Showing papers on "Ring (mathematics) published in 2009"

Moduli in Modern Mapping Theory

Abstract: and Notation.- Moduli and Capacity.- Moduli and Domains.- Q-Homeomorphisms with Q? Lloc1.- Q-homeomorphisms with Q in BMO.- More General Q-Homeomorphisms.- Ring Q-Homeomorphisms.- Mappings with Finite Length Distortion (FLD).- Lower Q-Homeomorphisms.- Mappings with Finite Area Distortion.- On Ring Solutions of the Beltrami Equation.- Homeomorphisms with Finite Mean Dilatations.- On Mapping Theory in Metric Spaces.

The Classification of Two-Dimensional Extended Topological Field Theories

Abstract: We provide a complete generators and relations presentation of the 2-dimensional extended unoriented and oriented bordism bicategories as symmetric monoidal bicategories. Thereby we classify these types of 2-dimensional extended topological field theories with arbitrary target bicategory. As an immediate corollary we obtain a concrete classification when the target is the symmetric monoidal bicategory of algebras, bimodules, and intertwiners over a fixed commutative ground ring. In the oriented case, such an extended topological field theory is equivalent to specifying a (non-commutative) separable symmetric Frobenius algebra. The text is divided into three chapters. The first develops a variant of higher Morse theory and uses it to obtain a combinatorial description of surfaces suitable for the higher categorical language used later. The second chapter is an extensive treatment of the theory of symmetric monoidal bicategories. We introduce several stricter variants on the notion of symmetric monoidal bicategory, and give a very general treatment of the notion of presentation by generators and relations. Finally we provide a host of strictification and cohernece results for symmetric monoidal bicategories. The final chapter focuses on extended tqfts. We give a precise treatment of the extended bordism bicategory equipped with additional structure (such as framings or orientations). We apply the results of the previous two chapters to obtain a simple presentation of both the oriented and unoriented bordism bicategories, and describe the general method to obtain such classifications for other choices of structure. We examine the consequences of our classification when the target is the bicategory of algebras, bimodules, and maps, over a fixed commutative ground ring.

On Hecke Algebras

Abstract: We will start this chapter by giving the definition and the classification of complex reflection groups. We will also define the braid group and the pure braid group associated to a complex reflection group. We will then introduce the generic Hecke algebra of a complex reflection group, which is a quotient of the group algebra of the associated braid group defined over a Laurent polynomial in a finite number of indeterminates. Under certain assumptions, which have been verified for all but a finite number of cases, we prove (Theorem 28) that the generic Hecke algebras of complex reflection groups are essential. Therefore, all results obtained in Chapter 3 apply to the case of the generic Hecke algebras. A cyclotomic Hecke algebra is obtained from the generic Hecke algebra via a cyclotomic specialization (Definition 26). We prove (Theorem 30) that any cyclotomic specialization is essentially an adapted morphism. Thus, we can use Theorem 21 in order to obtain the Rouquier blocks of a cyclotomic Hecke algebra (i.e., its blocks over the Rouquier ring, defined in Section 4.4), which are a substitute for the families of characters that can be applied to all complex reflection groups. We will see that the Rouquier blocks have the property of semi-continuity, thus depending only on some “essential” hyperplanes for the group, which are determined by the generic Hecke algebra. The theory developed in this chapter will allow us to determine the Rouquier blocks of the cyclotomic Hecke algebras of all (irreducible) complex reflection groups in the next chapter.

Secure Arithmetic Computation with No Honest Majority

Abstract: We study the complexity of securely evaluating arithmetic circuits over finite rings. This question is motivated by natural secure computation tasks. Focusing mainly on the case of two-party protocols with security against malicious parties, our main goals are to: (1) only make black-box calls to the ring operations and standard cryptographic primitives, and (2) minimize the number of such black-box calls as well as the communication overhead. We present several solutions which differ in their efficiency, generality, and underlying intractability assumptions. These include: An unconditionally secure protocol in the OT-hybrid model which makes a black-box use of an arbitrary ring R ,but where the number of ring operations grows linearly with (an upper bound on) log|R |. Computationally secure protocols in the OT-hybrid model which make a black-box use of an underlying ring, and in which the number of ring operations does not grow with the ring size. The protocols rely on variants of previous intractability assumptions related to linear codes. In the most efficient instance of these protocols, applied to a suitable class of fields, the (amortized) communication cost is a constant number of field elements per multiplication gate and the computational cost is dominated by O (logk ) field operations per gate, where k is a security parameter. These results extend a previous approach of Naor and Pinkas for secure polynomial evaluation (SIAM J. Comput. , 2006). A protocol for the rings *** m = ***/m *** which only makes a black-box use of a homomorphic encryption scheme. When m is prime, the (amortized) number of calls to the encryption scheme for each gate of the circuit is constant. All of our protocols are in fact UC-secure in the OT-hybrid model and can be generalized to multiparty computation with an arbitrary number of malicious parties.

Global Gorenstein dimensions

Abstract: In this paper, we prove that the global Gorenstein projective dimension of a ring R is equal to the global Gorenstein injective dimension of R and that the global Gorenstein flat dimension of R is smaller than the common value of the terms of this equality.

Quiver varieties and cluster algebras

Abstract: Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452], we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the Grothendieck ring R of the category of representations of quantum loop algebras U_q(Lg) of a symmetric Kac-Moody Lie algebra g, studied earlier by the author via perverse sheaves on graded quiver varieties [arXiv:math/9912158]. 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 positivity and linearly independence (and probably many other) conjectures of cluster monomials [arXiv:math/0104151] 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.

The Gersten conjecture for Milnor K-theory

Abstract: We prove that the n-th Milnor K-group of an essentially smooth local ring over an infinite field coincides with the (n,n)-motivic cohomology of the ring. This implies Levine’s generalized Bloch–Kato conjecture.

Exact chiral ring of AdS(3) / CFT(2)

Abstract: We carry out an exact worldsheet computation of tree level three-point correlators of chiral operators in string theory on AdS(3) x S^3 x T^4 with NS-NS flux. We present a simple representation for the string chiral operators in the coordinate basis of the dual boundary CFT. Striking cancelations occur between the three-point functions of the H3+ and the SU(2) WZW models which result in a simple factorized form for the final correlators. We show, by fixing a single free parameter in the H3+ WZW model, that the fusion rules and the structure constants of the N=2 chiral ring in the bulk are in precise agreement with earlier computations in the boundary CFT of the symmetric product of T^4 at the orbifold point in the large N limit.

On Free Lie Rings

Abstract: Let Σ be a commutative associative ring with unit, let R={a α } be some set of symbols, and let \( \mathfrak{A}_{\Sigma R} \) be the free associative algebra over Σ with free generating set R. In the ring \( \mathfrak{A}_{\Sigma R} \), the set R generates a Lie subring \( \mathfrak{A}_{\Sigma R}^{\left( - \right)} \) with respect to the operations x o y=xy−yx, addition, and scalar multiplication by elements of Σ.

Quantum Pieri rules for isotropic Grassmannians

Abstract: We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew-symmetric form We establish Pieri rules for the classical cohomology and the small quantum cohomology ring of these varieties, which give a combinatorial formula for the product of any Schubert class with certain special Schubert classes We also give presentations of these rings, with integer coefficients, in terms of special Schubert class generators and relations

The black di-ring: an inverse scattering construction

Abstract: We use the inverse scattering method (ISM) to derive concentric non-supersymmetric black rings. The approach used here is fully five dimensional, and has the modest advantage that it generalizes readily to the construction of more general axi-symmetric solutions.

The single ring theorem

Abstract: We study the empirical measure $L_{A_n}$ of the eigenvalues of non-normal square matrices of the form $A_n=U_nD_nV_n$ with $U_n,V_n$ independent Haar distributed on the unitary group and $D_n$ real diagonal. We show that when the empirical measure of the eigenvalues of $D_n$ converges, and $D_n$ satisfies some technical conditions, $L_{A_n}$ converges towards a rotationally invariant measure on the complex plan whose support is a single ring. In particular, we provide a complete proof of Feinberg-Zee single ring theorem \cite{FZ}. We also consider the case where $U_n,V_n$ are independent Haar distributed on the orthogonal group.

The twisted Eilenberg-Zilber Theorem

Abstract: The purpose of this paper is to give a simpler proof of a theorem of E.H. Brown [Bro59], that if F → E → B is a fibre space, then there is a differential on the graded group X = C(B) ⊗Λ C(F ) such that X with this differential is chain equivalent to to C(E) (where C(E) denotes the normalised singular chains of E over a ring Λ). We work in the context of (semi-simplicial) twisted cartesian products (thus we assume as do the proofs of the theorem given in [Gug60, Shi62, Szc61] the results of [BGM59] on the relation between fibre spaces and twisted cartesian products). In fact we prove a general result on filtered chain complexes; this result applies to give proofs not only of Brown’s theorem but also of a theorem of G. Hirsch, [Hir53]. Our proof is suggested by the formulae (1) of [Shi62, Ch. II, §1]. Let (X, d), (Y, d) be chain complexes over a ring Λ. Let

Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms

Abstract: An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

Breaking RSA Generically Is Equivalent to Factoring

Abstract: We show that a generic ring algorithm for breaking RSA in *** N can be converted into an algorithm for factoring the corresponding RSA-modulus N . Our results imply that any attempt at breaking RSA without factoring N will be non-generic and hence will have to manipulate the particular bit-representation of the input in *** N . This provides new evidence that breaking RSA may be equivalent to factoring the modulus.

Stark-heegner points and the cohomology of quaternionic shimura varieties

Abstract: Let F be a totally real field of narrow class number one, and let E/F be a modular, semistable elliptic curve of conductor N � (1) .L etK/F be a non-CM quadratic extension with (Disc K, N) = 1 such that the sign in the functional equation of L(E/K, s) is −1. Suppose further that there is a prime p|N that is inert in K .W e describe a p-adic construction of points on E which we conjecture to be rational over ring class fields of K/F and satisfy a Shimura reciprocity law. These points are expected to behave like classical Heegner points and can be viewed as new instances of the Stark-Heegner point construction of [5]. The key idea in our construction is a reinterpretation of Darmon’s theory of modular symbols and mixed period integrals in terms of group cohomology.

On the irreducible representations of a finite semigroup

Abstract: Work of Clifford, Munn and Ponizovskii parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees's theorem characterizing 0-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovskii result based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple semigroups. A novelty of this approach is that it works over any base ring.

Rings Over Which the Class of Gorenstein Flat Modules is Closed Under Extensions

Abstract: A ring R is called left “GF-closed”, if the class of all Gorenstein flat left R-modules is closed under extensions The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak dimension In this article, we investigate the Gorenstein flat dimension over left GF-closed rings Namely, we generalize the fact that the class of all Gorenstein flat left modules is projectively resolving over right coherent rings to left GF-closed rings Also, we generalize the characterization of Gorenstein flat left modules (then of Gorenstein flat dimension of left modules) over right coherent rings to left GF-closed rings Finally, using direct products of rings, we show how to construct a left GF-closed ring that is neither right coherent nor of finite weak dimension

Some ring-theoretic properties of almost P -frames

Abstract: Almost P-frames generalize almost P-spaces, and indeed transcend them. In this article we give several characterizations of these frames, mostly in terms of certain ring-theoretic properties of $${\mathcal{R}}L$$ , the ring of real-valued functions on L.

The sum-product phenomenon in arbitrary rings

Abstract: The \emph{sum-product phenomenon} predicts that a finite set $A$ in a ring $R$ should have either a large sumset $A+A$ or large product set $A \cdot A$ unless it is in some sense ``close'' to a finite subring of $R$. This phenomenon has been analysed intensively for various specific rings, notably the reals $\R$ and cyclic groups $\Z/q\Z$. In this paper we consider the problem in arbitrary rings $R$, which need not be commutative or contain a multiplicative identity. We obtain rigorous formulations of the sum-product phenomenon in such rings in the case when $A$ encounters few zero-divisors of $R$. As applications we recover (and generalise) several sum-product theorems already in the literature.

The rationality of Stark-Heegner points over genus fields of real quadratic fields

Abstract: We study the algebraicity of Stark-Heegner points on a modular elliptic curve E. These objects are /?-adic points on E given by the values of certain /?-adic integrals, but they are conjecturally defined over ring class fields of a real quadratic field K. The present article gives some evidence for this algebraicity conjecture by showing that linear combinations of Stark-Heegner points weighted by certain genus characters of K are defined over the predicted quadratic extensions of K. The non- vanishing of these combinations is also related to the appropriate twisted Hasse-Weil L-series of E over K, in the spirit of the Gross-Zagier formula for classical Heegner points.

Amalgamated algebras along an ideal

Abstract: Let $f:A \to B$ be a ring homomorphism and $J$ an ideal of $B$. In this paper, we initiate a systematic study of a new ring construction called the "amalgamation of $A$ with $B$ along $J$ with respect to $f$". This construction finds its roots in a paper by J.L. Dorroh appeared in 1932 and provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced and studied by D'Anna and Fontana in 2007, and other classical constructions such as the $A+ XB[X]$ and $A+ XB[[X]]$ constructions, the CPI-extensions of Boisen and Sheldon, the $D+M$ constructions and the Nagata's idealization.

What precisely are E∞ring spaces and E∞ring spectra?

Abstract: E∞ ring spectra were defined in 1972, but the term has since acquired several alternative meanings. The same is true of several related terms. The new formulations are not always known to be equivalent to the old ones and even when they are, the notion of “equivalence” needs discussion: Quillen equivalent categories can be quite seriously inequivalent. Part of the confusion stems from a gap in the modern resurgence of interest in E∞ structures. E∞ ring spaces were also defined in 1972 and have never been redefined. They were central to the early applications and they tie in implicitly to modern applications. We summarize the relationships between the old notions and various new ones, explaining what is and is not known. We take the opportunity to rework and modernize many of the early results. New proofs and perspectives are sprinkled throughout.

Dissipative ring solitons with vorticity

Abstract: We study dissipative ring solitons with vorticity in the frame of the (2+1)-dimensional cubic-quintic complex Ginzburg-Landau equation. In dissipative media, radially symmetric ring structures with any vorticity m can be stable in a finite range of parameters. Beyond the region of stability, the solitons lose the radial symmetry but may remain stable, keeping the same value of the topological charge. We have found bifurcations into solitons with n-fold bending symmetry, with n independent on m. Solitons without circular symmetry can also display (m + 1)-fold modulation behaviour. A sequence of bifurcations can transform the ring soliton into a pulsating or chaotic state which keeps the same value of the topological charge as the original ring.