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In the study of discrete dynamical systems, we typically start with a function from a space into itself, and ask questions about the properties of sequences of iterates of the function. In this paper we reverse the direction of this study.…
There is a deformation of the ordinary differential calculus which leads from the continuum to a lattice (and induces a corresponding deformation of physical theories). We recall some of its features and relate it to a general framework of…
We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose…
Direct methods for the simulation of optimal control problems apply a specific discretization to the dynamics of the problem, and the discrete adjoint method is suitable to calculate corresponding conditions to approximate an optimal…
In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to…
In this work, we propose a convenient framework for infinite-dimensional analysis (including both real and complex analysis in infinite dimensions), in which differentiation (in some weak sense) and integration operations can be easily…
We propose a new class of fundamental solutions for the numerical analysis of boundary value problems for the Maxwell equations. We prove completeness of systems of such fundamental solutions in appropriate Sobolev spaces on a smooth…
Classical spectral theory provides powerful tools for analyzing linear operators, but does not extend naturally to nonlinear or compositional settings. In particular, there is no general way to transport spectral invariants in a functorial…
We develop a general framework for construction and analysis of discrete extension operators with application to unfitted finite element approximation of partial differential equations. In unfitted methods so called cut elements intersected…
In the past decade, we had developed a series of splitting contraction algorithms for separable convex optimization problems, at the root of the alternating direction method of multipliers. Convergence of these algorithms was studied under…
We develop a discrete extension operator for trimmed spline spaces consisting of piecewise polynomial functions of degree $p$ with $k$ continuous derivatives. The construction is based on polynomial extension from neighboring elements…
In this paper we present a new theory of calculus over $k$-dimensional domains in a smooth $n$-manifold, unifying the discrete, exterior, and continuum theories. The calculus begins at a single point and is extended to chains of finitely…
Application of intersection theory to construction of n-point finite-difference equations associated with classical integrable systems is discussed. As an example, we present a few exact discretizations of one-dimensional cubic and quintic…
We apply topological methods to obtain global continuation results for harmonic solutions of some periodically perturbed ordinary differential equations on a $k$-dimensional differentiable manifold $M \subseteq \mathbb{R}^m$. We assume that…
The connection of (split-)division algebras with Clifford algebras and supersymmetry is investigated. At first we introduce the class of superalgebras constructed from any given (split-)division algebra. We further specify which real…
Fourier transforms are ubiquitous mathematical tools in basic and applied sciences. We here report classical and quantum optical realizations of the discrete fractional Fourier transform, a generalization of the Fourier transform. In the…
The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary…
The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how…
The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the…
In this thesis I discuss combinatorial optimization problems, from the statistical physics perspective. The starting point are the motivations which brought physicists together with computer scientists and mathematicians to work on this…