Let X and Y be sets, let λ be a bounded positive measure on X, and let μ be a bounded positive measure on Y. Furthermore let M be a subalgebra of L∞(λ), let p ∈ (0, ∞), and let A be a linear transformation of M into Lp(μ) such that

for all f in M.

In § 2 of this paper we will prove the following theorem.

Les systémes différentiels dont il est question dans ce travail, sont du type suivant:

The universal enveloping algebra, $U(\mathfrak{g})$, of a Lie algebra $\mathfrak{g}$ supports some norms and seminorms that have arisen naturally in the context of heat kernel analysis on Lie groups. These norms and seminorms are investigated here from an algebraic viewpoint. It is shown that the norms corresponding to heat kernels on the associated Lie groups decompose as product norms under the natural isomorphism $U({{\mathfrak{g}}_{1\,}}\oplus \,{{\mathfrak{g}}_{2}})\,\cong \,U({{\mathfrak{g}}_{1}})\,\otimes \,U({{\mathfrak{g}}_{2}})$. The seminorms corresponding to Green's functions are examined at a purely Lie algebra level for $\text{sl(2,}\,\mathbb{C}\text{)}$. It is also shown that the algebraic dual space ${U}'$ is spanned by its finite rank elements if and only if $\mathfrak{g}$ is nilpotent.

In the past few years, much progress have been made on several open problems in infinite dimensional Banach space theory. Here are some of the most recent results:

1. 1) The existence of boundedly complete basic sequences in a large class of Banach spaces including the ones with the so-called Radon-Nikodym property ([G-M2], [G-M4]).

2. 2) The embedding of separable reflexive Banach spaces into reflexive spaces with basis (fZl).

3. 3) The existence of long sequences of projections and hence of locally uniformly convex norms in the duals of Asplund spaces. ([F-G])

Dirichlet's Theorem says that for any real α and for N ≧ 1, there exists a natural n ≦ N with

where || || denotes the distance to the nearest integer. Heilbronn [2], improving estimates of Vinogradov [3], showed that for α, N as above and for ϵ ≧ 0, there exists an n ≦ N with

Let 1 < p < ∞ and let Lp be the usual Banach Space of complex valued functions on a σ-finite measure space. Let (Tn), n ≧ 1, be a sequence of positive linear contractions on Lp. Hence and , where is the part of Lp that consists of non-negative Lp functions. The adjoint of Tn is denoted by which is a positive linear contraction of Lq with q = p/(p — 1).

Our purpose in this paper is to show that the alternating sequences associated with (Tn), as introduced in [2], converge almost everywhere. Complete definitions will be given later. When applied to a non negative function, however, this result is reduced to the following theorem.

(1.1) THEOREM. If (Tn) is a sequence of positive contractions of Lp then (1.2)

exists a.e. for all.

Introduction. An element x is said to be right-quasi-regular (r.q.r.) if there exists an element y such that x + y + xy = 0. This concept had its inception in the fact that (for rings with unity) if 1 + x has an inverse, written as 1 + y, then (1 + x)(l + y) = 1, x + y + xy = 0. Thus in rings without unity elements it seemed (1; 3; 12) profitable to consider this latter equation.

The purpose of this paper is to show that most results concerning polyadic algebras can be generalized to transformation algebras. The results of this paper will clearly indicate that a great deal can be done in polyadic algebras without ever mentioning the quantifier structure (for instance, terms and operations can be characterized without the help of the quantifier structure, at least in the case where an equality is present). In § 1, we develop the elementary theory; in § 2, we study the different ways of extending a (locally finite) transformation algebra (of infinite degree) to a polyadic algebra; in § 3, we study equality transformation algebras; finally, in § 4, we show how terms and operations can be defined in equality transformation algebras.

1. Introduction. Let S denote the family of functions f(z) regular and univalent in ∣z∣ < 1, with the expansion f(z) = z + a2z2 + … about z = 0, and let Af denote the area of the intersection of the open circle ∣ω∣ < 1 with Df, the image of ∣z∣ < 1 under f(z). A few years ago one of the authors (1) proved that if

1

then

2.

In a previous paper (3) generalizations of M. Riesz's theorem by the method of asymptotic approximation have been given. The present paper is concerned with further generalizations for even and odd functions. In Section 3, we consider a generalization of a theorem due to Zygmund: If (10, p. 254). The results in this paper include generalizations of results by K. K. Chen (2) and T. M. Flett (4, Theorems 1, 2) as important special cases.

In the mathematical justification of the formal calculations of axiomatic quantum field theory and the theory of dispersion relations, a strategic role is played by a theorem on analytic functions of several complex variables which has been given the euphonious name of the edge of the wedge theorem. The statement of the theorem seems to be due originally to N. Bogoliubov (cf. 3, Mathematical Appendix, pp. 654-673) but no complete proof which is fully satisfactory from the mathematical point of view has yet appeared in the literature.

We shall consider the classification problem for space forms of (Riemannian manifolds which are covered by) real, complex, and quaternionic Grassmann manifolds. In the particular case of the real Grassmann manifold of oriented 1-dimensional subspaces of a real Euclidean space, this is the classical "spherical space form problem" of Clifford and Klein. We shall not consider space forms of the Cayley projective plane because it is easy to see that there are no non-trivial ones.

Let be a valuation ring with unit element, i.e., is a commutative ring such that for any a and b in , either a divides b or b divides a. We assume 2 is a unit of . V is an n-ary nonsingular quadratic module over , O(V) or On(V) is the orthogonal group on V, and S is the set of symmetries in O(V). We define l(σ) to be the minimal number of factors in the expression of a of O(V) as a product of symmetries on V. For the case where is a field, l(σ) has been determined by P. Scherk [6] and J. Dieudonné [1]. In [3] I have generalized the results of Scherk to orthogonal groups over valuation domains. In the present paper I generalize my results of [3] to orthogonal groups over valuation rings.

Since is a valuation ring, it is a local ring with the maximal ideal A which consists of all nonunits of .

A set of ordered pairs of integers {(ai, mi)} is said to cover the integers if each integer x satisfies the congruence x ≡ ai (mod mi) for some i. We may assume that the mi are positive. Trivially {(0, 1)} covers, as does {(0, m), (1, m), (2, m), … , (m — 1, m)}. In order to arrive at some non-trivial problems concerning covers, the following definition is given: A finite set of ordered pairs of integers with mi > 1 and mi ≠ mj if i ≠ j, is called a covering class of residues if every integer x satisfies the congruence x ≡ ai (mod mi) for some i.

Let G be a locally compact abelian group with Bohr compactification Ga. Then [3, Theorem 1.2] any subset F of G compact in Ga is necessarily compact in G; alternatively, any closed non-compact subset F of G has its closure F– in Ga ≠ F; hence F –\F ≠ ø. One of our aims in the present note is to give a result (Corollary 6) which asserts that F –\F has no points which are Gδs, so that F–\F is a perfect set. Another aim is to give an extension of a cited result of [3] in which commutativity and local compactness are essentially irrelevant, and to unify the proofs.

Block design games have been developed by Richardson (12) and by Hoffman and Richardson (8), who proved a number of theorems concerning such games by studying the number of elements in a blocking coalition. Hoffman and Richardson listed as unsolved (except for PG(2, 3)) the following problem: What is the minimum number of elements in a blocking coalition of a block design game?

This note considers blocking coalitions in those games that are dual to block designs having λ = 1 and r — k > 0. For such games certain blocking coalitions are shown to be related to sets of mutually disjoint blocks in the design to which the game is dual. In particular, for Steiner triple systems the largest odd-numbered set of mutually disjoint triples is shown to yield a minimum blocking coalition in the dual.

A classical theorem states that if a square matrix B over an algebraically closed field F commutes with all matrices X over F which commute with a matrix A over F, then B must be a polynomial in A with coefficients in F (2). Recently Marcus and Khan (1) generalized this theorem to double commutators. Our purpose is to complete the generalization to commutators of any order.

Let F be an algebraically closed field and let Fn be the ring of all n by n matrices with elements in F. We define ΔYZ — = [Z, Y] = ZY — YZ for all Y, Z in Fn.

Let $F$ be a $p$-adic field of characteristic 0, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $\Pi _{i=1}^{r}\,\text{S}{{\text{L}}_{ni}}\,\subseteq \,M\,\subseteq \,\Pi _{i=1}^{r}\,\text{G}{{\text{L}}_{ni}}$ for positive integers $r$ and ${{n}_{i}}$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M\left( F \right)$ is identically transferred under the local Jacquet–Langlands type correspondence between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of Muić and Savin (2000) for Siegel Levi subgroups of the groups $\text{S}{{\text{O}}_{4n}}$ and $\text{S}{{\text{p}}_{4n}}$ under the local Jacquet–Langlands correspondence. It can be applied to a simply connected simple $F$-group of type ${{E}_{6}}$ or ${{E}_{7}}$, and a connected reductive $F$-group of type ${{A}_{n}},\,{{B}_{n}},\,{{C}_{n}}$ or ${{D}_{n}}$.

A category is made up of its class of objects , of a function M (or ) which assigns to each (A,B) ∊ × the class M(A, B) of all morphisms from A to B, of an operation μ which is a class of maps

A, B, C ∊ , and of a family of maps

In this article we prove some new results on projective normality, normal presentation and higher syzygies for surfaces of general type, not necessarily smooth, embedded by adjoint linear series. Some of the corollaries of more general results include: results on property ${{N}_{p}}$ associated to ${{K}_{S}}\,\otimes \,{{B}^{\otimes n}}$ where $B$ is base-point free and ample divisor with $B\,\otimes \,{{K}^{*}}\,nef,$ results for pluricanonical linear systems and results giving effective bounds for adjoint linear series associated to ample bundles. Examples in the last section show that the results are optimal.