I wish to present here some of the results of a research in the Theory of Fibrations initiated some time ago by Peter Booth, Philip Heath, and myself. The philosophy behind the work is to approach certain aspects of the Theory of Fibrations in a unified way through the systematic use of the sections of suitable fibrations; this yields general theorems, of which some well-known results are eventually particular cases.

In this paper, an elementary proof is given of the following proposition:

Theorem 1. If Qp is an arbitrary field of p-adic numbers, then it contains normal subfields Ln(2 ≤ n ≤ p) which have symmetric groups Sn as their respective Galois groups over Q, the field of rational numbers. Furthermore, each Ln may be chosen to be totally real.

In the Simpson moduli space $M$ of semi-stable sheaves with Hilbert polynomial $dm\,\text{-}\,\text{1}$ on a projective plane we study the closed subvariety ${{M}^{'}}$ of sheaves that are not locally free on their support. We show that for $d\ge 4$, it is a singular subvariety of codimension 2 in $M$. The blow up of $M$ along ${{M}^{'}}$ is interpreted as a (partial) modification of $M\backslash {{M}^{'}}$ by line bundles (on support).

Simple structure theorems or representation theorems make their appearance at a very elementary level in subjects such as algebra, but one infrequently encounters such theorems in elementary topology. In this note we offer one such theorem, for finite topological spaces, which involves only the most elementary concepts: discrete space, indiscrete space, product space and homeomorphism. Our theorem describes the structure of those finite spaces which are homogeneous.

It is known [4] that the finite two- dimensional unitary group U(2, p2n) is generated by two reflections if p ≠ 2. The present note completes that result by giving two generating reflections for U(2, 22n), n > 1. As in [4] this implies that the points of the "unit circle " in the unitary plane over GF(22n), n > 1, are the vertices of a "regular unitary polygon" whose group of auto- 2n morphisms is U(2, 22n).

The final section gives abstract definitions for the particular groups U(2,24) and U(2,52) in terms of their generating reflections.

A sequential procedure for the exploration of response surfaces is proposed. The procedure, which is for experiments with two factors, uses the kiss-precise configuration, i.e., the design points are on circles in mutual contact at each stage. Only three points need be added at each stage and the design points form a first-order rotatable design. A second-degree surface may be fitted when a near stationary region is reached.

W-groups are treated in the more general context of near-ring modules. It is shown that they share properties of d.g. modules with respect to central series. In particular, results of Magomaev are obtained and used to show that the group-theoretic and module-theoretic central series coincide for W-groups.

In this note we prove what we believe to be a new result concerning matrices, namely, that if an n × n matrix with rational entries has a finite order then this order is bounded. We also give an estimate for this bound and an application.

Let T be a non-singular, conservative, ergodic automorphism of a Lebesgue space. We study a kind of weighted cocycles called H-cocycles. We introduce the notions of H-superrecurrence and H-supertransience. We use skew products to give necessary and sufficient conditions for H-superrecurrence.

Equivalence problems in Riesz-summability lead to the problem of finding the zeros of special power series in the unit circle, i.e.

Let R be an integral domain with quotient field K. If R has an overling S ≠ K, such that S[X] is integrally closed, then the "algebraic degree" of K((X)) over the quotient field of R[X] is infinite. In particular, it holds for completely integrally closed domain or Noetherian domain R.

The Fefferman–Stein type inequalities for strongmaximal operators and directional maximal operators are verified with an additional composition of the Hardy–Littlewood maximal operator in the plane.

We study metaplectic coverings of the adelized group of a split connected reductive group $G$ over a number field $F$. Assume its derived group ${G}'$ is a simply connected simple Chevalley group. The purpose is to provide some naturally defined sections for the coverings with good properties which might be helpful when we carry some explicit calculations in the theory of automorphic forms on metaplectic groups. Specifically, we

1. 1. construct metaplectic coverings of $G(\mathbb{A})$ from those of ${G}'\,(\mathbb{A})$;

2. 2. for any non-archimedean place $v$, show the section for a covering of $G\,({{F}_{v}})$ constructed from a Steinberg section is an isomorphism, both algebraically and topologically in an open subgroup of $G\,({{F}_{v}})$;

3. 3. define a global section which is a product of local sections on a maximal torus, a unipotent subgroup and a set of representatives for the Weyl group.

The purpose of this note is to give a simple combinatorial construction of the map from the canonically compactified moduli spaces of punctured complex projective lines to the moduli spaces ${{P}_{r}}$ of polygons with fixed side lengths in the Euclidean space ${{\mathbb{E}}^{3}}$. The advantage of this construction is that one can obtain a complete set of linear relations among the cycles that generate homology of ${{P}_{r}}$. We also classify moduli spaces of pentagons.

Let $X$ be a smooth complex projective variety with a holomorphic vector field with isolated zero set $Z$. From the results of Carrell and Lieberman there exists a filtration ${{F}_{0}}\subset {{F}_{1}}\subset \cdot \cdot \cdot$ of $A\left( Z \right)$, the ring of $\mathbb{C}$-valued functions on $Z$, such that $\text{Gr }A\left( Z \right)\cong {{H}^{*}}\left( X,\mathbb{C} \right)$ as graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a 1-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra of $X$.

The following proof of Caratheodory's Theorem, while not essentially new, seems to be natural and therefore of interest.

LEMMA 1. Let P denote a supporting hyperplane of the convex polytope K. Then P∩K is a convex polytope whose vertices are vertices of K.

For each pair (m, n) of integers such that m+1 ≦ n ≦ 2m+1, we give, by original and straightforward methods, some examples of rings A' such that dim A' = m and dim A'[X] = n.

In this article it is shown that every bounded linear operator on a complex, infinite dimensional, separable Hilbert space is a sum of at most eighteen unilateral (alternatively, bilateral) weighted shifts. As well, we classify products of weighted shifts, as well as sums and limits of the resulting operators.

Analytic multivalued functions are a powerful tool in spectral theory. In this paper we study analytic multivalued functions λ → K(λ) whose values are segments. Under the additional assumption that K is continuous, we prove that the endpoints of K vary locally holomorphically.

Applications of minimal surface methods are made to obtain information about univalent harmonic mappings. In the case where the mapping arises as the Poisson integral of a step function, lower bounds for the number of zeros of the dilatation are obtained in terms of the geometry of the image.