Let G be a finite additive abelian group, and suppose that A and B are subsets of G. We say that G = A⊕B if every element g ∈ G can be uniquely written in the form g = a + b, where a ∈ A, b ∈ B. The study of such decompositions (usually called factorizations in the literature) was initiated by G. Hájos [3] in connection with his solution to a problem of Minkowski in the geometry of numbers.

The purpose of this paper is twofold. In [6] Tomiuk gives a representation theorem for a topologically simple right complemented algebra that is also an annihilator algebra. We strengthen this and then give a converse, so as to characterise right complemented algebras among respectively primitive Banach algebras and primitive annihilator Banach algebras. Our second aim is to investigate the relationship between the different annihilator conditions—left annihilator, right annihilator, annihilator, and dual—when imposed on a complemented algebra. Tomiuk [6] has already shown that a right complemented semisimple algebra that is a left annihilator algebra is an annihilator algebra; further, a topologically simple bi-complemented algebra that is also an annihilator algebra is dual. We show that for a topologically simple right complemented algebra all four annihilator conditions are equivalent. Further, for a semi-simple Banach algebra the first three are equivalent provided it is right complemented, and if it is also left complemented, then they are equivalent to duality.

We present several conjectures which would describe the Nielsen equivalence classes of generating pairs for the groups SL(2,q) and PSL(2,q). The Higman invariant, which is the union of the conjugacy classes of the commutator of a generating pair and its inverse, and the trace of the commutator play key roles. Combining known results with additional work, we clarify the relationships between the conjectures, and obtain various partial results concerning them. Motivated by the work of Macbeath (A. M. Macbeath, Generators of the linear fractional groups, in Number theory (Proc. Sympos. Pure Math., vol. XII, Houston, TX, 1967) (American Mathematical Society, Providence, RI, 1969), 14–32), we use another invariant defined using traces to develop algorithms that enable us to verify the conjectures computationally for all q up to 101, and to prove the conjectures for a highly restricted but possibly infinite set of q.

Mathematicians have studied the diophantine equation of the title ever since the days of Fermat, Leibniz and Euler. In this paper, we review the history of this problem, present several new classes of values of d for which the equation has only trivial solutions, and find a nontrivial solution for d = 85 (a case Euler missed). With these results, the question of whether

has nontrivial solutions is now answered for all d, 0≤d ≤100.

This paper is a continuation of [3] in which some inequalities for the Schatten p-norm were considered. The purpose of the present paper is to improve some inequalities in [3] as well as to give more inequalities in the same spirit.

Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators acting on H. Let K(H) denote the closed two-sided ideal of compact operators on H. For any compact operator A, let |A| = (A*A)½ and s1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeated according to multiplicity. A compact operator A is said to be in the Schatten p-class Cp(1 ≤ p < ∞), if Σ s1(A)p < ∞. The Schatten p-norm of A is defined by ∥A∥p = (Σ si(A)p)1/p. This norm makes Cp into a Banach space. Hence C1 is the trace class and C2 is the Hilbert-Schmidt class. It is reasonable to let C∞ denote the ideal of compact operators K(H), and ∥.∥∞ stand for the usual operator norm.

A ring $R$ with identity is called strongly clean if every element of $R$ is the sum of an idempotent and a unit that commute. For a commutative local ring $R$, $n=3,4$, and $m, k, s \in {\mathbb N}$ it is proved that ${\mathbb M}_n(R)$ is strongly clean if and only if ${\mathbb M}_n(R[[x]])$ is strongly clean if and only if ${\mathbb M}_n(R[[x_1, x_2, \ldots, x_m]])$ is strongly clean if and only if $ {\mathbb M}_n(\frac{R[x]}{(x^{k})})$ is strongly clean if and only if $ {\mathbb M}_n(\dfrac{R[x_{1}, x_{2}, \ldots , x_{s}]}{(x^{n_1}_{1}, x^{n_{2}}_{2}, \ldots , x^{n_{s}}_{s})}) $ is strongly clean if and only if ${\mathbb M}_n(R \propto R)$ is strongly clean where $ R\propto R=\{\scriptsize(\begin{array}{@{}c@{\quad}c@{}} a& b \\ 0& a \end{array}): a, b \in R \}$ is the trivial extension of $R$. This extends a result of J. Chen, X. Yang and Y. Zhou [$\mathbf{5}$] from $n=2$ to 3 and 4.

In this paper, we classify 3-dimensional Lagrangian Willmore submanifolds of the nearly kaehler 6-sphere $S^6(1)$ with constant scalar curvature.

We present Bäcklund transformations for the non-commutative anti-self-dual Yang–Mills equations where the gauge group is G = GL(2) and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasi-determinants and belong to a non-commutative version of the Atiyah–Ward ansatz. In the commutative limit, our results coincide with those by Corrigan, Fairlie, Yates and Goddard.

A group G is said to be a minimal non-FC group, if G contains an infinite conjugacy class, while every proper subgroup of G merely has finite conjugacy classes. The structure of imperfect minimal non-FC groups is quite well-understood. These groups are in particular locally finite. At the other end of the spectrum, a perfect locally finite minimal non-FC group must be a p-group. And it has been an open question for quite a while now, whether such groups exist or not.

We characterise singularities of focal surfaces of wave fronts in terms of differential geometric properties of the initial wave fronts. Moreover, we study relationships between geometric properties of focal surfaces and geometric invariants of the initial wave fronts.

The object of this note is to study the regular coverings of the closed orientable surface of genus 2.

Let the closed orientable surface Fh of genus h be a covering of F2 and let and f be the fundamental groups respectively. Then is a subgroup of f of index n = h − 1. A covering is called regular if is normal in f.

Conversely, let be a normal subgroup of f of finite index. Then there is a uniquely determined regular covering Fh such that is the fundamental group of Fh. The covering Fh is an orientable surface. Since the index n of in f is supposed to be finite, Fh is closed, and its genus is given by n = h − 1.

The fundamental group f can be defined by

.

The α-regular classes of any finite group G are important since they are those classes on which the projective characters of G with factor set α take non-zero value, and thus a knowledge of the α-regular classes gives the number of irreducible projective representations of G with factor set α (see [4]). Here we look at the particular case of the generalized symmetric group Cm wr Sl. The analogous problem of constructing the irreducible projective representations of Cm wr Sl has been dealt with in [6] by generalizing Clifford's theory of inducing from normal subgroups, but unfortunately, it is not in general possible to determine the irreducible projective characters (and hence the α-regular classes) by this method.

A ring is called clean if every element is the sum of an idempotent and a unit. It is an open question whether the tensor products of two clean algebras over a field is clean. In this note we study the tensor product of clean algebras over a field and we provide some examples to show that the tensor product of two clean algebras over a field need not be clean.

We show that if R is a ring such that each minimal left ideal is essential in a (direct) summand of RR, then the dual of each simple right R-module is simple if and only if R is semiperfect with Soc(RR)=Soc(RR) and Soc(Re) is simple and essential for every local idempotent e of R. We also show that R is left CS and right Kasch if and only if R is a semiperfect left continuous ring with Soc(RR)⊆eRR. As a particular case of both results we obtain that R is a ring such that every (essential) closure of a minimal left ideal is summand (R is then said to be left strongly min-CS) and the dual of each simple right R-module is simple if and only if R is a semiperfect left continuous ring with Soc(RR)=Soc(RR)⊆eRR. Moreover, in this case R is also left Kasch, Soc(eR)≠0 for every local idempotent e of R, and R admits a (Nakayama) permutation of a basic set of primitive idempotents. As a consequence of this result we characterise left PF rings in terms of simple modules over the 2×2 matrix ring by showing that R is left PF if and only if M2(R) is a left strongly min-CS ring such that the dual of every simple right module is simple.

We consider the divisibility of the class numbers of imaginary quadratic fields , where q is an odd prime number, k and n are positive integers. Suppose that k ≡ 1 mod 2 or n ≢ 3 mod 6. We show that the class numbers of imaginary quadratic fields ≠ are divisible by n for q ≡ 3 mod 8. This is a generalization of the result of Kishi for imaginary quadratic fields when k ≡ 1 mod 2 or n ≢ 3 mod 6. We also show that the class numbers of imaginary quadratic fields ≠ are divisible by n for q ≡ 1 mod 4 and the class numbers of imaginary quadratic fields ≠ are divisible by n for q ≡ 7 mod 8.

In this paper, we continue the work initiated by Morris [5] and Saeed-ul-Islam [6,7] and determine complete sets of inequivalent irreducible projective representations (which we shall write as i.p.r.) of finite Abelian groups with respect to some additional factor sets.

We consider an Abelian group

which will be referred to as an Abelian group of type (a1, …, am).

The purpose of this short note is to make an observation about Dunford–Pettis operators from L1[0, 1] to C0. Recall that an operator T:E→F (where E and F are Banach spaces) is called Dunford–Pettis if T takes weakly convergent sequences of E into norm convergent sequences of F. A Banach space F has the Compact Range Property (CRP) if every operator T:L1]0, 1]→F is Dunford–Pettis. Talagrand shows in his book [2] that C0 does not have the CRP. It is of interest (especially in mathematical economics [3]) to note that every positive operator from L1[0, 1] to C0 is Dunford–Pettis.

In this paper, we investigate various “arithmetical” functions associated with the factorisation of polynomials in GF[q, X1, …, Xk], where k ≥ 1 and GF[q]is the finite field of order q. We shall assume throughout that all polynomials discussed are non-zero and have been normalised by selecting one polynomial from each equivalence class with respect to multiplication by non-zero elements of GF[q]. The constant polynomial will be denoted by 1. With this normalisation, GF[q, X1, …, Xk] becomes a unique factorisation domain. When k = 1, normalisation is achieved by considering only monic polynomials. By the degree of a polynomial A(X1, …, Xk) will be understood the ordered set (m1, …, mk), where m1 is the degree of A(X1, …, Xk) in X1,(i = 1, …, k).

The joint spectrum for a commuting n-tuple in functional analysis has its origin in functional calculus which appeared in J. L. Taylor's epoch-making paper [19] in 1970. Since then, many papers have been published on commuting n-tuples of operators on Hilbert spaces (for example, [3], [4], [5], [8], [9], [10], [21], [22]).

Let $\rho:G\hookrightarrow {\rm GL}(n, {\bb F})$ be a representation of a finite group over the field ${\bb F}, V = {\bb F}^n$ the corresponding $G$-module, and ${\bb F}[V\hspace*{1.5pt}]$ the algebra of polynomial functions on $V$. The action of $G$ on $V$ extends to ${\bb F}[V\hspace*{1.5pt}]$, and ${\bb F}[V\hspace*{1.5pt}]^G$, respectively ${\bb F}[V\hspace*{1.5pt}]_G$, denotes the ring of invariants, respectively coinvariants. The theorem of Steinberg referred to in the title says that when ${\bb F} = {\bb C}$, dim$_{\bb C}$ (Tot$({\bb C}[V\hspace*{1.5pt}]_G))=|G|$ if and only if $G$ is a complex reflection group. Here Tot$({\bb F}[V\hspace*{1.5pt}]_G)$ denotes the direct sum of all the homogeneous components of the graded algebra ${\bb F}[V\hspace*{1.5pt}]_G$ and $|G|$ is the order of $G$. Chevalley's theorem tells us that the ring of invariants of a complex pseudoreflection representation $G\hookrightarrow {\rm GL}(n, {\bb C})$ is polynomial algebra, and the theorem of Shephard and Todd yields the converse. Combining these results gives: dim$_{\bb F}$(Tot$({\bb C}[V\hspace*{1.5pt}]_G) = |G|$ if and only if ${\bb C}[V\hspace*{1.5pt}]^G$ is a polynomial algebra. The purpose of this note is to show that the two conditions

(i) dim$_{\bb F}$(Tot(${\bb F}[V\hspace*{1.5pt}]_G))\,{=}\,|G|$,

(ii) ${\bb F}[V\hspace*{1.5pt}]^G$ is a polynomial algebra

are equivalent regardless of the ground field; in particular in the modular case.