Algebraic classification of physical structures with zero. I

An algebraic version is considered of the axiomatization of physical structures. An arbitrary set R with a distinguished element O (zero) is taken as a set of measurements. Under an additional condition, understood to be an analog of the requirement that a physical structure is one-metric, the structure of a topological skew field with zero O is introduced on R; and on the object sets $\mathcal{M}$ and $\mathcal{N}$ , the structure of finite-dimensional vector spaces over the skew field is introduced. This leads to a complete classification of the corresponding physical structures. The classification theorem can be considered also as a variant of the axiomatics connected with a bilinear form on a pair of finite-dimensional vector spaces over a skew field; i.e., the variant which uses, as axioms, only the combinatorial properties of a bilinear form as a map $\mathcal{M} \times \mathcal{N} \to R$ (i.e., without the axioms of addition and multiplication).     

Phenomenologically symmetric local lie groups of transformations of the space <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript>

In this paper we define a phenomenologically symmetric local Lie group of transformations of an arbitrary-dimensional space. We take as a basis the axiom scheme of the theory of physical structures. Phenomenologically symmetric groups of transformations are nondegenerate both with respect to coordinates and to parameters. We obtain a multipoint invariant of this group of transformations and relate it with Ward quasigroups. We define a substructure of a physical structure as a certain phenomenologically symmetric subgroup of transformations. We establish a criterion for the phenomenological symmetry of the Lie group of transformations and prove the uniqueness of a structure with the minimal rank. We also introduce the notion of a phenomenologically symmetric product of physical structures.     

Some Normality-Type Properties,Topological Products,and Classes of Continuous Mappings

Some properties generalizing the normality separation axiom and their behavior under basic topological operations and continuous mappings are discussed. One of the general problems considered in the paper concerns the deduction of “nice” topological properties of spaces from generalized normality of certain topological products, their subspaces, and other related structures; in particular, extensions of the well-known theorems of Katetov and Tamano are suggested. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 2, pp. 205–238, 2003.     

Polar spaces of arbitrary rank

The incidence systems which satisfy the main axiom of Buekenhout and Shult, other than those related to generalized quadrangles, now have a satisfactory classification. Here the ones of infinite rank with no lines of cardinality 2 are characterized. They are the natural analogues of those of finite rank, and the nondegenerate ones arise from polarities or pseudoquadratic forms in the usual way. The treatment quotes some fundamental theorems of Tits, who handled the finite rank case, but is otherwise self-contained. A full exposition of the relevant concepts, with a new development of some of the elementary theory, is also given.Dedicated to Professor Jacques Tits on the occasion of his sixtieth birthday     

A Generalization of Rao's Covariance Structure with Applications to Several Linear Models

This paper presents a generalization of Rao's covariance structure. In a general linear regression model, we classify the error covariance structure into several categories and investigate the efficiency of the ordinary least squares estimator (OLSE) relative to the Gauss–Markov estimator (GME). The classification criterion considered here is the rank of the covariance matrix of the difference between the OLSE and the GME. Hence our classification includes Rao's covariance structure. The results are applied to models with special structures: a general multivariate analysis of variance model, a seemingly unrelated regression model, and a serial correlation model.     

Generalized Topological Index

This paper deals with an index which is different in general from the topological index defined by Atiyah and Singer because we loosen the normalization property. The intrinsic relation of this new index with operations in K-theory is explained. It is also shown that if we change the normalization axiom, the corresponding index is well-defined and may be expressed in terms of the topological index.     

Oriented matroids from smooth manifolds

Results of Folkman and Lawrence and Mandel on representations of oriented matroids by topological spheres are used to prove a method of constructing oriented matroids from intersections of smooth topological hyperplanes. A class of such constructions is given corresponding to non real-representable matroids of rank ϱ on 2ϱ + 1 elements, ϱ ≥ 4.     

Centroaffine First Order Invariants of Surfaces in IR4

An investigation of the centroaffine geometry of surfaces in IR⁴ leads to the centroaffine first order invariants: the vector bundle valued second fundamental form, the affine semiconformal structure, the h³-semiconformal structure and the centroaffine metric. A classification of surfaces by their semiconformal structures according to signature and rank is given. This involves the study of the orbits of two pencils of symmetric bilinear forms on IR² under a change of basis. Combined with previous results ([Nomizu-Sasaki 93]) a complete classification of the zero-degenerate surfaces is obtained and examples of the other surface types are constructed.     

Completeness theorems for continuous functions and product topologies

In this paper we formulate a first order theory of continuous functions on product topologies via generalized quantifiers. We present an axiom system for continuous functions on product topologies and prove a completeness theorem for them with respect to topological models. We also show that if a theory has a topological model which satisfies the Hausdorff separation axiom, then it has a 0-dimensional, normal topological model. We conclude by obtaining an axiomatization for topological algebraic structures, e.g. topological groups, proving a completeness theorem for the analogue with countable conjunctions and disjunctions, and presenting counterexamples to interpolation and definability.     

A short note on fuzzy neighbourhood spaces

Lowen introduced the concept of fuzzy neighbourhood spaces (fns's), and proved that the family of level topologies of a fns is an antichain. Wuyts characterized fns's as being those fully stratified fuzzy topological spaces (fts's) whose level topologies form antichains and whose fuzzy topologies are the finest possible for thier level topologies. Mashhour et al. introduced the axiom of structure A on fts's, and derived for it several characterizations and fuzzy topological consequences. We use those characterizations of structure A and Wuyts' characterization of fns's to prove that a fts is a fns iff it is of structure A and fully stratified.     

Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations

udy the perturbation theory of structured matrices under structured rank one perturbations, and then focus on several classes of complex matrices. Generic Jordan structures of perturbed matrices are identified. It is shown that the perturbation behavior of the Jordan structures in the case of singular J-Hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one perturbation as it has been studied in [18, 28, 30, 31]. Thus a generic structured perturbation would not be generic if considered as an unstructured perturbation. In other settings of structured matrices, the generic perturbation behavior of the Jordan structures, within the confines imposed by the structure, follows the pattern of that of unstructured perturbations.     

Spectral methods for studying degenerate differential-operator equations. I

A review of the results on degenerate differential-operator equations mainly obtained by Khar’kov and Donetsk mathematicians starting from the 1970s is presented. Applications to evolutionary problems of electrodynamics and mathematical circuit theory, including the construction of models of the corresponding physical systems, are considered. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 35, Voronezh Conference-2, 2005.     

Abstract Logics, Logic Maps, and Logic Homomorphisms

What is a logic? Which properties are preserved by maps between logics? What is the right notion for equivalence of logics? In order to give satisfactory answers we generalize and further develop the topological approach of [4] and present the foundations of a general theory of abstract logics which is based on the abstract concept of a theory. Each abstract logic determines a topology on the set of theories. We develop a theory of logic maps and show in what way they induce (continuous, open) functions on the corresponding topological spaces. We also establish connections to well-known notions such as translations of logics and the satisfaction axiom of institutions [5]. Logic homomorphisms are maps that behave in some sense like continuous functions and preserve more topological structure than logic maps in general. We introduce the notion of a logic isomorphism as a (not necessarily bijective) function on the sets of formulas that induces a homeomorphism between the respective topological spaces and gives rise to an equivalence relation on abstract logics. Therefore, we propose logic isomorphisms as an adequate and precise notion for equivalence of logics. Finally, we compare this concept with another recent proposal presented in [2]. This research was supported by the grant CNPq/FAPESB 350092/2006-0.     

Classification of rotational motions for the baryon model “triangle”

A baryon model with three particles (quarks) pairwise connected by relativistic strings forming a curvilinear triangle is considered. The classical analytic solutions corresponding to a uniform plane rotation of the system with arbitrary quark masses m_i are found for this model. The sections of the related world surfaces by the plane t=const are curvilinear triangles composed of segments of a hypocycloid. A complete classification of the types of motion is suggested, based on differences in topological configuration and the presence and number of interior massless points on the string that move at the velocity of light. The classification results from investigating the limiting states as m_i»0. The calculated energy dependence of the angular momentum of the motions creates possibilities to model baryon states on Regge trajectories using these motions.     

拓扑空间关于子基的分离性

在粗糙集理论研究中,覆盖方法的应用越来越受到重视,其中拓扑空间的子集关于子基的内部和闭包两个概念尤为重要.本文在由它们导人的关于子基的开集,闭集的基础上,给出了拓扑空间关于子基的分离性概念,并研究它们的性质,得到分离性公理定义的一般拓扑空间的进一步分类.     

Two-dimensional almost-Riemannian structures with tangency points

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-two vector bundle over the surface which defines the structure. We analyse the generic case including the presence of tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a classification of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss–Bonnet formula for almost-Riemannian structures with tangency points.     

三次系统E_3~1存在二角形双曲线分界线环的充要条件

得到了三次系统 E1 3存在二角形双曲线分界线环的充要条件 .并给出了它们的拓扑分类和各种拓扑结构的参数条件 .     

Topological transformation groups of manifolds over non-Archimedean fields,their representations,and quasi-invariant measures. I

Diffeomorphism groups and loop groups of manifolds on Banach spaces over non-Archimedean fields are defined. Moreover, for these groups, finite-and infinite-dimensional manifolds over the corresponding fields are considered. The group structure, the differential-geometric structure, and also the topological structure of diffeomorphism groups and loops groups are studied. We prove that these groups do not locally satisfy the Campbell-Hausdorff formula. The principal distinctions in the structure for the Archimedean and classical cases are found. The quasi-invariant measures on these groups with respect to dense subgroups are constructed. Stochastic processes on topological transformation groups of manifolds and, in particular, on diffeomorphism groups and on loop groups and also the corresponding transition probabilities are constructed. Regular, strongly continuous, unitary representations of dense subgroups of topological transformation groups of manifolds, in particular, those of diffeomorphism groups and loop groups associated with quasi-invariant measures on groups and also on the corresponding configurational spaces are constructed. The conditions imposed on the measure and groups under which these unitary representations are irreducible are found. The induced representations of topological groups are studied by using quasi-invariant measures on topological groups. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 39, Functional Analysis, 2006.     

The Liouville Foliation of Nonconvex Topological Billiards

Together with the classical plane billiards, topological billiards can be considered, where the motion occurs on a locally flat surface obtained by isometrically gluing together several plane domains along their boundaries, which are arcs of confocal quadrics. A point moves inside each of the domains along straight line segments; when it reaches the boundary of a domain, it passes to another domain. Previously, the author gave a Liouville classification of all topological billiards obtained by gluing along convex boundaries. In the present paper, all topological integrable billiards obtained by gluing along convex or nonconvex boundaries from elementary billiards bounded by arcs of confocal quadrics are classified. For some of such nonconvex topological billiards, the Fomenko–Zieschang invariants (marked molecules W*) for Liouville equivalence are calculated.