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This is an elementary introduction to exterior differential systems motivated by two examples: minimal submanifolds and the isometric embedding problem. The two main goals of the lectures are: 1. To explain how to find an appropriate…
We present a spectral mapping theorem for semigroups on any Banach space $E$. From this, we obtain a characterization of exponential dichotomy for nonautonomous differential equations for $E$-valued functions. This characterization is given…
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
The nonlinear sigma model (NLSM) epitomises a field-theoretical approach to (interacting) electrons in disordered media. These lectures are aimed at the audience who might have vaguely heard about its existence but know very little of what…
These notes were compiled as lecture notes for a course developed and taught at the University of the Southern California. They should be accessible to a typical engineering graduate student with a strong background in Applied Mathematics.…
We describe a method to model nonlinear dynamical systems using periodic solutions of delay-differential equations. We show that any finite-time trajectory of a nonlinear dynamical system can be loaded approximately into the initial…
High dimensional classification has been highlighted for last two decades and much research has been conducted in order to circumvent challenges encountered in high dimensions. While existing methods have focused mainly on developing…
In this note, we address the following question: Why certain nonassociative algebra structures emerge in the regularity theory of elliptic type PDEs and also in constructing nonclassical and singular solutions? The aim of the paper is…
The stability analysis of a class of discontinuous discrete-time systems is studied in this paper. The system under study is modeled as a feedback interconnection of a linear system and a set-valued nonlinearity. An equivalent…
This paper develops sufficient conditions for the existence of global exponential observers for two classes of nonlinear systems: (i) the class of systems with a globally asymptotically stable compact set, and (ii) the class of systems that…
This paper develops a new approach to the estimation of the degree of boundedness or stability of multidimensional nonlinear systems with time-dependent nonperiodic coefficients-an essential task in various engineering and natural science…
The linear complete differential resultant of a finite set of linear ordinary differential polynomials is defined. We study the computation by linear complete differential resultants of the implicit equation of a system of $n$ linear…
The purpose of the paper is to review a variety of recent developments in the theory of positive solutions of general linear elliptic and parabolic equations of second-order on noncompact Riemannian manifolds, and to point out a number of…
In this paper we consider a reduction of a non-homogeneous linear system of first order operator equations to a totally reduced system. Obtained results are applied to Cauchy problem for linear differential systems with constant…
The work relates to a new way for analysis of one-dimensional stochastic systems, based on consideration of its higher order difference structure. From this point of view, the deterministic and random processes are analyzed. A new numerical…
In this paper, we study representation learning for multi-task decision-making in non-stationary environments. We consider the framework of sequential linear bandits, where the agent performs a series of tasks drawn from distinct sets…
038<p type="texpara" tag="Body Text" >The identification theory and realization of the dynamic systems is a medullary aspect in the modern control theory that consists fundamentally in that, starting from the knowledge of the behavior…
A differential algebra of nonlinear generalized functions is presented as a tool for a wide range of nonsmooth nonlinear problems. The power of the differential algebra is used to do mathematical calculations or proofs; then the final…
This mini-course of 20 lectures aims at highlights of spectral theory for self-adjoint partial differential operators, with a heavy emphasis on problems with discrete spectrum. Part I: Discrete Spectrum (ODE preview, Laplacian - computable…
One of the most difficult topics in the subject of Discrete Mathematics is the subject of Propositional Logic, therefore the present work had as objective to facilitate the learning of Propositional Logic through the implementation of…