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...

First, a quick review of intervals. We define the following four cases: • The closed interval [a, b] = {t : a ≤ t ≤ b}; The half-open interval (a, b] = {t : a < t ≤ b}; • The half-open interval [a, b) = {t : a ≤ t < b}; The open interval (a, b) = {t : a < t < b}. If a or b is ±∞, use a half-open or open interval. Never write (even implicitly) “t = ∞”–this is uncouth! Now let x(t) be a continuou...

We give an involution on the set of lattice paths from (0, 0) to (a, b) with steps N = (0, 1) and E = (1, 0) that lie between two boundaries T and B, which proves that the statistics ‘number of E steps shared with T ’ and ‘number of E steps shared with B’ have a symmetric joint distribution on this set. This generalizes a result of Deutsch for the case of Dyck paths.

Consider an interval I ⊆ Q . Set S(I) = {m ∈ N : ∃ n ∈ N , mn ∈ I}. This turns out to be a numerical semigroup, and has been the subject of considerable recent investigation (see Chapter 4 of [2] for an introduction). Special cases include modular numerical semigroups (see [4]) where I = [mn , m n−1 ] (m,n ∈ N ), proportionally modular numerical semigroups (see [3]) where I = [mn , m n−s ] (m,n...

and Applied Analysis 3 (a, b), and let g󸀠(x) ̸ = 0 on (a, b). If f󸀠(x)/g󸀠(x) is increasing (decreasing) on (a, b), then so are f (x) − f (a) g (x) − g (a) , f (x) − f (b) g (x) − g (b) . (11) If f󸀠(x)/g󸀠(x) is strictly monotone, then the monotonicity in the conclusion is also strict. Lemma 6 (see [11, Lemma 1.1]). Suppose that the power series f(x) = ∑ ∞ n=0 a n x n and g(x) = ∑∞ n=0 b n x n hav...

In [1], Agarwal & Andrews studied partitions with "a copies of a," and in [2], Agarwal & Mullen studied partitions with "d(a) copies of a" (where d is the divisor function). In this note, partitions with "M(a) copies of a" are considered; the maximum exponent function, M, is defined by M(a) = max(e l5. .. , e T) if the integer a > 1 has canonical prime-power form a = p^ 1 ... p &r , and Af(l) =...

In a biased weak (a, b) polyform achievement game, the maker and the breaker alternately mark a, b previously unmarked cells on an infinite board, respectively. The maker’s goal is to mark a set of cells congruent to a polyform. The breaker tries to prevent the maker from achieving this goal. A winning maker strategy for the (a, b) game can be built from winning strategies for games involving f...