Related papers: Continuous Distribution Arising From the Three Gap…
Let x be a quadratic irrational and let P be the set of prime numbers. We show the existence of an infinite subset S of P such that the statistics of the period of the continued fraction expansions along the sequence {px: p\in S} approach…
In the paper, I consider appearance of unit's digits in minor totals of a few integer sequences. The sequences include the sequence of even integers, sequence of odd integers and Faulhaber polynomial at $p = 2$. Application of difference…
The Gauss map (or continued fraction map) is an important dissipative one-dimensional discrete-time dynamical system that exhibits chaotic behaviour and which generates a symbolic dynamics consisting of infinitely many different symbols.…
We prove an analog of Lagrange's Theorem for continued fractions on the Heisenberg group: points with an eventually periodic continued fraction expansion are those that satisfy a particular type of quadratic form, and vice-versa.
We consider the average-consensus problem in a multi-node network of finite size. Communication between nodes is modeled by a sequence of directed signals with arbitrary communication delays. Four distributed algorithms that achieve…
We determine the distributional behavior for products of free random variables in a general infinitesimal triangular array. In the case of positive variables, the main theorem extends a result proved earlier for arrays with identically…
We study the growth of random networks under a constraint that the diameter, defined as the average shortest path length between all nodes, remains approximately constant. We show that if the graph maintains the form of its degree…
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…
Conventional statistics begins with a model, and assigns a likelihood of obtaining any particular set of data. The opposite approach, beginning with the data and assigning a likelihood to any particular model, is explored here for the case…
An interesting characterization of the Fibonacci numbers is that, if we write them as $F_1 = 1$, $F_2 = 2$, $F_3 = 3$, $F_4 = 5, ...$, then every positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. This is…
Our purpose is to give an account of the $r$-tuple problem on the increasing sequence of reduced fractions having denominators bounded by a certain size and belonging to a fixed real interval. We show that when the size grows to infinity,…
The distribution of waiting times until the occurrence of a critical event is a crucial statistical problem across several disciplines in Science. In this work we present a statistical model in which a relevant quantity X accumulates until…
Katugampola's 2015 study of generalized fractional differential operators produced triangular arrays of integer coefficients indexed by a fractional order r and by dimensions n and k, but no combinatorial interpretation has been established…
We ask, for which $n$ does there exists a $k$, $1 \leq k < n$ and $(k,n)=1$, so that $k/n$ has a continued fraction whose partial quotients are bounded in average by a constant $B$? This question is intimately connected with several other…
Capsets are subsets of $\mathbb{F}_3^n$ with no three points on a line and a capset is complete if it is not a subset of a larger capset. We study some new constructions of capsets via algebraic equations over extensions of $\mathbb{F}_3$.…
In a recent work Friedlander studied the problem of how large consecutive prime gaps should be in order that the sum of the reciprocals should be divergent. Supposing a very deep Hypothesis, a generalization of the Hardy--Littlewood prime…
Order statistics theory is applied in this paper to probabilistic robust control theory to compute the minimum sample size needed to come up with a reliable estimate of an uncertain quantity under continuity assumption of the related…
We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. For $m = 1$, this is the classical…
We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches…
In this paper we investigate the sums of reciprocals to an arithmetic progression taken modulo one, that is sums of $\{n\alpha-\gamma\}^{-1}$, where $\alpha$ and $\gamma$ are real parameters and $\{\,\cdot\,\}$ is the fractional part of a…