Showing papers on "Ring (mathematics) published in 2016"

Speeding up the Number Theoretic Transform for Faster Ideal Lattice-Based Cryptography.

Abstract: The Number Theoretic Transform (NTT) provides efficient algorithms for cyclic and nega-cyclic convolutions, which have many applications in computer arithmetic, e.g., for multiplying large integers and large degree polynomials. It is commonly used in cryptographic schemes that are based on the hardness of the Ring Learning With Errors (R-LWE) problem to efficiently implement modular polynomial multiplication.

Speeding up the Number Theoretic Transform for Faster Ideal Lattice-Based Cryptography

Abstract: The Number Theoretic Transform (NTT) provides efficient algorithms for cyclic and nega-cyclic convolutions, which have many applications in computer arithmetic, e.g., for multiplying large integers and large degree polynomials. It is commonly used in cryptographic schemes that are based on the hardness of the Ring Learning With Errors (R-LWE) problem to efficiently implement modular polynomial multiplication.

Sets of Lengths

Abstract: Oftentimes the elements of a ring or semigroup can be written as finite products of irreducible elements. An element a can be a product of k irreducibles and a product of l irreducibles. The set L(a) of all possible factorization lengths of a is called the set of lengths of a, and the system consisting of all these sets L(a) is a well-studied means of describing the nonuniqueness of factorizations of a ring or semigroup. We provide a friendly introduction, which is largely self-contained, to what is known about systems of sets of lengths for rings of integers of algebraic number fields and for transfer Krull monoids of finite type as their generalization.

The moduli space of twisted canonical divisors

Abstract: The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the moduli spaces of canonical divisors.In case the differentials have at least one pole (the strictly meromorphic case), the moduli spaces of twisted canonical divisors on genus curves are of pure codimension in . In addition to the closure of the canonical divisors on nonsingular curves, the moduli spaces have virtual components. In the Appendix A, a complete proposal relating the sum of the fundamental classes of all components (with intrinsic multiplicities) to a formula of Pixton is proposed. The result is a precise and explicit conjecture in the tautological ring for the weighted fundamental class of the moduli spaces of twisted canonical divisors.As a consequence of the conjecture, the classes of the closures of the moduli spaces of canonical divisors on nonsingular curves are determined (both in the holomorphic and meromorphic cases).

The Ring of Algebraic Functions on Persistence Bar Codes

Abstract: We study the ring of algebraic functions on the space of persistence barcodes, with applications to pattern recognition.

Holomorphic field realization of SH c and quantum geometry of quiver gauge theories

Abstract: In the context of 4D/2D dualities, SH c algebra, introduced by Schiffmann and Vasserot, provides a systematic method to analyse the instanton partition functions of $$ \mathcal{N}=2 $$ supersymmetricgaugetheories. Inthispaper,werewritetheSH c algebrainterms of three holomorphic fields D 0(z), D ±1(z) with which the algebra and its representations are simplified. The instanton partition functions for arbitrary $$ \mathcal{N}=2 $$ super Yang-Mills theories with A n and A (1) type quiver diagrams are compactly expressed as a product of four building blocks, Gaiotto state, dilatation, flavor vertex operator and intertwiner which are written in terms of SH c and the orthogonal basis introduced by Alba, Fateev, Litvinov and Tarnopolskiy. These building blocks are characterized by new conditions which generalize the known ones on the Gaiotto state and the Carlsson-Okounkov vertex. Consistency conditions of the inner product give algebraic relations for the chiral ring generating functions defined by Nekrasov, Pestun and Shatashvili. In particular we show the polynomiality of the qq-characters which have been introduced as a deformation of the Yangian characters. These relations define a second quantization of the Seiberg-Witten geometry, and, accordingly, reduce to a Baxter TQ-equation in the Nekrasov-Shatashvili limit of the Omega-background.

On the algebraic K-theory of higher categories

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.

Landau level quantization and almost flat modes in three-dimensional semimetals with nodal ring spectra

Abstract: We investigate Landau level structures of semimetals with nodal ring dispersions. When the magnetic field is applied parallel to the plane in which the ring lies, there exist almost nondispersive Landau levels at the Fermi level (${E}_{F}=0$) as a function of the momentum along the field direction inside the ring. We show that the Landau levels at each momentum along the field direction can be described by the Hamiltonian for the graphene bilayer with fictitious interlayer couplings under a tilted magnetic field. Near the center of the ring where the in-terlayer coupling is negligible, we have Dirac Landau levels which explain the appearance of the zero modes. Although the interlayer hopping amplitudes become finite at higher momenta, the splitting of zero modes is exponentially small and they remain almost flat due to the finite artificial in-plane component of the magnetic field. The emergence of the density of states peak at the Fermi level would be a hallmark of the ring dispersion.

A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra

Abstract: This is a set of working notes which give a second proof of the Shareshian--Wachs conjecture, the first (and recent) proof being by Brosnan and Chow in November 2015. The conjecture relates some symmetric functions constructed combinatorially out of unit interval graphs (their $q$-chromatic quasisymmetric functions), and some symmetric functions constructed algebro-geometrically out of Tymoczko's representation of the symmetric group on the equivariant cohomology ring of a family of subvarieties of the complex flag variety, called regular semisimple Hessenberg varieties. Brosnan and Chow's proof is based in part on the idea of deforming the Hessenberg varieties. The proof given here, in contrast, is based on the idea of recursively decomposing Hessenberg varieties, using a new Hopf algebra as the organizing principle for this recursion. We hope that taken together, each approach will shed some light on the other, since there are still many outstanding questions regarding the objects under study.

Optimal design of corona ring on HV composite insulator using PSO approach with dynamic population size

Abstract: This paper deals with the use of corona ring at the HV end fitting for improving the electric field and potential distributions and then for minimizing the corona discharges on 230 kV AC transmission line composite insulator. A single corona ring is installed at the energized end side. Three-dimensional finite element method (FEM) software is employed to compute the electric field. As the performance of high voltage insulator strings closely depends on designs and locations of corona ring, the effects of the corona ring radius, the ring tube radius and the ring vertical position are examined. The minimization of the electric field necessitates the optimization of corona ring. For this purpose, new nonlinear mathematical objective function linking the electric field strength to the corona ring structure parameters is established. The optimization problem is achieved by minimizing the objective function using a modified particles swarm optimization (PSO) algorithm with a dynamic population size. The algorithm adjusts the size of population for each iteration. Based on the average value and the best solution of the objective function, we propose a new mathematical model to update the population size. This algorithm enables the population size reduction leading to computing time decrease. According to the results, FEM-PSO hybridization technique could be very helpful in optimization of corona ring design.

A categorification of Grassmannian cluster algebras

Abstract: We describe a ring whose category of Cohen-Macaulay modules provides an additive categorication the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. More precisely, there is a cluster character dened on the category which maps the rigid indecomposable objects to the cluster variables and the maximal rigid objects to clusters. This is proved by showing that the quotient of this category by a single projective-injective object is Geiss-Leclerc-Schroer's category Sub Qk, which categories the coordinate ring of the big cell in this Grassmannian.

Equations of tropical varieties

Abstract: We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as T=(R∪{−∞},max,+) by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-Archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of T-points this reduces to Kajiwara–Payne’s extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of T-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.

New Algebraic Properties of an Amalgamated Algebra Along an Ideal

Abstract: Let f: A → B be a ring homomorphism, and let J be an ideal of B. In this article, we study the amalgamation of A with B along J with respect to f (denoted by A ⋈fJ), a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced by D'Anna and Fontana in 2007, and other classical constructions (such as the A + XB[X], the A + XB[[X]] and the D + M constructions). In particular, we completely describe the prime spectrum of the amalgamation A ⋈fJ and, when it is a local Noetherian ring, we study its embedding dimension and when it turns to be a Cohen–Macaulay ring or a Gorenstein ring.

Fixed-point arithmetic in SHE schemes

Abstract: The purpose of this paper is to investigate fixed-point arithmetic in ring-based Somewhat Homomorphic Encryption (SHE) schemes. We provide three main contributions: firstly, we investigate the representation of fixed-point numbers. We analyse the two representations from Dowlin et al., representing a fixed-point number as a large integer (encoded as a scaled polynomial) versus a polynomial-based fractional representation. We show that these two are, in fact, isomorphic by presenting an explicit isomorphism between the two that enables us to map the parameters from one representation to another. Secondly, given a computation and a bound on the fixed-point numbers used as inputs and scalars within the computation, we achieve a way of producing lower bounds on the plaintext modulus p and the degree of the ring d needed to support complex homomorphic operations. Finally, as an application of these bounds, we investigate homomorphic image processing.

Rings in which every element is a sum of two tripotents

Abstract: Let be a ring. The following results are proved. Every element of is a sum of an idempotent and a tripotent that commute if and only if has the identity if and only if , where is Boolean with a group of exponent and is zero or a subdirect product of . Every element of is either a sum or a difference of two commuting idempotents if and only if , where is Boolean with or and is zero or a subdirect product of . Every element of is a sum of two commuting tripotents if and only if , where is Boolean with a group of exponent , is zero or a subdirect product of , and is zero or a subdirect product of .

Further results on the inverse along an element in semigroups and rings

Abstract: In this paper, we introduce a new notion in a semigroup as an extension of Mary’s inverse. Let . An element is called left (resp. right) invertible along if there exists such that (resp. ) and (resp. ). An existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right) -regularity and left (right) -regularity are given in a semigroup. Further, another existence criterion of this type inverse is given by means of a left (right) invertibility of certain elements in a ring. Finally, we study the (left, right) inverse along a product in a ring, and, as an application, Mary’s inverse along a matrix is expressed.

How (Not) to Instantiate Ring-LWE.

Abstract: The learning with errors over rings Ring-LWE problem--or more accurately, family of problems--has emerged as a promising foundation for cryptography due to its practical efficiency, conjectured quantum resistance, and provable worst-case hardness: breaking certain instantiations of Ring-LWE is at least as hard as quantumly approximating the Shortest Vector Problem on any ideal lattice in the ring. Despite this hardness guarantee, several recent works have shown that certain instantiations of Ring-LWE can be broken by relatively simple attacks. While the affected instantiations are not supported by worst-case hardness theorems and were not ever proposed for cryptographic purposes, this state of affairs raises natural questions about what other instantiations might be vulnerable, and in particular whether certain classes of rings are inherently unsafe for Ring-LWE. This work comprehensively reviews the known attacks on Ring-LWE and vulnerable instantiations. We give a new, unified exposition which reveals an elementary geometric reason why the attacks work, and provide rigorous analysis to explain certain phenomena that were previously only exhibited by experiments. In all cases, the insecurity of an instantiation is due to the fact that the error distribution is insufficiently "well spread" relative to the ring. In particular, the insecure instantiations use the so-called non-dual form of Ring-LWE, together with spherical error distributions that are much narrower and of a very different shape than the ones supported by hardness proofs. On the positive side, we show that any Ring-LWE instantiation which satisfies or only almost satisfies the hypotheses of the "worst-case hardness of search" theorem is provably immune to broad generalizations of the above-described attacks: the running time divided by advantage is at least exponential in the degree of the ring. This holds for the ring of integers in any number field, so the rings themselves are not the source of insecurity in the vulnerable instantiations. Moreover, the hypotheses of the worst-case hardness theorem are nearly minimal ones which provide these immunity guarantees.

On certain pure sextic fields related to a problem of Hasse

Abstract: An algebraic number ring is monogenic, or one-generated, if it has the form Z[α] for a single algebraic integer α. It is a problem of Hasse to characterize, whether an algebraic number ring is monogenic or not. In this note, we prove that if m is a square-free rational integer, m ≡ 1(mod 4) and m≢ ± 1(mod 9), then the pure sextic field L = Q(m6) is not monogenic. Our results are illustrated by examples.

Cluster algebras and category O for representations of Borel subalgebras of quantum affine algebras

Abstract: Let $\mathcal{O}$ be the category of representations of the Borel subalgebra of a quantum affine algebra introduced by Jimbo and the first author. We show that the Grothendieck ring of a certain monoidal subcategory of $\mathcal{O}$ has the structure of a cluster algebra of infinite rank, with an initial seed consisting of prefundamental representations. In particular, the celebrated Baxter relations for the 6-vertex model get interpreted as Fomin-Zelevinsky mutation relations.

K-theoretic obstructions to bounded t-structures

Abstract: Schlichting conjectured that the negative K-groups of small abelian categories vanish and proved this for noetherian abelian categories and for all abelian categories in degree $-1$. The main results of this paper are that $K_{-1}(E)$ vanishes when $E$ is a small stable $\infty$-category with a bounded t-structure and that $K_{-n}(E)$ vanishes for all $n\geq 1$ when additionally the heart of $E$ is noetherian. It follows that Barwick's theorem of the heart holds for nonconnective K-theory spectra when the heart is noetherian. We give several applications, to non-existence results for bounded t-structures and stability conditions, to possible K-theoretic obstructions to the existence of the motivic t-structure, and to vanishing results for the negative K-groups of a large class of dg algebras and ring spectra.

Quantum fault tolerance in small experiments

Abstract: I discuss a variety of issues relating to near-future experiments demonstrating fault-tolerant quantum computation. I describe a family of fault-tolerant quantum circuits that can be performed with 5 qubits arranged on a ring with nearest-neighbor interactions. I also present a criterion whereby we can say that an experiment has succeeded in demonstrating fault tolerance. Finally, I discuss the possibility of using future fault-tolerant experiments to answer important questions about the interaction of fault-tolerant protocols with real experimental errors.

Squarefree polynomials and möbius values in short intervals and arithmetic progressions

Abstract: We calculate the mean and variance of sums of the Mobius function and the indicator function of the squarefrees, in both short intervals and arithmetic progressions, in the context of the ring of polynomials over a finite field of $q$ elements, in the limit $q\to \infty$. We do this by relating the sums in question to certain matrix integrals over the unitary group, using recent equidistribution results due to N. Katz, and then by evaluating these integrals. In many cases our results mirror what is either known or conjectured for the corresponding problems involving sums over the integers, which have a long history. In some cases there are subtle and surprising differences. The ranges over which our results hold is significantly greater than those established for the corresponding problems in the number field setting.

Topological modular forms with level structure

Abstract: The cohomology theory known as $$\mathrm{Tmf}$$ , for “topological modular forms,” is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to restrict to the cusps, providing multiplicative maps from $$\mathrm{Tmf}$$ with level structure to forms of $$K$$ -theory. In particular, this allows us to construct a connective spectrum $$\mathrm{tmf}_0(3)$$ consistent with properties suggested by Mahowald and Rezk. This is accomplished using the machinery of logarithmic structures. We construct a presheaf of locally even-periodic elliptic cohomology theories, equipped with highly structured multiplication, on the log-etale site of the moduli of elliptic curves. Evaluating this presheaf on modular curves produces $$\mathrm{Tmf}$$ with level structure.

Summation Theory II: Characterizations of $\boldsymbol{R\Pi\Sigma^*}$-extensions and algorithmic aspects

Abstract: Recently, $R\Pi\Sigma^*$-extensions have been introduced which extend Karr's $\Pi\Sigma^*$-fields substantially: one can represent expressions not only in terms of transcendental sums and products, but one can work also with products over primitive roots of unity. Since one can solve the parameterized telescoping problem in such rings, covering as special cases the summation paradigms of telescoping and creative telescoping, one obtains a rather flexible toolbox for symbolic summation. This article is the continuation of this work. Inspired by Singer's Galois theory of difference equations we will work out several alternative characterizations of $R\Pi\Sigma^*$-extensions: adjoining naively sums and products leads to an $R\Pi\Sigma^*$-extension iff the obtained difference ring is simple iff the ring can be embedded into the ring of sequences iff the ring can be given by the interlacing of $\Pi\Sigma^*$-extensions. From the viewpoint of applications this leads to a fully automatic machinery to represent indefinite nested sums and products in such $R\Pi\Sigma^*$-rings. In addition, we work out how the parameterized telescoping paradigm can be used to prove algebraic independence of indefinite nested sums. Furthermore, one obtains an alternative reduction tactic to solve the parameterized telescoping problem in basic $R\Pi\Sigma^*$-extensions exploiting the interlacing property.

Coregular spaces and genus one curves

Abstract: A coregular space is a representation of an algebraic group for which the ring of polynomial invariants is free. In this paper, we show that the orbits of many coregular irreducible representations where the number of invariants is at least two, over a (not necessarily algebraically closed) field k, correspond to genus one curves over k together with line bundles, vector bundles, and/or points on their Jacobians. In forthcoming work, we use these orbit parametrizations to determine the average sizes of Selmer groups for various families of elliptic curves.