A functorial and categorical defined cyclotomic trace is given, extending the usual one for rings to ring spectra. There are two ingredients to this: first a cyclotomic trace is needed that accepts a categorical input with few restrictive assumptions. This is important in its own right, since this allows one to transport rich structures through the cyclotomic trace. Secondly, a sufficiently nice model is needed for the category of finitely generated free modules which is functorial in the ring spectrum.

To a finitely generated profinite group $G$, a formal Dirichlet series $P_G(s)\,{=}\,\sum_{n}{a_n}/{n^s}$ is associated, where $a_n \,{=}\,\sum_{|G:H|=n} \mu_G(H)$. It is proved that $G$ is prosoluble if and only if the sequence $\{a_n\}_{n \in \mathbb N}$ is multiplicative, that is, $a_{rs}\,{=}\,a_ra_s$ for any pair of coprime positive integers $r$ and $s$. This extends the analogous result on the probabilistic zeta function of finite groups.

The normalized positive definite class functions $\C G\,{\longrightarrow}\,\C$ are determined for all those direct limits $G$ of finite alternating groups $\Alt(\Omega_i)$ for which the embeddings $\Alt(\Omega_i)\,{\longrightarrow}\,\Alt(\Omega_j)$ are natural in the sense that every non-trivial $\Alt(\Omega_i)$-orbit in $\Omega_j$ is natural.

A coordinate-system called $\lambda$-lengths is constructed for an SL$(2,{\Bbb C})$ representation space of punctured surface groups. These $\lambda$-lengths can be considered as complexification of R. C. Penner's $\lambda$-lengths for decorated Teichmüller spaces of punctured surfaces. Via the coordinates the mapping class group acts on the representation space as a group of rational transformations. This fact is applied to find hyperbolic 3-manifolds which fibre over the circle.

A class of harmonic function spaces is introduced and studied, namely the spaces $H^p_\sigma(G)$ of $\sigma$-harmonic $L^p$ functions on a locally compact group $G$, for $1\leq p \leq \infty$ and a given complex measure $\sigma$ on $G$ of unit norm. It is shown that there is a contractive projection from $L^p(G)$ onto $H^p_\sigma(G)$, for $1\lt p\leq \infty$, and structural results for $H^p_\sigma (G)$ are deduced. Given an adapted probability measure $\sigma$ on $G$, a uniqueness result is proved, that the space $H^p_\sigma(G)$ contains only constant functions, for $1\leq p\,{\lt}\,\infty$. For any $\sigma$, a result on the dimension of $H^1_\sigma (G)$ is proved.

The paper describes the Garnier systems as isomonodromic deformation equations of a linear system with a simple pole at 0 and a Poincaré rank 1 singularity at infinity. The extension of Okamoto's birational canonical transformations to the Garnier systems in more than one variable and to the Schlesinger systems is discussed.

A version of interpolation inequalities for derivatives in logarithmic Orlicz spaces is obtained where the first gradient of $u$ is estimated in terms of $u$ and its second gradient. One of the Orlicz functions considered is supposed to be ${\backslash\lambda^p}$. The motivation, examples and applications are discussed.

Riemann–Stieltjes integrals are considered as linear operators on weighted Bloch and Bergman spaces of the open unit ball in several complex variables. For weighted Bloch spaces, boundedness, compactness and weak compactness of Riemann–Stieltjes operators are characterized by means of certain growth properties of holomorphic symbols. For weighted Bergman spaces, some criteria are given for Riemann–Stieltjes operators with holomorphic symbols to be bounded, compact and of Schatten–von Neumann's ideal.

An explicit asymptotic formula is obtained for the norms of the spectral projections of the non-self-adjoint harmonic oscillator $H$. It is deduced that the spectral expansion of $\rme^{-Ht}$ is norm convergent if and only if $t$ is greater than a certain explicit positive constant.

A numerical quasi-isometry invariant $R(\Gamma)$ of a finitely generated group $\Gamma$ is defined whose values parametrize the difference between $\Gamma$ being uniformly embeddable in a Hilbert space and $C^{*}_{r}(\Gamma)$ being exact.

The paper investigates the vectorial Dirichlet problem defined by $$\begin{cases}\sigma_j(\nabla u(x))=1, \backslash, x\in\Omega \text{a.e.},\, j=1,\ldots, n u(x)=\varphi(x),\,x\in\partial\Omega. \end{cases}$$ Here $\Omega$ is an open bounded subset of $\mathbb{R}^n$ with boundary $\partial\Omega$, and $\sigma_j(A)$ ($j=1, \ldots , n$) denote the singular values of the gradient $\nabla u(x)$. The existence of solutions is established under one of the following assumptions: $\varphi: \overline{\Omega} \longrightarrow \mathbb{R}^n$ is continuous on $\overline{\Omega}$ and locally contractive on $\Omega$, or $\varphi: \partial\Omega \longrightarrow \mathbb{R}^n$ is contractive on $\partial\Omega$. This extends a result due to Dacorogna and Marcellini. The approach is based on the Baire category method developed earlier by the authors.

Several new multiple-integral representations are proved for well-poised hypergeometric series and integrals. The results yield, in particular, transformations of the multiple integrals that cannot be achieved by evident changes of variable. All this generalizes some classical results of Whipple and Bailey in analysis and, on the other hand, certain analytic constructions with known connection to irrationality proofs for values of Riemann's zeta function at positive integers.

The statistical properties of endomorphisms under the assumption that the associated Perron–Frobenius operator is quasicompact are considered. In particular, the central limit theorem, weak invariance principle and law of the iterated logarithm for sufficiently regular observations are examined. The approach clarifies the role of the usual assumptions of ergodicity, weak mixing, and exactness.

Sufficient conditions are given for quasicompactness of the Perron–Frobenius operator to lift to the corresponding equivariant operator on a compact group extension of the base. This leads to statistical limit theorems for equivariant observations on compact group extensions.

Examples considered include compact group extensions of piecewise uniformly expanding maps (for example Lasota–Yorke maps), and subshifts of finite type, as well as systems that are nonuniformly expanding or nonuniformly hyperbolic.

A procedure is given to reduce the interpolation spaces on an ordered pair generated by the function parameter $t^{\theta}(1+|\hspace{-2pt}\log t|)^{-b}$ to the classical real interpolation spaces. Applications are given for Lorentz–Zygmund function spaces, Besov spaces of generalized smoothness and Lorentz–Zygmund operator spaces.

The boundedness of Calderón–Zygmund operators is proved in the scale of the mixed Lebesgue spaces. As a consequence, the boundedness of the bilinear null forms $Q_{i j} (u,v) \,{=}\,\p_i u\p_j v \,{-}\, \p_j u\p_i v$, $Q_0(u,v)\,{=}\,u_t v_t \,{-}\,\nabla_x u\,{\cdot}\, \nabla_x v$ on various space–time mixed Sobolev–Lebesgue spaces is shown.

It is proved that there are two minimal $\Pi_1^0$ classes that are not automorphic.

It is considered whether $ L\,{=}\,\limsup_{n\to\infty} n \snormo{T^{n+1}-T^n}\,\lt\,\infty$ implies that the operator $T$ is power-bounded. It is shown that this is so if $L\lt1/e$, but it does not necessarily hold if $L\,{=}\,1/e$. As part of the methods, a result of Esterle is improved, showing that if $\sigma(T)\,{=}\,\{1\}$ and $T\,{\ne}\,I$, then $\liminf_{n\to\infty} n \snormo{T^{n+1}-T^n} \ge 1/e$. The constant $1/e$ is sharp. Finally, a way to create many generalizations of Esterle's result is described, and also many conditions are given on an operator which imply that its norm is equal to its spectral radius.

Consider the Sobolev embedding operator from the space of functions in $W^{1,p}(I)$ with average zero into $L^p$, where $I$ is a finite interval and $p\,{>}\,1$. This operator plays an important role in recent work. The operator norm and its approximation numbers in closed form are calculated. The closed form of the norm and approximation numbers of several similar Sobolev embedding operators on a finite interval have recently been found. It is proved in the paper that most of these operator norms and approximation numbers on a finite interval are the same.

Let $T$ be a sublinear operator such that $(T\chi_E)^*(t)\,{\le}\, h(t, |E|)$ for some positive function $h(t,s)$ and every measurable set $E$. Then it is shown that under some conditions on the operator $T$, this restricted weak type estimate can be extended to the set of all functions $f\,{\in}\, L^1$ such that $\n f.\infty\,{\le}\, 1$, in the sense that $(Tf)^*(t)\,{\le}\, h(t, \n f.1)$. This inequality allows strong type estimates for $T$ to be obtained on several classes of spaces, such as logarithmic spaces and Lorentz spaces. A similar problem for operators $T$ acting on radial functions is also studied.

It is shown that the matrix spectral factorization mapping is sequentially continuous from $\LL^p$ to $\HH^{2p}$ (where $1\,{\le}\, p\,{<}\,\infty$), under the additional assumption of uniform integrability of the logarithms of the spectral densities to be factorized. It is shown, moreover, that this condition is necessary for continuity, as well as sufficient.