We prove that, under suitable conditions, a Jacobi Poincaré series of exponential type of integer weight and matrix index does not vanish identically. For the classical Jacobi forms, we construct a basis consisting of the ‘first’ few Poincaré series, and also give conditions, both dependent on and independent of the weight, that ensure the nonvanishing of a classical Jacobi Poincaré series. We also obtain a result on the nonvanishing of a Jacobi Poincaré series when an odd prime divides the index.

Several pairwise concepts have been studied for bitopological spaces. In this note an attempt has been made to see how much ‘bitopological’ these pairwise concepts are. For example pairwise T1 is purely a topological concept whereas pairwise normality is very much ‘bitopological’. Several questions pertaining to this theme are dealt with.

Generalised wreath products of permutation groups were discussed in a paper by Bailey and us. This note determines the orbits of the action of a generalised wreath product group on m–tuples (m ≥ 2) of elements of the product of the base sets on the assumption that the action on each component is m–transitive. Certain related results are also provided.

Various continuity conditions (in norm, in measure, weakly etc.) for the nonlinear superposition operator F x(s) = f(s, x(s)) between spaces of measurable functions are established in terms of the generating function f = f(s, u). In particular, a simple proof is given for the fact that, if F is continuous in measure, then f may be replaced by a function f which generates the same superposition operator F and satisfies the Carathéodory conditions. Moreover, it is shown that integral functional associated with the function f are proved.

N(G) denotes the near-ring of all continuous selfmaps of the topological group G (under composition and the pointwise induced operation) and N0(G) is the subnear-ring of N(G) consisting of all functions having the identity element of G fixed. It is known that if G is discrete then (a) N0(G) is simple and (b) N(G) is simple if and only if G is not of order 2. We begin a study of the ideal structure of these near-rings when G is a disconnected group.

Let C[[t]] denote the algebra of all formal power series over the complex field C in a commutative indeterminate t with the weak topology determined by the projections pj: Σαiti ↦αj. A subalgebra A of C[[t]] is a Banach algebra of power series if it contains the polynomials and is a Banach algebra under a norm such that the inclusion map A ⊂ C[[t]] is continuous. Such algebras were first introduced in [13] when considering algebras with one generator, and studied, in a special case, in [23]. For a partial bibliography of their subsequent study and application see the references of [9] (note that the usage of the term Banach algebra of power series in [9] differs from that here), and also [2], [3], [11]. Indeed, an examination of their use in [11], under more general topological conditions than here, led the present author to the results of [14], [15], [16], [17].

Let (х) = (x1, x2, … xn) = Σi Σiaij, xixf (aij = aij) be a positive quadratic form with determinant D, and let M be the minimum of for integral x ≠ 0. Then attains the value M for a finite number of integral x = ±mk ( k = 1, …, s) called its minimal vectors.

This paper deals with the Schrödinger equation $i{\partial }_{s} u(\mathbf{z} , t; s)- \mathcal{L} u(\mathbf{z} , t; s)= 0, $ where $ \mathcal{L} $ is the sub-Laplacian on the Heisenberg group. Assume that the initial data $f$ satisfies $\vert f(\mathbf{z} , t)\vert \lesssim {q}_{\alpha } (\mathbf{z} , t), $ where ${q}_{s} $ is the heat kernel associated to $ \mathcal{L} . $ If in addition $\vert u(\mathbf{z} , t; {s}_{0} )\vert \lesssim {q}_{\beta } (\mathbf{z} , t), $ for some ${s}_{0} \in \mathbb{R} \setminus \{ 0\} , $ then we prove that $u(\mathbf{z} , t; s)= 0$ for all $s\in \mathbb{R} $ whenever $\alpha \beta \lt { s}_{0}^{2} . $ This result holds true in the more general context of $H$-type groups. We also prove an analogous result for the Grushin operator on ${ \mathbb{R} }^{n+ 1} . $

Certain permutation representations of free groups are constructed by finite approximation. The first is a construction of a cofinitary group with special properties, answering a question of Tim Wall published by Cameron. The second yields, via a method of Kepert and Willis, a totally disconnected locally compact group which is compactly generated and uniscalar but has no compact open normal subgroup. Finally, an oligomorphic group of automorphisms of the random graph is built, all of whose non-trivial subgroups have just finitely many orbits.

Although the harmonic series diverges, there is a sense in which it “nearly converges”. Let N denote the set of all positive integers, and S a subset of N. Then there are various sequences S for which converges, but for which the “omitted sequence” N–S is, in intuitive sense, sparse, compared with N. For example, Apostol [1] (page 384) quotes, without proof the case where S is the set of all Positive integers whose decimal representation does not invlove the digit zero (e.g. 7∈S but 101 ∉ S); then (1) converges, with T < 90.

In a previous paper of this series – hereinafter to be referred to as [1] – the author introduced a new method for a large class of boundary value problems connected with the flexure analysis of elastic plates of arbitrary shape where the concept of ‘Lines of Equal Deflection’, i.e. lines which are obtained by intersecting the bent plate by planes parallel to the original plane of the plate, was introduced. The present paper extends this analysis to the buckling analysis of thin elastic plates with various forms of boundary conditions. It is shown that the proposed method appears to be a powerful tool for the investigation of those problems of elastic stability which could not be solved by conventional methods because of the difficulty of the mathematical treatment.

It is a fascinating problem in the axiomatics of any mathematical system to reduce the number of axioms, the number of variables used in each axiom, the length of the various identities, the number of concepts involved in the system etc. to a minimum. In other words, one is interested finding systems which are apparently ‘of different structures’ but which represent the same reality. Sheffer's stroke operation and. Byrne's brief formulations of Boolean algebras [1], Sholander's characterization of distributive lattices [7] and Sorkin's famous problem of characterizing lattices by means of two identities are all in the same spirit. In groups, when defined as usual, we demand a binary, unary and a nullary operation respectively, say, a, b →a·b; a→a−1; the existence of a unit element). However, as G. Rabinow first proved in [6], groups can be made as a subvariety of groupoids (mathematical systems with just one binary operation) with the operation * where a * b is the right division, ab−1. [8], M. Sholander proved the striking result that a mathematical system closed under a binary operation * and satisfying the identity S: x * ((x *z) * (y *z)) = y is an abelian group. Yet another identity, already known in the literature, characterizing abelian groups is HN: x * ((z * y) * (z * a;)) = y which is due to G. Higman and B. H. Neumann ([3], [4])*. As can be seen both the identities are of length six and both of them belong to the same ‘bracketting scheme’ or ‘bracket type’.

The paper presents a proof of the Brylinski conjecture for compact Kähler orbifolds. The result is a corollary of the foliated version of the Mathieu theorem on symplectic harmonic representations of de Rham cohomology classes. The proofs are based on the idea of representing an orbifold as the leaf space of a Riemannian foliation and on the correspondence between foliated and holonomy invariant objects for foliated manifolds.

Maximal estimates are considered for solutions to an initial value problem for the Schrödinger equation. The initial value function is assumed to be radial in ℝn, n≥2.

A permutation π of the set {1, 2, …, n} is four-discordant if π(i) ≢ i, i+ 1, i + 2, i +3 (mod n) for 1 ≦i ≦ n. Generating functions for rook polynomials associated with four-discordant permutations are derived. Hit polynomials associated with four-discordant permutations are studied. Finally, it is shown that the leading coefficients of these rook polynomials form a “tribonacci” sequence which is a generalized Fibonacci sequence.

The structure of Kronecker class of an extension K: k of algebraic number fields of degree |K: k| ≤ 8 is investigated. For such classes it is shown that the width and socle number are equal and are at most 2, and for those of width 2 the Galois group is given. Further, if |K: k | is 3 or 4, or if 5 ≤ |K: k| ≤ 8 and K: k is Galois, then the groups corresponding to all “second minimal” fields in K are determined.

We present a symmetric version of a normed algebra of quotients for each ultraprime normed algebra. In addition, a C*-a1gebra of quotients of an arbitrary C*-a1gebra is introduced.

Let G be a locally finite group, let k be a field of characteristic p ≧ 0, and let V be a (right) kG-module, not necessarily of finite dimension over k. We say that V is an Mc-modle over kG if, for each p′-subgroup H of G, the set of centrarlizers in V of subgroups of H satisfies the minimal condition under the relation of setheoretic inclusion. Here, p′ denotes the set of all peimes different from p, and in particular 0' denotes the set of all primes. It is straightforward to verify that V is an Mc-module over kG if and only if each p′-subgroup H of G contains a finite subgroup F such that CV(F) = CV(H).

Frames have been defined as a certain type of generalization of Room square. Frames have proven useful in the construction of Room squares, in particular, skew Room squares.

We generalize the definition of frame and consider the construction of Room squares and skew Room squares using these more general frames.

We are able to construct skew Room squares of three previously unknown sides, namely 93, 159, and 237. This reduces the number of unknown sides to four: 69, 87, 95 and 123. Also, using this construction, we are able to give a short proof of the existence of all skew Room squares of (odd) sides exceeding 123.

Finally, this frame construction is useful for constructing Room squares with subsquares. We can also construct Room squares “missing” subsquares of sides 3 and 5. The “missing” subsquares of sides 3 and 5 do not exist, so these incomplete Room squares cannot be completed to Room squares.

Under the standard assumptions, the formula for integration by parts is obtained by a straightforward calculation.