Let $A$ be a finite-dimensional division algebra containing a base field $k$ in its center $F$. $A$ is defined over a subfield $F_0$ if there exists an $F_0$-algebra $A_0$ such that $A = A_0 \bigotimes_{F_0} F$. The following are shown. (i) In many cases $A$ can be defined over a rational extension of $k$. (ii) If $A$ has odd degree $n \geq 5$, then $A$ is defined over a field $F_0$ of transcendence degree $\leq {\frac{1}{2}}(n-1)(n-2)$ over $k$. (iii) If $A$ is a $\mathbb{Z}/m \times \mathbb{Z}/2$-crossed product for some $m \geq 2$ (and in particular, if $A$ is any algebra of degree 4) then $A$ is Brauer equivalent to a tensor product of two symbol algebras. Consequently, ${\rm M}_m(A)$ can be defined over a field $F_0$ such that ${\rm trdeg}_k(F_0) \leq 4$. (iv) If $A$ has degree 4 then the trace form of $A$ can be defined over a field $F_0$ of transcendence degree $\leq 4$. (In (i), (iii) and (iv) it is assumed that the center of $A$ contains certain roots of unity.)

A new technique is presented which gives conditions under which perturbations of certain base potentials are uniquely determined from the location of eigenvalues and resonances in the context of a Schrödinger operator on a half line. The method extends to complex-valued potentials and certain potentials whose first moment is not integrable.Research supported in part by the US National Science Foundation under grants DMS-9970299 and DMS-0107492 and by the UK EPSRC.

Further properties of a group $\Gamma$ introduced by the first author in 1980 (sometimes called the first Grigorchuk group) are established. These are notably that any two infinite finitely generated subgroups of $\Gamma$ have isomorphic subgroups of finite index and that the generalized word problem for $\Gamma$ is soluble.

Let $\mu$ be the self-similar measure for a linear function system $S_jx=\rho x+b_j$ ($j=1,2,\ldots,m$) on the real line with the probability weight $\{p_j\}_{j=1}^m$. Under the condition that $\{S_j\}_{j=1}^m$ satisfies the finite type condition, the $L^q$-spectrum $\tau(q)$ of $\mu$ is shown to be differentiable on $(0,\infty)$; as an application, $\mu$ is exact dimensional and satisfies the multifractal formalism.

The existence of $2\pi$-periodic solutions of the second-order differential equation \[ x''+f(x)x'+ax^+-bx^-+g(x)=p(t), \qquad n\in \mathbb{N},\] where $a, b$ satisfy $1/\sqrt{a}+1/\sqrt{b}=2/n$ and $p(t)=p(t+2\pi)$, $t\in \mathbb{R}$, is examined. Assume that limits $\lim_{x\to\pm\infty}F(x)=F(\pm\infty)$ ($F(x)=\int_0^xf(u) du$) and $\lim_{x\to\pm\infty}g(x)=g(\pm\infty)$ exist and are finite. It is proved that the equation has at least one $2\pi$-periodic solution provided that the zeros of the function $\Sigma_1$ are simple and the zeros of the functions $\Sigma_1, \Sigma_2$ are different and the signs of $\Sigma_2$ at the zeros of $\Sigma_1$ in $[0,2\pi/n)$ do not change or change more than two times, where $\Sigma_1$ and $\Sigma_2$ are defined as follows: \[\Sigma_1(\theta)=\frac{n}{\pi}\left[\frac{g(+\infty)}{a}-\frac{g(-\infty)}{b} \right]-\frac {1}{2\pi}\int_0^{2\pi}p(t)\varphi(t+\theta)\,dt,\qquad \theta\in [0, 2\pi/n],\]\[\Sigma_2(\theta)=\frac{n}{\pi}[F(+\infty)-F(-\infty)]-\frac{1}{2\pi} \int_0^{2\pi}p(t)\varphi'(t+\theta)\,dt, \qquad\theta\in [0, 2\pi/n].\] Moreover, it is also proved that the given equation has at least one $2\pi$-periodic solution provided that the following conditions hold: \begin{eqnarray*} -\infty&<&\liminf_{x\to\pm\infty}\frac{F(x)}{|x|^{p-2}x}\le \limsup_ {x\to\pm\infty}\frac{F(x)}{|x|^{p-2}x}<+\infty,\\ 0&<&\liminf_{x\to\pm\infty}\frac{g(x)}{|x|^{q-2}x}\leq \limsup_{x\to\pm\infty} \frac{g(x)}{|x|^{q-2}x}<+\infty, \end{eqnarray*} with $1\leq p<q<2$.

The coefficients in asymptotics of regularized traces and associated trace (spectral) distributions for Schrödinger operators with short and long range potentials are computed. A kernel expansion for the Schrödinger semigroup is derived, and a connection with non-commutative Taylor formulas is established.

Acyclic groups of low dimension are considered. To indicate the results simply, let $G^{\prime}$ be the nontrivial perfect commutator subgroup of a finitely presentable group $G$. Then $\mathrm{def}(G)\leq1$. When $\mathrm{def}(G)=1$, $G^{\prime}$ is acyclic provided that it has no integral homology in dimensions above 2 (a sufficient condition for this is that $G^{\prime}$ be finitely generated); moreover, $G/G^{\prime}$ is then $Z$ or $Z^{2}$. Natural examples are the groups of knots and links with Alexander polynomial 1. A further construction is given, based on knots in $S^{2}\times S^{1}$. In these geometric examples, $G^{\prime}$ cannot be finitely generated; in general, it cannot be finitely presentable. When $G$ is a 3-manifold group it fails to be acyclic; on the other hand, if $G^{\prime}$ is finitely generated it has finite index in the group of a $\mathbb{Q}$-homology 3-sphere.

In this study of the behaviour of the number of conjugacy classes in finite $p$-groups using pro-$p$ groups, the conjugacy growth function $r_n(G)= \max\{r(G/N)\,{|}\,N\triangleleft_o G,|G\,{:}\,N|=n\}$ is introduced. It is proved that there are no infinite pro-$p$ groups of linear conjugacy growth (that is, there is no $c$ such that $r_n(G)\le c\log_2 n$ for all $n>1$) and it is shown that many known pro-$p$ groups are of exponential conjugacy growth (that is, there exists $\epsilon\,{>}\,0$ such that $r_n(G)\,{\ge}\,n^\epsilon$ for infinitely many values of $n$).

By critical point theory, a new approach is provided to study the existence of periodic and subharmonic solutions of the second order difference equation $$\Delta^2 x_{n-1} +f(n, x_n)\,{=}\,0,$$ where $f\in C({\bf R}\times {\bf R}^m, {\bf R}^m), f(t+M,z)=f(t,z)$ for any $(t, z)\in {\bf R}\times{\bf R}^m$ and $M$ is a positive integer. This is probably the first time critical point theory has been applied to deal with the existence of periodic solutions of difference systems.This project is supported by TRATOYT of China and by the State Education Commission Trans-Century Training Program Foundation for the Talents.

The paper shows that, for subanalytic stratifications, Lipschitz equisingularity as defined by Mostowski is preserved after intersection with generic wings, that is, $L$-regularity implies $L^*$-regularity. This was one of the conditions required of a good equisingularity notion by Teissier in his foundational 1974 Arcata paper.

Previous authors have shown that Lipschitz equisingularity is generic, implies bilipschitz triviality, and hence topological triviality, and implies equimultiplicity.

The paper first proves a result on the upper semi-continuity of the entropy map of random bundle maps, and then proves some large deviation results for random maps perturbations of an Axiom A diffeomorphism.

For a holomorphic function $f$ in the open unit disc $D$, the $N$th partial sum of its Taylor series with center $\zeta \in D$ is denoted by $S_N(f,\zeta)(z)=$${\sum\nolimits^N_{n=0}}({{f^{(n)}(\zeta)}/n!})(z-\zeta)^n$. Generically, all functions $f$ in $H(D)$ satisfy the following. For every compact set $K\subset\Bbb C$ with $K{\cap}\,D=\varnothing$ and $K^c$ connected and every polynomial $h$, there exists a sequence of positive integers $\{\lambda_n\}^{\infty}_{n=1}$ such that, for every $l \in \{ 0,1,2, \ldots \}$, \[ \sup_{z \in K}\Big\vert {{\partial^l}\over {\partial z^l}} S_{\lambda_n}(f,0)(z)-h^{(l)}(z)\Big\vert \,{\to}\,0 \quad {\rm as} \; n\,{\to}\,{+}\,\,\infty.\]

The heat content asymptotics on a Riemannian manifold with smoooth boundary defined by Dirichlet, Neumann, transmittal and transmission boundary conditions are studied.

It is proved that the duality map $\langle\,,\rangle:(\ell^\infty,\hbox{weak})\times((\ell^\infty)^*,\hbox{weak}^* )\longrightarrow {\bf R}$ is not Borel. More generally, the evaluation $e:(C(K),\wk)\times K\longrightarrow{\bf R}$, $e(f,x) = f(x)$, is not Borel for any function space $C(K)$ on a compact $F$-space. It is also shown that a non-coincidence of norm-Borel and weak-Borel sets in a function space does not imply that the duality map is non-Borel.

The cohomology groups of some sequence algebras are considered. In particular, the amenability of the James space $J_p$ is investigated; this space was first studied by R. C. James and was shown to be a Banach algebra for the pointwise product. Then a full classification of the closed ideals in $J_p$ is given, extending a result of N. J. Laustsen. Finally, some related algebras $\alp$ are considered, and it is established when these Banach algebras are amenable.

A new large class of rearrangement-invariant (symmetric) spaces is introduced which contains most classical spaces used in applications. It consists of spaces with arbitrary fundamental function $\f(t)$ which are interpolation spaces with respect to the corresponding extreme spaces $\Lambda_\varphi$ and $M_\varphi$ (the quasinorm case is also permitted). The (quasi)norms for these new spaces are of the form $\|\varphi(t)f^*(t)\|_{\widetilde E}$, where $\widetilde E$ is an arbitrary rearrangement-invariant function space with respect to the measure $dt/t$. Thus the considered spaces include and generalize Lorentz, Lorentz–Zygmund and other similar spaces as well as the spaces $L_{p,\alpha, E}$, previously studied by Pustylnik. In spite of their generality, these spaces can be investigated deeply and in detail with rather sharp results that open a simple way to various applications, extending results about their classical examples. For instance, their dual spaces, sharp conditions for separability and mutual embeddings, dilation functions, Boyd indices and so on are found. A theorem on optimal weak type interpolation in these spaces is also proved.

This research was partially supported by the National Security Agency. It is well known from work of Bruhat and Tits that an affine building has a spherical building at infinity. The paper studies the structure at infinity for an affine {\em twin} building. It is shown that the twinning restricts the structure at infinity in a natural way, producing two smaller spherical buildings that are canonically isomorphic to one another. In the process it is shown that to each wall and panel of these buildings at infinity is attached a twin tree.

Let $\phi$ be a bounded linear functional on $A$, where $A$ is a commutative Banach algebra, then the bilinear functional $\skew5\check{\phi}$ is defined as $\skew5\check{\phi} (a,b)\,{=}\,\phi (ab)-\phi (a) \phi (b)$ for each $a$ and $b$ in $A$. If the norm of $\skew5\check{\phi}$ is small then $\phi$ is approximately multiplicative, and it is of interest whether or not $\|\skew5\check{\phi}\|$ being small implies that $\phi$ is near to a multiplicative functional. If this property holds for a commutative Banach algebra then $A$ is an AMNM algebra (approximately multiplicative functionals are near multiplicative functionals). The main result of the paper shows that ${\text{C}}^N {[0,1]}^M$ (the complex valued functions defined on ${[0,1]}^M$ with all $N$th order partial derivatives continuous) is AMNM. It is also shown that a similar proof can be applied to certain Lipschitz algebras.

A sufficient condition is given under which the sum, product and indeed any polynomial combination of a well-bounded operator and a commuting real scalar-type spectral operator is well-bounded. This generalizes a result of Gillespie for Hilbert space operators. It is shown in particular that if $X$ is a UMD space, then the sum of finitely many commuting real scalar-type spectral operators acting on $X$ is a well-bounded operator (a result which fails on general reflexive Banach spaces).

Projections are constructed in the rotation algebra that are orthogonal to their Fourier transform and which are fixed under the flip automorphism. Such projections are expected in a construction of an inductive limit structure for the irrational rotation algebra that is invariant under the Fourier transform (namely, as two circle algebras of the same dimension, which are swapped by the Fourier transform, plus a few points). The calculation is based on Rieffel's construction of the Schwartz space as an equivalence bimodule of rotation algebras as well as on the theory of theta functions.