In (3), J. Lewittes establishes a connection between the number of fixed points of an automorphism of a compact Riemann surface and Weierstrass points on the surface; Lewittes′ techniques are analytic in nature. In (4), D. L. McQuillan proved the result by purely algebraic methods and extended it to arbitrary algebraic function fields in one variable over algebraically closed ground fields, but with restriction to tamely ramified places. In this paper we will give a different proof of the theorem and show that it is an elementary consequence of the Riemann-Hurwitz relative genus formula. Moreover, we can remove the tame ramification restriction.

Let $F_{2^n}$ be the Frobenius group of degree $2^n$ and of order $2^n ( 2^n-1)$ with $n \ge 4$ . We show that if $K/\mathbb {Q} $ is a Galois extension whose Galois group is isomorphic to $F_{2^n}$ , then there are $\dfrac {2^{n-1} +(-1)^n }{3}$ intermediate fields of $K/\mathbb {Q} $ of degree $4 (2^n-1)$ such that they are not conjugate over $\mathbb {Q}$ but arithmetically equivalent over $\mathbb {Q}$ . We also give an explicit method to construct these arithmetically equivalent fields.

This paper concerns an inverse problem in the calculus of variations, namely, when a two-dimensional symmetric connection is globally a Riemannian or pseudo-Riemannian connection. Two new local characterizations of such connections in terms of the Ricci tensor and the Riemann curvature tensor respectively are given, together with a solution to the global problem. As an application, the problem of whether the characteristic curves of a connection on an SO(3)-bundle on a surface are the geodesies of a Riemannian metric on the surface is studied. Some applications to non-holonomic dynamics are discussed.

Fuglede's commutativity theorem for normal operators is an easy consequence of the result that: For T normal, denoting the range of T - λ by R(T - λ), ∩ {R(T - λ) : all λ} = {0}:

Compact complex homogeneous manifolds have been studied in great detail by Borel, Goto, Remmert and Wang (cf., (13)): it was shown that every compact, connected complex homogeneous manifold M is a holomorphic fiber bundle over a projective algebraic homogeneous manifold B with a connected, complex parallelizable fiber F.

For $T$ a compact torus and $E_{T}^{*}$ a generalized $T$-equivariant cohomology theory, we provide a systematic framework for computing $E_{T}^{*}$ in the context of equivariantly stratified smooth complex projective varieties. This allows us to explicitly compute $H_{T}^{*}\left( X \right)$ as an $H_{T}^{*}\left( pt \right)$-module when $X$ is a direct limit of smooth complex projective ${{T}_{\mathbb{C}}}$-varieties. We perform this computation on the affine Grassmannian of a complex semisimple group.

An explicit construction of a pre-quantum line bundle for the moduli space of flat $G$-bundles over a Riemann surface is given, where $G$ is any non-simply connected compact simple Lie group. This work helps to explain a curious coincidence previously observed between Toledano Laredo's work classifying central extensions of loop groups $LG$ and the author's previous work on the obstruction to pre-quantization of the moduli space of flat $G$-bundles.

We prove that the composition of a minimal usco map, defined on a Baire space, with a lower semicontinuous function is single valued and usco at each point of a dense G$ subset of its domain. This extends earlier results of Kenderov and Fitzpatrick. As a first consequence, we prove that a Banach space, with the property that there exists a strictly convex, weak* lower semicontinuous function on its dual, is a weak Asplund space. As a second consequence, we present a short proof of the fact that a Banach space with separable dual is an Asplund space.

It is shown that a soluble-by-finite product G = AB of a nilpotent-by-noetherian group A and a noetherian group B is nilpotentby- noetherian. Moreover, a bound for the torsion-free rank of the Fitting factor group of G is given, in terms of the torsion-free rank of the Fitting factor group of A and the torsion-free rank of B.

A commutative ring R is said to have the n-generator property if each ideal of R can be generated by n elements. Rings with the n-generator property have Krull dimension at most one. In this paper we consider the problem of determining when a one-dimensional monoid ring R[S] has the n-generator property where R is an artinian ring and S is a commutative cancellative monoid. As an application, we explicitly determine when such monoid rings have the three-generator property.

Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees.

We provide an explicit formula for the generalized cyclic shuffle map for cylindrical modules. Using this formula we give a combinatorial proof of the generalized cyclic Eilenberg–Zilber theorem.

The maximal ring of quotients (in the sense of Johnson [4] and Utumi [5[) of the ring C(X) of real valued continuous functions on a completely regular Hausdorff space X has been studied in [1] and [2]. The aim of the present paper is to provide some additional results to those, and to study the relevant extensions on the absence of the real maximal ideals.

The purpose of this note is to give a direct proof of Koh's main theorem in [2, p. 31] applying well-known theorems of Faith and Utumi. While doing so, we shall mention a general theorem from which Koh's theorem may be obtained as a corollary.

Some formulas for multiplicities of pure cubic discriminants are generalized to the case of a pure field of arbitrary odd prime degree.

Let D be a division ring with center Z. Suppose for all xϵD, there exists a monic polynomial, fx(t), with integer coefficients such that fx(x)ϵZ. Then D is commutative.

In a recent note [1], M. R. Embry proved that if A is an operator on a Banach space then, under a certain condition on the spectrum of A, each operator commuting with An also commutes with A, where n is a fixed positive integer. It turns out that, when A is a finite matrix, Embry′s conditions imply that A is a polynomial in An and hence plainly each operator commuting with An also commutes with A.

Throughout this paper rings are understood to be commutative with unity. In this paper we prove the general approximation theorem for valuations whose infinite ideals have large Jacobson radicals. We give an example in which it is shown that approximation theorems for Manis valuations do not hold in the general case. Also we prove that every valuation pair (Rv, Pv) of a total quotient ring T(R) whose infinite ideal has large Jacobson radical is a Prüfer valuation pair.

We define certain arithmetic derivatives on $\mathbb {Z}$ that respect the Leibniz rule, are additive for a chosen equation $a+b=c$ , and satisfy a suitable nondegeneracy condition. Using Geometry of Numbers, we unconditionally show their existence with controlled size. We prove that any power-saving improvement on our size bounds would give a version of the $abc$ Conjecture. In fact, we show that the existence of sufficiently small arithmetic derivatives in our sense is equivalent to the $abc$ Conjecture. Our results give an explicit manifestation of an analogy suggested by Vojta in the eighties, relating Geometry of Numbers in arithmetic to derivatives in function fields and Nevanlinna theory. In addition, our construction formalizes the widespread intuition that the $abc$ Conjecture should be related to arithmetic derivatives of some sort.

Let S be a finite set of primes. We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of $\mathbb {Z}_{S}$ -points on integral models of Hilbert modular varieties, extending a result of D. Helm and F. Voloch about modular curves. Let L be a totally real field. Under (a special case of) the absolute Hodge conjecture and a weak Serre’s conjecture for mod $\ell $ representations of the absolute Galois group of L, we prove that the same holds also for the $\mathcal {O}_{L,S}$ -points.