Consider the elliptic integral of the first kind

or, alternatively,

where x = sin ϕ. It is customary in the theory of elliptic integrals to let k = sin α, k' = cos α, sot that k'2 + k2 = 1. For convenience we shall also introduce the parameter

For a (bounded, linear) operator A in a (complex, infinite-dimensional, separable) Hilbert space ℋ, the inner derivation DA as an operator on ℬ(ℋ), is defined by DAX = AX – XA. Johnson and Williams [4] showed that, when A is a normal operator, range inclusion DBℬ(ℋ)⊆DA(ℋ)⊆ is equivalent to the condition that B = f(A), where f is a Lipschitz function on σ(A) such that t(z, w)(f(z)–f(w))/(z–w) is a trace class kernel on L2(μ) whenever t(z, w) is such a kernel. (Here μ is the dominating scalar valued spectral measure of A constructed in multiplicity theory). This result is deep and its proof is difficult. In the present paper, we establish the following analogous result which is easier to prove: for a normal operator A, range inclusion DB℘2(ℋ) holds if and only if B = f(A) for some Lipschitz function f on σ(A). Here ℘(ℋ) stands for the Hilbert-Schmidt class of operators on ℋ. As by-products of our argument, we generalize some results in [4], [8], [9] concerning the non-existence of a one-sided ideal contained in certain derivation ranges; for example, we show that if A is hyponormal and if the point spectrum σP(A*) of A* is empty, then DAℬ(ℋ) does not contain any nonzero right ideal.

We provide a new computation of the K-theory of the group C*-algebra of the solvable Baumslag–Solitar group BS(1, n) (n ≠ 1); our computation is based on the Pimsner–Voiculescu 6-terms exact sequence, by viewing BS(1, n) as a semi-direct product ℤ[1/n] ⋊ ℤ. We deduce from it a new proof of the Baum–Connes conjecture with trivial coefficients for BS(1, n).

R. A. Rankin [3] considered the problem of finding, for each integer n ≧ 3, a sequence of positive integers containing no n−term geometric progression. He constructed such sets Bn having asymptotic density

For example A3 ≑ 0·71975, A4 ≑ 0·8626, and An→1 as n → ∞.

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. We show that B(X) is finite if and only if no proper, complemented subspace of X is isomorphic to X, and we show that B(X) is properly infinite if and only if X contains a complemented subspace isomorphic to X[oplus ]X. We apply these characterizations to find Banach spaces X1, X2, and X3 such that B(X1) is finite, B(X2) is infinite, but not properly infinite, and B(X3) is properly infinite. Moreover, we prove that every unital, properly infinite ring has a continued bisection of the identity, and we give examples of Banach spaces D1 and D2 such that B(D1) and B(D2) are infinite without being properly infinite, B(D1) has a continued bisection of the identity, and B(D2) has no continued bisection of the identity. Finally, we exhibit a unital $C^\ast$-algebra which is finite and has a continued bisection of the identity.

Let G be a polycyclic, metabelian or soluble of type (FP)∞ group such that the class Rat(G) of all rational subsets of G is a Boolean algebra. Then, G is virtually abelian. Every soluble biautomatic group is virtually abelian.

We study the existence of solutions for a class of nonuniformly degenerate elliptic systems in N, N ≥ 3, of the formwhere hi ∈ L1loc(N), hi(x) ≧ γ0|x|α with α ∈ (0, 2) and γ0 > 0, i = 1, 2. The proofs rely essentially on a variant of the Mountain pass theorem (D. M. Duc, Nonlinear singular elliptic equations, J. Lond. Math. Soc. 40(2) (1989), 420–440) combined with the Caffarelli–Kohn–Nirenberg inequality (First order interpolation inequalities with weights, Composito Math. 53 (1984), 259–275).

All groups considered are finite. In recent years a number of generalizations of the classic Jordan-Hölder Theorem have been obtained (see [7], Theorem A.9.13): in a finite group G a one-to-one correspondence as in the Jordan-Holder Theorem can be defined preserving not only G-isomorphic chief factors but even their property of being Frattini or non-Frattini chief factors. In [2] and [13] a new direction of generalization is presented: the above correspondence can be defined in such a way that the corresponding non-Frattini chief factors have the same complement (supplement).

Let k be an algebraically closed field. By an algebra is meant an associative finite dimensional k-algebra A with an identity. We are interested in studying the representation theory of Λ, that is, in describing the category mod Λ of finitely generated right Λ-modules. Thus we may, without loss of generality, assume that Λ is basic and connected. For our purpose, one strategy consists in using covering techniques to reduce the problem to the case where the algebra is simply connected, then in solving the problem in this latter case. This strategy was proved efficient for representation-finite algebras (that is, algebras having only finitely many isomorphism classes of indecomposable modules) and representation-finite simply connected algebras are by now well-understood: see, for instance [5], [7],[8]. While little is known about covering techniques in the representation-infinite case, it is clearly an interesting problem to describe the representation-infinite simply connected algebras. The objective of this paper is to give a criterion for the simple connectedness of a class of (mostly representationinfinite) algebras.

Two subgroups $M$ and $S$ of a group $G$ are said to permute, or $M$permutes with$S$, if $MS = SM$. Furthermore, $M$ is a permutable subgroup of $G$ if $M$ permutes with every subgroup of $G$. In this article, we provide necessary and sufficient conditions for a subgroup of $G\times H$, whose intersections with the direct factors are normal, to be a permutable subgroup.

Let M be an n-dimensional connected submanifold in an mdimensional Euclidean space Em. Denote by δ the Laplacian of M associated with the induced metric. Then the position vector x and the mean curvature vector H of Min Em satisfy

This yields the following fact: a submanifold M in Em is minimal if and only if all coordinate functions of Em, restricted to M, are harmonic functions. In other words, minimal submanifolds in Emare constructed from eigenfunctions of δ with one eigenvalue 0. By using (1. 1), T. Takahashi proved that minimal submanifolds of a hypersphere of Em are constructed from eigenfunctions of δ with one eigenvalue δ (≠0). In [3, 4], Chen initiated the study of submanifolds in Em which are constructed from harmonic functions and eigenfunctions of δ with a nonzero eigenvalue. The position vector x of such a submanifold admits the following simple spectral decomposition:

for some non-constant maps x0and xq, where A is a nonzero constant. He simply calls such a submanifold a submanifold of null 2-type.

Let H be a complex Hilbert space. For any operator (bounded linear transformation) T on H, we denote the spectrum of T by σ(T). Let T = (T1, …, Tn) be an n-tuple of commuting operators on H. Let Sp(T) be the Taylor joint spectrum of T. We refer the reader to [8] for the definition of Sp(T). A point v = (v1, …, vn) of ℂn is in the joint approximate point spectrum σπ(T) of T if there exists a sequence {xk} of unit vectors in H such that

.

A point v = (v1, …, vn) of ℂn is in the joint approximate compression spectrum σs(T) of T if there exists a sequence {xk} of unit vectors in H such that

A point v=(v1, …, vn) of ℂn is in the joint point spectrum σp(T) of T if there exists a non-zero vector x in H such that (Ti-vi)x = 0 for all i, 1 ≤ j ≤ n.

We present a classification of the nilpotent primitive subgroups of $\mathrm{GL}(n,q)$, up to conjugacy in $\mathrm{GL}(n,q)$. Groups in the classification are specified explicitly by generating sets of matrices, where each generating set has size at most 2. Additionally we give an algorithm designed to provide electronic access to the classification for any $n$ and $q$.

In a very recent paper [1], Basil Gordon discusses generalizations of Jacobi's identity

where x and z are complex numbers and |x| <1. He notes that some of its consequences, inter alia Euler's formula

are of interest in number theory and combinatory analysis. He proves the apparently new and striking result

where |s|<1, and also considers the possibility of generalizations. His methods are algebraic and quite simple, but perhaps do not make obvious what underlies such formulae. It may be worth while to do so, especially since the details become simpler and the presentation more perspicuous. The method given here assumes no more knowledge than his does, although the new proof is expressed in terms of theta-functions, in simple properties of which, formulae such as (3) have their origin. Further, (3) appears in a slightly more symmetrical form.

During the last few years several articles on asymptotic martingales (amarts) have appeared. The first unified treatment was given by Edgar and Sucheston in [7], where further references can be found. The purpose of this paper is to add some further results to the theory of amarts.

The following work establishes a new proof of the theorem: Every archimedean ordered group is abelian. This theorem has been proved differently by many authors. It was first proved by O. Hölder [2]. A second proof has been given by H. Cartan [1]: he uses the topology which is naturally introduced in the group by its order.

The purpose of this paper is to establish some basic properties of the Wielandt subgroup of a polycyclic group. The Wielandt subgroup of a group G is defined to be the intersection of the normalisers of all the subnormal subgroups of G and is denoted by ω(G). In 1958 Wielandt [9] showed that any minimal normal subgroup with the minimum condition on subnormal subgroups is contained in the Wielandt subgroup: it follows that the Wielandt subgroup of an artinian group is nontrivial. In contrast, the Wielandt subgroup of a polycyclic group can be trivial; an easy example is given by the infinite dihedral group. We will show that the Wielandt subgroup of a polycyclic group is close to being central.

This paper is concerned with three basic transforms

The first of these has been studied by Widder [1], who points out that f(t) can be interpreted as the temperature u(0, t) on the time axis, where u(x, t) is the solution of the heat equation withsymmetric initial temperature u(x, 0) = g(|x|). The second has also been studied by Widder [2], where it is pointed out that f(t) can be interpreted as the value of the harmonic function u(x, t) on the t-axis arising from the boundary data u(x, 0) = g(|x|).

The Lindenstrauss formula

which states a strong relationship between the (Clarkson) modulus of uniform convexity δx of a Banach space X and the modulus of uniform smoothness px* of the conjugate space X*, is well known. Following the idea of the definitions of nearly uniform smooth space by S. Prus and modulus of uniform smoothness we define a modulus of nearly uniform smoothness and prove some Lindenstrauss type formulae concerning this modulus and the modulus of nearly uniform convexity for some measures of noncompactness.

If G is the group of holomorphic automorphisms of a bounded symmetric domain, then G has a distinguished class of irreducible unitary representations called the holomorphic discrete series of G. These representations have been studied by Harish-Chandra in [7]. On the Lie algebra level, the Harish-Chandra modules corresponding to the holomorphic discrete series representations are highest weight modules. Even for G as above, it turns out that not all the unitary highest weight modules belong to the holomorphic discrete series but there exists a condition on the highest weight which characterizes the holomorphic discrete series among the unitary highest weight representations. They can be defined as those unitary highest weight representations with square integrable matrix coefficients.