The norm on a Banach space gives rise to a subharmonic function on the complex plane for which the distributional Laplacian gives a Riesz measure. This measure is calculated explicitly here for Lebesgue ${{L}^{p}}$ spaces and the von Neumann-Schatten trace ideals. Banach spaces that are $q$-uniformly $\text{PL}$-convex in the sense of Davis, Garling and Tomczak-Jaegermann are characterized in terms of the mass distribution of this measure. This gives a new proof that the trace ideals ${{c}^{p}}$ are 2-uniformly $\text{PL}$-convex for $1\,\le \,p\,\le \,2$.

On se donne un intervalle ouvert non vide $\omega $ de $\mathbb{R}$, un ouvert connexe non vide $\Omega $ de ${{\mathbb{R}}_{5}}$ et unpolynôme unitaire

$${{P}_{m}}\left( z,\text{ }\!\!\lambda\!\!\text{ } \right)={{z}^{m}}+{{a}_{1}}\left( \text{ }\!\!\lambda\!\!\text{ } \right){{z}^{m-1}}=+\cdot \cdot \cdot +{{a}_{m-1}}\left( \text{ }\!\!\lambda\!\!\text{ } \right)z+{{a}_{m}}\left( \text{ }\!\!\lambda\!\!\text{ } \right)$$

de degré $m>0$, dépendant du paramètre $\lambda \in \Omega $. Un tel polynôme est dit $\omega $-hyperbolique si, pour tout $\lambda \in \Omega $, ses racines sont réelles et appartiennent à $\omega $.

On suppose que les fonctions ${{a}_{k}},k=1,\cdot \cdot \cdot ,m$, appartiennent à une classe ultradifférentiable ${{C}_{M}}\left( \Omega \right)$. On s‘intéresse au problème suivant. Soit $f$ appartient à ${{C}_{M}}\left( \Omega \right)$, existe-t-il des fonctions ${{Q}_{f}}$ et ${{R}_{f,k}},k=0,\cdot \cdot \cdot ,m-1$, appartenant respectivement à ${{C}_{M}}\left( \omega \,\times \,\Omega \right)$ et à ${{C}_{M}}\left( \Omega \right)$, telles que l’on ait, pour $\left( x,\,\lambda \right)\,\in \,\omega \,\times \,\Omega$,

$$f\left( x \right)={{P}_{m}}\left( x,\text{ }\!\!\lambda\!\!\text{ } \right){{Q}_{f}}\left( x,\text{ }\!\!\lambda\!\!\text{ } \right)+\sum\limits_{k=0}^{m-1}{{{x}^{k}}{{R}_{f,k}}\left( \text{ }\!\!\lambda\!\!\text{ } \right)?}$$

On donne ici une réponse positive dès que le polynôme est $\omega$-hyperbolique, que la class untradifférentiable soit quasi-analytique ou non ; on obtient alors, des exemples d’idéaux fermés dans ${{C}_{M}}\left( {{\mathbb{R}}^{n}} \right)$ . On complète ce travail par une généralisation d’un résultat de C. L. Childress dans le cadre quasi-analytique et quelques remarques.

We introduce a new device of measuring the degree of the failure of convergence in the ergodic theorem along subsequences of integers. Relations with other types of bad behavior in ergodic theory and applications to weighted averages are also discussed.

Let $X$ be a locally compact non-compact Hausdorff topological space. Consider the algebras $C\left( X \right),{{C}_{b}}\left( X \right),{{C}_{0}}\left( X \right),\,\text{and}\,{{C}_{00}}\left( X \right)$ of respectively arbitrary, bounded, vanishing at infinity, and compactly supported continuous functions on $X$. Of these, the second and third are ${{C}^{*}}$-algebras, the fourth is a normed algebra, whereas the first is only a topological algebra (it is indeed a pro-${{C}^{*}}$- algebra). The interesting fact about these algebras is that if one of them is given, the others can be obtained using functional analysis tools. For instance, given the ${{C}^{*}}$-algebra ${{C}_{0}}\left( X \right)$, one can get the other three algebras by ${{C}_{00}}\left( X \right)=K\left( {{C}_{0}}\left( X \right) \right),{{C}_{b}}\left( X \right)=M\left( {{C}_{0}}\left( X \right) \right),C\left( X \right)=\Gamma \left( K\left( {{C}_{0}}\left( X \right) \right) \right)$, where the right hand sides are the Pedersen ideal, the multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of ${{C}_{0}}\left( X \right)$, respectively. In this article we consider the possibility of these transitions for general ${{C}^{*}}$-algebras. The difficult part is to start with a pro-${{C}^{*}}$-algebra A and to construct a ${{C}^{*}}$-algebra ${{A}_{0}}$ such that $A=\Gamma \left( K\left( {{A}_{0}} \right) \right)$. The pro-${{C}^{*}}$-algebras for which this is possible are called locally compact and we have characterized them using a concept similar to that of an approximate identity.

Constants with formulae of the form treated by D. Bailey, P. Borwein, and S. Plouffe ($b$ formulae to a given base $b$) have interesting computational properties, such as allowing single digits in their base $b$ expansion to be independently computed, and there are hints that they should be normal numbers, i.e., that their base $b$ digits are randomly distributed. We study a formally limited subset of BBP formulae, which we call Machin-type BBP formulae, for which it is relatively easy to determine whether or not a given constant $K$ has a Machin-type BBP formula. In particular, given $b\,\in \,\mathbb{N},\,b\,>\,2,\,b$ not a proper power, a $b$-ary Machin-type BBP arctangent formula for $K$ is a formula of the form $k\,=\,{{\Sigma }_{m}}\,{{a}_{m}}\,\arctan \,(-{{b}^{-m}}),\,{{a}_{m}}\,\in \,\mathbb{Q}$ , while when $b\,=\,2$, we also allow terms of the form ${{a}_{m}}\,\arctan \,(1/1\,-\,{{2}^{m}}))$ . Of particular interest, we show that $\pi$ has no Machin-type BBP arctangent formula when $b\,\ne \,2$. To the best of our knowledge, when there is no Machin-type BBP formula for a constant then no BBP formula of any form is known for that constant.

Let $\overline{{{\mathbb{Q}}_{2}}}$ be an algebraic closure of ${{\mathbb{Q}}_{2}}$ and $K$ be an unramified finite extension of ${{\mathbb{Q}}_{2}}$ included in $\overline{{{\mathbb{Q}}_{2}}}$. Let $E$ be an elliptic curve defined over $K$ with additive reduction over $K$, and having an integral modular invariant. Let us denote by ${{K}_{nr}}$ the maximal unramified extension of $K$ contained in $\overline{{{\mathbb{Q}}_{2}}}$. There exists a smallest finite extension $L$ of ${{K}_{nr}}$ over which $E$ has good reduction. We determine in this paper the degree of the extension $L/{{K}_{nr}}$.

We classify normal affine surfaces with trivial Makar-Limanov invariant and finite Picard group of the smooth locus, realizing them as open subsets of weighted projective planes. We also show that such a surface admits, up to conjugacy, one or two ${{G}_{a}}$-actions.

Nous étudions la structure du centralisateur d'un élément unipotent régulier d'un sous-groupe de Levi d'un groupe réductif, ainsi que la structure du groupe des composantes de ce centralisateur en relation avec la notion de système local cuspidal définie par Lusztig. Nous déterminons son radical unipotent, montrons l'existence d'un complément de Levi et étudions la structure de son groupe de Weyl. Comme application, nous démontrons des résultats qui étaient annoncés dans un précédent article de l'auteur sur les éléments unipotents réguliers.

This paper deals with generalizations of the notion of a polytope to infinite dimensions. The most general definition is the following: a bounded closed convex subset of a Banach space is called a polytope if each of its finite-dimensional affine sections is a (standard) polytope.

We study the relationships between eight known definitions of infinite-dimensional polyhedrality. We provide a complete isometric classification of them, which gives solutions to several open problems. An almost complete isomorphic classification is given as well (only one implication remains open).

We study the range of the gradients of a ${{C}^{1,\alpha }}$ -smooth bump function defined on a Banach space. We find that this set must satisfy two geometrical conditions: It can not be too flat and it satisfies a strong compactness condition with respect to an appropriate distance. These notions are defined precisely below. With these results we illustrate the differences with the case of ${{C}^{1}}$-smooth bump functions. Finally, we give a sufficient condition on a subset of ${{X}^{*}}$ so that it is the set of the gradients of a ${{C}^{1,1}}$-smooth bump function. In particular, if $X$ is an infinite dimensional Banach space with a ${{C}^{1,1}}$-smooth bump function, then any convex open bounded subset of ${{X}^{*}}$ containing 0 is the set of the gradients of a ${{C}^{1,1}}$-smooth bump function.

In this paper, we develop techniques for solving ternary Diophantine equations of the shape $A{{x}^{n}}+B{{y}^{n}}=C{{z}^{2}}$, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters $A,\,B\,\text{and}\,\text{C}$. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan–Nagell type.

We study the $\text{K}$-homology of the rotation algebras ${{A}_{\theta }}$ using the six-term cyclic sequence for the $\text{K}$-homology of a crossed product by $Z$. In the case that $\theta $ is irrational, we use Pimsner and Voiculescu's work on $\text{AF}$-embeddings of the ${{A}_{\theta }}$ to search for the missing generator of the even $\text{K}$-homology.

We study the Hilbert functions of fat points in ${{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$. If $Z\,\subseteq \,{{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$ is an arbitrary fat point scheme, then it can be shown that for every $i$ and $j$ the values of the Hilbert function ${{H}_{Z}}(l,\,j)$ and ${{H}_{Z}}(i,\,l)$ eventually become constant for $l\,\gg \,0$. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in ${{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$. This enables us to compute all but a finite number values of ${{H}_{Z}}$ without using the coordinates of points. We also characterize the $\text{ACM}$ fat point schemes using our description of the eventual behaviour. In fact, in the case that $Z\,\subseteq \,{{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$ is $\text{ACM}$, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.

It is shown that the singular values of the operator $aP\,-\,Pa$, where $P$ is Bergman's projection over a bounded domain $\Omega $ and $a$ is a function analytic on $\bar{\Omega }$, detect the length of the boundary of $a\left( \Omega \right)$. Also we point out the relation of that operator and the spectral asymptotics of a Hankel operator with an anti-analytic symbol.

For most of the finite subgroups of $\text{SL(3,}\,\text{C)}$ we give explicit formulae for the Molien series of the coinvariant algebras, generalizing McKay's formulae $\text{ }\!\![\!\!\text{ McKay99 }\!\!]\!\!\text{ }$ for subgroups of $\text{SU(2)}$. We also study the $G $-orbit Hilbert scheme $\text{Hil}{{\text{b}}^{G}}({{\mathbf{C}}^{3}})$ for any finite subgroup $G $ of $\text{SO(3)}$, which is known to be a minimal (crepant) resolution of the orbit space ${{\mathbf{C}}^{3}}/G$ . In this case the fiber over the origin of the Hilbert-Chow morphism from $\text{Hil}{{\text{b}}^{G}}({{\mathbf{C}}^{3}})$ to ${{\mathbf{C}}^{3}}/G$ consists of finitely many smooth rational curves, whose planar dual graph is identified with a certain subgraph of the representation graph of $G $. This is an $\text{SO(3)}$ version of the McKay correspondence in the $\text{SU(2)}$ case.

Let $\sigma $, $\theta $ be commuting involutions of the connected semisimple algebraic group $G$ where $\sigma$, $\theta $ and $G$ are defined over an algebraically closed field $\underset{\scriptscriptstyle-}{k},$ char $\underline{k}$=0. Let $H:={{G}^{\sigma }}$ and $K:={{G}^{\theta }}$ be the fixed point groups. We have an action $\left( H\,\times \,K \right)\,\times \,G\,\to \,G$, where $\left( \left( h,\,k \right),\,g \right)\,\mapsto \,hg{{k}^{-1}},\,h\,\in \,H$ , $k\,\in \,K,g\,\in \,G$. Let $G\,//\,\left( H\,\times \,K \right)$ denote the categorical quotient Spec $\mathcal{O}{{(G)}^{H\times K}}$ . We determine when this quotient is smooth. Our results are a generalization of those of Steinberg [Ste75], Pittie [Pit72] and Richardson [Ric82] in the symmetric case where $\sigma \,=\,\theta $ and $H\,=K$.

In this paper, Hinkkanen's problem (1984) is completely solved, i.e., it is shown that any meromorphic function $f$ is determined by its zeros and poles and the zeros of ${{f}^{\left( j \right)}}$ for $j=1,2,3,4$.

We provide the first unconditional proof that the ring $\mathbb{Z}[\sqrt{14}]$ is a Euclidean domain. The proof is generalized to other real quadratic fields and to cyclotomic extensions of $\mathbb{Q}$. It is proved that if $K$ is a real quadratic field (modulo the existence of two special primes of $K$) or if $K$ is a cyclotomic extension of $\mathbb{Q}$ then:

The proof is a modification of the proof of a theorem of Clark and Murty giving a similar result when $K$ is a totally real extension of degree at least three. The main changes are a new Motzkin-type lemma and the addition of the large sieve to the argument. These changes allow application of a powerful theorem due to Bombieri, Friedlander and Iwaniec in order to obtain the result in the real quadratic case. The modification also allows the completion of the classification of cyclotomic extensions in terms of the Euclidean property.

We consider the $s$-energy $E({{Z}_{n}};\,s)={{\Sigma }_{i\ne j}}K(\parallel {{z}_{i,n}}\,-\,{{z}_{j,n}}\parallel \,;\,s)$ for point sets $Zn\,=\,\{{{z}_{k,n}}\,:\,k\,=\,0,\,\ldots \,,\,n\} $ on certain compact sets $\Gamma $ in ${{\mathbb{R}}^{d}}$ having finite one-dimensional Hausdorff measure,where

$$K(t;\,s)\,=\,\left\{ _{-\ln \,t,\,\,\,\text{if}\,s\,=\,0,\,}^{{{t}^{-s}},\,\,\,\,\,\,\,\text{if}\,s\,>\,0,} \right\}$$

is the Riesz kernel. Asymptotics for the minimum $s$-energy and the distribution of minimizing sequences of points is studied. In particular, we prove that, for $s\,\ge \,1$, the minimizing nodes for a rectifiable Jordan curve Γ distribute asymptotically uniformly with respect to arclength as $n\,\to \,\infty $.

We study the connectedness of the moduli space of gauge equivalence classes of flat $G$-connections on a compact orientable surface or a compact nonorientable surface for a class of compact connected Lie groups. This class includes all the compact, connected, simply connected Lie groups, and some non-semisimple classical groups.