For each positive integer k, we consider the set A k of all ordered pairs [a, b] such that in every k-graph with n vertices and m edges some set of at most am + bn vertices meets all the edges. We show that each A k with k > 2 has infinitely many extreme points and conjecture that, for every positive e, it has only finitely many extreme points [a, b] with a _> e. With the extreme points ordered...

2. Let G be a set with binary operation (a, b) 7→ ab and consider the following possible axioms: (1) ∀ a, b ∈ G, a(bc) = (ab)c. (2) ∃ e ∈ G,∀ a ∈ G, ae = ea = a. (3) ∀ a ∈ G,∃ b ∈ G, ab = ba = e. (3’) ∀ a ∈ G,∃ b ∈ G, ab = e. Prove that the axioms (3) and (3’) are equivalent. That is, show that (1), (2), and (3) hold if and only if (1), (2), and (3’) hold. (One direction is easy. For the other ...

In this article we consider the existence and uniqueness of local weak solutions to the Emden-Fowler type wave equation tutt − uxx = |u|p−1u in [1, T ]× (a, b) with initial-boundary value conditions in a finite time interval.

In the recent papers [8] and [10] a class of Carathéodory functions f(t, η, ξ) rapidly sign-changing near the boundary point t = a, has been constructed so that the equation −(|y′|p−2y′)′ = f(t, y, y′) in (a, b) admits continuous bounded solutions y whose graphs G(y) do not possess a finite length. In this paper, the same class of functions f(t, η, ξ) will be given, but with slightly different ...

We derive mathematically a class of metrics for signals and images, considered as elements of R , that are based upon the structural similarity (SSIM) index. The important feature of our construction is that we consider the two terms of the SSIM index, which are normally multiplied together to produce a scalar, as components of an ordered pair. Each of these terms is then used to produce a norm...

A k-signed r-set on [n] = {1, ..., n} is an ordered pair (A, f), where A is an rsubset of [n] and f is a function from A to [k]. Families A1, ...,Ap are said to be cross-intersecting if any set in any family Ai intersects any set in any other family Aj . Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of [n]. Our aim is to generalise Hilton's bound t...

It is well known that the system ∂xf = a, ∂yf = b on R 2 has a solution if and only if the closure condition ∂xb = ∂ya holds. In this case the solution f is the work done by the force U = (a, b) from the origin to the point (x, y). This paper deals with a similar problem, where the vector fields ∂x, ∂y are replaced by the Heisenberg vector fields X1, X2. In this case the subRiemannian system X1...

Problems from Review of Set Theory Notes 3 (c) We prove that A× (B −C) = (A×B)− (A×C). We show that A× (B −C) ⊂ (A×B)−(A×C), and (A×B)−(A×C) ⊂ A×(B−C). For the first containment, let x ∈ A× (B − C). Then x = (a, b) where a ∈ A and b ∈ B − C. We certainly have x ∈ A × B, and since the second coordinate of x is not in C, we conclude that x / ∈ A× C. For the second containment, let x ∈ (A× B)− (A×...