Showing papers on "Ring (mathematics) published in 2005"
TL;DR: Haglund and Ulyanov as discussed by the authors conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support their conjecture, including Schur positive.
Abstract: Author(s): Haglund, J; Haiman, M; Loehr, N; Remmel, J B; Ulyanov, A | Abstract: Let Rn be the ring of coinvariants for the diagonal action of the symmetric group Sn. It is known that the character of Rn as a doubly graded S-module can be expressed using the Frobenius characteristic map as nabla en, where en is the n-th elementary symmetric function and nabla is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. These results make use of the theory of ribbon tableau generating functions of Lascoux, Leclerc, and Thibon. We also show that a variety of earlier conjectures and theorems on nabla en are special cases of our conjecture. Finally, we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en.
336 citations
TL;DR: In this article, the notions of S-topologies, S-sites, and stacks over sites were introduced, and a model category of pre-stacks over a Grothendieck site was introduced.
267 citations
TL;DR: In this paper, it was shown that every skew-polynomial ring with generating set X and binomial relations is an Artin-Schelter regular domain of global dimension | X |.
249 citations
Journal Article•
TL;DR: The generalized knapsack function is defined as f a (x)= Σ i a i x i, where a = (a 1,...,a m ) consists of m elements from some ring R, and x = (x i,...,x m ) consist of m coefficients from a specified subset S C R as mentioned in this paper.
Abstract: The generalized knapsack function is defined as f a (x)= Σ i a i x i , where a = (a 1 ,...,a m ) consists of m elements from some ring R, and x = (x i ,...,x m ) consists of m coefficients from a specified subset S C R. Micciancio (FOCS 2002) proposed a specific choice of the ring R and subset S for which inverting this function (for random a, x) is at least as hard as solving certain worst-case problems on cyclic lattices. We show that for a different choice of S ⊂ R, the generalized knapsack function is in fact collision-resistant, assuming it is infeasible to approximate the shortest vector in n-dimensional cyclic lattices up to factors 0(n). For slightly larger factors, we even get collision-resistance for any m > 2. This yields very efficient collision-resistant hash functions having key size and time complexity almost linear in the security parameter n. We also show that altering S is necessary, in the sense that Micciancio's original function is not collision-resistant (nor even universal one-way). Our results exploit an intimate connection between the linear algebra of n-dimensional cyclic lattices and the ring Z[α]/(α n -1), and crucially depend on the factorization of a n - 1 into irreducible cyclotomic polynomials. We also establish a new bound on the discrete Gaussian distribution over general lattices, employing techniques introduced by Micciancio and Regev (FOCS 2004) and also used by Micciancio in his study of compact knapsacks.
211 citations
TL;DR: For a ring endomorphism α and an α-derivation δ, the authors introduced α-compatible rings which are a generalization of α-rigid rings, and study on the relationship between the quasi Baerness and p.q.-Baer property of a ring R and those of the polynomial extensions (including formal skew power series, skew Laurent polynomials and skew Laurent series).
Abstract: For a ring endomorphism α and an α-derivation δ, we introduce α-compatible rings which are a generalization of α-rigid rings, and study on the relationship between the quasi Baerness and p.q.-Baer property of a ring R and those of the polynomial extensions (including formal skew power series, skew Laurent polynomials and skew Laurent series). As a consequence we obtain a generalization of [6], [8] and [16].
181 citations
TL;DR: The SL(2,Z) representation on the center of the restricted quantum group U_{q}sl(2) at the primitive 2p-th root of unity is shown to be equivalent to the SL( 2,Z)-representation on the extended characters of the logarithmic (1,p) conformal field theory model in this article.
Abstract: The SL(2,Z) representation $\pi$ on the center of the restricted quantum group U_{q}sl(2) at the primitive 2p-th root of unity is shown to be equivalent to the SL(2,Z) representation on the extended characters of the logarithmic (1,p) conformal field theory model. The multiplicative Jordan decomposition of the U_{q}sl(2) ribbon element determines the decomposition of $\pi$ into a ``pointwise'' product of two commuting SL(2,Z) representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2,Z) representation on the (1,p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of U_{q}sl(2) at the primitive 2p-th root of unity is shown to coincide with the fusion algebra of the (1,p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of~U_{q}sl(2).
163 citations
22 Aug 2005
TL;DR: A new implementation of a reflexive tactic which solves equalities in a ring structure inside the Coq system is presented, which shows that such reflective methods can be effectively used in symbolic computation.
Abstract: We present a new implementation of a reflexive tactic which solves equalities in a ring structure inside the Coq system. The efficiency is improved to a point that we can now prove equalities that were previously beyond reach. A special care has been taken to implement efficient algorithms while keeping the complexity of the correctness proofs low. This leads to a single tool, with a single implementation, which can be addressed for a ring or for a semi-ring, abstract or not, using the Leibniz equality or a setoid equality. This example shows that such reflective methods can be effectively used in symbolic computation.
123 citations
TL;DR: In this article, the authors describe the Hall algebra H_X of an elliptic curve X defined over a finite field and show that the group SL(2,Z) of exact auto-equivalences of the derived category D^b(Coh(X)) acts on the Drinfeld double DH_X by algebra automorphisms.
Abstract: In this article we describe the Hall algebra H_X of an elliptic curve X defined over a finite field and show that the group SL(2,Z) of exact auto-equivalences of the derived category D^b(Coh(X)) acts on the Drinfeld double DH_X of H_X by algebra automorphisms. Next, we study a certain natural subalgebra U_X of DH_X for which we give a presentation by generators and relations. This algebra turns out to be a flat two-parameter deformation of the ring of diagonal invariants C[x_1^{\pm 1}, ..., y_1^{\pm 1},...]^{S_{\infty}}, i.e. the ring of symmetric Laurent polynomials in two sets of countably many variables under the simultaneous symmetric group action.
95 citations
Patent•
12 May 2005TL;DR: In this article, a variety of features relating to such things as holder shape, handle attachment structures, securement of a ring to the holder, and release of the ring from the holder are discussed.
Abstract: Holders for releasably holding annuloplasty rings prior to and during the implantation of the rings employ any of a variety of features relating to such things as holder shape, handle attachment structures, securement of a ring to the holder, and release of the ring from the holder.
88 citations
TL;DR: In this article, the authors give a series of criteria for the existence of strong ring solutions, in particular in terms of finite mean oscillation majorants for tangential dilatations, and derive an extension of the well-known Lehto existence theorem.
Abstract: In this chapter we give a series of criteria for the existence of strong ring solutions, in particular, in terms of finite mean oscillation majorants for tangential dilatations. Moreover, we derive an extension of the well-known Lehto existence theorem and show that the latter implies the main known and many advanced results on the existence of ACL homeomorphic solutions for the Beltrami equations with degeneration.
83 citations
TL;DR: Previous studies are extended and it is shown that the writhe distribution broadens as the confining sphere narrows, and the dependence of other geometrical indicators, such as writhe and chirality, in terms of R and N is characterized.
Abstract: Stochastic simulations are used to characterize the knotting distributions of random ring polymers confined in spheres of various radii. The approach is based on the use of multiple Markov chains and reweighting techniques, combined with effective strategies for simplifying the geometrical complexity of ring conformations without altering their knot type. By these means we extend previous studies and characterize in detail how the probability to form a given prime or composite knot behaves in terms of the number of ring segments, $N$, and confining radius, $R$. For $ 50 \le N \le 450 $ we show that the probability of forming a composite knot rises significantly with the confinement, while the occurrence probability of prime knots are, in general, non-monotonic functions of 1/R. The dependence of other geometrical indicators, such as writhe and chirality, in terms of $R$ and $N$ is also characterized. It is found that the writhe distribution broadens as the confining sphere narrows.
Proceedings Article•
01 Jan 2005TL;DR: In this article, an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations is given, which is consistent with the ring axioms and many properties of division.
Abstract: We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that 0−1 = 0, an interesting equation consistent with the ring axioms and many properties of division. The existence of an equational specification of the rationals without hidden functions was an open question. We also give an axiomatic examination of the divisibility operator, from which some interesting new axioms emerge along with equational specifications of algebras of rationals, including one with the modulus function. Finally, we state some open problems, including: Does there exist an equational specification of the field operations on the rationals without hidden functions that is a complete term rewriting systemq
TL;DR: In this paper, necessary conditions for various algebraic types of the Weyl tensor in higher dimensions are determined, and these conditions are then used to find Weyl aligned null directions for the black ring solution.
Abstract: Necessary conditions for various algebraic types of the Weyl tensor in higher dimensions are determined. These conditions are then used to find Weyl aligned null directions for the black ring solution. It is shown that the black ring solution is algebraically special, of type Ii, while locally on the horizon the type is II. One exceptional subclass – the Myers-Perry solution – is of type D.
TL;DR: For locally compact groups G and H, this article showed that if G is amenable then any completely bounded homomorphism Φ : A (G ) → B (H ) is of this form; and this theorem fails if G contains a discrete nonabelian free group.
TL;DR: In this paper, the ring structure of Floer cohomology groups of Lagrangian torus fibers in some toric Fano manifolds has been studied and it has been shown that these rings are isomorphic to Clifford algebras, whose quadratic forms are given by the Hessians of functions W, which turn out to be the superpotentials of Landau-Ginzburg mirrors.
Abstract: We compute the ring structure of Floer cohomology groups of Lagrangian torus fibers in some toric Fano manifolds continuing the study of [CO]. Related A∞-formulas hold for a transversal choice of chains. Two different computations are provided: a direct calculation using the classification of holomorphic discs by Oh and the author in [CO], and another method by using an analogue of divisor equation in Gromov-Witten invariants to the case of discs. Floer cohomology rings are shown to be isomorphic to Clifford algebras, whose quadratic forms are given by the Hessians of functions W, which turn out to be the superpotentials of Landau-Ginzburg mirrors. In the case of Open image in new window, this proves the prediction made by Hori, Kapustin and Li by B-model calculations via physical arguments. The latter method also provides correspondence between higher derivatives of the superpotential of LG mirror with the higher products of the A∞(or L∞)-algebra of the Lagrangian submanifold.
TL;DR: In this paper, a new type of quotient for any triangulated category which generalizes Verdier's construction is introduced, and the derived category of an almost ring is shown to be of this form.
Abstract: The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier's construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S , a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic K -theory and demonstrates the relevance of the telescope conjecture for derived categories. Another application leads to a derived analogue of an almost module category in the sense of Gabber-Ramero. It is shown that the derived category of an almost ring is of this form.
TL;DR: In this article, it was shown that the ring Open image in new window is finitely generated as a module over its centre, and that the number of such modules is equal to the number nn(q) of irreducible k-representations of the Weil group WF of dimension n.
Abstract: The motivation of this paper is the search for a Langlands correspondence modulo p. We show that the pro-p-Iwahori Hecke ring Open image in new window of a split reductive p-adic group G over a local field F of finite residue field Fq with q elements, admits an Iwahori-Matsumoto presentation and a Bernstein Z-basis, and we determine its centre. We prove that the ring Open image in new window is finitely generated as a module over its centre. These results are proved in [11] only for the Iwahori Hecke ring. Let p be the prime number dividing q and let k be an algebraically closed field of characteristic p. A character from the centre of Open image in new window to k which is “as null as possible” will be called null. The simple Open image in new window-modules with a null central character are called supersingular. When G=GL(n), we show that each simple Open image in new window-module of dimension n containing a character of the affine subring Open image in new window is supersingular, using the minimal expressions of Haines generalized to Open image in new window, and that the number of such modules is equal to the number of irreducible k-representations of the Weil group WF of dimension n (when the action of an uniformizer pF in the Hecke algebra side and of the determinant of a Frobenius FrF in the Galois side are fixed), i.e. the number Nn(q) of unitary irreducible polynomials in Fq[X] of degree n. One knows that the converse is true by explicit computations when n=2 [10], and when n=3 (Rachel Ollivier).
Patent•
24 Mar 2005TL;DR: In this paper, a service broker intercepts a call invitation message generated by a calling phone and intended for the simultaneous ring set and determines the status of each of the plurality of called phones that are members of the ring set.
Abstract: The present invention provides a method for providing a simultaneous ring and check presence prior to terminating service for a plurality of called phones in a simultaneous ring set. The simultaneous ring set includes a plurality wireline or wireless units. A service broker intercepts a call invitation message generated by a calling phone and intended for the simultaneous ring set. Upon determining that the simultaneous ring set has simultaneous ring service, the service broker determines the status of each of the plurality of called phones that are members of the simultaneous ring set. The service broker sends an invitation accept message for the responding unit of the simultaneous ring set to the calling phone, thereby completing the call between the calling phone and the responding unit of the simultaneous ring set.
TL;DR: In this article, the stable range of quasi-duo corner rings was studied and it was shown that for any semisplit corner ring R, the same is true for any semi-plit ring of R. This contrast with earlier results of Vaserstein and Warfield, which showed that the stable ranges can increase unboundedly upon descent to (even) Peirce corner rings.
TL;DR: In this paper, the authors introduced (, )-skew Armendariz rings, which are a generalization of -rigid rings and Armenderiz rings and investigated their properties.
Abstract: For a ring endomorphism and an -derivation , we introduce (, )-skew Armendariz rings which are a generalization of -rigid rings and Armendariz rings, and investigate their properties. A semi prime left Goldie ring is -weak Armendariz if and only if it is -rigid. 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; , ] in case R is (, )-skew Armendariz. As a consequence we obtain a generalization of [11], [14] and [16].
TL;DR: In this paper, the mod 2 motivic cohomology of a reduced Cech complex of a splitting variety according to Orlov, Vishik and Voevodsky is computed and compared with the results of the real cobordism theory of Hu and Kriz.
Abstract: In this paper, we study applications of Atiyah?Hirzebruch spectral sequences for motivic cobordism recently found by Hopkins and Morel. For example, we compute the mod 2 motivic cohomology of a reduced Cech complex of a splitting variety according to Orlov, Vishik and Voevodsky. The cobordism ring $MGL / 2^{*,*} (Spec(\mathbb{R}))$ is computed and compared with the results of the real cobordism theory of Hu and Kriz. Moreover, we study algebraic cobordism of the classifying spaces $BG$ for algebraic groups over $\mathbb{C}$. For example, the Chow ring $CH^* (BG_2)_(2)$ for the exceptional Lie group $G_2$ is determined by using motivic cobordism and motivic cohomology.
TL;DR: In this article, the authors give a presentation of the ring of multisymmetric functions that holds for any commutative ring R, thereby answering a classical question coming from works of F. Junker in the late nineteen century and then implicitly in H. Weyl book ''The classical groups''.
Abstract: We give a presentation (in terms of generators and relations) of the ring of multisymmetric functions that holds for any commutative ring R, thereby answering a classical question coming from works of F. Junker in the late nineteen century and then implicitly in H. Weyl book ``The classical groups."
04 Apr 2005
TL;DR: These algorithms for inferring for every program point u, affine relations between program variables valid at u are presented and can be implemented by using the same modular integer arithmetic as the target language to be analyzed.
Abstract: We consider integer arithmetic modulo a power of 2 as provided by mainstream programming languages like Java or standard implementations of C. The difficulty here is that the ring ℤm of integers modulo m = 2w, w > 1, has zero divisors and thus cannot be embedded into a field. Not withstanding that, we present intra- and inter-procedural algorithms for inferring for every program point u, affine relations between program variables valid at u. Our algorithms are not only sound but also complete in that they detect all valid affine relations. Moreover, they run in time linear in the program size and polynomial in the number of program variables and can be implemented by using the same modular integer arithmetic as the target language to be analyzed.
TL;DR: In this article, it was shown that if Ext R n (D R, R ) = 0 for all n > 0, then R is Gorenstein, and proved this in several significant cases.
Posted Content•
TL;DR: In this paper, it was shown that the integral cohomology algebra of the moment-angle complex Z_K (equivalently, of the complement U(K) of the coordinate subspace arrangement) determined by a simplicial complex K is isomorphic to the Tor-algebra of the face ring of K. And the formality of this algebra was analyzed by using a generalisation of Hochster's theorem.
Abstract: In this survey article we present several new developments of `toric topology' concerning the cohomology of face rings (also known as Stanley-Reisner algebras). We prove that the integral cohomology algebra of the moment-angle complex Z_K (equivalently, of the complement U(K) of the coordinate subspace arrangement) determined by a simplicial complex K is isomorphic to the Tor-algebra of the face ring of K. Then we analyse Massey products and formality of this algebra by using a generalisation of Hochster's theorem. We also review several related combinatorial results and problems.
21 Oct 2005
TL;DR: In this paper, it was shown that the restriction of a Jordan derivation from a ring A into an A-bimodule M to the ideal of A generated by certain higher commutators of A is a derivation.
Abstract: Let d be a Jordan derivation from a ring A into an A-bimodule M. Our main result shows that the restriction of d to the ideal of A generated by certain higher commutators of A is a derivation. This general statement is used for proving that under various additional conditions d must be a derivation on A. Furthermore, several examples of proper Jordan derivations are given, C -algebras admitting proper additive Jordan derivations are characterized, and the connections with the related problems on Jordan homomorphisms and Jordan A-module homomorphisms are discussed.
TL;DR: In this paper, it was shown that a discrete polymatroid satisfying the strong exchange property is, up to an affinity, of Veronese type, and that all polymatroids of fiber type are fiber type.
Abstract: In the first half of this paper, we complement the theory on discrete polymatroids. More precisely, (i) we prove that a discrete polymatroid satisfying the strong exchange property is, up to an affinity, of Veronese type; (ii) we classify all uniform matroids which are level; (iii) we introduce the concept of ideals of fiber type and show that all polymatroidal ideals are of fiber type. On the other hand, in the latter half of this paper, we generalize the result proved by Stefan Blum that the defining ideal of the Rees ring of a base sortable matroid possesses a quadratic Grobner basis. For this purpose we introduce the concept of ``$l$-exchange property'' and show that a Grobner basis of the defining ideal of the Rees ring of an ideal $I$ can be determined and that $I$ is of fiber type if $I$ satisfies the $l$-exchange property. Ideals satisfying the $l$-exchange property include strongly stable ideals, polymatroid ideals of base sortable discrete polymatroids, ideals of Segre-Veronese type and certain ideals related to classical root systems.
TL;DR: A computer algebra library for basic operations in an arbitrary Ore ring can be used as a basis for various algorithms in Ore rings, in particular, in differential, shift, and q-shift rings.
Abstract: We present some algorithms related to rings of Ore polynomials (or, briefly, Ore rings) and describe a computer algebra library for basic operations in an arbitrary Ore ring. The library can be used as a basis for various algorithms in Ore rings, in particular, in differential, shift, and q-shift rings.
Patent•
02 May 2005
TL;DR: In this article, a method of navigating, organizing, browsing, searching, and selecting information objects and/or content in a ring based structure is provided wherein the relation between rings are clear and easily grasped.
Abstract: A method of navigating, organising, browsing, searching, and selecting information objects and/or content in a ring based structure is provided wherein the relation between rings are clear and easily grasped. The method is particularly well suited for navigating in media content. The method comprises locating a number of information objects, displaying one or more of the information objects on a display in a ring based structure comprising one or more rings, and changing the size of a specific ring by selecting the specific ring. By this enlargement, the visibility of information objects comprised in the specific ring is increased, and the individual information objects visually present in the ring, may readily be selected from the enlarged ring.
TL;DR: It is shown that the familiar existence of a left adjoint to the functor from the category of frames to the categories of archimedean commutative f-rings with unit provided by the rings of pointfree continuous real-valued functions is already a consequence of a minimal amount of entirely obvious information.
Abstract: It is shown that the familiar existence of a left adjoint to the functor from the category of frames to the category of archimedean commutative f-rings with unit provided by the rings of pointfree continuous real-valued functions is already a consequence of a minimal amount of entirely obvious information, and this is then used to obtain unexpectedly simple proofs for a number of results concerning these function rings, along with their counterparts for the rings of integer-valued continuous functions in this setting. In addition, two different concrete descriptions are given for the left adjoint in question, one in terms of generators and relations motivated by the propositional theory of l-ring homomorphisms into R, and the other based on a new notion of support specific to f-rings.