Related papers: Generalized incidence theorems, homogeneous forms,…

Discrete forms of the mean and directed curvature are constructed on piecewise flat manifolds, providing local curvature approximations for smooth manifolds embedded in both Euclidean and non-Euclidean spaces. The resulting expressions take…

We obtain nontrivial exponents for Erd\H os-Falconer type problems. Let $T_k(E)$ denote the set of distinct congruent $k$-dimensional simplexes determined by $(k+1)$-tuples of points from $E$. We prove that there exists $s_0(d)<d$ such…

Generalizing the well-known relations on characteristic functions on a plane to the case of a one-dimensional regular surface (curve) with compact support, we establish implicit equations for these functions. Introducing an approximation,…

It is shown that $n$ points and $e$ lines in the complex Euclidean plane ${\mathbb C}^2$ determine $O(n^{2/3}e^{2/3}+n+e)$ point-line incidences. This bound is the best possible, and it generalizes the celebrated theorem by Szemer\'edi and…

This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions,…

Scattering and transition amplitudes with three-hadron final states play an important role in nuclear and particle physics. However, predicting such quantities using numerical Lattice QCD is very difficult, in part because of the effects of…

We asymptotically analyse the volume-random variables of general, symmetric and cyclically symmetric plane partitions fitting inside a box. We consider the respective symmetry class equipped with the uniform distribution. We also prove area…

We establish a sharp point-sphere incidence bound in finite fields for point sets exhibiting controlled additive structure. Working in the framework of \((4,s)\)-Salem sets, which quantify pseudorandomness via fourth-order additive energy,…

This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called {\it the flow of finite type}, are in one-to-one correspondence with discrete structures such as…

A compact set $E\subset {\Bbb R}^d$ is said to be arithmetically thick if there exists a positive integer $n$ so that the $n$-fold arithmetic sum of $E$ has non-empty interior. We prove the arithmetic thickness of $E$, if $E$ is uniformly…

We present a new general class of four-dimensional effective field theories with interesting global symmetry groups. These theories arise from purely gravitational actions for (3+1)-dimensional branes embedded in higher dimensional spaces…

We study the sum-product problem for the planar hypercomplex numbers: the dual numbers and double numbers. These number systems are similar to the complex numbers, but it turns out that they have a very different combinatorial behavior. We…

We prove optimal convergence rates for the discretization of a general second-order linear elliptic PDE with an adaptive vertex-centered finite volume scheme. While our prior work Erath and Praetorius [SIAM J. Numer. Anal., 54 (2016), pp.…

Accurately depicting multiphysics interactions in interfacial systems requires computational frameworks capable of reconciling geometric adaptability with strict conservation fidelity. However, traditional spatiotemporal discretisation…

Many results in mass partitions are proved by lifting $\mathbb{R}^d$ to a higher-dimensional space and dividing the higher-dimensional space into pieces. We extend such methods to use lifting arguments to polyhedral surfaces. Among other…

We present Finite Volume methods for diffusion equations on generic meshes, that received important coverage in the last decade or so. After introducing the main ideas and construction principles of the methods, we review some literature…

We prove sharp upper bounds on the volume and the number of lattice points on edges of higher-dimensional reflexive simplices. These convex-geometric results are derived from new number-theoretic bounds on the denominators of unit fractions…

We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs.…

We study the log-rank conjecture from the perspective of point-hyperplane incidence geometry. We formulate the following conjecture: Given a point set in $\mathbb{R}^d$ that is covered by constant-sized sets of parallel hyperplanes, there…

We study the probability for a random line to intersect a given plane curve, defined over a finite field, in a given number of points defined over the same field. In particular, we focus on the limits of these probabilities under successive…