Showing papers on "Ring (mathematics) published in 2011"
TL;DR: In this article, the authors embed a Fomin-Zelevinsky cluster algebra into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric Kac-Moody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties.
Abstract: Motivated by a recent conjecture by Hernandez and Leclerc, we embed a Fomin-Zelevinsky cluster algebra into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric Kac-Moody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig’s dual canonical base. The conjectures that cluster monomials are positive and linearly independent (and probably many other conjectures) of Fomin and Zelevinsky follow as consequences when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize into tensor products of “prime” simple ones according to the cluster expansion.
164 citations
TL;DR: In this article, the authors studied the empirical measure LAn of the eigenvalues of non-normal square matrices of the form An = UnTnVn with Un;Vn independent Haar distributed on the unitary group and Tn real diagonal.
Abstract: We study the empirical measure LAn of the eigenvalues of non-normal square matrices of the form An = UnTnVn with Un;Vn independent Haar distributed on the unitary group and Tn real diagonal. We show that when the empirical measure of the eigenvalues of Tn converges, and Tn satisfies some technical conditions, LAn converges towards a rotationally invariant measure µ on the complex plane whose support is a single ring. In particular, we provide a complete proof of Feinberg-Zee single ring theorem [6]. We also consider the case where Un;Vn are independent Haar distributed on the orthogonal group.
138 citations
TL;DR: Weakly nonlinear theory and numerical simulations demonstrate how a ring can bifurcate to more complex equilibria including triangular shapes, annuli, and spot patterns with N-fold symmetry.
Abstract: Pairwise particle interactions arise in diverse physical systems ranging from insect swarms to self-assembly of nanoparticles. In the presence of long-range attraction and short-range repulsion, such systems can exhibit bound states. We use linear stability analysis of a ring equilibrium to classify the morphology of patterns in two dimensions. Conditions are identified that assure the well-posedness of the ring. In addition, weakly nonlinear theory and numerical simulations demonstrate how a ring can bifurcate to more complex equilibria including triangular shapes, annuli, and spot patterns with N-fold symmetry. Many of these patterns have been observed in nature, although a general theory has been lacking, in particular how small changes to the interaction potential can lead to large changes in the self-organized state.
137 citations
TL;DR: This paper introduces and study soft subrings and soft ideals of a ring by using Molodtsov's definition of the soft sets, and introduces soft subfields of a field and soft submodule of a left R-module.
Abstract: Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainties. In this paper, we introduce and study soft subrings and soft ideals of a ring by using Molodtsov's definition of the soft sets. Moreover, we introduce soft subfields of a field and soft submodule of a left R-module. Some related properties about soft substructures of rings, fields and modules are investigated and illustrated by many examples.
134 citations
TL;DR: In this article, the authors study cocycles of discrete countable groups with values in ε 2 G and the ring of affiliated operators UG and obtain strong results about the existence of free subgroups and subgroup structure.
Abstract: In this article we study cocycles of discrete countable groups with values in � 2 G and the ring of affiliated operators UG. We clarify properties of the first cohomology of a group G with coefficients in � 2 G and answer several questions from De Cornulier et al. (Transform. Groups 13(1):125-147, 2008). Moreover, we obtain strong results about the existence of free subgroups and the subgroup structure, provided the group has a positive first � 2 -Betti num- ber. We give numerous applications and examples of groups which satisfy our assumptions.
134 citations
TL;DR: In this article, it was shown that a hyperring of the form R / G (where R is a ring and G ⊂ R × is a subgroup of its multiplicative group) is a commutative hyperring extension of a global field K if and only if G ∪ { 0 }, where G ∈ R × denotes a subfield of R. This result applies to the adele class space which thus inherits the structure of the hyperring H K of K.
108 citations
TL;DR: In this article, the connections between recollements of the derived category D (Mod-R ) of a ring R and tilting theory were studied, and the authors provided constructions of tilting objects from given recollements.
90 citations
TL;DR: A nonlinear model for the dual-delay-path ring oscillator and the analysis of the stability of each operation mode are proposed to confirm both simulation and measurement results.
Abstract: A dual-operation-mode ring oscillator that employs dual-delay paths is presented. The two operation modes, referred to as the differential and common modes, have different output waveform characteristics and oscillation frequencies. A nonlinear model for the dual-delay-path ring oscillator and the analysis of the stability of each operation mode are proposed to confirm both simulation and measurement results. Furthermore, a method for operation-mode selection is presented. The differential four-stage dual-delay-path ring oscillator is fabricated in a 0.18-μm CMOS technology. Measurements show that the tuning ranges are from 1.77 to 1.92 GHz and from 1.01 to 1.055 GHz for differential- and common-mode operations, respectively.
80 citations
TL;DR: In this article, a method for constructing a Leavitt path algebra L R (E ) whose coefficients are in a commutative unital ring R was described, and the Graded Uniqueness Theorem and Cuntz-Krieger Theorem for these path algebras were proved.
74 citations
TL;DR: In this paper, it was shown that the category of modules over a separable ring object in a tensor triangulated category admits a unique structure of triangulation which is compatible with the original one.
64 citations
01 Jun 2011
TL;DR: In this article, the authors consider functors which are defined on larger classes of rings (such as the class of ring spectra which arise in algebraic topology), and sketch some applications to deformation theory.
Abstract: In algebraic geometry, it is common to study a geometric object X (such as a scheme) by means of the functor R 7! Hom(SpecR,X) represented by X. In this paper, we consider functors which are defined on larger classes of rings (such as the class of ring spectra which arise in algebraic topology), and sketch some applications to deformation theory.
14 Feb 2011
TL;DR: This paper proposes the first secure traceable ring signature schemes without random oracles in the common reference string model and has a signature size of O(√N), where N is the number of users in the ring.
Abstract: Traceable ring signatures, proposed at PKC'07, are a variant of ring signatures, which allow a signer to anonymously sign a message with a tag behind a ring, ie, a group of users chosen by the signer, unless he signs two messages with the same tag However, if a signer signs twice on the same tag, the two signatures will be linked and the identity of the signer will be revealed when the two signed messages are different Traceable ring signatures can be applied to anonymous writein voting without any special voting authority and electronic coupon services The previous traceable ring signature scheme relies on random oracles at its security and the signature size is linear in the number of ring members This paper proposes the first secure traceable ring signature schemes without random oracles in the common reference string model In addition, the proposed schemes have a signature size of O(√N), where N is the number of users in the ring
TL;DR: In this article, a detailed model for possible vibration effects on MEMS degenerate gyroscopes represented by vibratory ring gyroscope is presented, which includes four vibration modes needed to describe vibration-induced errors: two flexural modes for gyro operation and two translation modes (excited by external vibration).
Abstract: This paper presents a detailed model for possible vibration effects on MEMS degenerate gyroscopes represented by vibratory ring gyroscopes. Ring gyroscopes are believed to be relatively vibration-insensitive because the vibration modes utilized during gyro operation are decoupled from the modes excited by environmental vibration. Our model incorporates four vibration modes needed to describe vibration-induced errors: two flexural modes (for gyro operation) and two translation modes (excited by external vibration). The four-mode dynamical model for ring gyroscopes is derived using Lagrange's equations. The model considers all elements comprising a ring gyroscope, namely the ring structure, the support-spring structures, and the electrodes that surround the ring structure. Inspection of this model demonstrates that the output of a ring gyroscope is fundamentally insensitive to vibration due to the decoupled dynamics governing ring translation versus ring flexure, however, becomes vibration-sensitive in the presence of non-proportional damping and/or capacitive nonlinearity at the sense electrodes.
TL;DR: In this paper, the bound-state spectrum of an Aharonov Bohm ring in a two-dimensional topological insulator using the four-band model of HgTe quantum wells was analyzed theoretically.
Abstract: We analyze theoretically the bound-state spectrum of an Aharonov Bohm ring in a two-dimensional topological insulator using the four-band model of HgTe quantum wells as a concrete example. We calculate analytically the circular helical edge states and their spectrum as well as the bound states evolving out of the bulk spectrum as a function of the applied magnetic flux and dimension of the ring. We also analyze the spin-dependent persistent currents, which can be used to measure the spin of single electrons. We further take into account the Rashba spin-orbit interaction that mixes the spin states and derive its effect on the ring spectrum. The flux tunability of the ring states allows for coherent mixing of the edge and the spin degrees of freedom of bound electrons, which could be exploited for quantum information processing in topological insulator rings.
TL;DR: In this paper, it was shown that 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersec- tions computed in the (equivariant) K-theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch and Tamvakis.
Abstract: We show that (equivariant) K-theoretic 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersec- tions computed in the (equivariant) K-theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we show that the Gromov-Witten variety of curves passing through 3 general points is irreducible and rational. Our applications include Pieri and Giambelli formulas for the quantum K-theory ring of a Grassmannian, which determine the multiplication in this ring. We also compute the dual Schubert basis for this ring, and show that its structure constants satisfy S3-symmetry. Our for- mula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin.
TL;DR: In this article, the basic theory of generalized Witt vectors is developed from the point of view of commuting Frobenius lifts and their universal properties, which is a new approach even for the classical Witt vectors.
Abstract: We give a concrete description of the category ofetale algebras over the ring of Witt vectors of a given nite length with entries in an arbitrary ring. We do this not only for the classical p-typical and big Witt vector functors but also for certain analogues over arbitrary local and global elds. The basic theory of these generalized Witt vectors is developed from the point of view of commuting Frobenius lifts and their universal properties, which is a new approach even for the classical Witt vectors. The larger purpose of this paper is to provide the ane foundations for the algebraic geometry of generalized Witt schemes and arithmetic jet spaces. So the basics here are developed somewhat fully, with an eye toward future applications.
TL;DR: In this article, it was shown that a tilting module T over a ring R admits an exact sequence 0 → R → T0 → T1 → 0 such that T_0,T_1\in\text{Add}(T)\) and HomR(T1,T0) = 0 if and only if T has the form S ⊕ S/R for some injective ring epimorphism λ : R → S with the property that \(\text{Tor}_1^R(S,S)=0\) and p
Abstract: We show that a tilting module T over a ring R admits an exact sequence 0 → R → T0 → T1 → 0 such that \(T_0,T_1\in\text{Add}(T)\) and HomR(T1,T0) = 0 if and only if T has the form S ⊕ S/R for some injective ring epimorphism λ : R → S with the property that \(\text{Tor}_1^R(S,S)=0\) and pdSR ≤ 1. We then study the case where λ is a universal localization in the sense of Schofield (1985). Using results by Crawley-Boevey (Proc Lond Math Soc (3) 62(3):490–508, 1991), we give applications to tame hereditary algebras and hereditary noetherian prime rings. In particular, we show that every tilting module over a Dedekind domain or over a classical maximal order arises from universal localization.
TL;DR: 1-Lipschitz measure-preserving and ergodic functions for arbitrary prime p are described and an algebraic structure induced by coordinate functions with partially frozen variables is used.
Abstract: Theory of dynamical systems in fields of p-adic numbers is an important part of algebraic and arithmetic dynamics. The study of p-adic dynamical systems is motivated by their applications in various areas of mathematics, e.g., in physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc.In particular, p-adic dynamical systems found applications in cryptography, which stimulated the interest to nonsmooth dynamical maps. An important class of (in general) nonsmooth maps is given by 1-Lipschitz functions.In this thesis we restrict our study to the class of 1-Lipschitz functions and describe measure-preserving (for the Haar measure on the ring of p-adic integers) and ergodic functions.The main mathematical tool used in this work is the representation of the function by the van der Put series which is actively used in p-adic analysis. The van der Put basis differs fundamentally from previously used ones (for example, the monomial and Mahler basis) which are related to the algebraic structure of p-adic fields. The basic point in the construction of van der Put basis is the continuity of the characteristic function of a p-adic ball.Also we use an algebraic structure (permutations) induced by coordinate functions with partially frozen variables.In this thesis, we present a description of 1-Lipschitz measure-preserving and ergodic functions for arbitrary prime p.
TL;DR: In this paper, an axiomatic setup for algorithmic homological algebra of Abelian categories is presented, where all existential quantifiers entering the definition of an Abelian category need to be turned into constructive ones.
Abstract: In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring R, i.e. a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over R. For a finitely generated maximal ideal 𝔪 in a commutative ring R, we show how solving (in)homogeneous linear systems over R𝔪 can be reduced to solving associated systems over R. Hence, the computability of R implies that of R𝔪. As a corollary, we obtain the computability of the category of finitely presented R𝔪-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a byproduct, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings, we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm.
TL;DR: In this article, the authors proved equicontinuity and normality of families R of the so-called ring Q(x)homeomorphisms with integral constraints of the type Φ(Q(x)) dm (x) < ∞ in a domain D ⊂ R, n ≥ 2.
Abstract: It is stated equicontinuity and normality of families R of the so-called ring Q(x)homeomorphisms with integral constraints of the type Φ(Q(x)) dm(x) < ∞ in a domain D ⊂ R, n ≥ 2. It is shown that the found conditions on the function Φ are not only sufficient but also necessary for equicontinuity and normality of such families of mappings. It is also given applications of these results to families of mappings in the Sobolev class W 1,n loc .
TL;DR: In this paper, the catenary degree c(H) of a Krull monoid with finite class group G such that every class contains a prime divisor has been characterized under a mild condition on the Davenport constant.
Abstract: Let H be a Krull monoid with finite class group G such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree c(H) of H is the smallest integer N with the following property: for each a 2 H and each two factorizations z, z 0 of a, there exist factorizations z = z0, . . . , zk = z 0 of a such that, for each i 2 (1, k), zi arises from zi−1 by replacing at most N atoms from zi−1 by at most N new atoms. Under a very mild condition on the Davenport constant of G, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between c(H) and the set of distances of H and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on c(H) and characterize when c(H) � 4.
TL;DR: In this article, a generalized Schubert calculus was developed for the ordinary and Borel-equivariant cohomology of the Peterson variety Y in type A n−1, with respect to a natural S 1 -action arising from the standard action of the maximal torus on flag varieties.
Abstract: Peterson varieties are a special class of Hessenberg varieties that have been extensively studied, for example, by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley–Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety Y in type A n−1 , with respect to a natural S 1 -action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculus beyond the realm of Kac–Moody flag varieties G/P. Our main results are as follows. First, we identify a computationally convenient basis of H * S1 (Y), which we call the basis of Peterson Schubert classes. Second, we derive a manifestly positive, integral Chevalley–Monk formula for the product of a cohomology-degree-2 Peterson Schubert class with an arbitrary Peterson Schubert class. Both H * S1 (Y) and H*(Y) are generated in degree 2. Finally, by using our Chevalley–Monk formula we give explicit descriptions (via generators and relations) of both the S 1 -equivariant cohomology ring H * S1 (Y) and the ordinary cohomology ring H*(Y) of the type A n−1 Peterson variety. Our methods are both directly from and inspired by those of the GKM (Goresky–Kottwitz–MacPherson) theory and classical Schubert calculus. We discuss several open questions and directions for future work.
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TL;DR: The Green ring of the Taft algebra is a commutative ring generated by two elements subject to certain relations defined recursively as discussed by the authors, where n is a positive integer greater than 1 and q is the root of unity.
Abstract: We compute the Green ring of the Taft algebra $H_n(q)$, where $n$ is a positive integer greater than 1, and $q$ is an $n$-th root of unity. It turns out that the Green ring $r(H_n(q))$ of the Taft algebra $H_n(q)$ is a commutative ring generated by two elements subject to certain relations defined recursively. Concrete examples for $n=2,3,..., 8$ are given.
TL;DR: In this article, the authors studied the homotopy category of bounded complexes with bounded homologies and its quotient category, and showed that the above quotient categories are triangle equivalent to the stable module category of Cohen-Macaulay T2(R)-modules.
Abstract: We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. We show the existence of a recollement of the above quotient category and it has the homotopy category of acyclic complexes as a triangulated subcategory. In the case of the homotopy category of finitely gen- erated projective modules over an Iwanaga-Gorenstein ring, we show that the above quotient category are triangle equivalent to the stable module category of Cohen-Macaulay T2(R)-modules.
TL;DR: In this paper, a Schubert calculus for Bott-Samelson resolutions in the algebraic cobordism ring of a complete flag variety G/B was established, extending the results of Bressler-Evens to the algebro-geometric setting.
Abstract: We establish a Schubert calculus for Bott-Samelson resolutions in the algebraic cobordism ring of a complete flag variety G/B extending the results of Bressler–Evens [4] to the algebro-geometric setting .
01 Jan 2011
TL;DR: F fuzzy soft ring is defined and some of its algebraic properties are studied, and a few of its properties are discussed.
Abstract: In this paper, we define fuzzy soft ring and study some of its algebraic properties. Thereafter, we define fuzzy soft ideal and discuss a few of its properties.
TL;DR: In this paper, the pushforward of a matrix factorization along a ring morphism is described in terms of an idempotent defined using relative Atiyah classes, and this construction is used to study the convolution of kernels defining integral functors between categories of matrix factorisations.
Abstract: We describe the pushforward of a matrix factorisation along a ring morphism in terms of an idempotent defined using relative Atiyah classes, and use this construction to study the convolution of kernels defining integral functors between categories of matrix factorisations. We give an elementary proof of a formula for the Chern character of the convolution generalising the Hirzebruch-Riemann-Roch formula of Polishchuk and Vaintrob.
TL;DR: In this paper, the Schr?dinger equation with the Hulth?n potential plus ring-shape potential was solved in terms of special functions using the conventional Nikiforov?Uvarov method.
Abstract: We present the solutions of the Schr?dinger equation with the Hulth?n potential plus ring-shape potential for ? ? 0 states within the framework of an exponential approximation of the centrifugal potential. Solutions to the corresponding angular and radial equations are obtained in terms of special functions using the conventional Nikiforov?Uvarov method. The normalization constant for the Hulth?n potential is also computed.
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TL;DR: The Faber-Zagier relations were defined using g and a single series in one variable with coefficients (6i)!/(3i)!(2i)!) as mentioned in this paper.
Abstract: These notes cover our series of three lectures at Humboldt University in Berlin for the October 2010 conference "Intersection theory on moduli space" (organized by G. Farkas). The topic concerns relations among the kappa classes in the tautological ring of the moduli space of genus g curves. After a discussion of classical constructions in Wick form, we derive an explicit set of relations obtained from the virtual geometry of the moduli space of stable quotients. In a series of steps, the stable quotient relations are transformed to simpler and simpler forms. Our final result establishes a previously conjectural set of tautological relations proposed a decade ago by Faber-Zagier.
The Faber-Zagier relations are defined using g and a single series in one variable with coefficients (6i)!/(3i)!(2i)!. Whether these relations span the complete set of relations among the kappa classes on the moduli space of genus g curves is an interesting question.
TL;DR: In this paper, the co-representability of non-connective K-theory by the base ring in the universal localizing motivator has been shown for free higher Chern characters, respectively higher trace maps, from cyclic homology to topological Hochschild homology.
Abstract: In this article, we further the study of higher K-theory of differential graded (dg) categories via universal invariants, initiated in [G. Tabuada, Higher K-theory via universal invariants, Duke Math. J. 145 (2008), 121–206]. Our main result is the co-representability of non-connective K-theory by the base ring in the ‘universal localizing motivator’. As an application, we obtain for free higher Chern characters, respectively higher trace maps, from non-connective K-theory to cyclic homology, respectively to topological Hochschild homology.