pinecone/core-2020-05-10-deduplication huggingface dataset:
Generalized Szegö Theory in Frequency Analysis

AbstractA sequence of positive Borel measures with infinite support on the unit circle is given, converging in the weak star topology to a measure with support consisting of a finite numbern0of mass points. The sequences of monic rational functions orthogonal with respect to inner products determined by the measures are studied, with the focus being on the asymptotic behavior of the zeros. The main result is twofold: For a fixedn>n0,n0of the zeros (“interesting zeros”) tend to the mass points of the limiting measure, while the remainingn−n0zeros (“uninteresting zeros”) may not converge at all. However, by considering subsequences, limits may be obtained and the remaining zeros (and their limits) will be located in a closed disk |z|≤R<1, provided that a certain boundedness condition for the norm of the monic orthogonal functions is satisfied. As an application it is proved that this theory may be applied in frequency analysis, where the interesting zeros tend to the frequency points while the uninteresting zeros are bounded away from the unit circle
Weak Convergence and Non-linear Ergodic Theorems for Reversible Semigroups of Non-Lipschitzian Mappings

AbstractLetGbe a semitopological semigroup. LetCbe a closed convex subset of a uniformly convex Banach spaceEwith aFréchetdifferentiable norm and T={Tt:t∈G} be a continuous representation ofGas asymptotically nonexpansive type mappings ofCinto itself such that the common fixed point setF(T) of (T) inCis nonempty. We prove in this paper that ifGis right reversible, then for every almost-orbitu(·) of T,[formula]consists of at most one point. Further,[formula]is nonempty for eachx∈Cif and only if there exists a nonexpansive retractionPofContoF(T) such thatPTs=TsP=Pfor alls∈GandP(x) is in the closed convex hull of {Tsx:s∈G},x∈C. This result is applied to study the problem of weak convergence of the net {u(t):t∈G} to a common fixed point of T
On the Magnetostatic Integral Operator for Ellipsoids

AbstractThe magnetostatic integral operator in R3is of considerable interest in both scattering and potential theory. As well as for its scalar pendant, the electrostatic integral operator, its eigenvalues, and its eigenfunctions play a fundamental role in the theory of permanent magnetization. While the eigenvalues of both operators are closely related, explicit expressions for the eigenfunctions of the magnetostatic operator are known only in the case when the underlying surface is a sphere. For the cases of ellipsoids and spheroids, we present explicit expressions for the eigenvalues in terms of Lamé functions and associated Legendre functions and expressions for eigenfunctions in terms of surface ellipsoidal and surface spheroidal harmonics, respectively. As an application the permanent magnetization of ellipsoids is discussed
On the Complex Oscillation of Some Linear Differential Equations

AbstractWe treat the linear differential equation (∗)f(k)+A(z)f=0, wherek≧2 is an integer andA(z) is a transcendental entire function of order σ(A). It is shown that any non-trivial solution of the equation (∗) satisfies λ(f)≧σ(A), where λ(f) is the exponent of convergence of the zero-sequence off, under the conditionKN̄(r,1/A)≦T(r,A),r∉Efor aK>2kand an exceptional setEof finite linear measure. The second order equationf″+(eP1(z)+eP2(z)+Q(z))f=0, whereP1(z),P2(z) are non-constant polynomials andQ(z) is an entire function, is also studied
A Sharper Condition for the Solvability of a Three-Point Second Order Boundary Value Problem

AbstractLetf: [0,1]×R2→Rbe a function satisfying Caratheodory's conditions ande(t)∈L1[0,1]. Let η∈(0,1), α∈R, α>1, αη≠1 be given. This paper is concerned with the problem of existence of a solution for the three-point boundary value problem - FORMULA OMITTED -This problem was studied earlier by Gupta, Ntouyas, and Tsamatos when α≤1 and when α>1 with αη<1. In the general case this problem was studied by Gupta as a multi-point boundary value problem. In this paper sharper existence conditions are obtained for the solvability of the above boundary value problem in the general case
Singularly Perturbed Differential/Algebraic Equations. I: Asymptotic Expansion of Outer Solutions

AbstractSingularly perturbed nonlinear differential/algebraic equations (DAE's) are considered, which are decomposed into two auxiliary problems, called the outer and inner problems, respectively. The structure of solutions of the singularly perturbed DAE's is determined by the outer and inner solutions, both of which are proved to exist. Asymptotic expansions for outer and inner solutions are obtained and proved to be uniformly convergent. This generalizes known results about asymptotic expansions of singularly perturbed ordinary differential equations.The presentation of this work is separated into two parts because of the limitation of space. The first part concerns the derivation of outer and inner problem, and the existence and asymptotic expansion of outer solutions, while the second part mainly focuses on the inner problem
Quasi-Diagonalization of Linear Impulsive Systems and Applications

AbstractThis work is concerned with the quasi-diagonalization of the impulsive linear systemx′=A(t)x,x(t+k)=Bkx(t−k), where the functionA(t) is bounded and piecewise uniformly continuous, and (Bk)∞k=1is a bounded sequence of impulse matrices. Let Λ(t) andDkbe the diagonal matrices of eigenvalues ofA(t) andBk. We prove that there exists a transformationx=T(t)ywhich reduces this impulsive system toy′=[Λ(t)+F(t)+Δ(t,σ)+R(t)]y,y(tk)=[Dk+Δk]y(t−k), whereF(t), Δ(t,σ), and (Δk)∞k=1are functions with small norms inL1,L∞, andl∞, respectively, andR(t)=−T−1(t)T′(t). An estimate for ∫tsR(u)duis given. We apply these results to the problem of the existence of periodic solutions of impulsive systems and to the problem of stability of the singularly perturbed linear impulsive system εx′=A(t)x,x(t+k)=Bkx(t−k)
On Neutral Functional–Differential Equations with Proportional Delays

AbstractIn this paper we develop a comprehensive theory on the well-posedness of the initial-value problem for the neutral functional-differential equationy′(t)=ay(t)+∑i=1∞biy(qit)+∑i=1∞ciy′(pit),t>0,y(0)=y0,and the asymptotic behaviour of its solutions. We prove that the existence and uniqueness of solutions depend mainly on the coefficientsci,i=1,2,…, and on the smoothness of functions in the solution space. As far as the asymptotic behaviour of analytic solutions is concerned, thecihave little effect. We prove that if Rea>0 then the solutiony(t) either grows exponentially or is polynomial. The most interesting result is that if Rea≤0 anda≠0 then the asymptotic behaviour of the solution depends mainly on the characteristic equationa+∑i=1∞biqλi=0.These results can be generalized to systems of equations. Finally, we present some examples to illustrate the change of asymptotic behaviour in response to the variation of some parameters. The main idea used in this paper is to express the solution in either Dirichlet or Dirichlet–Taylor series form
Overtaking Optimal Solutions for Convex Lagrange Problems with Time Delay

AbstractIn this paper we study infinite horizon convex problems of Lagrange in which hereditary effects in the state are present. The objective functional in these problems is an improper integral defined over the time interval [0,+∞). Dictated by applications arising in mathematical economics we do not assumea priorithat these improper integrals converge and therefore we consider overtaking optimal solutions. This concept arose in the economics literature in the 1960s. We extend known results for the analogous nondelay problem providing sufficient conditions for the existence of an overtaking optimal solution. Our proof rests on the assumption that a particular delay differential inclusion enjoys a strong asymptotic stability property (called property (S)). Examples are given to indicate the applicability of these results
Topological Properties of the Approximate Subdifferential

AbstractThe approximate subdifferential introduced by Mordukhovich has attracted much attention in recent works on nonsmooth optimization. Potential advantages over other concepts of subdifferentiability might be related to its nonconvexity. This is motivation to study some topological properties more in detail. As the main result, it is shown that any weakly compact subset of any Hilbert space may be obtained as the Kuratowski–Painlevé limit of approximate subdifferentials from a one-parametric family of Lipschitzian functions. Sharper characterizations are possible for strongly compact subsets. As a consequence, in any Hilbert space the approximate subdifferential of a suitable Lipschitzian function may be homeomorphic (both in the strong and weak topology) to the Cantor set. Further results relate the approximate subdifferential to specific topological types, to the one-dimensional case (which is extraordinary in some sense), and to the value function of a C1-optimization problem
Harmless Delays for Permanence in a Class of Population Models with Diffusion Effects

AbstractThis paper studies a class of time-delay reaction–diffusion systems modeling the dynamics of single or interacting populations. In the logistic equation, we prove that when the magnitude of the instantaneous term is larger than that of the delay terms, the population growthuhas the same asymptotic limit as in the case of no delay. For the predator–prey model, a condition on the interaction rates is given to ensure the permanence effect in the ecosystem regardless of the length of delay intervals. A permanence condition is also obtained in theN-species competition system with time delays. It is shown that when the natural growth rate (a1,a2,…,aN) is in an unbounded parameter set Λ, the reaction–diffusion system has a positive global attractor. Finally, long-term behavior of the solutions for those time-delay systems is numerically demonstrated through finite-difference approximations and compared with the corresponding systems without delays
Entire Solutions of Higher Order Abstract Cauchy Problems

AbstractWe consider the abstract Cauchy problem for the higher order differential equationu(n)(t)+∑i=0n−1Aiu(i)(t)=0,t≥0 (∗)u(j)(0)=uj,0≤j≤n−1,(∗)whereA0,…,An−1are closed linear operators in a complex Banach spaceE. It is shown that if there exist θ∈(0,π/2), φ∈(−π,π],r>0 and a positive integerhsuch that λnRλand λ−hAkRλ(0≤k≤n−1) are bounded in {λ∈C; |arg(e−iφλ)|≤π/2+θ, |λ|≥r}, whereRλ=(λn+∑n−1i=0λiAi)−1, then there exists a dense subsetGofEnsuch that for every initial value (u0,…,un−1)∈G, (∗) has a unique entire solution, that is, a solution which can be extended analytically to the whole complex plane. As a corollary, we get that the Cauchy problems for the parabolic higher order linear differential equations have a unique entire solution for a dense set of initial data
General Equilibrium in Asset Markets With or Without Short-Selling

AbstractIn this paper we extend the results of Cheng [J. Math. Econom.20(1991), 137–152] and Brown and Werner [Rev. Economic Studies62(1995), 101–114] on the existence of equilibrium in infinite dimensional asset markets. We do not assume that each agent's preferred sets have a uniform direction of improvement, but assume that the preferred sets of attainable allocations have nonempty interiors. We then deduce existence theorems for asset markets without short-selling and for the capital asset pricing model
Positive Solutions and Asymptotic Behavior of Delay Differential Equations with Nonlinear Impulses

AbstractConsider the delay differential equation (DDE) with nonlinear impulsesẋ(t)+∑i=1npi(t)x(t−τi)=0,t≠tj,t≥t0,x(t+j)−x(tj)=Ij(x(tj)),j=1,2,…,(∗)wherepi∈C([t0,∞),R+), τi≥0 fori=1,2,…,n, andIj∈C(R,R) forj=1,2,…. The purpose of this paper is to obtain a necessary and sufficient condition for the existence of positive solutions of DDE without impulses and to establish a kind of order persistence of solutions of Eq. (∗) and criteria of the asymptotic behavior of Eq. (∗), which can be used to improve and develop some of the known results in the literature
The Difference between Finite Dimensional Linear Programming Problems and Infinite Dimensional Linear Programming Problems

AbstractThis paper studies the difference between finite-dimensional linear programming problems and infinite dimensional linear programming problems. We discuss a special class of continuous linear programming problems. We develop the structure of extreme points of feasible region for this problem. Under some conditions we can characterize all extreme points of this problem. We show that under some conditions the optimal value for this problem may be finite but there is no optimal solution for it. Finally, we construct an example for this problem which has an optimal solution but all extreme points for this problem are not optimal solutions
A Note on “Uniqueness of Limit Cycles in a Liénard-Type System”

Abstract5proposed a theorem to guarantee the uniqueness of limit cycles for the generalized Liénard systemdx/dt=h(y)−F(x),dy/dt=−g(x). We will give a counterexample to their theorem. It will be shown that their theorem is valid only ifF(x) is monotone on certain intervals. For this case we give a shorter proof and we also show that the limit cycle is hyperbolic. If the condition on the monotonicity ofF(x) is violated then we need an additional condition to guarantee the uniqueness of the limit cycle. Examples are provided to illustrate our results
Remarks on the Analytic Implicit Function Theorem

AbstractIn this paper, we focus attention on the problem of to what extent abstract Banach space concepts can be utilized in order to simplify well-known formulas for the power series expansion of an implicitly defined holomorphic mapping in the finite-dimensional case. The formulas obtained in the previous literature (roughly speaking, extensions of the Bürmann–Lagrange formula) are extremely lengthy and involved. Our main purpose here is to give a more condensed and more comprehensible version of such a formula. The basic idea of the proof consists in combining the technique of multiple residues with certain elementary concepts (multilinear algebra) from differential calculus in Banach spaces
Optimal Locations and Inner Products

AbstractIn a normed spaceX, we consider objective functions which depend on the distances between a variable point and the points of certain finite setsA. A point where such a function attains its minimum onXis generically called an optimal location. In this paper we obtain characterizations of inner product spaces with properties connecting optimal locations and the convex hull ofAor barycenters of points ofAwith well chosen weights. We thus generalize several classical results about characterization of inner product spaces
Admissible Wavelets Associated with the Affine Automorphism Group of the Siegel Upper Half-Plane

AbstractLetP=NAMbe the minimal parabolic subgroup ofSU(n+1,1), which can be regarded as the affine automorphism group of the Siegel upper half-planeUn+1,Palso acts on the Heisenberg groupHn, the boundary ofUn+1. ThereforePhas a natural representationUonL2(Hn). We decomposeL2(Hn) into the direct sum of the irreducible invariant closed subspaces underU. The restrictions ofUon these subspaces are square-integrable. We give the characterization of the admissible condition in terms of the Fourier transform and define the wavelet transform with respect to admissible wavelets. The wavelet transform gives isometric operators from the irreducible invariant closed subspaces ofL2(Hn) toL2(P)
On a System of Nonlinear PDEs with Temperature-Dependent Hysteresis in One-Dimensional Thermoplasticity

AbstractIn this paper, we develop a thermodynamically consistent description of the uniaxial behaviour of thermoelastoplastic materials that are characterized by a constitutive law of the form σ(x,t)=P[ε,θ(x,t)](x,t), where ε,σ,θ denote the fields of strain, stress, and absolute temperature, respectively, and where {P[·,θ]}θ>0denotes a family of (rate-independent) hysteresis operators of Prandtl–Ishlinskii type, parametrized by the absolute temperature. The system of state equations governing the space-time evolution of the material are derived. It turns out that the resulting system of two nonlinearly coupled partial differential equations involves partial derivatives of hysteretic nonlinearities at different places. It is shown that an initial-boundary value problem for this system admits a global weak solution. The paper can be regarded as a first step towards a thermodynamic theory of rate-independent hysteresis operators depending on temperature
Global Solvability of Nonlinear Diffusion Equations with Forcing at the Boundary

AbstractUnder the influence of a sufficiently “weak” nonlinear source term, it is by now well known that a degenerate diffusion equation is globally solvable. A similar result is known when the nonlinear source is present as a forcing term at the boundary. Such results are usually established via comparison with solutions of a related differential equation or some other form of the maximum principle. However, these techniques do not appear to apply in general situations when forcing occurs on only part of the boundary and convection is present. In this work, we obtain the global existence of solutions for such problems by deriving a differential inequality involving theLr+1norms of solutions. The result is similar to that described above for equations with a nonlinear source term and, furthermore, establishes the global existence of “small” solutions even for “strong” forcings. The technique also applies to a more general class of problems involving nonlinear reaction, diffusion, and convection
Waveform Relaxation Methods for Functional Differential Systems of Neutral Type

AbstractWe investigate continuous-time and discretized waveform relaxation iterations for functional differential systems of neutral type. It is demonstrated that continuous-time iterations converge linearly for neutral equations and superlinearly when the right hand side is independent of the history of the derivative of the solution. The error bounds for discretized iterations are also obtained and some implementation aspects are discussed. Numerical results are presented which indicate a potential speedup of this technique as compared with the classical approach based on discrete variable methods
On Finite Difference Fitted Schemes for Singularly Perturbed Boundary Value Problems with a Parabolic Boundary Layer

AbstractA method to construct grid approximations for singularly perturbed boundary value problems for elliptic and parabolic equations, whose solutions contain a parabolic boundary layer, is considered. The grid approximations are based on the fitted operator method. Finite difference schemes, finite element, or finite volume techniques are included in the term grid approximation methods. It is shown that there exists no grid approximation method on uniform grids from the class of fitted operator methods, whose solutions converge, in the discrete maximum norm, uniformly with respect to the perturbation parameter to the solution of the boundary value problem
On the Expected Number of Level Crossings of a Random Polynomial

AbstractThere are many known asymptotic estimates for the expected number ofK-level crossings of an algebraic polynomiala0+a1x+a2x2+···+anxnwith normally distributed coefficients. The present paper provides the estimate for the expected number of such level crossings when the coefficients are independent identically Cauchy distributed random variables. Using a numerical approach we show that in the interval (−1,1) by increasingKthe number ofK-level crossings decreases, while outside this interval this number is invariant, as long asK=o(n). Since the Cauchy distribution does not belong to the wide class of distributions of domain of attraction of the normal law, the polynomials with Cauchy distributed coefficients are interesting as they indicate the behaviour of polynomials for distribution outside this class
Hankel Matrices of Finite Rank with Applications to Signal Processing and Polynomials

AbstractIt is shown that certain sequences of Hankel matrices of finite rank obtained from a given sequence of complex numbers and powers of companion matrices are closely related. This relation is established by investigating the algebraic properties of combinations of polynomial multiples of powers of complex numbers. Among many applications, these properties are used to construct polynomials with zeros being a function of the zeros of given polynomials. For example, Hankel matrices of finite rank are used to develop a method for computing the least common multiple of a finite number of polynomials without factoring them, or computing a polynomial whose zeros are the product of the zeros of two polynomials. A method for computing a factor of the characteristic polynomial of a given matrix is also presented and is established by forming certain types of Hankel matrices whose entries are generated from linear combinations of powers of zeros of its characteristic polynomial. Applications of these ideas to signal processing and computational linear algebra are also given
On a Class of Regular Singular Two Point Boundary Value Problems

AbstractExistence of a unique solution for a class of regular singular two point boundary value problems−(p(x)y′)′=p(x)f(x,y),0<x≤b,limx→0+y′(x)=0,y(b)=B,has been established wherep(x) satisfies (i)p(x)>0 on (0,b), (ii)p(x)∈C1(0,r), and for somer>b, (iii)xp′(x)/p(x) is analytic in {z:|z|<r} with Taylor expansionxp′(x)/p(x)=b0+b1x+…,b0≥1and with quite general conditions onf(x,y). These conditions onf(x,y) are sharp, which is seen through one example. Regions for multiple solutions have also been determined
Integro-differential Equations and Generalized Hypergeometric Functions

AbstractThis paper is concerned with the integro-differential equationsy′(t)=ay(t)+∫01y(qt)dμ(q)+∫01y′(qt)dν(q)andy(t)+∫01y(qt)dμ(q)+∫01y′(qt)dν(q)=0,whereais a complex constant, and μ and ν are complex-valued functions of bounded variation on [0, 1]. The main motivation is that the generalized hypergeometric functionAFB(α1,…,αA;β1,…,βB;t) satisfies the first equation whenA≤Band the second equation whenA=B+1 for appropriately chosen constantaand smooth functions μ and ν. The first equation also includes as a special case the well-known pantograph equation and many of its generalisations. The main objects of this paper are well-posedness of initial value problems, Dirichlet and Dirichlet–Taylor series expansions, and asymptotic behaviour of the solutions
Oscillatory Radial Solutions of Semilinear Elliptic Equations

AbstractWe study the oscillatory behavior of radial solutions of the nonlinear partial differential equation Δu+f(u)+g(|x|,u)=0 inRn, wherefandgare continuous restoring functions,uf(u)>0 andug(|x|,u)>0 foru≠0. We assume that for fixedqlimu→0(|f(u)|/|u|q)=B>0, for 1<q<n/(n−2), and, additionally, that 2F(u)≥(1−2/n)uf(u) whenn/(n−2)≤q<(n+2)/(n−2), whereF(u)=∫u0f(s)ds. We give conditions that guarantee that the solution oscillates infinitely and tends to zero asr→∞. Finally, we give bounds for the amplitude of the oscillations and show that the period of the oscillations increases asr→∞
A Fuzzy Algebraic System Based on the Theory of Falling Shadows

AbstractBased on the concept of a falling shadow, a theoretical approach of the fuzzy algebraic system is established. A fuzzy subalgebraic system of an algebraic system is considered as a falling shadow of the “cloud of the subalgebraic systems.” When an algebraic system is a ring, fuzzy subrings (fuzzy ideas, fuzzy prime ideas) of a ring are seen as falling shadows of “the cloud of subrings” (“cloud of ideas,” “cloud of prime ideas”). We shall show that fuzzy algebraic systems defined int-norms are consequences of our definition under certain conditions