Related papers: $LS$-sequences of points in the unit square
The main purpose of this master thesis is to study the $LS$-sequences of points introduced by Carbone in \cite{Carbone} and find two generalizations of them to the unit square. Here we also present a new algorithm proposed by the same…
We investigate the number of squares in a very broad family of binary recurrence sequences with $u_{0}=1$. We show that there are at most two distinct squares in such sequences (the best possible result), except under such very special…
We prove an infinite family of lacunary recurrences for the Lucas numbers using combinatorial means.
The $LS$-sequences are a parametric family of sequences of points in the unit interval. They were introduced by Carbone, who also proved that under an appropriate choice of the parameters $L$ and $S$, such sequences are low-discrepancy. The…
In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in…
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein together with theorems corollaries, formulae, examples, mathematical criteria, etc. (about integer sequences, numbers, quotients, residues,…
We construct a good sequence with uncountable spectrum. As an application, we answer to a question of Lesigne, Quas Rosenblatt and Wierdl.
We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also…
We study convergence almost everywhere of sequences of Schr\"odinger means. We also replace sequences by uncountable sets.
In this paper 101 new integer sequences, sub-sequences, and sequences of sequences, together with related unsolved problems and conjectures, are presented. Also, definitions, examples, solved or open questions, and references for each…
We argue that the low-lying scalar-meson nonet makes part of a subset of a family of infinitely many scalar-meson nonets, which in turn makes part of a family of infinitely many quark-antiquark bound states and resonances. We outline the…
We introduce and study a family of lattice equations which may be viewed either as a strongly nonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka-Volterra chain. Their deceptively simple form supports…
We exhibit a three parameter infinite family of quadratic recurrence relations inspired by the well known Somos sequences. For one infinite subfamily we prove that the recurrence generates an infinite sequence of integers by showing that…
It has been shown that for a given set of points in a plane, the least-squares regression line pivots about a fixed point when any single point in the set is repeated. We consider what happens when more than one point is repeated.…
We show that there exist real quadratic maps of the interval whose attractors are computationally intractable. This is the first known class of such natural examples.
A family of sequences produced by a non-homogeneous linear recurrence formula derived from the geometry of circles inscribed in lenses is introduced and studied. Mysterious ``underground'' sequences underlying them are discovered in this…
Let $X$ be a Banach space. We study the circumstances under which there exists an uncountable set $\mathcal A\subset X$ of unit vectors such that $\|x-y\|>1$ for distinct $x,y\in \mathcal A$. We prove that such a set exists if $X$ is…
Let $(L_n)_{n \geq 1}$ be the sequence of Lucas numbers, defined recursively by $L_1 := 1$, $L_2 := 3$, and $L_{n + 2} := L_{n + 1} + L_n$, for every integer $n \geq 1$. We determine the asymptotic behavior of $\log \operatorname{lcm} (L_1…
There exists "a square problem": in a unit square is there a point with four rational distances to the vertices? This problem is still regarded as unproved. Yang showed proofs for several special cases of the square problem. By the…
For a given family of similar shapes, what we call a "unit shape" strongly analogizes the role of the unit circle within the family of all circles. Within many such families of similar shapes, we present what we believe is naturally and…