Showing papers on "Ring (mathematics) published in 2017"

Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas

Abstract: A new topological gapless ring of nondegenerate states, characterized by both a quantized Chern number and a quantized Berry phase, is predicted to occur in 3D dissipative systems.

Canonical bases for cluster algebras

Abstract: In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalisations of holomorphic discs). Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock-Goncharov dual basis conjecture, [FG06], Conjecture 4.3. In particular, under suitable hypotheses, for each Y the partial compactification of an affine cluster variety U given by allowing some frozen variables to vanish, we obtain canonical bases for H0(Y,OY ) extending to a basis of H0(U,OU ). Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell U in the basic affine space Y we obtain a canonical basis of each irreducible representation of SLr, parameterized by a set which each choice of seed identifies with integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation theoretic considerations. Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type.

Light-Ring Stability for Ultracompact Objects.

Abstract: We prove the following theorem: axisymmetric, stationary solutions of the Einstein field equations formed from classical gravitational collapse of matter obeying the null energy condition, that are everywhere smooth and ultracompact (i.e., they have a light ring) must have at least two light rings, and one of them is stable. It has been argued that stable light rings generally lead to nonlinear spacetime instabilities. Our result implies that smooth, physically and dynamically reasonable ultracompact objects are not viable as observational alternatives to black holes whenever these instabilities occur on astrophysically short time scales. The proof of the theorem has two parts: (i) We show that light rings always come in pairs, one being a saddle point and the other a local extremum of an effective potential. This result follows from a topological argument based on the Brouwer degree of a continuous map, with no assumptions on the spacetime dynamics, and, hence, it is applicable to any metric gravity theory where photons follow null geodesics. (ii) Assuming Einstein's equations, we show that the extremum is a local minimum of the potential (i.e., a stable light ring) if the energy-momentum tensor satisfies the null energy condition.

Pseudorandomness of ring-LWE for any ring and modulus

Abstract: We give a polynomial-time quantum reduction from worst-case (ideal) lattice problems directly to decision (Ring-)LWE. This extends to decision all the worst-case hardness results that were previously known for the search version, for the same or even better parameters and with no algebraic restrictions on the modulus or number field. Indeed, our reduction is the first that works for decision Ring-LWE with any number field and any modulus.

Pseudorandomness of Ring-LWE for Any Ring and Modulus.

Abstract: We give a polynomial-time quantum reduction from worst-case (ideal) lattice problems directly to decision (Ring-)LWE. This extends to decision all the worst-case hardness results that were previously known for the search version, for the same or even better parameters and with no algebraic restrictions on the modulus or number field. Indeed, our reduction is the first that works for decision Ring-LWE with any number field and any modulus.

Efficient Low Rank Tensor Ring Completion

Abstract: Using the matrix product state (MPS) representation of the recently proposed tensor ring decompositions, in this paper we propose a tensor completion algorithm, which is an alternating minimization algorithm that alternates over the factors in the MPS representation. This development is motivated in part by the success of matrix completion algorithms that alternate over the (low-rank) factors. In this paper, we propose a spectral initialization for the tensor ring completion algorithm and analyze the computational complexity of the proposed algorithm. We numerically compare it with existing methods that employ a low rank tensor train approximation for data completion and show that our method outperforms the existing ones for a variety of real computer vision settings, and thus demonstrate the improved expressive power of tensor ring as compared to tensor train.

Self-dual Codes

Abstract: In this chapter, we describe self-dual codes over Frobenius rings. We give constructions of self-dual codes over any Frobenius ring. We describe connections to unimodular lattices, binary self-dual codes and to designs. We also describe linear complementary dual codes and make a new definition of a broad generalization encompassing both self-dual and linear complementary codes.

Refinement Monoids, Equidecomposability Types, and Boolean Inverse Semigroups

Abstract: For an action of a group G on a set Ω, preserving a ring B of subsets of Ω, the commutative monoid freely generated by elements [X], for X ∈ B, subjected to the relations [∅] = 0, [gX] = [X] (for g ∈ G), and [X Y ] = [X] + [Y ] (where denotes disjoint union), is called the monoid of equidecomposability types of elements of B, with respect to G, and denoted by Z + B/ /G. It is well known that Z + B/ /G is a conical refinement monoid. We observe, as an easy consequence of known results, that every countable conical refinement monoid appears as Z + B/ /G, and we develop the underlying algebraic theory, discussing in detail the quotients of refinement monoids by special sorts of congruences called V-congruences. Having in mind representation problems in nonstable K-theory of rings and operator algebras, we are naturally led to type monoids of Boolean inverse semigroups. Observing that those monoids are identical to monoids of equi-decomposability types, and formally similar to those appearing in nonstable K-theory of von Neumann regular rings, we investigate various similarities and differences between those theories. In the process, we prove that Boolean inverse semigroups form a congruence -permutable variety in the sense of universal algebra. We deduce from this that they encode a large number of embedding problems of (not necessarily Boolean) inverse semigroups into involutary rings and C*-algebras.

Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem

Abstract: We extend the so-called 'single ring theorem' (Feinberg and Zee 1997 Nucl. Phys. B 504 579), also known as the Haagerup–Larsen theorem (Haagerup and Larsen 2000 J. Funct. Anal. 176 331). We do this by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non-hermitian matrix X, being the spectral density weighted by the squared eigenvalue condition number, is given by a simple formula involving only the radial spectral cumulative distribution function of X. We show that this object allows the calculation of the conditional expectation of the squared eigenvalue condition number. We give examples and provide a cross-check of the analytic prediction by the large scale numerics.

Recollements of derived categories III: finitistic dimensions

Abstract: In this paper, we study homological dimensions of algebras linked by recollements of derived module categories, and establish a series of new upper bounds and relationships among their finitistic or global dimensions. This is closely related to a longstanding conjecture, the finitistic dimension conjecture, in representation theory and homological algebra. Further, we apply our results to a series of situations of particular interest: exact contexts, ring extensions, trivial extensions, pullbacks of rings, and algebras induced from Auslander-Reiten sequences. In particular, we not only extend and amplify Happel’s reduction techniques for finitistic dimenson conjecture to more general contexts, but also generalise some recent results in the literature.

Exotic matter on singular divisors in F-theory

Abstract: We analyze exotic matter representations that arise on singular seven-brane configurations in F-theory. We develop a general framework for analyzing such representations, and work out explicit descriptions for models with matter in the 2-index and 3-index symmetric representations of SU(N) and SU(2) respectively, associated with double and triple point singularities in the seven-brane locus. These matter representations are associated with Weierstrass models whose discriminants vanish to high order thanks to nontrivial cancellations possible only in the presence of a non-UFD algebraic structure. This structure can be described using the normalization of the ring of intrinsic local functions on a singular divisor. We consider the connection between geometric constraints on singular curves and corresponding constraints on the low-energy spectrum of 6D theories, identifying some new examples of apparent “swampland” theories that cannot be realized in F-theory but have no apparent low-energy inconsistency.

Duality and deformations of stable Grothendieck polynomials

Abstract: Stable Grothendieck polynomials can be viewed as a K-theory analog of Schur polynomials. We extend stable Grothendieck polynomials to a two-parameter version, which we call canonical stable Grothendieck functions. These functions have the same structure constants (with scaling) as stable Grothendieck polynomials and (composing with parameter switching) are self-dual under the standard involutive ring automorphism. We study various properties of these functions, including combinatorial formulas, Schur expansions, Jacobi---Trudi-type identities, and associated Fomin---Greene operators.

Strongly 2-nil-clean rings

Abstract: A ring R is strongly 2-nil-clean if every element in R is the sum of two idempotents and a nilpotent that commute. Fundamental properties of such rings are obtained. We prove that a ring R is strongly 2-nil-clean if and only if for all a ∈ R, a − a3 ∈ R is nilpotent, if and only if for all a ∈ R, a2 ∈ R is strongly nil-clean, if and only if every element in R is the sum of a tripotent and a nilpotent that commute. Furthermore, we prove that a ring R is strongly 2-nil-clean if and only if R/J(R) is tripotent and J(R) is nil, if and only if R≅R1,R2 or R1 × R2, where R1/J(R1) is a Boolean ring and J(R1) is nil; R2/J(R2) is a Yaqub ring and J(R2) is nil. Strongly 2-nil-clean group algebras are investigated as well.

Recent development of ring rolling theory and technique

Abstract: Rings are basic mechanical components. Ring rolling is a well-known advanced plastic forming technique to manufacture high-performance seamless rings. Since the 21st century, the fast development of global manufacturing has raised an urgent request to high-performance rings, thus, comprehensive and in-depth studies on ring rolling technique were carried out. Based on long-term investigations and attentions on ring rolling, important developments of ring rolling theory and technique in recent 15 years were simply summarised, including the traditional radial ring rolling and radial-axial ring rolling, as well as the novel combined ring rolling. It is expected to give a valuable reference to the research and application of ring rolling.

Continuous functions on the plane regular after one blowing-up

Abstract: We study rational functions admitting a continuous extension to the real affine space. First of all, we focus on the regularity of such functions exhibiting some nice properties of their partial derivatives. Afterwards, since these functions correspond to rational functions which become regular after some blowings-up, we work on the plane where it suffices to blow-up points and then we can count the number of stages of blowings-up necessary. In the latest parts of the paper, we investigate the ring of rational continuous functions on the plane regular after one stage of blowings-up. In particular, we prove a Positivstellensatz without denominator in this ring.

The Structure of Chariklo’s Rings from Stellar Occultations

Abstract: Two narrow and dense rings (called C1R and C2R) were discovered around the Centaur object (10199) Chariklo during a stellar occultation observed on 2013 June 3. Following this discovery, we planned observations of several occultations by Chariklo's system in order to better characterize the physical properties of the ring and main body. Here, we use 12 successful occulations by Chariklo observed between 2014 and 2016. They provide ring profiles (physical width, opacity, edge structure) and constraints on the radii and pole position. Our new observations are currently consistent with the circular ring solution and pole position, to within the ±3.3 km formal uncertainty for the ring radii derived by Braga-Ribas et al. The six resolved C1R profiles reveal significant width variations from ∼5 to 7.5 km. The width of the fainter ring C2R is less constrained, and may vary between 0.1 and 1 km. The inner and outer edges of C1R are consistent with infinitely sharp boundaries, with typical upper limits of one kilometer for the transition zone between the ring and empty space. No constraint on the sharpness of C2R's edges is available. A 1σ upper limit of ∼20 m is derived for the equivalent width of narrow (physical width <4 km) rings up to distances of 12,000 km, counted in the ring plane.

Efficient Ring Signatures in the Standard Model

Abstract: A ring signature scheme allows one party to sign messages on behalf of an arbitrary set of users, called the ring. The anonymity of the scheme guarantees that the signature does not reveal which member of the ring signed the message. The ring of users can be selected “on-the-fly” by the signer and no central coordination is required. Ring signatures have made their way into practice in the area of privacy-enhancing technologies and they build the core of several cryptocurrencies. Despite their popularity, almost all ring signature schemes are either secure in the random oracle model or in the common reference string model. The only candidate instantiations in the plain model are either impractical or not fully functional.

Models for homotopy categories of injectives and Gorenstein injectives

Abstract: A natural generalization of locally noetherian and locally coherent categories leads us to define locally type FP∞ categories. They include not just all categories of modules over a ring, but also ...

Coulomb Branch Operators and Mirror Symmetry in Three Dimensions.

Abstract: We develop new techniques for computing exact correlation functions of a class of local operators, including certain monopole operators, in three-dimensional $\mathcal{N} = 4$ abelian gauge theories that have superconformal infrared limits. These operators are position-dependent linear combinations of Coulomb branch operators. They form a one-dimensional topological sector that encodes a deformation quantization of the Coulomb branch chiral ring, and their correlation functions completely fix the ($n\leq 3$)-point functions of all half-BPS Coulomb branch operators. Using these results, we provide new derivations of the conformal dimension of half-BPS monopole operators as well as new and detailed tests of mirror symmetry. Our main approach involves supersymmetric localization on a hemisphere $HS^3$ with half-BPS boundary conditions, where operator insertions within the hemisphere are represented by certain shift operators acting on the $HS^3$ wavefunction. By gluing a pair of such wavefunctions, we obtain correlators on $S^3$ with an arbitrary number of operator insertions. Finally, we show that our results can be recovered by dimensionally reducing the Schur index of 4D $\mathcal{N} = 2$ theories decorated by BPS 't Hooft-Wilson loops.

Reconstruction of graded groupoids from graded Steinberg algebras

Abstract: We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies C∗-isomorphism of C∗-algebras for graphs E and F in which every cycle has an exit.