Power Series In Noncommuting Variables Research Articles

Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability

We study a particular group law on formal power series in non-commuting variables induced by their interpretation as linear forms on a suitable graded connected word Hopf algebra. This group law is left-linear and is therefore associated to a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu's theory of free probability.

An application of the Goulden–Jackson cluster theorem

Let A be an alphabet and let F be a set of words with letters in A. We show that the sum of all words with letters in A with no consecutive subwords in F, as a formal power series in noncommuting variables, is the reciprocal of a series with all coefficients 0, 1 or -1. We also explain how this result is related to a result of Curtis Greene on lattices with M\"obius function 0, 1, or -1.

Analytic mappings between noncommutative pencil balls

In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems (Helton et al. (2009) [10], de Oliviera et al. (2009) [8]). In the earlier paper (Helton et al. (2009) [9]) we characterized NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. In this paper we turn to a more general dimension-free ball BL, called a “pencil ball”, associated with a homogeneous linear pencilL(x):=A1x1+⋯+Agxg,Aj∈Cd′×d. For X=col(X1,…,Xg)∈(Cn×n)g, define L(X):=∑Aj⊗Xj and letBL:=({X∈(Cn×n)g:‖L(X)‖<1})n∈N. We study the generalization of NC ball maps to these pencil balls BL, and call them “pencil ball maps”. We show that every BL has a minimal dimensional (in a certain sense) defining pencil L˜. Up to normalization, a pencil ball map is the direct sum of L˜ with an NC analytic map of the pencil ball into the ball. That is, pencil ball maps are simple, in contrast to the classical result of D'Angelo (1993) [7, Chapter 5] showing there is a great variety of such analytic maps from Cg to Cm when g≪m. To prove our main theorem, this paper uses the results of our previous paper (Helton et al. (2009) [9]) plus entirely different techniques, namely, those of completely contractive maps. What we do here is a small piece of the bigger puzzle of understanding how Linear Matrix Inequalities (LMIs) behave with respect to noncommutative change of variables.

Noncommutative ball maps

In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, “NC ball maps” are very simple, in contrast to the classical result of D'Angelo on such analytic maps in C . Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables.

Rationality of the Möbius function of a composition poset

We consider the zeta and Möbius functions of a partial order on integer compositions first studied by Bergeron, Bousquet-Mélou, and Dulucq. The Möbius function of this poset was determined by Sagan and Vatter. We prove rationality of various formal power series in noncommuting variables whose coefficients are evaluations of the zeta function, ζ , and the Möbius function, μ . The proofs are either directly from the definitions or by constructing finite-state automata. We also obtain explicit expressions for generating functions obtained by specializing the variables to commutative ones. We reprove Sagan and Vatter's formula for μ using this machinery. These results are closely related to those of Björner and Reutenauer about subword order, and we discuss a common generalization.

The Kleene–Schützenberger Theorem for Formal Power Series in Partially Commuting Variables

Kleene's theorem on the coincidence of regular and rational languages in free monoids has been generalized by Schützenberger to a description of the recognizable formal power series in noncommuting variables over arbitrary semirings and by Ochmański to a characterization of the recognizable languages in trace monoids. We will describe the recognizable formal power series over arbitrary semirings and in partially commuting variables, i.e. over trace monoids. We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product, and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet. The converse is true if the underlying semiring is commutative. Moreover, if in addition the semiring is idempotent then the same result holds with a star restricted to series for which the elements in the support have connected (possibly different) alphabets. It is shown that these assumptions over the semiring are necessary. This provides a joint generalization of Kleene's, Schützenberger's and Ochmański's theorems.

The complexity of language recognition by neural networks

Neural networks are frequently used as adaptive classifiers. This research represents an attempt to measure the “neural complexity” of any regular set of binary strings, that is, to quantify the size of a recurrent continuous-valued neural network that is needed for correctly classifying the given regular set. Our estimate provides a predictor that is superior to the size of the minimal automaton that was used as an upper bound so far. Moreover, it is easily computable, using techniques from the theory of rational power series in non-commuting variables.

Formal moduli of modules over local $k$-algebras

We determine explicitly the formal moduli space of certain complete topological modules over a topologically finitely generated local $k$-algebra $R$, not necessarily commutative, where $k$ is a field. The class of topological modules we consider include all those of finite rank over $k$ and some of infinite rank as well, namely those with a Schauder basis in the sense of $\S 1$. This generalizes the results of [Sh], where the result was obtained in a different way in case the ring $R$ is the completion of the local ring of a plane curve singularity and the module is ${k^n}$. Along the way, we determine the ring of infinite matrices which correspond to the endomorphisms of the modules with Schauder bases. We also introduce functions called "growth functions" to handle explicit epsilonics involving the convergence of formal power series in noncommuting variables evaluated at endomorphisms of our modules. The description of the moduli space involves the study of a ring of infinite series involving possibly infinitely many variables and which is different from the ring of power series in these variables in either the wide or the narrow sense. Our approach is beyond the methods of [Sch] which were used in [Sh] and is more conceptual.

Ranking and formal power series

We prove that the ranking problem for unambiguous context-free languages is NC 1-reducible to the value problem for algebraic formal power series in noncommuting variables. In the particular case of regular languages we show that the problem of ranking is NC 1-reducible to the problem of counting the number of strings of given length in suitable regular languages. As a consequence ranking problems for regular languages are NC 1-reducible to integer division and hence computable by log-space uniform boolean circuits of polynomial size and depth O(log n log log n), or by P-uniform boolean circuits of polynomial size and depth O(log n).

Counting problems and algebraic formal power series in noncommuting variables

In this work we study the complexity of certain counting functions related to formal power series in noncommuting variables. We prove that, for every algebraic formal power series in Z 《 ∑》, the problem of computing the corresponding counting function is NC 1-reducible to integer division. As a consequence, for every unambiguous context-free language L⊆ ∑ *, the problem of computing #{ xϵ L:| x|= n} is also NC 1-reducible to integer division. Therefore all t hese counting problems are solvable by families of log-space uniform boolean circuits of depth O(log n log log n) and polynomial size.

On meta-normal forms for algebraic power series in noncommuting variables

On meta-normal forms for algebraic power series in noncommuting variables

On formal power series defined by infinite linear systems

We show that each formal power series in noncommuting variables may be obtained by an infinite linear system as those considered by Kuich and Urbanek (1983).

Cyclic derivation of noncommutative algebraic power series

We show that the cyclic derivative of any algebraic formal power series in noncommuting variables is again algebraic.

Un théorème de factorisation des produits d'endomorphismes de Kn

Let H be a finite set of endomorphisms of a finite dimension vector space K n over a field K. We prove that if φ is a product φ 1 φ 2 ··· φ m of endomorphisms of K n belonging to H , we can extract of it a product φ i φ i + 1 ··· φ j = ψ which is a pseudoregular endomorphism. That is: ψ has a representation as a square matrix over K which is diagonal sum of a null matrix and of a regular one. This result holds also if K is a skew-field. As a consequence of this result, we obtain two theorems about the rational power series in noncommuting variables. The first concerns the “regularity” of the support of the rational power series. The second generalizes a theorem of Polyà [12] and another one of Cantor [2], about rational power series with algebraic integer coefficients.

A representation theorem for algebraic and context-free power series in noncommuting variables

Algebraic power series in noncommuting variables are obtained as solutions of polynomial systems of equations. These systems can be regarded as context-free grammars with weights. Some properties of algebraic power series are derived, in particular their representation as transductions of certain canonical Dyck sets.

Generating functions for formal power series in noncommuting variables.

Generating functions for formal power series in noncommuting variables.