A way to the geometry of rings

This paper is concerned with the problem how to characterize geometrically the affine structure of free modules over rings (associative, with unit-element). The way to solve this problem leads to the concept of the so called affine line-geometry.     

A combinatorial approach to coarse geometry

Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding maps and by taking infinite subsequences. This embedding can be realized by either Rips complexes or analogs of Roe?s anti-?ech approximations of spaces.In this model coarse n-connectedness of K={K₁→K₂→?} means that for each k there is m>k such that the bonding map from Kk to Km induces trivial homomorphisms of all homotopy groups up to and including n.The asymptotic dimension being at most n means that for each k there is m>k such that the bonding map from Kk to Km factors (up to contiguity) through an n-dimensional complex.Property A of G. Yu is equivalent to the condition that for each k and for each ?>0 there is m>k such that the bonding map from |Kk| to |Km| has a contiguous approximation g:|Kk|→|Km| which sends simplices of |Kk| to sets of diameter at most ?.     

A combinatorial approach to p-adic geometry

An important tool in p-adic geometry is the process of p-adic completion. It is shown that this process can be performed on the level of valuated matroids, that is, certain structures which appear to capture much of the essence of p-adic geometry in a coordinate-free, combinatorial manner.     

Some aspects of the group-theoretic approach in problems of nonlinear mechanics

Two aspects of the application of the asymptotic decomposition method to the decomposition and investigation of nonlinear systems are considered. The first one is the extension of the class of functions in terms of which to form the asymptotic decomposition by applying special functions of mathematical physics. The other aspect is the extension of the asymptotic decomposition method to a new class of differential systems, namely, Pfaffian systems in involution.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 1, pp. 22–33, January, 1992.     

A unifying approach to descriptive geometry and mechanisms

Common polymatroidal structures are found in line drawings of polyhedra and two-dimensional frameworks composed of rods and joints. Moreover, those polymatroids are characterized by certain submodular functions, which are useful for practical calculation of the ranks of those polymatroids. A new start point is thus obtained for the unifying study on those subjects. Three different types of treatments of those subjects based on this unifying point of view are also presented.     

A constrained interval approach to the generalized distance geometry problem

This paper presents a new approach to the generalized distance geometry problem, based on a model that uses constraint interval arithmetic. In addition to theoretical results, we give some computational experiments that illustrate the better performance of the proposed approach, compared to others from the literature.     

Some Koszul rings from geometry

We give examples of Koszul rings that arise naturally in algebraic geometry. In the first part, we prove a general result on Koszul property associated to an ample line bundle on a projective variety. Specifically, we show how Koszul property of multiples of a base point free ample line bundle depends on its Castelnuovo–Mumford regularity. In the second part, we give examples of Koszul rings that come from adjoint line bundles on minimal irregular surfaces of general type.     

A dynamic programming approach to the optimization of elastic trusses

The optimal design of elastic trusses is discussed from a dynamic programming point of view. Emphasis is placed on minimum volume design of statically determinate trusses with displacement and stress constraints in the discrete case, i.e., when the cross-sectional areas of the bars are available from a discrete set of values. This, a design constraint usually very difficult to handle with standard nonlinear programming algorithms, is naturally incorporated in the present formulation. In addition, the functional equation approach is shown to furnish a direct solution to the problem of determining a design, among all possible ones satisfying certain volume and displacement constraints, for which the maximum stress is a minimum. A successive approximation approach is briefly indicated as an extension of the method to solve statically indeterminate trusses. Finally, several numerical examples are presented and the main features of the methods are briefly exposed.     

关于Goppa 几何码最小距离的一个新方法

引进一个关于Goppa几何码(代数几何码)最小距离界的一个新方法.应用Maharaj的思想(即用显示基来近似表达Riemann-Roch空间)到Goppa几何码的最小距离的界上去.通过厄米特曲线上的代数几何码的一类例子,来证明标准的几何码的下界在某些情形下可以被显著地改进.进一步地,我们给出了这些码的最小距离上界,并说明了我们的下界非常接近这个上界.     

A story-based dynamic geometry approach to improve attitudes toward geometry and geometric proof

This work is a part of a larger study, which presents geometry through a daily life story using dynamic geometry software. It aimed in particular to enable students to feel the importance of geometry in daily life, to share in the process of formulating geometric statements and conjectures, to experience the geometric proof more than validating the correctness of geometric statements and to start with a real-life situation and go through seven steps to geometric proof. The content of the suggested approach was organized so that every activity was a prerequisite for entering the next one, either in the structure of geometric concepts or in the geometric story context. Some indications will be presented according to three Likert-type questionnaires, which were prepared by the researcher with the purpose of assessing students?? attitudes toward geometry and geometric proof, using computers in mathematics learning and the suggested approach. The analysis of single responses to questionnaire items showed significant changes in students?? beliefs about geometry, importance and functions of geometric proof and toward using the suggested approach.     

A transform approach to polycyclic and serial codes over rings

In this paper, a transform approach is used for polycyclic and serial codes over finite local rings in the case that the defining polynomials have no multiple roots. This allows us to study them in terms of linear algebra and invariant subspaces as well as understand the duality in terms of the transform domain. We also make a characterization of when two polycyclic ambient spaces are Hamming-isometric.     

Quasi-ordered rings - A uniform approach to orderings and valuations

A quasi-order on a set S is a binary, reflexive and transitive relation on S. In [3 Fakhruddin, S. M. (1987). Quasi-ordered fields. J. Pure Appl. Algebra 45:207–210.[Crossref], [Web of Science ®] , [Google Scholar]], Fakhruddin introduced the notion of (totally) quasi-ordered fields and showed that each such field is either an ordered field or else a valued field. Hence, quasi-ordered fields are very well suited to treat ordered and valued fields simultaneously. The aim of the present paper is to prove that an analogous dichotomy holds for commutative rings with 1 as well.     

A non-canonical approach to arithmetic spin geometry and physical applications

A basic requirement of adelic physics is the principle of invariance of the fundamental physical laws under a change of the underlying number field proposed by I.V. Volovich (cf. [20]). In this paper, we develop a manifestly number field invariant approach to Yang-Mills theory, which is formulated within the framework of arithmetic geometry. As well source fields as the Higgs mechanism are incorporated. For this purpose a non-canonical approach to arithmetic spin geometry is proposed, and its physical applications are analyzed. The associated bundle construction is performed in the setting of arithmetic geometry. Furthermore the arithmetic analogue of the following well-known differential geometric fact is proven: Every covariant derivation on a torsor induces a canonical covariant derivation on the associated object.     

A projection approach to optimal control of elastic beam dynamics

Controlled lateral motions of an elastic beam – extended body with a cuboid shape – are studied. A generalized statement of the initial-boundary value problem is given in the frame of 3D linear elasticity based on the integral formulation of constitutive relations. A numerical approach relying on the Fourier method and semi-discretization technique is proposed to design the optimal boundary control that moves the beam to a given final state and minimizes a quadratic cost function. Some results of numerical simulation and quality analysis are presented and discussed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A distributional approach to the geometry of 2D dislocations at the continuum scale

This paper develops a geometrical model of dislocations and disclinations in single crystals at the continuum scale, with use of the distribution theory to represent concentrated effects in the defect lines which in turn form the branching lines of the multiple-valued elastic displacement and rotation fields. Fundamental identities relating the incompatibility tensor to the dislocation and disclination densities are proved in the case of locally countably many parallel defect lines, under global 2D strain assumptions relying on the geometric measure theory. Our approach provides a new understanding of the continuum theory of defects and introduces the appropriate objective internal variables and the required mathematical framework for a rigorous homogenization of all relevant fields from mesoscopic to macroscopic scale.     

A cone programming approach to the bilinear matrix inequality problem and its geometry

We discuss an approach for solving the Bilinear Matrix Inequality (BMI) based on its connections with certain problems defined over matrix cones. These problems are, among others, the cone generalization of the linear programming (LP) and the linear complementarity problem (LCP) (referred to as the Cone-LP and the Cone-LCP, respectively). Specifically, we show that solving a given BMI is equivalent to examining the solution set of a suitably constructed Cone-LP or Cone-LCP. This approach facilitates our understanding of the geometry of the BMI and opens up new avenues for the development of the computational procedures for its solution. Research supported in part by the National Science Foundation under Grant CCR-9222734.