Algebraic Series Research Articles

Exceptional Periodicity and Magic Star algebras: Generalized roots, gradings, Hermitian Vinberg T-algebras and their derivations

We introduce countably infinite series of finite dimensional generalizations of the exceptional Lie algebras: in fact, each exceptional Lie algebra (but g2) is the first element of an infinite series of finite dimensional algebras, which we name Magic Star algebras. All these algebras (but the first elements of the infinite series) are not Lie algebras, but nevertheless they have remarkable similarities with many characterizing features of the exceptional Lie algebras; they also enjoy a kind of periodicity (inherited by Bott periodicity), which we name Exceptional Periodicity. We analyze the graded algebraic structures arising in a certain projection (named Magic Star projection) of the generalized root systems pertaining to Magic Star algebras, and we highlight the occurrence of a class of rank-3, Hermitian matrix (special Vinberg T)-algebras (which we call H algebras) on each vertex of such a projection. We then focus on the Magic Star algebra f4(n), which generalizes the non-simply laced exceptional Lie algebra f4, and deserves a treatment apart. Finally, we compute the Lie algebra of the inner derivations of the H algebras, pointing out the enhancements occurring for each first element of the series of Magic Star algebras, thus retrieving the result known for the derivations of cubic simple Jordan algebras.

Pointwise Convergence of Noncommutative Fourier Series

This paper is devoted to the study of pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this paper studies the refined counterpart of pointwise convergence of these Fourier series. As a key ingredient, we develop a noncommutative bootstrap method and establish a general criterion of maximal inequalities for approximate identities of noncommutative Fourier multipliers. Based on this criterion, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to 1 1 pointwise, so that the associated Fourier multipliers on noncommutative L p L_p -spaces satisfy the pointwise convergence for all p > 1 p>1 . In a similar fashion, we also obtain results for a large subclass of groups (as well as quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fejér and Bochner-Riesz means in the noncommutative setting. Our approach heavily relies on the noncommutative ergodic theory in conjunction with abstract constructions of Markov semigroups, inspired by quantum probability and geometric group theory. Finally, we also obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.

Theta Series for Quantum Loop Algebras and Yangians

Theta Series for Quantum Loop Algebras and Yangians

Hyperpolyadic Structures

We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element, we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley–Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, which is proportional to the intermediate arities, and which are not isomorphic to those obtained by the previous constructions. Second, a new polyadic product of vectors in any vector space is defined, which is consistent with the polyadization procedure using vectorization. Endowed with this introduced product, the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process (we call it the “imaginary tower”), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call “half-quaternions” and “half-octonions”. The latter are not the subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced “half-quaternion” norm, we obtain the ternary analog of the sum of two squares identity. We show that the ternary division algebra of imaginary “half-octonions” is unitless and totally associative.

A new method for solving the linearized 1D Vlasov–Poisson system yielding a new class of solutions

We describe a new method for solving the linearized 1D Vlasov–Poisson system by using properties of Cauchy-type integrals. Our method remedies critical flaws of the two standard methods, reveals a previously unrecognized Gaussian-in-time-like decay, and can also account for an externally applied electric field. The Landau approximation involves deforming the Bromwich contour around the poles closest to the real axis due to the analytically continued dielectric function, finding the long-time behavior for a stable system: Landau damping. Jackson's generalization encircles all poles while sending the contour to infinity, assuming its contribution vanishes, which is not true in general. This gives incorrect solutions for physically reasonable configurations and can exhibit pathological behavior, of which we show examples. The van Kampen method expresses the solution for a stable equilibrium as a continuous superposition of waves, resulting in an opaque integral. Case's generalization includes unstable systems and predicts a decaying discrete mode for each growing discrete mode, an apparent contradiction to both the Jackson solution and ours. We show, without imposing additional constraints, that the decaying modes are never present in the time evolution due to an exact cancellation with part of the continuum. Our solution is free of integral expressions, is obtained using algebra and Laurent series expansions, does not rely on analytic continuations, and results in a correct asymptotically convergent form in the case of infinite sums. The analysis used can be readily applied in higher-dimensional, electromagnetic systems and also provides a new technique for evaluating certain inverse Laplace transforms.

Difference of Hilbert series of homogeneous monoid algebras and their normalizations

Let Q be an affine monoid, k[Q]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Bbbk [Q]$$\\end{document} the associated monoid k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Bbbk $$\\end{document}-algebra, and k[Q¯]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Bbbk [\\overline{Q}]$$\\end{document} its normalization, where we let k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Bbbk $$\\end{document} be a field. We discuss a difference of the Hilbert series of k[Q]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Bbbk [Q]$$\\end{document} and k[Q¯]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Bbbk [\\overline{Q}]$$\\end{document} in the case where k[Q]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Bbbk [Q]$$\\end{document} is homogeneous (i.e., standard graded). More precisely, we prove that if k[Q]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Bbbk [Q]$$\\end{document} satisfies Serre’s condition (S2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(S_2)$$\\end{document}, then the degree of the h-polynomial of k[Q]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Bbbk [Q]$$\\end{document} is always greater than or equal to that of k[Q¯]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Bbbk [\\overline{Q}]$$\\end{document}. Moreover, we also show counterexamples of this statement if we drop the assumption (S2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(S_2)$$\\end{document}.

Models of representations for classical series of Lie algebras

By a model of representations of a Lie algebra we mean a representation which is a direct sum of all irreducible finite-dimensional representations taken with multiplicity $1$. An explicit construction of a model of representations for all classical series of simple Lie algebras is given. This construction is generic for all classical series of Lie algebras. The space of the model is constructed as the space of polynomial solutions of a system of partial differential equations, where the equations are constructed form relations between minors of matrices taken from the corresponding Lie group. This system admits a simplification very close to the GKZ system, which is satisfied by $A$-hypergeometric functions.

Models of representations for classical series of Lie algebras

Моделью представлений алгебры Ли называется представление, являющее прямой суммой всех еe конечномерных неприводимых представлений, взятых с кратностью $1$. В работе даeтся явная конструкция некоторой модели, единообразная для всех классических серий алгебр Ли. Пространство модели строится как пространство полиномиальных решений некоторой системы уравнений в частных производных, уравнения в которой строятся по соотношениям между минорами матриц из соответствующей группы Ли. При этом данная система имеет упрощение, которое родственно системе ГКЗ, которой удовлетворяют $A$-гипергеометрические функции.

Graded trace generating functions of Aut(ΓKn) acting on A(ΓKn)

In this paper, we describe the algebras [Formula: see text] associated with the Hasse graph of the poset of the faces of the complete graph at n vertices [Formula: see text]. Present the relations that define this algebra, calculate the Hilbert series of quadratic grade algebra [Formula: see text] and the graded trace generating functions of [Formula: see text] acting on [Formula: see text] and show that [Formula: see text] is a Koszul algebra. The methodology adopted in this paper consists of building the algebra [Formula: see text] using [I. Gelfand, V. Retakh, S. Serconek and R. L. Wilson, On a class of algebras associated to directed graphs, Selecta Math. 11(2) (2005) 281], giving a presentation based on what was established by [V. Retakh, S. Serconek and R. L. Wilson, On a class of koszul algebras associated to directed graphs, J. Algebra 304(2) (2006) 1114–1129] in terms of the vertices of the graph [Formula: see text]. After that, we use [V. Retakh, S. Serconek and R. L. Wilson, Hilbert series of algebras associated to directed graphs, J. Algebra 312(1) (2007) 142–151] to determine the Hilbert series of [Formula: see text] and use [9] again to show that this algebra is a Koszul algebra. We use [J. Caldeira, A. De Souza Lima and J. Eder Salvador De Vasconcelos, Representations of automorphism groups of algebras associated to star polygons, J. Algebra Appl. 18(10) (2019) 1950197; C. Duffy, Representations of [Formula: see text] acting on homogeneous components of [Formula: see text] and [Formula: see text], Adv. Appl. Math. 42(1) (2009) 94–122] to determine the generating functions of the graduated trace of [Formula: see text] acting on [Formula: see text].

Formula omitted]-Stirling numbers in type [formula omitted

Stirling numbers of the first and second kind, which respectively count permutations in the symmetric group and partitions of a set, have found extensive application in combinatorics, geometry, and algebra. We study analogues and q-analogues of these numbers corresponding to the Coxeter group of type B. In particular, we show how they are related to complete homogeneous and elementary symmetric polynomials; demonstrate how they q-count signed partitions and permutations; compute their ordinary, exponential, and q-exponential generating functions; and prove various identities about them. Ordered analogues of the q-Stirling numbers of the second kind have recently appeared in conjectures of Zabrocki and of Swanson–Wallach concerning the Hilbert series of certain super coinvariant algebras. We provide conjectural bases for these algebras and show that they have the correct Hilbert series.

VaR Estimation with Quantum Computing Noise Correction Using Neural Networks

In this paper, we present the development of a quantum computing method for calculating the value at risk (VaR) for a portfolio of assets managed by a finance institution. We extend the conventional Monte Carlo algorithm to calculate the VaR of an arbitrary number of assets by employing random variable algebra and Taylor series approximation. The resulting algorithm is suitable to be executed in real quantum computers. However, the noise affecting current quantum computers renders them almost useless for the task. We present a methodology to mitigate the noise impact by using neural networks to compensate for the noise effects. The system combines the output from a real quantum computer with the neural network processing. The feedback is used to fine tune the quantum circuits. The results show that this approach is useful for estimating the VaR in finance institutions, particularly when dealing with a large number of assets. We demonstrate the validity of the proposed method with up to 139 assets. The accuracy of the method is also proven. We achieved an error of less than 1% in the empirical measurements with respect to the parametric model.

On Abel’s Problem and Gauss Congruences

Abstract A classical problem due to Abel is to determine if a differential equation $y^{\prime}=\eta y$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $\eta $ is a given algebraic function over $\mathbb C(x)$. Risch designed an algorithm that, given $\eta $, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when $\eta $ admits a Puiseux expansion with rational coefficients at some point in $\mathbb C\cup \{\infty \}$, which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of $y^{\prime}=\eta y$ if and only if the coefficients of the Puiseux expansion of $x\eta (x)$ at $0$ satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations $y^{\prime}=\eta y$ with an algebraic solution when $x\eta (x)$ is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present two other applications, namely to diagonals of rational fractions and to directed two-dimensional walks.

Walks avoiding a quadrant and the reflection principle

We continue the enumeration of plane lattice walks with small steps avoiding the negative quadrant, initiated by the first author in 2016. We solve in detail a new case, namely the king model where all eight nearest neighbour steps are allowed. The associated generating function is proved to be the sum of a simple, explicit D-finite series (related to the number of walks confined to the first quadrant), and an algebraic one. This was already the case for the two models solved by the first author in 2016. The principle of the approach is also the same, but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree.We expect a similar algebraicity phenomenon to hold for the seven Weyl step sets, which are those for which walks confined to the first quadrant can be counted using the reflection principle. With this paper, this is now proved for three of them. For the remaining four, we predict the D-finite part of the solution, and in three of the four cases, give evidence for the algebraicity of the remaining part.

Trigonometric Lie algebras, affine Kac-Moody Lie algebras, and equivariant quasi modules for vertex algebras

In this paper, we study a family of infinite-dimensional Lie algebras XˆS, where X stands for the type: A,B,C,D, and S is an abelian group, which generalize the A,B,C,D series of trigonometric Lie algebras. Among the main results, we identify XˆS with what are called the covariant algebras of the affine Lie algebra LSˆ with respect to some automorphism groups, where LS is an explicitly defined associative algebra viewed as a Lie algebra. We then show that restricted XˆS-modules of level ℓ naturally correspond to equivariant quasi modules for affine vertex algebras related to LS. Furthermore, for any finite cyclic group S, we completely determine the structures of these four families of Lie algebras, showing that they are essentially affine Kac-Moody Lie algebras of certain types.

Metrics related to oscillator algebras

A Lie algebra is said to be metric if it admits a symmetric, invariant, and non-degenerate bilinear form. The harmonic oscillator algebra, which arises in the quantum mechanical description of a harmonic oscillator, is the smallest solvable non-abelian metric example. This algebra is the first step of a countable series of solvable Lie algebras that support invariant Lorentzian forms. Generalizing this situation, in this paper, we arrive at the oscillator Lie -algebras as double extensions of metric spaces. The aim of this paper is to present some structural features, invariant metrics, and derivations of this class of algebras and to explore their possibilities of being extended to mixed metric Lie algebras.

ETNOMATEMATIKA: EKSPLORASI KESENIAN MUSIK CALUNG BANYUMASAN SEBAGAI SUMBER PEMBELAJARAN MATEMATIKA

The selection of material that is contextual and based on student culture is essential for improving the quality of learning mathematics. This study aims to examine, explore, and explore Calung Banyumasan music as a source of learning mathematics that is contextual and easy to understand. This research is qualitative with an ethnographic approach because it examines a particular cultural system (Banyumasan art) from an ethnomathematics perspective. The subjects in this study were three humanists, practitioners, and mathematicians related to Calung Banyumasan musical arts, and the research object was Calung Banyumasan instruments. Data collection methods use in-depth interviews, observation, documentation, and field notes. The data analysis method was carried out descriptively based on the results of the meaning and translation of the phenomena found based on the results of the informant's conception, the results of observations combined with the researcher's language after an in-depth understanding was carried out. Triangulation and Forum Group, Discussion was used to test the data's validity and the effects of data analysis. The research results show that Calung Banyumasan music art has mathematical wealth, especially in Geometry (parallelism, congruence, plane shapes, and curved side shapes) and Algebra (compound functions, arithmetic sequences, series, and inverse comparisons of values). Besides that, Calung Banyumasan music also has a lot of valuable philosophical content for human life.

Cosimplicial meromorphic functions cohomology on complex manifolds

Developing ideas of [B. L. Feigin, Conformal field theory and cohomologies of the Lie algebra of holomorphic vector fields on a complex curve, in Proc. Int. Congress of Mathematicians (Kyoto, 1990), Vols. 1 and 2 (Mathematical Society of Japan, Tokyo, 1991), pp. 71–85], we introduce canonical cosimplicial cohomology of meromorphic functions for infinite-dimensional Lie algebra formal series with prescribed analytic behavior on domains of a complex manifold [Formula: see text]. Graded differential cohomology of a sheaf of Lie algebras [Formula: see text] via the cosimplicial cohomology of [Formula: see text]-formal series for any covering by Stein spaces on [Formula: see text] is computed. A relation between cosimplicial cohomology (on a special set of open domains of [Formula: see text]) of formal series of an infinite-dimensional Lie algebra [Formula: see text] and singular cohomology of auxiliary manifold associated to a [Formula: see text]-module is found. Finally, multiple applications in conformal field theory, deformation theory, and in the theory of foliations are proposed.

Forecasting methodology with structural auto-adaptive intelligent grey models

Accurate mid- and long-term petroleum products (PP) consumption forecasting is vital for strategic reserve management and energy planning. In order to address the issue of energy forecasting, a novel structural auto-adaptive intelligent grey model (SAIGM) is developed in this paper. To start with, a novel time response function for predictions that corrects the main weaknesses of the traditional grey model is established. Then, the optimal parameter values are calculated using SAIGM to increase adaptability and flexibility to deal with a variety of forecasting dilemmas. The viability and performance of SAIGM are examined with both ideal and real-world data. The former is constructed from algebraic series while the latter is made up Cameroon's PP consumption data. With its ingrained structural flexibility, SAIGM yields forecasts with RMSE of 3.10 and 1.54% MAPE. The proposed model performs better than competing intelligent grey systems that have been developed to date and is thus a valid forecasting tool that can be used to track the growth of Cameroon's PP demand.•The ability of SAIGM enhances the forecasting power of intelligent grey models to fully extracting the laws of a system, no matter the data specifications.•SAIGM is extended to include quasi-exponential series by addressing structural flexibility and parametrization concerns.•Input attributes determination and data preprocessing are not required for the proposed model.

Computation of Algebraic Expressions and Geometric Series with Radicals

Computation of Algebraic Expressions and Geometric Series with Radicals

Symmetries of Non-Linear ODEs: Lambda Extensions of the Ising Correlations

This paper provides several illustrations of the numerous remarkable properties of the lambda extensions of the two-point correlation functions of the Ising model, shedding some light on the non-linear ODEs of the Painlevé type they satisfy. We first show that this concept also exists for the factors of the two-point correlation functions focusing, for pedagogical reasons, on two examples, namely C(0,5) and C(2,5) at ν=−k. We then display, in a learn-by-example approach, some of the puzzling properties and structures of these lambda extensions: for an infinite set of (algebraic) values of λ these power series become algebraic functions, and for a finite set of (rational) values of lambda they become D-finite functions, more precisely polynomials (of different degrees) in the complete elliptic integrals of the first and second kind K and E. For generic values of λ these power series are not D-finite, they are differentially algebraic. For an infinite number of other (rational) values of λ these power series are globally bounded series, thus providing an example of an infinite number of globally bounded differentially algebraic series. Finally, taking the example of a product of two diagonal two-point correlation functions, we suggest that many more families of non-linear ODEs of the Painlevé type remain to be discovered on the two-dimensional Ising model, as well as their structures, and in particular their associated lambda extensions. The question of their possible reduction, after complicated transformations, to Okamoto sigma forms of Painlevé VI remains an extremely difficult challenge.