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MOTIVATION: Predict future population with a mathmatical model
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APPROACH: Ideas of Population Model
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$\frac{dP(t)}{dt}=γP(t)(\frac{P∞-P(t)}{P∞})$ #1 |$P(t)$ : population - The population growth rate in a given year is proportional to the population in that year.
- Population growth has a upper limit
- The population growth rate decreases as the population gets close to the upper limitation
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Analitical solution of Equation #1
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$P(t)=\frac{P∞}{1+(\frac{P∞}{P0}-1)e^-γt}$ #2 - The solution is a logistic function
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EXERCISE: Parameter estimation
- Determine
$P0, P∞$ , and$γ$ with real data
- Determine
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MOTIVATION: Predict and analyze processes of epidemics with a mathmatical model
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APPROACH: Ideas of SIR Model
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$\frac{dS(t)}{dt}=-βS(t)I(t)$ #3 |$S(t)$ : Susceptibles -
$\frac{dI(t)}{dt}=βS(t)I(t)-γI(t)$ #4 |$I(t)$ : Infectives -
$\frac{dR(t)}{dt}=γI(t)$ #5 |$R(t)$ : Recovered -
$β$ stands for infection rate and$γ$ stands for recovery rate -
$S(t)*I(t)$ represents contacts between the susceptibles and the infectives - New infetives are proportional to Infection rate
- Variation of the infectives is the difference between new infectives and new people who recovered
- New people who recovered are propotional to
$I(t)$
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EXERCISE: Simulation and analysis focusing on
$S(0)$ - SIR Model behaves in a different way depending on the initial value of
$S(t)$ -
$S(0)<=\frac{γ}{β}$ or$S(0)>\frac{γ}{β}$ ?
- SIR Model behaves in a different way depending on the initial value of
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MOTIVATION: Expalain population variation of organisms with a mathmatical model
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APPROACH: Ideas of Predator-prey Model
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$\frac{dx(t)}{dt}=ax(t)-bx(t)y(t)$ #6 |$x(t)$ : Preys -
$\frac{dy(t)}{dt}=-cy(t)+dx(t)y(t)$ #7 |$y(t)$ : Predators - Regarding Equation #6, the Preys increase exponentially without the predators
- Regarding Equation #6, the preys decrease depending on the numbers of chances to encounter the predators
- Regarding Equation #7, the predators exponentially decrease without the preys
- Regarding Equation #7, the predators increase depending on the numbers of chances to encounter the preys
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EXERCISE: Conduct simulations
- Check the trend on the numbers of the predators and preys
- What if both of the predators and preys are overfished simultaneously?
- What if both of the predators and preys are not fished so much simultaneously?
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Fractals are geometric shapes which have similar structure at a given scale (self similarity)
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EXERCISE #1: Generate Mandelbrot Set
- Mandelbrot Set is a fractal resulted from calculation for the divergence condition for a complex sequence
- Check the details of the figure at a given scale
- Find self similarity
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MOTIVATION: Depict complex fern leaves (シダの葉) with a few simple rules
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APPROACH: Repetition and randomization
- prepare four specific affine tranformations to get the values of x and y for the next step
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$f1$ is for stems -
$f2$ is for continuous smaller leaves -
$f3$ is for larger leaves on left -
$f4$ is for larger leaves on right
- Set probabilities for each transformation
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EXERCISE #2: Depict fern leaves on computer
- Generate Barnsley Fern
- Focus on self similarity
- Can you depict different types of fern leaves with different parameters?
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Chaos is complex system which is deterministic but hard to predict its future behavior due to sensitive dependence on the initial conditions (初期値鋭敏性)
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EXERCISE #1: Depict Lorenz attractor
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$\frac{dx(t)}{dt}=s(y-x)$ #8 -
$\frac{dy(t)}{dt}=rx-y-xz$ #9 -
$\frac{dz(t)}{dt}=xy-bz$ #10 - A famous set of values
$(s, r, b)$ is$(10, 28, 8/3)$ - Change the initial values by just a bit (e.g. x(0)=0 -> x(0)=0.0001)
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EXERCISE #2: Depict chaos of logistic equation
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$x(n+1)=ax(n)(1-x(n)), (n=0, 1, 2,... )$ #11 - Check different types of behavior for the value of
$x(n)$ , with different$a$ - Change the initial value of
$x(0)$ to check sensitive dependence on the initial conditions (e.g. x(0)=0.01 -> x(0)=0.010001) - Generate figures for the values of
$a$ and$x(n)$ (logistic maps)
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Twelve equal temperament is a musical note system deviding sounds into twelve notes
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When a note gets a one round, the frequency of the note doubles
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When the frequency of a note is
$f$ , that of the next note is$\sqrt[12]{2}f$ -
EXERCISE: Calculate frequency of notes
- What is the frequency for the next Do?
- What is the frequency for the Ti on the table?
- Hint: You can find La# between La and Ti
| Note | Do | Re | Mi | Fa | So | La | Ti |
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| Frequency (Hz) | 262 | 294 | 330 | 349 | 392 | 440 | ??? |