In a recent paper by Douglas Underwood [8] several definitions of “associated prime divisors” were discussed and shown to be unique. In this note we produce a fifth type, which is due to W. Krull, and is found in his classical paper [2] and further discussed by B. Banaschewski in [1]. Historically this characterization considerably predates the other four definitions.

Throughout this note, R denotes a commutative ring with unity, and all ideals and elements are assumed to be in such a ring. We shall let upper case letters, most frequently the beginning of the alphabet, denote ideals and lower case letters, elements of R. On the whole, our terminology will be that of [9]. We do, however, take exception with [9] in two instances, viz.

Let G be a finite graph topologically embedded in the interior of a 3-manifold M. Doyle (4) and Debrunner and Fox (3) have noted that the following local homotopy condition at each point p ∈ G is necessary in order for the embedding of G to be tame:

For each sufficiently small open set U containing p, there is an open set V such that p ∈ V ⊂ U and if W is any connected open set such that p ∈ W ⊂ V, then the image under the inclusion homomorphism i*: π1(W — G) → π1(U — G) is a free group on n — 1 generators.

Considerable advance has been made recently towards a systematic method of evaluating the “product” {μ} ꕕ {λ}, most notably in the methods of Robinson (3), Littlewood (2), and Todd (5) and the differential operator technique of H. O. Foulkes.

If we define the weight b of a Young diagram containing n nodes to be the number of removable p-hooks where n = a + bp, then three fundamental theorems stand out in the modular representation theory of the symmetric group Sn.

We show that p-groups of maximal class and order p5 are determined by their group algebras over the field of p elements. The most important information requisite for the proof is obtained from a detailed study of the unit group of a quotient algebra of the group algebra, larger than the small group algebra.

Following (2) we say that a measure μ on a ring is semifinite if

Clearly every σ-finite measure is semifinite, but the converse fails.

In § 1 we present several reformulations of semifiniteness (Theorem 2), and characterize those semifinite measures μ on a ring that possess unique extensions to the σ-ring generated by (Theorem 3). Theorem 3 extends a classical result for σ-finite measures (3, 13.A). Then, in § 2, we apply the results of § 1 to the study of product measures; in the process, we compare the “semifinite product measure” (1; 2, pp. 127ff.) with the product measure described in (4, pp. 229ff.), finding necessary and sufficient conditions for their equality; see Theorem 6 and, in relation to it, Theorem 7.

Ingham (2) has defined the following summation method. A series ∑ an will be called summable (I) to s if

where as usual [x] is the greatest integer ⩽ x. (An equivalent method was described somewhat earlier by Wintner (7), who called it “an Eratosthenian method“; however, the notation (I) and the name “Ingham summability” introduced by Hardy (1) seem to have become usual.)

Introduction. Certain aspects of the theory of subspaces of a Finsler space had been treated by the present author in earlier papers (7). These developments were based on an approach essentially different from the classical theory of Cartan (2) and subsequent writers, whose use of the element of support enables one to introduce the so-called “euclidean connection,” which effects the vanishing of the covariant derivative of the metric tensor.

Let C be the field of complex numbers and E a locally compact topological space. The authors' theory of C-bimeasures Λ and their Λ-integrals in (1; 2) leads to integral representation of bounded operators from A to B' where A and B are MT-spaces as defined in (3). These MT-spaces include the -spaces and Orlicz spaces as special cases.

Let $\mathbf{k}$ be an algebraically closed field complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let ${{D}_{1}},...,\,{{D}_{n}}$ be effective nef divisors intersecting transversally in an $n$-dimensional nonsingular projective variety $X$. We study the degeneracy of non-Archimedean analytic maps from $\mathbf{k}$ into $X\,\backslash \,\cup _{i=1}^{n}\,{{D}_{i}}$ under various geometric conditions. When $X$ is a rational ruled surface and ${{D}_{1}}$ and ${{D}_{2}}$ are ample, we obtain a necessary and sufficient condition such that there is no non-Archimedean analytic map from $\mathbf{k}$ into $X\,\backslash \,{{D}_{1}}\,\cup \,{{D}_{2}}$. Using the dictionary between non-Archimedean Nevanlinna theory and Diophantine approximation that originated in earlier work with T. T. H. An, we also study arithmetic analogues of these problems, establishing results on integral points on these varieties over $\mathbb{Z}$ or the ring of integers of an imaginary quadratic field.

The Weierstrass transform f(x) of a function ϕ(y) is defined by

1.1

where

whenever this integral exists (7, p. 174). It is also known as the Gauss transform (11; 12). Its basic properties have been developed and studied in (7) and in particular it has been shown that the symbolic operator

will invert this transform under suitable assumptions and with certain definitions of this operator. We propose to study the definition

for f(x) in C∞. This formula seems to have been first examined by Pollard (9) and later by Rooney (12). In so far as convergence of (1.2) is concerned, we will considerably improve the results (12).

Given vectors and (or functions f(x) and g(x)) we define the Hölder Quotient Hpq by

1

or in case of functions by

2

Here ‖·‖p and ‖·‖q are the usual Lp and Lq norms. We assume throughout that

If p and q are both greater than one then they are positive but if we allow p and q to be less than one then one of them must be positive and the other one must be negative. This may cause a problem if for example, some value ai is zero and p is negative. In this case we use the convention that and

Let denote the spaces of Bessel potentials as denned and discussed in [6]. When 1 < q < ∞ and α is an integer , the Sobolev space which consists of functions Fin Lq with α derivatives in Lq and with norm

If now we make the change of variables x = ev, f(x) = F (In x) it is easily seen that the ratio

Several authors have considered eigenvalue problems for differential equations where the eigenvalue parameter also appears in the boundary conditions. Such problems do not appear to arise from any spectral problem associated with a linear operator on a Hilbert space. However, it is possible to reset such problems in this context. This has been done for certain second order cases by Walter [4] using a special measure on the interval in question, and by Fulton [1, 2] using the type of space indicated in the title of this article.

It is our purpose here to consider a general class of operators on L2(I) ⊕ Cr, which are based on a differential expression τ of order n on I. We shall first investigate adjoints, boundary conditions, and self-adjointness for such operators. We shall then show that all eigenvalue problems of the form τy = λy, with boundary conditions which involve λ in a linear fashion, can be reset in the context of such operators.

The aim of this paper is to construct Calabi–Yau 4-folds as crepant resolutions of the quotients of a hyperkähler 4-fold $X$ by a non-symplectic involution $\unicode[STIX]{x1D6FC}$. We first compute the Hodge numbers of a Calabi–Yau constructed in this way in a general setting, and then we apply the results to several specific examples of non-symplectic involutions, producing Calabi–Yau 4-folds with different Hodge diamonds. Then we restrict ourselves to the case where $X$ is the Hilbert scheme of two points on a K3 surface $S$, and the involution $\unicode[STIX]{x1D6FC}$ is induced by a non-symplectic involution on the K3 surface. In this case we compare the Calabi–Yau 4-fold $Y_{S}$, which is the crepant resolution of $X/\unicode[STIX]{x1D6FC}$, with the Calabi–Yau 4-fold $Z_{S}$, constructed from $S$ through the Borcea–Voisin construction. We give several explicit geometrical examples of both these Calabi–Yau 4-folds, describing maps related to interesting linear systems as well as a rational $2:1$ map from $Z_{S}$ to $Y_{S}$.

Let denote the family of all functions of the form

that are analytic in the unit disc U, f′(z) ≠ 0 in U and f maps U onto a domain of boundary rotation at most . Recently Brannan, Clunie and Kirwan [2] and Aharonov and Friedland [1] have solved the problem of estimating |amp+1| for all , provided m = 1.

The spaces P(ω), 1 ⩽ p ⩽ ∞, ω real are defined to consist of those analytic functions ƒ(s), regular for Re s > ω and for which μP(ƒ;x) is bounded for x > ω where

(1.1), 1 ⩽ p ⩽ ∞and

(1.2)

G will denote a finite (or infinite) group of order n. In a normal multiplication table (n.m.t.) of G (7, 8, 9, 12) all entries in one diagonal are equal to e (the identity of G), and if the entry on the intersection of the ith column and jth row is gi,j ∈ G, then

The following is a n.m.t. of Z6 = {0, 1, 2, 3, 4, 5}:

Remark. The cyclic groups Zn will always be written in additive notation. The table is uniquely defined by every one of its columns, in particular by the first.

Introduction. Let K be an algebraically closed field, G = SL(3, K) the group of 3 × 3 matrices over K of determinant 1. Let denote the monoid of all 3 × 3 matrices over K. If e is an idempotent in , then

are opposite parabolic subgroups of G in the usual sense [1], [28]. However the map

does not exhaust all pairs of opposite parabolic subgroups of G. Now consider the representation ϕ:G → SL(6, K) given by

The following results of Whitehead and Wall are well-known applications of the algebraic K-theoretic functors K0 and K1 to basic homotopy questions in topology.

THEOREM 1 [20]. If f : X → Y is a homotopy equivalence between compact CW complexes, then there is a torsion τ(ƒ) in the algebraically-defined Whitehead group Wh π1(Y) which vanishes if and only if f is a simple homotopy equivalence.

THEOREM 2 [18]. If X is an arbitrary space which is finitely dominated (i.e., homotopically dominated by a compact polyhedron), then there is an obstruction σ(X) in the algebraically-defined reduced projective class group which vanishes if and only if X is homotopy equivalent to some compact polyhedron.

If we direct sum over components, then the above statements make good sense even if the spaces involved are not connected.