Application of Math in Real Life

Application of Math in Real Life : Application of Math in Real Life Second Year Intermediate Seminar Tao Hong Department of Physics and Astronomy The Jhons Hopkins University hongtao@pha.jhu.edu

Introduction : Introduction 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 Phenomena Model Construction Observation Analysis Compare Model Improvement

1. Unit Analysis : 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: u=f(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

Slide4 : 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 energy=mass·length2·time-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 αi=0, Z is dimensionless. The value of Z is unchanged during the change of different unit sets

Slide5 : Assume that the units of the variables w1, ..., wm are independent to each other, the units of wm+1, …, wn can be written as: And We can construct the dimensionless combinations: So the original function f can also be expressed as

Slide6 : Since π,π1,···πn-m are all dimensionless and w1,···wm are all independent, we can arbitrary change the scale of wi. It means that for all 1≤i≤m, we have So the original function can be written as: This is called the π 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 : 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 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 ρ0 and pressure P0. So Let us observe their units: [R]=length, [t]=time, [E]=mass·length2·time-2, [ρ0]=mass·length-3, [P0]=mass·length-1·time-2

Slide9 : It is easy to see that [t], [E] and [ρ0] are independent to each other, so totally two dimensionless variables can be constructed as According to π principle, we have: In CGS [g· cm· s], ρ0 =1.25*10-3 g/cm3, P0=106 g/cm·s2, the exhausted energy E is a very large number, produced energy E (by 1 thousand ton TNT) =9*1020 g·cm2/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, t<0.1, then π1<0.01. In this condition, F(π1) 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 : 2. Fourier Transform If f(x) is a complex function in the zone [0, 2π], 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)

Slide11 : 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,

Slide12 : 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(ξ,η). On the other hand, if we know the FT function F(ξ,η), we can also deduce its original function f (x, y) while using the inverse FT.

Image Reconstruction of Computer Tomography : 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.

Slide14 : 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

Slide15 : 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(φ,v), so our task is to deduce the absorb coefficient function f(x,y).

Slide16 : We can first do FT for v: Next we do a variable transformation: Then we can get: Actually p(φ,ρ) 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 : 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.

Slide18 : 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) : Real data on the high way in Canada (each point stands for the average value within 5 minutes)

Slide20 : 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 : 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 v=0,1,2,···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

Slide22 : Step 1: acceleration All cars that have not already reached the maximal velocity vmax accelerate by one unit: v→ v+1. 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→ xn+vn. All cars move according to their new velocity

One example: Assume vmax=2, p=1/3 : One example: Assume vmax=2, p=1/3 Status at time t Acceleration Safety distance Randomization Driving

Slide24 : 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

Slide25 : 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 : 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

Slide27 : 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 (n+s)th people has one wealth is 1-q

Slide28 : 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)s+n. So the wealth stable density distribution is: From this equation, we can learn:

Slide29 : 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 : 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

Slide31 :

Slide32 : 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 (s=+1) or down (s=-1). Here direction of the spin stands for the transaction tendency. s=+1 for buying, s=-1 for selling. We randomly choose two neighboring cells i, i+1, the market price is determined as:

Slide33 : If sisi+1=1, si-1 and si+2 are same as si (si+1) If sisi+1=-1, si-1 and si+2 are chosen randomly These two rules are called “United we Stand, Divided we Act Randomly”. People buy or sell product due to other’s 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.

Slide34 : In reality, there are others called “fundamentalist”. They are rational traders, know the demand and provision of the market. If provision>demand, 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 buy is | xt |; 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:

Slide35 : 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 : 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