On Lipschitz continuity and properties of composition operators acting on some hyperbolic classes

Let \Omega be an open subset of \mathbb{R}^n . Let f be a Borel measurable function from \mathbb{R} to \mathbb{R} . We prove necessary and sufficient conditions on f in order that the composite function T_f[g]=f\circ g belongs to a generalized Morrey space \mathcal{M}_p^w(\Omega) whenever g belongs to \mathcal{M}_p^w(\Omega) . Then we prove necessary conditions and sufficient conditions on f in order that the composition operator T_f[{\cdot}] be continuous, uniformly continuous, Hölder continuous and Lipschitz continuous in \mathcal{M}_p^w(\Omega) . We also consider its ‘vanishing’ generalized Morrey subspace \mathcal{M}_p^{w,0}(\Omega) and prove the related results for the composition operator as operator acting from \mathcal{M}_p^{w,0}(\Omega) to \mathcal{M}_p^w(\Omega) and also between the spaces \mathcal{M}_p^{w,0}(\Omega) . For the uniform, Hölder and Lipschitz continuity, we also have conditions that are both necessary and sufficient. We also have both necessary and sufficient conditions for the continuity under certain additional natural assumptions. We also consider the most commonly used Morrey classes that are related to power-type weights in the context of a discussion of some of the conditions that we impose on the weights.

Complexity of pattern classes and the Lipschitz property

Some Factorization Properties of Composition Operators

A function f : D ? R has Lipschitz constant c if dR(f(x),f(y)) = c· dD(x,y) for all x,y in D, where dR and dD denote the distance functions on the range and domain of f, respectively. We say a function is Lipschitz if it has Lipschitz constant 1. (Note that rescaling by a factor of 1/c converts a function with a Lipschitz constant c into a Lipschitz function.) In other words, Lipschitz functions are not very sensitive to small changes in the input. We initiate the study of testing and local reconstruction of the Lipschitz property of functions. A property tester has to distinguish functions with the property (in this case, Lipschitz) from functions that are e -far from having the property, that is, differ from every function with the property on at least an e fraction of the domain. A local filter reconstructs an arbitrary function f to ensure that the reconstructed function g has the desired property (in this case, is Lipschitz), changing f only when necessary. A local filter is given a function f and a query x and, after looking up the value of f on a small number of points, it has to output g(x) for some function g, which has the desired property and does not depend on x. If f has the property, g must be equal to f. We consider functions over domains {0,1}d, {1,..., n} and {1,..., n}d, equipped with l1 distance. We design efficient testers of the Lipschitz property for functions of the form f:{0,1}d? d Z, where d ? (0,1] and d Z is the set of integer multiples of d, and of the form f: {1,..., n} ? R, where R is (discretely) metrically convex. In the first case, the tester runs in time O(d· min{d,r}/d e ), where r is the diameter of the image of f, in the second, in time O((log n)/e ). We give corresponding lower bounds of O (d) and O (log n) on the query complexity (in the second case, only for nonadaptive 1-sided error testers). Our lower bound for functions over {0,1}dis tight for the case of the {0,1,2} range and constant e. The first tester implies an algorithm for functions of the form f:{0,1}d? R that distinguishes Lipschitz functions from functions that are e -far from (1+d )-Lipschitz. We also present a local filter of the Lipschitz property for functions of the form f: {1,..., n}d ? R with lookup complexity O((log n+1)d). For functions of the form {0,1}d, we show that every nonadaptive local filter has lookup complexity exponential in d. The testers that we developed have applications to programs analysis. The reconstructors have applications to data privacy. For the first application, the Lipschitz property of the function computed by a program corresponds to a notion of robustness to noise in the data. The application to privacy is based on the fact that a function f of entries in a database of sensitive information can be released with noise of magnitude proportional to a Lipschitz constant of f, while preserving the privacy of individuals whose data is stored in the database (Dwork, McSherry, Nissim and Smith, TCC 2006). We give a differentially private mechanism, based on local filters, for releasing a function f when a Lipschitz constant of f is provided by a distrusted client. We show that when no reliable Lipschitz constant of f is given, previously known differentially private mechanisms either have a substantially higher running time or have a higher expected error for a large class of symmetric functions f.

A necessary and sufficient condition is obtained for M‐quasi‐hyponormal composition operators. It has also been proved that the class of M‐quasi‐hyponormal composition operators coincides with the class of M‐paranormal composition operators. Existence of M‐hyponormal composition operators which are not hyponormal; and M‐quasihyponormal composition operators which are not M‐hyponormal and quasi‐hyponormal are also shown.

We study local filters for two properties of functions f:{0,1} d → ℝ: the Lipschitz property and monotonicity. A local filter with additive error a is a randomized algorithm that is given black-box access to a function f and a query point x in the domain of f. Its output is a value F(x), such that (i) the reconstructed function F(x) satisfies the property (in our case, is Lipschitz or monotone) and (ii) if the input function f satisfies the property, then for every point x in the domain (with high constant probability) the reconstructed value F(x) differs from f(x) by at most a. Local filters were introduced by Saks and Seshadhri (SICOMP 2010) and the relaxed definition we study is due to Bhattacharyya et al.(RANDOM 2010), except that we further relax it by allowing additive error. Local filters for Lipschitz and monotone functions have applications to areas such as data privacy.We show that every local filter for Lipschitz or monotone functions runs in time exponential in the dimension d, even when the filter is allowed significant additive error. Prior lower bounds (for local filters with no additive error, i.e., with a = 0) applied only to more restrictive class of filters, e.g., nonadaptive filters. To prove our lower bounds, we construct families of hard functions and show that lookups of a local filter on these functions are captured by a combinatorial object that we call a c-connector. Then we present a lower bound on the maximum outdegree of a c-connector, and show that it implies the desired bounds on the running time of local filters. Our lower bounds, in particular, imply the same bound on the running time for a class of privacy mechanisms.

Cell array faults in random-access memories (RAMs) are usually represented by Mealy automata. In such a model, multiple faults should also be representable by automata; in fact, it should be possible to compute the automaton representing a multiple fault from the automata representing the single faults that make up the multiple fault. In this paper we study properties of binary composition operations on automata that are appropriate for the representation of multiple faults in RAMs. First, we derive a set of generic conditions that every composition operation must satisfy. Second, we develop a set of physical conditions that the composition must satisfy in order to apply to stuck-at, transition and coupling faults in RAMs. Third, we represent the transition table rules used by van de Goor and Smit by a composition operation and prove that this operation satisfies both the generic and physical conditions. Fourth, we point out that in some circumstances, it is appropriate to use a different composition operation (defined by us in a previous paper) to permit a different handling of coupling faults in the presence of stuck-at or transition faults. We compare and relate the properties of the two algebras.

In this paper, we investigate the boundedness of composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We prove that the boundedness of a composition operator strongly limits the behavior of the original map, and it provides an effective method to investigate the properties of composition operators using the theory of dynamical system. Consequently, we prove that only affine maps can induce bounded composition operators on any quasi-Banach space continuously included in the space of entire functions of one variable if the function space contains a nonconstant function. We also prove that any polynomial automorphisms except affine transforms cannot induce bounded composition operators on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under several mild conditions.

Global exponential periodicity and global exponential stability of a class of recurrent neural networks with various activation functions and time-varying delays

In this chapter we are concerned with the basic properties of the superposition operator in so-called ideal spaces which are, roughly speaking, Banach spaces of measurable functions with monotone norm. To formulate our results in a sufficiently general framework, we must introduce a large number of auxiliary notions which will be justified by the results in concrete function spaces given in subsequent chapters; we request the reader's indulgence until then. First, we give conditions for the local and global boundedness of the superposition operator F between ideal spaces X and Y which are typically ensured by special properties of the “source space” X . Second, special properties, such as absolute boundedness and compactness, are treated. Afterwards, we give conditions for the continuity and uniform continuity of F which are now typically ensured by special properties of the “target space” Y . For example, F is “always” continuous if Y is regular, and “never” continuous if Y is completely irregular (see the definitions below). Weak continuity of F between ideal spaces is also considered; here we mention the surprising fact that, loosely speaking, only linear superposition operators are weakly continuous. Next, we give necessary and sufficient conditions under which F satisfies a Lipschitz or Darbo condition. It turns out that in many spaces these two conditions are in fact equivalent. Finally, the last part of this chapter is concerned with differentiability conditions for the superposition operator between ideal spaces.

In this paper, we study quasicompact and Riesz composition operators on Banach spaces of Lipschitz–Holder functions on pointed metric spaces. For a composition operator T on these spaces, we give an upper bound for $$r_e(T)$$ , the essential spectral radius of T, and establish a formula for $$r_e(T)$$ whenever metric spaces are compact. We also give some necessary and some sufficient conditions that a composition operator T on these spaces to be quasicompact or Riesz. Finally, we get a relation for the set of eigenvalues and the spectrum of a quasicompact and Riesz composition operator on these spaces.

Suppose $\\varphi$ is an analytic self-map of open unit disk $\\mathbf{D}$ and $\\psi$ is an analytic function on $\\mathbf{D}$. Then a weighted composition operator induced by $\\varphi$ with weight $\\psi$ is given by $(W_{\\psi, \\varphi}f)(z)= \\psi(z)f(\\varphi(z))$, for $z \\in \\mathbf{D}$ and $f$ analytic on $\\mathbf{D}$. Necessary and sufficient conditions are given for the boundedness and compactness of the weighted composition operators $W_{\\psi, \\varphi}$. In terms of fixed points in the closed unit disk $\\overline{\\mathbf{D}}$, conditions under which $W_{\\psi, \\varphi}$ is compact are given. Necessary conditions for the compactness of $C_{\\varphi}$ are given in terms of the angular derivative $\\varphi^{\\prime}(\\zeta)$ where $\\zeta$ is on the boundary of the unit disk. Moreover, we present sufficient conditions for the membership of composition operators in the Schatten $p$-class $S_{p}(H^s(\\beta_1), H^q(\\beta_2))$, where the inducing map has supremum norm strictly smaller than~$1$.

This paper investigates global asymptotic stability (GAS) and global exponential stability (GES) of a class of continuous-time recurrent neural networks. First, we introduce a necessary and sufficient condition for the existence and uniqueness of equilibrium of the neural networks with Lipschitz continuous activation functions. Next, we present two sufficient conditions to ascertain the GAS of the neural networks with globally Lipschitz continuous and monotone nondecreasing activation functions. We then give two GES conditions for the neural networks whose activation functions may not be monotone nondecreasing. We also provide a Lyapunov diagonal stability condition, without the nonsingularity requirement for the connection weight matrices, to ascertain the GES of the neural networks with globally Lipschitz continuous and monotone nondecreasing activation functions. This Lyapunov diagonal stability condition generalizes and unifies many of the existing GAS and GES results. Moreover, two higher exponential convergence rates are estimated.

Extensions of vector-valued functions with preservation of derivatives

A non-linear unknown input observer (UIO) is designed for systems with non-linearities having lower bonds on their slopes. The observer is constructed such that the non-linearity in the observer error system satisfies a sector property. Based on this property rather than the Lipschitz property, an algorithm not limited by Lipschitz constant is proposed. Further, all the parameters in the proposed non-linear UIO can be calculated by solving a linear matrix inequality. The UIO is also extended to an adaptive one for a class of uncertain non-linear systems in which the arguments of non-linear terms are perturbed by unknown parameters. Finally, simulations of electric driving system are used to illustrate the effectiveness of the UIOs.