Average convergence rate of the first return time

Geon Choe; Dong Kim

Displaying similar documents to “Average convergence rate of the first return time”

Relative entropy and waiting times for continuous-time Markov processes.

Chazottes, Jean-René, Giardina, Cristian, Redig, Frank (2006)

Electronic Journal of Probability [electronic only]

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An arithmetic function arising from Carmichael’s conjecture

Florian Luca, Paul Pollack (2011)

Journal de Théorie des Nombres de Bordeaux

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Let $φ$ denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every $n$ , the equation $φ (n) = φ (m)$ has a solution $m \neq n$ . This suggests defining $F (n)$ as the number of solutions $m$ to the equation $φ (n) = φ (m)$ . (So Carmichael’s conjecture asserts that $F (n) \geq 2$ always.) Results on $F$ are scattered throughout the literature. For example, Sierpiński conjectured, and Ford proved, that the range of $F$ contains every natural number $k \geq 2$ . Also, the maximal order of $F$ has been investigated by Erdős and Pomerance....

On an arithmetic function considered by Pillai

Florian Luca, Ravindranathan Thangadurai (2009)

Journal de Théorie des Nombres de Bordeaux

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For every positive integer $n$ let $p (n)$ be the largest prime number $p \leq n$ . Given a positive integer $n = n_{1}$ , we study the positive integer $r = R (n)$ such that if we define recursively $n_{i + 1} = n_{i} - p (n_{i})$ for $i \geq 1$ , then $n_{r}$ is a prime or $1$ . We obtain upper bounds for $R (n)$ as well as an estimate for the set of $n$ whose $R (n)$ takes on a fixed value $k$ .

A constant in pluripotential theory

Zbigniew Błocki (1992)

Annales Polonici Mathematici

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We compute the constant sup $(1 / d e g P) (m a x_{S} l o g | P | - \int_{S} l o g | P | d σ)$ : P a polynomial in $ℂ^{n}$ , where S denotes the euclidean unit sphere in $ℂ^{n}$ and σ its unitary surface measure.

A sharp form of an embedding into multiple exponential spaces

Robert Černý, Silvie Mašková (2010)

Czechoslovak Mathematical Journal

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Let $Ω$ be a bounded open set in $ℝ^{n}$ , $n \geq 2$ . In a well-known paper , 20, 1077–1092 (1971) Moser found the smallest value of $K$ such that $sup \{\int_{Ω} exp ({(\frac{|f (x)|}{K})}^{n / (n - 1)}) : f \in W_{0}^{1, n} (Ω), {∥ \nabla f ∥}_{L^{n}} \leq 1\} < \infty .$ We extend this result to the situation in which the underlying space $L^{n}$ is replaced by the generalized Zygmund space $L^{n} {log}^{n - 1} L {log}^{α} log L$ $(α < n - 1)$ , the corresponding space of exponential growth then being given by a Young function which behaves like $exp (exp (t^{n / (n - 1 - α)}))$ for large $t$ . We also discuss the case of an embedding into triple and other multiple exponential cases.

Explicit lower bounds for linear forms in two logarithms

Nicolas Gouillon (2006)

Journal de Théorie des Nombres de Bordeaux

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We give an explicit lower bound for linear forms in two logarithms. For this we specialize the so-called Schneider method with multiplicity described in []. We substantially improve the numerical constants involved in existing statements for linear forms in two logarithms, obtained from Baker’s method or Schneider’s method with multiplicity. Our constant is around $5 . 10^{4}$ instead of $10^{8}$ .