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...

Selfadjoint Sturm-Liouville operators Hω on L2(a, b) with random potentials are considered and it is proven, using positivity conditions, that for almost every ω the operator Hω does not share eigenvalues with a broad family of random operators and in particular with operators generated in the same way as Hω but in L2(ã, b̃) where (ã, b̃) ⊂ (a, b).

Let A, B, be finite subsets of an abelian group, and let G ⊂ A × B be such that #A, #B, #{a + b : (a, b) ∈ G} ≤ N. We consider the question of estimating the quantity #{a − b : (a, b) ∈ G}. In [2] Bourgain obtained the bound of N 2− 1 13 , and applied this to the Kakeya conjecture. We improve Bourgain's estimate to N

Sudoku has risen in popularity over the past few years. The rules are simple, yet the solutions are often less than trivial. Mathematically, these puzzles are interesting in their own right. This paper will generalize the idea of a sudoku puzzle to define a new kind of n× n array. We define a latin square of order n as an n×n array where every row and every column contain every symbol 1, 2, . ....

Let ex(n, P ) be the maximum possible number of ones in any 0-1 matrix of dimensions n×n that avoids P . Matrix P is called minimally non-linear if ex(n, P ) 6= O(n) but ex(n, P ′) = O(n) for every proper subpattern P ′ of P . We prove that the ratio between the length and width of any minimally non-linear 0-1 matrix is at most 4, and that a minimally non-linear 0-1 matrix with k rows has at mo...