Showing papers on "Ring (mathematics) published in 1996"
TL;DR: In this paper, Soergel et al. showed that the block of the Bernstein-Gelfand-gelfand category O that corresponds to any fixed central character is a Koszul ring and the dual of that ring governs a certain subcategory of the category O again.
Abstract: The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to repre- sentation theory. The paper consists of three parts relatively independent of each other. The first part gives a reasonably selfcontained introduction to Koszul rings and Koszul duality. Koszul rings are certain Z-graded rings with particularly nice homological properties which involve a kind of duality. Thus, to a Koszul ring one associates naturally the dual Koszul ring. The second part is devoted to an application to representation theory of semisimple Lie algebras. We show License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use KOSZUL DUALITY PATTERNS 527 that the block of the Bernstein-Gelfand-Gelfand category O that corresponds to any fixed central character is governed by the Koszul ring. Moreover, the dual of that ring governs a certain subcategory of the category O again. This generalizes the selfduality theorem conjectured by Beilinson and Ginsburg in 1986 and proved by Soergel in 1990. In the third part we study certain cate- gories of mixed perverse sheaves on a variety stratified by affine linear spaces. We provide a general criterion for such a category to be governed by a Koszul ring. In the flag variety case this reduces to the setup of part two. In the more general case of affine flag manifolds and affine Grassmannians the criterion should yield interesting results about representations of quantum groups and affine Lie algebras. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 E-mail address: sasha@math.mit.edu Department of Mathematics, The University of Chicago, Chicago, Illinois 60637 E-mail address: ginzburg@math.uchicago.edu Max-Planck-Institut fur Mathematik, Gottfried-Claren-Strase 26, D-53 Bonn 3, Germany Current address: Mathematisches Institut, Universitat Freiburg, Albertstrase 23b, D-79104 Freiburg, Germany E-mail address: soergel@sun1.mathematik.uni-freiburg.de License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
1,119 citations
Book•
01 Jan 1996
TL;DR: Semialgebraic geometry has been studied extensively in the literature as mentioned in this paper, where the main result is that real algebra can be viewed as a real algebra of excellent rings, and real spaces of signs of rings.
Abstract: I. A First Look at Semialgebraic Geometry.- 1. Real Closed Fields and Transfer Principles.- 2. What is Semialgebraic Geometry?.- 3. Real Spaces.- 4. Examples.- II. Real Algebra.- 1. The Real Spectrum of a Ring.- 2. Specializations, Zero Sets and Real Ideals.- 3. Real Valuations.- 4. Real Going-Up and Real Going-Down.- 5. Abstract Semialgebraic Functions.- 6. Cylindrical Decomposition.- 7. Real Strict Localization.- Notes.- III. Spaces of Signs.- 1. The Axioms.- 2. Forms.- 3. SAP-Spaces and Fans.- 4. Local Spaces of Signs.- 5. The Space of Signs of a Ring.- 6. Subspaces.- Notes.- IV. Spaces of Orderings.- 1. The Axioms Revisited.- 2. Basic Constructions.- 3. Spaces of Finite Type.- 4. Spaces of Finite Chain Length.- 5. Finite Type = Finite Chain Length.- 6. Local-Global Principles.- 7. Representation Theorem and Invariants.- Notes.- V. The Main Results.- 1. Stability Formulae.- 2. Complexity of Constructible Sets.- 3. Separation.- 4. Real Divisors.- 5. The Artin-Lang Property.- Notes.- VI. Spaces of Signs of Rings.- 1. Fans and Valuations.- 2. Field Extensions: Upper Bounds.- 3. Field Extensions: Lower Bounds.- 4. Algebras.- 5. Algebras Finitely Generated over Fields.- 6. Archimedean Rings.- 7. Coming Back to Geometry.- Notes.- VII. Real Algebra of Excellent Rings.- 1. Regular Homomorphisms.- 2. Excellent Rings.- 3. Extension of Orderings Under Completion.- 4. Curve Selection Lemma.- 5. Dimension, Valuations and Fans.- 6. Closures of Constructible Sets.- 7. Real Going-down for Regular Homomorphisms.- 8. Connected Components of Constructible Sets.- Notes.- VIII. Real Analytic Geometry.- 1. Semianalytic Sets.- 2. Semianalytic Set Germs.- 3. Cylindrical Decomposition of Germs.- 4. Rings of Global Analytic Functions.- 5. Hilbert's 17th Problem and Real Nullstellensatz.- 6. Minimal Generation of Global Semianalytic Sets.- 7. Topology of Global Semianalytic Sets.- 8. Germs at Compact Sets.- Notes.
173 citations
TL;DR: In this paper, an exactly soluble model for a two-dimensional ring is proposed, which can also describe quantum dots, anti-dots, one-dimensional rings and straight twodimensional wires, which provides an integrated picture for the electron states and their magnetic field response.
Abstract: An exactly soluble model for a two-dimensional ring is proposed. Using this model, we have obtained analytically the energy spectrum and wavefunctions for a ring in the presence of a uniform magnetic field and a thin magnetic flux. The model can also describe quantum dots, anti-dots, one-dimensional rings and straight two-dimensional wires, which provides an integrated picture for the electron states and their magnetic field response in these geometries. The simplicity and the flexibility of the model make it an ideal tool for the modelling of the Aharonov - Bohm effects and the persistent currents in quantum rings.
141 citations
TL;DR: In this paper, it was shown that for any Schur ring S over a cyclic group G, if every subgroup is an S -subgroup, then S is either a wedge product of Schur rings over smaller cyclic groups, or every S -principal subset is an orbit of an element under a fixed subgroup of Aut G.
93 citations
TL;DR: It is found that, for such a weakly coupled ring in a uniform magnetic field, not only do the electron states in different subbands of the ring produce different AB frequencies, the clockwise and anticlockwise moving states in the same subband also lead to two differentAB frequencies.
Abstract: We propose an exactly soluble model for a ring with finite width. Exact energy spectra and wave functions are obtained analytically for a ring in the presence of both a uniform perpendicular magnetic field and a thin magnetic flux through the ring center. We use the model to study the Aharonov-Bohm (AB) effect in an ideal annular ring that is weakly coupled to both the emitter and the collector. It is found that, for such a weakly coupled ring in a uniform magnetic field, not only do the electron states in different subbands of the ring produce different AB frequencies, the clockwise and anticlockwise moving states in the same subband also lead to two different AB frequencies. Therefore, when many subbands in the ring are populated, the large number of different AB frequencies generally result in an aperiodic AB oscillation. More striking is that, even when only one subband is populated, the two AB frequencies corresponding to the states moving in opposite directions also cause beating in the AB oscillations. We have obtained explicit expressions for all these AB frequencies. Our results produce a clear explanation for the recent experimental observation of Liu and co-workers. \textcopyright{} 1996 The American Physical Society.
83 citations
TL;DR: In this article, the representation ring of the quantum double of a finite group over fields of arbitrary characteristic is decompositions into ideals involving Green rings of subgroups and given characters of such Green rings, all characters arise in this fashion.
83 citations
TL;DR: The existence of the stack of micro-differential modules on an arbitrary contact manifold was shown in this paper, although we cannot expect the global existence of a ring of microdifferential operators.
Abstract: We show the existence of the stack of micro-differential modules on an arbitrary contact manifold, although we cannot expect the global existence of the ring of micro-differential operators.
80 citations
80 citations
TL;DR: In this article, it was shown that if R is a right hereditary non-right perfect ring, then the existence of p-test modules is independent of ZFC + GCH.
Abstract: A (right R-) module N is said to be a Whitehead test module for projectivity (shortly: a p-test module) provided for each module M , ExtR(M,N) = 0 implies M is projective. Dually, i-test modules are defined. For example, Z is a p-test abelian group iff each Whitehead group is free. Our first main result says that if R is a right hereditary non-right perfect ring, then the existence of p-test modules is independent of ZFC + GCH. On the other hand, for any ring R, there is a proper class of i-test modules. Dually, there is a proper class of p-test modules over any right perfect ring. A non-semisimple ring R is said to be fully saturated (κ-saturated) provided that all non-projective (≤ κ-generated non-projective) modules are i-test. We show that classification of saturated rings can be reduced to the indecomposable ones. Indecomposable 1-saturated rings fall into two classes: type I, where all simple modules are isomorphic, and type II, the others. Our second main result gives a complete characterization of rings of type II as certain generalized upper triangular matrix rings, GT (1, n, p, S, T ). The four parameters involved here are skew-fields S and T , and natural numbers n, p. For rings of type I, we have several partial results: e.g. using a generalization of Bongartz Lemma, we show that it is consistent that each fully saturated ring of type I is a full matrix ring over a local quasi-Frobenius ring. In several recent papers, our results have been applied to Tilting Theory and to the Theory of ∗-modules. In modern algebra, the structure of rings, R, is studied by means of properties of corresponding module categories, Mod-R. In most cases, it is not possible to characterize Mod-R fully. Nevertheless, there are important subclasses of Mod-R that can be treated in detail and that shed light on the whole of Mod-R. Among the prominent ones are the classes of all projective and all injective modules. Recall that a module M is said to be projective (injective) provided that the functorHomR(M,−) (HomR(−,M)) preserves short exact sequences. There is also a universal algebraic aspect: each module is a factor module of a projective module, and a submodule of an injective module. So a possible strategy to investigate ModR consists in describing all injective modules, and for each injective module, I, all its submodules. The first step is usually relatively easy, but the second may be quite hard. For example, using this strategy for abelian groups, one meets serious difficulties already for I = Q⊕Q (see e.g. [E, Theorem 2]). Received by the editors March 17, 1995. 1991 Mathematics Subject Classification. Primary 16E30; Secondary 03E35, 20K35.
73 citations
TL;DR: Using necessary and sufficient conditions to ensure feasible linear and integral flows in the network, and the special structure of the ring topology, efficient algorithms to route all the demands are constructed.
Abstract: In this paper, we consider the problem of multicommodity flows in a ring network. Using necessary and sufficient conditions to ensure feasible linear and integral flows in the network, and the special structure of the ring topology, we construct efficient algorithms to route all the demands.
56 citations
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01 Jan 1996
TL;DR: In this paper, the Chow ring for 3-dimensional Q-factorial toric varieties having one bacl isolated singularity was calculated for the linear equivalence relation, and the authors described explic- itly the ChowRing for a 3-dimensions of a toric variety as the Stanley-Reisner ring for the corresponding fan.
Abstract: The properties of a toric variety have strong connection with the combinatorial structure of the corresponding fan and the rela- tions among the generators. Using this fact, we have described explic- itly the Chow ring for a Q-factorial toric variety as the Stanley-Reisner ring for the corresponding fan modulo the linear equivalence relation. In this paper, we calculate the Chow ring for 3-dimensional Q-factorial toric varieties having one bacl isolated singularity.
TL;DR: In this article, the authors investigated commutativity of the ring R, when the mapping G is assumed to be a derivation or an endomorphism of R, and they proved that G admits mappings F and G such that [F(x), G(y)] = [x,y] for all x, y ∈ S.
Abstract: Let R be a ring, and S a non-empty subset of R. Suppose that R admits mappings F and G such that [F(x), G(y)] = [x,y] for all x, y ∈ S. In the present paper, we investigate commutativity of the ring R, when the mapping G is assumed to be a derivation or an endomorphism of R.
TL;DR: It is proved that the ring of Siegel modular forms in any genus is determined by doubly even self-dual codes and the theta relations and codes are uniquely determined by their weight polynomials.
TL;DR: In this paper, it was shown that every M-definable ring without zero divisors is definably isomorphic to R, R( √ −1) or the ring of quaternions over R.
Abstract: Let 〈R,<,+, ·〉 be a real closed field and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in M. We show that every M-definable ring without zero divisors is definably isomorphic to R, R( √ −1) or the ring of quaternions over R. One corollary is that no model of Texp is interpretable in a model of Tan.
TL;DR: In this article, the authors apply Dwork's p-adic methods to study the meromorphic continuation and rationality of various L-functions arising from 7r-adic Galois representations, Drinfeld modules and (p-sheaves).
Abstract: In this paper, we apply Dwork's p-adic methods to study the meromorphic continuation and rationality of various L-functions arising from 7r-adic Galois representations, Drinfeld modules and (p-sheaves. As a consequence, we prove some conjectures of Goss about the rationality of the local L-function and the meromorphic continuation of the global L-function attached to a Drinfeld module. Let Fq be a finite field of q elements with characteristic p. Let 7r be a prime of the polynomial ring A = Fq [t]. Let A, be the completion of the ring A at 7r. This is an analogue of the classical ring 7p of p-adic integers. Let X be an irreducible algebraic variety defined over Fq and let 7r, (X) be the arithmetic fundamental group of X/IFq with respect to some base point. The group 7r, (X) may be regarded as the Galois groiip of a separable closure of the function field of X/Fq modulo the inertia groups at the closed points of X/lq. Suppose now that we are given a continuous 7r-adic representation
TL;DR: A very fast routine for ring perception is found that is orders of magnitude faster than depth-first ring detection, a result expected on the basis of recent work that establishes polynomial order for BFS.
Abstract: Combining breadth-first search with new ideas for uncovering embedded rings in complex systems1 yields a very fast routine for ring perception. With large structures, the new routine is orders of magnitude faster than depth-first ring detection, a result expected on the basis of recent work that establishes polynomial order for BFS.2
TL;DR: The use of thin rings is considered as a means of measuring two orthogonal forces and a moment applied in the plane of the ring, with appropriately mounted strain gauge bridges.
TL;DR: In this paper, it was shown that a line bundle on an elliptic-ruled surface is normally generated if the natural maps SmH0(X,L)! H0( X,L m) are surjective for all m 2.
TL;DR: In this paper, the authors identify the exactly solvable theory of the conformal fixed point of (0, 2) Calabi-Yau σ-models and their Landau-Ginzburg phases.
Patent•
14 May 1996
TL;DR: In this paper, a neuro-chip comprising a plurality of neuron operation circuits, a broadcast bus terminal for supplying data in parallel to the neuron operation circuit and receiving data from the broadcast bus, a program data bus for supplying a common program externally input, and a ring bus connected to the ring bus for transferring data among the neurons operation circuits.
Abstract: A neuro-chip comprising a plurality of neuron operation circuits, a broadcast bus terminal for supplying data in parallel to the neuron operation circuits and receiving data in parallel therefrom, a program data bus connected to the neuron operation circuits, a program data bus terminal for supplying a common program externally input, to the neuron operation circuits through the program data bus, a ring bus connecting the neuron operation circuits, and a plurality of ring bus terminals connected to the ring bus, for transferring data among the neuron operation circuits.
TL;DR: This work presents the first stabilizing, unidirectional, deterministic token ring where each process has a constant number of states, and this ring is an attractive candidate for hardware implementation.
01 Oct 1996
TL;DR: In this article, a ring of invariants is identified for some automorphisms θ of certain iterated skew polynomial rings R, including the enveloping algebra of sl2(k), the Weyl algebra A1 and their quantizations.
Abstract: Rings of invariants are identified for some automorphisms θ of certain iterated skew polynomial rings R, including the enveloping algebra of sl2(k), the Weyl algebra A1 and their quantizations. We investigate how finite-dimensional simple R-modules split over the ring of invariants Rθ and how finite-dimensional simple Rθ-modules extend to R.
Journal Article•
TL;DR: In this paper, the authors obtained an exact analogue of fuzzy ideals for near-ring which was discussed in [5, 11] and [12], and applied it to semigroups, distributive lattices, artinian rings, BCK-algebras, and near-rings.
Abstract: W. Liu [11] has studied fuzzy ideals of a ring, and many researchers [5,6,7,16] are engaged in extending the concepts. The notion of fuzzy ideals and its properties were applied to various areas: semigroups [8,9,10,13,15], distributive lattices [2], artinian rings [12], BCK-algebras [14], near-rings [1]. In this paper we obtained an exact analogue of fuzzy ideals for near-ring which was discussed in [5, 11].
TL;DR: In the case that R is a field of characteristic 0, Noether constructed a finite set of R -algebra generators of the invariants of G when | G |! is invertible in R as discussed by the authors.
TL;DR: In this paper, the multiplicative structure of rings of coinvariants for finite groups is studied and the authors develop methods that give rise to natural monomial bases for such rings over their ground fields and explicitly determine precisely which monomials are zero in the ring of coin-variants.
Abstract: We study the multiplicative structure of rings of coinvariants for finite groups. We develop methods that give rise to natural monomial bases for such rings over their ground fields and explicitly determine precisely which monomials are zero in the ring of coinvariants. We apply our methods to the Dickson, upper triangular and symmetric coinvariants. Along the way, we recover theorems of Steinberg [17] and E. Artin [1]. Using these monomial bases we prove that the image of the transfer for a general linear group over a finite field is a principal ideal in the ring of invariants.
17 Jun 1996
TL;DR: In this article, two new topologies for rejecting the harmonics in microstrip ring resonator filters were introduced. First one uses 50 /spl Omega/ spur line filter structures placed at the input and output of the ring to suppress the first harmonic.
Abstract: This paper introduces two new topologies for rejecting the harmonics in microstrip ring resonator filters. First one uses 50 /spl Omega/ spur line filter structures placed at the input and output of the ring to suppress the first harmonic. Second topology incorporates low-pass structures into the ring for elimination of the harmonics.
TL;DR: In this paper, it was shown that the moduli spaces of rank-two (parabolic) bundles on an irreducible projective curve of genus g over an algebraically closed field of positive characteristic + 2, 3.
Abstract: Let X be an irreducible projective curve of genus g over an algebraically closed field of positive characteristic + 2, 3. In Part I, we prove, adapting the degeneration arguments of (N-TR), that moduli spaces of rank- two (parabolic) bundles on X are Frobenius split (M-R) for generic smooth X. (A similar result holds for X nodal. A consequence is the Verlinde formula in positive characteristic.) In Part II, we give a direct proof of the fact that the local rings of the moduli spaces are F-split, and further, that they are Cohen- Macaulay. This involves showing that the ring of invariants of g copies of 2 x 2 matrices (under the adjoint action of SL(2)) is F-split and Cohen-Macaulay.
Patent•
06 Nov 1996
TL;DR: In this paper, the authors proposed a method and an apparatus for routing a data packet through a series of ring stations and routing elements, such as a middle switch, a mid switch, and an end switch.
Abstract: The present invention allows a data packet to reach its destination by automatically routing the data packet around failed components by means of a simple and efficient routing algorithms. The present invention is directed to a method and apparatus for routing a data packet through a series of ring stations, a series of mid-switch routing elements, and end-switch routing elements. An apparatus according to the invention includes a first series of ring stations and routing elements that are adapted to communicate with each other, and formed in a ring. Each ring is paired with another ring that includes a second series of ring stations and routing elements that are also adapted to communicate with each other as described above to form a second ring, and thus the two rings forming a ring pair. Each routing element, mid-switch and end-switch, is adapted to switch the path of the data packet between the two rings. The present invention further requires vertical links connecting a routing element from one ring pair to another routing element from another ring pair. The vertical links are further adapted to pass the data packet between the ring pairs.
TL;DR: In this paper, it was shown that a ring R has an epic C-envelope if and only if C is closed under direct products and submodules, and if R is left T-coherent and pure injective as a right R-module.
Abstract: We prove the following results for a ring R . (a) If C is a class of right R -modules closed under direct summands and isomorphisms, then every right R -module has an epic C -envelope if and only if C is closed under direct products and submodules. (b) If R is left T -coherent and pure injective as a right R -module, then every T -finitely presented right R -module has a T -flat envelope, (c) Let R be a left T -coherent ring and injective right R -modules be T -flat. If every finitely presented left R -module has a flat envelope, then every T -finitely presented right R -module has a projective cover.
TL;DR: In this paper, the problem of approximating the input-output maps of nonlinear discrete-time approximately finite-memory systems is studied, where the focus is on the linear dynamical parts of the approximating structures, and examples showing that these linear parts can be derived from asingle prespecified functions that meet certain conditions.
Abstract: We give results concerning the problem of approximating the input-output maps of nonlinear discrete-time approximately finite-memory systems. Here the focus is on the linear dynamical parts of the approximating structures, and we give examples showing that these linear parts can be derived from asingle prespecified function that meets certain conditions. This is done in the context of an approximation theorem in which attention is focused on what we call "basic sets." We also consider the related but very different problem of approximating nonlinear functionals using lattice operations or the usual linear ring operations. For this problem we give criteria, not just sufficient conditions, for approximation on compact subsets of reflexive Banach spaces (any Hilbert space is a reflexive Banach space).