Showing papers on "Ring (mathematics) published in 2021"
TL;DR: In this article, the monotonicity properties of the ratio between generalized elliptic integral of the first kind and its approximation for generalized Grotzsch ring function were studied, and the convexity properties of their difference were shown.
Abstract: In this paper, we present the monotonicity properties of the ratio between generalized elliptic integral of the first kind $${\mathcal {K}}_a(r)$$
and its approximation $$\log [1+2/(ar')]$$
, and also the convexity (concavity) of their difference for $$a\in (0,1/2]$$
. As an application, we give new bounds for generalized Grotzsch ring function $$\mu _a(r)$$
and a upper bound for $${\mathcal {K}}_a(r)$$
.
131 citations
TL;DR: In this paper, the authors investigated the skin effect in a ring resonator array which can be mapped into the square root of a SSH model with non-Hermitian asymmetric coupling.
Abstract: We investigate the topological skin effect in a ring resonator array which can be mapped into the square root of a Su-Schrieffer-Heeger (SSH) model with non-Hermitian asymmetric coupling. The asymmetric coupling is realized by integrating the same amount of gain and loss into the two half perimeters of linking rings that effectively couple two adjacent site rings. Such a square-root topological insulator inherits the properties from its parent Hamiltonian, which has the same phase transition points and exhibits non-Bloch features as well. We show the band closing points for open chain are different from that of periodic chain as a result of the skin effect. Moreover, the square-root insulator supports multiple topological edge modes as the number of band gaps is doubled compared to the original Hamiltonian. The full-wave simulations agree well with the theoretical analyses based on a tight-binding model. The study provides a promising approach to investigate the skin effect by utilizing ring resonators and may find potential applications in light trapping, lasers, and filters.
44 citations
20 Jul 2021
TL;DR: In this article, the authors provide a pedagogical introduction to the theoretical approaches in describing the dynamic modulated ring resonator system and then review experimental methods in building such a system.
Abstract: The concept of synthetic dimensions in photonics has attracted rapidly growing interest in the past few years. Among a variety of photonic systems, the ring resonator system under dynamic modulation has been investigated in depth both in theory and experiment and has proven to be a powerful way to build synthetic frequency dimensions. In this Tutorial, we start with a pedagogical introduction to the theoretical approaches in describing the dynamically modulated ring resonator system and then review experimental methods in building such a system. Moreover, we discuss important physical phenomena in synthetic dimensions, including nontrivial topological physics. This Tutorial provides a pathway toward studying the dynamically modulated ring resonator system and understanding synthetic dimensions in photonics and discusses future prospects for both fundamental research and practical applications using synthetic dimensions.
43 citations
TL;DR: In this paper, the generalized Grotzsch ring function was studied in the theory of the Ramanujan generalized modular equation and new inequalities were presented for the generalized ring function with respect to the generalized generalized generalized Grozsch ring functions.
Abstract: In this paper, we deal with the generalized Grotzsch ring function $$\mu _a(r)$$
for $$r\in (0,1)$$
in the theory of the Ramanujan generalized modular equation and present new inequalities for $$\mu _a(r)$$
.
43 citations
TL;DR: The main objective of this paper is to introduce some algebraic properties of finite linear Diophantine fuzzy subsets of group, ring and field, and to establish the homomorphic images and preimages of the emerged linear Diophile fuzzy algebraic structures.
Abstract: The main objective of this paper is to introduce some algebraic properties of finite linear Diophantine fuzzy subsets of group, ring and field. Relatedly, we define the concepts of linear Diophantine fuzzy subgroup and normal subgroup of a group, linear Diophantine fuzzy subring and ideal of a ring, and linear Diophantine fuzzy subfield of a field. We investigate their basic properties, relations and characterizations in detail. Furthermore, we establish the homomorphic images and preimages of the emerged linear Diophantine fuzzy algebraic structures. Finally, we describe linear Diophantine fuzzy code and investigate the relationships between this code and some linear Diophantine fuzzy algebraic structures.
35 citations
TL;DR: The main idea of RTGBO is to simulate the behaviour of players and rules of the ring toss game in the design of the proposed algorithm, which indicates the high exploitation ability ofRTGBO in solving optimization problems.
Abstract: There are many optimization problems in different scientific disciplines that should be solved and optimized using appropriate techniques. Population-based optimization algorithms are one of the most widely used techniques to solve optimization problems. This paper is focused on presenting a new population-based optimization approach called Ring Toss Game-Based Optimization (RTGBO) algorithm. The main idea of RTGBO is to simulate the behaviour of players and rules of the ring toss game in the design of the proposed algorithm. The main feature of the proposed RTGBO algorithm is the lack of control parameters. Steps of implementing RTGBO are described in detail and the proposed algorithm is mathematically modeled. The ability of RTGBO to solve optimization problems is evaluated on a set of twenty-three standard objective functions. These functions are selected from three different groups including unimodal, high-dimensional multimodal, and fixed-dimensional multimodal. The performance of RTGBO is also compared with eight other well-known optimization algorithms including Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Gravitational Search Algorithm (GSA), Teaching Learning-Based Optimization (TLBO), Gray Wolf Optimizer (GWO), Emperor Penguin Optimizer (EPO), Hide Objects Game Optimization (HOGO), and Shell Game Optimization (SGO). The results of optimization of objective functions of unimodal type indicate the high exploitation ability of RTGBO in solving optimization problems. On the other hand, the results of optimizing the multi-model type objective functions indicate the acceptable exploration ability of RTGBO. The results also confirm the superiority of the proposed RTGBO algorithm over mentioned optimization techniques.
30 citations
TL;DR: In this paper, a pedagogical introduction to the theoretical approaches in describing the dynamically modulated ring resonator system, and then review experimental methods in building such a system is provided.
Abstract: The concept of synthetic dimensions in photonics has attracted rapidly growing interest in the past few years. Among a variety of photonic systems, the ring resonator system under dynamic modulation has been investigated in depth both in theory and experiment, and has proven to be a powerful way to build synthetic frequency dimensions. In this tutorial, we start with a pedagogical introduction to the theoretical approaches in describing the dynamically modulated ring resonator system, and then review experimental methods in building such a system. Moreover, we discuss important physical phenomena in synthetic dimensions, including nontrivial topological physics. Our tutorial provides a pathway towards studying the dynamically modulated ring resonator system, understanding synthetic dimensions in photonics, and discusses future prospects for both fundamental research and practical applications using synthetic dimensions.
30 citations
TL;DR: In this paper, it was shown that any stationary, axisymmetric, and asymptotically flat spacetime in $1+3$ dimensions with an ergoregion must have at least one light ring outside the ergorescion.
Abstract: We present a novel theorem regarding light rings in a stationary spacetime with an ergoregion. We prove that any stationary, axisymmetric, and asymptotically flat spacetime in $1+3$ dimensions with an ergoregion must have at least one light ring outside the ergoregion. A possible extension of the proof for asymptotically de Sitter and anti--de Sitter spherically symmetric black holes is also discussed.
29 citations
TL;DR: Zhang et al. as mentioned in this paper proposed a Bayesian low rank tensor ring completion method for image recovery by automatically learning the low-rank structure of data, where sparsity-inducing hierarchical prior is placed over horizontal and frontal slices of core factors.
Abstract: Low rank tensor ring based data recovery can recover missing image entries in signal acquisition and transformation. The recently proposed tensor ring (TR) based completion algorithms generally solve the low rank optimization problem by alternating least squares method with predefined ranks, which may easily lead to overfitting when the unknown ranks are set too large and only a few measurements are available. In this article, we present a Bayesian low rank tensor ring completion method for image recovery by automatically learning the low-rank structure of data. A multiplicative interaction model is developed for low rank tensor ring approximation, where sparsity-inducing hierarchical prior is placed over horizontal and frontal slices of core factors. Compared with most of the existing methods, the proposed one is free of parameter-tuning, and the TR ranks can be obtained by Bayesian inference. Numerical experiments, including synthetic data, real-world color images and YaleFace dataset, show that the proposed method outperforms state-of-the-art ones, especially in terms of recovery accuracy.
28 citations
TL;DR: In this article, a notion of universal quasi-isomorphism was proposed and the Grothendieck ring of bounded double complexes over a field with finite cohomologies up to such quasi-Isomorphism.
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.
28 citations
01 Jun 2021
TL;DR: In this article, the authors investigate different radicals (Wedderburn radical, lower nil radical, Levitzky radical, upper nil radical) of the noncommutative rings known as skew Poincare-Birkhoff-Witt extensions and prove that the Kothe's conjecture holds for these extensions.
Abstract: The aim of this paper is to investigate different radicals (Wedderburn radical, lower nil radical, Levitzky radical, upper nil radical, the set of all nilpotent elements, the sum of all nil left ideals) of the noncommutative rings known as skew Poincare–Birkhoff–Witt extensions. We characterize minimal prime ideals of these rings and prove that the Kothe’s conjecture holds for these extensions. Finally, we establish the transfer of several ring-theoretical properties (reduced, symmetric, reversible, 2-primal) from the coefficients ring of a skew PBW extension to the extension itself.
TL;DR: In this paper, the Eisenstein part of the p-adic Hecke algebra for Γ 0 (N ) is studied and it is shown that it is a local complete intersection and isomorphic to a pseudodeformation ring.
16 Aug 2021
TL;DR: This work introduces a novel generic ring signature construction, called DualRing, which can be built from several canonical identification schemes (such as Schnorr identification), and achieves the shortest lattice-based ring signature, named DualRing-LB, when the ring size is between 4 and 2000.
Abstract: We introduce a novel generic ring signature construction, called DualRing, which can be built from several canonical identification schemes (such as Schnorr identification). DualRing differs from the classical ring signatures by its formation of two rings: a ring of commitments and a ring of challenges. It has a structural difference from the common ring signature approaches based on accumulators or zero-knowledge proofs of the signer index. Comparatively, DualRing has a number of unique advantages.
09 Jun 2021
TL;DR: In this article, the authors propose a ring indexing scheme that supports worst-case optimal joins over graphs within compact space, where each triple is indexed as a set of cyclic bidirectional strings of length 3.
Abstract: We present an indexing scheme that supports worst-case optimal (wco) joins over graphs within compact space. Supporting all possible wco joins using conventional data structures - based on B(+)-Trees, tries, etc. - requires 6 index orders in the case of graphs represented as triples. We rather propose a form of index, which we call a ring, that indexes each triple as a set of cyclic bidirectional strings of length 3. Rather than maintaining 6 orderings, we can use one ring to index them all. This ring replaces the graph and uses only sublinear extra space on top of the graph; in order words, the ring supports worst-case optimal graph joins in almost no space beyond storing the graph itself. We perform experiments using our representation to index a large graph (Wikidata) in memory, over which wco join algorithms are implemented. Our experiments show that the ring offers the best overall performance for query times while using only a small fraction of the space when compared with several state-of-the-art approaches.
TL;DR: In this paper, the design of an antenna based on nested square shaped ring fractal geometry with circular ring elements for multi-band wireless applications has been presented, which achieves the enhanced bandwidth greater than 3 GHz at three resonant frequency bands and exhibits additional frequency band at 2.4 GHz.
Abstract: This manuscript presents the design of an antenna based on nested square shaped ring fractal geometry with circular ring elements for multi-band wireless applications. The impedance bandwidth and reflection coefficient of the antenna are improved with the design of different iterations from the 0th to 2nd. The performance parameters of the antenna like reflection coefficient, VSWR, bandwidth, bandwidth ratio, and current density are improved in the final iteration. It also achieves the enhanced bandwidth greater than 3 GHz at three resonant frequency bands and exhibits additional frequency band at 2.4 GHz. Likewise, the frequency band of designed fractal antenna shifts towards the lower end and helps in achieving the miniaturization of antenna. The proposed fractal antenna is designed and fabricated on a low-cost FR4 glass epoxy substrate and investigated using HFSS software. The proposed antenna is optimized for generating different parameters, and the last geometry is fabricated and tested. Further, these parameters are compared with the experimental results and found in good agreement with each other. Due to the multi-band behaviour and improved bandwidth, the proposed fractal antenna can be considered as a good candidate for several wireless standards.
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23 Jan 2021TL;DR: In this paper, a cycle map from geometrized bundles of modules over a ring of integers in number fields to differential algebraic K-theory was constructed and some non-trivial consequences were derived by independent means.
Abstract: We develop differential algebraic K-theory for rings of integers in number fields and we construct a cycle map from geometrized bundles of modules over such a ring to the differential algebraic K-theory. We also treat some of the foundational aspects of differential cohomology, including differential function spectra and the differential Becker-Gottlieb transfer. We then state a transfer index conjecture about the equality of the Becker-Gottlieb transfer and the analytic transfer defined by Lott. In support of this conjecture, we derive some non-trivial consequences which are provable by independent means.
01 Mar 2021
TL;DR: In this article, the notion of McCoy ring over non-commutative rings of polynomial type known as skew Poincare-Birkhoff-Witt extensions was studied.
Abstract: In this paper, we study the notion of McCoy ring over the class of non-commutative rings of polynomial type known as skew Poincare–Birkhoff–Witt extensions. As a consequence, we generalize several results about this notion considered in the literature for commutative rings and Ore extensions.
TL;DR: In this article, bases for the morphism spaces of the Frobenius Heisenberg categories associated to a symmetric graded FH algebra were described and proved using a categorical comultiplication and generalized cyclotomic quotients of the category.
TL;DR: It is observed that some quantum codes constructed from skew cyclic codes over the ring R are MDS quantum codes, and they are observed to contain its dual.
TL;DR: In this article, a detailed proof of Gabber's result on lifting of quasi-excellent rings is given, and it is shown that an ideal-adic completion of an excellent ring is also an excellent (resp., quasiexcellent) ring.
Abstract: We give a detailed proof of a result of Gabber (unpublished) on the lifting problem of quasi-excellent rings, extending the previous work of Nishimura and Nishimura. As a corollary, we establish that an ideal-adic completion of an excellent (resp., quasi-excellent) ring is excellent (resp., quasi-excellent).
01 Oct 2021
TL;DR: In this paper, the Poisson structures on the finitary incidence algebra F I (P, R ) of an arbitrary poset P over a commutative unital ring R are described.
TL;DR: Two new constructions of codes over the ring F p + u F p , u 2 = u by using down-sets are given, and the Lee weight distributions of two classes of codes are computed when the down- sets are generated by a single maximal element.
TL;DR: In this paper, the authors used a Serre spectral sequence to determine the ring structure of H∗(BPU n; ℤ) up to degree 10 in BPUn.
Abstract: Let BPUn be the classifying space of PUn, the projective unitary group of order n, for n > 1. We use a Serre spectral sequence to determine the ring structure of H∗(BPU n; ℤ) up to degree 10, as we...
06 Sep 2021
TL;DR: This paper gives an algebraic complete axiomatisation of ZX-calculus in the sense that there are only ring operations involved for phases, without any need of trigonometry functions such as sin and cos, in contrast to previous universallycomplete axiomatisations.
Abstract: ZX-calculus is a graphical language for quantum computing which is complete in the sense that calculation in matrices can be done in a purely diagrammatic way. However, all previous universally complete axiomatisations of ZX-calculus have included at least one rule involving trigonometric functions such as sin and cos which makes it difficult for application purpose. In this paper we give an algebraic complete axiomatisation of ZX-calculus instead such that there are only ring operations involved for phases. With this algebraic axiomatisation of ZX-calculus, we are able to establish for the first time a simple translation of diagrams from another graphical language called ZH-calculus and to derive all the ZX-translated rules of ZH-calculus. As a consequence, we have a great benefit that all techniques obtained in ZH-calculus can be transplanted to ZX-calculus, which can't be obtained by just using the completeness of ZX-calculus.
TL;DR: In this article, the authors introduce two new bases of polynomials and study their relations to known bases, the first is the quasi-Lascoux basis, which is simultaneously both a -analogues mirror the relationships among their cohomological counterparts.
Abstract: We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a -analogues mirror the relationships among their cohomological counterparts. We make several “alternating sum” conjectures that are suggestive of Euler characteristic calculations.
TL;DR: First, the existence and uniqueness of solutions are proved using the Banach contraction principle and Krasnoselskii's fixed point theorem and different kinds of Ulam-type stability for the proposed problem are investigated.
Abstract: In this paper, we study a nonlinear fractional boundary value problem on a particular metric graph, namely, a circular ring with an attached edge. First, we prove existence and uniqueness of solutions using the Banach contraction principle and Krasnoselskii's fixed point theorem. Further, we investigate different kinds of Ulam-type stability for the proposed problem. Finally, an example is given in order to demonstrate the application of the obtained theoretical results.
TL;DR: In this paper, the authors studied the geometry of units and ideals of cyclotomic rings and derived an algorithm to find a mildly short vector in any given cyclotome ideal lattice in quantum polynomial time, under some plausible number-theoretic assumptions.
Abstract: In this article, we study the geometry of units and ideals of cyclotomic rings and derive an algorithm to find a mildly short vector in any given cyclotomic ideal lattice in quantum polynomial time, under some plausible number-theoretic assumptions. More precisely, given an ideal lattice of the cyclotomic ring of conductor m, the algorithm finds an approximation of the shortest vector by a factor exp (O(√ m)). This result exposes an unexpected hardness gap between these structured lattices and general lattices: The best known polynomial time generic lattice algorithms can only reach an approximation factor exp (O(m)). Following a recent series of attacks, these results call into question the hardness of various problems over structured lattices, such as Ideal-SVP and Ring-LWE, upon which relies the security of a number of cryptographic schemes. NOTE. This article is an extended version of a conference paper [11]. The results are generalized to arbitrary cyclotomic fields. In particular, we also extend some results of Reference [10] to arbitrary cyclotomic fields. In addition, we prove the numerical stability of the method of Reference [10]. These extended results appeared in the Ph.D. dissertation of the third author [46].
TL;DR: In this article, the generalized n-strong Drazin inverse and pseudo-n-strong inverse for arbitrary n∈N were introduced, where n is the number of inverses in a ring.
Abstract: We introduce new classes of generalized Drazin inverses in a ring, which are called the generalized n-strong Drazin inverse and pseudo n-strong Drazin inverse for arbitrary n∈N. Some basic properti...
TL;DR: In this article, a numerical framework combining several existing techniques to study mechanics of elastic networks consisting of thin strips is proposed, where each strip is modeled as a Kirchhoff rod, and the entire strip network is formulated as a two-point boundary value problem (BVP).
Abstract: In this study, we explore the mechanics of a bigon and a bigon ring from a combination of experiments and numerical simulations. A bigon is a simple elastic network consisting of two initially straight strips that are deformed to intersect with each other through a fixed intersection angle at each end. A bigon ring is a novel multistable structure composed of a series of bigons arranged to form a loop. We find that a bigon ring usually contains several families of stable states and one of them is a multiply-covered loop, which is similar to the folding behavior of a bandsaw blade. To model bigons and bigon rings, we propose a numerical framework combining several existing techniques to study mechanics of elastic networks consisting of thin strips. Each strip is modeled as a Kirchhoff rod, and the entire strip network is formulated as a two-point boundary value problem (BVP) that can be solved by a general-purpose BVP solver. Together with numerical continuation, we apply the numerical framework to study static equilibria and bifurcations of the bigons and bigon rings. Both numerical and experimental results show that the intersection angle and the aspect ratio of the strip’s cross section contribute to the bistability of a bigon and the multistability of a bigon ring; the latter also depends on the number of bigon cells in the ring. The numerical results further reveal interesting connections among various stable states in a bigon ring. Our numerical framework can be applied to general elastic rod networks that may contain flexible joints, naturally curved strips of different lengths, etc. The folding and multistable behaviors of a bigon ring may inspire the design of novel deployable and morphable structures.