Related papers: Vandermonde Matrices with Nodes in the Unit Disk a…
In default of a fundamental MOND theory -- a FUNDAMOND -- I advocate that, alongside searching for one, we should try to identify predictions that follow from wide classes of MOND theories, if not necessarily from all. In particular,…
In this thesis we study the principle that extremal objects in differential geometry correspond to stable objects in algebraic geometry. In our introduction we survey the most famous instances of this principle with a view towards the…
For finite difference discretizations with linear complexity and provably convergent to weak solutions of the second boundary value problem for the Monge-Amp\`ere equation, we give the first proof of uniqueness. The boundary condition is…
We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result…
We extend the Ax-Katz theorem for a single polynomial from finite fields to the rings Z_m with m composite. This extension not only yields the analogous result, but gives significantly higher divisibility bounds. We conjecture what computer…
We provide a monotone non increasing sequence of upper bounds $f^H_k$ ($k\ge 1$) converging to the global minimum of a polynomial $f$ on simple sets like the unit hypercube. The novelty with respect to the converging sequence of upper…
We consider three different questions related to the Steklov and mixed Steklov problems on surfaces. These questions are connected by the techniques that we use to study them, which exploit symmetry in various ways even though the surfaces…
We introduce an arbitrary order, stabilized finite element method for solving a unique continuation problem subject to the time-harmonic elastic wave equation with variable coefficients. Based on conditional stability estimates we prove…
We show that there is generically non-uniqueness for the anisotropic Calder\'on problem at fixed frequency when the Dirichlet and Neumann data are measured on disjoint sets of the boundary of a given domain. More precisely, we first show…
In this paper, we generalize the algorithm described by Rump and Graillat, as well as our previous work on certifying breadth-one singular solutions of polynomial systems, to compute verified and narrow error bounds such that a slightly…
Let $\mathfrak{M}_n$ be the multiplicative monoid of $n \times n$ matrices over a finite field. The monoid algebra $\mathbf{C}[\mathfrak{M}_n]$ has been studied for several decades. One of the important early results is Kov\'acs' theorem…
We develop a unified approach to bounding the largest and smallest singular values of an inhomogeneous random rectangular matrix, based on the non-backtracking operator and the Ihara-Bass formula for general random Hermitian matrices with a…
Given a reproducing kernel $k$ on a nonempty set $X$, we define the reproductive boundary of $X$ with respect to $k$. Furthermore, we generalize the well known nontangential and horocyclic approach regions of the unit circle to this new…
The theory of polar forms of polynomials is used to provide for sharp bounds on the radius of the largest possible disc (absolute stability radius), and on the length of the largest possible real interval (parabolic stability radius), to be…
Matrices with displacement structure such as Pick, Vandermonde, and Hankel matrices appear in a diverse range of applications. In this paper, we use an extremal problem involving rational functions to derive explicit bounds on the singular…
In this paper, we study the perturbation of the extreme singular values of a matrix in the particular case where it is obtained after appending an arbitrary column vector. Such results have many applications in bifurcation theory, signal…
We prove a special case of Erd\H{o}s' unit distance problem using a corollary of the subspace theorem bounding the number of solutions of linear equations from a multiplicative group. We restrict our attention to unit distances coming from…
There are many application papers that solve elliptic boundary value problems by meshless methods, and they use various forms of generalized stiffness matrices that approximate derivatives of functions from values at scattered nodes…
The scalar wave equation is solved using higher order immersed finite elements. We demonstrate that higher order convergence can be obtained. Small cuts with the background mesh are stabilized by adding penalty terms to the weak…
We consider a strongly damped wave equation on compact manifolds, both with and without boundaries, and formulate the corresponding inverse problems. For closed manifolds, we prove that the metric can be uniquely determined, up to an…