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Showing papers in "Mathematische Zeitschrift in 1989"



Journal ArticleDOI
TL;DR: In this article, the authors consider the problems of Riemannian evolution and demontre l'existence d'une solution globale faible, on the assumption that there exists a global solution.
Abstract: On considere le probleme d'evolution ∂ t u−Δ M u+λγ N (u)=0 dans M×R, u(•,0)=u 0 , sur M ou M et N sont des varietes de Riemann lisses compactes et u 0 l'application harmonique de M dans N. On demontre l'existence d'une solution globale faible

319 citations



Journal ArticleDOI
TL;DR: On etudie des applications symplectiques non lineaires as mentioned in this paper, construction d'une capacite symplectique, Problemes de plongement, and Probleme de rigidite
Abstract: On etudie des applications symplectiques non lineaires. Capacites symplectiques. Construction d'une capacite symplectique. Problemes de plongement. Problemes de rigidite

264 citations



Book ChapterDOI
TL;DR: The theory of non-commutative symmetric spaces was introduced by Gohberg and Krein this paper, who considered the problem of symmetrically normed ideals of bounded linear operators in Hilbert space.
Abstract: In this paper we survey some aspects of the theory of non-commutative Banach function spaces, that is, spaces of measurable operators associated with a semi- finite von Neumann algebra. These spaces are also known as non-commutative symmetric spaces. The theory of such spaces emerged as a common generalization of the theory of classical (“commutative”) rearrangement invariant Banach function spaces (in the sense of W.A.J. Luxemburg and A.C. Zaanen) and of the theory of symmetrically normed ideals of bounded linear operators in Hilbert space (in the sense of I.C. Gohberg and M.G. Krein). These two cases may be considered as the two extremes of the theory: in the first case the underlying von Neumann algebra is the commutative algebra L ∞ on some measure space (with integration as trace); in the second case the underlying von Neumann algebra is B (ℌ), the algebra of all bounded linear operators on a Hilbert space ℌ (with standard trace). Important special cases of these non-commutative spaces are the non-commutative L p-spaces, which correspond in the commutative case with the usual L p-spaces on a measure space, and in the setting of symmetrically normed operator ideals they correspond to the Schatten p-classes \( \mathfrak{S}_p \) .

172 citations


Journal ArticleDOI
TL;DR: In this paper, a fonction caracteristique non commutative was introduced for calculing le spectre de #7B-A, on obtient des resultats sur la generation de semi-groupes par des matrices operateur.
Abstract: Sur l'espace produit R×L 2 (R + ) on considere l'operateur #7B-A:=(a δ 0 ) ou D=d/dx est defini sur D(A)=W 1 (R + ), δ 0 est la mesure de Dirac en 0, c∈L 2 (R + ) est un operateur de R dans L 2 (R + ) et #7B-A a un domaine D(#7B-A)=R×W 1 (R + ). On introduit une fonction caracteristique non commutative donnant une facon efficace pour calculer le spectre de #7B-A. On obtient des resultats sur la generation de semi-groupes par des matrices operateur

143 citations


Journal ArticleDOI
TL;DR: On demontre l'existence de solutions faibles globales a des problemes d'evolution des applications harmoniques de varietes de Riemann compactes (sans bord) dans des spheres.
Abstract: On demontre l'existence de solutions faibles globales a des problemes d'evolution des applications harmoniques de varietes de Riemann compactes (sans bord) dans des spheres

122 citations


Journal ArticleDOI

98 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the uniqueness of the potential q(t, x) depends on t and satisfies the following conditions: (i) qe C ~ (~t x ~-~:~), (ii) there exist some positive constants p, C, N such that q( t, x ) = 0 for Ixl >P and Iq(t, x)l < C(1 + Itl) N.
Abstract: te]R, x~N\", n > 3 odd. The potential q(t, x) depends on t and satisfies the following conditions: (i) qe C ~ (~t x ~-~:~), (ii) there exist some positive constants p, C, N such that q(t, x ) = 0 for Ixl >P and Iq(t, x)l < C(1 + Itl) N. The aim of this paper is to prove that the potential q(t, x) is uniquely determined by the scattering data. The inverse scattering problem for stationary (time independent) potentials q(x) has been attacked by many authors. The reader should consult [2], [,10], [18] and the references given there for the history and the recent progress in the analysis of this problem. Most of the works in this direction deal with the Schr6dinger equation. Nevertheless it is clear that the results obtained in these papers are applicable to the wave equation u , A u + q ( x ) u=0. It is known that for stationary potentials the uniqueness holds. The proof is based on the Born approximation of the scattering amplitude A(k, co', og) as k ~ (see [-7], [,16], [-22]). The situation considerably changes when we deal with time dependent potentials. The techniques used in the papers cited above are not available in this case. One of the reasons is that we cannot use the tools from the spectral theory. The local energy decay for (1.1) was examined by Tamura [,28], while the existence of the scattering operator was proved in [,1], [8], [19], [20] for certain classes of potentials q(t, x). In [-8] Ferreira and Perla Menzala proved that if the scattering operators related to qi(t, x), i= 1, 2 coincide and if ql >~-~q2 then ql=q2, provided that q~ are non-negative, \"small\" in a suitable sense and qt=O(Itl -~) as l t l ~ o% 0

98 citations


Journal ArticleDOI
TL;DR: On associe a un enlacement L dans S 3 un groupe O(L), appele le groupe de Π-variete d'orbite de L.
Abstract: On associe a un enlacement L dans S 3 un groupe O(L), appele le groupe de Π-variete d'orbite de L. On etudie certaines proprietes du groupe O(L) reliees a la geometrie de l'enlacement L



Journal ArticleDOI
TL;DR: In this article, a natural homomorphism A from 7h (Sym (P, co)) to G (p, co) is defined, which measures the average over P (with respect to the symplectic measure) of the action integral around the trajectories of a loop of symplectomorphisms.
Abstract: Let (P, co) be a compact, simply connected symplectic manifold, Sym (P, co) its group of automorphisms. The period group F(P, co)c R is defined to be the image of [co] x H2 (P; Z) under the integration pairing H 2 (P; R) x H2 (P; Z) ~ R. Since [co] 4:0, the quotient G(P, co)=R/F(P, co) is either a circle or the quotient of a circle by a countable dense subgroup. The purpose of this paper is to define and study a natural homomorphism A from 7h (Sym (P, co)) to G (P, co). Simply stated, A measures the average over P (with respect to the symplectic measure) of the action integral around the trajectories of a loop of symplectomorphisms. In terms of hamiltonians, we give a formula for A which becomes especially simple when the loop is a 1-parameter subgroup, in which case we can make some concrete computations. These are related to the work of Guillemin-Sternberg [9], and Duistermaat-Heckmann [7] on momentum mappings. We also give an interpretation of A in terms of lifting symplectomorphisms from P to a prequantization, i.e. a G(P, co) bundle over P with a connection whose curvature form is co. Here, we extend the theory from the usual case where G(P, co) is a circle by using the concept of diffeological structure introduced by Souriau [16]. In a sequel to this paper [19], taking as a starting point the observation that the graphs of symplectomorphisms are lagrangian submanifolds in a product manifold, we shall study the possible extension of A to an invariant on loops in the space of lagrangian submanifolds in a symplectic manifold. It turns out that this invariant should be defined as the holonomy of a connection whose definition involves choosing a probability measure on each lagrangian submanifold. The connection is flat only for certain ways of choosing these measures, and the curvature in the general case is closely related to the "geometric phase" introduced by Berry [4], which is a subject of current interest among theoretical and experimental physicists as well as mathematicians [1, 6, I0, 11, 13, 173.

Journal ArticleDOI
TL;DR: In this article, the authors studied mixed multiplicities of ideals and used them to calculate multiplicity of certain homogeneous ideals of extended Rees algebras and to give a complete characterization of those parameter ideals whose extended rees algesbras are Cohen-Macaulay with minimal multiplicity at their maximal homogenous ideals.
Abstract: The aim of this paper is to study mixed multiplicities of ideals and use them to calculate multiplicities of certain homogeneous ideals of extended Rees algebras and to give a complete characterization of those parameter ideals whose extended Rees algebras are Cohen-Macaulay with minimal multiplicity at their maximal homogeneous ideals. Let (R, m) be a local ring of dimension d. Let I be an m-primary ideal. It is well known that the length of the artinian ring R/F, I(R/F), is a polynomial of degree d in r for all large r. The coefficient of nd/d ! in this polynomial, called the multiplicity of I, denoted by e(I), is a well-understood and useful invariant of/. Suppose J is another m-primary ideal. Then it is natural to consider l (R/FJ s) for positive integers r and s. It is proved in [B] that for large values of r and s, I(R/FJ ~) is given by a polynomial P(r, s) of total degree d in r and s. Moreover, the terms of total degree d in P(r, s) have the form






Journal ArticleDOI
TL;DR: In this article, it was shown that the probability of finding the value a(n) in the continuous spectrum of a quantum system is at most 2.l 2, and at most slowly increasing (plane waves for V = 0).
Abstract: Let a quantum mechanical system be in the normalized state f; measuring the observable quantity which is represented by H, we have the probability I~.l 2 to find the value a(n) and the probability density to find a(2) in the continuous spectrum is Ic~(2)l 2. b) Let for example H = A + V be a one-body Hamiltonian (i.e. V(x) ~ 0 for [xJ--.oo). We expect the T~ to be bounded or at most slowly increasing (plane waves for V=0). On the other hand, for values E$a(H) the solutions T of (--A + V) T = E 7 ~ should grow fast (exponentially) at infinity. These conjectures can be summarized to

Journal ArticleDOI
TL;DR: In this paper, l'existence des solutions periodiques for une classe de systemes hamiltoniens obtenus comme des perturbations de l'hamiltonien keplerien K(p,q)=1/2.
Abstract: On considere l'existence des solutions periodiques pour une classe de systemes hamiltoniens obtenus comme des perturbations de l'hamiltonien keplerien K(p,q)=1/2.|p| 2 −|q| − α, α>0





Journal ArticleDOI
TL;DR: On considere l'existence locale en temps des solutions for un systeme hyperbolique quasilineaire avec des conditions aux limites de type Neumann.
Abstract: On considere l'existence locale en temps des solutions pour un systeme hyperbolique quasilineaire avec des conditions aux limites de type Neumann

Journal ArticleDOI
TL;DR: In this article, an infinite column R consisting of a homogeneous isotropic incompressible solid with the semilinear constitutive equation is considered, where the reader is referred to [-20] for background material.
Abstract: Here A is the two-dimensional Laplacian, g is a scalar function, and Uo, u~, f are given functions. Equation (1) occurs in the description of antiplane shear motions of certain viscoelastic solids; for background material, the reader is referred to [-20]. Let y(~, t) denote the position of a particle with reference position ~ at time t, and consider an infinite column ~2 • R consisting of a homogeneous isotropic incompressible solid with the semilinear constitutive equation


Journal ArticleDOI
TL;DR: In this paper, the authors consider complex manifolds M which are holomorphically separable, i.e., the global holomorphic functions (9 (M)) separate the points of M. The most obvious examples of such actions are given by Reinhardt domains in ~", with the usual action O.z=(ei~..., e i ~).
Abstract: Here we consider complex manifolds M which are holomorphically separable, i.e., the global holomorphic functions (9 (M) separate the points of M. We investigate T n (n-torus) actions on M, where T"={(ei~ ei~ 0eN"}, and n is the complex dimension of M. We assume that the action is effective, i.e., if 0ET ~ is nonzero, then there exists z e M with O.z+z. Our final assumption on the action is one of smoothness; the map T n x M ~ M given by (0, z ) , O . z is C 1 in both variables and holomorphic in z. The most obvious examples of such actions are given by Reinhardt domains in ~", with the usual action O.z=(ei~ . . . . , e i ~ The standard Reinhardt action may also be changed as follows: if A is an algebraic automorphism of T n, then O.az=(AO).z. An obvious question that arises is whether all T"-actions can arise in this way. Our main results are contained in the following four theorems.