The use of graphs in the study of groups is well-established. In this paper, we wish to indicate how certain graph-like objects may be used in a similar way. A diagram is a pseudograph which may have some free edges, i.e. edges with just one end.

A classical result in potential theory is the Schwarz reflection principle for solutions of Laplace's equation which vanish on a portion of a spherical boundary. The question naturally arises whether or not such a property is also true for solutions of the Helmholtz equation. This has been answered in the affirmative by Diaz and Ludford ([4]; see also [10]) in the limiting case of the plane. It is the purpose of this paper to show that a reflection principle is also valid for spheres of finite radius. As an application of this result we shall study the problem of the analytic continuation of solutions to the Helmholtz equation defined in the exterior of a bounded domain in three-dimensional Euclidean space ℝ3 We shall show that through the use of the reflection principle derived in this paper, this problem can be reduced to the problem of the analytic continuation of an analytic function of two complex variables, which in turn can be performed through a variety of known methods (cf. [7]).

A group is called a T-group if all its subnormal subgroups are normal. It is proved here that if G is a periodic (generalized) soluble group in which all subnormal subgroups of infinite rank are normal, then either G is a T-group or it has finite rank. It follows that if G is an arbitrary group whose Fitting subgroup has infinite rank, then G has the property T if and only if all its subnormal subgroups of infinite rank are normal.

We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to ‘some remarks on legendrian rectiable currents’, Manuscripta Math. 113(3) (2004), 397–401) result: ‘If Ω is a set of locally finite perimeter in ℝ2, then there is no function f ∈ C1(ℝ2) such that ∇f(x1, x2) = (x2, 0) at a.e. (x1, x2) ∈ Ω’. We also prove that every measurable set can be approximated arbitrarily closely in L1 by subsets that do not contain enhanced density points. Finally, we provide a new proof of a Poincaré-type lemma for locally finite perimeter sets, which was first stated by Delladio (S. Delladio, Functions of class C1 subject to a Legendre condition in an enhanced density set, to appear in Rev. Mat. Iberoamericana).

An integer n\geq2 is said to be a genus of a finite group G if there is a compact Riemann surface of genus n on which G acts as a group of automorphisms. In this paper, formulae are given for the minimum genus, minimum stable genus and the gap sequence, i.e., the (finite) set of non-genera, for a split metacyclic group of order pq, where p and q are primes. This information completely determines the genus spectrum for such groups.

D. Brannan's conjecture says that for 0 <α,β≤1, |x|=1, and n∈N one has |A2n−1(α,β,x)|≤|A2n−1(α,β,1)|, where We prove this for the case α=β, and also prove a differentiated version of the Brannan conjecture. This has applications to estimates for Gegenbauer polynomials and also to coefficient estimates for univalent functions in the unit disk that are ‘starlike with respect to a boundary point’. The latter application has previously been conjectured by H. Silverman and E. Silvia. The proofs make use of various properties of the Gauss hypergeometric function.

Let $G$ be a compact $p$-adic analytic group and let $\Lambda_G$ be its completed group algebra with coefficient ring the $p$-adic integers $\mathbb{Z}_p$. We show that the augmentation ideal in $\Lambda_G$ of a closed normal subgroup $H$ of $G$ is localisable if and only if $H$ is finite-by-nilpotent, answering a question of Sujatha. The localisations are shown to be Auslander-regular rings with Krull and global dimensions equal to dim $H$. It is also shown that the minimal prime ideals and the prime radical of the $\mathbb{F}_p$-version $\Omega_G$ of $\Lambda_G$ are controlled by $\Omega_{\Delta^+}$, where $\Delta^+$ is the largest finite normal subgroup of $G$. Finally, we prove a conjecture of Ardakov and Brown [1].

We study the eigenvalue problem = λ|u|q(x)−2u in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝN with smooth boundary ∂Ω, λ is a positive real number, and p1,⋅ ⋅ ⋅, pN, q are continuous functions satisfying the following conditions: 2 ≤ pi(x) < N, 1 < q(x) for all x ∈ Ω, i ∈ {1,. . .,N}; there exist j, k ∈ {1,. . .,N}, j ≠ k, such that pj ≡ q in Ω, q is independent of xj and maxΩq < minΩpk. The main result of this paper establishes the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that every λ ∈(λ1, ∞) is an eigenvalue, while no λ ∈ (0, λ0) can be an eigenvalue of the above problem.

Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. Let K(H) denote the ideal of compact operators on H. For any compact operator A let |A|=(A*A)1,2 and S1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeatedaccording to multiplicity. If, for some 1<p<∞, si(A)p <∞, we say that A is in the Schatten p-class Cp and ∥A∥p=1/p is the p-norm of A. Hence, C1 is the trace class, C2 is the Hilbert–Schmidt class, and C∞ is the ideal of compact operators K(H).

Let k be an algebraic number field and Ck its ideal class group (in the wider sense). Suppose K is a finite extension of k. Then we say that an ideal class of k capitulates in K if this class is in the kernel of the homomorphism

induced by extension of ideals from k to K (See Section 2 below). In [4], Iwasawa gives examples of real quadratic number fields, with distinct primes Pi ≡ 1 (mod 4), for which all the ideal classes of the 2-class group, Ck,2 (the 2-Sylow subgroup of Ck), capitulate in an unramified quadratic extension of k. In these examples, Ck,2 is abelian of type (2,2), i.e. isomorphic to ℤ/2ℤ×ℤ/2ℤ and so all four ideal classes capitulate.

For X a complex Banach space and U an open subset of the complex plane С, let O (U, X) denote the space of analytic X- valued functions defined on U. This is a Frechet space when endowed with the topology of uniform convergence on compact subsets, and the space X may be viewed as simply the constants in O(U, X). Every bounded operator T on X induces a continuous mapping TU on O(U, X) given by (Tuf)(λ) = (λ – T)f(λ) for every f e O(U, X) and λ e U. Corresponding to each closed F ⊂ С there is also an associated analytic subspace XT(F) = X ∩ ran(7c//F). For an arbitrary T e L(X), the spaces XT(F) are T-invariant, generally non-closed linear manifolds in X.

A positive integer nis called a square-full integer if p2 divides n whenever p is a prime divisor of n. For x > 1 we denote by Q(x) the number of square-full integers not exceeding x. Bateman and Grosswald [1] proved that

Let R→U be an associative ring epimorphism such that U is a flat left R-module. Assume that the related Gabriel topology $\mathbb{G}$ of right ideals in R has a countable base. Then we show that the left R-module U has projective dimension at most 1. Furthermore, the abelian category of left contramodules over the completion of R at $\mathbb{G}$ fully faithfully embeds into the Geigle–Lenzing right perpendicular subcategory to U in the category of left R-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an associative ring R, we consider the induced topology on every left R-module and, for a perfect Gabriel topology $\mathbb{G}$, compare the completion of a module with an appropriate Ext module. Finally, we characterize the U-strongly flat left R-modules by the two conditions of left positive-degree Ext-orthogonality to all left U-modules and all $\mathbb{G}$-separated $\mathbb{G}$-complete left R-modules.

The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue is investigated. Connections with discrete and continuous isothermic surfaces and generalised Clifford configurations are recorded. An interpretation of the Schwarzian Davey-Stewartson II flows as integrable deformations of conformally immersed surfaces is given.

The purpose of this paper is a further investigation on the 2-nilpotent multiplier, $\mathcal{M}$(2)(G), when G is a non-abelian p-group. Furthermore, taking G in the class of extra-special p-groups, we will get the explicit structure of $\mathcal{M}$(2)(G) and will classify 2-capable groups in that class.

In a previous note [3], Mennicke and I showed that the relations

(8, 7|2, 3): A8=B7=(AB)2=(AB)2=(A-1B)2=(A-1B)3=E

define a group of order 10752. As we remarked, the results of §§ 2, 3 of that note are not restricted in their application to this group; they apply to the group

[3, 7]+: B7=(AB)2=(A-1B)3=E

and to any factor group of this group which in turn has Klein's simple group of order 168, defined by

(4,7|2, 3): A4=B7=(AB)2=(A-1B)3=E,

as a factor group. In this note I use these results to establish alternative “weaker” definitions for Klein's group and for two groups discussed by Sinkov [4], namely (8, 7|2, 3) defined above and a factor group of this group of order 1344. These latter groups are eloquently discussed by Coxeter [1].

A compact bordered Klein surface X of algebraic genus g ≥ 2 has maximal symmetry [6] if its automorphism group A(X) is of order 12(g — 1), the largest possible. The bordered surfaces with maximal symmetry are clearly of special interest and have been studied in several recent papers ([6] and [9] among others).

In ([6]; pages 36–41), Lambek constructs the maximal ring of quotients Q(R) of a commutative ring R by denning a multiplication on Homr(D, R) where D ranges over all the dense ideals of R, and this generalizes the classical construction of ring of quotients, (cf. [6] for all the references on the subject.)

This programme is carried over, in the first section of this article, to the categoryof commutative reductive semigroups. Examples show that the maximal semigroup of quotients of a commutative monoid can be different from the classical one.

Fixed point and random fixed point theorems are presented for weakly inward maps. Also a continuation theorem for weakly inward maps is presented.

Let γ and γ' be non-negative integers. We say that the graph G is (γ, γ') bi-embeddable if G can be embedded in a surface of genus γ and the complement Ḡ of G can be embedded in a surface of genus γ'. Let N(γ, γ') be the least integer such that every graph with at least N(γ, γ') points is not (γ, γ') bi-embeddable. It has been shown in [1] and [5] that N(0, 0) = 9; this result was also obtained by John R. Ball of the Carnegie Institute of Technology. Our object here is to obtain upper and lower bounds for N(γ, γ').