We present a sufficient and necessary condition for a function module space $X$ to have the approximate hyperplane series property (AHSP). As a consequence, we have that the space ${\mathcal{C}}_{0}(L,E)$ of bounded and continuous $E$-valued mappings defined on the locally compact Hausdorff space $L$ has AHSP if and only if $E$ has AHSP.

We consider the algebra of holomorphic functions on L∞ that are symmetric, i.e. that are invariant under composition of the variable with any measure-preserving bijection of [0, 1]. Its spectrum is identified with the collection of scalar sequences such that is bounded and turns to be separable. All this follows from our main result that the subalgebra of symmetric polynomials on L∞ has a natural algebraic basis.

Let ${{C}^{M}}$ denote a Denjoy–Carleman class of ${{C}^{\infty }}$ functions (for a given logarithmically-convex sequence $M\,=\,\left( {{M}_{n}} \right))$. We construct: (1) a function in ${{C}^{M}}\left( \left( -1,\,1 \right) \right)$ that is nowhere in any smaller class; (2) a function on $\mathbb{R}$ that is formally ${{C}^{M}}$ at every point, but not in ${{C}^{M}}\left( \mathbb{R} \right)$; (3) (under the assumption of quasianalyticity) a smooth function on ${{\mathbb{R}}^{p}}\,\left( p\,\ge \,2 \right)$ that is ${{C}^{M}}$ on every ${{C}^{M}}$ curve, but not in ${{C}^{M}}\left( {{\mathbb{R}}^{p}} \right)$.

The precise condition on a completely regular space $X$ for every character on $C\left( X \right)$ to be an evaluation at some point in $X$ is that $X$ be realcompact. Usually, this classical result is obtained by relying heavily on involved (and even nonconstructive) extension arguments. This note provides a direct proof that is accessible to a large audience.

We describe how to compute the ideal class group and the unit group of an order in a number field in subexponential time. Our method relies on the generalized Riemann hypothesis and other usual heuristics concerning the smoothness of ideals. It applies to arbitrary classes of number fields, including those for which the degree goes to infinity.

For a Lie group $G$, we show that the map $C_{c}^{\infty }\,\left( G \right)\,\times \,C_{c}^{\infty }\,\left( G \right)\,\to \,C_{c}^{\infty }\,\left( G \right),\,\left( \gamma ,\,\eta \right)\mapsto \,\gamma \,*\,\eta $ taking a pair of test functions to their convolution, is continuous if and only if $G$ is $\sigma -$compact. More generally, consider $r,\,s,\,t\,\in {{\mathbb{N}}_{0}}\,\cup \,\left\{ \infty \right\}$ with $t\,\le \,r\,+\,s$, locally convex spaces ${{E}_{1}}\,,\,{{E}_{2}}$ and a continuous bilinear map $b:\,{{E}_{1}}\,\times \,{{E}_{2}}\,\to \,F$ to a complete locally convex space $F$. Let $\beta :\,C_{c}^{r}\,\left( G,\,{{E}_{1}} \right)\,\times \,C_{c}^{S}\,\left( G,\,{{E}_{2}} \right)\,\to$$C_{c}^{t}\,\left( G,\,F \right),\,\left( \gamma ,\,\eta \right)\,\mapsto \,\gamma \,*\,b\,\eta$ be the associated convolution map. The main result is a characterization of those $\left( G,\,r,s,t,b \right)$ for which $\beta$ is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed as well as convolution of compactly supported ${{L}^{1}}$-functions and convolution of compactly supported Radon measures.

We show that for a normal locally-$\mathscr{P}$ space $X$ (where $\mathscr{P}$ is a topological property subject to some mild requirements) the subset ${C}_{\mathscr{P}} (X)$ of ${C}_{b} (X)$ consisting of those elements whose support has a neighborhood with $\mathscr{P}$, is a subalgebra of ${C}_{b} (X)$ isometrically isomorphic to ${C}_{c} (Y)$ for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$ is explicitly constructed as a subspace of the Stone–Čech compactification $\beta X$ of $X$ and contains $X$ as a dense subspace. Under certain conditions, ${C}_{\mathscr{P}} (X)$ coincides with the set of those elements of ${C}_{b} (X)$ whose support has $\mathscr{P}$, it moreover becomes a Banach algebra, and simultaneously, $Y$ satisfies ${C}_{c} (Y)= {C}_{0} (Y)$. This includes the cases when $\mathscr{P}$ is the Lindelöf property and $X$ is either a locally compact paracompact space or a locally-$\mathscr{P}$ metrizable space. In either of the latter cases, if $X$ is non-$\mathscr{P}$, then $Y$ is nonnormal and ${C}_{\mathscr{P}} (X)$ fits properly between ${C}_{0} (X)$ and ${C}_{b} (X)$; even more, we can fit a chain of ideals of certain length between ${C}_{0} (X)$ and ${C}_{b} (X)$. The known construction of $Y$ enables us to derive a few further properties of either ${C}_{\mathscr{P}} (X)$ or $Y$. Specifically, when $\mathscr{P}$ is the Lindelöf property and $X$ is a locally-$\mathscr{P}$ metrizable space, we show that

$$\begin{eqnarray*}\dim C_{\mathscr{P}}(X)= \ell \mathop{(X)}\nolimits ^{{\aleph }_{0} } ,\end{eqnarray*}$$

$\ell (X)$

$X$

$\mathscr{P}$

$X$

$$\begin{eqnarray*}Y= {\mathrm{int} }_{\beta X} \upsilon X\end{eqnarray*}$$

$\upsilon X$

$X$

In the spectrum of the algebra of symmetric analytic functions of bounded type on ℓp, 1 ≤ p < +∞, and along the same lines as the general non-symmetric case, we define and study a convolution operation and give a formula for the ‘radius’ function. It is also proved that the algebra of analytic functions of bounded type on ℓ1 is isometrically isomorphic to an algebra of symmetric analytic functions on a polydisc of ℓ1. We also consider the existence of algebraic projections between algebras of symmetric polynomials and the corresponding subspace of subsymmetric polynomials.

This paper provides a survey of both old and new results about minimal surfaces and submanifolds.

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 give a positive answer to a question of Horst Tietz. A theorem of his that is related to the Mittag-Leffler theorem looks like a duality restult under some locally convex topology on the space of meromorphic functions. Tietz has posed the problem of finding such a topology. It is shown that a topology introduced by Holdguün in 1973 solves the problem. The main tool in the study of this topology is a projective description of it that is derived here. We also argue that Holdgrün's topology is the natural locally convex topology on the space of meromorphic functions.

In this paper we study the approximation of vector valued continuous functions defined on a topological space and we apply this study to different problems. Thus we give a new proof of Machado's Theorem. Also we get a short proof of a Theorem of Katětov and we prove a generalization of Tietze's Extension Theorem for vector-valued continuous functions, thereby solving a question left open by Blair.

Let A be an algebra of continuous real functions on a topological space X. We study when every nonzero algebra homomorphism φ: A → R is given by evaluation at some point of X. In the case that A is the algebra of rational functions (or real-analytic functions, or Cm-functions) on a Banach space, we provide a positive answer for a wide class of spaces, including separable spaces and super-reflexive spaces (with nonmeasurable cardinal).