Showing papers on "Ring (mathematics) published in 2003"

A Taste of Jordan Algebras

Abstract: In this book, Kevin McCrimmon describes the history of Jordan Algebras and he describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim Zel'manov. To keep the exposition elementary, the structure theory is developed for linear Jordan algebras, though the modern quadratic methods are used throughout. Both the quadratic methods and the Zelmanov results go beyond the previous textbooks on Jordan theory, written in the 1960's and 1980's before the theory reached its final form. This book is intended for graduate students and for individuals wishing to learn more about Jordan algebras. No previous knowledge is required beyond the standard first-year graduate algebra course. General students of algebra can profit from exposure to nonassociative algebras, and students or professional mathematicians working in areas such as Lie algebras, differential geometry, functional analysis, or exceptional groups and geometry can also profit from acquaintance with the material. Jordan algebras crop up in many surprising settings and can be applied to a variety of mathematical areas. Kevin McCrimmon introduced the concept of a quadratic Jordan algebra and developed a structure theory of Jordan algebras over an arbitrary ring of scalars. He is a Professor of Mathematics at the University of Virginia and the author of more than 100 research papers.

Hecke Algebras With Unequal Parameters

Abstract: Introduction Coxeter groups Partial order on $W$ The algebra ${\mathcal H}$ The bar operator The elements $c_w$ Left or right multiplication by $c_s$ Dihedral groups Cells Cosets of parabolic subgroups Inversion The longest element for a finite $W$ Examples of elements $D_w$ The function $\mathbf{a}$ Conjectures Example: The split case Example: The quasisplit case Example: The infinite dihedral case The ring $J$ Algebras with trace form The function ${\mathbf{a}}_E$ Study of a left cell Constructible representations Two-sided cells Virtual cells Relative Coxeter groups Representations A new realization of Hecke algebras Bibliography Other titles in this series.

Algebras of p-adic distributions and admissible representations

Abstract: Let G be a compact, locally L-analytic group, where L is a finite extension of Qp. Let K be a discretely valued extension field of L. We study the algebra D(G,K) of K-valued locally analytic distributions on G, and apply our results to the locally analytic representation theory of G in vector spaces over K. Our objective is to lay a useful and powerful foundation for the further study of such representations. We show that the noncommutative, nonnoetherian ring D(G,K) "behaves" like the ring of functions on a rigid Stein space, and that (at least when G is Qp-analytic) it is a faithfully flat extension of its subring K\otimes Zp[[G]], where Zp[[G]] is the completed group ring of G. We use this point of view to describe an abelian subcategory of D(G,K) modules that we call coadmissible. We say that a locally analytic representation V of G is admissible if its strong dual is coadmissible as D(G,K)-module. For noncompact G, we say V is admissible if its strong dual is coadmissible as D(H,K) module for some compact open subgroup H. In this way we obtain an abelian category of admissible locally analytic representations. These methods allow us to answer a number of questions raised in our earlier papers on p-adic representations; for example we show the existence of analytic vectors in the admissible Banach space representations of G that we studied in "Banach space representations ...", Israel J. Math. 127, 359-380 (2002). Finally we construct a dimension theory for D(G,K), which behaves for coadmissible modules like a regular ring, and show that smooth admissible representations are zero dimensional.

Quasi-Frobenius Rings

Abstract: A ring is called quasi-Frobenius if it is right or left selfinjective, and right or left artinian (all four combinations are equivalent). The study of these rings grew out of the theory of representations of a finite group as a group of matrices over a field, and the subject is intimately related to duality, the duality from right to left modules induced by the hom functor and the duality related to annihilators. The present extent of the theory is vast, and this book makes no attempt to be encyclopedic; instead it provides an elementary, self-contained account of the basic facts about these rings at a level allowing researchers and graduate students to gain entry to the field.

Short rational generating functions for lattice point problems

Abstract: . We prove that for any fixed d the generating function of the projectionof the set of integer points in a rational d-dimensional polytope can be computed inpolynomial time. As a corollary, we deduce that various interesting sets of latticepoints, notably integer semigroups and (minimal) Hilbert bases of rational cones,have short rational generating functions provided certain parameters (the dimensionand the number of generators) are fixed. It follows then that many computationalproblems for such sets (for example, finding the number of positive integers notrepresentable as a non-negative integer combination of given coprime positive integersa 1 ,... ,a d ) admit polynomial time algorithms. We also discuss a related problem ofcomputing the Hilbert series of a ring generated by monomials. 1. Introduction and Main ResultsOur main motivation is the following question which goes back to Frobenius andSylvester.(1.1) The Frobenius Problem. Let a 1 ,... ,a d be positive coprime integers andletS =nµ

New 3d-scroll attractors in hyperchaotic chua's circuits forming a ring

Abstract: This paper presents an approach for generating new hyperchaotic attractors in a ring of Chua's circuits. By taking a closed chain of three circuits and exploiting sine functions as nonlinearities, the proposed technique enables 3D-scroll attractors to be generated. In particular, the paper shows that 3D-scroll dynamics can be designed by modifying six parameters related to the circuit nonlinearities.

Ring dark solitons and vortex necklaces in Bose-Einstein condensates.

Abstract: We introduce the concept of ring dark solitons in Bose-Einstein condensates. We show that relatively shallow rings are not subject to the snake instability, but a deeper ring splits into a robust ringlike cluster of vortex pairs, which performs oscillations in the radial and azimuthal directions, following the dynamics of the original ring soliton.

On Skew Armendariz Rings

Abstract: For a ring endomorphism α, we introduce α-skew Armendariz rings which are a generalization of α-rigid rings and Armendariz rings, and investigate their properties. Moreover, we study on the relationship between the Baerness and p.p.-property of a ring R and these of the skew polynomial ring R[x; α] in case R is α-skew Armendariz.

Poisson brackets and two-generated subalgebras of rings of polynomials

Abstract: Let A = F [x1, x2, . . . , xn] be a ring of polynomials over a field F on the variables x1, x2, . . . , xn. It is well known (see, for example, [11]) that the study of automorphisms of the algebra A is closely related with the description of its subalgebras. By the theorem of P. M. Cohn [4], a subalgebra of the algebra F [x] is free if and only if it is integrally closed. The theorem of A. Zaks [13] says that the Dedekind subalgebras of the algebra A are rings of polynomials in a single variable. A. Nowicki and M. Nagata [8] proved that the kernel of any nontrivial derivation of the algebra F [x, y], char(F ) = 0, is also a ring of polynomials in a single generator. An original solution of the occurrence problem for the algebra A, using the Groebner basis, was given by D. Shannon and M. Sweedler [9]. However, the method of the Groebner basis does not give any information about the structure of concrete subalgebras. Recall that the solubility of the occurrence problem for rings of polynomials over fields of characteristic 0 was proved earlier by G. Noskov [7]. The present paper is devoted to the investigation of the structure of twogenerated subalgebras of A. In the sequel, we always assume that F is an arbitrary field of characteristic 0. Let us denote by f the highest homogeneous part of an element f ∈ A, and by 〈f1, f2, . . . , fk〉 the subalgebra of A generated by the elements f1, f2, . . . , fk ∈ A. Definition 1. A pair of polynomials f1, f2 ∈ A is called ∗-reduced if they satisfy the following conditions: 1) f1, f2 are algebraically dependent; 2) f1, f2 are algebraically independent; 3) f1 / ∈ 〈f2〉, f2 / ∈ 〈f1〉. Recall that a pair f1, f2 with condition 3) is usually called reduced. Condition 1) means that we exclude the trivial case when f1, f2 are algebraically independent. We do not consider the case when f1, f2 are algebraically dependent. Concerning this case, recall the well-known theorem of S. S. Abhyankar and T. -T. Moh [1], which says that if f, g ∈ F [x] and 〈f, g〉 = F [x], then f ∈ 〈ḡ〉 or ḡ ∈ 〈f〉.

On the Erdős-Volkmann and Katz-Tao ring conjectures

Abstract: Abstract. ((without abstract))

System and method for fault recovery for a two line bi-directional ring network

Abstract: The present invention provides a protection protocol for fault recovery, such as a ring wrap, for a network, such as a two line bi-directional ring network. An embodiment of the present invention works in conjunction with a ring topology network in which a node in the network can identify a problem with a connection between the node and a first neighbor. The present invention provides a protection protocol which simplifies the coordination required by the nodes in a ring network. The nodes do not need to maintain a topology map of the ring, identifying and locating each node on the ring, for effective protection. Additionally, independently operating ring networks can be merged and the protection protocol will appropriately remove a protection, such as a ring wrap, to allow the formation of a single ring. It also provides for multiple levels of protection priority so that protection for a high priority failure, such as a physical break in a connection, would remove protection for a low priority failure, such as a signal degrade, on another link.

The Alexander polynomial of a plane curve singularity via the ring of functions on it

Abstract: We prove two formulae that express the Alexander polynomial $\Delta\sp C$ of several variables of a plane curve singularity $C$ in terms of the ring $\mathscr {O}\sb C$ of germs of analytic functions on the curve. One of them expresses $\Delta\sp C$ in terms of dimensions of some factors corresponding to a (multi-indexed) filtration on the ring $\mathscr {O}\sb C$. The other one gives the coefficients of the Alexander polynomial $\Delta\sp C$ as Euler characteristics of some explicitly described spaces (complements to arrangements of projective hyperplanes).

On the Algebraic Structure of Quasi-cyclic Codes II: Chain Rings

Abstract: The ring decomposition technique of part I is extended to the case when the factors in the direct product decomposition are no longer fields but arbitrary chain rings. This includes not only the case of quasi-cyclic codes over rings but also the case of quasi-cyclic codes over fields whose co-index is no longer prime to the characteristic of the field. A new quaternary construction of the Leech lattice is derived.

Solving Polynomial Equation Systems I

Abstract: In this fourth and final volume the author extends Buchberger's Algorithm in three different directions. First, he extends the theory to group rings and other Ore-like extensions, and provides an operative scheme that allows one to set a Buchberger theory over any effective associative ring. Second, he covers similar extensions as tools for discussing parametric polynomial systems, the notion of SAGBI-bases, Grobner bases over invariant rings and Hironaka's theory. Finally, Mora shows how Hilbert's followers - notably Janet, Gunther and Macaulay - anticipated Buchberger's ideas and discusses the most promising recent alternatives by Gerdt (involutive bases) and Faugere (F4 and F5). This comprehensive treatment in four volumes is a significant contribution to algorithmic commutative algebra that will be essential reading for algebraists and algebraic geometers.

Cosimplicial resolutions and homotopy spectral sequences in model categories

Abstract: We develop a general theory of cosimplicial resolutions, homotopy spectral sequences, and completions for objects in model categories, extending work of Bouseld{Kan and Bendersky{Thompson for ordinary spaces. This is based on a generalized cosimplicial version of the Dwyer{Kan{Stover theory of resolution model categories, and we are able to construct our homotopy spectral sequences and completions using very flexible weak resolutions in the spirit of relative homological algebra. We deduce that our completion functors have triple structures and preserve certain ber squares up to homotopy. We also deduce that the Bendersky{Thompson completions over connective ring spectra are equivalent to Bouseld{Kan completions over solid rings. The present work allows us to show, in a subsequent paper, that the homotopy spectral sequences over arbitrary ring spectra have well-behaved composition pairings.

High-order lifting and integrality certification

Abstract: Reductions to polynomial matrix multiplication are given for some classical problems involving a nonsingular input matrix over the ring of univariate polynomials with coefficients from a field. High-order lifting is used to compute the determinant, the Smith form, and a rational system solution with about the same number of field operations as required to multiply together two matrices having the same dimension and degree as the input matrix. Integrality certification is used to verify correctness of the output. The algorithms are space efficient.

Symmetry in the vanishing of Ext over Gorenstein rings

Abstract: We investigate symmetry in the vanishing of Ext for finitely generated modules over local Gorenstein rings. In particular, we define a class of local Gorenstein rings, which we call AB rings, and show that for finitely generated modules $M$ and $N$ over an AB ring $R$, $\mathrm{Ext}^i_R(M,N)=0$ for all $i\gg 0$ if and only if $\mathrm{Ext}^i_R(N,M)=0$ for all $i\gg 0$.

(0,2) Duality

Abstract: We construct dual descriptions of (0, 2) gauged linear sigma models. In some cases, the dual is a (0, 2) Landau-Ginzburg theory, while in other cases, it is a non-linear sigma model. The duality map defines an analogue of mirror symmetry for (0, 2) theories. Using the dual description, we determine the instanton corrected chiral ring for some illustrative examples. This ring defines a (0, 2) generalization of the quantum cohomology ring of (2, 2) theories.

Higher algebraic K-theory for actions of diagonalizable groups

Abstract: We study the K-theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces: our main result shows how to reconstruct the K-theory ring of such an action from the K-theory rings of the loci where the stabilizers have constant dimension. We apply this to the calculation of the equivariant K-theory of toric varieties, and give conditions under which the Merkurjev spectral sequence degenerates, so that the equivariant K-theory ring determines the ordinary K-theory ring. We also prove a very refined localization theorem for actions of this type.

On Reasoning About Rings

Abstract: Distributed protocols are often composed of similar processes connected in a unidirectional ring network. Processes communicate by passing a token in a fixed direction; the process that holds the token is allowed to perform certain actions. Usually, correctness properties are expected to hold irrespective of the size of the ring. We show that the question of checking many useful correctness properties for rings of all sizes can be reduced to checking them on ring of sizes up to a small cutoff size. We apply our results to the verification of a mutual exclusion protocol and Milner's scheduler protocol.

Exact Nonstationary Probabilities in the Asymmetric Exclusion Process on a Ring

Abstract: The complete solution of the master equation for a system of interacting particles of finite density is presented. By using a new form of the Bethe ansatz, the totally asymmetric exclusion process on a ring is solved for arbitrary initial conditions and time intervals.

Efficient multi-party computation over rings

Abstract: Secure multi-party computation (MPC) is an active research area, and a wide range of literature can be found nowadays suggesting improvements and generalizations of existing protocols in various directions. However, all current techniques for secure MPC apply to functions that are represented by (boolean or arithmetic) circuits over finite fields. We are motivated by two limitations of these techniques: - GENERALITY. Existing protocols do not apply to computation over more general algebraic structures (except via a brute-force simulation of computation in these structures). - EFFICIENCY. The best known constant-round protocols do not efficiently scale even to the case of large finite fields. Our contribution goes in these two directions. First, we propose a basis for unconditionally secure MPC over an arbitrary finite ring, an algebraic object with a much less nice structure than a field, and obtain efficient MPC protocols requiring only a black-box access to the ring operations and to random ring elements. second, we extend these results to the constant-round setting, and suggest efficiency improvements that are relevant also for the important special case of fields. We demonstrate the usefulness of the above results by presenting a novel application of MPC over (non-field) rings to the round-efficient secure computation of the maximum function.

Skew category, Galois covering and smash product of a $k$-category

Abstract: In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In case of the smash product for an infinite group our construction specialized for a ring agrees with M. Beattie's construction of a ring with local units in \cite{be}. We recover in a categorical generalized setting the Duality Theorems of M. Cohen and S. Montgomery in \cite{cm}, and we provide a unification with the results on coverings of quivers and relations by E. Green in \cite{g}. We obtain a confirmation in a quiver and relations free categorical setting that both constructions are mutual inverses, namely the quotient of a free action category and the smash product of a graded category. Finally we describe functorial relations between the representation theories of a category and of a Galois cover of it.

Efficient Multi-Party Computation over Rings.

Abstract: Secure multi-party computation (MPC) is an active research area, and a wide range of literature can be found nowadays suggesting improvements and generalizations of existing protocols in various directions. However, all current techniques for secure MPC apply to functions that are represented by (boolean or arithmetic) circuits over finite fields. We are motivated by two limitations of these techniques: - GENERALITY. Existing protocols do not apply to computation over more general algebraic structures (except via a brute-force simulation of computation in these structures). - EFFICIENCY. The best known constant-round protocols do not efficiently scale even to the case of large finite fields. Our contribution goes in these two directions. First, we propose a basis for unconditionally secure MPC over an arbitrary finite ring, an algebraic object with a much less nice structure than a field, and obtain efficient MPC protocols requiring only a black-box access to the ring operations and to random ring elements. Second, we extend these results to the constant-round setting, and suggest efficiency improvements that are relevant also for the important special case of fields. We demonstrate the usefulness of the above results by presenting a novel application of MPC over (non-field) rings to the round-efficient secure computation of the maximum function.

Induced Representations of Affine Hecke Algebras and Canonical Bases of Quantum Groups

Abstract: A criterion of irreducibility for induction products of evaluation modules of type A affine Hecke algebras is given. It is derived from multiplicative properties of the canonical basis of a quantum deformation of the Bernstein—Zelevinsky ring.

(0,2) Duality

Abstract: We construct dual descriptions of (0,2) gauged linear sigma models. In some cases, the dual is a (0,2) Landau-Ginzburg theory, while in other cases, it is a non-linear sigma model. The duality map defines an analogue of mirror symmetry for (0,2) theories. Using the dual description, we determine the instanton corrected chiral ring for some illustrative examples. This ring defines a (0,2) generalization of the quantum cohomology ring of (2,2) theories.