Related papers: Partial Dynamical Symmetry as an Intermediate Symm…
We consider several aspects of conjugating symmetry methods, including the method of invariants, with an asymptotic approach. In particular we consider how to extend to the stochastic setting several ideas which are well established in the…
A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in $\mathbb{R}^n$ is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for…
Apparently, all partial differential equations that describe physical phenomena in space-time can be cast into a universal quasilinear, first-order form. In this paper, we do two things. First, we describe some broad features of systems of…
A new type of sectional curvature is introduced. The notion is purely algebraic and can be located in linear algebra as well as in differential geometry.
Finding correspondences between 3D shapes is an important and long-standing problem in computer vision, graphics and beyond. A prominent challenge are partial-to-partial shape matching settings, which occur when the shapes to match are only…
The purpose of this article is to show that on an open and dense set, complete integrability implies the existence of symmetry.
To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop…
This article is focused on two related topics within the study of partial differential equations (PDEs) that illustrate a beautiful connection between dynamics, topology, and analysis: stability and spatial dynamics. The first is a property…
Partially ordered sets (posets) play a universal role as an abstract structure in many areas of mathematics. For finite posets, an explicit enumeration of distinct partial orders on a set of unlabelled elements is known only up to a…
Various versions of the Dynamical Systems Method (DSM) are proposed for solving linear ill-posed problems with bounded and unbounded operators. Convergence of the proposed methods is proved. Some new results concerning discrepancy principle…
The concept of spontaneous symmetry breaking (SSB) generally lacks a simple and intuitive introduction in the literature. This gap is filled by defining SSB in a universal context beyond its usual applications in physics and by discussing…
Dynamical Symmetry Breaking (DSB) is a concept theorists rely on very often in the discussions of strong dynamics, model building, and hierarchy problems. In this talk, I will discuss why this is such a permeating concept among theorists…
Perhaps the most important aspect of symmetry in physics is the idea that a state does not need to have the same symmetries as the theory that describes it. This phenomenon is known as spontaneous symmetry breaking. In these lecture notes,…
We consider nonconforming methods for symmetric elliptic problems and characterize their quasi-optimality in terms of suitable notions of stability and consistency. The quasi-optimality constant is determined and the possible impact of…
We solve the problem of best approximation by partial isometries of given rank to an arbitrary rectangular matrix, when the distance is measured in any unitarily invariant norm. In the case where the norm is strictly convex, we parametrize…
We introduce the notion of a topological symmetry as a quantum mechanical symmetry involving a certain topological invariant. We obtain the underlying algebraic structure of the Z_2-graded uniform topological symmetries of type (1,1) and…
A fractional generalization of variations is used to define a stability of non-integer order. Fractional variational derivatives are suggested to describe the properties of dynamical systems at fractional perturbations. We formulate…
Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for…
Partial differential equations (PDEs) are at the heart of many mathematical and scientific advances. While great progress has been made on the theory of PDEs of standard types during the last eight decades, the analysis of nonlinear PDEs of…
In this paper, the problem of partial stabilization of nonlinear systems along a given trajectory is considered. This problem is treated within the framework of stability of a family of sets. Sufficient conditions for the asymptotic…