Upper and lower bounds are established for the entropy numbers of certain diagonal operators between Banach sequence spaces. These diagonal operators are isomorphisms between the spaces considered in the paper and weighted sequence spaces considered by Leopold so that the entropy numbers in question coincide with those considered by Leopold. The results in the paper improve the previous results in at least two ways. The estimates in the paper are ‘almost’ sharp in the sense that the upper and lower estimates differ only by logarithmic factors for a much wider range of parameters. Moreover, all the upper estimates are improvements on the previous ones, the improvement being quite significant in some cases.

Criteria are found which classify Schatten class membership for operators on the Hardy space in terms of their action on the reproducing kernels and their derivatives. The Hilbert–Schmidt conditions are shown to be both necessary and sufficient for arbitrary operators. For other Schatten classes, the criteria are shown to be either necessary or sufficient for arbitrary operators, depending upon the exponent of the class. However, using known results of Peller and Luecking, the criteria are shown to be both necessary and sufficient for Hankel operators and Carleson embeddings.

The paper gives simple necessary and sufficient conditions for a closed 4-manifold to be homotopy equivalent to the mapping torus of a self homotopy equivalence of a PD3-complex. This is a homotopy analogue of the Stallings and Farrell fibration theorems available in other dimensions. The paper also considers 4-manifolds which admit a geometry of Euclidean factor type and complex surfaces which fibre over S1.

Two main results are shown: (1) a characterisation of the symmetric sequence spaces which embed into C(ωω), and (2) a characterisation, in terms of the Orlicz function M, of the Orlicz sequence spaces hM which embed into C(K) for some countable compact Hausdorff space K. The proofs make use of the Schreier families of sets, which have found wide applications in Banach space theory.

We prove that simple, thick hyperbolic P-manifolds of dimension at least three exhibit Mostow rigidity. We also prove a quasi-isometry rigidity result for the fundamental groups of simple, thick hyperbolic P-manifolds of dimension at least three. The key tool in the proof of these rigidity results is a strong form of the Jordan separation theorem, for maps from $S^n\rightarrow S^{n+1}$ which are not necessarily injective.

The paper deals with locally finite groups $G$ having an involution $\phi$ such that $C_G(\phi)$ is of finite rank. The following theorem gives a very detailed description of such groups.

Let $G$ be a locally finite group having an involution $\phi$ such that $C_G(\phi)$ is of finite rank. Then $G/[G,\phi]$ has finite rank. Furthermore, $[G,\phi]'$ contains a characteristic subgroup $B$ such that the following hold.

(1) $B$ is a product of finitely many subgroups normal in $[G,\phi]$ isomorphic to either PSL$(2,K)$ or SL$(2,K)$ for some infinite locally finite fields $K$ of odd characteristic.

(2) $[G,\phi]'/B$ has finite rank.

Singly-periodic monogenic cotangent and cosecant functions are important to Clifford analysis because they are the building blocks of the Bergman and Szegö reproducing kernels for strip domains, that is, rectangular domains with a single bounded dimension. The paper establishes a wide range of explicit formulas for these functions, in terms of derivatives, of one-dimensional integrals, and of Fourier and plane wave multidimensional integrals. These results indicate how the elementary trigonometric functions $\cot(z)$, $\csc(z)$ and $\csc^2(z)$ are ramified into different entities when the setting is switched from complex analytic to Clifford monogenic.

We compare the classical semidirect product V[midast ] W of pseudovarieties of semigroups with the recently found ‘regular’ semidirect product [Vscr ] [midast ]e [Wscr ] of e-pseudovarieties of regular semigroups. Applications of our findings include a substantial class of additions to the list of pseudovarieties generated by classes of regular semigroups and for which the membership problem is solved. For example, we compute the pseudovarieties generated by all locally inverse and locally orthodox semigroups, respectively, thereby answering a question raised by Almeida and Kaďourek.

Let X and Y be pointed spaces. A phantom map from X to Y is a map whose restriction to any finite skeleton of X is null-homotopic. Let Ph (X, Y) denote the set of homotopy classes of phantom maps from X to Y. As a pointed set it is isomorphic to the lim1 term of the tower of groups

formula here

where Y(n) denotes the Postnikov approximation of Y through dimension n. The homomorphisms in this tower are induced by the projections ΩY(n)← ΩY(n+1)). The groups in this tower are not abelian in general; however they do have some nice algebraic properties.

We consider singular exact Poisson structures on an oriented manifold of dimension 4. Using normal forms for divergence-free vector fields on ${\mathbb R}^3$ and for 1-parameter deformations of functions on ${\mathbb R}^3$ we produce normal forms for two classes of such Poisson structures around the singular point.

Hoh's calculus for pseudo-differential operators with negative-definite symbols is extended to the case of parameter-dependent symbols. Then this parameter-dependent calculus is applied to the study of subordinate sub-Markovian semigroups.

Let G be a group endowed with its profinite topology, then G is called product separable if the profinite topology of G is Hausdorff and, whenever H1, H2, [ctdot ], Hn are finitely generated subgroups of G, then the product subset H1H2 [ctdot ] Hn is closed in G. In this paper, we prove that if G=F×Z is the direct product of a free group and an infinite cyclic group, then G is product separable. As a consequence, we obtain the result that if G is a generalized free product of two cyclic groups amalgamating a common subgroup, then G is also product separable. These results generalize the theorems of M. Hall Jr. (who proved the conclusion in the case of n=1, [3]), and L. Ribes and P. Zalesskii (who proved the conclusion in the case of that G is a finite extension of a free group, [6]).

Let $(M_n)$ be a sequence of positive numbers satisfying $M_0\,{=}\,1$ and \[\displaystyle \frac{M_{n+m}}{M_n M_m}\,{\geq}\,{{n+m}\choose{m}} \] for all non-negative integers $m$, $n$. Let \[D([0,1], M)=\left\{f\,{\in}\,C^{\infty}([0,1]):\|f\|_{D}=\sum_{n=0}^{\infty} \frac{\|f^{(n)}\|_{\infty}}{M_n}\,{<}\,\infty\right\}.\] With pointwise addition and multiplication, $D([0,1],M)$ is a unital commutative semisimple Banach algebra. If $\lim_{n\to\infty} (n!/M_n)^{1/n}\,{=}\,0,$ then the maximal ideal space of the algebra is $[0,1]$, and every non-zero endomorphism $T$ has the form $Tf(x)\,{=}\,f(\phi(x))$ for some selfmap $\phi$ of the unit interval. The authors have previously shown for a wide class of $\phi$ mapping the unit interval to itself that if $\|\phi'\|_\infty\,{<}\,1$, then $\phi$ induces a compact endomorphism. The paper investigates the extent to which this condition is necessary, and the spectra of all compact endomorphisms of $D([0,1],M)$ are determined. Some of the authors' earlier results on general endomorphisms of $D([0,1],M)$ are simplified and strengthened.

A locally projective amalgam is formed by the stabilizer $G(x)$ of a vertex $x$ and the global stabilizer $G\{x,y\}$ of an edge containing $x$ in a group $G$, acting faithfully and locally finitely on a connected graph $\Gm$ of valency $2^n\,{-}\,1$ so that (i) the action is 2-arc-transitive, (ii) the sub-constituent $G(x)^{\Gm(x)}$ is the linear group ${\rm SL}_n(2) \,{\cong}\, {\rm L}_n(2)$ in its natural doubly transitive action, and (iii) $[t,G\{x,y\}] \,{\le}\, O_2(G(x) \,{\cap}\, G\{x,y\})$ for some $t \,{\in}\, G\{x,y\} \setminus G(x)$. Djoković and Miller used the classical Tutte theorem to show that there are seven locally projective amalgams for $n\,{=}\,2$. Trofimov's theorem was used by the first author and Shpectorov to extend the classification to the case $n \,{\ge}\, 3$. It turned out that for $n\,{\geq}\,3$, besides two infinite series of locally projective amalgams (embedded into the groups ${\rm AGL}_n(2)$ and $O_{2n}^+(2)$), there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups $M_{22}$, $M_{23}$, $Co_2$, $J_4$ and $\hbox{\it BM}$. For a locally projective amalgam $\cA$, the minimal degree $m\,{=}\,m(\cA)$ of its complex representation (which is a faithful completion into ${\rm GL}_m(\CC)$) is calculated. The minimal representations are analysed and three open questions on exceptional locally projective amalgams are answered. It is shown that

$\cA_4^{(1)}$ possesses ${\rm SL}_{20}(13)$ as a faithful completion in which the third geometric subgroup is improper;

$\cA_4^{(2)}$ possesses the alternating group ${\rm Alt}_{64}$ as a completion constrained at levels 2 and 3;

$\cA_4^{(5)}$ possesses ${\rm Alt}_{256}$ as a completion which is constrained at level 2 but not at level 3.

The results here generalise [2, Proposition 4.3] and [9, Theorem 5.11]. We shall prove the following.

THEOREM A. Let R be a Noetherian PI-ring. Let P be a non-idempotent prime ideal of R such that PRis projective. Then P is left localisable and RPis a prime principal left and right ideal ring.

We also have the following theorem.

THEOREM B. Let R be a Noetherian PI-ring. Let M be a non-idempotent maximal ideal of R such that MRis projective. Then M has the left AR-property and M contains a right regular element of R.

A model (M, <, …) is κ-like if M has cardinality κ but, for all α ∈ M, the cardinality of {x ∈ M [ratio ] x < a} is strictly less than κ. In this paper we shall give constructions of κ-like models of arithmetic satisfying an arbitrarily large finite part of PA but not PA itself, for various singular cardinals κ. The main results are: (1) for each countable nonstandard M [models ] Π2−Th(PA) with arbitrarily large initial segments satisfying PA and each uncountable κ of cofinality ω there is a cofinal extension K of M which is κ-like; also hierarchical variants of this result for Πn−Th(PA); and (2) for every n [ges ] 1, every singular κ and every M [models ] B[sum ]n+ exp+¬ I[sum ]n there is a κ-like model K elementarily equivalent to M.

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.

It is proved that a Tychonoff topological group is locally compact if and only if it is of pointwise countable type and its left uniformity is cofinally complete. From this result a characterization is derived of those T0 paratopological groups (X, τ) of pointwise countable type for which (X, τ ∨ τ−1) is locally compact and also a characterization is deduced of locally pseudocompact topological groups in terms of cofinal completeness. Also characterized are the Tychonoff topological groups of pointwise countable type for which their left uniformity has property U. Finally, cofinal completeness of the Hausdorff–Bourbaki uniformity of a topological group is studied.

In the appendix to [20] Waldhausen discussed a trace map tr:K(R)&→HH(R), from the algebraic K-theory of a ring to its Hochschild homology, which can be used to obtain information about K(R) from HH(R). In [1] Bökstedt described a factorization of this trace map. The intermediate functor THH(HR) is called the topological Hochschild homology of the Eilenberg–MacLane spectrum HR associated with R, because it is constructed similarly to Hochschild homology with the tensor product replaced by the smash product of spectra.

We consider a variation of the list colouring problem in which the lists are required to be sets of consecutive integers, and the colours assigned to adjacent vertices must differ by at least a fixed integer $s$. We introduce and investigate a new parameter $\tau(G)$ of a graph $G$, called the consecutive choosability ratio and defined to be the ratio of the required list size to the separation $s$ in the limit as $s\to\infty$.

We show that the above limit exists and that, for finite graphs $G,\ \tau(G)$ is rational and is a refinement of the chromatic number $\chi(G)$. We provide general bounds on $\tau(G)$, and determine its value for various classes of graphs including bipartite graphs, circuits, wheels and balanced complete multipartite graphs. Finally, we explore relationships between $\tau(G)$ and the circular chromatic number $\chi_{\mathrm{c}}(G)$.