Related papers: Novel structures in Stanley sequences
When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the set's underlying structure. We begin by investigating finite sets of…
Spiking neural networks, also often referred to as the third generation of neural networks, carry the potential for a massive reduction in memory and energy consumption over traditional, second-generation neural networks. Inspired by the…
In the course of studies of the measure of chaos for the distribution of the prime numbers among the positive integers N arched structures have been found. It is given a brief description of the fine structure of the positive integers…
Rationals are known to form interesting and computationally rich structures, such as Farey sequences and infinite trees. Little attention is being paid to more general, systematic exposition of the basic properties of fractions as a set.…
This paper is a continuation of our previous work \cite{St} where we have studied the Stokes phenomenon for a particular family of equation \eqref{initial} with \eqref{form-0}-\eqref{npe} from a perturbative point of view. Here we focus on…
New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a…
The set splittability problem is the following: given a finite collection of finite sets, does there exits a single set that contains exactly half the elements from each set in the collection? (If a set has odd size, we allow the floor or…
In this work we extend our study on a link between automaticity and certain algebraic power series over finite fields. Our starting point is a family of sequences in a finite field of characteristic $2$, recently introduced by the first…
Based on Euclid's algorithm, we find a kind of special sequences which play an interesting role in the study of primes. We call them W Sequences. They not only ties up the distribution of primes in short interval but also enables us to give…
A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which are known…
Regular sequences generalize the extensively studied automatic sequences. Let $S$ be an abstract numeration system. When the numeration language $L$ is prefix-closed and regular, a sequence is said to be $S$-regular if the module generated…
Regular sequences are natural generalisations of fixed points of constant-length substitutions on finite alphabets, that is, of automatic sequences. Using the harmonic analysis of measures associated with substitutions as motivation, we…
The last advances in sequence modeling are mainly based on deep learning approaches. The current state of the art involves the use of variations of the standard LSTM architecture, combined with several tricks that improve the final…
Number sequences defined by a linear recursion relation are studied by means of generating functions. Indices of the terms in the recursion relation have arbitrary differenses. In addition to formulas for the nth term an algorithm is…
We introduce tree linear cascades, a class of linear structural equation models for which the error variables are uncorrelated but need not be Gaussian nor independent. We show that, in spite of this weak assumption, the tree structure of…
A new family of sequences is proposed. An example of sequence of this family is more accurately studied. This sequence is composed by the integers $n$ for which the sum of binary digits is equal to the sum of binary digits of $n^2$. Some…
We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating…
We propose a multi-scale analysis method for studying arithmetic properties of integer sets, such as primality. Our approach organizes information through a hierarchy of nested sequences, where each level enables a hierarchical expression…
In the first part we associate a periodic sequence to a partition and study the connection the distribution of elements of uniform limit of the sequences. Then some facts of statistical independence of these limits are proved
A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of…