An interesting class of submanifolds of Hermitian manifolds is the class of slant submanifolds which are submanifolds with constant Wirtinger angle. In [1–4,7,8] slant submanifolds of complex projective and complex hyperbolic spaces have been investigated. In particular, it was shown that there exist many proper slant surfaces in CP^2 and inCH^2 and many proper slant minimal surfaces in C2. In contrast, in the first part of this paper we prove that there do not exist proper slant minimal surfaces inCP^2 and in CH^2. In the second part, we present a general construction procedure for obtaining the explicit expressions of such slant submanifolds. By applying this general construction procedure, we determine the explicit expressions of special slant surfaces of CP^2 and of CH^2. Consequently, we are able to completely determine the slant surface which satisfies a basic equality. Finally, we apply the construction procedure to prove that special θ-slant isometric immersions of a hyperbolic plane into a complex hyperbolic plane are not unique in general.

Following, for example, Kurošs [8], we define the (transfinite) upper central series of a group G to be the series

such that Zα + 1/Za is the centre of G/Zα, and if β is a limit ordinal, then If α is the least ordinal for which Zα =Zα+1=…, then we say that the upper central series has length α, and that Zα= His the hypercentre of G. As usual, we call G nilpotent if Zn= Gfor some finite n.

Potential problems in which different conditions hold over two different parts of the same boundary can often be conveniently reduced to the solution of a pair of dual integral equations. In some problems, however, the boundary condition is such that different conditions hold over three different parts of the boundary and, in such cases, the integral equations involved are frequently of the form

where f(r), g(r) are specified functions of r, p = ± ½ and ø(u) is to be found. Such equations might well be called triple integral equations and, in this note, I point out certain special cases which I have found to be capable of solution in closed form.

Denote by

the Euler product, and by

the partition generating function. More generally, if k is any integer, put

so that p(n) = p−1(n). In [3], Atkin proved the following theorem.

In this paper we shall be concerned with the derivation of simple expressions for the sums of some infinite series involving the zeros of Bessel functions of the first kind. For instance, if we denote by γv, n (n = l, 2, 3,…) the positive zeros of Jv(z), then, in certain physical applications, we are interested in finding the values of the sums

and

where m is a positive integer. In § 4 of this paper we shall derive a simple recurrence relation for S2m,v which enables the value of any sum to be calculated as a rational function of the order vof the Bessel function. Similar results are given in § 5 for the sum T2m,v.

We define the notion of lightcone Gauss maps, lightcone pedal curves and lightcone developables of spacelike curves in Minkowski 3-space and establish the relationships between singularities of these objects and geometric invariants of curves under the action of the Lorentz group.

We approach the analysis of the extent of the projectivity of modules from a fresh perspective as we introduce the notion of relative subprojectivity. A module M is said to be N-subprojective if for every epimorphism g : B → N and homomorphism f : M → N, there exists a homomorphism h : M → B such that gh = f. For a module M, the subprojectivity domain of M is defined to be the collection of all modules N such that M is N-subprojective. We consider, for every ring R, the subprojective profile of R, namely, the class of all subprojectivity domains for R modules. We show that the subprojective profile of R is a semi-lattice, and consider when this structure has coatoms or a smallest element. Modules whose subprojectivity domain is as smallest as possible will be called subprojectively poor (sp-poor) or projectively indigent (p-indigent), and those with co-atomic subprojectivy domain are said to be maximally subprojective. While we do not know if sp-poor modules and maximally subprojective modules exist over every ring, their existence is determined for various families. For example, we determine that artinian serial rings have sp-poor modules and attain the existence of maximally subprojective modules over the integers and for arbitrary V-rings. This work is a natural continuation to recent papers that have embraced the systematic study of the injective, projective and subinjective profiles of rings.

In [2], R. C. James proved that a weakly closed subset X of a real Banach space is weakly compact if and only if each continuous linear form attains its supremum on X. He also extended the result to the locally convex case, and, in [5], J. D. Pryce gave a simplified proof of the general result that is recorded below for reference in the sequel.

In this paper, we study Lp-boundedness properties for higher order Littlewood-Paley g-functions in the Bessel setting. We use the Calderón-Zygmund theory in a homogeneous-type space (in the sense of Coifman and Weiss) ((0, ∞), d, γα), where d represents the usual metric on (0, ∞) and γα denotes the doubling measure on (0, ∞) with respect to d defined by dγα(x) = x2α+1dx, with α > −1/2.

Given a group presentation (or more generally† a 2-complex) one can associate with it an object which has variously been called the co-initial graph, star-graph, star-complex, and which has proved useful in several contexts [2], [6], [7], [8], [9], [10], [12]. For certain mappings of 2-complexes φ: ⃗ℒ (”strong mappings”) one gets an induced mapping φst: st⃗ℒst of the associated star-complexes. Then st is a covariant functor from the category of 2-complexes (where the morphisms are strong mappings) to the category of 1-complexes, and this functor behaves very nicely with respect to coverings (Theorem 1).

Many q-identities have been proved combinatorially. For example

It has long been known that there is a strong connection between the class numbers of quadratic fields and the distribution of quadratic residues. This connection is exemplified, for instance, by the class number formulae of Dirichlet, which have formed the basis of much of the work on the subject of class numbers.

In 1927 Schreier [8] proved the Nielsen-Schreier Theorem that a subgroup H of a free group F is a free group by selecting a left transversal for H in F possessing a certain cancellation property. Hall and Rado [5] call a subset T of a free group F a Schreier system in F if it possesses this cancellation property, and consider the existence of a subgroup H of F such that a given Schreier system T is a left transversal for H in F.

We introduce the concept of left APP-rings which is a generalization of left p.q.-Baer rings and right PP-rings, and investigate its properties. It is shown that the APP property is inherited by polynomial extensions and is a Morita invariant property.

Let $\alpha$ be an irrational number. We determine the Hausdorff dimension of sets of real numbers which are close to infinitely many elements of the sequence $(\{n\alpha\})_{n\,{\ge}\,1}$.

The hierarchy structure of a derivative nonlinear Schrödinger equation is investigated in terms of the Sato-Segal-Wilson formulation. Special solutions are constructed as ratios of Wronski determinants. Relations to the Painlevé IV and the discrete Painlevé I are discussed by applying a similarity reduction.

The equi-attraction properties concerning the global attractors $\a_\lam$ of dynamical systems $S_\lam(t)$ with parameter $\lam\in\Lam$, where $\Lam$ is a compact metric space, are investigated. In particular, under appropriate conditions, it is shown that the equi-attraction of the family $\{\a_\lam\}$ is equivalent to the continuity of $\a_\lam$ in $\lam$ with respect to the Hausdorff distance.

Given the series ,the n-th Casáro sum of order k is defined by the relation

where is the binomial coefficient . Let Then Σan is said to be summable (C; K) to the sum s if, as n → ∞, The series is said to be absolutely summable (C; k), or summable | is convergent. The series is said to be strongly summable (C; k) with index p, or summable [Ck, p], to the sum s if

It is assumed that k and p are positive.

Let G be a non-nilpotent group in which all proper subgroups are nilpotent. If G is finite then G is soluble [18], and a classification of such groups is given in [14]. The paper [12]. of Newman and Wiegold discusses infinite groups with this property. Clearly such a group is either finitely generated or locally nilpotent. Many interesting results concerning the finitely generated case are established in [12]. Since the publication of that paper there have appeared the examples due to Ol'shanskii and Rips (see [13]) of finitely generated infinite simple p-groups all of whose proper nontrivial subgroups have order p, a prime. Following [12], let us say that a group G is an AN-group if it is locally nilpotent and non-nilpotent with all proper subgroups nilpotent. A complete description is given in Section 4 of [12] of AN-groups having maximal subgroups. Every soluble AN-gvoup has locally cyclic derived factor group and is a p-group for some prime p ([12; Lemma 4.2]). The only further information provided in [12] on AN-groups without maximal subgroups is that they are countable. Four years or so after the publication of [12], there appeared the examples of Heineken and Mohamed [5]: for every prime p there exists a metabelian, non-nilpotent p-group G having all proper subgroups nilpotent and subnormal; further, G has no maximal subgroups and so G/G' is a Prüfer p-group in each case.

We develop a general theory of Hopf image of a Hopf algebra representation, with the associated concept of inner faithful representation, modelled on the notion of faithful representation of a discrete group. We study several examples, including group algebras, enveloping algebras of Lie algebras, pointed Hopf algebras, function algebras, twistings and cotwistings, and we present a Tannaka duality formulation of the notion of Hopf image.