Apr 23

paper folding

Category: Video
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Mar 18

Do u know Euler ?

Category: Books

Euler….i am proud of you….

Time

People

 

Place

 

 

1748

 

Born: 15 April 1707 in Basel, Switzerland
Died: 18 Sept 1783 in St Petersburg, Russia

Leonhard Euler was a Swiss mathematician who was tutored by Johann Bernoulli. He worked at the Petersburg Academy and Berlin Academy of Science. By 1740 Euler had a very high reputation, having won the Grand Prize of the Paris Academy in 1738 and 1740. He won the Paris Academy Prize 12 times. He worked at the Petersburg Academy and Berlin academy of Science.

 

The place is in Berlin in the 18th century

In 1701, the King of Berlin is Frederick I. Berlin was the capital of Prussia at that time. In 1740-1786, Frederick II (The Great) brought Berlin to be under the rule of philosophically oriented Frederick II.

In 1871, Berlin became the capital of newly founded German Empire. The population and economy expanded and Berlin be the economic center of Germany during 19 century.

 

 

Math

History (Book)

 

Motivation

 

It talks about Exponentials and logarithms.

Let the exponential to be considered be y = az where a is a constant and the exponent z is a variable. The base a can also be a variable.

If we give z is integer such as

  • z =0 then az=1
  • z is negative exponents then az is decreasing
  • z is positive exponents then az is increasing
  • z goes to infinity then az increase to infinity

and the value of y depends on the value a

 

For the Logarithms :

  • we always have log 1 =0
  • since in the equation y = az, it corresponds to z = log y
  • log yn = n z
  • log v.y = log v + log y
  • log a/b = log a – log b
  • if we have P = am and Q=an then log P = m and log Q= n.

Log P/log Q = n/m

 

Frederick William who was known as the “Great Elector” interested to ask and invite Euler to come and work at the Academy of Science Berlin because the very good reputation of Euler. Euler wrote the book “Introduction to analysis of infinite” in 1748 in Berlin then He was the leader of the Berlin Academy.

 

The motivation of Euler is to give more understandable description of exponents and logarithms to students and develop the students ability of exponents and logarithmics

 

 

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Mar 18

Leslie model population

Category: Math

One major thing that is important in Leslie model is that the construction of matrix is depends on the Survival rate and The female offspring.

Furthermore about the Leslie models are described below :

Leslie matrix model is a model based on several assumption that can be considered both of a limitation and a feature of the model . Of course no model is ever 100 % accurate.

· The population is considered in fixed age groups depending on the lifetime of a population.

· Leslie model is a discrete model therefore the model doesn’t give further information of what happening between the discrete time

· The rates are assumed to be constant. It means that everything that happened in environment is assumed constant. For example the disaster and the health problem that effect the changing of Survival rate are ignored in this model.

· The Leslie model is mostly based on female. The dynamic of the population are determined by one sex independent

· In practice, the estimation of female offspring is very difficult.

To look into the dynamics of the growth process, we need the role of Linear Algebra in it that are the eigenvalue and eigenvector of the Leslie matrix. The eigenvalue and the eigenvector are used to learn about the change of population in dynamical system. The goal is to decide the long term dynamics of the population for example to show whether the population is increasing, decreasing or constant.

How about Indonesia’s population?.To forecast population, Indonesia Statistics bureau usually use the Logistic function formula and Micro Computer Programs for Demographics Analysis (Method of UN, 1956). Actually the Leslie matrix model can be applied in Indonesia data if the data that is needed to construct the Leslie matrix is reliable. Unfortunately, because of the limitation of the data that make the construction of Leslie matrix is not reliable, we can not make projection which is similar to the result of using Logistic function formula. It also because of the difficult to estimate the female offspring based on data that produce by Indonesia statistics bureau.

The Leslie matrix model is represented below :

 

Note that each Si is assumed to be constant over time and independent of the total number of the population. Those are in the first sub diagonal of the matrices. Thus the model does not take crowding into account. Let Fi be the average number of female offspring, born to individual in age class i. Fi is determined not only by the average number of female births per mother per time period, but also infant survival rates. Those are in the first row of the matrices.The Lesli matrices is the square matrices.

 

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