Non-completely simple bisimple semigroups S which are generated by a finite number of idempotents are studied by means of Rees matrix semigroups over local submonoids eSe, e = e2 ∈ S. If under the natural partial order on the set Es of idempotents of such a semigroup S the sets ω(e) = {ƒ ∈ Es: ƒ ≤ e} for each e ∈ Es are well-ordered, then S is shown to contain a subsemigroup isomorphic to Sp4, the fundamental four-spiral semigroup. A non-completely simple hisimple semigroup is constructed which is generated by 5 idempotents but which does not contain a subsemigroup isomorphic to Sp4.

In this article, we give new proofs of two of Ramanujan’s $1/\unicode[STIX]{x1D70B}$ formulae

$$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70B}}=\frac{2\sqrt{2}}{99^{2}}\mathop{\sum }_{m=0}^{\infty }(26390m+1103)\frac{(4m)!}{396^{4m}(m!)^{4}}\end{eqnarray}$$

$$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70B}}=\frac{2}{84^{2}}\mathop{\sum }_{m=0}^{\infty }(21460m+1123)\frac{(-1)^{m}(4m)!}{(84\sqrt{2})^{4m}(m!)^{4}}\end{eqnarray}$$

$1/\unicode[STIX]{x1D70B}$

Let (X, T, π) be a topological transformation group, where X is a Hausedorff continuum. We will say that X is irreducibly T-invariant if no proper subcontinuum of X is T-invariant. Wallace, [6], has shown that if T is abelian and X is irreducibly T-invariant, then X has no cut point; he then asked if this statement remains true if “abelian” is replaced by “compact”. In this paper we answer this question in the affirmative, and prove a related result when T satisfies a recursive property.

We consider the motion of a particle of mass m and electrical charge e, moving in a constant magnetic field Bk, where k is a unit vector, and acted upon by a force mf(t). The position vector r(t) of this particle is governed by the differential equation where .

Let λ0 be a semisimple eigenvalue of an operator T0. Let Γ0 be a circle with centre λs0 containing no other spectral value of T0. Some lower bounds are obtained for the convergence radius of the power series for the spectral projection P(t) and for trace T(t)P(t) associated with linear perturbation family T(t) = T0 + tV0 and the circle Γ0. They are useful when T0 is a member of a sequence (Tn) which approximates an operator T in a collectively compact manner. These bounds result from a modification of Kato's method of majorizing series, based on an idea of Redont. I λ0 is simple, it is shown that the same lower bound are valid for the convergence radius of a power series yielding an eigenvector of T(t).

Characterizations of collectively precompact and collectively semi-precompact sets of operators on topological vector spaces are obtained. These lead to the characterization of totally bounded sets of semi-precompact operators on locally convex spaces.

1980 Mathematics subject classification (Amer. Math. Soc): primary 47 B 05, 47 D 15; secondary 46 A 05, 46 A 15.

Throughout this note Rn denotes the n-dimensional Euclidean space. Addition and multiplication in Rn are defined component-wise. If k ≦ n is a positive integer and x ∈ Rn, we write xk for the k-th component of x. The set {x ∈ Rn: xk ≠ 0 for each k ≦ n} is designated by R#.

It is well known that Thom–Boardman symbols are realized by nonincreasing sequences of nonnegative integers. A natural question is whether the converse is also true. In this paper we answer this question affirmatively, that is, for any nonincreasing sequence of nonnegative integers, there is at least one map-germ with the prescribed sequence as its Thom–Boardman symbol.

Consider the simultaneous real differential equations , where the dot indicates differentiation with respect to time, and a, b, c, d are functions about which the only thing that we know is that they are uniformly bounded, .

For numerical semigroups with a specified list of (not necessarily minimal) generators, we describe the asymptotic distribution of factorization lengths with respect to an arbitrary modulus. In particular, we prove that the factorization lengths are equidistributed across all congruence classes that are not trivially ruled out by modular considerations.

Gaschütz has introduced the concept of a product of a Schunck class and a (saturated) formation (differing from the usual product of classes) and has shown that this product is a Schunck class provided that both of its factors consist of finite soluble groups. We investigate the same question in the context of arbitrary finite groups.

In the controversy which has been continuing over the “clock paradox”, it has been claimed by supporters ([1]) of the orthodox interpretation that the difference in behaviour of clocks carried by observers in motion relative to each other can be traced to differencesin the motion relative to the universe. Whilst A remains unaccelerated relative to the universe, B for a portion of his journey does not. Professor Dingle, in reply ([2]), remarked that he would “not dispute a possible slight effect” but could not understand why, if the acceleration of B were so important, it did not appear explicitly in the answer.

We give a formula for the Dixmier-Douady class of a continuous-trace groupoid crossed product that arises from an action of a locally trivial, proper, principal groupoid on a bundle of elementary C*-algebras that satisfies Fell's condition.

Barnes (1978, 1979) introduced the concept of a -extreme form, which is a Minkowski-reduced positive definite quadratic form having prescribed diagonal coefficients α1, α2, …, αn and providing a local minimum of the determinant of the form over all such forms. Here a class of forms which are -extreme for all α and all n is described.

There is an old conjecture that every integer can be decomposed into four (positive or negative) perfect cubes. More specifically one would like to know the asymptotic number of solutions of when a large bound N is placed on the parts mi. Using the circle method it is shown that the number of such representations of n when N → ∞ is asymptotically equal to C(n). N for a certain positive constant C(n), provided that the contribution of the minor arcs can be neglected.

Given a group G, we may ask whether it is the commutator subgroup of some group G. For example, every abelian group G is the commutator subgroup of a semi-direct product of G x G by a cyclic group of order 2. On the other hand, no symmetric group Sn(n>2) is the commutator subgroup of any group G. In this paper we examine the classical linear groups over finite fields K of characteristic not equal to 2, and determine which can be commutator subgroups of other groups. In particular, we settle the question for all normal subgroups of the general linear groups GLn(K), the unitary groups Un(K) (n≠4), and the orthogonal groups On(K) (n≧7).

Let X be a weakly complete proper cone contained in a weak space E and h(E) the Riesz space generated by the continuous linear forms on E. A positive conical measure μ on X is a positive linear form on h(E)|x. G. Choquet has proved μ is a Daniell integral on E when E is weakly complete, but μ is not generally a Daniell integral on X. However we give an integration theory for functions on X and compare this theory with the classical Daniell theory. The case where μ is maximal in the sense of G. Choquet is remarkable.

In an elementary topos if R is a ring and X is a decidable object then there exists a canonical homomorphism from the coproduct of an X-family of R-modules to the product of the same family. In this paper it is shown that this homomorphisms is a monomorphism.

Let $M$ be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup ${\mathcal{M}}_{\unicode[STIX]{x1D6E5}}f(x):=\sup _{t>0}|e^{-t\unicode[STIX]{x1D6E5}}f(x)|$ where $\unicode[STIX]{x1D6E5}$ is the Laplace–Beltrami operator acting on $M$. Then, by combining the subordination formula with the previous result, we obtain the weak type $(1,1)$ and $L^{p}$ boundedness of the maximal function ${\mathcal{M}}_{\sqrt{L}}^{k}f(x):=\sup _{t>0}|(t\sqrt{L})^{k}e^{-t\sqrt{L}}f(x)|$ on $L^{p}(M)$ for $1<p\leq \infty$ where $k$ is a nonnegative integer and $L$ is a nonnegative self-adjoint operator satisfying a suitable heat kernel upper bound. An interesting thing about the results is the lack of both doubling condition of $M$ and the smoothness of the operators’ kernels.

First, we generalize the definition of a locally compact topology given by Paterson and Welch for a sequence of locally compact spaces to the case where the underlying spaces are $T_{1}$ and sober. We then consider a certain semilattice of basic open sets for this topology on the space of all paths on a graph and impose relations motivated by the definitions of graph C*-algebra in order to recover the boundary path space of a graph. This is done using techniques of pointless topology. Finally, we generalize the results to the case of topological graphs.