Basic Algebra Research Articles

The flow method for the Baker-Campbell-Hausdorff formula: exact results

Leveraging techniques from the literature on geometric numerical integration, we propose a new general method to compute exact expressions for the Baker-Campbell-Hausdorff (BCH) formula. In its utmost generality, the method consists in embedding the Lie algebra of interest into a subalgebra of the algebra of vector fields on some manifold by means of an isomorphism, so that the BCH formula for two elements of the original algebra can be recovered from the composition of the flows of the corresponding vector fields. For this reason we call our method the flow method. Clearly, this method has great advantage in cases where the flows can be computed analytically. We illustrate its usefulness on some benchmark examples where it can be applied directly, and discuss some possible extensions for cases where an exact expression cannot be obtained.

The Intuitive Way to Solve Construction Problems in the Dynamic Geometry Environment

The ability to solve geometric construction problems is justly regarded as an essential component of mathematical culture. The dynamic, general nature of objects provided by dynamic geometry systems allows the development of intuitive methods for solving construction problems. The core mathematical concept underlying this approach is the loci method, in which the main element of the required object forms as the intersection of two loci, each one obtained by purposely ignoring part of the conditions of the problem. With GeoGebra, the relevant locus may be visualized and detected in trace mode. Using dynamic geometry this way for problem solving replaces the single object by the infinite locus and is similar to the introduction of variables for solving equations in school algebra. This paper presents some examples of how this approach may be realized along with the results of a small-scale experiment with pre-service mathematics teachers.

Characterization of simple smooth modules

In this paper, we characterize simple smooth modules over some infinite-dimensional Z-graded Lie algebras. More precisely, we prove that if one specific positive root element of a Z-graded Lie algebra g locally finitely acts on a simple g-module V, then V is a smooth g-module. These infinite-dimensional Z-graded Lie algebras include the Virasoro algebra, affine-Virasoro algebras, the (twisted, mirror) Heisenberg-Virasoro algebras, the planar Galilean conformal algebra, and many others. This result for untwisted affine Kac-Moody algebras holds unless we change the condition from “locally finitely” to “locally nilpotently”. We also show that these are not the case for the Heisenberg algebra.

Development of the Method of Averaging in Clifford Geometric Algebras

We develop the method of averaging in Clifford (geometric) algebras suggested by the author in previous papers. We consider operators constructed using two different sets of anticommuting elements of real or complexified Clifford algebras. These operators generalize Reynolds operators from the representation theory of finite groups. We prove a number of new properties of these operators. Using the generalized Reynolds operators, we give a complete proof of the generalization of Pauli’s theorem to the case of Clifford algebras of arbitrary dimension. The results can be used in geometry, physics, engineering, computer science, and other applications.

One dimensional representations of finite W-algebras, Dirac reduction and the orbit method

In this paper we study the variety of one dimensional representations of a finite W-algebra attached to a classical Lie algebra, giving a precise description of the dimensions of the irreducible components. We apply this to prove a conjecture of Losev describing the image of his orbit method map. In order to do so we first establish new Yangian-type presentations of semiclassical limits of the W-algebras attached to distinguished nilpotent elements in classical Lie algebras, using Dirac reduction.

Automorphism groups of the constituent graphs of integral distance graphs

In this paper, we consider the automorphism groups of Cayley graphs which are a basis of a complete Boolean algebra of strongly regular graphs, one of such graph is the integral distance graph Γ . The automorphism groups of the integral distance graphs Γ were determined by Kovács and Ruff in 2014 and by Kurz in 2009. Our point of interest will be restricted to the one determined by Kovács and Ruff. Here we consider the automorphism groups of subgraphs Γ 1 and Γ 2 , which inherit the properties of Γ . It will be shown that Aut Γ is isomorphic to Aut Γ 2 and that Aut Γ 1 is isomorphic to S F q 2 ≀ S [ 2 ] .

Countably Generated Algebras of Analytic Functions on Banach Spaces

In the paper, we study various countably generated algebras of entire analytic functions on complex Banach spaces and their homomorphisms. Countably generated algebras often appear as algebras of symmetric analytic functions on Banach spaces with respect to a group of symmetries, and are interesting for their possible applications. Some conditions of the existence of topological isomorphisms between such algebras are obtained. We construct a class of countably generated algebras, where all normalized algebraic bases are equivalent. On the other hand, we find non-isomorphic classes of such algebras. In addition, we establish the conditions of the hypercyclicity of derivations in countably generated algebras of entire analytic functions of the bounded type. We use methods from the theory of analytic functions of several variables, the theory of commutative Fréchet algebras, and the theory of linear dynamical systems.

SCHEME OF FUNCTION STUDY AND METHODOLOGICAL APPROACH OF SKETCHING GRAPHS OF THE FUNCTIONS IN THE SCHOOL COURSE OF MATHEMATICAL ANALYSIS

The article examines one of the most important topics of the high school course "Elements of Algebra and Mathematical Analysis": scheme of function study and methodological approaches to graph construction. In RA the methods of sketching graphs of function are studied during the high school course "Elements of Algebra and Mathematical Analysis" in two stages: without using the concept of derivative (10th grade) and with the use of the derivative (11th grade) [1, 2]. The derivative is introduced in the 11th grade during the step-by-step study of a function, which facilitates the study of the properties of the function. However, after analyzing the function with the recommended steps, the students sometimes find it difficult to sketch the graph of the function because they encounter a number of difficulties that are insurmountable due to the lack of relevant knowledge. This article recommends introducing the concepts of function convexity (concavity), the inflection point during the teaching of second-order derivatives in the 11th grade in basic mathematics streams of the high school course "Elements of Algebra and Mathematical Analysis", which will significantly simplify the process of sketching the graph of the function. As a result, the course will be more comprehensive, and the outline of the function study and sketching graphs will be more understandable to high school students

Free special Gelfand–Dorfman algebra

A Gelfand–Dorfman algebra is called special if it can be embedded into a differential Poisson algebra. We find a new basis of the free Novikov algebra. With its help, we construct the monomial basis of the free special Gelfand–Dorfman algebra.

A structure theorem for streamed information

We identify the free half shuffle algebra of Schützenberger [31] with an algebra of real-valued functionals on paths, where the half shuffle emulates the integration of a functional against another. We then provide two, to our knowledge, new identities in arity 3 involving its commutator (area), and show that these are sufficient to recover the Zinbiel and Tortkara identities introduced by Dzhumadil'daev [11]. We then use these identities to provide a simple proof of the main result of Diehl et al. [8], namely that any element of the free half shuffle algebra can be expressed as a polynomial over iterated areas.Moreover, we consider minimal sets of Hall iterated integrals defined through the recursive application of the half shuffle product to Hall trees. Leveraging the duality between this set of Hall integrals and classical Hall bases of the free Lie algebra, we prove using combinatorial arguments that any element of the free half shuffle algebra can be written uniquely as a polynomial over Hall integrals. We interpret this result as a structure theorem for streamed information, loosely analogous to the unique prime factorisation of integers, allowing to split any real valued function on streamed data into two parts: a first that extracts and packages the streamed information into recursively defined atomic objects (Hall integrals), and a second that evaluates a polynomial function in these objects without further reference to the original stream. The question of whether a similar result holds if Hall integrals are replaced by Hall areas is left as an open conjecture.Finally, we construct a canonical, but to our knowledge, new decomposition of the free half shuffle algebra as shuffle power series in the greatest letter of the original alphabet with coefficients in a sub-algebra freely generated by a new alphabet with an infinite number of letters. We use this construction to provide a second proof of our structure theorem.

Biunit pairs in semiheaps and associated semigroups

The notion of neutral element generalizes to a pair of elements in ternary algebras. Biunit pairs are introduced as pairs of elements in a semiheap that generalize the notion of Mal’cev element. In order to generalize the known correspondences between semiheaps and certain kinds of semigroups, families of functions generalizing involutions and conjugations, called switches and warps, are investigated. The main theorem establishes that there is a one-to-one correspondence between monoids equipped with a particular switch and semiheaps with a fixed biunit pair. This generalizes the celebrated result in semiheap theory that gives a one-to-one correspondence between involuted monoids and semiheaps with a fixed biunit element. A novel, previously undocumented, algebra is motivated by this result: diheaps are introduced as semiheaps whose elements belong to biunit pairs, which generalize the well-known case of heaps. Diheaps are of great interest since they are shown to be isomorphic to heaps only when they are heaps themselves and explicit non-heap examples are constructed from abelian groups and hypermatrices.

On Some Classes of Bases in Finite-Dimensional Lie Algebras

On Some Classes of Bases in Finite-Dimensional Lie Algebras

On the need to build an alternative financial system

Subject. The article underpins of the need to build a national financial system which will fully guarantee addressing socio-economic problems. Objectives. The purpose is to create fundamental institutions for development of national concepts of economic and financial models, pricing methodologies, approaches designed to build a State model of a socially oriented economy that guarantees the functioning of all branches of financial and economic activity, and military, economic, and other types of national integrated security. Methods. The paper uses tensor notation involving the use of elements of linear algebra, including operations with matrices and vectors, interval mathematics, and a set of methods and principles of scientific research and complex logical analysis and modeling of product and financial flows. Results. We show the possibility of obtaining the country's resources for free when creating a shortage of means of payment under implementation of the basic principle of a market economy, i.e. free pricing. The paper offers financial and economic models of the main participants of financial and economic activity: legal entities, individuals, and State-owned enterprises and organizations. On their basis, we demonstrate closed cycles of money circulation, which formed the basis for building a national financial model. We propose to construct financial models using interval analysis, which correlates with the specifics of measured parameters in the financial and economic sphere. Conclusions. The suggested approach is confirmed by the results of trial calculations. It has practical significance not only for the Russian economy, but can also be used in other countries.

A note on saturated distinguished chains of length one

Let [Formula: see text] be a Henselian valued field. In this paper, we investigate algebraic elements [Formula: see text] having saturated distinguished chains of length one over [Formula: see text]. We provide some characterizations for them. In particular, we characterize all irreducible polynomials over [Formula: see text] whose roots have saturated distinguished chains of length one. Moreover, using the properties of such elements we give various necessary, sufficient, or both conditions for the metric invariants of algebraic elements to be equal.

Cohomology and deformations of compatible Hom-Lie algebras

In this paper, we consider compatible Hom-Lie algebras as a twisted version of compatible Lie algebras. Compatible Hom-Lie algebras are characterized as Maurer-Cartan elements in a suitable bidifferential graded Lie algebra. We also define a cohomology theory for compatible Hom-Lie algebras generalizing the recent work of Liu, Sheng and Bai. As applications of cohomology, we study abelian extensions and deformations of compatible Hom-Lie algebras.

Rebalance Your Portfolio Without Selling

Summary How do you bring your assets as close as possible to your target allocation by only investing a fixed amount of additional funds, and not selling any assets? We look at two versions of this problem which have simple, closed form solutions revealed by basic calculus and algebra.

A Method for Improving the Caching Strategy for Computing Systems with Shared Memory

This paper considers the problem of increasing the software efficiency in terms of reducing the costs of their development and operation in the process of solving production and research tasks. We have analysed the existing approaches to solving this problem by example of parameterized algorithms for implementing mVm (matrix—vector multiplication) and MMM (matrix—matrix multiplication)) BLAS (Basic Linear Algebra Subroutines) operations. To achieve the goal of increasing the software efficiency, we proposed a new design method, designed to improve data caching algorithms in the software development for computing systems with a hierarchical memory structure. Using the proposed design procedure, we developed an analytical approach to evaluating the software effectiveness from the point of view of using a memory subsystem with a hierarchical structure is implemented. We applied the proposed method to the two-sided Householder transformation for the task of reducing the general form matrix to the Hessenberg form. Then we presented new algorithms for solving the problem, which are optimized variants of the Householder classical transformation: Row-Oriented Householder and Single-Pass Householder. The use of these algorithms can significantly reduce the software execution time. Computational experiments were carried out on a parallel computing system with shared memory, which is one of the nodes of the computing cluster of the Volgograd State Technical University. We made a comparison of the software execution time that reduce general-form matrices to Hessenberg form, written using the proposed algorithms and using the LAPACKE_dgehrd() function of the Intel MKL library. The conclusions made in the work are confirmed by the results of the conducted computational experiments.

Polynomial solutions of generalized quartic anharmonic oscillators

This paper deals with the partial solution of the energy eigenvalue problem for generalized symmetric quartic oscillators. Algebraization of the problem is achieved by expressing the Schrödinger operator in terms of the generators of a nilpotent group, which we call the quartic group. Energy eigenvalues are then seen to depend on the values of the two Casimir operators of the group. This dependence exhibits a scaling law which follows from the scaling properties of the group generators. Demanding that the potential gives rise to polynomial solutions in a particular Lie algebra element puts constraints on the four potential parameters, leaving only two of them free. For potentials satisfying such constraints, at least one of the energy eigenvalues and the corresponding eigenfunctions can be obtained in closed analytic form by pure algebraic means. With our approach, we extend the class of quasi-exactly solvable quartic oscillators which have been obtained in the literature by means of the more common textrm{sl}(2,{mathbb {R}}) algebraization. Finally, we show how solutions of the generalized quartic oscillator problem give rise to solutions for a charged particle moving in particular non-constant electromagnetic fields.

Another Geometric Interpretation of Cramer’s Rule

Summary We develop a geometric interpretation of Cramer’s rule as a generalization of projection onto orthogonal basis vectors using the rows of the adjugate. This interpretation makes connections between elementary linear algebra concepts like the solution to linear equations, inner products, and projections. Such connections are useful for introducing broader concepts related to Hilbert spaces and geometric algebras like Grassman algebra. Such connections were essential for the author’s mathematical education as an engineer.

Revisiting Takahashi's inversion theorem in discrete symmetry-based dual frameworks

The so-called Takahashi's Inversion Theorem, the reconstruction of a given spinor based on its bilinear covariants [1], are re-examined, considering alternative dual structures. In contrast to the classical results, where the Dirac dual structure plays the central role, new duals are built using the discrete symmetries C,P,T. Their combinations are also taken into account. Furthermore, the imposition of a new adjoint structure led us also to re-examine the representation of the Clifford algebra basis elements, uncovering new bilinear structures and a new Fierz aggregate. Those results might help construct theories for new beyond standard model fields.