Related papers: Parikh Matrices for Powers of Words
This study is devoted to the polynomial representation of the matrix $p$th root functions. The Fibonacci-H\"orner decomposition of the matrix powers and some techniques arisen from properties of generalized Fibonacci sequences, notably the…
We develop new polynomial methods for studying systems of word equations. We use them to improve some earlier results and to analyze how sizes of systems of word equations satisfying certain independence properties depend on the lengths of…
We study matrix three term relations for orthogonal polynomials in two variables constructed from orthogonal polynomials in one variable. Using the three term recurrence relation for the involved univariate orthogonal polynomials, the…
We show that the Parikh image of the language of an NFA with n states over an alphabet of size k can be described as a finite union of linear sets with at most k generators and total size 2^{O(k^2 log n)}, i.e., polynomial for all fixed k…
The $\imath$-divided powers (depending on a parity) form the $\imath$canonical basis for the split rank 1 $\imath$quantum group and they are a basic ingredient for $\imath$quantum groups of higher rank. We obtain closed formulae for the…
In this paper we discourse basises of representable algebras. This question lead to arithmetic problems. We prove algorithmical solvability of exponential-Diophantine equations in rings represented by matrices over fields of positive…
We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…
Combinatorial properties of maximal repetitions (runs) in formal words are studied. We classify all maximal repetitions in a word as primary and secondary where the set of all primary repetitions determines all the other repetitons in the…
We use recent results on algorithms for Markov decision problems to show that a canonical form for a generalized P-matrix can be computed, in some important cases, by a strongly polynomial algorithm.
Reasoning about implied relationships (e.g., paraphrastic, common sense, encyclopedic) between pairs of words is crucial for many cross-sentence inference problems. This paper proposes new methods for learning and using embeddings of word…
In this article, we revisit some block matrix construction methods and use them to derive various general expansion formulas for calculating the ranks of matrix expressions. As applications, we derive a variety of interesting rank…
All current investigations to analyze the derivational complexity of term rewrite systems are based on a single termination method, possibly preceded by transformations. However, the exclusive use of direct criteria is problematic due to…
Ranking words is an important way to summarize a text or to retrieve information. A word graph is a way to represent the words of a sentence or a text as the vertices of a graph and to show the relationship among the words. It is also…
It has been conjectured that the Parikh (commutative) image of every language over an infinite alphabet recognized by an automaton with registers is defined by a rational expression. This conjecture is known to hold for all languages…
Zimin words are very special finite words which are closely related to the pattern-avoidability problem. This problem consists in testing if an instance of a given pattern with variables occurs in almost all words over any finite alphabet.…
In this article we give two different ways of representations of circular words. Representations with tuples are intended as a compact notation, while representations with trees give a way to easily process all conjugates of a word. The…
We introduce the notion of unavoidable (complete) sets of word patterns, which is a refinement for that of words, and study certain numerical characteristics for unavoidable sets of patterns. In some cases we employ the graph of pattern…
Under categorial grammars that have powerful rules like composition, a simple n-word sentence can have exponentially many parses. Generating all parses is inefficient and obscures whatever true semantic ambiguities are in the input. This…
In this paper polynomial maps are represented by the use of matrices whose entries are numbered by pair of multiindices and a new product of such matrices is introduced. A matrix representation of composition of polynomial maps is given. In…
We characterize matrices whose powers coincide with their Hadamard powers.