pinecone/core-2020-05-10-deduplication huggingface dataset:
Uniqueness of Algebraic Limit Cycles for Quadratic Systems

AbstractWe know five different families of algebraic limit cycles in quadratic systems, one of degree 2 and four of degree 4. Moreover, if there are other families of algebraic limit cycles for quadratic systems, then their degrees must be larger than 4. It is known that if a quadratic system has an algebraic limit cycle of degree 2, then this is the unique limit cycle of the system. The first main goal of this paper is to prove that if a quadratic system has an algebraic limit cycle of degree ≤4, then this is the unique limit cycle of the system. For the Yablonskii and Filiptsov algebraic limit cycles the problem of proving their uniqueness has been open since 1966 and 1973, respectively. The second main goal is to provide sufficient conditions in order that all the limit cycles of a quadratic system to be algebraic
Viability for Semilinear Differential Inclusions via the Weak Sequential Tangency Condition

AbstractLet X be a reflexive and separable Banach space, A:D(A)⊂X→X the generator of a C0-semigroup S(t):X→X,t≥0,D a locally weakly sequentially closed subset in X, and F:D→2X a nonempty, closed, convex, and bounded valued mapping which is weakly-weakly upper semi-continuous. The main result of the paper is:Theorem.Under the general assumptions above a necessary and sufficient condition in order that for each ξ∈D there exists at least one mild solution u ofdudtt∈Aut+Futsatisfying u(0)=ξ is the so-called “weak sequential tangency condition” below.(WSTC) For each ξ∈D there exists y∈F(ξ) and two sequences (tn)n∈Nin R+and (pn)n∈Nin X such that tn→0,pn→0 and satisfying S(tn)ξ+tn(y+pn)∈D
Stability of the Shifts of Global Supported Distributions

AbstractFor a tempered distribution with ℓ1 decay, we characterize its stable shifts via its Fourier transform and via a shift-invariant space of summable sequences. Also we show that if the tempered distribution with ℓ1 decay has stable shifts, then we can recover all distributions in V∞, the space of all linear combinations of its shifts using bounded sequences, in a stable way using C∞ dual functions with ℓ1 decay at infinity. If, additionally, that tempered distribution is compactly supported, then the above C∞ dual functions can be chosen to have exponential decay at infinity
Efficiency and Approachability of Nonconvex Bicriteria Programs

AbstractIn this paper we investigate equivalences between an efficient solution of a bicriteria program and a lower envelope point of a certain image set of the bicriteria program. We also employ various kinds of approachabilities to characterize efficiency and proper efficiency for nonconvex bicriteria programs. In particular, nonlinear Lagranian functions are applied to construct dual problems and to study stability for the corresponding constrained scalar optimization problems. Under certain conditions we show that the finite approachability, proper efficiency, stability, and exact penalization of the relevant constrained scalar optimization problems are equivalent
Hyperbolic Phase-Field Dynamics with Memory

AbstractWe consider a non-conserved phase-field system which consists of two nonlinearly coupled hyperbolic integrodifferential equations. This model is derived from two basic assumptions: the heat flux law is of Gurtin–Pipkin type and the response of the order parameter to the variation of the free energy functional is delayed. These hypotheses might be a reasonable attempt to describe, for instance, the melt of He4 crystals. A suitable initial and boundary value problem is then associated with the system and its well-posedness is analyzed in detail
On Roughness of Exponential Dichotomy

AbstractWe present new perturbation theorems on the roughness of exponential dichotomy, which improve previous results. The proofs given here are also much simpler compared with previous ones. The new results provide significant improvements of existing results in the case where the operator A(t) is unbounded for t∈J. This is precisely the situation that is of interest from the point of view of the applications in lobe dynamics. The results are also valid in Banach spaces and useful for general purposes
Fredholm, Semi-Fredholm Perturbations, and Essential Spectra

AbstractThis paper is devoted to the investigation of the stability of the various essential spectra of closed densely defined linear operators under perturbations belonging to any two-sided ideal of the algebra of bounded linear operators contained in the set of Fredholm perturbations. Our results generalize many known ones in the literature and, in particular, extend and unify those obtained by K. Latrach (1999, J. Math. Anal. Appl.233, 607–623) and K. Latrach and A. Jeribi (1998, J. Math. Anal. Appl.225, 461–485). They are used to describe the essential spectra of one-dimensional transport equations with anisotropic scattering and abstract boundary conditions
Lie Subalgebras, Reduction of Order, and Group-Invariant Solutions

AbstractOne of the great utilities of Lie symmetries of differential equations is in their use to reduce the order of ordinary differential equations and partial differential equations to ordinary differential equations. This process is guided by the Lie algebra of the symmetries admitted by the equation under study. It is important in these reductions to use appropriate symmetries. This will ensure that a sufficient number remain after the initial reduction to allow subsequent reductions. By considering two- and three-dimensional subalgebras of the Lie algebra of the symmetries of differential equations, we indicate the fate of these symmetries under various reductions. In doing so, it is clear as to which symmetries need be prioritized in reduction
Analysis of a Three Species Eco-Epidemiological Model

AbstractThis paper formulates and analyzes a predator–prey model with disease in the prey. Mathematical analyses of the model equations with regard to invariance of nonnegativity, boundedness of solutions, nature of equilibria, permanence, and global stability are analyzed. It is also shown that for some parameter values, the positive equilibrium is asymptotically stable, but for other parameter values, it is unstable and a surrounding periodic solution appears by Hopf bifurcation. A concluding discussion with numerical simulation is then presented
Strong Convergence of Averaged Approximants for Lipschitz Pseudocontractive Maps

AbstractLet T be a Lipschitzian pseudocontractive self-mapping of a closed convex and bounded subset K of a Banach space E which is both uniformly convex and q-uniformly smooth such that the set F(T) of fixed points of T is nonempty. Then F(T) is a sunny nonexpansive retract of K. If U is the sunny nonexpansive retraction of K onto F(T), ω is any point of K, and {an}∞n=0 a real sequence in (0,1], then the sequence {xn}∞n=0 in K defined byxn=anω+1−an1n+1∑j=0n1−ajI+ajTxnfor n=0,1,2,…, converges strongly to Uω. No compactness assumption is made on K
The Discrete Spectrum in the Spectral Gaps of Semibounded Operators with Non-sign-definite Perturbations

AbstractGiven two self-adjoint operators A and V=V+−V−, we study the motion of the eigenvalues of the operator A(t)=A−tV as t increases. Let α>0 and let λ be a regular point for A. We consider the quantities N+(V;λ,α),N−(V;λ,α), and N0(V;λ,α) defined as the number of eigenvalues of the operator A(t) that pass point λ from the right to the left, from the left to the right, or change the direction of their motion exactly at point λ, respectively, as t increases from 0 to α>0. We study asymptotic characteristics of these quantities as α→∞. In the present paper, the results obtained previously [O. L. Safronov, Comm. Math. Phys.193 (1998), 233–243] are extended and given new applications to differential operators
Asymptotic Stability for Tracking Control of Nonlinear Uncertain Dynamical Systems Described by Differential Inclusions

AbstractIn this paper, we will study the tracking control problem for nonlinear uncertain dynamical systems. Two generalized feedback control inputs have been proposed such that the feedback-controlled systems satisfy the complete tracking control property with exponential asymptotic stability and the trajectories of the systems are steered to the pre-specified observation map with an exponential convergence rate. Moreover, an estimate of the tracking time of the trajectories attaining the observation map has also been given. An example inspired from a guided missile problem illustrates the use of our main results
Spectral Properties of Non-selfadjoint Difference Operators

AbstractLet L denote the operator generated in ℓ2(Z) by the difference expression (ℓy)n=an−1yn−1+bnyn+anyn+1,n∈Z={0,±1,±2,…}, where {an}n∈Z and {bn}n∈Z are complex sequences. In this paper we investigated the spectrum, the spectral singularities, and the properties of the principal vectors corresponding to the spectral singularities of L. We also studied similar problems for the discrete Dirac operator M generated in ℓ2(Z,C2) by the system of difference expressionΛyn=(Λ1y)n(Λ2y)n=Δy(2)n+pny(1)n−Δy(1)n−1+qny(2)n,n∈Z,wherey=y(1)ny(2)nn∈Z,Δ is the forward difference operator, i.e., Δy(i)n=y(i)n+1−y(i)n, i=1,2, and {pn}n∈Z,{qn}n∈Z are complex sequences
A Domain Decomposition Method Based on BEM and FEM for Linear Exterior Boundary Value Problems

AbstractWe develop the finite dimensional analysis of a new domain decomposition method for linear exterior boundary value problems arising in potential theory and heat conductivity. Our approach uses a Dirichlet-to-Neumann mapping to transform the exterior problem into an equivalent boundary value problem on a bounded domain. Then the domain is decomposed into a finite number of annular subregions and the local Steklov–Poincaré operators are expressed conveniently either by BEM or FEM in order to obtain a symmetric interface problem. The global Steklov–Poincaré problem is solved by using both a Richardson-type scheme and the preconditioned conjugate gradient method, which yield iteration-by-subdomain algorithms well suited for parallel processing. Finally, contractivity results and finite dimensional approximations are provided
Permanence and Stability of a Stage-Structured Predator–Prey Model

AbstractA predator–prey model with a stage structure for the predator which improves the assumption that each individual predator has the same ability to capture prey is proposed. It is assumed that immature individuals and mature individuals of the predator are divided by a fixed age and that immature predators do not have the ability to attack prey. We obtain conditions that determine the permanence of the populations and the extinction of the populations. Furthermore, we establish necessary and sufficient conditions for the local stability of the positive equilibrium of the model. By exploiting the monotonicity of one equation of the model, we obtain conditions for the global attractivity of the positive equilibrium, which allow for long delay as long as the predator birth rate is large or the death rate of immature predators is small. By constructing Liapunov functionals, we also obtain conditions under which the positive equilibrium is globally stable when the delay is small
Feedback Stabilization of Linear Retarded Systems in Banach Spaces

AbstractFeedback stabilizability is studied for linear retarded systems in Banach spaces. Under the assumptions that the control is finite dimensional and the corresponding instantaneous free system generates a compact semigroup, the rank condition for exponential stabilizability is established based on the stabilizability result due to Bhat and Wonham. The condition is expressed in terms of eigenvectors and controllers, and the equivalent spectral mode controllability and observability conditions are given. An application is given to a class of parabolic partial functional differential equations with mixed boundary conditions
Complex Zeros of Trigonometric Polynomials with Standard Normal Random Coefficients

AbstractIn this paper, we obtain an exact formula for the average density of the distribution of complex zeros of a random trigonometric polynomial η0+η1cosθ+η2cos2θ+⋯+ηncosnθ in (0,2π), where the coefficients ηj=aj+ιbj, and {aj}j=1n and {bj}j=1n are sequences of independent normally distributed random variables with mean 0 and variance 1. We also provide the limiting behaviour of the zeros density function as n tends to infinity. The corresponding results for the case of random algebraic polynomials are known
Bessel Wavelets and the Galerkin Analysis of the Bessel Operator

AbstractWe construct a complete orthonormal system {ψj,k}j∈Z,k∈N for L2(R+,dx) such that each Bessel wavelet ψj,k has a compactly supported Hankel transform. We apply this system in the Galerkin solution of the differential equation Lu=f on R+, where L is the Bessel operator. The associated linear system can be preconditioned using a simple diagonal preconditioning matrix to give a system which is sparse and has condition number which is bounded independent of the Galerkin subspace. Thus we obtain the same advantages for the singular operator L that have been obtained for uniformly elliptic operators using standard wavelets. The main step is a characterization of the Bessel–Sobolev norm of a function in terms of its Bessel wavelet coefficients. We also present numerical work which shows that the condition numbers are small enough for this method to be practical
Complete Symmetry Groups of Ordinary Differential Equations and Their Integrals: Some Basic Considerations

AbstractThe concept of the complete symmetry group of a differential equation introduced by J. Krause (1994, J. Math. Phys.35, 5734–5748) is extended to integrals of such equations. This paper is devoted to some aspects characterising complete symmetry groups. The algebras of the symmetries of both differential equations and integrals are studied in the context of equations for which the elements are represented by point or contact symmetries so that there is no ambiguity about the group. Both algebras and groups are found to be nonunique
(p,r)-Invex Sets and Functions

AbstractNotions of invexity of a function and of a set are generalized. The notion of an invex function with respect to η can be further extended with the aid of p-invex sets. Slight generalization of the notion of p-invex sets with respect to η leads to a new class of functions. A family of real functions called, in general, (p,r)-pre-invex functions with respect to η (without differentiability) or (p,r)-invex functions with respect to η (in the differentiable case) is introduced. Some (geometric) properties of these classes of functions are derived. Sufficient optimality conditions are obtained for a nonlinear programming problem involving (p,r)-invex functions with respect to η
On the Regularity of Harmonic Functions and Spherical Harmonic Maps Defined on Lattices

AbstractA growth lemma for certain discrete symmetric Laplacians defined on a lattice Zδd=δZd⊂Rd with spacing δ is proved. The lemma implies a De Giorgi theorem, that the harmonic functions for these Laplacians are equi-Hölder continuous, δ→0. These results are then applied to establish regularity properties for the harmonic maps defined on Zdδ and taking values in an n-dimensional sphere Sn, uniform in δ. Questions of the convergence δ→0 and the Dirichlet problem for these discrete harmonic maps are also addressed
Symmetry Properties for Positive Solutions to Some Elliptic Equations in Sector Domains with Large Amplitude

AbstractIn this paper we study symmetry properties for positive solutions of semilinear elliptic equation Δu+f(u)=0 with mixed boundary condition in a spherical sector Σ(α,R), where α, the amplitude of the sector, is between π and 2π. Under certain conditions on f(u), we prove that all positive solutions are radially symmetric about the origin. Unlike well-known results of B. Gidas et al. (1979, Comm. Math. Phys.68, 209–243) on Dirichlet problem, or early results of Berestycki and Pacella on the same problem for Σ(α,R) with an acute angle, extra conditions on f(u) are needed, as pointed out early by H. Berestycki and F. Pacella (1989, J. Funct. Anal.87, 177–211)
Convergence in Almost Periodic Competition Diffusion Systems

AbstractThe paper deals with the convergence of positive solutions for almost-periodic competition diffusion systems. The asymptotic almost periodicity of a positive solution for such a system is described by the almost periodicity of the ω-limit set of the corresponding positive motion in the associated skew-product flow. In the framework of the skew-product flow, it will be proved that the ω-limit set of any spatially homogeneous positive motion contains at most two minimal sets which are both almost automorphic. It will also be proved that if each spatially homogeneous positive solution is asymptotically almost periodic and each spatially homogeneous positive almost periodic solution is lower (upper) asymptotically Lyapunov stable, then every positive solution converges to a spatially homogeneous almost periodic solution. Several important special cases are described where every positive solution converges to a spatially homogeneous almost-periodic solution
Some Further Generalizations of the Hyers–Ulam–Rassias Stability of Functional Equations

AbstractIn this paper we study the Hyers–Ulam–Rassias stability theory by considering the cases where the approximate remainder φ is defined byfx∗y−fx−fy=φx,y∀x,y∈G,1fx∗y−gx−hy=φx,y∀x,y∈G,22fx∗y1/2−fx−fy=φx,y∀x,y∈G,3where (G,∗) is a certain kind of algebraic system, E is a real or complex Hausdorff topological vector space, and f,g,h are mappings from G into E. We prove theorems for the Hyers–Ulam–Rassias stability of the above three kinds of functional equations and obtain the corresponding error formulas
Strict Efficiency in Vector Optimization

AbstractThe notion of strict minimum of order m for real optimization problems is extended to vector optimization. Its properties and characterization are studied in the case of finite-dimensional spaces (multiobjective problems). Also the notion of super-strict efficiency is introduced for multiobjective problems, and it is proved that, in the scalar case, all of them coincide. Necessary conditions for strict minimality and for super-strict minimality of order m are provided for multiobjective problems with an arbitrary feasible set. When the objective function is Fréchet differentiable, necessary and sufficient conditions are established for the case m=1, resulting in the situation that the strict efficiency and super-strict efficiency notions coincide
Operators with Dense Images Everywhere

AbstractIn this paper, the authors introduce the dense-image operators T as those with a wild behavior near the boundary of a domain G, via certain subsets. The relationship with other kinds of operators with wild behavior is studied, proving that the new concept generalizes the earlier idea of omnipresence, but there is no good relationship with the strongly omnipresent operators. We obtain, among other results, that the following kinds of operators are dense-image: onto linear operators, operators with local dense range satisfying soft conditions, Volterra complex integral operators plus infinite order differential operators, and multiplication operators. In addition, holomorphic self-mappings and entire functions generating dense-image right or left composition operators are completely characterized
The Commutant Modulo Cp of Co-prime Powers of Operators on a Hilbert Space

AbstractLet H be a separable infinite-dimensional complex Hilbert space and let A,B∈B(H), where B(H) is the algebra of operators on H into itself. Let δA,B: B(H)→B(H) denote the generalized derivation δAB(X)=AX−XB. This note considers the relationship between the commutant of an operator and the commutant of co-prime powers of the operator. Let m,n be some co-prime natural numbers and let Cp denote the Schatten p-class, 1≤p<∞. We prove (i) If δAmBm(X)=0 for some X∈B(H) and if either of A and B* is injective, then a necessary and sufficient condition for δAB(X)=0 is that ArXBn−r−An−rXBr=0 for (any) two consecutive values of r,1≤r<n. (ii) If δAmBm(X) and δAnBn(X)∈Cp for some X∈B(H), and if m=2 or 3, then either δABn(X) or δABn+3(X)∈Cp; for general m and n, if A and B* are normal or subnormal, then there exists a natural number t such that δAB(X)∈C2tnp. (iii) If δAmBm(X) and δAnBn(X)∈Cp for some X∈B(H), and if either A is semi-Fredholm with ind A≤0 or 1−A*A∈Cp, then δAB(X)∈Cp
Generalized Anderson's Inequality

AbstractLet H be a separable infinite-dimensional complex Hilbert space and let B(H) denote the algebra of operators on H into itself. We study the elementary operator Φ:B(H)→B(H) defined by Φ(X)=AXB−CXD, where A and C (respectively, B and D) are nonzero normal commuting operators. We prove that•(i)‖Φ()+‖≥‖‖ for all ∈N(Φ) (the kernel of Φ) and for all ∈B() or•(ii)‖Φ()+‖≥‖‖ for all ∈N(Φ)∩C (the von Neumann–Schatten class), 1≤<∞, ≠2, and for all ∈B() such that Φ()∈C if and only if N()∩N()=N(*)∩N(*)={0}
A Trapping Principle for Discontinuous Elliptic Systems of Mixed Monotone Type

AbstractWe consider discontinuous semilinear elliptic systems, with boundary conditions on the individual components of Dirichlet/Neumann type. The system is a divergence form generalization of Δu=f(·,u). The components of f are required to satisfy monotonicity conditions associated with competitive or cooperative species. The latter model defines a system of mixed monotone type. We also illustrate the theory via higher order mixed monotone systems which combine competitive and cooperative subunits. We seek solutions on special intervals defined by lower and upper solutions associated with outward pointing vector fields. It had been shown by Heikkilä and Lakshmikantham that the general discontinuous mixed monotone system does not necessarily admit solutions on an interval defined by lower and upper solutions. Our result, obtained via the Tarski fixed-point theorem, shows that solutions exist for the models described above in the sense of a measurable selection (in the principal arguments) from a maximal monotone multivalued mapping. We use intermediate variational inequalities in the proof. Applications involving quantum confinement and chemically reacting systems with change of phase are discussed. These are natural examples of discontinuous systems
Weighted Composition Operators on Hardy Spaces

AbstractLet ϕ, ψ be analytic functions defined on D, such that ϕ(D)⊆D. The operator given by f↦ψ(f∘ϕ) is called a weighted composition operator. In this paper we deal with the boundedness, compactness, weak compactness, and complete continuity of weighted composition operators on Hardy spaces Hp (1≤p<∞). In particular, we prove that such an operator is compact on H1 if and only if it is weakly compact on this space. This result depends on a technique which passes the weak compactness from an operator T to operators dominated in norm by T
Pointwise Blow-up of Sequences Bounded in L1

AbstractGiven a sequence of functions bounded in L1([0,1]), is it possible to extract a subsequence that is pointwise bounded almost everywhere? The main objective of this note is to present an example showing that this is not possible in general. We will also prove a pair of positive results. We show that if the sequence of functions consists of multiples of characteristic functions of measurable sets, the answer is yes. We also show that it is always possible to extract a subsequence that is pointwise bounded on a countable, dense set of points
Precise Coefficient Estimates for Close-to-Convex Harmonic Univalent Mappings

AbstractThe class SH consists of harmonic, univalent, and sense-preserving functions f in the open unit disk U={z:|z|<1}, such that f=h+g, where h(z)=z+∑n=2∞anzn and g(z)=∑n=1∞a−nzn. Let S0H, CH, and C0H denote the subclass of SH with a−1=0, the subclass of SH with f being a close-to-convex mapping, and the intersection of S0H and CH, respectively. In this paper, for f∈C0H and f∈CH, we prove that the harmonic analogue of the Bieberbach conjecture and the generalization of the Bieberbach conjecture are true