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(2006-01-03)  
From ants to elephants, following the father of modern physics.

For a given material, the strength of a structure depends on its variouscross-sections and is thus proportional to the square of the overall size. As weight is proportional to the cube of size, such a structurewould therefore collapse if scaled-up beyond a certain size. This subject was first discussed by Galileo Galilei (1564-1642)who put the following words in the mouth ofSalviati,on the "second day" of the Dialogues Concerning Two New Sciences  (1638).

We may reasonably expect the dynamic forces which make a creaturejump to be proportional to the cross-section of its muscles,just like static forces are expected to be proportional tothe cross-sections of its bones. At the very least, this is a much better starting point thanthe popular misguided assumption discussed in thefollowing article, which wouldhave us believe that humans built like scaled-up fleascould jump over skycrapers!

(2006-01-03)  
Are fleasreally much better jumpers than people?

Les puces peuvent sauter 135 fois leur taille. C'est comme si
un homme sautait aussi haut qu'un immeuble de65 étages.
CaramBarInfo  [inside candy wrapping]

French kiddy sensationalism notwithstanding,the performance of creatures havingvastly different sizes should not be compared usingthe linear scaling implied by the abovecomparison between men and fleas...

Takingthe above at face value, a jumping creature would be expectedto release an energy roughly proportional to its volume (limbs apply a force proportional to the square of thesize, along a launching trajectory proportional to the size). This mechanical energy is thus expected to be proportional to the creature'svolume or its mass [since all living tissues have roughly the same density]. Neglecting air resistance, this would mean that all jumping creatures areexpected to jump to about the same height, not very different heights proportional to their sizes...

Fair comparisons of the jumping performances of various animalsare best based on the ratio of the aforementioned mechanical energyto the mass of the creature (this ratio is equal to half the square of the speed reached at liftoff).

People can jump up only slightly more than a foot in height. Gifted athletes can do significantly better, butan athlete who clears a bar several feet off the ground does so partly becausehis center of gravity is already about 3 feet high to begin with,and also because the center of gravity of his bent bodymay stayunder the bar...

The human flea (pulex irritans) is commonlyquoted as being able to jump abouta foot in length, or a few inches in height. This is commensurate with human performance, as predicted. The size of the flea is essentially irrelevant...

(2012-12-07)  
A  ¾  power law.

Max Kleiber  (1893-1976)  graduated from theETH Zürich in 1920as an agricultural chemist and went on to obtain a doctorate  (1924) with a dissertation on the Energy Concept in Nutrition. He joined the Animal Husbandry Department  of UC Davis in 1929,where he constructed respiration chambers and researched the energy metabolismin animals of various sizes.

In 1932, Max Kleiber concluded experimentally that the metabolic rate of animals variesroughly as their mass raised to the power of  3/4.

Naively, one could have expected the exponent of that power law to be to  2/3  as would be the case if energy was simply producedproportionally to the mass  (or volume)  of a warm-blooded animaland lost at a rate proportional to the surface area of its outer skin.

The experimental law obtained by Kleiber implies mathematically that the circulatory and/or respiratorysystem of animals has afractal structure. This is, of course, consistent with anatomical observations. A simple justification of Kleiber's original exponent  (k=3/4)  would be obtained if energy wasproduced by the bulk of a  d-dimensional system  (d being 3 or less) and lost through a fractal object of dimension  k.d  which has to be  2 or more. That would give the inner surface of the lungs and/or the circulatory system below the skin an effective combined fractal dimension  2.25  (9/4)  which seems about right...

The body of a  14.4 ggrey mouseis approximately  60 mm  long and 20 mm  wide  (with a 70 mm tail  of 2 mm  width). Ablack rat  would be roughly3 times as big and 20 times more massive.

A male adultAfrican savanna elephanttypically weighs  5500 kg, and is thus about  400000  times asmassive as a common mouse  (or 75 times as big, take your pick). A  30 m blue whale  would weigh about180000 kg  (33 times the mass of an elephant or 3 times its size). In the video quoted in the footnotes below, Pr. Woolley (Oxford University)  doesn't seem to have any clear quantitative idea of thesizes of the animals he uses as examples...

(2007-08-03)  
On the resistive force exerted by a fluid on a sphere at constant velocity.

In 1883,Osborne Reynolds(1842-1912)  introduced a dimensionless parameteras he investigated the transition from laminar to turbulent flow for fluids in pipes.

That parameter  R  was first called "Reynolds number"by Arnold Sommerfeld as he used it in what's now known as the Orr-Sommerfeld equation  which he introduced in the paperentitled "Ein Beitrag zur hydrodynamischen Erklärung der turbulentenFlüssigkeitsbewegung" presented in Rome in 1908,at the 4thInternational Congress of Mathematicians  (3, 116-124).

The uniform motion of a sphere through afluid involvesthe following quantities:

• The mass of the sphere:  m.
• The radius of the sphere:  r.
• Thedynamic viscosityof the fluid:  .
• The density of the fluid:  .
• The speed of the sphere relative to the fluid:  v.
• The resistive force:  F.

Those form four relevant quantities: r  (in m), v  (in m/s),  F/m  (acceleration, in m/s² ) and    (kinematic viscosity,in m²/s).

As two units are involved, there must be two dimensionless parameters whichare functions of each other.  One is the drag coefficient (C)  the other is the aforementioned Reynolds number  (R). Other such pairs of parameters would be acceptable,but this is the traditional choice which we do retain.

(2016-04-14)  
The leading decimal digit is  d  with probability  log₁₀ (1+1/d).
Equivalently, the first digit is less than  d  with probability  log₁₀ (d).

That the ten digits do not occur with equal frequency must be evident
to anyonemaking much use of logarithmic tables, and noticing
how much faster the first ones wear out than the last ones.
Simon Newcomb  (1881)

Mathematically, Benford's law  is a property which may or may not apply toan infinite dataset  (a random variable with infinitely many values). The qualifier Benford  applies to datasets verifying said property, which is the case when one of the following four equivalent criteria is satisfied.

1. ("Uniform mantissae") :  The mantissae (fractional parts of the logarithms)  are uniformly distributed in the interval  [0,1[.
2. ("Scale Invariance") :  If a positive constant  u  and the base  b  aren't powers of the same integer, then the leading digits of  X  and  u X  form two identically distributed  random variables.
3. ("Leading-digits law") :  The probability that the leading radix-b digitsare the  radix-b  digits of the integer  n  is equal to  log_b (1+1/n).
4. ("First-digit law") :  In any base of numeration  b ≥ 3 the most significant digit is  d  (1≤d≤b-1) with probability  log_b (1+1/d).

The first-digit law is often presented in the decimal case as a definition of Bendford's law.  However,  this restricted criterion is not equivalent to theother three.  There are distributions which obey the first-digit law ina particular base but fail to do so in any other base.

We'll prove the fruitful equivalence of these criteria,  using a few lemma:

Lemma :  If  q  is irrational,  then the fractional parts  (np mod 1) of the sequence  0, q, 2q, 3q, 4q, ... is uniformly disributed in the interval  [0,1[. (Therefore, the sequence of the powers aⁿ  is Benford.)

Lemma :  If the random variable  Y  is uniformly distributed in the interval  [0,1[ then so is the random variable  (q+Y) mod 1  when  q  is constant.

Uniform mantissae (1) implies (2), (3) and (4) criteria:

Equivalence of the single and multiple-digits criteria :

A sequence of  q  radix-b digits   d₁ ... d_q   may be uniquelyidentified by the integer formed by their concatenation using  b  as the base of numeration.

N   =   d₁ b^q-1 +  d₂ b^q-2 +  ... +  d_q-1 b¹ +  d_q b⁰

N  must be greater than or equal to  b^q-1 for this to represent a legitimate sequence of leading digits  (d₁ must be nonzero). The  q  radix-b digits so specified are the leading digits with the following probability:

P   =   log_b( 1 + 1/N )

The probability that a given digit  d  occurs at position  k ≥ 2  is the sum:

Pb(d)   =		logb ( 1 + 1/(bp+d) )

For all practical purposes,  that's just  10%  when  k  is  3  or  4  or more: