AI for Science

A data-driven inverse optimization problem (DDIOP) is the problem of estimating the objective-function parameters (weights) that explain observed optimal-solution data, and it arises in many applications, including mixed integer linear programming (MILP). In inverse optimization for MILPs, the prediction error of the features is discontinuous with respect to the weights, so applying gradient-based optimization directly is difficult. In this paper we focus on the suboptimality loss. This loss attains its minimum value, zero, if and only if the weights are exactly consistent with the observed data. We reveal a geometric structure of this loss -- it is convex and piecewise linear, and moreover the set of weights that are exactly consistent with the observed data has a positive ``thickness'' rather than being a single point or a thin boundary -- and use it to show the following. First, a broad class of gradient-based optimization methods, including projected subgradient descent, reaches exact consistency with the observed data in finitely many iterations (an exact solution is obtained in finite time). Second, for projected subgradient descent we give an explicit upper bound on the number of iterations needed to reach exact consistency. Third, when the forward problem is an integer linear program (ILP), we give this upper bound as a fully explicit iteration count determined solely by the number of samples, the dimension of the features, and the structure of the constraint coefficient matrix. Through numerical experiments, we confirm this finite-step attainment behavior.

Trajectory mutual information is frequently used to quantify information transfer in biochemical systems. Tractable solutions of the trajectory mutual information can be obtained via the widely used Linear-Noise Approximation (LNA) using Gaussian channel theory. This approach is expected to be accurate for sufficiently large systems. However, recent observations show that there are cases, where the mutual information obtained this way differs qualitatively from results derived using an exact Markov jump process formalism, and that the differences remain even in the large copy number regime. In this letter, we show that these differences can be explained by introducing the notion of reaction- versus state-based descriptions of trajectories. In chemical systems, the information is encoded in the sequence of reaction events, and the reaction-based trajectories of Markov jump processes capture this information. We show that within the Gaussian formalism, trajectories can be defined either based on individual reaction channels, or on a state-based level, where different reaction channels are summarised into a single noise term. While both definitions agree in terms of copy number fluctuations, state-based trajectories contain in general less information than reaction-based trajectories. The commonly used Gaussian mutual information via the Linear-Noise Approximation is consistent with a state-based trajectory notion, which causes a systematic loss of information independent of system size. We show that an alternative, reaction-based variant of the Gaussian mutual information prevents this loss of information. We illustrate the consequences of different trajectory descriptions for two common cellular reaction motifs and discuss their connection with Berg-Purcell and Maximum-Likelihood sensing.

We provide a superselection theory of symmetry defects in 2+1D symmetry enriched topological (SET) order in the infinite volume setting. For a finite symmetry group $G$ with a unitary on-site action, our formalism produces a $G$-crossed braided tensor category $G\mathsf{Sec}$. This superselection theory is a direct generalization of the usual superselection theory of anyons, and thus is consistent with this standard analysis in the trivially graded component $G\mathsf{Sec}_1$. This framework also gives us a completely rigorous understanding of symmetry fractionalization. To demonstrate the utility of our formalism, we compute $G\mathsf{Sec}$ explicitly in both short-range and long-range entangled spin systems with symmetry and recover the relevant skeletal data.

The concept of geodesic-like curves was introduced by Chen in 2010 as a method for estimating shortest paths (geodesics) on parametric surfaces, with its convergence established theoretically. However, an efficient numerical computational framework has not yet been developed. In this paper, we propose an elegant and efficient approach for computing geodesic-like curves by leveraging deep learning and Physics-Informed Neural Networks (PINNs). Under the proposed framework, not only can single parametric surfaces be handled efficiently, but a broad class of complex parametric surfaces including multi-surface systems with $C^0$ or higher continuity and surfaces of revolution can also be robustly addressed.

We consider an ideal Fermi gas in a container subject to a uniform gravitational field at absolute zero temperature. Under a semiclassical approximation, we examine the density profile of the particles and derive an expression for the chemical potential. A critical value of the chemical potential separates the weak- and strong-gravity regimes, and the kinetic and potential energies of the Fermi gas are determined in both regimes.

Considered and analyzed below are fully non-homogeneous initial-boundary-value problems for the celebrated Dirac system, formulated on the spatial half-line. Analytical solution formulae are derived formally via suitable implementation of the well-known Fokas' unified transform methodology, and rigorously verified a posteriori. The latter substantial task relies on complex-analytic tools and careful interpretation of the obtained integral representations. These valid solutions are then used for investigating qualitative properties. These include boundary behavior near the axes of the domain as well as long-range asymptotics and long-time (eventual) periodicity. Notably, smoothness of the solution, both within and upto the boundary of the domain, depends heavily on certain compatibility conditions between initial, boundary and forcing data. Further results pertaining to solution's regularity and uniqueness are thence established based on the qualitative theory. The closed-form expressions reported here are also useful in the study of non-linear counterparts.

Stochastic oscillations can emerge from a two-population model as triggered by endogenous finite size fluctuations. Here, an extended dynamical scenario is considered in which a third fluctuating species is added to a proto-typical scheme of neuronal interaction. As we shall prove both analytically and numerically, the third added species can enhance or even suppress the emergence of quasi-cycles, namely the coherent oscillations of the two original populations, as instigated by the demographic noise component. In general, investigating the coupled dynamics of noisy oscillators of the type considered could yield an extended framework for synchronization studies, beyond the pioneering setting introduced by Kuramoto.

Using the energy function on the extended Siegel-Jacobi upper half space of order $n$, $\tilde{\mathcal{X}}^J_n$, with $n\in \mathbb{N}$, the equations of motion in the variables $(x,y,q,p,κ)$ attached to linear Hamiltonians in the generators of the real Jacobi group $G^J_n(\mathbb{R})$ are presented, where $x,y$ are symmetric matrices in $\mathcal{M}(n,\mathbb{R})$ and $p,q$ are real $n$-vectors. The case $n=1$ is presented separately.

For compressible nonlinear isotropic elasticity it is well known that rank-one convexity, polyconvexity and the monotonicity of the Cauchy stress tensor with respect to the logarithmic stretch tensor (the true stress-true strain monotonicity, TSTS-M$^+$) are independent constitutive conditions which should, however, all together be satisfied for a physically meaningful description of idealized elastic materials. In the incompressible case, TSTS-M$^+$ turns into Hill's inequality since the Cauchy stress $σ$ reduces to the Kirchhoff stress $τ$. Hill's inequality requires then monotonicity of the Kirchhoff stress in terms of the logarithmic stretch tensor evaluated for incompressible response. In this paper we clarify how the a priori independent notions of Legendre-Hadamard ellipticity (LH), polyconvexity and Hill's inequality are nevertheless intimately connected. More precisely, by providing several alternative proofs, we show that both LH-ellipticity (rank-one convexity) and polyconvexity imply the weak Hill inequality in the incompressible two-dimensional case.

Let $X_r\sim N_r(0,1)$ be the centered unit-scale generalized Gaussian random variable with density proportional to $\exp(-|x|^r/2)$. We prove that, for $p,q>0$, there exists a strictly positive random variable $V$, independent of $X_q$, such that $X_p\stackrel{d}{=}VX_q$ if and only if $p\le q$. Moreover, the law of $V$ is unique. For $p<q$, put $a=1/p$, $b=1/q$, and $α=b/a=p/q$. If $S_α$ is a positive $α$-stable random variable with Laplace transform $\mathbb{E}\exp(-uS_α)=\exp(-u^α)$, set $W_0=S_α^{-b}$, let $W$ be the $W_0$-size-biased version of $W_0$, and define $V_{p,q}=2^{a-b}W$. Then $X_p\stackrel{d}{=}V_{p,q}X_q$. For $p>q$, the required Mellin quotient, viewed as the candidate characteristic function of $\log V$, is unbounded by Stirling's formula, and hence cannot be a characteristic function. The factor laws form a multiplicative cocycle, $V_{p,r}\stackrel{d}{=}V_{p,q}V_{q,r}$, for $p\le q\le r$, where the factors on the right-hand side are independent copies. Thus the Mellin quotient isolated by Dytso, Bustin, Poor and Shamai is realized constructively throughout the $p<q$ branch. In particular, $Φ_{p,q}$ is positive definite exactly in the range $p\le q$, and the inverse Fourier--Mellin candidate density in the remaining $p<q$ branch is a genuine nonnegative probability density. The known Gaussian-base and bounded-parameter product cases are recovered as parts of a single positive scale-mixture classification.

Color graphs and their subgraphs, referred to as bubble graphs, correspond bijectively to the simplicial complexes of pseudomanifolds and their subsimplices, respectively. In this paper, we introduce matrix representations for colored graphs and their associated bubble graphs. By using this correspondence, we define simplicial-complex matrices and subsimplex matrices that encode the simplicial complexes of pseudomanifolds and their subsimplices. Moreover, we formulate mutation and crossover operations on colored graphs. Through the established correspondence among simplicial complexes, colored graphs, and simplicial-complex matrices, we extend these operations to simplicial complexes and simplicial-complex matrices. We further implement an algorithm generating simplicial-complex matrices and a genetic algorithm performing mutation and crossover of them to produce pseudomanifolds exhibiting diverse topologies. In addition, we implement procedures for decomposing the generated simplicial-complex matrices into simplex matrices, reconstructing the simplicial complexes of the associated pseudomanifolds from this information, and computing geometric quantities such as the volume, circumcenter, and dual-simplex volume of each simplex.

We study finite-order meromorphic functions representable as absolutely convergent sums of squares of Cauchy kernels and having only finitely many zeros. By earlier work of Baranov and the author, such functions admit a representation $f=P/g^2$, where $P$ is a polynomial and $g$ is entire, satisfying the differential equation $ Pg''-P'g'+Qg=0, $ where $Q$ is a polynomial. We show that the zeros of $g$ asymptotically accumulate along the Stokes rays. If $\mathrm{deg}\ Q>\mathrm{deg}\ P$, they approach these rays in the Euclidean metric, whereas in the borderline case $\mathrm{deg}\ Q=\mathrm{deg}\ P$ one obtains in general only localization in logarithmic neighborhoods of the Stokes rays, and this is sharp. We then characterize the existence of a decomposition $ P/g^2=\sum c_n (z-t_n)^{-2} $ in terms of the sectorial behavior of $g$ and, equivalently, in terms of the Laine condition for the corresponding Schwarzian equation. Finally, for fixed $P$ and fixed order, we identify the resulting families, modulo the natural equivalence relation, with finite-dimensional affine algebraic varieties.

Von Neumann's Hilbert-space formalism of quantum mechanics constitutes a logico-physical theory of observed or measured reality. Imposing the logical constraint of Booleanity, essential for objectively shareable descriptions among observers, reveals the physical meaning of classicality inherently embedded within the formalism itself. Starting from this consideration, the present work reformulates the quantum-classical transition via Hilbert-space classical mechanics (HCM), grounding classicality not in the dynamical limit ($\hbar \to 0$), but in the logical constraint of Booleanity (i.e., the mutual commutativity of preparable states). Within this state-centric framework, applying the Einstein-Podolsky-Rosen (EPR) criterion alongside locality and measurement independence reduces standard quantum mechanics to the HCM model. Thus, the EPR argument reveals not quantum incompleteness, but the implicit classical boundaries of its own premises. To resolve this impasse, we articulate a nuanced quantum ontology grounded in a fundamental structural bipartition between the observational environment and the observed object, which accommodates three categorical distinctions: ontic, processional, and tropos-existential. Building on this, we propose a criterion of objective reality wherein descriptive objectivity is treated as merely a sufficient condition for physical reality. This addresses the historical Bohr-Einstein ambiguity, enabling the quantum formalism to ontologically unify objective measured phenomena and non-objective observed interference within a context-dependent framework.

This is the second part of a two--part series investigating bifurcation phenomena in autonomous Lagrangian systems and geodesic flows on Finsler and Riemannian manifolds. Building upon the abstract bifurcation theorems established in earlier work and the results of Part I, this study makes contributions in two main directions. In Part A, we focus on bifurcations of generalized periodic solutions in autonomous Lagrangian systems. By employing Morse index and nullity techniques within the normal space to the $\mathbb{R}$-orbits of solutions, we derive necessary and sufficient conditions for bifurcation, encompassing scenarios of both Fadell--Rabinowitz and Rabinowitz type. In Part B, we extend these results to the geometric setting of geodesic bifurcations in Finsler and Riemannian manifolds. A principal achievement is the significant refinement of the classical Gauss lemma and its generalizations by Morse-Littauer and Savage, providing a precise description of geodesic behavior near critical points of the exponential map. The sharpness of these theoretical results is rigorously tested and confirmed through explicit counterexamples, such as the round sphere. The work is technically rigorous, leveraging a specialized technique developed by the author to establish novel bifurcation theorems. These findings have profound theoretical implications and potential applications in related fields such as the Zermelo navigation problem and the study of stationary spacetimes.

This paper aims to develop a non-commutative geometrical version of the theory of Yang--Mills--Scalar--Matter fields. To accomplish this purpose, we will dualize the geometrical formulation of this theory, in which principal $G$--bundles, principal connections, and linear representations play the most important role. In addition, we will present the non-commutative geometrical Lagrangian of the system as well as non-commutative geometrical associated field equations. At the end of this work, we show some examples

In this paper, we prove a decomposition lemma for symmetric matrix fields on bounded domains: $D+\mathrm{Sym}\nablaΦ=\sum_i a_i^2ξ_i\otimesξ_i$ with uniform control on $Φ$ and $a_i^2$, using fewer than the usual $n(n+1)/2$ rank-one symmetric terms. Except possibly in dimensions $n=8,16$, the decomposition is shown to be optimal through algebraic arguments. This reduces the number of steps in convex integration for a nonlinear PDE system, improving Hölder regularity of flexible solutions in dimension $n\ge3$. This PDE is a partial linearization of the codimension-one local isometric embedding equation in the Nash--Kuiper theorem, and also yields improved regularity for very weak solutions of related 2D Monge--Ampére and $2$-Hessian systems. The improved Hölder exponent is any $α<(n^2+1)^{-1}$ for $n=2,4,8,16$ and any $α<(n^2+n-2ρ(n/2)-1)^{-1}$ otherwise, where $ρ$ is the Radon--Hurwitz number, related to Bott periodicity. The proof involves novel applications of algebraic geometry and topology that yield the optimality of decomposition, including Adams' theorem on vector fields on spheres, intersections of projective varieties, and projective duality, combined with an elliptic method that avoids loss of differentiability.

A step-reinforced random walk is a discrete-time stochastic process with long-range dependence. At each step, with a fixed probability $α$, the so-called positively step-reinforced random walk repeats one of its previous steps, chosen randomly and uniformly from its entire history. Alternatively, with probability $1-α$, it makes an independent move. For the so-called negatively step-reinforced random walk, the process is similar, but any repeated step is taken with its direction reversed. These random walks have been introduced respectively by Simon (1955) and Bertoin (2024) and are sometimes refered to the self-confident step-reinforced random walk and the counterbalanced step-reinforced random walk respectively. In this work, we introduce a new class of unbalanced step-reinforced random walks for which we prove the strong law of large numbers and the central limit theorem. In particular, our work provides a unified treatment of the elephant random walk introduced by Schutz and Trimper (2004) and the positively and negatively step-reinforced random walks.

In this paper, we prove a uniform estimate for the modulus of continuity of solutions to degenerate complex Monge--Ampère equation in big cohomology classes. This improves the previous results of Di Nezza--Lu and of the first author.

Let $G$ be a cyclic $p$-group or generalized quaternion group, $X\in π_0 S_G$ be a virtual $G$-set, and $V$ be a fixed point free complex $G$-representation. Under conditions depending on the sizes of $G$, $X$, and $V$, we construct a self map $v\colonΣ^V C(X)_{(p)}\rightarrow C(X)_{(p)}$ on the cofiber of $X$ which induces an equivalence in $G$-equivariant $K$-theory. These are transchromatic $v_{1,\vec{0}}$-self maps, in the sense that they are lifts of classical $v_1$-self maps for which the telescope $C(X)_{(p)}[v^{-1}]$ can have nonzero rational geometric fixed points.