The Morse index,repeller-attractor pairs and the connection index for semiflows on noncompact spaces

This paper extends the Morse index theory of C. C. Conley to semiflows π on a noncompact meric space X. π is assumed to satisfy a hypothesis related to conditional α-contraction. We collect background material, define quasi-index pairs and the Morse index of a compact, isolated invariant set K, and prove that the Morse index is a connected simple system. We study repeller-attractor pairs in K, define index triples, and prove their existence and several properties leading to the concepts of the connection index, the connection map and the splitting class. Finally, we consider paths (continuous families) of pairs (π, K) and study continuations of the Morse and the connection indices along such paths. The present paper is a sequel to the author's previous work: On the homotopy index for infinite-dimensional semiflows (Trans. Amer. Math. Soc.269 (1982), 351–382).     

CH, a problem of Rolewicz and bidiscrete systems

We give a construction under CH of a non-metrizable compact Hausdorff space K such that any uncountable ‘nice’ semi-biorthogonal sequence in C(K) must be of a very specific kind. The space K has many nice properties, such as being hereditarily separable, hereditarily Lindelöf and a 2-to-1 continuous preimage of a metric space, and all Radon measures on K are separable. However K is not a Rosenthal compactum.We introduce the notion of a bidiscrete system in a compact space K. These are subsets of K² which determine biorthogonal systems of a special kind in C(K) that we call nice. We note that for every infinite compact Hausdorff space K, the space C(K) has a bidiscrete system and hence a nice biorthogonal system of size d(K), the density of K.     

Semigroup-valued solutions of some composite equations

Let X be a linear space over the field K of real or complex numbers and (S, °) be a semigroup. We determine all solutions of the functional equation $$f(x+g(x)y)=f(x)\circ f(y)\quad \text{for}\quad x,y\in X$$ in the class of pairs of functions (f,g) such that f : X → S and g : X → K satisfies some regularity assumptions. Several consequences of this result are presented.     

Berezin-Toeplitz quantization on Lie groups

Let K be a connected compact semisimple Lie group and KC its complexification. The generalized Segal-Bargmann space for KC is a space of square-integrable holomorphic functions on KC, with respect to a K-invariant heat kernel measure. This space is connected to the “Schrödinger” Hilbert space L²(K) by a unitary map, the generalized Segal-Bargmann transform. This paper considers certain natural operators on L²(K), namely multiplication operators and differential operators, conjugated by the generalized Segal-Bargmann transform. The main results show that the resulting operators on the generalized Segal-Bargmann space can be represented as Toeplitz operators. The symbols of these Toeplitz operators are expressed in terms of a certain subelliptic heat kernel on KC. I also examine some of the results from an infinite-dimensional point of view based on the work of L. Gross and P. Malliavin.     

A remark on CD 0(K)-spaces

A representation of the CD ₀(K)-space is given in [1, 2] for a compact Hausdorff space K without isolated points. We generalize this to an arbitrary countably compact space K without any assumption on isolated points.     

Contractive type non-self mappings on metric spaces of hyperbolic type

Let (X,d) be a metric space of hyperbolic type and K a nonempty closed subset of X. In this paper we study a class of mappings from K into X (not necessarily self-mappings on K), which are defined by the contractive condition (2.1) below, and a class of pairs of mappings from K into X which satisfy the condition (2.28) below. We present fixed point and common fixed point theorems which are generalizations of the corresponding fixed point theorems of ?iri? [L.B. ?iri?, Quasi-contraction non-self mappings on Banach spaces, Bull. Acad. Serbe Sci. Arts 23 (1998) 25-31; L.B. ?iri?, J.S. Ume, M.S. Khan, H.K.T. Pathak, On some non-self mappings, Math. Nachr. 251 (2003) 28-33], Rhoades [B.E. Rhoades, A fixed point theorem for some non-self mappings, Math. Japon. 23 (1978) 457-459] and many other authors. Some examples are presented to show that our results are genuine generalizations of known results from this area.     

Existence of log canonical flips and a special LMMP

Let (X/Z,B+A) be a Q-factorial dlt pair where B,A??0 are Q-divisors and K _X +B+A?? _Q 0/Z. We prove that any LMMP/Z on K _X +B with scaling of an ample/Z divisor terminates with a good log minimal model or a Mori fibre space. We show that a more general statement follows from the ACC for lc thresholds. An immediate corollary of these results is that log flips exist for log canonical pairs.     

Separable subspaces of affine function spaces on convex compact sets

Let K be a compact convex subset of a separated locally convex space (over R) and let Ap(K) denote the space of all continuous real-valued affine mappings defined on K, endowed with the topology of pointwise convergence on the extreme points of K. In this paper we shall examine some topological properties of Ap(K). For example, we shall consider when Ap(K) is monolithic and when separable compact subsets of Ap(K) are metrizable.     

Commutative cochains of germs of differential forms for Euclidean compacta

The purpose of the paper is for any compactum K⊂Rn to construct a space Cp(K) of commutative germs of C^∞-differential p-forms, which is a reflexive locally convex topological vector space with the Cp(K) as its dual.     

q-Inverting pairs of shape 1, 2, 1

Let V denote a vector space with finite positive dimension over an algebraically closed field 𝔽. Let K, K* be a q-inverting pair on V, an ordered pair of invertible linear transformations on V satisfying certain conditions. In this article we study the q-inverting pairs with shape 1, 2, 1. We define a pair of scalars called the parameter pair for K, K*. We give six bases for V and give the action of K, K ^?1, K* and (K*)^?1 on each of these bases, respectively. We classify the q-inverting pairs of shape 1, 2, 1 in terms of the parameter pairs. We conclude with some trace formulae involving the parameter pair.     

Universal space in the Cartesian product of Peano curves without free arcs

We prove that there is the universal space for the class of n-dimensional separable metric spaces in the Cartesian product K₁×?×Kn+1 of Peano curves without free arcs. It is also shown that the set of embeddings of any n-dimensional separable metric space X into this universal space is a residual set in C(X,K₁×?×Kn+1). Other properties of product of Peano curves without free arcs are also proved.     

Fixed point properties of semigroups of non-expansive mappings

In recent years, there have been considerable interests in the study of when a closed convex subset K of a Banach space has the fixed point property, i.e. whenever T is a non-expansive mapping from K into K, then K contains a fixed point for T. In this paper we shall study fixed point properties of semigroups of non-expansive mappings on weakly compact convex subsets of a Banach space (or, more generally, a locally convex space). By considering the classes of bicyclic semigroups we answer two open questions, one posted earlier by the first author in 1976 (Dalhousie) and the other posted by T. Mitchell in 1984 (Virginia). We also provide a characterization for the existence of a left invariant mean on the space of weakly almost periodic functions on separable semitopological semigroups in terms of fixed point property for non-expansive mappings related to another open problem raised by the first author in 1976.     

Extension of Gaussian cylinder set measures

Let E be a one-to-one continuous map of the real and separable Hilbert space H into the real and separable Hilbert space K, with E having dense range. One considers Gaussian cylinder set measures on H defined by weak covariance operators. Such cylinder set measures may be used to induce, through E, Gaussian cylinder set measures on K. The result of this paper extends a result of Sato: it characterizes the norm of the spaces K for which the induced measure extends to a probability measure on the Borel sets of K. Such a result is of interest in the robustness study of signal detection.     

Quadratic forms over polynomial rings over global fields

It is proved that a quadratic space over the polynomial extension of a global field K is extended from K if it is extended from K_v for every completion K_v of K.     

Banach spaces of universal disposition

In this paper we present a method to obtain Banach spaces of universal and almost-universal disposition with respect to a given class M of normed spaces. The method produces, among others, the only separable Banach space of almost-universal disposition with respect to the class F of finite-dimensional spaces (Gurari? space G); or the only, under CH, Banach space with density character the continuum which is of universal disposition with respect to the class S of separable spaces (Kubis space K). We moreover show that K is isomorphic to an ultrapower of the Gurari? space and that it is not isomorphic to a complemented subspace of any C(K)-space. Other properties of spaces of universal disposition are also studied: separable injectivity, partially automorphic character and uniqueness.     

Submaximal and door compactifications

In this paper, a characterization is given for compact door spaces. We, also, deal with spaces X such that a compactification K(X) of X is submaximal or door.Let X be a topological space and K(X) be a compactification of X.We prove, here, that K(X) is submaximal if and only if for each dense subset D of X, the following properties hold:

(i)

D is co-finite in K(X);

(ii)

for each x∈K(X)?D, {x} is closed.

If X is a noncompact space, then we show that K(X) is a door space if and only if X is a discrete space and K(X) is the one-point compactification of X.     

On convex bodies containing a given number of lattice points

Given a convex body K in Euclidean space, a necessary and sufficient condition is established in order that for each n there exists a homothetic copy of K containing exactly n lattice points. Similar theorems are proved for congruent copies of K and for discrete sets other than lattices.     

Duality of metric entropy in Euclidean space

Let K be a convex body in a Euclidean space, K° its polar body and D the Euclidean unit ball. We prove that the covering numbers N(K,tD) and N(D,tK°) are comparable in the appropriate sense, uniformly over symmetric convex bodies K, over t>0 and over the dimension of the space. In particular this verifies the duality conjecture for entropy numbers of linear operators, posed by Pietsch in 1972, in the central case when either the domain or the range of the operator is a Hilbert space. To cite this article: S. Artstein et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Coupled connections on a compact Riemann surface

We consider holomorphic differential operators on a compact Riemann surface X whose symbol is an isomorphism. Such a differential operator of order n on a vector bundle E sends E to K^⊗n_X⊗E, where K_X is the holomorphic cotangent bundle. We classify all those holomorphic vector bundles E over X that admit such a differential operator. The space of all differential operators whose symbol is an isomorphism is in bijective correspondence with the collection of pairs consisting of a flat vector bundle E over X and a holomorphic subbundle of E satisfying a transversality condition with respect to the connection.