Let $π:(X,L)\rightarrow \mathbb D^*$ be the Fermat family of cubic curves in $\mathbb P^2$. For each $k\geq 1$, we construct a valuatively independent basis for $H^0(X,L^k)$. The construction uses a canonical cost function determined by a Hessian structure on the essential skeleton $\op{Sk}(X,π)$.

Modelling how populations respond to habitat loss is crucial for understanding ecosystem stability, especially when positive interactions among resource species, such as plant-plant facilitation, play a key role. Habitat loss not only reduces available organic nutrients and space for primary producers but also disrupts the positive feedbacks that sustain resource populations, thereby affecting consumer persistence and the overall system's stability. We analyse a cubic planar model describing resource-consumer dynamics with facilitation under progressive habitat loss. Our study characterizes the parameter space and enumerates all the phase portraits within the Poincaré disk under ecologically relevant conditions. We show that the system has a unique stable limit cycle and characterize analytically the heteroclinic bifurcation curve involving the collapse of the resource and the consumer, enabling us to determine how the parameter region sustaining coexistence oscillations narrows under habitat destruction. To further explore these dynamics, we construct a piecewise-linear (PWL) approximation that preserves the system's qualitative behaviour, allowing us to obtain an explicit expression for the heteroclinic bifurcation. Finally, we investigate how extrinsic noise affecting the resource species impacts the overall dynamics, showing that stochasticity can anticipate the onset of the heteroclinic bifurcation causing earlier co-extinctions.

Let $M$ be a complete hyperbolic $n$-manifold, $n\geq 2$. Via integration over geodesic simplices, any closed bounded differential 2-form on $M$ defines a bounded cohomology class in $H^2_b(M)$. It was proved by Barge and Ghys (for $n=2$) and by Battista et al. (for $n>2$) that, if $M$ is closed, then this procedure defines an injective embedding of the (infinite-dimensional) space of closed differential $2$-forms on $M$ into $H^2_b(M)$. We extend this result to the case when the fundamental group of $M$ is of the first kind, i.e. its limit set is equal to the whole boundary at infinity of hyperbolic space (this holds, for example, when $M$ has finite volume). Our argument is different from Barge and Ghys' original one, and relies on the following fact of independent interest: an $L^\infty$ function on the hyperbolic plane is determined by its integrals over all ideal triangles. We prove this fact by way of Fourier analysis on the hyperbolic plane.

We study phase transitions for repulsive-attractive mean-field free energies on the circle. For a $\frac{1}{n+1}$-periodic interaction whose Fourier coefficients satisfy a certain decay condition, we prove that the critical coupling strength $K_c$ coincides with the linear stability threshold $K_\#$ of the uniform distribution and that the phase transition is continuous, in the sense that the uniform distribution is the unique global minimizer at criticality. The proof is based on a sharp coercivity estimate for the free energy obtained from the constrained Lebedev--Milin inequality. We apply this result to three motivating models for which the exact value of the phase transition and its (dis)continuity in terms of the model parameters was not fully known. For the two-dimensional Doi--Onsager model $W(θ)=-|\sin(2πθ)|$, we prove that the phase transition is continuous at $K_c=K_\#=3π/4$. For the noisy transformer model $W_β(θ)=(e^{β\cos(2πθ)}-1)/β$, we identify the sharp threshold $β_*$ such that $K_c(β) = K_\#(β)$ and the phase transition is continuous for $β\leq β_*$, while $K_c(β)<K_\#(β)$ and the phase transition is discontinuous for $β> β_*$. We also obtain the corresponding sharp dichotomy for the noisy Hegselmann--Krause model $W_{R}(θ) = (R-2π|θ|)_{+}^2$ .

It is well-known that any two pure quantum states (in the same Hilbert space) can be mapped to any other using unitary transformations. However, previous approaches to this problem required two explicit bases for the Hilbert space, one each for the initial and target states, and thus their complexity necessarily scales with the dimension of the Hilbert space. In this Letter, we show how to utilize novel algebraic methods to construct a closed-form exponential unitary transformation which achieves this in general, using only a single unitary generator. This construction is independent of any bases and agnostic to the dimension of the Hilbert space. We highlight the usefulness of this tool for studying relationships between systems of pure states in quantum information theory, as well in elementary analyses of quantum circuits and unitary operators.

Stochastic dynamical systems with slow or metastable behavior evolve, on long time scales, on an unknown low-dimensional manifold in high-dimensional ambient space. Building a reduced simulator from short-burst ambient ensembles is a long-standing problem: local-chart methods like ATLAS suffer from exponential landmark scaling and per-step reprojection, while autoencoder alternatives leave tangent-bundle geometry poorly constrained, and the errors propagate into the learned drift and diffusion. We observe that the ambient covariance~$Λ$ already encodes coordinate-invariant tangent-space information, its range spanning the tangent bundle. Using this, we construct a tangent-bundle penalty and an inverse-consistency penalty for a three-stage pipeline (chart learning, latent drift, latent diffusion) that learns a single nonlinear chart and the latent SDE. The penalties induce a function-space metric, the $ρ$-metric, strictly weaker than the Sobolev $H^1$ norm yet achieving the same chart-quality generalization rate up to logarithmic factors. For the drift, we derive an encoder-pullback target via Itô's formula on the learned encoder and prove a bias decomposition showing the standard decoder-side formula carries systematic error for any imperfect chart. Under a $W^{2,\infty}$ chart-convergence assumption, chart-level error propagates controllably to weak convergence of the ambient dynamics and to convergence of radial mean first-passage times. Experiments on four surfaces embedded in up to $201$ ambient dimensions reduce radial MFPT error by $50$--$70\%$ under rotation dynamics and achieve the lowest inter-well MFPT error on most surface--transition pairs under metastable Müller--Brown Langevin dynamics, while reducing end-to-end ambient coefficient errors by up to an order of magnitude relative to an unregularized autoencoder.

We provide a method for parallelizing the branch-and-bound with isomorphism pruning algorithm developed by Margot [Symmetric ILP: Coloring and small integers, Discrete Optimization (4) (2007), 40-62]. We apply our method to classify orthogonal arrays. For classifying all non-OD- equivalent OA(128, 9, 2, 4) and OA(144, 9, 2, 4) our method results in linear speedups. Finally, our method enables classifying all non-OD-equivalent OA(192, k, 2, 4) for k = 9, 10, 11 for the first time.

The study of spanning trees and related structures is central in graph theory, closely connected to understanding functions between finite sets. This paper generalizes the established relationship between rooted trees and eventually constant endomorphisms to a wider context including $k$-tuples of functions among $k$ disjoint vertex sets. We derive a weighted count of eventually constant $k$-tuples, which are characterized by their stabilization to constancy upon iterated composition. This construction is the set-theoretic analogue of the nilpotent cone and offers new insight into the combinatorial structure of cyclic digraphs. By identifying these $k$-tuples with their induced digraphs, we construct explicit formulas for their generating polynomials and analyze the cardinality of the set of eventually constant $k$-tuples. These polynomials are multisymmetric in $k$ sets of variables and can be re-expressed as the character of a representation of the product of general linear groups. We extend the ideas to the more general structures of eventually $N$-cyclic and $λ$-cyclic $k$-tuples, which we define and provide similar theorems for their generating functions and cardinality.

We identify a universal finite-$N$ structure underlying Wilson loop expectations in lattice Yang-Mills, in any dimension $d\geq 2$, for gauge group $\mathrm{U}(N)$, and for arbitrary smooth central plaquette actions. The starting point is a state-sum expansion in plaquette labels by irreducible representations, in which each term factorizes into an action-dependent spectral weight and an action-independent topological coefficient. We then analyze these coefficients in three exact ways: as a gauge/string expansion over decorated spanning surfaces, as a local spin-foam/channel model on the dual incidence graph, and as a universal finite-$N$ master loop equation that closes on the coefficient side. As a consequence, several recent Wilson-action results are recovered as specializations of our broader action-agnostic framework.

We show that jets of initial data can be approximated up to arbitrary order by finite-gap solutions for classes of so-called BKM systems of PDEs introduced by Bolsinov--Konyaev--Matveev, which include classical PDEs such as KdV, Kaup--Boussinesq and Camassa--Holm. Finite-gap solutions are obtained via a finite-reduction map, defined algebraically, which sends solutions of a Stäckel system to solutions of the BKM PDE. For the classes containing KdV and Kaup--Boussinesq we obtain full jet-surjectivity via a triangular structure, whereas for the class containing Camassa--Holm we establish jet-surjectivity on an open set of initial data over $\mathbb{R}$ and a Zariski-open (dense) set over $\mathbb{C}$.

We construct cubic scalar field theory on $λ$-Minkowski space by combining the Batalin-Vilkovisky formalism with harmonic analysis, and produce two inequivalent noncommutative quantum field theories. The braided theory is based on a braided $L_\infty$-algebra whereby covariance dictates a spectral decomposition into cylindrical Bessel functions that diagonalise the angular Drinfel'd twist; in this theory we find the usual logarithmic ultraviolet divergences and confirm the absence of UV/IR mixing. The standard noncommutative theory is based on a classical $L_\infty$-algebra; in this theory we relate the spectral decompositions into plane wave and cylindrical harmonic eigenmodes of the Klein-Gordan operator, we verify the planar equivalence theorem, and we demonstrate a periodic form of UV/IR mixing in which non-planar correlators are generically ultraviolet finite but become non-analytic on an infinite lattice of exceptional momenta.

For time series with long-range temporal dependence, inference for covariance and precision matrices is non-trivial. We propose a Berry-Esseen type Gaussian approximation result that gives a finite-sample bound for the Kolmogorov distance between the infinity norms of the estimation error of sample covariance matrix and the corresponding Gaussian approximation. The method utilizes martingale and m-dependent approximation and relies on constructing triadic blocks. We also establish a bootstrapping result with block sampling method, which preserves validity despite strong temporal dependence. Our results on covariance allow ultra-high-dimensional settings where the dimension of time series can grow sub-exponentially with sample size. Similar results can be built for precision matrix under low-dimensional settings. No assumption is required on the structure of covariance and precision matrices.

Let $(X,Δ)$ be a smooth complex projective simple normal crossing pair of dimension $n\geq 3$ endowed with an everywhere nondegenerate logarithmic conformal tensor. If $K_X+Δ$ is not nef, then precisely one of the following mutually exclusive alternatives occurs: either $Δ=\varnothing$ and $X\simeq Q^n$; or $X\simeq \mathbb{P}^n$ and $Δ$ is a hyperplane; or $n=2m$ is even and $(X,Δ)$ admits a rational maximal isotropic fibration whose geometric generic fibre is the log pair $(\mathbb{P}^m,H)$. If $K_X+Δ\equiv 0$, then, under a Bochner extension principle and an irreducibility assumption on the restricted holonomy of a complete Ricci-flat Kähler metric on $M:=X\setminus Δ$, the existence of a logarithmic conformal tensor with trivial conformal line bundle forces $M$ to be semi-abelian and $(X,Δ)$ to be its toroidal compactification.

We propose a method for the prediction of stationary max--stable random fields with $α$-Fréchet marginal distribution $H_α$. The method is suitable to cope with heavy tails for $α\in(0,2)$ and is (approximately) exact in marginal distributions. It is based on a recent extrapolation approach via level sets which requires no moment assumptions. An explicit connection between the excursion metric and the Davis-Resnick distance is established. The existence of the predictor is proven. The non-uniqueness of the forecast is demonstrated on several examples. The method is tested on multiple simulated time series and random fields as well as applied to real data of annual maximum precipitation.

This paper takes an empirical look at asymptotic runtime growth rates for the most widely used algorithms for solving linear programming (LP) problems across a set of six optimization application areas that are known to produce large and difficult LP models. On the algorithm side, we consider the simplex method, interior-point methods, and PDHG. On the model side, we use a large language model (LLM) to create families of instances in different application areas, allowing us to study model types and sizes that are simultaneously synthetic and realistic. The results indicate that simple regression models typically predict observed runtimes quite well within a model class, and that asymptotic behavior can vary significantly between the different algorithms. This may have a significant impact on which algorithms will be most effective for solving large LP models in the future.

Let $k \in \mathbb{N}$ and let $H_1, H_2, \ldots, H_k$ be simple graphs such that for each $j \in \{ 1, 2, \ldots, k \}$, the vertex set of $H_j$ is $\{ 0, 1, 2, \ldots, n_j - 1 \}$ for some $n_j \in \mathbb{N}$. The ordered Ramsey number $R_\mathrm{ord}(H_1, H_2, \ldots, H_k)$ is the smallest $n \in \mathbb{N}$ for which every $k$-edge-coloring of the complete graph on the vertex set $\{ 0, 1, 2, \ldots, n - 1 \}$ contains $H_j$ as a monochromatic subgraph of color $j$ for some $j \in \{ 1, 2, \ldots, k \}$, with the vertices appearing in the same order as in $H_j$. Inspired by the work of Poljak, we apply the Kissat SAT solver to determine new small two-color ordered Ramsey numbers of various classes of graphs: monotone paths, monotone cycles, alternating paths, stars, complete graphs and nested matchings. In addition, we introduce the cyclic Ramsey numbers $R_\mathrm{cyc}(H_1, H_2, \ldots, H_k)$ as a natural relaxation of the ordered Ramsey numbers, and once again use Kissat to determine various such numbers for the two-color case. By observing structural patterns in the computational results, we determine all ordered or cyclic Ramsey numbers for several pairs of classes of graphs. Furthermore, we obtain some bounds on ordered and cyclic Ramsey numbers where one argument is a connected graph, while the other is a monotone path or a monotone cycle. We also explore how reinforcement learning can be used through the recently developed Reinforcement Learning for Graph Theory (RLGT) framework to obtain lower bounds on ordered and cyclic Ramsey numbers. Finally, we introduce the permutational Ramsey numbers to show how the different Ramsey-type formulations involving standard, ordered and cyclic Ramsey numbers can be unified within a group-theoretic framework.

Let $G = V, E$ be a simple connected undirected graph. A set $X \subseteq V$ is \emph{geodesically convex} if for any pair of vertices $x, y \in X$, all vertices on all shortest paths in $G$ from $x$ to $y$ are contained in $X$. A set $H \subseteq V$ is said to be a {halfspace} if both $H$ and its complement (denoted by $H^c$) are convex. Given two sets $A, B \subseteq V$, the { halfspace separation} problem asks if there exist complementary halfspaces $H, H^c$ such that $A \subseteq H$ and $B \subseteq H^c$. The halfspace separation problem is known to be NP-complete for the geodesic convexity of general graphs. We show that geodesic halfspace separation is polynomial for weakly bridged graphs, pseudo-modular graphs, and the basis graphs of matroids.

The binding number $b(G)$ of a graph, introduced by Woodall [J. Combin. Theory, Ser. B, 1973], is a central topic of both structural and extremal graph theory. It is closely related to fundamental combinatorial and structural properties of graphs. The graphs with $b(G)\geq1$ exhibit strong expansion properties and a highly connected global structure. In contrast, the structure for graphs with $b(G)<1$ remains far less well understood. Kane et al. [J. Graph Theory, 1981] proved that if $b(G)<1$, then every binding set of $G$ is independent. Goddard and Swart [Quaest. Math., 1990] showed that if $b(G)\leq1$, then the toughness $τ(G)\leq b(G).$ This makes it particularly interesting to investigate extremal problems for graphs with \(b(G)<1\). For any integer $r\geq1,$ we completely characterize the unique extremal graph that maximizes the size (spectral radius) among all graphs of order $n$ satisfying $b(G)<\frac{1}{r}.$ For any bipartite graph $G=(X,Y)$ on $n$ vertices, it is readily seen that $b(G)\leq\min\{|X|/|Y|,|Y|/|X|\}\leq1.$ Notably, the complete balanced bipartite graph $K_{\frac{n}{2}, \frac{n}{2}}$ achieves the maximum size (spectral radius) among all bipartite graphs with $b(G)=1$. In this paper, we completely determine the extremal graphs maximizing the size or the spectral radius among all bipartite graphs with $b(G)<\frac{1}{r}$, where $r\geq1$ is an integer.

Let I be a finite partially ordered set and let (Sym(Δi),Δi)i be a sequence of symmetric groups indexed by I. Construct the generalised wreath product (F, Δ) on this sequence of permutation groups. We determine the minimum number d(F) of generators required for this generalised wreath product.

We prove that for a generic family of circle diffeomorphisms every parameter value that corresponds to an irrational rotation number is approximated by parameter values for which the diffeomorphisms have arbitrarily large finite numbers of periodic orbits. This phenomenon implies that families where irrational rotation numbers appear are not weakly structurally stable. Moreover, we prove that any locally residual set of one-parameter families with nonconstant rotation number yields a continuum of weak equivalence classes of families.

This paper addresses the numerical optimization of proportional-integral-derivative (PID) controllers for linear time-invariant systems with delays, where the derivative action is implemented using a low-pass filter. While performance assessment is often based on the spectral abscissa of the ideal PID-controlled system, the inclusion of a derivative filter fundamentally alters the closed-loop spectral properties and cannot be treated as a post-processing step. In particular, the spectral abscissa of the filtered closed-loop system may differ significantly from that of its unfiltered counterpart, potentially affecting both stability and performance. We propose a systematic numerical design framework in which the PID gains and the filter constant are optimized simultaneously by directly minimizing the spectral abscissa of the filtered closed-loop system. Treating the filter as an integral part of the control design allows us to reconcile robustness at high frequencies, in the sense of mitigating fragility issues due to approximate identities, with performance at low frequencies, in addition to counter measurement noise amplification. At the end of the presentation, numerical examples illustrate the proposed approach and highlight the benefits of controller-filter co-design. The results apply to general linear systems with input and/or state delays and are valid for both single-input single-output (SISO) and multi-input multi-output (MIMO) configurations.

In the winter semester of 1890--1891 Adolf Hurwitz delivered a lecture course at the Albertina University in Königsberg entitled -Theorie der algebraischen Gleichungen-. These lectures contain a particularly clear presentation of the ideas of Evariste Galois and, in particular, a proof of the fundamental theorem of Galois theory formulated in the language of substitutions. The present paper analyzes Hurwitz's treatment of this result on the basis of his lecture notes preserved in the ETH Library in Zurich (Hs 582:66), together with material from his Mathematisches Tagebuch 23 (Hs 582:23). After placing the Königsberg lectures in their historical context, we give an overview of their mathematical content and reconstruct in detail Hurwitz's argument leading to the fundamental theorem.

We consider an iterative eigensolver for Schrödinger equations that constructs low-rank approximations of eigenfunctions with accuracy-adapted ranks, with particular focus on fermionic Schrödinger equations in second-quantized form and on matrix product state approximations enforcing particle number conservation. We provide a complete analysis of a solver based on preconditioned inverse iteration combined with rank truncation and propose a generalization to subspace iteration for the joint approximation of several eigenspaces. The practical performance of the method is illustrated by numerical tests for several model problems.

We propose a concept of dissipative weak (DW) solutions for the Navier-Stokes-Korteweg (NSK) system and prove conditional convergence of a structure-preserving finite volume scheme towards such a solution. DW solutions provide a generalized solution concept in computational fluid dynamics and have recently attracted significant attention. They provide an extension of the famous Lax Equivalence Theorem to nonlinear problems, i.e. consistency and stability of a numerical scheme imply convergence. Our work builds on recent advances where convergence towards DW solutions of structure-preserving schemes has been established for the Euler and Navier-Stokes equations. Indeed, we prove convergence of a recently proposed FV scheme by leveraging its conservation and dissipation properties as well as its consistency.

We establish an isoperimetric type inequality for the level sets of functions in fractional Sobolev spaces. This answers a question posed by the first author in a previous paper. To obtain it, we work out a slight modification of some estimates for nonlocal interaction functionals established by Savin and Valdinoci. We also show how said isoperimetric inequality leads to the Hölder continuity of functions in (weak) fractional De Giorgi classes.

We find an exact nonequilibrium steady state of an open dissipatively driven XXZ spin-1/2 chain with source or sink spin bath at one end and an arbitrary boundary field at the other end.

We show that the Hilbert space of the Koopman--von Neumann formulation of Landau--Ginzburg theory is parametrised by a real Monge--Ampère domain, which carries a natural pre-Frobenius. Restricting to finite-dimensional (dually flat) exponential families, the parameter space becomes a Monge--Ampère domain and a pre-Frobenius manifold. Our main theorem proves that for every Berglund--Hübsch--Krawitz mirror pair of Calabi--Yau orbifolds arising from an invertible polynomial, this Monge--Ampère domain (the open probability simplex) is the base of a Lagrangian torus fibration on both the original and the mirror hypersurface, with dual fibres in the sense of Strominger--Yau--Zaslow. The construction recovers the SYZ picture from the Landau--Ginzburg--Koopman--von Neumann framework. In particular, this proves the Kontsevich--Soibelman conjecture (2001) for all Berglund--Hübsch--Krawitz mirror pairs: the base of the SYZ fibration is a Monge--Ampère domain (the open simplex), and the torus fibrations on the mirror pair are dual. A toy model of cones of positive definite matrices illustrates the geometric structures.

Using the LP algebraic toolkit, Conway's original topograph is rethought of as a cluster construction, paving the way for a wider topography based on mutation-type local rules. As a remarkable application of such cluster-driven upgrade, both the process of analytic continuation for Painlevé VI and the reduction algorithm for quadratic forms are endowed with the Laurent phenomenon. En passant, the rattlesnake is defined so to complete the bijection between snake graphs and rationals to the whole of $\mathbb{Q}$.

We present a geometric framework for reconstruction problems based on Vaisman foliations and Atiyah--Molino sequences. Independent projections induce transverse foliations and dual connections; vanishing torsion and curvature duality guarantee unique, path-independent reconstruction, while obstructions yield non-associative quasigroupoids. Toric symmetry provides equivariant uniqueness. Applications to generative AI imputation and cryo-electron microscopy demonstrate the framework's practical power, unifying differential geometry with data-driven inverse problems.

In this article, we derive explicit combinatorial formulas, depending only on $q$, for the Wedderburn decomposition of the rational group algebras of the finite linear groups $\operatorname{SL}_2(q)$ and $\operatorname{PSL}_2(q)$. Furthermore, we also determine the number of pairwise non-isomorphic simple $\mathbb Q G$-modules of each possible dimension for $G$ being either $\operatorname{SL}_2(q)$ or $\operatorname{PSL}_2(q)$.