Papers

We express the local-equivalence classes of real Seiberg–Witten Floer homotopy types for spin Seifert $3$-manifolds with certain odd involutions in terms of the Fukumoto-Furuta $w$-invariant. The proof uses real orbifold Bauer-Furuta invariant for a class of spin $4$-orbifolds constructed by Fukumoto-Furuta-Ue. For torus knots, we prove that the local-equivalence classes of real Floer homotopy types associated with their $2^k$-fold cyclic branched covers are eventually periodic in $k$. Finally, we prove that the integer-valued concordance homomorphisms $\{8\delta_R^{(k)}\}_{k\geq1}$ are linearly independent.

We analyze pattern formation from a homogeneous steady state in a one-component nonlocal adhesion model with population pressure and degenerate mobility. First, using linear stability analysis, we derive the instability threshold and a selection rule for the fastest-growing mode, and elucidate the mechanism by which the selected wavenumber shifts toward lower wavenumbers as the mean density increases. We then perform a weakly nonlinear analysis near the critical adhesion strength and derive an explicit expression for the Landau coefficient in the Stuart–Landau equation. This expression shows that the critical bifurcation is classified as supercritical or subcritical according to the mean density, the nonlinear exponent, and the second-harmonic response ratio of the kernel. Furthermore, we show that a large nonlinear exponent promotes a transition to subcriticality and confirm, through numerical bifurcation analysis and time-dependent simulations, a bifurcation structure with a fold point and the formation of mesa patterns. Finally, through the energy limit as \(m\to\infty\), we relate the observed mesa profiles to a capacity-constrained limiting structure.

Semi-supervised classifiers are commonly trained from samples in which all features are observed but some class labels are missing. When label missingness is independent of the observed data, unavailable class memberships reduce Fisher information relative to a completely classified sample. We study a different regime in which the probability of label missingness depends on posterior classification uncertainty, so that the observed missing-label indicators can themselves carry information about the Bayes decision boundary. Building on the conditionally weighted information decomposition of Ahfock and McLachlan, we develop this phenomenon for a two-component exponential mixture. Although the exponential model is non-Gaussian, asymmetric, and supported on the positive half-line, its log-posterior odds remain linear in the feature. We derive Bayes' rule and its exact error rate, formulate entropy-logistic and squared-discriminant missingness mechanisms, and obtain the full partially classified likelihood. We then derive a decomposition of the Fisher information into the complete-data information, the conditionally weighted loss due to missing labels, and the information contributed by the missing labels. Numerical quadrature identifies regions in which the full likelihood classifier has asymptotic relative efficiency above or below one. Monte Carlo experiments with finite training samples broadly support the population calculations, with the largest departures from the asymptotic predictions occurring near the transition at which the relative efficiency crosses one.

A textbook result in information theory is that linear encoders achieve the entropy for lossless compression of Bernoulli source with parameter $p$. For lossy compression, however, linearity is known to incur strict suboptimality compared to the rate-distortion function. Massey asked whether the optimal rate for linear encoding is achieved simply by compressing a fraction of the bits linearly and losslessly and estimating the rest by zero \cite{Massey1978}. For $p=\frac12$, Ancheta answered this question affirmatively \cite{Ancheta1978}. This note extends Ancheta's result to all $p<\frac12$. The key argument is to bound the entropy of the posterior distribution conditioned on an affine subspace in terms of its marginals. The proof was discovered by GPT-5.6 Sol in an interactive process guided by the author. The purpose of the present note is to communicate a simplified version of this proof and to make connections with the existing literature on coding theory and spin glass theory.

In this paper, we construct a class of infinite graphs, called substitution graphs. The vertex set consists of all finite words over a finite alphabet. A directed graph is formed by adding vertical edges connecting each word to its children and horizontal edges defined recursively by two finite directed graphs G and J: edges among vertices with the same parent follow G, while edges between vertices whose parents are horizontally linked follow J. The substitution graph is defined as its underlying graph. Substitution graphs provide a purely combinatorial model of self-similar structures, independent of any underlying geometric structure. Furthermore, we establish a necessary and sufficient condition for substitution graphs to be hyperbolic, formulated in terms of the vanishing of path matrices associated with sufficiently long shortest horizontal paths. Based on this characterization, we further derive several conditions that are either necessary or sufficient for hyperbolicity, depending only on the generators G and J.

We derive convergence guarantees for mirror descent and proximal mirror descent with a logarithmic barrier as the distance-generating function. When the solution lies on the boundary, the Bregman divergence blows up and classical guarantees become vacuous. While existing remedies recover $O(\log k/k)$ rates by adapting classical bounds, we propose a more direct route in which the Bregman distance to the solution never appears, and which achieves the same rate with constant stepsize. We also construct an example showing that this rate cannot be improved in general. Finally, we compare the proposed approach with interior-point methods.

Stone conjectured in 1981 that every fully semimonotone $\mathbf{Q}_0$-matrix is a $\mathbf{P}_0$-matrix; Murthy and Parthasarathy (1995) proved this for orders up to $4$ and for several special classes. We study the principal minors of fully semimonotone matrices $A\in\Ezerof$ without using $\Qzero$. Let $\G(A)$ be the graph whose vertices are the index sets $\alpha$ with $\det A_{\alpha\alpha}\neq 0$ (including $\alpha=\emptyset$), two vertices being adjacent when they differ in exactly one index. We prove that the sign of the principal minor is constant on every connected component of $\G(A)$, and positive on the component of $\emptyset$. Consequences: an $\Ezerof$-matrix with no vanishing principal minor is a $\mathbf{P}$-matrix, and any $\Ezerof$-matrix satisfying a simple “descending” condition on its nonvanishing minors is a $\mathbf{P}_0$-matrix. We show that a sign change across two steps through vanishing minors forces strictly positive coupling in the corresponding principal pivotal transform, and that a principal submatrix of minimal order with negative determinant is an almost-$\mathbf{P}_0$ matrix whose inverse lies in $\Ezerof\cap\Nzero$ and has zero diagonal. Explicit examples show that the component structure can contain negative components, that $\det A>0$ does not force $A\in\Pzero$ in $\Ezerof$, that $\Ezerof$ is not invariant under $A\mapsto A+\varepsilon I$, and that $\Ezerof\cap\Pzero$ contains matrices with $\det A>0$ outside $\Qzero$. Stone's conjecture itself remains open.

We determine the Hilbert series of ideals generated by $d$'th powers of $n+2$ general linear forms in $n$ variables, to give upper bounds on the degree of the Hilbert series of ideals generated by $d$'th powers of $n+k$ general linear forms for $k>2$. This allows us to show that the Iarrobino-Fröberg Conjecture fails for all $n$ large enough. We also determine the degree of the Hilbert series for the ideal generated by $d$'th powers of $n+3$ general linear forms, for some values of $n$, and give counterexamples to a conjecture on the failure of the Weak Lefschetz Property for ideals generated by sufficiently large powers of general linear forms. Moreover, we determine the Hilbert series of the ideal generated by two generic quadratic forms in the exterior algebra on an even number of generators.

We study the reverse-mathematical strength of frame definability in modal logic. The central principle is the Valuation Extension Lemma (VEL), which asserts that every assignment of propositional variables on a frame extends to a full valuation. We show that, over $\mathrm{RCA}_0$, VEL is equivalent to $\mathrm{ACA}^{+}_0$, and as are frame-definability principles for Geach axioms and for $\mathbf{GL}$. We also obtain analogous $\mathrm{ACA}^{+}_0$-equivalences for the Barcan and Converse Barcan formulas in modal predicate logic. Finally, we examine variants of VEL for $\mathbf{CTL}$ and $\mathbf{LTL}$ and locate their strengths between familiar subsystems of second-order arithmetic.

We consider the finite horizon continuous-time Principal–Agent problem under deterministic discount factors. Following the Sannikov reduction to a stochastic control problem, we provide a further characterization of the Principal's value function in terms of a backward SDE inducing the corresponding optimal contract. In particular, this allows to bypass the fully nonlinear HJB equation satisfied by the Principal value function in the Markovian setting. This new approach allows to handle a new class of Principal-Agent problems which was not accessible with the existing method, namely the setting where the Agent faces a regime-switching control problem.

Let $0\le \mu_1\le \mu_2 \le \cdots \le \mu_n$ and $\delta > 0$. Samuels' conjecture states that if $X_1,\dots,X_n$ are independent non-negative random variables with $\mathbb{E}[X_i] = \mu_i$, then $$ \mathbb{P}\left( \sum_{i=1}^n X_i < \delta + \sum_{i=1}^n \mu_i \right) \ge \min_{1\le i\le n} \prod_{j=i}^n \left(1-\frac{\mu_j}{\delta + \sum_{k=i}^n \mu_k}\right).$$ This conjecture was recently proved by Ling. In this note, we prove a refinement of Samuels' inequality via an operation called the cap move. Our refined inequality is sharp, with equality attained for a strictly larger class of distributions than in the original inequality.

We study mixed-integer quadratic optimization (MIQO) problems with indicator variables. We propose a unified framework, based on decision diagrams, that serves both to solve the associated optimization problems and to construct ideal conic quadratic extended formulations of the closure of the convex hull of the underlying mixed-integer set. The construction applies to arbitrary quadratics and to any combinatorial constraints admitting a tractable dynamic programming representation. The resulting diagrams and the ensuing convex hull descriptions are of polynomial size when the quadratic is low-rank, or when the support graph of the Hessian or of its inverse is a tree, recovering and generalizing several results from the literature. For structured sparse and inverse-sparse quadratics, we show that approximate decision diagrams have size linear in the dimension while yielding solutions with arbitrarily low optimality gap. Computational experiments demonstrate the effectiveness of the proposed approach.

A triangulation is a graph on a surface where every face is triangular. It is well-known that a planar triangulation $G$ is $3$-chromatic if and only if $G$ is even, that is, all the vertices of $G$ have even degree. We focus on even triangulations of non-spherical surfaces. It is known that there is an invariant of even triangulations, called a monodromy. This concept was firstly introduced by Hutchinson et al. [J. Combin. Theory Ser. B 84 (2002) 225–239], and the congruence class of monodromies was defined by Kawarabayashi et al. [J. Combin. Theory Ser. B 99 (2009) 229–246]. For a lower genus surface $F^2$, the number of congruence classes of monodromies have already been shown, $1$ if $F^2$ is the sphere, $2$ if $F^2$ is the projective plane by Mohar [Discrete Math. 244 (2002) 339–343] and $3$ if $F^2$ is the torus by Higuchi et al. [Discrete Math. 311 (2011) 1128–1135]. In this paper, we count the number of congruence classes of monodromies of even triangulations of general closed surface $F^2$.

In this paper we prove that the Eaton–Moretó conjecture holds true for all $p$-solvable groups.

We prove that the Toeplitz operator $T_{\bar{z}+h}$ is invertible on the Bergman space provided that $|\bar{z}+h|$ is bounded below by some positive constant on the open unit disk, where $h$ is in the disk algebra. This provides a general class of harmonic functions for which the answer to the Douglas question on the Bergman space is affirmative.

Orthonormal Strichartz estimates for the free Schrödinger propagator take the form \begin{equation*} \left\Vert\sum_j\lambda_j\left|e^{it\Delta}f_j\right|^2\right\Vert_{L_t^\frac{q}{2}L_x^\frac{r}{2}(\mathbb{R}\times\mathbb{R}^d)}\lesssim\Vert\lambda\Vert_{\ell^\alpha(\mathbb{C})} \end{equation*}for arbitrary orthonormal systems $(f_j)_j$ in the homogeneous Sobolev space $\dot{H}^s(\mathbb{R}^d)$. In the admissible region, the optimal range of $\alpha$ has been established when $q\geq r/d'$. For $d\geq2$ and $2<q<r/d'$, it has been an open question as to determine whether the estimate holds when $\alpha=q/2$. We prove that the estimate holds in this critical case whenever $q=4$ and $r<\infty$. Our approach is robust and we illustrate this by extending the result to fractional Schrödinger propagators.

Based on work with Greenfeld and with Ziegler, Tao showed a concatenation result that if a map $f : \mathbb{F}^2 \to \mathbb{F}$ is affine-linear on every line parallel to the coordinate axes, and on all lines with a fixed nonzero slope (where the field $\mathbb{F}$ has size $> 2$), then $f$ is affine-linear on $\mathbb{F}^2$. We extend this from $\mathbb{F}^2$ to $\mathbb{F}^n$ and obtain a tight minimum number of additional lines needed – $N = \binom{n}{\lfloor n/2 \rfloor}$ – for every field $\mathbb{F}$ with $3 \leqslant n < |\mathbb{F}|$. The proof is constructive and shows a stronger result: the existence of a universal family of $0$-$1$ matrices of size $\binom{n}{k} \times \binom{n}{k}$ (one for each pair $0 \leqslant k \leqslant n$), which are indexed by ballot sets and are unimodular over all unital commutative rings. We also show a second tightness: of the assumption $n < |\mathbb{F}|$. Else, there exist multi-affine maps $f$ which are affine-linear on every line through the origin, but not affine-linear globally on $\mathbb{F}^n$. More strongly, we prove this dichotomy – including the bound of $N$ – over all integral domains, or Noetherian (e.g.\ finite or Artinian) rings, or products of these. This yields a novel numerical invariant for affine-linearity, for every product of Noetherian rings and integral domains.

Suppose that $M$ is a closed manifold of dimension greater than two. We show that there exists a $C^1$-diffeomorphism $f_0:M\longrightarrow M$ with a wild affine blender-horseshoe $\Lambda_{f_0}$ such that any element $f$ of $\mathrm{Diff}^1(M)$ sufficiently $C^1$-close to $f_0$ is strongly pluripotent for the continuation $\Lambda_f^{(\mathrm{mj})}$ of $\Lambda_{f_0}^{(\mathrm{mj})}$, where $\Lambda_{f_0}^{(\mathrm{mj})}$ is the dense subset of $\Lambda_{f_0}$ consisting of elements with majority condition.

Let d=2g+1>4 and let A^{d-1} be the coefficient space parametrizing hyperelliptic families C_a: y^2 = x^d + a_{d-1}x^{d-1} + ... + a_1x+t over the t-line. We show that for a Zariski-generic coefficient vector a=(a_1,...,a_{d-1}) in \bar{Z}^{d-1}, for every prime p large enough, C_a has generically ordinary reduction at every prime above p. For arbitrary a in \bar{Z}^{d-1}, the set of ordinary primes of the hyperelliptic curve C: y^2=x^d+a_{d-1}x^{d-1}+...+a_1x+t over Q(a)(t) has natural density at least 1/[Q(a,\zeta_d):Q(a)]; after base change to Q(a,\zeta_d)(t), it has natural density 1.

Lovász and Szegedy asked whether a nonconstant continuous, or even smooth, finitely forcible kernel exists. For every $0<\lambda\leq1/128$, we construct an explicit nonconstant real-analytic kernel $W_\lambda$ taking values in $(1/4,7/8)$. A single finite family of simple graphs, independent of $\lambda$, forces every $W_\lambda$ among all bounded symmetric real-valued kernels on $[0,1]^2$.

Let \[ \mu(r)=\frac{\pi}{2}\frac{\Kc(r')}{\Kc(r)}, \qquad r'=\sqrt{1-r^2},\qquad 0<r<1, \] be the Grötzsch modulus function. Problems 3.38(a)–(b) in a survey of Vuorinen ask whether Newton's iteration for $\mu^{-1}(y)$, initialized by $x_0=1/\cosh y$, converges for every $y>\pi/2$, and whether it is strictly increasing when $y>\pi$. We prove both assertions. The main point is that $\mu$ has exactly one inflection point $a\in(1/2,1/\sqrt2)$. If the zero lies in the convex region, the Newton iterates increase to it from the left. If the zero lies in the concave region, the orbit crosses the inflection point, overshoots the zero at most once, and then decreases to the zero. For $y>\pi$ we obtain the stronger estimate $0<x_n<x_{n+1}<\mu^{-1}(y)<3-2\sqrt2<1.$

In this paper, we propose an inexact Riemannian gradient descent algorithm on the Stiefel manifold (IRGS-StieONS) using an adaptive step size, where the “inexact” refers to the inexactness of retraction. It is proven that one single Newton-Schulz iteration for the retraction is sufficient for global convergence and local linear convergence. Compared to the landing and augmented Lagrangian-based algorithms, the proposed algorithm is the first infeasible algorithm that permits adaptive step sizes with a practical initial step size and guarantees global convergence and local linear convergence under mild assumptions. Moreover, we show that the local convergence rate depends on the condition number of the Riemannian Hessian, which matches the Riemannian steepest descent algorithm. This result implies that the infeasibility in the proposed algorithm does not influence the local convergence rate. Furthermore, a stochastic gradient version of IRGD-StieONS is proposed and is shown to achieve the same convergence rate as Riemannian stochastic gradient descent with decreasing step size. Numerical experiments demonstrate that both IRGD-StieONS and its stochastic counterpart exhibit superior performance and robustness.

Coupled-dynamics environments expose the one-step outcomes that would follow from several possible counterfactual actions under a common realization of exogenous randomness. The ordinary Markov decision process formalism allows one to reason about the marginal law of each action but discards dependence across these counterfactual outcomes. The Joint Markov decision process (JMDP) formalism preserves that dependence. Prior work established the formalism and solved the fixed-policy joint moment evaluation problem in JMDPs. This paper develops optimal-control methods. We define a nonparametric distributional Bellman optimality operator for JMDPs, and prove that when the induced marginal MDP has a unique optimal policy, its iterates converge in Wasserstein distance to the optimal joint return law. For the first two moments, we establish convergence under a weaker condition that permits several mean-optimal actions as long as their tie resolutions share a second-moment fixed point. We also derive sampled targets for neural approximation.

Let $(V,0)$ be an isolated complex hypersurface singularity defined by $f\in\mathbb{C}\{x_1,\ldots,x_n\}$. Let $A(V)$ be the moduli(Tjurina) algebra, let $A^*(V)$ be the generalized moduli algebra, and let $$ L(V)=\mathrm{Der}_{\mathbb{C}}(A(V),A(V)),\qquad L^*(V)=\mathrm{Der}_{\mathbb{C}}(A^*(V),A^*(V)) $$ be the Yau algebra and the new Yau algebra of $V$, respectively. Write $\lambda(V)=\dim_{\mathbb{C}}L(V)$ and $\lambda^*(V)=\dim_{\mathbb{C}}L^*(V)$. We prove that the difference between these two dimensions is determined by the Hessian corank. In the Morse case, the generalized moduli algebra is zero, and hence both derivation Lie algebras are zero. We have $\lambda^*(V)=\lambda(V)-1$ if $\mathrm{corank}(\mathrm{Hess}(f)(0))=1$, and $\lambda^*(V)=\lambda(V)$ otherwise. In particular, $\lambda^*(V)=\lambda(V)$ when $n\ge2$ and $\mathrm{mult}(f)\ge3$. This proves Conjecture~1.1 proposed in \cite{ChenHussainYauZuo2020}. We construct an exact sequence whose two end terms are copies of the socle of the Milnor algebra. This homological approach reveals the detailed algebraic structure underlying the dimension count. We also give an alternative proof that pushes the original arguments in the proof of \cite[Theorem~C]{ChenHussainYauZuo2020} further. The general case is obtained by combining Saito's criterion for the non-quasi-homogeneous case.

Associated with an even nilpotent element $e$ in a basic classical Lie superalgebra $\mathfrak{g}$, we study, in full generality, two constructions of finite $W$-superalgebras, defined via Whittaker models and isotropic subspaces, respectively. We prove that both formulations are independent of the various choices made in their constructions, thereby yielding, for a fixed good grading for $e$, at most two isomorphism classes of $W$-superalgebras. In the case when there are two non-isomorphic versions, we establish that they differ precisely by a Clifford extension. Consequently, when the two $W$-superalgebras are non-isomorphic, their module categories are equivalent up to a Clifford twist. Building on this equivalence and utilizing the Skryabin equivalence, we classify their irreducible representations in terms of generalized Whittaker modules over $\mathfrak{g}$

We establish a priori interior curvature estimates for the scalar curvature equation in dimension 4. Our approach relies on the doubling method introduced by Shankar and Yuan [SY25]. Key to the proof are an a priori doubling inequality under a small gradient assumption and the Alexandrov regularity for viscosity solutions; the latter is obtained via a new iterative rotation argument. Furthermore, we establish interior regularity for $C^1$ viscosity solutions.

This paper studies the Sinkhorn distributionally robust hypothesis testing (SDRHT) problem, seeking a robust detector against least-favorable distributions in Sinkhorn discrepancy-based ambiguity sets centered at the empirical distributions. Existing approaches solve this problem by solving large-scale conic programs, which are not scalable. To overcome this, we propose a generative framework that learns least-favorable distributions and supports efficient training and end-to-end sampling. For the Sinkhorn discrepancy-based ambiguity sets, we first derive an equivalent conditional-KL-divergence representation with respect to kernel-smoothed reference distributions. This property allows us to prove strong duality for both constrained and unconstrained minimax SDRHT formulations. Based on the closed-form optimal detector and Brenier's theorem, we reformulate the max-min dual formulation as a maximization problem over convex potentials whose gradients characterize invertible transport maps between kernel-smoothed distributions and their least-favorable counterparts. We efficiently approximate these potentials using Hyper Input Convex Neural Networks (HyCNNs) equipped with stochastic gradient estimators and prove the representation power of HyCNNs and the distributional universality of their induced transport maps. Numerical results show that the proposed method achieves superior accuracy and robustness across different sample sizes and dimensions, while avoiding the scalability limitations of classical SDRHT methods.

Given graphs $H$ and $F$, the generalized Turán number ${\rm ex}(n,H,F)$ is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Alon and Shikhelman (J. Combin. Theory Ser. B, 2016) initiated the systematic study of generalized Turán problems. Recently, Gao, Wu and Xue (J. Graph Theory, 2026) asked whether every graph $F$ with chromatic number $\chi(F)=r\geq3$ and treewidth ${\rm tw}(F)\geq r$ satisfies ${\rm ex}(n,K_r,F)=\Omega(n^{r-1})$. In this note, we give a negative answer to this question for every $r\geq3$. More precisely, we prove that the graph $F_r=K_{r-3}\vee H$, where $H$ is obtained from $K_4$ by subdividing one edge once, satisfies $\chi(F_r)={\rm tw}(F_r)=r$ and \[ n^{r-1}e^{-O(\sqrt{\log n})}\leq {\rm ex}(n,K_r,F_r)=o(n^{r-1}). \] This result also disproves Conjecture 6.3 in the recent survey of Gerbner and Palmer (Electron. J. Combin., 2026).

We study the cone $\mathcal{M}_{n,2d}$ of nonnegative mean polynomials—real $n$-variate forms of degree $2d$ that can be expressed as weighted power means $M_{q,p}(Y,w)$ with $q>p$. This cone simultaneously generalises the cone of sums of squares $\Sigma_{n,2d}$ and the cone of sums of nonnegative circuit polynomials $\mathcal{C}_{n,2d}$. We prove that every square of an arbitrary polynomial belongs to the mean polynomial preprime $T_{\mathrm{mean}}$, that $T_{\mathrm{mean}}$ is strongly generating, and consequently that every polynomial strictly positive on a compact semialgebraic set admits a representation with mean polynomial certificates. We exhibit the Robinson form $\hat{R}$ as a separating example that lies in $\mathcal{M}_{4,4}$ but outside $\mathrm{SOSONC}_{4,4}$. Finally, we outline a convergent hierarchy of lower bounds for polynomial optimization based on the mean polynomial cone and discuss tractable depth-truncated approximations via signomial programming.

We study congruences between special values of zeta functions and modular $L$-functions in cyclotomic towers. The underlying mechanism is an integral norm congruence for finite-level Iwasawa-theoretic elements, which becomes stronger as one ascends the tower. In the $\mathrm{GL}_1$ setting, this gives congruences for Dedekind zeta values over totally real fields, with applications to higher $K$-groups, generalized Bernoulli numbers, and Euler-Poincaré characteristics. In the $\mathrm{GL}_2$ setting, we use Mazur-Tate elements and their distribution relations to control the primitive twisted special values appearing at each cyclotomic level. Via Artin formalism, these give congruences for the corresponding base-change $L$-values over successive cyclotomic fields.