Showing papers on "Ring (mathematics) published in 1986"

Bieberbach Groups and Flat Manifolds

Abstract: I. Bierberbach's Three Theorems.- 1. Rigid Motions.- 2. Examples.- 3. Bierberbach's First Theorem.- 4. Bierberbach's Second Theorem.- 5. Digression - Group Extensions.- 6. Digression - Integral Repesentations of Finite Groups.- 7. Bieberbach's Third Theorem and Some Remarks.- II. Flat Riemannian Manifolds.- 1. Introduction.- 2. A Tiny Bit of Differential Topology.- 3. Connections and Curvature.- 4. Riemannian Structures.- 5. Flat Manifolds.- 6. Conjectures and Counterexamples.- III. Classification Theorems.- 1. The Algebraic Structure of Bieberbach Groups.- 2. A General Classification Scheme for Bieberbach Groups.- 3. Digression - Cohomology of Groups.- 4. Examples.- 5. Holonomy Groups.- IV. Holonomy Groups of Prime Order.- 1. Introduction.- 2. Digression - Some Algebraic Number Theory.- 3. Modules over the Cyclotomic Ring.- 4. Modules over Groups of Prime Order.- 5. The Cohomology of Modules over Groups of Prime Order.- 6. The Classification Theorem.- 7. ?p-manifolds.- 8. An Interesting Example.- 9. The Riemannian Structure of Some ?p manifolds.- V. Automorphisms.- 1. The Basic Diagram.- 2. The Hochschild-Serre Exact Sequence.- 3. 9-Diagrams.- 4. Automorphisms of Group Extensions.- 5. Automorphisms of Bieberbach Groups.- 6. Automorphisms of Flat Manifolds.- 7. Automorphisms of ?p-manifolds.

The nil Hecke ring and cohomology of G/P for a Kac-Moody group G.

Abstract: Let G be the group with Borel subgroup B, associated to a Kac-Moody Lie algebra [unk] (with Weyl group W and Cartan subalgebra [unk]). Then H*(G/B) has, among others, four distinguished structures (i) an algebra structure, (ii) a distinguished basis, given by the Schubert cells, (iii) a module for W, and (iv) a module for Hecke-type operators Aw, for w [unk] W. We construct a ring R, which we refer to as the nil Hecke ring, which is very simply and explicitly defined as a functor of W together with the W-module [unk] alone and such that all these four structures on H*(G/B) arise naturally from the ring R.

The ring of invariants of matrices

Abstract: This action of G defines an action of G on an algebra C[W] = C[xo(l), • , Xij(l)] of all polynomial functions on W. We denote by C[W] the subalgebra of G invariant polynomials. This is a finitely generated subalgebra of C[W]. If Z = 1 it is a classical result that this ring of invariants is a polynomial ring in n variables. In fact the coefficients of characteristic polynomial of the matrix X(ΐ) — (x^(l)) are algebraically independent invariants and the ring of invariants is generated by them. By the Newton's formula all coefficients of characteristic polynomial of X(ί) are expressed by n traces Tr(X(l)), Ίx(X\\ΐj), . . . ,Tr(X(l)»),

Logarithmic depth circuits for algebraic functions

Abstract: This paper describes circuits for computation of a large class of algebraic functions on polynomials, power series, and integers, for which, it has been a long standing open problem to compute in depth less than $\Omega (\log n)^2 $.Algebraic circuits assume unit cost for elemental addition and multiplication. This paper describes $O(\log n)$ depth algebraic circuits which given as input the coefficients of n degree polynomials (over an appropriate ring), compute the product of $n^{O(1)} $ polynomials, the symmetric functions, as well as division and interpolation of real polynomials. Also described are $O(\log n)$ depth algebraic circuits which are given as input the first n coefficients of a power series (over an appropriate ring) compute the product of $n^{O(1)} $ power series, as well as division, reciprocal and reversion of real power series.Furthermore this paper describes boolean circuits of depth $O(\log n(\log \log n))$ which, given n-bit binary numbers, compute the product of n numbers and integ...

The Curzon singularity. I: Spatial sections

Abstract: Some new properties of geodesies and other curves lying in the spatial sectionst = const. of the Curzon solution are derived. These are shown to allow one to build up a new coordinate system in which the singularity appears unambiguously as a ring. A new region of spacelike infinity is also revealed on the “other side” of this ring, which can be approached by spatial geodesies threading through the ring.

Ring galaxies – II. Classification and statistics

Abstract: Resultats de l'etude morphologique de 69 systemes decouverts sur une plaque de Schmidt IIIa-J et classes par Arp et Madore comme «galaxies annulaires»

An object-oriented approach to algebra system design

Abstract: This paper describes a new computer algebra system design based on the object-oriented style of programming and an implementation of this design, called Views, written in Smalltalk-80. The design is similar in goals to other 'new' generation computer algebra systems, by allowing the runtime creation of computational domains and providing a way to view these domains as members of categories such as 'group', 'ring' or 'field'. However, Views introduces several unique features. The most notable is the strong distinction made between a domain and its view as a member of a particular category. This distinction between the implementation of a domain and its adherence to a set of algebraic laws allows a great degree of flexibility when choosing the algebraic structures that are to be active during a computation. It also allows for a richer variety of interrelationships among categories than exhibited in other systems.

The theory of Möbius functions

Abstract: CONTENTS § 1. Introduction § 2. Distributive lattices and partially ordered sets § 3. Valuation cones § 4. The valuation ring of a distributive lattice § 5. The Mobius function § 6. The Mobius algebra § 7. The Mobius function on semimodular lattices § 8. The Tutte-Grothendieck ring § 9. Geometric application References

Generating the ring of matrix invariants

Abstract: The ring of invariants of a set of n×n matrices over a field of characteristic zero is a graded ring. The Procesi-Razmyslov theory of trace identities gives criteria for it to be generated by its elements of degree ≤r. These criteria are used to reprove the Procesi-Razmyslov result that it is generated by its elements of degree ≤n2 and to show that it is not generated by its elements of degree ≤n2/8.

Tensor products of dimension groups and $Ksb 0$ of unit-regular rings

Abstract: We study direct limits of finite products of matrix algebras (i.e., locally matricial algebras), their ordered Grothendieck groups (K 0), and their tensor products. Given a dimension group G, a general problem is to determine whether G arises as K 0 of a unit-regular ring or even as K 0 of a locally matricial algebra. If G is countable, this is well known to be true. Here we provide positive answers in case (a) the cardinality of G is ℵ1, or (b) G is an arbitrary infinite tensor product of the groups considered in (a), or (c) G is the group of all continuous real-valued functions on an arbitrary compact Hausdorff space. In cases (a) and (b), we show that G in fact appears as K 0 of a locally matricial algebra. Result (a) is the basis for an example due to de la Harpe and Skandalis of the failure of a determinantal property in a non-separable AF C*-algebra [18, Section 3].

Quantum interference effects in lithium ring arrays.

Abstract: We report detailed measurements on the weak-localization corrections in well-defined and -characterized arrays of quasi one-dimensional normal-metal rings. The Bohm-Aharonov oscillations are very well resolved. Measurements on sets of samples of varying ring size and geometry determine the distinct effects of these parameters and allow an unambiguous, quantitative comparison to the weak localization theory for ring geometries.

Hopfian rings and Hopfian modules

Abstract: Let R be a ring with identity. R is said to be Hopfian if every homomorphism of R onto R is an automorphism. Similarly, an R-module M is Hopfian if every endomorphism of M is an automorphism. The author investigates rings and modules which are Hopfian. For example he shows that if R is a right semiperfect ring, then a projective R-module P is Hopfian if and only if P is finitely generated.

Probabilistic solitude verification on a ring

Abstract: Matching upper and lower bounds for the bit complexity of a problem on asynchronous unidirectional rings are established, assuming that algorithms must reach a correct conclusion with probability 1-e, for some e > 0. Processors can have identities, but the identities are not necessarily distinct. The problem is that of a distinguished processor verifying that it is the only distinguished processor. The complexity depends on the processors' knowledge of the size a of the ring. When no upper bound is known, only nondistributive termination is possible, and O(nlog(1/e)) bits are necessary and sufficient. When only an upper bound N is known, distributive termination is possible, but the complexity of achieving distributive termination is e(nVlog(N/n) + alog(1/~)). When processors know that 3N/4 < a < N~ then the bound drops to e(aloglog(1/~)), even for distributive termination, for sufficiently large N. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the AC M copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission o f the Association for Comput ing Machinery. To copy otherwise, or to republish, requires a fee and /o r specfic permission. © 1986 ACM 0-89791-198-9/86/0800-0161 75¢ 161 1. I n t r o d u c t i o n An asynchronous unidirectional ring of processors is one of the simplest of network topologies. Nonetheless, rings exhibit features which can be expected to show up in many topologies. Consequently, rings serve as a suitable test-bed for studies in distributed computing. This paper is concerned with the bit complexity (total number of bits transmitted) of a fundamental problem, solitude verification, on asynchronous unidirectional rings. We are primarily concerned with versions of solitude verification which cannot be solved with certainty, but can be solved probabilisticaUy, with arbitrarily small positive probability of error. One of a few recognized fundamental problems on rings is that of electing a leader. It was pointed out in [IR], and later expanded upon in [AAGHK], that leader election is composed of two more fundamental problems: attrition and solitude verification. The attrition problem is that of reducing a set of contenders to just one contender. Solitude verification is the problem of determining that only one contender exists. The division of leader election into attrition and solitude verification gains validity from the fact that both attrition and solitude verification reduce to leader election in O(a) bits, where n is the size of the ring. For distributively terminating algorithms, the reduction is a natural one. Though it is less obvious, an O(a) bit reduction exists in the case of nondistrihutively terminating probabilistic algorithms. The reduction is not described here, but the reader might find it amusing to derive one. The complexity, or indeed the solvabillty, of a problem on a ring depends on features of the processors and on the nature of the desired solution. Features which are relevant to an asynchronous ring are: Kaowledge. What does each processor know about the size of the ring? To what extent can an algorithm exploit identities assigned to the processors? Type of algorithm. Is the desired algor i thm deterministic, randomized (correct with probabili ty one) or probabilistic (correct with probability 1 e ) ? Type of terminatioa. Must the algorithm terminate distributively, or is nondistriburive termination acceptable? An algor i thm terminates distributively if each processor, upon reaching a conclusion, will not subsequently revoke its conclusion on receipt of further messages. Deterministic [AAHK, DKR, Pe, PKR] and randomized [AAGHK, IR] solutions to leader election and solitude verification have been considered elsewhere. Pachl [Pa] considers the closely related problem of maximum finding when processors have distinct identities, and shows that prohabiliztic algorithms are not significantly bet ter than deterministic algorithms in that case. This paper establishes complementary upper and lower bounds on the bit complexity of probabilistic solitude verification in two situations where processors do not necessarily have distinct identities, and a solution with correctness probability one is impossible. The two situations of interest here are 1) each processor knows an upper hound on the size of the ring, and termination is distributive, and 2) each processor knows nothing about the size of the ring, and termination is nondiztributive. Additionally, we consider one situation where solution with certainty is possible, but admitting error can decrease the complexity. Tha t is the situation in which each processor knows that the ring size n lies in the range (1/2 + p)N< n < N, where p > 0. Results for solitude verification are summarized in section 2. Section 3 contains descriptions and correctness proofs of solitude verification algorithms, assuming various properties of the ring. The corresponding lower bounds are proved in section 4. Section 5 contains a description of some related results. 2 . O v e r v i e w o f R e s u l t s A solitude verification algorithm runs on a ring of processors of two kinds: initiators and non-initiators. Our algorithms assume that processors of each kind are indistinguishable from one another. On the other hand, our lower bounds apply when processors can have arbit rary identities, and algorithms may rely on identities for achieving efficiency. Algorithms must not, however, rely on either the distinctness or the distribution of identities for their correctness. There is always at least one initiator, and only initiators are able to send a message without first receiving one. The problem is for each initiator to determine whether it is the only initiator. We are concerned with algorithms which may make errors on one side only. If there is only one initiator, then a solitude verification algorithm must, with probability one, cause that initiator to reach the conclusion "alone" . If there are two or more initiators, then with probability at least 1 e no initiator should reach the conclusion t h a t i t is alone. T h e

On a construction in bordism theory

Abstract: In [2], R. Arthan and S. Bullett pose the problem of representing generators of the complex bordism ring MU* by manifolds which are totally normally split; i.e. whose stable normal bundles are split into a sum of complex line bundles. This has recently been solved by Ochanine and Schwartz [8] who use a mixture of J-theory and surgery theory to establish several results, including the following.

Implementing branch-and-bound in a ring of processors

Abstract: A set of personal computers is connected to form a ring structured parallel system: Each processor has access to its local memory and can exchange messages with its two ring neighbors.

Products of idempotents in regular rings

Abstract: The problem of describing the subsemigroup generated by the idempotents in various natural semigroups has received the attention of several semigroup theorists ([1], [2], [3], [5], [7]). However, in those cases where the parent semigroup is in fact the multiplicative semigroup of a natural ring, the known ring structure has not been exploited. When this ring structure is taken into account, proofs can often be streamlined and can lead to more general arguments (such as not requiring that the elements of the semigroup be already transformations of some known structure).

Computation of the Variance of the Waiting Time for Token Rings

Abstract: The main new result in this paper is a simple expression for the variance of the waiting time for a token ring with an infinite number of stations, nonsymmetric Poisson traffic, deterministic ring delays, and virtually identical service distributions. These results can be considered as an approximation to the large ring where no individual station dominates the traffic. In the process, the exact equations for obtaining the variance of the waiting time for arbitrary traffics and service densities are found. In general, if there are n nonsymmetric stations, about n3equations need to be solved.