In the literature there are two different notions of lovely pairs of a theory T, according to whether T is simple or geometric. We introduce a notion of lovely pairs for an independence relation, which generalizes both the simple and the geometric case, and show how the main theorems for those two cases extend to our general notion.

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.

The origin of this paper is in a question that was asked of the author by Michael Wellman, a computer scientist who works in artificial intelligence at Wright Patterson Air Force Base in Dayton, Ohio. He wanted to know if, starting with R and its usual topology and product partial order, he could linearly reorder every finite subset and still obtain a continuous function from R into R that was ...

We study boundary value problems of the type Ax = r, φ(x) = φ(b) (φ ∈ M ⊆ E∗) in ordered Banach spaces. MSC classification (2000): 47A50, 47B60

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.

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.