14,251 research outputs found

### Efficient prime counting and the Chebyshev primes

The function \epsilon(x)=\mbox{li}(x)-\pi(x) is known to be positive up to
the (very large) Skewes' number. Besides, according to Robin's work, the
functions \epsilon_{\theta}(x)=\mbox{li}[\theta(x)]-\pi(x) and
\epsilon_{\psi}(x)=\mbox{li}[\psi(x)]-\pi(x) are positive if and only if
Riemann hypothesis (RH) holds (the first and the second Chebyshev function are
$\theta(x)=\sum_{p \le x} \log p$ and $\psi(x)=\sum_{n=1}^x \Lambda(n)$,
respectively, \mbox{li}(x) is the logarithmic integral, $\mu(n)$ and
$\Lambda(n)$ are the M\"obius and the Von Mangoldt functions). Negative jumps
in the above functions $\epsilon$, $\epsilon_{\theta}$ and $\epsilon_{\psi}$
may potentially occur only at $x+1 \in \mathcal{P}$ (the set of primes). One
denotes j_p=\mbox{li}(p)-\mbox{li}(p-1) and one investigates the jumps $j_p$,
$j_{\theta(p)}$ and $j_{\psi(p)}$. In particular, $j_p<1$, and
$j_{\theta(p)}>1$ for $p<10^{11}$. Besides, $j_{\psi(p)}<1$ for any odd p \in
\mathcal{\mbox{Ch}}, an infinite set of so-called {\it Chebyshev primes } with
partial list $\{109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313,
317, 443, 449, 461, 463, \ldots\}$. We establish a few properties of the set
\mathcal{\mbox{Ch}}, give accurate approximations of the jump $j_{\psi(p)}$
and relate the derivation of \mbox{Ch} to the explicit Mangoldt formula for
$\psi(x)$. In the context of RH, we introduce the so-called {\it Riemann
primes} as champions of the function $\psi(p_n^l)-p_n^l$ (or of the function
$\theta(p_n^l)-p_n^l$ ). Finally, we find a {\it good} prime counting function
S_N(x)=\sum_{n=1}^N \frac{\mu(n)}{n}\mbox{li}[\psi(x)^{1/n}], that is found
to be much better than the standard Riemann prime counting function.Comment: 15 pages section 2.2 added, new sequences added, Fig. 2 and 3 are ne

### Extreme values of the Dedekind $\Psi$ function

Let $\Psi(n):=n\prod_{p | n}(1+\frac{1}{p})$ denote the Dedekind $\Psi$
function. Define, for $n\ge 3,$ the ratio $R(n):=\frac{\Psi(n)}{n\log\log n}.$
We prove unconditionally that $R(n)< e^\gamma$ for $n\ge 31.$ Let $N_n=2...p_n$
be the primorial of order $n.$ We prove that the statement
$R(N_n)>\frac{e^\gamma}{\zeta(2)}$ for $n\ge 3$ is equivalent to the Riemann
Hypothesis.Comment: 5 pages, to appear in Journal of Combinatorics and Number theor

### HRM and Value Creation

Itâs conceptually attractive to look for connection between performance, HRM and economic situation. How measure epiphenomenonâs impact when we canât isolate that from global strategy? If casual relations maybe established, event can be interpreted in several ways (e.g. its chicken and egg situationĂąâŹÂŠ). This paper presents the results of a research on corporate performance measured by the creation of shareholder value. To do that we test empirically forced rankingâs performance versus all other classic human resource managementsâ result first with a statistical comparison of share based on fortune 100 (from 1996 to 2000); second with Standard & Poorâs (S&P) 500 value creation (from 1997 to 2000) with ĂąâŹĆMarakon AssociatesĂąâŹ (the growth between Market-to-book values ratio and the ROE spread (ROE â Cost of equity capital)Forced Ranking, Classic HRM, Value Creation

### Garside families and Garside germs

Garside families have recently emerged as a relevant context for extending
results involving Garside monoids and groups, which themselves extend the
classical theory of (generalized) braid groups. Here we establish various
characterizations of Garside families, that is, equivalently, various criteria
for establishing the existence of normal decompositions of a certain type

### Chebyshev's bias and generalized Riemann hypothesis

It is well known that $li(x)>\pi(x)$ (i) up to the (very large) Skewes'
number $x_1 \sim 1.40 \times 10^{316}$ \cite{Bays00}. But, according to a
Littlewood's theorem, there exist infinitely many $x$ that violate the
inequality, due to the specific distribution of non-trivial zeros $\gamma$ of
the Riemann zeta function $\zeta(s)$, encoded by the equation
$li(x)-\pi(x)\approx \frac{\sqrt{x}}{\log x}[1+2 \sum_{\gamma}\frac{\sin
(\gamma \log x)}{\gamma}]$ (1). If Riemann hypothesis (RH) holds, (i) may be
replaced by the equivalent statement $li[\psi(x)]>\pi(x)$ (ii) due to Robin
\cite{Robin84}. A statement similar to (i) was found by Chebyshev that
$\pi(x;4,3)-\pi(x;4,1)>0$ (iii) holds for any $x<26861$ \cite{Rubin94} (the
notation $\pi(x;k,l)$ means the number of primes up to $x$ and congruent to
$l\mod k$). The {\it Chebyshev's bias}(iii) is related to the generalized
Riemann hypothesis (GRH) and occurs with a logarithmic density $\approx 0.9959$
\cite{Rubin94}. In this paper, we reformulate the Chebyshev's bias for a
general modulus $q$ as the inequality $B(x;q,R)-B(x;q,N)>0$ (iv), where
$B(x;k,l)=li[\phi(k)*\psi(x;k,l)]-\phi(k)*\pi(x;k,l)$ is a counting function
introduced in Robin's paper \cite{Robin84} and $R$ resp. $N$) is a quadratic
residue modulo $q$ (resp. a non-quadratic residue). We investigate numerically
the case $q=4$ and a few prime moduli $p$. Then, we proove that (iv) is
equivalent to GRH for the modulus $q$.Comment: 9 page

### Meeting the Challenge of Interdependent Critical Networks under Threat : The Paris Initiative

NARisques Ă grande Ă©chelle;Gestion des crises internationale;InterdĂ©pendances;Infrastructures critiques;Anthrax;Initiative collective;StratĂ©gie;PrĂ©paration des Etats-majors

### Understanding and modeling the small-world phenomenon in dynamic networks

The small-world phenomenon first introduced in the context of static graphs consists of graphs with high clustering coefficient and low shortest path length. This is an intrinsic property of many real complex static networks. Recent research has shown that this structure is also observable in dynamic networks but how it emerges remains an open problem. In this paper, we propose a model capable of capturing the small-world behavior observed in various real traces. We then study information diffusion in such small-world networks. Analytical and simulation results with epidemic model show that the small-world structure increases dramatically the information spreading speed in dynamic networks

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