Willis's structure theory of totally disconnected locally compact groups is investigated in the context of permutation actions. This leads to new interpretations of the basic concepts in the theory and also to new proofs of the fundamental theorems and to several new results. The treatment of Willis's theory is self-contained and full proofs are given of all the fundamental results.

We investigate exceptional sets associated with various additive problems involving sums of cubes. By developing a method wherein an exponential sum over the set of exceptions is employed explicitly within the Hardy-Littlewood method, we are better able to exploit excess variables. By way of illustration, we show that the number of odd integers not divisible by 9, and not exceeding $X$, that fail to have a representation as the sum of 7 cubes of prime numbers, is $O\left( {{X}^{23/36+\varepsilon }} \right)$. For sums of eight cubes of prime numbers, the corresponding number of exceptional integers is $O\left( {{X}^{11/36+\varepsilon }} \right)$.

We regard a system of left invariant vector fields $X=\,\{{{X}_{1}},\,\ldots ,\,{{X}_{k}}\}$ satisfying the Hörmander condition and the related Carnot-Carathéodory metric on a unimodular Lie group $G$. We define Besov spaces corresponding to the sub-Laplacian $\Delta \,=\,\sum X_{i}^{2}$ both with positive and negative smoothness. The atomic decomposition of the spaces is given. In consequence we get the distributional characterization of the Hausdorff dimension of Borel subsets with the Haar measure zero.

We present a concise explicit expression for the heat trace coefficients of spheres. Our formulas yield certain combinatorial identities which are proved following ideas of D. Zeilberger. In particular, these identities allow to recover in a surprising way some known formulas for the heat trace asymptotics. Our approach is based on a method for computation of heat invariants developed in $\left[ \text{P} \right]$.

Let $\pi $ be an irreducible generalized principal series representation of $G\,=\,\text{Sp}\left( 2,\,\mathbb{R} \right)$ induced from its Jacobi parabolic subgroup. We show that the space of algebraic intertwining operators from $\pi $ to the representation induced from an irreducible admissible representation of $\text{SL}\left( 2,\,\mathbb{C} \right)$ in $G$ is at most one dimensional. Spherical functions in the title are the images of $K$-finite vectors by this intertwining operator. We obtain an integral expression of Mellin-Barnes type for the radial part of our spherical function.

We give conditions which determine if cat of a map go up when extending over a cofibre. We apply this to reprove a result of Roitberg giving an example of a $\text{CW}$ complex $Z$ such that $\text{cat}(Z)\,=\,2$ but every skeleton of $Z$ is of category 1. We also find conditions when $\text{cat}(f\,\times \,g)\,<\,\text{cat}(f)\,+\,\text{cat}(g)$. We apply our result to show that under suitable conditions for rational maps $f,\,\text{mcat}(f)\,<\,\text{cat}(f)$ is equivalent to $\text{cat(}f)\,=\,\text{cat(}f\,\times \,\text{i}{{\text{d}}_{{{S}^{n}}}})$. Many examples with $\text{mcat}(f)\,<\,\text{cat}(f)$ satisfying our conditions are constructed. We also answer a question of Iwase by constructing $p$-local spaces $X$ such that $\text{cat(}X\ \times \,{{S}^{1}}\text{)}\,\text{=}\,\text{cat(}X\text{)}\,\text{=2}$. In fact for our spaces and every $Y\,\not{\simeq }\,*,\,\text{cat}(X\,\times \,Y)\,\le \,\text{cat}(Y)\,+\,1\,\text{cat}(Y)\,+\,\text{cat}(X)$. We show that this same $X$ has the property $\text{cat}(X)=\,\text{cat}(X\,\times \,X)\,=\,\text{cl}(X\,\times \,X)\,=\,2$.

We consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson’s theorem to apply to frames generated by the action of the group. Within this setup we use Stone’s theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multiresolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on ${{L}^{2}}(\mathbb{R})$.

Continued fractions associated with $\text{G}{{\text{L}}_{3}}\left( \mathbf{Z} \right)$ are introduced and applied to find fundamental units in a two-parameter family of complex cubic fields.

In this paper, we investigate projectivity in the category of operator spaces. In particular, we show that the Fourier algebra of a locally compact group $G$ is operator biprojective if and only if $G$ is discrete.

Consider the sixth Painlevé equation $({{\text{P}}_{6}})$ below where $\alpha ,\beta ,\gamma$ and $\delta$ are complex parameters. We prove the necessary and sufficient conditions for the existence of rational solutions of equation $({{\text{P}}_{6}})$ in term of special relations among the parameters. The number of distinct rational solutions in each case is exactly one or two or infinite. And each of them may be generated by means of transformation group found by Okamoto [7] and Bäcklund transformations found by Fokas and Yortsos [4]. A list of rational solutions is included in the appendix. For the sake of completeness, we collected all the corresponding results of other five Painlevé equations $({{\text{P}}_{1}})-({{\text{P}}_{5}})$ below, which have been investigated by many authors [1]–[7].

Let $X$ be a separated finite type scheme over a noetherian base ring $\mathbb{K}$. There is a complex ${{\hat{C}}^{\cdot }}\left( X \right)$ of topological ${{\mathcal{O}}_{X}}$-modules, called the complete Hochschild chain complex of $X$. To any ${{\mathcal{O}}_{X}}$-module $\mathcal{M}$—not necessarily quasi-coherent—we assign the complex $Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{{\hat{C}}}^{\cdot }}\left( X \right),\,\mathcal{M} \right)$ of continuous Hochschild cochains with values in $\mathcal{M}$. Our first main result is that when $X$ is smooth over $\mathbb{K}$ there is a functorial isomorphism

$$Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{C}^{\cdot }}\left( X \right),\,M \right)\,\cong \,\text{R}\,Hom_{{{\mathcal{O}}_{X}}^{2}}^{{}}\,\left( {{\mathcal{O}}_{X}},\,M \right)$$

in the derived category $\text{D}\left( \text{Mod}\,{{\mathcal{O}}_{{{X}^{2}}}} \right)$, where ${{X}^{2}}\,:=\,X\,{{\times }_{\mathbb{K}}}\,X$.

The second main result is that if $X$ is smooth of relative dimension $n$ and $n!$ is invertible in $\mathbb{K}$, then the standard maps $\text{ }\!\!\pi\!\!\text{ }\,\text{:}\,{{\hat{C}}^{-q}}\left( X \right)\,\to \,\Omega _{X/\mathbb{K}}^{q}$ induce a quasi-isomorphism

$$Hom_{{{\mathcal{O}}_{X}}}^{{}}\,\left( \underset{q}{\mathop \oplus }\,\,\Omega _{X/\mathbb{K}}^{q}\,\left[ q \right],\,M \right)\,\,\to \,Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{{\hat{C}}}^{\cdot }}\left( X \right),\,M \right).$$

When $M\,=\,{{\mathcal{O}}_{X}}$ this is the quasi-isomorphism underlying the Kontsevich Formality Theorem.

Combining the two results above we deduce a decomposition of the global Hochschild cohomology

$$\text{Ext}_{{{\mathcal{O}}_{{{X}^{2}}}}}^{i}\,\left( {{\mathcal{O}}_{X}}\,,\,M \right)\,\cong \,\underset{q}{\mathop \oplus }\,\,\,{{\text{H}}^{i-q}}\,\left( X,\,\left( \underset{{{\mathcal{O}}_{X}}}{\overset{q}{\mathop \Lambda }}\,\,{{T}_{X/\mathbb{K}}} \right)\,{{\otimes }_{{{\mathcal{O}}_{X}}}}\,M \right),$$

where ${{T}_{X/\mathbb{K}}}$ is the relative tangent sheaf.