Papers

We prove that deciding whether there is a surjective homomorphism from an arbitrary finitely presented inverse monoid onto the bicyclic monoid is $\mathsf{NP}$-complete. As part of the proof, we show that an extension of existential Presburger arithmetic which involves greatest common divisors on $n$ arguments is in $\mathsf{NP}$, extending a recent result of Défossez, Haase, Mansutti, and Pérez (SODA 2024).

We introduce three homology theories for hypergraphs, namely $\Gamma$-homology, $\Box$-homology, and $\times$-homology, and show that they are pairwise non-isomorphic and distinct from the embedded homology of hypergraphs. We further introduce a notion of homotopy for hypergraphs that extends the discrete homotopy theory of graphs. Among the homology theories considered, we prove that $\Box$-homology is invariant under this homotopy, whereas $\Gamma$-homology and $\times$-homology fail to satisfy homotopy invariance. Based on excision, we also identify a distinctive structural behavior exhibited by $\Gamma$-homology that further differentiates it from $\Box$-homology.

We study C*-algebras generated by representations of product systems over unital subsemigroups of amenable discrete groups. In the first part, we show that exactness of the generated C*-algebra is equivalent to exactness of the cores over the constructible ideals. Nuclearity of the generated C*-algebra is equivalent to nuclear embeddability of the cores in the fixed point algebra. In the second part, we further study injective equivariant Fock covariant representations, and whether the properties of the cores are inherited from properties of the coefficient C*-algebra, as in the case of compactly aligned saturated product systems over right LCM semigroups. If the semigroup is finitely aligned and the product system is saturated and compactly aligned, then exactness of the cores is equivalent to exactness of the coefficient C*-algebra. If in addition the semigroup is strongly finitely aligned, then nuclear embeddability of the cores is equivalent to nuclear embeddability of the coefficient C*-algebra. These results do not hold without the compact alignment hypothesis, even for right LCM semigroups.

In this paper, we establish necessary and sufficient representation theorems for the classes $\widetilde{\mathcal{U}}_{\mathrm{top}}$, $\widetilde{\mathcal{U}}_{\mathrm{bot}}$, $\widetilde{\mathcal{U}}_{\mathrm{max}}^{r}$, and $\widetilde{\mathcal{U}}_{\mathrm{min}}^{r}$ of uninorms on bounded trellises, extending the corresponding bounded-lattice results. For a non-extremal neutral element $e$ belonging to no non-trivial cycle, we use a common decomposition of the elements incomparable with $e$; every neutral element of a uninorm is shown to be middle-transitive. Each uninorm in $\widetilde{\mathcal{U}}_{\mathrm{top}}$ (respectively, $\widetilde{\mathcal{U}}_{\mathrm{bot}}$) is represented by an interior operator (respectively, a closure operator), a component uninorm, and an increasing, associative, and commutative operation, with all components uniquely determined. Two further conditions characterize the subclasses $\widetilde{\mathcal{U}}_{\mathrm{top}}^{\star}$ and $\widetilde{\mathcal{U}}_{\mathrm{bot}}^{\star}$. For $\widetilde{\mathcal{U}}_{\mathrm{max}}^{r}$ (respectively, $\widetilde{\mathcal{U}}_{\mathrm{min}}^{r}$), we identify the pairs in $I_e^3\times I_e^3$ whose values lie in $[0,e[\cup I_e^1$ (respectively, $]e,1]\cup I_e^2$) or equal $e$. The pairs producing $e$ satisfy a symmetric unique-partner condition, while the remaining values are encoded by a partial operation. Regional order compatibility and explicit associativity conditions then yield necessary and sufficient representations. Finally, we compare the four classes, relate them to the corresponding bounded-psoset classes, prove their bounded-lattice specialization under transitivity, and provide finite examples on proper trellises illustrating the additional compatibility conditions.

We determine the first extension groups between simple restricted modules for the restricted simple Lie algebras of Cartan type over an algebraically closed field of characteristic $p\ge5$, which answers the restricted case of a problem of Benkart and Feldvoss. Extensions from parabolically induced modules are computed by finite restricted-cohomology complexes of the grading parabolic, and for the exceptional modules we obtain closed formulas in all four families. Neither the trivial module nor any nontrivial exceptional simple has self-extensions. In characteristic seven we determine the quiver of $S(3;1)^{(1)}$, which has 49 vertices and 566 arrows.

We propose a new variant of a biased random walk on a Galton-Watson tree, in which some vertices have infinitely many children. These vertices induce a natural regeneration structure, which allows to study differentiability and also monotonicity of the speed of the random walk in the bias parameter. Additionally, we show that the speed in this model is the limit of the speed of random walks on appropriate locally finite trees.

The Worley-Sagan insertion and Haiman's mixed insertion are insertion algorithms for shifted Young tableaux, and each of them gives a Robinson-Schensted-type correspondence between the symmetric group of degree $n$ and a set consisting of certain pairs of same-shape shifted Young tableaux with $n$ cells. It is a known fact that these two insertions are dual to each other. Our purpose is to give an extension of these two insertions without losing the duality relationship. The extended ones will be insertions producing pairs of shifted tableaux from colored permutations. Our extension of the Worley-Sagan insertion is different from the restriction of Sagan's own "Knuth version" to colored permutations. In proving the duality between our extended insertions, we "embed" them into Shimozono and White's doubly mixed insertion for unshifted tableaux by "doubling" shifted tableaux and use the self-duality of the doubly mixed insertion shown by Shimozono and White.

Kuperberg conjectured in 1983 that every convex body $K$ in the plane is contained in a quadrilateral of area at most $\frac{3}{\sqrt5}\,|K|$, the extremal bodies being the affine-regular pentagons. On the basis of numerical experiments we propose a stronger conjecture of a purely polygonal nature: every convex polygon $P$ is contained in a quadrilateral of area at most $\frac{3}{\sqrt5}\,|P|$ whose sides are parallel to sides or diagonals of $P$. This chord conjecture implies the inequality conjectured by Kuperberg. For pentagons it is a theorem of Hong, Ismailescu, Kwak and Park, of which we give a short proof by area identities. We also show that the classical bound $\sqrt2$ remains valid for quadrilaterals with sides parallel to chords; the proof is similar to Ismailescu's proof of this bound, but it starts from an inscribed quadrilateral of maximal area instead of a circumscribed quadrilateral of minimal area. We close with a discussion of the difficulties in reaching the constant $\frac{3}{\sqrt5}$.

We study weighted coefficient estimates for the Laguerre transform in terms of moduli of smoothness generated by the Laguerre translation. For every integer $r\ge1$ and $1\le p\le2$, a weighted $\ell^{p'}$ norm of $\min\{1,nt\}^{r}\widehat f_\alpha(n)$ is bounded by the modulus of order $r$; for $2\le p\le\infty$ the reverse direction holds on the same weighted sequence scale. At $p=2$ these estimates yield a two-sided equivalence, and testing individual Laguerre polynomials proves optimality of the multiplier weight up to constants. The proof combines Hausdorff–Young inequalities with an averaged lower bound for the translation multiplier. As consequences we recover Jackson- and Bernstein-type estimates and the known $L^2$ approximation characterisation of the Lipschitz classes for $0<\beta<r$. We also give the corresponding one-sided coefficient-tail criteria for general $p$, together with saturation and endpoint statements.

Adaptively Implicit-Explicit (AdImEx) time stepping provides stability for large time steps for mass-conservative transport, making it attractive for the numerical representation of advection in weather and climate prediction. Existing AdImEx transport schemes become first-order accurate in the large-time-step, implicit regime and may require multiple sparse matrix solutions per time step. This work introduces AdHImEx, a new AdImEx scheme that addresses both limitations simultaneously. AdHImEx provides second-order accuracy for large Courant numbers while requiring only a single matrix solution per time step. It is a Runge-Kutta scheme that blends third-order accurate explicit time stepping and Crank-Nicolson implicit time stepping to provide numerically verified stability for Courant numbers up to 100. AdHImEx is designed for flows that are predominantly explicit, activating implicit time stepping only where large Courant numbers occur. It retains the full efficiency and third-order accuracy of the explicit time stepping in regions with small Courant numbers. In this work, it is combined with a fifth-order accurate finite-volume discretisation in space for mass conservation. A second contribution of this work is a novel stage-dependent treatment of the divergence operator that preserves constancy despite different implicit and explicit Runge-Kutta stage time steps and spatially varying implicitness, removing spurious divergence. This innovation is applicable to other AdImEx time stepping schemes. Convergence, transport, and efficiency tests demonstrate substantial improvements over previous first-order AdImEx schemes, little phase and amplitude error, accurate transport on locally refined meshes with large Courant numbers, and promising efficiency characteristics for atmospheric transport.

In this paper, we prove a uniform approximation theorem with interpolation for complete non-orientable minimal surfaces of finite total curvature in the Euclidean space $\mathbb{R}^n$ $(n\ge 3)$. In particular, we establish the existence of such surfaces with the complex structure of any open conformal non-orientable surface of finite topology and parabolic type.

In time-varying D-optimal subset selection, the number of replaced coordinates does not measure physical displacement: when the optimal selection for a set of mobile sensors changes, a single sensor may have to cross the entire graph. We therefore measure movement by a bottleneck matching distance on a fixed mobility graph, and define the distance margin at radius P as the smallest old-objective loss among configurations farther than P from the current maximizer. Our main result shows that if the projector drift is small and the distance margin at a chosen radius exceeds an explicit threshold, then every maximizer of the new objective lies within that radius, without tie-breaking rules or uniform conditioning assumptions over subsets. Moreover, we show that separation information of this kind is genuinely needed: a weighted-path construction shows that, at a fixed baseline eigengap, an arbitrarily small perturbation can move an optimal sensor from one end of the graph to the other, so no bound depending only on perturbation size and eigengap can vanish with the perturbation. We also give a finite certificate, conditions under which local and Voronoi restrictions retain a global optimum, and oversampled and regularized extensions, and we identify the learned dictionary criteria to which the results apply.

We give a general principle that converts fixed-uniformity bounds for forbidden traces into bounds with polynomial dependence on the uniformity. More precisely, let $H$ be a fixed set system on $h\ge1$ vertices, and suppose that, for some $\alpha\ge0$, $\operatorname{ex}_j(m,\operatorname{Tr}(H))=O_{H,j}(m^\alpha)$ for every fixed integer $j\ge1$. Then, for every $\varepsilon>0$, there is a constant $C_{H,\alpha,\varepsilon}$ such that \[ \operatorname{ex}_r(m,\operatorname{Tr}(H)) \le C_{H,\alpha,\varepsilon} r^{h-1-\alpha+\varepsilon}m^\alpha \] for all $m\ge r\ge2$. In particular, for trace-$C_4$-free hypergraphs and every $\varepsilon>0$ there is a constant $C_\varepsilon$ such that \[ \operatorname{ex}_r(m,\operatorname{Tr}(C_4)) \le C_\varepsilon r^{3/2+\varepsilon}m^{3/2} \] for all $m\ge r\ge2$. We also construct trace-$C_4$-free $r$-graphs showing that \[ \operatorname{ex}_r(m,\operatorname{Tr}(C_4)) \ge c r^{1/2}m^{3/2} \] for an absolute constant $c>0$, for every fixed $r\ge3$ and all sufficiently large $m$ (depending on $r$).

We prove that, on a Galton-Watson tree with almost surely finite offspring and infinite offspring mean, the random interchange process has critical parameter zero for infinite cycles, conditionally on survival. Combined with the finite-mean strict inequality established in arXiv:2503.03319, this gives a dichotomy for locally finite Galton-Watson trees with offspring mean in $(1,\infty]$: the loop and link critical parameters coincide exactly in the infinite-mean case. The proof reduces the tree problem to a finite permutation estimate derived from peeling estimates for random surfaces.

We study linear Maxwell's equations on a cuboid that is split into two parts by a planar interface, across which the tangential magnetic field jumps proportionally to the tangential electric field. Conditions of this type arise in the modeling of atomically thin metamaterials, where the sheet is replaced by an effective surface current. Working in a state space that admits a surface charge on the interface, we show that the problem is governed by a contraction semigroup. To this end, we provide density results, Weber-type inequalities, and trace estimates for the underlying function spaces. We further prove a hidden regularity result, stating that the fields in the domain of the Maxwell operator are piecewise Sobolev regular of first order. Since the damping acts on the interface only, the energy does not decay for all initial fields. We decompose the dynamics into a unitary part, carried by the eigenmodes that are invisible to the damping, and a strongly stable part. In the non-refracting case with a non-resonant interface position, we characterize the decaying initial fields by two moment conditions.

Let M be a compact orientable simple 3-manifold, and let F be a component of the boundary of M with genus at least two. Suppose that alpha and beta are separating slopes on F and that both M(alpha) and M(beta) are reducible. We prove that the distance between alpha and beta is zero. Thus the reducible-reducible case of the conjecture of Qiu and Zhang on separating degenerating slopes holds in arbitrary boundary genus.

Autoformalisation is increasingly used to verify mathematical texts, including those generated by AI, as in OpenAI's announced proof of blow-up of solutions to the Navier-Stokes equations. In this process, an AI system translates the text from a natural language (NL) into a formal language such as Lean. Once this translation is done, the argument expressed in the formal language can easily be mechanically verified. The purpose of this article is to demonstrate why this process may offer no confidence in the original NL argument, owing to the various difficulties in performing the translation semantically faithfully. In particular, we highlight that the problem of resolving ambiguities in mathematical NL text, which is necessary in order to provide semantically faithful translation, is arbitrarily high up in the Solvability Complexity Index (SCI) hierarchy/arithmetical hierarchy (the SCI $= \infty$). Hence, informally, providing semantically faithful AI autoformalisation is harder than any computational problem including the Halting problem (which has SCI $= 1$). To demonstrate the effect of this result we provide several examples of AI mistranslations of NL statements and proofs into Lean in practice, resulting in mismatches between NL proofs and their Lean `verifications'. These include OpenAI's announced Navier-Stokes proof. In particular, we show that the formalised Lean proof does not correspond to the NL proof of blow-up of solutions to the Navier-Stokes equations.

We study a one-dimensional multispecies TASEP model with each particle type having an associated speed parameter and faster particles overtaking slower ones with the rate being the positive part of their speed difference. After expressing the sorting time of a finite system given a reverse-speed initial condition as the maximum of sorting times for all different fast-slow cuts, we prove that forbidding fast-fast and slow-slow overtakes does not alter the law of the sorting time. This allows for an LPP formulation and an analytic expression of the sorting time.

We consider the adjacency matrix of a uniformly random simple $d$-regular graph on $N$ vertices. For every fixed $d\geq3$ and every fixed bulk energy, we prove that the rescaled eigenvalue point process converges to the $\mathrm{Sine}_1$ process with intensity $1/\pi$. We also establish universality of consecutive gaps at deterministic bulk labels and of fixed-energy correlation measures. The eigenvector input consists of polynomial estimates for mixed fourth moments on a uniformly sampled star. These estimates prevent loss of variance in the qualitative joint Gaussian limits of Backhausz–Szegedy, yielding independent variance-one Gaussian waves for fixed tuples at nearby energies. Exact switching identities then give a stationary marked flow and the microscopic loop hierarchy, which identifies the point-process limit. The appendices prove the stronger correlation and gap statements using counting-moment bounds and conditional log-gas laws for tagged limits.

We prove that the centered Hardy–Littlewood maximal operator is not of weak type $(p,p)$ for any $1\le p<\infty$ in three solvable settings: the lamplighter group $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}$ with the switch–walk–switch word metric, the Baumslag–Solitar groups $BS(1,n)$, $n\ge2$, with their digit word metrics, and the three-dimensional Sol group with any left-invariant Riemannian metric and its Riemannian volume. To the best of our knowledge, we give the first examples of finitely generated groups, equipped with word metrics and counting measure, for which the centered Hardy–Littlewood maximal operator fails to be of weak type $(1,1)$.

We study optimal control problems governed by a system of semilinear reaction-diffusion equations. The assumptions on the governing system are formulated so that our results apply to several models, including the Brusselator, Schnakenberg, and Gray–Scott systems. We prove the existence and uniqueness of a solution to the governing system without imposing any non-negativity assumption on the data. This allows us to study unconstrained, control-constrained, and state-constrained optimal control problems. For all these problems, we derive first-order necessary and second-order sufficient optimality conditions. The control acts on the activator equation through the boundary. Numerical examples demonstrate that both the inhibitor and the activator can be efficiently controlled.

We prove that if a compact Kähler manifold admits a Hermitian metric with quasi-negative holomorphic sectional curvature for the Chern connection, then its canonical bundle is ample. The Hermitian metric realizing this curvature need not be Kähler. This proves the Kähler case of the quasi-negative conjecture of Yang and Zheng.

In this paper we investigate biconservative and biharmonic hypersurfaces in product spaces $L^m(\varepsilon)\times {\mathbb R}$, where $L^m(\varepsilon)$ denotes the space form ${\mathbb S}^m$ or ${\mathbb H}^m$ of constant sectional curvature $\varepsilon=1$ or $\varepsilon=-1$ respectively. We obtain their complete description in the case of isoparametric hypersurfaces and hypersurfaces of constant angle. Moreover, we prove some classification results for biconservative and biharmonic hypersurfaces for which the gradient of the mean curvature function is a principal direction. In this context we obtain new examples of complete, non-cylindrical, non-totally umbilical biconservative hypersurfaces in ${\mathbb H}^m\times {\mathbb R}$. Then we prove a rigidity theorem related to these examples. Finally, we prove a global non-existence theorem for complete proper biharmonic hypersurfaces in ${\mathbb H}^m \times {\mathbb R}$ such that the square of the mean curvature function is bounded and the Ricci curvature is bounded from below.

We introduce mean-field Gibbs–Ornstein–Uhlenbeck (MFGOU) processes, a class of discrete-time Markov processes with increments of order $1/n$, whose conditional means approximate a smooth vector field. If the field has a unique attracting zero and satisfies a global Lyapunov condition, we prove \[ \tau_{\mathrm{mix}} = \frac{1}{\lambda_1}n\log n+O(n), \] where $\lambda_1$ is the smallest eigenvalue of the drift Jacobian at the attractor. Mean-field block measures form an important subclass encompassing mean-field spin systems and projected Gibbs samplers for Latent Dirichlet Allocation (LDA) and related mixture models. The leading mixing constant has a spectral expression involving the free-energy Hessian and the covariance of single-step dynamics. A block-refinement argument transfers projected-chain results to the full chain under a mild spectral condition, preserving this constant. The theory recovers known cutoff locations for Curie–Weiss Ising and Potts models and complete multipartite Ising models, gives an external-field formula for Curie–Weiss Ising, and covers regular-graph interactions. It yields explicit $n\log n$ asymptotics for multinomial mixtures and partially collapsed LDA. For full LDA, mixing is determined conditionally on the stationary-point structure of the free energy, distinguishing metastability from topic-label switching. The framework also applies to continuous-state Gibbs sampling, constant-step stochastic gradient descent, and frequently refreshed Hamiltonian Monte Carlo, rigorously for truncated Gaussian increments and heuristically without truncation. The proof combines variance and mean estimates with path coupling allowing slight expansion in expected distance. After rescaling, the coupling distance is a nonnegative supermartingale whose quadratic variation gives a positive probability of coalescence.

Nonexistence of the maximum likelihood estimate (MLE) under separation is treated as a solved problem for binary logistic regression and as a scattered collection of model-specific results everywhere else. We show that it is one phenomenon with one criterion. In a latent polyhedral categorical response model, every observed outcome corresponds to a polyhedral event in latent variables whose faces shift linearly with the parameter. Random-utility choice, cumulative-link, ranking, multivariate binary and ordinal, sequential, adjacent-category logit, and fixed-score stereotype models belong to this family. The likelihood sees each observed factor only through the columns of its threshold map, the structure vectors. A finite MLE exists if and only if the pooled structure vector set has overlap. Sufficiency requires only continuity, necessity requires only strictly threshold-increasing probabilities, and neither likelihood concavity, exchangeability, nor full design rank is needed. Positive strictly log-concave latent densities and threshold-identifiable polyhedra then yield uniqueness on the estimable span. A single linear program determines whether overlap holds, and convex cone geometry measures the dimension of separation. An application using cumulative-link models for willingness to share health data identifies observations and model terms associated with nonexistence and illustrates how the diagnostics inform model revision and sensitivity analysis.

We define an intrinsic unital completely positive (ucp) map from the Cuntz algebra to the Pythagorean with finitely many generators. This permits to lift states and cyclic representations. This process restricts to the Cuntz-Toeplitz algebras and the contraction algebras. Taking inductive limits we obtain an effective machinery for constructing states on the infinite Cuntz algebra $\mathcal O_\infty$. Combining this approach with previous techniques of the first author we obtain large families of pairwise inequivalent irreducible representations of $\mathcal O_\infty$: one family being in an obvious bijection with the unit sphere of Hilbert's space.

For two bigraded spaces $k^{m||n}$ and $k^{n\wedge m}$ equipped with appropriate quadratic Poisson structures, Koszul duality identifies the mixed Poisson differential calculi and, in the unimodular case, their Poisson cohomologies as Batalin–Vilkovisky algebras. In the current paper we prove that Koszul duality induces an isomorphism of Batalin–Vilkovisky algebras between their Poisson cohomologies when the modular derivation of one algebra is diagonalizable, including non-unimodular cases.

Phase-field equations are fundamental examples of gradient-flow systems and require numerical solvers that preserve intrinsic energy dissipation. In this work, we propose a neural network-based method that combines an SAV time discretization with a mesh-free neural representation. We establish the unconditional energy stability of the underlying SAV semi-discretization, quantify the energy defect caused by solver residuals, and derive a conditional high-probability error estimate for the Allen–Cahn equation. Numerical experiments show that our approach consistently exhibits energy-stable behavior, while outperforming existing neural network-based solvers.

This paper studies an entropy Bregman alternating direction method of multipliers (BADMM) with logarithmic dual update for solving discrete optimal transport (OT) problems. The algorithm uses simple row and column normalizations without inner iterations, allowing efficient CPU and GPU implementations. However, the convergence of this method for general OT problems has not been established. Starting from a strictly positive feasible primal point and a zero multiplier, here we establish the $R$-linear convergence of the primal–dual sequence for general OT costs with any fixed multiplier stepsize in $(0,2)$ and a sufficiently large fixed penalty, even when the primal limit lies on the boundary. Our analysis further characterizes the primal limit as the entropy Bregman projection of the initial point onto the optimal solution set and establishes strict complementarity of the limiting OT primal–dual solution. For separable costs on two-dimensional grids, we exploit the separable structure to accelerate matrix–vector products and reduce storage requirements. Numerical experiments show competitive performance of our BADMM for general cost matrices with substantial speedups over state-of-the-art solvers, including HOT and HALO, for squared Euclidean costs on two-dimensional-grid marginals. The largest tested problem, with $2048\times2048 \approx 4.19$M grid points in each marginal, is solved in approximately 100 seconds on a single GPU.

We study the Cauchy problem for a semilinear heat equation driven by the fractional harmonic oscillator. We establish local and global well-posedness in Orlicz and Morse–Transue spaces, based on sharp semigroup estimates and an integral compatibility condition between the Young function and the nonlinearity. We identify a critical Young function separating local well-posedness from instantaneous nonexistence for suitable data. Finally, we prove a confinement effect: under a Dini condition at the origin, small data yield global solutions decaying at the sharp exponential rate, with no Fujita-type restriction, in contrast with the corresponding free equation.