Algebraic Series Research Articles

On the automaticity of the Hankel determinants of a family of automatic sequences

Hankel determinants and automatic sequences are two classical subjects widely studied in Mathematics and Theoretical Computer Science. However, these two topics were considered totally independently, until in 1998, when Allouche, Peyrière, Wen and Wen proved that all the Hankel determinants of the Thue-Morse sequence are nonzero. This property allowed Bugeaud to prove that the irrationality exponents of the Thue-Morse-Mahler numbers are exactly 2. Since then, the Hankel determinants of several other automatic sequences, in particular, the paperfolding sequence, the Stern sequence, the period-doubling sequence, are studied by Coons, Vrbik, Guo, Wu, Wen, Bugeaud, Fu, Han, Fokkink, Kraaikamp, and Shallit. On the other hand, it is known that the Hankel determinants of a rational power series are ultimately zero, and the Hankel determinants of a quadratic power series over finite fields are ultimately periodic. It is therefore natural to ask if we can obtain similar results about the Hankel determinants of algebraic series. In the present paper, we provide a partial answer to this question by establishing the automaticity of the reduced Hankel determinants modulo 2 of a family of automatic sequences. As an application of our result, we give upper bounds for the irrationality exponent of a family of automatic numbers.

Noncommutative algebras, context-free grammars and algebraic Hilbert series

In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of unambiguous context-free grammars for this purpose. We also provide examples of finitely presented graded algebras whose corresponding leading monomial algebras belong to the proposed class and hence possess algebraic Hilbert series.

NONREPRESENTABLE RELATION ALGEBRAS FROM GROUPS

AbstractA series of nonrepresentable relation algebras is constructed from groups. We use them to prove that there are continuum many subvarieties between the variety of representable relation algebras and the variety of coset relation algebras. We present our main construction in terms of polygroupoids.

The ascending central series of nilpotent Lie algebras with complex structure

We obtain several restrictions on the terms of the ascending central series of a nilpotent Lie algebra g \mathfrak {g} under the presence of a complex structure J J . In particular, we find a bound for the dimension of the center of g \mathfrak {g} when it does not contain any non-trivial J J -invariant ideal. Thanks to these results, we provide a structural theorem describing the ascending central series of 8-dimensional nilpotent Lie algebras g \mathfrak {g} admitting this particular type of complex structure J J . Since our method is constructive, it allows us to describe the complex structure equations that parametrize all such pairs ( g , J ) (\mathfrak {g}, J) .

Computing noncommutative Hilbert series

We propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory of regular languages, we provide conditions when the methods are effective and hence the Hilbert series have a rational sum. Efficient variants of the methods are also developed for the truncations of infinite-dimensional algebras which provide approximations of possibly irrational Hilbert series. Moreover, we provide a characterization of the finite-dimensional algebras in terms of the nilpotency of a key matrix involved in the computations. Finally, we present a well-tested and complete implementation for the computation of graded and multigraded Hilbert series which has been developed in the kernel of the computer algebra system Singular (for the details, see preprint[1]).

TH YAU NUMBER OF ISOLATED HYPERSURFACE SINGULARITIES AND AN INEQUALITY CONJECTURE

AbstractLet $V$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)$. The Yau algebra $L(V)$ is defined to be the Lie algebra of derivations of the moduli algebra $A(V):={\mathcal{O}}_{n}/(f,\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{1},\ldots ,\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{n})$, that is, $L(V)=\text{Der}(A(V),A(V))$. It is known that $L(V)$ is finite dimensional and its dimension $\unicode[STIX]{x1D706}(V)$ is called the Yau number. We introduce a new series of Lie algebras, that is, $k$th Yau algebras $L^{k}(V)$, $k\geq 0$, which are a generalization of the Yau algebra. The algebra $L^{k}(V)$ is defined to be the Lie algebra of derivations of the $k$th moduli algebra $A^{k}(V):={\mathcal{O}}_{n}/(f,m^{k}J(f)),k\geq 0$, that is, $L^{k}(V)=\text{Der}(A^{k}(V),A^{k}(V))$, where $m$ is the maximal ideal of ${\mathcal{O}}_{n}$. The $k$th Yau number is the dimension of $L^{k}(V)$, which we denote by $\unicode[STIX]{x1D706}^{k}(V)$. In particular, $L^{0}(V)$ is exactly the Yau algebra, that is, $L^{0}(V)=L(V),\unicode[STIX]{x1D706}^{0}(V)=\unicode[STIX]{x1D706}(V)$. These numbers $\unicode[STIX]{x1D706}^{k}(V)$ are new numerical analytic invariants of singularities. In this paper we formulate a conjecture that $\unicode[STIX]{x1D706}^{(k+1)}(V)>\unicode[STIX]{x1D706}^{k}(V),k\geq 0.$ We prove this conjecture for a large class of singularities.

On the new k-th Yau algebras of isolated hypersurface singularities

Let V be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $$f: ({\mathbb {C}}^n, 0)\rightarrow ({\mathbb {C}}, 0)$$. The Yau algebra L(V) is defined to be the Lie algebra of derivations of the moduli algebra $$A(V):= {\mathcal {O}}_n/(f, \frac{\partial f}{\partial x_1},\cdots , \frac{\partial f}{\partial x_n})$$, i.e., $$L(V)=\text {Der}(A(V), A(V))$$ and plays an important role in singularity theory. It is known that L(V) is a finite dimensional Lie algebra and its dimension $$\lambda (V)$$ is called Yau number. In this article, we generalize the Yau algebra and introduce a new series of k-th Yau algebras $$L^k(V)$$ which are defined to be the Lie algebras of derivations of the moduli algebras $$A^k(V) = {\mathcal {O}}_n/(f, m^k J(f)), k\ge 0$$, i.e., $$L^k(V)=\text {Der}(A^k(V), A^k(V))$$ and where m is the maximal ideal of $${\mathcal {O}}_n$$. In particular, it is Yau algebra when $$k=0$$. The dimension of $$L^k(V)$$ is denoted by $$\lambda ^k(V)$$. These numbers i.e., k-th Yau numbers $$\lambda ^k(V)$$, are new numerical analytic invariants of an isolated singularity. In this paper we studied these new series of Lie algebras $$L^k(V)$$ and also compute the Lie algebras $$L^1(V)$$ for fewnomial isolated singularities. We also formulate a sharp upper estimate conjecture for the $$\lambda ^k(V)$$ of weighted homogeneous isolated hypersurface singularities and we prove this conjecture in case of $$k=1$$ for large class of singularities.

The quasi-AKNS and quasi-[formula omitted] type AKNS hierarchies as well as a gallery of simple Lie algebras contained in [formula omitted

This paper provides a new representation of sl(3;ℂ), i.e., a high-dimensional special linear Lie algebra A, which contributes to the generation of a series of simple Lie algebras owning different bases with inequable commutators. Their counterparts are obtained in succession by using some special invertible linear transformations. As applications, the quasi-AKNS and quasi-so(3) type AKNS Hamiltonian hierarchies as well as their nonlinear continuous integrable couplings with the Hamiltonian structures are established, respectively.

The external labelling problem and Clebsch–Gordan series of semisimple Lie algebras

It is shown that the external labelling problem for simple Lie algebras can be solved in a satisfactory manner taking into account the decomposability properties of the Casimir operators of . This enables us to construct a complete set of commuting labelling operators for irreducible representations of , allowing us in particular to separate the degeneracies in the Clebsch–Gordan series of the simple Lie algebra .

ALTERNATIVE INDICATORS OF POWER OF ELECTRIC ENERGY IN A SINGLE-PHASE CIRCUIT WITH POLYHARMONIC CURRENT AND VOLTAGE

Introduction. Many electrical engineering issues use a power balance. It is compiled from averaged power values, and equivalent power is used to characterize power of transient processes. To account electricity, both mono- and polyharmonic currents and voltages use active and reactive power, the quality of electricity is not taken into account. Problem . A number of works are declared a certain number of power components that reflect indicators of quantity and quality of electrical energy. These components of power are subject to criticism. The order of determining power components requires algorithmization, as well the task of determining indicators that will reflect poor quality of energy. Goal . Development of a technique for determining the components of power in single-phase circuits with polyharmonic current and voltage, for definition electrical energy transmission indicators. Methodology . Based on analysis of power components determined in known papers and order of their calculation, the features of taking sign of sine and cosine orthogonal components are marked, depending on combination of numbers a current and voltage harmonics. Using Fourier theory of series and elements of the logic algebra, an algorithm for determining components of electric power energy is developed. Results . Highlighting active and reactive powers of the fundamental harmonic of current and voltage; active and reactive power; canonical power components; non-canonical power components, and proposed indicators of quality of transmission of electrical energy. Originality . Based on analysis of power represented by trigonometric Fourier series, the specific calculation of canonical and non-canonical components with use of a number of indicators of electric energy transmission is proposed that reflect its quality. Practical value . The proposed power components of transmission of electrical energy can be used in technical accounting systems.

About algebraic Puiseux series in several variables

We deal with the algebraicity of an iterated Puiseux series in several variables in terms of the properties of its coefficients. Our aim is to generalize to several variables the results from [8]. We show that the algebraicity of such a series for given bounded degrees is determined by a finite number of explicit universal polynomial formulas. Conversely, given a vanishing polynomial, there is a closed-form formula for the coefficients of the series in terms of the coefficients of the polynomial and of a bounded initial part of the series.

Auslander–Gorenstein algebras from Serre-formal algebras via replication

We introduce a new family of algebras, called Serre-formal algebras. They are Iwanaga–Gorenstein algebras for which applying any power of the Serre functor on any indecomposable projective module, the result remains a stalk complex. Typical examples are given by (higher) hereditary algebras and self-injective algebras; it turns out that other interesting algebras such as (higher) canonical algebras are also Serre-formal. Starting from a Serre-formal algebra, we consider a series of algebras – called the replicated algebras – given by certain subquotients of its repetitive algebra. We calculate the self-injective dimension and dominant dimension of all such replicated algebras and determine which of them are minimal Auslander–Gorenstein, i.e. when the two dimensions are finite and equal to each other. In particular, we show that there exist infinitely many minimal Auslander–Gorenstein algebras in such a series if, and only if, the Serre-formal algebra is twisted fractionally Calabi–Yau. We apply these results to a construction of algebras from Yamagata [29], called SGC extensions, given by iteratively taking the endomorphism ring of the smallest generator-cogenerator. We give a sufficient condition so that the SGC extensions and replicated algebras coincide. Consequently, in such a case, we obtain explicit formulae for the self-injective dimension and dominant dimension of the SGC extension algebras.

Detecting coupling directions with transcript mutual information: A comparative study

<p style='text-indent:20px;'>Causal relationships are important to understand the dynamics of coupled processes and, moreover, to influence or control the effects by acting on the causes. Among the different approaches to determine cause-effect relationships and, in particular, coupling directions in interacting random or deterministic processes, we focus in this paper on information-theoretic measures. So, we study in the theoretical part the difference between directionality indicators based on transfer entropy as well as on its dimensional reduction via transcripts in algebraic time series representations. In the applications we consider specifically the lowest dimensional case, i.e., 3-dimensional transfer entropy, which is currently one of the most popular causality indicators, and the (2-dimensional) mutual information of transcripts. Needless to say, the lower dimensionality of the transcript-based indicator can make a difference in practice, where datasets are usually small. To compare numerically the performance of both directionality indicators, synthetic data (obtained with random processes) and real world data (in the form of biomedical recordings) are used. As happened in previous related work, we found again that the transcript mutual information performs as good as, and in some cases even better than, the lowest dimensional binned and symbolic transfer entropy, the symbols being ordinal patterns.

Quiver theories and formulae for Slodowy slices of classical algebras

We utilise SUSY quiver gauge theories to compute properties of Slodowy slices; these are spaces transverse to the nilpotent orbits of a Lie algebra g. We analyse classes of quiver theories, with Classical gauge and flavour groups, whose Higgs branch Hilbert series are the intersections between Slodowy slices and the nilpotent cone S∩N of g. We calculate refined Hilbert series for Classical algebras up to rank 4 (and A5), and find descriptions of their representation matrix generators as algebraic varieties encoding the relations of the chiral ring. We also analyse a class of dual quiver theories, whose Coulomb branches are intersections S∩N; such dual quiver theories exist for the Slodowy slices of A algebras, but are limited to a subset of the Slodowy slices of BCD algebras. The analysis opens new questions about the extent of 3d mirror symmetry within the class of SCFTs known as Tσρ(G) theories. We also give simple group theoretic formulae for the Hilbert series of Slodowy slices; these draw directly on the SU(2) embedding into G of the associated nilpotent orbit, and the Hilbert series of the nilpotent cone.

On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2

The ground field in the text is of characteristic 2. The classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finite-dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated, see arXiv:1407.1695. No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson-Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than 3 indeterminates. Here we completely describe gradings modulo 2 for several series of Lie algebras and their simple relatives: of special linear series, its projectivizations, and projectivizations of the derived Lie algebras of two inequivalent orthogonal series (except for ${\mathfrak{o}}_\Pi(8)$). The classification of gradings is new, but all of the corresponding superizations are known. For the simple derived Zassenhaus algebras of height $n>1$, there is an $(n-2)$-parametric family of modulo 2 gradings; all but one of the corresponding simple Lie superalgebras are new. Our classification also proves non-triviality of a deformation of a simple $3|2$-dimensional Lie superalgebra (new result).

Algebras of linear growth and the dynamical Mordell–Lang conjecture

Ufnarovski remarked in 1990 that it is unknown whether any finitely presented associative algebra of linear growth is automaton, that is, whether the set of normal words in the algebra form a regular language. If the algebra is graded, then the rationality of the Hilbert series of the algebra follows from the affirmative answer to Ufnarovski's question. Assuming that the ground field has a positive characteristic, we show that the answer to Ufnarovskii's question is positive for graded algebras if and only if the basic field is an algebraic extension of its prime subfield. Moreover, in the “only if” part we show that there exists a finitely presented graded algebra of linear growth with irrational Hilbert series. In addition, over an arbitrary infinite basic field, the set of Hilbert series of the quadratic algebras of linear growth with 5 generators is infinite.Our approach is based on a connection with the dynamical Mordell–Lang conjecture. This conjecture describes the intersection of an orbit of an algebraic variety endomorphism with a subvariety. We show that the positive answer to Ufnarovski's question implies some known cases of the dynamical Mordell–Lang conjecture. In particular, the positive answer for a class of algebras is equivalent to the Skolem–Mahler–Lech theorem which says that the set of the zero elements of any linear recurrent sequence over a zero characteristic field is the finite union of several arithmetic progressions. In particular, the counter-examples to this theorem in the positive characteristic case give examples of algebras with irrational Hilbert series.

The Variety of Two-dimensional Algebras Over an Algebraically Closed Field

AbstractThe work is devoted to the variety of two-dimensional algebras over algebraically closed fields. First we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of certain algebra series in the variety of two-dimensional algebras. Finally, we apply our results to obtain analogous descriptions for the subvarieties of flexible and bicommutative algebras. In particular, we describe rigid algebras and irreducible components for these subvarieties.

Multigraded Hilbert series of noncommutative modules

In this paper, we propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory of regular languages, we provide conditions when the methods are effective and hence the sum of the Hilbert series is a rational function. Moreover, a characterization of finite-dimensional algebras is obtained in terms of the nilpotency of a key matrix involved in the computations. Using this result, efficient variants of the methods are also developed for the computation of Hilbert series of truncated infinite-dimensional algebras whose (non-truncated) Hilbert series may not be rational functions. We consider some applications of the computation of multigraded Hilbert series to algebras that are invariant under the action of the general linear group. In fact, in this case such series are symmetric functions which can be decomposed in terms of Schur functions. Finally, we present an efficient and complete implementation of (standard) graded and multigraded Hilbert series that has been developed in the kernel of the computer algebra system Singular. A large set of tests provides a comprehensive experimentation for the proposed algorithms and their implementations.

Counting Algorithms for Recognizable and Algebraic Series

Counting Algorithms for Recognizable and Algebraic Series

Two problems from the Polishchuk and Positselski book on quadratic algebras

In the book Quadratic algebras by Polishchuk and Positselski [23], algebras with a small number of generators (n=2,3) are considered. For some number r of relations possible Hilbert series are listed, and those appearing as series of Koszul algebras are specified. The first case, where it was not possible to do, namely the case of three generators n=3 and six relations r=6 is formulated as an open problem. We give here a complete answer to this question, namely for quadratic algebras with \mathrm {dim} A_1=\mathrm {dim} A_2=3 , we list all possible Hilbert series, and find out which of them can come from Koszul algebras, and which can not. As a consequence of this classification, we found an algebra, which serves as a counterexample to another problem from the same book [23, Chapter 7, Sec. 1, Conjecture 2], saying that Koszul algebra of finite global homological dimension d has \mathrm {dim} A_1 \geq d . Namely, the 3-generated algebra A given by relations xx+yx=xz=zy=0 is Koszul and its Koszul dual algebra A^! has Hilbert series of degree 4: H_{A^!}(t)= 1+3t+3t^2+2t^3+t^4 , hence A has global homological dimension 4.