Eikonal Blog

2010.02.09

A Knichin’s theorem on limit of geometric means for continued fraction form of (the most) real numbers

Filed under: mathematics — Tags: , — sandokan65 @ 15:57

Theorem: (Aleksandr Yakovlevich Khinchin; 1894-1959)
Let $\{a_n\in{\Bbb R}|n\in{\Bbb N}\}$ be a series of numbers. Define the impromptu geometric means $GM_n\{a\} :\equiv \left(a_1\cdot a_2 \cdots a_n\right)^{\frac1n}$. Then

$\lim_{n\rightarrow \infty} GM_n\{a\} = \prod_{k=1}^\infty \left(1+ \frac1{k(k+2)} \right)^{\frac{\ln(k)}{\ln(2)}} = 2.685,452,001,\cdots$

for almost all series $\{a\}$.

Since $\forall x\in{\Bbb R}$ one can write $x = [a_0; a_1, a_2, \cdots]$ (expansion of $x$ in the continued-fraction form), the above theorem is valid for almost all real $x$, except for:

• algebraic numbers with definite patterns for denominators,
• some transcendental numbers (e.g. $e$).

Source:

• Calvin C. Clawson “Mathematical Sorcery” (Revealing the secrets of numbers); ISBN 0-7382-0496-X; Perseus Publishing

Related:

Products

• $\prod_{n=1}^\infty \frac{(2n)^2}{(2n-1)(2n+1)} = \frac\pi2$, (John Wallis; 1616-1703)
• $\prod_{n=1}^\infty \frac{(4n)^2}{(4n-1)(4n+1)} = \frac{\pi\sqrt{2}}4$,
• $\prod_{n=1}^\infty \frac{(4n+2)^2}{(4n+1)(4n+31)} = \sqrt{2}$,
• Question: what is the value of $P_m:\equiv \prod_{n=1}^\infty \frac{(mn)^2}{(mn-1)(mn+1)}$? Clearly $P_2= \frac\pi2$ and $P_4= \frac{\pi\sqrt{2}}4$.
• $\prod_{n=1}^\infty \left(1+\frac{(-)^{n+1}}{2n-1}\right) = \sqrt{2}$,
• $\prod_{n=1}^\infty \left(1-\frac1{n^2}\right) = \frac12$,
• $\prod_{n=1}^\infty \left(1+\frac1{n^2}\right) = \frac{\sinh(\pi)}\pi = 3.0\cdots$,
• $x \prod_{n=1}^\infty \left(1- \frac{x^2}{n^2\pi^2}\right) = \sin(x)$,
• $x \prod_{n=1}^\infty \left(1+ \frac{x^2}{n^2\pi^2}\right) = \sinh(x)$,
• $\prod_{n=1}^\infty \left(1- \frac{4x^2}{(2n-1)^2\pi^2}\right) = \cos(x)$,
• $\prod_{n=1}^\infty \left(1+ \frac{4x^2}{(2n-1)^2\pi^2}\right) = \cosh(x)$,
• $\prod_{n=3}^\infty \left(1-\frac4{n^2}\right) = \frac16$,
• $\prod_{n=1}^\infty \left(1+\frac1{n^3}\right) = \frac{\cosh(\\pi\sqrt{3}/2)}\pi = 2.428,\cdots$,
• $\prod_{p\in{\Bbb P}} \left(1-\frac1{p^s}\right) = \frac1{\zeta(s)}$ (Leonard Euler), where ${\Bbb P}$ is the set of prime numbers, and $\zeta(s):\equiv\sum_{n=1}^\infty \frac1{n^s}$ is the Riemann’s zeta function.
• $\prod_{p\in{\Bbb P}} \left(1-\frac1{p^2}\right) = \frac6{\pi^2}$.

Strange forms:

• $\frac{\sqrt{2}}2 \frac{\sqrt{2+\sqrt{2}}}2 \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}2 \cdots = \frac{2}{\pi}$, (Francois Viete; 1540-1603).
• $\sqrt{x \cdot \sqrt{x\cdot \sqrt{x \cdots}}} = x$,
• $\sqrt[n+1]{x \cdot \sqrt[n+1]{x \cdots}} = \sqrt[n]{x}$,
• $\prod_{n=1}^\infty$ $\begin{pmatrix} \frac1n & 1 \\ 0 & 1\end{pmatrix} = \begin{pmatrix} 0 & e \\ 0 & 1\end{pmatrix}$,
• $\prod_{n=1}^\infty$ $\begin{pmatrix} 1 & \frac1{n!} \\ 0 & 1\end{pmatrix} = \begin{pmatrix} 1 & e \\ 0 & 1\end{pmatrix}$, (which is quite trivial).

Sources:

• Calvin C. Clawson “Mathematical Sorcery” (Revealing the secrets of numbers); ISBN 0-7382-0496-X; Perseus Publishing

Religion is derived from moral instinct

Filed under: religion — Tags: , — sandokan65 @ 11:21

Source: “Morality research sheds light on the origins of religion” (PhysOrg.com; February 8, 2010): http://www.physorg.com/news184857515.html

This article summarizes the paper published in “Cell Press in the journal Trends in Cognitive Sciences” (Feb 2010) that discusses the current studies in relation of religiosity and morality. These paragraphs summarize the paper’s main conclusion:

Citing several studies in moral psychology, the authors highlight the finding that despite differences in, or even an absence of, religious backgrounds, individuals show no difference in moral judgments for unfamiliar moral dilemmas. The research suggests that intuitive judgments of right and wrong seem to operate independently of explicit religious commitments.

“This supports the theory that religion did not originally emerge as a biological adaptation for cooperation, but evolved as a separate by-product of pre-existing cognitive functions that evolved from non-religious functions,” says Dr. Pyysiainen. “However, although it appears as if cooperation is made possible by mental mechanisms that are not specific to religion, religion can play a role in facilitating and stabilizing cooperation between groups.”

Perhaps this may help to explain the complex association between morality and religion. “It seems that in many cultures religious concepts and beliefs have become the standard way of conceptualizing moral intuitions. Although, as we discuss in our paper, this link is not a necessary one, many people have become so accustomed to using it, that criticism targeted at religion is experienced as a fundamental threat to our moral existence,” concludes Dr. Hauser.