pinecone/core-2020-05-10-deduplication huggingface dataset:
Simply Connected Compact Subsets of the Plane

AbstractIf the condition of path-connectedness in the definition of simple-connectedness is relaxed, the concept of a set with no holes arises. A natural rank function is defined on the set NH of all compact sets with no holes in the plane. The rank function is obtained by associating a sequence of well-founded trees with each set in NH. It provides a natural measure of the complexity of these sets. It is also shown that the rank function is unbounded in ω1 on the set SC of all compact simply connected sets in the plane. From this it will follow that NH and SC are not Borel subsets of the Polish space K(R2) of all compact sets in the plane
Capacitary Estimates for the Scattering Phase

AbstractLet H≔−12Δ acting in L2(Rn), n≥3. Given a compact obstacle K, let HY be the operator obtained from H by imposing Dirichlet boundary conditions on K; here Y is the complement of K. Let θY(λ) be the scattering phase for the scattering system {HY,H;J}, where J denotes restriction to Y. We show that the variation in θY(λ) (modulo integer multiples of π/2) as K varies may be estimated in terms of the capacity of the symmetric difference of the obstacles. Suppose that K has smooth boundary. Let Hβ≔H+β1K (β>0) and θβ(λ) be the scattering phase for the scattering system {Hβ,H}. We show furthermore that θβ(λ) converges to θY(λ) (again modulo integer multiples of π/2) roughly at the rate β−1/2 as β→∞
Existence of Bounded Solutions of a Second-Order System with Dissipation

AbstractIn this article, we study the following second-order system of ordinary differential equations with dissipationu″+cu′+dAu+kHu=Pt,u∈Rn,t∈R,where c, d, and k are positive constants, H: Rn→Rn is a locally Lipschitz function, and P: R→Rn is a continuous and bounded function. A is a n×n matrix whose eigenvalues are positive. Under these conditions, we prove that for some values of c, d, and k this system has a bounded solution which is exponentially asymptotically stable. Moreover; if P(t) is almost periodic, then this bounded solution is also almost periodic. These results are applied to the spatial discretization of very well-known second-order partial differential equations
Composition Operators Belonging to Operator Ideals

AbstractWith each composition operator Cϕf=f∘ϕ acting between classical Bergman spaces Ap and Aq, 1≤p, q<∞, we associate in a natural way a sequence of non-negative scalars ξk, k∈N. Our main result states that the composition operator belongs to a prescribed quasi-Banach ideal A if and only if the diagonal operator Dξ: lp→lq given by (ak)↦(akξk) belongs to A. Conversely, we show that for any λ>max(2/q−2/p,0) and κ∈R there exists a composition operator Cϕ: Ap→Aq whose associated sequence is (equivalent to) k−λ(logk)κ, k≥2. This provides an effective tool to separate quasi-Banach ideals by means of composition operators. The results can be generalized to standard weighted Bergman spaces and beyond
Invexity and the Kuhn–Tucker Theorem

AbstractIt is pointed out that Type 1 invex functions are the most general class of functions relevant to necessary and sufficient conditions for Kuhn–Tucker optimality in nonlinear programming. Linear programming duality is used to show an equivalence between the concept of invexity and the Kuhn–Tucker conditions for optimality. The invexity kernel η and the Lagrange multiplier y in the Kuhn–Tucker theory are dual variables. The Kuhn–Tucker conditions are necessary conditions for optimality provided that certain constraint qualifications apply. A particular result given here is that invexity in itself constitutes an appropriate constraint qualification
A Kinetic Model for Vehicular Traffic: Existence of Stationary Solutions

AbstractIn this paper the kinetic model for vehicular traffic developed by Klar and Wegener (SIAM J. Appl. Math.59 (1999), 983–1001) is considered, and theoretical results for the space homogeneous kinetic equation are presented. In particular, the stationary kinetic equation is considered. An investigation of this equation leads to a boundary value problem for an ordinary differential equation. Existence of the solution and some properties are proved. A numerical investigation of the stationary equation is included
Spectral Properties of a Streaming Operator with Diffuse Reflection Boundary Condition

AbstractIn this paper, the spectrum of a streaming operator with diffuse reflection boundary condition arising in transport theory is observed in L1 space. We associate this unbounded operator with bounded integral operators which are determined by the boundary condition. It is shown that the spectrum of the streaming operator is completely determined by these integral operators and consists of countable isolated eigenvalues, each of which has finite algebraic multiplicity. Furthermore, under the assumption that the diffusion on the boundary is of Maxwell type, it is proved that the streaming operator has only one real eigenvalue which is simple and that each of the complex eigenvalues has geometry multiplicity one. A formula for computing these eigenvalues is also derived, by means of which the algebraic multiplicity of the complex eigenvalues is discussed
The Generalized Quasi-linearization Method for Reaction Diffusion Equations on an Unbounded Domain

AbstractThe method of generalized quasi-linearization has been well developed for ordinary differential equations. In this paper, we extend the method of generalized quasi-linearization to reaction diffusion equations on an unbounded domain. The iterates, which are solutions of linear equations starting from lower and upper solutions, converge uniformly and monotonically to the unique solution of the nonlinear reaction diffusion equation in an unbounded domain. Initially an existence theorem for the linear nonhomogeneous reaction diffusion equation in an unbounded domain has been proved under improved conditions. The quadratic convergence has been proved by using a comparison theorem of reaction diffusion equations with ordinary differential equations. This avoids the computational complexity of the quasi-linearization method, since the computation of Green's function at each stage of the iterates is avoided
Grid Approximation for the Solution of the Singularly Perturbed Heat Equation with Concentrated Capacity

AbstractMixed boundary-value problems for the heat equation are considered. The diffusion coefficient is small, i.e., it is multiplied by a parameter ε∈(0,1]. The coefficients in the equation are discontinuous at some points. We confine our attention to the case in which at these points heat capacity is concentrated. First, some properties of the solution (regularity, large-time behavior, estimates for the derivatives) are studied. Then, we show that it is impossible to achieve uniform convergence on the discrete maximum norm of difference schemes on classical meshes. For the problems described we construct grid approximations based on standard weighted difference schemes and condensed (Shishkin's) mesh. We prove the ε-uniformly O(N−2log2N+τmσ) order of convergence for the error in the discrete solution. Here N is the number of the nodes in the space mesh, τ is the time step, and 1≤mσ≤2 is a weight parameter
Triple Positive Solutions and Dependence on Higher Order Derivatives

AbstractIn this paper, we consider the Lidstone boundary value problem, y(2m)(t)=f(y(t),…,y(2j)(t),…y(2(m−1))(t)), 0≤t≤1, y(2i)(0)=0=y(2i)(1), 0≤i≤m−1, where (−1)mf>0. Growth conditions are imposed on f and inequalities involving an associated Green's function are employed which enable us to apply the Leggett–Williams Fixed Point Theorem to cones in ordered Banach spaces. This in turn yields the existence of at least three positive symmetric concave solutions. The emphasis here is that f depends on higher order derivatives
On Integral Operators

AbstractLet fn(z)=z/(1−z)n+1, n∈No, and f(−1)n be defined such that fn∗f(−1)n=z1−z, where ∗ denotes convolution (Hadamard product). Let f be analytic in the unit disc E. We define a new operator Inf=f(−1)n∗f which is analogous to one defined by Ruscheweyh. Using this operator, the classes M*(n) are defined. A function f, analytic in E, is in M*(n) if and only if Inf is close-to-convex. The properties of f∈M*(n) are discussed in some detail. It is shown that M*(n)⊂M*(n+1) for n∈No and for n=0,1, M*(n) consists entirely of univalent functions. Closure properties of some integral operators defined on M*(n) are also given
On Certain Families of Generalized Nörlund Methods and Power Series Methods

AbstractWe consider families {Aα} of summability methods which have similar features in their construction as the family of Cesàro methods (C,α). Abel-type power series methods will be added to those families and inclusion and Tauberian theorems will be proved. The Tauberian and inclusion theorems proved in the papers [R. Kiesel, Math. Z.214 (1993), 273–286; R. Kiesel and U. Stadtmüller, Canad. J. Math.46, (1994), 982–994; R. Kiesel, Math. Nachr.176 (1995), 129–138] for certain families of generalized Nörlund methods and Abel-type power series methods are combined with the results of the papers [A. Tali, Tartu Riikl. Ül. Toimetised960 (1993), 117–138; M. Müristaja and A. Tali, Acta et Comment. Univ. Tartuensis Math.1 (1996), 93–103] in order to extend them, to improve the arguments used in [R. Kiesel, Math. Z.214 (1993), 273–286; R. Kiesel and U. Stadtmüller, Canad. J. Math.46 (1994), 982–994] and to get more general convex families of summability methods. The main theorems deal with two-parameter families {Aαβ} of generalized Nörlund methods (Theorems 3.1–3.3)
Finite-Rank Perturbations of the Dirac Operator

AbstractIn this paper we determine those relativistic point interactions for which renormalization of the coupling constant occurs when the corresponding potentials are approximated by local, short-range perturbations of the free Dirac operator in one dimension. We also find a general formula for the renormalization constant for the entire four-parameter family of relativistic point interactions. Our results, which unify and extend earlier work on relativistic point interactions, include perturbations of the Dirac operator by finitely many δ or δ′ potentials, as well as relativistic scalar and magnetic point interactions. In addition, we show that similar methods may be used to define more general perturbations of the Dirac operator, including time-dependent ones. These operators are shown to depend continuously on the perturbation in the sense of strong resolvent convergence
Solutions to Nonlinear Partial Differential Equations from Symmetry-Enhancing and Symmetry-Preserving Constraints

AbstractWe show how solutions to practical partial differential equations can be found by classical symmetry reductions of a larger system of equations including constraints that result in the enlarged system having a larger symmetry group or an identical symmetry group compared to the original target equation. Symmetry-enhancing constraints provide additional similarity solutions beyond those of the standard Lie algorithm for scalar equations. Symmetry-preserving constraints enable solutions to be found more easily, after reduction of variables. Examples are given for the cylindrical boundary layer equations and for a Navier–Stokes formulation of Schrödinger wave mechanics
Some Families of Generating Functions for the Jacobi and Related Orthogonal Polynomials

AbstractBy making use of the familiar group-theoretic (Lie algebraic) method of Louis Weisner (1899–1988), many authors recently proved various single- and multiple-series generating functions for the so-called extended Jacobi polynomials. The main object of the present sequel to these earlier works is to show how easily each of such generating functions can be derived from the corresponding known result for the classical Jacobi polynomials. Many general families of bilinear, bilateral, or mixed multilateral generating functions for the Jacobi and related orthogonal polynomials, which are seemingly relevant to the present investigation, are also considered here
Spectral Theory of Second-Order Vector Difference Equations

AbstractThis paper is concerned with spectral problems of second-order vector difference equations with two-point boundary value conditions, where the matrix-valued coefficient of the leading term may be singular. A concept of self-adjointness of the boundary value conditions is introduced. The self-adjointness of the corresponding difference operator is discussed on a suitable admissible function space, and fundamental spectral results are obtained. The dual orthogonality of eigenfunctions is shown in a special case. Rayleigh's principles and the minimax theorems in two linear spaces are given. As an application, a comparison theorem for eigenvalues of two Sturm–Liouville problems is presented
Stochastic Controls with Terminal Contingent Conditions

AbstractThis paper considers a nonlinear stochastic control problem where the system dynamics is a controlled nonlinear backward stochastic differential equation and the state must coincide with a given random vector at the terminal time. A necessary condition of optimality in the form of a global maximum principle as well as a sufficient condition of optimality are presented. The general result is also applied to backward linear-quadratic control problem and an optimal control is obtained explicitly as a feedback of the solution to a forward–backward equation. Finally, a nonlinear problem with additional integral constraints is discussed and it is shown that the duality gap is zero under the Slater condition
Characterization of L1-Closed Decomposable Sets in L∞

AbstractIn this paper the set of essentially bounded measurable selections of a measurable set-valued map is considered. One of the main results states that only those subsets of L∞ that are L1-closed in L∞ and decomposable can be represented in this way. The connection between topological and convexity properties of the set-valued map and its L∞-selection set is also studied in details. The adjacent and Clarke's tangent cones to L1-closed and decomposable sets and their polar cones are described in terms of the corresponding set-valued map as well
On the Characterization of the Fréchet Derivative with Respect to a Lipschitz Domain of the Acoustic Scattered Field

AbstractIn this paper, we consider the Fréchet differentiability with respect to a given domain of the solution of direct acoustic scattering problems with sound-soft, sound-hard, or lossy boundary conditions. We propose a new proof for characterizing the derivative of the acoustic scattered field as the solution of a particular direct scattering problem. This proof assumes that the boundary of the scatterer is only Lipschitzian and therefore extends this well-known characterization to obstacles with rough surfaces. Our result can be of practical interest because the characterization of the derivative of the acoustic scattered field has a great potential for the solution of inverse acoustic scattering problems by Newton-type methods
On the Existence of Solutions to Non-parametric Fairing Problems

AbstractLet s be the graph of a function s: Ω→R over the bounded set Ω⊂R2. The fairness of s can be measured by means of functionals depending on the principal curvatures κ1,κ2 and the area a of s. Here we consider functionals of type Fψ,p(s)≔∫Ω(κ21+κ22)pndμ+ψ(a), where ψ is a non-negative lower semi continuous function, n≔(1+‖∇s‖2)1/2, and 1<p<∞. Given s0∈H1,∞(Ω) and some constant M>0, we show that there exists a function s*∈H2,2p(Ω) which minimizes Fψ,p on the set {s∈H2,2p(Ω):‖s−s0‖1,∞≤M}. The set of competing functions can also be restricted by additional constraints involving in particular point and gradient data
New Closedness Results for Efficient Sets in Multiple Objective Mathematical Programming

AbstractThere are various theoretical, algorithmic, and practical reasons for developing necessary and sufficient conditions for the efficient set of a multiple objective mathematical programming problem (P) to be closed. Yet only a small number of results of this type, limited to special cases of Problem (P), have been developed. In this article we present a necessary condition and several sufficient conditions for the closedness of the efficient sets of more general cases of Problem (P) than have heretofore been studied. Our approach relies in part upon generalizing the concepts of quasi-concavity and strict quasi-concavity for real-valued functions to vector-valued functions. Our approach also relies upon some new characterizations of an efficient solution for Problem (P) that we develop in the article
A Note on Determining the Range of Optimality for Decision Analysis Problems

AbstractThe method for determining the range over which a state probability may change without affecting the optimal solution is well documented in introductory management science text books for problems in which there are only two states of nature. For more realistic problems in which there are more than two states of nature, the methodology does not work. In this paper, we extend the approach taken for the two state problem to handle n states and present a simple algorithm for determining the range
Two-Scale Homogenization of Non-Linear Degenerate Evolution Equations

AbstractUsing the notion of two-scale convergence developed by Allaire, the homogenization of a degenerate non-linear evolution equation with periodically oscillating coefficients is presented. A two-scale homogenized system is obtained as the limit of the periodic problem. Monotone operator methods and two-scale convergence are employed to show that the solutions of the periodic problem converge to the unique solution of the homogenized system. Homogenized initial conditions are also obtained and the sense in which they hold for the homogenized initial value problem is made specific
Rational Filter Wavelets

AbstractIn this paper, we introduce two types of rational filters for the construction of wavelets. The rational filters are a natural development of the polynomial filters. We use rational filters to derive a large family of the wavelets which can include Daubechies wavelets and Battle–Lemarié wavelets. Especially the II-type rational filter wavelets among the family have linear phases. Furthermore, we analyze the regularity of the rational filter wavelets and estimate their regularity indices. Some examples are also given
Boundary and Distributed Controllability of the Damped Wave Equation: Reduction of Control Time

AbstractWe consider the zero controllability problem for the distributed parameter system governed by the damped wave equationutt+2dxut−1ρ(x)ddxp(x)dudx+qxu=Fx,t,where x∈[0,a], a<∞ with the boundary conditionsux+kut0=0,ux+huta=ϕt,h,k∈C∪∞.We consider two cases: (a) F(x,t)=g(x)f(t) and ϕ(t) is not identically zero; (b) ϕ(t)=0 and F(x,t)=∑mj=1g(j)(x)fj(t). The functions f(t), fj(t), and ϕ(t) are the distributed and boundary controls respectively. The problem with a nontrivial function f(t) and ϕ(t)=0 was studied in our recent (joint) work using the spectral decomposition method. The approach was based on the recent results of one of the authors on the spectral analysis of the nonselfadjoint dynamics generator of the system. It was shown that the system is controllable in a time T1 which is equal to twice the time it takes for a wave to travel between the ends of the interval [0,a]. In the present paper, we investigate the conditions on the force profile functions g(x),g(j)(x), j=1,2,…,m, under which the control time can be reduced. Namely, in case (a), we give the sufficient conditions on g(x) under which the combination of two controls f(t) and ϕ(t) allows us to reduce the control time twice. In case (b), we give the sufficient conditions on the force profile functions g(j)(x), j=1,2,…,m, under which the control time is equal to T1/m. In the proofs of these results, we use the generalization of the approach suggested by Russell for the boundary controllability of the undamped wave equations
An Initial Boundary Value Problem for Parabolic Differential-Operator Equations

AbstractIn this paper we give, for the first time, an abstract interpretation of such initial boundary value problems for parabolic equations that a part of boundary value conditions contains also a differentiation on the time t. Initial boundary value problems for parabolic equations are reduced to the Cauchy problem for a system of parabolic differential-equations [see below problems (1)–(3)]. A solution of this system is not a vector-function but one function. At the same time, the system is not overdetermined
Optimal Vibration Quenching for an Euler–Bernoulli Beam

AbstractThe quenching of the vibration of an Euler–Bernoulli beam under tension with general linear homogeneous boundary conditions is studied using a distributed control. A method for determining the control that quenches a finite number of modes is given and it is shown that the method can be extended theoretically to determine a control to quench all modes of the vibration. In general there is more than one control that can be used to quench the same modes. It is shown that of all controls that quench specified modes of vibration at a given time and are square integrable the method described yields the unique control whose mean square is minimum. A method is given for determining how many modes are sufficient to be quenched if the residual position and velocity of the beam are both to remain within a restricted band after the control is removed. Numerical results are given in graphical form
Integration and Microlocal Analysis in Colombeau Algebras of Generalized Functions

AbstractWe study integration and Fourier transform in the Colombeau algebra Gτ of tempered generalized functions using a general damping factor. This unifies different settings described earlier by J. F. Colombeau, M. Nedeljikov, S. Pilipović, and M. Damsma (for a simplified version). Further we prove characterizations of regularity for generalized functions in two situations: compactly supported or in the image of S′ inside Gτ. Finally we investigate the notion of wave front set in the Colombeau algebra G(Ω), Ω an open subset of Rn, and show that it is in fact independent of the damping measure used for Fourier transform
Truncated Trigonometric Moment Problems and Determinate Measures

AbstractWe consider the problem of deciding whether or not a given positive Borel measure μ on the n-dimensional torus Tn is S-determinate, which means that μ is entirely determined by the subset of its Fourier coefficients {μ̂(k), k∈S}, where S is an arbitrary subset of Zn containing 0 and with the property that −k∈S whenever k∈S. We give necessary and sufficient conditions for S-determinacy and show that, when S is a finite set, the validity of this property can be verified algebraically by an algorithmic procedure involving trigonometric polynomials with spectrum in S vanishing on the support of μ and satisfying certain positivity properties
Smooth Attractors Have Zero “Thickness”

AbstractA finite-dimensional global attractor A can be embedded, using some linear map L, into a Euclidean space Rk of sufficiently high dimension. The Hölder exponent of L−1 depends upon k and upon τ(A), the “thickness exponent” of A. We show that global attractors which are uniformly bounded in the Sobolev spaces Hs for all s>0 have τ(A)=0. It follows, using a result of B. R. Hunt and V. Y. Kaloshin, that the Hölder constant of the inverse of a typical linear embedding into Rk (or rank k orthogonal projection) can be chosen arbitrarily close to 1 if k is large enough