C. Widland [14] has defined Weierstrass points on integral, projective Gorenstein curves. We show here that the Weierstrass points on a generic integral rational nodal curve have the minimal possible weights or, equivalently, that such a curve has the maximum possible number of distinct nonsingular Weierstrass points. Rational curves with g nodes arise in degeneration arguments involving smooth curves of genus g and they have also recently arisen in connection with g-soliton solutions to certain nonlinear partial differential equations [11], [13].

Let f(z) be an entire function (of several variables). We define the function

which is increasing. The orderof f(z) is the constant (perhaps infinite)

In the analysis of mixed boundary value problems in the plane, we encounter dual integral equations of the type

If we make the substitutions cos we obtain a pair of dual integral equations of the Titchmarsh type [1, p. 334] with α = − 1, v = − ½ (in Titchmarsh's notation). This is a particular case which is not covered by Busbridge's general solution [2], so that special methods have to be derived for the solution.

In [1], a condition was sought on a commutative Noetherian ring R such that whenever R satisfied the condition and R was cyclically pure in an R-algebra S, then R was necessarily pure in S. Such a condition was found and turned out not only to be a sufficient condition but also a necessary condition. Rings satisfying this condition were called approximately Gorenstein. In this paper, we give new equivalent conditions for a ring to be approximately Gorenstein. Our conditions involve a certain metric topology defined on the lattice of ideals of a commutative Noetherian ring as well as the concept of a principal system. The notion of principal systems was defined in [9] and was suggested by Macaulay's theory of inverse systems [7]. These ideas have proved to be useful in many research probems in commutative algebra.

In this paper, we establish a new (but equivalent) definition of projective Hjelmslev planes of level n. This shows that the nth floor of a triangle building is a projective Hjelmslev plane of level n (a result already announced in [9], but left unproved). This will allow us to characterize Artmann-sequences by means of their inverse limits and to construct new ones. We also deduce a new existence theorem for level n projective Hjelmslev planes. All results hold in the finite as well as in the infinite case.

Potential problems in which different conditions hold over two different parts of the same boundary can often be conveniently reduced to the solution of a pair of dual integral equations. In some problems, however, the boundary condition is such that different conditions hold over three different parts of the boundary and, in such cases, the integral equations involved are frequently of the form

where f(r), g(r) are specified functions of r, p = ± ½ and ø(u) is to be found. Such equations might well be called triple integral equations and, in this note, I point out certain special cases which I have found to be capable of solution in closed form.

For regular semigroups, the appropriate analogue of the concept of a variety seems to be that of an e(xistence)-variety, developed by Hall [6,7,8]. A class V of regular semigroups is an e-variety if it is closed under taking direct products, regular subsemigroups and homomorphic images. For orthodox semigroups, this concept has been introduced under the term “bivariety” by Kaďourek and Szendrei [12]. Hall showed that the collection of all e-varieties of regular semigroups forms a complete lattice under inclusion. Further, he proved a Birkhoff-type theorem: each e-variety is determined by a set of identities. For e-varieties of orthodox semigroups a similar result has been proved by Kaďourek and Szendrei. At variance with the case of varieties, prima facie the free objects in general do not exist for e-varieties. For instance, there is no free regular or free orthodox semigroup. This seems to be true for most of the naturally appearing e-varieties (except for cases of e-varieties which coincide with varieties of unary semigroups such as the classes of all inverse and completely regular semigroups, respectively). This is true if the underlying concept of free objects is denned as usual. Kaďourek and Szendrei adopted the definition of a free object according to e-varieties of orthodox semigroups by taking into account generalized inverses in an appropriate way. They called such semigroups bifree objects. These semigroups satisfy the properties one intuitively expects from the “most general members” of a given class of semigroups. In particular, each semigroup in the given class is a homomorphic image of a bifree object, provided the bifree objects exist on sets of any cardinality. Concerning existence, Kaďourek and Szendrei were able to prove that in any class of orthodox semigroups which is closed under taking direct products and regular subsemigroups, all bifree objects exist and are unique up to isomorphism. Further, similar to the case of varieties, there is an order inverting bijection between the fully invariant congruences on the bifree orthodox semigroup on an infinite set and the e-varieties of orthodox semigroups. Recently, Y. T. Yeh [22] has shown that suitable analogues to free objects exist in an e-variety V of regular semigroups if and only if all members of V are either E-solid or locally inverse.

Recently H.-E. Richert [10] introduced a new method of summability, for which he completely solved the “summability problem” for Dirichlet series, and which led also to an extension of our knowledge of the relations between the abscissae of ordinary and absolute Rieszian summability. This non-linear method, which may best be characterized by the notion “strong Rieszian summability” †, depends on three parameters, on the order k;, the type λ, and the index p;. While Richert's paper deals almost exclusively with the application of that method of summability in a specialized form (namely the case p = 2, λn=log n) to Dirichlet series, it is the object of the present paper, to consider the general theory of strong Rieszian summability.

The structure of semigroups whose subsemigroups form a chain under inclusion was determined by Tamura [9]. If we consider the analogous problem for inverse semigroups it is immediate that (since idempotents are singleton inverse subsemigroups) any inverse semigroup whose inverse subsemigroups form a chain is a group. We will therefore, continuing the approach of [5, 6], consider inverse semigroups whose full inverse subsemigroups form a chain: we call these inverse ▽-semigroups.

The aim of the present note is to describe the possible products when taking all the nonzero elements of a finite ring in some sequence. Compared with the analogous situation for finite groups, where the set of products of all elements has been shown in [2] to be a whole coset of the derived group, for rings the set of the above mentioned products will be proved either to be as large as possible or to consist of one or two elements only.

A Banach space E is said to be regular if every bounded linear operator from E into E′ is weakly compact. This property was studied in [7, 9] under the name Property (w). In [7], using James type spaces as constructed in [4], examples were given of regular Banach spaces which fail to have weakly sequentially complete duals, answering a question raised in [9]. In this paper, we present some more results concerning the regularity of James type spaces.

Prime rings came into prominence when Posner characterized prime rings satisfying a polynomial identity [9]. The scarcity of invertible central elements made it difficult to generalize results from central simple and primitive algebras to prime rings. For example, we do not automatically have tensor products at our disposal. In [5], the first author introduced the Martindale ring of quotients Q(R) of a prime ring R in his theorem characterizing prime rings satisfying a generalized polynomial identity (GPI). Q(R) is a prime ring containing R whose center C is a field called the extended centroid of R. The central closure of R is the subring RC of Q(R) generated by R and C. RC is a closed prime ring since its extended centroid equals its center C. Hence we have a useful procedure for proving results about an arbitrary prime ring R. We first answer the question for closed prime rings and then apply to R the information obtained from RC. It should be noted that simple rings and free algebras of rank at least 2 are closed prime rings. For these reasons, closed prime rings are natural objects to study.

Let E be a locally compact space which can be expressed as the union of an increasing sequence of compact subsets Kn (n =1, 2, …) and let μ be a positive Radon measure on E. Ω is the space of equivalence classes of locally integrable functions on E. We denote the equivalence class of a function f by and if is an equivalence class then f denotes any function belonging to f. Provided with the topology defined by the sequence of seminorms

Ω is a Fréchet space. The dual of Ω is the space φ of equivalence classes of measurable, p.p. bounded functions vanishing outside a compact subset of E. For a subset Γ of Ω, the collection Λ of all ∊Ω, such that for each g∊Γ the product fg is integrable, is called a Köthe space and Γ is said to be the denning set of Λ. The Köthe space Λx which has Λ as a denning set is called the associated Kothe space of Λ. Λ and Λx are put into duality by the bilinear form

Quadratic forms associated with graphs were introduced over a century ago by Jordan [4]. We are concerned with the optimisation of such quadratic forms, following Motzkin and Straus [5], and we use the setting of categories and functors to express the nice interplay between the algebra and the graph theory. Applications to interchange graphs are also obtained.

C. Eberhart and W. Williams [3] showed that the least inverse semigroup congruence , on an orthodox semigroup S, plays a very important role in determining the structure of the lattice of congruences on S. In this note we show that their results can be applied to give an explicit construction for the idempotent separating congruences on S in terms of idempotent separating congruences on S/.

Let γ and γ' be non-negative integers. We say that the graph G is (γ, γ') bi-embeddable if G can be embedded in a surface of genus γ and the complement Ḡ of G can be embedded in a surface of genus γ'. Let N(γ, γ') be the least integer such that every graph with at least N(γ, γ') points is not (γ, γ') bi-embeddable. It has been shown in [1] and [5] that N(0, 0) = 9; this result was also obtained by John R. Ball of the Carnegie Institute of Technology. Our object here is to obtain upper and lower bounds for N(γ, γ').

Recently there have appeared papers ([7], [8]; also see [9]) in which integral equations with kernels involving the confluent hypergeometric function

have been studied. These equations are mainly Volterra equations of the first kind except that they have infinite domain (0, ∞). The rest are of the related type with integrals over (x, ∞) instead of (0, x); and all are convolution equations.

Ramanujan's short third notebook contains 33 pages of unorganized material. On page 386 of the third notebook, in the pagination of the Tata Institute's publication [3], Ramanujan offers the following remarkable identity.

Several definitions of a semigroup of quotients have been proposed and studied by a number of authors. For a survey, the reader may consult Weinert's paper [8]. The motivation for many of these concepts comes from ring theory and the various notions of rings of quotients. We are concerned in this paper with an analogue of the classical ring of quotients, introduced by Fountain and Petrich in [3].