Related papers: Multiplicative automatic sequences
Near-vector spaces extend linear algebra tools to non-linear algebraic structures, enabling the study of non-linear problems. However, explicit constructions remain rare. This paper introduces a broad computable family of near-vector…
We prove that the uniform recurrence of morphic sequences is decidable. For this we show that the number of derived sequences of uniformly recurrent morphic sequences is bounded. As a corollary we obtain that uniformly recurrent morphic…
We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the…
We derive various weighted summation identities, including binomial and double binomial identities, for Tribonacci numbers. Our results contain some previously known results as special cases.
Certain famous combinatorial sequences, such as the Catalan numbers and the Motzkin numbers, when taken modulo a prime power, can be computed by finite automata. Many theorems about such sequences can therefore be proved using Walnut, which…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
Complex risk is a critical factor for both intelligent systems and risk management. In this paper, we consider a special class of risk statistics, named complex risk statistics. Our result provides a new approach for addressing complex…
Many combinatorial sequences (for example, the Catalan and Motzkin numbers) may be expressed as the constant term of $P(x)^k Q(x)$, for some Laurent polynomials $P(x)$ and $Q(x)$ in the variable $x$ with integer coefficients. Denoting such…
The classification of separable operator spaces and systems is commonly believed to be intractable. We analyze this belief from the point of view of Borel complexity theory. On one hand we confirm that the classification problems for…
This note is a survey and collection of results, as well as presenting some original research. For Bessel sequences and frames, the analysis, synthesis and frame operators as well as the Gram matrix are well-known, bounded operators. We…
In this note a general result is proved that can be used to evaluate exactly a class of highly oscillatory integrals.
Continued fraction expansions and Hankel determinants of automatic sequences are extensively studied during the last two decades. These studies found applications in number theory in evaluating irrationality exponents. The present paper is…
We derive weighted summation identities involving the second order recurrence sequence $\{w_n\} =\{ w_n(a,b; p, q)\}$ defined by $w_0 = a,\,w_1 = b;\,w_n = pw_{n - 1} - qw_{n - 2}\, (n \ge 2)$, where $a$, $b$, $p$ and $q$ are arbitrary…
Clones are specializations of operads forming powerful instruments to describe varieties of algebras wherein repeating variables are allowed in their equations. They allow us in this way to realize and study a large range of algebraic…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
We study sets of recurrence, in both measurable and topological settings, for actions of $(\mathbb{N},\times)$ and $(\mathbb{Q}^{>0},\times)$. In particular, we show that autocorrelation sequences of positive functions arising from…
We improve lower bounds on the $k$th-order nonlinear complexity of pseudorandom sequences over finite fields and we establish a probabilistic result on the behavior of the $k$th-order nonlinear complexity of random sequences over finite…
We consider two algorithms which can be used for proving positivity of sequences that are defined by a linear recurrence equation with polynomial coefficients (P-finite sequences). Both algorithms have in common that while they do succeed…
We prove the existence of a ternary sequence of factor complexity $2n+1$ for any given vector of rationally independent letter frequencies. Such sequences are constructed from an infinite product of two substitutions according to a…
In an earlier paper, the notion of integrality known from algebraic number fields and fields of algebraic functions has been extended to D-finite functions. The aim of the present paper is to extend the notion to the case of P-recursive…