In this note we prove that an isomorphism of p-adic group rings of finite p-groups maps class sums onto class sums. For integral group rings this is a well known theorem of Glauberman (see [3; 7]). As an application, we show that any automorphism of the p-adic group ring of a finite p-group of nilpotency class 2 is composed of a group automorphism and a conjugation by a suitable element of the p-adic group algebra. This was proved for integral group rings of finite nilpotent groups of class 2 in [5]. In general this question remains open. We also indicate an extension of a theorem of Passman and Whitcomb. The following notation is used.

G denotes a finite p-group.

Z denotes the ring of (rational) integers.

ZP denotes the ring of p-adic integers.

Qp denotes the p-adic number field.

In [1], a number of “multilinear” functors Lpq, denned for finitely generated free modules, were introduced. They arose as cycles in a generic Koszul complex, and in turn gave rise to a large family of other generic complexes. One of the things we will do in this paper is study some of these new complexes in order to obtain new multilinear functors on free modules which appear as their cycles.

We examine the problem of finding rational points defined over solvable extensions on algebraic curves defined over general fields. We construct non-singular, geometrically irreducible projective curves without solvable points of genus $g$, when $g$ is at least 40, over fields of arbitrary characteristic. We prove that every smooth, geometrically irreducible projective curve of genus 0, 2, 3 or 4 defined over any field has a solvable point. Finally we prove that every genus 1 curve defined over a local field of characteristic zero with residue field of characteristic $p$ has a divisor of degree prime to $6p$ defined over a solvable extension.

We study the semi-classical behavior as $h\,\to \,0$ of the scattering amplitude $f(\theta ,\,\omega ,\,\lambda ,\,h)$ associated to a Schrödinger operator $P(h)\,=\,-\,\frac{1}{2}{{h}^{2}}\Delta \,+\,V\,(x)$ with short-range trapping perturbations. First we realize a spatial localization in the general case and we deduce a bound of the scattering amplitude on the real line. Under an additional assumption on the resonances, we show that if we modify the potential $V(x)$ in a domain lying behind the barrier $\left\{ x\,:\,V(x)\,>\,\lambda \right\}$, the scattering amplitude $f(\theta ,\,\omega ,\,\lambda ,\,h)$ changes by a term of order $\mathcal{O}({{h}^{\infty }})$. Under an escape assumption on the classical trajectories incoming with fixed direction $\omega $, we obtain an asymptotic development of $f(\theta ,\,\omega ,\,\lambda ,\,h)$ similar to the one established in the non-trapping case.

Bivariate polynomials with a fixed leading term ${{x}^{m}}{{y}^{n}}$ , which deviate least from zero in the uniform or ${{L}^{2}}$-norm on the unit disk $D$ (resp. a triangle) are given explicitly. A similar problem in ${{L}^{p}}$ , $1\,\le \,p\,\le \,\infty $ is studied on $D$ in the set of products of linear polynomials.

In this paper we obtain a partial answer in graph-theoretic form to a question raised by Ryser (2, p. 68) concerning the minimal number of interchanges required to transform equivalent (0, l)-matrices into each other.

For given positive integers m and n we consider the collection of m × n (0, 1)-matricesA = {aij}, i.e. aij = 0 or 1 for 1 ≤ i ≤ m, 1 ≤ j ≤ n. We say the (0, 1)-matrices A = {aij} and B = {bij} are equivalent and write A ~ B if and only if they have the same row and column sums, that is, if and only if.

Certain finite groups H do not occur as a regular subgroup of a uniprimitive (primitive but not doubly transitive) group G. If such a group H occurs as a regular subgroup of a primitive group G, it follows that G is doubly transitive. Such groups H are called B-groups (8) since the first example was given by Burnside (1, p. 343), who showed that a cyclic p-group of order greater than p has this property (and is therefore a B-group in our terminology).

Burnside conjectured that all abelian groups are B-groups. A class of counterexamples to this conjecture due to W. A. Manning was given by Dorothy Manning in 1936 (3).

1. Let E be a Banach space (by this we shall mean, for simplicity, a real Banach space) and (xn,fn) ({xn} ⊂ E, {fn} ⊂ E*) a biorthogonal system, such that {fn} is total on E (i.e. the relations x ∈ E,fn(x) = 0, n = 1, 2, …, imply x = 0). Then it is natural to consider the cone

1

which we shall call “the cone associated with the biorthogonal system (xn,fn)”. In particular, if {xn} is a basis of E and {fn} the sequence of coefficient functional associated with the basis {xn}, this cone is nothing else but

2

and we shall call it “the cone associated with the basis {xn}”.

Throughout this note K will denote a field of characteristic p > 0. Let I be the set {1,2, … , m}, and a finite additive group of functions on I with values in K. We assume that is total in the sense that, for any λ1, … , λm in K Σi=imλiσ(i) = 0 for all a in G implies all σi = 0. It is clear that is an elementary p-group. Let pn be the order of . A generalized Witt algebra is defined as an algebra over K with basis elements {e(σ, i)∣ σ ∈ , i ∈ I} and the multiplication table

is a simple Lie algebra except when p = 2, m = 1.

1. Introduction. For define C(n) to be the integer j such that , where denotes the j th iterate of the Euler ø-function. Define C(l) = C(2) = 0. This function has been studied by S. S. Pillai [1], with the notation R(n) for 1 + C(n) if , and R(l) = 0. H. Shapiro [2] has also investigated this function, proving the basic relations

(1)

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and set

The Bergman-Whittaker operator defined by

Let G be an abelian group of order [G] ≤ ∞. Let A = {a}, B = {b}, … denote non-empty finite complexes in G. Let [A] be the number of elements of A. Finally put

A + B = {a + b}.

1. Introduction. The literature of the Laplace transformation contains many examples of inversion operators. Particular attention has been given in this literature to the so-called “real” inversion operators, that is, those operators which make use of values of the generating function arising only from real values of the independent variable.

We show that the multiplier algebra of the Fourier algebra on a locally compact group $G$ can be isometrically represented on a direct sum on non-commutative ${{L}^{p}}$ spaces associated with the right von Neumann algebra of $G$. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the noncommutative ${{L}^{p}}$ spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca–Herz algebra built out of these non-commutative ${{L}^{p}}$ spaces, say ${{A}_{p}}(\hat{G})$. It is shown that ${{A}_{2}}(\hat{G})$ is isometric to ${{L}^{1}}(G)$, generalising the abelian situation.

Let dndn–1 … d0 be the b-ary representation of a positive integer N. Call the polynomial obtained from N base b. In the case the base is 10, f(x) will be called the polynomial obtained from N. Pólya and Szegö attribute the following theorem to A. Cohn [2, b. 2, VIII, 128]:

THEOREM 1. A polynomial obtained from a prime is irreducible.

This theorem was generalized in two different ways by John Brillhart, Andrew Odlyzko, and myself [1]. One way was by proving the theorem remains true regardless of the base being used. The second way was by permitting the coefficients of f(x) to be different from digits. Thus, for example, if , where 0 ≦ dk ≦ 167 for all k, and if f(10) is prime, then f(x) is irreducible. In this paper, Theorem 1 will be generalized in another way by considering composite N.

Soit f: [0,1] → [0,1] une fonction continue. C[0,1] est l'espace (de Banach) des fonctions continues à valeurs complexes définies sur [0, 1], muni de la norme définie par le suprémum. L'opérateur de composition [f] se définit très naturellement sur cet espace:

In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let $\pi $ be a minimal or next-to-minimal automorphic representation of G. We prove that any $\eta \in \pi $ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro–Shalika formula for cusp forms on $\operatorname {GL}_n$ . We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient, in terms of these Whittaker coefficients. A consequence of our results is the nonexistence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for G of type $D_5$ and $E_8$ with a view toward applications to scattering amplitudes in string theory.

Given a $p$-dimensional oriented foliation of an $n$-dimensional compact manifold ${{M}^{n}}$ and a transversal invariant measure $\tau$, Sullivan has defined an element of ${{H}_{p}}\left( {{M}^{n}},\,R \right)$. This generalized the notion of a $\mu$-asymptotic cycle, which was originally defined for actions of the real line on compact spaces preserving an invariant measure $\mu$. In this one-dimensional case there was a natural 1–1 correspondence between transversal invariant measures $\tau$ and invariant measures $\mu$ when one had a smooth flow without stationary points.

For what we call an oriented action of a connected Lie group on a compact manifold we again get in this paper such a correspondence, provided we have what we call a positive quantifier. (In the one-dimensional case such a quantifier is provided by the vector field defining the flow.) Sufficient conditions for the existence of such a quantifier are given, together with some applications.

We consider the problem of finding a unique canonical form for complex matrices under unitary transformation, the analogue of the Jordan form (1, p. 305, §3), and of determining the transforming unitary matrix (1, p. 298, 1. 2). The term “canonical form” appears in the literature with different meanings. It might mean merely a general pattern as a triangular form (the Jacobi canonical form (8, p. 64)). Again it might mean a certain matrix which can be obtained from a given matrix only by following a specific set of instructions (1). More generally, and this is the sense in which we take it, it might mean a form that can actually be described, which is independent of the method used to obtain it, and with the property that any two matrices in this form which are unitarily equivalent are identical.

Let ${{M}_{n}}$ be the variety of spatial polygons $P\,=\,({{a}_{1}},\,{{a}_{2}},...,{{a}_{n}})$ whose sides are vectors ${{a}_{i}}\,\in \,{{\mathbf{R}}^{3}}$ of length $\left| {{a}_{i}} \right|\,=\,1\,(1\,\le \,i\,\le \,n)$, up to motion in ${{\mathbf{R}}^{3}}$. It is known that for odd $n$, ${{M}_{n}}$ is a smooth manifold, while for even $n$, ${{M}_{n}}$ has cone-like singular points. For odd $n$, the rational homology of ${{M}_{n}}$ was determined by Kirwan and Klyachko [6], [9]. The purpose of this paper is to determine the rational homology of ${{M}_{n}}$ for even $n$. For even $n$, let ${{\tilde{M}}_{n}}$ be the manifold obtained from ${{M}_{n}}$ by the resolution of the singularities. Then we also determine the integral homology of ${{\tilde{M}}_{n}}$.