Showing papers on "Ring (mathematics) published in 2018"

Quadratic transformation inequalities for Gaussian hypergeometric function.

Abstract: In the article, we present several quadratic transformation inequalities for Gaussian hypergeometric function and find the analogs of duplication inequalities for the generalized Grotzsch ring function

Experimental realization of a Weyl exceptional ring

Abstract: Weyl points are isolated degeneracies in reciprocal space that are monopoles of the Berry curvature This topological charge makes them inherently robust to Hermitian perturbations of the system However, non-Hermitian effects, usually inaccessible in condensed matter systems, are an important feature of photonics systems, and when added to an otherwise Hermitian Weyl material have been predicted to spread the Berry charge of the Weyl point out onto a ring of exceptional points, creating a Weyl exceptional ring and fundamentally altering its properties Here, we observe the implications of the Weyl exceptional ring using real-space measurements of an evanescently-coupled bipartite optical waveguide array by probing its effects on the Fermi arc surface states, the bulk diffraction properties, and the output power ratio of the two constituent sublattices This is the first realization of an object with topological Berry charge in a non-Hermitian system

Wide Compression: Tensor Ring Nets

Abstract: Deep neural networks have demonstrated state-of-the-art performance in a variety of real-world applications. In order to obtain performance gains, these networks have grown larger and deeper, containing millions or even billions of parameters and over a thousand layers. The tradeoff is that these large architectures require an enormous amount of memory, storage, and computation, thus limiting their usability. Inspired by the recent tensor ring factorization, we introduce Tensor Ring Networks (TR-Nets), which significantly compress both the fully connected layers and the convolutional layers of deep neural networks. Our results show that our TR-Nets approach is able to compress LeNet-5 by 11A— without losing accuracy, and can compress the state-of-the-art Wide ResNet by 243A— with only 2.3% degradation in Cifar10 image classification. Overall, this compression scheme shows promise in scientific computing and deep learning, especially for emerging resource-constrained devices such as smartphones, wearables, and IoT devices.

LAC: Practical Ring-LWE Based Public-Key Encryption with Byte-Level Modulus.

Abstract: We propose an instantiation of public key encryption scheme based on the ring learning with error problem, where the modulus is at a byte level and the noise is at a bit level, achieving one of the most compact lattice based schemes in the literature. The main technical challenges are a) the decryption error rates increases and needs to be handled elegantly, and b) we cannot use the Number Theoretic Transform (NTT) technique to speed up the implementation. We overcome those limitations with some customized parameter sets and heavy error correction codes. We give a treatment of the concrete security of the proposed parameter set, with regards to the recent advance in lattice based cryptanalysis. We present an optimized implementation taking advantage of our byte level modulus and bit level noise. In addition, a byte level modulus allows for high parallelization and the bit level noise avoids the modulus reduction during multiplication. Our result shows that LAC is more compact than most of the existing (Ring-)LWE based solutions, while achieving a similar level of efficiency, compared with popular solutions in this domain, such as Kyber.

The grin of Cheshire cat resurgence from supersymmetric localization

Abstract: First we compute the S2 partition function of the supersymmetric CPN−1 model via localization and as a check we show that the chiral ring structure can be correctly reproduced. For the CP1 case we provide a concrete realisation of this ring in terms of Bessel functions. We consider a weak coupling expansion in each topological sector and write it as a finite number of perturbative corrections plus an infinite series of instantonanti-instanton contributions. To be able to apply resurgent analysis we then consider a non-supersymmetric deformation of the localized model by introducing a small unbalance between the number of bosons and fermions. The perturbative expansion of the deformed model becomes asymptotic and we analyse it within the framework of resurgence theory. Although the perturbative series truncates when we send the deformation parameter to zero we can still reconstruct non-perturbative physics out of the perturbative data in a nice example of Cheshire cat resurgence in quantum field theory. We also show that the same type of resurgence takes place when we consider an analytic continuation in the number of chiral fields from N to r ∈ R. Although for generic real r supersymmetry is still formally preserved, we find that the perturbative expansion of the supersymmetric partition function becomes asymptotic so that we can use resurgent analysis and only at the end take the limit of integer r to recover the undeformed model.

Trusses: Between braces and rings

Abstract: In an attempt to understand the origins and the nature of the law binding together two group operations into a {\em skew brace}, introduced in [L.\ Guarnieri \& L.\ Vendramin, Math.\ Comp.\ \textbf{86} (2017), 2519--2534] as a non-Abelian version of the {\em brace distributive law} of [W.\ Rump, J.\ Algebra {\bf 307} (2007), 153--170] and [F.\ Cedo, E.\ Jespers \& J.\ Okninski, Commun.\ Math.\ Phys.\ {\bf 327} (2014), 101--116], the notion of a {\em skew truss} is proposed. A skew truss consists of a set with a group operation and a semigroup operation connected by a modified distributive law that interpolates between that of a ring and a brace. It is shown that a particular action and a cocycle characteristic of skew braces are already present in a skew truss; in fact the interpolating function is a 1-cocycle, the bijecitivity of which indicates the existence of an operation that turns a truss into a brace. Furthermore, if the group structure in a two-sided truss is Abelian, then there is an associated ring -- another feature characteristic of a two-sided brace. To characterise a morphism of trusses, a {\em pith} is defined as a particular subset of the domain consisting of subsets termed {\em chambers}, which contains the kernel of the morphism as a group homomorphism. In the case of both rings and braces piths coincide with kernels. In general the pith of a morphism is a sub-semigroup of the domain and, if additional properties are satisfied, a pith is a $\mathbb{N}_+$-graded semigroup. Finally, giving heed to [I.\ Angiono, C.\ Galindo \& L.\ Vendramin, Proc.\ Amer.\ Math.\ Soc.\ {\bf 145} (2017), 1981--1995] we linearise trusses and thus define {\em Hopf trusses} and study their properties, from which, in parallel to the set-theoretic case, some properties of Hopf braces are shown to follow.

Fluctuations of TASEP on a Ring in Relaxation Time Scale

Abstract: We consider the totally asymmetric simple exclusion process on a ring with flat and step initial conditions. We assume that the size of the ring and the number of particles tend to infinity proportionally and evaluate the fluctuations of tagged particles and currents. The crossover from the KPZ dynamics to the equilibrium dynamics occurs when the time is proportional to the 3/2 power of the ring size. We compute the limiting distributions in this relaxation time scale. The analysis is based on an explicit formula of the finite-time one-point distribution obtained from the coordinate Bethe ansatz method.© 2017 Wiley Periodicals, Inc.

Application of an Innovative SMA Ring Spring System for Self-Centering Steel Frames Subject to Seismic Conditions

Abstract: This paper presents an innovative shape memory alloy (SMA) ring spring system used for high-performance steel beam-to-column connections. The ring spring system, which includes a series of ...

Group rings, G -codes and constructions of self-dual and formally self-dual codes

Abstract: We describe G-codes, which are codes that are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a G-code is also a G-code. We give constructions of self-dual and formally self-dual codes in this setting and we improve the existing construction given in Hurley (Int J Pure Appl Math 31(3):319–335, 2006) by showing that one of the conditions given in the theorem is unnecessary and, moreover, it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes must have an automorphism group that contains G as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative [72, 36, 16] Type II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48. We define quasi-G codes and give a construction of these codes.

On semisimplification of tensor categories

Abstract: We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic $p$ in terms of representations of the normnalizer of its Sylow $p$-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group $S_{n+p}$ in characteristic $p$, where $0\le n\le p-1$, and of the Deligne category $\underline{\rm Rep}^{\rm ab}S_t$, where $t\in \Bbb N$. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of $\mathfrak{sl}_2$. Finally, we study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). In the appendix, we classify categorifications of the Grothendieck ring of representations of $SO(3)$ and its truncations.

Singular compactness and definability for $\Sigma$-cotorsion and Gorenstein modules

Abstract: We introduce a general version of singular compactness theorem which makes it possible to show that being a $\Sigma$-cotorsion module is a property of the complete theory of the module. As an application of the powerful tools developed along the way, we give a new description of Gorenstein flat modules which implies that, regardless of the ring, the class of all Gorenstein flat modules forms the left-hand class of a perfect cotorsion pair. We also prove the dual result for Gorenstein injective modules.

A dynamical mechanism for the origin of nuclear rings

Abstract: We develop a dynamical theory for the origin of nuclear rings in barred galaxies. In analogy with the standard theory of accretion discs, our theory is based on shear viscous forces among nested annuli of gas. However, the fact that gas follows non circular orbits in an external barred potential has profound consequences: it creates a region of reverse shear in which it is energetically favourable to form a stable ring which does not spread despite dissipation. Our theory allows us to approximately predict the size of the ring given the underlying gravitational potential. The size of the ring is loosely related to the location of the Inner Lindblad Resonance in the epicyclic approximation, but the predicted location is more accurate and is also valid for strongly barred potentials. By comparing analytical predictions with the results of hydrodynamical simulations, we find that our theory provides a viable mechanism for ring formation if the effective sound speed of the gas is low ($\cs\lesssim1\kms$), but that nuclear spirals/shocks created by pressure destroy the ring when the sound speed is high ($\cs\simeq10\kms$). We conclude that whether this mechanism for ring formation is relevant for real galaxies ultimately depends on the effective equation of state of the ISM. Promising confirmation comes from simulations in which the ISM is modelled using state-of-the-art cooling functions coupled to live chemical networks, but more tests are needed regarding the role of turbulence driven by stellar feedback. If the mechanism is relevant in real galaxies, it could provide a powerful tool to constrain the gravitational potential, in particular the bar pattern speed.

Special Arithmetic of Flavor

Abstract: We revisit the classification of rank-1 4d $$ \mathcal{N}=2 $$ QFTs in the spirit of Diophantine Geometry, viewing their special geometries as elliptic curves over the chiral ring (a Dedekind domain). The Kodaira-Neron model maps the space of non-trivial rank-1 special geometries to the well-known moduli of pairs (e, F∞) where E is a relatively minimal, rational elliptic surface with section, and F∞ a fiber with additive reduction. Requiring enough Seiberg-Witten differentials yields a condition on (e, F∞) equivalent to the “safely irrelevant conjecture”. The Mordell-Weil group of E (with the Neron-Tate pairing) contains a canonical root system arising from (−1)-curves in special position in the Neron-Severi group. This canonical system is identified with the roots of the flavor group F: the allowed flavor groups are then read from the Oguiso-Shioda table of Mordell-Weil groups. Discrete gaugings correspond to base changes. Our results are consistent with previous work by Argyres et al.

On the chiral algebra of Argyres-Douglas theories and S-duality

Abstract: We study the two-dimensional chiral algebra associated with the simplest Argyres-Douglas type theory with an exactly marginal coupling, i.e., the (A3, A3) theory. Near a cusp in the space of the exactly marginal deformations (i.e., the conformal manifold), the theory is well-described by the SU(2) gauge theory coupled to isolated Argyres-Douglas theories and a fundamental hypermultiplet. In this sense, the (A3, A3) theory is an Argyres-Douglas version of the $$ \mathcal{N} $$ = 2 SU(2) conformal QCD. By studying its Higgs branch and Schur index, we identify the minimal possible set of chiral algebra generators for the (A3, A3) theory, and show that there is a unique set of closed OPEs among these generators. The resulting OPEs are consistent with the Schur index, Higgs branch chiral ring relations, and the BRST cohomology conjecture. We then show that the automorphism group of the chiral algebra we constructed contains a discrete group G with an S3 subgroup and a homomorphism G → S4 × Z2. This result is consistent with the S-duality of the (A3, A3) theory.

Incomplete Tambara functors

Abstract: For a “genuine” equivariant commutative ring spectrum R , π 0 ( R ) admits a rich algebraic structure known as a Tambara functor. This algebraic structure mirrors the structure on R arising from the existence of multiplicative norm maps. Motivated by the surprising fact that Bousfield localization can destroy some of the norm maps, in previous work we studied equivariant commutative ring structures parametrized by N ∞ operads. In a precise sense, these interpolate between “naive” and “genuine” equivariant ring structures. In this paper, we describe the algebraic analogue of N ∞ ring structures. We introduce and study categories of incomplete Tambara functors, described in terms of certain categories of bispans. Incomplete Tambara functors arise as π 0 of N ∞ algebras, and interpolate between Green functors and Tambara functors. We classify all incomplete Tambara functors in terms of a basic structural result about polynomial functors. This classification gives a conceptual justification for our prior description of N ∞ operads and also allows us to easily describe the properties of the category of incomplete Tambara functors.

Graphs in Perturbation Theory: Algebraic Structure and Asymptotics

Abstract: This thesis provides an extension of the work of Dirk Kreimer and Alain Connes on the Hopf algebra structure of Feynman graphs and renormalization to general graphs. Additionally, an algebraic structure of the asymptotics of formal power series with factorial growth, which is compatible with the Hopf algebraic structure, is introduced. The Hopf algebraic structure on graphs permits the explicit enumeration of graphs with constraints for the allowed subgraphs. In the case of Feynman diagrams a lattice structure, which will be introduced, exposes additional unique properties for physical quantum field theories. The differential ring of factorially divergent power series allows the extraction of asymptotic results of implicitly defined power series with vanishing radius of convergence. Together both structures provide an algebraic formulation of large graphs with constraints on the allowed subgraphs. These structures are motivated by and used to analyze renormalized zero-dimensional quantum field theory at high orders in perturbation theory. As a pure application of the Hopf algebra structure, an Hopf algebraic interpretation of the Legendre transformation in quantum field theory is given. The differential ring of factorially divergent power series will be used to solve two asymptotic counting problems from combinatorics: The asymptotic number of connected chord diagrams and the number of simple permutations. For both asymptotic solutions, all order asymptotic expansions are provided as generating functions in closed form. Both structures are combined in an application to zero-dimensional quantum field theory. Various quantities are explicitly given asymptotically in the zero-dimensional version of $\varphi^3$, $\varphi^4$, QED, quenched QED and Yukawa theory with their all order asymptotic expansions.

Regularity and cohomology of determinantal thickenings

Abstract: We consider the ring S=C[xij] of polynomial functions on the vector space Cm×n of complex m×n matrices. We let GL=GLm(C)×GLn(C) and consider its action via row and column operations on Cm×n (and the induced action on S). For every GL-invariant ideal I⊆S and every j⩾0, we describe the decomposition of the modules ExtSj(S/I,S) into irreducible GL-representations. For any inclusion I⊇J of GL-invariant ideals we determine the kernels and cokernels of the induced maps ExtSj(S/I,S)⟶ExtSj(S/J,S). As a consequence of our work, we give a formula for the regularity of the powers and symbolic powers of generic determinantal ideals, and in particular we determine which powers have a linear minimal free resolution. As another consequence, we characterize the GL-invariant ideals I⊆S for which the induced maps ExtSj(S/I,S)⟶HIj(S) are injective. In a different direction we verify that Kodaira vanishing, as described in work of Bhatt–Blickle–Lyubeznik–Singh–Zhang, holds for determinantal thickenings.

Dynamic characterizations of quasi-isometry, and applications to cohomology

Abstract: We build a bridge between geometric group theory and topological dynamical systems by establishing a dictionary between coarse equivalence and continuous orbit equivalence. As an application, we show that group homology and cohomology in a class of coefficients, including all induced and coinduced modules, are coarse invariants. We deduce that being of type FPn (over arbitrary rings) is a coarse invariant, and that being a (Poincare) duality group over a ring is a coarse invariant among all groups which have finite cohomological dimension over that ring. Our results also imply that every coarse self-embedding of a Poincare duality group must be a coarse equivalence. These results were only known under suitable finiteness assumptions, and our work shows that they hold in full generality.

The primitive idempotents and weight distributions of irreducible constacyclic codes

Abstract: Let $${\mathbb {F}}_q$$ be a finite field with q elements such that $$l^v||(q^t-1)$$ and $$\gcd (l,q(q-1))=1$$ , where l, t are primes and v is a positive integer. In this paper, we give all primitive idempotents in a ring $$\mathbb F_q[x]/\langle x^{l^m}-a\rangle $$ for $$a\in {\mathbb {F}}_q^*$$ . Specially for $$t=2$$ , we give the weight distributions of all irreducible constacyclic codes and their dual codes of length $$l^m$$ over $${\mathbb {F}}_q$$ .

On the Structure of Double Complexes.

Abstract: We study consequences and applications of the folklore statement that every double complex over a field decomposes into so-called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences easy to understand. We describe a notion of `universal' quasi-isomorphism, investigate the behaviour of the decomposition under tensor product and compute the Grothendieck ring of the category of bounded double complexes over a field with finite cohomologies up to such quasi-isomorphism (and some variants). Applying the theory to the double complexes of smooth complex valued forms on compact complex manifolds, we obtain a Poincare duality for higher pages of the Frolicher spectral sequence, construct a functorial three-space decomposition of the middle cohomology, give an example of a map between compact complex manifolds which does not respect the Hodge filtration strictly, compute the Bott-Chern and Aeppli cohomology for Calabi-Eckmann manifolds, introduce new numerical bimeromorphic invariants, show that the non-Kahlerness degrees are not bimeromorphic invariants in dimensions higher than three and that the $\partial\overline{\partial}$-lemma and some related properties are bimeromorphic invariants if, and only if, they are stable under restriction to complex submanifolds.

The Grothendieck Ring of Varieties

Abstract: In this chapter, we define the Grothendieck ring of varieties over an arbitrary base scheme. This is a ring of virtual varieties up to cut-and-paste relations; it takes a central place in the theory of motivic integration, because (after a suitable localization and/or completion) it serves as the ring where motivic integrals take their values. After the basic definitions in section 1, we define the notion of motivic measures, which are ring morphisms from the Grothendieck ring to other rings with a more explicit structure. Motivic measures are fundamental both for the understanding of Grothendieck ring itself and for extracting geometric information from its elements. Among the motivic measures, we develop in sections 3 and 5 the cohomological and motivic realizations. In sections 5 and 6, we study the main structure theorems for the Grothendieck ring over a field of characteristic zero: the theorems of Bittner and Larsen-Lunts. Bittner’s theorem gives a presentation of the Grothendieck ring in terms of smooth projective varieties and blow-up relations, which is quite useful to construct motivic measures. The theorem of Larsen and Lunts relates equalities in the Grothendieck ring to the notion of stable birational equivalence. In section 4 we discuss a process of dimensional completion for the Grothendieck ring of varieties.

A geometric approach to 1-singular Gelfand–Tsetlin gln-modules

Abstract: This paper is devoted to an elementary new construction of 1-singular Gelfand–Tsetlin modules using complex geometry. We introduce a universal ring D v together with the vector space S = S ( D v ) with basis B v = B ( D v ) consisted of some local distributions such that S is a natural D v -module. For any homomorphism of rings U ( h ) → D v , where h is a Lie algebra, it follows that S is also an h -module. We observe that the homomorphism of rings constructed in [8] is a homomorphism of type U ( gl n ( C ) ) → D v . Using this observation we obtain a construction of the universal 1-singular Gelfand–Tsetlin gl n ( C ) -module from [7] .