A note on a formula for the fractional powers of infinitesimal generators of semigroups

Celso Martinez; Miguel Sanz

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Displaying similar documents to “A note on a formula for the fractional powers of infinitesimal generators of semigroups”

On the fractional parts of $x / n$ and related sequences. II

Bahman Saffari, R. C. Vaughan (1977)

Annales de l'institut Fourier

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As promised in the first paper of this series (Ann. Inst. Fourier, 26-4 (1976), 115-131), these two articles deal with the asymptotic distribution of the fractional parts of $x h (x)$ where $h$ is an arithmetical function (namely $h (n) = 1 / n$ , $h (n) = log n$ , $h (n) = 1 / log n$ ) and $n$ is an integer (or a prime order) running over the interval $[y (x), x)]$ . The results obtained are rather sharp, although one can improve on some of them at the cost of increased technicality. Number-theoretic applications will be given later on.

On the $L^{1}$ norm of exponential sums

S. K. Pichorides (1980)

Annales de l'institut Fourier

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The $L^{1}$ norm of a trigonometric polynomial with $N$ non zero coefficients of absolute value not less than 1 exceeds a fixed positive multiple of $C (\log N) / (\log \log N)^{2} .$

Linear forms in the logarithms of three positive rational numbers

Curtis D. Bennett, Josef Blass, A. M. W. Glass, David B. Meronk, Ray P. Steiner (1997)

Journal de théorie des nombres de Bordeaux

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In this paper we prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear independence conditions on the coefficients of the linear forms. Let $Λ = b_{2} log α_{2} - b_{1} log α_{1} - b_{3} log α_{3} \neq 0$ with $b_{1}, b_{2}, b_{3}$ positive integers and $α_{1}, α_{2}, α_{3}$ positive multiplicatively independent rational numbers greater than $1$ . Let $α_{j 1} = α_{j 1} / α_{j 2}$ with $α_{j 1}, α_{j 2}$ coprime positive integers $(j = 1, 2, 3)$ . Let $α_{j} \geq max {α_{j 1}, e}$ and assume that gcd $(b_{1}, b_{2}, b_{3}) = 1 .$ Let $b^{'} = (\frac{b_{2}}{log α_{1}} + \frac{b_{1}}{log α_{2}}) (\frac{b_{2}}{log α_{3}} + \frac{b_{3}}{log α_{2}})$ and assume that $B \geq max {10, log b^{'}} .$ We prove that either ${b_{1}, b_{2}, b_{3}}$ is $(c_{4}, B)$ -linearly dependent over $ℤ$ (with respect to $(a_{1}, a_{2}, a_{3})$ )...

On the distribution of $p^{α}$ modulo one

Xiaodong Cao, Wenguang Zhai (1999)

Journal de théorie des nombres de Bordeaux

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In this paper, we give a new upper-bound for the discrepancy $D (N) : = sup_{0 \leq γ \leq 0} | \sum_{\binom{p \leq / N}{\{p^{α}\} \leq γ}} 1 - π (N) γ |$ for the sequence $(p^{α})$ , when $5 / 3 \leq α > 3$ and $α \neq 2$ .

Linear independence of linear forms in polylogarithms

Raffaele Marcovecchio (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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For $x \in ℂ$ , $| x | < 1$ , $s \in ℕ$ , let ${Li}_{s} (x)$ be the $s$ -th polylogarithm of $x$ . We prove that for any non-zero algebraic number $α$ such that $| α | < 1$ , the $ℚ (α)$ -vector space spanned by $1, {Li}_{1} (α), {Li}_{2} (α), \dots$ has infinite dimension. This result extends a previous one by Rivoal for rational $α$ . The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Padé approximation problem.

Regularity properties of solutions of elliptic equations in $R^{2}$ in limit cases

Angela Alberico, Vincenzo Ferone (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this paper the Dirichlet problem for a linear elliptic equation in an open, bounded subset of $R^{2}$ is studied. Regularity properties of the solutions are proved, when the data are $L^{1}$ -functions or Radon measures. In particular sharp assumptions which guarantee the continuity of solutions are given.

On the almost Goldbach problem of Linnik

Jianya Liu, Ming-Chit Liu, Tianze Wang (1999)

Journal de théorie des nombres de Bordeaux

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Under the Generalized Riemann Hypothesis, it is proved that for any $k \geq 200$ there is $N_{k} > 0$ depending on $k$ only such that every even integer $\geq N_{k}$ is a sum of two odd primes and $k$ powers of $2$ .

The complex sum of digits function and primes

Jörg M. Thuswaldner (2000)

Journal de théorie des nombres de Bordeaux

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Canonical number systems in the ring of gaussian integers $ℤ [i]$ are the natural generalization of ordinary $q$ -adic number systems to $ℤ [i]$ . It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number $b$ . In this paper we investigate the sum of digits function $ν_{b}$ of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the $f$ -th power of a prime. Furthermore, we establish an Erdös-Kac type...

Plurisubharmonic functions with logarithmic singularities

E. Bedford, B. A. Taylor (1988)

Annales de l'institut Fourier

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To a plurisubharmonic function $u$ on $C^{n}$ with logarithmic growth at infinity, we may associate the Robin function $ρ_{u} (z) = \underset{λ \to \infty}{lim sup} u (λ z) - log (λ z)$ defined on $P^{n - 1}$ , the hyperplane at infinity. We study the classes $L_{+}$ , and (respectively) $L_{p}$ of plurisubharmonic functions which have the form $u = log (1 + | z |) + O (1)$ and (respectively) for which the function $ρ_{u}$ is not identically $- \infty$ . We obtain an integral formula which connects the Monge-Ampère measure on the space $C^{n}$ with the Robin function on $P^{n - 1}$ . As an application we obtain a criterion...