Polygons with primitive normal and additive theories

Previously, primitive normal, primitive connected, and additive theories of S-polygons were studied. In particular, it was proved that the class of all S-polygons is primitive normal iff S is a linearly ordered monoid. The present paper is a continuation of this research. Here, Spolygons with primitive normal, additive, and antiadditive theories are described in the language of a primitive equivalence structure. It is shown that the class of all S-polygons is antiadditive only for a linearly ordered monoid S, that is, this class is antiadditive iff it is primitive normal.     

Elementary Pairs of Primitive Normal Theories

The main objective of the paper is proving that classes of primitive normal, primitive bound, antiadditive, and additive theories are closed under P-expansions. This phenomenon is quite remarkable, for the main structure classes of theories studied within model theory (such as stable, totally transcendental, etc.) do not possess such a property. Furthermore, it is proved that primitive bound theories are P-stable, and we furnish an example of a primitive bound theory with models that are not primitive bound.     

Primitive connected theories

We prove the quantifier-elimination theorem for so-called primitive connected theories, exemplified by theories of modules. The theorem generalizes the well-known Baur-Monk-Garavaglia theorem on the elimination of quantifiers in the model theory of modules. The definition of a class of primitive connected theories, as distinct from modules. is not supposed to impose any conditions on a type of axioms that would specify those theories. Dedicated to the 60th birthday of Academician Yu. L. Ershov Supported by RFFR grant No. 99-01-00600. Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 145–169, March–April, 2000.     

Primitive connected and additive theories of polygons

We study into monoids S the class of all S-polygons over which is primitive normal, primitive connected, or additive, that is, the monoids S the theory of any S-polygon over which is primitive normal, primitive connected, or additive. It is proved that the class of all S-polygons is primitive normal iff S is a linearly ordered monoid, and that it is primitive connected iff S is a group. It is pointed out that there exists no monoid S with an additive class of all S-polygons. __________ Translated from Algebra i Logika, Vol. 45, No. 3, pp. 300–313, May–June, 2006.     

On Primitive Cellular Algebras

First we define and study the exponentiation of a cellular algebra by a permutation group that is similar to the corresponding operation (the wreath product in primitive action) in permutation group theory. Necessary and sufficient conditions for the resulting cellular algebra to be primitive and Schurian are given. This enables us to construct infinite series of primitive non-Schurian algebras. Also we define and study, for cellular algebras, the notion of a base, which is similar to that for permutation groups. We present an upper bound for the size of an irredundant base of a primitive cellular algebra in terms of the parameters of its standard representation. This produces new upper bounds for the order of the automorphism group of such an algebra and in particular for the order of a primitive permutation group. Finally, we generalize to 2-closed primitive algebras some classical theorems for primitive groups and show that the hypothesis for a primitive algebra to be 2-closed is essential. Bibliography: 16 titles.     

Some Decidability Results for ℤ[G]-Modules when G is Cyclic of Squarefree Order

We extend the analysis of the decision problem for modules over a group ring ?[G] to the case when G is a cyclic group of squarefree order. We show that separated ?[G]-modules have a decidable theory, and we discuss the model theoretic role of these modules within the class of all ?[G]-modules. The paper includes a short analysis of the decision problem for the theories of (finitely generated) modules over ?[ζ_m], where m is a positive integer and ζ_m is a primitive mth root of 1. Mathematics Subject Classification: 03C60, 03B25.     

Progress in physical concepts of string and superstring theory—thirty years of string theory

In this paper we discuss the evolution of physical concepts which led to the generation and development of string theories. The paper is conceived with the intention of summarizing and extending with new aspects the specific characteristics of strings which refer to the physical intuition and experiment. We hope to present new insights into the physics of strings and make it understandable from the point of view of a non-string theorist. Even if there exist some opinions that the (super)string theory appertains to the twenty-first or twenty-second century or that there are no concrete new predictions of string theory at low energies, we believe that string theory presents a rich field of research and a source of physical intuition not only for mathematicians but also for theoretical and experimental physicists. We offer as an example an atomic electron cloud which can also be interpreted in terms of a fixed point in a string theory We propose also an experiment to verify the fundamental hypotheses. Finally we deduce that the number of dimensions of spacetime must be infinite by virtue of the axiom of universality of motion.     

A Counterfactual Analysis of Infinite Regress Arguments

I propose a counterfactual theory of infinite regress arguments. Most theories of infinite regress arguments present infinite regresses in terms of indicative conditionals. These theories direct us to seek conditions under which an infinite regress generates an infinite inadmissible set. Since in ordinary language infinite regresses are usually expressed by means of infinite sequences of counterfactuals, it is natural to expect that an analysis of infinite regress arguments should be based on a theory of counterfactuals. The Stalnaker–Lewis theory of counterfactuals, augmented with some fundamental notions from metric-spaces, provides a basis for such an analysis of infinite regress arguments. Since the technique involved in the analysis is easily adaptable to various analyses, it facilitates a rigorous comparison among conflicting philosophical analyses of any given infinite regress.     

Vertex Cuts

Given a connected graph, in many cases it is possible to construct a structure tree that provides information about the ends of the graph or its connectivity. For example Stallings' theorem on the structure of groups with more than one end can be proved by analyzing the action of the group on a structure tree and Tutte used a structure tree to investigate finite 2‐connected graphs, that are not 3‐connected. Most of these structure tree theories have been based on edge cuts, which are components of the graph obtained by removing finitely many edges. A new axiomatic theory is described here using vertex cuts, components of the graph obtained by removing finitely many vertices. This generalizes Tutte's decomposition of 2‐connected graphs to k‐connected graphs for any k, in finite and infinite graphs. The theory can be applied to nonlocally finite graphs with more than one vertex end, i.e. ends that can be separated by removing a finite number of vertices. This gives a decomposition for a group acting on such a graph, generalizing Stallings' theorem. Further applications include the classification of distance transitive graphs and k‐CS‐transitive graphs.     

Weakly semisimple modulesand density theory

The objective of this article is to describe the theory of a class of semiprime rings which bear the same relationship to weakly primitive rings that semiprimitive rings have to primitive rings. In this theory, the role of semisimple modules is assumed by modules which are locally finite dimen- sional, polyform and weakly compressible.     

A theory of rules for enumerated classes of functions

We define an applicative theoryCL ₂ similar to combinatory logic which can be interpreted in classes of functions possessing an enumerating function. In contrast to the models of classical combinatory logic, it is not necessarily assumed that the enumerating function itself belongs to that function class. Thereby we get a variety of possible models including e. g. the classes of primitive recursive, recursive, elementary, polynomial-time comptable of ₀-recursive functions.We show that inCL ₂ a major part of the metatheory of enumerated classes of functions can be developed. Namely, a kind of -abstraction can be defined and abstract versions of theS _n ^m - and (Primitive) Recursion Theorems are proved. Thereby, a closer analysis of the phenomenon of the different recursion theorems is achieved.A theory closely related toCL ₂ can be used to replace the applicative part of Feferman's theories for explicit mathematics. So this work can be seen as an answer to Feferman's question to formulate a theory for explicit mathematics in which operations can be interpreted as primitive recursive or even more feasible ones.Finally it is shown that the proof-theoretical strength of various theoreies for explicit mathematics is preserved when replacing the applicative part of the theories by our theory together with an operation for primitive recursion.     

Regular S-acts with primitive normal and antiadditive theories

In this work, we investigate the commutative monoids over which the axiomatizable class of regular S-acts is primitive normal and antiadditive. We prove that the primitive normality of an axiomatizable class of regular S-acts over the commutative monoid S is equivalent to the antiadditivity of this class and it is equivalent to the linearity of the order of a semigroup R such that an S-act sR is a maximal (under the inclusion) regular subact of the S-act sS.     

An infinite dimensional Morse theory with applications

In this paper we construct an infinite dimensional (extraordinary) cohomology theory and a Morse theory corresponding to it. These theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals (by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension). Several applications are given to Hamiltonian systems, the one-dimensional wave equation (of vibrating string type) and systems of elliptic partial differential equations.

     

Decision making under uncertainty: A semi-infinite programming approach

The subject of this paper is the formulation and discussion of a semi-infinite linear vector optimization problem which extends multiple objective linear programming problems to those with an infinite number of objective functions and constraints. Furthermore it generalizes in some way semi-infinite programming. Besides the statement of some immediately derived results which are related to known results in semi-infinite linear programming and vector optimization, the problem mentioned above is interpreted as a decision model, under risk or uncertainty containing continuous random variables. Thus we treat the case of an infinite number of occuring states of nature. These types of problems frequently occur within aspects of decision theory in management science.     

Minimal algebras of infinite representation type with preprojective component

Let A be a finite dimensional, basic and connected algebra (associative, with 1) over an algebraically closed field k. Denote by e₁,...,e_n a complete set of primitive orthogonal idempotents in A and by A_i= A/Ae_iA. A is called a minimal algebra of infinite representation type provided A is itself of infinite representation type,whereas all A_i, 1≤i≤n,are of finite representation type. The main result gives the classification of the minimal algebras having a preprojective component in their Auslander-Reiten quiver. The classification is obtained by realizing that these algebras are essentially given by preprojective tilting modules over tame hereditary algebras.     

A Hereditary Torsion Theory for Modules Over Integral Domains and Its Applications

In this article, we study the hereditary torsion theory defined by the set of associated primes of principle ideals of an integral domain, which is called the g-torsion theory. We first discuss some general properties of g-torsion theories, and after that give some applications of them. For example, we generalize a characterization of reflexive modules over quasi-normal domains to a class of non-Noetherian domains. Among other things, a characterization of coherent domains of weak Gorenstein global dimension at most two is also given in terms of Gorenstein projectivity (or Gorenstein flatness) of injective modules relative to the g-torsion theory.     

Characterization of eventually periodic modules in the singularity categories

The singularity category of a ring makes only the modules of finite projective dimension vanish among the modules, so that the singularity category is expected to characterize a homological property of modules of infinite projective dimension. In this paper, among such modules, we deal with eventually periodic modules over a left artin ring, and, as our main result, we characterize them in terms of morphisms in the singularity category. As applications, we first prove that, for the class of finite dimensional algebras over a field, being eventually periodic is preserved under singular equivalence of Morita type with level. Moreover, we determine which finite dimensional connected Nakayama algebras are eventually periodic when the ground field is algebraically closed.     

Groups with T1 primitive ideal spaces

We give a simple necessary and sufficient condition for the group C¹-algebra of a connected locally compact group to have a T₁ primitive ideal space, i.e., to have the property that all primitive ideals are maximal. A companion result settles the same question almost entirely for almost connected groups. As a by-product of the method used, we show that every point in the primitive ideal space of the group C¹-algebra of a connected locally compact group is at least locally closed. Finally, we obtain an analog of these results for discrete finitely generated groups; in particular the primitive ideal space of the group C¹-algebra of a discrete finitely generated solvable group is T₁ if and only if the group is a finite extension of a nilpotent group.     

Bilinear spaces over a fixed field are simple unstable

We study the model theory of vector spaces with a bilinear form over a fixed field. For finite fields this can be, and has been, done in the classical framework of full first-order logic. For infinite fields we need different logical frameworks. First we take a category-theoretic approach, which requires very little set-up. We show that linear independence forms a simple unstable independence relation. With some more work we then show that we can also work in the framework of positive logic, which is much more powerful than the category-theoretic approach and much closer to the classical framework of full first-order logic. We fully characterise the existentially closed models of the arising positive theory. Using the independence relation from before we conclude that the theory is simple unstable, in the sense that dividing has local character but there are many distinct types. We also provide positive version of what is commonly known as the Ryll-Nardzewski theorem for ω-categorical theories in full first-order logic, from which we conclude that bilinear spaces over a countable field are ω-categorical.