Integrable Functions Research Articles

Observer-based resilient cooperative output regulation for nonlinear multiagent systems with time-varying parameters and DoS attacks

This paper aims to address the cooperative output regulation (COR) problem of nonlinear multiagent systems (MASs) with time-varying parameters under denial-of-service (DoS) attacks. First, to address the challenges of unavailable leader signals and non-differentiability issue caused by the DoS attacks, a class of first-order low-pass filters is employed to develop a novel distributed resilient observer, which accurately estimate the exosystem state with the existence of high-order derivatives. Then, by viewing the output of observer as the reference signal, a distributed resilient controller is designed for each follower to address the reference-tracking problem. By introducing a smooth function with positive integrable time-varying function, novel adaptive laws are developed and incorporated with the controller design to effectively compensate for the unknown time-varying parameters. By utilising Lyapunov stability theory, the asymptotic convergence of output regulation error can be ensured. Finally, simulation results are conducted to verify the validity of the designed method.

A closure theorem for $\Gamma$-convergence and $H$-convergence with applications to non-periodic homogenization

In this work we examine the stability of some classes of integrals, in particular with respect to homogenization. The prototypical case is the homogenization of quadratic energies with periodic coefficients perturbed by a term vanishing at infinity, which has recently been examined in the framework of elliptic PDEs. We use localization techniques and higher-integrability Meyers-type results to provide a closure theorem by \Gamma -convergence within a large class of integral functionals. From such a result we derive stability theorems in homogenization which comprise the case of perturbations with zero average on the whole space. The results are also extended to the stochastic case, and specialized to the G -convergence of operators corresponding to quadratic forms. A corresponding analysis is also carried out for non-symmetric operators using the localization properties of H -convergence. Finally, we treat the case of perforated domains with Neumann boundary condition, and their stability.

On the speed of convergence in the ergodic theorem for shift operators

Abstract Given a probability space $(X,\mu )$ , a square integrable function f on such space and a (unilateral or bilateral) shift operator T, we prove under suitable assumptions that the ergodic means $N^{-1}\sum _{n=0}^{N-1} T^nf$ converge pointwise almost everywhere to zero with a speed of convergence which, up to a small logarithmic transgression, is essentially of the order of $N^{-1/2}$ . We also provide a few applications of our results, especially in the case of shifts associated with toral endomorphisms.

Stochastic differential formula and solution of control problem

Exact solution of finite-dimensional linear stochastic differential system with control is derived. Its uniqueness up to stochastic modification is proved. In order to achieve this, detailed grounding of existence for stochastic integral over a process with orthogonal increments is provided. Particularly, density of step-functions set in the set of all integrable functions is proved. Integration by parts formula for stochastic integral is derived. Analogue of Fubini theorem is proved in the case, when one measure is linear and another one is a random orthogonal measure. Stochastic differential formula for finite-dimensional integral functional is established. Formulation of Ito’s stochastic differential formula and its comparison with the result is presented. The solution formula for controlled linear stochastic differential system was conditionally used in applied sciences. Its rigorous proof provides the necessary background for further research.

Efficiency criteria and dual techniques for some nonconvex multiple cost minimization models

In this study, we investigate a class of multi-objective variational control problems governed by nonconvex simple integral functionals. Concretely, we establish and prove (necessary and sufficient) efficiency criteria and dual techniques for some nonconvex multiple cost minimization models. To this aim, we extend and use the concept of (Ψ,ω)-invexity to the case of multi-objective variational control problems. Thereafter, by assuming (Ψ,ω)-invexity, (strictly) (Ψ,ω)-pseudoinvexity and/or (Ψ,ω)-quasiinvexity of the involved functionals, we state the sufficient efficiency criteria and associate a dual problem for the considered model.

Estimation for conditional moment models based on martingale difference divergence

We provide a new estimation method for conditional moment models via the martingale difference divergence (MDD). Our MDD‐based estimation method is formed in the framework of a continuum of unconditional moment restrictions. Unlike the existing estimation methods in this framework, the MDD‐based estimation method adopts a non‐integrable weighting function, which could capture more information from unconditional moment restrictions than the integrable weighting function to enhance the estimation efficiency. Due to the nature of shift‐invariance in MDD, our MDD‐based estimation method can not identify the intercept parameters. To overcome this identification issue, we further provide a two‐step estimation procedure for the model with intercept parameters. Under regularity conditions, we establish the asymptotics of the proposed estimators, which are not only easy‐to‐implement with expectation‐based asymptotic variances, but also applicable to time series data with an unspecified form of conditional heteroskedasticity. Finally, we illustrate the usefulness of the proposed estimators by simulations and two real examples.

A class of Hausdorff–Berezin operators on the unit ball

Abstract The article introduces and studies Hausdorff–Berezin operators on the unit ball in a complex space. These operators are a natural generalization of the Berezin transform. In addition, the class of such operators contains, for example, the invariant Green potential, and some other operators of complex analysis. Sufficient and necessary conditions for boundedness in the space of p – integrable functions with Haar measure (invariant with respect to involutive automorphisms of the unit ball) are given. We also provide results on compactness of Hausdorff–Berezin operators in Lebesgue spaces on the unit ball. Such operators have previously been introduced and studied in the context of the unit disc in the complex plane. Present work is a natural continuation of these studies.

Exponential Sampling Type Kantorovich Max-Product Neural Network Operators

This article deals with the study of approximation behavior of the exponential sampling type Kantorovich max-product neural network operators based on sigmoidal activation function. We study point-wise and uniform convergence theorems in the space of continuous functions for these operators and determine the rate of convergence by means of logarithmic modulus of continuity. We also investigate the approximation of p -th ( 1 ≤ p < ∞ ) Lebesgue integrable functions by the aforementioned operators and study the rate of convergence exploiting Peetre’s K − functional.

Regularity for minimizers of scalar integral functionals with (p, q)-growth conditions

Regularity for minimizers of scalar integral functionals with (p, q)-growth conditions

Pointwise ergodic theorems for nonconventional bilinear polynomial averages along prime orbits

Pointwise ergodic theorems for nonconventional bilinear polynomial averages along prime orbits Following our proof of the first Goldbach result in number field setting after 60 years (Mitsui 1960s), We give new information on density ternary Goldbach problem after 10 years (Shao 2013), and also design a new strategy to prove density version of representing a number as sum of “certain” primes. We also apply Rosser Sieve arguments to prove represtation of integers into sum of primes coming from a zero density subset. From Ergodic theory side, we give the first pointwise bilinear theorem along the primes, which is the natural pointwise convergence along prime result after Wierdl (1990) and a natural follow up after recent pointwise bilinear break-through by Krause Mirek-Tao (2021). In another project, the sharpest pointwise convergence average along primes for “near” integrable function is given, which is sharp up to Generalized Riemann Hypothesis.

Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory

Bishop's measure theory (BMT) is an abstraction of the measure theory of a locally compact metric space $X$, and the use of an informal notion of a set-indexed family of complemented subsets is crucial to its predicative character. The more general Bishop-Cheng measure theory (BCMT) is a constructive version of the classical Daniell approach to measure and integration, and highly impredicative, as many of its fundamental notions, such as the integration space of $p$-integrable functions $L^p$, rely on quantification over proper classes (from the constructive point of view). In this paper we introduce the notions of a pre-measure and pre-integration space, a predicative variation of the Bishop-Cheng notion of a measure space and of an integration space, respectively. Working within Bishop Set Theory (BST), and using the theory of set-indexed families of complemented subsets and set-indexed families of real-valued partial functions within BST, we apply the implicit, predicative spirit of BMT to BCMT. As a first example, we present the pre-measure space of complemented detachable subsets of a set $X$ with the Dirac-measure, concentrated at a single point. Furthermore, we translate in our predicative framework the non-trivial, Bishop-Cheng construction of an integration space from a given measure space, showing that a pre-measure space induces the pre-integration space of simple functions associated to it. Finally, a predicative construction of the canonically integrable functions $L^1$, as the completion of an integration space, is included.

Order-boundary characterization of the linear lattice of Riemann 𝜇-integrable functions as a certain completion of the linear lattice of bounded continuous functions

The linear lattice of Riemann μ \mu -integrable functions on a completely regular space with a bounded positive Radon measure μ \mu is viewed as an extension of the linear lattice of bounded continuous functions. To characterize this Riemann extension, the new functional analysis category of c c -latlineals with refinements ( ≡ c r \equiv cr -latlineals) is introduced. On this basis, the procedure of c r cr -completion of certain order-boundary type is defined. The μ \mu -Riemann extension is shown to be the result of applying this procedure to the c r μ cr_{\mu } -latlineal of bounded continuous functions.

On the strong summability of the Fourier–Walsh series in the Besov space

UDC 517.5 The Fourier–Walsh series of even continuous functions may be divergent at some points. Moreover, among integrable functions, there are functions such that their Fourier–Walsh series diverge everywhere on [ 0,1 ) . In this connection, it becomes necessary to consider various summation methods that would allow us to restore the function according to its Fourier–Walsh series. We also investigate the Besov space on a dyadic group in terms of strong summability. Finally, we present necessary information about the Fourier–Walsh transform.

The A-integral and Calderon-Zygmund operators

It is known that the function obtained by the action of the Calderon-Zygmund operator on a Lebesgue integrable function can be non-Lebesgue integrable. In this paper, we prove that the function obtained by the action of the Calderon-Zygmund operator on a Lebesgue integrable function is A-integrable and derive an analogue of the Riesz equality.

Laplace-Stieltjes Transform and its Asymptotic Expansion

In this paper we will present the Laplace-Stieltjes transform and its asymptotic expansion.Asymptotic Transforms which comes to further improvement the extension of the Laplace-Stieltjes Transforms to all locally integrable function as proposed by Watson's Lemma. It gives us a way to obtain the complete asymptotic expansion of integrals on the real axis that have an exponential type of kernel.. To obtain the asymptotic expansion of the transform, we follow the technique of Handelsman and Blestein.

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(Two-scale) gradient Young measures in Orlicz–Sobolev setting are introduced and characterized providing also an integral representation formula for non convex energies arising in homogenization problems with nonstandard growth.

Publisher Correction: Differentiation with Respect to Domains of Boundary Integral Functionals Involving Support Functions

Publisher Correction: Differentiation with Respect to Domains of Boundary Integral Functionals Involving Support Functions

Lipschitz bounds for nonuniformly elliptic integral functionals in the plane

We study local regularity properties of local minimizer of scalar integral functionals with controlled ( p , q ) (p,q) -growth in the two-dimensional plane. We establish Lipschitz continuity for local minimizer under the condition 1 &gt; p ≤ q &gt; ∞ 1&gt;p\leq q&gt;\infty with q &gt; 3 p q&gt;3p which improve upon the classical results valid in the regime q &gt; 2 p q&gt;2p . Along the way, we establish an L ∞ L^\infty - L 2 L^2 -estimate for solutions of linear uniformly elliptic equations in the plane which is optimal with respect to the ellipticity contrast of the coefficients.

Global existence and Blow-up for the 1D damped compressible Euler equations with time and space dependent perturbation

In this paper, we consider the 1D Euler equation with time and space dependent damping term −a(t,x)v. It has long been known that when a(t,x) is a positive constant or 0, the solution exists globally in time or blows up in finite time, respectively. In this paper, we prove that those results are invariant with respect to time and space dependent perturbations. We suppose that the coefficient a satisfies the following condition |a(t,x)−μ0|≤a1(t)+a2(x),where μ0≥0 and a1 and a2 are integrable functions with t and x. Under this condition, we show the global existence and the blow-up with small initial data, when μ0>0 and μ0=0 respectively. The key of the proof is to divide space into time-dependent regions, using characteristic curves.

Revisited convexity notions for $$L^\infty $$ variational problems

AbstractWe address a detailed study of the convexity notions that arise in the study of weak* lower semicontinuity of supremal functionals, as well as those arising by the $$L^p$$ L p -approximation, as $$p \rightarrow +\infty $$ p → + ∞ of such functionals. Our quest is motivated by the knowledge we have on the analogous integral functionals and aims at establishing a solid groundwork underlying further research in the $$L^\infty $$ L ∞ context.