Mathematics

We investigate a modal extension of the infinitary classical logic with countable meets and joins, formulated with an eye toward measure-theoretic work in dynamical systems and in point-free ergodic theory. We define a modal formalism in…

A coequivalence relation over a modal logic L is a formula in two tuples of propositional variables of the same length such that the logic L proves it to be an equivalence relation. They were introduced by Ghilardi and Zawadowski in the…

We investigate logics that generalize both intuitionistic logic and quantum logic. In earlier work, we introduced Ex-logic, an extension of Holliday's fundamental logic that coincides with the intersection of orthologic and the…

Optimal spline subspaces are an elegant and efficient tool to remove spurious outliers in isogeometric Galerkin discretizations for the approximation of the spectrum of the Laplace operator. For practical purposes, it is valuable to have a…

We present a numerical study of eigenvector deflation as a means of accelerating the WaveHoltz method for solving the Helmholtz equation. For energy-conserving (Dirichlet or Neumann) boundary conditions the WaveHoltz fixed-point iteration…

Complementary families of polynomials are introduced to generate $C^m$ finite element basis functions of order $p \geq 2m+2$ for arbitrary $m \ge 0$. One family consists of the Hermite splines that serve as the nodal basis functions by…

We propose Andrews-Gordon type series for certain level 2 standard modules of type $A^{(2)}_{\textrm{odd}}$, and prove the corresponding sum-product identities except for $A^{(2)}_{6n+3}$. These identities generalize the identities of…

We mainly consider a Liouville-type problem for the three dimensional stationary fractional Navier-Stokes equations with arbitrary asymptotic state $u_\infty$ at infinity. When $u_\infty\neq 0$ and $\frac{1}{2}\leq s<1$, we prove a complete…

We study a family of Hopf algebras arising as liftings of the Jordan plane over the infinite cyclic group. We determine their centres, prime and primitive spectra, and automorphism groups. We show that every prime ideal is completely prime…

In this paper, we introduce the concept of block sub-additive potential. The topological and measure-theoretic pressures are then defined for the space of average pseudo-orbits relative to any block sub-additive potential and any open cover…

Katz conjectured in a 2018 lecture that the family of curves $y^2=x^d-dx+t$ over the $t$-line is generically ordinary for all sufficiently large primes $p$. We prove that, for every $g\ge 2$ and every nonzero algebraic integer $\alpha$, the…

We explore identifying partial differential equations (PDEs) from noisy observations of single time-space trajectories. Recent developments show the benefits of identifying PDEs in their weak forms. We investigate the use of differential…

We show that if $G$ is a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture with vanishing first $L^{2}$-Betti number, then the map that assigns to each surjective integral character the first $L^2$-Betti number…

The fast multipole method (FMM) is an important component for the boundary element method (BEM), because with the FMM the efficiency and feasibility of the BEM can be enhanced to a large degree. Part of the FMM is grouping the elements of…

We study the large-scale behavior of the coincidence set of perturbations of global solutions to the classical obstacle problem in $\mathbb{R}^n\setminus B_1$, with blow-down invariant in the $e_n$ direction. In dimensions $n\geq 3$, we…

M.V. Zaicev and S.K. Segal, as well as S. D\u{a}sc\u{a}lescu, B. Ion, C. N\u{a}st\u{a}sescu, and D. Raios Montes studied certain gradings on matrix rings and algebras - 'elementary' gradings. However, examples of gradings on a matrix ring…

We propose an efficient algorithm for approximating the prime counting function $\pi(x)$ using a structured non-uniform partition derived from generalized triangular numbers. The method yields an incremental estimator whose updates require…

We prove that the special linear Lie algebra $\mathfrak{sl}_n(\textbf{F}_q)$ over a finite field of characteristic $p$ is generated by two random elements with high probability as $|\mathfrak{sl}_n(\textbf{F}_q)|$ tends to infinity,…

We propose a regularized algorithm, Regularized Newton-SLRA (RN-SLRA), for local manifold--affine intersection problems under weak intersection conditions, motivated in particular by structured low-rank approximation (SLRA). Newton-SLRA is…

In this paper, we establish a completed pre-Lie bialgebra structure on the tensor product of a Leibniz-dendriform bialgebra and a quadratic $\mathbb{Z}$-graded Zinbiel algebra. We also obtain such a structure on the tensor product of a…