## Dec 22, 2013

### On the Mutual Languages of States

Languages lie beyond the boarder of nations. Normally, if two states speak the same language, cultural connection is implied. The notable example is the US. and UK, both are built by English speaking people. In the real world, one state may speak various languages. Each language is spoken by some portion of the population. To model the language similarity between two states, we introduce a function $\phi(state1, state2)$ such that: 1) $\phi = 1$ if state1 = state2; 2) $\phi(state1, state2) = \phi(state2, state1)$; 3) if no mutual language spoken between state1 and state2, then $\phi(state1, state2) = 0$.

A very intuitive definition of $\phi$ could be the fraction of people speaking any mutual language of two states. Let $p_1, p_2$ be the population of the two states, respectively. $f_i^{(1)}, f_i^{(2)}$ be the population fractions of the  $i$-th mutual language. Then
$\phi(state1, state2) := \frac{p_1 \sum_i f_i^{(1)} + p_2 \sum_i f_i^{(2)}}{p_1+p_2}.$ For example, the mutual native languages spoken in the US and UK are English and Angloromani. In the US, the fractions of these two languages are: 89% and 0.043%. In the UK, the numbers are 98% and 0.16%. Taken into account the population of the two states, 308.476 millions and 61.899 millions, the similarity function $phi$ yields 0.9085.

Another interesting definition of $\phi$ (but 1). is not satified) would be the probability of two men from the two states respectively speak the same tongue.  It can be shown the probability is $\phi(state1, state2) := \sum_i f_i^{(1)} f_i^{(2)}.$ In the case of US and UK, $\phi$ yields 0.8722.

#### Asia

With this definition, let's explore the international language connection. Fig. 1 shows the adjacent matrix of Asia states. Fig. 2 shows the graph representation of it wit the edge colored according to the language similarity.

 Fig. 1, the language connection among Asian states: the adjacent matrix.

 Fig. 2, the language connections among Asian states: the graph (We have removed the self-loops). The edges are colored according to their values.
Fig. 3 again shows the language connections among Asian states. But the edges are dyed by the type of the primary mutual language shared by the two vertex states. The red edges in Fig. 3 represent English. Compare with Fig. 2, English is not very popular in Asia but still widely used.
 Fig. 3, the language connections among Asian states: the graph edges are colored according to the type of primary mutual languages shared by each pair of states.
Notice that some states enjoy closer language relation with their neighbours than others. We can define a language capacity of a state within some group: $LC(s) := \frac{1}{|G|-1}\sum_{s' \in G, s'\ne s} \phi(s, s')$ Fig. 4 shows the language capcities of Asian states.

 Fig. 4, the Language Capacity of the states in Asia. The two countries that have the largest two language capacities are China and Singapore.
We can gather further information from the language relation graph. We can split the vertices into graph communities. Using the FindGraphCommunities function in Mathematica, we can identify 7 communities. The first community is Southeast Asia including China, Singapore, Brunei etc. The most popular language is Chinese, the most widely used language is English. The second and third communities are the crossroad countries including Russia, the central asia stans, Iran, Iraq, Syria, Isreal etc. The most popular and most widely used languages include Russia and Kurdish. The fourth zone is south asia, including India, Pakistan, Bangladesh Nepal Bhutan and Myanmar. The most popular language is Hindi and the most widely used language is Tibetan. The fifth region is the Arabic states. The most popular and most widely used language is Arabic. Next region is Japan and the Koreas. The most popular language is Japanese and the most widely used language is Korean. The last region is Turkey and Uzbekistan. Both countries speak Turkish.

 Fig. 5, the language communities of the Asia states. The edges are colored by the communities.

#### Europe

Europe is another continent with flourish civilizations. The average language capacity in Europe is higher than in Asia.

The languages spoken in Asia may be very diverse, it is not true in other continent. Europe, for example,  is dominated by French, German, Italian, English and Romanian etc.
Finally, the world is dominated by four languages: English, French, Spanish and Portuguese - they may not be the most popular ones - they are widely used in communicating with other states. Apparently, a language beyond the boundary of a country is a proof of the cultural influence. As we know, the four dominant "foreign languages" are the results of the colonism.

## Dec 7, 2013

### Frenet Tube

Given a 3D curve $\vec{r}(t) = (x(t), y(t), z(t)), t \in I=[0,1]$,  Tubify[ {x, y, z},  a] turns it into a tube around the curve. The Mathematica Tube[] has similar function (works on lines). The essential thing is to find the unit circle in the normal plane for each point (Shown in Fig. 1). The tangential vector is simply $\vec{\alpha} = \vec{r}'/|\vec{r}'|$, where $\vec{r}' = \mathrm{d} \vec{r}/\mathrm{d}t$. The normal vector is $\vec \beta = \vec{\alpha}' / |\vec{\alpha}'| = \frac{\vec{r}'' |\vec{r}'|^2 - \vec{r}' \vec{r}'\cdot\vec{r}''}{|\vec{r}'| \sqrt{|\vec{r}''|^2 |\vec{r}'|^2 - (\vec{r}'\cdot\vec{r}'')^2} }$. The binormal vector is $\vec{\gamma} = \vec{\alpha} \times \vec{\beta} = \frac{\vec{r}' \times \vec{r}'' }{\sqrt{|\vec{r}''|^2 |\vec{r}'|^2 - (\vec{r}'\cdot\vec{r}'')^2} }$.

 Fig. 1: black, $\vec{\alpha}$, green $\vec{\beta}$, blue $\vec{\gamma}$.

Tubify[{x_, y_, z_}, r_] :=
({x[#1], y[#1], z[#1]}
+ (r[#1] Cos[#2])/
Norm[(Norm[{x'[#1], y'[#1], z'[#1]}]^2 {x''[#1], y''[#1],
z''[#1]} - {x'[#1], y'[#1], z'[#1]}.{x''[#1], y''[#1],
z''[#1]} {x'[#1], y'[#1], z'[#1]})] (Norm[{x'[#1], y'[#1],
z'[#1]}]^2 {x''[#1], y''[#1],
z''[#1]} - {x'[#1], y'[#1], z'[#1]}.{x''[#1], y''[#1],
z''[#1]} {x'[#1], y'[#1], z'[#1]})
+ (r [#1] Sin[#2] )/
Norm[{-z'[#1]*y''[#1] + z''[#1]*y'[#1],
z'[#1]*x''[#1] - z''[#1]*x'[#1], -y'[#1]*x''[#1] +
x'[#1]*y''[#1]}] {-z'[#1]*y''[#1] + z''[#1]*y'[#1],
z'[#1]*x''[#1] - z''[#1]*x'[#1], -y'[#1]*x''[#1] +
x'[#1]*y''[#1]}) &


All the arguments are functions.

Example 1:
Show[ParametricPlot3D[
Tubify[{(2 + Cos[3 #]) Cos[ #] &, (2 + Cos[3 #]) Sin[#] &,
Sin[3 #] &}, 0.25 &][u, v],
{u, 0, 2 Pi}, {v, 0, 2 Pi}, PlotStyle -> Opacity[0.2],
Mesh -> None, Boxed -> True, Axes -> False, BoxRatios -> Automatic,
PerformanceGoal -> "Quality", PlotPoints -> 80, MaxRecursion -> 0,
PlotLabel ->
Style[TraditionalForm /@ {(2 + Cos[3 t]) Cos[
t], (2 + Cos[3 t]) Sin[t], Sin[3 t]}, 20]
],
ParametricPlot3D[{(2 + Cos[3 u]) Cos[ u], (2 + Cos[3 u]) Sin[u],
Sin[3 u]}, {u, 0, 2 Pi},
PlotStyle -> Directive[{Red, Opacity[1], Thick}]], ImageSize -> 600
]


A similar code using Tube[]:

ParametricPlot3D[
{(2 + Cos[3 u]) Cos[ u], (2 + Cos[3 u]) Sin[u],
Sin[3 u]}, {u, 0, 2 Pi}, PlotStyle -> Opacity[0.2],
Mesh -> None, Boxed -> True, Axes -> False, BoxRatios -> Automatic,
PerformanceGoal -> "Quality", PlotPoints -> 80, MaxRecursion -> 0,
PlotLabel ->
Style[TraditionalForm /@ {(2 + Cos[3 t]) Cos[
t], (2 + Cos[3 t]) Sin[t], Sin[3 t]}, 20]
]/.Line[pts_, rest___] :> Tube[pts, 0.1, rest]


 Fig. 2

Example 2, trefoil knot:

Show[ParametricPlot3D[
Tubify[{Sin[#] + 2 Sin[2 #] &, Cos[#] - 2 Cos[2 #] &, -Sin[3 #] &},
0.25 &][u, v],
{u, 0, 2 Pi}, {v, 0, 2 Pi}, PlotStyle -> Opacity[0.3],
Mesh -> None, Boxed -> True, Axes -> False, BoxRatios -> Automatic,
PerformanceGoal -> "Quality", PlotPoints -> 100, MaxRecursion -> 0,
PlotLabel -> Style[TraditionalForm /@ trefoil[t], 20]
], ImageSize -> 600
]


 Fig. 3 Trefoil Knot

Similarly, we can also define the Frenet's Ribbon:
Ribbonize[{x_, y_, z_}, r_] :=
({x[#1], y[#1], z[#1]}
+ r [#1] #2 /
Norm[{-z'[#1]*y''[#1] + z''[#1]*y'[#1],
z'[#1]*x''[#1] - z''[#1]*x'[#1], -y'[#1]*x''[#1] +
x'[#1]*y''[#1]}] {-z'[#1]*y''[#1] + z''[#1]*y'[#1],
z'[#1]*x''[#1] - z''[#1]*x'[#1], -y'[#1]*x''[#1] +
x'[#1]*y''[#1]}) &


 Fig. 4, Trefoil Ribbon

 Fig. 5, Helix Ribbon

#### more examples:

 Fig. 6, A Seashell Surface
 Fig. 7, Double Helix (cf. DNA)
 Fig. 8, A Figure-eight Knot

ParametricPlot3D[
Tubify[{#/Pi Cos[#] &, #/Pi Sin[#] &, #/Pi &}, Sin[#/4] &][u, u*80],
{u, 0, 4 Pi}, PlotStyle -> Directive[{Red, Thick, Opacity[0.8]}],
Mesh -> None, Boxed -> True, Axes -> False, BoxRatios -> Automatic,
PlotRange -> All, PerformanceGoal -> "Quality", MaxRecursion -> 0,
PlotPoints -> 2560, PlotLabel -> Style["", 20],
ImageSize -> 500
]
 Fig. 9, a coilfied 3D curve

## Dec 4, 2013

### Chang'e 3's Unmanned Moon Landing Mission

Chang'e 3 is a Chinese soft-landing moon mission. It features a moon lander (Chang'e 3) and a moon rover (Yutu).

 Fig. 1 (left) the artist's impression of the moon goddess Chang'e; (right) a yutu or a jade rabbit
Chang'e was the wife of Yi the Archer (not be confused with Lord Yi, Xia dynasty). Yi the Archer and his wife lived in the dawn of the Hua Xia civilization. As a hero of his time, Yi killed 9 monsters and managed to see the Goddess Wangmu on the Kunlun Montain. Wangmu gave him the elixir of immortality. Chang'e stole the elixir and became the goddess of the moon.

The rabbit was actually a different myth regarding the moon shadow. Later on, the two myths merged and the rabbit became Chang'e's pet.

 Fig. 2, the ancients' impression of the lunar shadow as a rabbit.
So much about the names and myths. The Chang'e 3 mission is planned to land in Sinus Iridum. The coordinate of the center of Sinus Iridum is 44.1°N , 31.5°W. The nearest historical lunar site is the former Soviet Union's Luna 17. The final landing coordinate however is 44.12°N 19.51°W, in the Mare Imbrium, ~100 km from Sinus Iridum, ~400km from the Luna 17 site. It has been reported as in the planned 91 km x 356 km Sinus Iridum landing area.

 Fig. 2 the designated landing area.

 Fig. 3, Chang'E 3 mission shown together with the historical luna missions.
The closest crater near the site is Laplace F, a 3 mile crate.

## Dec 2, 2013

### The Hierarchy of Knowledge

In a conversation, my advisor categorized human knowledge with the following hierarchy:

### wisdom      insight knowledge information data

Each set of data comprises noise and signal. Information can be abstracted from data. But in a broader sense, information has been quantified with entropy by Shannon.

Knowledge sits above information. For example, Schroedinger's equation (plus the information of boundary conditions) itself contains little information. Yet by solving it, we can obtain the information of the electron inside Hydrogen atom.

Beyond knowledge, there is insight and wisdom, which are particularly important for researchers. But they are also very difficult to quantify.

With the hierarchy of knowledge, one may also find out the corresponding intellectuals:

### thinker scholar teacher analyst labour

 Fig. 1, After WWII, the number of scientific publications doubles in about every 25 years. If knowledge is measured by the number of scientific publications, then human knowledge increases 15 times in one century.  (Data from World Bank, World Development Indicators.  Scientific and technical journal articles refer to publications in the following fields: physics, biology, chemistry, mathematics, clinical medicine, biomedical research, engineering and technology, and earth and space sciences.)