We determine all triples (a, b, n) of positive integers such that a and b are relatively prime and nk divides an+bn (respectively, an bn), when k is the maximum of a and b (in fact, we answer a slightly more general question). As a by-product, we see that, for m,n 2 N+ with n 2, nm divides mn + 1 if and only if (m,n) = (2, 3) or (1, 2), which generalizes problems from the 1990 and 1999 editions...

The issue of shared use urban freight facilities first received attention during the 1970's when it was observed that, while inter-urban freight movements were becoming increasingly efficient, there were significant diseconomies in the movement of freight via truck within urban areas. Early research suggested that shared urban freight facilities should be constructed so that trucking companies ...

2. Cycles and their terminology. Let L be an n Xn Latin square or, alternatively, let Q be a quasigroup of order n. Let M be the set of n2 ordered triplets ijk, where k is the entry in the ith row and jth column of L. If 5 is the set of n2 ordered pairs ij, and if 7r,:Af—>S is the projection parallel to the ith coordinate (for example, ir2(ijk) = ik), then for each i, l^i^3, t< is onto S (or eq...

This paper analyzes the determinants of life satisfaction in a pooled data set of representative individual survey data from seven East European countries collected during the early phase of economic and political transformation using ordered logit models. Those core sociodemographic and economic variables known to be relevant from studies on the US and West European countries (u-shaped age eff...

This thesis explores the relationship between voter income and political ideology preference at the regional and state level. Traditional theories of voter preference contend that richer Americans tend to favor more conservative policy agendas in an effort to minimize income redistribution. The existing empirical literature supports this claim at the national level by finding a significant posi...

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

a f(x, y) dy, it is often important to know when F is differentiable and when F (x) = ∫ b a f1(x, y) dy. A sufficient condition for differentiating under the integral sign is that ∫ b a f1(x, y) dy converges uniformly; see [6, p. 260]. When we have absolute convergence, the condition |f1(x, y)| ≤ g(y) with ∫ b a g(y) dy < ∞ suffices (Weierstrass M-test and Lebesgue Dominated Convergence). If we...

We calculate the B-model on the mirror pair of X2N−2(2, 2, · · · , 2, 1, 1) , which is an (N − 2)-dimensional Calabi-Yau manifold and has two marginal operators i.e. h(X2N−2(2, 2, · · · , 2, 1, 1)) = 2. In [1] we have discussed about mirror symmetry on XN (1, 1, · · · , 1) and its mirror pair. However, XN (1, 1, · · · , 1) had only one moduli. In this paper we extend its methods to the case wit...

Let M and N be a compact differentiable manifolds of dimension d and d + 1 respectively. Given a compact interval I ⊂ R, we suppose that we have a (fixed) embedding I ×M ⊂ N . See figure 1. Let N denote the complement of {0} ×M . Let h be a Riemannian metric on M , and let ρ be a positive function on R that is positively homogeneous of degree 1 and smooth away from (0, 0). Given (a, b) ∈ R, we ...

It is known that if (A,A∗) is a Leonard pair, then the linear transformations A, A∗ satisfy the Askey-Wilson relations A 2 A ∗ − βAA ∗ A + A∗A2 − γ (AA∗+A∗A) − ̺A∗ = γ∗A2 + ωA + η I, A ∗2 A− βA ∗ AA ∗+ AA∗2− γ∗(A∗A+AA∗) − ̺∗A = γA∗2+ ωA∗+ η∗I, for some scalars β, γ, γ∗, ̺, ̺∗, ω, η, η∗. The problem of this paper is the following: given a pair of Askey-Wilson relations as above, how many Leonard pai...