Given a holomorphic self-map $\varphi $ of $\mathbb {D}$ (the open unit disc in $\mathbb {C}$ ), the composition operator $C_{\varphi } f = f \circ \varphi $ , $f \in H^2(\mathbb {\mathbb {D}})$ , defines a bounded linear operator on the Hardy space $H^2(\mathbb {\mathbb {D}})$ . The model spaces are the backward shift-invariant closed subspaces of $H^2(\mathbb {\mathbb {D}})$ , which are canonically associated with inner functions. In this paper, we study model spaces that are invariant under composition operators. Emphasis is put on finite-dimensional model spaces, affine transformations, and linear fractional transformations.

A new concept—that of pseudoconsistency—which seems to be particularly appropriate for the estimation of a quantile is introduced. It is shown without any conditions whatsoever on the underlying distributionthat the sample quantile is strongly pseudoconsistent for the corresponding population quantile. The asymptotic distribution of the sample quantiles and order statistics is derived when the underlying distribution is discrete.

In [7,3.1 ] the authors show that if a space X is realcompact and locally compact, then X* is a P′-space. In this paper we show that the hypothesis of realcompactness can be weakened. We also look at other conditions on X that are sufficient to guarantee that X* is a P′-space.

Let Rm denote a m dimensional Euclidean space. When x ∊ Rm will write x = (x1, x2,..., xm). Let R+m ={x: x ∊ Rm, xi < 0 for all i} and R-m ={x: x ∊ Rm, xi < 0 for all i}. In this paper we consider a class of functions which consists of mappings, Er(K) and Hr(K) of Rm into R which are indexed by K ∊ R+m and K ∊ R-m respectively, and defined at any point α ∊ Rm by

1.1

We consider the problem of determining the best possible bounds on the eigenvalues of an nth order positive definite matrix B, when the determinant (D) and trace (T) are given. A large variety of bounds on the eigenvalues are known when different information concerning B is available (see, for example, [1], [2]). Since D and T simply provide the geometric mean and arithmetic mean of the positive, real eigenvalues of B, the solution to the problem involves certain inequalities satisfied by these means (see [3] for such inequalities in a more general setting). A related problem in which the largest and smallest eigenvalue are known, and inequalities involving D and T are obtained, is described in [4].

This note is based on Hu's Homotopy Theory (New York, 1959), pp. 115 f.

The theorem that the homotopy sequence is exact splits into six statements. The fifth in Hu's exposition can be stated as follows: Let f:In, In-1, Jn-1→X, A, x0,

1

Then there exists a map

2

such that f~j g: In, In-1, Jn-1→X, A, x0.

An A-group is a finite solvable group all of whose Sylow subgroups are abelian. In this paper, we are interested in bounding the derived length of an A-group G as a function of the number of distinct sizes of the conjugacy classes of G. Although we do not find a specific bound of this type, we do prove that such a bound exists. We also prove that if G is an A-group with a faithful and completely reducible G-module V, then the derived length of G is bounded by a function of the number of distinct orbit sizes under the action of G on V.

The method for computing the $p$-localization of the group $\left[ X,\,\text{U}\left( n \right) \right]$, by Hamanaka in 2004, is revised. As an application, an explicit description of the self-homotopy group of $\text{Sp}\left( 3 \right)$ localized at $p\,\ge \,5$ is given and the homotopy nilpotency of $\text{Sp}\left( 3 \right)$ localized at $p\,\ge \,5$ is determined.

Identities containing the Gaussian binomial coefficient are obtained. From these results, we deduce some identities of the combinatorial analysis which contain the binomial coefficient

In [2] we showed how, for a certain class of outer measures on a metric space, a measurable cover could be constructed for each subset A of the space. The function is a closure operator, and in this note some of the properties of the resulting topology are investigated. In particular, we obtain a sufficient condition for the space to be connected.

In 1962, J. M. G. Fell [5] indicated the important role played by certain topological spaces which, though locally compact in a specialized sense, do not, in general, satisfy even the weakest separation axiom. He called them "locally compact". These were called "punktal kompakt" by Flachsmeyer [6] and to avoid confusion, we shall call them pointwise compact spaces.

We show that the sequence of integers which have nearly the typical number of distinct prime factors forms a Poisson process. More precisely, for $\delta$ arbitrarily small and positive, the nearest neighbor spacings between integers $n$ with $\left| \omega \left( n \right)\,-\,\log \log n \right|\,<\,{{\left( \log \log n \right)}^{\delta }}$ obey the Poisson distribution law.

In the problem of finding the motion of a classical particle one has the choice of dealing with a system of second order ordinary differential equations (Lagrange's equations) or a single first order partial differential equation (the Hamilton-Jacobi equation, henceforth referred to as the H-J equation). In practice the latter method is less frequently used because of the difficulty in finding complete integrals. When these are obtainable, however, the method leads rapidly to the equations of the trajectories. Furthermore it is of fundamental theoretical importance and it provides a basis for quantum mechanical analogues.

Let $\omega ^*(n)$ be the number of primes p such that $p-1$ divides n. Recently, M. R. Murty and V. K. Murty proved that

$$ \begin{align*}x(\log\log x)^3\ll\sum_{n\le x}\omega^*(n)^2\ll x\log x.\end{align*} $$

They further conjectured that there is some positive constant C such that

$$ \begin{align*}\sum_{n\le x}\omega^*(n)^2\sim Cx\log x,\end{align*} $$

as $x\rightarrow \infty $ . In this short note, we give the correct order of the sum by showing that

$$ \begin{align*}\sum_{n\le x}\omega^*(n)^2\asymp x\log x.\end{align*} $$

Given a domain D in R” and two specified points P0 and P1 in D we consider the problem of minimizing u(p1) over all functions harmonic in D with values between 0 and 1 normalised by the requirement u(P0) = 1/2. We show that when D is suitably regular the problem has a unique solution u* which necessarily takes on boundary values 0 or 1 almost everywhere on the boundary. In the process we prove that it is possible to separate P0 and P1by a harmonic function whose boundary value is supported in an arbitrary set of positive measure. These results depend on the fact that (under suitable regularity conditions) a harmonic function which vanishes on an open subset of the boundary has a normal derivative which is almost everywhere non-vanishing in that set.

We provide a novel proof of the existence of regulator indecomposables in the cycle group $C{{H}^{2}}\left( X,\,1 \right)$, where $X$ is a sufficiently general product of two elliptic curves. In particular, the nature of our proof provides an illustration of Beilinson rigidity.