Algebraic Theory Research Articles

Multi-fuzzy [formula omitted]-covering fusion based accuracy and self-information for feature subset selection

Granular structures are mathematical representations of knowledge used in granular computing. As a new type of granular structure, fuzzy β-covering has attracted widespread attention in recent years. One of the most critical related studies is measuring the uncertainty of fuzzy β-coverings. It is the foundation for studying fuzzy β-covering applications, such as classification and clustering. In this paper, we investigate the uncertainty measures of fuzzy β-coverings from the viewpoints of algebra and information theory, and propose the corresponding methods of fuzzy β-covering reduction for feature selection. Firstly, a generalized fuzzy β-neighborhood and the associated fuzzy β-covering rough sets are presented. Secondly, three types of accuracy measures of fuzzy β-coverings are constructed via fuzzy β-covering rough sets. On this basis, the self-information of fuzzy β-coverings is defined by fusing accuracy and roughness measures. Based on the proposed uncertainty measures, two heuristic methods of fuzzy β-covering reduction are put forward and the corresponding algorithms are designed for monotonic feature selection. Finally, the performance of the proposed methods is compared with several mainstream feature selection methods and experimental results demonstrate the rationality and superiority of our proposed methods.

Chemoreactomic analysis of the antioxidant properties of vitamin B12 derivatives

Background. Synthetic derivatives of vitamin B12 exhibit various physical, chemical and pharmacological properties. The development of methods for predicting the properties of these molecules based on their chemical structure is important for the targeted organic synthesis of corrins with the desired properties and range of applications in pharmacology.Objective: chemoreactomic assessment of the antioxidant effects of vitamin B12 and its derivatives: aquacobalamin, diaquacobinamide, aquacyano-forms of heptaethanolamine-, heptaethylenediamine-, heptamethyl- and heptabutylcobyrinates.Material and methods. The study was conducted using the method of chemoreactomic analysis, implemented within the framework of the algebraic theory of recognition and topological data analysis and based on the theory of isomorphism of labeled graphs and modern methods for predicting numerical target variables. To carry out chemoinformatic chemoreactome analysis, a special problem-oriented theory was developed within the boundaries of the combinatorial theory of solvability and the antioxidant properties of vitamin B12 derivatives were assessed (140 activities in total).Results. Significant differences were found in the properties of the studied substances in relation to oxidative stress. In test systems based on the oxidant 1,1-diphenyl-2-picrylhydrazyl, at different exposure times to molecules (15–60 min) in different concentrations (10–125 μM, 50–3000 μg/ml), aquacobalamin exhibited the most pronounced antioxidant properties at lower concentrations (up to 100 µM). At higher concentrations of substances (125 μM), the antioxidant activity of other vitamin B12 derivatives was higher. All studied molecules had a moderate antimicronutrient effect (total score of about 3.0; for most synthetic drugs, this score is higher than 3.6). Aquacobalamin was characterized by еhe least pronounced antimicronutrient effect (total score less than 0.8), which indicates an almost complete absence of antivitamin and antimineral action, corresponding to an average increase in the risk of a particular micronutrient excretion by no more than 5%.Conclusion. All the studied compounds exert antioxidant properties to one degree or another. Regardless of the choice of simulated test systems for assessing oxidative stress, aquacobalamin demonstrated antioxidant effects to the greatest extent and practically did not stimulate the loss of other micronutrients.

Unifying the Three Algebraic Approaches to the CSP via Minimal Taylor Algebras

This paper focuses on the algebraic theory underlying the study of the complexity and the algorithms for the Constraint Satisfaction Problem (CSP). We unify, simplify, and extend parts of the three approaches that have been developed to study the CSP over finite templates -- absorption theory that was used to characterize CSPs solvable by local consistency methods (JACM'14), and Bulatov's and Zhuk's theories that were used for two independent proofs of the CSP Dichotomy Theorem (FOCS'17, JACM'20). As the first contribution we present an elementary theorem about primitive positive definability and use it to obtain the starting points of Bulatov's and Zhuk's proofs as corollaries. As the second contribution we propose and initiate a systematic study of minimal Taylor algebras. This class of algebras is broad enough that it suffices to verify the CSP Dichotomy Theorem on this class only, but still is unusually well behaved. In particular, many concepts from the three approaches coincide in this class, which is in striking contrast with the general setting. We believe that the theory initiated in this paper will eventually result in a simple and more natural proof of the Dichotomy Theorem that employs a simpler and more efficient algorithm, and will help in attacking complexity questions in other CSP-related problems.

Succinct ordering and aggregation constraints in algebraic array theories

We discuss two extensions to a recently introduced theory of arrays, which are based on considerations coming from the model theory of power structures. First, we discuss how the ordering relation on the index set can be expressed succinctly by referring to arbitrary Venn regions. Second, we show how to add general aggregators to the calculus. The result is a logic that subsumes four previous fragments discussed in the literature and is distinct from array fold logic, in that it can express summations, while its satisfiability problem remains in non-deterministic polynomial time.

TypeQL: A Type-Theoretic & Polymorphic Query Language

Relational data modeling can often be restrictive as it provides no direct facility for modeling polymorphic types, reified relations, multi-valued attributes, and other common high-level structures in data. This creates many challenges in data modeling and engineering tasks, and has led to the rise of more flexible NoSQL databases, such as graph and document databases. In the absence of structured schemas, however, we can neither express nor validate the intention of data models, making long-term maintenance of databases substantially more difficult. To resolve this dilemma, we argue that, parallel to the role of classical predicate logic for relational algebra, contemporary foundations of mathematics rooted in type theory can guide us in the development of powerful new high-level data models and query languages. To this end, we introduce a new polymorphic entity-relation-attribute (PERA) data model, grounded in type-theoretic principles and accessible through classical conceptual modeling, with a near-natural query language: TypeQL. We illustrate the syntax of TypeQL as well as its denotation in the PERA model, formalize our model as an algebraic theory with dependent types, and describe its stratified semantics.

Order-six complex hadamard matrices constructed by Schmidt rank and partial transpose in operator algebra

Hadamard matrices play a key role in the study of algebra and quantum information theory, and it is an open problem to characterize 6 6 Hadamard matrices. In this paper, we investigate the problem in terms of the Schmidt rank. The primary achievement of this paper lies in establishing a systematic approach to generate 6 6 Hadamard matrices and H-2 reducible matrices through partial transpose. First, if the Schmidt rank of a Hadamard matrix is at most three, then the partial transpose of the Hadamard matrix is also a Hadamard matrix. Conversely, if the Schmidt rank is four, then the partial transpose is no longer a Hadamard matrix. Second, we discuss the relationship between Schmidt rank and H-2 reducible matrices. We prove Hadamard matrices with Schmidt-rank-one are all H-2 reducible, and prove that some Schmidt-rank-two matrices are H-2 reducible. Finally, we confirm that the partial transpose of an H-2 reducible Schmidt-rank-one or two Hadamard matrix remains H-2 reducible.

Conditions for The Saturation of a Matrix Inequality

Matrix inequalities play a vital role in the study of linear algebra and information theory. In this paper, the author investigates the condition when a proven matrix inequality is saturated, and Pi’s are linearly independent matrices of the same size, and Si’s are also linearly independent matrices of the same size. The author constructs the conditions when Pi is a column vector or a 2 2 matrix, and Si is an arbitrary square matrix. The author finds a very potential result when investigating the column situation. The author also gets massive result in the 2 2 situation. In particular, the inequality is not saturated when Pi is a 2 2 matrix and K=2 or 3 and it could be saturated under the situation when K=4 if only if two certain matrices can enable the product of and form two established matrices. At the end of the essay, the author tries some more complex cases such as 2 3 matrices Pi when K=2.

Deformations, cohomologies and abelian extensions of compatible 3-Lie algebras

In this paper, first we give the notion of a compatible 3-Lie algebra and construct a bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible 3-Lie algebras. We also obtain the bidifferential graded Lie algebra that governs deformations of a compatible 3-Lie algebra. Then we introduce a cohomology theory of a compatible 3-Lie algebra with coefficients in itself and show that there is a one-to-one correspondence between equivalent classes of infinitesimal deformations of a compatible 3-Lie algebra and the second cohomology group. We further study 2-order 1-parameter deformations of a compatible 3-Lie algebra and introduce the notion of a Nijenhuis operator on a compatible 3-Lie algebra, which could give rise to a trivial deformation. At last, we introduce a cohomology theory of a compatible 3-Lie algebra with coefficients in arbitrary representation and classify abelian extensions of a compatible 3-Lie algebra using the second cohomology group.

The expanding universe of the geometric mean

AbstractIn this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital $$C^*$$ C ∗ -algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the $$C^*$$ C ∗ -algebra of operators on a Hilbert space, even to general unital $$C^*$$ C ∗ -algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.

Signed graphs with exactly two distinct main eigenvalues

An eigenvalue λ of a signed graph S of order n is a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector j. Characterizing signed graphs with exactly k(1≤k≤n) main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Z. Stanić has noticed that a signed graph has exactly one main eigenvalue if and only if it is net-regular, and in this paper, we study signed graphs with exactly two distinct main eigenvalues by studying (0,1,2)-multi-graphs with exactly two distinct main eigenvalues. We partially characterize the case when the basic graphs of the (0,1,2)-multi-graphs are trees, and propose some problems for further research.

On the Dimensions of Hermitian Subfield Subcodes from Higher-Degree Places.

The focus of our research is the examination of Hermitian curves over finite fields, specifically concentrating on places of degree three and their role in constructing Hermitian codes. We begin by studying the structure of the Riemann-Roch space associated with these degree-three places, aiming to determine essential characteristics such as the basis. The investigation then turns to Hermitian codes, where we analyze both functional and differential codes of degree-three places, focusing on their parameters and automorphisms. In addition, we explore the study of subfield subcodes and trace codes, determining their structure by giving lower bounds for their dimensions. This presents a complex problem in coding theory. Based on numerical experiments, we formulate a conjecture for the dimension of some subfield subcodes of Hermitian codes. Our comprehensive exploration seeks to deepen the understanding of Hermitian codes and their associated subfield subcodes related to degree-three places, thus contributing to the advancement of algebraic coding theory and code-based cryptography.

Understanding the dynamics of hepatitis B transmission: A stochastic model with vaccination and Ornstein-Uhlenbeck process

In this paper, we explore and analyze a stochastic epidemiological model with vaccination and two mean-reverting Ornstein-Uhlenbeck processes to describe the dynamics of HBV transmission. Our study begins by deriving a sufficient condition for the extinction of hepatitis B. Additionally, we determine the presence of a stationary distribution using the Markov process stability theory. Next, we utilize algebraic equation theory and the associated Fokker-Planck equation to derive an explicit expression for the unique probability density function. To validate our theoretical findings, we conduct several numerical simulations using hepatitis B data provided by the World Health Organization (WHO). The results of these simulations support and complement our analytical approach.

Из опыта преподавания. XIII. Имя кристаллического полиэдра. К 130-летию со дня рождения А. Ф. Лосева и 100-летию «Философии имени»

The article is timed to the 130th anniversary of the birth of the outstanding Russian philosopher A. F. Losev (1893—1988) and the 100th anniversary of his work «The Philosophy of Name». Against the background of his ideas, the article considers the nomenclature of polyhedral crystalline simple forms, developed by the scientific school of the «Fedorov Institute» under the leadership of A. K. Boldyrev in the walls of the Leningrad Mining Institute. A general algorithmic approach to the nomenclature of convex polyhedra, convenient for computer data processing, is proposed. It opens an extensive research programme on the frontier of combinatorial theory of convex polyhedra, linear algebra and number theory. The aim of the article is to improve the teaching of crystallography in Russian universities and to broaden the outlook of students in related fields of knowledge.

Quasi-free Isomorphisms of Second Quantization Algebras and Modular Theory

Using Araki–Yamagami’s characterization of quasi-equivalence for quasi-free representations of the CCRs, we provide an abstract criterion for the existence of isomorphisms of second quantization local von Neumann algebras induced by Bogolubov transformations in terms of the respective one particle modular operators. We discuss possible applications to the problem of local normality of vacua of Klein-Gordon fields with different masses.

Computational Construction of Weierstrass Sections for Semi-Invariant Polynomial Functions on Lie Algebras

In this paper, we present a computational approach to construct Weierstrass sections for semi-invariant polynomial functions on Lie algebras, extending the foundational work of Bourbaki and Popov. We focus on simple Lie algebras of type B, C, or D, and their associated parabolic subalgebras, particularly those with Levi factors composed of successive blocks of size two. Our method extends the notion of Weierstrass sections introduced by Popov, enabling us to explicitly construct these sections and establish their polynomiality. Furthermore, we demonstrate how these sections facilitate the linearization of semi-invariant generators. Central to our approach is the construction of an adapted pair, akin to a principal sl_2-triple in the non-reductive case. We provide computational algorithms and implementations for constructing these Weierstrass sections, offering a novel avenue for research in Lie algebra theory and algebraic geometry.

Łukasiewicz fuzzy filters of Sheffer stroke Hilbert algebras

Using the notion of the Łukasiewicz fuzzy set, we study the filter theory of Sheffer stroke Hilbert algebras. Here’s what we’re trying to do. 1. We first introduce the Łukasiewicz fuzzy filter of Sheffer stroke Hilbert algebras. 2. We provide an example to illustrates the Łukasiewicz fuzzy filter. 3. We examine the various properties of the Łukasiewicz fuzzy filter. 4. We discuss characterizations of the Łukasiewicz fuzzy filter. 5. We explore the conditions under which Łukasiewicz fuzzy set can be Łukasiewicz fuzzy filter. 6. We discuss the relationship between fuzzy filter and Łukasiewicz fuzzy filter. 7. We use the given filter to creates a Łukasiewicz fuzzy filter. 8. We present conditions for the three subsets, called ∈-set, q-set and O-set, to be filters.

Homotopy Theory

The workshop brought together experts in homotopy theory from many areas, including chromatic homotopy theory, algebraic K-theory, derived algebra and equivariant homotopy theory. We had a lecture series on the recent disproof of the telescope conjecture. In addition to a total of 24 research talks we had two gong-shows where participants presented their research in 10-minute talks.

Crossed product algebras and generalized entropy for subregions

An early result of algebraic quantum field theory is that the algebra of any subregion in a QFT is a von Neumann factor of type III_{1}1, in which entropy cannot be well-defined because such algebras do not admit a trace or density states. However, associated to the algebra is a modular group of automorphisms characterizing the local dynamics of degrees of freedom in the region, and the crossed product of the algebra with its modular group yields a type II_∞∞ factor, in which traces and hence von Neumann entropy can be well-defined. In this work, we generalize recent constructions of the crossed product algebra for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs, formally paving the way to the study of entanglement entropy without UV divergences. In contrast to previous works, we emphasize that this construction is independent of gravity. In this sense, the crossed product construction represents a refinement of Haag’s assignment of nets of observable algebras to spacetime regions by providing a natural construction of a type II factor. We present several concrete examples: a QFT in Rindler space, a CFT in an open ball of Minkowski space, and arbitrary boundary subregions in AdS/CFT. In the holographic setting, we provide a novel argument for why the bulk dual must be the entanglement wedge, and discuss the distinction arising from boundary modular flow between causal and entanglement wedges for excited states and disjoint regions.

On the solutions of some Lebesgue–Ramanujan–Nagell type equations

Denote by [Formula: see text] the class number of the imaginary quadratic field [Formula: see text] with [Formula: see text] prime. It is well known that [Formula: see text] for [Formula: see text]. Recently, all the solutions of the Diophantine equation [Formula: see text] with [Formula: see text] were given by Chakraborty et al. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, Publ. Math. Debrecen 97(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation [Formula: see text] in unknown integers [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. To do this, we use the known results from the modularity of Galois representations associated with Frey–Hellegoaurch elliptic curves, the symplectic method and elementary methods of classical algebraic number theory. The aim of this paper is to extend the above results of Chakraborty et al.

Simplicity and tracial weights on non-unital reduced crossed products

We extend theorems of Breuillard–Kalantar–Kennedy–Ozawa on unital reduced crossed products to the non-unital case under mild assumptions. As a result simplicity of [Formula: see text]-algebras is stable under taking reduced crossed products over discrete [Formula: see text]-simple groups, and a similar result for uniqueness of tracial weight. Interestingly, our analysis on tracial weights involves von Neumann algebra theory. Our generalizations have two applications. The first is to locally compact groups. We establish stability results of (non-discrete) [Formula: see text]-simplicity and the unique trace property under discrete group extensions. The second is to the twisted crossed product. Thanks to the Packer–Raeburn theorem, our results lead to (generalizations of) the results of Bryder–Kennedy by a different method.