Showing papers on "Ring (mathematics) published in 1997"

Complexity and Real Computation

Abstract: 1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem "P ? NP?".- 8 Newton's Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bezout's Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.

Conformal Invariants, Inequalities, and Quasiconformal Maps

Abstract: Basic functions: Hypergeometric Functions Gamma and Beta Functions Complete Elliptic Integrals The Arithmetic-Geometric Mean Quotients of Elliptic Integrals Jacobian Elliptic Functions and Conformal Maps. Conformal and quasiconformal mappings: Geometry of M/bius Transformations Conformal Invariants Quasiconformal Mappings Distortion Functions in the Plane. N-dimensional functions: The Gr/tzsch Ring Capacity Estimates for the Gr/tzsch Ring Constant Bounds for Distortion Functions. Applications: Quadruples and Quasiconformal Maps Distances and Quasiconformal Maps Inequalities for Conformal Invariants.

Calabi-Yau fourfolds for M- and F-Theory compactifications

Abstract: We investigate topological properties of Calabi-Yau fourfolds and consider a wide class of explicit constructions in weighted projective spaces and, more generally, toric varieties. Divisors which lead to a non-perturbative superpotential in the effective theory have a very simple description in the toric construction. Relevant properties of them follow just by counting lattice points and can be also used to construct examples with negative Euler number. We study nets of transitions between cases with generically smooth elliptic fibres and cases with ADE gauge symmetries in the N=1 theory due to degenerations of the fibre over codimension one loci in the base. Finally we investigate the quantum cohomology ring of this fourfolds using Frobenius algebras.

Quantum Schubert polynomials

Abstract: where In is the ideal generated by symmetric polynomials in x1,... ,xn without constant term. Another, geometric, description of the cohomology ring of the flag manifold is based on the decomposition of Fln into Schubert cells. These are even-dimensional cells indexed by the elements w of the symmetric group Sn. The corresponding cohomology classes oa, called Schubert classes, form an additive basis in H* (Fln 2) . To relate the two descriptions, one would like to determine which elements of 2[xl, ... , Xn]/In correspond to the Schubert classes under the isomorphism (1.1). This was first done in [2] (see also [8]) for a general case of an arbitrary complex semisimple Lie group. Later, Lascoux and Schiitzenberger [22] came up with a combinatorial version of this theory (for the type A) by introducing remarkable polynomial representatives of the Schubert classes oa called Schubert polynomials and denoted Gw. Recently, motivated by ideas that came from the string theory [31, 30], mathematicians defined, for any Kahler algebraic manifold X, the (small) quantum cohomology ring QH* (X, 2), which is a certain deformation of the classical cohomology ring (see, e.g., [28, 19, 14] and references therein). The additive structure of QH* (X , 2) is essentially the same as that of ordinary cohomology. In particular, QH* (Fln , Z) is canonically isomorphic, as an abelian group, to the tensor product H* (Fln , 2) (0 Z[ql,..., qn-1], where the qi are formal variables (deformation parameters). The multiplicative structure of the quantum cohomology is however

Simplicity of rings of differential operators in prime characteristic

Abstract: Let be a finite dimensional representation of a linearly reductive group . Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under has a simple ring of differential operators.In this paper, we show that this is true in prime characteristic. Indeed, if is a graded subring of a polynomial ring over a perfect field of characteristic splits, then is a simple ring. In the last section of the paper, we discuss how one might try to deduce the characteristic zero case from this result. As yet, however, this is a subtle problem and the answer to the question of Levasseur and Stafford remains open in characteristic zero.http://www.luc.ac.be/Research/Algebra1991 Mathematics Subject Classification: 16S32, 16G60, 13A35.

The Ziegler Spectrum of a Locally Coherent Grothendieck Category

Abstract: The general theory of locally coherent Grothendieck categories is presented. To each locally coherent Grothendieck category a topological space, the Ziegler spectrum of is associated. It is proved that the open subsets of the Ziegler spectrum of are in bijective correspondence with the Serre subcategories of This is a Nullstellensatz for locally coherent Grothendieck categories. If -modules and its Ziegler spectrum is quasi-compact, a property used to construct large (not finitely generated) indecomposable modules over an artin algebra. Two kinds of examples of locally coherent Grothendieck categories are given: the abstract category theoretic examples arising from torsion and localization and the examples that arise from particular modules over the ring The Nullstellensatz is used to give a proof of the result of Crawley-Boevey that every character -linear combination of irreducible characters.1991 Mathematics Subject Classification: 16D90, 18E15.

Type II codes over Z/sub 4/

Abstract: The conditions satisfied by the weight enumerator of self-dual codes, defined over the ring of integers module four, have been studied by Klemm (1989), then by Conway and Sloane (1993). The MacWilliams (1977) transform determines a group of substitutions, each of which fixes the weight enumerator of a self-dual code. This weight enumerator belongs to the ring of polynomials fixed by the group of substitutions, called the ring R of invariants. Among all of the quaternary self-dual codes, some have the property that all euclidean weights are multiples of 8. These codes are called type II codes by analogy with the binary case. An upper bound on their minimum euclidean weight is given, thereby leading to a natural notion of extremality akin to similar concepts for type II binary codes and type II lattices. The most interesting examples of type II codes are perhaps the extended quaternary quadratic residue codes. This class of codes includes the octacode [8, 4, 6] and the lifted Golay [24, 12, 12]. Other classes of interest comprise a multilevel construction from binary Reed-Muller and lifted double circulant codes.

Serre duality for noncommutative projective schemes

Abstract: We prove the Serre duality theorem for the noncommutative projective scheme proj A when A is a graded noetherian PI ring or a graded noetherian AS-Gorenstein ring.

Rings of SL2(C)-characters and the Kauman bracket skein module

Abstract: Let M be a compact orientable 3-manifold. The set of characters of SL2( )- representations of 1(M) forms a closed ane algebraic set. We show that its coordinate ring is isomorphic to a specialization of the Kauman bracket skein module, modulo its nilradical. This is accomplished by realizing the module as a combinatorial analog of the ring in which tools of skein theory are exploited to illuminate relations among characters. We conclude with an application, proving that a small manifold's specialized module is necessarily nite dimensional. Mathematics Subject Classication (1991). 57M99.

On the ring of hurwitz series

Abstract: This paper introduces the ring of Hurwitz series over a commu- tative ring with identity, and examines its structure and applications, especially to the study of differential algebra. In particular, we see that rings of Hurwitz series bear a resemblance to rings of formal power se- ries, and that for rings of positive characteristic, the structure of the ring of Hurwitz series closely mirrors that of the ground ring.

Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach

Abstract: Foreword. Preface. Introduction. 1. Algebraic Preliminaries. 2. The Quantum Yang-Baxter Equation (QYBE). 3. Categories of Quantum Yang-Baxter Modules. 4. More on the Bialgebra Associated to the QYBE. 5. The Fundamental Example of a Quantum Group. 6. Quasitriangular Structures and the Double. 7. Coquasitriangular Structures. 8. Some Classes of Solutions. 9. Categorical Constructions. Appendices: A-Prerequisites. A.1. The Ground Ring k and Basic k-Linear Maps. A.2. Algebras, Coalgebras, and Their Representations. A.3. Various Notations Related to the QYBE. A.4. Some Results from Linear Algebra. References. Index.

An Engel condition with derivation for left ideals

Abstract: We generalize a number of results in the literature by proving the following theorem: Let R be a semiprime ring, D a nonzero derivation of R, L a nonzero left ideal of R, and let [x, y] = xy yx. If for some positive integers tooth, ... , tn, and all x E L, the identity [[... [[D(xtO),xtl],xt2], . .],Xtn] -0 holds, then either D(L) = 0 or else the ideal of R generated by D(L) and D(R)L is in the center of R. In particular, when R is a prime ring, R is commutative. In this paper we prove a theorem generalizing several results, principally [20] and [9], which combine derivations with Engel type conditions. Before stating our theorem we discuss the relevant literature. If one defines [x, y]0 = x and [x, y]1 = [x, y] = xy yx, then an Engel condition is a polynomial [x, Y]n+l = [[X, Y]n, y] in noncommuting indeterminates. A commutative ring satisfies any such polynomial, and a nilpotent ring satisfies one if n is sufficiently large. The question of whether a ring is commutative, or nilpotent, if it satisfies an Engel condition goes back to the well known work of Engel on Lie algebras [15, Chapter 2], and has been considered, with various modifications, by many since then (e.g. [2] or [7]). The connection of Engel type conditions and derivations appeared in a well known paper of E. C. Posner [23] which showed that for a nonzero derivation D of a prime ring R, if [D(x),x] is central for all x C R, then R is commutative. This result has led to many others (see [19] for various references), and in particular to a result of J. Vukman [25] showing that if [D(x), X]2 is central for all x E R, a prime ring with char R + 2,3, then again R is commutative. We extended this result [20] by proving that if [D(x), X]n = 0 for all x c I, an ideal of the prime ring R, then R is commutative, and if instead, this Engel type condition holds for all x C U, a Lie ideal of R, then R embeds in M2(F) for F a field with char F 2. Recently, [9] proved that for a left ideal L of a semiprime ring R, either D(L) = 0 or R contains a nonzero central ideal if either: R is 6-torsion free and [D(x), x]2 is central for all x E L; or if [D(x), Xn] is central for all x C L and R is n!-torsion free. The first of these conditions generalized [1, Theorem 3, p. 99], which assumed that [D(x), x] is central for all x E L, with no restriction on torsion. The second, involving powers, is related to both [12], which showed that a prime ring R is commutative if D(Xk) = 0 for all x C R, and to [8], a significant extension of [12], showing that R is commutative if it contains no nonzero nil ideal and [D(xk(x)), Xk(x)]n = 0 on Received by the editors August 2, 1995. 1991 Mathematics Subject Classification. Primary 16W25; Secondary 16N60, 16U80. ?D1997 American Mathematical Society

Algebraically constructible functions

Abstract: An algebraic version of Kashiwara and Schapira's calculus of constructible functions is used to describe local topological properties of real algebraic sets, including Akbulut and King's numerical conditions for a stratified set of dimension three to be algebraic. These properties, which include generalizations of the invariants modulo 4, 8 and 16 of Coste and Kurdyka, are defined using the link operator on the ring of constructible functions.

WDVV Equations from Algebra of Forms

Abstract: A class of solutions to the WDVV equations is provided by period matrices of hyperelliptic Riemann surfaces, with or without punctures. The equations themselves reflect associativity of explicitly described multiplicative algebra of (possibly meromorphic) one-differentials, which holds at least in the hyperelliptic case. This construction is direct generalization of the old one, involving the ring of polynomials factorized over an ideal, and is inspired by the study of the Seiberg–Witten theory. It has potential to be further extended to reveal algebraic structures underlying the theory of quantum cohomologies and the prepotentials in string models with N=2 supersymmetry.

Arbitrary termination impedances, arbitrary power division, and small-sized ring hybrids

Abstract: If a ring hybrid is terminated by arbitrary impedances, design equations cannot be derived with conventional methods because symmetry planes for even- and/or odd-mode excitation are not available. Therefore, under these conditions, new design equations for ring hybrids were derived. They can be applied to ring hybrids with both arbitrary termination impedances and arbitrary power-division ratios. Also, new design equations for small-sized ring hybrids have been developed. They allow the design of arbitrary power division, arbitrary termination impedances, and especially small-sized ring hybrids. On the basis of these derived equations, a theoretical evaluation was made using microstrip ring hybrids, and experiments are demonstrated using a coplanar ring hybrid.

Design of dual-mode ring resonators with transmission zeros

Abstract: A set of design equations is presented for the implementation of dual-mode ring filters with transmission zeros. These expressions also give insight into the design limitations in practical situations. The simulated and measured results of a 1.4 GHz microstrip dual-mode ring filter are shown.

The group inverse of a companion matrix

Abstract: A complete characterization is given for the group inverse of a companion matrix over an arbitrary ring to exist. Formulae are given for the actual group inverse and some consequences are drawn.

Rankings of partial derivatives

Abstract: Let rn be a nonnegative integer, n a positive integer, N = {0,1,2, } and N,, = {1,..., n}. Aranking~is a total order of N“i x N,, such that (a, i) ~ (b, j) implies (a +c, Z) S (I)+ c,j) for a, b, c E INmand i,j E N=. We describe an approach to such rankings and a theorem which gives an explicit construction of an arbitrary ranking using finite real data. The case n = 1 corresponds to term-orderings of monomials which are crucial inputs for Buchbrrger’s Crobner Basis algorithm for polynomial rings. The case n > 1 corresponds to rankings of partial derivatives which are inputs in algorithms in differential algebra and Buchberger’s algorithm for free modules over polynomial rings. .4 subclass of such rankings determined by finite integer data is given which is sufficient for effective implementation of such rankings. This hm been implemented in the symbolic language Maple. The rankings considered by Riquier are a special case of those considered here. Examples including applications to initial value problems are given.

Fusion of Positive Energy Representations of LSpin(2n)

Abstract: Building upon the Jones-Wassermann program of studying Conformal Field Theory using operator algebraic tools, and the work of A. Wassermann on the loop group of LSU(n) (Invent. Math. 133 (1998), 467-538), we give a solution to the problem of fusion for the loop group of Spin(2n). Our approach relies on the use of A. Connes' tensor product of bimodules over a von Neumann algebra to define a multiplicative operation (Connes fusion) on the (integrable) positive energy representations of a given level. The notion of bimodules arises by restricting these representations to loops with support contained in an interval I of the circle or its complement. We study the corresponding Grothendieck ring and show that fusion with the vector representation is given by the Verlinde rules. The computation rests on 1) the solution of a 6-parameter family of Knizhnik-Zamolodchikhov equations and the determination of its monodromy, 2) the explicit construction of the primary fields of the theory, which allows to prove that they define operator-valued distributions and 3) the algebraic theory of superselection sectors developed by Doplicher-Haag-Roberts.

A complete infrared Einstein ring in the gravitational lens system B1938+666

Abstract: We report the discovery, using NICMOS on the Hubble Space Telescope, of an arcsecond-diameter Einstein ring in the gravitational lens system B1938+666. The lensing galaxy is also detected, and is most likely an early-type. Modelling of the ring is presented and compared with the radio structure from MERLIN maps. We show that the Einstein ring is consistent with the gravitational lensing of an extended infrared component, centred between the two radio components.

On non-vanishing of twisted symmetric and exterior square l-functions for gl(n)

Abstract: Let π be a cuspidal representation of GLn(AF), where AF is the ring of adeles of a number field F.

Categorical logic of names and abstraction in action calculi

Abstract: Milner's action calculus implements abstraction in monoidal categories, so that familiar λ-calculi can be subsumed together with the π-calculus and the Petri nets. Variables are generalised to names, which allow only a restricted form of substitution.In the present paper, the well-known categorical semantics of the λ-calculus is generalised to the action calculus. A suitable functional completeness theorem for symmetric monoidal categories is proved: we determine the conditions under which the abstraction is definable. Algebraically, the distinction between the variables and the names boils down to the distinction between the transcendental and the algebraic elements. The former lead to polynomial extensions, like, for example, the ring Zlxr; the latter lead to algebraic extensions like Zl√2r or Zlir.Building upon the work of P. Gardner, we introduce action categories, and show that they are related to the static action calculus in exactly the same way as cartesian closed categories are related to the λ-calculus. Natural examples of this structure arise from allegories and cartesian bicategories. On the other hand, the free algebras for any commutative Moggi monad form an action category. The general correspondence of action calculi and Moggi monads will be worked out in a sequel to this work.

Architectural analysis of multiple fiber ring networks employing optical paths

Abstract: Analyzes the performance of various types of multiple fiber ring networks employing optical paths (OP's). The multiple fiber ring network architecture is suitable for achieving failure resilient networks that have extremely large bandwidth but are still upgradable against future increases in traffic. This architecture will overcome the limitation of conventional WDM rings in terms of network expansion capabilities, the number of nodes within the ring, and the number of OP's accommodated in the network. The generic node architecture suitable for multiple fiber ring networks is presented and functionality requirements are identified. The OP accommodation design algorithms that minimize the required node system scale are proposed. Based on the generic node architecture and proposed OP accommodation design algorithms, we evaluated the performance of several types of multiple fiber rings in terms of the required node system scale for rings under various conditions. The effect of the ring architecture (uni-/bidirectional rings), optical path schemes (wavelength path/virtual wavelength path), and different node connectivity patterns are demonstrated for the first time. The obtained results elucidate the criteria for selecting the most suitable multiple fiber ring architecture.