Equisingularity of families of map germs between curves

Given a finite map germ f : (X, 0) → (Y, 0) between complex analytic reduced space curves, we look at invariants which control the topological triviality and the Whitney equisingularity in families of this type of map germs. In the case that (Y, 0) is smooth, the main invariant is the Milnor number of a function on a curve. We deduce some applications to the equisingularity of families of finitely determined map germs ${f : (\mathbb{C}^2, 0) \to (\mathbb{C}^2, 0)}$ and ${f : (\mathbb{C}^2, 0) \to (\mathbb{C}^3, 0)}$ .     

Limits of tangents and minimality of complex links

We show that the complex link of a large class of space germs (X,x₀) is characterized by its “simplicity”, among the Milnor fibres of functions with isolated singularity on X. This amounts to the minimality of the Milnor number, whenever this number is defined. Such a phenomenon has been first pointed out in case (X,x₀) is an isolated hypersurface singularity, by Teissier (Cycles évanescents, sections planes et conditions de Whitney, in: Singularités à Cargèse 1972, Asterisque, Nos. 7 et 8, Soc. Math. France, Paris, 1973, pp. 285-362).     

Analytic mappings between LB-spaces and applications in infinite-dimensional Lie theory

We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: The group DiffGerm (K, X) of germs of analytic diffeomorphisms around a compact set K in a Banach space X and the group ${\bigcup_{n\in\mathbb {N}}G_n}We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: The group DiffGerm (K, X) of germs of analytic diffeomorphisms around a compact set K in a Banach space X and the group è_{n ? \mathbb N}G_n{\bigcup_{n\in\mathbb {N}}G_n} where the G _n are Banach Lie groups.     

The Goldstine Theorem for asymmetric normed linear spaces

It is well known that if (X,q) is an asymmetric normed linear space, then the function qs defined on X by qs(x)=max{q(x),q(−x)}, is a norm on the linear space X. However, the lack of symmetry in the definition of the asymmetric norm q yields an algebraic asymmetry in the dual space of (X,q). This fact establishes a significant difference with the standard results on duality that hold in the case of locally convex spaces. In this paper we study some aspects of a reflexivity theory in the setting of asymmetric normed linear spaces. In particular, we obtain a version of the Goldstine Theorem to these spaces which is applied to prove, among other results, a characterization of reflexive asymmetric normed linear spaces.     

Classifying Hilbert bundles

A Hilbert bundle (p, B, X) is a type of fibre space p:B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying (m, n)-bundles over (X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [A, G_m($\text{C}$ⁿ)]/[X, U(n)] ¦ A, and the isomorphism classes of (m, n)-bundles over (X, A) which are trivial over X, A.     

Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces

Let X be a non-empty set and F:X×X→X be a given mapping. An element (x,y)∈X×X is said to be a coupled fixed point of the mapping F if F(x,y)=x and F(y,x)=y. In this paper, we consider the case when X is a complete metric space endowed with a partial order. We define generalized Meir-Keeler type functions and we prove some coupled fixed point theorems under a generalized Meir-Keeler contractive condition. Some applications of our obtained results are given. The presented theorems extend and complement the recent fixed point theorems due to Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379-1393].     

Classifying Hilbert bundles. II

A Hilbert bundle (p, B, X) is a type of fibre space p: B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X, even when X is connected. We give two “homotopy” type classification theorems for Hilbert bundles having primarily finite dimensional fibres. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle over (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. As a special case, we show that if X is a compact metric space, C₊X the upper cone of the suspension SX, then the isomorphism classes of (m, n)-bundles over (SX, C₊X) are in one-to-one correspondence with the members of [X, V_m($\text{C}$ⁿ)] where V_m($\text{C}$ⁿ) is the Stiefel manifold. The results are all applicable to the classification of separable, continuous trace C¹-algebras, with specific results given to illustrate.     

On an internal property of absolute retracts,II

Let X be a (metrizable) space. A mixer for X is, roughly speaking, a map μ:X³→X such that μ(x, x, y) = μ(x, y, x) = μ(y, x, x) = x for all x, y ∈ X. We show that each AR has a mixer and that a finite dimensional path connected space with a mixer is an AR. Our main result is that each separable space with a mixer and having an open cover by sets contractible within the whole space, is LEC.     

The Ptolemy and Zb?ganu constants of normed spaces

In every inner product space H the Ptolemy inequality holds: the product of the diagonals of a quadrilateral is less than or equal to the sum of the products of the opposite sides. In other words, ‖x−y‖‖z−w‖≤‖x−z‖‖y−w‖+‖z−y‖‖x−w‖ for any points w,x,y,z in H. It is known that for each normed space (X,‖⋅‖), there exists a constant C such that for any w,x,y,z∈X, we have ‖x−y‖‖z−w‖≤C(‖x−z‖‖y−w‖+‖z−y‖‖x−w‖). The smallest such C is called the Ptolemy constant of X and is denoted by C_P(X). We study the relationships between this constant and the geometry of the space X, and hence with metric fixed point theory. In particular, we relate the Ptolemy constant C_P to the Zb?ganu constant C_Z, and prove that if X is a Banach space with , then X has (uniform) normal structure and therefore the fixed point property for nonexpansive mappings. We derive general lower and upper bounds for both C_P and C_Z, and calculate the precise values of these two constants for several normed spaces. We also present a number of conjectures and open problems.     

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 Peano's theorem in Banach spaces

We show that if X is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping f:X→X such that the autonomous differential equation x^′=f(x) has no solution at any point.     

Linearizability and integrability of vector fields via commutation

In this paper, we consider complex smooth and analytic vector fields X in a neighborhood of a nondegenerate singular point. It is proved the equivalence between linearizability and commutation, i.e., the existence of a commuting vector field Y such that the Lie brackets [X,Y]≡0. For complex smooth and analytic vector fields in the plane and in a neighborhood of a nondegenerate singular point, it is also proved the equivalence between integrability and the existence of a smooth vector field Y, such that Y is a normalizer of X, i.e., [X,Y]=μX.     

Noethérianité et privilège en géométrique analytique p-adique

Let k be a non-Archimedean field, X a k-affinoid space and f an analytic function over X. We describe precisely how the geometric connected components of the spaces $\{x\in X||f(x)|?\epsilon \}$ behave with regards to ε. We also obtain a result concerning privileged neighbourhoods and adapt a theorem from complex analytic geometry about Noetherianity for germs of analytic functions. To cite this article: J. Poineau, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A glance at spaces with closure-preserving local bases

Call a space X (weakly) Japanese at a pointx∈X if X has a closure-preserving local base (or quasi-base respectively) at the point x. The space X is (weakly) Japanese if it is (weakly) Japanese at every x∈X. We prove, in particular, that any generalized ordered space is Japanese and that the property of being (weakly) Japanese is preserved by σ-products; besides, a dyadic compact space is weakly Japanese if and only if it is metrizable. It turns out that every scattered Corson compact space is Japanese while there exist even Eberlein compact spaces which are not weakly Japanese. We show that a continuous image of a compact first countable space can fail to be weakly Japanese so the (weak) Japanese property is not preserved by perfect maps. Another interesting property of Japanese spaces is their tightness-monolithity, i.e., in every weakly Japanese space X we have for any set A⊂X.     

An extension of the Aumann-shapley value concept to functions on arbitrary banach spaces

LetX denote a linear space of real valued functions defined on a subset of a Banach space such thatX containsE′ the dual space ofE as a subspace. Given a distinguished vectorx ₀ inE anx ₀-value (onX) is defined to be a projectionP fromX ontoE′ which satisfies the following two hypotheses: (VA) (PF)(x₀)=Fx₀ for allF inX; (VB) IfT is a continuous isomorphism fromE intoE such thatTx ₀=x ₀ thenP(F?T) = (PF) ? T for allF inX. The existence and uniqueness of a value is established for two choices ofX, one of which is the space of polynomials in functional onE. The existence and partial uniqueness of a value is established on a third choice forX.     

A discrete homotopy theory for binary reflexive structures

We present a simple combinatorial construction of a sequence of functors σ_k from the category of pointed binary reflexive structures to the category of groups. We prove that if the relational structure is a poset P then the groups are (naturally) isomorphic to the homotopy groups of P when viewed as a topological space with the topology of ideals, or equivalently, to the homotopy groups of the simplicial complex associated to P. We deduce that the group σ_k(X,x₀) of the pointed structure (X,x₀) is (naturally) isomorphic to the kth homotopy group of the simplicial complex of simplices of X, i.e. those subsets of X which are the homomorphic image of a finite totally ordered set.     

Hyperbolicity in median graphs

If X is a geodesic metric space and x ₁,x ₂,x ₃?∈?X, a geodesic triangle T?=?{x ₁,x ₂,x ₃} is the union of the three geodesics [x ₁ x ₂], [x ₂ x ₃] and [x ₃ x ₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e., ${\delta}(X)=\inf\{{\delta}\ge 0: \, X \, \text{is $\delta$-hyperbolic}\}. $ In this paper we study the hyperbolicity of median graphs and we also obtain some results about general hyperbolic graphs. In particular, we prove that a median graph is hyperbolic if and only if its bigons are thin.     

On 2-dimensional Nonaspherical Cell-like Peano Continua: A Simplified Approach

We construct a functor AC(?, ?) from the category of path connected spaces X with a base point x to the category of simply connected spaces. The following are the main results of the paper: (i) If X is a Peano continuum then AC(X, x) is a cell-like Peano continuum; (ii) If X is n-dimensional then AC(X, x) is (n + 1)?dimensional; and (iii) For a path connected space X, π ₁(X, x) is trivial if and only if π ₂(AC(X, x)) is trivial. As a corollary, AC(S ¹, x) is a 2-dimensional nonaspherical cell-like Peano continuum.     

The Cartesian product of a compactum and a space is a bifunctor in shape

In 2003 the author has associated with every cofinite inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution R(X,K) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then R(X,K) consists of spaces having the homotopy type of polyhedra. In two subsequent papers the author proved that R(X,K) is a covariant functor in each of its variables X and K. In the present paper it is proved that R(X,K) is a bifunctor. Using this result, it is proved that the Cartesian product X×Z of a compact Hausdorff space X and a topological space Z is a bifunctor SSh(Cpt)×Sh(Top)→Sh(Top) from the product category of the strong shape category of compact Hausdorff spaces SSh(Cpt) and the shape category Sh(Top) of topological spaces to the category Sh(Top). This holds in spite of the fact that X×Z need not be a direct product in Sh(Top).     

Some properties of the injective tensor product of L[0,1] and a Banach space

Let X be a (real or complex) Banach space and 1<p,p′<∞ such that 1/p+1/p′=1. Then , the injective tensor product of L^p[0,1] and X, has the Radon-Nikodym property (resp. the analytic Radon-Nikodym property, the near Radon-Nikodym property, contains no copy of c₀, is weakly sequentially complete) if and only if X has the same property and each continuous linear operator from L^p′[0,1] to X is compact.