A Baire space is a topological space which satisfies the Baire Category Theorem, i.e., in which the intersection of countably many dense open sets is dense. In this note we shall be interested in the size of Baire spaces, so to avoid trivialities we shall consider only non-atomic spaces, that is, spaces X whose regular open algebras ro(X) are non-atomic. All natural examples of Baire spaces, such as complete metric spaces or compact spaces, seem to have sizes at least 2ℵ0. So a natural question, asked first by W. Fleissner and K. Kunen [5], is whether there exists a Baire space of the minimal possible size ℵ1. The purpose of this note is to show that such a space need not exist by proving the following result.

In this paper, we extend to the non-Hille–Yosida case a variation of constants formula for a nonautonomous and nonhomogeneous Cauchy problems first obtained by Gühring and Räbiger. By using this variation of constants formula, we derive a necessary and sufficient condition for the existence of an exponential dichotomy for the evolution family generated by the associated nonautonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally, we illustrate our results by considering two examples. The first example is a parabolic equation with nonlocal and nonautonomous boundary conditions, and the second example is an age-structured model that is a hyperbolic equation.

In [1] Arhangel'skiĭ announced that any σ-paracompact p-space could be mapped onto a Moore space by a perfect map. However Burke [3] recently showed that this is not true in general and he gave an example of a T2, locally compact, σ-paracompact space which cannot be mapped onto a Moore space by a perfect map.

In this article we examine conditions for the appearance or nonappearance of the two extra-special 2-groups of order 32 as Galois groups over a field F of characteristic not 2. The groups in question are the central products DD of two dihedral groups of order 8, and DQ of a dihedral group with the quaternion group, obtained by identifying the central elements of order 2 in each factor group. It is shown that the realizability of each of these groups as Galois groups over F implies the realizability of other 2-groups (which are not their quotient groups), and in turn that realizability of certain other 2-groups implies the realizability of DD and DQ. We conclude by providing an explicit construction of field extensions with Galois group DD.

We extend the min-max methods used in the critical point theory of differentiable functionals on smooth manifolds to the case of continuous functionals on a complete metric space. We study the topological properties of the min-max generated critical points in this new setting by adopting the methodology developed by Ghoussoub in the smooth case. Many old and new results are extended and unified and some applications are given.

Let ƒ(z) be a complex function analytic in some neighbourhood of the origin with ƒ(0) = 1. It is known that ƒ(z) admits a unique "power product" expansion of the form convergent near zero. We derive a simple direct bound for the radius of convergence of this product expansion in terms of the coefficients of ƒ(z). In addition we show that the same bound holds in the case of "inverse power product" expansions Examples are given for which these bounds are sharp. We show also that products with nonnegative coefficients have the same radius of convergence as their corresponding series.

In [4] N. Steinmetz used Nevanlinna theory to establish remarkably versatile theorems on the factorization of ordinary differential equations which implied numerous previous results of various authors. (Here factorization is taken in the sense of function composition as introduced by F. Gross in [2].) The thrust of Steinmetz’ central results on factorization is that if g(z) is entire and f(z) is meromorphic in C such that the composite fog satisfies an algebraic differential equation, then so do f(z) and, degenerate cases aside, g(z). In addition, the more one knows about the equation for fog (e.g. degree, weight, autonomy), the more one can conclude about the equations for f and g.

In this note we generalize Steinmetz’ work to show the following:

• a) Steinmetz’ two basic results, Satz 1 and Korollar 1 of [4] can be seen as one-variable specializations of a single two variable result, and

• b) the function g(z) can itself be allowed to be a function of several variables.

B. Sz.-Nagy and C. Foiaş gave the first examples of quasisimilar operators with different spectra [25]. Indeed, quasisimilar operators can even have different spectral radius [19] (see also [15]). Nevertheless, T. B. Hoover has shown in [19] that if T and S are quasisimilar, then σ(T)∩σ(S) ≠ ∅, where σ(R) denotes the spectrum of the (bounded linear) operator R. In [12], the author improved Hoover's result by showing that each component of σ(S) intersects σ(T), and viceversa. This, in turn, was further improved by L. A. Fialkow in [7]. (Actually, Fialkow's results were obtained independently of [12].)

L. A. Fialkow [7] and L. R. Williams [27] independently proved that σe(T)∩σe(S) ≠ ∅, where σe(R) denotes the essential spectrum of R. Several authors have raised the following natural question:

Is it also true that each component of σe(S) intersects σe(T), and viceversa? [7] [21] [26].

We introduce the oriented Brauer–Clifford and degenerate affine oriented Brauer–Clifford supercategories. These are diagrammatically defined monoidal supercategories that provide combinatorial models for certain natural monoidal supercategories of supermodules and endosuperfunctors, respectively, for the Lie superalgebras of type Q. Our main results are basis theorems for these diagram supercategories. We also discuss connections and applications to the representation theory of the Lie superalgebra of type Q.

Two complex hermitian n × n matrices A and B are congruent if S*AS = B for some invertible n × n matrix S (with complex entries). The matrix S is called a congruence matrix. Given congruent hermitian matrices A and B, a congruence matrix is, of course, not unique. For instance, if A = B then one can take S = αI with |α| = 1, as well as any other matrix satisfying S*AS = A. However, here the choice S = I seems naturally to be best possible in the sense that when applied to an n-dimensional column vector it produces no distortion or movement of the vector at all. We shall measure the distortion (or movement) of the vector x ∊ Cn under an n × n invertible matrix A in terms of ║x − Ax║, where the norm is euclidean. Then the distortion produced by A is ║I − A║, with the induced operator norm.

A nonstandard hull of a TVS (locally convex topological vector space) is a standard TVS constructed from a nonstandard model for [3]. If the nonstandard hulls of a TVS are independent of the non-standard model, we say that the TVS has invariant nonstandard hulls. This is (for complete spaces) the property that every finite element is inflnitesimally close to a standard point. We build on the work of Henson and Moore [4], to show that invariance of nonstandard hulls is a self dual property equivalent to bounded sets being precompact, for F and DF spaces, (see Theorem 4.4).

Positive, copositive, onesided and intertwining (co-onesided) polynomial and spline approximations of functions are considered. Both uniform and pointwise estimates, which are exact in some sense, are obtained.

Let $Y$ be an infinite covering space of a projective manifold $M$ in ${{\mathbb{P}}^{N}}$ of dimension $n\ge 2$. Let $C$ be the intersection with $M$ of at most $n-1$ generic hypersurfaces of degree $d$ in ${{\mathbb{P}}^{N}}$. The preimage $X$ of $C$ in $Y$ is a connected submanifold. Let $\phi$ be the smoothed distance from a fixed point in $Y$ in a metric pulled up from $M$. Let ${{\mathcal{O}}_{\phi }}(X)$ be the Hilbert space of holomorphic functions $f$ on $X$ such that ${{f}^{2}}{{e}^{-\phi }}$ is integrable on $X$, and define ${{\mathcal{O}}_{\phi }}(X)$ similarly. Our main result is that (under more general hypotheses than described here) the restriction ${{\mathcal{O}}_{\phi }}(Y)\to {{\mathcal{O}}_{\phi }}(X)$ is an isomorphism for $d$ large enough.

This yields new examples of Riemann surfaces and domains of holomorphy in ${{\mathbb{C}}^{n}}$ with corona. We consider the important special case when $Y$ is the unit ball $\mathbb{B}$ in ${{\mathbb{C}}^{n}}$, and show that for $d$ large enough, every bounded holomorphic function on $X$ extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on $\mathbb{B}$. Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from $X$ to $\mathbb{B}$.

For two entire functions f(z) and g(z) the composition f(g(z)) may or may not be periodic even though g(z) is not periodic. For example, when f(u) = cos √u and g(z) = z2, or f(u) = eu and g(z) = p(z) + z, where p(z) is a periodic function of period 2πi, f(g(z)) will be periodic. On the other hand, for any polynomial Q(u) and any non-periodic entire function f(z) the composition Q(f(z)) is never periodic (2).

The general problem of finding necessary and sufficient conditions for f(g(z)) to be periodic is a difficult one and we have not succeeded in solving it. However, we have found some interesting related results, which we present in this paper.

This paper establishes the following combinatorial result concerning the automorphisms of a modular lattice.

THEOREM. Let M be a modular lattice and let G be a finite subgroup of the automorphism group of M. If the sublattice, MG, of (common) fixed points (under G) satisfies any of a large class of chain conditions, then M satisfies the same chain condition. Some chain conditions in this class are the following: the ascending chain condition; the descending chain condition; Krull dimension; the property of having no uncountable chains, no chains order-isomorphic to the rational numbers; etc.

We continue in this paper our study of conjugacy classes of a reductive monoid $M$. The main theorems establish a strong connection with the Bruhat-Renner decomposition of $M$. We use our results to decompose the variety ${{M}_{\text{nil}}}$ of nilpotent elements of $M$ into irreducible components. We also identify a class of nilpotent elements that we call standard and prove that the number of conjugacy classes of standard nilpotent elements is always finite.

Let KM = {z : \z\ < M}, 1 ≦ M < ∞ and K = K1. Let S denote the collection of functions f(z) = z + a2z2 + a3s3 + … that are regular and univalent in K. We write, for 1 < M < ∞,

S(M) = ﹛f : f ∞ S, f (K) ⊂ KM ﹜,

S*(M) = ﹛f : f ∞ S(M), f(K) is starlike with respect to the origin﹜.

In this paper we develop a variational technique for slit domains and give some applications with respect to finding the

and the

for any nonconstant entire function ᶲ(w) and a given z ∊ K.

It is well known now that certain spherical methods in k (≧2) dimensions are rather poor for reconstructing a function from its Fourier transform. Consider a function f in L1(Rk), k ≧ 2,

and

where both integrals are integrals in Rk, the first over the whole space the second over the ball of radius R; x • y is the usual Euclidean inner product of x and y in Rk and \z\2 = z • z.

In a recent paper [2] we defined four classes of infinite dimensional simple Lie algebras over a field of characteristic 0 which we called W*, S*, H*, and K*. As the names suggest, these classes generalize the Lie algebras of Cartan type. A second paper [3] investigates the derivations of the algebras W* and S*, and the possible isomorphisms between these algebras and the algebras defined by Block [1]. In the present paper we investigate the automorphisms of the algebras of type W*.

A neofield of order v, Nv( + , •), is an algebraic system of v elements including 0 and 1,0 ≠ 1, with two binary operations + and • such that (Nv, + ) is a loop with identity element 0; (Nv*, •) is a group with identity element 1 (where Nv* = Nv\﹛0﹜) and every element of Nv is both right and left distributive (i.e., (y + z)x = yx + zx and x(y + z) = xy + xz for all y, z∈ Nv).