If A and B are torsion-free groups, and W is a cyclically reduced word of even length in A*B, it is generally conjectured that a Freiheitssatz holds, namely that each of A and B are embedded via the natural map into the one-relator product group G = (A*B)/N(W), where N denotes normal closure. If W has length 2, then G is a free product of A and B with infinite cyclic amalgamation, and the result is obvious. The purpose of this note is to prove the Freiheitssatz in some special cases.

We define model structures on a triangulated category with respect to some proper classes of triangles and give a general study of triangulated model structures. We look at the relationship between these model structures and cotorsion pairs with respect to a proper class of triangles on the triangulated category. In particular, we get Hovey's one-to-one correspondence between triangulated model structures and complete cotorsion pairs with respect to a proper class of triangles. Some applications are given.

An n-tournament is a set of n labelled points, each pair A, B of which is joined either by the oriented line AB or by the oriented line BA. There are N = n(n –1)/2 such pairs and so Fn different n-tournaments, where Fn = 2N. A tournament is reducible if the points can be separated into two non-empty subsets and ℬ, such that every line joining a point in to a point in ℬ is directed towards the point in ℬ. Rado [3] showed that an irreducible tournamentis strongly connected; i.e. for every ordered pair of points A, B, there is a sequence of correctly oriented lines AC1, C1C2, …, ChB in the tournament, and conversely that a strongly connected tournament is irreducible.

Our aim in this paper is to investigate the restrictions placed on the structure of a finite group if it can be generated by subnormal T-subgroups (a T-group is a group in which every subnormal subgroup is normal). For notational convenience we denote by the class of finite groups that can be generated by subnormal T-subgroups and by the subclass of of those finite groups generated by normal T-subgroups; and for the remainder of this paper we will only consider finite groups.

Let Rn denote real Euclidean space of n dimensions. If

define , and (as usual) so that, by the inequality of arithmetic and geometric means,

Let Λ0 be the integer lattice, consisting of those points in Rn whose co-ordinates are integers. A non-singular n × n matrix M will be called a Minkowski matrix, if, for any point a ∈ Rn, there exists a point x ∈ Λ0 such that

It was shown by Minkowski that, when n = 2, every non-singular matrix is a Minkowski matrix, and that, for general n, every rational non-singular matrix is a Minkowski matrix. Minkowski is also said to have conjectured that every non-singular matrix is a Minkowski matrix, whatever the value of n. For n = 3, this was proved by Remak [5], and a much simpler proof was given later by Davenport [2]. For n = 4, it was proved by Dyson [3], who used a method similar to that of Remak and Davenport, but required the methods of algebraic topology to deal with some of the complications which arise in the higher dimension. Since this method depends also on the reduction of quadratic forms, it is quite likely that it might fail for higher values of n even if the topological difficulties could be overcome. Therefore, an alternative proof for n = 3, due to Birch and Swinnerton-Dyer [1], is of some interest, though in this dimension it is more complicated than Davenport's proof. A good deal of their analysis applies to general n, and they showed that for all n there is a neighbourhood of the unit matrix I that consists entirely of Minkowski matrices.

It is well known that a compact embedded hypersurface of the Euclidean space without boundary is a round sphere if one of mean curvature functions is constant. In this note, we show that a compact embedded hypersurface of the Euclidean space (and other constant curvature spaces) without boundary is a round sphere if the ratio of some two mean curvature functions is constant.

1991 Mathematics Subject Classification 53C40, 53C20.

Throughout this paper R will be an associative ring with unity and all R-modules are unitary. The right (resp. left) annihilator in R of a subset X of a module is denoted by r(X)(resp. I(X)). The Jacobson radical of R is denoted by J(R), the singular ideals are denoted by Z(RR) and Z(RR) and the socles by Soc(RR) and Soc(RR). For a module M, E(M) and PE(M) denote the injective and pure-injective envelopes of M, respectively. For a submodule A ⊆ M, the notation A ⊆⊕M will mean that A is a direct summand of M.

In this paper, we mainly provide a categorical view on the braided structures appearing in the Hom-quantum groups. Let $\mathcal{C}$ be a monoidal category on which F is a bimonad, G is a bicomonad, and ϕ is a distributive law, we discuss the necessary and sufficient conditions for $\mathcal{C}^G_F(\varphi)$, the category of mixed bimodules to be monoidal and braided. As applications, we discuss the Hom-type (co)quasitriangular structures, the Hom–Yetter–Drinfeld modules, and the Hom–Long dimodules.

Recently some inversion integrals for integral equations involving Legendre, Chebyshev, Gegenbauer and Laguerre polynomials in the kernel have been obtained [1, 2, 3, 5, 6]. In this note, two inversion integrals for integral equations involving Whittaker's function in the kernel are obtained. We shall make use of the following known integral [4, p. 402]

The results of this note are based on the following two integrals, which are derived from (1) by writing u – t = (v – t)x.

for m + 1 > 2v > – 1;

for m + 1 > 2v > – 1.

Let $\alpha$ be an irrational number and $\varphi$: $\mathbb{N} \to \mathbb{R^+}$ be a decreasing sequence tending to zero. Consider the set \[E_{\varphi}(\alpha)=\{\beta \in \mathbb{R}: \ \|n \alpha- \beta\|<\varphi(n)\ {\rm {holds\ for\ infinitely\ many}} \ n \in \mathbb{N}\}\], where $\|{\cdot}\|$ denotes the distance to the nearest integer. We show that for general error function $\varphi$, the Hausdorff dimension of $E_{\varphi}(\alpha)$ depends not only on $\varphi$, but also heavily on $\alpha$. However, recall that the Hausdorff dimension of $E_{\varphi}(\alpha)$ is independent of $\alpha$ when $\varphi(n) = n^{-\gamma}$ with $\gamma >1$.

In this paper we generalize the formula of Frobenius-Stickelberger (see (0.1) below) and the formula of Kiepert (see (0.2) below) to the genus-two case.

Motivated by a classical theorem of Schoenberg, we prove that an n + 1 point finite metric space has strict 2-negative type if and only if it can be isometrically embedded in the Euclidean space $\mathbb{R}^{n}$ of dimension n but it cannot be isometrically embedded in any Euclidean space $\mathbb{R}^{r}$ of dimension r < n. We use this result as a technical tool to study ‘roundness’ properties of additive metrics with a particular focus on ultrametrics and leaf metrics. The following conditions are shown to be equivalent for a metric space (X,d): (1) X is ultrametric, (2) X has infinite roundness, (3) X has infinite generalized roundness, (4) X has strict p-negative type for all p ≥ 0 and (5) X admits no p-polygonal equality for any p ≥ 0. As all ultrametric spaces have strict 2-negative type by (4) we thus obtain a short new proof of Lemin's theorem: Every finite ultrametric space is isometrically embeddable into some Euclidean space as an affinely independent set. Motivated by a question of Lemin, Shkarin introduced the class $\mathcal{M}$ of all finite metric spaces that may be isometrically embedded into ℓ2 as an affinely independent set. The results of this paper show that Shkarin's class $\mathcal{M}$ consists of all finite metric spaces of strict 2-negative type. We also note that it is possible to construct an additive metric space whose generalized roundness is exactly ℘ for each ℘ ∈ [1, ∞].

In this note, we find all the solutions of the Diophantine equation x2 + 5a 13b = yn in positive integers x, y, a, b, n≥ 3 with x and y coprime.

In his fundamental paper, “On the structure of semigroups” [6], J. A. Green has examined certain important minimal conditions which may be satisfied bya semigroup S.We say that S satisfies the minimal condition on principal left ideals if every set of principal left ideals of S contains a minimal member with respect to inclusion:this condition is denoted by ℳ1. The corresponding conditions on principal rightideals and principal two-sided ideals are denoted by ℳr and ℳ1 respectively. The purpose of the present paper is to give some further results concerning these three conditions.Extensive use is made of the work of A. H. Clifford ([3] and [4]) onminimal ideals.

Let ζ(s) = σn-s (Res >1) denote the Riemann zeta function; then, as is well known, , where Bm denotes the mth Bernoulli number, In this paper we investigate the possibility of similar evaluations of the Epstein zeta function ζq(s) at the rational integers s = k> 2. Let

be a positive definite quadratic form and

where the summation is over all pairs of integers except (0, 0). In attempting to evaluate ζq(k) we are guided by Kronecker's first limit formula [11]

where γ is Euler's constant,

is the Dedekind eta-function, and τ is the complex number in the upper half plane, ℋ, associated with Q by the formula

On the basis of (1.3) we would expect a formula involving functions of τ. This formula is stated in Theorem 1, (2.13).

The word problem for free bands (idempotent semigroups) was solved by Green and Rees [4] in an early paper. They also established certain properties of the free band. This was followed by McLean [6] who provided a general structure theory for bands with some indication as to the structure of the free band. Since then a great many papers have appeared dealing with various aspects of the topic of bands and their varieties. A different solution of the word problem for free bands was recently given by Siekmann and Szabó [9]. For a discussion of bands, see the books [5] and [8].

We make the convention that if a is an element of a semigroup Q then by writing a–1 it is implicit that a lies in a subgroup of Q and has inverse a–1 in this subgroup; equivalently, a ℋ a2 and a–1 is the inverse of a in Ha.

A subsemigroup S of a semigroup Q is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be written as a–1b where a, b ∈ S and, in addition, every element of S satisfying a weak cancellation condition which we call square-cancellable lies in a subgroup of Q. The notions of right order and semigroup of right quotients are defined dually; if S is both a left order and a right order in Q then S is an order in Q and Q is a semigroup of quotients of S.

A graph is called weakly perfect if its chromatic number equals its clique number. In this paper a new class of weakly perfect graphs arising from rings are presented and an explicit formula for the chromatic number of such graphs is given.

We prove a general result on Bailey pairs and show that two Bailey pairs of Bringmann and Kane are special cases. We also show how to use a change of base formula to pass from the pairs of Bringmann and Kane to pairs used by Andrews in his study of Ramanujan's seventh order mock theta functions. We derive several more Bailey pairs of a similar type and use these to construct a number of new q-hypergeometric double sums which are mock theta functions. Finally, we prove identities between some of these mock theta double sums and classical mock theta functions.

We study Gevrey classes of holomorphic functions of several variables on a polysector, and their relation to classes of Gevrey strongly asymptotically developable functions. A new Borel-Ritt-Gevrey interpolation problem is formulated, and its solution is obtained by the construction of adequate linear continuous extension operators. Our results improve those given by Haraoka in this context, and extend to several variables the one-dimensional versions of the Borel-Ritt-Gevrey theorem given by Ramis and Thilliez, respectively. Some rigidity properties for the constructed operators are stated.