数学

We design and analyse a family of hypocoercivity-preserving fully discrete Galerkin methods for the (inhomogeneous) kinetic Fokker--Planck (kFP) equations, a class of evolution PDEs with degenerate diffusion. The proposed methods mimic…

We study qualitative and quantitative homogenization for a Cahn--Hilliard system with a nonconservative source term in a periodically perforated domain. Using the periodic unfolding method, we derive uniform energy estimates and prove…

Cumulants are classical statistics associated with a random variable, defined as polynomial functions of its moments and distinguished by their additivity under convolution of distributions. A statistic is the name given to a function of a…

We prove a global \(C^{1+\text{H\"older}}\)-rigidity theorem for Anosov diffeomorphisms of tori with irreducible linearization. Let \(f:\mathbb T^N\to\mathbb T^N\) be a \(C^2\) Anosov diffeomorphism with linearization \(A\in GL(N,\mathbb…

Mossel and Peres (2005) established a comprehensive framework for designing Bernoulli factories. Notably, they demonstrated that a single-variable function admits a finite-automata Bernoulli factory if and only if it is a rational function.…

We construct the quantum geometric Langlands functor in the Betti setting via Whittaker coefficients. We show that the functor is compatible with the 2-Fourier-Mukai equivalence between sheaves of categories over 2-stacks…

We study Turn-Based Deterministic Forward Games (TBDFGs), the subclass of turn-based deterministic zero-sum games in which no directed cycle contains actions controlled by both players. This forward condition is strictly weaker than…

We prove that, for almost all even integers $N>0$, the set of Goldbach primes $\mathbb{P} \cap (N-\mathbb{P})$ has a level of distribution $1/6$. As applications, we show that almost all even integers $N>0$ can be written as the sum of two…

The paper studies the graph algebra whose monomial basis is naturally indexed by simple graphs on a fixed set of vertices. This algebra is at the same time the algebra of pseudo-Boolean functions on the Boolean cube and a natural object of…

We prove propagation of chaos for a finite-dimensional system of N interacting density matrix-valued diffusions. The interaction enters through a mean-field Hamiltonian, and each particle is driven by independent Brownian noises. The…

We record a Weyl-positive reduction and certificate framework for the Riemann phase kernel associated with the even Riemann kernel $\Phi$. The manuscript does not present a complete proof of the Riemann hypothesis. Its immediate analytic…

Automated conjecturing programs scan collections of graphs for inequalities between invariants that no stored graph violates, then offer the survivors for proof or refutation. TxGraffiti, one such program, conjectured that every nontrivial…

We prove that every entire smooth plurisubharmonic solution of the self-shrinking $J$-equation on $\mathbb{C}^n$ is a quadratic polynomial. This removes the asymptotic lower bound assumption on the complex Hessian in \cite[Theorem 4]{HJ}.…

Given a 3-term perfect complex E over a quasi-projective variety X and a nonnegative integer r, we define two virtual cycles and their refinements supported over the r-th degeneracy loci of E. This is done by modifying the complex E after…

We study the $p$-colored top-$m$-to-random shuffle on $C_p\wr S_n$ when the block size $m=m_n$ grows with $n$. Let $E_{k_n}^{(m_n)}$ be the number of labels never touched after $k_n$ independent uniform $m_n$-subset draws, and set…

In this paper, we consider operators that are compositions of an isometric isomorphism and a left multiplier on a Banach algebra, and we provide necessary and sufficient conditions for these operators to have a dense set of periodic…

In this paper, we introduce the class of Stable Positive Integral Deferred Correction (SPIDeC) methods for the numerical integration of positive dynamical systems. The proposed framework embeds a deferred correction mechanism within an…

In this paper, we investigate the multiplicity of normalized solutions to a nonlinear Dirac equation with localized nonlinearities on noncompact metric graphs under the \(L^2\)-constraint, as well as the asymptotic behavior of these…

Theorems on uniqueness and continuous dependence on the initial condition of solutions to stochastic Navier-Stokes equations driven by Wiener processes and Poisson martingale measures are presented. These theorems generalise some results…

For a fixed integer $k \ge 3$, we study the multiplicative functions $f\colon\mathbb{N}\to\mathbb{C}$ satisfying \[ f\Bigl(\sum_{i=1}^{2k} x_i^2\Bigr) = \sum_{j=1}^{k} f\bigl(x_{2j-1}^2 + x_{2j}^2\bigr) \] for all positive integers…