The notion of almost alternating links was introduced by C. Adams et al ([1]). We give a sufficient condition for an almost alternating link diagram to represent a non-splittable link. This solves a question asked in [1, 3]. A partial solution for special almost alternating links has been obtained by M. Hirasawa ([5]). As an application, Theorem 2.2 gives us a simple finite algorithm to decide if a given almost alternating link diagram represents a splittable link without increasing the number of crossings of diagrams in the process. Moreover, we show that almost alternating links with more than two components are non-trivial. In Section 2, we state these results in detail. To prove our theorem, we use a technique invented by W. Menasco (see [6, 7]), this is reviewed in Section 3. We also apply “the charge and discharge method” to our graph-theoretic argument, which was used to prove the four color theorem in [4], we explain this in Section 4.

An integral inequality for a compact Lorentzian manifold which admits a timelike conformal vector field and has no conjugate points along its null geodesics is given. Moreover, equality holds if and only if the manifold has nonpositive constant sectional curvature. The inequality can be improved if the timelike vector field is assumed to be Killing and, in this case, the equality characterizes (up to a finite covering) flat Lorentzian $n(\geq3)$-dimensional tori. As an indirect application of our technique, it is proved that a Lorentzian $2-$torus with no conjugate points along its timelike geodesics and admitting a timelike Killing vector field must be flat.

A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this paper is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt–Eccles operad. We also prove that differential graded algebras over the Barratt–Eccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt–Eccles operad which is equivalent to the classical little squares operad.

We study strong triviality of 2-bridge knots via Dehn surgery techniques and prove that a 2-bridge knot is strongly $n$-trivial for $n\geq 1$ if and only if it is the trivial knot, the trefoil knot or the figure-eight knot.

In the metastable range $3 \leq p < q < 2p - 3$, and modulo the action of the group of homotopy spheres, we classify diffeomorphism types of smooth manifolds that are homotopy equivalent to the products $S^{p} \times S^{q}$ of spheres.

We denote by $B_Y$ the unit ball of a normed linear space $Y$.

Definition. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed space $X$ is called a sufficient enlargement for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P{:}\,Y\,{\to}\,X$ such that $P(B_Y)\,{\subset}\,A$.

Minimal-volume sufficient enlargements are determined for two-dimensional spaces. The main results are:

(i) Each minimal-volume sufficient enlargement for a two-dimensional space is a parallelogram or a hexagon.

(ii) If a two-dimensional normed space $X$ has a minimal-volume sufficient enlargement that is not a parallelogram, then $B_X$ is linearly equivalent to the regular hexagon.

Given a diagram $D$ of a knot $K$, we consider the number $c(D)$ of crossings and the number $b(D)$ of overpasses of $D$. We show that, if $D$ is a diagram of a nontrivial knot $K$ whose number $c(D)$ of crossings is minimal, then $1+\sqrt{1+c(D)} \leq b(D)\leq c(D)$. These inequalities are sharp in the sense that the upper bound of $b(D)$ is achieved by alternating knots and the lower bound of $b(D)$ is achieved by torus knots. The second inequality becomes an equality only when the knot is an alternating knot. We prove that the first inequality becomes an equality only when the knot is a torus knot.

We prove the toral rank conjecture of Halperin in some new cases. Our results apply to certain elliptic spaces that have a two-stage Sullivan minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work.

In this paper we consider topological properties of ordinary as well as graph directed iterated function systems. First we give the basic definitions.

We describe probably the simplest 3-manifold which contains closed separating incompressible surfaces of arbitrarily large genus. Two applications of this observation are given. (1) For any closed, orientable 3-manifold $M$ and any integer $m\,{>}\,0$, a surgery on a link in $M$ of at most $2m\,{+}\,1$ components will provide a closed, orientable, irreducible 3-manifold containing $m$ disjoint, non-parallel, separating, incompressible surfaces of arbitrarily high genus. (2) There exists a 3-manifold $M$ containing separating incompressible surfaces $S_n$ of genus $g(S_n)$ arbitrarily large, such that the amalgamation of minimal Heegaard splittings of two resulting 3-manifolds cutting along $S_n$ can be stabilized $g(S_n)-3$ times to a minimal Heegaard splitting of $M$.

We propose a method of constructing examples in operator ergodic theory which unifies and extends some previously known examples. It also allows us to answer several questions that have been open for some time (including a question of Allan [1]).

Painlevé differential equations have been an important topic of research in complex differential equations during the last century, and the last two decades in particular, with many applications not only to pure mathematics but also to physics and engineering. In this paper, we prove that any transcendental solution of the second Painlevé equation $w''\,{=}\,2w^{3}+zw+\alpha$ is of order at least $3/2$, and that any transcendental solution of the fourth Painlevé equation $2ww''\,{=}\,(w')^{2}+3w^{4}+8zw^{3}+4(z^{2}-\alpha )w^{2}+2\beta$ is of order at least $2$.

We are interested in the dynamics that can arise close to rotational group orbits after forced symmetry-breaking to discrete symmetries. In particular we ask how simple or complicated the dynamics induced by symmetric linear vector fields can be. We look in detail at related tetrahedral and dihedral examples, and there we find precise conditions for a linear field to exhibit homoclinic orbits.

With a view to generalising rectifiability and density results to more general spaces we prove the following: let $H^s$ denote Hausdorff $s$ measure in $l^n_{\infty}$. Let $s\in(0,2]$. Let $S\subset l^n_{\infty}$ be a subset of positive locally finite Hausdorff $s$-measure with the property $$\lim_{r\rightarrow 0} \frac{H^s(B_r(x)\cap S)}{\alpha(s)2^{-s}r^s}=1\;\;\;\mathrm{for}\;\;H^{s}\;a.e.\;x\in S,$$ then $s$ is an integer and $S$ has a weak tangent at almost every point.

Consider a system of diagonal equations \begin{equation}\sum_{j=1}^sa_{ij}x_j^k=0\quad (1\le i\le r),\end{equation} satisfying the property that the (fixed) integral coefficient matrix $(a_{ij})$ contains no singular $r\times r$ submatrix. A recent paper of the authors [3] establishes that whenever $k\ge 3$ and $s>(3r+1)2^{k-2}$, then the expected asymptotic formula holds for the number $N(P)$ of integral solutions ${\bf x}$ of ($1{\cdot}1$) with $|x_i|\le P$$(1\le i\le s)$.

We prove that every Lie derivation on a symmetrically amenable semisimple Banach algebra can be uniquely decomposed into the sum of a derivation and a centre-valued trace.

In a recent paper, we gave a topological description of Colombeau type algebras introducing algebras of sequences with exponential weights. Embeddings of Schwartz spaces into the Colombeau algebra $\G$ are well known, but for ultradistribution and periodic hyperfunction type spaces we give new constructions. We show that the multiplication of regular enough functions (smooth, ultradifferentiable or quasianalytic), embedded into corresponding algebras, is the ordinary multiplication.

The main theorem states that all Lie and Jordan derivations from a von Neumann algebra $\frak{A}$ into any Banach $\frak{A}$-bimodule are standard. Moreover, this statement is proved for some other classes of algebras. Our approach is based on the algebraic theory of functional identities and the strong degree, and is combined with analytic tools.

Spacetimes with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry are shown to be timelike and null geodesically complete in the expanding direction, provided the data satisfy a certain size restriction.

We study the distribution of closed geodesics in homology on a finite area hyperbolic surface. We obtain an estimate which is uniform as the homology class varies, refining an asymptotic formula due to C. Epstein.