frames (classical or quantum). Usually transformations among classical reference frames have sharply defined parameters; geometrically they can be considered as pure states in the parameters' space, and they form a group. It is however possible that the distributions in the parameters' space are mixed state; such states form a semigroup. Similarly, transformations among quantum reference frames can be either pure or mixed. This gives rise to interesting consequences, in particular, the state of a system S can be pure with respect to a reference frame and mixed with respect to another. We argue that these nonpure transformations are natural, and give an application to the connections of time and (inverse) temperature for thermal states.