pinecone/core-2020-05-10-deduplication huggingface dataset:
Regularized Solutions for Abstract Volterra Equations

AbstractWe introduce a two-kernel dependent family of strong continuous operators defined in a Banach space, which allows us to consider in an unified treatment the notions of, among others, C0-semigroups of operators, cosine families, n-times integrated semigroups, resolvent families and k-generalized solutions.The results are applied to the study of existence and uniqueness of solutions for the Volterra equation of convolution type u=f+a∗Au, in the case A is not necessarily densely defined. Examples for equations defined in Lp spaces are also given
A Class of Globally Controllable Semilinear Heat Equations with Superlinear Terms

AbstractIt is well known nowadays that a rather general semilinear parabolic equation, governed in a bounded domain by boundary or locally distributed controls, can be made globally approximately controllable in Lp-type spaces, provided the nonlinearity is globally Lipschitz. However, the question whether this property can possibly take place in the superlinear case in several space dimensions seems to be open. The goal of this article is to describe a special class of semilinear heat equations with superlinear terms for which this property indeed holds. Our technique combines the asymptotic method of A. Y. Khapalov (1995, J. Math. Anal. Appl.,194, 858–882) with certain a priori estimates for the corresponding truncated spectral problem
Inequalities Involving Hadamard Products of Positive Semidefinite Matrices

AbstractAn inequality established by G. P. H. Styan (1973, Linear Algebra Appl.,6, 217–240) is on the Hadamard product and a correlation matrix. An inequality obtained by B.-Y. Wang and F. Zhang (1997, Linear and Multilinear Algebra, 43, 315–326) involves the Hadamard product and Schur complements. These two inequalities hold in the positive definite matrix case. Based on Albert's theorem, we present their extensions to cover the positive semidefinite matrix case. We also give relevant inequalities
Jacobi Approximations in Certain Hilbert Spaces and Their Applications to Singular Differential Equations

AbstractJacobi approximations in certain Hilbert spaces are investigated. Several weighted inverse inequalities and Poincaré inequalities are obtained. Some approximation results are given. Singular differential equations are approximated by using Jacobi polynomials. This method keeps the spectral accuracy. Some linear problems and a nonlinear logistic equation are considered. The stabilities and the convergences of proposed schemes are proved strictly. The main idea and techniques used in this paper are also applicable to other singular problems in multiple-dimensional spaces
Observability of Compactly Perturbed Systems

AbstractWe propose a general and constructive method for the study of observability of weakly coupled or, more generally, compactly perturbed linear distributed systems. This approach allows us to avoid the usual indirect compactness–uniqueness arguments in many cases. Our method is based on two observations. First, in many cases there exists a Riesz basis formed by ordinary and generalized eigenfunctions of the underlying operator of the evolutionary equation. Second, in the usual estimates some constants depend on the initial data in a rather particular way. This allows us to write the solution as a trigonometric sum with vector coefficients and then to apply a generalization of an estimation method introduced by Haraux for the study of scalar nonharmonic series. Applying this approach we improve and generalize some former theorems of Lions, Haraux, and Jaffard
Convergence of Weighted Averages of Martingales in Banach Function Spaces

AbstractLet f=(fn)n≥1 be a martingale and (wn)n≥1 a sequence of positive numbers such that Wn=∑nk=1wk→∞. Kazamaki and Tsuchikura proved that f converges in Lp (1<p<∞) if and only if the weighted average (σn(f))n≥1 of f converges in Lp, where σn(f) are given byσnf=1Wn∑k=1nwkfk,n=1,2,….We shall investigate the convergence of f and σn(f) in general Banach function spaces X. Our main result can be applied to the case where X is a rearrangement-invariant space, or X is a weighted Lp-space with a weight function satisfying the condition Ap introduced by Izumisawa and Kazamaki
Behavior of Solutions of Impulsively Perturbed Non-Halflinear Oscillator Equations

AbstractIntermittently and instantaneously perturbed oscillator equations play an important role in theory and application. In this paper, we investigate the asymptotic behavior of solutions of the impulsive system (φβ(x′))′+f(x)=0 for t≠tn, x′(tn+0)=bnx′(tn), where n=1,2,…, and φβ(u)=|u|βsgnu for β>0. In the special case f(u)=φβ(u), we obtain the so-called half-linear system, which exhibits similar behavior to the linear case. First, we prove attractivity results, and then apply our theorems to the nonautonomous equation (φβ(x′))′+q(t)f(x)=0, where q(t) is a step-function
Parameter Differentiability of the Solution of a Nonlinear Abstract Cauchy Problem

AbstractIn this article, an abstract nonlinear evolution equation with a parameter appearing in the nonlinear part is considered. By using the theory of analytic semigroups and a generalization of Gronwall's lemma for singular kernels, sufficient conditions to ensure differentiability of the solution with respect to the parameter are derived in terms of the smoothness of the nonlinear part of the equation. The results are applied to a nonlinear system of partial differential equations modeling the dynamics of shape memory alloys
On Passive Stabilization in Critical Cases

AbstractThe attitude of a satellite is often controlled by reactive forces requiring some additional energy. But it can also be stabilized by means of some subsystem of the satellite moving in a nonideal fluid as an oscillator with damping. This does not require additional energy and is called “passive stabilization.” Moreover the relative motion tends asymptotically to zero together with the satellite finding the desired position. Here we consider passive stabilization for hamiltonian systems from a mathematical point of view and show that stabilization can sometimes be obtained by nonlinear terms. As an example, we consider passive stabilization of a simple pendulum
Solving Nonstationary Infinite Horizon Dynamic Optimization Problems

AbstractMonotonicity of optimal solutions to finite horizon dynamic optimization problems is used to prove the existence of a forecast horizon, i.e., a long enough planning horizon that ensures that a first-period optimal action for the infinite horizon and the finite horizon problem agree, regardless of problem parameter changes in the tail. The existence of extremal monotone optimal solutions motivates a stopping rule that is ensured to detect the minimal forecast horizon to be used in a rolling horizon procedure to exactly solve the problem
Existence and Non-existence Results for a Quasilinear Problem with Nonlinear Boundary Condition

AbstractWe study the problem−divax|∇u|p−2∇u=λ1+|x|α1|u|q−2u−hx|u|r−2uin Ω⊂RN,ax|∇u|p−2∇u·n+bx·|u|p−2u=θgx,uon Γ,u≥0in Ω,where Ω is an unbounded domain with smooth boundary Γ, n denotes the unit outward normal vector on Γ, and λ>0, θ are real parameters. We assume throughout that p<q<r<p*=pNN−p, 1<p<N, −N<α1<q·N−pp−N, while a, b, and h are positive functions. We show that there exist an open interval I and λ*>0 such that the problem has no solution if θ∈I and λ∈(0,λ*). Furthermore, there exist an open interval J⊂I and λ0>0 such that, for any θ∈J, the above problem has at least a solution if λ≥λ0, but it has no solution provided that λ∈(0,λ0). Our paper extends previous results obtained by J. Chabrowski and K. Pflüger
Eigenvalue Asymptotics in Isothermal Forced Elongation

AbstractIn this paper, we consider the formal linearization of the governing equations of forced elongation in the isothermal regime. We give an asymptotic characterization of the spectrum of the associated strongly continuous semigroup. To this end, we develop rigorous asymptotic techniques to determine the asymptotic behavior of the eigenvalues by solving a characteristic integral equation in a generalized sector of the complex plane. The spectral mapping theorem, relating the spectrum of the semigroup to the spectrum of its generator, proves essential. Numerical evidence will be given to demonstrate the accuracy of our asymptotic predictions
On the Spectrum of the Transport Operator with Abstract Boundary Conditions in Slab Geometry

AbstractThe paper is concerned with the spectral properties of the one-dimensional transport operator with general boundary conditions where an abstract boundary operator relates the incoming and the outgoing fluxes. We first give some existence and nonexistence results of eigenvalues in the half-plane {λ∈C:Reλ>–λ*} where –λ* is the type of the semigroup generated by the streaming operator. Next, we discuss the irreducibility of the transport semigroup. In particular, we show that the transport semigroup is irreducible if the boundary operator is strictly positive. We end the paper by investigating the strict monotonicity of the leading eigenvalue of the transport operator with respect to different parameters of the equation. Finally, an open problem is indicated
On the Semilinear Elliptic Equations Δu+β(1+|x|)μup−γ(1+|x|)νuq=0 ((1.1)) in Rn

AbstractIn this paper, we consider the semilinear elliptic equationΔu+β1+|x|μup−γ1+|x|νuqinRn,where n≥3, Δ=∑ni=1(∂2/∂x2i), β and γ are two positive constants, and p,q,μ,ν are constants with q>p>1 and μ≥ν>2. We note that if β=0, γ>0, and ν>2, then the complete classification of all possible positive solutions was conducted by Cheng and Ni [Indiana Univ. Math. J.41 (1992), 261–278]. If γ=0 and β>0, then (1.1) is the so-called Matukuma-type equation, and the solution structures were classified by Li and Ni [Duke Math. J.53 (1985), 895–924] and Ni and Yotsutani [Japan J. Appl. Math.5 (1988), 1–32]. If β>0 and γ>0, then some results about the structure of positive solutions of (1.1) were derived by the first author [Nonlinear Analysis, TM&A 28 (1997), 1741–1750]. The purpose of this paper is to discuss the uniqueness and properties of unbounded positive solutions and investigate some further structures of the positive solutions of Eq. (1.1)
Rates of Convergence of Adaptive Step-Size of Stochastic Approximation Algorithms

AbstractWe propose a new adaptive algorithm with decreasing step-size for stochastic approximations. The use of adaptive algorithms in various applications is widely spread across fields such as system identification and adaptive control. We analyze the rate of convergence of the proposed algorithms. An averaging algorithm, on its optimality of the rate of convergence, is used to control the step sizes. Our proofs are based on recent results in stochastic approximations and Gaussian Approximation Theorem
On Energy Inequalities

AbstractWe consider the Schrödinger operator H=−Δ2+q in a bounded domain D in Rd, d≥3, with q in the Kato class Kd(D) and finite gauge g(x)=Ex[exp∫τ0q(Xs)ds], where (Xt) is a Brownian motion and τ is the first exit time of the Brownian motion (Xt) from the domain D. Let K and G denote the Green operators in D for H and Δ2, respectively. We prove that there is a positive constant α such that1αGf,f≤Kf,f≤αGf,f,where f is a real, measurable function for which (G|f|,|f|) is finite. As a direct consequence of this double inequality, we have that the potential Gf is in the Sobolev space H10(D) if and only if Kf∈H10(D)
Picard Iterations for Solution of Nonlinear Equations in Certain Banach Spaces

AbstractLet E be a real uniformly smooth Banach space and let A: D(A)⊂E↦E be locally Lipschitzian and strongly quasi-accretive. It is proved that a Picard recursion process converges strongly to the unique solution of the equation Ax=f, f∈R(A), with the convergence being at least as fast as a geometric progression. Related results deal with the convergence of Picard iterations to the fixed point of locally Lipschitzian strong hemicontractions T and to the solutions of nonlinear equations of the forms x+Ax=f and x−λAx=f, where A is an accretive-type map
Differential Equations Satisfied by the Components with Respect to the Cyclic Group of Order n of Some Special Functions

AbstractLet f be a complex function of the variable z admitting a Laurent expansion in an annulus C with center in the origin. For an arbitrary positive integer n, Ricci's theorem asserts that the function f can be written as the sum of n functions f[n,k], k∈{0,1,…,n−1}, defined byf[n,k]z=1n∑l=0n−1exp−2iπklnfzexp2iπln,z∈C.In this paper, we present a technique which, starting from a suitable differential equation satisfies by the function f, provides a differential equation satisfied by the functions f[n,k]. Some known special functions are treated for illustration
A Characterization of Banach Spaces with Separable Duals via Weak Statistical Convergence

AbstractLet B be a Banach space. A B-valued sequence 〈 xk〉 is weakly statistically null provided limn1n |{k≤n:|f(xk)|>ε}|=0 for all ε>0 and every continuous linear functional f on B. A Banach space is finite dimensional if and only if every weakly statistically null B-valued sequence has a bounded subsequence. If B is separable, B* is separable if and only if every bounded weakly statistically null B-valued sequence contains a large weakly null sequence. A characterization of spaces containing an isomorphic copy of l1 is given, and it is also shown that l2 has a “statistical M-basis” which is not a Schauder basis
A Mathematical Model for Gamma Ray Transport in the Cardiac Region

AbstractWe study gamma ray transport in a medium composed of three time-dependent regions. The inner region V1 is bounded by the time-dependent surface Σ1. The second region V2 is bounded by Σ1 and by the surface Σ2 which is homothetic to Σ1. This is used to model radionuclide imaging of the cardiac region: V1 is, roughly, the blood within the heart of a patient, V2 represents the heart walls, and V3 is the rest of the body. We prove that the quasi-stationary solution of the corresponding photon transport problem, which satisfies a stationary-like equation, is a good approximation to the exact solution because the speed of the heart walls is much smaller than the speed of light. Furthermore, we show that the quasi-stationary solution n(t) and its properties may be used to convert the gamma ray densities n(t+δ),n(t+2δ),…, detected at later times, to the first detection n(t)
On the Solvability of Second Kind Integral Equations on the Real Line

AbstractThis paper considers general second kind integral equations of the formφs−∫Rks,tφtdt=ψs(in operator form φ−Kkφ=ψ), where the functions k and ψ are assumed known, with ψ∈Y, the space of bounded continuous functions on R, and k such that the mapping s→k(s,·), from R to L1(R), is bounded and continuous. The function φ∈Y is the solution to be determined. Conditions on a set W⊂BC(R,L1(R)) are obtained such that a generalised Fredholm alternative holds: If W satisfies these conditions and I−Kk is injective for all k∈W then I−Kk is also surjective for all k∈W and, moreover, the inverse operators (I−Kk)−1 on Y are uniformly bounded for k∈W. The approximation of the kernel in the integral equation by a sequence (kn) converging in a weak sense to k is also considered and results on stability and convergence are obtained. These general theorems are used to establish results for two special classes of kernels: k(s,t)=κ(s−t)z(t) and k(s,t)=κ(s−t)λ(s−t,t), where κ∈L1(R), z∈L∞(R), and λ∈BC((R\{0})×R). Kernels of both classes arise in problems of time harmonic wave scattering by unbounded surfaces. The general integral equation results are here applied to prove the existence of a solution for a boundary integral equation formulation of scattering by an infinite rough surface and to consider the stability and convergence of approximation of the rough surface problem by a sequence of diffraction grating problems of increasingly large period
Nonlinearly Weighted L1-Solvability of the First Kind Fredholm Integral Equation

AbstractThe Fredholm integral equation of the first kind is a well known example of linear ill-posed problems and of functional equations that do not admit of a direct analytical solution in terms of their data. We show that a variational perturbation of the kernel, K(v,x), (v,x)∈Ω, in an inverse-problem space, dual to the solution space, can yield for the integral equation a maximum a posteriori direct primal solution ψ(x), x∈I. This dual space L1C(Ω;γ(K)) happens to be a certain restriction of a nonlinearly (covariantly) weighted L1-space. It is proved that the emerging analytical primal solution must satisfy an equivalent regularizing Urysohn nonlinear integral equation, with a movable bifurcation point, and that it belongs also in a certain nonlinearly weighted L1(I;ρ(ψ)) space invoked by L1C(Ω;γ(K)). An analytical explanation is advanced for a possible autotransformability between the first-kind and second-kind Fredholm integral equations as a result of perturbations of K only in the dual space
Bifurcation of Reaction–Diffusion Systems: Application to Epidemics of Many Species

AbstractThis article considers a system of reaction–diffusion equations describing the spread of several bacterial infections among many interacting species. Conditions concerning the growth, infection, and interaction rates such that positive steady states can exist are obtained. The stabilities of the steady states are also studied. Bifurcation methods are used to show the existence of positive steady-state solutions, and the semigroup method is used to analyze the stability of the steady states
Hierarchical Production Control in a Stochastic N-Machine Flowshop with Limited Buffers

AbstractThis paper is concerned with an asymptotic analysis of hierarchical production planning in a stochastic manufacturing system consisting of machines that are subject to breakdown and repair. The system produces a single product whose rate of demand over time is constant. The problem is formulated as a continuous-time dynamic programming problem in which the objective is to minimize the cost of production, inventories, and backlogs. The size of the buffer between the two adjacent machines is assumed to have lower and upper bound constraints. As the rates of change of the machine states approach infinity, the analysis results in a limiting problem in which the stochastic machine availability is replaced by its equilibrium mean availability. The value function for the original problem converges to the value function of the limiting problem. A method of “shrinking,” “total lifting,” and “modification” is introduced in order to construct near-optimal controls for the original problem by using near-optimal controls of the limiting problem. The convergence rate of the value function 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
The Global Dynamic Behavior of the Competition Model of Three Species

AbstractIn this paper, we decompose the dynamic behavior of the competitive Lotka–Volterra (LV) model ẋi=xi(1−xi−αixi+1−βixi+2), xi(0)>0, αi>0, βi>0, i=1,2,3, with x4=x1, x5=x2, into the dynamic behavior of two two-dimensional manifolds, and completely analyse the global asymptotic behavior of the competitive LV model. We obtain the necessary and sufficient conditions for the equilibrium of the competitive LV model to be positive and globally asymptotically stable in IntR3+, the necessary and sufficient conditions for the model having a family of limit cycle solutions and a heteroclinic cycle, both of which are the ω-limit set of some other trajectories of the competitive LV model
Steepest Descent Method for Equilibrium Points of Nonlinear Systems with Accretive Operators

AbstractLet E be a real normed linear space and let A: E↦2E be a bounded uniformly continuous φ-strongly accretive multi-valued map with nonempty closed values such that the inclusion 0∈Ax has a solution x*∈E. It is proved that both the Ishikawa and the Mann iteration processes are bounded and converge strongly to x*. Related results deal with the convergence of the iteration processes to fixed points of φ-strongly pseudocontractive maps and the iterative solution of the equation 0∈x+Fx for an accretive map F. Furthermore, the Ishikawa and Mann iteration methods with errors are discussed and proofs are sketched on how our theorems extend to these methods. Our method of proof is of independent interest
Some simple algorithms for the evaluations and representations of the Riemann zeta function at positive integer arguments

Many interesting solutions of the so-called Basler problem of evaluating the Riemann zeta function ζ(s) when s = 2, which was of vital importance to Euler and the Bernoulli brothers (Jakob and Johann Bernoulli), have appeared in the mathematical literature ever since Euler first solved this problem in the year 1736. The main object of the present paper is to investigate rather systematically several interesting evaluations and representations of ζ(s) when s ∈ ℕ¼ {1}. In one of many computationally useful special cases considered here, it is observed that ζ(3) can be represented by means of a series which converges much more rapidly than that in Euler's celebrated formula as well as the series used recently by Apéry in his proof of the irrationality of ζ(3). Symbolic and numerical computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of this series are capable of producing an accuracy of seven decimal places
Lie Symmetry Analysis and Approximate Solutions for Non-linear Radial Oscillations of an Incompressible Mooney–Rivlin Cylindrical Tube

AbstractThe non-linear, second-order differential equation derived by Knowles (1960, Quart. Appl. Math.18, 71–77) which governs the axisymmetric radial oscillations of an infinitely long, hyperelastic cylindrical tube of Mooney–Rivlin material is considered. It is shown that if the boundary conditions are time dependent, then the Knowles equation has no Lie point symmetries, while if the boundary conditions are constant it has one Lie point symmetry corresponding to time translational invariance. The derivation by Knowles (1962, J. Appl. Mech.29, 283–286) of bounds on the period of the oscillation for the heaviside step loading boundary condition is extended to obtain limiting oscillations that exhibit periods that bound the exact period above and below. The Knowles equation for a Mooney–Rivlin material is expanded in powers of a dimensionless parameter, μ, defined in terms of the thickness of the tube wall. To zero order in μ an Ermakov–Pinney equation is obtained which has three Lie point symmetries. It is shown that the differential equation which is correct to first order in μ also has three Lie point symmetries which disappear at second order in μ. For time independent boundary conditions, the three Lie point symmetries of the order μ equation are derived explicitly and the associated first integrals are obtained. The general solution is derived in terms of the three first integrals and it is illustrated for free oscillations and the heaviside step loading boundary condition. The non-autonomous first order in μ equation is transformed to an autonomous Ermakov–Pinney equation and a non-linear superposition principle for the solution to first order in μ is derived and applied to a blast loaded applied pressure that decays linearly with time. The solutions to first order in μ are compared with numerical solutions of the Knowles equation for a thick-walled cylinder and are found to be more accurate than the zero order solutions described by the Ermakov–Pinney equation