Related papers: Lyapunov's Theorem for continuous frames

We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS…

In this paper we study the Feichtinger Conjecture in frame theory, which was recently shown to be equivalent to the 1959 Kadison-Singer Problem in $C^{*}$-Algebras. We will show that every bounded Bessel sequence can be decomposed into two…

We consider a class of one--dimensional non--convex non--coercive problems in the Calculus of Variations. We prove an existence result for this class of problems using a Liapunov type theorem on the range of non--atomic measures.

We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the…

We study the construction of exponential frames and Riesz sequences for a class of fractal measures on ${\mathbb R}^d$ generated by infinite convolution of discrete measures using the idea of frame towers and Riesz-sequence towers. The…

In this article, we prove a general viability theorem for continuity inclusions in Wasserstein spaces, and provide an application thereof to the existence of exponentially stable trajectories obtained via the second method of Lyapunov.

The paper is concerned with the development of Lyapunov methods for the analysis of equilibrium stability in a dynamical system on the space of probability measures driven by a non-local continuity equation. We derive sufficient conditions…

We prove a robust converse barrier function theorem via the converse Lyapunov theory. While the use of a Lyapunov function as a barrier function is straightforward, the existence of a converse Lyapunov function as a barrier function for a…

This paper is concerned with stability analysis of nonlinear time-varying systems by using Lyapunov function based approach. The classical Lyapunov stability theorems are generalized in the sense that the time-derivative of the Lyapunov…

We give for the first time a detailed proof of the Palamodov's total instability conjecture in Lagrangian dynamics. This proves an older related Lyapunov instability conjecture posed by Lyapunov and Arnold and reduces the Lagrange-Dirichlet…

In 1998, Bou\'e and Dupuis proved a variational representation for exponentials of bounded Wiener functionals. Since their proof involves arguments related to the weak convergence of probability measures, the boundedness of functionals…

Urysohn's Lemma is a crucial property of normal spaces that deals with separation of closed sets by continuous functions. It is also a fundamental ingredient in proving the Tietze Extension Theorem, another property of normal spaces that…

In this paper, we investigate the almost surely pointwise convergence problem of free KdV equation, free wave equation, free elliptic and non-elliptic Schr\"odinger equation respectively. We firstly establish some estimates related to the…

In this paper, we rigorously construct $2d$ Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov "Quantum Geometry of bosonic strings". We also establish some of its fundamental properties…

We consider extensions of the Rattray theorem and two Makeev's theorems, showing that they hold for several maps, measures, or functions simultaneously, when we consider orthonormal $k$-frames in $\R^n$ instead of orthonormal basis (full…

For a complete noncompact connected Riemannian manifold with bounded geometry, we prove a compactness result for sequences of finite perimeter sets with uniformly bounded volume and perimeter in a larger space obtained by adding limit…

The classical Ka\v{c}anov scheme for the solution of nonlinear variational problems can be interpreted as a fixed point iteration method that updates a given approximation by solving a linear problem in each step. Based on this observation,…

We apply a Lyapunov function to obtain conditions for the existence and uniqueness of small classical time-periodic solutions to first order quasilinear 1D hyperbolic systems with (nonlinear) nonlocal boundary conditions in a strip. The…

This paper presents a proof that existence of a polynomial Lyapunov function is necessary and sufficient for exponential stability of sufficiently smooth nonlinear ordinary differential equations on bounded sets. The main result states that…

Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension. We prove a vanishing result for the continuously controlled algebraic K-theory of bounded…