text:
Open university

The Free Software Foundation (FSF; initiated by Students for Free Culture (SFC)) define an open university as one in which:

 * 1) The research produced is open access;
 * 2) The course materials are open educational resources;
 * 3) The university embraces free software and open standards;
 * 4) The university’s patents are readily licensed for free software, essential medicine, and the public good;
 * 5) The university’s network reflects the open nature of the Internet.

The FSF goes further, to list specific criteria for each of these open university principles and university-rankings based on their degree of openness: Open University Report Cards.

title: Open university


text:
Boundary Value Problems/Review: ODEs/BVP-solution-1

Return Review: ODEs

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


title: File:Egm6341.s10.mtg26.djvu


title: File:Egm6341.s10.mtg27.djvu


text:
User:Egm6341.s10.team4.anandankala/HW4-3

Question

 * {| style="width:70%" border="0" align="center"

 * Discuss the pros and cons of different numerical methods.
 * 1) Taylor's series.
 * 2) Composite Trapezoidal rule.
 * 3) Composite Simpson's rule.
 * 4) Romberg Table (Richardson Extrapolation).
 * 5) Corrected Trapezoidal rule.
 * }.
 * }.

Solution
The pros and cons of the above mentioned methods are discussed below.

Taylors Series:

Advantages:
 * 1) The solution is highly accurate for a small order compared to Trapezoidal and Simpson's rule.
 * 2) It is one step and explicit.

Disadvantages:
 * 1) It needs the explicit form of derivatives of the function.

Composite Trapezoidal rule:

Advantages:
 * 1) The method is quite simple in implementing, compared to other methods.
 * 2) It takes a piecewise linear approximation of the function and hence execution of the solution is faster.
 * 3) The solution is very rapidly convergent.
 * 4) It consists of the fact that weighting coefficients are nearly equal to each other.

Disadvantages:
 * 1) The error is much higher compared to the other methods.
 * 2) This method is less accurate for non linear functions since it has straight line approximation.

Composite Simpson's rule:

Advantages:
 * 1) It assumes piecewise quadratic interpolation and hence the accuracy is much higher compared to trapezoidal rule.
 * 2) It can integrate polynomials up to third order accurately.
 * 3) The error is less compared to that of trapezoidal rule.
 * 4) Weighting coefficients are simple and do not fluctuate in terms of magnitude.

Disadvantages:
 * 1) A large number of ordinates are needed in between the interval.

Romberg Table (Richardson Extrapolation):

Advantages:
 * 1) Error is reduced subsequently as we go from TK(n) to TK+1(n).
 * 2) It is efficient in the sense that TK+1(n) is calculated from the already computed values of TK(n) and TK(2n).
 * 3) The above condition reduces the time and space requirement.

Disadvantages:
 * 1) In Richardson extrapolation, We are neglecting the higher orders of aih2i.

Corrected Trapezoidal rule:

Advantages:
 * 1) When integrals of periodic functions are approximated numerically, Corrected Trapezoidal functions are the best choice.

Disadvantages:
 * 1) This method demands derivatives of the function at the end points of the interval.


text:
User:Egm6341.s10.Team4.yunseok/HW4

= Problem 11: Application of Engineering Orbital Mech. =

Given
Refer Lecture slide 25-2 for problem statement

Polar form relative to focus
 * {| style="width:100%" border="0" align="left"

r(\theta) = \frac{1-e^2}{1-e \, cos(\theta)} $$
 * [[Image:Ellipse Polar.svg|200px]]
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

where, eccentricity $$\displaystyle e = \sin(\frac{\pi}{12}) $$

Find
'''Find Arc length of ellipse$$\displaystyle (I_n)$$ (orbital)using below integration method.

compute $$\displaystyle I_n$$ to the error $$\displaystyle 10^{-10}$$ and

compute time using tic/toc matlab commend

1) Composit Trapozoidal rule

2) Composit Simpson's rule

3) Romberg Table

4) Chebfun, sum commend

5) matlab commend : Trapz, quad, clencurt

Solution
The circumference $$C$$ of an ellipse is $$\displaystyle 4 a E(e)$$, where the function $$E$$ is the complete elliptic integral of the second kind. Here, a=1:

The complete elliptic integral of the second kind E is defined as

 * $$E(e) = \int_0^{\pi/2}\sqrt {1-e^2 \sin^2\theta}\ d\theta\!$$

So,

 * {| style="width:100%" border="0" align="left"

I= 4\int_0^{\pi/2}\sqrt {1-e^2 \sin^2\theta}\ d\theta $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.

using the equation (1), the arc length of ellipse was calculated by several numerical method.

The result is summarized.

Ramanujan approximation is 's:
 * $$C \approx \pi \left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]= \pi(3(a+b)-\sqrt{10ab+3(a^2+b^2)})$$

where, a = longest radius, b= shortest radius in ellipse

'''(1) Composit Trapoziodal rule

 Matlab Code:

 subfunction ff(x)

'''(2) Composit Simpson's rule

 Matlab Code:

'''(3) Romberg Table

 Matlab Code:

'''(4) Chebfun

 Matlab Code:

'''(5) Trapz (matlab commend)

 Matlab Code:

'''(6) quad (matlab commend)

 Matlab Code:

'''(7) Clencurt (matlab commend)

 Matlab Code:

title: User:Egm6341.s10.Team4.yunseok/HW4


text:
Portal talk:Persian

this flag is not persian symbol at all! this is such a disgusting insulting thing!

This flag was adopted some 30 years ago ! You may disagree with the regime but this flag is official.--AC'hM 20:05, 5 March 2010 (UTC)

title: Portal talk:Persian


text:
User:Egm6341.s10.Team4.andy/HW4

= Problem 1: Higher Order Corrected Trapezoidal Rule =

Given
Refer Lecture slide 20-1 for problem statement.

Error for Higher Order Error for Trapezoidal rule on slide 18-2, is given by:

 * {| style="width:100%" border="0" align="left"

$$ $$
 * $$\displaystyle E_n^1 := I - I_n
 * $$\displaystyle E_n^1 := I - I_n
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.

where $$\displaystyle I_n = T(n) $$ is the Composite Trapezoidal approximation of the Integral $$\displaystyle I = \int_{a}^{b} f(x)dx $$ with $$\displaystyle n $$ intervals and $$\displaystyle n+1 $$ points $$\displaystyle \forall x \in [a,b] $$.

 * {| style="width:100%" border="0" align="left"

T(n)=\frac{h}{2}(f_0+2f_1+2f_2+ \cdots +2f_{n-1}+f_n) $$ $$
 * $$ \displaystyle\
 * $$ \displaystyle\
 * $$\displaystyle \longrightarrow (2)
 * }.
 * }.

where $$\displaystyle\ f_i=f(x_i)$$.

Equation (1) is a Euler-Maclaurin series given by Refer 18-3:

 * {| style="width:100%" border="0" align="left"

$$ $$
 * $$\displaystyle E_n^1 = \sum_{q=1}^{\infty} a_q h^{2q}
 * $$\displaystyle E_n^1 = \sum_{q=1}^{\infty} a_q h^{2q}
 * $$\displaystyle \longrightarrow (3)
 * }.
 * }.

where $$\displaystyle h = \frac{b-a}{n} $$.

Find
1. To develop a general higher order corrected Trapezoidal Rule.

2. To find $$\displaystyle n, I_n, E_n^1 $$ using $$\displaystyle CT_1, CT_2, CT_3 $$ for which $$\displaystyle I - I_n $$ is of the order $$\displaystyle \Theta(10^{-6}) $$ for $$\displaystyle I = \int_{0}^{1} \frac{e^x-1}{x}dx $$.

Solution
1. To develop a general Higher Order Corrected Trapezoidal Rule:

Equation (3),

 * {| style="width:100%" border="0" align="left"

\begin{align} E_n^1 &= \sum_{q=1}^{\infty} a_q h^{2q} \\ \Rightarrow     E_n^1 &= I - T(n) = \sum_{q=1}^{\infty} a_q h^{2q} &= I - T(n) = a_1h^{2} + \sum_{q=2}^{\infty} a_q h^{2q} \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

\begin{align} \Rightarrow     E_n^1 &= I - \left[\underbrace{T(n) - a_1h^{2}}_{CT_1}\right] = \sum_{q=2}^{\infty} a_q h^{2q} \\ &= I - CT_1(n) = a_2h^{4} + \sum_{q=3}^{\infty} a_q h^{2q} \\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

\begin{align} \Rightarrow     E_n^1 &= I - \left[\underbrace{CT_1 + a_2h^{4}}_{CT_2}\right] = \sum_{q=3}^{\infty} a_q h^{2q} \\ &= I - CT_2(n) = a_3h^{6} + \sum_{q=4}^{\infty} a_q h^{2q} \\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

\begin{align} \Rightarrow     E_n^1 &= I - \left[\underbrace{CT_2 + a_3h^{6}}_{CT_3}\right] = \sum_{q=4}^{\infty} a_q h^{2q} \\ &= I - CT_3(n) = a_4h^{8} + \sum_{q=5}^{\infty} a_q h^{2q} \\ &.\\ &.\\ &.\\ &.\\ &.\\ \Rightarrow     E_n^1 &= I - \left[\underbrace{CT_{(k-1)} + a_kh^{2k}}_{CT_k}\right] = \sum_{q=(k+1)}^{\infty} a_q h^{2q} \\ &= I - \underbrace{CT_k(n)}_{I_n} = a_{(k+1)}h^{2(k+1)} + \sum_{q=k+2}^{\infty} a_q h^{2q} \\
 * $$\displaystyle
 * $$\displaystyle

\end{align} $$
 * }.
 * }.

Hence the Higher Order Corrected Trapezoidal rule is given by,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

\Rightarrow     I_n = CT_k(n) = CT_{(k-1)}(n) + a_kh^{2k}

$$ $$
 * style= |
 * $$\displaystyle \longrightarrow (4)

 * }.
 * }.

2. To find $$\displaystyle I_n $$ using $$\displaystyle CT_1, CT_2, CT_3 $$:

From Equation (4), $$\displaystyle I_n $$ using $$\displaystyle CT_1, CT_2, CT_3 $$ can be given by:

 * {| style="width:100%" border="0" align="left"

\begin{align} \Rightarrow     I_n &= CT_3 \\ &= CT_2 + a_3h^{6}\\ &= CT_1 + a_2h^{4} + a_3h^{6}\\ &= T(n) + a_1h^{2} + a_2h^{4} + a_3h^{6} \end{align} $$
 * $$\displaystyle

 * }.
 * }.

Here,

 * {| style="width:100%" border="0" align="left"

\begin{align} a_i &= d_i \left[ f^{(2i-1)}(b) - f^{(2i-1)}(a)\right]\\ d_i &= - \frac{B_{2i}}{2i}; \ \ \ B_{2i} \rightarrow Bernoulli\ Numbers;\ \ \ d_1 = \frac{-1}{12}; \ \ \ d_2 = \frac{1}{720}; \ \ \ d_3 = \frac{-1}{30240} \end{align}
 * $$\displaystyle

$$

 * }.
 * }.

At $$\displaystyle n = 8$$, the Error is reduced to an order of $$\displaystyle \Theta(10^{-6}) $$.

 Matlab Code:

Author
Solved and typed by - Egm6341.s10.Team4.andy 08:44, 3 March 2010 (UTC)

= Problem 7: Theorem of Higher Order Error for Trap. rule [Step:2a - Part2] =

Given
Refer Lecture Slide Refer 21-2 for problem statement.

Higher Order Error for Trap. rule is given by [ Refer 21-1 ]:

 * {| style="width:100%" border="0" align="left"

E_n^1=\frac{h}{2} \sum_{k=0}^{n-1} \left[ \underbrace{\int_{-1}^{1}g_k(t)dt-(g_k(-1)+g_k(+1))}_{E}\right] $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.

where $$\displaystyle E $$ can be written as,

 * {| style="width:100%" border="0" align="left"

E =\int_{-1}^{1}(-t)g^{(1)}(t)dt $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (2)
 * }.
 * }.

Find
Show that,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle E =\int_{-1}^{1}\underbrace{(-t)}_{p_{1}(t)}g^{(1)}(t)dt = \left[p_2(t)g^{(1)}(t) \right]_{-1}^{+1} - \int_{-1}^{+1}p_2(t)g^{(2)}(t)dt $$

$$
 * $$\displaystyle \longrightarrow (3)
 * }.
 * }.

where $$\displaystyle p_2(t) = \int p_1(t)dt$$.

Solution
Integrating by Parts,

 * {| style="width:100%" border="0" align="left"

\int u dv = uv - \int v du $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

where,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \begin{align} u &= g^{(1)}(t) \\ \Rightarrow du &= g^{(2)}(t)dt \end{align} $$

 * }.
 * }.

and,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \begin{align} dv &= p_1(t)dt \\ &= -tdt \\ & \\ \Rightarrow v &= \int p_1(t)dt &=p_2(t) \end{align} $$

 * }.
 * }.

Equation (3) is then,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \begin{align} \Rightarrow E &=\int_{-1}^{1}\underbrace{(-t)}_{p_{1}(t)}g^{(1)}(t)dt \\ &= \left[p_2(t)g^{(1)}(t) \right]_{-1}^{+1} - \int_{-1}^{+1}p_2(t)g^{(2)}(t)dt \end{align} $$

 * }.
 * }.

Author
Solved and typed by - Egm6341.s10.Team4.andy 08:44, 3 March 2010 (UTC)

title: User:Egm6341.s10.Team4.andy/HW4


text:
User:Egm6341.s10.team3.heejun/HW4

=  6.1 b) Comparison 2nd column of Romberg Table to Composite Simpson's rule =

Problem Statement
$$ \displaystyle Compare\, the\, second\, column\, of\, Romberg\, Table\, Romberg\, to\, Simpson's\, rule\, and \, derive any relationship,\, $$

Ref: Lecture Notes [[media:Egm6341.s10.mtg25.djvu|p.25-1]]

Solution
$$ \displaystyle a)\, Formula\, for\, Composite\, Trap.\, rule\, is\, given\, by\, $$ $$ \int_{a}^{b}f(x)dx=\frac{b-a}{2n}*[f(x_{0})+2f(x_{1})+2f(x_{2})+.........+2f(x_{n-1})+f(x_{n})] $$

Ref: Lecture Notes [[media:Egm6341.s10.mtg7.pdf|p.7-1]]

$$ \displaystyle b)\, Formaula\, for.\, Romberg\, table\, is\, given\, by\, $$ $$ T_{k}(n)=\frac{2^{2k}T_{k-1}(n)-T_{k-1}(\frac{n}{2})}{2^{2k}-1} $$

Ref: Lecture Notes [[media:Egm6341.s10.mtg19.pdf|p.19-2]]

$$ \displaystyle Since\, the\, 2nd\, column\, of\, Romberg\, table\, corresponds\, to\, k=1\, $$ $$ T_{1}(n)=\frac{4T_{0}(n)-T_{0}(\frac{n}{2})}{3} $$

$$ \displaystyle T_{0}(n)\, and\, T_{0}(\frac{n}{2})\, are\, composite\, trap.\, rule\, $$

$$ \displaystyle i)\, T_{0}(\frac{n}{2})= \frac{b-a}{\frac{n}{2}}[\frac{1}{2}f_{0}+f_{2}+...+f_{n-2}+\frac{1}{2}f_{n}]\, =\, \frac{b-a}{n}[f_{0}+2f_{2}+...+2f_{n-2}+2f_{n-1}+f_{n}] $$ $$ \displaystyle Where\, the\, number\, of\, pannels\, =\, \frac{n}{2}\,, \,\, h=\frac{b-a}{\frac{n}{2}}\, $$

$$ \displaystyle ii)\, T_{0}(n)= \frac{b-a}{n}[\frac{1}{2}f_{0}+f_{1}+f_{2}+...+f_{n-2}+f_{n-1}+\frac{1}{2}f_{n}] $$ $$ \displaystyle Where\, the\, number\, of\, pannels\, =\, n\,, \,\, h=\frac{b-a}{n}\, $$

$$ \displaystyle iii)\, T_{1}(n)= \frac{1}{3}\underbrace{[4T_{0}(n)-T_{0}(\frac{n}{2})]}_{call\, p} $$

$$ \displaystyle Where\, p\, =\, 4T_{0}(n)-T_{0}(\frac{n}{2})\, =\, \frac{b-a}{n}[f_{0}+4f_{1}+2f_{2}+...+2f_{n-2}+4f_{n-1}+f_{n}] $$

$$ \begin{align} \displaystyle Therefore,\, T_{1}(n)&=\frac{1}{3}p \\ &=\frac{1}{3}(\frac{b-a}{n}[f_{0}+4f_{1}+2f_{2}+...+2f_{n-2}+4f_{n-1}+f_{n}]) \\ &=\underbrace{\frac{h}{3}[f_{0}+4f_{1}+2f_{2}+...+2f_{n-2}+4f_{n-1}+f_{n}])}_{Exactly same with the formula of Comp. Simp. rule} \\\end{align}\ $$

=  6.1 e) Time Comparison and Results =

Problem Statement
$$ \displaystyle Compare\, the\, time\, for\, above\, each\, method\, with\, comment\, on\, the\, results\, $$

Ref: Lecture Notes [[media:Egm6341.s10.mtg25.djvu|p.25-1 & 25-2]]

Solution
As above table shown, Romberg Table has the shortest computational time for  $$\displaystyle I = \int_{0}^{1}\frac{e^{x}-1}{x}dx$$

However, note that Comp. Simpson's rule and Clencurt have almost same computational time with the Romberg Table even though the Romberg Table has slightly shorter computational time.

Therefore, ''' The Romberg Table, Comp. Simpson's rule and Clencurt are efficient ways to get a numerical integration value for the given function.'''

=  7.1 e) Time Comparison and Results =

Problem Statement
$$ \displaystyle Compare\, the\, time\, for\, above\, each\, method\, with\, comment\, on\, the\, results\, $$

Ref: Lecture Notes [[media:Egm6341.s10.mtg25.djvu|p.25-2]]

Solution
As above table shown,''' The Comp. Simpson's rule has the shortest computational time for ''' $$\displaystyle I = \int_{-5}^{5}\frac{1}{1+x^2}dx$$ However, note that Romberg Table has almost same computational time with Simpson's rule even though Simpson's rule has slightly shorter computational time than Romberg Table. Therefore,  both Simpson's rule and Romberg Table are efficient ways to get a numerical integration value for the given function.

=  8.1 e) Time Comparison and Results =

Problem Statement
$$ \displaystyle Compare\, the\, time\, for\, above\, each\, method\, with\, comment\, on\, the\, results\, $$

Ref: Lecture Notes [[media:Egm6341.s10.mtg25.djvu|p.25-2]]

Solution
As above table shown,''' Comp. Simpson's Rule has the shortest computational time for ''' $$\displaystyle I = \int_{0}^{2\pi}\frac{1-sin^{2}(\frac{\pi}{12})}{1-sin(\frac{\pi}{12})cos\theta}d\theta$$

However, we should note that the computational time does not really different from each other for three method. Especially, Comp. Trap Rule has a shorter computational time than Romberg Table (in general, computational time of Romberg Table is much shorter than that of Comp. Trap. Rule.). Since the given function $$\displaystyle f(\theta) = \frac{1-sin^{2}(\frac{\pi}{12})}{1-sin(\frac{\pi}{12})cos\theta}$$ is a periodic function, the numerical integration of this function is converged very fast by Comp. Trap. rule.

Therefore, ''' Comp. Trap. rule is an efficient way to numerically integrate the given function $$\displaystyle f(\theta) = \frac{1-sin^{2}(\frac{\pi}{12})}{1-sin(\frac{\pi}{12})cos\theta}$$ because this function is a periodic function.'''

title: User:Egm6341.s10.team3.heejun/HW4


text:
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User:Egm6341.s10.Team4.andy/HW4-1

= Problem 1: Higher Order Corrected Trapezoidal Rule =

Given
Refer Lecture slide 20-1 for problem statement.

Error for Higher Order Error for Trapezoidal rule on slide 18-2, is given by:

 * {| style="width:100%" border="0" align="left"

$$ $$
 * $$\displaystyle E_n^1 := I - I_n
 * $$\displaystyle E_n^1 := I - I_n
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.

where $$\displaystyle I_n = T(n) $$ is the Composite Trapezoidal approximation of the Integral $$\displaystyle I = \int_{a}^{b} f(x)dx $$ with $$\displaystyle n $$ intervals and $$\displaystyle n+1 $$ points $$\displaystyle \forall x \in [a,b] $$.

 * {| style="width:100%" border="0" align="left"

T(n)=\frac{h}{2}(f_0+2f_1+2f_2+ \cdots +2f_{n-1}+f_n) $$ $$
 * $$ \displaystyle\
 * $$ \displaystyle\
 * $$\displaystyle \longrightarrow (2)
 * }.
 * }.

where $$\displaystyle\ f_i=f(x_i)$$.

Equation (1) is a Euler-Maclaurin series given by Refer 18-3:

 * {| style="width:100%" border="0" align="left"

$$ $$
 * $$\displaystyle E_n^1 = \sum_{q=1}^{\infty} a_q h^{2q}
 * $$\displaystyle E_n^1 = \sum_{q=1}^{\infty} a_q h^{2q}
 * $$\displaystyle \longrightarrow (3)
 * }.
 * }.

where $$\displaystyle h = \frac{b-a}{n} $$.

Find
1. To develop a general higher order corrected Trapezoidal Rule.

2. To find $$\displaystyle n, I_n, E_n^1 $$ using $$\displaystyle CT_1, CT_2, CT_3 $$ for which $$\displaystyle I - I_n $$ is of the order $$\displaystyle \Theta(10^{-6}) $$ for $$\displaystyle I = \int_{0}^{1} \frac{e^x-1}{x}dx $$.

Solution
1. To develop a general Higher Order Corrected Trapezoidal Rule:

Equation (3),

 * {| style="width:100%" border="0" align="left"

\begin{align} E_n^1 &= \sum_{q=1}^{\infty} a_q h^{2q} \\ \Rightarrow     E_n^1 &= I - T(n) = \sum_{q=1}^{\infty} a_q h^{2q} &= I - T(n) = a_1h^{2} + \sum_{q=2}^{\infty} a_q h^{2q} \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

\begin{align} \Rightarrow     E_n^1 &= I - \left[\underbrace{T(n) - a_1h^{2}}_{CT_1}\right] = \sum_{q=2}^{\infty} a_q h^{2q} \\ &= I - CT_1(n) = a_2h^{4} + \sum_{q=3}^{\infty} a_q h^{2q} \\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

\begin{align} \Rightarrow     E_n^1 &= I - \left[\underbrace{CT_1 + a_2h^{4}}_{CT_2}\right] = \sum_{q=3}^{\infty} a_q h^{2q} \\ &= I - CT_2(n) = a_3h^{6} + \sum_{q=4}^{\infty} a_q h^{2q} \\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

\begin{align} \Rightarrow     E_n^1 &= I - \left[\underbrace{CT_2 + a_3h^{6}}_{CT_3}\right] = \sum_{q=4}^{\infty} a_q h^{2q} \\ &= I - CT_3(n) = a_4h^{8} + \sum_{q=5}^{\infty} a_q h^{2q} \\ &.\\ &.\\ &.\\ &.\\ &.\\ \Rightarrow     E_n^1 &= I - \left[\underbrace{CT_{(k-1)} + a_kh^{2k}}_{CT_k}\right] = \sum_{q=(k+1)}^{\infty} a_q h^{2q} \\ &= I - \underbrace{CT_k(n)}_{I_n} = a_{(k+1)}h^{2(k+1)} + \sum_{q=k+2}^{\infty} a_q h^{2q} \\
 * $$\displaystyle
 * $$\displaystyle

\end{align} $$
 * }.
 * }.

Hence the Higher Order Corrected Trapezoidal rule is given by,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

\Rightarrow     I_n = CT_k(n) = CT_{(k-1)}(n) + a_kh^{2k}

$$ $$
 * style= |
 * $$\displaystyle \longrightarrow (4)

 * }.
 * }.

2. To find $$\displaystyle I_n $$ using $$\displaystyle CT_1, CT_2, CT_3 $$:

From Equation (4), $$\displaystyle I_n $$ using $$\displaystyle CT_1, CT_2, CT_3 $$ can be given by:

 * {| style="width:100%" border="0" align="left"

\begin{align} \Rightarrow     I_n &= CT_3 \\ &= CT_2 + a_3h^{6}\\ &= CT_1 + a_2h^{4} + a_3h^{6}\\ &= T(n) + a_1h^{2} + a_2h^{4} + a_3h^{6} \end{align} $$
 * $$\displaystyle

 * }.
 * }.

Here,

 * {| style="width:100%" border="0" align="left"

\begin{align} a_i &= d_i \left[ f^{(2i-1)}(b) - f^{(2i-1)}(a)\right]\\ d_i &= - \frac{B_{2i}}{2i}; \ \ \ B_{2i} \rightarrow Bernoulli\ Numbers;\ \ \ d_1 = \frac{-1}{12}; \ \ \ d_2 = \frac{1}{720}; \ \ \ d_3 = \frac{-1}{30240} \end{align}
 * $$\displaystyle

$$

 * }.
 * }.

At $$\displaystyle n = 8$$, the Error is reduced to an order of $$\displaystyle \Theta(10^{-6}) $$.

 Matlab Code:

Author
Solved and typed by - Reviewed by -

= Problem 7: Theorem of Higher Order Error for Trap. rule [Step:2a - Part2] =

Given
Refer Lecture Slide Refer 21-2 for problem statement.

Higher Order Error for Trap. rule is given by [ Refer 21-1 ]:

 * {| style="width:100%" border="0" align="left"

E_n^1=\frac{h}{2} \sum_{k=0}^{n-1} \left[ \underbrace{\int_{-1}^{1}g_k(t)dt-(g_k(-1)+g_k(+1))}_{E}\right] $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.

where $$\displaystyle E $$ can be written as,

 * {| style="width:100%" border="0" align="left"

E =\int_{-1}^{1}(-t)g^{(1)}(t)dt $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (2)
 * }.
 * }.

Find
Show that,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle E =\int_{-1}^{1}\underbrace{(-t)}_{p_{1}(t)}g^{(1)}(t)dt = \left[p_2(t)g^{(1)}(t) \right]_{-1}^{+1} - \int_{-1}^{+1}p_2(t)g^{(2)}(t)dt $$

$$
 * $$\displaystyle \longrightarrow (3)
 * }.
 * }.

where $$\displaystyle p_2(t) = \int p_1(t)dt$$.

Solution
Integrating by Parts,

 * {| style="width:100%" border="0" align="left"

\int u dv = uv - \int v du $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

where,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \begin{align} u &= g^{(1)}(t) \\ \Rightarrow du &= g^{(2)}(t)dt \end{align} $$

 * }.
 * }.

and,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \begin{align} dv &= p_1(t)dt \\ &= -tdt \\ & \\ \Rightarrow v &= \int p_1(t)dt &=p_2(t) \end{align} $$

 * }.
 * }.

Equation (3) is then,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \begin{align} \Rightarrow E &=\int_{-1}^{1}\underbrace{(-t)}_{p_{1}(t)}g^{(1)}(t)dt \\ &= \left[p_2(t)g^{(1)}(t) \right]_{-1}^{+1} - \int_{-1}^{+1}p_2(t)g^{(2)}(t)dt \end{align} $$

 * }.
 * }.

Author
Solved and typed by - Reviewed by -

title: User:Egm6341.s10.Team4.andy/HW4-1


text:
Ruby/Flow Control

Overview
By the end of this lesson you should be able to:
 * Use dot-times-loop.
 * Use while and until loops.
 * Use if.

Loops
First of all, you should know what a loop is. Therefore, I will tell you:

A loop is the thing you use, when you want to do something again and again. For example, let's say that you wanted to greet five persons: Peter, Jones, Bill, Sophie and Anne. In ruby you would write:

# puts is same as print, but appends a newline. puts "Hello, Peter" puts "Hello, Jones" puts "Hello, Bill" puts "Hallo, Sophie"      # I've typed 'Hallo' not 'Hello' puts "Hello, Anne"

But, whith a loop you could write:

["Peter", "Jones", "Bill", "Sophie", "Anne"].each do |name| puts "Hello, #{name}" end

Which gives the same, but without any typos--much smarter. So, lets learn something about loops:

The dot-times loop
I don't really know what you call this one, but dot-times is a good name, isn't? Anyways, this is used like:

5.times do     print "Again and " end

This will print:

Again and Again and Again and Again and Again and

Not perfect but still, the idea is there, right? So, now I am supposed to tell you how that works... and I will, in a moment, but first let's get something clear. The do and ends makes a block, a block is some code that is sticked together. The dot means that we want to do something with the number 5. And the times is what we want to do with the number. Everything that is inside the do and the end is a block. And what is in the block will be executed each time the loop runs.

The while loop
Ruby can also do while-loops. While loops are found in many other programming languages. They are a bit of an old programming tradition. An example:

count = 0 while count < 5 print 'The count is now ', count, '. '      count = count + 1 end

Result: The count is now 0. The count is now 1. The count is now 2. The count is now 3. The count is now 4.

The until loop
The until loop is a variation on the while loop.

count = 0 until count == 5 print 'The count is now ', count, '. '      count = count + 1 end Result: same as above.

Pay attention to the double equal signs. In Ruby and many other programming languages a == 5 means Compare a to 5, and a = 5 means Make a point to a value of 5.

If
Ruby call also execute code only if certain conditions are met, this is the 'if' function. For example:

if(count == 5) print 'The count is equal to five' end

What this means is really "if count is equal to five the execute this code." this will also work with other mathematical operators.

else
'else' is another part of the 'if' function this will execute if the 'if' condition(s) are not met. For example:

if(count == 5) print 'The count is equal to five' else print 'The count is not equal to five' end

elsif
'elsif' is a variation on the 'else' function it is really a contraction of the 'else' and 'if' functions

if(count == 5) print 'The count is equal to five' else if(count < 5) print 'The count is less than five' end end

The above code can be modified to do the same thing with the 'elsif' function

if(count == 5) print 'The count is equal to five' elsif(count < 5) print 'The count is less than five' end

This code will do the same as the first piece but it is shorter and tidier than the first peice

title: Ruby/Flow Control


text:
User:Wikitanvir

The purpose of this user page is to point you to the right direction where you can contact me in need. Please do not leave your message here, otherwise in most cases, they will not be noticed in time and eventually be removed.

However, you are encouraged to contact me or say hello. :-)

Please note that I can communicate in either Bangla or English. My apologies for not knowing your language.

If you prefer to leave me a message, please choose any of the following talk pages I maintain frequently.
 * Meta-Wiki talk page
 * Bangla Wikipedia talk page (my home wiki)
 * English Wikipedia talk page
 * Wikimedia Commons talk page

If you have something to say to me in private, e-mail is a good option.
 * If you are logged in to your Wikimedia account you can probably e-mail me by clicking here.
 * Otherwise, you can e-mail me directly to At sign.svg.com.

Finally, if you are interested to know about my work or myself, they can be found on my Meta-Wiki user page.

title: User:Wikitanvir


text:
User talk:Wikitanvir

The purpose of this user page is to point you to the right direction where you can contact me in need. Please do not leave your message here, otherwise in most cases, they will not be noticed in time and eventually be removed.

However, you are encouraged to contact me or say hello. :-)

Please note that I can communicate in either Bangla or English. My apologies for not knowing your language.

If you prefer to leave me a message, please choose any of the following talk pages I maintain frequently.
 * Meta-Wiki talk page
 * Bangla Wikipedia talk page (my home wiki)
 * English Wikipedia talk page
 * Wikimedia Commons talk page

If you have something to say to me in private, e-mail is a good option.
 * If you are logged in to your Wikimedia account you can probably e-mail me by clicking here.
 * Otherwise, you can e-mail me directly to At sign.svg.com.

Finally, if you are interested to know about my work or myself, they can be found on my Meta-Wiki user page.

title: User talk:Wikitanvir


text:
Help talk:Abaqus

Hi all,

I'm working with contact definition between shell and solid and I don't know how to maintane a gap between two in-contact surfaces to represent the shell thickness. So what I see in gradual drop of the loaded solid items untill they reach the shell visible surface and it pick up contact from this point. It seems like it does not take into accoutn for the typed-in shell thickness,I've already confirmed the thickness offset direction to match with Normal direction. Any advise?

Thanks,

Omid

title: Help talk:Abaqus


text:
MediaWiki:Gadget-edittop.js

// ********************************************************************** // **                ***WARNING GLOBAL GADGET FILE***                 ** // **            changes to this file affect many users. ** // **          please discuss on the talk page before editing         ** // **                                                                 ** // ********************************************************************** // Imported from http://en.wikipedia.org/wiki/MediaWiki:Gadget-edittop.js, 2007-12-18 if (mw.config.get('wgAction') == 'view' && mw.config.get('wgNamespaceNumber') >=0) $(function{ var h2s = document.getElementsByTagName('H2'); var h2 = h2s[0]; if (!h2) return; if (h2.parentNode.id == 'toctitle') h2 = h2s[1]; if (!h2) return; var span = h2.firstChild; if (!span || span.className != 'editsection') return; var zero = span.cloneNode(true); if (document.getElementById('featured-star')) zero.style.marginRight = '25px'; var parent = document.getElementsByTagName('H1')[0]; parent.insertBefore(zero, parent.firstChild); var a = zero.getElementsByTagName('A')[0]; a.title = a.title.replace(/:.*$/,': 0'); a.setAttribute('href', a.href.replace(/&section=1/,'&section=0')); });

title: MediaWiki:Gadget-edittop.js


text:
MediaWiki:Gadget-edittop

Add a "edit" link for the first section of a page

title: MediaWiki:Gadget-edittop


text:
User:Lessouth

About Me
My name is Leslie Southerland and I am a student in the New Media and Global Education program at Appalachian State University in North Carolina.

Education
 * I have a BA in Education from UNC-Chapel Hill (1997), with a major in Secondary English and a Minor in French.
 * I hope to graduate with my MA from App State in May of 2010.

Professional Background
 * I currently serve as Dean of Academics at Queen's Grant Preparatory High School in Mint Hill, North Carolina.
 * I worked in the Charlotte-Mecklenburg School system for nine years, teaching at Independence High School from 1997-2002 and at Mint Hill Middle School from 2004-2008.
 * I also do adjunct work for South Piedmont Community College, teaching developmental writing. I have taught the same course at Queens University of Charlotte in the past.

title: User:Lessouth


text:
Creative writing for teens

Creative Writing Activity
Sometimes it's difficult to get students started with writing. Here's a fun activity that can reinforce the elements of fiction and give students a chance to brainstorm using each other as resources:

Story-Go-Round

 * First, place students in circular groups. Groups of 4-6 work well.
 * Explain to the students that they will each start a story and that they will have a specific amount of time to complete each step before passing their paper to the right or to the left (that part doesn't matter, as long as everyone passes the same way).
 * Time each step carefully. For beginners, a minute to a minute and a half is sufficient for step 1.  Add about ten seconds to each subsequent step to give them time to read.  Don't tell the students you are adding the time, as they will never notice because they will be too busy working!
 * Depending on your audience, you may want to set some ground rules about what topics are off-limits. Topics that are typically off limits are illegal activities, drugs and weapons, profanity, and sexual references.
 * Write the steps on the board or project them using an overhead, LCD projector or Smartboard. The steps are as follows:

 * 1) Describe a setting (time and place)
 * 2) Introduce a protagonist (if your students are younger, you may want to use the term "main character")
 * 3) Introduce a second character/antagonist
 * 4) Describe a conflict
 * 5) Bring the conflict to its climax
 * 6) Resolve the conflict

 * Make sure you explain that students may ONLY work on ONE step at a time. If they jump ahead and introduce the conflict while they are describing the antagonist, the next person who gets the paper won't know what to do.
 * Once the activity is complete, have each group share the best story and explain which member came up with which element(s).
 * As an extension activity, you can have each student revise their original paper and turn in the final copy.

This is a fun activity that takes little prep time and students usually really enjoy it, even at the college level. I did this in my teaching methods class when I was a senior in college, and we all had a lot of fun with our crazy stories!

--Lessouth 14:06, 3 March 2010 (UTC)

title: Creative writing for teens


text:
User:Monaliza990

hello; how are you?i am fine.

title: User:Monaliza990


text:
User:Sennadacosta

This page intends to present a summary of my research activities on academic collaboration, specially phenomena related to co-authorship activities.... Feedbacks, suggestions, inputs, insights, questions or criticisms are very welcome.

My current research project at the Humboldt University of Berlin has the following scope:

 Bibliometrics / Scientometrics and the 'commoditization of knowledge' in the era of academic capitalism

Author: Mariano Senna da Costa (MSc)

Affiliations: Institut für Bibliotheks-und Informationswissenschaft / Berlin School of Library and Information Science, Humboldt-Universität zu Berlin, Dorotheenstrasse 26, Berlin, 10117 (Germany)

Type of work: Preliminary report from a PhD dissertation

Keywords: academic activities, academic capitalism, academic communication, co-authorship, collective writing, co-publication, e-science, information management, knowledge monopolies, networking processes, open access, open science, papermania, philosophy of science, publish or perish, research policy, scientific collaboration, scientific openness, scientific networks, virtual research environment, wikitopia

Focus of Interest: Academic publishing policy, Co-authorship of scientific texts and works, Research assessment, Scientific collaboration

Reviewers: Celia Querol

Introduction
"'What is true of the species that live together in a wood is also true of the groupings and sorts of people in a society, who are similarly in an uneasy balance of dependency and competition'. (Gregory Bateson, 'Steps to an Ecology of Mind', Chicago, 1972 - p.437)"

Bibliometrics is a set of quantitative techniques applied to assess scientific papers and publications according to their statistical output. These techniques are used to determine the influence of a writer (e.g. H-index), or a publication (impact factor) in relation to their citation records, number of articles, periodicity of a publication etc, currently being the standard of assessment for most of the scientific institutions in the world (Palmquist, 1996; NIHL, 2010).

There is a trend as well in which these Research Performance Measurements (RPM) are becoming central to the judgement of academic work. As globally measurable and comparable, it is comprehensible to see them as a main method to justify the allocation of research funding and infrastructure investments. The growing importance of university rankings is an evidence and a driver of such tendency (Dempsey, 2010).

On the other hand, these measurements do not consider many factors involved in writing and publishing, such as bureaucratic requirements, institutional context and individual motivations or interests, much less the impact of research to society. Furthermore, the "network metrics" (Yao, 2009) are inaccurate to describe and understand core structures, processes and problems of scientific authoring and publishing (Liang & Zhong, 2013).

According to a report prepared in 2007 on behalf of the Higher Education Funding Council for England, some of the problems of bibliometrics as main form of academic output assessment are:
 * Citations per paper - presents critical aspects regarding accuracy and appropriateness of citation.
 * Citations among disciplines - very different approaches according to each discipline, specially between those from engineering and humanities.
 * Contextual information - metrics do not take into account information about the researchers, do not consider the differences of context across disciplines and countries, as well as they have no proper means to evaluate interdisciplinary works. (Lipsett, 2007)

Most important, taking into consideration the general socio-economic context, in which profit and accumulation of material means are pursued as a fundamental dogma, it is possible to foresee the main risk represented by the almost exclusive application of quantitative techniques to assess the work of scientists. 'Bibliometrics' or 'scientometrics' play a decisive role to certify expert publication (Hilgartner, 2002), strengthening the process in which knowledge and information are turned into a commodity (Levins, 2010).

Quantitative assessment methods are not directly responsible for turning scientific studies into products, but they definitely help to attribute to these attempts a certain aura of objective and universal truth. As a consequence, the entire structure of the scientific establishment is shaped accordingly, with certain areas, topics and projects being favoured in detriment of others with less impressive quantitative records (Münch, 2011).

In other words, Bibliometrics do not even grasp what has been left aside from the current scientific establishment. It does not add any improvement to the scientific endeavour itself, it just reinforces the current way of centralization and institutional control over scientists' work. (NIHL, 2010)

The present paper gathers arguments and evidences showing the early development of such a systemic bias in academic communication, and what risks it involves. It tries as well to indicate how academics can contribute for a better system to assess their works.

Academic Capitalism
It is undeniable that citation index represents a huge advancement brought about by communication technology in recent years (Belbachir & Harik, 2013). “Electronic data systems” allow the quantitative assessment of scientific texts in a global scale. Similarly, it is important to admit that the progress by now is still a tiny part of the potential for improvement, specially regarding the qualitative aspect of communication at research and learning environments (Nentwich, 2003). This assumption is directly proportional to our very limited comprehension of many of the aspects of scientists' behaviour when producing knowledge (Foster et al., 2007).

Furthermore, considering the historical importance of science, it is convenient to ask what kind of metrics would fit better to assess the knowledge production of educational institutions. Trust and credibility are decisive factors for scientific progress, as promotion and tenure are fundamental to a scientist's career advancement. Are they being addressed accordingly? Instead of new “metrics”, we should start seeking new social norms in order to qualify and foster scientific collaboration (Bollier, 2011).

Institutional context is fundamental to the present discussion. It helps to understand why the publication of scientific articles is still in most areas an activity dominated by big publishing conglomerates. Even if the number of publications and personal websites presented a huge incrementation, it is a matter of fact that such pluralization of information sources has been systematically boycotted by institutional and corporate agenda, which seems threatened by the direction of the communication within the WWW (Gatti, 2012).

According to recent literature, the ongoing initiatives and debates about the transformation of the way scientists communicate has had almost no significant impact on fundamental business models of the academic publishing world (PRWEB, 2013). Once more, the current publishing practices and structures have to be considered in order to explain this phenomenon (Young, Ioannidis, & Al-Ubaydli, 2008).

Academic institutions have adopted almost exclusively quantitative assessment mechanisms as a mean to justify administrative decisions regarding infrastructure and funding of research. Also the scientist's career advancement is mainly based on the number of articles published, among other quantitative outputs. Inevitably, the pursuit of certain numerical records is becoming the main reason for academics to write an article.

At the same time, commercial publishers are responsible for the most important titles of academic publications nowadays. They manage the editorial process with focus on the maintenance of loyal readers, who are the root of their credibility. However, at the decision making level, publishers have to balance pure scientific goals with issues regarding the management of the publication (Bhat, 2009 /Goldacre, 2012). The abundant evidence of a "citation game" taking place among editors is one of the effects of this current culture. An analogy between few economic terms used currently to analyse and describe practices of academic publishing illustrates the extension of the problem: (Young, Ioannidis, & Al-Ubaydli, 2008).

Taking the “central importance of context”, we may grasp how this whole mechanism, including the quantification of research, promotes the phenomena of “instrumental thought” in science as described by Gregory Bateson (1942 / 1972). At this point, it is possible to make an analogy to the "Quantophrenia" phenomena. The tendency of social sciences of expressing everything in numbers. It is a way to look serious, like the natural sciences, but it is also a way of avoiding going deeper into the analysis and the understanding of many problems (Askwith Forum, 2013).

The idea is quite simple. Universities, scientists and their publications are increasingly being judged according to their “numbers”. As the common saying goes: numbers speak for themselves. In the sense that, giving too much importance to numerical outputs, and becoming too dependent of merely quantitative assessements, academics naturally get used to put aside their pure scientific goals. Furthermore, in such environment a quantitative standard of evaluation may easily serve as an instrument of manipulation (European Network of Scientists for Social and Environmental Responsibility (ENSSER) & French Fondation Sciences Citoyennes (FSC), 2010).

Accusations of this kind of manipulation have fallen upon scientists with increasingly frequency all over the world. There are many examples in the literature on how this whole process is influencing our institutions, education system and society. They were already named as “Private Science”, “Academic Capitalism” or “Ökonomisierung der Forschung”. It makes generally reference to the conflict in which scientists are keen to sacrifice their very scientific ideals in exchange of a funding grant, or a chance of progression in the career (Kohlenberg & Musharbash, 2013).

Ben Goldacre (TEDMED, 2012) shows clearly how corporate interests, for instance, created a bias where researches pointing good results on drug tests are being favoured in detriment of other investigations presenting side effects of the same substances. At the end the conclusion is clear: even doctors do not know all problems caused by the use of the drugs they prescribe. This has been clearly a matter of editorial choice rather than technical or scientific limitation.

Knowledge Monopolies
The discussion is indeed recurrent. The Canadian professor Harold Innis (The Bias of Communication, 1951) wrote a seminal work about the role of communication and media in shaping culture and the development of civilization. Innis pointed to our “historical lack of interest in educational philosophy” and the “institutional conservatism” traditionally verified in the Western society, as the roots of many challenges facing the academic establishment nowadays.

He coined the term “knowledge monopolies” to explain the trend towards social polarization (knowledge elite X mass of ignorants) and centralization of power (who defines reality and truth). In academic environments this trend is manifested as:
 * Hierarchy struggle
 * Difficulties to reach the sources of information
 * Lack of professionals carrying out communication tasks
 * Conflicts among different areas of knowledge
 * Differences among professional timings.
 * (McGinn, 1991; Soules, 2004)

Some of Innis' statements concerning adult education, universities and scientific texts seem prophetic: "'Mechanical devices become concerned with useless knowledge of useful facts. Complaints of duplication and confusion are inevitable. But I do see all the injury to the higher literature inflicted by the torrential multiplication of printed stuff coinciding with the legal enforcement of mechanical reading - absurdly misnamed Education... the materialised state of mind and morals of modern anarchy, without beliefs, or ideals, or principles, or duties - this is to inaugurate a millennium of vapid commonplace and vulgar realism... The incessant education drill, the deluge of printed matter, asphyxiate the brain, dull beauty of thought, and chill genius into lethargic sleep.... a direct result of our mechanical schooling... is the gradual deterioration of literature into dry specialism and monotonous commonplace ... today we may say - the school has been the death of literature...” (Harold A. Innis, The Bias of Communication, Toronto, 1951 – p. 205)"

The idea of Science "'...science is not a perfect instrument, but by no means ceases to be an excellent and invaluable utensil, which only causes damage when it is taken as an end in itself' (C.G. Jung, introduction to the book The Secret of the Golden Flower: A Chinese Book of Life, 1931 - p. 24)"

The American professor, Neil Postman, (The End of Education, 1996) invokes the very idea of science in order to approach the issue. According to him, one of the main sins of the current scientific establishment is to give a misleading idea that it holds something called "ultimate truth". Postman argue as follows about this: "'Such a belief is, in itself, an instance of the sin of pride, and no self-respecting scientist will admit to holding it...science is a moral imperative drawn from a larger narrative whose purpose is to give perspective, balance and humility to learning.' (p. 68)"

Postman's argument relates to "acceptance of uncertainty" in any scientific endeavour. According to him, the quest for certainty usually leads to the instrumentalization of science in form of dogmas or ultimative formulas ("the knowledge of gods", p. 69), opposite to the idea that human knowledge is always limited. The discussion refers to the biblical legend of fallen angels (Ge 6.1-4), or in analogy to the myth of Sisyphus, the perception that whole truth can not be revealed to our limited human condition, although we keep pursuing it.

Few decades before, Thomas Kuhn (The Structure of Scientific Revolutions, 1962) made a similar point explaining how science advances from one paradigm to another. While admitting that new knowledge may be an advance, he pointed out that it can never be absolute. Only human arrogance could be responsible for such a belief. Evoking the perception of science as a "subjective and irrational enterprise", the former professor emeritus of linguistics and philosophy at the MIT admits his "relativistic" view, while calling upon "the problem of choice" and of "the problems of translation" in order to explain communication barriers among academics (Kuhn, 1970 - p.175).

Depending on the perspective, the sin may be called "pride", "hubris", or "dogma". But in any case, it is the kind of thing that scientists must get rid of it.

The idea of Education

The book "The End of Education" (Postman, 1996) aims to promote a serious debate about the reasons for education, which will afterwards be embedded in the policies and matters of organizations. Nevertheless Postman admits that professors and schools administrators seem the least interested in debating the reasons for education.

According to Postman, the main barrier to the improvement of the academic institutions is that they are structured in separated, and much independent "major subjects". These subjects act and operate as bureaucratic and institutional entities, defining and shaping the way teachers, faculties, publishers, national organizations etc work. Where the "entity" of one subject is licensed to teach that one subject, but not another; (e.g. students are entitled to attend one course, but not another, even if the last one has better answers to the very questions any student can come about during an academic semester). This inflexible structure puts administrative and bureaucratic control above the learning process in a scale of importance of academic activities.

Again, the issue gets back to the current idea of education, where texts books are seen as "enemies of education, promoting dogmatism and trivial learning". In that regard, Postman testifies: "'...there is no sense of frailty or ambiguity of human judgement, no hint of the possibilities of error. Knowledge is presented as a commodity to be acquired, never as a human struggle to understand, to overcome falsity, to stumble toward the truth'. (p. 116)"

Quantity ≠ Quality

Libraries and writing centres, or community-based social enterprises may be the seed of a solution for such a dilemma. But these activities barely count to a formal evaluation system. The British economist, Tim Jackson, connects the dots in his book Prosperity Without Growth (2009). Jackson asserts about the way whole society is locked in a economic cage as follows: "'They (social enterprises) represent a kind of Cinderella economy that sits neglected at the margins of consumer society... These activities are usually labour intensive. So if they contribute anything at all to GDP, their labour productivity is of course dismal - in the language of the dismal science. '(p. 131)"

The former Economics Commissioner for the Sustainable Development Commission (SDC/UK) goes further, explaining why "de-materialized services" are not counted for performance evaluation, and what is the effect. And it is related to our obsession with productivity: "'It's because, in most cases, human input is what constitutes the value in them. The pursuit of labour productivity in activities whose integrity depends on human interaction systematically undermines the quality of the output. Secondly, work itself is one of the ways in which humans participate meaningfully in society. Reducing our ability to do that - or reducing the quality of our experience in doing so - is a direct hit on flourishing.' (Jackson, 2009 - p. 132 - 133)"

Obstacles of Knowledge

"As Harold Laski (1930:102) observed, expertise can be a major obstacle to understanding: 'Expertise, it may be argued, sacrifices the insight of common sense to intensify experience. It breeds an inability to accept new views from the very depth of its preoccupation with its own conclusions. It too often fails to see round its subject. It sees its results out of perspective by making them the center of relevance to which all other results must be related... it has, also, a certain caste-spirit about it, so that experts neglect all evidence that does not come from those who belong to their own ranks. Above all, perhaps,... the expert fails to see that every judgement he makes not purely factual in nature brings with it a scheme of values that has no special validity about it. He tends to confuse the importance of his facts with the importance of what he proposes to do about them'... ...So far, I have emphasized the constraints on learning imposed by academic institutions, disciplinary conventions, and cautious faculty members. Reluctance to deal with core issues or to mount cross-disciplinary efforts are readily explained as the legacies of graduate school indoctrination, professionalized social solidarity, and a peer review system that promotes narrow intellectual cloning over broad critical thinking." (p. 51) I call it the "exclusive character of expertise"

"...teaching is being short changed in order to perpetuate an intellectual caste system that caters to publishable forms of microspecialization, most of it in isolation from broad social and environmental needs... ...The point is that our pedagogy is embedded in a vast system of institutions whose prevailing standards are based on the professional and economic profit from research rather than the social profit from teaching. Instructors who attempt to challenge these standards with innovative courses quickly discover that their professional status may suffer." (p. 55) (Maniates, 2003)

END OF PART TO BE EDITED

Papermania
Evidence of such deterioration can be verified in different places, although, considering the ubiquity and instantaneous character of the WWW as a global communication platform, it is reasonable to see similar effects everywhere.

//TO BE EDITED//

A critical point of view of the publishing system

The laureate with the Nobel prize of medicine 2013, Randy Wayne Schekman, accused some of the most important journals in the world (Nature, Science and Cell) of impeding the development of science in order to guarantee their editorial interests. Among the pernicious practices the publications were accused there are the "artificial reduction" of the quantity of articles accepted, hyped criteria for selection of topics and texts, and a complete disengagement with the qualification of the scientific debate. "The science must get rid of the tyranny of the "luxury journals", declared the scientist in an article to the Guardian newspaper after the announcement of the Nobel prize. He defended a research and publishing system that serves better to the interests of society.

A critical point of view of the ranking system

The CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), responsible for the accreditation and evaluation of graduate programs all over Brazil, attributes a decisive weight to the publication of articles in well ranked journals like the Nature. On Anthropology, for instance, 40% of the entire assessment weight is due to publishing. It is to say that, in order to receive a good grade, any program must pressure its staff to publish the results of research on publications ranked on the Qualis index as A1, A2, or B1. In numbers, each scientist must publish at least two papers in such journals, not to mention other kinds of publications like books, technical reports and conference posters. The main problem to the Brazilian scientists in relation this system is that in order to be indexed as A1 or A2, the publications must be classified on international indexes. And this represents a weird form of dependence from the dynamics of the international publishing market.

According to scientists in many scientific disciplines, this is a wrong way to promote the internationalization of the Brazilian research, privileging foreign publications greatly financed by private interests pegged to intellectual property defenders, over the national publishing system constituted mostly by journals published by public universities and associations.

Productivism

Some effects of this policy are: - Deterioration of intellectual culture oriented towards scientific progress and innovation (i.e. collaborative and participative practices of investigation); - Obliteration or inferiorization of certain kinds of knowledge in front of others, considered more efficient, productive or simply appropriate in a specific context or perspective (i.e. indigenous methods, concepts and knowledge). The phenomenon also called "Epistemicide"; - Reduction of the diversity of ideas and the intellectual creativity, with consequent legitimization of hegemonic strategies and policies intended to justify economic and political power (i.e. distribution of resources and funding); - Maintenance of a colonial vision of the science, where the new and the authentic tend, without almost no exception, to be imported (i.e. publishing indexes and evaluation criteria); - Publishing to publish, or production for the production, disconnected to any compromise to solve real problems of the society;

(Martins Moraes, 2014)

Increased academic productivity = increased scientific bias

Besides the "publish or perish" culture, academic environments are pervaded by a context where individual achievement and success is practically the only way to assess research work....

The author of the article concludes: "...competitive academic environments increase not only scientist's productivity but also their bias. The same phenomenon might be observed in other countries where academic competition and pressures to publish are high".

Notes

"Te objectivity and integrity of contemporary science faces many threats. A cause of particular concern is the growing competition for research funding and academic positions, which combined with an increasing use of bibliometric parameters to evaluate careers (e.g. number of publications and the impact factor of the journals they appeared in), pressures scientists into continuously producing "publishable" results."

Theoretically competition is a vector to efficiency and productivity on science... but: "...In many fields of research, papers are more likely to be published, to be cited by colleagues and to be accepted by high-profile journals if they report results that are "positive"..."

Regarding the philosophical background of the article's critic, the text states: "Words like "positive", "significant", "negative" or "null" are common scientific jargon, but are obviously misleading, because all results are equally relevant to science, as long as they have been produced by sound logic and methods."

The article attributes the mentioned "publication bias" to the roots of science's philosophy and sociology, but also to a fact that is commonly ignored: "Like all human beings, scientists are confirmation-biased (i.e. tend to select information that supports their hypotheses about the world), and they are far from indifferent to the outcome of their own research: positive results make them happy and negative ones disappointed. This bias is likely to be reinforced by a positive feedback from the scientific community"... this statement seems to underestimate the influential power of the context, since the community can be the major influential factor to such behaviour...

"...careers are evaluated by counting the number of papers listed in their CVs and the impact factor of the journals they are published in..."

Another arguments can be to a certain extent linked to the idea of "Knowledge Monopolies": "Quantitative studies studies have repeatedly shown that financial interests can influence the outcome of biomedical research but they appear to have neglected the much more widespread conflict of interest created by scientists' need to publish. Yet, fears that the professionalization of research might compromise its objectivity and integrity had been expressed already in the 19th. century. Since then, the competitiveness and precariousness of scientific careers have increased, and evidence that this might encourage misconduct has accumulated".

One of the weaknesses of the study is its exclusive focus on quantitative techniques. This is causing the author to almost dismiss many important qualitative arguments and critiques of certain practices: "Surveys suggest that competitive research environments decrease the likelihood to follow scientific ideals... no direct quantitative study has verified the connection between pressures to publish and bias in the scientific literature, so the existence and gravity of the problem are still a matter of speculation and debate. Results

"The probability of papers to support the tested hypothesis increased significantly with the per capita academic productivity of the state of the corresponding author... "

Productivity is a statistically significant predictor, controlling for expenditure...

"...in US states where researchers publish more papers per capita were significantly more likely to report positive results, independently of their discipline, methodology and research expenditure... results supports the hypothesis that competitive academic environments increase not only the productivity of researchers, but also their bias against "negative" results.... the effect is truly cross-disciplinary. "

The study would have problems if any one asks about the definition of certain terms, like "intellectual environment": "An unavoidable confounding factor in this study is the quality and prestige of academic institutions, which is intrinsically linked to the productivity of their resident researchers. Indeed, official rankings of universities often include parameters measuring publication rates (although the validity of such rankings is controversial. Therefore, it could be argued that the more productive states are also the ones hosting the "best" universities, which provide better academic structures (laboratories, libraries, etc...) and more advanced and stimulating intellectual environments".

It is about a self-reinforced system: "Separating this quality-of-institution effect from that of bias induced by pressures to publish is difficult, because the two factors are strictly linked: the best universities are also the most competitive, and thus presumably the ones where pressures to produce are highest."

Cultural aspect with 2 different bias: 1) regarding missing negative results 2) regarding prestigious researchers and institutions: "... journal editors tend to reject papers from low-income countries unless they have particularly "good" results. If there were a similar editorial bias favouring highly prestigious universities in the US - and some studies suggest that there is - then the more productive states (prestigious institutions) should be allowed to publish more negative results.... the missing negative results? ... went completely unpublished or were somehow turned into positive through selective reporting, post-hoc re-interpretation, and alteration of methods, analysis and data. The relative frequency of these behaviours remains to be established, but the simple non-publication of results is unlikely to be the only explanation".

The article suggests that pressure to compulsory publishing has clear effects on research work itself and the communication of its results: "... positive results should be treated with the same scrutiny and rigour applied to negative ones, but with all likelihood they are not. This latter form of neglect is probably one of the main sources of bias in science".

"...the detrimental effects of the publish-or-perish culture could be manifest in other countries around the world". (Fanelli, 2010)

The luxury-journals tyranny

Despite ideological misinterpretation, the argument of Randy Schekman is clear and can be put as follows: - Place of publication is not a proxy for quality of science; - Outstanding research do not only appear in outstanding publications; - Well ranked journals target primarily the selling of subscriptions than stimulating research work; - Papers and journals cited more often (impact factor) are not necessarily the better ones; - Papers', Authors', Journals' and Institutions' score say little about the quality of research itself; - "Sexy" subjects or challenging claims are preferred by editors for editorial interests (attraction of readers attention), not for their contributions to any scientific subject; - This publishing system is influencing the science that scientists do;

Possible way out "There is a better way, through the new breed of open-access journals that are free for anybody to read, and have no expensive subscriptions to promote. Born on the web, they can accept all papers that meet quality standards, with no artificial caps. Many are edited by working scientists, who can assess the worth of papers without regard for citations. As I know from my editorship of eLife, an open access journal funded by the Wellcome Trust, the Howard Hughes Medical Institute and the Max Planck Society, they are publishing world-class science every week.

Funders and universities, too, have a role to play. They must tell the committees that decide on grants and positions not to judge papers by where they are published. It is the quality of the science, not the journal's brand, that matters. Most importantly of all, we scientists need to take action. Like many successful researchers, I have published in the big brands, including the papers that won me the Nobel prize for medicine, which I will be honoured to collect tomorrow.. But no longer. I have now committed my lab to avoiding luxury journals, and I encourage others to do likewise.

Just as Wall Street needs to break the hold of the bonus culture, which drives risk-taking that is rational for individuals but damaging to the financial system, so science must break the tyranny of the luxury journals. The result will be better research that better serves science and society." (Schekman, 2013)

//END OF PART AWAITING FOR EDITION//

An investigation conducted by me during the first semester of 2012 showed the effects of the enforcement of international quantitative standards as the main method to evaluate academic work in a specific environment. The target group in this case was the staff of the School of Forest Engineering (CIFLOMA) at the Federal University of Paraná / Southern Brazil.

The answers of a survey in combination with an in-depth interview applied to 46 researchers were almost unanimous: the prevailing form of research assessment (mandatory requirement of a minimal number of articles published in well ranked journals according to impact factors) is promoting a decrease in the significance of the research output. Several interviewees refer to this phenomenon as “papermania” or “publishing to publish” (in allusion to the “publish or perish” trend (Wikipedia, 2010)).

While submitting their texts almost exclusively to traditional publications with higher impact factors, the scientists at the CIFLOMA declared to be aware that these publications make digital copies of the articles to be available at libraries, institutional portals and other sites on the Internet.

Therefore, the online publication is only due to the publication on traditional journals, and at the end, this renders another reason for focusing the publishing efforts on these publications. In other words, the publication of articles has a compulsory, though exclusive character, and does not target the democratization or popularization of expert knowledge. Academic publishing is currently done almost exclusively to attend bureaucratic and administrative requirements. The obvious main reason is the competitive standard of assessment for promotion and tenure based in the number of published papers (CAPES, 2004).

Another evidence is the lack of conviction that knowledge must be shared (Stallbaum, 2005), represented by the resistance in publishing work in progress publicly (Table 1).

Table 1. Do you publish your "work in progress" in its early stages?

Note that, apparently the interviewees are not afraid of the openness embedded in digital communication. Only few of them justify this “resistance” against a more open approach towards data and research with “fear” in front of plagiarism threats, or any copyright infringement (Table 2). Most attributed the attitude to the mandatory requirement of submitting original papers, as an archaic demand from all relevant publications in the field. “It is simply the standard procedure not to expose the work or material under construction or analysis”, commented one of the interviewees.

Table 2. To what extent do you agree that other researchers are likely to copy or steal your ideas and work if you publish them online? The testimony of professor Nivaldo Rizzi, who was dean at the UFPR and manager of research funding policy for the Parana State for 10 years, addresses directly the issue concerning the lack of institutional policy towards communication services and a more efficient collaborative process among academics: "I am critical to any publication ranking being used nowadays. First of all, they do not represent our country's reality. Even if you take a single area, avoiding the comparison among them, you will see that it is structured in a corporative way. The list QUALIS, for instance, in our field (agricultural sciences), you will find there publications with lower quality, in terms of content, but with a better position in the ranking. In my view some of reasons for this bias is that the most ranking systems are based on a calculation system (index) that prioritizes data-bases conglomerates, and this is so because it is done by a private company... it means, it is unavoidable to have there a corporative bias... the magazines on the CAPES data-base, for example, are indexed according to the ISI index system, which is developed by a big private conglomerate. It is easy to see that if they do a business to it, they would inevitably manipulate their product in order to attend their interests. Our institutions nowadays work with the idea of "knowledge dominance", and not with the principle of knowledge sharing or collectivization of knowledge... it is a competitive standard... among individuals and institutions. This creates an entire set of behaviours accordingly, including the principle of competition which generates an attitude of secrecy towards data and information... the policy regarding research evaluation in Brazil nowadays reinforces this obsession for a competitive edge, it does not try to change this vicious circle... it is a philosophical matter, but it has a huge impact on how we work and are used to relate to each other. We miss a policy guideline to counter the current academic culture". (Prof. Nivaldo Rizzi, CIFLOMA, UFPR )

New Media = New Culture
The Internet presents a structural change on how texts, information and knowledge are produced, managed and distributed. Thus, it is reasonable to think about a structural transformation of the academic communication, including a cultural change on the production and assessment of texts. The Open Science Movement (Waldrop, 2008) seems to suggest a first step towards such change. Since communication is the heart of science, making it more open, logically appears to be the best way to improve connection with colleagues, create new knowledge and promote scientific progress (Mittler, 2007).

However, in many areas scholars remain sceptics regarding the so called scientific openness. The main reason again points to the competitive framework verified in those fields (e.g. ranking systems). It is to say that scepticism towards open science is directly proportional to academic rivalry, represented also by the importance of patents and intellectual property for funding accessibility, promotion and tenure. A report published by the Joint Information Systems Committee (JISC-UK) is clear about the issue: "'The traditional journal publishing model... does not reward data sharing, social Web contributions or peer production approaches to data curation... it is clear that the lack of incentives for data sharing and participatory methodologies, are a barrier to the wider adoption of the open science agenda.'(Liz Lyon, Open Science at Web Scale, JISC-UK, 2009 – p. 08)"

Institutions play a decisive role in this process, and normally act by limiting people's empathic feeling according to social, cultural or ethnic differences, introducing the base for separation and competition among peoples and groups. Therefore, currently the challenge is to de-naturalize the “social egoism” imposed by our institutions (Azzi, 2013).

Scholars have been challenged to transform their approach toward knowledge and information. This should be a new way, beyond political and economic convenience of established institutions. “We need to re-shape the entire academic culture”, defended the German mathematician and professor Martin Groetschel, during the Open Access Week (October, 2012) in Berlin, Germany. As an experienced OA activist, he believes that the solution for the bottle neck of the scientific production, today mostly concentrated in the hands of big publishers, depends on the scientists themselves. They have to find more transparent and democratic ways to deal with their own production, and these must be ways of true connection, real progress and unavoidable sustainability.

title: User:Sennadacosta


text:
Boundary Value Problems/Lesson 2

Boundary Value Problems

--First Order Differential Equation -- If the function $$ F(t,y) $$ is continuous in the variable $$ t \in \{t_0-\delta,t_0 + \delta \} $$ and uniformly Lipschitz in the variable $$ y $$ $$ \frac{ d y(t)}{dt} = F(t,y(t)) $$

title: Boundary Value Problems/Lesson 2


text:
Informative Writing

S3 Unit Plan
Informative Writing

Contents

 * /Lesson 1 - What are we looking for?/
 * /Lesson 2 - How will we find this?/
 * /Lesson 3 - Finding out what we are looking for./
 * /Lesson 4 - What have we found?/
 * /Lesson 5 - What can we make from this?/
 * /Lesson 6 - Can we improve this?/
 * /Lesson 7 - How can we improve?/
 * /Lesson 8 - What effects have our improvements had?/

title: Informative Writing


text:
Informative Writing/Lesson 1 - What are we looking for?

In this lesson we will...

 * Find out what we already know about Wikipedia.

 * Discover Wikiversity.

 * Decide what a Wikipedia entry needs to be successful.

 * Make a list of all the headings we will need in our article.


text:
Informative Writing/Lesson 2 - How will we find this?

In this lesson we will...
 * Learn how to cite sources correctly.
 * Decide on some useful sources to use.
 * Discuss possible research methods.
 * Divide the work effectively between group members.


text:
Informative Writing/Lesson 3 - Finding out what we are looking for.

In this lesson we will...
 * Gather relevant information.
 * Work effectively as a team.
 * Cite our sources.


text:
Informative Writing/Lesson 4 - What have we found?

In this lesson we will...
 * Share the information we have gathered.
 * Decide on the quality of this information.
 * Check what a Wikipedia entry needs to be successful.
 * Decide the best way to organise our information.


text:
Informative Writing/Lesson 5 - What can we make from this?

In this lesson we will...
 * Arrange our information into an entry.
 * Check all our sources.
 * Check that our spelling is perfect.


text:
Informative Writing/Lesson 6 - Can we improve this?

In this lesson we will...
 * Share the completed sections of our entry.
 * Make a list of any changes that we need to make.
 * Check our progress so far.


text:
Informative Writing/Lesson 7 - How can we improve?

In this lesson we will...
 * Make the changes we identified in the last lesson.
 * Complete our entry.
 * Check our entry against the Five pillars of Wikipedia.


text:
Informative Writing/Lesson 8 - What effects have our improvements had?

In this lesson we will...
 * Submit our Wikipedia entry.
 * Decide how well we worked throughout this unit.
 * Think about anything we might have done differently.
 * Discover how to make changes or improvements to our entry.


text:
Boundary Value Problems/Lesson 4

Return to Boundary Value Problems

Solution of a BVP
This page is copied from "http://en.wikiversity.org/wiki/Fourier_series"

The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. The weights, or coefficients, of the modes, are a one-to-one mapping of the original function. Generalizations include generalized Fourier series and other expansions over orthonormal bases.

Fourier series serve many useful purposes, as manipulation and conceptualization of the modal coefficients are often easier than with the original function. Areas of application include electrical engineering, vibration analysis, acoustics, optics, signal and image processing, and data compression. Using the tools and techniques of spectroscopy, for example, astronomers can deduce the chemical composition of a star by analyzing the frequency components, or spectrum, of the star's emitted light. Similarly, engineers can optimize the design of a telecommunications system using information about the spectral components of the data signal that the system will carry. See also spectrum analyzer.

The Fourier series is named after the French scientist and mathematician Joseph Fourier, who used them in his influential work on heat conduction, Théorie Analytique de la Chaleur (The Analytical Theory of Heat), published in 1822.

General form
Given a complex-valued function f of real argument t, f: R → C, where f(t) is piecewise smooth and continuous, periodic with period T, and square-integrable over the interval from $$ t_1 $$ to $$ t_2 $$ of length T, that is,

 * $$ \int_{t_1}^{t_2} |f(t)|^2\, dt<+\infty $$

where

 * $$ T = t_2 - t_1 $$ is the period,
 * $$ t_1 $$ and $$ t_2 $$ are integration bounds.

The Fourier series expansion of f is
 * $$ f(t) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty}[a_n \cos(\omega_n t) + b_n \sin(\omega_n t)] $$

where, for any non-negative integer n,

 * $$ \omega_n = n\frac{2\pi}{T}$$    is the nth harmonic (in radians) of the function f,
 * $$a_n = \frac{2}{T}\int_{t_1}^{t_2} f(t) \cos(\omega_n t)\, dt$$    are the even Fourier coefficients of f, and
 * $$b_n = \frac{2}{T}\int_{t_1}^{t_2} f(t) \sin(\omega_n t)\, dt $$    are the odd Fourier coefficients of f.

Equivalently, in complex exponential form,

 * $$f(t) = \sum_{n=-\infty}^{+\infty} c_n e^{i \omega_n t}$$

where:
 * $$c_n = \frac{1}{T}\int_{t_1}^{t_2} f(t) e^{-i \omega_n t}\, dt,$$
 * $$i$$ is the imaginary unit, and
 * $$e^{i \omega_n t} = \cos(\omega_n t) + i \sin(\omega_n t)$$    in accordance with Euler's formula.

For a formal justification, see Modern derivation of the Fourier coefficients below.

Canonical form
In the special case where the period T = 2π, we have

 * $$\omega_n = n \, $$

In this case, the Fourier series expansion reduces to a particularly simple form:

 * $$f(t) = \frac{1}{2} a_0 +\sum_{n=1}^{\infty}[a_n \cos(nt) + b_n \sin(nt)] $$

where
 * $$a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(t) \cos(nt)\, dt$$
 * $$b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(t) \sin(nt)\, dt$$

for any non-negative integer n.

or, equivalently:

 * $$f(t) = \sum_{n=-\infty}^{+\infty} c_n e^{i nt}$$

where
 * $$c_n = \frac{1}{2 \pi}\int_{-\pi}^{\pi} f(t) e^{-i nt}\, dt = \frac{1}{2}(a_n-ib_n).$$

Choice of the form
The form for period T can be easily derived from the canonical one with the change of variable defined by $$x=\frac{2\pi}{T}t$$. Therefore, both formulations are equivalent. However, the form for period T is used in most practical cases because it is directly applicable. For the theory, the canonical form is preferred because it is more elegant and easier to interpret mathematically, as will later be seen.

Simple Fourier series
Let f be periodic of period $$2\pi$$, with $$ f(x) = x $$ for x from −π to π. Note that this function is a periodic version of the identity function.

We will compute the Fourier coefficients for this function.

 * $$\begin{align}

a_n &{}= \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)\,dx \\ &{}= \frac{1}{\pi}\int_{-\pi}^{\pi}x \cos(nx)\,dx \\ &{}= 0. \end{align} $$

 * $$\begin{align}

b_n &{}= \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(nx)\,dx \\ &{}= \frac{1}{\pi}\int_{-\pi}^{\pi} x \sin(nx)\, dx \\ &{}= \frac{2}{\pi}\int_{0}^{\pi} x\sin(nx)\, dx \\ &{}= \frac{2}{\pi} \left(\left[-\frac{x\cos(nx)}{n}\right]_0^{\pi} + \left[\frac{\sin(nx)}{n^2}\right]_0^{\pi}\right) \\ &{}= 2\frac{(-1)^{n+1}}{n}.\end{align}$$

Notice that an are 0 because the $$x\mapsto x\cos(nx)$$ are odd functions. Hence the Fourier series for this function is:

 * $$f(x)=\frac{a_0}{2} + \sum_{n=1}^{\infty}\left[a_n\cos\left(nx\right)+b_n\sin\left(nx\right)\right] $$

 * $$=2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} \sin(nx), \quad \forall x\in [-\pi,\pi].$$

One application of this Fourier series is to compute the value of the Riemann zeta function at s = 2; by Parseval's theorem, we have:
 * $$ \frac{1}{2\pi} \int_{-\pi}^\pi x^2 dx=\frac{1}{2}\sum_{n>0}\left[2\frac{(-1)^n}{n}\right]^2$$

which yields: $$\sum_{n>0}\frac{1}{n^2}=\frac{\pi^2}{6}$$.

Modern derivation of the Fourier coefficients
The method used by Fourier to derive the coefficients of the series is very practical and well-suited to the problem he was dealing with (heat propagation). However, this method has since been generalized to a much wider class of problems: writing a function as a sum of periodic functions.

More precisely, if f:R → C is a function, we would like to write this function as a sum of trigonometric functions, i.e. $$f(x)=\sum c_n e^{inx}$$. We have to restrict our choice of functions in order for this to make sense. First of all, if f has period T, then by changing variables, can study $$x\mapsto f\left(\frac{T}{2\pi}x\right)$$ which has period 2π. This simplifies notations a lot and allows us to use a canonical (standard) form. We can restrict the study of $$x\mapsto f\left(\frac{T}{2\pi}x\right)$$ to any interval of length 2π, [-π,π], say.

We will take the functions f:R → C in the set of piecewise continuous, 2π periodic functions with $$\int_{-\pi}^\pi |f(x)|^2 \, dx<+\infty$$. Technically speaking, we are in fact taking functions from the Lp space L2(μ), where μ is the normalized Lebesgue measure of the interval [-π,π] (i.e. such that $$\int_{[-\pi,\pi]}f \, d\mu=\frac{1}{2\pi}\int_{-\pi}^\pi f(x)\,dx$$.

Complex Fourier coefficients
We can make L2(μ) into a Hilbert space, which is well-suited for orthogonal projections, by defining the scalar product:

 * $$\langle f, g \rangle = \int_{[-\pi,\pi]} f \overline{g} \,d\mu=\frac{1}{2\pi}\int_{-\pi}^\pi f(x)\overline{g(x)}\,dx,$$

where $$\overline{f(x)}$$ denotes the conjugate of f(x). We will denote by $$\| \cdot \|$$ the associated norm.

$$E=\{t\mapsto e^{i n t},n\in\mathbb{Z}\}$$ is an orthonormal basis of L2(μ), which means we can write

 * $$f(x)=\sum_{n\in\mathbb{Z}}\left\langle f,e^{i n x}\right\rangle e^{i n x}.$$

We usually define $$\forall n\in\mathbb{Z}, c_n=\left\langle f,e^{i n x}\right\rangle$$. These numbers are called complex Fourier coefficients. Their expression is

 * $$c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-i n x}\,dx.\,$$

An equivalent formulation is to write f as a sum of sine and cosine functions.

Real Fourier coefficients
The sum in the previous section is symmetrical around 0: indeed, except for n=0, a c-n coefficient corresponds to every cn coefficient. This reminds one of the formulae

 * $$\cos(x)=\frac{e^{ix}+e^{-ix}}{2}{\rm~and~}\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}.$$

It is therefore possible to express the Fourier series with real-valued functions. To do this, we first notice that

 * $$f(x)=\sum_{n\in\mathbb{Z}}c_n e^{i n x}=c_0+\sum_{n>0}\left[c_{-n}e^{-i n x}+c_n e^{i n x}\right].$$

After replacing cn by its expression and simplifying the result we get

 * $$f(x)=c_0+\sum_{n>0}\left[\frac{1}{\pi}\left(\int_{-\pi}^\pi f(t)\cos\left(n t\right)\, dt\right)\cos\left(n x\right)+\frac{1}{\pi}\left(\int_{-\pi}^\pi f(t)\sin\left(n t\right)\, dt\right)\sin\left(n x\right)\right].$$

If, for a non-negative integer n, we define the real Fourier coefficients an and bn by

 * $$a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \cos\left(n x\right)\, dx,$$

 * $$b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \sin\left(n x\right)\, dx, $$

we get:

 * $$f(x)=\frac{a_0}{2}+\sum_{n>0}\left[a_n\cos\left(n x\right)+b_n\sin\left(n x\right)\right].$$

Properties

 * The following properties can be easily derived from Euler's formula: $$e^{ix} = \cos x + i\sin x. \!$$

 * $$a_n=c_n+c_{-n}\mbox{ and }b_n=i(c_n-c_{-n})\mbox{ for all } n \mbox{ and }\,$$
 * $$c_n=\frac{a_n-ib_n}{2} \mbox{ and }c_{-n}=\frac{a_n+ib_n}{2}\mbox{ for all } n.$$

 * If f is an odd function, then $$a_n=0$$ for all $$n$$ because $$f(x)\cos\left(n\pi\frac{x}{T}\right)$$ is then also odd, so its integral on $$[-T, T]$$ is zero. If f is an even function, then $$b_n=0$$ for a similar reason.

 * If f is piecewise continuous, $$\lim_{n\rightarrow +\infty}c_n(f)=0$$, $$\lim_{n\rightarrow +\infty}c_{-n}(f)=0$$, $$\lim_{n\rightarrow +\infty}a_n(f)=0$$ and $$\lim_{n\rightarrow +\infty}b_n(f)=0.$$

 * If f is k-times piecewise continuously differentiable, then we can easily compute the Fourier coefficients of $$f^{(k)}$$ given those of f:

 * $$c_n\left(f^{(k)}\right)=(in)^k c_n(f),$$

where $$f^{(k)}$$ denotes the kth derivative of f.

 * For any positive integer k, if f is Ck − 1 and piecewise Ck, then
 * $$\lim_{n\rightarrow +\infty}|n^kc_n(f)|=0$$ because $$n^kc_n(f)=i^{-k}c_n\left(f^{(k)}\right)\rightarrow 0.$$

This means that the sequence $$c_n(f)$$ is rapidly decreasing.

General case
Fourier series take advantage of the periodicity of a function f but what if f is periodic in more than one variable, or for that matter, what if f is not periodic? These problems led mathematicians and theoretical physicists to try to define Fourier series on any group G. The advantage of this is that it allows us, for example, to define Fourier series for functions of several variables. Fourier series and Fourier transforms usually used in signal processing then become special cases of this theory and are easier to interpret.

If G is a locally compact Abelian group and T is the unit circle, we can define the dual of G by $$ \widehat{G} = \{\chi:G\rightarrow\mathbb{T} \mbox{ homomorphism}\}$$. This is the set of rotations on the unit circle and its elements are called characters. We can define a scalar product $$\langle\cdot,\cdot\rangle$$ on C[G] by: $$\langle\chi_1, \chi_2\rangle=\int_{G}\chi_1(g) \overline{\chi_2(g)}\,dg$$. $$\widehat{G}$$ is then an orthonormal basis of C[G] with respect to this scalar product. Let f :G → C. The Fourier coefficients of f are defined by: $$\widehat{f}(\chi)=\langle f,\chi\rangle$$ and we have $$ f(g) = \int_{\widehat{G}} \widehat{f} (\chi)\chi(g)\,d\chi$$. If the group is discrete, then the integral reduces to an ordinary sum.

For example, the Fourier series of this article are obtained by taking G=R/2πZ. We get

 * $$\widehat{G}=\{\chi_n:t\mapsto e^{i n t}, n\in\mathbb{Z}\}

$$

and

 * $$c_n(f) = \widehat{f}(\chi_n) = \int_G f(g)\overline{\chi(g)}\,dg = \frac{1}{2 \pi}\int_{-\pi}^{\pi} f(t) e^{-i nt}\,dt.$$

Periodic functions in n dimensions can be defined on an n-dimensional torus (the function taking a value at each point on the torus). Such a torus is defined by Tn=Rn/(2πZ)n. For n=1 we get a circle, for n=2 the cartesian product of two circles, i.e. a torus in the usual sense. Choosing G=Tn gives the corresponding Fourier series.

Definition of a Fourier series
Let $$\chi_n(x)=e^{in\pi \frac{x}{T}}$$. We call Fourier series of the function f the series $$\sum c_n \chi_n$$. For any positive integer N, we call $$f_N(x)=\sum_{n=-N}^Nc_n \chi_n(x)$$ the N-th partial sum of the Fourier series of this function.

Approximation with the partial sums
Say we want to find the best approximation of f using only the functions $$\chi_n$$ for n from $$-N$$ to N. Let $$\mathcal{T}_N=\left\{p=\sum_{n=-N}^N x_n \chi_n, x_n\in\mathbb{C}\right\}$$. We are trying to find coefficients $$(x_{-N},\dots,x_{N})$$ such that $$\|f-p\|$$ is minimum (where $$\| \cdot \|$$ denotes the norm).

We have $$\|f-p\|^2=\|f\|^2-2\mbox{Re}\langle f,p\rangle+\|p\|^2$$, where Re(z) denotes the real part of z.

 * $$\langle f,p\rangle=\sum_{n=-N}^N\overline{x_n}\langle f,\chi_n\rangle.$$

Parseval's theorem (which can be derived independently from Fourier series) gives us

 * $$\|p\|^2=\sum_{n=-N}^N|x_n|^2.$$

By definition, $$c_n=\langle f,\chi_n\rangle$$; therefore

 * $$\|f-p\|^2=\|f\|^2+\sum_{n=-N}^N\left[|c_n-x_n|^2-|c_n|^2\right].$$

It is clear that this expression is minimum for $$x_n=c_n$$ and for this value only.

This means that there is one and only one $$f_N\in\mathcal{T}_N$$ such that

 * $$\|f-f_N\|=\min_{p\in\mathcal{T}_N}\left\{\|f-p\|,p\in\mathcal{T}_N\right\},$$

it is given by

 * $$f_N(x)=\sum_{n=-N}^N c_n \chi_n(x),$$

where

 * $$c_n=\frac{1}{2T}\int_{-T}^T f(t)\chi_{-n}(t)\,dt.$$

This means that the best approximation of f we can make using only the functions $$\chi_n(x)=e^{in\pi \frac{x}{T}}$$ for n from $$-N$$ to N is precisely the Nth partial sum of the Fourier series. An illustration of this is given on the animated plot of example 1.

Convergence
While the Fourier coefficients an and bn can be formally defined for any function for which the integrals make sense, whether the series so defined actually converges to f(x) depends on the properties of f.

The simplest answer is that if f is square-integrable then

 * $$\lim_{N\rightarrow\infty}\int_{-\pi}^\pi\left|f(x)-\sum_{n=-N}^{N}

c_n\,\chi_n(x)\right|^2\,dx=0.$$

This is convergence in the norm of the space L2. The proof of this result is simple, unlike Lennart Carleson's much stronger result that the series actually converges almost everywhere.

There are many known tests that ensure that the series converges at a given point x, for example, if the function is differentiable at x. Even a jump discontinuity does not pose a problem: if the function has left and right derivatives at x, then the Fourier series will converge to the average of the left and right limits (but see Gibbs phenomenon). However, a fact that many find surprising, is that the Fourier series of a continuous function need not converge pointwise.

This unpleasant situation is counter-balanced by a theorem by Dirichlet which states that if f is $$2T$$-periodic and piecewise continuously differentiable function, then its Fourier series converges pointwise and $$\sum_{n\in\mathbb{Z}} c_n \chi_n(x)=\frac{f(x^+)+f(x^-)}{2}$$, where $$f(x^+)=\lim_{t\rightarrow x, t>x} f(x)$$ and $$f(x^-)=\lim_{t\rightarrow x, t<x} f(x)$$. If f is continuous as well as piecewise continuously differentiable, then the Fourier series converges uniformly.

In 1922, Andrey Kolmogorov published an article entitled Une série de Fourier-Lebesgue divergente presque partout in which he gave an example of a Lebesgue-integrable function whose Fourier series diverges almost everywhere. This function is not in $$L^2(\mu)$$.

Plancherel's and Parseval's theorems
Another important property of the Fourier series is the Plancherel theorem. Let $$f,g\in L^2(\mu)$$ and $$c_n(f), c_n(g)$$ be the corresponding complex Fourier coefficients. Then

 * $$\sum_{n\in\mathbb{Z}} c_n(f)\overline{c_n(g)} = \frac{1}{2T} \int_{-T}^T f(x)\overline{g(x)}\,dx$$

where $$\overline{z}$$ denotes the conjugate of z.

Parseval's theorem, a special case of the Plancherel theorem, states that:

 * $$\sum_{n\in\mathbb{Z}} |c_n(f)|^2 = \frac{1}{2T} \int_{-T}^T |f(x)|^2 \,dx$$

which can be restated with the real Fourier coefficients:

 * $$\frac{a_0^2}{4} + \frac{1}{2} \sum_{n=1}^\infty \left( a_n^2 + b_n^2 \right) = \frac{1}{2T} \int_{-T}^T |f(x)|^2\, dx.$$

These theorems may be proven using the orthogonality relationships. They can be interpreted physically by saying that writing a signal as a Fourier series does not change its energy.

title: Boundary Value Problems/Lesson 4


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