Given an irreducible algebraic curves $A$ in ${ℂ}^N$, let $m_d$ be the dimension of the complex vector space of all holomorphic polynomials of degree at most $d$ restricted to $A$. Let $K$ be a nonpolar compact subset of $A$, and for each $d=1,2,...,$ choose $m_d$ points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$ in $K$. Finally, let ${\Lambda }_d$ be the $d$-th Lebesgue constant of the array $\left\{A_{dj}\right\}$; i.e., ${\Lambda }_d$ is the operator norm of the Lagrange interpolation operator $L_d$ acting on $C\left(K\right)$, where $L_d\left(f\right)$ is the Lagrange interpolating polynomial for $f$ of degree $d$ at the points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$. Using techniques...

Assume $f_1,...,f_N$ a finite set of functions in $H^{\infty }\left(D\right)$, the space of bounded analytic functions on the open unit disc. We give a sufficient condition on a function $f$ in $H^{\infty }\left(D\right)$ to belong to the norm-closure of the ideal $I(f_1,...,f_N)$ generated by $f_1,...,f_N$, namely the property $$\left|f\right(z\left)\right|\le \alpha \left(\right|f_1\left(z\right)|+...+|f_N\left(z\right)\left|\right)\text{for}z\in D$$ for some function $\alpha$ : ${\mathbf{R}}_{+}\to {\mathbf{R}}_{+}$ satisfying ${lim}_{t\to 0}\alpha \left(t\right)/t=0.$ The main feature in the proof is an improvement in the contour-construction appearing in L. Carleson’s solution of the corona-problem. It is also shown that the property $$\left|f\left(z\right)\right|\le C\underset{1\le j\le N}{max}\left|f_j\left(z\right)\right|\text{for}z\in D$$ ...

The tempered fundamental group of a $p$-adic analytic space classifies covers that are dominated by a topological cover (for the Berkovich topology) of a finite étale cover of the space. Here we construct cospecialization homomorphisms between $\left(p^{\prime }\right)$ versions of the tempered fundamental groups of the fibers of a smooth family of curves with semistable reduction. To do so, we will translate our problem in terms of cospecialization morphisms of fundamental groups of the log fibers of the log...

We prove: If $A\left(n\right)$ and $G\left(n\right)$ denote the arithmetic and geometric means of the first $n$ positive integers, then the sequence $n\mapsto nA\left(n\right)/G\left(n\right)-(n-1)A(n-1)/G(n-1)$ $(n\ge 2)$ is strictly increasing and converges to $e/2$, as $n$ tends to $\infty$.

Let λ denote Carmichael’s function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler’s φ-function, but we show here that the image of λ is much denser than the image of φ. In particular the number of λ-values to x exceeds $x/{\left(logx\right)}^{.36}$ for all large x, while for φ it is equal to $x/{\left(logx\right)}^{1+o\left(1\right)}$, an old result of Erdős. We also improve on an earlier result of the first author and Friedlander giving an upper bound for the distribution of λ-values.

Let We prove that $F$ is completely monotonic on $(0,\mathrm{\infty })$. This complements a result of Miller and Moskowitz (2006), who proved that $F$ is positive and strictly decreasing on $(0,\mathrm{\infty })$. The sequence $\{S(k)\}$ $(k=1,2,\mathrm{\dots })$ plays a role in information theory.

We present a method that in certain sense stores the inverse of the stiffness matrix in $O(NlogN)$ memory places, where $N$ is the number of degrees of freedom and hence the matrix size. The setup of this storage format requires $O\left(N^{3/2}\right)$ arithmetic operations. However, once the setup is done, the multiplication of the inverse matrix and a vector can be performed with $O(NlogN)$ operations. This approach applies to the first order finite element discretization of linear elliptic and parabolic problems in triangular...

Pour les fonctions $f\left(x\right)$ dont les coefficients de Fourier $c_n$ satisfont à $\Sigma |c_n{|}^2{\lambda }_n\<\infty$, la capacité est évaluée pour l’ensemble où la fonction maximale satisfait à $f^{*}\left(x\right)\ge \lambda$.