The problem of existence of closed knight’s tours in [n]d, where [n] = {0, 1, 2, . . . , n− 1}, was recently solved by Erde, Golénia, and Golénia. They raised the same question for a generalised, (a, b) knight, which is allowed to move along any two axes of [n]d by a and b unit lengths respectively. Given an even number a, we show that the [n]d grid admits an (a, 1) knight’s tour for sufficient...

Let σ(A) denote the number of automorphisms of a model A of power ω1 . We derive a necessary and sufficient condition in terms of trees for the existence of an A with ω1 < σ(A) < 2 ω1 . We study the sufficiency of some conditions for σ(A) = 21 . These conditions are analogous to conditions studied by D.Kueker in connection with countable models. The starting point of this paper was an attempt t...

Let m + n particles be thrown randomly, independently of each other into N cells, using the following two-stage procedure. 1. The first m particles are allocated equiprobably, that is, the probability of a particle falling into any particular cell is 1/N . Let the ith cell contain mi particles on completion. Then associate with this cell the probability ai = mi/m and withdraw the particles. 2. ...

Let c be a constant and (e1, f1), (e2, f2), . . . , (ecn, fcn) be a sequence of ordered pairs of edges from the complete graph Kn chosen uniformly and independently at random. We prove that there exists a constant c2 such that if c > c2 then whp every graph which contains at least one edge from each ordered pair (ei, fi) has a component of size Ω(n) and if c < c2 then whp there is a graph conta...

In this paper, we look at those pairs of integers (a, b) for which the Lucas sequence of general term un = un(a, b) has the property that 4 | φ(|un|) for almost all positive integers n.

Current paper defines the concept of the conceptual difference of knowledge and presents the methods to evaluate it numerically. For that purpose, knowledge is viewed as ordered pairs formed from the notation and denotation. Lorents’s metrics is used for measuring the conceptual difference of knowledge.

We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture holds for a finite number of pairs, for any given height. Moreover, we propose a natural generalization of the conjecture to the case of skew shapes.

A general phenomenon puzzles all investors is that on one hand, most individual investors believe they need to construct the portfolio consisting of 15 or more stocks to prevent risk because that large investment companies frequently get high returns is due to they obey the existing investment theory to make the portfolio consisting of more than 100 stocks, but the individual investors loss the...

We construct classes of Bailey pairs where the exponent of q in αn is an indefinite quadratic form. As an application we obtain families of q-hypergeometric mock theta multisums.

highly self-ordered alumina nanopore arrays were fabricated using hard anodization technique in different mixtures of oxalic/phosphoric acid. the phosphoric acid concentration was varied from 0.05 to 0.3 m while the oxalic acid concentration was changed between 0.3 and 0.4 m. the self ordered nanoporous arrays were obtained in anodization voltage changing from 130 to 200 v. the interpore distan...