By a theorem of Hurwitz [3], an algebraic curve of genus g ≧ 2 cannot have more than 84(g − l) birational self-transformations, or, as we shall call them, automorphisms. The bound is attained for Klein's quartic

of genus 3 [4]. In studying the problem whether there are any other curves for which the bound is attained, I was led to consider the universal covering space of the Riemann surface, which, as Siegel observed, relates Hurwitz's theorem to Siegel's own result [7] on the measure of the fundamental region of Fuchsian groups. Any curve with 84(g − 1) automorphisms must be uniformized by a normal subgroup of the triangle group (2, 3, 7), and, by a closer analysis of possible finite factor groups of (2, 3, 7), purely algebraic methods yield an infinite family of curves with the maximum number of automorphisms. This will be shown in a later paper.

Suppose that is a newform of weight k on Г1(N). Thus f is in particular a cusp form on Г1(N), satisfying

for all n≥1. Associated with f is a Dirichlet character

such that

for all, .

Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.

Among such forms, let . The Epstein zeta function of f is denned to be

Rankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,

We prove a corresponding result for theta functions. For real α > 0, let

This function satisfies the functional equation

(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

Let $R$ be a ring. An $R$-module $M$ is called a (weak) duo module provided every (direct summand) submodule of $M$ is fully invariant. It is proved that if $R$ is a commutative domain with field of fractions $K$ then a torsion-free uniform $R$-module is a duo module if and only if every element $k$ in $K$ such that $kM$ is contained in $M$ belongs to $R$. Moreover every non-zero finitely generated torsion-free duo $R$-module is uniform. In addition, if $R$ is a Dedekind domain then a torsion $R$-module is a duo module if and only if it is a weak duo module and this occurs precisely when the $P$-primary component of $M$ is uniform for every maximal ideal $P$ of $R$.

A generalized tensor product of groups was introduced by R. Brown and J.-L. Loday [6], and has led to a substantial algebraic theory contained essentially in the following papers: [6, 7, 1, 5, 11, 12, 13, 14, 18, 19, 20, 23, 24] ([9, 27, 28] also contain results related to the theory, but are independent of Brown and Loday's work). It is clear that one should be able to develop an analogous theory of tensor products for other algebraic structures such as Lie algebras or commutative algebras. However to do so, many non-obvious algebraic identities need to be verified, and various topological proofs (which exist only in the group case) have to be replaced by purely algebraic ones. The work involved is sufficiently non-trivial to make it interesting.

By a ‘representation’ we shall mean throughout a representation by n × n matrices with entries from an arbitrary field. Elsewhere [9] the author has introduced the concept of a principal representation of a semigroup S (see § 3 below for the definition) and has shown that if S satisfies the minimal condition on principal ideals then every irreducible representation is of this type. Moreover, if S satisfies the minimal conditions on both principal left and right ideals, which together imply the minimal condition on principal two-sided ideals [6, Theorem 4], the irreducible representations of S can ultimately be expressed explicitly in terms of group representations.

A subgroup H of a group G is said to be c-supplemented inG if there exists a subgroup K of G such thatHK=G and H \cap K is contained in Core_G (H). We follow Hall's ideas to characterize the structure of the finite groups in which every subgroup is c-supplemented. Properties of c-supplemented subgroups are also applied to determine the structure of some finite groups.

Throughout this note all rings considered will be commutative and noetherian and will have non-zero identity elements. A will always denote such a ring and the category of all A-modules and all A-homomorphisms will be denoted by A.

The oscillatory and asymptotic behaviour of the positive solutions of the autonomous neutral delay logistic equation

with r, c, T, K ∈ (0, ∞) has been recently investigated in [2]. More recently the dynamics of the periodic delay logistic equation

in which r, K are periodic functions of period τ and m is a positive integer is considered in [6]. The purpose of the following analysis is to obtain sufficient conditions for the existence and linear asymptotic stability of a positive periodic solution of a periodic neutral delay logistic equation

in which Ṅ denotes and r, K, c are positive continuous periodic functions of period τ at and m is a positive integer. For the origin and biological relevance of (1.3) we refer to [2].

In some recent work by D. G. Kendall and the author † on the number of points of a lattice which lie in a random circle the mean value of the variance emerged as a constant multiple of the value of the Epstein zeta-function Z(s) associated with the lattice, taken at the point s=. Because of the connexion with the problems of closest packing and covering it seemed likely that the minimum value of Z() would be attained for the hexagonal lattice; it is the purpose of this paper to prove this and to extend the result to other real values of the variable s.

First we recall that a (real) quasi-Banach space X is a complete metrizable real vector space whose topology is given by a quasi-norm satisfying

where C is some constant independent of x1 and x2. X is said to be p-normable (or topologically p-convex), where 0 < p ≤ l, if for some constant B we have

for any x1, …, xn, є X. A theorem of Aolci and Rolewicz (see [18]) asserts that if in C = 21/p-1, then X is p-normable. We can then equivalently re-norm X so that in (1.4) B = 1.

The paper deals with local existence, blow-up and global existence for the solutions of a wave equation with an internal nonlinear source and a nonlinear boundary damping. The typical problem studied is\cases{u_{tt}-\Delta u=|u|^{p-2}u \hfill & \rm{~}OPEN~7~in <hsp sp=0.25>[0,\rm{inf}ty )\times \Omega ,}\hfill \cr u=0 \hfill & \rm{~}OPEN~6~on <hsp sp=0.25>[0,\rm{inf}ty )\times \Gamma _0,}\hfill \cr \frac {\partial u}{\partial \nu }=-\alpha (x)|u_t|^{m-2}u_t \hfill & \rm{~}OPEN~2~on <hsp sp=0.25>[0,\rm{inf}ty )\times \Gamma _1,}\cr u(0,x)=u_0(x),u_t(0,x)=u_1(x) & \rm{~}OPEN~1~on<hsp sp=0.25>\Omega ,}\hfill }

where \Omega \subset ℝ^n (n\ge 1) is a regular and bounded domain, \partial \Omega =\Gamma _0\cup \Gamma _1, \lambda _{n-1}(\Gamma _0)>>;0, 2<>;p\le 2(n-1)/(n-2) (when n\ge 3), m>>;1, \alpha \in L^\rm{inf}ty (\Gamma _1), \alpha \ge 0, and the initial data are in the energy space. The results proved extend the potential well theory, which is well known when the nonlinear damping acts in the interior of \Omega, to this problem.

Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B (X) (K(X)). Let σ(A) and σa(A) denote, respectively, the spectrum and approximate point spectrum of an element A of B(X). Set

σem(A)and σeb(A) are respectively Schechter's and Browder's essential spectrum of A ([16], [9]). σea (A) is a non-empty compact subset of the set of complex numbers ℂ and it is called the essential approximate point spectrum of A ([13], [14]). In this note we characterize σab(A) and show that if f is a function analytic in a neighborhood of σ(A), then σab(f(A)) = f(σab(A)). The relation between σa(A) and σeb(A), that is exhibited in this paper, resembles the relation between the σ(A) and the σeb(A), and it is reasonable to call σab(A) Browder's essential approximate point spectrum of A.

Let M2 be a (connected) surface in Euclidean 3-space E3, and let G:M2→S2(1) ⊂ E3 be its Gauss map. Then, according to a theorem of E. A. Ruh and J. Vilms [3], M2 is a surface of constant mean curvature if and only if, as a map from M2 to S2(1), G is harmonic, or equivalently, if and only if

where δ is the Laplace operator on M2 corresponding to the induced metric on M2 from E3 and where G is seen as a map from M2to E3. A special case of (1.1) is given by

i.e., the case where the Gauss map G:M2→E3 is an eigenfunction of the Laplacian δ on M2.

We extend the well-known Paley and Paley-Kahane-Khintchine inequalities on lacunary series to the unit polydisk of . Then we apply them to obtain sharp estimates for the mean growth in weighted spaces h(p, α), h(p, log(α)) of Hardy–Bloch type, consisting of functions n-harmonic in the polydisk. These spaces are closely related to the Bloch and mixed norm spaces and naturally arise as images under some fractional operators.

For a class of second order nonlinear differential equations, sufficient conditions are presented to ensure that some, respectively all solutions are asymptotic to lines.

Let $\{x_i\}_{i=1}^{\infty}$ be an arbitrary strictly increasing infinite sequence of positive integers. For an integer $n\ge 1$, let $S_n=\{x_1,\ldots,x_n\}$. Let $\varepsilon$ be a real number and $q\ge 1$ a given integer. Let \smash{$\lambda _n^{(1)}\le \cdots\le \lambda _n^{(n)}$} be the eigenvalues of the power GCD matrix $((x_i, x_j)^{\varepsilon})$ having the power $(x_i,x_j)^{\varepsilon}$ of the greatest common divisor of $x_i$ and $x_j$ as its $i,j$-entry. We give a nontrivial lower bound depending on $x_1$ and $n$ for \smash{$\lambda _n^{(1)}$} if $\varepsilon>0$. Especially for $\varepsilon>1$, this lower bound is given by using the Riemann zeta function. Let $x\ge 1$ be an integer. For a sequence \smash{$\{x_i\}_{i=1}^{\infty }$} satisfying that $(x_i, x_j)=x$ for any $i\ne j$ and \smash{$\sum_{i=1}^{\infty }{1\over {x_i}}=\infty$}, we show that if $0<\varepsilon\le 1$, then \smash{${\rm lim}_{n\rightarrow \infty }\lambda _n^{(1)}=x_1^{\varepsilon}-x^{\varepsilon }$}. Let $a\ge 0, b\ge 1$ and $e\ge 0$ be any given integers. For the arithmetic progression \smash{$\{x_{i-e+1}=a+bi\}_{i=e}^{\infty}$}, we show that if $0<\varepsilon\le 1$, then \smash{${\rm lim}_{n\rightarrow \infty }\lambda _n^{(q)}=0$}. Finally, we show that for any sequence \smash{$\{x_i\}_{i=1}^{\infty}$} and any \smash{$\varepsilon>0$, $\lambda_n^{(n-q+1)}$} approaches infinity when $n$ goes to infinity.

Let S be a semigroup whose set E of idempotents is non-empty. We define a partial ordering ≧ on E by the rule that e ≧ f and only if ef = f = fe. If E = {ei: i∈ N}, where N denotes the set of all non-negative integers, and if the elements of E form the chain

then S is called an ω-semigroup.

When the theory of Hankel transforms is applied to the solution of certain mixed boundary value problems in mathematical physics, the problems are reduced to the solution of dual integral equations of the type

where α and ν are prescribed constants and f(ρ) is a prescribed function of ρ [1]. The formal solution of these equations was first derived by Titchmarsh [2]. The method employed by Titchmarsh in deriving the solution in the general case is difficult, involving the theory of Mellin transforms and what is essentially a Wiener-Hopf procedure. In lecturing to students on this subject one often feels the need for an elementary solution of these equations, especially in the cases α = ± 1, ν = 0. That such an elementary solution exists is suggested by Copson's solution [3] of the problem of the electrified disc which corresponds to the case α = –l, ν = 0. A systematic use of a procedure similar to Copson's has in fact been made by Noble [4] to find the solution of a pair of general dual integral equations, but again the analysis is involved and long. The object of the present note is to give a simple solution of the pairs of equations which arise most frequently in physical applications. The method of solution was suggested by a procedure used by Lebedev and Uflyand [5] in the solution of a much more general problem.

The equations. We shall solve the equation

giving the solution in two forms, and give a new solution of

originally solved by Carlemann.

The latter will be extended to the case where the limits are a and b with a < x <b.