pinecone/core-2020-05-10-deduplication huggingface dataset:
Applications of a Theorem of Grothendieck to Vector Measures

AbstractLet R be a ring of subsets of a nonempty set Ω and Σ(R) the Banach space of uniform limits of sequences of R-simple functions in Ω. LetXbe a quasicomplete locally convex Hausdorff space (briefly, lcHs). Given a boundedX-valued vector measuremon R, the concepts ofm-integrability of functions in Σ(R) and of representing measure of a continuous linear mappingu:Σ(R)→Xare introduced. Based on these concepts and a theorem of Grothendieck on the range of the biadjointu** ofu∈L(Σ(R),X), it is shown that such a mappinguis weakly compact if and only if its representing measure is strongly additive. The result subsumes the range theorems of12and9. Also the theorem on extension is deduced. The method of proof for all these results in vector measures is more natural than the known ones
Nonlinear Evolution of the Kelvin–Helmholtz Instability of Supersonic Tangential Velocity Discontinuities

AbstractA nonlinear stability analysis using a multiple-scales perturbation procedure is performed for the instability of two layers of immiscible, inviscid, arbitrarily compressible fluids in relative motion. Such configurations are of relevance in a variety of astrophysical and space configurations. For modes ofallwavenumbers on, or in the stable neighborhood of, the linear neutral curve, the nonlinear evolution of the amplitude of the linear fields on the slow first-order scales is shown to be governed by a complicated nonlinear Klein–Gordon equation. Both the spatially dependent and space-independent versions of this equation are considered to obtain the regimes of physical parameter space where the linearly unstable solutions either evolve to final permanent envelope wave patterns resembling the ensembles of interacting vortices observed empirically, or are disrupted via nonlinear modulation instability
Vanishing Viscosity in the Barotropic β-Plane

AbstractThe initial boundary value problems associated with the inviscid barotropic potential vorticity equation in the β-plane and its viscous analogue are considered. It is shown that the solution velocity to the viscous equation converges to the inviscid solution in aC1sense for finite times and that, under additional smoothness assumptions on the inviscid flow, this convergence can be extended toC3. Moreover, this convergence occurs as O(ε), where ε is the viscous parameter. This particular form of vanishing viscosity is of relevance in analysing viscosity induced advection for barotropic models
Continuous Dependence on Modeling for Related Cauchy Problems of a Class of Evolution Equations

AbstractSolutions of a class of Cauchy problems are compared with solutions of related perturbed problems. Hölder continuous dependence on the perturbation parameter is established for the difference of these solutions using the logarithmic convexity method. Results are also obtained under weaker restrictions for a special class of linear equations by employing the Lagrange identity method. Studies of this kind attempt to regularize problems that may be ill posed against errors made in formulating the governing equations of mathematical models
Simple Waves and Simple States inR2

AbstractThe so-called “solutions, constructed by means of Riemann invariants” for quasi-linear systems of PDEs, play an important role in gas dynamics, since they describe the propagation and the superposition of waves. The existing methods for finding solutions (simple waves and simple states) of this type usually deal with systems whose coefficients do not depend on the independent variables, and have a very limited application to non-homogeneous systems. In this paper a method for finding such solutions in the case of general quasi-linear systems of PDEs inR2is presented. This method is based on a detailed investigation of the respective overdetermined system and on a special construction, involving Pfaff systems
Hierarchical Production Control in a Stochastic Manufacturing System with Long-Run Average Cost

AbstractThis paper presents an asymptotic analysis of a stochastic manufacturing system consisting of parallel machines subject to breakdown and repair and facing a constant demand, as the rates of change of the machine states approach infinity. This situation gives rise to a limiting problem in which the stochastic machine availability is replaced by its equilibrium mean availability. The long-run average cost for the original problem converges to the long-run average cost of the limiting problem. Open-loop and feedback controls for the original problem are constructed from optimal controls of the limiting problem in a way that guarantees their asymptotic optimality. The convergence rate of the long-run average cost for the original problem to that of the limiting problem is established. This helps in providing an error estimate for the constructed open-loop asymptotic optimal control
E-Optimality for Regression Designs of Supplementary Experiments

AbstractFor the regression model whose linear functional of unknown parameters is estimable, the existence of anE-optimal design for supplementary experiments on the set of all design matrices whose Euclid norm does not exceed a given constant is obtained. TheE-optimal designs for supplementary experiments are found in some reasonable regions by spectral decomposition of a matrix. Also, the relationship between theE-optimal design for supplementary experiments and theE-optimal design is obtained. A simple numerical example is given to illustrate the procedure for finding theE-optimal design matrices of supplementary experiments
On Gap Functions and Duality of Variational Inequality Problems

AbstractWe extend the definition of the gap function defined by Auslender for a more general class of variational inequality problems involving some convex function. A study of the duality of the extended variational inequality problem and its dual sheds new light on the meaning of gap functions. Convexity and differentiability of the gap function are also studied and sufficient conditions are derived. We also show how the gap functions for the primal and the dual are related by dual Fenchel optimization problems
Exponentially Stable Feedback Control of Discrete-Time Nonlinear Distributed Parameter Systems on Banach Spaces

AbstractPractical engineering systems require feedback control implementation in discrete-time; however, stable control of an infinite-dimensional distributed parameter system (DPS) is not always guaranteed. We introduce two nonlinear controllers, a DPS one and a finite-dimensional one with residual mode filter (RMF) compensation, which produce local exponential closed-loop stability for nonlinear discrete-time DPS on Banach spaces under appropriate local Lipschitz continuity conditions. Both controllers operate in discrete-time and our results place no restrictions on the smoothness of the initial data in the Banach space
Sobolev Type Spaces Associated with Bessel Operators

AbstractA Sobolev type spaceGμs,pis defined and its properties including completeness and inclusion are investigated using the theory of distributional Hankel transform. The Hankel potentialHμsis defined. It is shown that the Hankel potentialHμsis a continuous linear mapping of the Zemanian spaceHμinto itself. TheLp-space of all such Hankel potentials,Wμs,p(0,∞) is defined. It is shown thatWμs,pis a Banach space with respect to the norm ‖ ‖s,p,μ. It is also shown that the Hankel potential is an isometry ofWμs,p. AnLp-boundedness result for the Hankel potential is proved. It is shown that solutions of certain nonhomogeneous equations involving Bessel differential operators belong to these spaces
Removable Sets in the Oscillation Theory of Complex Differential Equations

AbstractLetf1,f2be two linearly independent solutions of the linear differential equationf″+A(z)f=0, whereA(z) is transcendental entire, and assume that the exponents of convergence for the zero-sequences off1,f2satisfy max(λ(f1),λ(f2))=∞. Our main result proves that the zeros ofE≔f1f2are uniformly distributed in the sense that quite arbitrary large areas of the complex plane can be removed in such a way that if only zeros outside of these areas will be counted for the exponents of convergences, their maximum still remains infinite
Mehler Integral Transforms Associated with Jacobi Functions with Respect to the Dual Variable

AbstractWe prove a Mehler representation for Jacobi functions ϕλ(α,β)(t) with respect to the dual variable λ. We exploit this representation to define a pair of dual integral transforms χα,βand its transposedtχα,β. We define two second order difference operatorsPα,βandQsuch that ϕλ(α,β)(t) is an eigenfunction ofPα,βwith respect to the dual variable λ, and χα,βandtχα,βare permutation operators betweenPα,βandQ. Next we give some spaces of functions on which χα,βandtχα,βare isomorphisms and we establish inversion formulas for these transforms
The Solution by Iteration of Nonlinear Equations in Uniformly Smooth Banach Spaces

AbstractLetEbe a uniformly smooth Banach space and letT:D(T)⊂E↦Ebe a strong pseudocontraction with an open domainD(T) inEand a fixed pointx*∈D(T). We establish the strong convergence of the Mann and Ishikawa iterative processes (with errors) to the fixed point ofT. Related results deal with the iterative solution of operator equations of the formsf∈Txandf∈x+λTx, λ>0, whenTis a set-valued strongly accretative operator. Our theorems include the cases in which the operatorTis defined only locally. Explicit error estimates are also given
Spectral Properties of Schrödinger Operators with Irregular Magnetic Potentials, for a Spin 12 Particle

AbstractThe two-dimensional Schrödinger operatorH̃(a) for a spin 12 particle is considered. The magnetic fieldbgenerated byadoes not grow in some directions and stabilizes to a positively homogeneous function. It is shown that the spectrum σ(H̃(a)) consists of σdisc(H̃(a)) and {0}, the latter being an isolated eigenvalue of infinite multiplicity, the former accumulating to +∞ only. The principal term of the asymptotics of σdisc(H̃(a)), and of σ(H(a)+V), wherebandVdo not grow in some directions, is computed
Optimality Conditions for Multiobjective Programming with Generalized (I,ρ,θ)-Convex Set Functions

AbstractNecessary conditions for Pareto optimality in multiobjective programming with subdifferentiable set functions are established in Theorem 12 of H. C. Lai and L. J. Lin (J. Math. Anal. Appl.132,1988, 558–571). In this paper, we establish some sufficient conditions under which a feasible solution of such a problem will be Pareto optimal provided that a weaker convexity requirement is satisfied; for instance (I,ρ,θ)-convexity is assumed for both objective and constraint set functions. Some duality models are also discussed. Wolfe-type and Mond–Weir-type duality theorems are proved
Classical Solutions for a Generalized Euler Equation in Two Dimensions

AbstractIt is well known that the Euler equations in two spatial dimensions have global classical solutions. We provide a new proof which is analytic rather than geometric. It is set in an abstract framework that applies to the so-called lake and the great lake equations describing weakly non-hydrostatic effects of bottom topography on the motion of shallow water. The key ingredient is a new Lpestimate on the nonlinear term. The estimate is used to develop a global Hmtheory for bounded domains in R2which is similar in spirit to a 1975 paper by R. Temam. It also provides explicit bounds on the Hmnorm which grow like exp(expt)
Finite-Dimensional Control of Parabolic PDE Systems Using Approximate Inertial Manifolds

AbstractThis paper introduces a methodology for the synthesis of nonlinear finite-dimensional output feedback controllers for systems of quasi-linear parabolic partial differential equations (PDEs), for which the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Combination of Galerkin's method with a novel procedure for the construction of approximate inertial manifolds for the PDE system is employed for the derivation of ordinary differential equation (ODE) systems (whose dimension is equal to the number of slow modes) that yield solutions which are close, up to a desired accuracy, to the ones of the PDE system, for almost all times. These ODE systems are used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed-loop system to follow up to a desired accuracy, a prespecified response for almost all times
Solvability Theorems for Linear Equations of Tricomi Type

AbstractFor a large class of linear mixed type partial differential equations, theorems on local and semi-global solvability are stated and rely on a prior result of the author about the non-trapping of null-bicharacteristics over compact sets and the theory of Hörmander. A global solvability result modulo smooth error is demonstrated for a slightly reduced class which includes the equations found most commonly in the literature. The key point is the proof of an additional global convexity property for the bicharacteristic flow which results from the construction of a type of convex hull. All of the results are independent of lower order terms as they depend only on the Hamiltonian system associated to the principal symbol. This independence allows for the interpretation of the results in the context of general relativity and semi-Riemannian geometry
On the Inequality of I. Schur

AbstractDenote by πnthe set of all real algebraic polynomials of degree at mostn. The classical inequality of I. Schur asserts that the transformed Chebyshev polynomialTn(x)=Tn(xcos(π/2n)) has the greatest uniform norm of its first derivative on [−1,1] among allf∈Sn, where[formula]Here we extend this result to thekth derivative by proving the inequality[formula]Fork≥2 we prove the same inequality in the larger class[formula]This extension is in the spirit of the refinement of the Markov inequality found by Duffin and Schaeffer
On Continuity Properties of the Modulus of Local Contractibility

AbstractLetMXbe the set of all metrics compatible with a given topology on a locally contractible spaceXand let for each triplez=(ρ,x,ε)∈MX×X×(0,∞), Δ(z) be the set of all positive δ such that the open δ-neighborhood ofxis contractible in the open ε-neighborhood ofxin metric ρ. We prove several continuity properties of the map Δ:MX×X×(0,∞)→(0,∞) and then, using a selection theorem for non-lower semicontinuous mappings, show that Δ admits a continuous singlevalued selection. Similar, but somewhat different properties are also demonstrated for the modulus Δnof localn-connectedness
Generalized Area Operators on Hardy Spaces

AbstractWe show that if 0<p<∞ then the operatorGf(ζ)=∫Γ(ζ)|f(z)|dμ/(1−|z|) maps the Hardy spaceHptoLp(|dζ|) if and only if μ is a Carleson measure. Here Γ(ζ) is the usual nontangential approach region with vertex ζ on the unit circle Γ(ζ)={z∈D:|1−z|≤1−|z|2}, and |dζ| is arclength measure on the circle. We also show that if 0<p≤1, β>0, and 1−βp>0 then the operatorGfmaps the Hardy–Sobolev spaceHpβintoLp(|dζ|) if and only if the functionGμ(ζ)=∫Γ(ζ)dμ/(1−|z|) belongs to the Morrey spaceLp,1−βp. In casep=1, this condition is equivalent to the condition that μ(T(I))≤C|I|1−βfor all arcsIcontained in the circle, whereT(I) is the tent overIcontained in the unit disk
On the Generalized Implicit Quasivariational Inequalities

AbstractIn the present paper, we introduce and study a new class of generalized nonlinear implicit quasivariational inequalities for set-valued mappings and construct some new iterative algorithms. We prove the existence of solutions for this class of generalized nonlinear implicit quasivariational inequalities for set-valued mappings without compactness and the convergence of iterative sequences generated by the algorithms. We also study a new perturbed iterative algorithm for solving a class of general strongly nonlinear quasivariational inequalities
Plasticity Models and Nonlinear Semigroups

AbstractThe evolution of an elastic-plastic material is modeled as an initial boundary value problem consisting of the dynamic momentum equation coupled with a constitutive law for which the hysteretic dependence between stress and strain is described by a system of variational inequalities. This system is posed as an evolution equation in Hilbert space for which is proved the existence and uniqueness of three classes of solutions which are distinguished by their regularity. Weak solutions are obtained in a very general situation, strong solutions arise in the presence of kinematic work-hardening or viscosity, and the solution is even more regular under a stability assumption connecting the constraint set with the divergence operator
Oscillation Results of Certain Higher Order Linear Differential Equations with Periodic Coefficients in the Complex Plane

Abstract1has given a characterization of homogeneous linear differential equations with certain analytic periodic coefficients which admits a solution with finite exponent of convergence. However, her method seems too general and in most cases too complicated for applications. We give, in this paper, a direct approach to the problem and obtain several such characterizations which do not seem to follow from those of Baesch. In particular, an explicit, necessary and sufficient condition to the problem is given for certain third order equations. The results again do not seem to follow from those of Baesch. Our method is based on that of8which in turn depends on basic representations of solutions given by Bank, Laine, and Langley
Exponential Energy Decay Estimate for the Solutions of Internally Damped Wave Equation in a Bounded Domain

AbstractHere we are concerned about the stability of the solution of internally damped wave equationy″=Δy+μΔy′ with small damping constant μ>0, in a bounded domain Ω inRnunder mixed undamped boundary conditions. A uniform exponential energy decay rateE(t)≤Me−βtE(0) whereM≥1 and β=μ/μ0, μ0being a quadratic function of μ, is proved for the solution of this type of boundary value problem. Earlier authors have considered the undamped wave equation with certain forms of damped boundary conditions proving similar and faster energy decay rates
Dynamics on Spaces of Compact Subsets with Application to Brain Modeling

AbstractLetEbe an open, bounded subset ofRnand let P(E) be the collection of all subsets ofE. The theory of random sets deals with random processes whose outcomes are elements of P(E). Due to the infinite-dimensional nature of P(E), this theory is very technical. In this note we introduce a finite dimensional class of compact subsets ofE,Kn(E), which is dense in P(E) yet sufficiently rich for many applications.We study dynamical systems on the spaceKn(E) by considering transformations τ:Kn(E)→Kn(E) which are constructed from image source data such as occur in the dynamics of the brain. In particular, we establish sufficient conditions for the existence of invariant measures onKn(E). Under certain conditions these measures are absolutely continuous. We attempt to give meaning to the notion of expansiveness in brain dynamics
Quadratic Pencil of Schrödinger Operators with Spectral Singularities: Discrete Spectrum and Principal Functions

AbstractIn this article we investigated the spectrum of the quadratic pencil of Schrödinger operatorsL(λ) generated inL2(R+) by the equation[formula]and the boundary condition[formula] whereU,Vare complex valued functions andUis absolutely continuous in each finite subinterval of R+. Discussing the spectrum, we proved thatL(λ) has a finite number of eigenvalues and spectral singularities with finite multiplicities, if the conditions [formula]and[formula] hold. It is shown that the principal functions corresponding to eigenvalues ofL(λ) are in the spaceL2(R+), and the principal functions corresponding to spectral singularities are in another Hilbert space, which containsL2(R+). Some results about the spectrum ofL(λ) have also been applied to radial Klein–Gordon and one-dimensional Schrödinger equations
On the Boundedness of Global Solutions of Abstract Semilinear Parabolic Equations

AbstractIn this paper we consider the question of the long time behavior of solutions of the initial value problem for evolution equations of the formdu/dt+Au(t)=F(u(t)) in a Banach space whereAis the infinitesimal generator of an analytic semigroup andFis a nonlinear function such that the initial value problem possesses nonglobal solutions for some initial data. We give sufficient conditions to ensure that global solutions [on [0,∞)] are necessarily bounded in a suitable norm. We give several examples of problems to which the abstract result applies
Finite-Dimensional H∞Control of a Class of Infinite-Dimensional Linear Time-Varying Systems

AbstractIn this paper, we consider H∞control of infinite-dimensional linear time-varying (IDLTV) systems whose homogeneous parts are of modal type, by using a finite-dimensional controller. The aim is to construct a finite-dimensional stabilizing controller for the IDLTV system that makes the norm of the closed-loop operator less than a given positive number δ. To this end, a finite-dimensional model is firstly derived for the IDLTV system, and a stabilizing controller that makes the norm of the closed-loop operator less than another positive number γ(<δ) is constructed for the finite-dimensional model. Then it is shown that a controller, which consists of a residual mode filter and the controller mentioned above, becomes a finite-dimensional stabilizing controller for the given IDLTV system such that the closed-loop operator has norm less than δ, if the order of the residual mode filter is chosen sufficiently large
Pressure-based Semi-analytical Solution of Radial Stokes Flows

AbstractA semi-analytical methodology is proposed and implemented to approximate the solution of the partial differential equations that model the radial Stokes flow between two parallel disks. Two traditional approaches for solving partial differential equations of mathematical physics, namely, eigenfunction expansions and Green's function representations, are used to obtain directly the pressure distribution for a variety of inflow boundary conditions. We combine these standard techniques with a functional analytic framework to propose a proper representation for the pressure. This methodology has been implemented first in a symbolic software (Mathematica®) and then numerically. We provide both validation of the methodology when an exact solution exists and comparison with a spectral element solver for a test case where inflow boundary conditions render the existence of a recirculation zone close to the entrance of the domain. We briefly discuss how to extend the methodology to handle the Navier–Stokes equation in radial flow—a subject that will be presented in detail in a following paper
Global Bifurcations of Solutions of Elliptic Differential Equations

AbstractThe aim of this article is to prove global bifurcation theorems forS1-equivariant potential operators of the form “compact perturbation of identity.” As an application we prove that components of the set of nontrivial solutions of system[formula]which bifurcate from the set of trivial solutions are unbounded in suitably chosen space. The important point to note here is that our results apply only to systems with an even number of equations. As the main tool we use degree theory forS1-equivariant potential operators constructed by the author in [16] and developed in [17]
Parabolic Systems in Unbounded Domains I. Existence and Dynamics

AbstractIn this paper we investigate the existence, uniqueness, and asymptotic behavior of a solution for a class of coupled nonlinear parabolic equations in a general unbounded domain that includes the whole space Rn, the exterior of a bounded domain, and a half space in Rn. The asymptotic behavior of the solution is with respect to a pair of quasi-solutions of the corresponding elliptic system, and when these two quasi-solutions coincide the solution of the parabolic system converges to a unique solution of the elliptic system