Showing papers on "Ring (mathematics) published in 2000"

Algebras and Modules in Monoidal Model Categories

Abstract: In recent years the theory of structured ring spectra (formerly known as A$_{\infty}$- and E$_{\infty}$-ring spectra) has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products. Now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra. In this paper we provide a general method for constructing model category structures for categories of ring, algebra, and module spectra. This provides the necessary input for obtaining model categories of symmetric ring spectra, functors with smash product, $\Gamma$-rings, and diagram ring spectra. Algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra. 1991 Mathematics Subject Classification: primary 55U35; secondary 18D10.

Determinants of block matrices

Abstract: Let us first consider the 2 x 2 matrices and Their sum and product are given by Here the entries a, b, c, d, e, f, g, h can come from a field, such as the real numbers, or more generally from a ring, commutative or not.

Entangled Rings

Abstract: Consider a ring of N qubits in a translationally invariant quantum state We ask to what extent each pair of nearest neighbors can be entangled Under certain assumptions about the form of the state, we find a formula for the maximum possible nearest-neighbor entanglement We then compare this maximum with the entanglement achieved by the ground state of an antiferromagnetic ring consisting of an even number of spin-1/2 particles We find that, though the antiferromagnetic ground state does not maximize the nearest-neighbor entanglement relative to all other states, it does so relative to other states having zero z-component of spin

On the Structure of Linear and Cyclic Codes over a Finite Chain Ring

Abstract: We generalise structure theorems of Calderbank and Sloane for linear and cyclic codes over ℤ pa to a finite chain ring. Our results are more detailed and do not use non-trivial results from Commutative Algebra.

Support varieties and cohomology over complete intersections

Abstract: Quillen’s geometric approach to the cohomology of finite groups [30] has revolutionized modular representation theory. The ideology and techniques involved have been generalized and extended to representations of various Hopf algebras, culminating in the recent work of Friedlander and Suslin [17] on finite group schemes. In this paper we develop geometric methods for the study of finite modules over a local complete intersection R. If R is such a ring and m is its maximal ideal, then the m-adic completion R has the form Q/(f), where f is a regular sequence and Q is a regular local ring that can be taken to be a ring of formal power series over a field or a discrete valuation ring. The least number of equations needed to cut out R from a regular local ring equals the codimension of R, where codim R = νR(m)− dim R and νR(M) denotes the minimal number of generators of an R-module M. In [5] a cone, that is, a homogeneous algebraic set VR(M) is attached to each finite R-module M and used to study its minimal free resolution. Here we prove

A New Cohomology Theory for Orbifold

Abstract: Motivated by orbifold string theory, we introduce orbifold cohomology group for any almost complex orbifold and orbifold Dolbeault cohomology for any complex orbifold. Then, we show that our new cohomology group satisfies Poincare duality and has a natural ring structure. Some examples of orbifold cohomology ring are computed.

A canonical bundle formula

Abstract: A higher dimensional analogue of Kodaira’s canonical bundle formula is obtained. As applications, we prove that the log-canonical ring of a klt pair with κ ≤ 3is finitely generated, and that there exists an effectively computable natural number M such that | MK X | induces the Iitaka fibering for every algebraic threefold X with Kodaira dimension κ =1 .

On the Hamming distance of linear codes over a finite chain ring

Abstract: Let R be a finite chain ring (e.g., a Galois ring), K its residue field, and C a linear code over R. We prove that d(C), the Hamming distance of C, is d((~C~:~/spl alpha/~)~), where (C:/spl alpha/) is a submodule quotient, /spl alpha/ is a certain element of R, and denotes the canonical projection to K. These two codes also have the same set of minimal codeword supports. We explicitly construct a generator matrix/polynomial of (~C~:~/spl alpha/~)~ from the generator matrix/polynomials of C. We show that in general d(C)/spl les/d(C~) with equality for free codes (i.e., for free R-submodules of R/sup n/) and in particular for Hensel lifts of cyclic codes over K. Most of the codes over rings described in the literature fall into this class. We characterize minimum distance separable (MDS) codes over R and prove several analogs of properties of MDS codes over finite fields. We compute the Hamming weight enumerator of a free MDS code over R.

Energy and velocity of a forming vortex ring

Abstract: It is known that vortex rings formed by large stroke ratios (in a piston/cylinder arrangement) pinch off from their generating jets at a fairly constant universal time scale. In this paper we show that the hypothesis that at the pinch off the translational velocity of the ring equals the jet flow velocity near the ring is equivalent to the recently proposed idea based on a variational principle by Kelvin and Benjamin that the pinch off occurs when the apparatus is no longer able to deliver energy at a rate required for steady vortex ring existence. A formula for the propagation velocity of a thick vortex ring is also proposed and compared with available experimental data and empirical relations.

Packet switching fabric having a segmented ring with token based resource control protocol and output queuing control

Abstract: A packet switching fabric includes means forming a data ring, means forming a control ring, and means forming a plurality of data communication network links each having at least one network node coupled thereto. The fabric further includes a plurality of output queuing controlled switching devices coupled together by the data ring means and the control ring means so that the network links can be selectively communicatively coupled. Each of the output queuing controlled switching devices includes control ring processing means operative to develop, transmit and receive control messages to and from adjacent ones of the devices via the control ring means. The control messages provide for controlling packet transfer operations including transmitting associated selected ones of the received data packets from the associated source device to the associated destination device via an associated source-destination channel path including associated ones of the data ring segments and an associated one of the memory unit links. A channel resource patrol message is passed by each one of the devices, the patrol message carrying channel bandwidth information indicative of bandwidth available on the data ring means and bandwidth available on memory unit links. The control ring processing means is responsive to the channel bandwidth information and operative to read and modify the channel bandwidth information for the purpose of managing data transfer via the data ring means and via each of the memory unit links.

The Enhancon and N=2 Gauge Theory/Gravity RG Flows

Abstract: We study the family of ten dimensional type IIB supergravity solutions corresponding to renormalisation group flows from N=4 to N=2 supersymmetric Yang-Mills theory. Part of the solution set corresponds to a submanifold of the Coulomb branch of the gauge theory, and we use a D3-brane probe to uncover details of this physics. At generic places where supergravity is singular, the smooth physics of the probe yields the correct one-loop form of the effective low energy gauge coupling. The probe becomes tensionless on a ring at finite radius. Supergravity flows which end on this ``enhancon'' ring correspond to the vacua where extra massless degrees of freedom appear in the gauge theory, and the gauge coupling diverges there. We identify an SL(2,Z) duality action on the enhancon ring which relates the special vacua, and comment on the massless dyons within them. We propose that the supergravity solution inside the enhancon ring should be excised, since the probe's tension is unphysical there.

Localization in Categories of Complexes and Unbounded Resolutions

Abstract: In this paper we show that for a Grothendieck category and a complex in there is an associated localization endofunctor in . This means that is idempotent (in a natural way) and that the objects that go to 0 by are those of the smallest localizing (= triangulated and stable for coproducts) subcategory of that contains . As applications, we construct -injective resolutions for complexes of objects of and derive Brown representability for from the known result for , where is a ring with unit.

F-regular and F-pure rings vs. log terminal and log canonical singularities

Abstract: In this paper, we investigate the relationship of F-regular (resp. F-pure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of F-regularity and F-purity to "F-singularities of pairs." The notions of F-regular and F-pure rings in characteristic $p>0$ are characterized by a splitting of the Frobenius map, and define some classes of rings having "mild" singularities. On the other hand, there are notions of log terminal and log canonical singularities defined via resolutions of singularities in characteristic zero. These are defined also for pairs of a normal variety and a $\Bbb Q$-divisor $\Delta$ on it, and play an important role in birational algebraic geometry. As an analog of these singularities of pairs in characteristic zero, we define the notions of "F-singularities of pairs," namely strong F-regularity, divisorial F-regularity and F-purity for a pair $(A,\Delta)$ of a normal ring $A$ of characteristic $p > 0$ and an effective $\Bbb Q$-divisor $\Delta$ on $Y = Spec A$. The main theorem of this paper asserts that, if $K_Y + \Delta$ is $\Bbb Q$-Cartier, then the above three variants of F-singularitiesof pairs imply KLT, PLT and LC properties, respectively. We also prove some results for F-singularities of pairs which are analoguous to singularities of pairs in characteristic zero.

Logarithmic series and Hodge integrals in the tautological ring. With an appendix by Don Zagier.

Abstract: 0.1. Overview. Let Mg be the moduli space of Deligne–Mumford stable curves of genus g ≥ 2. The study of the Chow ring of the moduli space of curves was initiated by Mumford in [Mu]. In the past two decades, many remarkable properties of these intersection rings have been discovered. Our first goal in this paper is to describe a new perspective on the intersection theory of the moduli space of curves that encompasses advances from both classical degeneracy studies and topological gravity. This approach is developed in Sections 0.2–0.7. The main new results of the paper are computations of basic Hodge integral series in A∗(Mg) encoding the canonical evaluations of κg−2−iλi . The motivation for the study of these tautological elements and the series results are given in Section 0.8. The body of the paper contains the Hodge integral derivations.

Structure of hyperbolic unitary groups i. elementary subgroups

Abstract: This is the first in a series of papers dedicated to the structure of hyperbolic unitary groups over form rings and their subgroups. In this part, we recall foundations of the theory and study the elementary subgroup of the hyperbolic unitary group over a form ring. In particular, using a variant of Suslin’s patching method, we prove standard commutator formulae for relative elementary subgroups.

The enhancon and Script N = 2 gauge theory/gravity RG flows

Abstract: We study the family of ten-dimensional type-IIB supergravity solutions corresponding to renormalisation group flows from = 4 to = 2 supersymmetric SU(N) Yang-Mills theory. Part of the solution set corresponds to a submanifold of the Coulomb branch of the gauge theory, and we use a D3-brane probe to uncover details of this physics. At generic places where supergravity is singular, the smooth physics of the probe yields the correct one-loop form of the effective low energy gauge coupling. The probe becomes tensionless on a ring at finite radius. Supergravity flows which end on this ``enhancon'' ring correspond to the vacua where extra massless degrees of freedom appear in the gauge theory, and the gauge coupling diverges there. We identify an SL(2,) duality action on the enhancon ring which relates the special vacua, and comment on the massless dyons within them. We propose that the supergravity solution inside the enhancon ring should be excised, since the probe's tension is unphysical there.

Logarithmic series and Hodge integrals in the tautological ring (with an appendix by D. Zagier)

Abstract: We describe a new perspective on the intersection theory of the moduli space of curves involving both Virasoro constraints and Gorenstein conditions. The main result of the paper is the computation of a basic 1-point Hodge integral series occurring in the tautological ring of the moduli space of nonsingular curves. In the appendix by D. Zagier, "Polynomials arising from the tautological ring", a detailed study is made of certain polynomials whose coefficients are intersection numbers on moduli space. The paper and the appendix together provide proofs of all previously conjectured formulas for 1-point integrals in the tautological ring, and of natural extensions of these formulas as well.

A model complete theory of valued d-fields

Abstract: The notion of a D-ring, generalizing that of a dierential or a dierence ring, is introduced. Quantifier elimination and a version of the Ax- Kochen-Ershov principle is proven for a theory of valued D-fields of residual characteristic zero.

Putting T-Invertibility to Use

Abstract: This article gives a survey of how the notion of t-invertibility has, in recent years, been used to develop new concepts that enhance our understanding of the multiplicative structure of commutative integral domains. The concept of t-invertibility arises in the context of star operations. However, in general terms a (fractional) ideal A, of an integral domain D, is t-invertible if there is a finitely generated (fractional) ideal F ⊆ A and a finitely generated fractional ideal G ⊆ A -l such that (FG)-1 = D. In a more specialized context the notion of t-invertibility has to do with the t-operation which is one of the so called star operations. There seems to be no book other than Gilmer’s [Gil] that treats star operations purely from a ring theoretic view point. But a lot has changed since Gilmer’s book was published. So I have devoted a part of section 1. to an introduction to star operations, *-invertibility in general, and t-invertibility in particular.

Symmetric groups and the cup product on the cohomology of Hilbert schemes

Abstract: The integral cohomology ring of the Hilbert scheme of n-tuples on the affine plane is shown to be isomorphic to the graded ring associated to a filtration of the ring of integral class functions on the symmetric group.

An introduction to rings and modules with K-theory in view

Abstract: This book, first published in 2000, is a concise introduction to ring theory, module theory and number theory, ideal for a first year graduate student, as well as an excellent reference for working mathematicians in other areas Starting from definitions, the book introduces fundamental constructions of rings and modules, as direct sums or products, and by exact sequences It then explores the structure of modules over various types of ring: noncommutative polynomial rings, Artinian rings (both semisimple and not), and Dedekind domains It also shows how Dedekind domains arise in number theory, and explicitly calculates some rings of integers and their class groups About 200 exercises complement the text and introduce further topics This book provides the background material for the authors' companion volume Categories and Modules, soon to appear Armed with these two texts, the reader will be ready for more advanced topics in K-theory, homological algebra and algebraic number theory

Tangential Structures on Toric Manifolds, and Connected Sums of Polytopes

Abstract: We extend work of Davis and Januszkiewicz by considering {\it omnioriented} toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples $B_{i,j}$, which are toric manifolds over products of simplices, and verify that their natural stably complex structure is induced by an omniorientation. Studying connected sums of products of the $B_{i,j}$ allows us to deduce that every complex cobordism class of dimension >2 contains a toric manifold, necessarily connected, and so provides a positive answer to the toric analogue of Hirzebruch's famous question for algebraic varieties. In previous work, we dealt only with disjoint unions, and ignored the relationship between the stably complex structure and the action of the torus. In passing, we introduce a notion of connected sum $#$ for simple $n$-dimensional polytopes; when $P^n$ is a product of simplices, we describe $P^n# Q^n$ by applying an appropriate sequence of {\it pruning operators}, or hyperplane cuts, to $Q^n$.

Littlewood-Richardson coefficients via Yang-Baxter equation

Abstract: This paper presents an interpretation for the Littlewood-Richardson coefficients in terms of a system of quantum particles. Our approach is based on a certain scattering matrix that satisfies a Yang-Baxter-type equation. The corresponding piecewise-linear transformations of parameters give a solution to the tetrahedron equation. These transformation maps are naturally related to the dual canonical bases for modules over the quantum enveloping algebra Uq(sln). A byproduct of our construction is an explicit description for Kashiwara’s parametrizations of dual canonical bases. This solves a problem posed by Berenstien and Zelevinsky. We present a graphical interpretation of the scattering matrices in terms of web functions, which are related to honeycombs of Knutson and Tao. The aim of this paper is to further investigate the Grothendieck ring KN of polynomial representations of the general linear group GL(N). Let Vλ be the irreducible representation of GL(N) with highest weight λ.The structure constants c ν of the Grothendieck ring in the basis of irreducible representations are given by

δ-derivations of prime alternative and mal’tsev algebras

Abstract: We give a description of δ-derivations of prime alternative and non-Lie Mal’tsev Φ-algebras, with some restrictions on an operator ring Φ. For algebras in these classes, every δ-derivation is proved trivial.

Parameter variation probing technique

Abstract: A technique for probing parameter variations in an integrated circuit chip includes providing a group of ring oscillators disposed over the integrated circuit chip, each oscillator producing an output representative of an integrated circuit chip parameter or group of parameter. A controller is provided to selectively enable one of the group of ring oscillators and a multiplexer is provided to output an output of one of the group of ring oscillators enabled by the controller. Each ring oscillator may include a gated inverter, having an enable/disable input, connected to a group of inverters arranged in a ring, the total number of inverters being an odd number. The group of ring oscillators may include a shift register chain consisting of a group of shift registers, an output of each shift register being respectively connected to an enable/disable input of the gated inverter of one of the group of ring oscillators. The multiplexer may include an exclusive-OR gate having inputs respectively connected to an output of each of the group of ring oscillators. An inverter may be disposed between each output of the group of ring oscillators and its' respective input to the multiplexer.

On Polynomial Extensions of Principally Quasi-Baer Ring

Abstract: A ring R is called a right principally quasi-Baer (or simply right pq-Baer) ring if the right annihilator of a lprincipal right ideal is generated by an idempotent Weshow that a ring R is right pq-Baer if and only if R[x]is right pq-Baer This result allow us to generalize results of E P Armendariz and S J