pinecone/core-2020-05-10-deduplication huggingface dataset:
Generalized Esclangon–Landau Conditions for Differential–Difference Equations

AbstractLet J=[α,∞) for some α∈R or J=R and denote byXa Banach space. We study the solutions on J of the equation (*) Ω(n)(t)+∑n−1k=0∑mj=1aj,k(t)Ω(k)(t+tj)=b(t), wheret1<t2···<tm,aj,k: J→L(X) andb: J→X. We prove that ifband all of the coefficients are bounded and Ω is a bounded solution of (*), then Ω′,Ω″,…,Ω(n)are bounded, provided one of the following holds: (i)a(p)j,kare bounded for all 1≤p≤k≤n−1; (ii) J=[α,∞) andtm≤0; (iii) J=[α,∞) and ‖a(p)j,k(t)‖=O(eρ|t|) for some 0<ρr≤n/(n−1),r=max{tm,0}, and all 0≤p≤k≤n−1, 1≤j≤m. An analogous result for uniform continuity instead of boundedness is obtained. More generally, in the delay casetm≤0, we give estimates on the growth of the Ω(k)in terms of the growth of Ω,aj,k, andb. We also investigate neutral equations. We prove that an Esclangon–Landau theorem is still valid for neutral equations, if one has small perturbations of (*) or if the characteristic function has positive distance from 0. Our results extend the classical Esclangon–Landau theorem; they are needed in the study of asymptotic properties (for example, almost periodicity) of solutions of (*) and neutral equations
Generalized Kronig–Penney Hamiltonians

AbstractIn this article, we extend the study of generalized point interactions to the periodic case, and we investigate analogues of the Kronig–Penney model for the class of point interactions introduced in joint work with P. Chernoff. In addition to periodic δ- and δ′-potentials, which follow as special cases, several examples are studied in detail. In general, the models under consideration are shown to have purely absolutely continuous spectrum, and in some cases all gaps in the spectrum close
Restricted Subadditivity of Homogeneous Means

AbstractWe study Minkowski's inequality[formula] (I⊆R is an interval) and its reverse for the means[formula]satisfying the generalized homogeneity equation[formula]By a result of Aczél and Daróczy [1], these homogeneous means belong to three classes of two-parameter families of means. For one class (the class called Gini means), Minkowski's inequality (or subadditivity) has been studied extensively by Beckenbach [2], Dresher [4], Danskin [3], Losonczi [7,8], and Páles [9]. The aim of this paper is to study the sub- and superadditivity of the other two classes
A Note on General Algorithm for Variational Inequalities

AbstractIn this note we introduce and study a new general strongly quasivariational inequality problem which includes, as special cases, a number of known classes of variational inequalities studied previously by many authors in this field. A general and unified iterative algorithm for finding the approximate solution to this problem is constructed by the projection method. We prove the existence of the solution for this problem and the convergence of the iterative sequence generated by this algorithm
Bi-Complementarity and Duality: A Framework in Nonlinear Equilibria with Applications to the Contact Problem of Elastoplastic Beam Theory

AbstractNonlinear complementarity problems and variational inequalities in nonlinear equilibrium problems are studied within a unified framework. Based on the generalized Rockafellar–Tonti diagram, a bi-complementarity problem with both internal and external nonlinear complementarity conditions is proposed. A general duality theory in variational inequality is established and the Mosco dual variational inequality has been generalized to the nonsmooth systems. In order to study the frictional contact problem of beam theory, a two-dimensional elastoplastic beam model is proposed. The external complementarity condition provides the free boundary of contact region, while the internal complementarity condition gives the interface of the elastic–plastic regions. Our results shown that in nonsmooth equilibrium problems, the dual approaches are much easier than the primal problems
Optimal Stopping Inequalities for the Integral of Brownian Paths

AbstractLetB=(Bt)t≥0be a Brownian motion started atx∈R. Given a stopping time τ forBand a real valued mapF, we show how one can optimally bound: [formula]in terms ofE(τ). The method of proof relies upon solving the optimal stopping problem where one minimizes or maximizes[formula]over all stopping times τ forBwithc>0. By Itô's formula and the optional sampling theorem, this problem simplifies to the form where explicit computations are possible. The method is quantitatively demonstrated through the example ofF(x)=xr, withr>−1. This yields some new sharp inequalities
Heat Transfer in a Viscoelastic Fluid over a Stretching Sheet

AbstractThis paper presents an analytical solution for the heat transfer in the steady laminar flow of an incompressible viscoelastic fluid past a semi-infinite stretching sheet with power-law surface temperature or power-law surface heat flux, including the effects of viscous dissipation and internal heat generation or absorption. Important physical quantities such as the coefficients of skin friction and heat transfer are obtained. Asymptotic results for the temperature function for large Prandtl numbers are presented
Geometry of Spaces of Polynomials

AbstractConnections between the shape of the unit ball of a Banach space and analytic properties of the Banach space have been studied for many years. In this article, some geometric properties of spaces related ton-homogeneous polynomials are considered. In particular, the rotundity and smoothness of spaces of continuousn-homogeneous polynomials and its preduals are studied. Furthermore, an inequality relating the product of the norms of linear functionals on a Banach space with the norm of the continuousn-homogeneous polynomial determined by the product of the linear functionals is derived. This inequality is used to study the strongly exposed points of the predual of the space of continuous 2-homogeneous polynomials
A Trace Formula for Variable-Coefficient Toeplitz Matrices with Symbols of Bounded Variation

AbstractIn this paper we derive second-order asymptotic results for matrices[formula]asn→∞, where the functionUis given by an explicit formula, and the symbol σ(x,θ) is a function defined on [0,1]×T. More precisely, the main result of this paper describes the second-order asymptotics of the trace off(opnσ), wherefbelongs to a suitable class of functions. Here opnσ is the matrix given by[formula]Matrices of the above form can be thought of as variable-coefficient Toeplitz matrices, or a discrete analogue of a pseudodifferential operator. Ideas from pseudodifferential operator theory are used in the proof
Balanced Exponential Growth of Operator Semigroups

AbstractAC0-semigroupS(t),t≥0, on a Banach spaceX(weakly, strongly, uniformly) approachesbalanced (or asynchronous) exponential growthif there exists somes∈R such that[formula]exists (in the weak, strong, uniform operator topology) andPis not the 0 operator. In this paper, the strong and uniform approach to balanced exponential growth is characterized and applicable sufficient conditions are derived. The results will be applied to models for age-size-structured population dynamics in a forthcoming publication
Practical Stability and Exponential Estimates of Trajectory Bounds for Retarded Systems with Bounded Disturbances

AbstractThe properties of practical stability and exponential estimates of trajectory bounds for retarded systems with uncertain external disturbances are investigated. Our approach is adaptable to general regarded functional differential equations (RFDEs) with bounded disturbances. By applying our results to the LQ regulator problem with uncertain input delay and bounded external disturbances, it can be found that our results provide an effective method for quantitative stability analysis of retarded systems
On a Polygon Equality Problem

AbstractBernius and Blanchard of Bielefeld University in Germany have conjectured the followingpolygon inequality: for any two sets of vectorsx1,…,xnandy1,…,ynin Rm,[formula]in the 2-norm and that, moreover, equality holds in (∗) if and only if there exists a permutation π on {1,2,…,n} such thatyi=xπ(i),i=1,…,n. That (∗) is valid is a consequence of an inequality that holds in certain Banach spaces and which was recently proved by Lennard, Tongue, and Weston. We therefore characterize here the case of equality in (∗), actually for vectors in the spaceX=L1(Ω,μ), and subsequently use this characterization to complete the proof of the Bernius–Blanchard conjecture concerning the equality case in a Hilbert space
A Model of Following

AbstractIn the discrete-time models of learning in [5] (based on the Kaczmarz projection algorithm [3]) there were obtained estimates (Theorems 4.1. and 4.2) of the sum of the errors and the length of the learning path of a learning process. These depended on the size of a time steph, a parameter of the model, and they would yield limits ash→0 but the resulting asymptotic estimates were not realizable in any particular model (with fixedh). Here we define a continuous-time analogue of our former discrete-time model of learning and we call it a model of following. It is shown that the asymptotic forms of the main results about learning are true in the models of following
Implications of Sharp Trace Regularity Results on Boundary Stabilization of the System of Linear Elasticity

AbstractUniform exponential decay of solution is established for the elastodynamic system of elasticity using boundary feedback control. Energy dissipation is introduced via linear velocity feedbacks acting through a portion of the boundary as traction forces. Two primary goals are achieved: First, these results are proven without the imposition of strong geometric assumptions on the controlled portion of the boundary, thus extending earlier work which required that the domain be “star shaped.” Second, the feedback is only a function of velocity, as opposed to also containing the tangential derivative of the displacement, resulting in a physically viable feedback. Proof is based on the multiplier method and relies critically on sharp trace estimates for the tangential derivative of the displacement on the boundary as well as on unique continuation results for the corresponding static system
The Length of the de Rham Curve

AbstractThe lengthLof the de Rham curve is the common limit of two monotonic sequences of lengths (ln) and (Ln) of inscribed and circumscribed polygons, respectively. Numerical computations show that their convergence is linear with the same convergence rate. This result is easy to prove for the parabola. For arbitrary de Rham curves, we prove two nearby results. First, the existence of a limitq∈]0,1[ of the sequence of ratios (Ln+1−L)/(Ln−L) implies the convergence to the same limit of the two sequences (ln+1−L)/(ln−L) and (Ln+1−ln+1)/(Ln−ln). Second, the sequence (Ln+1−Ln) is bounded by a convergent geometric sequence. In practice, this allows us to accelerate the convergence of both sequences by standard extrapolation algorithms
The Corona Theorem and the Canonical Factorization of Triangular AP Matrix Functions—Effective Criteria and Explicit Formulas

AbstractUsing results of the companion paper, the authors obtain effective criteria for the existence of canonical factorization for new classes of triangular almost periodic matrix functions of the form[formula]. The important role in the proof of necessity in these criteria is played by a result on the stability of canonical AP factorizability obtained in the present paper. By finding corona solutions for a pair of binomial almost periodic functions, it is possible to calculate explicitly the factors of a canonical factorization for a class of considered matrix functions with trinomialf
State-Constrained Relaxed Problems for Semilinear Elliptic Equations

AbstractIn this paper we study optimal control problems governed by semilinear elliptic equations in the presence of pointwise state constraints. Since no convexity condition is assumed on data of the problem, we define a relaxed control problem, prove the existence of relaxed solutions, and give some relaxation results. By adapting the penalty method of Berkovitz, we prove a Pontryagin's minimum principle for relaxed solutions in nonqualified form and in qualified form under a stability condition
Analysis of Carleman Representation of Analytical Recursions

AbstractWe study a general method to map a nonlinear analytical recursion onto a linear one. The solution of the recursion is represented as a product of matrices whose elements depend only on the form of the recursion and not on initial conditions. First we consider the method for polynomial recursions of arbitrary degree and then the method is generalized to analytical recursions. Some properties of these matrices, such as the existence of an inverse matrix and diagonalization, are also studied
Ishikawa and Mann Iterative Processes with Errors for Nonlinear Strongly Accretive Operator Equations

AbstractThe purpose of this paper is to revise the definitions of Ishikawa and Mann iterative processes with errors, by using a new inequality to study the unique solution of the nonlinear strongly accretive operator equationTx=fand the convergence problem of Ishikawa and Mann iterative sequences for strongly pseudo-contractive mappings without the Lipschitz condition. The results presented in this paper improve and extend the corresponding results in [4,5,7–10,12,15,16] in the more general setting. In particular, the open problem mentioned by Chidume in [5] has been given an affirmative answer
Variational Inequalities and Fixed Point Theorems for PM-Maps

AbstractWe study mapsTfor whichJ−Tis pseudo-monotone; we call suchTa PM-map. This includes compact maps in suitable spaces and pseudo-contractive maps in Hilbert spaces. We study variational inequalities in a situation which was previously done only whenTis compact. We show that our variational inequalities are well suited to treating existence of fixed points for generalized inward (in particular, weakly inward) PM-maps in Hilbert spaces. Our new results on existence of fixed points generalizes many earlier results obtained by using other methods. An application to homogeneous integral equations is provided
Stabilizing Stationary Linear Discrete Systems: Minimal and Balanced Expansions in Any Real Base

AbstractLetabe a real number strictly greater than 1 and letDbe a finite interval of Z containing 0. In this paper we give necessary and sufficient conditions that guarantee the existence, for any real, of an expansion in baseawith coefficients inD. Then we turn to balanced expansions, for which the sums of the digits of any initial segment are uniformly bounded. These issues yield necessary and sufficient conditions for stabilizing stationary linear discrete systems with a particular family of control laws
On Sobolev Orthogonal Polynomials with Coherent Pairs: The Laguerre Case, Type 1

AbstractWe study the asymptotic properties of orthogonal polynomials with Sobolev inner product[formula]The pair }dμ,dν{ is called a coherent pair if there exists nonzero constantsDnsuch that[formula] wherePn(x) andQn(x) are thenth monic orthogonal polynomials with respect todμ anddν, respectively. One can divide the coherent pairs into two cases: the Jacobi case and the Laguerre case. There are two types in each case. We consider thenth root asymptotics and the zero distribution for the Laguerre case, type 1
Iterative Approximations of Fixed Points and Solutions for Strongly Accretive and Strongly Pseudo-Contractive Mappings in Banach Spaces

AbstractIn this article, we prove some new convergence theorems of the Ishikawa and Mann iteration sequences for strongly accretive and strongly pseudo-contractive mappings by using the new inequality and the new approximation methods. Our main results improve and extend the corresponding results of [J. Math. Anal. Appl.151(1990), 453–461;Proc. Amer. Math. Soc.120(1994), 545–551;J. Math. Appl.192(1995), 502–518;J. Math. Anal. Appl.174(1993), 441–449;Acta Appl. Math.32(1993), 183–196;Nonlinear Anal. TMA24 (1995), 981–987;J. Math. Anal. Appl.178(1993), 9–21;Proc. Amer. Math. Soc.125(6) (1997), 1705–1709]
Isomorphism between Systems of Equivariant Singularities

AbstractIn this article isomorphisms between systems of singularities equivariant under different Lie group actions are investigated and a sufficient condition for two systems to be isomorphic is given. With this sufficiency theorem we show that the system ofO(n)-equivariant singularities in its irreducible representation on Rnis isomorphic to that of one-dimensional Z2-equivariant singularities and the system of[formula]-dimensionalO(n)-equivariant singularities is isomorphic to that ofn-dimensionalSn-equivariant singularities
A Birth and Death Process Related to the Rogers–Ramanujan Continued Fraction

AbstractTime-dependent system size probabilities of a birth and death process related to the Rogers–Ramanujan continued fraction are obtained. The range for the parameter in this continued fraction is obtained to ensure the positivity of the recursively defined birth and death rates. The general behavior of the birth and death rates is described and the asymptotic behavior of the transition probabilities and the (quasi)-stationary distribution is determined. For the transient case the birth and death process can be seen as a model in queuing theory where the length of the queue encourages customers to join the queue (the more the merrier)
Measurability of Fixed Point Sets of Multivalued Random Operators

AbstractLet (M,d) be a complete separable metric space, (Ω,Σ) a measurable space with Σ a σ-algebra of subsets of Ω, andT: Ω×M→CB(M) a multivalued random operator. The measurability of the functionFof fixed point sets ofTdefined byF(ω)≔{x∈M:x∈T(ω,x)} is studied. In particular, it is proved thatFis measurable providedTis a random contraction, orMis a weakly compact convex separable subset of a Banach space andTis a random multivalued nonexpansive mapping such thatI−T(ω,·) is demiclosed at 0 for every ω∈Ω. The same result is also verified for a single-valued random nonexpansive mapping in a uniformly smooth Banach space
Convergence of Nonmonotone Sequence of Sub-σ-Fields and Convergence of Associated SubspacesLp(Bn) (p∈[1,+∞])

AbstractKudō [Ann. Probab.2(1) (1974), 76–83] introduced a notion of convergence of sub-σ-fields in terms ofL1-convergence of conditional expectation. We study relationships between convergence of sub-σ-fields and convergence of associated subspacesLp(Bn). In particular, we show that the Slice convergence is not adapted to theL1-case in contrast to the Mosco convergence. Notions of upper and lower limits are also studied. This enables us to obtain measurability results regarding those limits. We also give examples and counterexamples in order to exhibit characteristic behaviours of the sequences (Bn)n, (Lp(Bn))n, and (EBn)n
A Random Fixed Point Theorem and the Random Graph Transformation

AbstractIn the first part of this article we formulate a random fixed point theorem for random dynamical systems where a contraction condition is formulated as an expectation of a particular expression. In the following we are going to formulate a random graph transformation. This transformation defines a random dynamical system. Under certain conditions this random dynamical system has a random fixed point which can be found by the mentioned random fixed point theorem. In one application we show that random dynamical systems given by particular differential equations have a random unstable invariant manifold
On Convergence Classes inL-fuzzy Topological Spaces

AbstractConvergence theory is a primary topic in topology. In fact, topology and so-called convergence class are characterized by each other. In fuzzy topology (L-fuzzy topology), more than 40 papers published in the last ten years were concerned with convergence theory. Among these papers, the problem of convergence class was solved for the case ofL=[0,1] [7]. Since the neighbor structure, so called “quasi-coincident neighborhood system,” of anL-fuzzy point in anL-fuzzy topological space is in general not directed under the inclusion order, the conditions of convergence class in [0,1]-fuzzy topology will not be valid any longer in the case of lattice. Moreover, quite different from the cases of {0,1}-fuzzy topology (i.e., ordinary topology) and [0,1]-fuzzy topology, the so called Bolzano–Weierstrass property does not hold, i.e., a net with a cluster point in anL-fuzzy topological space is not still necessary to have a subnet converging to the point. In this paper, a necessary and sufficient condition for the Bolzano–Weierstrass property is produced, the result is also used in a satisfactory theory of convergence classes inL-fuzzy topological spaces, and the associated characterization theorem betweenL-fuzzy topologies and convergence classes is established
Stable Feedback Control of Linear Distributed Parameter Systems: Time and Frequency Domain Conditions

AbstractLarge space structures, or any mechanically flexible structures, are inherently distributed parameter systems (DPSs) whose dynamics are modeled by partial, rather than ordinary, differential equations. Such DPSs are described by operator equations on an infinite-dimensional Hilbert (or Banach) space. However, any feedback controller for such a DPS must be a finite-dimensional (and discrete-time) system to be implemented with on-line digital computers and a finite (small) number of actuators and sensors. There are many ways to synthesize such controllers; we will emphasize the Galerkin or finite-element approach. Although the overall performance of finite-dimensional controllers is important, the first consideration is their stability in closed-loop with the actual DPS. The analysis of DPSs makes use of the theory of semigroups on the infinite-dimensional state space. We will present stability bounds in both the time and frequency domains for infinite-dimensional systems. Currently, the frequency domain approach appears to yield more easily tested stability conditions than the time domain approach; however, we will show some relationships between these two methods and emphasize the role played by the DPS semigroup and its properties. It seems to us that such stability conditions are essential for the planning and successful operation of complex systems like large aerospace structures
Bipolar Isothermal Non-Newtonian Compressible Fluids

AbstractThe existence and uniqueness of a solution to isothermal non-Newtonian bipolar fluid is proved. We consider a special case, where the stress tensor is expressed in the form of potentials depending oneijand (∂eij/∂xk). The second part is devoted to the problem of the stability of the rest state. It is proved that the perturbations of velocity and density tend to zero in appropriate norms along a chosen time sequence. Moreover, we obtain the asymptotic stability of the rest state under the assumption of the regularity of the potential forces
The State of Dynamical Input–Output Systems as an Operator

AbstractA causal input–output system operating for all time from the indefinite past to the indefinite future may be described by a function space for inputs, a function space for outputs, and a causal operator mapping the input space into the output space. The state of such a system at any instant is defined here as an operator from the space of possible future inputs to that of future outputs. This operator is called the natural state. The output space is taken to be a time-shift-invariant normed linear function space, and the input space is either also such a space or a time-shift-invariant subset thereof. There is flexibility allowed in the choice of these spaces. Both the input–output operator and the operator giving the natural state are themselves taken to be elements of normed linear spaces with one of a particular family of norms calledN-power norms. The general development applies to nonlinear and time-varying systems. Continuity and boundedness of the natural state (as an operator) and properties of the natural state and its trajectory as related to the input–output description of the system are investigated. Two examples are presented