On semi-I-regular sets, AB I -sets and decompositions of continuity, R I C-continuity, A I -continuity

Summary We introduce the notion of a semi-I-regular set and investigate some of its properties. We show that it is weaker than the notion of a regular-I-closed set. Additionally, we also introduce the notion of an AB_I -set by using the semi-I-regular set and study some of its properties. We conclude that a subset A of an ideal topological space (X,τ,I) is open if and only if it is an AB_I -set and a pre-I-open set.     

Semi-exactitude du bifoncteur de Kasparov équivariant

We define a notion of strong K-theoretic amenability for a locally compact group G. This notion coincides with the K-theoretic amenability of many groups. We prove that all results obtained concerning the behavior of KK(.,.) with respect to exact sequences are generalized to the case of KK _G (.,.) for G strongly K-amenable.     

Vanishing conditions for the simplicial volume of compact complex varieties

Gromov has defined a notion of simplicial volume: it is a topological invariant for compact manifolds which is closely related to the fundamental group. We investigate here the relevance of this notion in the realm of complex varieties.

     

On some notions of convergence for n‐tuples of operators

The aim of this paper is to show that we can extend the notion of convergence in the norm‐resolvent sense to the case of several unbounded noncommuting operators (and to quaternionic operators as a particular case) using the notion of S‐resolvent operator. With this notion, we can define bounded functions of unbounded operators using the S‐functional calculus for n‐tuples of noncommuting operators. The same notion can be extended to the case of the F‐resolvent operator, which is the basis of the F‐functional calculus, a monogenic functional calculus for n‐tuples of commuting operators. We also prove some properties of the F‐functional calculus, which are of independent interest. Copyright © 2013 John Wiley & Sons, Ltd.     

Graph-theoretic characterization of the matrix property of full irreducibility without using a transversal

In this paper we introduce the notion of strongly connected polygons in the line graph of a bipartite graph. We use this notion to give a necessary and sufficient condition for a matrix Y to be fully irreducible without the need to construct a transversal of Y. In addition, we show that the notion of strongly connected polygons forms the basis of a general theory that may be used for finding all the cycles in the directed graph of a fully irreducible matrix and for constructing all the nonzero transversals of a fully irreducible matrix.     

About q-approximations and q-hulls over a Noetherian ring,some refinements of the Auslander-Bridger theory

We revisit the notion of q-approximations for a module over a Noetherian ring, originally due to Auslander and Bridger without a name and rediscovered later by Evans and Griffith. We introduce a somewhat symmetric notion of q-hull and provide existence theorems for both q-approximations and q-hulls. The main feature here are existence theorems and characterizations of minimal such ones when the ring is local. The q-hulls being close to the morphism obtained by Auslander and Bridger in their “approximation theorem”, we also obtain for the latter a minimal statement in the case when the ring is local.     

SMYTH COMPLETENESS IN TERMS OF NETS: THE GENERAL CASE

Abstract

Smyth completeness is the appropriate notion of completeness for quasi-uniform spaces carrying an additional topology to serve as domains of computation [2, 3]. The goal of this paper is to provide a better understanding of Smyth completeness by giving a characterization in terms of nets. We develop the notion of computational Cauchy net and an appropriate notion of strong convergence to get the result that a space is Smyth complete if and only if every computational Cauchy net strongly converges. As we are dealing with typically non-symmetric spaces, this is not an instance of the classical net-filter translation in general topology.     

Bounded orbits and measures on a group

We consider a definable group G acting on the space of complete types over G, in a monster model of a theory T. We discuss the notion of a bounded orbit of this action. We prove that some boundedness assumptions imply definable amenability of G.     

A Characterization of the Tutte Polynomial via Combinatorial Embeddings

We give a new characterization of the Tutte polynomial of graphs. Our characterization is formally close (but inequivalent) to the original definition given by Tutte as the generating function of spanning trees counted according to activities. Tutte’s notion of activity requires a choice of a linear order on the edge set (though the generating function of the activities is, in fact, independent of this order). We define a new notion of activity, the embedding-activity, which requires a choice of a combinatorial embedding of the graph, that is, a cyclic order of the edges around each vertex. We prove that the Tutte polynomial equals the generating function of spanning trees counted according to embedding-activities. This generating function is, in fact, independent of the embedding. Received March 15, 2006     

On Subnormality and Formal Subnormality for Tuples of Unbounded Operators

We introduce the notion of (joint) formal normality for a collection of unbounded linear operators on a separable Hilbert space H which is, in some sense, a natural generalization of the notion of formal normality for a single operator. We give some relations between this new notion and (joint) subnormality and hyponormality. We adapt, in particular, a proof of Stochel and Szafraniec to give necessary and sufficient conditions for a tupleof unbounded operators with invariant domain to be (jointly) subnormal.     

Champs affines

The purpose of this work is to introduce a notion of affine stacks, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and p-adic homotopy theory. This gives a first solution to A. Grothendieck’s schematization problem described in [18]. We also use affine stacks in order to introduce a notion of schematic homotopy types. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyond rational and p-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in order to construct various homotopy types of algebraic varieties corresponding to various co-homology theories (Betti, de Rham, l-adic, ...), extending the well known constructions of the various fundamental groups. Finally, just as algebraic stacks are obtained by gluing affine schemes we define $$ \infty $$-geometric stacks as a certain gluing of affine stacks. Examples of $$ \infty $$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms) and Hodge theory (non-abelian periods) are given.     

On the nonexistence of exceptional branch points for weak relative minima in Plateau’s problem

We define the notion of an exceptional branch point for a minimal surface in \mathbb R³{{\mathbb R}^3} and show that it is closely related to the notion of false branch point. We prove that even in the presence of an exceptional branch point we may, under certain conditions, reduce Dirichlet’s energy and therefore also reduce area.     

Champs affines

The purpose of this work is to introduce a notion of affine stacks, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and p-adic homotopy theory. This gives a first solution to A. Grothendieck’s schematization problem described in [18]. We also use affine stacks in order to introduce a notion of schematic homotopy types. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyond rational and p-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in order to construct various homotopy types of algebraic varieties corresponding to various co-homology theories (Betti, de Rham, l-adic, ...), extending the well known constructions of the various fundamental groups. Finally, just as algebraic stacks are obtained by gluing affine schemes we define $$ \infty $$-geometric stacks as a certain gluing of affine stacks. Examples of $$ \infty $$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms) and Hodge theory (non-abelian periods) are given.     

Cardinalities of definable sets in superstructures over models

We introduce the notion of a superstructure over a model. This is a generalization of the notion of the hereditarily finite superstructure ℍ$ \mathbb{F}\mathfrak{M} $ \mathbb{F}\mathfrak{M} over a model $ \mathfrak{M} $ \mathfrak{M} . We consider the question on cardinalities of definable (interpretable) sets in superstructures over λ-homogeneous and λ-saturated models.     

S-Storage Operators

In 1990, J. L. Krivine introduced the notion of storage operator to simulate, for Church integers, the “call by value” in a context of a “call by name” strategy. In the present paper we define for every λ-term S which realizes the successor function on Church integers the notion of S-storage operator. We prove that every storage operator is an S-storage operator. But the converse is not always true.     

On BF-algebras

In this paper we introduce the notion of BF-algebras, which is a generalization of B-algebras. We also introduce the notions of an ideal and a normal ideal in BF-algebras. We investigate the properties and characterizations of them.      

Sur les extensions purement inséparables

If K/k is a finite purely inseparable extension of fields, we are interested in the factorizations of K as a tensor product over k of intermediates fields of K/k. We introduce the notion of e-factorization that generalizes the notion of modular factorization. Contrary to modular factorization, K/k has always an e-factorization and its factors, when their number is maximum, are quasi-invariants.Received: 4 April 2002     

Preservation theorems for Kripke models

There are several ways for defining the notion submodel for Kripke models of intuitionistic first‐order logic. In our approach a Kripke model A is a submodel of a Kripke model B if they have the same frame and for each two corresponding worlds A^α and B^α of them, A^α is a subset of B^α and forcing of atomic formulas with parameters in the smaller one, in A and B, are the same. In this case, B is called an extension of A. We characterize theories that are preserved under taking submodels and also those that are preserved under taking extensions as universal and existential theories, respectively. We also study the notion elementary submodel defined in the same style and give some results concerning this notion. In particular, we prove that the relation between each two corresponding worlds of finite Kripke models A ≤ B is elementary extension (in the classical sense) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some Remarks on the Class of Continuous (Semi-) Strictly Quasiconvex Functions

We introduce the notion of variational (semi-) strict quasimonotonicity for a multivalued operator T  : X⇉X ^* relative to a nonempty subset A of X which is not necessarily included in the domain of T. We use this notion to characterize the subdifferentials of continuous (semi-) strictly quasiconvex functions. The proposed definition is a relaxation of the standard definition of (semi-) strict quasimonotonicity, the latter being appropriate only for operators with nonempty values. Thus, the derived results are extensions to the continuous case of the corresponding results for locally Lipschitz functions.     

Pluripotential Energy

We discuss a notion of the energy of a compactly supported measure in \mathbbCⁿ \mathbb{C}^n for n > 1 which we show is equivalent to that defined by Berman, Boucksom, Guedj and Zeriahi. This generalizes the classical notion of logarithmic energy of a measure in the complex plane \mathbbC \mathbb{C} ; i.e., the case n = 1.