Eugen J. Ionascu

k-Dependence and Domination in Kings Graphs.

For most choices of k = 0, ... , 8, there is a tidy solution: an upper bound can be proved by a short elementary argument, and an arrangement of kings can be con structed to show that the upper bound is tight. These limiting densities are given in Section 6. However, tight upper bounds are not yet known for either k = 4 or k ? 5. It is easy to construct arrangements of kings (on arbitrarily large boards) that achieve the densities of 3/5 and 9/13 for k ? 4 and 5, respectively. We conjecture that these are indeed the maximum limiting densities. The story in the present article concerns the struggle to support this conjecture by good upper bounds, as well as the variety of rival techniques used for different val ues of k. Along the way, we make elementary use of graph theory, number theory, group theory, real analysis, and integer linear programming. We believe the methods of the present paper can provide the basis for undergraduate research projects on re lated problems.

Bisecting binomial coefficients

In this paper, we deal with the problem of bisecting binomial coefficients. We find many (previously unknown) infinite classes of integers which admit nontrivial bisections, and a class with only trivial bisections. As a byproduct of this last construction, we show conjectures Q2 and Q4 of Cusick and Li. We next find several bounds for the number of nontrivial bisections and further compute (using a supercomputer) the exact number of such bisections for n <= 51.

Some Unexpected Connections Between Analysis and Combinatorics

We go through a series of results related to the k-signum equation \(\pm 1^k\pm 2^k\pm\cdots\pm n^k=0\). We are investigating the number S k (n) of possible writings and the asymptotic behavior of these numbers, as k is fixed and \(n\to \infty\). The results are presented in connections with the Erdos–Suranyi sequences. Analytic methods and algebraic ones are employed in order to predict the asymptotic behavior in general and to study in detail various situations for small values of k. Some simplifications and further ramifications are discussed in the end about the recent proof of Andrica–Tomescu conjecture.

Wandering sets for a class of borel isomorphisms of [0,1)

A wandering set for a map ϕ is a set containing precisely one element from each orbit of ϕ. We study the existence of Borel wandering sets for piecewise linear isomorphisms. Such sets need not exist even when the parameters involved are rational, but they do exist if in addition all the slopes are powers of 2. For ϕ having at most one discontinuity, the existence of a Borel wandering set is equivalent to rationality of the Poincaré rotation number. We compute the rotation numbers for a special class of such functions. The main result provides a concrete method of connecting certain pairs of wavelet sets.

A variation on bisecting the binomial coefficients

Abstract In this paper, we present an algorithm which allows us to search for all the bisections for the binomial coefficients { n k } k = 0 , … , n and include a table with the results for all n ≤ 154 . Connections with previous work on this topic are included. We conjecture that the probability of having only trivial solutions is 5 ∕ 6 .

On the number of polynomials with coefficients in (n)

Abstract In this paper we introduce several natural sequences related to polynomials of degree s having coefficients in {1,2, …,n} (n ∈ N) which factor completely over the integers. These sequences can be seen as generalizations of A006218. We provide precise methods for calculating the terms and investigate the asymptotic behavior of these sequences for s ∈ {1, 2, 3}.

Cubes in {0,1,...,n}3

Abstract. The main aim of this paper is to describe a procedure for calculating the number of cubes that have coordinates in the set {0,1,...,n}. For this purpose we continue and, at the same time, revise some of the work begun in a sequence of papers about equilateral triangles and regular tetrahedra all having integer coordinates for their vertices. We adapt the code that was included in a paper by the first author and was used to calculate the number of regular tetrahedra with vertices in {0,1,...,n}3. The idea is based on the theoretical results obtained by the first author with A. Markov. We then extend the sequence A098928 in the Online Encyclopedia of Integer Sequences to the first one hundred terms.

Minimal Niven Numbers

Deflne ak to be the smallest positive multiple of k such that the sum of its digits in base q is equal to k. The asymptotic behavior, lower and upper bound estimates of ak are investigated. A characterization of the minimality condition is also considered.

Remarks on a sequence of minimal niven numbers

In this short note we introduce two new sequences defined using the sum of digits in the representation of an integer in a certain base. A connection to Niven numbers is proposed and some results are proven.