pinecone/core-2020-05-10-deduplication huggingface dataset:
A Schwarz Lemma for Convex Domains in Arbitrary Banach Spaces

AbstractIn this note the following new version of the Schwarz lemma is proved: Iffis a holomorphic function mapping a bounded convex domain D1of a complex Banach space into a convex domain D2of another complex Banach space andf(a)=b, then the image byfof the set of points in D1lying at a distance greater thanrfrom the frontier of D1is at a positive distance from the frontier of D2. This distance depends only upona,b, andr, and it can be estimated specifically in terms of the norms of the Banach spaces. Our result extends several earlier theorems
Sequence Spaces with Oscillating Properties

AbstractIn this note we consider various types of oscillating properties for a sequence spaceEbeing motivated by an oscillating property introduced by Snyder and by recent papers dealing with theorems of Mazur–Orlicz type and gliding hump properties. Our main tools, two summability theorems, allow us to identify two such oscillating properties for a sequence spaceEone of which provides a sufficient condition forE⊂Fto implyE⊂WFwhile the other affords a sufficient condition forE⊂Fto implyE⊂SF. HereFis anyLϕ-space, a class of spaces which includes the class of separable FK-spaces,SFdenotes the elements ofFhaving sectional convergence, andWFdenotes the elements ofFhaving weak sectional convergence. This, in turn, is applied to yield improvements on some other theorems of Mazur–Orlicz type and to obtain a general consistency theorem. Furthermore, combining the above observations with the work of Bennett and Kalton we obtain the first oscillating property on a sequence spaceEas a sufficient condition forEβ, the β-dual ofE, to be σ(Eβ,E) sequentially complete whereas the second assures both the weak sequential completeness ofEβand the AK-property forEwith the Mackey topology of the dual pair (E,Eβ)
Reflexivity and the Fixed-Point Property for Nonexpansive Maps

AbstractConnections between reflexivity and the fixed-point property for nonexpansive self-mappings of nonempty, closed, bounded, convex subsets of a Banach space are investigated. In particular, it is shown thatl1(Γ) for uncountable sets Γ andl∞cannot even be renormed to have the fixed-point property. As a consequence, if an Orlicz space on a finite measure space that is not purely atomic is endowed with the Orlicz norm, the Orlicz space has the fixed-point property exactly when it is reflexive
Associative Dynamic Programs

AbstractIn this paper we consider finite-stage stochastic dynamic programs with associative reward systems. An invariant imbedding technique yields a dynamic program with a terminal reward system on an augmented state space. According to the associativity, we clarify the validity of three possible recursive equations—one parametrized equation and two non-parametrized ones. Each of the additive, multiplicative, and multiplicative-additive dynamic programs admits the three recursive equations. Further, the corresponding composite relations among three optimum value functions hold. However, minimum, maximum, and fractional dynamic programs admit only the parametrized recursive equation. They do not admit the two non-parametrized equations. Further, the corresponding relations among the three optimum value functions do not hold
Distribution Theory for Discontinuous Test Functions and Differential Operators with Generalized Coefficients

AbstractInvestigation of the differential operators with the generalized coefficients having singular support on a disjoint set of points requires the consideration of the distribution theory with the set of discontinuous test functions. Such a distribution theory for test functions having discontinuity at one point is developed. A four-parameter family of Schrödinger operators, formed by the operators with singular potential, singular metrics and singular gauge field, is considered. It is proved that this family of singular interactions describes all possible selfadjoint extensions of the second derivative operator defined on the functions vanishing in a neighbourhood of the point. Approximation by operators with smooth coefficients is discussed
An Abstract Form of Maximum and Anti-maximum Principles of Hopf's Type

AbstractWe consider an abstract linear elliptic boundary value problemAu−λu=−f≤0 in a strongly ordered Banach spaceX. The resolvent (λI−A)−1of the closed linear operatorA:X→Xis assumed to be strongly positive and compact for all λ>λ1, where λ1denotes the principal eigenvalue ofA. We prove that there exists a constant δ≡δ(f)>0 depending uponf∈X+\{0} such that −u=−(λI−A)−1f∈X+holds for all λ∈(λ1−δ,λ1). Here,X+={x∈X:x≥0} denotes the positive cone inXwith the topological interiorX+≠∅. We also present nearly sharp sufficient conditions forAguaranteeing independence of δ>0 fromf, i.e., −(λI−A)−1is strongly positive for all Λ∈(λ1−δ,λ1). In particular, for an elliptic Dirichlet boundary value problem, or for a strictly cooperative system of such problems, the strong maximum and boundary point principles (for λ>λ1) yield an anti-maximum principle of Hopf's type (for λ∈(λ1−δ,λ1) depending uponf): If 0≤f∈Lp(Ω),N<p<∞, andf≢0 in Ω, a boundedC2-domain in RN, thenu<0 in Ω and ∂u/∂ν>0 on ∂Ω whenever λ∈(λ1−δ,λ1)
Constrained Controllability of Nonlinear Systems

AbstractIn this paper infinite-dimensional dynamical systems described by nonlinear abstract differential equations are considered. Using the generalized open mapping theorem sufficient conditions for constrained exact local controllability are formulated and proved. It is generally assumed that the values of controls are in a convex and closed cone with vertex at zero. As an illustrative example a constrained exact local controllability problem for a nonlinear delayed dynamical system is solved in detail. Some remarks and comments on the existing results for controllability of nonlinear dynamical systems are also presented
On Time Periodic Solutions of the Dirichlet Problem for Degenerate Parabolic Equations of Nondivergence Type

AbstractIn this paper, we are concerned with the existence of periodic solutions of a quasilinear parabolic equation[formula]with the Dirichlet boundary condition, where Ω is a smoothly bounded domain inRNandfis a given function periodic in time defined on[formula]. Our results depend on the first eigenvalue λ1of −Δ in Ω with the Dirichlet boundary condition. If λ1>1, then there exists a unique positive periodic solution for a positivef(γ∈R). In the case of λ1<1, we construct a nonnegative periodic solution for a negativef(1≤γ<3)
Directed Perturbation Analysis of an Integer Program

AbstractIn this paper, we examine the largest tolerance levels in integer programming (IP) problems when the perturbation occurs on the right hand sides of the constraints in positive or negative directions. Based on the properties of defined stepsizes, we have not only revealed the nested and the inverted nested structures of a parametric IP with a developed algorithm, but also analyzed the configuration of a non-parametric IP with a comparative study. The complexity analysis is incorporated with numerical examples
Structure of Solutions to Linear Evolution Equations: Extensions of d'Alembert's Formula

AbstractThe d'Alembert formula expresses the general solution of the factored equation ∏Nj=1(d/dt−Aj)u=0 asu(t)=∑Nj=1exp(tAj)fj. HereA1,…,ANare (linear) commuting semigroup generators, andAi−Ajis injective fori≠j. The analogue of this fails whenAjdepends ont. But in this nonautonomous case we show that the general solution has the formu(t)=∫P∫Xexp{∫t0Cν(r)dr}fμ(df)λ(dν), where ν: [0,∞)→{1,…,N} is locally Riemann integrable,Cν(r)=Aν(r)(r), and μ (resp. λ) is a finite measure onX(resp. the space P of these functions ν). In addition we discuss the general solution of the inhomogeneous equation ∏Nj=1(d/dt−Aj)u=h(t) for a rather general right-hand sideh
TheN-Laplacian Elliptic Equation: Variational versus Entropy Solutions

AbstractTheN-Laplacian equation −∇·(|∇u|N−2∇u)=fis a kind of limit case in the existence theory of nonlinear elliptic equations when the second member is assumed to be merely integrable (L1theory). Here we consider for definiteness the solutions of the homogeneous Dirichlet problem in a bounded domain Ω∈RNand compare the standard concept of a variational (or energy) solution with the recently introduced concept of an entropy solution, which seems natural in theL1theory. For our equation both concepts have slightly different domains of application. We also discuss the conditions onfunder which the solutions are bounded, a slightly smaller class
Analysis of Bifurcation Due to Loss of Linear Independence and Strict Complementarity for Penalty Methods for Solving Constrained Optimization Problems, II

AbstractIn this article we investigate the structure and persistence of critical point solutions of the nonlinear programming problem obtained from the quadratic penalty function, the logarithmic-barrier function, and the multiplier method. The analysis focuses on singularities arising from the loss of the linear independence constraint qualification and the loss of strict complementarity. The programming problem is first formulated as a system of equations using the Fritz John first order necessary conditions and a nonstandard normalization of the multipliers. The singularities of this system are then classified and solutions are investigated at each type of singularity of codimension zero and one in terms of the bifurcation behavior and persistence of minima of the critical point curves
Pseudo Almost Periodic Solutions of Some Delay Differential Equations

AbstractThe purpose of this paper is to show how the decomposition theory of linear autonomous and pseudo almost periodic systems can be used to obtain results for “perturbed” linear systems with pseudo almost periodic coefficients. This theory together with a natural adaptation of the methods for ordinary differential equations yields the results in an elementary manner. No attempt is made to present the results in the most general form since our primary concern is to indicate the variety of problems to which the methods are applicable. The essential mean used is the exponential dichotomy method
On the Existence of Global Analytic Conjugations for Polynomial Mappings of Yagzhev Type

AbstractConsider a mappingf:Cn→Cnof the form identity plus a term with polynomial components that are homogeneous of the third degree, and suppose that the Jacobian determinant of this mapping is constant throughout Cn(polynomial mapping of Yagzhev type). As a stronger version of the classical Jacobian conjecture, the question has been posed whether for some values of λ∈C\{0} there exists a global change of variables (“conjugation”) on Cnthrough which the mapping λfbecomes its linear part at the origin. Van den Essen has recently produced a simple Yagzhev mapping for which no suchpolynomialconjugation exists. We show here that van den Essen's example still admitsglobal analyticconjugations. The question on the existence of global conjugations for general Yagzhev maps is then still open
On the Topology of the Space of Hankel Convolution Operators

AbstractLet Hμbe the Zemanian space of Hankel transformable functions, let O′μ,#be its space of convolution operators, and let Oμ,#be the predual of O′μ,#. We prove that the topology of uniform convergence on bounded subsets of Hμand the strong dual toplogy coincide on O′μ,#. Our technique, involving Mackey topologies, differs from, and is simpler than, those usually employed with the same purpose for other spaces of convolution operators, to which it is also applicable. As a consequence, the properties of Oμ,#being reflexive, complete, nuclear, and Montel are established
Generalized Proper Efficiency and Duality for a Class of Nondifferentiable Multiobjective Variational Problems withV-Invexity

AbstractA Mond–Weir type dual for a class of nondifferentiable multiobjective variational problems in which every component of the objective function contains a term involving the square root of a certain positive semidefinite quadratic form, is considered and various duality results, viz. weak, strong, and converse duality theorems, are developed for conditionally properly efficient solutions. These results are obtained underV-invexity assumptions and its generalizations on objective and constraint functions. This work extends many results on variational problems established earlier
Weak Asymptotic Stabilizability of Discrete-Time Systems Given by Set-Valued Operators

AbstractThe paper deals with the weak asymptotic stabilizability property of discrete-time inclusions, whose right-hand sides are set-valued operators in Banach spaces. Such inclusions include control systems with constrained controls. Questions of existence of positive eigenvalues of convex closed operators are discussed. Based on the method of decomposition of the state space we derive sufficient conditions for weakly asymptotic stabilizability of discrete-time convex processes in terms of controllability-like conditions. The results are applied in weak asymptotic stabilizability of linear discrete and continuous-time systems with constrained controls in Banach spaces
A Fixed Point Theorem in Probabilistic Metric Spaces and an Application

AbstractThe notion of a (Ψ,C)-contraction type multivalued mapping is introduced. This notion is a generalization of the notion ofC-contraction introduced by T. L. Hicks (Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat.13, 1983, 63–72). A fixed point theorem for (Ψ,C)-contraction is proved. An application on the existence of a random fixed point for random operatorf:M×Ω→M, where (M,d) is a separable metric space and (Ω,A,m) a measure space with a decomposable measure of (NSA)-type, is given
Sensitivity Analysis in Parametrized Convex Vector Optimization

AbstractThis paper provides some results concerning sensitivity analysis in parametrized convex vector optimization. We consider three types of perturbation maps (i.e., perturbation map, proper perturbation map, and weak perturbation map) according to three kinds of solution concepts (i.e., minimality, proper minimality, and weak minimality with respect to a fixed ordering cone) for a vector optimization problem. As for general vector optimization, authors have already established the behavior of the above three types of perturbation maps by using the concept of contingent derivatives for set-valued maps in finite dimensional Euclidean spaces. In this paper we concentrate on convex vector optimization and provide quantitative properties of the perturbation maps under some convexity assumptions. Namely, we investigate the relationships between the contingent derivatives of the perturbation maps and those of the feasible set map in the objective space
Optimization of a Class of Distributed Parameter Systems with Deviating Argument

AbstractIn this paper we consider an optimal control problem described by a system of nonlinear first order hyperbolic partial differential equations with deviating argument, including integral inequality constraints. The control variables are assumed to be measurable, with the corresponding state variables in Lp. We introduce the adjoint equations, derive an integral representation of the increments of the functionals involved, and use separation theorems of functional analysis to obtain new necessary optimality conditions in the form of the Pontryagin maximum principle
Full InformationH∞-Control for Discrete-Time Infinite Markov Jump Parameter Systems

AbstractIn this paper we consider the full information discrete-timeH∞-control problem for the class of linear systems with Markovian jumping parameters. The state-space of the Markov chain is assumed to take values in a countably infinite set. Full information here means that the controller has access to both the state-variables and jump-variables. A necessary and sufficient condition for the existence of a feedback controller that makes thel2-induced norm of the system less than a prespecified bound is obtained. This condition is written in terms of a set of infinite coupled algebraic Riccati equations
Approximate Solutions, Existence, and Uniqueness of the Cauchy Problem of Fuzzy Differential Equations

AbstractThe Cauchy problem of fuzzy differential equations is investigated by the use of the concept ofH-differentiability due to M. L. Puri and D. A. Ralescu (J. Math. Anal. Appl.91,1983, 552–558). The existence and uniqueness theorem is obtained for the solution of the Cauchy problemx′(t)=f(t,x(t)),x(t0)=x0, for the fuzzy-valued mappings of a real variable whose values are normal, convex, upper semicontinuous, and compactly supported fuzzy sets inRn, where the functionfsatisfies the generalized Lipschitz condition
Decay Rates for a Two-Dimensional Viscous Ocean of Finite Depth

AbstractIn 1888, A. M. Basset (“A Treatise on Hydrodynamics,” Deighton, Bell and Co., Cambridge, 1888) gave a formal argument to show that slow modes of decay do not exist for the free surface and velocity field of a linearized, two dimensional, viscous, finite-depth ocean with small viscosity. His dispersion relation for waves of the formeikx+ωtwas ω≈O(k2). His argument did not apply to very small wavelengths (largek) which are important to consider since nonlinear effects can cause small wavelengths to focus and form singularities. In Lamb's book on hydrodynamics, there is a formal argument which shows that slow modes of decay exist for small wavelengths on the free surface of a linearized, two dimensional, viscous ocean of infinite depth if surface tension is neglected. There, Lamb showed that ω≈O(1/k). The lack of boundary conditions on a fixed bottom makes the analysis somewhat easier than for the finite depth case. Lamb's analysis, as well as Basset's, relies on physical intuition rather than careful mathematical analysis—the smallness of certain quantities is assumed before the dispersion relation is derived. More importantly, both Basset and Lamb rely on a particular decomposition (we will be specific below) of the stream function which becomes degenerate in the small wavelength limit. They obtain dispersion relations by setting to zero a determinant, but the zeros of that determinant do not all correspond to physically nontrivial pressures or velocities. Some zeros merely reflect the degeneracy of the decomposition they used. The purpose of this article is to warn others of this pitfall, while pointing out a simple direct method of calculation which avoids it. Our approach applies equally well to the cases of finite or infinite depth, with or without surface tension. In addition, our method yields a complete asymptotic expansion for the dispersion relation for small wavelengths. In order to obtain a complete expansion from other methods, one would have to decide at which point in the expansion small terms that were dropped would need to be reinserted, etc
Measure Driven Differential Inclusions

AbstractMeasure driven differential inclusions arise when we attempt to derive necessary conditions of optimality for optimal impulsive control problems with nonsmooth data. We introduce the concept of a robust solution to a measure driven inclusion, which extends to a multifunction setting interpretations of solutions to measure driven differential equations provided by Dal Maso and Rampazzo and others. Closure properties of sets of robust solutions are established, and notions of relaxation investigated. Implications for optimality conditions for impulsive control problems are pursued in a companion paper
On the convergence of quadrature formulas connected with multipoint Padé-type approximants

29 pages, no figures.-- MSC2000 codes: 41A55, 41A21.MR#: MR1408352 (97e:41066)Zbl#: Zbl 0856.41027^aLet $I(F)= \int^1_{- 1} F(x)\omega(x) dx$, where $\omega$ is a complex valued integrable function. We consider quadrature formulas for $I(F)$ which are exact with respect to rational functions with prescribed poles contained in $\overline{\bbfC}\backslash [- 1, 1]$. Their rate of convergence is studied.The research by the first three authors (P.G.-V., M.J.P., R.O.) was partially supported by the HCM project ROLLS, under Contract CHRX-CT93-0416. Research by the fourth author (G.L.L.) was carried out while on a visit at Universidad de La Laguna. This visit was made possible by a travel grant from CDE-IMU.Publicad
Boundary Value Problems and Hardy Spaces Associated to the Helmholtz Equation in Lipschitz Domains

AbstractIn this paper we introduce and discuss, in the Clifford algebra framework, certain Hardy-like spaces which are well suited for the study of the Helmholtz equation Δu+k2u=0 in Lipschitz domains of Rn+1. In particular, in the second part of the paper, these results are used in connection with the classical boundary value problems for the Helmholtz equation in Lipschitz domains in arbitrary space dimensions. In this setting, existence, uniqueness, and optimal estimates are obtained by inverting the corresponding layer potential operators onLpfor sharp ranges ofp's. Also, a detailed discussion of the Helmholtz eigenvalues of Lipschitz domains is presented
On the Spectrum of a One-Velocity Transport Operator with Maxwell Boundary Condition

AbstractThe spectrum of a one-velocity transport operator with Maxwell boundary condition is discussed inL1space. First, it is proved that the spectrum of a streaming operator associated with the transport operator consists of infinitely isolated eigenvalues, each of which is simple; furthermore, a formula for computation of these eigenvalues is obtained. Second, it is proved that the essential spectrum of the transport operator is empty and it is pointed out that the dominant eigenvalue of the transport operator exists. Finally, a necessary and sufficient condition for the existence of a complex eigenvalue of the transport operator in a right half plane is given
Uniform Boundary Stabilization of a Nonlinear Shallow and Thin Elastic Spherical Cap

AbstractA coupled nonlinear system describing the vibrations of a shallow thin spherical shell is considered. It is shown that if the “thickness” parameterh/ρ0is large with respect to the “shallowness” parameterQ0=ρ0/R, then the dissipative feedback applied at the boundary of the shell causes the energy of the system to decay to zero at an uniform rate. The limitations on the values of these physical parameters are confirmed by the static nonlinear theory which predicts, for small values of the thickness parameter, the existence of multiple equilibrium states (everted states)