Related papers: Stability of non-proper functions

We prove oscillation stability for the Banach space $\ell_\infty$: every weak-* Borel, uniformily continuous map from the unit sphere of this space to a compact metric space can be made arbitrarily close to a constant map when restricted to…

Let $f$ be a partially hyperbolic diffeomorphism. $f$ is called has the quasi-shadowing property if for any pseudo orbit $\{x_k\}_{k\in \mathbb{Z}}$, there is a sequence $\{y_k\}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ lies in the…

We develop a theory of generically stable and smooth Keisler measures in NIP metric theories, generalizing the case of classical logic. Using smooth extensions, we verify that fundamental properties of (Borel)-definable measures and the…

This work considers the question: what convergence guarantees does the stochastic subgradient method have in the absence of smoothness and convexity? We prove that the stochastic subgradient method, on any semialgebraic locally Lipschitz…

Let $G \subset {\mathbb R}^{n}$ be an open convex set which is either bounded or contains a translation of a convex cone with nonempty interior. It is known that then, for every modulus $\omega$, every function on $G$ which is both…

As consequence of the VC theorem, any pseudo-finite measure over an NIP ultraproduct is generically stable. We demonstrate a converse of this theorem and prove that any finitely approximable measure over an ultraproduct is itself…

As well-known, the $C^\infty$ stability of proper $C^\infty$ maps is characterized by the infinitesimal $C^\infty$ stability. In the present paper we study the counterpart in real analytic context. In particular, we show that the…

The leading-order approximation to a Filippov system $f$ about a generic boundary equilibrium $x^*$ is a system $F$ that is affine one side of the boundary and constant on the other side. We prove $x^*$ is exponentially stable for $f$ if…

Around 1970, Mather established a significant theory on the stability of $C^\infty$ mappings and gave a characterization of the density of proper stable mappings in the set of all proper mappings. The result yields a characterization of the…

We consider Gomory and Johnson's infinite group model with a single row. Valid inequalities for this model are expressed by valid functions and it has been recently shown that any valid function is dominated by some nonnegative valid…

A mathematical smooth function means that the function has continuous derivatives to a certain degree C(k). We call it a k-smooth function or a smooth function if k can grow infinitively. Based on quantum physics, there is no such smooth…

We study the stability properties of a class of time-varying nonlinear systems. We assume that non-strict input-to-state stable (ISS) Lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our…

We consider the incompressible Euler equations in $R^2$ when the initial vorticity is bounded, radially symmetric and non-increasing in the radial direction. Such a radial distribution is stationary, and we show that the monotonicity…

Reeb spaces of (continuous) real-valued functions on (nice) topological spaces are the spaces whose underlying sets consist of all connected components (contours) of their level sets and seen naturally as quotient spaces of the spaces. They…

We introduce a notion of "local stability in permutations" for finitely generated groups. If a group is sofic and locally stable in our sense, then it is also locally embeddable into finite groups (LEF). Our notion is weaker than the…

In this paper, we claim the availability of deterministic noises for stabilization of the origins of dynamical systems, provided that the noises have unbounded variations. To achieve the result, we first consider the system representations…

We show that if the rotation set of a homeomorphism of the torus is stable under small perturbations of the dynamics, then it is a convex polygon with rational vertices. We also show that such homeomorphisms are $C^0$-generic and have…

Linear stability of stratified two-phase flows in horizontal channels to arbitrary wavenumber disturbances is studied. The problem is reduced to Orr-Sommerfeld equations for the stream function disturbances, defined in each sublayer and…

We study the notion of stochastic stability with respect to diffusive perturbations for flows with smooth invariant measures. We investigate the question fully for non-singular flows on the circle. We also show that volume-preserving flows…

Let $G$ be a locally compact group. For every $G$-flow $X$, one can consider the stabilizer map $x \mapsto G_x$, from $X$ to the space $\mathrm{Sub}(G)$ of closed subgroups of $G$. This map is not continuous in general. We prove that if one…