数学前沿

Let $\overline{G}$ denote the wonderful compactification of a semisimple algebraic group $G$ of adjoint type. We use contractive flows to show that the totally nonnegative part of every $G \times G$ orbit closure of $\overline{G}$ is homeomorphic to a closed Euclidean ball. In particular, the totally nonnegative part of $\overline{G}$ is homeomorphic to a closed ball.

The usual settings for discussing the notion of stochastic objects are very different from Kolmogorov's original structure-free setting, in which there is no predefined external structure on the objects (such as the lengths of binary strings). A natural modification of stochasticity, obtained by replacing finite sets as models with probability measures, is quasi-stochasticity. The structure-free setting is common in studies of quasi-stochasticity, and known results show that the universal machine generates non-quasi-stochastic objects with exponentially small probability. This paper shows that, on the other hand, the universal machine generates nonstochastic objects surprisingly often. It also systematically explores numerous connections between stochasticity and quasi-stochasticity in the structure-free setting. In particular, the definition of quasi-stochasticity can be modified in a natural way modelled on Kolmogorov's definition so that non-quasi-stochastic objects are produced as often as nonstochastic ones; the results for this modification imply all the results of this paper showing that nonstochastic objects are frequent.

We consider the homogenization problem of the stationary Stokes equations in a periodically perforated domain whose solid structure consists of a thin truss. That is, the width $a_\epsilon$ of the truss tends to zero faster than the period size $\epsilon$. We identify the different scaling regimes and derive the corresponding effective equations. In the critical regime, we obtain the Brinkman equations, while in the subcritical regime, we recover the unchanged Stokes equations. In a subrange of the supercritical regime, we derive Darcy's law as the effective equation. Furthermore, we explicitly compute the effective coefficients in the Darcy and Brinkman limits. Moreover, as a tool in the analysis, we establish the analogous homogenization result for a periodic forest of cylinders. Finally, we also consider the homogenization result for a fence of cylinders.

We introduce the class $LW_+$ of $L$-tauberian operators and its dual counterpart, the class $MW_-$ of $M$-cotauberian operators. We obtain sequential, perturbative, and algebraic characterizations of these classes. As a consequence, every $T\in LW_+(E,Y)$ is disjointly non-singular, and the converse implication holds if and only if every dispersed subspace of $E$ has equi-integrable unit ball. For $E=L_p$, the latter property depends on $p$. For $1\leq p<2$, equi-integrability of the unit ball is equivalent to dispersedness, and $L$-tauberian operators coincide with disjointly non-singular operators, while for $p\geq 2$ these equivalences fail. Moreover, $LW_+(L_p)$ is neither open nor stable under composition. Applying a result of Haydon, Odell, and Schlumprecht, we show that for $2<p<\infty$ every infinite dimensional dispersed subspace of $L_p$ contains an infinite dimensional subspace with equi-integrable unit ball. However, we give an example showing that this is not true for $p=2$.

In 1981 Guibas and Odlyzko showed that the set $A$ of all words of a given length $n$ that avoid a given word $p$, and the set $B$ of all words of length $n$ that avoid a given word $q$, have the same cardinality iff $p$ and $q$ have the same self-border lengths. In 2024 Carrigan et. al. asked if there is a "natural" bijection between $A$ and $B$. This paper makes partial progress toward answering that question. We provide such a bijection by imposing additional conditions on $p$ and $q$, but we also obtain a stronger conclusion: there is a "natural" bijection $\Phi$ from the set of all finite words to itself such that a word $\Omega$ has a $p$ or $q$ at a given position iff $\Phi(\Omega)$ has, repectively, a $q$ or a $p$ at that same position. An algorithm is given for converting $\Omega$ into $\Phi(\Omega)$.

We study the dynamic allocation of scarce medications in Liberia's community health program. A central warehouse supplies a health facility and the Community Health Assistants (CHAs) it supervises. When a CHA runs out of medication, patients are referred to the facility, incurring travel costs if they complete the referral and remain untreated otherwise. Allocating inventory to CHAs allows more patients to receive treatment without traveling, while retaining inventory centrally preserves flexibility to respond to uncertain demand. The allocation problem must balance these considerations while accounting for fairness across locations. We formulate this problem as a finite-horizon stochastic control problem whose objective combines the costs of referrals and untreated patients with a demand-weighted fairness utility. Because standard dynamic programming methods are computationally intractable in this setting, we develop a continuous-time approximation that involves controlling the drift rate of a reflected Brownian motion. We then adapt the deep splitting method, using deep neural networks to approximate the value function and its gradient and compute a dynamic allocation policy. We calibrate the model using data on malaria cases among children in the Fangonda region of Lofa County, Liberia, and evaluate the proposed policy in the original discrete-time model. In the baseline setting, our policy reduces operating costs by about 6% relative to the strongest benchmark, the fairness-only policy, while retaining approximately 98% of its fairness utility. Additional experiments show that higher referral costs shift inventory toward CHAs and reduce local unmet demand, while greater demand variability increases our policy's performance over the fairness-only benchmark. Our model and computational approach provide a framework for studying similar last-mile allocation problems.

A description of scattering in asymptotically flat spacetimes that includes both radiation and massive asymptotic states requires control of the relation between null and timelike infinity. Motivated by this problem, we use the properties of conformal geodesics to construct an extended representation of timelike infinity in Minkowski spacetime. The construction exploits the fact that every non-null metric geodesic in an Einstein spacetime is, up to an explicit local reparametrisation, a conformal geodesic. It replaces the point $i^+$ by a hyperboloidal boundary $\mathcal{H}^+$ meeting future null infinity $\mathscr{I}^+$ at a corner $\mathcal{C}^{+}$, while fixing the location of $\mathscr{I}^+$ in the associated conformal Gaussian gauge. In this gauge, we derive the equations governing massless scalar and spin-$2$ fields. In particular, we obtain a transport equation for the scalar field along future null infinity and analyse the asymptotic behaviour of its spherical-harmonic modes near timelike infinity. We also derive the explicit massless spin-$2$ equations in this gauge. Finally, we examine the relation to projective extensions and to the hyperbolic base of the extended Carrollian space $\mathrm{Ti}$. Our results provide a geometric framework for studying the connection between late-time and null asymptotics which is a prerequisite for scattering constructions relevant to flat space holography.

We characterize observability of strongly coupled parabolic systems from arbitrary space-time measurable sets of positive measure. For constant coercive diffusion matrices and arbitrary constant zeroth-order coupling and observation matrices, observability holds if and only if the generalized Kalman rank condition is satisfied at every Dirichlet eigenvalue. The diffusion matrix need not be diagonalizable, and the number of components is arbitrary. The observation is measured in $L^1$, yielding null controllability with bounded controls by duality. The proof combines a Remez-type inequality for exponential polynomials with a quantitative analysis of the mode-dependent Kalman condition. The resulting finite-dimensional observability estimate is uniform over the spatial modes up to a polynomial frequency loss. A spectral inequality and the Lebeau--Robbiano strategy then give observability of the full system.

For a smooth morphism $f: X \longrightarrow \Sigma$ of real analytic manifolds and an $\mathbb{R}$-constructible sheaf $F$ on $X$ satisfying some condition, we define a family of Lagrangian cycles parameterized by $\Sigma$ that we call the relative characteristic cycle of $F$ for $f$. In this way, the theory of characteristic cycles due to Kashiwara and Schapira is naturally extended to the relative setting. Based on it, we then prove a formula for the characteristic cycles of real nearby cycle sheaves. This leads us to obtain also formulas for the characteristic cycles of various constructible sheaves, such as specialization, microlocalization, and complex nearby and vanishing cycle sheaves, in a unified manner. In fact, our methods allow us to calculate not only their characteristic cycles but also their microlocal types in many situations. We will illustrate it by various examples.

Cross helicity, together with total energy, determines the Els\"asser energies and thus quantifies departures from Alfv\'enic alignment in incompressible MHD. Building on the dual-field method in [Mao and Xi, \emph{J. Comput. Phys.}, 2025], we construct a linear finite element method for periodic domains that reproduces the cross-helicity balance. We connect the discrete energy and cross-helicity conservation laws to Alfv\'enic alignment through a nonnegative Els\"asser-energy representation, establishing preservation of the corresponding exact relation when either Els\"asser energy is initially zero in the unforced ideal case. We reassign the two discrete approximations of the magnetic field to the integer and half-integer time levels and redesign the Lorentz and induction couplings within the Mao-Xi formulation. The resulting scheme uses linear staggered updates and retains the energy balances, strong and weak divergence-free properties, and magnetic- and fluid-helicity balances. Numerical experiments illustrate the role of the cross-helicity balance in preserving an aligned state in the ideal case and in alignment dynamics under weak dissipation.

We prove new monotonicity properties for joint and generalized spectral radius and their essential versions, operator norm, Hausdorff measure of non-compactness and numerical radius of products and sums of Hadamard weighted geometric means of positive kernel operators on appropriate Banach function spaces. Our results extend recent results on weighted geometric symmetrizations of bounded sets of such operators defined on $L^2$. To our knowledge, several proved properties in this generality are new even in the finite dimensional case and also in the case of singelton sets.

We examine the use of neural networks in conjunction with the Born approximation for solving the inverse scattering problem for a penetrable medium in the frequency domain. The goal is a fast method for determining the refractive index of a scatterer from multistatic far-field measurements at one frequency. We present a new neural network model that seeks to improve on our previous results in by sharpening the reconstructed images. In addition, we present two other networks motivated by the corrected Born method that avoid the need to invert the regularized Born operator. These latter algorithms will be important for future three dimensional applications when the inverse Born approximation is difficult to compute, or when near-field measurements are used. Computational results illustrate that the new corrections to the Born approximation can provide an accurate and flexible approach to the inverse scattering problem.

Solving high-dimensional stationary Fokker-Planck (SFP) equations remains challenging. Recently, deep learning methods have shown great potential for solving high-dimensional SFP equations due to their universal approximation and mesh-free training. However, the high-dimensional integrals involved in loss computation and normalization can be expensive to approximate accurately using Monte Carlo integration. To address this issue, we propose a weak adversarial tensor neural network (WATNN) method for solving high-dimensional SFP equations, which combines a weak adversarial (WA) formulation with the tensor neural network (TNN). The WA formulation avoids the cross terms introduced by squaring the strong residual, reducing the number of integrals to be computed. When both the primal network and the adversarial test network are set as TNNs, the TNN quadrature scheme provides accurate and efficient normalization and loss computation with relatively few fixed quadrature points, avoiding the statistical sampling error that arises when using Monte Carlo integration. We demonstrate the effectiveness numerically on four high-dimensional SFP equations. For the 6D unimodal, 6D multi-modal, 10D multi-modal and 20D multi-modal problems, WATNN achieves the mean relative errors of 1.50%, 1.13%, 1.20% and 7.93% respectively, in only a few minutes.

Let $\mathrm{rep}(\mathbb I)$ be the category of finitely generated persistence modules on the interval $\mathbb I\subseteq\mathbb R$. We study the maximal rigid subcategories of $\mathrm{rep}(\mathbb I)$. Maximal rigid interval consistent subcategories are classified via tilting modules over the corresponding discrete type $A$ quiver. We compare several related notions, (weak) tilting and maximal rigidity, in both the context of hereditary abelian categories with enough projective objects and $\mathrm{rep}(\mathbb I)$ and prove that every maximal rigid interval consistent subcategory is weak tilting and every maximal rigid consistent below subcategory is tilting. We also establish a continuous analogue of Ingalls-Thomas's bijection by showing a bijection between integrally supported maximal rigid consistent below subcategories and integrally generated torsion classes.

We analyze the long-time stability and convergence of a third-order backward differentiation formula (BDF3) for parabolic equations with exponentially fading memory. The BDF3 scalar resolvent has oscillatory coefficients, so the monotonicity argument used for lower-order formulas in [Math. Comp. 62 (1994) 477-496] does not apply. We instead control memory feedback by the absolute mass of its discrete Green sequence and establish the rigorous stability-constant bounds $23/20<\Lambda_3<579/500$. A causal third-order memory quadrature with nonnegative assembled weights then yields exponentially weighted stability without a parabolic step-size restriction. With third-order starting values, we prove a uniform-in-time third-order temporal error estimate and extend it to conforming finite elements. Numerical tests confirm the predicted convergence orders and the long-time behavior.

We study an inductive method of computing initial ideals and Gr\"obner bases for families of ideals in a polynomial ring. This method starts from a given set of pairs $(I,J)$ where $I$ is any ideal and $J$ is a monomial ideal contained in the initial ideal of $I$. These containments become a system of equalities if one can establish a particular transition recurrence among the chosen ideals. We describe explicit constructions of such systems in two motivating cases -- namely, for the ideals of matrix Schubert varieties and their skew-symmetric analogues. Despite many formal similarities with these examples, for the symmetric versions of matrix Schubert varieties, it is an open problem to construct the same kind of transition system. We present several conjectures that would follow from such a construction, while also discussing the special obstructions arising in the symmetric case.

On a connected compact Riemannian spin manifold with boundary, we prove an upper estimate for the smallest eigenvalue of the Dirac operator under the boundary condition associated with a chirality operator. This upper bound is expressed in terms of the hyper-hemispherical radius. Moreover, the upper bound is realized if and only if the manifold is a hemisphere. As an application, we get a scalar curvature rigidity result for the hemisphere in the case of Lipschitz maps whose differential has a prescribed trace-norm bound. Our estimate extends B\"ar's result for closed manifolds, where the upper bound is given in terms of hyperspherical radius.

We develop a stochastic mirror descent method for continuous-time McKean-Vlasov control with measure-valued controls and general Bregman regularisation. The dependence on the state law leads to a coupled Wasserstein-Bregman stability problem and to a McKean-Vlasov adjoint equation, which have no counterpart in classical stochastic control. We establish a first-order representation of the cost and prove its relative smoothness in two regimes: nonlinear dynamics with bounded cost derivatives, and affine state-law dynamics with linearly growing cost derivatives, the latter covering linear-quadratic (LQ) and more general nonquadratic models. Our regularity conditions in the control measure are formulated through directional finite increments and do not require second-order flat derivatives. We further introduce relative semiconvexity along admissible forward-backward trajectories, allowing compensation between the dynamics, running cost, and terminal cost beyond pointwise Hamiltonian convexity. This yields energy dissipation, an O(1/n) convergence rate at the convexity threshold, and geometric convergence when regularisation dominates relative nonconvexity. Explicit LQ criteria and numerical experiments illustrate the scope of the results.

We review recent work connected with the invariant subspace problem for operators, in particular new developments in the last 15 years. In particular, we include discussions of almost-invariant subspaces, universal operators, specific classes of operators and new results in the framework of Banach spaces.

In 1948, Erd\"{o}s and Straus formulated a conjecture on Egyptian fractions. One only needs to prove the conjecture for any prime number $n$ such that $n\equiv 1\;(\mbox{mod}\;24)$. In this article, we give a necessary and sufficient polynomial condition for a number of the form $24m+1$ to be a prime number. Then we present 56 solvability polynomials consisting of our earlier tame solutions, wild solutions and the polynomials for composite numbers of the form $24m+1$. Numeric test shows that they imply the conjecture. Moreover, we give a list of 36 exponential-polynomial functions consisting of our earlier 28 tame solutions and the polynomials for composite numbers, whose numeric test shows that Howerton's conjecture on fourteen wild primes holds. Furthermore, we find a version of Goldbach conjecture modulo 24.