Application of Math in Real Life

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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

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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

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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

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• 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

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• 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

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• 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.

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The Air Shock of Atomic Bomb

• Mushroom cloud formed by the explosion of atomic
  bomb
• (Truckee, June 9, 1962, Airdrop, 210kt)

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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

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• 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!!!

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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)

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• 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,

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• 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.

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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.

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• 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

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• 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).

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• 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)

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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.

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• 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.

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

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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

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• 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

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One example Assume vmax2, p1/3

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

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• 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

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• 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.

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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

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• 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

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• 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

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• 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.

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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

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• 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

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• 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.

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• 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

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• 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

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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