Considérons la fonction arithmétique $$D^*(n,t)\colonequal \sum_{\di{d|n,\,d'|n}{0<|\log (d'/d)|\le t}} 1\qquad (n\,{>}\,1,\,t\,{>}\,0),$$ qui généralise la fonction $T^*(n,\alpha)\colonequal D^*(n,(\log n)^{-\alpha})$, étudiée dans la première partie de cet article [7]. Les variations de $D^*(n,t)$ au voisinage de l'origine permettent de mesurer, en moyenne quadratique, la propension des diviseurs de l'entier $n$ à s'agglutiner autour de certaines valeurs. Nous nous proposons ici de donner une majoration essentiellement optimale de cette fonction de $t$ sur $[0,1]$ lorsque $n$ est astreint à parcourir un ensemble d'entiers non exceptionnels en un certain sens quantitativement précisé.

Bescovitch considered the Hausdorff dimensions of sets related to the sum of digits of real numbers represented in the dyadic expansions. In this paper, we provide an analogy for continued fraction expansions over the field of formal Laurent series. We also calculate the Hausdorff dimensions of sets of Laurent series which have given polynomial or exponential approximation orders. Such approximations converge faster than those of almost all Laurent series (with respect to the Haar measure).

The following theorem is proved. For any positive integers $k$ and $n$ there exists $t$ depending only on $k$ and $n$ such that the class of all groups $G$ having $\gamma_k(G)$ locally finite and satisfying the condition that the product of any $t$ commutators of the form $[x_1,x_2,{\dots}\,,x_k]$ is of order dividing $n$ is a variety.

In this paper, we shall prove that if $G$ is a countably infinite product of second countable amenable locally compact groups $G_i,i\,{=}\,0,1,2,\dots$ and each $G_i$ is a non-trivial compact group for $i>0,$ then the center of the second conjugate algebra of the Fourier algebra $A(G)$ of $G$ is precisely $A(G)$.

Throughout we let ($T,{\frak m},k$) denote a commutative Noetherian local ring with maximal ideal ${\frak m}$ and residue field $k$. We let $I\,{\subseteq}\,T$ be an ideal generated by a regular sequence of length $c$ and set $R\,{\colonequal}\,T/I$. In the important paper [1], Avramov addresses the following question. Given a finitely generated $R$-module $M$, when does $M$ have finite projective dimension over a ring of the form $T/J$, where $J$ is generated by part (or all) of a set of minimal generators for $I$? He gives a fairly complete answer to this question that is expressed in terms of the geometry of varieties in affine space defined by annihilators of certain graded modules derived from resolutions over $R$. In an attempt to understand these ideas more fully, we became interested in the idea that one might answer the question at hand by using data about $M$ (orits syzygies) coming from $T$, in particular, information gleaned from various Fitting ideals defined over $T$. The following theorem from Section 3 is one of our main results. We use $\Fitt_T(M)$ to denote the Fitting ideal of $M$.

We show that $S_3*_{Z/2Z}S_3$ satisfies Turaev's criterion to be the fundamental group of an orientable $PD_3$-complex. Since this group has infinitely many ends but is indecomposable as a free product this provides a negative answer to a question of Wall.

We prove that the Casson–Gordon invariants of the connected sum of two knots split when the Alexander polynomials of the knots are coprime. As one application, for any knot $K$, all but finitely many algebraically slice twisted doubles of $K$ are linearly independent in the knot concordance group.

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.

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 $S^3$.

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.

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.