pinecone/core-2020-05-10-deduplication huggingface dataset:
Estimates of Eigenfunctions and Localization of the Spectrum of Differential Operators

AbstractWe consider a differential equation with parameter λ,(−1)ny(2n)λ(x)+σk=02n−2qk(x)y(k)λ(x)=λyλ(x),x∈[−1,1],where λ∈G={λ∈C: |λ|≥1},qk(x)∈L1(− 1,1), andk=0,2n−2. LetT={yλ(x)}λ∈Gbe some set of solutions of (1). We show that the inequalitiesyλ(x)(j)C[−1,1]≤c(q)|λ|j/2nyλ(x)L2(−1,1),j=0,2n−1,yλ(x)∈T,can hold only if λ∈H. Here and below we denote byc(q) different positive constants, depending only onL1norms ‖qk‖L1(−1,1),k=0,2n−2, of coefficients of (1),H(c(q))=λ∈C:Reλ≥−c(q),(1 +|Reλ|)(2n−1)/2n≥1c(q)|Imλ|.In addition, if (2) holds, then for any λ∈H, |λ|⪢c(q), there is an integerssuch that λ=|sπ|2n[1+O(1/|s π|)]. Here the constant inO(·) depends only onL1norms of coefficients of (1), and the exponent of (1+|Reλ|) in the definition ofHcannot be lessened. As an example, we study the problem of localization of the spectrum of a class of semiregular boundary value problems
Random Fixed Points of 1-Set Contractive Random Maps in Frechet Spaces

AbstractA random fixed point theorem for more general 1-set-contractive random operators in Fréchet spaces is proved. As applications, some random fixed point theorems for nonexpansive, semicontractive, and LANE random operators are also derived. Thus the results of Itoh [J. Math. Anal. Appl.67(1979), 261–273], Lin [Proc. Amer. Math. Soc.123(1995), 1167–1176;Proc. Amer. Math. Soc.103(1988), 1129–1135], Shahzad [J. Math. Anal. Appl.203(1996), 712–718], Tan and Yuan [Stochastic Anal. Appl.15(1997), 103–123], and Xu [Proc. Amer. Math. Soc.110(1990), 495–500], are extended
Cubic and General Extensions of the Lorentz Transformation

AbstractThe Lorentz transformation involves essentially only two variables, one space and the other time, though space is three dimensional, because the changes in space variables relate only to direction of motion of one observer with respect to the other. The extension to a larger number of variables should not mean the increase in spatial dimensions or the dimensions of time as was attempted earlier for decades, leading to no meaningful results. To understand the meaning of “extension” we first enunciate the “principle of same behaviour” for two variables and we show that the Lorentz transformation obeys that principle. The extension to a larger number of variables using this principle then becomes logical, elegant, and consistent
Examples of Discrete Operators with a Pure Point Spectrum of Finite Multiplicity

AbstractOne constructs operators acting onl2(Zm) (orl2(Zm)p),m,p≥1, with a real pure point spectrum of finite multiplicity by perturbing diagonal matrices using a KAM procedure. The point spectrum can be dense on an interval or a Cantor set of measure zero. The basic fact here is to remark that for perturbations built up with an infinite number of block diagonals, regularly separated, it is possible to deal with eigenvalues of multiplicity strictly greater than 1. Examples of discrete operators associated with discretization of systems of partial differential equations are given
Bayliss–Turkel-like Radiation Conditions on Surfaces of Arbitrary Shape

AbstractThis paper addresses the extension of the Bayliss–Turkel second-order radiation condition to an arbitrarily shaped surface. The derivation is based mainly on the pseudo-differential calculus as well as on the introduction of a criterion providing a precise handling of the approximation process involved in the derivation of the radiation condition. The radiation condition then ranges among the most accurate of those of order two. As a by-product of the derivation, almost all known radiation conditions of order less than or equal to two are recovered and their respective accuracies are compared
On the Equivalence of Some Basic Principles in Variational Analysis

AbstractThe primary goal of this paper is to study relationships between certain basic principles of variational analysis and its applications to nonsmooth calculus and optimization. Considering a broad class of Banach spaces admitting smooth renorms with respect to some bornology, we establish an equivalence between useful versions of a smooth variational principle for lower semicontinuous functions, an extremal principle for nonconvex sets, and an enhanced fuzzy sum rule formulated in terms of viscosity normals and subgradients with controlled ranks. Further refinements of the equivalence result are obtained in the case of a Fréchet differentiable norm. Based on the new enhanced sum rule, we provide a simplified proof for the refined sequential description of approximate normals and subgradients in smooth spaces
On the Basic Representation Theorem for Convex Domination of Measures

AbstractA direct, constructive proof is given for the basic representation theorem for convex domination of measures. The proof is given in the finitistic case (purely atomic measures with a finite number of atoms), and a simple argument is then given to extend this result to the general case, including both probability measures and finite Borel measures on infinite-dimensional spaces. The infinite-dimensional case follows quickly from the finite-dimensional case with the use of the approximation property
Existence and Uniqueness of Solutions for a Dynamic One-Dimensional Damage Model

AbstractWe consider a one-dimensional dynamic model that describes the evolution of damage caused by tension in a viscoelastic material. The process is modeled by a coupled set of two differential inclusions for the elastic displacement and damage fields. We establish the existence of local weak solutions. The existence result is derived from the a priori estimates obtained for a sequence of regularized, truncated, and time-retarded approximations. We also establish the existence of the unique weak solution of a simplified version of the model
Extreme Polynomials and Multilinear Forms onl1

AbstractGiven a 2-homogeneous polynomialP(x,y)=ax2+by2+cxywith real coefficients, let ‖P‖rand ‖P‖cdenote the norms ofPon the real and complex Banach spacel21, respectively. We show ‖P‖r=‖P‖c, and obtain a sufficient and necessary condition on the coefficientsa,b, andcforPto have norm 1. Applying these results, we characterize extreme points of the unit ball of P(2l1) for the real Banach spacel21and examine them for the complex Banach spacel21. We apply them to find extreme points and strongly extreme points of the unit ball of P(2l1) and get an extremal 2-homogeneous polynomial onl1that is not an extreme point. We also characterize extreme points and strongly extreme points of the unit ball of L(ml1) or L(mL1[0,1])
Existence and Uniqueness of Positive Solutions to a Semilinear Elliptic Problem in RN

AbstractLetp∈Cαloc(RN) withp>0 and letf∈C1((0,∞),(0,∞)) be such that limu↘0f(u)/u=+∞,fis bounded at infinity, and the mappingu↦f(u)/(u+β) is decreasing on (0,∞), for some β>0. We prove that the problem −Δu=p(x)f(u) in RN,N>2, has a unique positiveC2+αloc(RN) solution that vanishes at infinity provided ∫∞0rΦ(r)dr<∞, where Φ(r)=max{p(x);|x|=r}. Furthermore, it is showed that this condition is nearly optimal. Our results extend previous works by Lair–Shaker and Zhang, while the proofs are based on two theorems on bounded domains, due to Brezis–Oswald and Crandall–Rabinowitz–Tartar
Necessary and Sufficient Conditions for the Unique Solvability of a Nonlinear Reaction-Diffusion Model

AbstractIt has been known for some time that a nonlinear reaction-diffusion model, with Dirichlet boundary conditions, is uniquely solvable if the reaction term satisfies an appropriate Lipschitz condition. However, as recently shown for an absorption model, such a condition is not necessary. We establish a uniqueness result which, in the case of reaction and diffusion governed by power laws, is in fact both necessary and sufficient for the unique solvability of the model. The improvement that is needed on the above-mentioned Lipschitz condition occurs in the so-called fast diffusion model
Robust Stability Conditions for Large-Scale Interconnected Systems with Structured and Unstructured Uncertainties

AbstractThe paper is mainly concerned with the problem of decentralized robust stability of large-scale interconnected systems with structured and unstructured uncertainties. A simple method is presented whereby some sufficient conditions are derived so that asymptotic stability of large-scale interconnected systems can be guaranteed in the presence of uncertain perturbations. The method is also extended to large-scale discrete-time systems and the corresponding robust stability conditions are established for uncertain large-scale discrete-time systems. Finally, two numerical examples are given to demonstrate that our robust stability conditions are less conservative than those reported in the control literature
An Algebraic Approach to the Classification of Centers in Polynomial Liénard Systems

AbstractWe consider the second-order Liénard systemẍ+f(x)ẋ+g(x)=0,wheref(x) andg(x) are polynomials, which we rewrite in the equivalent two-dimensional form,ẋ=y,ẏ=−g(x)−yf(x).We show that the local question of whether a critical point of this system is a center can be expressed in terms of global conditions onfandg. Using these results, we give a simple classification of all such centers. We also address the problem of the coexistence of centers, foci, and limit cycles for these systems. Systems with degenerate centers are also considered
Duality for a Minimax Programming Problem Containingn-Set Functions

AbstractParameter necessary and sufficient optimality conditions for the discrete minimax programming problem with differentiablen-set functions are established in Zalmai [Optimization20(1989) 377–395]. In this paper, we establish nonparameter necessary and sufficient optimality conditions. The nonparameter necessary optimality conditions suggest the establishment of two parameter-free duality models for the discrete minimax programming problem with differentiablen-set functions, and the weak, strong, and strict converse duality theorems have been derived
Singularly Perturbed Multidimensional Switching Diffusions with Fast and Slow Switchings

AbstractThis work is concerned with singularly perturbed switching diffusions with fast and slow components. The diffusion component of the joint process is multidimensional and the diffusive motion evolves an order of magnitude slower than that of the jumps or the switchings. The jump component may be decomposed into several groups. There is one group consisting of transient states, and all other groups involve recurrent states. Within each group, there are rapid switchings. Among different groups, the switchings occur relatively infrequently. Under simple conditions, an asymptotic expansion of the probability density is developed; error bounds and justification of the expansion are provided
On Duality in Semi-Infinite Programming and Existence Theorems for Linear Inequalities

AbstractLinear semi-infinite programming deals with the optimization of linear functionals on finite-dimensional spaces under infinitely many linear constraints. For such kind of programs, a positive duality gap can occur between them and their corresponding dual problems, which are linear programs posed on infinite-dimensional spaces. This paper exploits some recent existence theorems for systems of linear inequalities in order to obtain a complete classification of linear semi-infinite programming problems from the point of view of the duality gap and the viability of the discretization numerical approach. The elimination of the duality gap is also discussed
On Some Mathematical Aspects of the Direct-Interaction Approximation in Turbulence Theory

AbstractThe purpose of this paper is to clarify some mathematical issues associated with the direct-interaction approximation (DIA) given by Kraichnan [Phys. Rev.109(1958), 1407;J. Fluid Mech.5(1959), 497;Phys. Fluids7(1964), 1723;8(1965), 575;9(1966), 1728] via its application to Kraichnan's [J. Math. Phys.2(1961), 124] model equation. The latter process is shown to lead to a functional equation; the basic properties and construction of the solution of this equation are addressed. The relation of the DIA solution to the perturbative solution is discussed within the framework of a continued-fraction representation
Conditional Decision Processes with Recursive Function

AbstractWe consider stochastic optimization of not necessarily additive but recursive functions over multistage decision processes. Without assuming any monotonicity, we optimize a regular process by a direct dynamic programming approach. On the regular decision process, we propose two related conditional decision processes: an a posteriori conditional decision process and an a priori. When the Markov transition law degenerates into a deterministic dynamics, the two conditional processes reduce to the same deterministic decision process. The conditional processes with monotonicity are optimized by the usual backward dynamic programming. We show that under additional convexity the regular process dominates the a priori in maximum value function and the a priori does the a posteriori. We show that the a posteriori process illustrates Kreps and Porteus's dynamic choice problem. The numerical example also verifies the dominance relation in three optimal value functions
The Relationship Between Various Filter Notions on aGL-Monoid

AbstractThe notion of a generalised filter is extended to the setting of aGL-monoid. It is shown that there exists a one-to-one correspondence between the collection of generalised filters on a setXand the collection of strongly stratifiedL-filters onX. Specialising to the case whereLis the closed unit interval [0,c] viewed as a Heyting algebra, we show that any strongly stratified [0,c]-filter onXcan be uniquely identified with a saturated filter onIXwith characteristic valuec. In this way, the notion of a generalised filter unifies various filter notions. In particular, necessity measures and finitely additive probability measures are specific examples of generalised filters
Generalized Self-Similarity

AbstractWe prove the existence ofLpfunctions satisfying a kind of self-similarity condition. This is achieved by solving a functional equation by means of the construction of a contractive operator on an appropriate functional space. The solution, a fixed point of the operator, can be obtained by an iterative process, making this model very suitable to use in applications such as fractal image and signal compression. On the other hand, this “generalized self-similarity equation” includes matrix refinement equations of the typef(x)=∑ckf(Ax−k) which are central in the construction of wavelets and multiwavelets. The results of this paper will therefore yield conditions for the existence ofLp-refinable functions in a very general setting
Minimizing Risk Models in Markov Decision Processes with Policies Depending on Target Values

AbstractThis paper studies the minimizing risk problems in Markov decision processes with countable state space and reward set. The objective is to find a policy which minimizes the probability (risk) that the total discounted rewards do not exceed a specified value (target). In this sort of model, the decision made by the decision maker depends not only on system's states, but also on his target values. By introducing the decision-maker's state, we formulate a framework for minimizing risk models. The policies discussed depend on target values and the rewards may be arbitrary real numbers. For the finite horizon model, the main results obtained are: (i) The optimal value functions are distribution functions of the target, (ii) there exists an optimal deterministic Markov policy, and (iii) a policy is optimal if and only if at each realizable state it always takes optimal action. In addition, we obtain a sufficient condition and a necessary condition for the existence of finite horizon optimal policy independent of targets and we give an algorithm computing finite horizon optimal policies and optimal value functions. For an infinite horizon model, we establish the optimality equation and we obtain the structure property of optimal policy. We prove that the optimal value function is a distribution function of target and we present a new approximation formula which is the generalization of the nonnegative rewards cases. An example which illustrates the mistakes of previous literature shows that the existence of optimal policy has not been proved really. In this paper, we give an existence condition, which is a sufficient and necessary condition for the existence of an infinite horizon optimal policy independent of targets, and we point out that whether there exists an optimal policy remains an open problem in the general case
Iterated Function System and Ruelle Operator

AbstractWe consider a generalized iterated function system where the weights are variable functions. By using the Ruelle operator and a dynamical system consideration we prove that if the system is contractive and the weights are strictly positive functions and satisfy the Dini condition, then there exists a unique eigenmeasure (corresponding to the Ruelle operator) on the attractor. If in addition the maps are conformal and satisfy the open set condition, then we prove that they satisfy the strong open set condition, and by using this we can give a description of theLp-scaling spectrum and the multifractal structure of the eigenmeasure. The work extends some results of [Proc. London Math. Soc.73(1996), 105–154;Adv. Appl. Math.19(1997), 486–513;J. Statist. Phys.86(1997), 233–275;Indiana Univ. Math. J.42(1993), 367–411]
B-Convexity, the Analytic Radon–Nikodym Property, and Individual Stability ofC0-Semigroups

AbstractLetT={T(t)}t≥0be aC0-semigroup on a Banach spaceX, with generatorAand growth bound ω. Assume thatx0∈Xis such that the local resolvent λ↦R(λ,A)x0admits a bounded holomorphic extension to the right half-plane {Reλ>0}. We prove the following results:•(i) Ifhas Fourier type∈(1,2], then lim‖()(λ−)‖=0 for all β>1/and λ>ω.•(ii) Ifhas the analytic RNP, then lim‖()(λ−)‖=0 for all β>1 and λ>ω.•(iii) Ifis arbitrary, then weak-lim()(λ−)=0 for all β>1 and λ>ω.As an application we prove a Tauberian theorem for the Laplace transform of functions with values in aB-convex Banach space
Certain Classes of Series Involving the Zeta Function

AbstractThe authors apply the theory of the double gamma function, which was recently revived in the study of determinants of Laplacians, to evaluate some families of series involving the Riemann zeta function. Introducing a (presumably new) mathematical constant in the theory of the double gamma function, they also systematically evaluate a definite integral of the double gamma function and various classes of series associated with zeta functions. Some of these definite integrals are expressed in terms of quotients of double gamma functions
Fractional Order Continuity and Some Properties about Integrability and Differentiability of Real Functions

AbstractIn this paper a certain function spaceCα, 0≤α≤1, larger than the space of continuous functions, is introduced in order to study new properties and the extension of some already known results about the Riemann–Liouville fractional integral and derivative operators.Sufficient conditions for the continuity ofI1−αafare given. Furthermore, necessary conditions are given for the pointwise existence of fractional derivatives. The existence of a derivative of order β, from the existence of a certain derivative of order α, β<α, is also analyzed
Accumulation Point and C. T. Yang's Theorem inL-fuzzy Topological Spaces

AbstractThe investigations on accumulation point in fuzzy topological spaces (L-fuzzy topological spaces) have lasted for more than 16 years (e.g., see [9], [11], [6], [15], [17], [20], [13], [4], [1], [18], [16], [5]). Amongst these researches, the notion introduced in [9] even yielded the fuzzy form of the famous C. T. Yang's Theorem. But, however, the main basic problem, making the derivation operator derived from it preserve finite joins, is still unsolved; even only inclusion order of subsets cannot be preserved. Compared with the fundamental position of this concept standing at in topology, this shortage is obvious. With a new operation possessing many quite nice properties, the so-called “quasi-difference” between twoL-fuzzy subsets, a reasonable definition is introduced for this basic notion in this paper which possesses the wanted properties and, applying its nice property, theL-fuzzy form of the C. T. Yang's Theorem is proved
The Boundary-value Problem in Domains with Very Rapidly Oscillating Boundary

AbstractWe study the asymptotic behavior of the solution to boundary-value problem for the second order elliptic equation in the bounded domain Ωε⊂Rnwith a very rapidly oscillating locally periodic boundary. We assume that the Fourier boundary condition involving a small positive parameter ε is posed on the oscillating part of the boundary and that the (n−1)-dimensional volume of this part goes to infinity as ε→0. Under proper normalization conditions that homogenized problem is found and the estimates of the residual are obtained. Also, we construct an additional term of the asymptotics to improve the estimates of the residual. It is shown that the limiting problem can involve Dirichlet, Fourier or Neumann boundary conditions depending on the structure of the coefficient of the original problem
On the Global Convergence of a Class of Functional Differential Equations with Applications in Neural Network Theory

AbstractWe study a system of retarded functional differential equations which generalise both the Hopfield neural network model as well as hybrid network models of the cellular neural network type. Our main results give sufficient conditions for the global asymptotic stability of such systems and are milder than previously known conditions for the hybrid models. When specialised to neural networks, our models allow us to consider several different types of activation functions, including piecewise linear sigmoids and unbounded activations as well as the usualC′-smooth sigmoids. These issues are vital in the applications. We also study neural network models with nonconstant delaysr(t)
Fréchet Differentiability of Parameter-Dependent Analytic Semigroups

AbstractWe study the dependence on a vector-valued parameterqof a collection of analytic semigroups {T(t;q),t≥0} arising, for example, from a collection of diffusion-convection equations whose infinitesimal generators are abstract elliptic operators defined in terms of sesquilinear forms in a “Gelfand triple” or “pivot space” framework. Within a mathematical framework slightly more general than the one set forth below, Banks and Ito [Banks, H. T. and Ito, K., “A unified framework for approximation in inverse problems for distributed parameter systems,” Control—Theory and Advanced Technology,4(1988), pp. 73–90] have shown, as an application of the Trotter-Kato Theorem, that the mapq↦T(t;q) is continuous in the strong operator topology. In this paper, we establish the analyticity of this map in the uniform operator topology, and exhibit its Fréchet derivative both as a contour integral and as the solution of a particular initial-value problem