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.

Motived by an intrinisic necessary condition for the purifiability of subgroups of primary abelian groups due to K. Benabdallah and T. Okuyama we introduce new functors on the category of pairs (G, A),where A is a subgroup of G,to the category of Z/pZ-vector spaces. The vanishing of these functors leads to the notion of vertical subgroup which is a weakening of purity but also an essential component of the latter. In fact, a vertical subgroup is pure if and only if it is neat. We establish various facts about vertical subgroups and “maximal” vertical subgroups and apply the resulting theory to the problem of purifiability. We show that the class of quasi-complete groups is precisely the class of reduced groups in which every subgroup satisfying the intrinsic necessary condition for purifiability is in fact purifiable. This is also the class of reduced p-groups in which the maximal vertical subgroups are precisely the pure subgroups.

We consider two independent Dawson-Watanabe super-Brownian motions, Y1 and Y2. These processes are diffusions taking values in the space of finite measures on ℝd. We show that if d ≤ 5 then with positive probability there exist times t such that the closed supports of intersect; whereas if d > 5 then no such intersections occur. For the case d ≤ 5, we construct a continuous, non-decreasing measure–valued process L(Y1, Y2), the collision local time, such that the measure defined by , is concentrated on the set of times and places at which intersections occur. We give a Tanaka-like semimartingale decomposition of L(Y1, Y2). We also extend these results to a certain class of coupled measurevalued processes. This extension will be important in a forthcoming paper where we use the tools developed here to construct coupled pairs of measure-valued diffusions with point interactions. In the course of our proofs we obtain smoothness results for the random measures that are uniform in t. These theorems use a nonstandard description of Yi and are of independent interest.

The nonnegative weight function pairs u, v for which the operator maps the nonnegative nonincreasing functions in LP(v) boundedly into weak Lq(u) are characterized. This result is used, in particular, both to generalize and to provide an alternate proof of certain strong type inequalities recently obtained by Ariño and Muckenhouptfor the Hardy averaging operator restricted to nonnegative nonincreasing functions.

We consider a semilinear elliptic problem , (n > 2m). Under suitable conditions on f, we show the existence of a decaying positive solution. We do not employ radial arguments. Our main tools are weighted spaces, various applications of the Mountain Pass Theorem and LP regularity estimates of Agmon. We answer an open question of Kusano, Naito and Swanson [Canad. J. Math. 40(1988), 1281-1300] in the superlinear case: , and improve the results of Dalmasso [C. R. Acad. Sci. Paris 308(1989), 411-414] for the case .

The problem of characterizing the semigroups with Boolean congruence lattices has been solved for several classes of semigroups. Hamilton [9] and the author of this paper [1] studied the question for semilattices. Hamilton and Nordahl [10] considered commutative semigroups, Fountain and Lockley [7,8] solved the problem for Clifford semigroups and idempotent semigroups, in [1] the author generalized their results to completely regular semigroups. Finally, Zhitomirskiy [19] studied the question for inverse semigroups.

The problem of linear independence of the integer translates of μ * B, where μ is a compactly supported distribution and B is an exponential box spline, is considered in this paper. The main result relates the linear independence issue with the distribution of the zeros of the Fourier-Laplace transform, of μ on certain linear manifolds associated with B. The proof of our result makes an essential use of the necessary and sufficient condition derived in [12]. Several applications to specific situations are discussed. Particularly, it is shown that if the support of μ is small enough then linear independence is guaranteed provided that does not vanish at a certain finite set of critical points associated with B. Also, the results here provide a new proof of the linear independence condition for the translates of B itself.

A geometric version of the Vitali integral convergence theorem is established. Parameterized versions of results on uniform absolute continuity in spaces of measures suggested by the convergence theorem are studied.

Generalized (distribution-valued) Ornstein-Uhlenbeck processes, which by definition are solutions of generalized Langevin equations, arise in many investigations on fluctuation limits of particle systems (eg. Bojdecki and Gorostiza [1], Dawson, Fleischmann and Gorostiza [5], Fernández [7], Gorostiza [8,9], Holley and Stroock [10], Itô [12], Kallianpur and Pérez-Abreu [16], Kallianpur and Wolpert [14], Kotelenez [17], Martin-Löf [19], Mitoma [22], Uchiyama [25]). The state space for such a process is the strong dual Φ′ of a nuclear space Φ. A generalized Langevin equation for a Φ′-valued process X ≡ {Xt} is a stochastic evolution equation of the form

where {At} is a family of linear operators on Φ and Z ≡ {Zt} is a Φ'-valued semimartingale (in some sense) with independent increments. Equations of the type (1.1) where Z does not have independent increments also arise in applications (eg. [9,14,20]) but here we are interested precisely in the case when Z has independent increments (we restrict the term generalized Langevin equation to this case in accordance with the classical Langevin equation).

In this paper we make use of semigroup methods on the space of compactly supported probability measures to obtain a complete Lévy-Khinchin representation for negative definite functions on a commutative hypergroup. In addition we obtain representation theorems for completely monotone and completely alternating functions. The techniques employed here also lead to considerable simplification of the proofs of known results on positive definite and negative definite functions on hypergroups.

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).

We generalize the generic single-valuedness and continuity of monotone operators defined on open subsets of Banach spaces of class (S) and Asplund spaces to monotone operators defined on convex subsets of such spaces which may even fail to have non-support points. This yields differentiability theorems for convex Lipschitzian functions on such sets. From a result about minimal convex uscos which are densely single-valued we obtain generic differentiability results for certain Lipschitzian realvalued functions.

For a number field F with ring of integers OF the tame symbols yield a surjective homomorphism with a finite kernel, which is called the tame kernel, isomorphic to K2(OF). For the relative quadratic extension E/F, where and E ≠ F, let CS(E/ F)(2) denote the 2-Sylow-subgroup of the relative S-class-group of E over F, where S consists of all infinite and dyadic primes of F, and let m be the number of dyadic primes of F, which decompose in E.

We show that when a positive contraction of type (p, q) is equipped with a positive norming function having full support, then it is related in a natural way to operators on other Lp spaces.

We adopt the terminology and notations of [5]. If f ∈ Bτ is an entire function of exponential type τ bounded on the real axis then we have the complementary interpolation formulas [1, p. 142-143]

and

where t, γ are reals and

Let F be a skew field with a valuation (also called total) subring B, i.e. x in F\ B implies x-1 in B. Such rings are useful not only in the investigation and construction of division algebras (see for example [5],[6],[12]) but also in geometry ([15]).

Associated with B is an invariant subring R of F and a value group G. We investigate the relationship between properties like the distributivity of R and properties like being lattice ordered of G.

It is a well known fact that for rapidly increasing nkthe sequence , behaves like a sequence of independent random variables; in particular has a limiting Gaussian distribution as N → ∞. Under a certain critical speed this result breaks down and becomes strongly dependent. The purpose of this paper is to investigate the asymptotic behavior of normed sums in the strongly dependent domain; specifically, we construct a large class of nongaussian limit distributions of such sums.

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.

Strong convergence has been investigated in summability theory and Fourier analysis. This paper extends strong convergence to a topological property of sequence spaces E. The more general property of strong boundedness is also defined and examined. One of the main results shows that for an FK-space E which contains all finite sequences, strong convergence is equivalent to the invariance property E = ℓ ν0. E with respect to coordinatewise multiplication by sequences in the space ℓν0 defined in the paper. Similarly, strong boundedness is equivalent to another invariance E = ℓν.E. The results of the paper are applied to summability fields and spaces of Fourier series.

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.