Holder Continuous Functions Research Articles

On the Practical Compensator Design of Time-Varying Perturbed Systems in Hilbert Spaces

In this paper, we treat the problem of practical feedback stabilization for a class of nonlinear infinite-dimensional control systems by means of an observer. A compensator design is given under a restriction of the perturbed term that the perturbations is bounded by the sum of a Holder continuous function and a Lipschitz function where the nominal system is supposed to be exponentially stabilizable by a linear feedback. The results are illustrated with an example.

The Double Difference Property for the Class of Locally Hölder Continuous Functions

In this paper, we show that the pair of classes of locally Holder continuous functions (considered on $\mathbb{R}$ and $\mathbb{R}^{2}$, respectively) has the double difference property.

Distribution of Typical Orbits for Random Dynamics Generated by Finitely Many Rational Maps

In this paper, we consider the dynamics of a skew-product map defined on the Cartesian product of the symbolic one-sided shift space on N symbols and the complex sphere where we allow N rational maps, $$R_{1}, R_{2}, \ldots , R_{N}$$ , each with degree $$d_{i};\ 1 \le i \le N$$ and with at least one $$R_{i}$$ in the collection whose degree is at least 2. We obtain results regarding the distribution of pre-images of points and the periodic points in a subset of the product space (where the skew-product map does not behave normally). We further explore the ergodicity of the Sumi-Urbanski (equilibrium) measure associated to some real-valued Holder continuous function defined on the Julia set of the skew-product map and obtain estimates on the mean deviation of the behaviour of typical orbits, violating such ergodic necessities.

Convergence of delay equations driven by a Holder continuous function of order 1/3<β<1/2

In this article we show that, when the delay approaches zero, the solution of multidimensional delay differential equations driven by a Holder continuous function of order 1/3<β<1/2 converges with the supremum norm to the solution for the equation without delay. Finally we discuss the applications to stochastic differential equations For more information see https://ejde.math.txstate.edu/Volumes/2020/65/abstr.html

Long Time Behavior of Random and Nonautonomous Fisher–KPP Equations: Part I—Stability of Equilibria and Spreading Speeds

In the current series of two papers, we study the long time behavior of nonnegative solutions to the following random Fisher–KPP equation, 1 $$\begin{aligned} u_t =u_{xx}+a(\theta _t\omega )u(1-u),\quad x\in {{\mathbb {R}}}, \end{aligned}$$ where $$\omega \in \Omega $$ , $$(\Omega , {\mathcal {F}},{\mathbb {P}})$$ is a given probability space, $$\theta _t$$ is an ergodic metric dynamical system on $$\Omega $$ , and $$a(\omega )>0$$ for every $$\omega \in \Omega $$ . We also study the long time behavior of nonnegative solutions to the following nonautonomous Fisher–KPP equation, 2 $$\begin{aligned} u_t=u_{xx}+a_0(t)u(1-u),\quad x\in {{\mathbb {R}}}, \end{aligned}$$ where $$a_0(t)$$ is a positive locally Holder continuous function. In this first part of the series, we investigate the stability of positive equilibria and the spreading speeds. Under some proper assumption on $$a(\omega )$$ , we show that the constant solution $$u=1$$ of (1) is asymptotically stable with respect to strictly positive perturbations and show that (1) has a deterministic spreading speed interval $$[2\sqrt{{\underline{a}}}, 2\sqrt{{\bar{a}}}]$$ , where $${\underline{a}}$$ and $${\bar{a}}$$ are the least and the greatest means of $$a(\cdot )$$ , respectively, and hence the spreading speed interval is linearly determinate. It is shown that the solution of (1) with a nonnegative initial function which is bounded away from 0 for $$x\ll -1$$ and is 0 for $$x\gg 1$$ propagates at the speed $$2\sqrt{{\hat{a}}}$$ , where $${\hat{a}}$$ is the mean of $$a(\cdot )$$ . Under some assumption on $$a_0(\cdot )$$ , we also show that the constant solution $$u=1$$ of (2) is asymptotically stably and (2) admits a bounded spreading speed interval. It is not assumed that $$a(\omega )$$ and $$a_0(t)$$ are bounded above and below by some positive constants. The results obtained in this part are new and extend the existing results in literature on spreading speeds of Fisher–KPP equations. In the second part of the series, we will study the existence and stability of transition fronts of (1) and (2).

Towards Geometric Integration of Rough Differential Forms

We provide a draft of a theory of geometric integration of “rough differential forms” which are generalizations of classical (smooth) differential forms to similar objects with very low regularity, for instance, involving Holder continuous functions that may be nowhere differentiable. Borrowing ideas from the theory of rough paths, we show that such a geometric integration can be constructed substituting appropriately differentials with more general asymptotic expansions. This can be seen as the basis of geometric integration similar to that used in geometric measure theory, but without any underlying differentiable structure, thus allowing Lipschitz functions and rectifiable sets to be substituted by far less regular objects (e.g. Holder functions and their images which may be purely unrectifiable). Our construction includes both the one-dimensional Young integral and multidimensional integrals introduced recently by Zust, and provides also an alternative (and more geometric) view on the standard construction of rough paths. To simplify the exposition, we limit ourselves to integration of rough k-forms with $$k\le 2$$ .

The Relation between Hölder Continuous Function of Order α ∈ (0,1) and Function of Bounded Variation

The Cantor ternary function is the most famous example of a continuous function of bounded variation for which it satisfies the Holder continuous function of order α = log3 2, but does not satisfy for order α = 1. In this paper, based on previous work of Hölder continuous function of order α ∈ (0, 1) and using F α - calculus on fractal set F, we show the relation between the Hölder continuous function of order α ∈ (0, 1) and function of bounded variation. In particular, we give the necessary and sufficient condition for the variation of function satisfies Hölder condition or bi-Hölder condition.

Estimates of the Difference Between Two Probability Densities of Wiener Functionals and Its Application

This study investigates precise estimates of the difference between two probability densities of Wiener functionals in the space of continuously differentiable functions and the Holder continuous functions. As an application, the convergence rate of the density of the solution to non-Markovian stochastic differential equations is derived utilizing these precise estimates.

Perturbed finite-state Markov systems with holes and Perron complements of Ruelle operators

We consider a perturbed system ($$\Sigma_A^+$$,ϕ(ϵ, ·)) with topologically transitive subshift of finite type $$\Sigma_A^+$$ and Holder continuous functions ϕ(ϵ, ·) defined on $$\Sigma_A^+$$ endowed with small parameter ϵ > 0. Through our choice of ϕ(ϵ, ·), we realize the situation that the perturbed system has a unique Gibbs measure μϵ of the potential ϕ(ϵ, ·) for each ϵ > 0 and on the other hand the unperturbed system possesses several Gibbs measures μ1, μ2,…, μm of the limit potential at ϵ = 0. In this paper, we give a necessary and sufficient condition for convergence of the measure μϵ using the notion of Perron complements of Ruelle operators. Our results can be applied also to the problems of convergence of stationary distributions of perturbed Markov chains with holes.

Numerical methods for construction of value functions in optimal control problems with infinite horizon

The article is devoted to the analysis of optimal control problems with infinite time horizon. These problems arise in economic growth models and in stabilization problems for dynamic systems. The problem peculiarity is a quality functional with an unbounded integrand which is discounted by an exponential index. The problem is reduced to an equivalent optimal control problem with the stationary value function. It is shown that the value function is the generalized minimax solution of the corresponding Hamilton-Jacobi equation. The boundary condition for the stationary value function is replaced by the property of the Hölder continuity and the sublinear growth condition. A backward procedure on infinite time horizon is proposed for construction of the value function. This procedure approximates the value function as the generalized minimax solution of the stationary Hamilton-Jacobi equation. Its convergence is based on the contraction mapping method defined on the family of uniformly bounded and Holder continuous functions. After the special change of variables the procedure is realized in numerical finite difference schemes on strongly invariant compact sets for optimal control problems and differential games.

SPDE limit of weakly inhomogeneous ASEP

We study ASEP in a spatially inhomogeneous environment on a torus $ \mathbb {T}^{(N)} = \mathbf {Z}/N\mathbf {Z}$ of $ N $ sites. A given inhomogeneity $ \widetilde { \mathbb {a} } (x)\in (0,\infty ) $, $ x\in \mathbb {T}^{(N)} $, perturbs the overall asymmetric jumping rates $ r<\ell \in (0,1) $ at bonds, so that particles jump from site $x$ to $x+1$ with rate $r \widetilde { \mathbb {a} } (x)$ and from $x+1$ to $x$ with rate $\ell \widetilde { \mathbb {a} } (x)$ (subject to the exclusion rule in both cases). Under the limit $ N\to \infty $, we suitably tune the asymmetry $ (\ell -r) $ to zero like $N^{-\frac {1}{2}}$ and the inhomogeneity $ \widetilde { \mathbb {a} } $ to unity, so that the two compete on equal footing. At the level of the Gartner (or microscopic Hopf–Cole) transform, we show convergence to a new SPDE — the Stochastic Heat Equation with a mix of spatial and spacetime multiplicative noise. Equivalently, at the level of the height function we show convergence to the Kardar–Parisi–Zhang equation with a mix of spatial and spacetime additive noise. Our method applies to a general class of $ \widetilde { \mathbb {a} } (x) $, which, in particular, includes i.i.d., fractional-Brownian-motion like, and periodic inhomogeneities. The key technical component of our analysis consists of a host of estimates on the kernel of the semigroup $ \mathcal {Q} (t):=e^{t \mathcal {H} } $ for a Hill-type operator $ \mathcal {H} := \frac {1}{2}\partial _{xx} + \mathcal {A} '(x) $, and its discrete analog, where $ \mathcal {A} $ (and its discrete analog) is a generic Holder continuous function.

Decompositions of log-correlated fields with applications

In this article, we establish novel decompositions of Gaussian fields taking values in suitable spaces of generalized functions, and then use these decompositions to prove results about Gaussian multiplicative chaos. We prove two decomposition theorems. The first one is a global one and says that if the difference between the covariance kernels of two Gaussian fields, taking values in some Sobolev space, has suitable Sobolev regularity, then these fields differ by a Holder continuous Gaussian process. Our second decomposition theorem is more specialized and is in the setting of Gaussian fields whose covariance kernel has a logarithmic singularity on the diagonal—or log-correlated Gaussian fields. The theorem states that any log-correlated Gaussian field $X$ can be decomposed locally into a sum of a Holder continuous function and an independent almost $\star $-scale invariant field (a special class of stationary log-correlated fields with ’cone-like’ white noise representations). This decomposition holds whenever the term $g$ in the covariance kernel $C_{X}(x,y)=\log (1/|x-y|)+g(x,y)$ has locally $H^{d+\varepsilon }$ Sobolev smoothness. We use these decompositions to extend several results that have been known basically only for $\star $-scale invariant fields to general log-correlated fields. These include the existence of critical multiplicative chaos, analytic continuation of the subcritical chaos in the so-called inverse temperature parameter $\beta $, as well as generalised Onsager-type covariance inequalities which play a role in the study of imaginary multiplicative chaos.

Functions of bounded fractional variation and fractal currents

Extending the notion of bounded variation, a function $$u \in L_c^1(\mathbb {R}^n)$$ is of bounded fractional variation with respect to some exponent $$\alpha $$ if there is a finite constant $$C \ge 0$$ such that the estimate $$\begin{aligned} \biggl |\int u(x) \det D(f,g_1,\ldots ,g_{n-1})_x \, dx\biggr | \le C\text{ Lip }^\alpha (f) \text{ Lip }(g_1) \cdots \text{ Lip }(g_{n-1}) \end{aligned}$$ holds for all Lipschitz functions $$f,g_1,\ldots ,g_{n-1}$$ on $$\mathbb {R}^n$$ . Among such functions are characteristic functions of domains with fractal boundaries and Holder continuous functions. We characterize functions of bounded fractional variation as a certain subspace of Whitney’s flat chains and as multilinear functionals in the setting of Ambrosio–Kirchheim currents. Consequently we discuss extensions to Holder differential forms, higher integrability, an isoperimetric inequality, a Lusin type property and change of variables. As an application we obtain sharp integrability results for Brouwer degree functions with respect to Holder maps defined on domains with fractal boundaries.

Online Contextual Influence Maximization With Costly Observations

In the online contextual influence maximization problem with costly observations , the learner faces a series of epochs in each of which a different influence spread process takes place over a network. At the beginning of each epoch, the learner exogenously influences (activates) a set of seed nodes in the network. Then, the influence spread process takes place over the network, through which other nodes get influenced. The learner has the option to observe the spread of influence by paying an observation cost . The goal of the learner is to maximize its cumulative reward, which is defined as the expected total number of influenced nodes over all epochs minus the observation costs. We depart from the prior work in three aspects: 1) the learner does not know how the influence spreads over the network, i.e., it is unaware of the influence probabilities; 2) influence probabilities depend on the context; and 3) observing influence is costly. We consider two different influence observation settings: costly edge-level feedback, in which the learner freely observes the set of influenced nodes, but pays to observe the influence outcomes on the edges of the network; and costly node-level feedback, in which the learner pays to observe whether a node is influenced or not. Since the offline influence maximization problem itself is NP-hard, for these settings, we develop online learning algorithms that use an approximation algorithm as a subroutine to obtain the set of seed nodes in each epoch. When the influence probabilities are Holder continuous functions of the context, we prove that these algorithms achieve sublinear regret (for any sequence of contexts) with respect to an approximation oracle that knows the influence probabilities for all contexts. Our numerical results on several networks illustrate that the proposed algorithms perform on par with the state-of-the-art methods even when the observations are cost free.

Measures of maximal entropy for suspension flows over the full shift

We consider suspension flows with continuous roof function over the full shift $$\Sigma $$ on a finite alphabet. For any positive entropy subshift of finite type $$Y \subset \Sigma $$, we explicitly construct a roof function such that the measure(s) of maximal entropy for the suspension flow over $$\Sigma $$ are exactly the lifts of the measure(s) of maximal entropy for Y. In the case when Y is transitive, this gives a unique measure of maximal entropy for the flow which is not fully supported. If Y has more than one transitive component, all with the same entropy, this gives explicit examples of suspension flows over the full shift with multiple measures of maximal entropy. This contrasts with the case of a Holder continuous roof function where it is well known the measure of maximal entropy is unique and fully supported.

On the Uniqueness of a Solution to an Inverse Problem of Scattering by an Inhomogeneous Solid with a Piecewise Hölder Refractive Index in a Special Function Class

The problem of reconstructing a piecewise Holder continuous function describing the refractive index of an inhomogeneous obstacle scattering a monochromatic wave is considered. The boundary value scattering problem is reduced to a system of integral equations. The equivalence of the integral and differential formulations of the problem is proved. A two-step method for solving the inverse problem is proposed. A linear integral equation of the first kind is solved at the first step. Sufficient conditions for the uniqueness of its solution in the class of piecewise constant functions are obtained. At the second step, the unknown refractive index is explicitly expressed in terms of the solution obtained at the first step.

Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case

The problem of prescribing Gaussian curvature on Riemann surface with conical singularity is considered. Let $(\Sigma,\beta)$ be a closed Riemann surface with a divisor $\beta$, and $K_\lambda=K+\lambda$, where $K:\Sigma\rightarrow\mathbb{R}$ is a Holder continuous function satisfying $\max_\Sigma K= 0$, $K\not\equiv 0$, and $\lambda\in\mathbb{R}$. If the Euler characteristic $\chi(\Sigma,\beta)$ is negative, then by a variational method, it is proved that there exists a constant $\lambda^\ast>0$ such that for any $\lambda\leq 0$, there is a unique conformal metric with the Gaussian curvature $K_\lambda$; for any $\lambda$, $0 \lambda^\ast$, there is no certain conformal metric having $K_{\lambda}$ its Gaussian curvature. This result is an analog of that of Ding and Liu \cite{Ding-Liu}, partly resembles that of Borer, Galimberti and Struwe \cite{B-G-Stru}, and generalizes that of Troyanov \cite{Troyanov} in the negative case.

On maximal regularity for the Cauchy-Dirichlet mixed parabolic problem with fractional time derivative

In this seminar we illustrate some results of maximal regularity for the Cauchy-Dirichlet mixed problem, with a fractional time derivative of Caputo type in spaces of continuous and Holder continuous functions.

Estimates for Correlation in Dynamical Systems: From Hölder Continuous Functions to General Observables

For many dynamical systems that are popular in applications, estimates are known for the decay of correlation in the case of Holder continuous functions. In the present article, we suggest an approach that allows us to obtain estimates for correlation in dynamical systems in the case of arbitrary functions. This approach is based on approximation and estimates are obtained with the use of known estimates for Holder continuous functions. We apply our approach to transitive Anosov diffeomorphisms and derive the central limit theorem for the characteristic functions of certain sets with boundary of zero measure.

Approximation of Continuous Functions by Matrix Means of Hexagonal Fourier Series

Some estimates for the degree of approximation by matrix means of hexagonal Fourier series of $$\mathbf {H}$$ -periodic continuous functions are obtained. The degree of approximation of $$\mathbf {H}$$ -periodic Holder continuous functions by these means is investigated in uniform and Holder norms and some results about Riesz and Norlund means of hexagonal Fourier series are concluded.