数学

The generalized Burgers-Huxley (GBH) equation is a prototype model that describes the interplay among reaction, convection, and diffusion. In this article, we explore the controllability of this model by means of an interior control…

Let $k_r(G)$ denote the number of $r$-cliques in a graph $G$ and let $F_r(\cdot)$ be the Lov\'asz--Simonovits $r$-clique density function. For any integers $2\le s<t$, we determine the asymptotically sharp lower bound on $k_t(G)$ in an…

Motivated by applications of algebraic geometry to reconstruction problems in computer vision, we initiate a study of the equations of degeneracy loci associated with linearly dependent points on Segre varieties. When these points are…

We introduce the Beckmann boundary of a Boolean function \[ \mathsf{B}(f)=\inf_{\operatorname{div} V=Lf}\mathbb E\|V(x)\|_2. \] Here \[ L=\sum_iD_i,\qquad D_i f(x)=\frac{f(x)-f(x^{\oplus i})}{2}, \] and $\operatorname{div} V(x)=\sum_i…

The nonconvex $\ell_p$ quasi-norm with $0<p<1$ is a powerful sparsity surrogate but makes the proximity operator $\mathrm{prox}_{\lambda|\cdot|^p}$ nontrivial to evaluate robustly. We give an explicit characterization of the scalar proximal…

We give algebraic models for the tame homotopy type of the configuration spaces of certain algebraic varieties of Tate type. Such tame models carry information on the l-adic homotopy type. Our method uses the theory of weights in \'etale…

In this paper, we establish the relation between the Ekedahl-Kottwitz-Oort-Rapoport stratification and the Bruhat-Tits stratification on the unramified $\mathrm{GU}(1,n-1)$ Rapoport-Zink space with arbitrary parahoric level. More precisely,…

We develop maximal inequalities for empirical processes indexed by graph-dependent observations. Our bounds separate the complexity of the indexing class from two features specific to graph dependence: the geometry of the underlying graph…

This paper establishes the first rigorous superconvergence theory for semidiscrete and fully discrete central discontinuous Galerkin (CDG) methods for linear hyperbolic equations on overlapping meshes. While the optimal $L^2$ convergence of…

We introduce a class of separable II$_1$ factors $M$ admitting no non-trivial crossed product decompositions: $M\not\cong B\rtimes_\sigma G$, for any trace preserving action $G\curvearrowright^\sigma (B,\tau)$ of an infinite countable group…

We study the solvability of some linear nonhomogeneous elliptic problems and establish that under certain technical assumptions the convergence in $L^2$ of their right-hand sides yields the existence and the convergence in $H^4$ of the…

The familiar color wheel is a disk divided into six sectors, colored red, orange, yellow, green, blue, and purple, in circular order. Three of the colors can be obtained by blending the colors in the two neighboring sectors. One might…

We present an orbit--theoretic reformulation of Galois theory based on the natural action of automorphism groups on fields. Given a field $\mathbf{E}$ and a subgroup $H$ of the automorphism group $\mathrm{Aut}(\mathbf{E})$, we show that…

In this paper, we focus on a new type of non-linear kinetic Fokker-Planck equation where the non-linearity comes from a non-linear diffusion in the velocity variable. The existence of solutions in suitable Lebesgue spaces is proved,…

Let $(X,\Theta)$ be a very general principally polarized abelian variety of dimension $g$, and consider the minimal cohomology class $\theta_k=[\Theta]^k/k!$ for $k<g$. We show that the minimal positive multiple of $\theta_k$ which is…

In nonstandard analysis the halo of a point in a topological space is the intersection of the nonstandard extensions of all its open neighbourhoods. We define a parametric family of modal operators from the halo by varying which elements of…

We develop translations from relevant logics into normal modal logics, and use them to clarify structural connections between relevant and modal logic, obtain a few corollary results, and raise questions for future work.

We define the notion of IK-bisimulation between the relational semantics for the intuitionistic modal logic IK, and prove that IK arises as the IK-bisimulation-invariant fragment of intuitionistic first-order logic. En route, we provide an…

We study interpretations of modal logics in one another where the Boolean connectives are interpreted identically and the modal operator diamond is interpreted by an arbitrary formula A(p). Clearly, such a formula A(p) defines an…

We compute the Clarke subdifferential of the $k$th eigenvalue functional on the space of self-adjoint operators, obtaining a first-variation formula that remains valid even when the eigenvalue lies at the edge of the essential spectrum.…