Papers

A new family of Darboux integrable partial differential equations is constructed. The minimal orders of integrals in both characteristics can simultaneously be arbitrarily high for equations within this family. This suggests that the complete list of Darboux integrable equations may contain substantially more equations than are currently known.

We propose boundary theories, CFT$_{p}$ with $p\in \beta\mathbb{N}/\mathbb{N}$, which can be interpreted as an ensemble of theories in the context of AdS/CFT duality, when the boundary spacetime dimension is more than one. These theories are only emergent in the infinite $N$ limit of the boundary CFT and depend on $p$, which corresponds to the different ways one can take a generalized form of the infinite $N$ limit, in the presence of a chaotic or an oscillatory operator. This generalized infinite $N$ limit is called an ultralimit, and $p$ is a free ultrafilter on $\mathbb{N}$. We propose that the gravitational path integral computes an average of CFT$_{p}$ in each of its sectors; and in the appropriate cases, it produces baby universes and spacetime wormholes. We apply this proposal to the Antonini-Sasieta-Swingle (AS$^{2}$) like states.

We give a rigorous one-dimensional formulation of Kane's transport proposal for probing the Euler characteristic of a Fermi sea. For a clean, translation-invariant continuum Hamiltonian with real-analytic dispersion, a Fermi sea confined to a finite momentum range, and nonzero Fermi velocity at every Fermi point, we analyze a particular Abel-regularized transport response (obtained by tracing over the Fermi sea) and prove that its large-time limit equals the Euler characteristic. We derive an explicit finite-time formula, show that quantization requires the large-time limit, and obtain a convergence rate under additional nonstationarity assumptions. We also prove the lattice analog, where the sharp commutator is trace-class and a completely filled band contributes zero. Finite-volume calculations illustrate the prescribed order of the thermodynamic and large-time limits.

We introduce the Quantum Information Space at Infinity (Quinfinity) $\mathcal{Q}_\infty$ as the inverse limit of the symbolic Quantum $N$-Spaces $\mathcal{Q}_N$, identified with the complete local real algebra of formal power series under the adic topology. The pure state pro-variety is formalized as the inverse limit of $N$-level pure state varieties, driven by the algebraic colimit of a direct system of real radical ideals. This real architecture internalizes the imaginary unit, collapsing the metric landscapes onto an invariant Riemannian isometry, which identifies the pro-variety directly with the classical Fisher-Rao statistical manifold. Within this framework, the Heisenberg uncertainty principle undergoes a structural regularization at infinity being intrinsically manifested as a localized truncation obstruction governed by adic algebraic derivations. Concurrently, the continuous Liouville-von Neumann equation operates as an internal derivation tangent to the pro-variety, while the open Gorini-Kossakowski-Sudarshan-Lindblad asymptotic master equation is intrinsically obtained via a non-associative symmetric Jordan product. This dissipative flow acts as a contracting radial vector field that dampens higher-order jet configurations, driving the state trajectories down the hierarchical tree toward the absolute zero element as a non-singular universal attractor.

We give an all-level proof of the four-point $SU(2)$ AGT correspondence with four fundamental hypermultiplets at generic $\beta=-\epsilon_1/\epsilon_2$, by proving an all-level factorization formula for Selberg averages of generalized Jack polynomials. Taking a coefficientwise Jack limit of the generalized Macdonald Pieri rule, we obtain the required one-box matrix elements in the strict Cauchy dual basis. A rational corner-function identity then evaluates the sum over all parent double partitions, while an explicit total derivative of the Selberg kernel yields a triangular recursion on the Dotsenko-Fateev charge balance hyperplane. The unique solution of this recursion is the generalized Kadell formula previously verified only through finite level. Combining this result with the generalized Cauchy identity identifies each double partition term with the corresponding Nekrasov fixed-point contribution.

We construct Casimir operators of $4D, \,\mathcal{N}=2$ supersymmetry algebra in the harmonic superspace approach, and provide the explicit expressions for them in terms of covariant derivatives in the analytic basis. Specifically, the superisospin operator $C_I$ is expressed in terms of new 'long' harmonic derivatives $\nabla^{\pm\pm}, \nabla^0$ as $C_I = \frac{1}{4}\left(\nabla^0\right)^2 + \frac{1}{2}\left\{\nabla^{++},\nabla^{--}\right\}$. These derivatives involve non-local terms with the inverse Box operator $\Box^{-1}$ and satisfy modified $SU(2)$ algebra relations in which the eigenvalues of $\nabla^0$ on analytic superfields are shifted by -2 relative to the standard $U(1)$ harmonic charges. We also construct $\mathcal{N}=2$ superhelicity and super-isohelicity operators relevant to the massless case (with vanishing $\Box$) and find their eigenvalues for few instructive examples of $4D, \,\mathcal{N}=2$ superfield theories.

In several articles, the authors assume that the convex roof structure of the EoF and finite-dimensionality of subsystems $A$ and $B$ guarantee the existence of the (global) supporting affine functional for the EoF at any state of the system $AB$. This means that for any state $\rho$ of $AB$ there is a Hermitian operator $\Lambda_\rho$ on $\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B$ such that $E_F(\rho)=\mathrm{Tr}\Lambda_\rho\rho$ and $E_F(\sigma)\geq\mathrm{Tr}\Lambda_\rho\sigma$ for any state $\sigma$ of $AB$. We present an explicit example showing that, when $\rho$ is degenerate, this is not true even in the simplest case when $A$ and $B$ are qubit systems. The construction is based on the fact that the existence of a supporting affine functional for the EoF at a state $\rho$ is equivalent to the Lipschitz lower semicontinuity of the EoF at this state $\rho$. We use Wootters' formula and the help of Claude Fable 5 to find a state $\rho$ of the system $AB$ for which the latter property does not hold. We also describe conditions for the existence the local and global supporting affine functionals for the EoF at a given state of both finite and infinite-dimensional bipartite quantum systems. These conditions allow us to find Lipschitz lower semicontinuity bounds for the EoF at a given finite rank state $\rho$ (i.e. inequalities of the form $\,E_F(\rho)-E_F(\sigma)\leq C_\rho\|\rho-\sigma\|_1$) with and without restrictions on the support of the state $\sigma$.

Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.

Lie algebras describe how control Hamiltonians combine in closed quantum systems. In open Markovian systems, however, dissipation introduces irreversible directions that the Lie algebra alone does not retain. A dynamical Lie wedge preserves this information as a convex cone of locally admissible generators. The classical Levi decomposition separates a finite-dimensional Lie algebra into semisimple and solvable parts. In this work, we prove a corresponding decomposition and reconstruction theorem for dynamical Lie wedges. The theorem decomposes the wedge into semisimple and solvable data, records how these components are coupled, and provides a converse reconstruction of the wedge. The framework yields four structural types of the generated algebra. We further prove that the total dissipation strength of every admissible generator depends only on its coordinate in the solvable radical. In particular, if the generated Lie algebra is semisimple, every admissible generator is Hamiltonian. Together, these results extend Lie-algebraic structural methods to irreversible Markovian control and clarify how dissipation is encoded in the generator geometry.

We construct the first examples of one-dimensional long-range crystals exhibiting arbitrarily fast polynomial dispersive bounds. The Floquet functions of our Hamiltonians are highly oscillatory Weierstrass functions, whose rough autosimilar structure drives the fast dispersion. The proof develops a new Fourier decay theory for $C^\alpha$ images of the Lebesgue measure, based on a Dolgopyat-inspired transfer operator method, and yields a van der Corput lemma for Weierstrass functions. As a consequence, the local time of classical Weierstrass functions of sufficiently large lacunarity exists and is $C^k$, answering a question raised by Geman and Horowitz in 1980.

Building on our previous work on 2-Morita equivalence for $E_2$-algebras using topological pictures, we develop a systematic framework for Morita equivalence of topological orders in different dimensions in terms of $n$-Morita categories $\mathrm{Mrt}_{E_n}(\mathcal{C})$. In this framework, various notions of $n$-Morita equivalence are unified as equivalences of objects in $\mathrm{Mrt}_{E_n}(\mathcal{C})$. We also compare the constructions of higher Morita categories due to Haugseng and to Gwilliam–Scheimbauer. For $n=1,2$, we prove that the functor $\mathrm{Mod}_n:\mathrm{Mrt}_{E_n}(\mathcal{C})\to \mathrm{Mrt}_{E_{n-1}}(\mathrm{LMod}^{\mathrm{rep}}(\mathcal{C}))$ is an equivalence, relating the algebraic description of higher Morita theory to its realization in terms of module categories. In this formulation, the notion of a bi-bimodule arises naturally and provides a common framework for local and confined modules. Its explicit orientation data, together with the corresponding fusion rules, clarifies the relations among defects arising from condensation in topological orders.

We present an open-source Python package for sampling the Gaussian, Laguerre, Jacobi, and Circular $\beta$-ensembles of random matrix theory. The package implements the Dumitriu–Edelman and Killip–Nenciu constructions, allowing efficient generation of random spectra for general $\beta > 0$. In addition to spectrum generation, it includes tools for the analysis of spectral statistics, from standard nearest-neighbor spacings and spacing ratios to non-adjacent $k$-spacings and the spectral form factor. These tools can be applied to generic spectral data, allowing users to compare them with and fit them to $\beta$-ensemble predictions. In this note, we review the $\beta$-ensembles, describe the package interface, and illustrate its use through several numerical experiments motivated by applications to quantum chaos. Our numerical results include an analysis of $\beta$ as a continuous fitting parameter in spacing ratio statistics, an examination of the numerical evidence for the conjectured $k$-spacing ratio distributions, and a study of the spectral form factor for general values of $\beta$.

In this paper, we develop a deterministic, physically constrained generative framework based on the Madelung formulation of the free-particle Schrödinger equation. A reference Born probability density and a controllable initial phase function serve as initial data for the free Madelung system, which couples the Born probability density and phase function through the Bohm quantum potential, while the phase function determines the hydrodynamic velocity field. Provided that the Born probability density remains positive and the hydrodynamic velocity field generates a unique global characteristic flow, samples drawn from the reference density and transported along the characteristic flow are distributed according to the evolving Born probability density at every time. As a consequence, randomness enters only through the initial sampling; the subsequent generation is deterministic and involves neither stochastic dynamics nor an independently parameterized time-dependent velocity field. We formulate terminal-time distribution matching as a PDE-constrained phase-identification problem and derive the underlying Hamiltonian and Fisher-information structure. For isotropic Gaussian wave packets, we obtain explicit dynamics and a necessary and sufficient condition for exact reachability of isotropic Gaussian targets by quadratic initial phase functions, together with the corresponding sampling map. For a smooth prescribed potential initial velocity field, we further establish that the characteristic flow approximates the associated first-order transport map with an O(T^2) error, both uniformly and in the 1- and 2-Wasserstein distances. A numerical Gaussian benchmark validates the fully discrete forward solver, while full-grid PDE-constrained phase identification is demonstrated for asymmetric bimodal targets.

We consider the $\phi^4_2$ model. It has been called the simplest non- trivial super-renormalisable model. The method we use is the multiscale loop vertex expansion, an improvement of constructive field theory. We prove analyticity and Borel summability of the cumulants up to a finite order.

In this paper, we study the structures of single connected $k$-leaky $(r+1)$-completed Hurwitz numbers. We first prove that, for fixed genus, after extracting an explicit product of Pochhammer type factors, the stable connected leaky Hurwitz numbers for partition $\mu$ are polynomials in the quotients $[\mu_i]$ modulo $k+r$. We then give a closed formula for the generating function of single connected leaky Hurwitz numbers, which can be used to study the genus dependence of leaky Hurwitz numbers.

Building on the stochastic variational framework established in the companion paper, we investigate here the linearized stochastic water-wave system, consisting of a large-scale stochastic wave dynamics coupled to transport dynamics for the small-scale correlation modes. Within this framework, we develop, in the deep-water regime, a kinetic theory for surface gravity waves interacting with unresolved stochastic velocity fields. An energy analysis yields a wave-action kinetic equation exhibiting two distinct regimes: a diffusive scattering regime and a quartic interaction regime with structural similarities to Hasselmann–Zakharov theory. In the present framework, these effective quartic interactions arise through stochastic transport of unresolved fluctuations by the large-scale flow rather than through classical intrinsic resonant nonlinearity. Scaling laws are derived for the diffusion tensor and the effective growth rate, revealing a Miles-type production–dissipation mechanism. Using JONSWAP spectra, we then compare the strength of stochastic transport and classical Hasselmann interactions. For realistic oceanic values of unresolved velocity variance ($\sigma_u \approx 0.1\,\mathrm{m\,s^{-1}}$) and decorrelation time ($\tau_c \approx 10\,\mathrm{s}$), stochastic transport is found to compete with, and often exceed, classical four-wave interaction rates over broad spectral ranges. The transport intensity $S=\sigma_u^2\tau_c$ emerges as a key parameter controlling the transition between interaction regimes. These results suggest that unresolved stochastic transport may play a substantially larger role in spectral evolution than is commonly represented in operational wave models, and motivate the inclusion of transport-induced source terms alongside standard resonant interaction closures.

This paper develops a comprehensive stochastic variational framework for surface gravity waves. Starting from Luke's variational principle for irrotational, incompressible free-surface flow, we introduce a decomposition of the velocity potential into a large-scale deterministic component and a regularized stochastic noise term representing unresolved scales. A path-wise variational principle yields the stochastic counterparts of the Laplace, Bernoulli, and kinematic boundary conditions. To close the system, a second variational principle in expectation is formulated, providing evolution equations for the noise correlation functions. The resulting coupled system preserves the Hamiltonian structure of the Zakharov–Craig–Sulem formulation. We further analyze explicit solutions via WKB approximation and show that the noise correlation functions satisfy a Hamilton-Jacobi equation with ray-tracing dynamics. This framework provides a rigorous foundation for reduced stochastic models of ocean waves and for the kinetic theory developed in the companion paper.

Accurate and efficient simulation of ultrasound pressure fields and pulse-echo responses is essential for transducer design, beamforming optimization, and model-based imaging research. Conventional Spatial Impulse Response methods compute acoustic fields by explicitly sampling trapezoidal impulse responses for each rectangular aperture subdivision, which leads to high computational cost for large apertures, dense spatial grids, and high sampling frequencies. We introduce Sparse Delta Integration, a mathematical framework for the far-field SIR of rectangular apertures that expresses the trapezoidal spatial impulse response as the double integration of a sparse set of Dirac delta distributions. This formulation yields compact expressions for time-domain and frequency-domain SIRs and enables vectorized implementations whose computational cost is independent of the trapezoid duration. We further derive a spectral formulation for pulse-echo simulation that removes part of the conventional Fourier-domain convolution pipeline. Implemented in the open-source Python package eSDIva, these methods achieve speedups of up to 180x for temporal SIR computation and up to 20x for pulse-echo RF simulation relative to Field II, while maintaining high numerical accuracy with mean squared errors below 1e-5 and correlation coefficients close to unity.

The maximum energy dissipation principle is employed to nonlinear chemical thermodynamics in terms of distance variable (generalized displacement) from the global equilibrium, applying the optimal control interpretation to develop a variational formulation. The cost-like functional was chosen to support the suggestion that such a formulation corresponds to the maximum energy dissipation principle. Using this approach, the variational framework was proposed for a nonlinear chemical thermodynamics, including general cooperative kinetics model. The formulation is in good agreement with standard linear non-equilibrium chemical thermodynamics.

Many matrices in computational chemistry are neither sparse nor low-rank, making the design of low-scaling algorithms difficult. In this Perspective, we point out that many important matrices in computational chemistry, such as the Coulomb matrix, the electronic repulsion integral tensor, the density matrix, the localized molecular orbital (LMO) coefficient matrix, the Fock matrix, and the nuclear Hessian matrix share the same property: when their basis functions are suitably ordered, their off-diagonal blocks have low numerical ranks (despite that they as a whole have high numerical ranks). This property, termed off-diagonal low rank (ODLR), has been extensively studied in the mathematics community, but has surprisingly found very little use in computational chemistry. This Perspective reviews the existing mathematical literature on how to use the ODLR property of matrices to compactly store, as well as efficiently calculate or use them. Subsequently, we review the use of the ODLR property in computational chemistry, and point out possible future opportunities of devising new low-scaling methods for dense, full-rank matrices, exploiting the ODLR property. In particular, we prove for the first time that Fock matrices and LMO coefficient matrices satisfy the ODLR property, even if the system is gapless (in which case the matrices are dense). This paves the way to linear scaling electronic structure calculations of gapless systems at zero electronic temperature.

We investigate the form of the Saito metric on quotients of the reflection representation of an extended affine Weyl group, restricted to an arbitrary stratum of its discriminant. We compute the Saito determinant for all strata and Dynkin types and show that it is divisible by a product of q-analogues of restricted roots for the associated hyperplane arrangement. Our results simultaneously generalise the factorisation in the Weyl denominator formula to higher codimension strata, and provide q-analogue refinements of results of Antoniou-Feigin-Strachan for Coxeter groups.

It is well known that the Einstein equations are tensor equations for a Lorentzian metric. Hence, choosing a suitable gauge condition is crucial for solving them. As a powerful gauge condition, maximal foliations have played a pivotal role in two groundbreaking works in general relativity: the global stability of Minkowski spacetime and the bounded L^2 curvature conjecture. Nevertheless, if the initial hypersurface fails to be maximal, the usual elliptic approach for constructing maximal foliations encounters fundamental difficulties when solving the Einstein equations. The purpose of the paper is to provide a construction of maximal foliations around any initial asymptotically flat Cauchy surface satisfying vacuum Einstein constraint equations, under a suitable smallness condition on the mean curvature.

We study moderate deviations for the largest eigenvalue of the randomly deformed Gaussian unitary ensemble introduced by Johansson (Probab. Theory Relat. Fields, 138: 75–112, 2007). In the fixed-coupling regime, the rescaled largest eigenvalue converges to the convolution of the Tracy–Widom law and a Gaussian law arising from the random displacement of the spectral edge. We derive sharp logarithmic asymptotics for the right and left tails on growing Airy scales and obtain the corresponding phase diagram. The two tails have different transition scales. At criticality, the rate functions are nontrivial infimal convolutions of the Tracy–Widom and Gaussian rate functions. We establish a trace-norm Airy approximation that is uniform over typical diagonal configurations on a logarithmic window. A conditional convolution argument then combines the resulting tail estimates with the Gaussian moderate deviations of the edge displacement.

We study the emptiness formation probability (EFP) in the ground state of the repulsive one-dimensional Lieb-Liniger Bose gas. For a macroscopic empty interval of length $2R$, its leading asymptotic behavior is described by a rate function $f(\gamma_0)$, defined by $-\log P(R)\sim(\rho_0R)^2f(\gamma_0)$, where $\rho_0$ is the mean density and $\gamma_0$ is the dimensionless interaction strength. We propose a parameter-free integral equation for $f(\gamma_0)$. Starting from the exact dual-field Fredholm-determinant representation of the EFP, we show how the conjectured kernel arises formally and identify the uniform asymptotic statement that remains to be proven for a rigorous derivation. The conjecture reproduces the Tonks-Girardeau and weak-coupling limits, as well as the first correction obtained independently in both limits. It also agrees at the few-percent level with numerical minimization of the Lieb-Liniger hydrodynamic action over more than four orders of magnitude in coupling. The numerical calculation yields the corresponding emptiness instantons and their astroid-like vacuum regions.

We prove that for any given frequency resonance exponent and phase resonance exponent, there exist a frequency and a phase exactly realizing these values such that the Almost Mathieu operator exhibits Anderson localization for $\ln|\lambda|>\max\{\beta(\alpha),\delta(\alpha,\theta)\}$. This resolves the conjecture in \cite{MR4686650}.

In this paper we derive the time-dependent correlation functions of the renormalized Hartree NLS equation on the torus $\mathbb T^d$, $d=2,3$, from the corresponding bosonic many-body Gibbs dynamics. In contrast with the 1D problem studied earlier by Fröhlich, Knowles, Schlein and Sohinger, in higher dimensions the scaled quantum particle number is not uniformly bounded and the relevant classical fields require Wick renormalization. Our proof combines convergence of the quantum generators in weighted Hilbert spaces with uniqueness for positive solutions of the limiting Liouville equation that are dominated by the Gibbs measure.

We study graded arrays of highly dispersive subwavelength resonators through their volume-potential matrix reduction. The reduction gives a finite-dimensional nonlinear resonance problem whose diagonal entries encode the local dispersive response and whose off-diagonal entries encode volume-mediated coupling between distinct resonators. We show that monotone material grading turns this matrix into an analytically controllable frequency sorter. When graded detuning dominates coupling in the reduced characteristic matrix, the characteristic vectors localize near individual resonators; under an explicit half-lattice-site localization condition, this localization becomes a monotone frequency-position map. We also prove a constructive inverse-design statement: prescribed monotone target frequencies can be realized by choosing the local dispersive parameters, with robustness under sufficiently small grading errors. Numerical experiments, whose parameters and algorithms are recorded in an appendix, illustrate localized resonances, designed frequency-position maps, disorder stability, spacing dependence, and a resolved physical Lippmann-Schwinger validation beyond the leading logarithmic quadrature model.

This paper presents a macroscopic, or continuum, theory aimed at describing when, where, and why cracks nucleate and propagate in nominally elastic brittle materials under monotonic, quasi-static, but otherwise arbitrary mechanical loads. Motivated by recent insights, the proposed sharp theory posits that: cracks nucleate and propagate exclusively in regions where the strength surface of the material is exceeded, with their evolution dictated by the minimization of the sum of the potential — the elastic energy minus the work done by the externally applied forces — and surface energies. While the theory applies to materials with any elasticity (linear or nonlinear) and any material symmetry (isotropic or anisotropic), attention is restricted here to the most basic case of isotropic elastic brittle materials. For demonstration purposes, the theory is confronted with a set of nine tests that span the entire range of well-settled experimental knowledge on fracture nucleation and propagation — the so-called “Nine Circles of Elastic Brittle Fracture” — on both a hard material (a silicate glass) and a soft material (a synthetic rubber).

We introduce and study (fermionic and bosonic) invertible quasi-local algebras over uniformly locally finite metric spaces $X$ with infinite-dimensional local von Neumann algebras. We show that the group of Brauer equivalence classes of such algebras is isomorphic to both the group of phases of invertible states and the group of stable equivalence classes of quantum cellular automata over $X\times \mathbb{Z}$. Using K-theory of the symmetric monoidal category of invertible quasi-local algebras and bounded spread isomorphisms, we propose a definition of an $\Omega$-spectrum of invertible phases as conjectured by Kitaev. We then show that the $c=\frac{1}{2}$ chiral Majorana fermion net and the $(E_{8})_{1}$ conformal net provide Brauer non-trivial invertible quasi-local algebras, thus providing explicit constructions of non-trivial invertible states and quantum cellular automata on $\mathbb{Z}^{2}$. In addition, we show that the time-slice nets of rational diagonal conformal field theories admit lattice degrees of freedom, which implies the discretization of any holomorphic conformal net is invertible.

In the present work we consider the problem of the homogenization of the Cauchy problem for the equations of the elasticity. We assume the composite material is constructed as a periodic repetition along the chosen axis of the unit cell which models to different materials. The homogenization procedure is based on the operator separation of variables, which gives the new incites for the homogenization of the elasticity equations and provides new way of the homogenization and constructing the homogenized equation.