The modular group Г is the group of integral bilinear transformations of the extended complex plane which preserve the upper half-plane. It has the presentation 〈x, y:x2 = y3 = 1〉, and the generators can be chosen so that u = xy maps z to z + 1.

In this paper the author considers the problem of finding a formula of inversion for the integral transform defined by the equation

where a >0, k > 0 and r-1f(r) εL (a, ∞). This transform appeared in connection with an earlier investigation [4] in which an attempt was made to devise an integral transform that could be adapted to the solution of certain boundary value problems involving the space form of the wave equation and the condition of radiation:

In [5] Professor Hooley announced without proof the following result which is a variant of well-known work by Heilbronn [4]and Danicic [3] (see [1]).

Let k≥2 be an integer, b a fixed non-zero integer, and a an irrational real number. Then, for any ɛ> 0, there are infinitely many solutions to the inequality

Here

An inverse semigroup whose idempotents form an ω-chain e0 > e1 > e2 > … is called briefly an ω-semigroup. A structure theorem for simple ω-semigroups was established by Kočin [7]; a related structure theorem for simple, and also general, ω-semigroups was proved by Munn [10]. These results represent an extension of the structure theorem for bisimple a ω-semigroups due to Reilly [14].

During the past few years, some papers of P. Deligne and J.-P. Serre (see [2], [9], [10] and other references cited there) have included an investigation of certain properties of coefficients of modular forms, and in particular Serre [10] (see also [11]) obtained the divisibility property (1) below. Let

be a modular form of integral weight k ≧ 1 on a congruence subgroup of SL2(Z), and suppose that each cn belongs to the ring RK of integers of an algebraic number field K finite over Q. For c ∈ RK and m ≧ 1 an integer, write c ≡ 0 (mod m) if c ∈ m RK and c ≢ 0 (mod m) otherwise. Then Serre showed that there exists α > 0 such that

as x → ∞, where throughout this note N(n ≦ x: P) denotes the number of positive integers n ≦ x with the property P.

In this paper we show that the (p+l)st homotopy group of the p-spun trefoil knot is nontrivial. This result was obtained for p = 1 in [1] using duality arguments. Here we take a totally different approach via the algorithm given in [3] and a module representation giving a simpler and more natural argument.

In [9] L. Moser classified all manifolds obtained by Dehn surgery on torus knots. In particular she proved the following (see also [8, Chapter IV]).

Theorem 1 [9]. Nontrivial surgery with slope m/n on a nontrivial torus knot T(p, q) gives a manifold with cyclic fundamental group iff m = npq ± 1 and the manifold obtained is the lens space L(m, nq2).

We prove that if n>0 is an integer and r>0 is a real number, then

The upper bound is best possible. Inequality (*) is a converse of a result of G. Bennett who proved that Qn(r)>l.

We confine ourselves, for simplicity, to first-order algebraic differential equations (ADE's), although analogous considerations may be made for higher-order ADE's:

P(x, y(x), y'(x)) = 0. (*)

A motion of (*) is a change of independent variable that takes solutions to solutions, that is, a suitable map <p of the underlying interval I into itself so that if y is a solution of (*) then y ° φ is a solution of (*), i.e.

In previous papers [1, 2, 3] the sums of a number of series of products of E-functions have been found. For the definitions and properties of the E-functions the reader is referred to [4, pp. 348–358]. In § 3 a further series of this type is given. The proof is based on an integral of an E-function with respect to its parameters, to be established in § 2. Similar integrals were given in [5] and [6].

Let H be a group of characters on an (algebraic) abelian group G. In a natural way, we may regard G as a group of characters on H. In this way, we obtain a duality between the two groups G and H. One may pose several problems about this duality. Firstly, one may ask whether there exists a group topology on G for which H is precisely the set of continuous characters. This question has been answered in the affirmative in [1]. We shall say that such a topology is compatible with the duality between G and H. Next, one may ask whether there exists a locally compact group topology on G which is compatible with a given duality and, if so, whether there is more than one such topology. It is this second question (previously considered by other authors, to whom we shall refer below) which we shall consider here.

V-rings and their generalisations have been studied extensively in recent years [2], [3], [5],[6], [7]. All the rings we consider will be associative rings with 1 ≠ 0 and all the modules considered will be unitary left R-modules. All the concepts will be left-sided unless otherwise mentioned. Thus by an ideal in R we mean a left ideal of R. A ring R is called a V-ring (respectively a GV-ring) if every simple (resp. simple, singular) module is injective. An R-module M is called p-injective if any homomorphism f: I → M with I a principal left ideal of R can be extended to a homomorphism g: R → M. A ring R is called a p-V-ring (resp. a p-V'-ring) if every simple (resp. simple, singular) module over R is p-injective. The object of the present paper is to introduce torsion theoretic generalizations of p-V-rings and prove results similar to those obtained by Yue Chi Ming about p-V-rings and p-V'-rings [6], [7]. For any M ∈ R-mod, J(M) will denote the Jacobson radical of M and Z(M) the singular submodule of M. For any λ ∈ R, we denote the left annihilator { r ∈ R| rλ =0 } of λ in R by l(λ).

Several papers have appeared in the past few years which have explored the topic of the Riesz decomposition for amarts. Such a decomposition for amarts enjoys several special properties. See [5, p. 208–209]. While it has been proved in [6] that not every martingale in the limit has a Riesz decomposition “in the weakest form assuring uniqueness” it is the major objective of this paper to characterize a class of martingales in the limit which is strictly larger than the class of amarts but enjoys all the properties of the decomposition for amarts.

A function φ(p) is operationally related to h(t) when they satisfy the integral equation

provided that the integral is convergent and R(p)> 0.

As usual, we shall denote (1) by the symbolic expression

φ(p) ≑ h(t).

The object of this paper is to evaluate some infinite integrals involving E-functions by the methods of the operational calculus. Most of the results obtained are believed to be new.

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].

Congruences on a semigroup S such that the corresponding factor semigroups are of a special type have been considered by several authors. Frequently it has been difficult to obtain worthwhile results unless restrictions have been imposed on the type of semigroup considered. For example, Munn [6] has studied minimum group congruences on an inverse semigroup, R. R. Stoll [9] has considered the maximal group homomorphic image of a Rees matrix semigroup which immediately determines the smallest group congruence on a Rees matrix semigroup. The smallest semilattice congruence on a general or commutative semigroup has been studied by Tamura and Kimura [10], Yamada [12] and Petrich [8]. In this paper we shall study congruences ρ on a completely regular semigroup S such that S/ρ is a semilattice of groups. We shall call such a congruence an SG-congruence.

The main aims of this paper are to provide a device for constructing large families of complex-multiplication (CM) fields, and to examine the Galois groups of some related field extensions. We recall that an algebraic number field K (i.e. ) is called a CM-field if it is totally complex but is quadratic over some totally real field (see Section 1). CM fields are important in algebraic geomety, since the ring of endomorphisms of a simple abelian variety defined over an algebraic number field is either ℤ or a ℤ-order in a CM field. Moreover, CM fields figure prominently in classfield theory, since Shimura [15] has shown that “almost all” classfields over CM fields K can be generated by means of division points on abelian varieties admitting ℤ-orders in K as their endomorphism rings. Shimura's work can be regarded as a natural generalization of the classical method (due to Kronecker and H. Weber) of generating classfields of imaginary quadratic fields via division points on CM-elliptic curves.