Showing papers on "Ring (mathematics) published in 2002"
TL;DR: The vacuum Einstein equations in five dimensions are shown to admit a solution describing a stationary asymptotically flat spacetime regular on and outside an event horizon of topology S1xS2, which describes a rotating "black ring".
Abstract: The vacuum Einstein equations in five dimensions are shown to admit a solution describing a stationary asymptotically flat spacetime regular on and outside an event horizon of topology ${S}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{2}$. It describes a rotating ``black ring.'' This is the first example of a stationary asymptotically flat vacuum solution with an event horizon of nonspherical topology. The existence of this solution implies that the uniqueness theorems valid in four dimensions do not have simple five-dimensional generalizations. It is suggested that increasing the spin of a spherical black hole beyond a critical value results in a transition to a black ring, which can have an arbitrarily large angular momentum for a given mass.
1,099 citations
TL;DR: In this article, an explicit combinatorial formula for the structure constants of the Grothendieck ring of a Grassmann variety with respect to its basis of Schubert structure sheaves is given.
Abstract: We prove an explicit combinatorial formula for the structure constants of the Grothendieck ring of a Grassmann variety with respect to its basis of Schubert structure sheaves. We furthermore relate K-theory of Grassmannians to a bialgebra of stable Grothendieck polynomials, which is a K-theory parallel of the ring of symmetric functions.
280 citations
TL;DR: In this article, it was shown that a maximal Cohen-Macaulay local ring of finite Cohen-Maulay type has an isolated singularity and a complete local Gorenstein domain of positive characteristic p and dimension d is F-rational if and only if the number of copies of R splitting out of R 1/p e divided by p de has a positive limit.
Abstract: This paper contains two theorems concerning the theory of maximal Cohen-Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen-Macaulay modules M and N must have finite length, provided only finitely many isomorphism classes of maximal Cohen-Macaulay modules exist having ranks up to the sum of the ranks of M and N. This has several corollaries. In particular it proves that a Cohen-Macaulay local ring of finite Cohen-Macaulay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning when a ring is Gorenstein or a complete intersection on the punctured spectrum, and the recent theorem of Leuschke and Wiegand that the completion of an excellent Cohen-Macaulay local ring of finite Cohen-Macaulay type is again of finite Cohen-Macaulay type. The second theorem proves that a complete local Gorenstein domain of positive characteristic p and dimension d is F-rational if and only if the number of copies of R splitting out of R 1/p e divided by p de has a positive limit. This result generalizes work of Smith and Van den Bergh. We call this limit the F-signature of the ring and give some of its properties.
175 citations
TL;DR: An efficient algorithm for finding primitive rings in a topological network has been developed and the calculation of complete ring statistics, with virtually no upper limit on ring size, becomes feasible for very large network systems.
162 citations
TL;DR: In this paper, a duality theory between the continuous representations of a compact p-adic Lie group in Banach spaces over a given padic field and certain compact modules over the completed group ring is introduced.
Abstract: We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group ringoK[[G]]. We then introduce a “finiteness” condition for Banach space representations called admissibility. It will be shown that under this duality admissibility corresponds to finite generation over the ringK[[G]]: =K ⊗oK[[G]]. Since this latter ring is noetherian it follows that the admissible representations ofG form an abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the groupG: = GL2(ℤp).
157 citations
Journal Article•
TL;DR: In this article, a nouvelle theorie de Dieudonne is proposed, which associe a group of formels p-divisible X sur un anneau p-adique excellent R. A partir du display on peut exhiber des equations structurelles for le module de Cartier de X.
Abstract: Nous proposons une nouvelle theorie de Dieudonne qui associe a un groupe formel p-divisible X sur un anneau p-adique excellent R un objet d'algebre lineaire appele display. A partir du display on peut exhiber des equations structurelles pour le module de Cartier de X et retrouver son cristal de Grothendieck-Messing. Nous donnons des applications a la theorie des deformations des groupes formels p-divisibles.
155 citations
Posted Content•
TL;DR: In this article, it was shown that for any fixed d the generating function of the projection of the set of integer points in a rational d-dimensional polytope can be computed in polynomial time.
Abstract: We prove that for any fixed d the generating function of the projection of the set of integer points in a rational d-dimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice points, notably integer semigroups and (minimal) Hilbert bases of rational cones, have short rational generating functions provided certain parameters (the dimension and the number of generators) are fixed. It follows then that many computational problems for such sets (for example, finding the number of positive integers not representable as a non-negative integer combination of given coprime positive integers a_1 ... a_d admit polynomial time algorithms. We also discuss a related problem of computing the Hilbert series of a ring generated by monomials.
147 citations
Book•
01 Jan 2002
TL;DR: In this article, the Jacobson radical groups of units (RUGs) were used to construct ring constructions, and the Gradient Gradient Matrix Ring (GRR) was used for ring construction.
Abstract: Preliminaries Graded Rings Examples of Ring Constructions The Jacobson Radical Groups of Units Finiteness Conditions Pl-Rings and Varieties Gradings of Matrix Rings Examples of Applications Open Problems.
143 citations
TL;DR: In this paper, a ring is a clean ring if every element of a ring can be written uniquely as the sum of a unit and an idempotent, which is the definition of uniquely clean rings.
Abstract: As defined by Nicholson a (noncommutative) ring is a clean ring if every element of is a sum of a unit and an idempotent. Let be a commutative ring with identity. We define to be a uniquely clean ring if every element of can be written uniquely as the sum of a unit and an idempotent. Examples of clean rings (uniquely clean rings) include von Neumann regular rings (Boolean rings) and quasilocal rings (with residue field ). A ring is a clean ring or uniquely clean ring if and only if is. So every zero-dimensional ring is a clean ring, but a zero-dimensional ring is a uniquely clean ring if and only if is a Boolean ring.
132 citations
Posted Content•
TL;DR: In this paper, the superpotential of a certain class of N = 1 supersymmetric type II compactications with fluxes and D-branes was studied, and it has an important two-dimensional meaning in terms of a chiral ring of the topologically twisted theory on the world-sheet.
Abstract: We study the superpotential of a certain class ofN = 1 supersymmetric type II compactications with fluxes andD-branes. We show that it has an important two-dimensional meaning in terms of a chiral ring of the topologically twisted theory on the world-sheet. In the open-closed string B-model, this chiral ring is isomorphic to a certain relative cohomology group V , which is the appropriate mathematical concept to deal with both the open and closed string sectors. The family of mixed Hodge structures on V then implies for the superpotential to have a certain geometric structure. This structure represents a holomorphic, N = 1 supersymmetric generalization of the well-known N = 2 special geometry. It denes an integrable connection on the topological family of open-closed B-models, and a set of special coordinates on the spaceM of vev’s inN = 1 chiral multiplets. We show that it can be given a very concrete and simple realization for linear sigma models, which leads to a powerful and systematic method for computing the exact non-perturbativeN = 1 superpotentials for a broad class of toric D-brane geometries.
127 citations
TL;DR: In this paper, a decomposition of the global Hochschild cohomology where is the relative tangent sheaf has been derived for a separated finite type scheme over a noetherian base ring.
Abstract: Let be a separated finite type scheme over a noetherian base ring . There is a complex of topological -modules, called the complete Hochschild chain complex of . To any -module —not necessarily quasi-coherent—we assign the complex of continuous Hochschild cochains with values in . Our first main result is that when is smooth over there is a functorial isomorphism in the derived category , where . The second main result is that if is smooth of relative dimension and is invertible in , then the standard maps induce a quasi-isomorphism When this is the quasi-isomorphism underlying the Kontsevich Formality Theorem. Combining the two results above we deduce a decomposition of the global Hochschild cohomology where is the relative tangent sheaf.
Patent•
04 Mar 2002
TL;DR: In this article, a network configuration protocol and algorithm are described which resolve deficiencies with existing protocols, and a loop free topology is achieved by selectively blocking and unblocking data traffic in one of the ring ports of the single master bridge for the ring, while all other bridges in the ring keep their ports in non-blocked states.
Abstract: A network configuration protocol and algorithm are described which resolve deficiencies with existing protocols. A large network having many bridges may be built as a combination of smaller networks, many of which may each be arranged in a ring topology. Each ring may be monitored by a single master bridge regularly sending control packets, and each other bridge in the ring does not make decisions with respect to its status. A loop free topology is achieved by selectively blocking and unblocking data traffic in one of the ring ports of the single master bridge for the ring, while all other bridges in the ring keep their ports in non-blocked states. In multiple ring topologies, each ring has a single master bridge which chooses one of its ports to be blocking. When rings are connected through a shared link formed between two shared bridges, rings with higher priorities carry control packets of rings with lower priorities so that, in case of failure of the shared link, a single ring may be formed from the connected rings with only the master bridge of the higher priority ring aware of and monitoring the larger ring. In case of any link failure inside a given ring, the master bridge quickly detects the failure and automatically changes its blocking port to a non-blocking state in which traffic may flow and follow an alternate path, avoiding the failed link.
TL;DR: In this paper, the authors investigate commutativity of R satisfying any one of the properties (i)d([x,y]) = [x, y], (ii)d(x o y) = xoy, (iii)d (x) o d(y) = 0, or (iv) d(x) O d(Y) = X o y, for all x, y in some apropriate subset of R.
Abstract: Let R be a ring and d : R → R a derivation of R. In the present paper we investigate commutativity of R satisfying any one of the properties (i)d([x,y]) = [x,y], (ii)d(x o y) = xoy, (iii)d(x) o d(y) = 0, or (iv)d(x) o d(y) = x o y, for all x, y in some apropriate subset of R.
Journal Article•
TL;DR: In this paper, the authors develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra, and give the toric quiver varieties, in the sense of Nakajima.
Abstract: Extending work of Bielawski-Dancer (3) and Konno (12), we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions. A toric hyperkahler variety is a complete intersection in a Lawrence toric variety. Both varieties are non- compact, and they share the same cohomology ring, namely, the Stanley-Reisner ring of a matroid modulo a linear system of parameters. Familiar applications of toric geometry to combinatorics, including the Hard Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov (10), are extended to the hyperkahler setting. When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of Nakajima (15).
TL;DR: In this article, a finite element ring rolling simulation by conventional Lagrangian codes carries an excessive computational cost and the main reason for this is the large number of incremental stages typically required to complete a full simulation.
TL;DR: The theory of non-commutative rings is introduced to provide a basis for the study of nonlinear control systems with time delays and properties of closed submodules are developed to obtain a result on the accessibility of such systems.
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TL;DR: In this paper, a theory of toric hyperkahler varieties is developed, which involves toric geometry, matroid theory and convex polyhedra, and is extended to semi-projective toric varieties.
Abstract: Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions A toric hyperkahler variety is a complete intersection in a Lawrence toric variety Both varieties are non-compact, and they share the same cohomology ring, namely, the Stanley-Reisner ring of a matroid modulo a linear system of parameters Familiar applications of toric geometry to combinatorics, including the Hard Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov, are extended to the hyperkahler setting When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of Nakajima
TL;DR: In this article, it was shown that if the characteristic of k is zero, then K_0(V_k) is not a domain, i.e., if k is a field.
Abstract: Let k be a field. Let K_0(V_k) denote the quotient of the free abelian group generated by the geometrically reduced varieties over k, modulo the relations of the form [X]=[X-Y]+[Y] whenever Y is a closed subvariety of X. Product of varieties makes K_0(V_k) into a ring. We prove that if the characteristic of k is zero, then K_0(V_k) is not a domain.
17 Sep 2002
TL;DR: In this article, the Rees algebra is made into a functor on modules over a ring in a way that extends its classical definition for ideals, and the analytic spread and reductions of a module M can be determined from any embedding of M into a free module, and in characteristic 0-but not in positive characteristic!
Abstract: In this paper we show that the Rees algebra can be made into a functor on modules over a ring in a way that extends its classical definition for ideals. The Rees algebra of a module M may be computed in terms of a maximal map f from M to a free module as the image of the map induced by f on symmetric algebras. We show that the analytic spread and reductions of M can be determined from any embedding of M into a free module, and in characteristic 0-but not in positive characteristic!-the Rees algebra itself can be computed from any such embedding.
Patent•
11 Mar 2002
TL;DR: In this article, the authors proposed a two-ring disclosed network, where the first ring transmits data in a clockwise direction and the other ring is transmitted in a counterclockwise direction.
Abstract: The disclosed network includes two rings, wherein a first ring transmits data in a clockwise direction, and the other ring transmits data in a counterclockwise direction. The traffic is removed from the ring by the destination node. During normal operations (i.e., all spans operational), data between nodes can flow on either ring. Thus, both rings are fully utilized during normal operations. The nodes periodically test the bit error rate of the links (or the error rate is constantly calculated) to detect a fault in one of the links. The detection of such a fault sends a broadcast signal to all nodes to reconfigure a routing table within the node so as to identify the optimum routing of source traffic to the destination node after the fault.
TL;DR: In this article, it was shown that the set P(X, F ) is a tame combinatorial pattern if the base ring R 0 is semilocal and of dimension ⩽ 1.
TL;DR: In this article, the stability of a generalization of Bourgin's result on approximate ring homomorphisms was shown for the case of approximate ring-homomorphisms with respect to approximate rings.
TL;DR: A framework of algebraic structures in the proof assistant Coq is described, in which a constructive proof of the fundamental theorem of algebra has been formalized in Coq by applying a combination of labelled record types and coercions.
TL;DR: In this paper, the authors studied the graded ∂(V)-module structure of C(Δ) of rational functions generated by {1/α ∣ α ∈ Δ}.
TL;DR: In this paper, the Baum-Connes Conjecture implies the modified trace conjecture, which says that the image of the standard trace K0(C* r (G))→ℝ takes values in λ G.
Abstract: We prove a version of the L2-index Theorem of Atiyah, which uses the universal center-valued trace instead of the standard trace. We construct for G-equivariant K-homology an equivariant Chern character, which is an isomorphism and lives over the ring ℤ⊂λ G ⊂ℚ obtained from the integers by inverting the orders of all finite subgroups of G. We use these two results to show that the Baum-Connes Conjecture implies the modified Trace Conjecture, which says that the image of the standard trace K0(C* r (G))→ℝ takes values in λ G . The original Trace Conjecture predicted that its image lies in the additive subgroup of ℝ generated by the inverses of all the orders of the finite subgroups of G, and has been disproved by Roy [15].
TL;DR: In this article, it was shown that a bijective map φ form A, a standard operator algebra on a Banach space of dimension > 1, onto a ring that satisfies φ(AB+BA)=φ(A)φ (B)+ φ (A,B∈ A) is additive.
TL;DR: In this paper, the authors studied the localized coherent structures of a general non-integrable (2+1)-dimensional KdV equation via a variable separation approach, and showed that the integrable case possesses the breather-like localized excitations, and the similar static ring soliton solutions as in the general nonintegrability case.
Abstract: We study the localized coherent structures of a generally nonintegrable (2+1)-dimensional KdV equation via a variable separation approach. In a special integrable case, the entrance of some arbitrary functions leads to abundant coherent structures. However, in the general nonintegrable case, an additional condition has to be introduced for these arbitrary functions. Although the additional condition has been introduced into the solutions of the nonintegrable KdV equation, there still exist many interesting solitary wave structures. Especially, the nonintegrable KdV equation possesses the breather-like localized excitations, and the similar static ring soliton solutions as in the integrable case. Furthermore, in the integrable case, the interaction between two travelling ring solitons is elastic, while in the nonintegrable case we cannot find even the single travelling ring soliton solution.
TL;DR: In this article, the structural properties of stable integral domains of commutative rings with identity have been studied, and the first half of a two-part study has been published.
Abstract: Let be a commutative ring with identity. A regular ideal of is stable if is a projective over its ring of endomorphisms. If every regular ideal of is stable, then is said to be stable. In the first half of a two-part study, we describe the basic structural properties of stable integral domains.
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TL;DR: In this paper, it was shown that the F-signature of a local ring of characteristic p is positive if and only if the ring is strongly F-regular, defined by Huneke and Leuschke.
Abstract: We show that the F-signature of a local ring of characteristic p, defined by Huneke and Leuschke, is positive if and only if the ring is strongly F-regular.
TL;DR: In this paper, the structure sheaf of a quiver variety in the Grothendieck ring of its embedding variety is shown to have signs that alternate with degree.
Abstract: We prove a formula for the structure sheaf of a quiver variety in the Grothendieck ring of its embedding variety. This formula generalizes and gives new expressions for Grothendieck polynomials. Our formula is stated in terms of coefficients that are uniquely determined by the geometry and can be computed by an explicit combinatorial algorithm. We conjecture that these coefficients have signs that alternate with degree. The proof of our formula involves K-theoretic generalizations of several useful cohomological tools, including the Thom-Porteous formula, the Jacobi-Trudi formula, and a Gysin formula of P. Pragacz.