We introduce the concept of logarithmic dimension of a compact set. In terms of this magnitude, the extension property and the diametral dimension of spaces $\varepsilon \left( K \right)$ can be described for Cantor-type compact sets.

We describe the set of numbers ${{\sigma }_{k}}\left( {{z}_{1}},\cdot \cdot \cdot ,{{z}_{n+1}} \right)$, where ${{z}_{1}},\cdot \cdot \cdot ,{{z}_{n+1}}$ are complex numbers of modulus 1 for which ${{z}_{1}}{{z}_{2}}\cdot \cdot \cdot {{z}_{n+1}}=1$, and ${{\sigma }_{k}}$ denotes the $k$-th elementary symmetric polynomial. Consequently, we give sharp constraints on the coefficients of a complex polynomial all of whose roots are of the same modulus. Another application is the calculation of the spectrum of certain adjacency operators arising naturally on a building of type ${{\overset{\sim }{\mathop{\text{A}}}\,}_{n}}$.

Soit $G$ un groupe réductif connexe défini sur un corps $p$-adique $F$ et $\mathfrak{g}$ son algèbre de Lie. Les intégrales orbitales pondérées sur $\mathfrak{g}\left( F \right)$ sont des distributions ${{J}_{M}}\left( X,\,f \right)-f$ est une fonction test—indexées par les sous-groupes de Lévi $M$ de $G$ et les éléments semi-simples réguliers $X\,\in \,\mathfrak{m}\left( F \right)\,\cap \,{{\mathfrak{g}}_{\text{reg}}}$. Leurs analogues sur $G$ sont les principales composantes du côté géométrique des formules des traces locale et globale d’Arthur.

Si $M=G$, on retrouve les intégrales orbitales invariantes qui, vues comme fonction de $X$, sont borńees sur $\mathfrak{m}\left( F \right)\,\cap \,{{\mathfrak{g}}_{\text{reg}}}$ : c’est un résultat bien connu de Harish-Chandra. Si $M\subsetneq G$, les intégrales orbitales pondérées explosent au voisinage des éléments singuliers. Nous construisons dans cet article de nouvelles intégrales orbitales pondérées $J_{M}^{b}\left( X,f \right)$, égales à ${{J}_{M}}\left( X,f \right)$ à un terme correctif près, qui tout en conservant les principales propriétés des précédentes (comportement par conjugaison, développement en germes, etc.) restent borńees quand $X$ parcourt $\mathfrak{m}\left( F \right)\,\cap \,{{\mathfrak{g}}_{\text{reg}}}$. Nous montrons également que les intégrales orbitales pondérées globales, associées à des éléments semi-simples réguliers, se décomposent en produits de ces nouvelles intégrales locales.

We study convergence in weighted convolution algebras ${{L}^{1}}\left( \omega \right)$ on ${{R}^{+}}$, with the weights chosen such that the corresponding weighted space $M\left( \omega \right)$ of measures is also a Banach algebra and is the dual space of a natural related space of continuous functions. We determine convergence factor $\eta $ for which weak$*$-convergence of $\left\{ {{\text{ }\!\!\lambda\!\!\text{ }}_{n}} \right\}$ to $\text{ }\!\!\lambda\!\!\text{ }$ in $M\left( \omega \right)$ implies norm convergence of ${{\text{ }\!\!\lambda\!\!\text{ }}_{n}}*f$ to $\text{ }\!\!\lambda\!\!\text{ *}f$ in ${{L}^{1}}\left( \omega \eta \right)$. We find necessary and sufficent conditions which depend on $\omega$ and $f$ and also find necessary and sufficent conditions for $\eta$ to be a convergence factor for all ${{L}^{1}}\left( \omega \right)$ and all $f$ in ${{L}^{1}}\left( \omega \right)$. We also give some applications to the structure of weighted convolution algebras. As a preliminary result we observe that $\eta$ is a convergence factor for $\omega$ and $f$ if and only if convolution by $f$ is a compact operator from $M\left( \omega \right)$ (or ${{L}^{1}}\left( \omega \right)$) to ${{L}^{1}}\left( \omega \eta \right)$.

Let $B$ be the unit ball of ${{\mathbb{C}}^{n}}$ with respect to an arbitrary norm. We prove that the analog of the Carathéodory set, i.e. the set of normalized holomorphic mappings from $B$ into ${{\mathbb{C}}^{n}}$ of “positive real part”, is compact. This leads to improvements in the existence theorems for the Loewner differential equation in several complex variables. We investigate a subset of the normalized biholomorphic mappings of $B$ which arises in the study of the Loewner equation, namely the set ${{S}^{0}}\left( B \right)$ of mappings which have parametric representation. For the case of the unit polydisc these mappings were studied by Poreda, and on the Euclidean unit ball they were studied by Kohr. As in Kohr’s work, we consider subsets of ${{S}^{0}}\left( B \right)$ obtained by placing restrictions on the mapping from the Carathéodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of ${{S}^{0}}\left( B \right)$ as well as coefficient estimates, and consider various examples. Also we shall see that in higher dimensions there exist mappings in $S(B)$ which can be imbedded in Loewner chains, but which do not have parametric representation.

We study the cohomology of connected components of Shimura varieties ${{S}_{Kp}}$ coming from the group $\text{GS}{{\text{p}}_{2g}}$, by an approach modeled on the stabilization of the twisted trace formula, due to Kottwitz and Shelstad. More precisely, for each character $\bar{\omega }$ on the group of connected components of ${{S}_{Kp}}$ we define an operator $L(\omega )$ on the cohomology groups with compact supports $H_{c}^{i}\left( {{S}_{Kp,}}{{\overset{-}{\mathop{\mathbb{Q}}}\,}_{\ell }} \right)$, and then we prove that the virtual trace of the composition of $L(\omega )$ with a Hecke operator $f$ away from $p$ and a sufficiently high power of a geometric Frobenius $\Phi _{p}^{r}$, can be expressed as a sum of $\omega$-weighted (twisted) orbital integrals (where $\omega$-weighted means that the orbital integrals and twisted orbital integrals occuring here each have a weighting factor coming from the character $\bar{\omega }$). As the crucial step, we define and study a new invariant ${{\alpha }_{1}}\left( {{\gamma }_{0}};\gamma ,\delta \right)$ which is a refinement of the invariant $\alpha \left( {{\gamma }_{0}},\,\gamma ,\,\delta \right)$ defined by Kottwitz. This is done by using a theorem of Reimann and Zink.

In this paper, we present a smooth framework for some aspects of the “geometry of CW complexes”, in the sense of Buoncristiano, Rourke and Sanderson. We then apply these ideas to Morse theory, in order to generalize results of Franks and Iriye-Kono.

More precisely, consider a Morse function $f$ on a closed manifold $M$. We investigate the relations between the attaching maps in a $\text{CW}$ complex determined by $f$, and the moduli spaces of gradient flow lines of $f$, with respect to some Riemannian metric on $M$.

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 $O\left( {{X}^{11/36+\varepsilon }} \right)$.