All functions will be complex, periodic, integrable (on [0, 2π]) functions of a real variable x. Moreover, we shall require that every function have mean zero on [0, 2π], so that in particular non-zero constants are excluded.

1. Plessner's characterization of absolutely continuous functions. An old theorem of Plessner (4), generalized to arbitrary compact groups by Bochner (1), can be taken as our starting point. Consider the functions f of bounded variation on [0, 2π]. These f form a Banach space F when each f is normed by its total variation on [0, 2π]. And translations define a natural one-parameter group of isometries on F.

The theorems of the present paper deal with conditions which are necessary and sufficient in order that a solvable or nilpotent infinite group should have a completely reducible matrix representation over a given algebraically closed field.

It is known (17) that a locally nilpotent group of matrices is always solvable. Thus the first theorem of the present paper is a partial generalization of Theorem 1 of (16), which states:

If G is a locally nilpotent subgroup of the full linear group GL(n, P) over a perfect field P, then G is completely reducible if and only if each matrix of G is diagonizable (by a similarity transformation over some extension field of P).

In this paper all representations are over the complex field K. The generalized symmetric group S(n, m) of order n!mn is isomorphic to the semi-direct product of the group of n × n diagonal matrices whose rath powers are the unit matrix by the group of all n × n permutation matrices over K. As a permutation group, S(n, m) consists of all permutations of the mn symbols {1, 2, …, mn} which commute with

Obviously, S (1, m) is a cyclic group of order m, while S(n, 1) is the symmetric group of order n!. If ci = (i, n+ i, …, (m – 1)n+ i) and

then {c1, c2, …, cn} generate a normal subgroup Q(n) of order mn and {s1, s2, …, sn…1} generate a subgroup S(n) isomorphic to S(n, 1).

Soient f(z) et g (z) deux fonctions univalentes normalisées convexes dans |z|. Nous avons démontré récemment (3) que leur moyenne géométrique F = (fg)1/2 applique le disque sur un domaine convexe et l'estimation est précise.

Nous voulons ici considérer le même problème pour la moyenne arithmétique des deux fonctions f(z) et g (z).

THÉORÈME. Soient f(z) et g(z) deux fonctions univalentes normalisées convexes dans |z|. Si K représente la plus petite racine positive de l'équation

1

alors la moyenne arithmétique applique |z| < κ sur un domaine convexe.

Il nous faut, pour la preuve du théorème, le lemme suivant.

LEMME. Si la function

In [6], Pierce studied the modules over a commutative regular ring R by using the representation of R as the global sections of a sheaf which we call the Pierce sheaf. When the stalks of the Pierce sheaf are regular, Magid gave a Galois theory and some properties for a central separable R-algebra [4, (2.4), (2.5), (2.6) and (2.7)]. When the stalks of the Pierce sheaf are semi-local, DeMeyer presented a Galois theory for a central separable R-algebra [3, sections 2 and 3] and the author characterized the finitely generated and projective modules over a central separable R-algebra in terms of the R-modules in [7] and [8].

In general, the tensor product, $A\otimes B$, of the lattices $A$ and $B$ with zero is not a lattice (it is only a join-semilattice with zero). If $A\otimes B$ is a capped tensor product, then $A\otimes B$ is a lattice (the converse is not known). In this paper, we investigate lattices $A$ with zero enjoying the property that $A\otimes B$ is a capped tensor product, for every lattice $B$ with zero; we shall call such lattices amenable.

The first author introduced in 1966 the concept of a sharply transferable lattice. In 1972, H. Gaskill defined, similarly, sharply transferable semilattices, and characterized them by a very effective condition $\left( \text{T} \right)$.

We prove that a finite lattice $A$ is amenable iff it is sharply transferable as a join-semilattice.

For a general lattice $A$ with zero, we obtain the result: $A$ is amenable iff $A$ is locally finite and every finite sublattice of $A$ is transferable as a join-semilattice.

This yields, for example, that a finite lattice $A$ is amenable iff $A\otimes \text{F}\left( 3 \right)$ is a lattice iff $A$ satisfies $\left( \text{T} \right)$, with respect to join. In particular, ${{M}_{3}}\,\otimes \,\text{F}\left( 3 \right)$ is not a lattice. This solves a problem raised by R. W. Quackenbush in 1985 whether the tensor product of lattices with zero is always a lattice.

Let $C$ denote the Fermat curve over $\mathbb{Q}$ of prime exponent $l$ . The Jacobian $\text{Jac(}C\text{)}$ of $C$ splits over $\mathbb{Q}$ as the product of Jacobians $\text{Jac(}{{C}_{k}})$ , $1\,\le \,k\,\le \,\ell \,-\text{2}$ , where ${{C}_{k}}$ are curves obtained as quotients of $C$ by certain subgroups of automorphisms of $C$ . It is well known that $\text{Jac(}{{C}_{k}}\text{)}$ is the power of an absolutely simple abelian variety ${{B}_{k}}$ with complex multiplication. We call degenerate those pairs $(l,\,k)$ for which ${{B}_{k}}$ has degenerate $\text{CM}$ type. For a non-degenerate pair $(l,\,k)$ , we compute the Sato–Tate group of $\text{Jac(}{{C}_{k}}\text{)}$ , prove the generalized Sato–Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of whether $(l,\,k)$ is degenerate, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the $l$ -th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

Let $F$ be a non-separable $\text{LF}$-space homeomorphic to the direct sum ${{\sum }_{n\in \text{N}}}\,{{\ell }_{2}}\left( {{\tau }_{n}} \right)$, where ${{\aleph }_{0}}<{{\tau }_{1}}<{{\tau }_{2}}<\cdot \cdot \cdot $. It is proved that every open subset $U$ of $F$ is homeomorphic to the product $\left| K \right|\,\times \,F$ for some locally finite-dimensional simplicial complex $K$ such that every vertex $v\,\in \,{{K}^{\left( 0 \right)}}$ has the star $\text{St}\left( v,\,K \right)$ with card $\text{St}{{\left( v,K \right)}^{\left( 0 \right)}}<\tau =\sup {{\tau }_{n}}$ (and card ${{K}^{\left( 0 \right)}}\le \tau $), and, conversely, if $K$ is such a simplicial complex, then the product $\left| K \right|\,\times \,F$ can be embedded in $F$ as an open set, where $\left| K \right|$ is the polyhedron of $K$ with the metric topology.

We characterize the pairs of weights $\left( v,\,w \right)$ for which the operator $Tf\left( x \right)=g\left( x \right)\int{_{s\left( x \right)}^{h\left( x \right)}}\,f\,$ with $s$ and $h$ increasing and continuous functions is of strong type $\left( p,\,q \right)$ or weak type $\left( p,\,q \right)$ with respect to the pair $\left( v,\,w \right)$ in the case $0\,<\,q\,<\,p$ and $1\,<\,p\,<\,\infty$. The result for the weak type is new while the characterizations for the strong type improve the ones given by H. P. Heinig and G. Sinnamon. In particular, we do not assume differentiability properties on $s$ and $h$ and we obtain that the strong type inequality $\left( p,q \right),q\,<p$, is characterized by the fact that the function

$$\Phi \left( x \right)\,=\,\sup \,{{\left( \int_{c}^{d}{{{g}^{q}}w} \right)}^{1/p}}\left( \int_{s\left( d \right)}^{h\left( c \right)}{{{v}^{1-{p}'}}} \right){{\,}^{1/{p}'}}$$

belongs to ${{L}^{r}}\left( {{g}^{q}}w \right)$, where ${1}/{r\,=\,{\,1}/{q\,-\,{1}/{p}\;}\;}\;$ and the supremum is taken over all $c$ and $d$ such that $c\le x\le d$ and $s\left( d \right)\,\le \,h\left( c \right)$.

A matrix P with elements from an arbitrary field is called a cogredient automorph (c.a.) of a symmetric matrix A if P'AP = A, where P' is the transpose of P.

This paper discusses projective planes from the viewpoint of their classification into singly-generated and non-singly-generated planes. (Singly-generatedness, a property explicit in Hall (6) and Wagner (19), and implicit in Hughes (10), is defined in Section 5.) The elements (points and lines) of a singly-generated plane are expressible in four basic points called a quadrangle. A t'completion procedure" enables us to obtain expressions for the elements of a plane.

In this note we discuss how the saturation I X J, where I, J are k-complete ideals on a regular uncountable cardinal K, depends on the saturation of I and J. We show that if 2k = k+ then the saturation of I X J is completely determined by the saturation of I and J . A consequence of a negative saturation result is that NSk X NSk is not k+-saturated, where NSk is the nonstationary ideal on k (even though it is still open whether NSk can be K+- saturated). We also discuss the preservation of precipitousness under certain products, obtaining a simple example of an ideal on k that is precipitous but not k+-saturated.

Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let ${{A}_{\mathbb{R}}}\left( K \right)$ denote the real Banach algebra of all real symmetric continuous functions on $K$ that are analytic in the interior ${{K}^{\circ }}$ of $K$, endowed with the supremum norm. We characterize all unimodular pairs $\left( f,\,g \right)$ in ${{A}_{\mathbb{R}}}{{\left( K \right)}^{2}}$ which are reducible. In addition, for an arbitrary compact $K$ in $\mathbb{C}$, we give a new proof (not relying on Banach algebra theory or elementary stable rank techniques) of the fact that the Bass stable rank of $A\left( K \right)$ is 1. Finally, we also characterize all compact real symmetric sets $K$ such that ${{A}_{\mathbb{R}}}\left( K \right)$, respectively ${{C}_{\mathbb{R}}}\left( K \right)$, has Bass stable rank 1.

We determine the structure of representations induced from discrete series of parabolic subgroups of quasi-split p-adic groups G with G/G° a cyclic group of prime order. We attach to each such representation an R-group which extends the definition of the Knapp-Stein R-group. We show that this R-group has the properties predicted by Arthur. We apply our results to the case of Orthogonal groups.

Let MR be a right module over the associative ring R (with 1). Assume one has an expression for M as a colimit (direct limit) of a system

over the (directed) poset D. A natural way to get M as a colimit of the family ﹛F∞ —” Fβ|∞, fβ ∈ E﹜ for some subset £ of D is to take E cofinal in D. However, if one is concerned about the cardinality of the set E, cofinal subsets may be too large. Let us look at a specific example. Lazard [3] has shown that any flat MR is a direct limit of finitely generated free R-modules. The cardinality of his indexing set depends on the cardinality of M.

Let M be a. complete connected Riemannian manifold of dimension n and let £:M → Rn+k be an isometric immersion into the Euclidean space Rn+k. Let ∇ be the connection on Mn and let be the Euclidean connection on Rn+k. Also let

denote the second fundamental form B(X, Y) = (xY)→. Here TP(M) denotes the tangent space at p, NP(M) the normal space and (…)→ the normal component.

We prove a nonvanishing result for families of $\text{G}{{\text{L}}_{n}}\times \text{G}{{\text{L}}_{n}}$ Rankin–Selberg $L$-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on $\text{G}{{\text{L}}_{n}}$. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group.

The problem of convergence in distribution of a large class of generalized semimartingales to a continuous process is considerably simplified by a recent theorem of Aldous [1], in conjunction with a result of Cremers and Kadelka [3] on convergence of integral functional, and the results of Mitoma [15] and Fouque [8] for generalized processes. We will give a convenient convergence criterion in this setting. The proof amounts to a direct combination of the results of the abovementioned authors, requiring only a minor extension (of a special case) of the theorem of Cremers and Kadelka.

In [7] Varadarajan denned the notion of a cyclic map f : A → X. The collection of all homotopy classes of such cyclic maps forms the Gottlieb subset G(A, X) of [A, X]. If A = S1 this reduces to the group G(X, X0) of Gottlieb [5]. We show that a cyclic map f maps ΩA into the centre of ΩX in the sense of Ganea [4]. If A and X are both suspensions, we then show that if f : A → X maps ΩA into the centre of ΩX, then f is cyclic. Thus for maps from suspensions to suspensions, Varadarajan's cyclic maps are just those maps considered by Ganea. We also define G (Σ4, ΣX) in terms of the generalized Whitehead product [1], This gives the computations for G(Sn+k, Sn) in terms of Whitehead products in π2n+k(Sn).

We work in the category of spaces with base points and having the homotopy type of countable CW-complexes. All maps and homotopies are with respect to base points. For simplicity, we shall frequently use the same symbol for a map and its homotopy class.

Let A be the co-ordinate ring of a reduced curve over a field k. This means that A is an algebra of finite type over k, A has no nilpotent elements, and that if P is a minimal prime ideal of A, then A/P is an integral domain of Krull dimension one. Let M be a maximal ideal of A. Then G(A) (the graded ring of A relative to M) is defined to be . We get the same graded ring if we first localize at M, and then form the graded ring of AM relative to the maximal ideal MAM. That is

Let Ā be the integral closure of A. If P1, P2, …, Ps are the minimal primes of A then

where A/Pi is a domain and is the integral closure of A/Pi in its quotient field.