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Application of Math in Real Life

- Second Year Intermediate Seminar
- Tao Hong
- Department of Physics and Astronomy
- The Jhons Hopkins University
- hongtao_at_pha.jhu.edu

Introduction

Phenomena

Model Construction

Observation

Model Improvement

Analysis

Compare

- Using the math as a useful tool, we can better

understand complicated phenomena in our real

life. The application of math includes model

construction, model analysis and model

improvement - Several examples will be illustrated. Some of

them are mature, others are immature, needed

further study

1. Unit Analysis

- When setting up the model, we first try to find a

set of variables u, w1, w2, , wn to express

the phenomena what we are interested in. For

simplicity, assume that the variable the model

want to determine is called u, and u can be

expressed with a function f uf(w1, w2, , wn) - If we are only doing the pure math study, this

function f can be chosen arbitrarily. However in

reality, each variable has its own physical

meanings, it has an unique unit. Here we just

introduce how to take advantage of this

characteristics in Unit Analysis

- In classical mechanics, we usually use two kinds

of basic unit sets CGS g, cm, s and SI kg, m,

s. And unit of other physical quantities can be

deduced from the product of these basic units.

For example, the unit of the velocity is cm.s-1

or m.s-1, that of the acceleration is cm.s-2

or m.s-2 - The change between different unit sets is ,in

essence, the use of different calibrations during

the measurement. Although the values will change,

the phenomena is same. If we change CGS to SI,

mass should time 10-3, length should time 10-2

and energymasslength2time-2 should time 10 -7 - In general, assume that the basic unit is L1, ,

Lm. The unit of all variables u, w1, ,

wn are the product of these basic units. For

instance, if Z is a variable, the unit of Z can

be expressed - for the special case, when all ai0, Z is

dimensionless. The value of Z is unchanged during

the change of different unit sets

- Assume that the units of the variables w1, ...,

wm are independent to each other, the units of

wm1, , wn can be written as - And
- We can construct the dimensionless combinations
- So the original function f can also be expressed

as

- Since p,p1,pn-m are all dimensionless and

w1,wm are all independent, we can arbitrary

change the scale of wi. It means that for all

1im, we have - So the original function can be written as
- This is called the p principle.
- With the help of the application of the unit

analysis, we will study the evolution of the

radius of the atomic bomb after explosion.

The Air Shock of Atomic Bomb

- Mushroom cloud formed by the explosion of atomic

bomb - (Truckee, June 9, 1962, Airdrop, 210kt)

The process of the explosion of first atomic bomb

within first one second

- The process of atomic bomb explosion can be

simplified as such a model that lots of energy

has been produced at one point. Let the radius of

the strong shock is R, which increases with the

time. As we know, R is related to the time t, the

produced energy E, the around air density 0 and

pressure P0. So - Let us observe their units
- Rlength, ttime, Emasslength2tim

e-2, - 0masslength-3, P0masslength-1time-

2

- It is easy to see that t, E and 0 are

independent to each other, so totally two

dimensionless variables can be constructed as - According to p principle, we have
- In CGS g cm s, 0 1.2510-3 g/cm3, P0106

g/cms2, the exhausted energy E is a very large

number, produced energy E (by 1 thousand ton TNT)

91020 gcm2/s2, E by atomic bomb blast should

be much greater than that value. If E of the

atomic bomb is approximate same as 10 thousand

ton TNT, tF(p1) can be approximately expressed as F(0).

Then - On the other hand, F(0) can be determined by

little powder explosion. Combining this blast

scaling law with the explosion picture, we can

deduce E of the atomic bomb. Actually it had been

known by G. I. Taylor in 1941, four year earlier

than the explosion of first atomic bomb!!!

2. Fourier Transform

- If f(x) is a complex function in the zone 0,

2p, f(x) can be expressed as - Here
- It is called Fourier series. Here we transform

f(x) to a number array an, also from an we

can also get the original complex function f(x)

- If the complex function f(x) is defined in the

whole real axis and we assume - We can define the Fourier Transform (FT) of f(x)

as - In fact, if we also get the inverse

FT formula,

- For the multi-variable function, we can also deal

with one by one. For example, f (x, y) is

two-variable function, we can first do FT for x - Then do FT for y
- We also have inverse FT formula
- Above all, if we know the original function f (x,

y), we can deduce its FT function F(,). On the

other hand, if we know the FT function F(,), we

can also deduce its original function f (x, y)

while using the inverse FT.

Image Reconstruction of Computer Tomography

- Today computer tomography has been an important

tool in medical field. It can be used to find

some hidden illness which are difficult to be

determined in the past.

- In principle, CT technology is the combination of

physics and math. - The absorb coefficient of different tissues in

human body is different. Assume the absorb

coefficient is a function f (x, y), the signal

intensity when X-ray travels along a straight

line L through the human body to the detector can

be expressed as

- If the straight line satisfies the equation
- Let u is a parameter of straight line, then the

parameter equation of straight line can be

expressed as - So the signal the detector receives is
- From the hardware structure of CT, we can

physically measure the function p(f,v), so our

task is to deduce the absorb coefficient function

f(x,y).

- We can first do FT for v
- Next we do a variable transformation
- Then we can get
- Actually p(f,) is FT polar coordinate expression

of function f (x, y). So using inverse FT, we can

get the absorb coefficient f (x, y)

3. Math in Road Traffic

- The above two examples are relatively mature

examples. Following we will give some immature

examples, needed a lot for improvement - Traffic engineering mainly study the basic

principles of the traffic system, devise and

improve traffic network and control system.

Actually, it has not reach such a satisfactory

level that we can often encounter traffic jams in

the road.

- In one road, a basic function should be the

relationship between the vehicle density and the

maximum flux. Vehicle density is defined as the

average number of cars in unit distance, maximum

flux is defined as the uplimit of cars through

one point in unit time. The plot of this function

in traffic engineering is called basic plot.

Real data on the high way in Canada (each point

stands for the average value within 5 minutes)

- Right plot is from real measurement result. Here

every trajectory describes the motion of one car.

We can clearly see the jam which is caused

spontaneously

Traffic Flow Model

- The model suggested by Nagel and Schreckenberg

has been applied to traffic flow using cellular

automata. - Cellular automata (CA) are models that are

discrete in space, time and state variables. - To describe the state of a street using a CA, the

street is first divided into cells. Each cell can

now either be empty or occupied by exactly one

car. Each vehicle is characterized by its current

velocity v which can take the value of

v0,1,2,vmax. Here vmax corresponds to a speed

limit and is the same for all cars. A typical

configuration of the road is shown in the

following figure. - Now one need to specify rules that define the

temporal evolution of a given state. It consists

of 4 steps that have to applied at the same time

to all cars

- Step 1 acceleration
- All cars that have not already reached the

maximal velocity vmax accelerate by one unit v

v1. It describes the desire of the drivers to

drive as far as possible (or allowed) - Step 2 safety distance
- If a car has d empty cells in front of n

and is its velocity v (after step1) larger then

d, then it reduces the velocity to d v min d,

v. It encodes the interaction between the cars.

Here interactions only occur to avoid accidents - Step 3 randomization
- With probability p, the velocity is reduced

by one unit (if v after step 2) v v-1. It

corresponds to many complex effects that play an

important role in real traffic. - Step 4 driving
- After steps 1-3 the new velocity vn for

each car n has been determined forward by vn,

cells xn xnvn. All cars move according to

their new velocity

One example Assume vmax2, p1/3

- Status at time t
- Acceleration
- Safety distance
- Randomization
- Driving

- Both of the plots are simulated by computer while

applying the rules we discussed above - The left is the plot to describe spontaneous

jams, the right is the basic plot. However there

is no abrupt jump in the basic plot

- If we modify this model, change the first

acceleration step to such a rule (Slow-to-Start)

for the standing car only one empty cell ahead,

it will accelerate as possibility q, others keep

same. Now the simulation plot can fit well the

real data. - Even after correction the model can qualitatively

describe the traffic, it can not fit the real

data quantitatively.

4. Wealth, Bias Good Tendency Increase and Pareto

distribution

- Now we discuss economic problems. From

economists point, wealth increase is a typical

bias good tendency increase model. - Assume there n people, ith people has ki wealth.

Is the total wealth. The

basic rules are set as following - 1. When the wealth increase one, the probability

to be given to new people is 1-q the probability

to the previous people is q. Here for simplicity,

when the wealth increase 1, there will be a new

people joining into the model. - 2. The probability that the increase wealth to be

given to certain people is proportional to his

(her) current wealth ki

- Let p (k, i, s) is the probability that ith

people has the wealth k when the total wealth

increase s. - Initial condition
- Boundary condition
- It tell us that the probability that (ns)th

people has one wealth is 1-q

- The evolution equation of P (k , i, s) is
- When the total wealth increase s, the number of

people who have wealth are (1-q)sn. So the

wealth stable density distribution is - From this equation, we can learn

- Its deduced equation is
- If k is changing continuously, the above equation

can also be written as - The solution to this 1st order differential

equation is - This density distribution is called Pareto

distribution.

5. The fluctuation of stock price and Ising spin

chain

- The purpose of financial math is to make a

finance market model to predict the intendancy of

some finance problem - In 1900, Bachelier first use random change to

make the model. He thought the price of up and

down is caused by many independent random

factors. According to the middle limit theorem,

the distribution of price fluctuation should obey

Gauss distribution. The famous Black-Scholes

formula is based on this model - In 1953, Kendall first noticed that Gauss

distribution did not fit well with the real

financial data. For real financial data, the

price fluctuation obeys Pareto distribution

(No Transcript)

- For simplicity, assume we have one product, n

people. We consider the Ising chain, a string of

n neighbor spin sites. In each site, there is one

spin direction, up (s1) or down (s-1). - Here direction of the spin stands for the

transaction tendency. s1 for buying, s-1 for

selling. We randomly choose two neighboring cells

i, i1, the market price is determined as

- If sisi11, si-1 and si2 are same as si (si1)
- If sisi1-1, si-1 and si2 are chosen randomly
- These two rules are called United we Stand,

Divided we Act Randomly. People buy or sell

product due to others action. They are called

noise trader. If in whole market all of people

are noise trader, there are two equilibrium

parallel magnetic states. Actually these two

states do not exist.

- In reality, there are others called

fundamentalist. They are rational traders, know

the demand and provision of the market. If

provisiondemand, he buy, and vice versa. - Let xt defined as the difference between

provision and demand - The trade rule for fundamentalist is
- If xt 0, the probability to sell is xt .
- Since the attendance of fundamentalist, there

is no stable state in the market, it is always

changing. The computer simulation for this simple

asset pricing model is got as

- Left are Monte Carlo simulation results. Right

are real the exchange ratio between US dollar and

German Marks (Aug.9,1900 Aug 20, 1999) - The curve of price fluctuation
- The distribution curve of price fluctuation

Summary

- The core of application of math
- Model construction, how to use the math

language to - describe the phenomena
- Unfortunately, the phenomena we observe in

nature - are usually complicated, at least in surface.

There are four - directions needed improvement
- Many-body problem
- Uncertainty
- Multiscales
- Computation

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