We introduce complex differential geometry twisted by a real line bundle. This provides a new approach to understand the various real objects that are associated with an anti-holomorphic involution. We also generalize a result of Greenleaf about real analytic sheaves from dimension 2 to higher dimensions.

There are a number of well known theorems on the mutual independence of forms, either linear or quadratic, in normal variables. Some of these theorems can only hold when the system of variables is normal or degenerate and so the possibility of certain forms being independent characterises the normal distribution. The theorems on characterisation have usually been proved by consideration of the necessary properties of the characteristic function. Here we shall be considering the characteristic function of the variables but we shall make more use of cumulant theory than previous authors. To do so we have first to prove that the existence of cumulants of all orders is implied by the independence conditions. The basic theorems we use are those from general cumulant theory and the special theorems of Cramer and of Marcinkiewicz. An advantage of the methods of this paper is that it is possible to show that some of the characterisation theorems require neither of these special theorems. For example, spherical symmetry is a very strong condition and so neither theorem is required whereas both theorems are needed for the most general theorems on the independence of two linear forms. Throughout we take the class of normal distributions to include the degenerate normal, that is, the distribution of a sure variable.

A (ν, κ, λ)-graph is defined in [3] as a graph on ν points, each of valency κ, and such that for any two points P and Q there are exactly λ points which are joined to both. In other words, if Si is the set of points joined to Pi, then

Si has k elements for any i

SiSj has λ elements if i≠j

The sets Si are the blocks of a (v, k, λ)-configuration, so a necessary condition on v, k, and λ that a graph should exist is that a (v, k, λ)- configuration should exist. Another necessary condition, reported by Bose (see [1]) and others, is that there should be an integer m satisfying have equal parity. We shall prove that these conditions are not sufficient.

We speak of rigidity, if partial information about the prime decomposition in an extension of number fields K¦k determines the decomposition law completely (and hence the zeta function ζK), or even fixes the field K itself. Several concepts of rigidity, depending on the degree of information we start from, are introduced and studied. The strongest concept (absolute rigidity) was only known to hold for the ground field and all quadratic extensions. Here a complete list of all Galois quartic extensions which are absolutely rigid is given. For the weaker concept of rigidity, all rigid situations among the fields of degree up to 8 are determined.

Let N denote the discrete space of all natural numbers. A space X is N-compact if it is homeomorphic with some closed subspace of a product of copies of N. In this paper, N-compact spaces are characterized as homeomorphs of inverse limit space of inverse systems of copies of subsets of N. Also, it is shown that a space X is N-compact if and only if the space (X) of all non-empty compact subsets of X with the finite topology is N-compact.

Let G be a compact group. A sequence of functions in L∞ (G) is said to be a Rudin-Shapiro sequence (briefly, an RS-sequence) if the following conditions hold: (1) (2) (3) The main purpose here is to prove the following theorem: Theorem: Theorem. Let G be an infinite compact group. Then G has an RS-sequence consisting of trigonometric polynomials.

Groups, all proper factor-groups of which are hypercentral of finite torsion-free rank, are studied in this article.

Let F′ be the commutator subgroup of F and let Γ0 be the cyclic group generated by the first generator of F. We continue the study of the central sequences of the factor L(F′), and we prove that the abelian von Neumann algebra L(Γ0) is a strongly singular MASA in L(F). We also prove that the natural action of F on [0, 1] is ergodic and that its ratio set is {0} ∪ {2k; k ∞ Z}.

In this paper we investigate the following problem proposed by Lausch and Nöbauer: Let be a variety of universal algebras, B an algebra of and A a subalgebra of B. If a system of algebraic equations over A is solvable in , is it then solvable over B? We show that the answer is affirmative in certain varieties but negative in the general case.

In this paper, we introduce quotients of étale groupoids. Using the notion of quotients, we describe the abelianizations of groupoid C*-algebras. As another application, we obtain a simple proof that effectiveness of an étale groupoid is implied by a Cuntz–Krieger uniqueness theorem for a universal groupoid C*-algebra.

This paper is concerned with the problem of automatic continuity of derivations from group algebras L1(G) is a locally compact group, and convolution algebras L1(ω), where ω is a weight function. In the case of group algebras, it is shown that either the problem reduces to the case when G is the free group on a countably infinite number of generators or there is a non-discrete group G with a discontinuous l1(G)-bimodule homomorphism from L1(G). It is also shown that every derivation from L1(G) to a commutative L1(G)-bimodule is continuous. Similar results are obtained for weighted convolution algebras.

Let A and B be (not necessarily bounded) linear operators on a Banach lattice E such that |(s – B)-1x|≤ (s – A)-1|x| for all x in E and sufficiently large s ∈ R. The main purpose of this paper is to investigate the relation between the spectra σ(B) and σ(A) of B and A, respectively. We apply our results to study asymptotic properties of dominated C0-semigroups.

Homotopy theory folklore tells us that the sheaf defining the cohomology theory $\operatorname {\mathrm {Tmf}}$ of topological modular forms is unique up to homotopy. Here we provide a proof of this fact, although we claim no originality for the statement. This retroactively reconciles all previous constructions of $\operatorname {\mathrm {Tmf}}$.

In this paper we give new results concerning the maximal regularity of the strict solution of an abstract second-order differential equation, with non-homogeneous boundary conditions of Dirichlet type, and set in an unbounded interval. The right-hand term of the equation is a Hölder continuous function.

The set C(G) of conjugacy classes of subgroups of a group G has a natural partial order. We study p-groups G for which C(G) has antichains of prescribed lengths.

An infinite class of T-matrices is constructed using Golay sequences. A list is given with new Hadamard matrices of order 2t. q, q odd, q < 10000, improving the known values of t.

Finally T-matrices are given of order 2m + 1, for small values of m ≤ 12 which do not coincide with those generated by Turyn sequences.

It is determined when there exists a minimal essential ideal, or minimal essential left ideal, in the incidence algebra of a locally finite partially ordered set defined over a commutative ring. When such an ideal exists, it is described.

In the past a number of papers have appeared which give representations of abstract lattices as rings of sets of various kinds. We refer particularly to authors who have given necessary and sufficient conditions for an abstract lattice to be lattice isomorphic to a complete ring of sets, to the lattice of all closed sets of a topological space, or to the lattice of all open sets of a topological space. Most papers on these subjects give the conditions in terms of special elements of the lattice. We thus have completely join-irreducible elements — G. N. Raney [7]; join prime, completely join prime, and supercompact elements — V. K. Balachandran [1], [2]; N-sub-irreducible elements — J. R. Büchi [5]; and lattice bisectors — P. D. Finch [6]. Also meet-irreducible and completely meet-irreducible dual ideals play a part in some representations of G. Birkhoff & 0. Frink [4].

In this paper we characterize the ideals of the centralizer near-ring N = MR(R2), where R is a commutative principle ideal ring. The characterization is used to determine the radicals Jυ(N) and the quotient structures N/ Jv(N), v = 0, 1, 2.

We describe the group of all reflection-preserving automorphisms of an imprimitive complex reflection group. We also study some properties of this automorphism group.