Consider all planar walks, with positive unit steps (1, 0) and (0, 1), from the origin (0, 0) to a given point (a, b). Let L be the line joining the beginning to the end, i.e., the line b x a y = 0. Call the region below L "downtown," and the region above L "uptown," the line L being the border-line between downtown and uptown. Each such walk has a + b 1 points, not counting the endpoints. For ...

For A ⊆ Zm and n ∈ Zm, let σ1(A, n), σ2(A, n), σ3(A, n) denote the number of solutions of the equation n = a+a′ with ordered pairs (a, a′) ∈ A, unordered pairs (a, a′) ∈ A(a = a′) and unordered pairs (a, a′) ∈ A, respectively. In this paper, for any i ∈ {1, 2, 3}, we determine all subsets A of Zm such that σi(A, n) = σi(Zm\A, n) holds for all n ∈ Zm.

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → V that satisfy both conditions below: (i) There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A is irreducible tridiagonal. (ii) There exists a basis for V with respect to wh...

Piensik (1984) has given a sharp upper bound for the sum of the distance between all ordered pairs of nodes in a strong tournament Tn. We strengthen this result by deriving a bound that involves an additional parameter.

In what follows, an ordered pair will always be an ordered pair (x, y), where x # y. A transitive triple is a collection of three ordered pairs of the form (6, Y), (Y, z), (x, z)}, which we will always denote by (x, y, z). A transitive triple system (TTS(u)) is a pair (X, B), where X is a set containing v elements and B is a collection of transitive triples of elements of X such that every orde...

Let X be a finitistic space with non-trivial cohomology groups H (X;Z) ∼= Z with generators vi , where i = 0, 1, 2, 3. We say that X has cohomology type (a, b) if v 1 = av2 and v1v2 = bv3 . In this note, we determine the mod 2 cohomology ring of the orbit space X/G of a free action of G = Z2 on X , where both a and b are even. In this case, we observed that there is no equivariant map S → X for...

Introduction. Although derivatives need not be continuous, the inverse images of open intervals are heavy in another sense. Denjoy and Clarkson have shown that the inverse image of every open interval either is empty or has positive measure (see [l] and [3]). Zahorski refined this property to one of homogeneity. He proved that if x is in the inverse image of an open interval (a, b), and if { /„...

In this note, we study Fibonacci-like sequences that are defined by the recurrence Sk = a, Sk+1 = b, Sn+2 ≡ Sn+1 + Sn (mod n + 2) for all n ≥ k, where k, a, b ∈ N, 0 ≤ a < k, 0 ≤ b < k + 1, and (a, b) 6= (0, 0). We will show that the number α = 0.SkSk+1Sk+2 · · · is irrational. We also propose a conjecture on the pattern of the sequence {Sn}n≥k.

FEASIBILITY OF INTEGER KNAPSACKS∗ ISKANDER ALIEV† AND MARTIN HENK‡ Abstract. Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, we consider the set F(A) of all vectors b ∈ Zm such that the associated knapsack polytope P (A, b) = {x ∈ R≥0 : Ax = b} contains an integer point. When m = 1 the set F(A) is known to contain all consecutive integers greater than the Frobenius number ass...