Using computability to measure complexity of algebraic structures and classes of structures

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Automorphism groups of simple Kokoris algebras

In 1991, Nisan proved an exponential lower bound on the size of an algebraic branching program (ABP) that computes the determinant of a matrix in the noncommutative “free algebra” setting, in which there are no nontrivial relationships between the matrix entries. By contrast, when the matrix entries commute there are polynomial size ABPs for the determinant. This paper extends Nisan’s result to a much wider class of noncommutative algebras, including all nontrivial matrix algebras over any field of characteristic 0, group algebras of all nonabelian finite groups over algebraically closed fields of characteristic 0, the quaternion algebra, and the Clifford algebras. As a result, we obtain more compelling evidence for the essential role played by commutativity in the efficient computation of the determinant. The key to our approach is a characterization of noncommutative algebras by means of the polynomial identities that they satisfy. Extending Nisan’s lower bound framework, we find that any reduction in complexity compared to the free algebra must arise from the ability of the identities to reduce the rank of certain naturally associated matrices. Using results from the theory of algebras with polynomial identities, we are able to show that none of the identities of the above classes of algebras is able to achieve such a rank reduction.

In [Church 56, sect.07] mathematical logic or symbolic logic, or logistics, is defined as “the subject of formal logic when treated by the method of setting up a formalized language”. Among formalized languages there are purely logical (propositional and predicate) languages, languages of arithmetic and languages of set theory. Predicate languages — elementary (=first-order) as well as non-elementary (of higher orders) — are used for formal descriptions of properties of mathematical structures (first of all algebraic structures). We can use them for an axiomatic description of different classes of structures. Languages of arithmetic describe the set of natural numbers (which hardly can be described axiomatically and must anyway be regarded as existing before any axiomatic considerations). Languages of set theory have no clear semantics and are intended to formulate various axiomatic theories.

The wedge-direct sum of a sequence of semisimple algebras (over an arbitrary field) was introduced in [Xu, B. (2019). Wedge-direct sums of table algebras and applications to association schemes, I. Commun. Algebra 47(8):3309–3328]. The group algebras (over the complex field) of finite groups are a special class of semisimple algebras. In [Xu, B. (2020). Wedge-direct sums of table algebras and applications to association schemes, II. J. Pure Appl. Algebra 224(11):106402], the wedge-direct sum of a sequence of group algebras of finite p-groups is used to obtain the characterization and classification of a class of p-table algebras. In this article, we continue the research in above mentioned articles and characterize, in terms of table algebras, the wedge-direct sum of a sequence of group algebras of arbitrary finite groups. Applications to association schemes will also be discussed.

Let be a Hopf algebra and an -module algebra. Then one can form the smash product , which is a generalization of the ordinary tensor product (the latter occurs if the action of on is trivial). The case when satisfies a polynomial identity is studied. Appropriate delta sets are introduced and necessary conditions on the action of on in terms of these delta sets for a certain class of algebras are given. The main theorem treats the special case when is a group algebra acting on a Lie superalgebra of characteristic zero. In this case the results obtained on delta sets, in combination with known facts about group algebras and universal enveloping algebras, enable one to give necessary and sufficient conditions for the existence of a polynomial identity in .

In most cases, the process of adaptation of foreign students in a host country is difficult and includes many spheres which they have to adapt to: social, everyday, cultural, educational, linguistic, communication, legal, etc. The central part in this list is undoubtedly taken by the academic adaptation, However, one cannot ignore the need to become involved in a rich ethnocultural environment. The article reveals the role, influence and place of a university library in introducing foreign students of the Kazan (Volga) Federal University to the educational process exemplified by the Scientific Library named after N.I. Lobachevsky of the Kazan (Volga) Federal University. A university library is the intellectual core of the community, a modern information center, whose activities are concentrated on the operational and high-quality library and information support of the scientific and educational process, including students who have come from abroad.

Consider a set I which, in the text which follows, will play the role of a universal set. The different subsets of I form a set on which may be defined three operations: union, intersection and complementation. Endowed with these, the set of subsets of I then constitutes an ‘algebra’, usually known as an “algebra of sets” or an “algebra of classes”. From the definitions of set, subset and those of the three operations above the diverse properties of this algebra can be demonstrated. The more important of these properties will be stated later. They will be justified with the aid of Venn diagrams. However, by way of an exercise the reader can carry out rigorous abstract demonstrations based on the definitions in Chapter 1. It will be shown by an example (relationship E9 below) which type of demonstration can then be made.

PrefaceThe IX Republican Scientific Technical Conference “Low-temperature plasma during the deposition of functional coatings” took place from 5 to 8 November 2017 at the Academy of Sciences of the Republic of Tatarstan and the Kazan Federal University. The conference was chaired by a Member of the Academy of Sciences of the Republic of Tatarstan Nail Kashapov – Professor, Doctor of Technical Sciences– a member of the Scientific and Technical Council of the Ministry of Economy of the Republic of Tatarstan.At the conference, the participants discussed a wide range of issues affecting the theoretical and computational aspects of research problems in the physics and technology of low-temperature plasma. A series of works were devoted to the study of thin films obtained by low-temperature plasma. This year work dedicated to the related field of heat mass transfer in multiphase media and low-temperature plasma was also presented. Of special interest were reports on the exploration of gas discharges with liquid electrolytic electrodes and the study of dusty plasmas.Kashapov Nail, D.Sc., Professor (Kazan Federal University)

Modern logic begins with Boole's The Mathematical Analysis of Logic when the algebra of logic was developed so that classical logic syllogisms were proven as algebraic equations and the turn from the logic of classes to propositional logic was suggested. The emergence was incomplete as Boole algebraised classical logic. Frege in Begriffsschrift replaced Aristotelian subject-predicate propositions by function and argument and displaced syllogisms with an axiomatic propositional calculus using conditionals, modus ponens and the law of substitution. Further Frege provided the breakthrough to lay down the groundwork for the development of quantified logic as well as the logic of relations. He achieved all of this through his innovative formal notations which have remained underrated. Frege hence completed the emergence of modern logic. Both Boole and Frege mathematised logic, but Frege's goal was to logicise mathematics. However the emergence of modern logic in Frege should be detached from his logicism.

Protein engineering, the process by which novel proteins with desired properties are developed, has grown by leaps and bounds since the first examples of protein mutagenesis were described over three decades ago. In the last 30 years, protein engineers have successfully created a wide range of proteins tailored to specific industrial, medical, and research applications. Yet challenges remain that prevent the engineering of complex protein functions on demand. With the aim of creating a forum for the protein engineering community, the inaugural Protein Engineering Canada Conference was held on June 20–22, 2014, in Ottawa, Canada. Over the course of the 2-day conference, 115 protein scientists from over 30 institutions representing 5 countries shared ideas, networked with colleagues, and learned about exciting new research covering a broad array of topics such as enzyme engineering, computational protein design, X-ray crystallography, protein NMR, molecular modeling, and protein evolution. This special issue highlights the breadth of research that was covered during the Protein Engineering Canada Conference by including previously unpublished research articles submitted by speakers as well as some selections that were submitted directly to Protein Science.

Unions of conjunctive queries, also known as select-project-join-union queries, are the most frequently asked queries in relational database systems. These queries are definable by existential positive first-order formulas and are preserved under homomorphisms. A classical result of mathematical logic asserts that the existential positive formulas are the only first-order formulas (up to logical equivalence) that are preserved under homomorphisms on all structures, finite and infinite. The question of whether the homomorphism-preservation theorem holds for the class of all finite structures resisted solution for a long time. It was eventually shown that, unlike other classical preservation theorems, the homomorphism-preservation theorem does hold in the finite. In this article, we show that the homomorphism-preservation theorem holds also for several restricted classes of finite structures of interest in graph theory and database theory. Specifically, we show that this result holds for all classes of finite structures of bounded degree, all classes of finite structures of bounded treewidth, and, more generally, all classes of finite structures whose cores exclude at least one minor.

This paper presents and studies a class of algebras which includes the usual Volterra algebra. Roughly speaking, they relate to the Volterra algebra in the way a general locally compact group relates to R. We show that they can be viewed as quotients of some semigroup algebras introduced by BAKER and BAKER (1). Their sets of nilpotent elements are dense. We investigate the second duals of these algebras and find that most of the properties found in (7) for the biduals of the group algebras L 1 (G) for compact G are retained here.

Given a ternary relation C on a set U and an algebra A, we present a construction of a convolution algebra A(U, C) of U = (U, C) over A. This generalises bothmatrix algebras and algebras obtained from convolution of monoids. To any class of algebras corresponds a class of convolution structures. Our study cases are the classes of commutative, associative, Lie, and Jordan algebras. In each of these classes we give conditions on (U, C) under which A(U, C) is in the same class as A. It turns out that in some situations these conditions are even necessary.

Given a ternary relation C on a set U and an algebra A, we present a construction of a convolution algebra A(U, C) of U = (U, C) over A. This generalises bothmatrix algebras and algebras obtained from convolution of monoids. To any class of algebras corresponds a class of convolution structures. Our study cases are the classes of commutative, associative, Lie, and Jordan algebras. In each of these classes we give conditions on (U, C) under which A(U, C) is in the same class as A. It turns out that in some situations these conditions are even necessary.