Related papers: An Analytical Approach to Exponent-Restricted Mult…
These notes present an approach to obtaining the basic operations of addition and multiplication on the natural numbers in terms of elementary results about commutative monoids.
We construct closed forms that generate with repetitions all Mersenne primes, respectively all Fermat primes, all twin-prime pairs and all Sophie Germain primes. Also, we construct closed forms that count all Mersenne primes between $0$ and…
We introduce a new technique for the efficient management of large sequences of multidimensional data, which takes advantage of regularities that arise in real-world datasets and supports different types of aggregation queries. More…
In this paper we apply ideas from the theory of Uniform Distribution of sequences to Functional Analysis and then drawing inspiration from the consequent results, we study concepts and results in Uniform Distribution itself. So let $E$ be a…
This short note reports a master theorem on tight asymptotic solutions to divide-and-conquer recurrences with more than one recursive term: for example, T(n) = 1/4 T(n/16) + 1/3 T(3n/5) + 4 T(n/100) + 10 T(n/300) + n^2.
This short note presents a peculiar generalization of the Riemann hypothesis, as the action of the permutation group on the elements of continued fractions. The problem is difficult to attack through traditional analytic techniques, and…
A commutative semigroup of abstract factorials is defined in the context of the ring of integers. We study such factorials for their own sake, whether they are or are not connected to sets of integers. Given a subset X of the positive…
We define a class of "algebraic" random matrix channels for which one can generically compute the limiting Shannon transform using numerical techniques and often enumerate the low SNR series expansion coefficients in closed form. We…
The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These…
Simple methods permit to generalize the concepts of iteration and of recursive processes. We shall see briefly on several examples what these methods generate. In additive sequences, we shall encounter not only the golden or the silver…
We give recurrence relations for any family of generalized Appell polynomials unifying so some known recurrences of many classical sequences of polynomials. Our main tool to get our goal is the Riordan group. We use the product of Riordan…
In this note we provide a simple formula of general term of recurrent sequence.
Extensions of previous linear regression models for interval data are presented. A more flexible simple linear model is formalized. The new model may express cross-relationships between mid-points and spreads of the interval data in a…
Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles…
Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation…
We apply ideas from the cluster method to q-count the permutations of a multiset according to the number of occurrences of certain generalized patterns, as defined by Babson and Steingrimsson. In particular, we consider those patterns with…
The classical continued fraction is generalized for studying the rational approximation problem on multi-formal Laurent series in this paper, the construction is called m-continued fraction. It is proved that the approximants of an…
The summatory function of a $q$-regular sequence in the sense of Allouche and Shallit is analysed asymptotically. The result is a sum of periodic fluctuations for eigenvalues of absolute value larger than the joint spectral radius of the…
A common theme of enumerative combinatorics is formed by counting functions that are polynomials evaluated at positive integers. In this expository paper, we focus on four families of such counting functions connected to hyperplane…
In this paper the approach to obtaining nonrecurrent formulas for some recursively defined sequences is illustrated. The most interesting result in the paper is the formula for the solution of quadratic map-like recurrence. Also, some…