Given integers ν, a and b, when does a pairwise balanced design on ν elements with a triples and b quadruples exist? Necessary conditions are developed, and shown to be sufficient for all v ≥ 96. An extensive set of constructions for pairwise balanced designs is used to obtain the result.

Let L: [1, ∞) → [1, ∞) be a nondecreasing function such that limx→∞L(x) = +∞. Let f= fL be a strongly additive function determined by f(p) = L(p) on the set of primes. In what followsp, p1, p2, …, q, q1, q2, …,P, Q stand for prime numbers, P(n) denotes the largest prime divisor of n. The letters c, c1, c2, … denote suitable positive constants, not necessarily the same at each occurrence. As usual, π(x) denotes the number of primes p ≤ x, while π(x, k, ℓ) is the number of primes p ≤ x such that p ≡ ℓ (mod k).

A partially ordered set (T, ≤) is called a tree if it is semilinearly ordered, i.e. any two elements have a common lower bound but no two incomparable elements have a common upper bound, and contains an infinite chain and at least two incomparable elements. Let k ∈ ℕ. We say that a partially ordered set (T, ≤) is k-homogeneous, if each isomorphism between two k-element subsets of T extends to an automorphism of (T, ≤), and weakly k-transitive, if for any two k-element subchains of T there exists an automorphism of (T, ≤) taking one to the other.

Properties of fixed points of equivariant maps have been studied by several authors including A. Dold (cf. [2], 1982), H. Ulrich (cf. [9], 1988), A. Marzantowicz (cf. [7], 1975) and others. Closely related is the work of R. Rubinsztein (cf. [8], 1976) in which he investigated homotopy classes of equivariant maps between spheres. There have been many attempts to introduce and effectively apply these concepts to nonlinear problems. In particular we mention the work of E. Dancer (cf. [1], 1982) in which some applications to nonlinear problems are given.

Assume that A is a finite algebra contained in a variety that has the congruence extension property and that B is a subalgebra of A. If α ≺ β in Con A and α |B ≠ β |B, then we show that α |B ≺ β |B and that there is a close connection between the type labellings of the quotients 〈α, α〉 and 〈α|B, β|B〉.

Let P(z1,…, zn) be a polynomial with positive coefficients. For positive x define A classical diophantine problem is to describe the asymptotic behavior of N1(x) as x → ∞. More generally, one can introduce a second polynomial φ satisfying the condition (0.1) Sign φ (m) is constant for all m outside at most a finite subset of ℕn.

In his fundamental paper [1] J. L. Alperin introduced the idea of a weight in modular representation theory of finite groups G. Let p be a prime. A p-subgroup R is called a radical subgroup of G if R = Op(NG(R)). An irreducible character φ of NG(R) is called a weight character if φ is trivial on R and belongs to a p-block of defect zero of NG(R)/R. The G-conjugacy class of the pair (R, φ) is a weight of G. Let b be the p-block of NG(R) containing φ, and let B be p-block of G. A weight (R, φ) is a B-weight for the block B of G if B = bG, which means that B and b correspond under the Brauer homomorphism. Alperin's conjecture on weights asserts that the number l*(B) of B-weights of a p-block B of a finite group G equals the number l(B) of modular characters of B.

The purpose of this paper is to introduce the notion of a CW complex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category.

A brief description of the paper goes as follows: in Section 1 we introduce the homotopy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type homotopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method.

We give mild sufficient conditions on a nonlinear functional to have eigenvalues. These results are intended for the study of boundary value problems for semilinear elliptic equations.

Martin's axiom and the Continuum Hypothesis are studied here using the notion of a ccc partition i.e., a partition of the form where K0 has the following properties:

(a) K0 contains subsets of its elements as well as all singletons of X.

(b) Every uncountable subset of K0 contains two elements whose union is in K0.