Algebra Of Power Series Research Articles

DISCONTINUOUS HOMOMORPHISMS OF WITH

AbstractAssume that M is a transitive model of $ZFC+CH$ containing a simplified $(\omega _1,2)$ -morass, $P\in M$ is the poset adding $\aleph _3$ generic reals and G is P-generic over M. In M we construct a function between sets of terms in the forcing language, that interpreted in $M[G]$ is an $\mathbb R$ -linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on $\omega $ , into the Esterle algebra of formal power series. Therefore it is consistent that $2^{\aleph _0}>\aleph _2$ and, for any infinite compact Hausdorff space X, there exists a discontinuous homomorphism of $C(X)$ , the algebra of continuous real-valued functions on X.

CppTPSA/pyTPSA: a C++/Python package for truncated power series algebra

cppTPSA/pyTPSA: a C++/Python package for truncated power series algebra

ORTHOGONAL SYSTEMS AND PERMUTATION POLYNOMIAL VECTORS OVER MODULAR ALGEBRAS

Known results on orthogonal systems and permutation polynomials vectors over finite fields are extended to modular algebras of the form Lν = K[X]/(p(X)ν ), where K is a finite field, p(X) ∈ K[X] is an irreducible polynomial, ν = 1, 2, . . ., and to the algebra of formal power series L[[Z]], where L1 = K[X]/(p(X)) = L.

PERMUTATION POLYNOMIALS IN ONE INDETERMINATE OVER MODULAR ALGEBRAS

Known results about permutation polynomials in connection with coefficients in a finite field extend to algebras of the form Lv = K[X]/(p(X)v)\), where K is a finite field,(p(X) ∈ ‍ K[X] is an irreducible polynomial, and v = 1,2,..., and to the algebra of power series L[[Z]], where L = K[X]/(p(X)). Analogues of Dickson polynomials are also studied in this context.

Free limits of free algebras

Consider a diagram ⋯→F3→F2→F1 of algebraic systems, where Fn denotes the free object on n generators and the connecting maps send the extra generator to some distinguished trivial element. We prove that (a) if the Fi are free associative algebras over a fixed field then the limit in the category of graded algebras is again free on a set of homogeneous generators; (b) on the other hand, the limit in the category of associative (ungraded) algebras is a free formal power series algebra on a set of homogeneous elements, and (c) if the Fi are free Lie algebras then the limit in the category of graded Lie algebras is again free.

Classification of D-bialgebra structures on power series algebras

In this paper, we use algebro-geometric methods in order to derive classification results for so-called D-bialgebra structures on the power series algebra A〚z〛 for certain central simple non-associative algebras A. These structures are closely related to a version of the classical Yang-Baxter equation (CYBE) over A.If A is a Lie algebra, we obtain new proofs for pivotal steps in the known classification of non-degenerate topological Lie bialgebra structures on A〚z〛 as well as of non-degenerate solutions of the usual CYBE.If A is associative, we achieve the classification of non-triangular topological balanced infinitesimal bialgebra structures on A〚z〛 as well as of all non-degenerate solutions of an associative version of the CYBE.

Topological Manin pairs and (n,s)-type series

Lie subalgebras of L = {mathfrak {g}}(!(x)!) times {mathfrak {g}}[x]/x^n{mathfrak {g}}[x] , complementary to the diagonal embedding Delta of {mathfrak {g}}[![x]!] and Lagrangian with respect to some particular form, are in bijection with formal classical r-matrices and topological Lie bialgebra structures on the Lie algebra of formal power series {mathfrak {g}}[![x]!] . In this work we consider arbitrary subspaces of L complementary to Delta and associate them with so-called series of type (n, s) . We prove that Lagrangian subspaces are in bijection with skew-symmetric (n, s) -type series and topological quasi-Lie bialgebra structures on {mathfrak {g}}[![x]!] . Using the classificaiton of Manin pairs we classify up to twisting and coordinate transformations all quasi-Lie bialgebra structures. Series of type (n, s) , solving the generalized classical Yang-Baxter equation, correspond to subalgebras of L. We discuss their possible utility in the theory of integrable systems.

The structure group for quasi-linear equations via universal enveloping algebras

We replace trees by multi-indices as an index set of the abstract model space to tackle quasi-linear singular stochastic partial differential equations. We show that this approach is consistent with the postulates of regularity structures when it comes to the structure group, which arises from a Hopf algebra and a comodule. Our approach, where the dual of the abstract model space naturally embeds into a formal power series algebra, allows to interpret the structure group as a Lie group arising from a Lie algebra consisting of derivations on this power series algebra. These derivations in turn are the infinitesimal generators of two actions on the space of pairs (non-linearities, functions of space-time mod constants). We also argue that there exist pre-Lie algebra and Hopf algebra morphisms between our structure and the tree-based one in the cases of branched rough paths (Grossman-Larson, Connes-Kreimer) and of the stochastic heat equation.

Quantum queer supergroups via υ-differential operators

By using certain quantum differential operators, we construct a super representation for the quantum queer supergroup Uυ(qn). The underlying space of this representation is a deformed polynomial superalgebra in 2n2 variables whose homogeneous components can be used as the underlying spaces of queer q-Schur superalgebras. We then extend the representation to its formal power series algebra which contains a (super) submodule isomorphic to the regular representation of Uυ(qn). A monomial basis M for Uυ(qn) plays a key role in proving the isomorphism. In this way, we may present the quantum queer supergroup Uυ(qn) by another new basis L together with some explicit multiplication formulas by the generators. As an application, similar presentations are obtained for queer q-Schur superalgebras via the above mentioned homogeneous components.The existence of the bases M and L and the new presentation show that the seminal construction of quantum gln established by Beilinson–Lusztig–MacPherson thirty years ago extends to this “queer” quantum supergroup via a completely different approach.

On the Cumulants of the First Passage Time of the Inhomogeneous Geometric Brownian Motion

We consider the problem of the first passage time T of an inhomogeneous geometric Brownian motion through a constant threshold, for which only limited results are available in the literature. In the case of a strong positive drift, we get an approximation of the cumulants of T of any order using the algebra of formal power series applied to an asymptotic expansion of its Laplace transform. The interest in the cumulants is due to their connection with moments and the accounting of some statistical properties of the density of T like skewness and kurtosis. Some case studies coming from neuronal modeling with reversal potential and mean reversion models of financial markets show the goodness of the approximation of the first moment of T. However hints on the evaluation of higher order moments are also given, together with considerations on the numerical performance of the method.

Potentials for some tensor algebras

This paper generalizes former works of Derksen, Weyman and Zelevinsky about quivers with potentials. We consider semisimple finite-dimensional algebras E over a field F, such that E⊗FEop is semisimple. We assume that E contains a certain type of F-basis which is a generalization of a multiplicative basis. We study potentials belonging to the algebra of formal power series, with coefficients in the tensor algebra over E, of any finite-dimensional E-E-bimodule on which F acts centrally. In this case, we introduce a cyclic derivative and to each potential we associate a Jacobian ideal. Finally, we develop a mutation theory of potentials, which in the case that the bimodule is Z-free, it behaves as the quiver case; but allows us to obtain realizations of a certain class of skew-symmetrizable integer matrices.

A cumulant approach for the first-passage-time problem of the Feller square-root process

The paper focuses on an approximation of the first passage time probability density function of a Feller stochastic process by using cumulants and a Laguerre-Gamma polynomial approximation. The feasibility of the method relies on closed form formulae for cumulants and moments recovered from the Laplace transform of the probability density function and using the algebra of formal power series. To improve the approximation, sufficient conditions on cumulants are stated. The resulting procedure is made easier to implement by the symbolic calculus and a rational choice of the polynomial degree depending on skewness, kurtosis and hyperskewness. Some case studies coming from neuronal and financial fields show the goodness of the approximation even for a low number of terms. Open problems are addressed at the end of the paper.

Formal Functional Calculus for Weakly Locally Nilpotent Operators in Fréchet Spaces

We generalize the classical Riesz–Dunford holomorphic functional calculus with weakly locally nilpotent operators in a Frechet space to the case of the algebra of all formal power series. We obtain a counterpart of the spectral mapping theorem and formulas similar to the Taylor and Riesz–Dunford formulas. We show that the weak local nilpotency of an operator is necessary for constructing the formal functional calculus. Bibliography: 12 titles.

The regular representation of 𝑈_{𝜐}(𝔤𝔩_{𝔪|𝔫})

Using quantum differential operators, we construct a super representation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) on a certain polynomial superalgebra. We then extend the representation to its formal power series algebra which contains a U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) -submodule isomorphic to the regular representation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) . In this way, we obtain a presentation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) by a basis together with explicit multiplication formulas of the basis elements by generators.

The Laurent Coefficients of the Hilbert Series of a Gorenstein Algebra

By a theorem of R. Stanley, a graded Cohen-Macaulay domain A is Gorenstein if and only if its Hilbert series satisfies the functional equation:where, d is the Krull dimension and a is the -invariant of A. We reformulate this functional equation in terms of an infinite system of linear constraints on the Laurent coefficients of at t = 1. The main idea consists of examining the graded algebra of formal power series in the variable x that fulfill the condition . As a byproduct, we derive quadratic and cubic relations for the Bernoulli numbers. The cubic relations have a natural interpretation in terms of coefficients of the Euler polynomials. For the special case of degree these results have been investigated previously by the authors and involved merely even Euler polynomials. A link to the work of Gould and Carlitz on power sums of symmetric number triangles is established.

Inverse limits of Macaulay's inverse systems

Generalizing a result of Masuti and the second author, we describe inverse limits of Macaulay's inverse systems for Cohen–Macaulay factor algebras of formal power series or polynomial rings over an infinite field. On the way we find a strictness result for filtrations defined by regular sequences. It generalizes both a lemma of Uli Walther and the Rees isomorphism.

Global Homological Dimension of Radical Banach Algebras of Power Series

We show that the global dimension of a broad class of radical Banach algebras of power series is at least 3 and obtain applications to cohomology groups.

AN OPERATOR-BASED APPROACH FOR THE CONSTRUCTION OF CLOSED-FORM SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS

An operator-based approach for the construction of closed-form solutions to fractional differential equations is presented in this paper. The technique is based on the analysis of Caputo and Riemann-Liouville algebras of fractional power series. Explicit solutions to a class of linear fractional differential equations are obtained in terms of Mittag-Leffler and fractionally-integrated exponential functions in order to demonstrate the viability of the proposed technique.

On images of locally finite derivations and [formula omitted]-derivations

Some cases of the LFED Conjecture, proposed by the second author [15], for certain integral domains are proved. In particular, the LFED Conjecture is completely established for the field of fractions k(x) of the polynomial algebra k[x], the formal power series algebra k[[x]] and the Laurent formal power series algebra k[[x]][x−1], where x=(x1,x2,…,xn) denotes n commutative free variables and k a field of characteristic zero. Furthermore, the relation between the LFED Conjecture and the Duistermaat–van der Kallen Theorem [3] is also discussed and emphasized.

Skolem–Noether algebras

An algebra S is called a Skolem–Noether algebra (SN algebra for short) if for every central simple algebra R, every homomorphism R→R⊗S extends to an inner automorphism of R⊗S. One of the important properties of such an algebra is that each automorphism of a matrix algebra over S is the composition of an inner automorphism with an automorphism of S. The bulk of the paper is devoted to finding properties and examples of SN algebras. The classical Skolem–Noether theorem implies that every central simple algebra is SN. In this article it is shown that actually so is every semilocal, and hence every finite-dimensional algebra. Not every domain is SN, but, for instance, unique factorization domains, polynomial algebras and free algebras are. Further, an algebra S is SN if and only if the power series algebra S[[ξ]] is SN.