The asymptotic behaviour of entropy numbers of Trudinger–Strichartz embeddings of radial Besov spaces on ${\mathbb R}^n$ into exponential Orlicz spaces is calculated. Estimates of the entropy numbers as well as estimates of entropy numbers of Sobolev embeddings of radial Besov spaces are applied to spectral theory of certain pseudo-differential operators.

Let $l$ be an oriented link of $d$ components in a homology $3$-sphere. For any nonnegative integer $q$, let $l(q)$ be the link of $d-1$ components obtained from $l$ by performing $1/q$ surgery on its $d$th component $l_d$. The Mahler measure of the multivariable Alexander polynomial $\Delta_{l(q)}$ converges to the Mahler measure of $\Delta_l$ as $q$ goes to infinity, provided that $l_d$ has nonzero linking number with some other component. If $l_d$ has zero linking number with each of the other components, then the Mahler measure of $\Delta_{l(q)}$ has a well defined but different limiting behavior. Examples are given of links $l$ such that the Mahler measure of $\Delta_l$ is small. Possible connections with hyperbolic volume are discussed.

The minimum number of critical points of a small codimension smooth map between two manifolds is computed. Some partial results for the case of higher codimension when the manifolds are spheres are also given.

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 correspondence between the equivariant degree introduced by Ize, Massabó, and Vignoli and an unstable version of the equivariant fixed point index defined by Prieto and Ulrich is shown. With the help of conormal maps and properties of the unstable index, a sum decomposition formula is proved for the index and consequently also for the degree. As an application, equivariant homotopy groups are decomposed as direct sums of smaller groups of fixed orbit types, and a geometric interpretation of each summand is given in terms of conormal maps.

Let K be the field of real or complex numbers. Let $(X \cong {\bf K}^{2n}, \omega)$ be a symplectic vector space and take $0 < k < n,N = ({2n \atop 2k})$. Let $L_1,\ldots,L_N \subset X$ be $2k$-dimensional linear subspaces which are in a sufficiently general position. It is shown that if $F : X \longrightarrow X$ is a linear automorphism which preserves the form $\omega^k$ on all subspaces $L_1,\ldots,L_N$, then $F$ is an $\epsilon_k$-symplectomorphism (that is, $F^*\omega = \epsilon_k\omega$, where $\epsilon_k^k = 1$). In particular, if ${\bf K} = \R$ and $k$ is odd then $F$ must be a symplectomorphism. The unitary version of this theorem is proved as well. It is also observed that the set ${\cal A}_{l,2r}$ of all $l$-dimensional linear subspaces on which the form $\omega$ has rank $\le 2r$ is linear in the Grassmannian $G(l,2n)$, that is, there is a linear subspace $L$ such that ${\cal A}_{l,2r} = L \cap G(l, 2n)$. In particular, the set ${\cal A}_{l,2r}$ can be computed effectively. Finally, the notion of symplectic volume is introduced and it is proved that it is another strong invariant.

A necessary and sufficient condition for the almost sure existence of the exact packing measure on the boundary of a Galton–Watson tree is obtained with an explicit exact packing measure. In the case where the exact packing measure does not exist almost surely, a criterion which classifies functions $\ph$ according to whether the $\ph$ packing measure is zero or infinity is given.

Given a smooth family ${\hbox{\ac L}}$ of real or complex variable taking values within the class of Fredholm operators of index zero in a Banach space, there are some available definitions in the literature of the concept of algebraic multiplicity of the family ${\hbox{\ac L}}$ at a point $x_0$ of the parameter at which the operator ${\hbox{\ac L}}(x_0)$ becomes non-invertible. The purpose of the paper is to show that the algebraic multiplicity is uniquely determined by a few of its properties, independently of its construction. The main technical tools to obtain this uniqueness result are a Lyapunov–Schmidt reduction, the local Smith form and a new factorization result for general families at non-algebraic eigenvalues obtained in the paper.

The paper presents a construction of fibered links $(K, \Sigma)$ out of chord diagrams $\sL$. Let $\Gamma$ be the incidence graph of $\sL$. Under certain conditions on $\sL$ the symmetrized Seifert matrix of $(K, \Sigma)$ equals the bilinear form of the simply-laced Coxeter system $(W, S)$ associated to $\Gamma$; and the monodromy of $(K, \Sigma)$ equals minus the Coxeter element of $(W, S)$. Lehmer's problem is solved for the monodromy of these Coxeter links.

Sun and Wilson defined the notion of infinite determinacy of a smooth function germ singular along a line, and related this notion to some good geometric properties of derived objects related to the given function germ. The paper extends their results to a wider class of smooth function with prescribed non-isolated singularities. For this purpose, it was necessary to study the behaviour of the function germ along a transverse direction of the given singular set, and to relate these properties to geometric properties of the function and derived objects, expressed in terms of relative Łojasiewicz conditions.

It is shown that for every non-negative integer $n$, there is a real $n$-dimensional family of minimal Lagrangian tori in $\mathbb{CP}^2$, and hence of special Lagrangian cones in $\mathbb{C}^3$ whose link is a torus. The proof utilises the fact that such tori arise from integrable systems, and can be described using algebro-geometric (spectral curve) data.

Two functions on finitely additive probability spaces that behave well under products are introduced: discrepancy, which measures how close one space comes to extending another, and bi-discrepancy, which is a pseudo-metric on the collection of all spaces on a given set, and a metric on the collection of complete spaces. These are then applied to show that the Loeb space of the internal product of two internal finitely additive probability spaces depends only on the Loeb spaces of the two original internal spaces. Thus the notion of a Loeb product of two Loeb spaces is well defined. The Loeb operation induces an isometry from the nonstandard hull of the space of internal probability spaces on a given set to the space of Loeb spaces on that set, with the metric of bi-discrepancy.