Papers

Let $K$ be a totally real field of odd degree $m$ in which $2$ is inert. Let $l\geq5$ be a prime totally ramified in $K$ with $\gcd(m,l(l-1)/2)=1$ and $2^{l-1}\not\equiv1\pmod{l^2}$. We prove the asymptotic Fermat's Last Theorem over every layer $K_{n,l}$, $n\geq1$, of the cyclotomic $\mathbb{Z}_l$-extension of $K$. Using $S$-unit bounds, we also give criteria for the generalized Fermat equation $Ax^p+By^p=Cz^p$ to have no asymptotic solutions in $K_{n,l}^3$ when the coefficients $A,B,C$ are supported at the primes above $2$ or $2d$, with $d$ an odd prime. In particular, for every fixed integer $r\geq2$ other than $4$, the equation $x^p+y^p=2^rz^p$ has no asymptotic solutions in $K_{n,l}^3$ for each fixed $n\geq1$. Thorne's modularity theorem makes the resulting bounds effective in each layer $\mathbb{Q}_{n,l}$ to which our criteria apply.

In this article, we consider an algebraic version of the tame site of a pair $(X,\widetilde{X})$. With this definition, we provide a general machinery to construct a tame sheaf from the data of an étale sheaf on $X$ and a family of local tame sections. We apply this construction to the big de Rham–Witt sheaves with tame sections defined by log poles and, over a field, to reciprocity sheaves, and deduce some consequences. As an application, we compare tame syntomic cohomology with the Nygaard filtration on the tame de Rham–Witt complex.

In this paper, we first confirm the Hamilton-Tian conjecture for the Sasaki-Ricci flow in a compact transverse Fano quasi-regular Sasakian $5$-manifold with klt foliation singularities. Secondly, we derive the compactness theorem of Sasaki-Ricci solitons on transverse Fano quasi-regular Sasakian $5$-manifolds. Then,by the second Sasakian structure theorem, we confirm the Hamilton-Tian conjecture for a compact transverse Fano Sasakian $5$-manifold. With its applications, we show that the gradient Sasaki-Ricci soliton orbifold metric on a compact Sasakian $5$-manifold is Sasaki-Einstein if $M$ is transverse $K$-stable.

Operations Research (OR) serves as a core decision-support methodology for complex systems, with significant applications across mathematics, management science, and computer science. Traditional approaches heavily rely on expert knowledge and often struggle to efficiently solve large-scale and multi-constraint problems. The rapid advancement of Large Language Models (LLMs) in recent years has offered a novel research paradigm to address these challenges. This paper presents a systematic survey of Large Language Models for Operations Research (LLM4OR). We begin by introducing the definition of OR problems and the fundamental principles of LLMs. We then focus on analyzing the roles of LLMs in OR, specifically covering such as model formulation, algorithm design, and solution verification. In addition, we discuss practical applications in representative scenarios and summarize benchmark datasets in this field. Finally, we outline the key challenges and provide perspectives on future research directions. A collection of related literature is available at https://github.com/xianchaoxiu/LLM4OR.

We give two presentations for bordisms of $S^2$ in the 3-dimensional oriented bordism category $\operatorname{Cob}(3) $, encoding the algebraic structures on $S^2$. After passing through topological field theories, we define two kinds of monoids which we call P-monoids and L-monoids. In addition to both being commutative Frobenius monoids, P-monoids are equipped with a class of endomorphisms while L-monoids are equipped with a class of unit morphisms, all of which are labelled by closed oriented irreducible prime 3-manifolds. They turn out to be equivalent. The new prime structures satisfy some countable relations with the commutative Frobenius structure, the most notable of which we call "legs relations." We then restrict to the setting of algebras and show that the legs relations place strong constraints on the new prime endomorphisms which forces them to act by multiplications by prime units, rendering the additional prime structures remarkably simple. We also propose an $\infty$-operad which encodes these prime structures and contains the $\infty$-little 3-cube operad as a sub-operad.% We briefly discuss the relations between P/L-algebras and J-algebras which classify 3-dimensional TFTs.

We study the $2\frac12$-dimensional electron magnetohydrodynamics (EMHD) system on $\mathbb T^2$ with componentwise fractional dissipation: $\partial_t a+a_yb_x-a_xb_y=-\Lambda^\alpha a$ and $\partial_t b-a_y\Delta a_x+a_x\Delta a_y=-\Lambda^\beta b$, where $0<\alpha,\beta<2$. This system is a $2\frac12$-dimensional reduction of the magnetic equation in Hall–MHD/EMHD under the ansatz $B=\nabla\times(ae_z)+be_z$. We prove local well-posedness for initial data $(a_0,b_0)\in H^{s+1}(\mathbb T^2)\times H^s(\mathbb T^2)$ with $s\geq 2-\varepsilon$, provided that $\alpha+\beta>2$. Thus neither component is required to carry a full Laplacian dissipation; the smoothing effects of the two fractional dissipations can be combined to control the Hall nonlinearity. The proof is based on Littlewood–Paley energy estimates, commutator bounds, and cancellations between the leading low–high interactions.

Schneps \cite{Schneps2012,Schneps2025} and Enriquez-Furusho \cite{EF4} proved that the double shuffle Lie algebra $\mathfrak{dmr}_0$ embeds into the Kashiwara–Vergne Lie algebra $\mathfrak{krv}_2$. We give a Brunnian braid interpretation of a related embedding into the symmetric Kashiwara–Vergne Lie algebra $\mathfrak{krv}_2^{\mathrm{sym}}$. More precisely, the map \[ \varphi\longmapsto \bigl(\varphi(-x_0-x_1,x_0),\varphi(-x_0-x_1,x_1)\bigr) \] defines an injective Lie algebra homomorphism from the subalgebra of $\mathfrak{dmr}_0$ satisfying the condition \[ [x_0,\varphi(-x_0-x_1,x_0)] +[x_1,\varphi(-x_0-x_1,x_1)]=0 \] into $\mathfrak{krv}_2^{\mathrm{sym}}$. The proof reformulate the double shuffle and symmetric Kashiwara–Vergne relations through abelianizations of Brunnian Lie algebras associated with the disk and punctured disks. We generalize this inclusion in two directions. First, replacing these abelianizations by higher lower central series quotients yields generalizations of relations and implications among them. Second, we establish explicit identities relating the linear pentagon defect to the stuffle coproduct, the divergence map, and the necklace cobracket.

In this paper, we discuss the 2D convection-diffusion-reaction equation with variable smooth coefficients and the Dirichlet boundary condition on a complicated, thin, and curved domain. We propose the fourth-order compact FDM at every grid point with the uniform Cartesian mesh. For the regular stencil center, we utilize the fourth-order compact 9-point FDM to approximate the solution. According to the preliminary analysis, we use vertical and horizontal transformations to derive fourth-order compact FDMs in 10 cases for all irregular stencil centers. To obtain the left-hand side of the stencil of the fourth-order FDM in each case, we only need to solve an at most $6 \times 24$ linear system which is presented with the explicit formula. The right-hand side of the FDM is constructed in explicit expression for any irregular stencil centers too. To achieve the fourth-order consistency, up to second-order partial derivatives of convection, diffusion, reaction, and source terms are used for the FDM at the regular stencil center, and the FDM at an irregular stencil center only requires first-order partial derivatives of convection, diffusion, reaction, and source terms, and up to third-order derivatives of the Dirichlet boundary function and the parametric expression of the boundary curve. We test challenging domains with 100-leaf, high-curvature, high-frequency, sharply varying, and nearly overlapping boundary curves, the proposed FDM produces the high accuracy and the stable fourth-order convergence rate in $l_2$ and $l_{\infty}$ norms. All stencils of our FDMs have a simple desired structure by only keeping grid points inside $\Omega$ in the standard compact 9-point stencil for both regular stencils and boundary stencils, but without assuming any information outside the domain $\Omega$.

There exist initial segments of both the Dyment lattice and the Dyment-Muchnik lattice that yield Brouwer algebras modeling exactly the intuitionistic propositional calculus. For the Dyment-Muchnik lattice, this result is obtained by constructing a splitting class of enumeration degrees. In contrast, the full Dyment lattice and the full Dyment-Muchnik lattice model the intuitionistic propositional calculus plus the weak law of excluded middle. We also observe that certain naturally definable classes of enumeration degrees, which are downwards closed under enumeration reducibility, fail to form splitting classes.

In a classical paper [21] in 2000, Lutwak-Yang-Zhang established the $L^p$ analog of the Petty projection inequality and the $L^p$ analog of the Busemann-Petty centroid inequality. In Section 7 of [21], Lutwak-Yang-Zhang proposed the important $L^p$ projection centroid conjecture. We give a positive solution to the $L^p$ projection centroid conjecture in this work.

We consider an infinite graph with the vertex set $\mathbb{Z}^2$ and edges connecting the vertices iff the Euclidean distance between the respective points is an integer, and the points do not lie on the same horizontal or vertical. Equivalently, there must exist a Pythagorean triangle with the hypotenuse corresponding to the graph edge and the legs parallel to the axes. We prove that the diameter of this graph is $3$, but surprisingly it appears that the nodes at the maximal (graph) distance of $3$ apart seem to be only those that are geometrically very close to each other. It also appears that the paths of length $2$ connecting geometrically close nodes may need to go through geometrically very distant points. We prove a general relation that generates infinite series of length-$2$ paths, and present the results of our computer experiments. We conclude the paper with a general conjecture about the length-$2$ and length-$3$ paths. We have posed this conjecture to several of the current leading AI models. Remarkably, none of them managed to make any significant progress in proving it.

We study Hamiltonian and symplectic tensor structures in the T-product algebra. We define T-Hamiltonian and T-symplectic tensors and characterize them through their Fourier-domain slices. For T-Hamiltonian tensors we establish the standard block form and the spectral symmetry of T-eigenvalues, while for T-symplectic tensors we derive the inverse and exponential-map properties. Our main result is a constructive T-Williamson normal form for tensors whose Fourier-domain slices are real symmetric positive-definite matrices. We also show that, under the Hermitian symplectic convention adopted here, this decomposition does not extend directly to arbitrary Hermitian positive-definite Fourier-domain slices, and we derive a real-valued recovery criterion under Fourier conjugate symmetry. Numerical experiments verify the construction, exhibit runtime trends consistent with the slice-wise complexity $O(pn^3)$, and illustrate the framework on a Fourier-domain encoding of covariance-matrix families arising in continuous-variable quantum dynamics.

Compression based on asymmetric numeral systems (ANS) combines high encoding and decoding speeds with a compression ratio close to Shannon entropy, while forward modeling of the information source makes it possible to obtain an estimated compressed message size that is less than the entropy. This paper proposes combining these modeling and adaptive coding methods. In addition to ensuring high data processing speeds and compression ratios, this approach enables one to implement the adaptive ANS, which has long remained an important scientific and practical problem.

The Riemann-Roch formula is a cornerstone in the classical theory of algebraic curves. Here we present a novel approach to its proof, by answering a question posed in 2007 by Matthew Baker and Serguei Norine.

We formulate a framework of polynomial diagrams, which are a generalisation of power diagrams (PDs) and anisotropic power diagrams (APDs) allowing for boundaries between cells to be algebraic curves of a prescribed degree. We show that they arise naturally from rephrasing PDs (APDs) as first-degree (second-degree) instances of linear parametrised minimisation diagrams. We also develop an efficient GPU-accelerated framework for fitting polynomial diagrams to image data using Legendre polynomials and by maximising a regularised concave objective function adapted from classical logistic regression literature. A largely self-contained analysis of the optimisation algorithm is also provided, including identification of scale and gauge invariances and the limiting objective function as the regularisation parameter vanishes. We apply the algorithm to fit polynomial diagrams to electron backscatter diffraction images of steel.

We prove the almost sure weak convergence of a stochastic proximal point method for minimizing a convex integral function in the general nonlinear context of complete geodesic metric spaces of nonpositive curvature (so-called Hadamard spaces), solving a problem of M. Bačák. This method, formulated in the context of a mild growth condition on the function which generalizes Lipschitz continuity, was previously only considered in the context of strong metric regularity conditions or in the context of locally compact spaces (all of which immediately induce strong convergence). In particular, this result is novel already over Hilbert spaces. The proof is a combination of a weak almost sure convergence theorem for stochastic processes in Hadamard spaces which confine to a stochastic variant of quasi-Fejér monotonicity, due to previous work of the author, together with a new argument for proving the almost sure convergence of the mean function values of the process towards the minimal value.

In this paper, we consider the subcritical branching random walk in a random environment. We assume the branching and the step jump are independent; and the branching is in random envirenment, i.e., the particles in generation $n$ produce children according the probability measure $F_n\in \mathcal{P}\left(\N_0\right)$, and the $F_n$, $n=1,2,\cdots, $ are i.i.d under the $P_E$. “subcritical" means that $ a:=\E[X_1]\in (-\infty,0)$, where $X_1:=\log \overline{F}_1$ and $\overline{F}_1$ is the mean of $F_1$.

It is shown that the product of two nonscalar conjugacy classes of the special linear group SL$(n,K)$ contains matrices of arbitrary trace if $n \ge 4$ and $K$ is an abitrary field or $n=3$ and $K$ is finite.

A pseudo-Euclidean Novikov superalgebra $A$ is a Novikov superalgebra endowed with a non-degenerate symmetric bilinear form $\langle, \rangle$ such that all left multiplication operators are $\langle, \rangle$-antisymmetric. In this case, the associated Lie superalgebra $(A^{-},\langle, \rangle)$ is a flat pseudo-Euclidean Lie superalgebra. In this paper, we investigate the structure of pseudo-Euclidean Novikov superalgebras. In particular, we introduce a distinguished subclass, called Milnor superalgebras, and prove that any pseudo-Euclidean Novikov superalgebra whose two-sided ideal is non-degenerate belongs to this class. We provide a method for constructing pseudo-Euclidean Novikov superalgebras. We also introduce a double extension procedure for pseudo-Euclidean Novikov superalgebras and show that every such superalgebra with a degenerate two-sided ideal can be obtained via this method. Furthermore, we establish that any pseudo-Euclidean Novikov superalgebra is either a Milnor superalgebra or can be obtained by a sequence of double extensions starting from a Milnor superalgebra. As an application, we provide a complete classification of pseudo-Euclidean Novikov superalgebras of total dimension at most four over the field of real numbers.

In this paper, we derive the multi-peakon dynamical system of a class of Camassa-Holm-type equations with quadratic nonlinearities. We also consider the analytical properties for the Cauchy problem. Firstly, we establish local well-posedness of solutions in Besov spaces and then provide the blow-up criteria. Subsequently, we impose appropriate sufficient conditions on the initial data to guaranty that the corresponding solution either exists globally or blows up in a finite time. Finally, we prove the ill-posedness in the Besov space $B_{2,\infty}^{3/2}$ by utilizing the non-traveling wave solutions.

Computing posterior distributions in large-scale Bayesian linear inverse problems is challenging due to the high dimensionality of the parameter space. In this work, we develop a data-informed framework that shifts the computational focus from the parameter space to the data space. We rigorously characterize an intrinsically low-dimensional data space, establish its isometric embedding into the parameter space, and show that the prior-to-posterior update is confined to a data-informed subspace. This perspective allows posterior inference to be carried out in a reduced data-informed subspace. Based on this formulation, we propose a quotient-space Golub–Kahan bidiagonalization method to construct data-informed Krylov subspaces, and integrate empirical Bayesian inference into the iterative framework, enabling simultaneous hyperparameter estimation and posterior approximation in a matrix-free manner. Numerical experiments on representative problems support the theoretical framework and demonstrate the effectiveness of the resulting method.

We introduce a new propositional logic, called very weak subintuitionistic logic $\mathbf{VF}$, by adapting the relational semantics of Fitting, Marek, and Truszczyński for the pure logic of necessitation $\mathbf{N}$ to the propositional setting. We prove that $\mathbf{VF}$ and its closed negative extensions are sound and complete with respect to this semantics, and that they have the disjunction property and the finite frame property. We also prove that $\mathbf{VF}$ is strictly weaker than the weak subintuitionistic logic $\mathbf{WF}$ of Maleki and de Jongh. Finally, we study modal companions of $\mathbf{VF}$ and its closed negative extensions via Corsi's modified Gödel translation.

We establish a quantum Fisher information (QFI) duality for distributed quantum sensor networks with local phase encoding. For any $N$-qubit probe state, where $N$ denotes the number of sensors, $F_Q(\boldsymbol{w}^\top \boldsymbol{\theta}) + F_Q(\boldsymbol{v}^\top \boldsymbol{\theta}) \leq N$ for all unit orthogonal sensing directions $\boldsymbol{w}$ and $\boldsymbol{v}$, with equality for all equatorial states when $N=2$ and for Greenberger–Horne–Zeilinger (GHZ) states when $N\geq 2$. Heisenberg-limited precision for direction $\boldsymbol{w}$, $F_Q(\boldsymbol{w}^\top \boldsymbol{\theta})=N$, saturates the bound and simultaneously forces zero QFI for all other independent directions. This can be interpreted as the condition for parameter privacy in distributed quantum sensing: attaining Heisenberg-limited precision for the sensing target renders all alternative privacy-intrusive estimations impossible.

By using Brownian motion and stochastic calculus, we establish a second main theorem for holomorphic curves into a projective subvariety $V\subset\mathbb P^n(\mathbb C)$ with an arbitrary family $\mathcal Q$ of $q$ hypersurfaces $Q_1,\ldots,Q_q$ concerning its distributive constant $\Delta_{\mathcal Q,V}$. In our result, the counting functions are truncated to level $H_V(d)-1$, where $d=lcd(\deg Q_1,\ldots,\deg Q_d)$ and $H_V(d)$ is the Hilbert function of $V$. As an application of the second main theorem, we give a uniqueness theorem for holomorphic curves from $\mathbb C$ into $V$ sharing an arbitrary family of hypersurfaces regardless of multiplicity.

Preconditioned optimizers are central to language model training, but their stochastic update rules are usually treated as direct approximations to population preconditioned descent. We show that this view misses two finite-sample biases. First, the gradient and preconditioner are typically estimated from the same minibatch, introducing gradient–preconditioner coupling bias. Second, even when the preconditioner estimate is unbiased, its inverse or inverse-root is generally biased because inversion is nonlinear. We propose a single-batch bias-correction framework that addresses both effects: cross-fitted preconditioning estimates the numerator and preconditioner from independent microbatch groups, while variance-corrected inversion uses microbatch variability to subtract the leading delta-method bias term. The framework applies to diagonal moment, diagonal curvature, and matrix preconditioning methods, instantiated in AdamW, Sophia, and Shampoo. Bias correction reduces held-out pretraining loss on Qwen2.5-0.5B by $0.15$, $0.07$, and $0.11$ nats, respectively; the effects on mixed-quality pretraining and downstream instruction tuning are consistently neutral-to-positive. Together, these results establish bias correction as a practical mechanism for reducing finite-sample update bias and improving the performance of preconditioned optimizers.

We show that for every foliation $\mathcal{F}$ of rank $r$ on a normal projective variety, the optimal constant in the bend-and-break inequality for tangent rational curves is $r+1$. The proof combines the method of Bogomolov–McQuillan and the bend-and-shatter method developed by Jovinelly–Lehmann–Riedl. The proof of the main result of this paper substantially uses generative AI, particularly the Rethlas system.

This paper investigates an initial-boundary value problem for three-dimensional (3D) micropolar fluids in a strip domain, including both the compressible and the (homogeneous and inhomogeneous) incompressible cases in the absence of angular viscosity. The analysis is rendered difficult by two major obstacles: the degeneracy induced by vanishing angular viscosity, and the strong coupling between micro-rotation and velocity fields characterized by a non-dissipative anti-symmetric structure. Moreover, the presence of physical boundaries in the strip domain further compounds these obstacles. While the global well-posedness of the 2D incompressible Cauchy problem has been established in the literature, no results are available for the 3D system and the initial-boundary value problem in both two and three dimensions, particularly in the compressible case. By exploiting the intrinsic structure of the system and establishing delicate energy estimates, we overcome these difficulties and prove the global well-posedness of strong solutions near equilibrium in a strip domain.

In a recent work, Roelands and Tiersma proved that, for a compact convex set $K$, the space $A(K)$ of all real-valued continuous affine functions on $K$, is a JB-algebra if and only if there is a gauge-reversing bijection on $A_c(K)$, the set of positive real-valued continuous affine functions on $K$. In this paper, we show that every such gauge-reversing bijection on $A_c(K)$ is completely determined by the induced bijection on the set of extreme points of $K$.

For an oriented graph $G$, the oriented discrepancy problem concerns the existence of a spanning subgraph of $G$ with a large imbalance between its forward and backward edge orientations. Freschi and Lo proved the Dirac-type Hamilton cycle result in oriented graphs, and asked for an analogue for powers of Hamilton cycles under a minimum-degree condition. We show that, for sufficiently large $n$, every oriented graph $G$ on $n$ vertices with minimum degree $\delta(G)\geq 2n/3$ contains the square of a Hamilton cycle $H$ with $\sigma_{\max}(H)$ guaranteed to exceed a function depending on $\delta(G)$ and $n$.

Estimation under model misspecification arises in many signal processing problems, where the assumed observation model deviates from the true data-generating mechanism due to errors or simplifications. The misspecified Cramér-Rao bound (MCRB) is a widely recognized mean-squared-error (MSE) lower bound for this case, which has originally been used to describe the asymptotic behavior of the misspecified maximum likelihood (MML) estimator. Despite its widespread use, the MCRB lacks a rigorous characterization of the class of estimators for which it is valid. In this paper, we revisit the theory of parameter estimation under model misspecification and re-examine the foundations of the MCRB. We first demonstrate these limitations and examine a naive version of the MCRB, which relies only on local misspecified unbiasedness. We show that this bound is generally not tight and may be unattainable. To obtain a meaningful bound, we develop a new derivation based on the concept of pointwise equivalent models. By maximizing the naive bound for these models, we recover the classical MCRB, now supported by a constructive derivation, an explicit characterization of the associated estimator class, and an equality condition. This formulation establishes a formal link between local unbiasedness conditions and achievable bounds, offering new insights into the MCRB structure and its relevance to practical estimators. Finally, we define the notion of an efficient misspecified estimator and show that if it exists, it is achieved by the MML estimator.