Composition Theorem Research Articles

Some new results on bounded solutions to a semilinear integro-differential equation in Banach spaces

In this article, by a concept of Stepanov type $\mu$-pseudo almost automorphic functions developed recently, we investigate some new existence results on bounded solutions to a semilinear integro-differential equation in Banach spaces. We first establish a new composition theorem of such functions, and then we prove the main results via ergodicity and composition theorems of Stepanov type $\mu$-pseudo almost automorphic functions combined with theories of uniformly exponentially stable and strongly continuous family of operators. These bounded solutions can cover (weighted) pseudo almost automorphic solutions with a Stepanov type forcing term as special cases.

Composition Theorems of Stepanov $$\mu $$ μ -Pseudo Almost Automorphic Functions and Applications to Nonautonomous Neutral Evolution Equations

In this work, we present a new composition theorem of $$\mu $$ -pseudo almost automorphic functions in the sense of Stepanov satisfying some local Lipschitz conditions. Using this results, we establish an existence result of $$\mu $$ -pseudo almost automorphic solutions for some nonautonomous neutral partial evolution equation with Stepanov $$\mu $$ -pseudo almost automorphic nonlinearity. An example is shown to illustrate our results.

Existence and global attractiveness of a pseudo almost periodic solution in p-th mean sense for stochastic evolution equation driven by a fractional Brownian motion

In this work, we establish a new concept of pseudo almost periodic processes in p-th mean sense using the measure theory. We use the μ-ergodic process to define the spaces of μ-pseudo almost periodic process in the p-th mean sense. We establish many interesting results on the functional space of such processes like completeness and composition theorems. The main objective of this paper is to use those results and some stochastic analysis approaches to study the existence, the uniqueness and the global attractiveness for a μ-pseudo almost periodic mild solution to a class of abstract stochastic evolution equations driven by fractional Brownian motion. We provide an example to illustrate our results.

Pseudo-almost Periodic and Pseudo-almost Automorphic Solutions to Evolution Equations in Hilbert Spaces

In this work we prove some existence and uniqueness results for pseudo-almost periodic and pseudo-almost automorphic solutions to a class of semi-linear differential equations in Hilbert spaces using theoretical measure theory. The main technique is based upon some appropriate composition theorems combined with the Banach contraction mapping principle and the method of the invariant subspaces for unbounded linear operators. A few illustrative examples will be discussed at the end of the paper.

Sufficient conditions for the global rigidity of graphs

We investigate how to find generic and globally rigid realizations of graphs in Rd based on elementary geometric observations. Our arguments lead to new proofs of a combinatorial characterization of the global rigidity of graphs in R2 by Jackson and Jordán and that of body-bar graphs in Rd recently shown by Connelly, Jordán, and Whiteley. We also extend the 1-extension theorem and Connelly's composition theorem, which are main tools for generating globally rigid graphs in Rd. In particular we show that any vertex-redundantly rigid graph in Rd is globally rigid in Rd, where a graph G=(V,E) is called vertex-redundantly rigid if G−v is rigid for any v∈V.

Weighted pseudo almost automorphic solutions to a semilinear fractional differential equation with Stepanov-like weighted pseudo almost automorphic nonlinear term

In this paper, we investigate some existence results of weighted pseudo almost automorphic solutions to a semilinear fractional differential equation in Banach spaces with Stepanov-like weighted pseudo almost automorphic nonlinear term. Our main results are based upon ergodicity and composition theorems of Stepanov-like weighted pseudo almost automorphic functions combined with fixed point techniques.

PC-Almost Automorphic Solution of Impulsive Fractional Differential Equations

In this paper, we introduce a PC-almost automorphic function and establish the composition theorem, which is an important result from application point of view. As an application, we study the existence of almost automorphic solution to impulsive fractional functional differential equations with the assumption that the forcing term is almost automorphic. The results are established by fixed point methods and α-resolvent family of bounded linear operators. At the end, some examples are given to illustrate our analytic findings.

Weighted piecewise pseudo almost automorphic functions with applications to abstract impulsive ∇-dynamic equations on time scales

In the present paper, by introducing the concept of equipotentially almost automorphic sequence, the concept of weighted piecewise pseudo almost automorphic functions on time scales is proposed. Some first results about their basic properties are obtained and some composition theorems are established. Then we apply these to investigate the existence of weighted piecewise pseudo almost automorphic mild solutions to abstract impulsive ∇-dynamic equations on time scales. In addition, the exponential stability of weighted piecewise pseudo almost automorphic mild solutions is also considered. Finally, the obtained results are applied to the study of a class of ∇-partial differential equations on time scales.MSC:34N05, 35B15, 43A60, 12H20, 35R12.

Measure theory and $$S^{2}$$ S 2 -pseudo almost periodic and automorphic process: application to stochastic evolution equations

In this work, we use the measure theory to define and to study the spaces of Stepanov-like $$\mu $$ -pseudo almost ( $$S^{2}$$ - $$\mu $$ -pseudo almost) periodic and automorphic processes in the square mean sense. We establish many interesting results on the functional spaces of such processes like completeness and composition theorems. The main objective of this paper is to establish the existence, uniqueness and stability of square-mean $$\mu $$ -pseudo almost periodic (resp. automorphic) mild solution to a linear and semilinear case of the stochastic evolution equations in case when the functions forcing are both continuous and $$S^{2}$$ - $$\mu $$ -pseudo almost periodic (resp. automorphic) and verify some suitable assumptions. The technique analysis used to achieve the required results are the stochastic analysis theory, the composition theorems and the Banach fixed point theorem. We provide an example to illustrate ours results.

Multivariate McCormick relaxations

McCormick (Math Prog 10(1):147---175, 1976) provides the framework for convex/concave relaxations of factorable functions, via rules for the product of functions and compositions of the form $$F\circ f$$F?f, where $$F$$F is a univariate function. Herein, the composition theorem is generalized to allow multivariate outer functions $$F$$F, and theory for the propagation of subgradients is presented. The generalization interprets the McCormick relaxation approach as a decomposition method for the auxiliary variable method. In addition to extending the framework, the new result provides a tool for the proof of relaxations of specific functions. Moreover, a direct consequence is an improved relaxation for the product of two functions, at least as tight as McCormick's result, and often tighter. The result also allows the direct relaxation of multilinear products of functions. Furthermore, the composition result is applied to obtain improved convex underestimators for the minimum/maximum and the division of two functions for which current relaxations are often weak. These cases can be extended to allow composition of a variety of functions for which relaxations have been proposed.

Some properties of weighted Stepanov-like pseudo almost auto-morphic functions and applications to Volterra integral equations

This paper is concerned with weighted Stepanov-like pseudo-almost automorphic functions. We rst study the relationship between a weighted Stepanov-like pseudo-almost automorphic function and its Stepanov-like almost automorphic component. With the help of the relationship, previous composition theorems of these functions are improved. Secondly, we study the convolution operators generated by absolutely integrable functions in weighted Stepanov-like pseudo-almost automorphic function spaces. Finally, by using contraction mapping principle, we study the existence and uniqueness of weighted Stepanov-like pseudo-almost automorphic solutions for two classes of Volterra integral equations. These results are new and improve some known ones.

Composition results for strongly summing and dominated multilinear operators

AbstractIn this paper we prove some composition results for strongly summing and dominated operators. As an application we give necessary and sufficient conditions for a multilinear tensor product of multilinear operators to be strongly summing or dominated. Moreover, we show the failure of some possible n-linear versions of Grothendieck’s composition theorem in the case n ≥ 2 and give a new example of a 1-dominated, hence strongly 1-summing bilinear operator which is not weakly compact.

Discrete Weighted Pseudo Asymptotic Periodicity of Second Order Difference Equations

We define the concept of discrete weighted pseudo-S-asymptotically periodic function and prove some basic results including composition theorem. We investigate the existence, and uniqueness of discrete weighted pseudo-S-asymptotically periodic solution to nonautonomous semilinear difference equations. Furthermore, an application to scalar second order difference equations is given. The working tools are based on the exponential dichotomy theory and fixed point theorem.

Composition of semi-LTCs by two-wise tensor products

In this paper, we obtain a composition theorem that allows us to construct locally testable codes (LTCs) by repeated two-wise tensor products. This is the first composition theorem showing that repeating the two-wise tensor operation any constant number of times still results in a locally testable code, improving upon previous results which only worked when the tensor product was applied once. To obtain our results, we define a new tester for tensor products. Our tester uses the distribution of the "inner tester" associated with the base code to sample rows and columns of the product code. This construction differs from previously studied testers for tensor product codes which sampled rows and columns uniformly. We show that if the base code is any LTC or any expander code, then the code obtained by taking the repeated two-wise tensor product of the base code with itself is locally testable. In particular, this answers a question posed in the paper of Dinur et al. (2006) by expanding the class of allowed base codes to include all LTCs and not just so-called uniform LTCs whose associated tester queries all codeword entries with equal probability. One corollary of our composition theorem is a simple construction of high-rate LTCs with sublinear query complexity. More formally, we show that for every ?, ? > 0 there exists a family of LTCs over the binary field with query complexity n? and rate at least 1 ? ?.

GNUC: A New Universal Composability Framework

We put forward a framework for the modular design and analysis of multi-party protocols. Our framework is called “GNUC” (with the recursive meaning “GNUC’s Not UC”), already alluding to the similarity to Canetti’s Universal Composability (UC) framework. In particular, like UC, we offer a universal composition theorem, as well as a theorem for composing protocols with joint state. We deviate from UC in several important aspects. Specifically, we have a rather different view than UC on the structuring of protocols, on the notion of polynomial-time protocols and attacks, and on corruptions. We will motivate our definitional choices by explaining why the definitions in the UC framework are problematic, and how we overcome these problems. Our goal is to offer a framework that is largely compatible with UC, such that previous results formulated in UC carry over to GNUC with minimal changes. We exemplify this by giving explicit formulations for several important protocol tasks, including authenticated and secure communication, as well as commitment and secure function evaluation.

Weighted pseudo almost automorphic functions and WPAA solutions to semilinear evolution equations

In this paper, we reveal several basic properties about nonlinear vector-valued weighted pseudo almost automorphic functions, including equivalence, completeness, translation invariance, composition theorem, and convolution theorem of these functions. We also give some concrete examples to illustrate our results. Finally, we obtain a new existence theorem of nonlinear weighted pseudo almost automorphic solutions for semilinear evolution equations in Banach spaces.

Dilation bootstrap

We propose a methodology for combining several sources of model and data incompleteness and partial identification, which we call Composition Theorem. We apply this methodology to the construction of confidence regions with partially identified models of general form. The region is obtained by inverting a test of internal consistency of the econometric structure. We develop a dilation bootstrap methodology to deal with sampling uncertainty without reference to the hypothesized economic structure. It requires bootstrapping the quantile process for univariate data and a novel generalization of the latter to higher dimensions. Once the dilation is chosen to control the confidence level, the unknown true distribution of the observed data can be replaced by the known empirical distribution and confidence regions can then be obtained as in Galichon and Henry (2011) and Beresteanu et al. (2011).

Weighted pseudo almost automorphic mild solutions to semilinear integral equations with $S^{p}$-weighted pseudo almost automorphic coefficients

In this paper, we consider the existence of weighted pseudo almost automorphic solutions of the semilinear integral equation $x(t)= \int_{-\infty}^{t}a(t-s)[Ax(s) + f(s,x(s))]ds, t\in\mathbb{R}$ in a Banach space $\mathbb{X}$, where $a\in L^{1}(\mathbb{R}_{+})$, $A$ is the generator of an integral resolvent family of linear bounded operators defined on the Banach space $\mathbb{X}$, and $f : \mathbb{R}\times\mathbb{X} \rightarrow \mathbb{X}$ is a weighted pseudo almost automorphic function. The main results are proved by using integral resolvent families, suitable composition theorems combined with the theory of fixed points.

Composition of piecewise pseudo almost periodic functions and applications to abstract impulsive differential equations

In this paper, we introduce the concept of piecewise pseudo almost periodic functions on a Banach space and establish some composition theorems of piecewise pseudo almost periodic functions. We apply these composition theorems to investigate the existence of piecewise pseudo almost periodic (mild) solutions to abstract impulsive differential equations. In addition, the stability of piecewise pseudo almost periodic solutions is considered.

Mixing Multilinear Operators with or Without a Linear Analogue

As a natural extension of mixing linear operators we introduce new notions of mixing multilinear operators with or without a linear analogue. We prove Pietsch’s composition theorems in this new context extending recent results from the case of dominated operators.