Let R be a ring with identity and let Eij ∈ Mn(R) be the usual n X n matrix units, where n ≥ 2 and 1≤i, j≤N. Let En(R) be the subgroup of GLn(R) generated by all Tij(q where r ∈ R and i ≄ j. For each (two-sided) R-ideal q let En(R, q) be the normal subgroup of En(R) generated by Tij(q), where q ∈ q. The subgroup En(R, q) plays an important role in the theory of GLn(R). For example, Vaserˇstein has proved that, for a larger class of rings (which includes all commutative rings), every subgroup S of GLn(R), when R ∈ and n≥3, contains the subgroup En(R, q0), where q0 is the R-ideal generated by αij, rαij-αjjr (i ≄ j, r ∈ R), for all (αij) ∈ S. (See [13, Theorem 1].) In addition Vaseršstein has shown that, for the same class of rings, En(R, q) has a simple set of generators when n ≥ 3. Let Ên(R, q) be the subgroup of En(R, q) generated by Tij(r)Tij(q)Tij(−r), where r ∈ R, q ∈ q. Then Ên(R, q) = En(R, q), for all q, when R ∈ and n ≥ 3.(See [13, Lemma 8].)

Let S be a semigroup and let be an S-graded ring. Rs = 0 for all but finitely many elements s ∈ S1, then R is said to have finite support. In this paper we concern ourselves with the question of whether a graded ring R with finite support inherits a given ring theoretic property from the homogeneous subrings Re corresponding to idempotent semigroup elements e.

In the thirty years since it was proved that 0, 1 and 144 were the only perfect squares in the Fibonacci sequence [1, 9], several generalisations have been proved, but many problems remain. Thus it has been shown that 0, 1 and 8 are the only Fibonacci cubes [6] but there seems to be no method available to prove the conjecture that 0, 1, 8 and 144 are the only perfect powers.

A Banach space operator has property (δ) if and only if it is the quotient of a decomposable operator, equivalently, if and only if its adjoint has Bishop's property (β). Within this class of operators, it is shown that quasisimilarity preserves essential spectra.

The most classical sufficient condition for the fixed point property of non-expansive mappings FPP in Banach spaces is the normal structure (see [6] and [10]). (See definitions below). Although the normal structure is preserved under finite lp-product of Banach spaces, (1<p≤∞), (see Landes, [12], [13]), not too many positive results are known about the normal structure of an l1,-product of two Banach spaces with this property. In fact, this question was explicitly raised by T. Landes [12], and M. A. Khamsi [9] and T. Domíinguez Benavides [1] proved partial affirmative answers. Here we give wider conditions yielding normal structure for the product X1⊗1X2.

In this paper we consider examples of orders in restricted power semigroups, where for any semigroup Sthe restricted power semigroup is given by with multiplication XY = {xy:x ∈ X, y ∈ Y} for all X, Y ∈ . We use the notion of order introduced by Fountain and Petrich in [2] which first appears in the form used here in [3]. If S is a subsemigroup of Q then S is an order in Q and Q is a semigroup of quotients of S if any q ∈ Q can be written as q = a*b = cd* where a, b, c, d ∈ S is the inverse of a(d) in a subgroup of Q, and in addition, all elements of S satisfying a weak cancellability condition called square-cancellability lie in a subgroup of Q.

We present an elementary proof of the theorem, usually attributed to Noether, that if L/K is a tame finite Galois extension of local fields, then is a free -module where Γ=Gal(L/K. The attribution to Noether is slightly misleading as she only states and proves the result in the case where the residual characteristic of K does not divide the order of Γ [4]. In this case is a maximal order in KΓ which is not true for general groups Γ. There is an elegant proof in the standard reference [2], but this relies on a difficult result in representation theory due to Swan. Our proof depends on a close examination of the structure of tame local extensions, and uses only elementary facts about local fields. It also gives an explicit construction of a generator element, and the same proof works both for localizations of number fields and of global function fields.

The Mountain-Pass Theorem of Ambrosetti and Rabinowitz (see [1]) and the Saddle Point Theorem of Rabinowitz (see [21]) are very important tools in the critical point theory of C1-functional. That is why it is natural to ask us what happens if the functional fails to be differentiable. The first who considered such a case were Aubin and Clarke (see [6]) and Chang (see [12]),who gave suitable variants of the Mountain-Pass Theorem for locally Lipschitz functionals which are denned on reflexive Banach spaces. For this aim they replaced the usual gradient with a generalized one, which was firstly defined by Clarke (see [13], [14]).As observed by Brezis (see [12, p. 114]), these abstract critical point theorems remain valid in non-reflexive Banach spaces.

Suppose H is a Hilbert space and write ℒ(H) for the set of all bounded linear operators on H. If T ∈ ℒ(H) we write σ(T) for the spectrum of T; π0(T) for the set of eigenvalues of T; and π00(T) for the isolated points of σ(T) that are eigenvalues of finite multiplicity. If K is a subset of C, we write iso K for the set of isolated points of K. An operator T ∈ ℒ(H) is said to be Fredholm if both T−1(0) and T(H)⊥ are finite dimensional. The index of a Fredholm operator T, denoted by index(T), is defined by

If E is a Hausdorff locally convex space and M is an -dimensional subspace of the algebraic dual E* that is transverse to the continuous dual E′, then, according to [7], the Mackey topology τ(E, E′ + M) is a countable enlargement (CE) of τ(E, E′) [or of E]. Much is still unknown as to when CEs preserve barrelledness (cf. [14]). E is quasidistinguished (QD) if each bounded subset of the completion Ê is contained in the completion of a bounded subset of E [12]. Clearly, each normed space is QD, and Tsirulnikov [12] asked if each CE of a normed space must be a QDCE, i.e., must preserve the QD property. Since CEs preserve metrizability (but not normability), her question was whether metrizable spaces so obtained must be QD, and was moderated by Amemiya's negative answer (cf. [5, p. 404]) to Grothendieck's query, who had asked if all metrizable spaces are QD, having proved the separable ones are [4].

In this paper we calculate the localisation at the prime 3 of the integral cohomology ring of the Mathieu group M24, together with its mod-3 cohomology ring. The main results are

Theorem 1. The ring H*(M24, Z)(3)is the commutative graded Z(3)-algebra with generators

and relations v2 = 0 and βθ = 0. The Chern classes of the Todd representation in GL11F2

J. H. E. Cohn solved the diophantine equations x2 + 74 = yn and x2 + 86 = yn, with the condition 5 ∤ n, by more or less elementary methods. We complete this work by solving these equations for 5 | n, by less elementary methods.

According to the well-known Nash's theorem, every Riemannian n-manifold admits an isometric immersion into the Euclidean space En(n+1)(3n+11)/2. In general, there exist enormously many isometric immersions from a Riemannian manifold into Euclidean spaces if no restriction on the codimension is made. For a submanifold of a Riemannian manifold there are associated several extrinsic invariants beside its intrinsic invariants. Among the extrinsic invariants, the mean curvature function and shape operator are the most fundamental ones.

A number of theorems are established about positive definite functions and representations of certain topological semigroups. In particular we establish theorems which show that measurable positive definite functions and measurable representations can each be decomposed into the sum of two parts one of which is continuous and the other of which is “small”.

If Sis a regular semigroup then an inverse transversal of S is an inverse subsemigroup T with the property that |T ∩ V(x)| = 1 for every x ∈ S where V(x) denotes the set of inverses of x ∈ S. In a previous publication [1] we considered the similar concept of a subsemigroup T of S such that |T ∩ A(x)| = 1 for every x ∩ S where A(x) = {y∈ S;xyx = x} denotes the set of associates (or pre-inverses) of x ∈ S, and showed that such a subsemigroup T is necessarily a maximal subgroup Ha for some idempotent α ∈ S. Throughout what follows, we shall assume that S is orthodox and α is a middle unit (in the sense that xαy = xy for all x, y ∈ S). Under these assumptions, we obtained in [1] a structure theorem which generalises that given in [3] for uniquely unit orthodox semigroups. Adopting the notation of [1], we let T ∩ A(x) = {x*} and write the subgroup T as Hα = {x*;x ∈ S}, which we call an associate subgroup of S. For every x ∈ S we therefore have x*α = x* = αx* and x*x** = α = x**x*. As shown in [1, Theorems 4, 5] we also have (xy)* = y*x* for all x, y ∈ S, and e* = α for every idempotent e.

Let μ ≠ 0 be an ultradistribution of Beurling type with compact support in the space . We investigate the range of the convolution operator Tμ on the space of non-quasianalytic functions of Beurling type associated with a weight w, in the case the operator is not surjective. It is proved that the range of TM always contains the space of real-analytic functions, and that it contains a smaller space of Beurling type for a weight σ ≥ ω if and only if the convolution operator is surjective on the smaller class.