Subspaces of Lp Isometric to Subspaces of ℓ p

We present three results on isometric embeddings of a (closed, linear) subspace X of L_p=L_p[0,1] into _p . First we show that if p 2N, then X is isometrically isomorphic to a subspace of _p if and only if some, equivalently every, subspace of L_p which contains the constant functions and which is isometrically isomorphic to X, consists of functions having discrete distribution. In contrast, if p 2N; and X is finite-dimensional, then X is isometrically isomorphic to a subspace of _p , where the positive integer N depends on the dimension of X, on p , and on the chosen scalar field. The third result, stated in local terms, shows in particular that if p is not an even integer, then no finite-dimensional Banach space can be isometrically universal for the 2-dimensional subspaces of L_p .     

Example of Fitting Polynomials to Data

The table below shows the accumulated total number of reported casesof AIDS in the United States since 1983,based on Center for Disease Control statistics.At first thought,we might expect that the growth In the total number of reported cases of AIDSis exponential and so we would attempt to fit an exponential function to this set of data.The re-sulting best-fit exponential function,found with a calculator,is     

Trend to equilibrium of renormalized solutions to reaction–cross-diffusion systems

The convergence to equilibrium of renormalized solutions to reaction–cross-diffusion systems in a bounded domain under no-flux boundary conditions is studied. The reactions model complex balanced chemical reaction networks coming from mass-action kinetics and thus do not obey any growth condition, while the diffusion matrix is of cross-diffusion type and hence nondiagonal and neither symmetric nor positive semi-definite, but the system admits a formal gradient-flow or entropy structure. The diffusion term generalizes the population model of Shigesada, Kawasaki and Teramoto to an arbitrary number of species. By showing that any renormalized solution satisfies the conservation of masses and a weak entropy–entropyproduction inequality, it can be proved under the assumption of no boundary equilibria that all renormalized solutions converge exponentially to the complex balanced equilibrium with a rate which is explicit up to a finite dimensional inequality.     

Application of an idea of Voronoĭ to lattice packing

The geometric variant of a criterion of Vorono? says, a lattice packing of balls in \(\mathbb{E }^d\) has (locally) maximum density if and only if it is eutactic and perfect. This article deals with refinements of Vorono?’s result and extensions to lattice packings of smooth convex bodies. Versions of eutaxy and perfection are used to characterize lattices with semi-stationary, stationary, maximum and ultra-maximum lattice packing density, where ultra-maximality is a sharper version of maximality. Surprisingly, for balls, the lattice packings with maximum density have ultra-maximum density. To make the picture more complete, for \(d=2,3\) , we specify the lattices that provide lattice packings of balls with maximum properties. These lattices are related to Bravais types. Finally, similar results of a duality type are given.     

Addendum to ‘asymptotic eigenfunctions of mixed problems of Stekloff's type’: Extension to exterior problems

Summary The asymptotic eigenvalues and eigenfunctions of the problem: V=0 in the exterior domain to a regular closed curvec and V/=V(s)g(s) onc, are obtained. It is shown that _n =0(n) andV _n (s) becomes trigonometric.

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Résumé SoitD la région extérieure d'une courbe fermée régulière,c; on détermine le comportement asymptotique des valeurs propres et fonctions propres du problème: V=0 dansD et V/=V(s)g(s) surc. On peut démontrer que _n =0(n) et lesV _n (s) deviennent des fonctions trigonométriques.

     

Asymptotic Behavior of Solutions to a Class of Systems of Delay Differential Equations

The authors investigate the asymptotic behavior of solutions to a class of systems of delay differential equations. It is shown that every bounded solution of such a class of systems tends to a constant vector as t→∞. Our results improve and extend some corresponding ones already known.     

Application of an idea of Voronoĭ to lattice zeta functions

A major problem in the geometry of numbers is the investigation of the local minima of the Epstein zeta function. In this article refined minimum properties of the Epstein zeta function and more general lattice zeta functions are studied. Using an idea of Voronoĭ, characterizations and sufficient conditions are given for lattices at which the Epstein zeta function is stationary or quadratic minimum. Similar problems of a duality character are investigated for the product of the Epstein zeta function of a lattice and the Epstein zeta function of the polar lattice. Besides Voronoĭ type notions such as versions of perfection and eutaxy, these results involve spherical designs and automorphism groups of lattices. Several results are extended to more general lattice zeta functions, where the Euclidean norm is replaced by a smooth norm.     

Some of Melvin Henriksen?s contributions to spaces of ideals

The three of us have written this note to discuss Mel Henriksen?s joint paper with us, Joincompact spaces, continuous lattices andC^?-algebras. In this paper we learned that the space of closed primal ideals of a C^?-algebra is a continuous lattice, so it is joincompact when equipped with the lower and Scott topologies. In addition to its mathematics, we discuss how Mel brought us into this work, and his and its continuing influence on us.     

Application of an idea of Vorono? to John type problems

Using an idea of Vorono?, many John type and minimum position problems in dimension d can be transformed into more accessible geometric problems on convex subsets of the -dimensional cone of positive definite quadratic forms. In this way, we prove several new John type and minimum position results and give alternative versions and extensions of known results. In particular, we characterize minimum ellipsoidal shells of convex bodies and, in the typical case, show their uniqueness and determine the contact number. These results are formulated also in terms of the circumradius of convex bodies. Next, circumscribed ellipsoids of minimum surface area of a convex body and the corresponding minimum position problem are studied. Then we investigate John type characterizations of minimum positions of a convex body with respect to moments and the product of a moment and the moment of the polar body. The technique used in this context, finally, is applied to obtain corresponding results for the mean width and the surface area.     

The Asymptotic Property of Solutions to a Class of Integrodifferential Equations

The asymptotic property of solutions to a class of integrodifferential systems containing a small parameter singularly is studied.     

Kinematics of Mechanisms to the Second Order – Application to the Closed Mechanisms

The degree of freedom of a closed mechanism is the dimension of a subset M of R ⁿ , M being the inverse image of the unity by the closure function f : (q ₁, ..., q _n ) f(q ₁, ..., q _n ), where q ₁, ..., q _n are the articular coordinates. We first study the regular points for the mapping f from R ⁿ into the Lie group of displacements and, second, study the singularities of the mapping f. The classical theory of mechanisms considers, often implicitly, that f is a subimmersion. Here, the calculations are made in a larger case, up to second order, and the results are then slightly different. The case of such classical mechanisms as Bennett, Bricard, and Goldberg mechanisms, justify the considerations of this more general framework and the example of a Bricard mechanism is chosen as an application of the method.     

The Aysmptotic Behavior of Solution to a Class of Inhomogeneons Systems of Gas Dynamics

1.IntroductionInthispaper,weresearchsomeclassofinhomogeneonssystemsofgasdynamic:wherevisreciprocalofgasdensity,uisvelocity,6isabsolutetemperatureofgas.p,k,aareconstants.Suppose(t,x)e[0,+~)X[0,fiingeneral,thentheboundaryconditionareu(t,0)~u(t,l)~0,sx(t,0)~gx(t,1)~0,(l.2)theinitialconditionis(v,u,6)(0,x)~(v,,uo,0,)(x),(1.3)andsatisfiessuitableconditionsWheninI相似文献   

A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton–Jacobi equations

We investigate the large-time behavior of the value functions of the optimal control problems on the n-dimensional torus which appear in the dynamic programming for the system whose states are governed by random changes. From the point of view of the study on partial differential equations, it is equivalent to consider viscosity solutions of quasi-monotone weakly coupled systems of Hamilton–Jacobi equations. The large-time behavior of viscosity solutions of this problem has been recently studied by the authors and Camilli, Ley, Loreti, and Nguyen for some special cases, independently, but the general cases remain widely open. We establish a convergence result to asymptotic solutions as time goes to infinity under rather general assumptions by using dynamical properties of value functions.     

Constants related to operators of class Cρ

Let T be a Hilbert-space operator of class $$\left\| {S^{ - 1} TS} \right\| \leqslant 1\,\,\,\,and\,\,\left\| {S^{ - 1} } \right\| \cdot \left\| S \right\| \leqslant \max (1,\rho )$$ in the sense of Sz.-Nagy and Foia?. Then there exists an invertible operator S such that ∥S^?1TS∥≤1 and ∥S^?1∥·∥S∥≤max(1,ρ). The following estimate is valid for the bilateral limit: $$\mathop {lim}\limits_{n \to \infty } \{ \left\| {T^n h} \right\|^2 + \left\| {T*^n h} \right\|^2 \} \leqslant max(2,\rho )\left\| h \right\|^2 .$$ Here the constants max(1,ρ) and max(2,ρ) are the best possible in their respective cases.     

Existence of Positive Solutions to a Fractional-Kirchhoff System

Let Ω be a bounded smooth domain in R^N(N≥3).Assuming that 0 0 are constants,we consider the existence results for positive solutions of a class of fractional elliptic system below,■Under some assumptions of h_i(x,u,v,▽u,▽v)(i=1,2),we get a priori bounds of the positive solutions to the problem(1.1) by the blow-up methods and rescaling argument.Based on these estimates and degree theory,we establis...     

Several applications of the theory of random conjugate spaces to measurability problems

The central purpose of this paper is to illustrate that combining the recently developed theory of random conjugate spaces and the deep theory of Banach spaces can, indeed, solve some difficult measurability problems which occur in the recent study of the Lebesgue (or more general, Orlicz)-Bochner function spaces as well as in a slightly different way in the study of the random functional analysis but for which the measurable selection theorems currently available are not applicable. It is important that this paper provides a new method of studying a large class of the measurability problems, namely first converting the measurability problems to the abstract existence problems in the random metric theory and then combining the random metric theory and the relative theory of classical spaces so that the measurability problems can be eventually solved. The new method is based on the deep development of the random metric theory as well as on the subtle combination of the random metric theory with classical space theory.     

Long-Time Stabilization of Solutions to the Ginzburg–Landau Equations of Superconductivity

 The long-time dynamical properties of solutions (φ,A) to the time-dependent Ginzburg–Landau (TDGL) equations of superconductivity are investigated. The applied magnetic field varies with time, but it is assumed to approach a long-time asymptotic limit. Sufficient conditions (in terms of the time rate of change of the applied magnetic field) are given which guarantee that the dynamical process defined by the TDGL equations is asymptotically autonomous, i.e., it approaches a dynamical system as time goes to infinity. Analyticity of an energy functional is used to show that every solution of the TDGL equations asymptotically approaches a (single) stationary solution of the (time-independent) Ginzburg–Landau equations. The standard “φ = − ∇ · A” gauge is chosen.     

Long-Time Stabilization of Solutions to the Ginzburg–Landau Equations of Superconductivity

 The long-time dynamical properties of solutions (φ,A) to the time-dependent Ginzburg–Landau (TDGL) equations of superconductivity are investigated. The applied magnetic field varies with time, but it is assumed to approach a long-time asymptotic limit. Sufficient conditions (in terms of the time rate of change of the applied magnetic field) are given which guarantee that the dynamical process defined by the TDGL equations is asymptotically autonomous, i.e., it approaches a dynamical system as time goes to infinity. Analyticity of an energy functional is used to show that every solution of the TDGL equations asymptotically approaches a (single) stationary solution of the (time-independent) Ginzburg–Landau equations. The standard “φ = − ∇ · A” gauge is chosen. (Received 30 June 2000; in revised form 30 December 2000)