M-gap conjecture and m-normal theories

We find a small weakly minimal theory with an isolated weakly minimal type ofM-rank ∞ and an isolated weakly minimal type of arbitrarily large finiteM-rank. These examples lead to the notion of an m-normal theory. We prove theM-gap conjecture for m-normalT. In superstable theories with few countable models we characterize traces of complete types as traces of some formulas. We prove that a 1-based theory with few countable models is m-normal. We investigate generic subgroups of small superstable groups. We compare the notions of independence induced by measure (μ-independence) and category (m-independence). Research supported by KBN grant 2 P03A 006 09.     

Isogeny in superstable groups

We study and develop a notion of isogeny for superstable groups inspired by the notion in algebraic groups and differential algebraic notions developed by Cassidy and Singer. We prove several fundamental properties of the notion. Then we use it to formulate and prove a uniqueness results for a decomposition theorem about superstable groups similar to one proved by Baudisch. Connections to existing model theoretic notions and existing differential algebraic notions are explained.     

Espaces de Banach superstables,distances stables et homeomorphismes uniformes

We introduce here the notion of superstable Banach space, as the superproperty associated with the stability property of J. L. Krivine and B. Maurey. IfE is superstable, so are theL ^p (E) for eachp∈[1, +∞[. If the Banach spaceX uniformly imbeds into a superstable Banach space, then there exists an equivalent invariant superstable distance onX; as a consequenceX contains subspaces isomorphic tol ^p spaces (for somep∈[1, ∞[). We give also a generalization of a result of P. Enflo: the unit ball ofc ₀ does not uniformly imbed into any stable Banach space.     

Nonfacets for shellable spheres

We generalize the notion of a map on a 2-sphere to maps on then-sphere and then show that there exist combinatorial types of countries that cannot be the only type of country for a shellablen-sphere. This generalizes the well known theorem that there are no maps on the 2-sphere all of whose countries arek-gons for anyk≧6. Research supported by N.S.F. grant, number GP-42941     

Using state reduction for computing steady state vectors in Markov chains ofM/G/1 type

This paper presents a state reduction based algorithm for computing the steady state probability vectors of the embedded Markov chains ofM/G/1 type. The algorithm is based on the use of the notion of a state reduction box for streamlining compution. The computational details are linked directly to the theoretical results developed recently by Grassmann and Heyman [6]. Exploiting these connections and a method given in Neuts [18] for finding theG matrix forPH/PH/1 queues, we also propose an hybrid approach for solvingPH/PH/1 queues. Using several numerical examples, we report our computational experiences and present some observations about the relative merits of these approaches.     

Necessary and sufficient conditions forl-stability of games in constitutional form

This paper is devoted to the notion of game in constitutional form. For this game, we define three notions of cores: theo-core, thei-core and thej-core. For each core, we give a necessary and sufficient condition for a game to be stable. We finally prove that these theorems generalize Nakamura's theorems for stability of a simple game and Keiding's theorems for stability of an effectivity function.     

Superstable semigroups of operators

In [NR] the authors introduced the notion of superstable operators on a Banach space E using ultrapowers E_u of E. In [HR] this notion was extended to strongly continuous one-parameter semigroups again by means of ultrapowers.It is the aim of the present paper to give an equivalent intrinsic definition of superstability (without the reference to ultrapowers). This definition allows us to improve the results of [NR] as well as of [HR]. We apply our results to semigroups of positive linear operators on Banach lattices and C^*-algebras, respectively.     

Zeta functions of<Emphasis Type="Italic">q</Emphasis>-perfect numbers

We introduce a notion ofq-analogue of the perfect numbers. We also define a new zeta function which we call a zeta function ofq-perfect numbers. In this paper, the properties of theq-perfect numbers and the zeta functions are studied. Especially, we determine theq-perfect numbers whenq is a root of unity.     

Superstable cycles for antiferromagnetic Q-state Potts and three-site interaction Ising models on recursive lattices

We consider the superstable cycles of the Q-state Potts (QSP) and the three-site interaction antiferromagnetic Ising (TSAI) models on recursive lattices. The rational mappings describing the models’ statistical properties are obtained via the recurrence relation technique. We provide analytical solutions for the superstable cycles of the second order for both models. A particular attention is devoted to the period three window. Here we present an exact result for the third order superstable orbit for the QSP and a numerical solution for the TSAI model. Additionally, we point out a non-trivial connection between bifurcations and superstability: in some regions of parameters a superstable cycle is not followed by a doubling bifurcation. Furthermore, we use symbolic dynamics to understand the changes taking place at points of superstability and to distinguish areas between two consecutive superstable orbits.     

Finiteness of U‐rank implies simplicity in homogeneous structures

A (large) superstable homogeneous structure is said to be simple if every complete type over any set A has a free extension over any B ? A. In this paper we give a characterization for this property in terms of U‐rank. As a corollary we get that if the structure has finite U‐rank, then it is simple. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Superstable theories with few countable models

Summary We prove here: Theorem. LetT be a countable complete superstable non -stable theory with fewer than continuum many countable models. Then there is a definable groupG with locally modular regular generics, such thatG is not connected-by-finite and any type inG ^eq orthogonal to the generics has Morley rank. Corollary. LetT be a countable complete superstable theory in which no infinite group is definable. ThenT has either at most countably many, or exactly continuum many countable models, up to isomorphism.Supported by NSF grant DMS 90-06628     

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.     

Computing a centerpoint of a finite planar set of points in linear time

The notion of a centerpoint of a finite set of points in two and higher dimensions is a generalization of the concept of the median of a set of reals. In this paper we present a linear-time algorithm for computing a centerpoint of a set ofn points in the plane, which is optimal compared with theO(n log³ n) complexity of the previously best-known algorithm. We use suitable modifications of the hamsandwich cut algorithm in [Me2] and the prune-and-search technique of Megiddo [Me1] to achieve this improvement.     

Superstable groups

We initiate a study of superstable groups, generalizing previous work of Zil'ber and Cherlin. After having introduced the various tools of stability and model theory needed for that purpose we prove a general ‘Indecomposability Theorem’ and apply them to prove:(1) the definability of many subgroups of superstable groups (which has the consequence that, for superstable groups, “to be simple” is a first-order property);(2) the existence of ‘large’ abelian subgroups of all superstable groups; this allows us for example to give a transparent proof to the theorem of Cherlin stating that superstable division rings are commutative.This study of superstable groups is continued in [4].     

A Notion of Rank for Right Congruences on Semigroups

ABSTRACT

We introduce a new notion of rank for a semigroup S. The rank is associated with pairs (I,ρ), where ρ is a right congruence and I is a ρ-saturated right ideal. We allow I to be the empty set; in this case the rank of (?, ρ) is the Cantor-Bendixson rank of ρ in the lattice of right congruences of S, with respect to a topology we title the finite type topology. If all pairs have rank, then we say that S is ranked. Our notion of rank is intimately connected with chain conditions: every right Noetherian semigroup is ranked, and every ranked inverse semigroup is weakly right Noetherian.

Our interest in ranked semigroups stems from the study of the class ± b? _S of existentially closed S-sets over a right coherent monoid S. It is known that for such S the set of sentences in the language of S-sets that are true in every existentially closed S-set, that is, the theory T _S of ± b? _S , has the model theoretic property of being stable. Moreover, T _S is superstable if and only if S is weakly right Noetherian. In the present article, we show that T _S satisfies the stronger property of being totally transcendental if and only if S is ranked and weakly right Noetherian.     

On the class of flat stable theories

A new notion of independence relation is given and associated to it, the class of flat theories, a subclass of strong stable theories including the superstable ones is introduced. More precisely, after introducing this independence relation, flat theories are defined as an appropriate version of superstability. It is shown that in a flat theory every type has finite weight and therefore flat theories are strong. Furthermore, it is shown that under reasonable conditions any type is non-orthogonal to a regular one. Concerning groups in flat theories, it is shown that type-definable groups behave like superstable ones, since they satisfy the same chain condition on definable subgroups and also admit a normal series of definable subgroup with semi-regular quotients.     

On Preservation of Stability for Finite Extensions of Abelian Groups

We characterize preservation of superstability and ω-stability for finite extensions of abelian groups and reduce the general case to the case of p-groups. In particular we study finite extensions of divisible abelian groups. We prove that superstable abelian-by-finite groups have only finitely many conjugacy classes of Sylow p-subgroups. Mathematics Subject Classification: 03C60, 20C05.     

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 definability of types and relative stability

In this paper, we consider the question of definability of types in non‐stable theories. In order to do this we introduce a notion of a relatively stable theory: a theory is stable up to Δ if any Δ‐type over a model has few extensions up to complete types. We prove that an n‐type over a model of a theory that is stable up to Δ is definable if and only if its Δ‐part is definable.     

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.