Representations Of Algebras Research Articles

Comparison Of Two Encoding Methods in RAID6 Storage Architecture and Analysis of Improvement of EVENODD Encodings

This paper provides a comparative analysis of two widely used error correction codes: RS (Reed-Solomon) and EVENODD. The analysis covers principles, algebraic forms, computational complexities, minimum Hamming distances, decoding processes, and specific application scenarios. RS encoding is known for its versatility, allowing for the flexible addition of parity symbols. However, it comes with a high computational cost due to polynomial evaluations. On the other hand, EVENODD encoding offers simplicity and computational efficiency through mere XOR operations but has limitations on the number of parity symbols it can generate. The paper explores the algebraic representations of both codes, discusses their minimum Hamming distances, and evaluates their decoding processes. Additionally, it analyzes the computational requirements of EVENODD under small write operations and provides a comprehensive comparison of the computational complexities of RS and EVENODD, aiding practitioners in choosing the most suitable error correction scheme for specific applications. A seeming limitation in our building process is because of the number of information disks must fall into a prime number. Nevertheless, if the preferred quantity of data sectors isn't a prime number, we can easily posit the existence of additional disks filled entirely with zeros, and this won't impact the encoding and decoding procedures.

Quantum state complexity meets many-body scars

Scar eigenstates in a many-body system refers to a small subset of non-thermal finite energy density eigenstates embedded into an otherwise thermal spectrum. This novel non-thermal behaviour has been seen in recent experiments simulating a one-dimensional PXP model with a kinetically-constrained local Hilbert space realised by a chain of Rydberg atoms. We probe these small sets of special eigenstates starting from particular initial states by computing the spread complexity associated to time evolution of the PXP hamiltonian. Since the scar subspace in this model is embedded only loosely, the scar states form a weakly broken representation of the Lie algebra. We demonstrate why a careful usage of the forward scattering approximation (FSA), instead of any other method, is required to extract the most appropriate set of Lanczos coefficients in this case as the consequence of this approximate symmetry. Only such a method leads to a well defined notion of a closed Krylov subspace and consequently, that of spread complexity. We show this using three separate initial states, namely |Z2⟩,|Z3⟩ and the vacuum state, due to the disparate classes of scar states hosted by these sectors. We also discuss systematic methods of remedying the imperfections in the FSA setup stemming from these approximate symmetries.

Numerical representation of topological real algebras

We show that the isomorphism provided by the Gelfand-Mazur theorem for a commutative pre-ordered real Banach algebra A with unit defines a numerical representation which is compatible with the order structure.

More on characteristic polynomials of Lie algebras

In recent years, the notion of characteristic polynomials of representations of Lie algebras has been widely studied. This paper provides more properties of these characteristic polynomials. For simple Lie algebras, we characterize the linearization of characteristic polynomials. Additionally, we characterize nilpotent Lie algebras via characteristic polynomials of the adjoint representation.

On correlation functions in models related to the Temperley-Lieb algebra

We deal with quantum spin chains whose Hamiltonian arises from a representation of the Temperley-Lieb algebra, and we consider the mean values of those local operators which are generated by the Temperley-Lieb algebra. We present two key conjectures which relate these mean values to existing literature about factorized correlation functions in the XXZ spin chain. The first conjecture states that the finite volume mean values of the current and generalized current operators are given by the same simple formulas as in the case of the XXZ chain. The second conjecture states that the mean values of products of Temperley-Lieb generators can be factorized: they can expressed as sums of products of current mean values, such that the coefficients in the factorization depend neither on the eigenstate in question, nor on the selected representation of the algebra. The coefficients can be extracted from existing work on factorized correlation functions in the XXZ model. The conjectures should hold for all eigenstates that are non-degenerate with respect to the local charges of the models. We consider concrete representations, where we check the conjectures: the so-called golden chain, the QQ-state Potts model, and the trace representation. We also explain how to derive the generalized current operators from concrete expressions for the local charges.

Exact hierarchical reductions of dynamical models via linear transformations

Dynamical models described by ordinary differential equations (ODEs) are a fundamental tool in the sciences and engineering. Exact reduction aims at producing a lower-dimensional model in which each macro-variable can be directly related to the original variables, and it is thus a natural step towards the model’s formal analysis and mechanistic understanding. We present an algorithm which, given a polynomial ODE model, computes a longest possible chain of exact linear reductions of the model such that each reduction refines the previous one, thus giving a user control of the level of detail preserved by the reduction. This significantly generalizes over the existing approaches which compute only the reduction of the lowest dimension subject to an approach-specific constraint. The algorithm reduces finding exact linear reductions to a question about representations of finite-dimensional algebras. We provide an implementation of the algorithm, demonstrate its performance on a set of benchmarks, and illustrate the applicability via case studies. Our implementation is freely available at https://github.com/x3042/ExactODEReduction.jl.

Models of representations for classical series of Lie algebras

By a model of representations of a Lie algebra we mean a representation which is a direct sum of all irreducible finite-dimensional representations taken with multiplicity $1$. An explicit construction of a model of representations for all classical series of simple Lie algebras is given. This construction is generic for all classical series of Lie algebras. The space of the model is constructed as the space of polynomial solutions of a system of partial differential equations, where the equations are constructed form relations between minors of matrices taken from the corresponding Lie group. This system admits a simplification very close to the GKZ system, which is satisfied by $A$-hypergeometric functions.

Representations of quadratic Heisenberg-Weyl algebras and polynomials in the fourth Painlevé transcendent

<p>We provide new insights into the solvability property of a Hamiltonian involving the fourth Painlevé transcendent and its derivatives. This Hamiltonian is third-order shape invariant and can also be interpreted within the context of second supersymmetric quantum mechanics. In addition, this Hamiltonian admits third-order lowering and raising operators. We have considered the case when this Hamiltonian is irreducible, i.e., when no special solutions exist for given parameters $ \alpha $ and $ \beta $ of the fourth Painlevé transcendent $ P_{IV}(x, \alpha, \beta) $. This means that the Hamiltonian does not admit a potential in terms of rational functions (or the hypergeometric type of special functions) for those parameters. In such irreducible cases, the ladder operators are as well involving the fourth Painlevé transcendent and its derivative. An important case for which this occurs is when the second parameter (i.e., $ \beta $) of the fourth Painlevé transcendent $ P_{IV}(x, \alpha, \beta) $ is strictly positive, $ \beta > 0 $. This Hamiltonian was studied for all hierarchies of rational solutions that come in three families connected to the generalized Hermite and Okamoto polynomials. The explicit form of ladder, the associated wavefunctions involving exceptional orthogonal polynomials, and recurrence relations were also completed described. Much less is known for the irreducible case, in particular the excited states. Here, we developed a description of the induced representations based on various commutator identities for the highest and lowest weight type representations. We also provided for such representations a new formula concerning the explicit form of the related excited states from the point of view of the Schrödinger equation as two-variables polynomials that involve the fourth Painlevé transcendent and its derivative.</p>

On Representations of Groups and Algebras in Spaces with Indefinite Metric

On Representations of Groups and Algebras in Spaces with Indefinite Metric

Полная группа изометрий специальной трёхмерной группы Ли

We consider a special three-dimensional Lie algebra 3 of Bianchi type V. A matrix representation of this Lie algebra and the corresponding connected simply connected Lie group 𝑆3 is found and natural coordinates are introduced, which are determined by the matrix representation. Formulas for exponential mapping with respect to natural coordinates are found. Complete groups of autoisometries of the special Lie algebra with respect to the Euclidean or Lorentz scalar product are written out. The leftinvariant Riemannian metric of the Lie group 𝑆3 and the complete group of isometries of the resulting homogeneous manifold are found. This manifold turned out to be a space of constant Ricci curvature.

Enhancing Geometry Teaching in STEAM Education with Interactive Learning Environments

In confronting the challenge of effectively integrating technology into mathematics education, this study investigates the impact of GeoGebra, an interactive learning tool, on enhancing the understanding of geometry among high school students. Through a rigorous mixed-methods approach, involving 150 students segmented into control and experimental cohorts, this research meticulously assesses the differential outcomes attributable to GeoGebra engagement. Notably, our findings illuminate a significant augmentation in the experimental group’s grasp of geometric concepts, with a pronounced improvement in correlating algebraic and graphical representations of trigonometric functions. This advancement was not merely academic, it corresponded with elevated levels of student engagement and performance. Beyond the anticipated benefits of interactive tools in educational settings, our study contributes novel insights into the specific advantages of GeoGebra within the STEAM curriculum. The results advocate for a more pronounced integration of technology-enhanced learning environments in educational systems, aiming to not only enrich students’ learning experiences but also to prepare them more effectively for the complexities of modern scientific and mathematical problem-solving.

Cluster Synchronization of Finite-Field Networks.

This technical paper investigates the cluster synchronization of finite-field networks (FFNs) based on the algebraic state space representation. By resorting to the semi-tensor product of matrices, the cluster synchronization of an FFN can be completely converted into the set stability of state trajectory. Then, a necessary and sufficient condition for the cluster synchronization of FFNs is obtained based on the invariant subset and one-step transition matrix. In particular, the obtained results are applicable to check the leader-follower synchronization and group consensus of FFNs. Finally, a numerical example is given to illustrate the feasibility of the obtained results.

Analysis of Discrete-Time Switched Linear Systems Under Logical Dynamic Switching.

The control properties of discrete-time switched linear systems (SLSs) with switching signals generated by logical dynamical systems are studied using the semitensor product (STP) approach. With the algebraic state-space representation (ASSR), the linear modes and the logical generators are aggregated as a system with hybrid states, leading to the criteria of reachability, controllability, observability, and reconstructibility of the SLSs. Algorithms for checking these properties are given. Then, two kinds of realization problems concerning whether the logical dynamical systems can generate the desired switching signals are investigated, and necessary and sufficient conditions for the realizability of the desired switching signals are given with respect to the cases of fixed operating time (FOT) switching and finite reference signal switching.

Local-global compatibility for regular algebraic cuspidal automorphic representations when

Abstract We prove the compatibility of local and global Langlands correspondences for $\operatorname {GL}_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne [10] and Scholze [18]. More precisely, let $r_p(\pi )$ denote an n-dimensional p-adic representation of the Galois group of a CM field F attached to a regular algebraic cuspidal automorphic representation $\pi $ of $\operatorname {GL}_n(\mathbb {A}_F)$ . We show that the restriction of $r_p(\pi )$ to the decomposition group of a place $v\nmid p$ of F corresponds up to semisimplification to $\operatorname {rec}(\pi _v)$ , the image of $\pi _v$ under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of $\left .r_p(\pi )\right |{}_{\operatorname {Gal}_{F_v}}$ is ‘more nilpotent’ than the monodromy of $\operatorname {rec}(\pi _v)$ .

A Note on Secant-Defective Varieties and Clifford Modules

We generalise a construction of Landsberg, which associates certain Clifford algebra representations to Severi varieties. We thus obtain a new proof of Russo’s Divisibility Property for LQEL varieties whose secant varieties do not fill the ambient space. We also make an observation about the question of whether there exists a smooth, irreducible, non-degenerate, projective variety with secant deficiency delta > 8.

A Framework for Analyzing System Loadability With Multiple VSCs Using a Hybrid Model

Due to the increasing number of power electronics-based devices in modern power systems, there are concerns about the impact of higher-frequency interactions on system stability. The typically-used algebraic representation of networks fails to capture these newly emergent issues, but dynamically modelling the entire network with differential equations will lead to a prohibitively long simulation time. This paper proposes a computationally-efficient analysis framework to determine the system loadability in power electronics-rich, large networks with the consideration of multiple types of stabilities. The framework developed in this paper identifies the critical network elements based on the eigenvalue sensitivity in order to exploit a hybrid modelling approach. Within the hybrid model, only the critical network portion is modelled dynamically with the rest of the network represented by algebraic equations. Bifurcation theory is used to simultaneously analyze both small-disturbance rotor angle and small-disturbance voltage stability. The results obtained show that the high-frequency interactions are accurately captured with the proposed framework. Additionally, applying this methodology results in a small dimension for the system matrix and a reduction in the computational burden. Two test networks, a three-bus system and the IEEE 39-bus system, are used to illustrate and verify the analysis framework.

Self-Supervised Lie Algebra Representation Learning via Optimal Canonical Metric

Self-Supervised Lie Algebra Representation Learning via Optimal Canonical Metric

A full process algebraic representation of Ant Colony Optimization

We present a process algebra capable of specifying parallelized Ant Colony Optimization algorithms in full detail: PA2CO. After explaining the basis of three different ACO algorithms (Ant System, MAX-MIN Ant System, and Ant Colony System), we formally define PA2CO and use it for representing several types of implementations with different parallel schemes. In particular fine-grained and coarse-grained specifications, each one taking advantage of parallel executions at different levels of system granularity, are formalized.

Approximate equivalence of representations of AH algebras into semifinite von Neumann factors

In this paper, we prove a non-commutative version of the Weyl–von Neumann theorem for representations of unital, separable AH algebras into countably decomposable, semifinite, properly infinite, von Neumann factors, where an AH algebra means an approximately homogeneous C^{\ast} -algebra. We also prove a result for approximate summands of representations of unital, separable AH algebras into finite von Neumann factors.

Linear algebra concepts with SageMath for systems engineering students

This work presents an innovative proposal for the teaching and learning of systems of linear equations, graphical interpretation in the plane, in space and orthogonal projection, in the subject of Linear Algebra, mediated by the SageMath software as a technology of learning and knowledge. The proposal was carried out in some Linear Algebra courses in the Systems Engineering career at the University of Cauca, helping to increase its quality, reduce the retention rate and contribute to educational innovation in the university student population. The methodology for the development of this work consisted of the design and application of activities that comprise a set of problems which require the creation of functions in the Python programming language under the SageMath, environment to obtain possible solutions and correlate algebraic and geometric representation registers of the mathematical objects necessary for its solution. We conclude at the end of this educational research the relevance of the integration of technological tools within the classroom in conjunction with active methodological strategies to stimulate the understanding of some concepts of Linear Algebra; since the student can visualize, manipulate and observe these abstract mathematical objects, eliminating unnecessary manual actions and focusing on the analysis of logical deduction for the solution of the proposed activities.