Volume 22, 2016, Number 2

By means of Bienayme’s theorem of Statistics is found that the remainder term in Gauss’s problem about the lattice points in the circle is a function normally distributed with mean value zero and standard deviation 1,10368 multiplied by the fourth root of x. This result can not be improved.

Let n > 3 be arbitrary integer. In the present paper it is shown that if K is an arbitrary circle and M_i, i = 1,…,n are points on K, dividing K into n equal arcs, then for each point M on K, different from the mentioned above, at least of the distances |MM_i| are irrational numbers.

This note is to capture and extend some of Finucan’s ideas for further exploration by students. These ideas connect with several elementary concepts in combinatorial analysis which lend themselves to undergraduate research projects

Some new properties of the arithmetic function called “Restrictive factor” are formulated and studied.

A polynomial P(x) with integer coefficients is said to be transitive modulo m, if for every x, y ∈ ℤ there exists k ≥ 0 such that P^k(x) = y (mod m). In this paper, we construct new examples of transitive polynomials modulo prime powers and partially describe cubic and quartic transitive polynomials. We also study the orbit structure of affine maps modulo prime powers.

We provide an optimal strategy to solve the n × n × n points problem inside the box, considering only 90° turns, and at the same time a pattern able to drastically lower down the known upper bound. We use a very simple spiral frame, especially if compared to the previous plane by plane approach that significantly reduces the number of straight lines connected at their end-points necessary to join all the n³ dots. In the end, we combine the square spiral frame with the rectangular spiral pattern in the most profitable way, in order to minimize the difference h_u(n) − h_l(n) between the upper and the lower bound, proving that it is ≤ 0.5 ∙ n ∙ (n + 3), for any n > 1.

If s_k denotes the number of independent sets of cardinality k in graph G, and α(G) is the size of a maximum independent set, then

   

is the independence polynomial of G (I. Gutman and F. Harary, 1983, [8]). The Merrifield–Simmons index σ(G) (known also as the Fibonacci number) of a graph G is defined as the number of all independent sets of G. Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erdos (1987, [2]) conjectured that I(T,x) is unimodal whenever T is a tree, while, in general, they proved that for each permutation π of {1, 2, …, α} there is a graph G with α(G) = α such that s_π(1) < s_π(2) < … < s_π(α). By maximal tree on n vertices we mean a tree having a maximum number of maximal independent sets among all the trees of order n. B. Sagan proved that there are three types of maximal trees, which he called batons [24].

In this paper we derive closed formulas for the independence polynomials and Merrifield–Simmons indices of all the batons. In addition, we prove that I(T,x) is log-concave for every maximal tree T having an odd number of vertices. Our findings give support to the above mentioned conjecture.

We show that the Gregory series combined with Newton’s binomial expansion give a natural approach to the logarithmic mean inequalities.

Integers in class ̅3₄ of the modular ring Z₄ equal x² – y² but not x² + y² whereas integers in class ̅1₄ can equal both x² + y² and x² – y². This structure generates an infinity of sequences with neat curious patterns.

In this note we consider a family of finite and infinite alternating sums containing products of Fibonacci numbers. We derive closed-form expressions for this family of sums. As a consequence of this result we establish new algebraic relationships between certain alternating sums of reciprocals of products of Fibonacci numbers with integer power.

This paper considers the pair (GR_n(F),⚬) consisting of the set of all invertible rhotrices of size n over an arbitrary field F; and together with the binary operation of row-column method for rhotrix multiplication; in order to introduce it as the concept of “non-commutative general rhotrix group”. We identify a number of subgroups of (GR_n(F),⚬) and then advance to show that its particular subgroup is embedded in a particular subgroup of the well-known general linear group (GR_n(F),•). Furthermore, we shall investigate isomorphic relationship between some subgroups of (GR_n(F),⚬).