Relational Algebra Research Articles

Block-Supersymmetric Polynomials on Spaces of Absolutely Convergent Series

In this paper, we consider a supersymmetric version of block-symmetric polynomials on a Banach space of two-sided absolutely summing series of vectors in Cs for some positive integer s>1. We describe some sequences of generators of the algebra of block-supersymmetric polynomials and algebraic relations between the generators for the finite-dimensional case and construct algebraic bases of block-supersymmetric polynomials in the infinite-dimensional case. Furthermore, we propose some consequences for algebras of block-supersymmetric analytic functions of bounded type and their spectra. Finally, we consider some special derivatives in algebras of block-symmetric and block-supersymmetric analytic functions and find related Appell-type sequences of polynomials.

On the relative L-theory and the relative signature of PL manifolds with boundary

In this paper, we give a new description of the relative topological structure group of PL manifolds with boundary, and put the relative L-group into an exact sequence of groups. Then we define the relative index of PL manifolds with boundary, and show that this relative index will induce an additive map from the relative L-group to the K-theory of the relative Roe algebra.

Modern technological tools for creating multidimensional matrix database machines

The article discusses a group of tasks that require online analytical data processing. The authors analyze the components of OLAP technology: multidimensional cubes, implementation methods and methods for accessing them. Based on the analysis, it is concluded that truly multidimensional databases do not exist. An algebra of multidimensional matrices is proposed, which corresponds to relational algebra. It is shown that on the basis of this algebra it is possible to develop a DBMS that will provide an effective parallel implementation of OLAP technology. A method for constructing a multidimensional matrix database engine based on available software and hardware is proposed.

Relative Poisson bialgebras and Frobenius Jacobi algebras

Jacobi algebras, as the algebraic counterparts of Jacobi manifolds, are exactly the unital relative Poisson algebras. The direct approach of constructing Frobenius Jacobi algebras in terms of Manin triples is not available due to the existence of the units, and hence alternatively we replace it by studying Manin triples of relative Poisson algebras. Such structures are equivalent to certain bialgebra structures, namely, relative Poisson bialgebras. The study of coboundary cases leads to the introduction of the relative Poisson Yang–Baxter equation (RPYBE). Antisymmetric solutions of the RPYBE give coboundary relative Poisson bialgebras. The notions of \mathcal{O} -operators of relative Poisson algebras and relative pre-Poisson algebras are introduced to give antisymmetric solutions of the RPYBE. A direct application is that relative Poisson bialgebras can be used to construct Frobenius Jacobi algebras, and in particular, there is a construction of Frobenius Jacobi algebras from relative pre-Poisson algebras.

GENERAL CHARACTERISTICS AND SELECTION OF A DATABASE MANAGEMENT SYSTEM IN DISTANCE LEARNING

The relevance of studying the discipline "Databases" for students of computer and engineering specialties is determined and their special needs and emphasis in learning are shown.
 The definition of the data model stored in the database is given. The features of the relational data model are considered, the main advantage of which is simplicity and accessibility for the user to understand and the availability of a fairly simple and at the same time powerful mathematical apparatus of set theory and relational algebra. The main educational elements of studying databases based on the relational data model are highlighted.
 The client-server architecture of modern DBMSs is described. The advantages of client-server technologies in the process of learning to work with databases are noted.
 The results of the study of the functionality of existing software products for working with databases have shown the expediency of choosing MySQL Community Edition as a training tool, which is free and in its functionality is not inferior to commercial systems by any criteria. The disadvantages of the system, which include some limitations when executing complex queries and creating analytical reports that require a high load and a large number of simultaneous queries, are not critical for training purposes. The MySQL server can be accessed both from a console client running any of the modern operating systems and from many GUI clients. The paper describes the software product MySQL Workbench Community Edition, which is offered by Oracle for free use both as part of the MySQL server and as a separate GUI client.
 As a result of the study, it was concluded that the free MySQL Community Edition DBMS together with the MySQL Workbench Community Edition GUI client fully develops the competencies of students of both engineering and computer specialties in working with databases and can be used in the study of the educational component "Databases" in the context of distance education.

Polynomial algebras from commutants: Classical and Quantum aspects of 𝒜3

We review some aspects of the Racah algebra R(n), including the closure relations, pointing out their role in superintegrability, as well as in the description of the symmetry algebra for several models with coalgebra symmetry. The connection includes the generic model on the (n − 1) sphere. We discuss an algebraic scheme of constructing Hamiltonians, integrals of the motion and symmetry algebras. This scheme reduces to the Racah algebra R(n) and the model on the (n − 1) sphere only for the case of specific differential operator realizations. We review the method, which allows us to obtain the commutant defined in the enveloping algebra of 𝔰𝔩(n) in the classical setting. The related 𝒜3 polynomial algebra is presented for the case 𝔰𝔩(3). An explicit construction of the quantization of the scheme for 𝒜3 by symmetrization of the polynomial and the replacement of the Berezin bracket by commutator and symmetrization of the polynomial relations is presented. We obtain the additional quantum terms. These explicit relations are of interest not only for superintegrability, but also for other applications in mathematical physics.

Sample-Efficient Cardinality Estimation Using Geometric Deep Learning

In database systems, accurate cardinality estimation is a cornerstone of effective query optimization. In this context, estimators that use machine learning have shown significant promise. Despite their potential, the effectiveness of these learned estimators strongly depends on their ability to learn from small training sets. This paper presents a novel approach for learned cardinality estimation that addresses this issue by enhancing sample efficiency. We propose a neural network architecture informed by geometric deep learning principles that represents queries as join graphs. Furthermore, we introduce an innovative encoding for complex predicates, treating their encoding as a feature selection problem. Additionally, we devise a regularization term that employs equalities of the relational algebra and three-valued logic, augmenting the training process without requiring additional ground truth cardinalities. We rigorously evaluate our model across multiple benchmarks, examining q-errors, runtimes, and the impact of workload distribution shifts. Our results demonstrate that our model significantly improves the end-to-end runtimes of PostgreSQL, even with cardinalities gathered from as little as 100 query executions.

Cohomologies of relative Rota-Baxter Lie algebras with derivations and applications

The purpose of the present paper is to investigate cohomologies of relative Rota-Baxter Lie algebras with derivations and applications. First, we introduce a notion of relative Rota-Baxter LieDer pairs. Moreover, we construct cohomologies of relative Rota-Baxter LieDer pairs. Finally, we discuss abelian extensions, crossed modules and skeletal relative Rota-Baxter Lie 2-Der pairs in terms of cohomology groups.

Resource Description Framework Query Languages : Comparative Study

Over the period of time, the data on the web grows by leaps and bounds that becomes unmanageable to store and retrieve. As a result, the concept of semantic web comes into existence. Nowadays, various components of semantic web stack become mature such as Resource Description Framework (RDF). Due to the higher adoption rate of RDF in both industry and academia, it becomes crucial to design such systems that can store and retrieve RDF formatted data. Since last decade, various query languages for RDF data retrieval were proposed mainly based on either graph query language or on relational algebra. However, none of them become mature enough that can overcome majority of challenges. Therefore, there is an utmost need for the comparative study of available languages in literature. In this paper, a comparative study based on various parameters was conducted that provides a better insight about the literature.

Strongly stratifying ideals, Morita contexts and Hochschild homology

We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context is an algebra built on a data consisting of two algebras, two bimodules and two morphisms. For a strongly stratifying Morita context - or equivalently for a strongly stratifying ideal - we show that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. One of the main consequences is that Han's conjecture holds for an algebra admitting a strongly (co-)stratifying chain whose steps verify Han's conjecture.If Han's conjecture is true for local algebras and an algebra Λ admits a primitive strongly (co-)stratifying chain, then Han's conjecture holds for Λ.

Directed Ramsey and anti-Ramsey schemes and the Flexible Atom Conjecture

In this paper, we shed new light on the Flexible Atom Conjecture. We first give finite representation results for relation algebras [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Prior to our paper, only [Formula: see text] and [Formula: see text] were known to be finitely representable. We accomplish this by generalizing the notion of a relation algebra generated by a Ramsey scheme to the directed (antisymmetric) setting, and then showing that each of these algebras embeds into a finite directed anti-Ramsey scheme. The notion of a directed anti-Ramsey scheme may be of independent interest. We complement our upper bounds with some lower bounds. Namely, we show that any square representation of [Formula: see text] requires at least 14 points, any square representation of [Formula: see text] requires at least 11 points, and any square representation of [Formula: see text] requires at least 12 points. Our technique adapts previous work of Alm et al. [Algebra Univ. (2022)], in that we examine the combinatorial structure induced by the flexible atom.

On the variety generated by generalized subreducts of Tarski’s algebras of relations

On the variety generated by generalized subreducts of Tarski’s algebras of relations

Relative Entropy of Fermion Excitation States on the CAR Algebra

The relative entropy of certain states on the algebra of canonical anticommutation relations (CAR) is studied in the present work. The CAR algebra is used to describe fermionic degrees of freedom in quantum mechanics and quantum field theory. The states for which the relative entropy is investigated are multi-excitation states (similar to multi-particle states) with respect to KMS states defined with respect to a time-evolution induced by a unitary dynamical group on the one-particle Hilbert space of the CAR algebra. If the KMS state is quasifree, the relative entropy of multi-excitation states can be explicitly calculated in terms of 2-point functions, which are defined entirely by the one-particle Hilbert space defining the CAR algebra and the Hamilton operator of the dynamical group on the one-particle Hilbert space. This applies also in the case that the one-particle Hilbert space Hamilton operator has a continuous spectrum so that the relative entropy of multi-excitation states cannot be defined in terms of von Neumann entropies. The results obtained here for the relative entropy of multi-excitation states on the CAR algebra can be viewed as counterparts of results for the relative entropy of coherent states on the algebra of canonical commutation relations which have appeared recently. It turns out to be useful to employ the setting of a self-dual CAR algebra introduced by Araki.

An Extended Type-2 Fuzzy Relational Database Model for Aggregate and Grouping Operations

This paper introduces an extended type-2 fuzzy relational database model (ET-2FRDB) for aggregate and grouping operations that can represent and query uncertain and imprecise information in the real world applications. In ET-2FRDB, each fuzzy relation is represented by a type-2 fuzzy set whose membership degree of each tuple is a fuzzy number on [0, 1], the fuzzy aggregate functions, the fuzzy relational algebraic, aggregate and grouping operations are defined as extensions of those in the classical relational database model thereby the membership degree of tuples associated by using the minimum and maximum of fuzzy numbers. Also, some properties of the fuzzy relational algebraic, aggregate and grouping operations in ET-2FRDB are formulated and proven.

Typology and Modeling Profiles of Moving-Class Process-Entity Interactions. Framework for Multimodal Route Network

Configuration of routes is a rather comprehensive, necessary and at the same time self-sufficient component of an integrated process that models the displacement of objects. This article is focused on the development of the multimodal route network model that leverages on only the pertinent suite of source abstract object types, which preliminary differentiate according to the open architecture of categorized types. The main emphasis is on the detection of relevant associations, which take place into the transportation domain being modeled, and constraints on its populations. One of these restrictions is the base segment constraint, which guarantees correct instances of the specialization segment relationship. The model contains formal expressions for verifying data reliability in terms of segments specialization, coherence and sequence order of the latter in routes. Another group of constraints encompasses the terminal pool restrictions that comprise constraints of the sequence order, non-redundancy, and unique combinations of source and destination route points. All terminal constraints have been formalized by means of the relational algebra operators. Such decision is capable of supporting integrity constraints by using triggers or stored procedures of database management systems. We have also exploited the universal structural primitive for the multilevel representation of entity lists, as well as have demonstrated the role and function of dedicated datasets under the forming of links between entities.

Roll Stabilisation of a Flexible Spacecraft using Quantitative Feedback Theory

Quantitative Feedback Theory (QFT) has achieved the status as a powerful design technique for the achievement of assigned performance over the specified range of plant uncertainties. This is due to the relative simple algebra and the transparency of the design methodology. A QFT solution to a Single Input Single Output (SISO) problem of roll stabilization of a flexible satellite is presented in this paper. A simple controller is designed which ensures robustness against parameter variations and disturbances. The results of the study, are compared with those obtained using an H* controller.

The Complexity of Aggregates over Extractions by Regular Expressions

Regular expressions with capture variables, also known as regex-formulas, extract relations of spans (intervals identified by their start and end indices) from text. In turn, the class of regular document spanners is the closure of the regex formulas under the Relational Algebra. We investigate the computational complexity of querying text by aggregate functions, such as sum, average, and quantile, on top of regular document spanners. To this end, we formally define aggregate functions over regular document spanners and analyze the computational complexity of exact and approximate computation. More precisely, we show that in a restricted case, all studied aggregate functions can be computed in polynomial time. In general, however, even though exact computation is intractable, some aggregates can still be approximated with fully polynomial-time randomized approximation schemes (FPRAS).

A Tutorial on Visual Representations of Relational Queries

Query formulation is increasingly performed by systems that need to guess a user's intent (e.g. via spoken word interfaces). But how can a user know that the computational agent is returning answers to the "right" query? More generally, given that relational queries can become pretty complicated, how can we help users understand existing relational queries , whether human-generated or automatically generated? Now seems the right moment to revisit a topic that predates the birth of the relational model: developing visual metaphors that help users understand relational queries. This lecture-style tutorial surveys the key visual metaphors developed for visual representations of relational expressions. We will survey the history and state-of-the art of relationally-complete diagrammatic representations of relational queries, discuss the key visual metaphors developed in over a century of investigating diagrammatic languages, and organize the landscape by mapping their used visual alphabets to the syntax and semantics of Relational Algebra (RA) and Relational Calculus (RC).

Quasi-free states on a class of algebras of multicomponent commutation relations

Multicomponent commutations relations (MCRs) describe plektons, i.e. multicomponent quantum systems with a generalized statistics. In such systems, exchange of quasiparticles is governed by a unitary matrix [Formula: see text] that depends on the position of quasiparticles. For such an exchange to be possible, the matrix must satisfy several conditions, including the functional Yang–Baxter equation. The aim of the paper is to give an appropriate definition of a quasi-free state on an MCR algebra, and construct such states on a class of MCR algebras. We observe a significant difference between the classical setting for bosons and fermions and the setting of MCR algebras. We show that the developed theory is applicable to systems that contain quasiparticles of opposite type. An example of such a system is a two-component system in which two quasiparticles, under exchange, change their respective types to the opposite ones ([Formula: see text], [Formula: see text]). Fusion of quasiparticles means intuitively putting several quasiparticles in an infinitely small box and identifying the statistical behavior of the box. By carrying out fusion of an odd number of particles from the two-component system as described above, we obtain further examples of quantum systems to which the developed theory is applicable.

Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra

Let $q$ be a nonzero complex number that is not a root of unity. In the $q$-oscillator with commutation relation $ a a^+-qa^+ a =1$, it is known that the smallest commutator algebra of operators containing the creation and annihilation operators $a^+$ and $ a $ is the linear span of $a^+$ and $ a $, together with all operators of the form ${a^+}^l{\left[a,a^+\right]}^k$, and ${\left[a,a^+\right]}^k a ^l$, where $l$ is a nonnegative integer and $k$ is a positive integer. That is, linear combinations of operators of the form $ a ^h$ or $(a^+)^h$ with $h\geq 2$ or $h=0$ are outside the commutator algebra generated by $ a $ and $a^+$. This is a solution to the Lie polynomial characterization problem for the associative algebra generated by $a^+$ and $ a $. In this work, we extend the Lie polynomial characterization into the associative algebra $\mathcal{P}=\mathcal{P}(q)$ generated by $ a $, $a^+$, and the operator $e^{\omega N}$ for some nonzero real parameter $\omega$, where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$.