We introduce sectional projectivity, which appears to be the correct notion of projectivity when working with orthomodularlattices. We prove some positive results for varieties of OMLs satisfying various finiteness conditions, namely that every finite OML in such a variety is sectionally projective. In contrast, we prove that the eight element modular ortholattice, MO 3, is not projective in the variety of modular ortholattices.

If K is a triangulation of a closed 3-manifold M with E0(K) edges and F0(K) triangles, then the average edge order of K is defined to be

In [8], the relations between this quantity and the topology of M are investigated, especially in the case of μ0(K) being small (where the study relies on Oda's classification of triangulations of 𝕊2 up to eight vertices—see [9]). In the present paper, the attention is fixed upon the average edge order of coloured triangulations; surprisingly enough, the obtained results are perfectly analogous to Luo-Stong' ones, and may be proved with little effort by means of edge-coloured graphs representing manifolds.

Let R be an integral domain with quotient field K. If R has an overling S ≠ K, such that S[X] is integrally closed, then the "algebraic degree" of K((X)) over the quotient field of R[X] is infinite. In particular, it holds for completely integrally closed domain or Noetherian domain R.

A new proof is given for the weak amenability of the group algebras of locally compact groups.

Let k ≥ 2 and q = g(k) — G(k), where g(k) is the smallest possible value of r such that every natural number is the sum of at most r k-th powers and G(k) is the minimal value of r such that every sufficiently large integer is the sum of r k-th powers. For each positive integer r ≥ q, let Then for every ε > 0 and N ≥ N(r, ε), we construct a set A of k-th powers such that |A| ≤ (r(2 + ε)r + l)N1/(k+r) and every nonnegative integer n ≤ N is the sum of k-th powers in A. Some related results are also obtained.

Let g be an element of a group G and [g, G] = 〈g-1a-1ga | a ∊ G〉. We prove that if G is locally nilpotent then for each g,t ∊ G either g[g, G] = t[t, G] or g[g, G] ∩ t[t, G] = Ø. The converse is true if G is finite.

Some sufficient conditions are given for all solutions of the nonlinear differential equation y″(x) +p(x)f(y) = 0 to be nonoscillatory, where p is positive and

for a quotient γ of odd positive integers, γ > 1.

We obtain a representation for the supercritical Dawson-Watanabe superprocessin terms of a subcritical superprocess with immigration, where the immigration at a given time is governed by the state of an underlying branching particle system. The proof requires a new result on the laws of weighted occupation times for branching particle systems.

We consider a convolution singular integral operator associated to a kernel K(x) = b(x)Ω(x)|x|-n, and prove that if b ∊ L∞(ℝn) is a radial function and Ω ∊ H(Σn-1) with mean zero condition (1), then is a bounded linear operator in the space L2(ℝn).

The Modular Group M is PSL2(Z) the group of linear fractional transformations with integral entries and determinant one. M has been of great interest in many diverse fields of Mathematics, including Number Theory, Automorphic Function Theory and Group Theory. In this paper we give an effective algorithm to determine, for each integer d, a complete set of representatives for the trace classes in trace d. This algorithm depends on the combinatorial group theoretic structure of M. It has been subsequently extended by Sheingorn to the general Hecke groups. The number h(d) of trace classes in trace d is equal to the ideal class number of the field The algorithm mentioned above then provides a new straightforward computational procedure for determining h(d). Finally as an outgrowth of the algorithm we present a wide generalization of the Fermat Two-Square theorem. This last result can also be derived from classical work of Gauss.

We study a class of irreducible modules for Affine Lie algebras which possess weight spaces of both finite and infinite dimensions. These modules appear as the quotients of "imaginary Verma modules" induced from the "imaginary Borel subalgebra".

We show the existence of atomless lattice-ordered groups which have doubly transitive representations. In so doing, we answer a question of M. Giraudet from 1981 [4].

Problem 36 of the third edition of Birkhoff's Lattice theory [2] asks whether the MacNeille completion of uniquely complemented lattice is necessarily uniquely complemented. We show that the MacNeille completion of a uniquely complemented lattice need not be complemented.

In this paper, we consider all metacyclic groups of the type 〈a,b | an - 1, b2 = 1, ba = aib〉 and give a concrete description of their rational group algebras. As a consequence we obtain, in a natural way, units which generate a subgroup of finite index in the full unit group, for almost all such groups.

In this paper, under certain conditions on the orthogonal distribution T0, we give a characterization of real hypersurfaces of type A in a complex space form Mn(c), c ≠ 0.

The main purpose of this paper is to investigate isotropic immersions with low codimensions into a real space form.

Known results for strictly stable motions as finiteness of moments and local boundednessof sample-path variation are generalized to self-affine processes, i.e., self-similar processes with stationary increments. The proofs are new, even for stable motions, and are obtained by applying the ergodic theorem to powers of the (one-sided) increments.

In this note we are going to give a geometric proof, using the method of the extremal metric, of the following result of Beurling. For any analytic function f(z) in the unit disc Δ of the plane with a bounded Dirichlet integral, the set E on the boundary of the unit disc where the nontangential limits of f(z) do not exist has logarithmic capacity zero. Also, using an unpublished result of Beurling, we will prove different results on boundary values for different classes of functions.

We give a simple proof that, for any ॉ > 0, there is an outer measure μ* on a finite set X such that, for any measure Thus there is a non-zero outer (finitely subadditive) measure v* on the clopen subsets of the Cantor set such that, if v ≤ v* is a finitely additive measure on the clopen subsets of the Cantor set, then v ≡ 0.

We prove that with some weak chain conditions, left SF-rings are semisimple Artinian or regular. We also prove that MERT left SF-rings are really regular.