## 2009年11月26日 星期四

### Log Is Everywhere

1) $1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\Theta(\ln n)$

2) 一種不穩定的物質進行radioactive decay。若它在k秒內質量由1變成$\frac{1}{d}$，則該物質的half-year為$\frac{k}{\log_2 d}$

3) 絕大部分在現實使用將數字排序（sorting）的算法（algorithm），其運算時間為$\Theta(n\log n)$

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#### 6 則留言:

Pop 提到...

pH = -log [H+]

Pop 提到...

dB = 10*log(I/Io)

something 提到...

tobywhcheng 提到...

Hyperbolic Geometry:
metric of the half plane

Applied maths:
Logarithmic spiral

Thermodynamics or information theory:
Entropy and Stirling's approximation

Optics:
Optical thickness and skin depth

Astronomy:
Magnitude of stars
(and hence in general,
Stevens' power law)
Palermo Technical Impact Hazard Scale, or Torino Scale

Cartography:
Mercator projection

Accounting or Auditing:
Benford's law

Music:
frequency of pitches

basically whenever you have an exponential, you are associated with a log

P.S.
There's a matrix inversion method described in Numerical Recipe. The computational cost of which scales as N^{log_2 7}, compare to LU in N^3

Pop 提到...

play game, u need to read the rule first :)

Prime Number Theorem
number of primes < x ~ x/ln(x)

Fractals
In calculating the fractal dimension by "box dimension"

Thermodynamics/Statistical Mechanics
Boltzmann entropy, von Neumann entropy
(application:Clausius-Clapeyron equation, Chemical Reaction Rate, ...)