相关论文: Grzegorczyk sequence
In the article integer divisibility properties and related prime factors natural number representation concepts have been defined over the whole infinite hyperoperation hierarchy. The definitions have been made across and above of unique…
Two objects are independent if they do not affect each other. Independence is well-understood in classical information theory, but less in algorithmic information theory. Working in the framework of algorithmic information theory, the paper…
Let $(u(n))_{n\in\mathbb{N}}$ be an arithmetic progression of natural integers in base $b\in\mathbb{N}\setminus \{0,1\}$. We consider the following sequences: $s(n)=\overline{u(0)u(1)\cdots u(n) }^b$ formed by concatenating the first $n+1$…
The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r,p,n). As a first application, we give a…
It is known that $G$-functions solutions of a linear differential equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we…
In this paper, we classify the irreducible representations of the rational Cherednik algebras of rank 1 in characteristic p > 0. There are two cases. One is the "quantum" case, where "Planck's constant" is nonzero and generic irreducible…
We use fast-growing finite and infinite sequences of natural numbers and more complicated constructs to define models of hypercomputation and interpret non-arithmetic predicates, with the strongest extensions reaching full second order…
Given a finitely generated free monoid $X$ and a morphism $\phi : X\to X$, we show that one can construct an algebra, which we call an iterative algebra, in a natural way. We show that many ring theoretic properties of iterative algebras…
We study recurrence, and multiple recurrence, properties along the $k$-th powers of a given set of integers. We show that the property of recurrence for some given values of $k$ does not give any constraint on the recurrence for the other…
The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.
A program is a finite piece of data that produces a (possibly infinite) sequence of primitive instructions. From scratch we develop a linear notation for sequential, imperative programs, using a familiar class of primitive instructions and…
An algebraic notion of representational consistency is defined. A theorem relating it to free actions is proved. A metrizability problem of the quotient (a shape space) is discussed. This leads to a new algebraic variety with a…
In this paper we construct an entire function of two variables having the property that its values and its partial derivatives of any order at any distinct algebraic points are algebraically independent. Such an entire function is generated…
This paper introduces a robust class of functions from finite words to integers that we call Z-polyregular functions. We show that it admits natural characterizations in terms of logics, Z-rational expressions, Z-rational series and…
Let $G$ be a finite permutation group acting on a set $\Omega$. An ordered sequence $(\omega_1,\ldots,\omega_\ell)$ of elements of $\Omega$ is an irredundant base for $G$ if the pointwise stabilizer of the sequence is trivial and no point…
In this note, we present a characterization of sets definable in Skolem arithmetic, i.e., the first-order theory of natural numbers with multiplication. This characterization allows us to prove the decidability of the theory. The idea is…
We show that induction of covariant representations for C*-dynamical systems is natural in the sense that it gives a natural transformation between certain crossed-product functors. This involves setting up suitable categories of…
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…
Let $b \geq 3$ be a positive integer. A natural number is said to be a base-$b$ Zuckerman number if it is divisible by the product of its base-$b$ digits. Let $\mathcal{Z}_b(x)$ be the set of base-$b$ Zuckerman numbers that do not exceed…
We consider representing of natural numbers by arithmetical expressions using ones, addition, multiplication and parentheses. The (integer) complexity of n -- denoted by ||n|| -- is defined as the number of ones in the shortest expressions…