We show that for every $n\,{\geq}\,2$ there exists closed hyperbolic n-manifolds for which the full group of orientation preserving isometries is trivial.

In a previous paper, the authors introduced a filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ $(V,0)$ of a complex analytic variety defined by arcs on the singularity and called the arc filtration. The Poincaré series of this filtration were computed for simple surface singularities in the 3-space. Here they are computed for surface singularities from Arnold's lists including uni- and bimodular ones. The classification of the unimodular singularities by these Poincaré series turns out to be in accordance with their hierarchy defined by E. Brieskorn using the adjacency relations. We also give a general formula for the Poincaré series of the arc filtration for isolated surface singularities which are stabilizations of plane curve ones.

It is proved that $fg$ is Henstock–Kurzweil integrable on a compact interval ${\mathop{\prod}_{i=1}^{m}}[a_i, b_i]$ in ${\mathbb R}^m$ for each Henstock–Kurzweil integrable function $f$ if and only if there exists a finite signed Borel measure $\nu$ on ${\mathop{\prod}_{i=1}^{m}}[a_i, b_i)$ such that $g$ is equivalent to $\nu({\mathop{\prod}_{i=1}^{m}}[a_i, \,{\cdot}\,))$ on ${\mathop{\prod}_{i=1}^{m}}[a_i, b_i]$.

The Hecke algebra $H_n$ can be described as the skein $R_n^n$ of $(n,n)$-tangle diagrams with respect to the framed Homfly relations. This algebra $R_n^n$ contains well-known idempotents $E_{\lambda}$ which are indexed by Young diagrams $\lambda$ with $n$ cells. The geometric closures $Q_{\lambda}$ of the $E_{\lambda}$ in the skein of the annulus are known to satisfy a combinatorial multiplication rule which is identical to the Littlewood–Richardson rule for Schur functions in the ring of symmetric functions. This fact is derived from two skein theoretic lemmas by using elementary determinantal arguments. Previously known proofs depended on results for quantum groups.

We compare conical density properties and spherical density properties for general Borel measures on $\mathbb{R}^n$. As a consequence, we obtain results for packing and Hausdorff measures $\mathcal{P}_h$ and $\mathcal{H}_h$ provided that the gauge function $h$ satisfies certain conditions.

One consequence of our general results is the following: let $m, n\,{\in}\,\mathbb{N}, 0\,{\lt}\,s\,{\lt}\,m\,{\leq}\,n$, $0\,{\lt}\,\eta\,{\lt}\,1$, and suppose that $V$ is an $m$-dimensional linear subspace of $\mathbb{R}^n$. Let $\mu$ be either the $s$-dimensional Hausdorff measure or the $s$-dimensional packing measure restricted to a set $A$ with $\mu(A)\,{\lt}\,\infty$. Then for $\mu$-almost every $x\,{\in}\,\mathbb{R}^n$, there is $\theta\,{\in}\,V\,{\cap}\, S^{n-1}$ such that \[ \liminf\limits_{r\downarrow 0}r^{-s}\mu(B(x, r)\cap H(x,\theta,\eta))=0,\] where $H\left(x,\theta,\eta\right)\,{=}\,\{{y\in\mathbb{R}^n}{\ip{\br{y-x}}{\theta}\,{\gt}\,\eta|{y-x}|\}$.

A Haken sphere $S$ in a 3-manifold $M$ with a Heegaard splitting $V\cup_F W$ is a separating 2-sphere in $M$ which intersects the Heegaard surface $F$ only in an essential circle in $F$. We describe how any two Haken spheres in the genus 2 Heegaard splitting of connected sum of two lens spaces are related, which extends Scharlemann–Thompson's corresponding result on Haken spheres in genus 2 Heegaard splitting for $S^3$.

The main result of this paper gives an explicit computation of the $L_4$ norm of any cyclotomic polynomial of the form \[ f(x)=\Phi_{p_1}(\pm x)\Phi_{p_2}(\pm x^{p_1})\cdots \Phi_{p_r}(\pm x^{p_1 p_2\cdots p_{r-1}}). \] Here $\Phi_{p}$ is the $p$th cyclotomic polynomial and the $p_i$ are primes that are not necessarily distinct.

A corollary of this is the following theorem.

THEOREM. If\[ f(x)=\Phi_{p_1}(\e_1 x)\Phi_{p_2}(\e_2 x^{p_1})\cdots \Phi_{p_r}(\e_r x^{p_1 p_2\cdots p_{r-1}}), \]where N = p1p2···prand εj = ±, then \[ \frac{\| f\|_4^4}{N^2} & \ge & \frac{\big\|\Phi_2(-x)\Phi_2(-x^2)\cdots \Phi_2\big({-}x^{2^{r-1}}\big) \big\|_4^4}{4^r}\\& = & \frac{\big(\frac12 +\frac{5}{34}\sqrt{17}\big)(1+\sqrt{17})^r - \big({-}\frac12 +\frac{5}{34}\sqrt{17}\big)(1-\sqrt{17})^r}{4^r}. \] In particular the minimum possible normalized L4 norm of any polynomial of the above form is attained by \[ \Phi_2(-x)\Phi_2(-x^2)\cdots \Phi_2\big({-}x^{2^{r-1}}\big). \]

Cochran, Orr and Teichner introduced $L^2$-eta-invariants to detect highly non-trivial examples of non slice knots. Using a recent theorem by Lück and Schick we show that their metabelian $L^2$-eta-invariants can be viewed as the limit of finite dimensional unitary representations. We recall a ribbon obstruction theorem proved by the author using finite dimensional unitary eta-invariants. We show that if for a knot $K$ this ribbon obstruction vanishes then the metabelian $L^2$-eta-invariant vanishes too. The converse has been shown by the author not to be true.

Let $P$ be the self-similar probability corresponding to an iterated function system of contracting similitudes $S_1,\ldots,S_N$ with probabilities $p_1,\ldots,p_N$ and contraction constants $s_1,\ldots, s_N$. Let $D_r$ be the quantization dimension of $P$ of order $r\in(0,+\infty)$, and let $P_r$ be the self-similar probability corresponding to $S_1,\ldots,S_N$ with probabilities $(p_1s_1^r)^{D_r/(r+D_r)},\ldots,(p_Ns_N^r)^{D_r/(r+D_r)}$. It is shown that the quanitzation coefficient of $P$ of order $r$ exists and that the empirical measures on an arbitrary sequence of asymptotically optimal quantizing sets (of order $r$) weakly converges to $P_r$ provided $\log(p_1s_1^r),\ldots,\log(p_Ns_N^r)$ is not arithmetic and $S_1,\ldots,S_N$ satisfies the open set condition.

For every $N > 0$ there exists a group of deficiency less than ${-}N$ that arises as the fundamental group of a smooth homology 4-sphere and also as the fundamental group of the complement of a compact contractible submanifold of the 4-sphere. A group is the fundamental group of the complement of a contractible submanifold of the $n$-sphere, $n > 4$, if and only if it is the fundamental group of a homology $n$-sphere. There exist fundamental groups of homology $n$-spheres, $n > 4$, that cannot arise as the fundamental group of the complement of a contractible submanifold of the 4-sphere.

In this paper we prove that reduced components of the fundamental cycle of a sandwiched singularity give rise to Nash families of arcs.

Classical solutions of the spherically symmetric Nordström–Vlasov system are shown to exist globally in time. The main motivation for investigating the mathematical properties of the Nordström–Vlasov system is its relation to the Einstein–Vlasov system. The former is not a physically correct model, but it is expected to capture some of the typical features of the latter, which constitutes a physically satisfactory, relativistic model but is mathematically much more complex. We show that classical solutions of the spherically symmetric Nordström–Vlasov system exist globally in time for compactly supported initial data under the additional condition that there is a lower bound on the modulus of the angular momentum of the initial particle system. We emphasize that this is not a smallness condition and that our result holds for arbitrary large initial data satisfying this hypothesis.

We study the modified Helmholtz equation in a semi-strip with Poincaré type boundary conditions. On each side of the semi-strip the boundary conditions involve two parameters and one real-valued function. Using a new transform method recently introduced in the literature we show that the above boundary-value problem is equivalent to a $2\times 2$-matrix Riemann–Hilbert (RH) problem. If the six parameters specified by the boundary conditions satisfy certain algebraic relations this RH problem can be solved in closed form. For certain values of the parameters the solution is not unique, furthermore in some cases the solution exists only under certain restrictions on the functions specifying the boundary conditions. The asymptotics of the solution at the corners of the semi-strip is investigated. In the case that the $2\times 2$ RH problem cannot be solved in closed form, the Carleman–Vekua method for regularising it is illustrated by analysing in detail a particular case.

The results on the local existence and continuation criteria obtained by G. Rein are extended to the case with a non-zero cosmological constant. It is also shown that for the spherically symmetric case and a positive cosmological constant there is a large class of initial data with global existence and inflationary asymptotics in the future as in the case of plane or hyperbolic symmetry. Furthermore we analyze the behaviour of the energy-momentum tensor at late times.

We consider asymptotic values of continuous functions u from $\mathbb{R}^m$, or a suitable subset of $\mathbb{R}^m$, to the extended real numbers $\overline\mathbb{R}\,{=}\,\mathbb{R}\,{\cup}\,\{-\infty\}\,{\cup}\,\{\infty\}$, with the usual topologies in each case. Building on work of Rippon we obtain some rather unexpected results about such asymptotic values, in particular an intermediate value property.

For $a\subs b\subs\omega$ with $b{\setminus} a$ infinite, the set $D\,{=}\,\{x\in\reals\,{:}\, a\subs x\subs b\}$ is called a doughnut. Doughnuts are equivalent to conditions of Silver forcing, and so, a set $S\subs\reals$ is called Silver measurable, or completely doughnut, if for every doughnut $D$ there is a doughnut $D'\subs D$ which is contained in or disjoint from $S$. In this paper, we investigate the Silver measurability of $\bDelta$ and $\bSigma$ sets of reals and compare it to other regularity properties like the Baire and the Ramsey property and Miller and Sacks measurability.

Let $\mu$ be a Radon measure on ${{\rr}^d}$ which satisfies the growth condition only namely, there is a constant $C>0$ such that for all $x\in{{\rr}^d}$, $r>0$ and for some fixed $0<n\le d$, \[ \mu(B(x,r))\le Cr^n, \] where $B(x,r)$ is the ball centered at $x$ and having radius $r$. In this paper, we first give a new atomic characterization of the Hardy space $H^1(\mu)$ introduced by X. Tolsa. As applications of this new characterization, we establish the $(H^1(\mu),L^{1,\infty}(\mu))$ estimate of the commutators generated by $RBMO(\mu)$ functions with the Calderón–Zygmund operators whose kernels satisfy only the size condition and a certain minimum regularity condition. Using this endpoint estimate and a new interpolation theorem for operators which is also established in this paper and has independent interest, we further obtain the $L^p(\mu)$$(1<p<\infty)$ boundedness of these commutators.

A contact problem is called tangential when arbitrary tangential displacements are prescribed over a part of the boundary of a transversely isotropic layer, while the tangential stress is zero over the rest of the boundary; the normal stress vanishes all over the boundary. The Generalized Images method is used to give a complete elementary solution to the problem. A new governing integral equation is derived. A particular case of a circular domain of contact is studied in detail. The governing integral equation can be inverted in this case. Approximate formulae are derived for the resultant force and the torque. A direct relationship is established between the integral transform method and the Generalized Images method. A limiting case of general solution gives the solution for an isotropic layer.