It is well known that in any near-ring, any intersection of prime ideals is a semiprime ideal. The aim of this paper is to prove that any semiprime ideal I in a near-ring N is the intersection of all minimal prime ideals of I in N. As a consequence of this we have any seimprime ideal I is the intersectionof all prime ideals containing I.

The main purpose of this paper is to discuss the categories of the minimal topological spaces investigated in [1], [2], [7], and [8]. After these results are given, an application will be made to answer the following question: If is the lattice of topologies on a set X and is a Hausdorff (or regular, or completely regular, or normal, or locally compact) topology does there always exist a minimal Hausdorff (or minimal regular, or minimal completely regular, or minimal normal, or minimal locally compact) topology weaker than ?

After reviewing geometric quantisation of linear bosonic and fermionic systems, we study the holonomy of the projectively flat connection on the bundle of Hilbert spaces over the space of compatible complex structures and relate it to the Maslov index and its various generalisations. We also consider bosonic and fermionic harmonic oscillators parametrised by compatible complex structures and compare Berry’s phase with the above holonomy.

It is known that, for semigroups, the property of admitting a finite presentation is preserved on passing to subsemigroups and extensions of finite Rees index. The present paper shows that the same holds true for Malcev, cancellative, left-cancellative and right-cancellative presentations. (A Malcev (respectively, cancellative, left-cancellative, right-cancellative) presentation is a presentation of a special type that can be used to define any group-embeddable (respectively, cancellative, left-cancellative, right-cancellative) semigroup.)

A generalization of Karamata's theorem on integrals of regularly varying functions is proved. Using Laplace-Stieltjes transforms it is shown that any regularly varying function with exponent α (α + 1 ∉ N) is asymptotic to another regularly varying function all of whose derivations are regularly varying.

Sum-free sets may be used to colour the edges of a complete graph in such a way as to avoid monochromatic triangles. We discuss the automorphism groups of such graphs. Embedding of colourings is considered. Finally we illustrate a way of constructing colourings using block designs.

For the $(d+1)$-dimensional Lie group $G=\mathbb{Z}_{p}^{\times }\ltimes \mathbb{Z}_{p}^{\oplus d}$, we determine through the use of $p$-power congruences a necessary and sufficient set of conditions whereby a collection of abelian $L$-functions arises from an element in $K_{1}(\mathbb{Z}_{p}\unicode[STIX]{x27E6}G\unicode[STIX]{x27E7})$. If $E$ is a semistable elliptic curve over $\mathbb{Q}$, these abelian $L$-functions already exist; therefore, one can obtain many new families of higher order $p$-adic congruences. The first layer congruences are then verified computationally in a variety of cases.

Until 1965, when Janko [7] established the existence of his finite simple group J1, the five Mathieu groups were the only known examples of isolated finite simple groups. In 1951, R. G. Stanton [10] showed that M12 and M24 were determined uniquely by their order. Recent characterizations of M22 and M23 by Janko [8], M22 by D. Held [6], and M11 by W. J. Wong [12], have facilitated the unique determination of the three remaining Mathieu groups by their orders. D. Parrott [9] has so characterized M22 and M11, while this paper is an outline of the characterization of M23 in terms of its order.

For a given vector measure n, an important problem, but in practice a difficult one, is to give a concrete description of the dual space of L1(n). In this note such a description is presented for an important class of measures n, namely the spectral measures (in the sense of N. Dunford) and certain other vector and operator-valued measures that they naturally induce. The basic idea is to represent the L1-spaces of such measures as a more familiar space whose dual space is known.

We show that the rings described in the title are precisely the indecomposable injectives for the category whose objects are the associative rings and whose morphisms are the ring homomorphisms with accessible images. These rings are more or less completely known. Those of cardinality greater than that of the continuum are subdirectly irreducible but there are some nontrivial principal ideal domains in the class.

We consider the wreath product of two permutation groups G≤Sym Γ and H≤Sym Δ as a permutation group acting on the set Π of functions from Δ to Γ. Such groups play an important role in the O’Nan–Scott theory of permutation groups and they also arise as automorphism groups of graph products and codes. Let X be a subgroup of Sym Γ≀Sym Δ. Our main result is that, in a suitable conjugate of X, the subgroup of SymΓ induced by a stabiliser of a coordinate δ∈Δ only depends on the orbit of δ under the induced action of X on Δ. Hence, if X is transitive on Δ, then X can be embedded into the wreath product of the permutation group induced by the stabiliser Xδ on Γ and the permutation group induced by X on Δ. We use this result to describe the case where X is intransitive on Δ and offer an application to error-correcting codes in Hamming graphs.

If Ω is a ring region with starlike boundary components α and β, then we show for each λ > 0 there exists a ring region ω ⊂ Ω with ∂ ω = α ∪ ϒ, α ∩ ϒ = φ such that there is a harmonic function V in ω satisfying (a) V(z) = 0 for z ∈ α, (b) V(z) = 1 for z ∈ ϒ, (c) | grad V(z)| = λ for z ∈ ϒ ∩ Ω. Furthermore, we show when ω is not equal to Ω; that is, that is, there is a non-trival solution.

Let 1≤p <∞ and 1/p+1/q = 1. If φ ∈ Lq, we denote by Tφ the functional defined on the Hardy space Hp by . A function f in Hp, which satisfies Tpφ(f) = ‖Tpφ‖ and ‖f‖p ≤ 1, is called an extremal function. Also, φ is called an extremal kernel when ‖φ‖q =‖Tpφ‖. In this paper, using the results in the case of p = 1, we study extremal kernel and extremal functions for p > 1.

Stampfli and Embry have shown that a point of the numerical range of an operator is extreme if and only if a set of vectors corresponding to it is linear. This is generalized here to show that a point of the closure of the numerical range is extreme if and only if a corresponding set of sequences forms a linear space. A more geometric alternative proof is given for a theorem of Das and Garske concerning weak convergence to zero at the unattained extreme points of the closure of the numerical range.The result is shown to hold also for lone extreme points of the numerical range which lie on line segments on its boundary. Further, a bound is obtained on the norm of the weak limit of the weakly convergent sequences corresponding to points on a line segment on the boundary of numerical range.

The Stokes-Beltrami equations, being applicable to many classes of physical problems, have for a long time received a great deal of attention from authors investigating their various aspects. One may cite, for example, Weinstein [1] and more recently Ranger [2] as just two of many research papers into this topic. Baecklund transformations have, in previous work, been applied to hodograph-type equations (Loowrner ], Power, Rogers and Onborn [4], Rogers [5, 6]) and reduction to appropriate canonical forms in elliptic, parabolic, and hyprebolic regimes has been achieved.

Suppose that f(z) is a function of one complex variable satisfying

where ρ is an integer larger than 1 and a(z) and b(z) are rational functions. We consider f evaluated at the algebraic point a and develop a transcendence measure for f(α) under suitable conditions on f and α.

In this paper it is shown how nonpointed exactness provides a framework which allows a simple categorical treatment of the basics of Kurosh–Amitsur radical theory in the nonpointed case. This is made possible by a new approach to semi-exactness, in the sense of the first author, using adjoint functors. This framework also reveals how categorical closure operators arise as radical theories.

Since 1954, when Yang and Mills [7] presented their idea of isotopic gauge transformation, the method of introducing interactions into field theories by using general gauge invariance has been extensively studied.

A general formalism was presented by Utiyama [6]. Reference [6] also contains the first application of the formalism to the theory of gravitation. A more general approach to the Young-Mills formalism applied to general relativity was described by Kibble [3]. In a special case of interacting Dirac field the gauge invariance group can still be enlarged, leading to the possibility of describing short-range interactions together with gravitation and electromagnetism [5]. It is, therefore important to have a definite formulation of the common geometrical content of such theories.

Two sets of r + 2 points, Pi, P'i, each spanning a projective space of r + 1 dimensions, [r + 1], which has no solid ([3]) common with that spanned by the other, are said to be projective from an [r — 1], if here is an [r — 1] which meets the r + 2 joins Pi ′i. It is to be proved that the two sets are projective, if and only if the r + 2 intersections Ai of their corresponding [r]s lie in a line a. Ai are said to be the arguesian points and a the arguesian line of the sets. When r= 1, the proposition becomes the well- known Desargues' two-triangle theorem (3) in a plane. Therefore in analogy with the same we name it as the Desargues' theorem in [2r]. Following Baker (1, pp. 8—39), we may prove this theorem in the same synthetic style by making use of the axioms and the corresponding proposition of incidence in [2r + 1] or with the aid of the Desargues' theorem in a plane and the axioms of [2r] only. But the use of symbols makes its proof more concise; the algebraic approach adopted here is due to the referee (Arts. 2, 3, 5, 6, 7). Pairs of sets of r + p points each projective from an [r— 1] are also introduced to serve as a basis for a much more thorough investigation.

Given two $C^{\ast }$-correspondences $X$ and $Y$ over $C^{\ast }$-algebras $A$ and $B$, we show that (under mild hypotheses) the Cuntz–Pimsner algebra ${\mathcal{O}}_{X\otimes Y}$ embeds as a certain subalgebra of ${\mathcal{O}}_{X}\otimes {\mathcal{O}}_{Y}$ and that this subalgebra can be described in a natural way in terms of the gauge actions on ${\mathcal{O}}_{X}$ and ${\mathcal{O}}_{Y}$. We explore implications for graph algebras, crossed products by $\mathbb{Z}$, crossed products by completely positive maps, and give a new proof of a result of Kaliszewski, Quigg, and Robertson related to coactions on correspondences.