Algebraic Structure Research Articles

A Hybrid Technique for Generation of Highly Nonlinear Component Based on Elliptic Curves and Algebraic Group Structure

A Hybrid Technique for Generation of Highly Nonlinear Component Based on Elliptic Curves and Algebraic Group Structure

Robust finite-temperature many-body scarring on a quantum computer

Mechanisms for suppressing thermalization in disorder-free many-body systems, such as Hilbert space fragmentation and quantum many-body scars, have recently attracted much interest in foundations of quantum statistical physics and potential quantum information processing applications. However, their sensitivity to realistic effects such as finite temperature remains largely unexplored. Here, we have utilized IBM's Kolkata quantum processor to demonstrate an unexpected robustness of quantum many-body scars at finite temperatures when the system is prepared in a thermal Gibbs ensemble. We identify such robustness in the PXP model, which describes quantum many-body scars in experimental systems of Rydberg atom arrays and ultracold atoms in tilted Bose-Hubbard optical lattices. By contrast, other theoretical models which host exact quantum many-body scars are found to lack such robustness and their scarring properties quickly decay with temperature. Our study sheds light on the important differences between scarred models in terms of their algebraic structures, which impacts their resilience to finite temperature. Published by the American Physical Society 2024

Object-Oriented Fixpoint Programming with Datalog

Modern usages of Datalog exceed its original design purpose in scale and complexity. In particular, Datalog lacks abstractions for code organization and reuse, making programs hard to maintain. Is it possible to exploit abstractions and design patterns from object-oriented programming (OOP) while retaining a Datalog-like fixpoint semantics? To answer this question, we design a new OOP language called OODL with common OOP features: dynamic object allocation, object identity, dynamic dispatch, and mutation. However, OODL has a Datalog-like fixpoint semantics, such that recursive computations iterate until their result becomes stable. We develop two semantics for OODL: a fixpoint interpreter and a compiler that translates OODL to Datalog. Although the side effects found in OOP (object allocation and mutation) conflict with Datalog's fixpoint semantics, we can mostly resolve these incompatibilities through extensions of OODL. Within fixpoint computations, we employ immutable algebraic data structures (e.g. case classes in Scala), rather than relying on object allocation, and we introduce monotonically mutable data types (mono types) to enable a relaxed form of mutation. Our performance evaluation shows that the interpreter fails to solve fixpoint problems efficiently, whereas the compiled code exploits Datalog's semi-naïve evaluation.

Quantum Codes as an Application of Constacyclic Codes

The main focus of this paper is to analyze the algebraic structure of constacyclic codes over the ring R=Fp+w1Fp+w2Fp+w22Fp+w1w2Fp+w1w22Fp, where w12−α2=0, w1w2=w2w1, w23−β2w2=0, and α,β∈Fp∖{0}, for a prime p. We begin by introducing a Gray map defined over R, which is associated with an invertible matrix. We demonstrate its advantages over the canonical Gray map through some examples. Finally, we create new and improved quantum codes from constacyclic codes over R using Calderbank–Shore–Steane (CSS) construction.

Information Theoretic Evaluation of Raccoon's Side-Channel Leakage

Raccoon is a lattice-based scheme submitted to the NIST 2022 call for additional post-quantum signatures. One of its main selling points is that its design is intrinsically easy to mask against side-channel attacks. So far, Raccoon's physical security guarantees were only stated in the abstract probing model. In this paper, we discuss how these probing security results translate into guarantees in more realistic leakage models. We also highlight that this translation differs from what is usually observed (e.g., in symmetric cryptography), due to the algebraic structure of Raccoon's operations. For this purpose, we perform an in-depth information theoretic evaluation of Raccoon's most innovative part, namely the AddRepNoise function which allows generating its arithmetic shares on-the-fly. Our results are twofold. First, we show that the resulting shares do not enforce a statistical security order (i.e., the need for the side-channel adversary to estimate higher-order moments of the leakage distribution), as usually expected when masking. Second, we observe that the first-order leakage on the (large) random coefficients manipulated by Raccoon cannot be efficiently turned into leakage on the (smaller) coefficients of its long-term secret. Concretely, our information theoretic evaluations for relevant leakage functions also suggest that Raccoon's masked implementations can ensure high security with less shares than suggested by a conservative analysis in the probing model.

Optimised GRS coded cooperation: sliding relay selection design and improved joint decoding

ABSTRACT In this paper, the novel optimised generalised Reed-Solomon coded cooperative (OGRSCC) scheme is developed with a flexible selection of the code length and information dimension. In the OGRSCC scheme, two GRS codes with different dimensions are deployed at the source and relay, respectively, where the information symbols of the relay are dependent on those of the source. At the relay, different symbol selections may contribute to distinct joint codes at the destination. As a result, to achieve a maximum free distance (FD) and the optimised weight distribution of the final code, the global search method is first proposed to obtain the optimal selection pattern. However, as the code grows longer, its complexity rises dramatically. To make the selection more practical, the low-complexity sliding search method is then investigated to obtain the local optimal symbol selection pattern. Monte-Carlo simulated results indicate the OGRSCC scheme with two selection methods demonstrates almost identical performance. Furthermore, based on this special selection method and the unique algebraic structure of GRS code, the improved joint decoding algorithm is first proposed to boost the overall system performance. In addition, the theoretical upper bound is analysed, verifying the effectiveness of the proposed OGRSCC scheme. Numerical simulations reveal that the proposed OGRSCC scheme outperforms its corresponding non-cooperative scheme by a gain of over 3.2 dB and its existing comparable schemes by a margin of approximately more than 4 dB at high signal-to-noise ratio (SNR), respectively.

Segre products of cluster algebras

AbstractWe show that under mild assumptions the Segre product of two graded cluster algebras has a natural cluster algebra structure.

Geometric Algebra Framework Applied to Single-Phase Linear Circuits with Nonsinusoidal Voltages and Currents

We apply a well known technique of theoretical physics, known as geometric algebra or Clifford algebra, to linear electrical circuits with nonsinusoidal voltages and currents. We rederive from the first principles the geometric algebra approach to the apparent power decomposition. The important new point consists of endowing the space of Fourier harmonics with a structure of a geometric algebra (it is enough to define the Clifford product of two periodic functions). We construct a set of commuting invariant imaginary units which are used to define impedance and admittance for any frequency.

On a general notion of a polynomial identity and codimensions

Using the braided version of Lawvere's algebraic theories and Mac Lane's PROPs, we introduce polynomial identities for arbitrary algebraic structures in a braided monoidal category C as well as their codimensions in the case when C is linear over some field. The new cases include coalgebras, bialgebras, Hopf algebras, braided vector spaces, Yetter–Drinfel'd modules, etc. We find bases for polynomial identities and calculate codimensions in some important particular cases.

Topology of quasi divisor graphs associated with non-associative algebra

The visualization of graphs representing algebraic structures has increasingly gained traction in chemical engineering research, emerging as a significant scientific challenge in contemporary studies. Due to their practical applications, graphs associated with groups and various algebraic systems have garnered significant interest in academic literature. Chemical graph theory employs graphs to illustrate molecular structures, where vertices symbolize atoms and edges represent bonds. The structure of quasi divisor graphs linked to non-associative algebra, including quasigroups and loops, presents numerous potential implications and wider applications. These applications span multiple disciplines, such as algebra, combinatorics, computer science, and network theory. Latin squares are tabular representations of quasigroups, generalizations of group structure. Numerous classifications of quasigroups exist in academic literature, such as Moufang quasigroups, Bol quasigroups, and alternative quasigroups. However, the structural characteristics of quasigroups exhibiting the weak inverse property closely resemble those of groups. The overall purpose of this study is to see the interrelationship between non-isomorphic quasi divisor graphs and a particular class of G-loops. To achieve our results, we employ a variety of methods, including a combinatorial approach, degree counting techniques, vertex and edge partitioning methods, as well as graph-theoretical tools and analytical procedures. In this paper, we compute a few topological indices based on degree, distance, and degree-distance using the structural aspects of quasi divisor graphs of orders 2φ,22ϕ,23ϕ and 4ϕ1ϕ2 where φ is a positive integer and ϕ1,ϕ2 are odd primes with ϕ1<ϕ2. This work also includes polynomial forms and their geometrical interpretations connected to linear, quadratic and higher degree topological indices of quasi divisor graphs associated with a class of non-commutative weak inverse property quasigroups.

Exceptional Periodicity and Magic Star algebras: Generalized roots, gradings, Hermitian Vinberg T-algebras and their derivations

We introduce countably infinite series of finite dimensional generalizations of the exceptional Lie algebras: in fact, each exceptional Lie algebra (but g2) is the first element of an infinite series of finite dimensional algebras, which we name Magic Star algebras. All these algebras (but the first elements of the infinite series) are not Lie algebras, but nevertheless they have remarkable similarities with many characterizing features of the exceptional Lie algebras; they also enjoy a kind of periodicity (inherited by Bott periodicity), which we name Exceptional Periodicity. We analyze the graded algebraic structures arising in a certain projection (named Magic Star projection) of the generalized root systems pertaining to Magic Star algebras, and we highlight the occurrence of a class of rank-3, Hermitian matrix (special Vinberg T)-algebras (which we call H algebras) on each vertex of such a projection. We then focus on the Magic Star algebra f4(n), which generalizes the non-simply laced exceptional Lie algebra f4, and deserves a treatment apart. Finally, we compute the Lie algebra of the inner derivations of the H algebras, pointing out the enhancements occurring for each first element of the series of Magic Star algebras, thus retrieving the result known for the derivations of cubic simple Jordan algebras.

Cluster structures on braid varieties

We show the existence of cluster A \mathcal {A} -structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig’s coordinates. Several explicit seeds are provided and the quiver and cluster variables are readily computable. We prove that these upper cluster algebras equal their cluster algebras, show local acyclicity, and explicitly determine their DT-transformations as the twist automorphisms of braid varieties. The main result also resolves the conjecture of B. Leclerc [Adv. Math. 300 (2016), pp. 190–228] on the existence of cluster algebra structures on the coordinate rings of open Richardson varieties.

Pure Rickart Modules

This paper focuses on the study of pure Rickart modules (or PR-modules for short), which are a class of modules over a commutative ring with identity. The main objective is to investigate the properties and characterizations of these modules, as well as their relationship with other classes of modules such as free, projective, and flat modules. This paper also explores the connections between PR-modules and various algebraic structures such as rings. The results obtained in this study provide a deeper understanding of the structure and behavior of PR-modules, which can have important applications in algebraic geometry, representation theory, and other areas of mathematics. Some results about PR-module have been investigated in this paper, for example we demonstrate a module is PR-module if and only if for every , is pure submodule of (or for short). Also, some kind of generalization of these rings have been constructed and demonstrated in term of PR-modules.

&amp;lt;i&amp;gt;A&amp;lt;sub&amp;gt;∞&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;-algebra Structure on Connected Multiplicative Operad

This work develops the structure of &amp;lt;i&amp;gt;A&amp;lt;sub&amp;gt;&amp;infin;&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;-algebras on operad theory and also the preservation of this structure by a morphism of operads well defined. This structure defined here is motivated by the important role that play certain particular properties such as multiplication and connectivity on the operads. Another key ingredient used to develop this work is the brace operations; which, combined with the properties cited above allowed to better frame the study of this structure. Thus, this paper show explicitly the existence of an &amp;lt;i&amp;gt;A&amp;lt;sub&amp;gt;&amp;infin;&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;-algebra structure on any connected multiplicative operad endowed with its brace operations and that this structure is minimal if the operad is only multiplicative. Furthermore, the paper also shows the existence of an operads morphism from an unital associative operad, &amp;lt;i&amp;gt;A&amp;lt;sub&amp;gt;ss&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt; to any connected multiplicative operad &amp;Oscr; preserving the structure of &amp;lt;i&amp;gt;A&amp;lt;sub&amp;gt;&amp;infin;&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;-algebras existing on these two operads. And when the operad &amp;Oscr; is just multiplicative then there is rather a morphism of operads from the associative operad, Asto &amp;Oscr; preserving this time the minimal &amp;lt;i&amp;gt;A&amp;lt;sub&amp;gt;&amp;infin;&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;-algebras structure existing on these operads.

Algebraic structure of the renormalization group in the renormalizable QFT theories

In this paper, we consider the group formed by finite renormalizations as an infinite-dimensional Lie group. It is demonstrated that for the finite renormalization of the gauge coupling constant, its generators [Formula: see text] with [Formula: see text] satisfy the commutation relations of the Witt algebra and, therefore, form its subalgebra. The commutation relations are also written for the more general case when finite renormalizations are made for both the coupling constant and matter fields. We also construct the generator of the Abelian subgroup corresponding to the changes of the renormalization scale. The explicit expressions for the renormalization group generators are written in the case when they act on the [Formula: see text]-function and the anomalous dimension. It is explained how the finite changes of these functions under the finite renormalizations can be obtained with the help of the exponential map.

Stone Pseudovarieties

We extend the theory of profinite algebras, as it is used in the algebraic theory of regular languages, to the more general setting of Stone topological algebras with the ultimate goal of going beyond classes of regular languages. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at the dual Stone spaces of Boolean algebras of subsets of a given topological algebra, we find a simple characterization of when the dual space admits a natural structure of topological algebra; the characterization is given in terms of how the Boolean algebra behaves under the inverses of the operation evaluation mappings of each arity. This provides an alternative, which is presented in the form of an equivalence of categories, to the duality theory of M. Gehrke. As an application, a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is shown to be also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures, which was recently extended to topological signatures jointly by the authors and H. Goulet-Ouellet. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established and it is shown how Reiterman’s theorem can be derived from it.

Transferring algebra structures on complexes

With the goal of transferring dg algebra structures on complexes along contractions, we introduce a new condition on the associated homotopy, namely a generalized version of the Leibniz rule. We prove that, with this condition, the transfer works to yield a dg algebra (with vanishing descended higher A∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\infty $$\\end{document} products) and prove that it works also after an application of the Perturbation Lemma even though the new homotopy may no longer satisfy that condition. We also extend these results to the setting of A∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\infty $$\\end{document} algebras. Then we return to our original motivation from commutative algebra. We apply these methods to find a new method for building a dg algebra structure on a well-known resolution, obtaining one that is both concrete and permutation invariant. The naturality of the construction enables us to find dg algebra homomorphisms between these as well, enabling them to be used as inputs for constructing bar resolutions.

Biderivations, commuting linear maps, post-Lie algebra structure on solvable Lie algebras of maximal rank

In this paper, all biderivations of solvable Lie algebras of maximal rank g = Q ⊕ N are characterized. Namely, we consider a solvable Lie algebra of the form g = Q ⊕ N , where Q is the maximal torus subalgebra of g , N is the nilradical of g and dim Q = dim N / N 2 .In the case dim Q = N we characterize the form of skew-symmetric and symmetric biderivations of g . As applications, the forms of linear commuting (skew-commuting) maps and the commutative post-Lie algebra structures on g are given.

A general unified consistency framework of reciprocal preference relations over bipolar Abelian quasi-ordered monoids

Diverse categories of reciprocal preference relations (RPRs) have been put forward and extensively employed to quantify preference information. Consistency frameworks, which are profoundly interrelated with algebraic structures of RPRs, play a critical role in decision-making systems with RPRs. This paper introduces a novel concept of bipolar Abelian quasi-ordered monoids (BAQO-monoids) and regards any kind of RPRs as preference matrices over a given BAQO-monoid. Six BAQO-monoids are established to characterize algebraic structures of six types of imprecision-based RPRs: interval multiplicative RPRs, interval additive RPRs, triangular fuzzy multiplicative RPRs, triangular fuzzy additive RPRs, trapezoidal fuzzy multiplicative RPRs and trapezoidal fuzzy additive RPRs. Based on a general BAQO-monoid, a transitivity equation system with the binary mapping and parametric elements is developed. These parametric elements are then identified to propose a general unified consistency framework for RPRs over the BAQO-monoid. A distance-based computational formula is built to acquire inconsistency indices of RPRs over the general BAQO-monoid. The proposed general unified consistency framework and inconsistency measuring model are applied to set up consistency models and to calculate inconsistency indices for the six types of imprecision-based RPRs. Six numerical examples and important properties are provided to validate the proposed consistency frameworks and inconsistency measuring models.

Algebraic Structures and Combinatorial Properties of Unit Graphs in Rings of Integer Modulo with Specific Orders

Unit graph is the intersection of graph theory and algebraic structure, which can be seen from the unit graph representing the ring modulo n in graph form. Let R be a ring with nonzero identity. The unit graph of R, denoted by G(R), has its set of vertices equal to the set of all elements of R; distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this study, the unit graph, which is in the ring of integers modulo n, denoted by G(Zn). It turns out when n is 2^k, G(Zn) forms a complete bipartite graph for k∈N, whereas when n is prime, G(Zn) forms a complete (n+1)/2-partites graph. Additionally, the numerical invariants of the graph G(Zn), such as degree, chromatic number, clique number, radius, diameter, domination number, and independence number complement the characteristics of G(Zn) for further research.