Statistics detects cheat

Statistics & Truth [by CR Rao]:

RedCross hires local food contractors to supply rice, salt, water to N unknown survived victims of a recent Earthquake.

The bill shows they supply quantities R, S, W, resp, per day.
Assume average daily consumption per person be r, s, w resp.

To verify any cheating of overbill by contractors:

R/r = S/s = W/w = N

GM ≦ AM Trick

"General Manager smaller than Assistant Manager"

Put 27 cubes (Length =a, Width=b, Height =c), inside a big Cube of side (a+b+c), can't fill the big Cube completely.
$latex 27abc leq (a+b+c)^{3} $
$latex sqrt [3] {abc} leq frac {a+b+c}{3}$
Generalized:
GM ≦AM
$latex sqrt [n] {x_1 x_2 x_3 dots x_n} leq frac {x_1+x_2+x_3...+xn}{n}$

GM = Geometric Mean
AM = Arithmetic Mean

$latex mbox {Application}$
$latex mbox {Prove:} : 1003^{2005} > 2005!$
Solution: use 'Promote' technique to general case:
$latex sqrt[n] {1.23...n} < frac {1+2...+n}{n}$
Let n=2005
$latex sqrt[2005] {1.23...2005} < frac {1+2...+2005}{2005}$
$latex sqrt[2005] {2005!} < frac {(2005)(2006)/2}{2005}$
$latex sqrt[2005] {2005!} <1003$
$latex 1003^{2005} > 2005! $ [QED]

Cute Geometry Proof

Prove:  Any line L will cut a circle at most 2 points:

Let circle C (x,y) be unit circle defined by

C(x,y) : x² + y² = 1

 

Factorize C(x,y) : (x+iy) (x-iy) = 1 in the complex plane.
So C  = {L1} U {L2}
where L1 and L2 are two lines

L1= x+iy

L2= x - iy

L1 and L2 intersect at origin (0,0):

x+ iy = x-iy

We know that any line L will cut L1 at most 1 point, and L2 at most 1 point

Therefore,

L cuts the circle C at most (1+1=) 2 points. [QED]

Plato Solids

Why only 5 Plato solids ?

Plato Solid is: Regular Polyhedron 正多面体

• Each Face is n-sided polygon


• Each Vertex is common to m-edges (m ≥ 3)

Only 5 solids possible:
Tetrahedron (n,m)=(3,3) 正四面体
Hexahedron (or Cube) (n,m)=(4,3) 正六面体
Octahedron  (n,m)=(3,4)正八面体
Dodecahedron  (n,m)=(5,3)正十二面体
Icosahedron  (n,m)=(3,5)正二十面体

Proof:
Since each Edge (E) is common to 2 Faces (F)
=> n Faces counts double the edges
nF = 2E ...(1)

Since each Vertex has m Edges, each Edge has 2 end-points (Vertex).
=> m Vertex counts double the edges
mV = 2E ...(2)

(1) : E= n/2 F
(2): V= 2/m. E = n/m. F
(1) & (2) into Euler Formula: V -E + F = 2
(n/m. F) - (n/2.F) + F = 2
F.(2m + 2n - mn) = 4m

Since F>0 , m>0
=> (2m + 2n - mn) >0
=> - (mn -2n -2m) > 0
=> (mn -2n -2m) < 0
=> (mn -2m -2n) + 4 < 4
=> (m- 2).(n -2 ) < 4

(m,n) only 5 possibilities:

n= 3  3   3    4   5

m=3  4   5    3   3

Substitute into (1),(2):

F= 4  8   20  6  12
E= 6  12 30 12  30
V= 4  6   12  8   20

Tetrahedron 正四面体
(n,m)=(3,3) => (F,E ,V)=(4,6,4)

Cube or Hexahedron 正六面体(n,m)=(4,3) => (6,12,8)

Octahedron 正八面体
(n,m)=(3,4) => (8,12,6)

Dodecahedron 正十二面体
(n, m)=(5,3)=> (12,30,20)

Icosahedron 正二十面体
(n,m)=(3,5) => (20, 30,12)

The most complicated and the prettiest symmetric solid is:
Icosahedron 正二十面体

Icosahedron is the shape of the incurable HiV viruses.

Icosahedron is the symmetry of Galois Group, proved the unsolvable Quintic equations have no radical roots.

Maxwell Equation: Symmetry

Maxwell Equation

Maxwell boldy derived from Faraday experimental results by symmetry to get the Maxwell Equation for Electro-Magnetic Fields:

1) rot E = -1/c ∂H/∂t
div H = 0

By symmetry (swap E <-> H )

2) rot H = +1/c ∂E/∂t
div E = 0

Note:

E = Electric Field
H= Magnetic Field
c = Speed of light

Double-Entry Accounting = Algebraic 'Module'

Accounting Double-Entry Algebraic Structure: Module

Double-entry Accounting forms a structure Module Rⁿ, with scalars from an Abelian Unity Ring Z: (positive=credit, negative = debit).
Note: Change Vector Space's scalar over Field to over Ring => Module's scalar is over Ring.

Is '0' Natural Number?

Why '0' is not Natural (Counting) Number (N)?

A famous Polish Mathematician is counting 10 coins which his wife has given him. He never arrives at 10.
His wife sees how he counts: 0, 1, 2, 3, 4 ..., 9

Note: to avoid confusion: N* = N \ {0}