On Semiconvexity Properties of Rotationally Invariant Functions in Two Dimensions

Let f be a function defined on the set M ^2×2 of all 2 by 2 matrices that is invariant with respect to left and right multiplications of its argument by proper orthogonal matrices. The function f can be represented as a function of the signed singular values of its matrix argument. The paper expresses the ordinary convexity, polyconvexity, and rank 1 convexity of f in terms of its representation      

Rotationally Invariant Rank 1 Convex Functions

Let f be a function on the set M ⁿ xn of all n by n real matrices. If f is rotationally invariant with respect to the proper orthogonal group, it has a representation \tilde f through the signed singular values of the matrix argument ?∈ M^nxn. Necessary and sufficient conditions are given for the rank 1 convexity of f in terms of \tilde f . Accepted 20 December 2000. Online Publication 18 May, 2001.     

On the Oka principle in a Banach space,I

 Let X be a complex Banach space with a countable unconditional basis, Ω⊂X pseudoconvex open, G a complex Banach Lie group. We show that a Runge–type approximation hypothesis on X, G (which we also prove for G a solvable Lie group) implies that any holomorphic cocycle on Ω with values in G can be resolved holomorphically if it can be resolved continuously. Received: 1 March 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): 32L05, 32E30, 46G20 RID="*" ID="*" Kedves Szímuskának. RID="*" ID="*" To my dear Wife.     

A result of Ambrosetti–Prodi type for first-order ODEs with cubic non-linearities. Part II

In this paper we study the scalar equation x′=f(t,x), where f(t,x) has cubic non-linearities in x and we prove that this equation has at most three bounded separate solutions. We say that λ∈ℝ is a critical value of the equation x′=f(t,x)+λx if this equation has a degenerate bounded solution and we exhibit two classes of functions f such that the above equation has a unique critical value. Received: February 4, 2000; in final form: March 19, 2002?Published online: April 14, 2003 RID="*" ID="*"This paper was partially supported by CDCHT, Universidad de los Andes.     

The central limit theorem for Markov chains started at a point

 The aim of this paper is to prove a central limit theorem and an invariance principle for an additive functional of an ergodic Markov chain on a general state space, with respect to the law of the chain started at a point. No irreducibility assumption nor mixing conditions are imposed; the only assumption bears on the growth of the L ² -norms of the ergodic sums for the function generating the additive functional, which must be with . The result holds almost surely with respect to the invariant probability of the chain. Received: 17 October 2001 / Revised version: 5 April 2002 / Published Online: 24 October 2002 Mathematics Subject Classification (2000): 60F05, 60J05     

Differentiable functions defined in closed sets. A problem of Whitney

In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of ℝ ⁿ is the restriction of a function of class 𝒞 ^p . A necessary and sufficient criterion was given in the case n=1 by Whitney, using limits of finite differences, and in the case p=1 by Glaeser (1958), using limits of secants. We introduce a necessary geometric criterion, for general n and p, involving limits of finite differences, that we conjecture is sufficient at least if X has a “tame topology”. We prove that, if X is a compact subanalytic set, then there exists q=q _X (p) such that the criterion of order q implies that f is 𝒞 ^p . The result gives a new approach to higher-order tangent bundles (or bundles of differential operators) on singular spaces. Oblatum 21-XI-2001 & 3-VII-2002?Published online: 8 November 2002 RID="*" ID="*"Research partially supported by the following grants: E.B. – NSERC OGP0009070, P.M. – NSERC OGP0008949 and the Killam Foundation, W.P. – KBN 5 PO3A 005 21.     

On conditionally positive definite dot product kernels

Let m and n be fixed, positive integers and P a space composed of real polynomials in m variables. The authors study functions f ： R →R which map Gram matrices, based upon n points of R^m, into matrices, which are nonnegative definite with respect to P Among other things, the authors discuss continuity, differentiability, convexity, and convexity in the sense of Jensen, of such functions     

On Certain Densely Invariant Quantities of Linear Operators

Let L(X,Y) denote the class of linear transformations T:D(T) ? X → Y where X and Y are normed spaces. A quantity f is called densely invariant if for every system L(X, Y) and every operator T ? L(X,Y) we have f(T/E)= f(T) whenever E is a core of T. The density invariance of certain well known quantities is established. In case Y is complete and T is closable, a stronger property is shown to hold for some of these quantitites, namely invariance under restriction to dense subspaces.     

Morse theory on spaces of braids and Lagrangian dynamics

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a Morse-Conley homotopy index.?In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits. Oblatum 11-V-2001 & 13-XI-2002?Published online: 24 February 2003 RID="*" ID="*"The first author was supported by NSF DMS-9971629 and NSF DMS-0134408. The second author was supported by an EPSRC Fellowship. The third author was supported by NWO Vidi-grant 639.032.202.     

Invariant means

Let m and M be symmetric means in two and three variables, respectively. We say that M is type 1 invariant with respect to m if M(m(a,c),m(a,b),m(b,c))≡M(a,b,c). If m is strict and isotone, then we show that there exists a unique M which is type 1 invariant with respect to m. In particular, we discuss the invariant logarithmic mean L₃, which is type 1 invariant with respect to L(a,b)=(b−a)/(logb−loga). We say that M is type 2 invariant with respect to m if M(a,b,m(a,b))≡m(a,b). We also prove existence and uniqueness results for type 2 invariance, given the mean M(a,b,c). The arithmetic, geometric, and harmonic means in two and three variables satisfy both type 1 and type 2 invariance. There are means m and M such that M is type 2 invariant with respect to m, but not type 1 invariant with respect to m (for example, the Lehmer means). L₃ is type 1 invariant with respect to L, but not type 2 invariant with respect to L.     

A result of Ambrosetti–Prodi type for first-order ODEs with cubic non-linearities. Part I

In this paper, we prove a result of Ambrosetti–Prodi type for the problem x′=f(t,x)+λx, where f(t,x) is T-periodic in t, f(t,0)≡0 and f(t,x) has “cubic nonlinearities”. Received: February 4, 2000?Published online: April 14, 2003 RID="*" ID="*"This paper was partially supported by CDCHT, Universidad de los Andes.     

Proper actions of lattices on contractible manifolds

Every lattice Γ in a connected semi-simple Lie group G acts properly discontinuously by isometries on the contractible manifold G/K (K a maximal compact subgroup of G). We prove that if Γ acts on a contractible manifold W and if either?1) the action is properly discontinuous, or?2) W is equipped with a complete Riemannian metric, the action is by isometries and with unbounded orbits, G is simple with finite center and rank >1,?then dimW≥dimG/K. Oblatum 19-I-2001 & 24-IV-2002?Published online: 5 September 2002 RID="*" ID="*"The authors gratefully acknowledge support from the National Science Foundation.     

Multiple Solutions for a Class of Hemivariational Inequalities Involving Periodic Energy Functionals

In this paper we prove firstly that if f:X→ℝ is a locally Lipschitz function, bounded from below and invariant to a discrete group of dimension N is a suitable sense, acting on a Banach space X, then the problem: find u∈X such that o∈∂ f(u) (here ∂f(u) denotes Clarke's generalized gradient of f at x) admits at least N+1 orbits of solutions. Then, for a class of discrete groups G of isometries of a Hilbert space X, we establish an existence result for infinitely many G-orbits of eigensolutions to the problem: find u∈X such that λΛu∈∂f(u) for some λ∈ℝ, where Λ:X→X* stands for the duality map. The last two sections are devoted to applications of the abstract existence results to hemivariational inequalities possessing invariance properties. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

An obstruction to the positivity of relative Yamabe invariants

For a supergroup , we describe an obstruction to the existence of positive scalar curvature metrics with minimal boundary condition on a compact n-dimensional -manifold W with nonempty boundary M, , in terms of the bordism class [M] in the Stolz obstruction group associated to [St2]. In par ticular, when W is a 5-dimensional spin manifold and the -invariant of a connected component of M is nonzero, we prove that W does not admit a positive scalar curvature metric with minimal boundary condition. Received: 4 July 2001; in final form: 5 February 2002 / Published online: 8 November 2002 RID="*" ID="*" Partially supported by the Grants-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 11640070.     

Injective Hulls of T0 Topological Fibre Spaces

We investigate the existence of injective hulls (with respect to the class of embeddings) in the categories Top ₀/B of T₀ topological fibre spaces over B. We prove that, if f:AB has a restriction to the image injective in Top ₀/f(A), (A,f) has an injective hull in Top ₀/B if and only if f(A) is locally closed in B, where A denotes the union of non-indiscrete fibres of f.     

Homotopy invariance of AF-embeddability

We prove that AF-embeddability is a homotopy invariant in the class of separable exact C ^*-algebras. This work was inspired by Spielberg's work on homotopy invariance of AF-embeddability and Dadarlat's serial works on AF-embeddability of residually finite dimensional C ^*-algebras. Submitted: February 2002.     

A basic relation between invariants of matrices under the action of the special orthogonal group

The first fundamental theorem of invariant theory for the action of the special orthogonal group onm tuples of matrices by simultaneous conjugation is proved in [2]. In this paper, as a first step in the direction of establishing the second fundamental theorem, we study a basic identity betweenSO(n, K) invariants ofm matrices.     

Normal limit theorems for symmetric random matrices

Using the machinery of zonal polynomials, we examine the limiting behavior of random symmetric matrices invariant under conjugation by orthogonal matrices as the dimension tends to infinity. In particular, we give sufficient conditions for the distribution of a fixed submatrix to tend to a normal distribution. We also consider the problem of when the sequence of partial sums of the diagonal elements tends to a Brownian motion. Using these results, we show that if O _n is a uniform random n×n orthogonal matrix, then for any fixed k>0, the sequence of partial sums of the diagonal of O ^k _n tends to a Brownian motion as n→∞. Received: 3 February 1998 / Revised version: 11 June 1998