This is a study of relations between pure cubic fields and their normal closures. Explicit formula shows how the discriminant, regulator and class number of the normal closure can be expressed in terms of the cubic field.

Let R⊂S be two orders in a number field, and let ER and ES be the respective groups of units in each ring. Then ES/ER and S/R are both finite. We consider the problem of bounding the order of ES/ER in terms of the index of R in S. In this paper we solve this problem in the special case that S/R is cyclic as a module over Z.

Let K be the finite field of pn elements, and Zp its prime subfield. It was proved by Davenport [3] that if p > p0(n) and θ is any given generating element of K, then there exists an integer m such that θ + m is a primitive root of K.

A natural number $n$ is called abundant if the sum of the proper divisors of $n$ exceeds $n$. For example, $12$ is abundant, since $1+ 2+ 3+ 4+ 6= 16$. In 1929, Bessel-Hagen asked whether or not the set of abundant numbers possesses an asymptotic density. In other words, if $A(x)$ denotes the count of abundant numbers belonging to the interval $[1, x] $, does $A(x)/ x$ tend to a limit? Four years later, Davenport answered Bessel-Hagen’s question in the affirmative. Calling this density $\Delta $, it is now known that $0. 24761\lt \Delta \lt 0. 24766$, so that just under one in four numbers are abundant. We show that $A(x)- \Delta x\lt x/ \mathrm{exp} (\mathop{(\log x)}\nolimits ^{1/ 3} )$ for all large $x$. We also study the behavior of the corresponding error term for the count of so-called $\alpha $-abundant numbers.

I investigate what can be said about a set E in a probability space X when the “square” E x E can be covered by the squares of stochastically independent sets of given measure.

In what follows, R will denote a commutative domain with 1, and Q(≠R) its field of quotients, which is viewed here as an R-module. By RP we denote the localization of R at the maximal ideal P, and more generally, by MP = Rp⊗RM the localization of the R-module M at P, which we define to be the P-component of M. The symbol R* will mean the multiplicative monoid of nonzero elements of R. For a submonoid S of R*, Rs will denote the localization of R at S.

Let υ be a valuation on K with value group Gυ, residue field kυ, rank υ = t and K (x1, …, xn) be the field of rational functions over K with n variables. If G is the direct sum of G1 and d infinite cyclic groups where G1 is a totally ordered group containing Gυ as an ordered subgroup with [G1 : Gυ] < ∞ and k′ is a finite field extension of kυ then there exists a residual transcendental extension u of υ to K (x1, …, xn) such that rank u = t + d, Gu = G the algebraic closure of kυ in kυ is k′ and trans deg ku/kυ = n − d.

It is shown that every compact convex set K which is centrally symmetric and has a non-empty interior admits a packing of Euclidean 3-space with density at least 0.46421 … The best such bound previously known is 0.30051 … due to the theorem of Minkowski-Hlawka. It is probable that there is such a lower bound which is significantly greater than the one shown in this note, since there is a packing of congruent spheres which has density

Let l be a prime and let v ≥ 1 be an integer (when l = 2 we assume v ≥ 2). Any ring, A, with unit, possesses mod lv algebraic K-groups [B] denoted by Ki(A; Z/v) (i ≥ 0). For i ≥ 2, Ki(A; Z/lv) = [Pi(lv), BGLA +], the group of based homotopy classes of maps from the Moore space , to BGLA+, the classifying space of algebraic K-theory [G–Q ; W].

In 1933, K. Borsuk [1] established the well-known result that if n closed sets cover Sn−1 then at least one set contains antipodal points, where Sn−1 is the surface of the ball Tn of centre O and unit diameter in Rn. This result prompted H. Hadwiger [2] to make a still unresolved conjecture which, in the spirit of B. Griinbaum's survey [3], we state as follows: Let r be the largest integer such that whenever r closed sets cover Sn−1 at least one set realizes all distances between 0 and 1. Then r = n.

For a half-integral weight modular form $f=\sum _{n=1}^{\infty }a_{f}(n)n^{(k-1)/2}q^{n}$ of weight $k=\ell +1/2$ on $\unicode[STIX]{x1D6E4}_{0}(4)$ such that $a_{f}(n)$ ( $n\in \mathbb{N}$ ) are real, we prove for a fixed suitable natural number $r$ that $a_{f}(n)$ changes sign infinitely often as $n$ varies over numbers having at most $r$ prime factors, assuming the analog of the Ramanujan conjecture for Fourier coefficients of half-integral weight forms.

A long time ago Ju. E. Vapne ([2], [3]) and, independently, the author ([4], [5]) classified those standard and complete wreath products that have faithful representations of finite degree over (commutative) fields. See [6] pages 37 40 & 150–154 for an account of this. Recently, in connection with finitary linear groups, I needed a more general wreath product. Somewhat to my surprise neither the classification nor the proof for these generalized wreath products was a straightforward translation from the standard case. The situation is intrinsically more complex and it seems worthwhile recording it separately.

A major result of D. B. McAlister is that every inverse semigroup is an idempotent separating morphic image of an E-unitary inverse semigroup. The result has been generalized by various authors (including Szendrei, Takizawa, Trotter, Fountain, Almeida, Pin, Weil) to any semigroup of the following types: orthodox, regular, ii-dense with commuting idempotents, E-dense with idempotents forming a subsemigroup, and is-dense. In each case, a semigroup is a morphic image of a semigroup in which the weakly self conjugate core is unitary and separated by the homomorphism. In the present paper, for any variety H of groups and any E-dense semigroup S, the concept of an “H-verbal subsemigroup” of S is introduced which is intimately connected with the least H-congruence on S. What is more, this construction provides a short and easy access to covering results of the aforementioned kind. Moreover, the results are generalized, in that covers over arbitrary group varieties are constructed for any E-dense semigroup. If the given semigroup enjoys a “regularity condition” such as being eventually regular, group bound, or regular, then so does the cover.

For a metric space <Ω, ρ> and a ‘measure function’ h, the Hausdorff measure mh on Ω is denned by applying Method II to the premeasure defined by τ(E) = h(d(E)), E ⊆ Ω, where

with d(Φ) = 0, is the diameter of E. The set function mh is then a metric outer measure. There are many variations on this definition producing measures also associated with the name Hausdorff. Here we are concerned with those measures which arise when there is a restriction on the sets E for which τ is defined. Such measures arise, for example, as net measures, Rogers [1]. Also we might find it useful to have τ defined only on disks, or only on squares, or only on rectangles with a given relation between vertical and horizontal sides.

The incidence matrices for the finite projective planes, for the so-called λ-planes (where two lines meet in λ points instead of the usual 1) and for some other configurations, including those of Pappus and Desargues, are all cases of what are defined below as (v, k, t, λ)-matrices. These are shown here to possess arithmetical properties which reduce, in the case of cyclic projective planes, to properties remarked on by Marshall Hall [3]. And if a certain group hypothesis (which suggests itself in a natural way) is satisfied by a matrix of this type, the matrix is shown to be equivalent (by rearranging the rows and columns) to a direct sum of incidence matrices for λ-planes, each of which satisfies the same group hypothesis.

In all that follows, E denotes a separated locally convex space, E' its topological dual, and <x, x−> the bilinear form expressing the duality. We consider the differentiation of a function f:t → f(t) of a real variable t which takes its value in E; the domain of f will be an open interval which, without loss of generality, may be taken to be the entire real axis. There are various senses in which the derivative may or may not exist, and it is proposed to consider some relations between these senses.

A method of finding critical points in a half space is developed. It is then applied to the study of semilinear boundary value problems, and used to determine conditions which lead to multiple non-trivial solutions.

An (r,r+1)-factor of a graph G is a spanning subgraph H such that dH(v)∈{r,r+1} for all vertices v∈𝒱(G). If G is expressed as the union of edge-disjoint (r,r+1)-factors, then this expression is an (r,r+1)-factorization of G. Let μ(r) be the smallest integer with the property that if G is a regular loopless multigraph of degree d with d≥μ(r), then G has an (r,r+1)-factorization. It is shown that if r is even. The proof employs a novel list-coloring approach. Together with known results, this shows that μ(r)=r2+1 if r is odd and if r is even.