Showing papers on "Ring (mathematics) published in 2012"
TL;DR: The theory of FI-modules is introduced and developed in this paper, where the authors show that for any fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n.
Abstract: In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about:
- the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold
- the diagonal coinvariant algebra on r sets of n variables
- the cohomology and tautological ring of the moduli space of n-pointed curves
- the space of polynomials on rank varieties of n x n matrices
- the subalgebra of the cohomology of the genus n Torelli group generated by H^1 and more.
The symmetric group S_n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. In this framework, representation stability (in the sense of Church-Farb) for a sequence of S_n-representations is converted to a finite generation property for a single FI-module.
291 citations
Book•
20 Apr 2012
TL;DR: The Dirichlet Problem for Beltrami Equations has been studied in the context of ring Q-homeomorphisms at boundary points and strong ring solutions of BMO-and FMO-Quasiconformal mapping as discussed by the authors.
Abstract: 1. Introduction.- 2. Preliminaries.- 3. The Classical Beltrami Equation ||mu|| < 1.- 4. The Degenerate Case.- 5. BMO- and FMO-Quasiconformal Mappings.- 6. Ring Q-Homeomorphisms at Boundary Points.- 7. Strong Ring Solutions of Beltrami Equations.- 8. On the Dirichlet Problem for Beltrami Equations.- 9. On the Beltrami Equations with Two Characteristics.- 10. Alternating Beltrami Equation.- References.- Index.
134 citations
TL;DR: The transport and manipulation of qubits over distances of 6 µm within 40 ps are reported, in an Aharonov-Bohm ring connected to two-channel wires that have a tunable tunnel coupling between channels.
Abstract: Quantum bits defined in an Aharonov–Bohm ring are transported over long distances while being controlled with electric fields.
116 citations
TL;DR: The notion of derived Azumaya algebras over ring and schemes was introduced by Dix et al. as discussed by the authors, who showed that a quasi-compact and quasi-separated scheme ϕ defines a bijective correspondence.
Abstract: We introduce a notion of derived Azumaya algebras over ring and schemes generalizing the notion of Azumaya algebras of Grothendieck (Le groupe de Brauer. I. Algebres d’Azumaya et interpretations diverses. Dix Exposes sur la Cohomologie des Schemas, pp. 46–66, North-Holland, Amsterdam, 1968). We prove that any such algebra B on a scheme X provides a class ϕ(B) in \(H^{1}_{et}(X,\mathbb{Z})\times H^{2}_{et}(X,\mathbb{G}_{m})\). We prove that for X a quasi-compact and quasi-separated scheme ϕ defines a bijective correspondence, and in particular that any class in \(H^{2}_{et}(X,\mathbb{G}_{m})\), torsion or not, can be represented by a derived Azumaya algebra on X. Our result is a consequence of a more general theorem about the existence of compact generators in twisted derived categories, with coefficients in any local system of reasonable dg-categories, generalizing the well known existence of compact generators in derived categories of quasi-coherent sheaves of Bondal and Van Den Bergh (Mosc. Math. J. 3(1):1–36, 2003). A huge part of this paper concerns the treatment of twisted derived categories, as well as the proof that the existence of compact generator locally for the fppf topology implies the existence of a global compact generator. We present explicit examples of derived Azumaya algebras that are not represented by classical Azumaya algebras, as well as applications of our main result to the localization for twisted algebraic K-theory and to the stability of saturated dg-categories by direct push-forwards along smooth and proper maps.
114 citations
TL;DR: Tsai et al. as mentioned in this paper introduced filtered cohomologies of differential forms on symplectic manifolds, which can be associated with differential elliptic complexes and give a two-sided resolution of Lefschetz maps.
Abstract: Author(s): Tsai, Chung-Jun; Tseng, Li-Sheng; Yau, Shing-Tung | Abstract: We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered cohomologies give a two-sided resolution of Lefschetz maps, and thereby, they are directly related to the kernels and cokernels of the Lefschetz maps. We also introduce a novel, non-associative product operation on differential forms for symplectic manifolds. This product generates an A-infinity algebra structure on forms that underlies the filtered cohomologies and gives them a ring structure. As an application, we demonstrate how the ring structure of the filtered cohomologies can distinguish different symplectic four-manifolds in the context of a circle times a fibered three-manifold.
113 citations
TL;DR: In this article, the authors give an introduction to Joyce's construction of the motivic Hall algebra of coherent sheaves on a variety M and define a semi-classical integration map from a Poisson subalgebra of this Hall algebra to the ring of functions on a symplectic torus.
93 citations
[...]
TL;DR: The F-signature of R was first formally defined by C. Huneke and G. Leuschke (in Math. Ann. 324(2), 391-404, 2002) and has previously been shown only in special cases.
Abstract: Suppose R is a d-dimensional reduced F-finite Noetherian local ring with prime characteristic p>0 and perfect residue field. Let \(R^{1/p^{e}}\) be the ring of pe-th roots of elements of R for e∈ℕ, and let ae denote the maximal rank of a free R-module appearing in a direct sum decomposition of \(R^{1/p^{e}}\). We show the existence of the limit \(s(R) := \lim_{e \to\infty} \frac{a_{e}}{p^{ed}}\), called the F-signature of R. This invariant—which can be extended to all local rings in prime characteristic—was first formally defined by C. Huneke and G. Leuschke (in Math. Ann. 324(2), 391–404, 2002) and has previously been shown to exist only in special cases. The proof of our main result is based on the development of certain uniform Hilbert-Kunz estimates of independent interest. Additionally, we analyze the behavior of the F-signature under finite ring extensions and recover explicit formulae for the F-signatures of arbitrary finite quotient singularities.
88 citations
TL;DR: In this paper, the connection between supercharacter theories and Schur rings was made explicit, and supercharacter theory constructions which correspond to Schur ring products of Leung and Man [12], Hirasaka and Muzychuk [10], and Tamaschke [20] were constructed.
Abstract: Diaconis and Isaacs have defined the supercharacter theories of a finite group to be certain approximations to the ordinary character theory of the group [7]. We make explicit the connection between supercharacter theories and Schur rings, and we provide supercharacter theory constructions which correspond to Schur ring products of Leung and Man [12], Hirasaka and Muzychuk [10], and Tamaschke [20].
78 citations
TL;DR: In this article, it was shown that Waldhausen K-theory spaces admit canonical (connective) deloopings, and the Ktheory functor enjoys a universal property.
Abstract: We prove that Waldhausen K-theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, K-theory spaces admit canonical (connective) deloopings, and the K-theory functor enjoys a universal property. Using this, we give new, higher categorical proofs of both the additivity and fibration theorems of Waldhausen. As applications of this technology, we study the algebraic K-theory of associative ring spectra and spectral Deligne-Mumford stacks.
75 citations
21 May 2012
TL;DR: The results show that one does not need to resort to using any algebraic structure that is more complicated than polynomial rings in order to fully utilize the hardness of the Ring-LWE problem as a building block for cryptographic applications.
Abstract: The Ring-LWE problem, introduced by Lyubashevsky, Peikert, and Regev (Eurocrypt 2010), has been steadily finding many uses in numerous cryptographic applications. Still, the Ring-LWE problem defined in [LPR10] involves the fractional ideal R ∨, the dual of the ring R , which is the source of many theoretical and implementation technicalities. Until now, getting rid of R ∨, required some relatively complex transformation that substantially increase the magnitude of the error polynomial and the practical complexity to sample it. It is only for rings R =ℤ[X ]/(X n +1) where n a power of 2, that this transformation is simple and benign.
In this work we show that by applying a different, and much simpler transformation, one can transfer the results from [LPR10] into an "easy-to-use" Ring-LWE setting (i.e. without the dual ring R ∨), with only a very slight increase in the magnitude of the noise coefficients. Additionally, we show that creating the correct noise distribution can also be simplified by generating a Gaussian distribution over a particular extension ring of R , and then performing a reduction modulo f (X ). In essence, our results show that one does not need to resort to using any algebraic structure that is more complicated than polynomial rings in order to fully utilize the hardness of the Ring-LWE problem as a building block for cryptographic applications.
66 citations
TL;DR: It is demonstrated that the Schrödinger equation for two electrons on a ring, which is the usual paradigm to model quantum rings, is solvable in closed form for particular values of the radius and both polynomial and irrational solutions can be found.
Abstract: We demonstrate that the Schr\"odinger equation for two electrons on a ring, which is the usual paradigm to model quantum rings, is solvable in closed form for particular values of the radius. We show that both polynomial and irrational solutions can be found for any value of the angular momentum and that the singlet and triplet manifolds, which are degenerate, have distinct geometric phases. We also study the nodal structure associated with these two-electron states.
05 Sep 2012
TL;DR: Brakerski, Gentry and Vaikuntanathan as discussed by the authors proposed a ring-switching operation over polynomial rings of the form ρ{Z}[X]/(X^{2^n}+1) to speed up the homomorphic operations for the lower levels of the circuit.
Abstract: The security of BGV-style homomorphic encryption schemes over polynomial rings relies on rings of very large dimension. This large dimension is needed because of the large modulus-to-noise ratio in the key-switching matrices that are used for the top few levels of the evaluated circuit. However, larger noise (and hence smaller modulus-to-noise ratio) is used in lower levels of the circuit, so from a security standpoint it is permissible to switch to lower-dimension rings, thus speeding up the homomorphic operations for the lower levels of the circuit. However, implementing such ring-switching is nontrivial, since these schemes rely on the ring algebraic structure for their homomorphic properties.
A basic ring-switching operation was used by Brakerski, Gentry and Vaikuntanathan, over polynomial rings of the form $\mathbb{Z}[X]/(X^{2^n}+1)$, in the context of bootstrapping. In this work we generalize and extend this technique to work over any cyclotomic ring and show how it can be used not only for bootstrapping but also during the computation itself (in conjunction with the "packed ciphertext" techniques of Gentry, Halevi and Smart).
TL;DR: In this paper, the balance of the Gorenstein Tor-functor for modules of finite projective, injective and flat dimensions has been investigated for a class of rings with global and weak dimensions, in terms of certain cohomological invariants of the ring.
TL;DR: In this article, it was shown that the Jordan-Holder theorem fails even for stratifications by derived module categories, and also answers negatively an open problem by Angeleri-Hugel, Konig and Liu.
Abstract: Let $T$ be an infinitely generated tilting module of projective dimension at most one over an arbitrary associative ring $A$, and let $B$ be the endomorphism ring of $T$. In this paper, we prove that if $T$ is good then there exists a ring $C$, a homological ring epimorphism $B\ra C$ and a recollement among the (unbounded) derived module categories $\D{C}$ of $C$, $\D{B}$ of $B$, and $\D{A}$ of $A$. In particular, the kernel of the total left derived functor $T\otimes_B^{\mathbb L}-$ is triangle equivalent to the derived module category $\D{C}$. Conversely, if the functor $T\otimes_B^{\mathbb L}-$ admits a fully faithful left adjoint functor, then $T$ is a good tilting module. We apply our result to tilting modules arising from ring epimorphisms, and can then describe the rings $C$ as coproducts of two relevant rings. Further, in case of commutative rings, we can weaken the condition of being tilting modules, strengthen the rings $C$ as tensor products of two commutative rings, and get similar recollements. Consequently, we can produce examples (from commutative algebra and $p$-adic number theory, or Kronecker algebra) to show that two different stratifications of the derived module category of a ring by derived module categories of rings may have completely different derived composition factors (even up to ordering and up to derived equivalence),or different lengths. This shows that the Jordan-Holder theorem fails even for stratifications by derived module categories, and also answers negatively an open problem by Angeleri-Hugel, Konig and Liu.
15 Aug 2012
TL;DR: A survey of polyhedral divisors describing T-varieties is given in this paper, in parallel to the well established the-ory of toric varieties, including singularities, separatedness, properness, intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations.
Abstract: This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established the- ory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others.
Posted Content•
TL;DR: In this article, it was shown that without the dual ring R∨, one can transfer the results from [LPR10] into an easy-to-use ring-LWE setting with only a very slight increase in the magnitude of the noise coefficients, and that creating the correct noise distribution can also be simplified by generating a Gaussian distribution over a particular extension ring of R and then performing a reduction modulo f(X).
Abstract: The Ring-LWE problem, introduced by Lyubashevsky, Peikert, and Regev (Eurocrypt 2010), has been steadily finding many uses in numerous cryptographic applications. Still, the Ring-LWE problem defined in [LPR10] involves the fractional ideal R∨, the dual of the ring R, which is the source of many theoretical and implementation technicalities. Until now, getting rid of R∨, required some relatively complex transformation that substantially increase the magnitude of the error polynomial and the practical complexity to sample it. It is only for rings R = Z[X]/(X + 1) where n a power of 2, that this transformation is simple and benign. In this work we show that by applying a different, and much simpler transformation, one can transfer the results from [LPR10] into an “easyto-use” Ring-LWE setting (i.e. without the dual ring R∨), with only a very slight increase in the magnitude of the noise coefficients. Additionally, we show that creating the correct noise distribution can also be simplified by generating a Gaussian distribution over a particular extension ring of R, and then performing a reduction modulo f(X). In essence, our results show that one does not need to resort to using any algebraic structure that is more complicated than polynomial rings in order to fully utilize the hardness of the Ring-LWE problem as a building block for cryptographic applications.
Journal Article•
TL;DR: In this paper, the authors investigated the Green ring of the generalized Taft algebra Hn,d and showed that each nilpotent element can be expressed as a sum of indecomposable projective representations.
Abstract: In this paper, we investigate the Green ring r(Hn,d) of the generalized Taft algebra Hn,d, extending the results of Chen, Van Oys- taeyen and Zhang in (7). We shall determine all nilpotent elements of the Green ring r(Hn,d). It turns out that each nilpotent element in r(Hn,d) can be written as a sum of indecomposable projective representations. The Jacobson radical J(r(Hn,d)) of r(Hn,d) is generated by one element, and its rank is n n/d. Moreover, we will present all the finite dimen- sional indecomposable representations over the complexified Green ring R(Hn,d) of Hn,d. Our analysis is based on the decomposition of the ten- sor product of indecomposable representations and the observation of the solutions for the system of equations associated to the generating relations of the Green ring r(Hn,d).
TL;DR: In this paper, the open problem of developing K-theory for F1-schemes has been studied and a wide range of results have been obtained for algebraic geometry over F1, including normal morphisms and locally projective sheaves.
TL;DR: In this paper, a symmetric monoidal structure on the localizing motivator Motlocdg of dg categories was constructed for non-commutative mixed motives and a simple construction of the Chern character maps from non-connective algebraic K-theory to negative and periodic cyclic homology was obtained.
Abstract: In this article we further the study of non-commutative motives, initiated in [12, 43]. Our main result is the construction of a symmetric monoidal structure on the localizing motivator Motlocdg of dg categories. As an application, we obtain : (1) a computation of the spectra of morphisms in Motlocdg in terms of non-connective algebraic K-theory; (2) a fully-faithful embedding of Kontsevich's category KMMk of non-commutative mixed motives into the base category Motlocdg(e) of the localizing motivator; (3) a simple construction of the Chern character maps from non-connective algebraic K-theory to negative and periodic cyclic homology; (4) a precise connection between Toen's secondary K-theory and the Grothendieck ring of KMMk; (5) a description of the Euler characteristic in KMMk in terms of Hochschild homology.
TL;DR: The canonical simplifier for integro-differential polynomials is used for generating an automated proof establishing a canonical simplifiers for integrospecific operators in the Theorema system.
Abstract: We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic representation and computational realization is based on canonical forms in certain commutative and noncommutative polynomial domains. The first of these, the ring of integro-differential operators, is used for both stating and solving linear boundary problems. The other structure, called integro-differential polynomials, is the key tool for describing extensions of integro-differential algebras. We use the canonical simplifier for integro-differential polynomials for generating an automated proof establishing a canonical simplifier for integro-differential operators. Our approach is fully implemented in the Theorema system; some code fragments and sample computations are included.
TL;DR: In this paper, the concept of pseudopolar elements was introduced to define pseudo-Drazin inverse in associative rings and Banach algebras, which can be used exactly to define a pseudo Drazin inverses.
TL;DR: In this paper, an optical measurement method of radial displacement of a ring sample during its expansion with velocity of the order 172 m/s and estimation technique of plastic flow stress of ring material on basis of the obtained experimental data are presented.
Abstract: An optical measurement method of radial displacement of a ring sample during its expansion with velocity of the order 172 m/s and estimation technique of plastic flow stress of a ring material on basis of the obtained experimental data are presented in the work. To measure the ring motion during the expansion process, the Phantom v12 digital high-speed camera was applied, whereas the specialized TEMA Automotive software was used to analyze the obtained movies. Application of the above-mentioned tools and the developed measuring procedure of the ring motion recording allowed to obtain reliable experimental data and calculation results of plastic flow stress of a copper ring with satisfactory accuracy.
TL;DR: In this article, a functor from prequantized quasi-Hamiltonian G-spaces (M,ω,Φ) at level k to the fusion ring (Verlinde algebra) R k (G) is defined as a push forward in twisted equivariant K-homology.
Abstract: Let G be a compact, simply connected Lie group. We develop a ‘quantization functor’ from prequantized quasi-Hamiltonian G-spaces (M,ω,Φ) at level k to the fusion ring (Verlinde algebra) R k (G). The quantization is defined as a push-forward in twisted equivariant K-homology. It may be computed by a fixed point formula, similar to the equivariant index theorem for Spin c -Dirac operators. Using the formula, we calculate in several examples.
30 Sep 2012
TL;DR: In this article, a new type of ring oscillator PUF (RO-PUF) is introduced, in which the inverters composing the ring oscillators can be supplied by independent voltages.
Abstract: In this paper we introduce a new type of Ring Oscillator PUF (RO-PUF) in which the inverters composing the ring oscillators can be supplied by independent voltages. This new RO-PUF can improve the reliability of the PUF in case of temperature variations.
TL;DR: In this paper it was shown that for every p-divisible group over a p-adic ring one can associate a display by crystalline Dieudonne theory, which is a smooth morphism of smooth algebraic stacks.
Abstract: We show that to every p-divisible group over a p-adic ring one can associate a display by crystalline Dieudonne theory. For an appropriate notion of truncated displays, this induces a functor from truncated Barsotti-Tate groups to truncated displays, which is a smooth morphism of smooth algebraic stacks. As an application we obtain a new proof of the equivalence between infinitesimal p-divisible groups and nilpotent displays over p-adic rings, and a new proof of the equivalence due to Berthelot and Gabber between commutative finite flat group schemes of p-power order and Dieudonne modules over perfect rings.
TL;DR: In this article, the ring of all n × n matrices over a field K was considered and additive maps G : M n ( K ) → M n( K ) such that G ( x ) x = xG (x ) for all invertible (singular) x ∈ M n ∈ K was given.
17 Apr 2012
TL;DR: In this article, the authors consider the case of the modules M.............. R.............. whose endomorphism ring E := {End(M.............. R }) is a semilocal ring, that is, E/J(E) is a semiisimpleasure artinian ring.
Abstract: According to the classical Krull–Schmidt Theorem, any module of finite composition length decomposes as a direct sum of indecomposable modules in an essentially unique way, that is, unique up to isomorphism of the indecomposable summands and a permutation of the summands. Modules that do not have finite composition length can have completely different behaviors. In this survey, we consider in particular the case of the modules M
R
whose endomorphism ring E := End(M
R
) is a semilocal ring, that is, E/J(E) is a semisimple artinian ring. For instance, modules of finite composition length have a semilocal endomorphism ring, but several other classes of modules also have a semilocal endomorphism ring, for example artinian modules, finite direct sums of uniserial modules, finitely generated modules over commutative semilocal rings, and finitely presented modules over arbitrary semilocal rings. Several interesting phenomena appear in these cases. For instance, modules with a semilocal endomorphism ring have very regular direct-sum decompositions into indecomposables, their direct summands can be described via lattices, and direct-sum decompositions into indecomposables (=uniserial submodules) of finite direct sums of uniserial modules are described via their monogeny classes and their epigeny classes up to two permutations of the factors.
Journal Article•
TL;DR: In this paper, it was shown that skew pseudo-monoidal categories are skew monoidales with invertible associativity constraint when the lax unit is a one-sided R-modules.
Abstract: Kornel Szlachanyi [28] recently used the term skew-monoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized in terms of skew-monoidal structures on the category of one-sided R-modules for which the lax unit was R itself. We de ne skew monoidales (or skew pseudo-monoids) in any monoidal bicategory M . These are skew-monoidal categories when M is Cat. Our main results are presented at the level of monoidal bicategories. However, a consequence is that quantum categories [10] with base comonoid C in a suitably complete braided monoidal category V are precisely skew monoidales in Comod(V ) with unit coming from the counit of C. Quantum groupoids (in the sense of [6] rather than [10]) are those skew monoidales with invertible associativity constraint. In fact, we provide some very general results connecting opmonoidal monads and skew monoidales. We use a lax version of the concept of warping de ned in [3] to modify monoidal structures.
TL;DR: This paper showed that a module M is flat Mittag-Leffler if and only if M is ℵ 1 -projective in the sense of Eklof and Mekler (2002) [10].
TL;DR: In this paper, the authors give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga-Gorenstein ring.
Abstract: We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga-Gorenstein ring. We then apply this result to the Frobenius category of special Cohen-Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga-Gorenstein rings of finite GP type. We also apply our method to representation theory, obtaining Auslander-Solberg and Kong type results.