Théorème de Whitehead stratifié et invariants de nœuds

By considering homotopies that preserve the stratification, one obtains a natural notion of homotopy for stratified spaces. In this short note, we introduce invariants of stratified homotopy, the stratified homotopy groups. We show that they satisfy a stratified version of Whitehead's theorem. As an example, we introduce a complete knot invariant defined via the stratified homotopy groups.     

On a product formula for the Conley–Zehnder index of symplectic paths and its applications

Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley–Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.      

Methods for deriving differential equations of arbitrary dynamic processes

Using only the notion of extremum and neglecting any laws of nature, we develop a method for the derivation of differential equations describing arbitrary processes for which we only know their dependence on some set of parameters. This provides the possibility of theoretically not only the equations but also the underlying physical laws. In particular, we have found quite a few earlier-unknown differential equations with properties that may have good prospects of being used in various fields of engineering and especially in aircraft flight dynamics. On the basis of the suggested extremal theory of dimensions, we also show that it becomes possible to find the form of an arbitrary unknown differential equation and define arbitrary processes and basic physical constants with the use of singular extremals and their envelopes.     

Computing all solutions to polynomial systems using homotopy continuation

In a previous paper we described a new method for defining homotopies for finding all solutions to polynomial systems. A major feature of this new approach is that the start system for the homotopy need not be a “random” or “generic” system. Also, homotopy paths are strictly increasing in the homotopy parameter. In this paper we establish some principles of implementation and report on the performance of programs that use the new homotopies. A feature of our implementation is that we eliminate divergent paths entirely. We include performance statistics for homotopies derived from more traditional approaches for comparison. Generally, the new approach is faster and more reliable.     

A Note on Fibrations

In this paper we generalize a theorem of McAuley-Tulley [2] to show that small homotopies of any topological space can be lifted to small homotopies for any Hurewicz fibration between arbitrary topological spaces. Received: July 7, 2005. Revised: July 13, 2006.     

A note on reverse derivations

In this note, the notion of reverse derivation is studied. It is shown that in the class of semiprime rings, this notion coincides with the usual derivation when it maps a semiprime ring into its centre. However, we provide some examples to show that it is not the case in general.     

On linear representations of quasigroups

For homotopies of quasigroups, an analog of the fundamental theorem on homomorphisms does not hold in general. In this paper, we consider two approaches that allow one to obtain an analog of this theorem: the introduction of strict homotopies and the move from quasigroups to three-sorted quasigroups.     

可换环上严格上三角矩阵李代数的BZ导子

本文研究了严格上三角矩阵李代数的BZ导子.利用BZ导子在其基上的作用,获得了严格上三角矩阵李代数的任意一个BZ导子的具体形式.对导子的概念进行了推广.     

On the generalized derivations of bimodules

Recently, we introduced the notion of a generalized derivation from a bimodule to a bimodule. In this paper, we give a more general notion based on commutators which covers generalized derivations as a special case. Using it, we show that the separability of an algebra extension is characterized by generalized derivations.     

Une relation fonctionnelle nouvelle sur les cartes planaires pointées

A new functional relation, whose unique solution is the generating function of rooted planar maps, is shown. This new relation in conjunction with the well-known relation established by Tutte, enables the easy derivation of a system of parametric equations for the wanted generating function. As a consequence, we infer a closed formula counting the rooted planar maps as a function of their number of vertices and faces. The geometrical nature of the decomposition used in the derivation of this functional relation, leads to the definition of a natural notion of the inner map of a rooted planar map. Some questions related to this notion are treated.     

Homotopy and semisymmetry of quasigroups

The study of quasigroup homotopies reduces to the study of homomorphisms between semisymmetric quasigroups. In particular, the study of homotopies between central quasigroups reduces to the study of homomorphisms between entropic semisymmetric quasigroups. Received December 20, 1996; accepted in final form September 17, 1997.     

On consistency of bounding operations in deterministic global optimization

This technical comment refers to the discussion of strong consistency of several bounding procedures in Lemma 2.1 and Proposition 2.1 of Ref. 1. A necessary clarification is given of the notion of convergence _q in Lemma 2.1, and a derivation of Proposition 2.1 is presented that includes a new and simple consistency proof of the classical bounding by convex envelopes used in many branch-and-bound procedures.     

Non-abelian extensions of Lie 2-algebras

In this paper,we give the notion of derivations of Lie 2-algebras using explicit formulas,and construct the associated derivation Lie 3-algebra.We prove that isomorphism classes of non-abelian extensions of Lie 2-algebras are classified by equivalence classes of morphisms from a Lie 2-algebra to a derivation Lie 3-algebra.     

Homotopical algebra in homotopical categories

We develop here a version of abstract homotopical algebra based onhomotopy kernels andcokernels, which are particular homotopy limits and colimits. These notions are introduced in anh-category, a sort of two-dimensional context more general than a 2-category, abstracting thenearly 2-categorical properties of topological spaces, continuous maps and homotopies. A setting which applies also, at different extents, to cubical or simplicial sets, chain complexes, chain algebras, ... and in which homotopical algebra can be established as a two-dimensional enrichment of homological algebra.Actually, a hierarchy of notions ofh-,h1-, ...h4-categories is introduced, through progressive enrichment of thevertical structure of homotopies, so that the strongest notion,h4-category, is a sort of relaxed 2-category. After investigating homotopy pullbacks and homotopical diagrammatical lemmas in these settings, we introduceright semihomotopical categories, ash-categories provided with terminal object and homotopy cokernels (mapping cones), andright homotopical categories, provided also with anh4-structure and verifying second-order regularity properties forh-cokernels.In these frames we study the Puppe sequence of a map, its comparison with the sequence of iterated homotopy cokernels and theh-cogroup structure of the suspension endofunctor. Left (semi-) homotopical categories, based on homotopy kernels, give the fibration sequence of a map and theh-group of loops. Finally, the self-dual notion of homotopical categories is considered, together with their stability properties.Lavoro esequito nell'ambito dei progetti di ricerca del MURST.     

Stable Homotopy Monomorphisms

It is well known that the concept of monomorphism in a category can be defined using an appropriate pullback diagram. In the homotopy category of TOP pullbacks do not generally exist. This motivated Michael Mather to introduce another notion of homotopy pullback which does exist. The aim of this paper is to investigate the modified notion of homotopy monomorphism obtained by applying the pullback characterization using Mather's homotopy pullback. The main result of Section 1 shows that these modified homotopy monomorphisms are exactly those homotopy monomorphisms (in the usual sense) which are homotopy pullback stable, hence the terminology “stable” homotopy monomorphism. We also link these stable homotopy monomorphisms to monomorphisms and products in the track homotopy category over a fixed space. In Section 2 we answer the question: when is a (weak) fibration also a stable homotopy monomorphism? In the final section it is shown that the class of (weak) fibrations with this additional property coincides with the class of “double” (weak) fibrations. The double (weak) covering homotopy property being introduced here is a stronger version of the (W) CHP in which the final maps of the homotopies involved play the same role as the initial maps.     

Tensor products and homotopies for ω-groupoids and crossed complexes

Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed complexes, nonabelian chain homotopies between them and similar higher homotopies. The tensor product involves non-abelian constructions related to the commutator calculus and the homotopy addition lemma. This monoidal closed structure is derived from that on the equivalent category of ω-groupoids where the underlying cubical structure gives geometrically natural definitions of tensor products and homotopies.     

On the functoriality of cohomology of categories

In this paper we show that the Baues-Wirsching complex used to define cohomology of categories is a 2-functor from a certain 2-category of natural systems of abelian groups to the 2-category of chain complexes, chain homomorphisms and relative homotopy classes of chain homotopies. As a consequence we derive (co)localization theorems for this cohomology.     

Derivation radical subspace arrangements

In this note we study modules of derivations on collections of linear subspaces in a finite dimensional vector space. The central aim is to generalize the notion of freeness from hyperplane arrangements to subspace arrangements. We call this generalization ‘derivation radical’. We classify all coordinate subspace arrangements that are derivation radical and show that certain subspace arrangements of the Braid arrangement are derivation radical. We conclude by proving that under an algebraic condition the subspace arrangement consisting of all codimension c intersections, where c is fixed, of a free hyperplane arrangement are derivation radical.     

A Full and Faithful Nerve for 2-Categories

The notion of geometric nerve of a 2-category (Street, J. Pure Appl. Algebra 49 (1987), 283–335) provides a full and faithful functor if regarded as defined on the category of 2-categories and lax 2-functors. Furthermore, lax 2-natural transformations between lax 2-functors give rise to homotopies between the corresponding simplicial maps. These facts allow us to prove a representation theorem of the general non-abelian cohomology of groupoids (classifying non-abelian extensions of groupoids) by means of homotopy classes of simplicial maps.Mathematics Subject Classifications (2000) 18D05, 18G30, 55P15.     

Extension of the Leray-Schauder degree for abstract Hammerstein type mappings

We introduce a new extension of the classical Leray-Schauder topological degree in a real separable reflexive Banach space. The new class of mappings for which the degree will be constructed is obtained essentially by replacing the compact perturbation by a composition of mappings of monotone type. It turns out that the class contains the Leray-Schauder type maps as a proper subclass. The new class is not convex thus preventing the free application of affine homotopies. However, there exists a large class of admissible homotopies including subclass of affine ones so that the degree can be effectively used. We shall construct the degree and prove that it is unique. We shall generalize the Borsuk theorem of the degree for odd mappings and show that the ‘principle of omitted rays’ remains valid. To illuminate the use of the new degree we shall briefly consider the solvability of abstract Hammerstein type equations and variational inequalities.