Terms Of Algebra Research Articles

The Menger algebra of terms induced by order-decreasing transformations

Let τn be a type of algebras with the n-ary operation symbols, for a fixed integer In this article, we introduce terms of type τn called order-decreasing full terms. We prove that the set of all order-decreasing full terms of type τn is closed under the superposition operation; and hence it forms an algebra denoted by Moreover, we prove that is a Menger algebra of rank n. Finally, we introduce and study order-decreasing full hypersubstitutions and the related order-decreasing full closed identities and order-decreasing full closed varieties.

Length function and characteristic sequences of quadratic algebras

In this paper we study the relations between numerical characteristics of finite dimensional algebras and such classical combinatorial objects as additive chains. We study the behavior of the length function via so-called characteristic sequences of quadratic algebras. As one of our main results, we prove the sharp upper bound for the length of quadratic algebras in terms of the Fibonacci numbers depending on the dimension of the algebra. Moreover, we show that quadratic algebras have the extremal behavior with respect to this bound. In addition, we obtain the description of the set of characteristic sequences for quadratic algebras. Namely, we completely determine the set of combinatorial properties which are satisfied for characteristic sequences of quadratic algebras and show that they belong to the family of additive chains known in combinatorics. Conversely, for a given integer sequence being an additive chain and satisfying these combinatorial properties, we construct a quadratic algebra with a characteristic sequence equal to this sequence.The obtained information on characteristic sequences is then applied to investigate the problem of realizability for the length function. In particular, we determine certain subsets of integers which are not realizable as values of the length function on quadratic algebras.

Coarse median algebras: the intrinsic geometry of coarse median spaces and their intervals

This paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov’s concept of delta -hyperbolicity.

Two-dimensional topological order and operator algebras

We review recent interactions between mathematical theory of two-dimensional topological order and operator algebras, particularly the Jones theory of subfactors. The role of representation theory in terms of tensor categories is emphasized. Connections to two-dimensional conformal field theory are also presented. In particular, we discuss anyon condensation, gapped domain walls and matrix product operators in terms of operator algebras.

Semigroups of Terms, Tree Languages, Menger Algebra of n-Ary Functions and Their Embedding Theorems

The concepts of terms and tree languages are significant tools for the development of research works in both universal algebra and theoretical computer science. In this paper, we establish a strong connection between semigroups of terms and tree languages, which provides the tools for studying monomorphisms between terms and generalized hypersubstitutions. A novel concept of a seminearring of non-deterministic generalized hypersubstitutions is introduced and some interesting properties among subsets of its are provided. Furthermore, we prove that there are monomorphisms from the power diagonal semigroup of tree languages and the monoid of generalized hypersubstitutions to the power diagonal semigroup of non-deterministic generalized hypersubstitutions and the monoid of non-deterministic generalized hypersubstitutions, respectively. Finally, the representation of terms using the theory of n-ary functions is defined. We then present the Cayley’s theorem for Menger algebra of terms, which allows us to provide a concrete example via full transformation semigroups.

ЕЛЕКТРОМАГНІТНІ ПОЛЯ В ЦИЛІНДРИЧНІЙ ІНДУКТОРНІЙ СИСТЕМІ ІЗ ЗОВНІШНІМ КОАКСІАЛЬНИМ БІФІЛЯРНИМ СОЛЕНОЇДОМ

Purpose. Bifilar structures are widely used in modern electrical devices for various purposes. The specific interest is the using of inductor systems with external bifilar coils in the elements of modern metalworking equipment. In particular, it is very important to study the possibility of using such devices as elements of equipment for magnetic-pulse processing of metals. The aim of this research is a derivation of design expressions for theoretical analysis and numerical estimates of the characteristics of electromagnetic processes in a cylindrical inductor system. The case when the inductor is located inside a coaxial solenoid, the winding of which is made in the form of a bifilar with oppositely directed currents is considered. Methodology. Maxwell’s equations with appropriate boundary conditions and Laplace transforms are used to solve this problem. This made it possible to determine the expression for the z-th component of the magnetic field intensity excited in the considered inductor system. Results. It was found that the excited magnetic fluxes are subtracted outside the bifilar coil windings, which leads to a decrease in the resulting field compared to the magnetic field of each of the currents separately. Thus, it is possible to reduce the inductance of the coil as an element of the electrical circuit. It is shown that the formulas obtained for the fields and currents remain valid for the case of unidirectional currents when the sign of the corresponding algebraic term changes. Numerical estimates for the experimental model of the inductor system showed that the induced current as a percentage of the value of the exciting current does not exceed ~ 6.3 %. Originality. The novelty of this work lies in proposing the idea of constructive execution of the inductor system itself, as well as in considering its physical and mathematical model and obtaining calculated expressions for analyzing the ongoing electromagnetic processes with numerical estimates of the characteristics of the excited fields. Practical value. The obtained time dependences for voltages and currents induced in the bifilar winding are applicable depending on the design conditions for various specific designs of the inductor system

Absolute irreducibility of the binomial polynomials

In this paper we investigate the factorization behaviour of the binomial polynomials (xn)=x(x−1)⋯(x−n+1)n! and their powers in the ring of integer-valued polynomials Int(Z). While it is well-known that the binomial polynomials are irreducible elements in Int(Z), the factorization behaviour of their powers has not yet been fully understood. We fill this gap and show that the binomial polynomials are absolutely irreducible in Int(Z), that is, (xn)m factors uniquely into irreducible elements in Int(Z) for all m∈N. By reformulating the problem in terms of linear algebra and number theory, we show that the question can be reduced to determining the rank of, what we call, the valuation matrix of n. A main ingredient in computing this rank is the following number-theoretical result for which we also provide a proof: If n>10 and n, n−1, …, n−(k−1) are composite integers, then there exists a prime number p>2k that divides one of these integers.

Construction of Dirac spinors for electron vortex beams in background electromagnetic fields

Exact solutions of the Dirac equation, a system of four partial differential equations, are rare. The vast majority of them are for highly symmetric stationary systems. Moreover, only a handful of solutions for time dependent dynamics exists. Given the growing number of applications of high energy electron beams interacting with a variety of quantum systems in laser fields, novel methods for finding exact solutions to the Dirac equation are called for. We present a method for building up solutions to the Dirac equation employing a recently introduced approach for the description of spinorial fields and their driving electromagnetic fields in terms of geometric algebras. We illustrate the method by developing several stationary as well as non-stationary solutions of the Dirac equation with well defined orbital angular momentum along the electron's propagation direction. The first set of solutions describe free electron beams in terms of Bessel functions as well as stationary solutions for both a homogeneous and an inhomogeneous magnetic field. The second set of solutions are new and involve a plane electromagnetic wave combined with a generally inhomogeneous longitudinal magnetic field. Moreover, the developed technique allows us to derive general physical properties of the dynamics in such field configurations, as well as provides physical predictions on the self-consistent electromagnetic fields induced by the dynamics.

Wedge-direct sums of group algebras of finite groups

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.

Inflationary Solutions in the Simplest Gravity Model with Conformal Symmetry

We discuss a model of gravity with conformal symmetry appearing in the simplest extension of General Relativity with the Poincar\'e algebra terms. The nonlinear realization of symmetry causes the existence of five scalar fields. Therefore it looks desirable to use them for driving the inflation at the earliest stages of the Universe evolution. It is shown that the evolution of the scale factor doesn't imply accelerated expansion so that there are no inflationary solutions in this model. To drive inflation a more complicated model induced by extension of the Poincare algebra is required.

Variational Embedding for Quantum Many‐Body Problems

AbstractQuantum embedding theories are powerful tools for approximately solving large‐scale, strongly correlated quantum many‐body problems. The main idea of quantum embedding is to glue together a highly accurate quantum theory at the local scale and a less accurate quantum theory at the global scale. We introduce the first quantum embedding theory that is also variational, in that it is guaranteed to provide a one‐sided bound for the exact ground‐state energy. Our method, which we call the variational embedding method, provides a lower bound for this quantity. The method relaxes the representability conditions for quantum marginals to a set of linear and semidefinite constraints that operate at both local and global scales, resulting in a semidefinite program (SDP) to be solved numerically. The accuracy of the method can be systematically improved. The method is versatile and can be applied, in particular, to quantum many‐body problems for both quantum spin systems and fermionic systems, such as those arising from electronic structure calculations. We describe how the proper notion of quantum marginal, sufficiently general to accommodate both of these settings, should be phrased in terms of certain algebras of operators. We also investigate the duality theory for our SDPs, which offers valuable perspective on our method as an embedding theory. As a byproduct of this investigation, we describe a formulation for efficiently implementing the variational embedding method via a partial dualization procedure and the solution of quantum analogues of the Kantorovich problem from optimal transport theory. © 2021 Wiley Periodicals LLC.

Free products of finite-dimensional and other von Neumann algebras in terms of free Araki–Woods factors

We show that any free product of finite-dimensional von Neumann algebras equipped with non-tracial states is isomorphic to a free Araki–Woods factor with its free quasi-free state possibly direct sum a finite-dimensional von Neumann algebra. This gives a complete answer to questions posed by Dykema in [5] and Shlyakhtenko in [10], which had been partially answered by Houdayer in [7] and Ueda in [16]. We also extend this to suitable infinite-dimensional von Neumann algebras with almost periodic states.

On Renormalized Hamiltonian Nets

We propose an abstract framework describing energy-renormalized Hamiltonians in terms of local algebras. Within the framework, we examine the positivity improvingness of the semigroup generated by the renormalized Hamiltonian. As examples, we discuss the renormalized Nelson Hamiltonian and the renormalized Nelson Hamiltonian at fixed total momentum. The characteristic features of our approach are as follows: (i) in contrast with the probabilistic approach in the Schrödinger representation, our method works well in the Fock representation and (ii) the method covers the massless case.

Higher Kac–Moody algebras and symmetries of holomorphic field theories

We introduce a higher dimensional generalization of the affine Kac-Moody algebra using the language of factorization algebras. In particular, on any complex manifold there is a factorization algebra of currents associated to any Lie algebra. We classify local cocycles of these current algebras, and compare them to central extensions of higher affine algebras recently proposed by Faonte-Hennion-Kapranov. A central goal of this paper is to witness higher Kac-Moody algebras as symmetries of a class of holomorphic quantum field theories. In particular, we prove a generalization of the free field realization of an affine Kac-Moody algebra and also develop the theory of q-characters for this class of algebras in terms of factorization homology. Finally, we exhibit the large N behavior of higher Kac-Moody algebras and their relationship to symmetries of non-commutative field theories.

Combinatorics on words, facrordynamics and normal forms

Methods of symbolic dynamics play an essential role in the study of combinatorial properties of words, problems in number theory and the theory of dynamical systems. The paper is devoted to the problems of combinatorics on words, its applications in algebra and dynamical systems. Section 2.1 considers the one-dimensional case using the key example of Sturm’s words. The proof of the criterion for substitutionality of Sturm palindromes using the Rauzy induction is given, the case of one-dimensional facordynamics is considered. Section 2.2 discusses the shift of the torus and the Rauzy fractal that generates the word Tribonacci. The relationship between the periodicity of Rauzy’s schemes and the substitutionality of the word generated by this system is discussed. The implementation of the word Tribonacci through the rearrangement of line segments is given. An approach to the Pisot hypothesis is outlined. Section 2.3 talks about unipotent torus transformations and billiards in polygons. Chapter 3 talks about normal forms and the growth of groups and algebras. Chapter 4 is devoted to Rosie graphs, Gr¨obner bases and co-growth, and algebraic applications. Section 4.1 discusses the results in the combinatorics of multilinear words developed by V. N. Latyshev and the problems he posed. Section 4.2 talks about finitely defined objects and the problems of controlling the relationships that define them. Section 4.3 describes some monomial algebras in terms of uniformly recurrent words. Chapter 5 deals with the problem of height and normal forms.

On Rees closure in some classes of algebras with an operator

In this paper we introduce the concept of Rees closure for subalgebras of universal algebras. We denote by △𝐴 the identity relation on 𝐴. A subalgebra 𝐵 of algebra 𝐴 is called a Rees subalgebra whenever 𝐵2 ∪ △𝐴 is a congruence on 𝐴. A congruence 𝜃 of algebra 𝐴 is called a Rees congruence if 𝜃 = 𝐵2 ∪△𝐴 for some subalgebra 𝐵 of 𝐴. We define a Rees closure operator by mapping arbitrary subalgebra 𝐵 of algebra 𝐴 into the smallest Rees subalgebra that contains 𝐵. It is shown that in the general case the Rees closure does not commute with the operation ∧ on the lattice of subalgebras of universal algebra. Consequently, in the general case, a lattice of Rees subalgebras is not a sublattice of lattice of subalgebras. A non-one-element universal algebra 𝐴 is called a Rees simple algebra if any Rees congruence on 𝐴 is trivial. We characterize Rees simple algebras in terms of Rees closure. Universal algebra is called an algebra with operators if it has an additional set of unary operations acting as endomorphisms with respect to basic operations. We described Rees simple algebras in some subclasses of the class of algebras with one operator and a ternary basic operation. For algebras from these classes, the structure of lattice of Rees subalgebras is described. Necessary and sufficient conditions for the lattice of Rees subalgebras of algebras from these classes to be a chain are obtained.

Instantly Propagating States

The current work considers the sprefield wave functions received as special g-qubit solutions of Maxwell equations in the terms of geometric algebra. I will call such g-qubits spreons or sprefields. The purpose of this article is to analyze behavior of such wave functions in scattering and measurements. It is shown that sprefields are defined through the whole three-dimensional space at all values of the time parameter. They instantly change all their values when get scattered, that is subjected to Clifford translation. In “measurements”, when a sprefield acts on a static geometric algebra element through the Hopf fibration, sprefield collapses and new geometric algebra non static, rotating element is thereby created.

A Well Balanced Finite Volume Scheme for General Relativity

In this work we present a novel second order accurate well balanced (WB) finite volume (FV) scheme for the solution of the general relativistic magnetohydrodynamics (GRMHD) equations and the first order CCZ4 formulation (FO-CCZ4) of the Einstein field equations of general relativity, as well as the fully coupled FO-CCZ4 + GRMHD system. These systems of first order hyperbolic PDEs allow to study the dynamics of the matter and the dynamics of the space-time according to the theory of general relativity. The new well balanced finite volume scheme presented here exploits the knowledge of an equilibrium solution of interest when integrating the conservative fluxes, the nonconservative products and the algebraic source terms, and also when performing the piecewise linear data reconstruction. This results in a rather simple modification of the underlying second order FV scheme, which, however, being able to cancel numerical errors committed with respect to the equilibrium component of the numerical solution, substantially improves the accuracy and long-time stability of the numerical scheme when simulating small perturbations of stationary equilibria. In particular, the need for well balanced techniques appears to be more and more crucial as the applications increase their complexity. We close the paper with a series of numerical tests of increasing difficulty, where we study the evolution of small perturbations of accretion problems and stable TOV neutron stars. Our results show that the well balancing significantly improves the long-time stability of the finite volume scheme compared to a standard one.

Fractional-Order Retinex for Adaptive Contrast Enhancement of Under-Exposed Traffic Images

In this paper, a Fractional-order Retinex (FR) for the adaptive contrast enhancement of Under-Exposed Traffic Images (UETI) is proposed to be achieved by the fractional-order variational method. The disposable reconstructive results of the contrast enhancement of UETI play a significant role in traffic safety and are often taken as intermediate results for the traffic virtual reality and augmented reality of intelligent transportation systems. To this end, this paper proposes a state-ofthe-art application of a promising mathematical method, fractional calculus, to extend the classic integer-order Retinex to the fractional-order one, a FR, which leads to a fractional-order algebraic regularization term and contributes to better conditioning of the reconstruction problem. At first, the fractional-order isotropic equation related to a FR is implemented by the Fractional-order Steepest Descent Method (FSDM). Secondly, the corresponding restrictive fractional-order optimization is achieved. Finally, the capability of a FR to non-linearly preserve complex textural details as well as desired contrast enhancing is validated by experimental analysis, which is a major advantage superior to conventional contrast enhancement algorithms, especially for UETI rich in textural details. The paper gives a novel mathematical approach, fractional calculus, to the family of Retinex algorithms that differs from most of the previous approaches and as such, it represents an interesting theoretical contribution.

Topologies, Ranks, and Closures for Families of Theories. I

We describe topological properties, ranks, closures, and their dynamics for families of theories. Types of topologies for families of theories are characterized. A relationship is established between ranks and topologies for families of theories. Boolean combinations of s-definable families of theories are treated, ranks and degrees with respect to these families are found, and values of the characteristics in question are described. We study closures of families of theories with respect to s-definable subfamilies and their Boolean combinations, properties of closure operators, and also a condition for the existence of a least generating set. Rank values for families of theories are specified in terms of algebras of definable subfamilies.