pinecone/core-2020-05-10-deduplication huggingface dataset:
On Radially Symmetric Minima of Nonconvex Functionals

AbstractWe study a minimization problem in the space W1,10(BR) where BR is the ball of radius R with center at the origin; the functional considered is not necessarily convex. Under suitable assumptions, we prove the existence of a radially symmetric (decreasing) solution. By strengthening the assumptions we obtain uniqueness results. Finally, we study under which assumptions and in which sense the solutions found solve the corresponding Euler equation. The proofs are very direct and simple: they only make use of the functions T±n introduced by the author [Arch. Rational Mech. Anal., 1999]
Convex Stochastic Fluid Programs with Average Cost

AbstractWe consider stochastic fluid programs under the average cost criterion. These models have been introduced by the author (in press, Math. Oper. Res.) and are of the following type: suppose (Zt) is a continuous-time Markov chain with finite state space. As long as Zt=z, the dynamics of the system at time t are given by a linear function bz(a(·)), where a is a control we have to choose. A convex cost rate function c is given, depending on the state and the action. We want to control the system in such a way as to minimize the expected average cost. Such models typically appear in production and telecommunication systems. Using a vanishing discount approach and a discretization technique, we show that the relative value function satisfies a HJB equation and derive a verification theorem. Last but not least we apply our results to manufacturing systems and network problems
Uniqueness for Near-Constant Data in Fourth-Order Inverse Eigenvalue Problems

AbstractLet L(p)u=d4u/dx4−(d/dx)(p1(du/dx))+p2u, u(0)=u″(0)=u(1)=u″(1)=0 where p∈L2[0,1]×L2[0,1]. We show that for near constant coefficients, if p is even about 1/2 and L(p) and L(p̃) have the same eigenvalues, then knowledge of the first coefficient uniquely determines the second up to average value. Further, we show that knowledge of the second coefficient uniquely determines the first. We derive precise eigenfunction asymptotics using resolvent perturbation theory and prove the result using a simple perturbation of basis argument
Abstract Degenerate Cauchy Problems in Locally Convex Spaces

AbstractThis paper is concerned with the abstract degenerate Cauchy problem DCP,ddtBu(t)=Au(t)(t≥0),Bu(0)=Bu0, where A and B are closed linear operators in a sequentially complete locally convex space. A C-propagation family for (DCP) is introduced (C is an operator), leading to a general C-wellposedness result about (DCP). Moreover, conditions are given ensuring the existence of C-propagation families for those (DCP) with differential operators, on various function spaces with Fréchet topologies, as coefficient operators. These results are new even in the case of Banach spaces
Poincaré's Reversibility Condition

AbstractWe consider a real planar analytic vector field, X, such that the origin, O, is a centre for the linearization of X. Poincaré's condition of reversibility with respect to a line passing through O is then a sufficient condition for O to be a centre for the vector field X. We provide necessary and sufficient conditions, involving the vanishing of certain polynomials in the coefficients in the expansion of X, for reversibility. We also show that if the linearization, L(x), of the divergence of X is non-trivial, then the only possible reversibility line is given by L(x)=0; in such cases, this provides the basis for a simple test of reversibility. We examine the consequences of our various tests for quadratic and cubic vector fields; all non-Hamiltonian cases are discussed. When L(x)≡0 in cubic systems, it is possible for the reversibility line (if it exists) to be unique, but it is also possible for there to be two such lines. These possibilities are characterized algebraically, and a prescription is provided for determining the reversibility line(s) in each case
On M-Ideals of Compact Operators in Lorentz Sequence Spaces

AbstractLet X=d(v,p) and Y=d(w,q) be Lorentz sequence spaces. We investigate when the space K(X,Y) of compact linear operators acting from X to Y forms or does not form an M-ideal (in the space of bounded linear operators). We show that K(X,Y) fails to be a non-trivial M-ideal whenever p=1 or p>q. In the case when 1>p≤q, we establish a general (essential) condition guaranteeing that K(X,Y) is not an M-ideal. In contrast, we prove that non-trivial M-ideals K(X,Y) do exist whenever 1<p<q, and we give a description of them
Convergence of Spectra of Mesoscopic Systems Collapsing onto a Graph

AbstractLet M be a finite graph in the plane and let Mε be a domain that looks like the ε-fattened graph M (exact conditions on the domain are given). It is shown that the spectrum of the Neumann Laplacian on Mε converges when ε→0 to the spectrum of an ODE problem on M. The presence of an electromagnetic field is also allowed. Considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry. The results of the paper extend the ones previously obtained by J. Rubinstein and M. Schatzman
Duality in Nondifferentiable Vector Programming

AbstractIn this paper we study the saddle point optimality conditions and Lagrange duality in multiobjective optimization for generalized subconvex-like functions. We obtain results which will allow us to characterize the solutions for multiobjective programming problems from the saddle point conditions and allow us to relate them to the dual problem solutions which will be adequately defined. We also define a new dual problem for the multiobjective programming problem with the special property of being a scalar programming problem
Operators on Spaces of Analytic Functions Belonging to L(1,∞)

AbstractLet H be a separable Hilbert space and K be the ideal of compact operators on H. A T∈K is said to be in L(1,∞) if λn(T)=O(logn) for n≥2 or, equivalently, supN≥2(1/logN)∑N1λn(T)<∞, where λn(T) are the singular values (eigenvalues of |T|=(T*T)1/2). In this paper, we will give geometric conditions on several classes of operators, including Hankel and composition operators, belonging to L(1,∞). Specifically, we will show that the function space characterizing the symbols of these operators is a nonseparable Banach space which lies strictly between B1(D) and all the other holomorphic Besov spaces Bp(D)(p>1)
Hankel- and Toeplitz-Type Operators on the Unit Ball

AbstractLet Bm be the unit ball in the m-dimensional complex plane Cm with the weighted measuredμαz=α+1α+2···α+mπm1−|z|2αdmzα>−1.From the viewpoint of the Cauchy–Riemann operator we give an orthogonal direct sum decomposition for L2(Bm,dμα(z)), i.e., L2(Bm,dμα(z))=⊕n∈Z+,σ∈ΔAσn, where the components A(+,+,…,+)0 and A(−,−,…,−)0 are just the weighted Bergman and conjugate Bergman spaces, respectively. Using the simplex polynomials from T. H. Koornwinder and A. L. Schwartz (1997, Constr. Approx13, 537–567), we obtain an orthogonal basis for every subspace. As an application of the orthogonal decomposition, we define the Hankel- and Toeplitz-type operators and discuss Sp-criteria for these kinds of operators
Generalized Plane Delta-Shock Waves for n-Dimensional Zero-Pressure Gas Dynamics

AbstractWith the help of a generalized plane wave solution, we study a type of generalized plane delta-shock wave for the n-dimensional zero-pressure gas dynamics and refine its generalized Rankine–Hugoniot relation which is a system of ordinary equations. This relation describes accurately the character of the generalized plane delta-shock: location, propagation speed, and weight. Under a suitable entropy condition, four different explicit constructions of solutions for a kind of Riemann problem with Radon measure as initial data are established uniquely. The overtaking of two plane delta-shocks forming a new generalized plane delta-shock is also investigated. Finally, the 2-D Riemann problem with four pieces of initial data is solved in a simplified situation
The Stability Radius of Fredholm Linear Pencils

AbstractLet T and S be two bounded linear operators from Banach spaces X into Y, and suppose that T is Fredholm and dimN(T−λS) is constant in a neighborhood of λ=0. Let d(T;S) be the supremum of all r>0 such that dimN(T−λS) and codimR(T−λS) are constant for all λ with |λ|<r. It is a consequence of more general results due to H. Bart and D. C. Lay (1980, Studia Math.66, 307–320) that d(T;S)=limn→∞γn(T;S)1/n, where γn(T;S) are some non-negative (extended) real numbers. For X=Y and S=I, the identity operator, we have γn(T;S)=γ(Tn), where γ is the reduced minimum modulus. A different representation of the stability radius d(T;S) is obtained here in terms of the spectral radii of generalized inverses of T. The existence of generalized resolvents for Fredholm linear pencils is also considered
Parabolic Approximation to Variational Problems with Double Obstacles

AbstractIn this paper, we discuss an elliptic variational inequality with double obstacles in a finite interval which can be rewritten as a nonlinear boundary value problem. We use a parabolic initial-boundary value problem to approximate it and prove that every smooth solution of the variational problem can be regarded as a limit of a smooth solution of a parabolic problem. From the viewpoint of numerical computation, parabolic equations are easy and can be solved using extremely stable standard computer routines. Therefore, our result is useful both for the theory of differential equations and for the numerical computation
Asymptotic Representations for Root Vectors of Nonselfadjoint Operators and Pencils Generated by an Aircraft Wing Model in Subsonic Air Flow

AbstractThis paper is the second in a series of several works devoted to the asymptotic and spectral analysis of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works by A. V. Balakrishnan. The model is governed by a system of two coupled integrodifferential equations and a two parameter family of boundary conditions modeling the action of the self-straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter. This generalized resolvent operator is a finite-meromorphic function on the complex plane having the branch cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. In the first paper and in the present one, our main object of interest is the dynamics generator of the differential parts of the system. It is a nonselfadjoint operator in the energy space with a purely discrete spectrum. In the first paper, we have shown that the spectrum consists of two branches and have derived their precise spectral asymptotics. In the present paper, we derive the asymptotical approximations for the mode shapes. Based on the asymptotical results of these first two papers, in the next paper, we will discuss the geometric properties of the mode shapes such as minimality, completeness, and the Riesz basis property in the energy space
Asymptotic Results for Injection of Reactive Solutes from a Three-Dimensional Well

AbstractIn this paper we consider some asymptotic aspects related to the profile of a reactive solute, which is injected from a well (radius ϵ>0) into a three-dimensional porous medium. We present a convergence result for ϵ↓0 as well as the large time behaviour. Regarding the latter we show that the solute profile evolves in a self-similar way towards a stationary distribution and we give an estimate for the rate of the convergence. This paper extends earlier work of C. J. van Duijn and M. A. Peletier (1996, J. Reine Angew. Math.479, 77–98), where the two-dimensional case was treated
On Various Modes of Scalar Convergence in L0(X)

AbstractA sequence {fn} of strongly measurable functions taking values in a Banach space X is scalarly null a.e. (resp. scalarly null in measure) if x*fn→0 a.e. (resp. x*fn→0 in measure) for every x*∈X*. Let 1≤p≤∞. The main questions addressed in this paper are whether an LP(X)-bounded sequence that is scalarly null a.e. will converge weakly a.e. (or have a subsequence which converges weakly a.e.), and whether an Lp(X)-bounded sequence that is scalarly null in measure will have a subsequence that is scalarly null a.e. The answers to these and other similar questions often depend upon p and upon the geometry of X
Applications of Feller–Reuter–Riley Transition Functions

AbstractA Feller–Reuter–Riley function is a Markov transition function whose corresponding semigroup maps the set of the real-valued continuous functions vanishing at infinity into itself. The aim of this paper is to investigate applications of such functions in the dual problem, Markov branching processes, and the Williams-matrix. The remarkable property of a Feller–Reuter–Riley function is that it is a Feller minimal transition function with a stable q-matrix. By using this property we are able to prove that, in the theory of branching processes, the branching property is equivalent to the requirement that the corresponding transition function satisfies the Kolmogorov forward equations associated with a stable q-matrix. It follows that the probabilistic definition and the analytic definition for Markov branching processes are actually equivalent. Also, by using this property, together with the Resolvent Decomposition Theorem, a simple analytical proof of the Williams' existence theorem with respect to the Williams-matrix is obtained. The close link between the dual problem and the Feller–Reuter–Riley transition functions is revealed. It enables us to prove that a dual transition function must satisfy the Kolmogorov forward equations. A necessary and sufficient condition for a dual transition function satisfying the Kolmogorov backward equations is also provided
Low-Gain Integral Control of Well-Posed Linear Infinite-Dimensional Systems with Input and Output Nonlinearities

AbstractTime-varying low-gain integral control strategies are presented for asymptotic tracking of constant reference signals in the context of exponentially stable, well-posed, linear, infinite-dimensional, single-input–single-output, systems—subject to globally Lipschitz, nondecreasing input and output nonlinearities. It is shown that applying error feedback using an integral controller ensures that the tracking error is small in a certain sense, provided that (a) the steady-state gain of the linear part of the system is positive, (b) the reference value r is feasible in an entirely natural sense, and (c) the positive gain function t↦k(t) is ultimately sufficiently small and not of class L1. Under a weak restriction on the initial data it is shown that (a), (b), and (c) ensure asymptotic tracking. If, additionally, the impulse response of the linear part of the system is a finite signed Borel measure, the global Lipschitz assumption on the output nonlinearity may be considerably relaxed
Symplectic Structure of Discrete Hamiltonian Systems

AbstractThis paper is concerned with the symplectic structure of discrete nonlinear Hamiltonian systems. The results are related to an open problem that was first proposed by C. D. Ahlbrandt [J. Math. Anal. Appl.180 (1993), 498–517] discussed elsewhere in the literature. But we give a different statement and different proof. Under a solvable condition, we show that the solution operator of a discrete nonlinear Halmiltonian system is symplectic. Then its phase flow is a discrete one-parameter family of symplectic transformations and preserves the phase volume
Extension of a Theorem of Ferenc Lukács from Single to Double Conjugate Series

AbstractA theorem of Ferenc Lukács states that if a periodic function f is integrable in the Lebesgue sense and has a discontinuity of the first kind at some point x, then the mth partial sum of the conjugate series of its Fourier series diverges at x at the rate of logm. The aim of the present paper is to extend this theorem to the rectangular partial sum of the conjugate series of a double Fourier series when conjugation is taken with respect to both variables. We also consider functions of two variables which are of bounded variation over a rectangle in the sense of Hardy and Krause. As a corollary, we obtain that the terms of the Fourier series of a periodic function f of bounded variation over the square [−π,π]×[−π,π] determine the atoms of the finite Borel measure induced by f
Lyapunov Functional for Multiple Delay General Lurie Systems with Multiple Non-linearities

AbstractIn this paper, the absolute stability problem of a multiple delay general Lurie control system with multiple non-linearities is considered. Necessary and sufficient conditions are obtained for the existence of the Lyapunov functional of extended Lurie form with negative definite derivative. From those conditions, a very general algebraic criterion for absolute stability is obtained, which extends the previous results. A concise example illustrates the effectiveness of the present results
Schrödinger-Type Evolution Equations in Lp(Ω)

AbstractLet Ap be a strongly elliptic operator of order 2m(m∈N) in Lp(Ω)(1≤p≤∞,Ω a bounded domain of Rn) with Dirichlet or Neumann boundary conditions. Of concern is the Cauchy problem for Schrödinger-type evolution equations in Lp(Ω)[formula]By showing that iAp is the generator of a C0 group on a certain interpolation space, we obtain results of wellposedness for (∗), which are stronger than those derived from the regularized or integrated groups on Lp(Ω). As a by-product, it is shown that iAp is the generator of a (ω+Ap)−r-regularized group (ω>0) on Lp(Ω) for all[formula
Global Attractors for a Semilinear Hyperbolic Equation in Viscoelasticity

AbstractA semilinear partial differential equation of hyperbolic type with a convolution term describing simple viscoelastic materials with fading memory is considered. Regarding the past history (memory) of the displacement as a new variable, the equation is transformed into a dynamical system in a suitable Hilbert space. The dissipation is extremely weak, and it is all contained in the memory term. Longtime behavior of solutions is analyzed. In particular, in the autonomous case, the existence of a global attractor for solutions is achieved
Generalized Hyperfunctions on the Circle

AbstractWe give an embedding of the space B(T) of hyperfunctions on the unit circle T in a differential algebra H(T) whose elements are called generalized hyperfunctions. This allows us to define the product of two hyperfunctions without any restriction. We also define pointvalues of a hyperfunction: these pointvalues are elements of an algebra C whose set of invertible elements is denoted C*. In Section 2 we recall and make precise some basic results on classical spaces of functions on T. Section 3 is devoted to our main results: we characterize the set H*(T) of invertible elements of H(T), and, since a generalized hyperfunction may vanish at all classical points without being zero, we give a vanishing theorem. We conclude our work with the study of the Cauchy problem: u′+fu+gu2=0;u(z0)=μ, where f,g∈H(T),z0∈T, and μ∈C*, by giving an existence theorem for a solution u∈H*(T)
Existence of Weak Solutions for a Hyperbolic Model of Chemosensitive Movement

AbstractA hyperbolic model for chemotaxis and chemosensitive movement in one space dimension is considered. In contrast to parabolic models for chemotaxis the hyperbolic model allows us to take the dependence of the particle speed on external stimuli explicitly into account. This qualitatively covers recent experiments on chemotaxis in which it has been shown that particles adapt their speed to the surrounding environment. The model presented here consists of two hyperbolic differential equations of first order coupled with an elliptic equation. We assume that the speed depends on the external stimulus only (and not on its gradients). In that case solutions with steep gradients are expected which have the interpretation of moving swarms. A notion of weak solutions for this hyperbolic chemotaxis model is presented and the global existence of weak solutions is shown. The proof relies on the vanishing viscosity method; i.e., we obtain the weak solution as the limit of classical solutions of an associated parabolically regularized problem for vanishing viscosity parameter. Numerical simulations demonstrate phenomena like swarming behaviour and formation of steep gradients
A Note on a-Weyl's Theorem

AbstractIf T or T* is log-hyponormal then for every f∈H(σ(T)), Weyl's theorem holds for f(T), where H(σ(T)) denotes the set of all analytic functions on an open neighborhood of σ(T). Moreover, if T* is p-hyponormal or log-hyponormal or M-hyponormal then for every f∈H(σ(T)), a-Weyl's theorem holds for f(T). Also it is shown that if we let A≔{T∈B(H):T* is log-hyponormal} then the approximate point spectrum, the essential approximate point spectrum, and the Browder essential approximate point spectrum are continuous in A
Some Contributions to Stochastic Asymptotic Stability and Boundedness via Multiple Lyapunov Functions

AbstractMost of the existing results on stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in this paper to establish some sufficient criteria for locating the limit sets of solutions of stochastic differential equations. From them follow many useful results on stochastic asymptotic stability and boundedness, which enable us to construct the Lyapunov functions much more easily in applications. In particular, the well-known classical theorem on stochastic asymptotic stability is a special case of our more general results. These show clearly the power of our new results
On the Dirichlet Problem for the Nonlinear Diffusion Equation in Non-smooth Domains

AbstractWe study the Dirichlet problem for the parabolic equation ut=Δum,m>0, in a bounded, non-cylindrical and non-smooth domain Ω⊂RN+1,N≥2. Existence and boundary regularity results are established. We introduce a notion of parabolic modulus of left-lower (or left-upper) semicontinuity at the points of the lateral boundary manifold and show that the upper (or lower) Hölder condition on it plays a crucial role for the boundary continuity of the constructed solution. The Hölder exponent 12 is critical as in the classical theory of the one-dimensional heat equation ut=uxx
Periodic and Homoclinic Solutions of Extended Fisher–Kolmogorov Equations

AbstractIn this paper we study the existence of periodic solutions of the fourth-order equations uiv−pu″−a(x)u+b(x)u3=0 and uiv−pu″+a(x)u−b(x)u3=0, where p is a positive constant, and a(x) and b(x) are continuous positive 2L-periodic functions. The boundary value problems (P1) and (P2) for these equations are considered respectively with the boundary conditions u(0)=u(L)=u″(0)=u″(L)=0. Existence of nontrivial solutions for (P1) is proved using a minimization theorem and a multiplicity result using Clark's theorem. Existence of nontrivial solutions for (P2) is proved using the symmetric mountain-pass theorem. We study also the homoclinic solutions for the fourth-order equation uiv+pu″+a(x)u−b(x)u2−c(x)u3=0, where p is a constant, and a(x), b(x), and c(x) are periodic functions. The mountain-pass theorem of Brezis and Nirenberg and concentration-compactness arguments are used