数学前沿

In this paper, we study matrix valued orthogonal polynomials whose block Jacobi matrix is $\Theta(J)$, viewed in blocks of size $m=\deg\Theta$, where $J$ is a tridiagonal Jacobi matrix and $\Theta$ is a scalar polynomial. We identify their matrix measure of orthogonality intrinsically as the push-forward, under $\Theta$, of a matrix measure constructed from the original scalar orthogonal polynomials. We formulate reducibility in the push-forward variable and prove that the $2\times2$ Hermite family associated with $\Theta(x)=x^2+\alpha_1x$ is irreducible for $\alpha_1\neq0$ and reducible for $\alpha_1=0$. This problem is motivated by random walks with transition matrix $\Theta(J)$. For these models, we derive scalar and block Karlin--McGregor representations for the powers of $\Theta(J)$. We apply the construction to the mixed one- and two-ball Ehrenfest chain, obtaining explicit spectral measures, spectral multiplicities, and transition formulas, and we conclude with the corresponding $k$-ball and multi-ball extensions.

Stochastic Galerkin schemes (SGS) are used to approximate solutions to differential equations with stochastic parameters. In particular, they express the solution with an infinite orthogonal functional basis of the parameter space and derive a deterministic system of PDEs; the solution of this deterministic system can be used to produce a {surrogate} approximation of the original stochastic equations. When the parameters are correlated, the infinite series representation of the solution does not provably converge for all joint distributions; further, classic decorrelating transformations (e.g., Roseblatt) often introduce known numerical instabilities. Rahman (2017) established convergence in probability and distribution for correlated {Gaussian} parameters when the basis functionals are {multivariate Hermite polynomials} (MHPs), where these polynomials are only {weakly orthogonal} with respect to the underlying joint density. Due to weak orthogonality, the $N$th moments of MHPs introduced during Galerkin projection do not trivially decompose into products of one-dimensional expectations and have no closed-form expression for $N > 2$; consequently, these moments are typically estimated via brute-force sampling and quickly become prohibitively expensive to compute. In this paper, we introduce and prove several foundational results for MHPs. These results culminate in an equation that can be used to calculate the $N = 3$ moments (triple products) exactly, whose implementation dramatically reduces the computational bottleneck otherwise inherent to SGS with correlated Gaussian parameters.

We present a substitution on six marked rhombs, called Seabed, with inflation factor $\varphi^3$. We prove that its tiling space is mutually locally derivable with the unmarked Anti-Penrose pentagrid hull. Eleven vertex patches of depth at most four recognize the parent vertices: their acceptance domains coincide with the five internally contracted windows. Matching the fine tiles around each parent rhomb to a replacement patch recovers their markings uniquely. The inverse local maps have a uniform finite radius and extend to all singular limit tilings in the hull. The construction also gives a recognizable parent hierarchy.

After defining the concept of inevitability for statements with the modal logic of forcing, we raise the question of which statements are inevitable under $\mathsf{ZFC}$ and under large cardinal hypotheses. We show that $\mathsf{SOA}$ (defined as $\mathsf{ZFC}^-$ + $\mathsf V=\mathsf{HC}$) plus the $\boldsymbol\Sigma^1_2$-perfect set property is $\mathsf{ZFC}$-inevitable, and that Projective Determinacy is inevitable in the theory of $\mathsf{ZFC}$ with a proper class of Woodin cardinals. We then discuss the possible conceptual significance of these results for the foundations of set theory.

A recently proposed reaction-diffusion model for the tulip breaking virus (TBV) has been shown to generate striped pigmentation patterns on growing tulip petals. In this paper, we investigate the formation of localized spike patterns in this model in the asymptotic limit of small viral diffusivity. It is known that two distinct bifurcation mechanisms can occur which generate new spikes as the domain half-length $L$ slowly increases: spike self-replication and spike nucleation. Self-replication occurs through a saddle-node bifurcation of the inner spike core problem, whereas nucleation is associated with a fold of the nonlinear outer problem. Using matched asymptotic expansions, we derive critical conditions for these mechanisms and construct phase diagrams that predict which mechanism occurs first as $L$ increases, or whether nucleation is absent. We further combine these results with a Turing instability analysis of the homogeneous steady state to obtain a phase diagram of pattern-forming mechanisms. An interesting finding is that, for the TBV model, the Turing-unstable region lies entirely within the parameter regime where spike nucleation is possible. These results provide further insight into the mechanisms underlying the formation and evolution of pigmentation patterns in virus-infected tulips.

Let $\F$ be a finite field. Given $N$ distinct points $\bm{\alpha}_1,\ldots,\bm{\alpha}_N \in \F^m$ and associated values $\beta_1,\ldots,\beta_N \in \F$, we study the problem of computing an $m$-variate polynomial $f(\vy) \in \F[\vy]$ of individual degree less than $d$, with $d^m \ge N$, that satisfies $f(\bm{\alpha}_i)=\beta_i$ for all $i\in[N]$, whenever such an $f$ exists. This is known as the\textit{ multivariate polynomial interpolation} problem. In this work, given $N=d^m$, we present a deterministic algorithm for multivariate polynomial interpolation over an arbitrary finite field $\F$ that runs in near-quadratic time $N^{2+o(1)}\poly(m,d,\log|\F|)$, assuming $d$ is $\omega(1)$. This bound holds whenever the associated \emph{multivariate Vandermonde} matrix is nonsingular, has constant co-rank, or admits some special structure. In the general case, we give a randomized Las Vegas algorithm attaining the same near-quadratic running time, provided the finite field is sufficiently large and $d$ grows similar to the previous case. Unlike other multivariate interpolation strategies, we do not assume black-box access to the polynomial. In particular, we are not permitted to query arbitrary input points to obtain their evaluations. Our interpolation result follows by combining Wiedemann's classical approach~\cite{w86} for solving sparse linear systems with known bounds for fast \emph{multivariate multipoint evaluation} over finite fields~\cite{umans08, KU11, BGKM23, BGGKU24}. Specifically, our deterministic interpolation strategy relies on a deterministic variant of Wiedemann's approach, which is more commonly used in the randomized setting.

We show that rigid modules are $\tau$-locally free for a large family of 1-Iwanaga--Gorenstein algebras. This family includes many cases of the algebras introduced by Geiss--Leclerc--Schr\"oer in their Inventiones paper "Quivers with relations for symmetrizable Cartan matrices I: Foundations", hence answering a question posed by Geiss--Leclerc--Schr\"oer for many new cases. Our result is achieved by introducing coactions of Taft algebras and applying the recent theory developed by the first-named author and Coulembier.

We establish the conditioned stochastic stability of equilibrium states for H\"older potentials on uniformly hyperbolic sets. While standard stochastic stability characterises measures on attractors, we analyse the statistics of transient dynamics on non-attracting sets by conditioning small random perturbations of the dynamics to not escape from our regions of interest. We prove that as the noise intensity vanishes, the quasi-ergodic measure of the $e^\phi$-weighted process generated by $\e$-small random perturbations of the deterministic dynamics converges to the unique equilibrium state associated with the potential $\phi - \log \left|\det \left. D T\right|_{E^u}\right|$. The results are obtained via perturbative spectral analysis of transfer operators acting on anisotropic Banach spaces and topological hyperbolic dynamics arguments. Furthermore, we extend this framework globally to Axiom A diffeomorphisms with multiple basic sets using dynamical filtrations. This work provides a rigorous characterisation of natural measures on uniformly hyperbolic repellers, which are fundamental in the context of transient chaos. Our results have been formalised in Lean~4.

In this paper we study the orbits of the automorphism group on a toroidal spherical variety. In the affine case, we prove that two points lie in the same orbit of the identity component if and only if the kernels of the homomorphisms from the divisor class group to their local divisor class groups coincide. These kernels are described in terms of $B$-stable prime divisors. For a smooth complete toroidal spherical variety, we obtain a similar criterion using the monoids generated by the classes of $B$-stable prime divisors that do not contain a given $G$-orbit.

We solve a problem posed by Hu and Virtanen by characterizing, for every $0<p<\infty$, the bounded symbols $f$ for which $T_f^{(t)}T_g^{(t)}-T_{fg}^{(t)}$ tends to zero in the unnormalized Schatten $p$-norm or quasi-norm as $t\to0^+$ for every fixed bounded $g$. The setting is weighted Fock spaces over $\mathbb C^n$ with weights whose real Hessians are uniformly positive and bounded. Only constants satisfy the limit when $p<2n$; at $p=2n$, it is equivalent to the weak real gradient belonging to $L^{2n}(\mathbb R^{2n})$. When $p>2n$, the limit is characterized by relative weak compactness in $L^1(\mathbb R^{2n})$ of normalized local mean oscillation densities. For $p\ge2$, uniform convergence over $\|g\|_\infty\le1$ is equivalent to vanishing density mass. A continuous compactly supported counterexample for $p>2n$ shows that bounded density mass and convergence for $g=\bar f$ do not imply convergence for every fixed bounded $g$. The symbol classes are weight-independent unital algebras, yielding limits for finite Toeplitz products and commutators; quantitative H\"older estimates also give finite-rank approximation bounds. The Gaussian case transfers to anti-Wick quantization by unitary equivalence. On compact K\"ahler manifolds with positive line bundles, we prove a H\"older sufficient condition, including the endpoint, for each fixed bounded second symbol. A projective-line example shows that trace norm convergence can fail for a smooth first symbol and a fixed bounded second symbol. The proofs combine local holomorphic approximation, Schatten and Sobolev estimates, singular integral methods, and the Dunford--Pettis criterion.

We study the computation of mixed volumes of convex sets given as rational subspace sections of bodies described by membership oracles. The sets need not have nonempty interior. For positive multiplicities $\alpha_1+\cdots+\alpha_k=n$, we give a randomized algorithm that computes a relative $\varepsilon$-approximation with probability at least $1-\delta$. Its running time is polynomial in Gurvits' factor $\Gamma_\alpha=(n^n/n!)\prod_i\alpha_i!/\alpha_i^{\alpha_i}$, the input presentation length, and $\varepsilon^{-1}$, and linear in $1+\log\delta^{-1}$, with degree independent of $k$. The running-time bound holds for every choice of the random bits, and zero mixed volumes are detected exactly. We extend the capacity-scaling and sampling method of Narayanan--Roy by using rational projections and a smooth mixed Monge--Amp\`ere representation. In particular, the algorithm runs in randomized polynomial time when the number of distinct convex sets is bounded above by a constant.

Lipshitz and Sarkar asked in an ICM 2018 question whether their homotopy-theoretic refinement of Khovanov homology can detect mirror reflection when ordinary integral Khovanov homology cannot. We give an affirmative answer by presenting a prime knot whose integral Khovanov homology agrees with that of its mirror in every bidegree, including torsion, and yet over $\mathbb{F}_2$, the second Steenrod square has rank one for the knot and rank zero for its mirror in a specified bidegree. We also give a composite example. The construction, search and computations are done with the help of {\it chat-GTP 5.6 sol Pro} combing over the {\it Regina Census} of knots.

We study reflection symmetry in the equal-mass Newtonian $1+N$ coorbital problem for $N=5,6,7,8$. We prove that every collision-free solution of the angular central-configuration equations has a reflection axis. As a consequence, every non-coalescent limit of planar central configurations consisting of one dominant mass and $N$ equal small masses is reflection-symmetric about a line through the dominant body. Our proof combines analytic arguments with rigorous computer-assisted verification. For each possible reflection, we introduce variables that measure the deviation from that reflection and vanish exactly when the configuration has that symmetry. The central-configuration equations yield a symmetric matrix equation in these variables. Concavity of the force derivative and estimates of the resulting quadratic forms reduce the symmetry problem to finitely many rigorously verified inequalities. Although the details depend on $N$, the same general method applies in all four cases. We also obtain, as a separate result, the complete three-class classification for $N=5$, up to the natural equivalences.

A folklore result says that any parabolic double coset $W_{I} w W_{J}$ in a Coxeter group $W$ is an interval of the Bruhat order, provided that $W_{I}$ and $W_{J}$ are finite. We include a self-contained proof, without assuming that $W$ is finite. Furthermore, we prove that each "triple coset" $W_{I} u W_{J} v W_{K}$ has a Bruhat minimum and -- if $W_{I}, W_{J}, W_{K}$ are finite -- a Bruhat maximum, but (in general) is not a Bruhat interval. This proves and extends a conjecture on MathOverflow #472072.

For a $\Delta$-regular connected graph $H$ we obtain the upper tail large deviations speed and rate function for the number of copies of $H$ in $\mathbb{G}_n^d$, a random $d$-regular graph on $n$ vertices, in the regime $np^{\Delta/2} \gg (\log n)^{1/(v_H-2)}$ and $p \ll 1$, where $p:=d/n$ and $v_H$ is the number of vertices, and also identify the typical structure of $\mathbb{G}_n^d$ conditioned on this upper tail event. For $\Delta \geq 3$, the speed and the rate function are same with those in the Erd\H{o}s-R\'enyi setting. In contrast, for $\Delta=2$ (i.e.~in case of cycles), due to the degree constraint in $\mathbb{G}_n^d$, although speed remains the same the rate function turns out to be different. Moreover, conditioned on the upper tail event of cycle counts in $\mathbb{G}_n^d$, we show that the random graph contains a disjoint collection of clique and biclique like structures with probability approaching to one, a stark contrast compared to the Erd\H{o}s-R\'enyi setting. In the complementary regime, i.e.~for $1 \ll n^{v_H}p^{e_H} \ll (\log n)^{\alpha^*_H/(\alpha^*_H-1)}$, where $e_H$ is the number of edges in $H$, $\alpha^*_H$ is the fractional independence number of $H$, and $H$ is a strictly balanced graph (and thus including the regular case), we show that the upper tail large deviations speed and rate function for the number of copies of $H$ match with those in the Erd\H{o}s-R\'enyi setting. In this regime, we further show that almost all of the unlabeled copies of $H$ conditioned on the upper tail event are edge disjoint.

Fixed-reference entropy-regularized transport yields smooth distributionally robust optimization (DRO) objectives, but its value as a function of the adversarial marginal is generally minimized at a kernel-smoothed law rather than at the nominal distribution. Consequently, the smallest raw ambiguity set need not recover nominal risk. Building on prior information-geometric plan-KL orientations, we study the forward tangent-centered discrepancy obtained by subtracting the value and first variation at the nominal law. This is the unique affine recentering, up to the simplex gauge; for symmetric costs, its plan-KL representation coincides with the forward candidate noted by Amari et al. (2018). The representation shows that the discrepancy dominates marginal KL, is uniquely zero at the nominal law, and is strongly convex in total variation. The resulting DRO problem retains a scalar log-sum-exp dual and closed-form conditional adversarial weights. On finite supports with symmetric zero-diagonal costs, the discrepancy converges uniformly to the unregularized transport cost at low temperature; for any finite cost, its rescaled value converges uniformly to marginal KL at high temperature. We derive its exact local Hessian, an explicit finite-sample containment radius, and a generalized chi-square calibration for multinomial data. Controlled simulations verify nominal centering, both limiting regimes, and practical plug-in coverage.

We extend the ``Brownian bees'' model of Berestycki et al.\ (2021, 2022) to cooperative reproduction, $kA\to(k{+}1)A$, of a population of $N$ symmetric random walkers with removal, at each birth event, of the particle farthest from the origin. Working in the limit $N\to\infty$, we formulate a hydrodynamic free-boundary problem for this model. Using this formalism, we determine steady state population densities for all~$k$ and prove their linear stability for $k\le 2$ and instability for $k\ge 4$. In the marginal case $k=3$, there is a whole continuous family of steady states at a single critical ratio of the reproduction and diffusion rates. Above criticality the population undergoes an asymptotically self-similar finite-time collapse to the origin. Below the criticality the population spreads diffusively, but the reproduction remains quantitatively relevant. For $k\ge 4$, the unstable steady state separates regimes of a finite-time collapse and diffusive spreading. Here the collapse dynamics is asymptotically self-similar, and the population density exhibits a scale separation requiring a matched-asymptotic description. Our analytical predictions are confirmed by numerical solutions of the hydrodynamic free-boundary problem and by Monte Carlo simulations of the original microscopic model.

Let $\xi$ be a real algebraic irrational number and $p(n,\xi,b)$ the number of distinct blocks of length $n$ in its base-$b$ expansion. Bugeaud and Evertse proved in 2008 that $p(n,\xi,b)/(n(\log n)^\eta)$ is unbounded for every $\eta<1/11$. We prove this for every $\eta<1/4$. We also prove the analogous bound for the continued fraction expansion of an algebraic number of degree at least three, with every exponent $\delta<1/4$. Bugeaud announced this statement in 2013 with an unspecified $\delta>0$. Finally, the subword complexity of the steering word of $(3/2)^n$ exceeds $C\,k(\log k)^\eta$ infinitely often for every $\eta<1/4$ and every $C$. The three proofs share one count: $\varepsilon^{-1}$ classes times $\varepsilon^{-3}$ subspaces from the quantitative Subspace Theorem of Evertse and Ferretti. The subspaces are then disposed of at the cost of a logarithm, with no further power of $\varepsilon^{-1}$. All three results, and the quantitative Subspace Theorem they rest on, are machine-verified in Lean~4.

We propose and analyse a fully discrete finite element approximation of the three-dimensional incompressible rotating MHD-Boussinesq system with nonlinear power-law ($p$-Laplace) viscosity. Conforming elements are used for the velocity and temperature, $\mathbf H(\operatorname{curl})$-conforming N\'ed\'elec elements for the magnetic induction, and a semi-implicit backward Euler scheme in time. The discretisation preserves the principal energy structure: the Lorentz and induction terms cancel exactly, while the Coriolis force remains energy neutral. This yields well-posedness and mesh-independent energy stability. The nonlinear power-law diffusion creates two main analytical difficulties. Since a uniform time-derivative estimate for the discrete velocity is unavailable, strong compactness is obtained through a time-translate argument and Simon's compactness criterion. The weak limit of the nonlinear diffusion is then identified by a Minty argument based on a discrete energy $\limsup$ inequality. Compactness and passage to the limit hold throughout the shear-thickening regime, while identification of a genuine weak solution is obtained in the Newtonian case and, in three dimensions, above an explicit threshold dictated by the Lorentz force rather than by convection. We further establish a priori error estimates without any mesh-time coupling condition. Three-dimensional numerical experiments confirm the stability and convergence behaviour predicted by the analysis.

We prove the 3-Decomposition Conjecture: every finite connected cubic loopless multigraph decomposes into a spanning tree, a 2-regular subgraph, and a matching. The proof rests on a new theorem on matching complements in subcubic graphs. Let H be a finite connected bridgeless simple graph of maximum degree three, and let S be its set of degree-two vertices, with |S| = k >= 2. We show that H has a matching of size equal to its cyclomatic number, |E(H)| - |V(H)| + 1, whose deletion leaves a single tree containing all of S, together with cycles disjoint from S. The proof is by induction on k.