pinecone/core-2020-05-10-deduplication huggingface dataset:
Critical Exponent for the Bipolar Blowup in a Semilinear Parabolic Equation

AbstractThe blowup of solutions of the initial-boundary value problem[formula]is studied, where Ω is a cone inRN. We say that a solutionuexhibits bipolar blowup if[formula]for someT<∞. It is shown that there exists a critical exponentp*>1 for the bipolar blowup in the following sense. If 1<p≤p*, then there exist arbitrarily small initial data such that the solution exhibits the bipolar blowup, whereas ifp>p*, then the bipolar blowup does not occur for any sufficiently small initial data. The value ofp* is expressed in terms of the dimensionNand the second Dirichlet eigenvalue of the Laplace–Beltrami operator on Ω∩SN−1. In the case of Ω=R, we definek-polar blowup, and determine critical exponents for thek-polar blowup fork=1,2,3,…
An Application of a Browder-Type Fixed Point Theorem to Generalized Variational Inequalities

AbstractThe purpose of this paper is to provide an application of a non-compact version, due to Park, of Browder's fixed point theorem to generalized variational inequalities. In a non-compact setting, we establish a fairly general existence theorem on a generalized variational inequality using the result of Park. Our main result is not only a strict generalization of Ding and Tan's result without assuming any continuity of given functions, but also a purely infinite dimensional version of recent results of Yao and Guo. As an immediate consequence, we obtain an extension of a result of Browder concerned with generalized variational inequalities without assuming the local convexity of the underlying topological vector space
Asymptotically Isometric Copies ofc0in Banach Spaces

AbstractIf Γ is an uncountable set, then any equivalent renorming ofc0(Γ) contains an asymptotically isometric copy ofc0. IfYis an infinite dimensional closed subspace of (c0‖·‖∞), thenYcontains an asymptotically isometric copy ofc0. IfXandYare two infinite dimensional Banach spaces and ifXcontains an asymptotically isometric copy ofc0, then the injective tensor product ofXandY,X⊗̂εY, contains a complemented asymptotically isometric copy ofc0. Similarly, ifXcontains an asymptotically isometric copy ofc0, then the Lebesgue–Bochner spaceLp([0,1],X) contains a complemented asymptotically isometric copy ofc0
The Third Order Ordinary Differential Equations Admitting the Painlevé I and II as First Integrals

AbstractThird order ordinary differential equations admitting the Painlevé equations PI and PII as first integrals are completely characterized. This is done by determining necessary and sufficient conditions for these equations to belong to the equivalence class of certain canonical equations. This characterization also includes equations admitting first integrals which are equivalent to PI and PII. The general form of these equations is given. In some cases we determine the transformation taking the equation into a canonical one
A Stone Space Approach to the Existence of Bounded Common Extensions

AbstractLet A and B be fields of subsets of a setXand let μ: A→Rand ν: B→Rbe bounded, finitely additive measures (i.e., charges). We give global necessary and sufficient conditions on A and B under which any bounded, consistent charges μ and ν have a bounded common extension. Conditions on A and B are given under which any 0–1 valued charges μ and ν have an integer valued common extension. Our results, together with Example 4.4infra, serve to answer open questions left unresolved by A. Basile, K. P. S. Bhaskara Rao, and R. M. Shortt (Proc. Amer. Math. Soc.121,1994, 137–143), and thereby end a line of enquiry begun by Z. Lipecki (Czechoslovak Math. J.36,1986, 489–494)
Discrete Variable Transformations on Symplectic Systems and Even Order Difference Operators

AbstractChange of independent variablet=1/xmotivates variable step size discretizations of even order differential operators. We develop variable change methods for discrete symplectic (i.e., J-orthogonal) systems. This enables us to perform simultaneous change of independent and dependent variables on discrete linear Hamiltonian systems and on newly defined even order variable step size formally self adjoint difference operators. These variable changes yield a new system which is related to the original system by an operator identity. We generalize results of Bohner and Došlý on transformations of formally self-adjoint scalar difference operators. They only considered a change of dependent variable whereas these methods allowy(xn)=μ(xn)z(tn) wheretn=f(xn). These variable change results bring the subject of transformation theory for even order difference operators closer to the known transformation theory in the continuous case
On Some Weighted Hardy Type Classes of One-Parametric Holomorphic Functions. I. Function Properties and Mellin Transforms

AbstractSeveral classes of functions of two variables (one complex and one real) are considered. These functions belong to some weighted Hardy type classes with respect to the complex variable in a wedge or in a strip and are either essentially bounded, or bounded, or continuous in a sense with respect to the real parameter. Properties of such functions and of their Mellin images are investigated. Classes of functions of two real variables are also studied in order to describe boundary properties of the one-parametric holomorphic functions. The analysis is based on properties of the corresponding classes of functions of one variable, which are studied here too
Bessel Polynomial Expansions in Spaces of Holomorphic Functions

AbstractGeneral methods of nuclear Fréchet spaces in conjunction with the theory of Köthe sequence spaces have been used to obtain a basis criterion for power sequence spaces. How this basis criterion is thus yielding a refinement of Cannon's criterion in the case of space of holomorphic functions is illustrated here. Hence, we give a characterization of those infinite matrices for which the respective Bessel polynomials constitute a basis in the space of holomorphic functions. Consequently, an answer to the question posed by H. L. Krall and O. Frink (Trans. Amer. Math. Soc.65, 1949, 100–115) concerning the actual expansion of a holomorphic function in terms of Bessel polynomials is given. Also, we append a new criterion for basis transforms in such a space of holomorphic functions by means of order of magnitude of coefficients of polynomials
The Structure of the Fundamental Solution of the Time-Independent Transport Equation

AbstractThe structure of singularities of the fundamental solution of the time-independent one-speed transport equation with smooth coefficients is studied. This structure is more complicated than in the elliptic equation case: two first terms in the decomposition have a singular support and two others are regular unbounded distributions. The exact expressions for the first two terms and estimates for the others are presented. The regular parts of these singularities contain some significant information concerning the medium and hence the exact description of these singularities may be useful in nondestructive control
Compactness Framework ofLpApproximate Solutions for Scalar Conservation Laws

AbstractIn this paper, we study the strong convergence of a sequence of uniformLploc(R×R+) bounded approximate solutions {uϵ(x,t)} to the following scalar conservation laws[formula]with initial data[formula]Without the convexity assumption and growth condition at infinity forf(x,t,u), we prove strong convergence of a subsequence of {uε(x,t)}. Under a more general growth condition than those in the previous work, we prove the existence of weak solution for the equation. The result obtained here generalizes those in earlier work. Some applications of the results are also given at the end of this paper
On Some Weighted Hardy Type Classes of One-Parametric Holomorphic Functions. II. Partial Volterra Operators in Parameter

AbstractSome classes of partial Volterra operators acting (with respect to a real variable) on a function of one complex and one real variable are explored. The case when the operator kernels depend additionally on a complex parameter is also considered. It is proved that Volterra operators from these classes and the resolvent operators act in weighted Hardy type classes of one parametric holomorphic functions defined on a wedge or on a strip of the complex plane. Moreover, the Volterra operators commutate with the Mellin operators in these classes. The half line Paley–Wiener theorem for the resolvent of the Volterra operator is extended to operators with parameter. Volterra operators tending to operators of the convolution type as well as Volterra operators with fading memory are studied too
Persistence of Invariant Sets for Dissipative Evolution Equations

AbstractWe show that results concerning the persistence of invariant sets of ordinary differential equations under perturbation may be applied directly to a certain class of partial differential equations. Our framework is particularly well-suited to encompass numerical approximations of these partial differential equations. Specifically, we show that for a class of PDEs with aC1inertial form, certain natural numerical approximations possess an inertial form close to that of the underlying PDE in theC1norm
Reproducing Kernels and Conformal Mappings in Rn

AbstractIn this paper we look at the theory of reproducing kernels for spaces of functions in a Clifford algebra R0,n. A first result is that reproducing kernels of this kind are solutions to a minimum problem, which is a non-trivial extension of the analogous property for real and complex valued functions. In the next sections we restrict our attention to Szegö and Bergman modules of monogenic functions. The transformation property of the Szegö kernel under conformal transformations is proved, and the Szegö and Bergman kernels for the half space are calculated
A Representation of Projection Lattices and Their States in Euclidean Space

AbstractWe propose a representationr:L∪Ω→Rν, where L is the collection of closed subspaces of ann-dimensional real, complex, or quaternionic Hilbert space H, or equivalently, the projection lattice of this Hilbert space, where Ω is the set of all states ω:L→[0,1]. The value that ω∈Ω takes ina∈L is given by the scalar product of the representative points (r(a) andr(ω)). The representationr(a∨b) of the join of two orthogonal elementsa,b∈L is equal tor(a)+r(b). The convex closure of the representation of Σ, the set of atoms of L, is equal to the representation of Ω
Solution of the Ulam Stability Problem for Euler–Lagrange Quadratic Mappings

AbstractIn 1940 S. M. Ulam proposed at the University of Wisconsin theproblem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist.” In 1968 S. U. Ulam proposed the moregeneral problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?” In 1978 P. M. Gruber proposed theUlam type problem: “Suppose a mathematical object satisfies a certain property approximately. Is it then possible to approximate this object by objects, satisfying the property exactly?” According to P. M. Gruber this kind of stability problems is of particular interest in probability theory and in the case of functional equations of different types. In 1982–1996 we solved the above Ulam problem, or equivalently the Ulam type problem for linear mappings and established analogous stability problems. In this paper we first introduce newquadratic weighted meansandfundamental functional equationsand then solve theUlam stability problemfornon-linear Euler–Lagrange quadratic mappingsQ:X→Y, satisfying a mean equation and functional equation[formula]for all 2-dimensional vectors (x1,x2)∈X2, withXa normed linear space (Y≔ a real complete normed linear space), and any fixed pair (a1,a2) of realsaiand any fixed pair (m1,m2) of positive realsmi(i=1,2), [formula
Pattern Selection in Double-Diffusive Ferro-Convection

AbstractPattern formation due to double-diffusive convection in ferrofluids is investigated in the presence of an externally applied transverse magnetic field. The critical value of the Rayleigh numberRfor steady state bifurcation is found to be different from that for Hopf bifurcation in contrast to ordinary fluids where the two critical values are the same. The stability of steady state patterns—rolls, squares, and hexagons—has been discussed. For magnetic fluids, the rolls are found to be structurally stable on the square lattice, whereas on the hexagon lattice bifurcation may lead to either rolls or hexagons depending upon the value of the nondimensional magnetic parameterM
Smooth Solution of the Compressible Navier–Stokes Equations in an Unbounded Domain with Inflow Boundary Condition

AbstractThe barotropic compressible Navier–Stokes equations in an unbounded domain are studied. We prove the unique existence of the solution (u,p) of the system (1.1) in the Sobolev spaceHk+3×Hk+2provided that the derivatives of the data of the problem are sufficiently small, wherek≥0 is any integer. The proof follows from an analysis of the linearized problem, the solvability of the continuity equation, and the Schauder fixed point theory. Similar smoothness results are obtained for a linearized form of (1.1)
Inverse Backscattering Problem for the Acoustic Equation in Even Dimensions

AbstractWe show that the sound speedc(x) of the acoustic wave equation in any even dimension can be uniquely determined by the backscattering data provided that it is close to a constant. In the three-dimensional case, P. Stefanov and G. Uhlmann (SIAM J. Math. Anal.28,1997, 1191–1204) have proved a similar result. Their method takes advantage of the inversion formula for the Radon transform in odd dimensions being a local operator. This is not true in even dimensions. Moreover, the odd-dimensional Lax and Phillips modified Radon transform fails to work in even dimensions. In this paper, we overcome these difficulties and prove an even-dimensional version of Stefanov and Uhlmann's result
Global Weak Solutions for a Parabolic System Modeling a One-Dimensional Miscible Flow in Porous Media

AbstractWe consider an initial boundary value problem for a nonlinear differential system of two equations. Such a system is formed by the equations of compressible miscible flow in a one-dimensional porous medium. No assumption about the mobility ratio is involved. Under some reasonable assumptions on the data, we prove the existence of a global weak solution. Our basic approach is the semi-Galerkin method. We use the technique of renormalized solutions for parabolic equations in the derivation ofa prioriestimates
On a New Method for Solving Irregular Problems

AbstractWe suggest a new approach to the investigation of boundary value problems with a parameter in the case when the parameter appears in both the equation and the boundary conditions. The method is applied to the Regge problem. In spite of the fact that the Regge problem is Birkhoff-irregular (see G. D. Birkhoff,Trans. Amer. Math. Soc.9, 1908, 373–395), the problem is coercive but with a defect with respect to the spectral parameter. We also prove 2-fold completeness of root functions of the Regge problem
On Hilbert's Integral Inequality

AbstractIn this paper, we generalize Hilbert's integral inequality and its equivalent form by introducing three parameterst,a, andb.Iff,g∈L2[0,∞), then[formula]where π is the best value. The inequality (1) is well known as Hilbert's integral inequality, and its equivalent form is[formula]where π2is also the best value (cf. [1, Chap. 9]). Recently, Hu Ke made the following improvement of (1) by introducing a real functionc(x),[formula]wherek(x)=2/π∫∞0(c(t2x)/(1+t2))dt−c(x),1−c(x)+c(y)≥0, andf,g≥0 (cf. [2]). In this paper, some generalizations of (1) and (2) are given in the following theorems, which are other than those in [2]
Periodic Solutions for Non-linear Equations of Structured Populations

AbstractWe show the existence of non-trivial periodic solutions for a class of non-linear equations, model of age-structured populations. To this aim we use the theory of Centre Manifold for a class of abstract differential equations introduced by Desch and Schappacher, and show that a Hopf Bifurcation Theorem can be proved for this class of equations. Finally, we consider some special cases, and present an algorithm for the determination of the type of bifurcation, and show that in the considered cases the bifurcation is always supercritical
Absolute Euler Summability of Fourier Series

AbstractIn this paper we study an application of Euler means for the absolute summability of Fourier series of functions of bounded variations. Due to the incomparability of the Euler and the Cesáro methods, it is not apparent what results to expect for the Euler means under the hypotheses similar to those for which extensive studies with Cesáro and Riesz means have been done over the years. In this paper we prove theorems that generalise some results of (i) B. K. Ray and B. Patra and (ii) P. Chandra and also exhibit some sort of linkage between their seemingly unrelated results
Preservation of Stability in Delay Equations under Delay Perturbations

AbstractWe consider a class of linear delay equations with perturbed time lags and present conditions which guarantee that the asymptotic stability of the trivial solution of the equation at hand is preserved under these perturbations. As an example we show how our results can be used to obtain an estimate on the maximum allowable sampling interval in the stabilization of a hybrid system with feedback delays. We also present applications of our perturbation theorem to obtain stability conditions for delay equations with multiple delays
Inequalities and Asymptotics for Riccati Matrix Difference Operators

AbstractIn the first part inequalities for solutions of Riccati matrix difference equations are obtained which correspond to the linear Hamiltonian difference system[formula]whereAk,Bk,Ck,Xk,Ukaren×n-matrices with symmetricBkandCk. If the matricesXkare invertible, then the matricesQk=UkX−1ksolve the Riccati matrix difference equation[formula]In contrast to some recent papers dealing with these equations we do not assume that the matricesBkare invertible. The second part of the paper deals with the asymptotic behaviour of solutionsQk(λ), as|λ|→∞, of the special Riccati matrix difference equation which corresponds to the Sturm–Liouville equation[formula]of even order 2nwith constant coefficientsr0,…,rn
Pseudo-Differential Operators Associated with the Jacobi Differential Operator

AbstractWe introduce two classesSmandSm0, of symbols withSm⊂Sm0. We define pseudo-differential operators associated with symbols belonging to these classes. We prove that a pseudo-differential operator associated with a symbol inSm0is a continuous linear mapping from some subspace of the Schwartz space into itself. Next we consider symbols inSmand we give an integral representation of the pseudo-differential operators associated with these symbols. Finally, using the dual convolution associated with Jacobi functions, we show that these pseudo-differential operators satisfy a certainL1-norm inequality
Weak Convergence for Rectangle-Indexed Weighted Multivariate EmpiricalU-Statistic Processes under Mixing Conditions

AbstractIn this paper we consider the weak convergence of the weighted multivariate empiricalU-statistic indexed by rectangles under mixing conditions in the Skorohod topology. It is a generalization of a result of F. H. Ruymgaart and M. C. A. van Zuijlen (J. Statist. Plann. Inference32, 1992, 259–269) from the i.i.d. case to the dependent case. M. Harel, C. A. O'Cinneide and M. L. Puri (J. Multivariate Anal.48, 1994, 297–314) gave only a generalization to the univariate case of the weighted empiricalU-statistic indexed by point; a generalization to the multivariate case is obtained by B. Ragbi (J. Statist. Plann. Inference,61, 1997, 1–16). There are a number of interesting applications dealing with the Cramer–Von Mises statistic or the Anderson–Darling statistic given in Section 4; applications to a Markov process as well as to an ARMA process are also provided
On the Weyl Spectrum: Spectral Mapping Theorem and Weyl's Theorem

AbstractIt is shown that ifTis a dominant operator or an analytic quasi-hyponormal operator on a complex Hilbert space and iffis a function analytic on a neighborhood of σ(T), then σw(f(T))=f(σw(T)), where σ(T) and σw(T) stand respectively for the spectrum and the Weyl spectrum ofT; moreover, Weyl's theorem holds forf(T)+Fif “dominant” is replaced by “M-hyponormal,” whereFis any finite rank operator commuting withT. These generalize earlier results for hyponormal operators. It is also shown that there exist an operatorTand a finite rank operatorFcommuting withTsuch that Weyl's theorem holds forTbut not forT+F. This answers negatively a problem raised by K. K. Oberai (Illinois J. Math.21, 1977, 84–90). However, ifTis required to be isoloid, then the statement that Weyl's theorem holds forTwill imply it holds forT+F
A Necessary and Sufficient Condition for Global Existence for a Degenerate Parabolic Boundary Value Problem

AbstractConsider the degenerate parabolic boundary value problemut=Δϕ(u)+f(u) on Ω×(0,∞) in which Ω is a bounded domain inRNand theC([0,∞)) functionsfand φ are nonnegative and nondecreasing with ϕ(s)f(s)>0 ifs>0 and ϕ(0)=0. Assume homogeneous Neumann boundary conditions and an initial condition that is nonnegative, nontrivial, and continuous on Ω. Because the function ϕ is not sufficiently nice to allow this problem to have a classical solution, we consider generalized solutions in a manner similar to that of Benilan, Crandall, and Sacks [Appl. Math. Optim.17(1988), 203–224]. We show that this initial boundary value problem has such a nonnegative generalized solution if and only ifi∞0ds/(1+f(s))=∞
Approximation and Fixed-Point Theorems for Condensing Composites of Multifunctions

AbstractOne of the authors previously extended an interesting (best approximation) theorey of Ky Fan to condensing maps defined on a closed ball in a Banach space and defined on a closed convex subset in a Hilbert space. Recently there have appeared a number of new results on fixed points of so-called Kakutani factorizable multifunctions or composites of acyclic or other type of multifunctions defined on convex subsets of topological vector spaces. In this paper, we extend Lin's results to the multifunction (or set-valued) version for the above-mentioned factorizable multifunctions. Applications to fixed-point theorems are given
Abstract Convexity and Fixed Points

AbstractThe purpose of this paper is to extend Himmelberg's fixed-point theorem, replacing the usual convexity in topological vector spaces with an abstract topological notion of convexity that generalizes classical convexity as well as several metric convexity structures found in the literature. We prove the existence, under weak hypotheses, of a fixed point for a compact approachable map, and we provide sufficient conditions under which this result applies to maps whose values are convex in the abstract sense mentioned above