text:
User:EGM6341.s10.Team1.Kumanchik/HW1

Hello, We are Team 1. --Rjorjorjo 10:44, 26 January 2010 (UTC) =Problem 2-1=

Find
Determine $$\lim _{x\mapsto 0}f\left ( x \right )$$ and plot $$f\left ( x \right ), x\in \left [ 0,1 \right ]$$

Given

 * $$f\left ( x \right )=\frac{e^{x}-1}{x}$$

Solution [[media:HW1_P2_1.jpg|P2-1]]
Determine the limit of the numerator and denominator separately and then evaluate the quotient.

Numerator:
 * $$\frac{\mathrm{d} }{\mathrm{d} x}\left ( e^{x}-1 \right )|_{x=0}=e^{x}|_{x=0}=1$$

Demoninator:
 * $$\frac{\mathrm{d} }{\mathrm{d} x}\left ( x \right )|_{x=0}=1$$

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\lim _{x\mapsto 0}\frac{e^{x}-1}{x}=\frac{1}{1}=1 $$ $$
 * $$\displaystyle
 * }
 * }

=Problem 2-3=

Find
Determine $$P_{n}\left ( x \right )$$ and $$R_{n+1}\left ( x \right )$$ for $$e^{x} \,$$

Given

 * $$P_{n}\left ( x \right )=f\left ( x_{0} \right )+\frac{x-x_{0}}{1!}f^{(1)}\left ( x_{0} \right )+...+\frac{x-x_{0}}{n!}f^{(n)}\left ( x_{0} \right )$$

And
 * $$R_{n+1}\left ( x \right )=\frac{\left (x-x_{0} \right )^{n+1}}{\left ( n+1 \right )!}f^{(n+1)}\left ( \xi  \right ) for \xi \in [x_{0},x]$$

Solution [[media:HW1_P2_3.jpg|P2-3]]
Since $$x_{0}=0 \,$$ then $$f\left ( x_{0} \right )=e^{0}=1$$ In fact every derivative of $$e^{x} \,$$ is $$e^{x}\,$$ therefore $$f^{(n)}\left ( x_{0} \right )=1$$ Therefore,
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

P_{n}\left ( x \right )=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{3!}+...+\frac{x^{n}}{n!} $$ $$
 * $$\displaystyle
 * }
 * }

=Problem 3-3=

Find
Determine $$\left \| f \right \|_{\infty }$$, $$\left \| g \right \|_{\infty }$$, and $$\left \| f-g \right \|_{\infty }$$ Plot $$f\left ( x \right ), g\left ( x \right ) x\in [0,\pi ]$$

Given

 * $$f\left ( x \right )=\sin x$$
 * $$g\left ( x \right )=\sin \left ( x-\frac{\pi }{2} \right )=-\cos x$$

Solution [[media:HW1_P3_3.jpg|P3-3]]
$$\left \| f \right \|_{\infty }=\max _{x}\left | f\left ( x \right ) \right |$$, so we must find where $$\frac{\mathrm{df} }{\mathrm{d} x}=0$$

Start with $$f\left ( x \right )$$
 * $$\frac{\mathrm{d} }{\mathrm{d} x}\left ( \sin x \right )=\cos x=0$$ between $$[0,\pi ] \,$$
 * $$\Rightarrow x=\frac{\pi }{2}$$
 * $$\Rightarrow \sin \left ( \frac{\pi }{2} \right )=1$$
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\left \| \sin x \right \|_{\infty }=1,x=\frac{\pi }{2} $$ $$ Next do $$g\left ( x \right )$$
 * $$\displaystyle
 * }
 * }
 * $$\left \| -\cos x \right \|_{\infty }=\max _{x}\left | -\cos x \right |=\max _{x}\left ( \cos x \right )$$
 * $$\frac{\mathrm{d\left ( \cos x \right )} }{\mathrm{d} x}=-\sin x=0$$
 * $$\Rightarrow \cos x=1|_{x=0}$$
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\left \| -\cos x \right \|_{\infty }=1,x=0 $$ $$ Finally do $$f-g \,$$
 * $$\displaystyle
 * }
 * }
 * $$\left \| f-g \right \|_{\infty }=\max _{x}\left | \sin x- \left ( -\cos x \right ) \right |=\max _{x}\left | \sin x+\cos x \right |$$
 * $$\frac{\mathrm{d} }{\mathrm{d} x}\left ( \sin x+\cos x \right )=\cos x-\sin x=0$$
 * $$\Rightarrow \cos x=\sin x$$
 * $$\Rightarrow x=\frac{\pi }{4}$$
 * $$\sin \left ( \frac{\pi }{4} \right )+\cos \left ( \frac{\pi }{4} \right )=\sqrt{2}$$
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\left \| \sin x+\cos \right \|_{\infty }=\sqrt{2},x=\frac{\pi }{4} $$ $$
 * $$\displaystyle
 * }
 * }

=Problem 5-1a=

Find

 * 1) Prove the Intermediate Mean Value Theorem for $$w\left ( . \right )$$ being non-negative, i.e. $$w\geq 0$$
 * 2) Prove the IMVT for $$w\left ( x \right )\neq 0, \forall x\in [a,b]$$

Given
IMVT
 * $$\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx=f\left ( \xi \right )\int_{a}^{b}w\left ( x \right )dx$$

Solution [[media:HW1_P5_1_1.jpg|P5-1a]]
1. For $$w\left ( x \right )\geq 0, \forall x\in [a,b]$$, we aim to prove that there exists a $$\xi \,$$ within $$[a,b] \,$$ such that $$\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx=f\left ( \xi \right )\int_{a}^{b}w\left ( x \right )dx$$.

If the functions are continuous on this interval there exists both a minimum (m) and maximum (M) value of $$w\left ( x \right )f\left ( x \right )$$ within $$[a,b]$$.

Thus,
 * $$\int_{a}^{b}mdx\leq \int_{a}^{b}w\left ( x \right )f\left ( x \right )dx\leq \int_{a}^{b}Mdx, \forall x\in [a,b]$$
 * $$\Rightarrow m\leq \frac{1}{b-a}\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx\leq M$$

If the functions are continuous on $$[a,b]$$ there must exist a $$\xi \,$$ in $$[a,b]$$ where,
 * $$f\left ( \xi \right )\int_{a}^{b}w\left ( x \right )dx=\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx$$

2. $$w> 0 \,$$ was covered in 1, and for $$w< 0 \,$$ the result is similar. In the case of $$w\left ( x \right )\neq 0$$ there must exist a minimum(m) and maximum(M) value of $$w\left ( x \right )f\left ( x \right )$$ within $$[a,b]$$.

 * $$\int_{a}^{b}mdx\leq \int_{a}^{b}w\left ( x \right )f\left ( x \right )dx\leq \int_{a}^{b}Mdx, \forall x\in [a,b]$$

though in this case the minumim and/or maximum could be negative. Again,
 * $$m\leq \frac{1}{b-a}\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx\leq M$$

This integral could be negative, but like in part-1 there must exist a $$\xi \,$$ in $$[a,b] \,$$ where
 * $$f\left ( \xi \right )\int_{a}^{b}w\left ( x \right )dx=\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx$$

=Problem 5-1b=

Find
Use the Intermediate Mean Value Theorem to show Eq. (5) [http://upload.wikimedia.org/wikipedia/commons/b/b5/Egm6341.s10.mtg2.pdf Lec. 2-2] and Eq. (1) [http://upload.wikimedia.org/wikipedia/commons/b/b5/Egm6341.s10.mtg2.pdf Lec. 2-3]

Given
Eq. (5)
 * $$R_{n+1}\left ( x \right )=\frac{1}{n!}\int_{x_{0}}^{x}\left ( x-t \right )^{n}f^{(n+1)}\left ( t \right )dt$$

Eq. (1)
 * $$R_{n+1}\left ( x \right )=\frac{(x-x_{0})^{n+1}}{(n+1)!}f^{(n+1)}\left ( \xi \right )$$

IMVT
 * $$\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx=f\left ( \xi \right )\int_{a}^{b}w\left ( x \right )dx$$

Solution [[media:HW1_P5_1_2.jpg|P5-1b]]
Let $$w\left ( x \right )$$ be nonnegative in the integral on $$[a,b] \,$$ and let $$f\left ( x \right )$$ be continuous on $$[a,b] \,$$. Then,

 * $$\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx=f\left ( \xi \right )\int_{a}^{b}w\left ( x \right )dx$$

for
 * $$\xi \in [a,b]$$

Using a Taylor series expansion,
 * $$f\left ( x \right )=P_{n}\left ( x \right )+R_{n+1}\left ( x \right )$$

where
 * $$R_{n+1}\left ( x \right )=\frac{1}{n!}\int_{x_{0}}^{x}\left ( x-1 \right )^{n}f^{(n+1)}\left ( t \right )dt$$
 * $$\Rightarrow \frac{1}{n!}f^{(n+1)}\left ( \xi \right )\int_{x_{0}}^{x}\left ( x-t \right )^{n}dt$$
 * $$\Rightarrow \frac{1}{n!}f^{(n+1)}\left ( \xi \right )\left ( \frac{-1}{n+1} \right )\left ( x-t \right )^{n+1}|_{x_{0}}^{x}$$
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\Rightarrow \frac{(x-x_{0})^{n+1}}{(n+1)!}f^{(n+1)}\left ( \xi \right ) $$ $$
 * $$\displaystyle
 * }
 * }

=Problem 5-3=

Find
Integrate by parts to obtain
 * $$\frac{(x-x_{0})^{2}}{2!}f^{(2)}\left ( x_{0} \right )+\frac{(x-x_{0})^{3}}{3!}f^{(3)}\left ( x_{0} \right )$$

and show the remainder.

Then assume Eq. (4) and Eq. (5) [http://upload.wikimedia.org/wikipedia/commons/b/b5/Egm6341.s10.mtg2.pdf Lec. 2-2] are true and do integration by parts once more.

Given

 * $$f\left ( x \right )=f\left ( x_{0} \right )+\left ( x-x_{0} \right )f^{(1)}\left ( x_{0} \right )+\int_{x_{0}}^{x}\left ( x-t \right )f^{(2)}\left ( t \right )dt$$

Solution [[media:HW1_P5_3.jpg|P5-3]]
Integrate by parts,
 * $$\int_{x_{0}}^{x}\underbrace{\left ( x-t \right )}_{u'}\underbrace{f^{(2)}\left ( t \right )}_{v}dt=\left [ -\frac{(x-t)^{2}}{2} f^{(2)}\left ( t \right )\right ]_{x_{0}}^{x}-\int_{x_{0}}^{x}\frac{-(x-t)^{2}}{2}f^{(3)}\left ( t \right )dt$$
 * $$\Rightarrow 0+\frac{(x-x_{0})^{2}}{2}f^{(2)}\left ( x_{0} \right )+\int_{x_{0}}^{x}\frac{(x-t)^{2}}{2}f^{(3)}\left ( t \right )dt$$
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

f\left ( x \right )=f\left ( x_{0} \right )+\frac{(x-x_{0})}{1!}f^{(1)}\left ( x_{0} \right )+\frac{(x-x_{0})^{2}}{2!}f^{(2)}\left ( x_{0} \right )+\int_{x_{0}}^{x}\frac{(x-t)^{2}}{2}f^{(3)}\left ( t \right )dt $$ $$
 * $$\displaystyle
 * }
 * }

And the remainders
 * $$f\left ( x \right )=P_{n}\left ( x \right )+R_{n+1}\left ( x \right )$$
 * $$P_{n}\left ( x \right )=f\left ( x_{0} \right )+\frac{(x-x_{0})}{1!}f^{(1)}\left ( x_{0} \right )+...+\frac{(x-x_{0})^{n}}{n!}f^{(n)}\left ( x_{0} \right )$$
 * $$R_{n+1}\left ( x \right )=\frac{1}{n!}\int_{x_{0}}^{x}\underbrace{\left ( x-t \right )^{n}}_{u'}\underbrace{f^{(n+1)}\left ( t \right )}_{v}dt$$

Integrate by parts of $$R_{n+1}\left ( x \right )$$
 * $$\frac{1}{n!}\int_{x_{0}}^{x}\left ( x-t \right )^{n}f^{(n+1)}\left ( t \right )dt=\frac{1}{n!}\left \{ \left [ -\frac{(x-t)^{n}}{(n+1)}f^{(n+1)}\left ( t \right ) \right ]_{x_{0}}^{x} - \int_{x_{0}}^{x}\frac{-(x-t)^{n+1}}{(n+1)}f^{(n+1)}\left ( t \right )dt\right \}$$
 * $$\Rightarrow 0+\frac{(x-x_{0})^{n+1}}{(n+1)!}f^{(n+1)}\left ( x_{0} \right )+\int_{x_{0}}^{x}\frac{(x-t)^{n+1}}{(n+1)!}f^{(n+2)}dt$$
 * $$f\left ( x \right )=P_{n+1}\left ( x \right )+R_{n+2}\left ( x \right )$$

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

P_{n+1}\left ( x \right )=f\left ( x_{0} \right )+\frac{(x-x_{0})}{1!}f^{(1)}\left ( x_{0} \right )+...+\frac{(x-x_{0})^{n+1}}{(n+1)!}f^{(n+1)}\left ( x_{0} \right ) $$ $$
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

R_{n+2}\left ( x \right )=\frac{1}{(n+1)!}\int_{x_{0}}^{x}(x-t)^{n+1}f^{(n+2)}dt $$ $$
 * $$\displaystyle
 * }
 * }

=Problem 6-1=

Find
Construct the Taylor series of $$f(x) \,$$ around $$x_0=\frac{\pi}{4}$$ for $$n=0,1,...,10 \,$$.

Plot these series for each $$n \,$$. Then estimate the maximum of $$R(x) \,$$ at $$x=\frac{\pi}{2}$$

Given

 * $$f\left ( x \right )=\sin x$$ for $$x\in [0,\pi ]$$

Solution [[media:HW1_P6_1_1.jpg|P6-1]]

 * $$f\left ( x \right )=\sin x=P_n(x)+R_{n+1}(x)$$
 * $$P_n(x)=f(x_0)+\frac{(x-x_0)}{1!}f^{(1)}(x_0)+\frac{(x-x_0)^2}{2!}f^{(2)}(x_0)+...+\frac{(x-x_0)^n}{n!}f^{(n)}(x_0)$$
 * $$R_{n+1}(x)=\frac{(x-x_0)^{n+1}}{(n+1)!}f^{(n_1)}(\xi)$$

For
 * $$f(x)=\sin x, f^{(1)}(x)=\cos x, f^{(2)}(x)=-\sin x, f^{(3)}(x)=-\cos x, f^{(4)}(x)=\sin x \,$$

When $$n=0 \,$$
 * $$\Rightarrow \sin x=P_0(x)+R_1(x)=\frac{\sqrt{2}}{2}+(x-\frac{\pi}{4})\cos \xi$$

When $$n=1 \,$$
 * $$\Rightarrow \sin x=P_1(x)+R_2(x)=\frac{\sqrt{2}}{2}\left [ 1+\left ( x-\frac{\pi}{4} \right ) \right ]-\frac{(x-\frac{\pi}{4})^2}{2!}\sin \xi$$

When $$n=2 \,$$
 * $$\Rightarrow \sin x=P_2(x)+R_3(x)=\frac{\sqrt{2}}{2}\left [ 1+\left ( x-\frac{\pi}{4} \right )-\frac{(x-\frac{\pi}{4})^2}{2!} \right ]-\frac{(x-\frac{\pi}{4})^3}{3!}\cos \xi$$

When $$n=3 \,$$
 * $$\Rightarrow \sin x=P_3(x)+R_4(x)=\frac{\sqrt{2}}{2}\left [ 1+\left ( x-\frac{\pi}{4} \right )-\frac{(x-\frac{\pi}{4})^2}{2!}-\frac{(x-\frac{\pi}{4})^3}{3!} \right ]+\frac{(x-\frac{\pi}{4})^4}{4!}\sin \xi$$

...

When $$n=10 \,$$
 * $$\Rightarrow \sin x=P_{10}(x)+R_{11}(x)=\frac{\sqrt{2}}{2}\left [ 1+\left ( x-\frac{\pi}{4} \right )-\frac{(x-\frac{\pi}{4})^2}{2!}-\frac{(x-\frac{\pi}{4})^3}{3!}...- \frac{(x-\frac{\pi}{4})^{10}}{10!}\right ]-\frac{(x-\frac{\pi}{4})^{11}}{11!}\cos \xi$$

Finally, the maximum can be computed with,
 * $$R_{n+1}(x)=\frac{(x-x_0)^{n+1}}{(n+1)!}f^{(n+1)}(\xi)$$ for $$\xi \in [0,\pi] \,$$
 * $$\left | R_{n+1}(x) \right |\leq \frac{(x-x_0)^{n+1}}{(n+1)!}\max \left \{ f^{(n+1)}(\xi) \right \} \leq \frac{(x-x_0)^{n+1}}{(n+1)!}=\frac{(\frac{\pi}{4})^{n+1}}{(n+1)!}\leq \frac{\pi}{4}$$

=Problem 6-5=

Find
Use the following three methods to find the integral, $$I \, $$

1. Taylor series expansion of $$f_n\,$$

2 Composite Trapezoidal Rule

3 Composite Simposon Rule

Let $$ n=2,4,8,... \,$$ until the error is of magnitude $$10^{-6} \,$$

Given

 * $$I=\int_{0}^{1}\frac{e^{x}-1}{x}dx$$

Solution [[media:HW1_P6_5_1.jpg|P6-5a]], [[media:HW1_P6_5_2.jpg|P6-5b]], [[media:HW1_P6_5_3.jpg|P6-5c]]
1. Taylor Series
 * $$e^{x}=\sum _{j=0}^{\infty }\frac{x^j}{j!}\Rightarrow \frac{e^{x}-1}{x}=\sum _{j=1}^{\infty }\frac{x^{j-1}}{j!}$$
 * $$I_n=\int_{0}^{1}f_n(x)dx=\int_{0}^{1}\sum _{j=1}^{\infty} \frac{x^{j-1}}{j!}dx=\sum _{j=1}^{n}\frac{x^3}{j!j}|_0^1=\sum _{j=1}^n \frac{1}{j!j}$$
 * $$f(x)-f_n(x)=R_n(x)=\frac{(x-0)^n}{(n+1)!}f^{(n+1)}(\xi)$$ with $$\xi \in [0,x] \,$$ and $$x_0=0 \,$$
 * $$E_n=I-I_n=\int_{0}^{1}\left [ f(x)-f_n(x) \right ]dx=\int_{0}^{1}\overbrace{\frac{x^n}{(n+1)!}}^{w(x)}\overbrace{f^{(n+1)}\left (\xi(x) \right )}^{g(x)}dx$$
 * $$\Rightarrow g(\alpha)\int_{0}^{1}w(x)dx$$ for $$\alpha \in [0,1]$$
 * $$E_n =\,$$
 * $$\Rightarrow max = \frac{g(\alpha)}{(n+1)!(n+1)}$$, $$\alpha=1 \,$$
 * $$\Rightarrow min = \frac{g(\alpha)}{(n+1)!(n+1)}$$, $$\alpha=0 \,$$
 * $$n=3, E_n \leq 0.02 \,$$
 * $$n=5, E_n \leq 8x10^{-5} \,$$

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

n=6, E_n \leq 9x10^{-6} \, $$ $$
 * $$\displaystyle
 * }
 * }

2. Comp. Trap. Rule
 * $$\int_{x_0}^{x_1}f(x)dx=\int_{0}^{h}f(t+x_0)dt=\left [ (t+A)F(t+x_0) \right ]_0^h-\int_{0}^{h}(t+A)f'(t+x_0)dt$$
 * $$\int_{a}^{b}f(x)dx\approx \left ( \frac{b-a}{n} \right )\left [ \frac{f(a)+f(b)}{2}+\sum _{k=1}^{n-1}f\left (a+k\frac{b-a}{n} \right ) \right ]$$

We want $$\left [ (t+A)f(t+x_0) \right ]_0^h=\frac{(f(x_0)+f(x_1))}{2}(h)$$ with $$x_1=x_0+h \,$$,
 * $$\Rightarrow A=\frac{-h}{2}$$

We want our error bound small
 * $$\Rightarrow \left [ \left ( \frac{(t+A)^2}{2}+B \right )f'(t+x_0) \right ]_0^h=0$$
 * $$\left [\left ( \frac{\left ( \frac{h}{2} \right )^2}{2}+B \right )f'(h+x_0) \right ]-\left ( \frac{\left ( \frac{-h}{2} \right )^2}{2}+B \right )f'(x_0)=0$$
 * $$\Rightarrow B=\frac{-h^2}{8}$$
 * $$\int_{x_0}^{x_1}f(x)dx=\frac{h}{2}\left [ f(x_0)+f(x_1) \right ]+\int_{0}^{h}\left ( \tfrac{\left ( t-\frac{h}{2} \right )^2}{2}+\frac{h^2}{8} \right )f''(t+x_0)dt$$

The error in the Trap. Rule equals the sum of these:
 * $$E_T=E(0)+E(1)+...+E(n-1) \,$$
 * $$\int_{0}^{h}Qf(t+x_0)dt+\int_{0}^{h}Qf(t+x_1)dt+...+\int_{0}^{h}Qf''(t+x_0)dt$$
 * $$=\int_{0}^{h}Q\left (f(t+x_0)+...+f(t+x_{n-1}) \right )dt$$
 * $$E_T=\int_{0}^{h}\frac{\left (t-\frac{h}{2} \right )^2}{2}\frac{h^2}{8}\left [ f(t+x_0)+...+f(t+x_{n-1}) \right ]dt=\frac{-h^3}{12}$$

 * $$E_T=\frac{-h^3}{12n^3}f''(\xi)$$ with $$\xi \in [0,1] \,$$ and $$h=\frac{1}{n}$$
 * $$E_T=\frac{h^3}{12}\left ( \frac{\left ( x^2-2x+2 \right )e^x-2}{x^3} \right )$$

For $$\xi=1 \,$$ then $$E \,$$ is max
 * $$E_T=\frac{e-2}{12n^3}$$

For $$\xi=0 \,$$ then $$E \,$$ is min and this does not exist.

 * $$n=1, E_n \leq 0.06 \,$$
 * $$n=3, E_n \leq 0.002 \,$$
 * $$n=5, E_n \leq 0.0005 \,$$
 * $$n=7, E_n \leq 0.0002 \,$$
 * $$n=10, E_n \leq 6x10^{-5} \,$$

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

n=19, E_n \leq 9x10^{-6} \, $$ $$
 * $$\displaystyle
 * }
 * }

3. Simpson's Rule
 * $$I_n=\frac{b-a}{3n}\left ( f(a)+f(b)+4B_n + 2C_n \right )$$

where,
 * $$B_n=\sum _{i=1,odd}^{n-1}f(x_i)$$

and,
 * $$C_n=\sum _{i=2,even}^{n-2}f(x_i)$$

Then,
 * $$\int_{a}^{b}f(x)dx=I_n+E_n$$
 * $$E_n=-\frac{(1-0)^5}{180n^4}f^{(4)}(\xi)$$ with $$\xi \in [0,1] \,$$

For $$\xi=1 \,$$ yields the maximum $$E_n \,$$

 * $$n=1, E_n \leq 0.01 \,$$
 * $$n=3, E_n \leq 2x10^{-3} \,$$
 * $$n=5, E_n \leq 2x10^{-5} \,$$
 * $$n=7, E_n \leq 5x10^{-6} \,$$
 * $$n=6, E_n \leq 1x10^{-5} \,$$

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

n=7, E_n \leq 5x10^{-6} \, $$ $$
 * $$\displaystyle
 * }
 * }

=Problem 7-1=

Find
Expand $$e^x \,$$ in Taylor series w/ remainder.

Find the Taylor series expansion and remainder of $$f(x)\,$$

Given
$$f(x)=\frac{e^x-1}{x}=\frac{1}{x}[e^x-1]$$

Solution [[media:HW1_P7_1.jpg|P7-1]]
Expanding $$e^x\,$$ in Taylor series w/ remainder,
 * $$e^x=\sum _{j=0}^{\infty}\frac{x^j}{j!}=\frac{x^0}{0!}+\frac{x^1}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}=1+x+\frac{x^2}{2}+\frac{x^3}{6}+...+\frac{x^n}{n!}$$
 * $$R(x)=\frac{(x-0)^{n+1}}{(n+1)!}e^{\xi (x)}=\frac{x^{n+1}}{(n+1)!}e^{\xi(x)}$$

So then the Taylor series expansion and remainder of $$f(x)\,$$ is,

 * $$e^x=\sum _{j=0}^{\infty}\frac{x^j}{j!}=1+\sum _{j=1}^{\infty}\frac{x^j}{j!}$$
 * $$\Rightarrow e^x-1=\sum _{j=1}^{\infty}\frac{x^j}{j!}$$
 * $$\Rightarrow \frac{e^x-1}{x}=\sum _{j=1}^{\infty}\frac{x^{j-1}}{j!}$$

For $$e^x\,$$,
 * $$R(x)=\frac{x^{n+1}}{(n+1)!}e^{\xi(x)}$$

For $$e^x-1\,$$
 * $$R(x)=\frac{x^{n+1}}{(n+1)!}e^{\xi(x)}$$

For $$\frac{e^x-1}{x}\,$$
 * $$R(x)=\frac{x^n}{(n+1)!}e^{\xi(x)}$$

=Problem 8-2=

Find
Use Eq. (2) [http://upload.wikimedia.org/wikipedia/commons/c/c1/Egm6341.s10.mtg7.pdf Lec. 8-2] to obtain Eq. (1) [http://upload.wikimedia.org/wikipedia/commons/c/c1/Egm6341.s10.mtg7.pdf Lec. 7-1]

Given
Eq. (2)
 * $$I_1=\left ( \int_{a}^{b} l_0(xdx) \right )f(x_0)+\left ( \int_{a}^{b}l_1(x) \right )f(x_1)$$

Eq. (1)
 * $$I_1=\frac{b-a}{2}\left [ f(a)-f(b) \right ]$$

Solution [[media:HW1_P8_2.jpg|P8-2]]
Start with $$x_0=a \,$$ and $$x_1=b\,$$. Then,
 * $$l_0(x)=\frac{x-x_1}{x_0-x_1}=\frac{x-b}{a-b}$$

and
 * $$l_1(x)=\frac{x-x_0}{x_1-x_0}=\frac{x-a}{b-a}$$

Now,
 * $$\int _{a}^{b}l_0(x)dx=\frac{x^2}{2(a-b)}-\frac{bx}{a-b}=\frac{-(a-b)}{2}$$

And
 * $$\int _a^b l_1(x)=\frac{x^2}{2(b-a)}-\frac{ax}{b-a}=\frac{-(a-b)}{2}$$
 * $$\Rightarrow \left ( \frac{b-a}{2} \right )f(a)+\left ( \frac{b-a}{2} \right )f(b)$$
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

I_1=\frac{b-a}{2}\left [ f(a)+f(b) \right ] $$ $$
 * $$\displaystyle
 * }
 * }

Bold text=Problem 8-3=

Find
Prove $$P_2(x_j)=f(x_j)\,$$

Given
$$P_2(x_j)=\sum _{i=0}^{2}l_i(x_j)f(x_i)$$

Solution [[media:HW1_P8_3_1.jpg|P8-3]]

 * $$P_2(x_j)=\sum _{i=0}^{2}l_i(x_j)f(x_i)=l_0(x_j)f(x_0)+l_1(x_j)f(x_1)+l_2(x_j)f(x_2)$$

Also,
 * $$l_1(x_j)=\delta _{ij}=\left\{\begin{matrix}

1(i=j)\\ 0(i\neq j) \end{matrix}\right.$$ And
 * $$l_0(x_0)=1\,$$
 * $$l_0(x_1)=l_0(x_2)=0\,$$

 * $$l_1(x_1)=1\,$$
 * $$l_1(x_0)=l_1(x_2)=0\,$$

 * $$l_2(x_2)=1\,$$
 * $$l_2(x_0)=l_2(x_1)=0\,$$

Therefore,
 * $$P_2(x_0)=l_0(x_0)f(x_0)+l_1(x_0)f(x_1)+l_2(x_0)f(x_2)=f(x_0)\,$$
 * $$P_2(x_1)=l_0(x_1)f(x_0)+l_1(x_1)f(x_1)+l_2(x_1)f(x_2)=f(x_1)\,$$
 * $$P_2(x_2)=l_0(x_2)f(x_0)+l_1(x_2)f(x_1)+l_2(x_2)f(x_2)=f(x_2)\,$$

...

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\Rightarrow P_2(x_j)=\sum _{i=0}^{2}l_i(x_j)f(x_i)=f(x_j) $$ $$
 * $$\displaystyle
 * }
 * }

=Contributing members= EGM6341.s10.Team1.Kumanchik 17:07, 27 January 2010 (UTC) EGM6341.s10.Team1.Ya-Chiao Chang18:32, 27 January 2010 (UTC) Egm6341.s10.team1.lei 19:22, 27 January 2010 (UTC) Egm6341.s10.team1.toddmock 21:03, 27 January 2010 (UTC)

title: User:EGM6341.s10.Team1.Kumanchik/HW1


text:
User:EGM6341.s10.Team1.Kumanchik/HW2

=Homework 2=

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

title: User:EGM6341.s10.Team1.Kumanchik/HW2


text:
User:Egm6341.s10.team1.lei

Hi, I am Lei Wu from team 1.

HW1 HW2 HW3 HW4 HW5 HW6 HW7

title: User:Egm6341.s10.team1.lei


text:
User:Egm6341.s10.Team4.yunseok/HW2

= Problem 5: Comparison of Error between Taylor series and Newton-Cotes formula =

Given
Refer Lecture slide 11-1 for problem statement
 * {| style="width:100%" border="0" align="left"

f(x) = \sin(x) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle x \in [0,\pi] $$
 * $$\displaystyle \longrightarrow (1)

 * }.
 * }.

Find
'''Find n at
 * {| style="width:100%" border="0" align="left"

\left | f^T_n\Big(\frac{7\pi}{8}\Big)-f\Big(\frac{7\pi}{8}\Big) \right | \le \underbrace{\left | f^L_4\Big(\frac{7\pi}{8}\Big)-f\Big(\frac{7\pi}{8}\Big) \right |}_{\left | E^L_4(\frac{7\pi}{8})\right |} \le \frac{\left | q_5(\frac{7\pi}{8}) \right |}{5!} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (2)

 * }.
 * }.

Solution
$$\displaystyle f^L_4\Big(\frac{7\pi}{8}\Big)$$, $$\displaystyle \left | E^L_4(\frac{7\pi}{8})\right |$$, $$\displaystyle \frac{\left | q_5(\frac{7\pi}{8}) \right |}{5!}$$, is claculated using Matlab.

 * $$\displaystyle f^L_4\Big(\frac{7\pi}{8}\Big) = \sum_{i=0}^4 l_{i,4}\Big(\frac{7\pi}{8}\Big)\sin(x_i)= 0.3812$$
 * $$\displaystyle \left | E^L_4(\frac{7\pi}{8})\right | = 0.0014808$$
 * $$\displaystyle \frac{\left | q_5(\frac{7\pi}{8}) \right |}{5!} = 0.0081716$$

At first, we can confirm that the seconde inequality of Equation(2)

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

\left | E^L_4(\frac{7\pi}{8})\right | = 0.0014808 \le \displaystyle \frac{\left | q_5(\frac{7\pi}{8}) \right |}{5!} = 0.0081716 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (3)

 * }.
 * }.

Then, the Taylor series were calculated until the Error or taylor become smaller than that of Newton-Cotes formula at n=4. the result is like that

$$ \begin{array}\hline{|c||c|c|} n & E^T_n(t) & E^T_n(t) - E^L_4(t) \\ \hline 1&1.7128&1.7113\\ 2&0.34976&0.3483\\	3&0.54236&0.5409\\ 4&0.10444&0.1030\\ 5&0.067533&0.0661\\	6&0.011256&0.0098\\	7&0.0045295&0.0030\\	8&0.00065513&-0.0008\\ \hline \end{array} $$

title: User:Egm6341.s10.Team4.yunseok/HW2


text: User talk:Egm6341.s10.team2.madala

title: User talk:Egm6341.s10.team2.madala


text:
User:Egm6341.s2010.team1

Hello, We are Team 1.

title: User:Egm6341.s2010.team1


title: File:Egm6341.s10.Team2.hw1Problem1.jpg


text:
User:LawrenceF

=Analytical Writing=

Statement of Intent

This will be an in depth study of the Rastafarian One Love. What its actual meaning is where we believe it comes from and how it relates to us, and how it affects everyone. I will try to show and explain how a few totally different ancient cultures have always practiced this One Love and how it is the root of all our beginnings and existence.

Overview

Rationale

My hope is to get a better understanding of this One Love. I am also hoping that I can find out where I feel all these religions have gone wrong in believing that fear is what give them their connection with the Creator or God whatever you want to call it.

Publication

Schedule

 * 1) week 2
 * 2) week 3
 * 3) week 4
 * 4) etc.

Anticipated Problems

So far I am finding that I know where and what I more I have to research. It's just getting the last of the reading done. I've had the chance to talk to other Rasta Bretherin and wrote a bit on my history of being raised a Catholic. Now I still have to interview my Muslim friend and find the link with Islam to One Love.

Week 6 Project Review

Before Or During Week 6

 * Looking at world examples of the "One Love" idea

 * Hopi
 * Ancient Mayans
 * Moslem - interview?
 * Catholicism - personal story

 * Conclusion

After Week 6
“One Love”

This term One Love has become popular among the younger generation. Very much that same as people saying “Peace” to each other when departing or saying bye on the phone. But the origin of this One Love is much deeper than most people know. This message of One Love has been spreading throughout the world. Its vibe of positivity is touching us all and changing lives everyday. How is this getting to us all? It gets to us through life. Everything we do is affected by this Love. But there have been many who spread the message of One Love directly. The most commonly known people are Reggae Artists. Reggae music originates from Jamaica. These days there are different types of sub-genres of Reggae. Roots Reggae is the main stage for the artists of One Love. The most popular Reggae artist of all time is none other than Bob Marley. “One Love, One Heart lets get together and feel alright. One Love, One Heart give thanks and praise to the lord and I will feel alright” – Bob Marley - “One Love”.

Growing up I was raised in the Catholic religion. Every Sunday was the same routine waking up earlier than I wanted to and thinking how much I did not want to go. We went to church on an Army base called Ft. Lewis in Washington State. As much as I didn’t want to go those mornings my outlook always changed after the Priest began to speak. I was fortunate enough to have a couple of Priests that were able to keep the attention of everyone, even me a kid with the shortest attention span of anyone I’ve ever known. I would go to what was called CCD. CCD was a Sunday school for this Catholic Church. In class I remember learning mostly about Jesus and his unconditional love for God, people, and the world. I was taught about Love. Taught that love and faith in the Almighty was the path to heaven. Never did we concentrate on the Saints or the Trinity or even the technicalities of the Catholic rules or Vatican law. Catholics are viewed by those on the outside of this religion, as being a bit strange or even outright crazy. These days I can understand their feelings. Though growing up in I didn’t see it. Not because I was brain washed not too but because my teachers taught us to decipher the message ourselves and would say take from it what you believe it to mean though they knew that was not exactly how the Catholic church would have wanted us to view it. I always remembered that and that faith and love never left me. During my teenage years my faith was tested. Life for me like many others began to take a turn for the worse. I began to say, “Why is God letting this happen?” I wanted to blame the Most High and even said it out loud that it was Gods fault. It was never Gods fault; only mine. I became angry and full of hate. Doing bad things and making bad choices. But in the back of my mind I knew every time it was wrong and that was because I always had that understanding of love. The problem was in me. I needed to find myself again the person I knew I was that loving kid that I once was. After much struggle internally I was mad I no longer was or I should say I finally in control of it. And that is when my journey into One Love began. Always a fan of Reggae music I would listen to the songs of Bob Marley, Peter Tosh, Bunny Wailer, Burning Spear, Third World and many more. The music was beautiful and powerful. It’s beat sends out a vibration that rocks you to your soul. Many people don’t know why they like it but they do. Some say I like the beat or the rhythm or they say it makes me think happy thoughts. All these things are correct. Buy why? The reason is because One Love. Each song that is sung or played by these artists is a piece of their soul that is imprinted on every part of what you hear. It carries a message to the world that has been forgotten for an immeasurable amount of time. The message that everyone and everything on this planet are one, that everything you do, say, think, feel, see, hear, and speak are all linked and effect each other every second of the day. Eventually this message reached me again. In the book “Rasta Heart” written by Robert Roskind he would speak about Jamaica and the Rastafarians with One Love and that they are the source of its broadcast, like a drum beat being played out for the all of us in the world to hear. And we hear. Years ago, before reading a book were a normal thing for me I began to wonder exactly what is Bob Marley talking about. I knew I liked Reggae music but I didn’t fully understand why. Part of it was everyone I hung out with loved Reggae too. Being Samoan or any Pacific Islander at that, it is most likely that you will naturally get exposed to this style of music. I started listening to the lyrics and watched the documentaries on Bob and found myself drawn to him and all he did. That unfortunately was the extent of my research to know Bob and his Rasta roots. More years went by and I found myself spiritually weak and the rules of Christianity began to take its toll on me. I wanted to feel again. I began talking to people of a few different faiths.

title: User:LawrenceF


text:
User:KaitlinA

=Humanities I/The Commons=

title: User:KaitlinA


text:
User:Egm6341.s2010.Team1

HW1 HW1 updated HW2 HW2 updated HW3 HW4 HW4 updated HW5 HW5 updated HW6 HW7

title: User:Egm6341.s2010.Team1


text:
User:Egm6341.s2010.Team1/HW1

=Problem 2-1=

Find
Determine $$\lim _{x\mapsto 0}f\left ( x \right )$$ and plot $$f\left ( x \right ), x\in \left [ 0,1 \right ]$$

Given

 * $$f\left ( x \right )=\frac{e^{x}-1}{x}$$

Solution [[media:HW1_P2_1.jpg|P2-1]]
Determine the limit of the numerator and denominator separately and then evaluate the quotient.

Numerator:
 * $$\frac{\mathrm{d} }{\mathrm{d} x}\left ( e^{x}-1 \right )|_{x=0}=e^{x}|_{x=0}=1$$

Demoninator:
 * $$\frac{\mathrm{d} }{\mathrm{d} x}\left ( x \right )|_{x=0}=1$$

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\lim _{x\mapsto 0}\frac{e^{x}-1}{x}=\frac{1}{1}=1 $$ $$
 * $$\displaystyle
 * }
 * }

=Problem 2-3=

Find
Determine $$P_{n}\left ( x \right )$$ and $$R_{n+1}\left ( x \right )$$ for $$e^{x} \,$$

Given

 * $$P_{n}\left ( x \right )=f\left ( x_{0} \right )+\frac{x-x_{0}}{1!}f^{(1)}\left ( x_{0} \right )+...+\frac{x-x_{0}}{n!}f^{(n)}\left ( x_{0} \right )$$

And
 * $$R_{n+1}\left ( x \right )=\frac{\left (x-x_{0} \right )^{n+1}}{\left ( n+1 \right )!}f^{(n+1)}\left ( \xi  \right ) for \xi \in [x_{0},x]$$

Solution [[media:HW1_P2_3.jpg|P2-3]]
Since $$x_{0}=0 \,$$ then $$f\left ( x_{0} \right )=e^{0}=1$$ In fact every derivative of $$e^{x} \,$$ is $$e^{x}\,$$ therefore $$f^{(n)}\left ( x_{0} \right )=1$$ Therefore,
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

P_{n}\left ( x \right )=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{3!}+...+\frac{x^{n}}{n!} $$ $$
 * $$\displaystyle
 * }
 * }

=Problem 3-3=

Find
Determine $$\left \| f \right \|_{\infty }$$, $$\left \| g \right \|_{\infty }$$, and $$\left \| f-g \right \|_{\infty }$$ Plot $$f\left ( x \right ), g\left ( x \right ) x\in [0,\pi ]$$

Given

 * $$f\left ( x \right )=\sin x$$
 * $$g\left ( x \right )=\sin \left ( x-\frac{\pi }{2} \right )=-\cos x$$

Solution [[media:HW1_P3_3.jpg|P3-3]]
$$\left \| f \right \|_{\infty }=\max _{x}\left | f\left ( x \right ) \right |$$, so we must find where $$\frac{\mathrm{df} }{\mathrm{d} x}=0$$

Start with $$f\left ( x \right )$$
 * $$\frac{\mathrm{d} }{\mathrm{d} x}\left ( \sin x \right )=\cos x=0$$ between $$[0,\pi ] \,$$
 * $$\Rightarrow x=\frac{\pi }{2}$$
 * $$\Rightarrow \sin \left ( \frac{\pi }{2} \right )=1$$
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\left \| \sin x \right \|_{\infty }=1,x=\frac{\pi }{2} $$ $$ Next do $$g\left ( x \right )$$
 * $$\displaystyle
 * }
 * }
 * $$\left \| -\cos x \right \|_{\infty }=\max _{x}\left | -\cos x \right |=\max _{x}\left ( \cos x \right )$$
 * $$\frac{\mathrm{d\left ( \cos x \right )} }{\mathrm{d} x}=-\sin x=0$$
 * $$\Rightarrow \cos x=1|_{x=0}$$
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\left \| -\cos x \right \|_{\infty }=1,x=0 $$ $$ Finally do $$f-g \,$$
 * $$\displaystyle
 * }
 * }
 * $$\left \| f-g \right \|_{\infty }=\max _{x}\left | \sin x- \left ( -\cos x \right ) \right |=\max _{x}\left | \sin x+\cos x \right |$$
 * $$\frac{\mathrm{d} }{\mathrm{d} x}\left ( \sin x+\cos x \right )=\cos x-\sin x=0$$
 * $$\Rightarrow \cos x=\sin x$$
 * $$\Rightarrow x=\frac{\pi }{4}$$
 * $$\sin \left ( \frac{\pi }{4} \right )+\cos \left ( \frac{\pi }{4} \right )=\sqrt{2}$$
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\left \| \sin x+\cos \right \|_{\infty }=\sqrt{2},x=\frac{\pi }{4} $$ $$
 * $$\displaystyle
 * }
 * }

=Problem 5-1a=

Find

 * 1) Prove the Intermediate Mean Value Theorem for $$w\left ( . \right )$$ being non-negative, i.e. $$w\geq 0$$
 * 2) Prove the IMVT for $$w\left ( x \right )\neq 0, \forall x\in [a,b]$$

Given
IMVT
 * $$\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx=f\left ( \xi \right )\int_{a}^{b}w\left ( x \right )dx$$

Solution [[media:HW1_P5_1_1.jpg|P5-1a]]
1. For $$w\left ( x \right )\geq 0, \forall x\in [a,b]$$, we aim to prove that there exists a $$\xi \,$$ within $$[a,b] \,$$ such that $$\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx=f\left ( \xi \right )\int_{a}^{b}w\left ( x \right )dx$$.

If the functions are continuous on this interval there exists both a minimum (m) and maximum (M) value of $$w\left ( x \right )f\left ( x \right )$$ within $$[a,b]$$.

Thus,
 * $$\int_{a}^{b}mdx\leq \int_{a}^{b}w\left ( x \right )f\left ( x \right )dx\leq \int_{a}^{b}Mdx, \forall x\in [a,b]$$
 * $$\Rightarrow m\leq \frac{1}{b-a}\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx\leq M$$

If the functions are continuous on $$[a,b]$$ there must exist a $$\xi \,$$ in $$[a,b]$$ where,
 * $$f\left ( \xi \right )\int_{a}^{b}w\left ( x \right )dx=\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx$$

2. $$w> 0 \,$$ was covered in 1, and for $$w< 0 \,$$ the result is similar. In the case of $$w\left ( x \right )\neq 0$$ there must exist a minimum(m) and maximum(M) value of $$w\left ( x \right )f\left ( x \right )$$ within $$[a,b]$$.

 * $$\int_{a}^{b}mdx\leq \int_{a}^{b}w\left ( x \right )f\left ( x \right )dx\leq \int_{a}^{b}Mdx, \forall x\in [a,b]$$

though in this case the minumim and/or maximum could be negative. Again,
 * $$m\leq \frac{1}{b-a}\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx\leq M$$

This integral could be negative, but like in part-1 there must exist a $$\xi \,$$ in $$[a,b] \,$$ where
 * $$f\left ( \xi \right )\int_{a}^{b}w\left ( x \right )dx=\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx$$

=Problem 5-1b=

Find
Use the Intermediate Mean Value Theorem to show Eq. (5) [http://upload.wikimedia.org/wikipedia/commons/b/b5/Egm6341.s10.mtg2.pdf Lec. 2-2] and Eq. (1) [http://upload.wikimedia.org/wikipedia/commons/b/b5/Egm6341.s10.mtg2.pdf Lec. 2-3]

Given
Eq. (5)
 * $$R_{n+1}\left ( x \right )=\frac{1}{n!}\int_{x_{0}}^{x}\left ( x-t \right )^{n}f^{(n+1)}\left ( t \right )dt$$

Eq. (1)
 * $$R_{n+1}\left ( x \right )=\frac{(x-x_{0})^{n+1}}{(n+1)!}f^{(n+1)}\left ( \xi \right )$$

IMVT
 * $$\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx=f\left ( \xi \right )\int_{a}^{b}w\left ( x \right )dx$$

Solution [[media:HW1_P5_1_2.jpg|P5-1b]]
Let $$w\left ( x \right )$$ be nonnegative in the integral on $$[a,b] \,$$ and let $$f\left ( x \right )$$ be continuous on $$[a,b] \,$$. Then,

 * $$\int_{a}^{b}w\left ( x \right )f\left ( x \right )dx=f\left ( \xi \right )\int_{a}^{b}w\left ( x \right )dx$$

for
 * $$\xi \in [a,b]$$

Using a Taylor series expansion,
 * $$f\left ( x \right )=P_{n}\left ( x \right )+R_{n+1}\left ( x \right )$$

where
 * $$R_{n+1}\left ( x \right )=\frac{1}{n!}\int_{x_{0}}^{x}\left ( x-1 \right )^{n}f^{(n+1)}\left ( t \right )dt$$
 * $$\Rightarrow \frac{1}{n!}f^{(n+1)}\left ( \xi \right )\int_{x_{0}}^{x}\left ( x-t \right )^{n}dt$$
 * $$\Rightarrow \frac{1}{n!}f^{(n+1)}\left ( \xi \right )\left ( \frac{-1}{n+1} \right )\left ( x-t \right )^{n+1}|_{x_{0}}^{x}$$
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\Rightarrow \frac{(x-x_{0})^{n+1}}{(n+1)!}f^{(n+1)}\left ( \xi \right ) $$ $$
 * $$\displaystyle
 * }
 * }

=Problem 5-3=

Find
Integrate by parts to obtain
 * $$\frac{(x-x_{0})^{2}}{2!}f^{(2)}\left ( x_{0} \right )+\frac{(x-x_{0})^{3}}{3!}f^{(3)}\left ( x_{0} \right )$$

and show the remainder.

Then assume Eq. (4) and Eq. (5) [http://upload.wikimedia.org/wikipedia/commons/b/b5/Egm6341.s10.mtg2.pdf Lec. 2-2] are true and do integration by parts once more.

Given

 * $$f\left ( x \right )=f\left ( x_{0} \right )+\left ( x-x_{0} \right )f^{(1)}\left ( x_{0} \right )+\int_{x_{0}}^{x}\left ( x-t \right )f^{(2)}\left ( t \right )dt$$

Solution [[media:HW1_P5_3.jpg|P5-3]]
Integrate by parts,
 * $$\int_{x_{0}}^{x}\underbrace{\left ( x-t \right )}_{u'}\underbrace{f^{(2)}\left ( t \right )}_{v}dt=\left [ -\frac{(x-t)^{2}}{2} f^{(2)}\left ( t \right )\right ]_{x_{0}}^{x}-\int_{x_{0}}^{x}\frac{-(x-t)^{2}}{2}f^{(3)}\left ( t \right )dt$$
 * $$\Rightarrow 0+\frac{(x-x_{0})^{2}}{2}f^{(2)}\left ( x_{0} \right )+\int_{x_{0}}^{x}\frac{(x-t)^{2}}{2}f^{(3)}\left ( t \right )dt$$
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

f\left ( x \right )=f\left ( x_{0} \right )+\frac{(x-x_{0})}{1!}f^{(1)}\left ( x_{0} \right )+\frac{(x-x_{0})^{2}}{2!}f^{(2)}\left ( x_{0} \right )+\int_{x_{0}}^{x}\frac{(x-t)^{2}}{2}f^{(3)}\left ( t \right )dt $$ $$
 * $$\displaystyle
 * }
 * }

And the remainders
 * $$f\left ( x \right )=P_{n}\left ( x \right )+R_{n+1}\left ( x \right )$$
 * $$P_{n}\left ( x \right )=f\left ( x_{0} \right )+\frac{(x-x_{0})}{1!}f^{(1)}\left ( x_{0} \right )+...+\frac{(x-x_{0})^{n}}{n!}f^{(n)}\left ( x_{0} \right )$$
 * $$R_{n+1}\left ( x \right )=\frac{1}{n!}\int_{x_{0}}^{x}\underbrace{\left ( x-t \right )^{n}}_{u'}\underbrace{f^{(n+1)}\left ( t \right )}_{v}dt$$

Integrate by parts of $$R_{n+1}\left ( x \right )$$
 * $$\frac{1}{n!}\int_{x_{0}}^{x}\left ( x-t \right )^{n}f^{(n+1)}\left ( t \right )dt=\frac{1}{n!}\left \{ \left [ -\frac{(x-t)^{n}}{(n+1)}f^{(n+1)}\left ( t \right ) \right ]_{x_{0}}^{x} - \int_{x_{0}}^{x}\frac{-(x-t)^{n+1}}{(n+1)}f^{(n+1)}\left ( t \right )dt\right \}$$
 * $$\Rightarrow 0+\frac{(x-x_{0})^{n+1}}{(n+1)!}f^{(n+1)}\left ( x_{0} \right )+\int_{x_{0}}^{x}\frac{(x-t)^{n+1}}{(n+1)!}f^{(n+2)}dt$$
 * $$f\left ( x \right )=P_{n+1}\left ( x \right )+R_{n+2}\left ( x \right )$$

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

P_{n+1}\left ( x \right )=f\left ( x_{0} \right )+\frac{(x-x_{0})}{1!}f^{(1)}\left ( x_{0} \right )+...+\frac{(x-x_{0})^{n+1}}{(n+1)!}f^{(n+1)}\left ( x_{0} \right ) $$ $$
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

R_{n+2}\left ( x \right )=\frac{1}{(n+1)!}\int_{x_{0}}^{x}(x-t)^{n+1}f^{(n+2)}dt $$ $$
 * $$\displaystyle
 * }
 * }

=Problem 6-1=

Find
Construct the Taylor series of $$f(x) \,$$ around $$x_0=\frac{\pi}{4}$$ for $$n=0,1,...,10 \,$$.

Plot these series for each $$n \,$$. Then estimate the maximum of $$R(x) \,$$ at $$x=\frac{\pi}{2}$$

Given

 * $$f\left ( x \right )=\sin x$$ for $$x\in [0,\pi ]$$

Solution [[media:HW1_P6_1_1.jpg|P6-1]]

 * $$f\left ( x \right )=\sin x=P_n(x)+R_{n+1}(x)$$
 * $$P_n(x)=f(x_0)+\frac{(x-x_0)}{1!}f^{(1)}(x_0)+\frac{(x-x_0)^2}{2!}f^{(2)}(x_0)+...+\frac{(x-x_0)^n}{n!}f^{(n)}(x_0)$$
 * $$R_{n+1}(x)=\frac{(x-x_0)^{n+1}}{(n+1)!}f^{(n_1)}(\xi)$$

For
 * $$f(x)=\sin x, f^{(1)}(x)=\cos x, f^{(2)}(x)=-\sin x, f^{(3)}(x)=-\cos x, f^{(4)}(x)=\sin x \,$$

When $$n=0 \,$$
 * $$\Rightarrow \sin x=P_0(x)+R_1(x)=\frac{\sqrt{2}}{2}+(x-\frac{\pi}{4})\cos \xi$$

When $$n=1 \,$$
 * $$\Rightarrow \sin x=P_1(x)+R_2(x)=\frac{\sqrt{2}}{2}\left [ 1+\left ( x-\frac{\pi}{4} \right ) \right ]-\frac{(x-\frac{\pi}{4})^2}{2!}\sin \xi$$

When $$n=2 \,$$
 * $$\Rightarrow \sin x=P_2(x)+R_3(x)=\frac{\sqrt{2}}{2}\left [ 1+\left ( x-\frac{\pi}{4} \right )-\frac{(x-\frac{\pi}{4})^2}{2!} \right ]-\frac{(x-\frac{\pi}{4})^3}{3!}\cos \xi$$

When $$n=3 \,$$
 * $$\Rightarrow \sin x=P_3(x)+R_4(x)=\frac{\sqrt{2}}{2}\left [ 1+\left ( x-\frac{\pi}{4} \right )-\frac{(x-\frac{\pi}{4})^2}{2!}-\frac{(x-\frac{\pi}{4})^3}{3!} \right ]+\frac{(x-\frac{\pi}{4})^4}{4!}\sin \xi$$

...

When $$n=10 \,$$
 * $$\Rightarrow \sin x=P_{10}(x)+R_{11}(x)=\frac{\sqrt{2}}{2}\left [ 1+\left ( x-\frac{\pi}{4} \right )-\frac{(x-\frac{\pi}{4})^2}{2!}-\frac{(x-\frac{\pi}{4})^3}{3!}...- \frac{(x-\frac{\pi}{4})^{10}}{10!}\right ]-\frac{(x-\frac{\pi}{4})^{11}}{11!}\cos \xi$$

Finally, the maximum can be computed with,
 * $$R_{n+1}(x)=\frac{(x-x_0)^{n+1}}{(n+1)!}f^{(n+1)}(\xi)$$ for $$\xi \in [0,\pi] \,$$
 * $$\left | R_{n+1}(x) \right |\leq \frac{(x-x_0)^{n+1}}{(n+1)!}\max \left \{ f^{(n+1)}(\xi) \right \} \leq \frac{(x-x_0)^{n+1}}{(n+1)!}=\frac{(\frac{\pi}{4})^{n+1}}{(n+1)!}\leq \frac{\pi}{4}$$

=Problem 6-5=

Find
Use the following three methods to find the integral, $$I \, $$

1. Taylor series expansion of $$f_n\,$$

2 Composite Trapezoidal Rule

3 Composite Simposon Rule

Let $$ n=2,4,8,... \,$$ until the error is of magnitude $$10^{-6} \,$$

Given

 * $$I=\int_{0}^{1}\frac{e^{x}-1}{x}dx$$

Solution [[media:HW1_P6_5_1.jpg|P6-5a]], [[media:HW1_P6_5_2.jpg|P6-5b]], [[media:HW1_P6_5_3.jpg|P6-5c]]
1. Taylor Series
 * $$e^{x}=\sum _{j=0}^{\infty }\frac{x^j}{j!}\Rightarrow \frac{e^{x}-1}{x}=\sum _{j=1}^{\infty }\frac{x^{j-1}}{j!}$$
 * $$I_n=\int_{0}^{1}f_n(x)dx=\int_{0}^{1}\sum _{j=1}^{\infty} \frac{x^{j-1}}{j!}dx=\sum _{j=1}^{n}\frac{x^3}{j!j}|_0^1=\sum _{j=1}^n \frac{1}{j!j}$$
 * $$f(x)-f_n(x)=R_n(x)=\frac{(x-0)^n}{(n+1)!}f^{(n+1)}(\xi)$$ with $$\xi \in [0,x] \,$$ and $$x_0=0 \,$$
 * $$E_n=I-I_n=\int_{0}^{1}\left [ f(x)-f_n(x) \right ]dx=\int_{0}^{1}\overbrace{\frac{x^n}{(n+1)!}}^{w(x)}\overbrace{f^{(n+1)}\left (\xi(x) \right )}^{g(x)}dx$$
 * $$\Rightarrow g(\alpha)\int_{0}^{1}w(x)dx$$ for $$\alpha \in [0,1]$$
 * $$E_n =\,$$
 * $$\Rightarrow max = \frac{g(\alpha)}{(n+1)!(n+1)}$$, $$\alpha=1 \,$$
 * $$\Rightarrow min = \frac{g(\alpha)}{(n+1)!(n+1)}$$, $$\alpha=0 \,$$
 * $$n=3, E_n \leq 0.02 \,$$
 * $$n=5, E_n \leq 8x10^{-5} \,$$

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

n=6, E_n \leq 9x10^{-6} \, $$ $$
 * $$\displaystyle
 * }
 * }

2. Comp. Trap. Rule
 * $$\int_{x_0}^{x_1}f(x)dx=\int_{0}^{h}f(t+x_0)dt=\left [ (t+A)F(t+x_0) \right ]_0^h-\int_{0}^{h}(t+A)f'(t+x_0)dt$$
 * $$\int_{a}^{b}f(x)dx\approx \left ( \frac{b-a}{n} \right )\left [ \frac{f(a)+f(b)}{2}+\sum _{k=1}^{n-1}f\left (a+k\frac{b-a}{n} \right ) \right ]$$

We want $$\left [ (t+A)f(t+x_0) \right ]_0^h=\frac{(f(x_0)+f(x_1))}{2}(h)$$ with $$x_1=x_0+h \,$$,
 * $$\Rightarrow A=\frac{-h}{2}$$

We want our error bound small
 * $$\Rightarrow \left [ \left ( \frac{(t+A)^2}{2}+B \right )f'(t+x_0) \right ]_0^h=0$$
 * $$\left [\left ( \frac{\left ( \frac{h}{2} \right )^2}{2}+B \right )f'(h+x_0) \right ]-\left ( \frac{\left ( \frac{-h}{2} \right )^2}{2}+B \right )f'(x_0)=0$$
 * $$\Rightarrow B=\frac{-h^2}{8}$$
 * $$\int_{x_0}^{x_1}f(x)dx=\frac{h}{2}\left [ f(x_0)+f(x_1) \right ]+\int_{0}^{h}\left ( \tfrac{\left ( t-\frac{h}{2} \right )^2}{2}+\frac{h^2}{8} \right )f''(t+x_0)dt$$

The error in the Trap. Rule equals the sum of these:
 * $$E_T=E(0)+E(1)+...+E(n-1) \,$$
 * $$\int_{0}^{h}Qf(t+x_0)dt+\int_{0}^{h}Qf(t+x_1)dt+...+\int_{0}^{h}Qf''(t+x_0)dt$$
 * $$=\int_{0}^{h}Q\left (f(t+x_0)+...+f(t+x_{n-1}) \right )dt$$
 * $$E_T=\int_{0}^{h}\frac{\left (t-\frac{h}{2} \right )^2}{2}\frac{h^2}{8}\left [ f(t+x_0)+...+f(t+x_{n-1}) \right ]dt=\frac{-h^3}{12}$$

 * $$E_T=\frac{-h^3}{12n^3}f''(\xi)$$ with $$\xi \in [0,1] \,$$ and $$h=\frac{1}{n}$$
 * $$E_T=\frac{h^3}{12}\left ( \frac{\left ( x^2-2x+2 \right )e^x-2}{x^3} \right )$$

For $$\xi=1 \,$$ then $$E \,$$ is max
 * $$E_T=\frac{e-2}{12n^3}$$

For $$\xi=0 \,$$ then $$E \,$$ is min and this does not exist.

 * $$n=1, E_n \leq 0.06 \,$$
 * $$n=3, E_n \leq 0.002 \,$$
 * $$n=5, E_n \leq 0.0005 \,$$
 * $$n=7, E_n \leq 0.0002 \,$$
 * $$n=10, E_n \leq 6x10^{-5} \,$$

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

n=19, E_n \leq 9x10^{-6} \, $$ $$
 * $$\displaystyle
 * }
 * }

3. Simpson's Rule
 * $$I_n=\frac{b-a}{3n}\left ( f(a)+f(b)+4B_n + 2C_n \right )$$

where,
 * $$B_n=\sum _{i=1,odd}^{n-1}f(x_i)$$

and,
 * $$C_n=\sum _{i=2,even}^{n-2}f(x_i)$$

Then,
 * $$\int_{a}^{b}f(x)dx=I_n+E_n$$
 * $$E_n=-\frac{(1-0)^5}{180n^4}f^{(4)}(\xi)$$ with $$\xi \in [0,1] \,$$

For $$\xi=1 \,$$ yields the maximum $$E_n \,$$

 * $$n=1, E_n \leq 0.01 \,$$
 * $$n=3, E_n \leq 2x10^{-3} \,$$
 * $$n=5, E_n \leq 2x10^{-5} \,$$
 * $$n=7, E_n \leq 5x10^{-6} \,$$
 * $$n=6, E_n \leq 1x10^{-5} \,$$

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

n=7, E_n \leq 5x10^{-6} \, $$ $$
 * $$\displaystyle
 * }
 * }

=Problem 7-1=

Find
Expand $$e^x \,$$ in Taylor series w/ remainder.

Find the Taylor series expansion and remainder of $$f(x)\,$$

Given
$$f(x)=\frac{e^x-1}{x}=\frac{1}{x}[e^x-1]$$

Solution [[media:HW1_P7_1.jpg|P7-1]]
Expanding $$e^x\,$$ in Taylor series w/ remainder,
 * $$e^x=\sum _{j=0}^{\infty}\frac{x^j}{j!}=\frac{x^0}{0!}+\frac{x^1}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}=1+x+\frac{x^2}{2}+\frac{x^3}{6}+...+\frac{x^n}{n!}$$
 * $$R(x)=\frac{(x-0)^{n+1}}{(n+1)!}e^{\xi (x)}=\frac{x^{n+1}}{(n+1)!}e^{\xi(x)}$$

So then the Taylor series expansion and remainder of $$f(x)\,$$ is,

 * $$e^x=\sum _{j=0}^{\infty}\frac{x^j}{j!}=1+\sum _{j=1}^{\infty}\frac{x^j}{j!}$$
 * $$\Rightarrow e^x-1=\sum _{j=1}^{\infty}\frac{x^j}{j!}$$
 * $$\Rightarrow \frac{e^x-1}{x}=\sum _{j=1}^{\infty}\frac{x^{j-1}}{j!}$$

For $$e^x\,$$,
 * $$R(x)=\frac{x^{n+1}}{(n+1)!}e^{\xi(x)}$$

For $$e^x-1\,$$
 * $$R(x)=\frac{x^{n+1}}{(n+1)!}e^{\xi(x)}$$

For $$\frac{e^x-1}{x}\,$$
 * $$R(x)=\frac{x^n}{(n+1)!}e^{\xi(x)}$$

=Problem 8-2=

Find
Use Eq. (2) [http://upload.wikimedia.org/wikipedia/commons/c/c1/Egm6341.s10.mtg7.pdf Lec. 8-2] to obtain Eq. (1) [http://upload.wikimedia.org/wikipedia/commons/c/c1/Egm6341.s10.mtg7.pdf Lec. 7-1]

Given
Eq. (2)
 * $$I_1=\left ( \int_{a}^{b} l_0(xdx) \right )f(x_0)+\left ( \int_{a}^{b}l_1(x) \right )f(x_1)$$

Eq. (1)
 * $$I_1=\frac{b-a}{2}\left [ f(a)-f(b) \right ]$$

Solution [[media:HW1_P8_2.jpg|P8-2]]
Start with $$x_0=a \,$$ and $$x_1=b\,$$. Then,
 * $$l_0(x)=\frac{x-x_1}{x_0-x_1}=\frac{x-b}{a-b}$$

and
 * $$l_1(x)=\frac{x-x_0}{x_1-x_0}=\frac{x-a}{b-a}$$

Now,
 * $$\int _{a}^{b}l_0(x)dx=\frac{x^2}{2(a-b)}-\frac{bx}{a-b}=\frac{-(a-b)}{2}$$

And
 * $$\int _a^b l_1(x)=\frac{x^2}{2(b-a)}-\frac{ax}{b-a}=\frac{-(a-b)}{2}$$
 * $$\Rightarrow \left ( \frac{b-a}{2} \right )f(a)+\left ( \frac{b-a}{2} \right )f(b)$$
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

I_1=\frac{b-a}{2}\left [ f(a)+f(b) \right ] $$ $$
 * $$\displaystyle
 * }
 * }

=Problem 8-3=

Find
Prove $$P_2(x_j)=f(x_j)\,$$

Given
$$P_2(x_j)=\sum _{i=0}^{2}l_i(x_j)f(x_i)$$

Solution [[media:HW1_P8_3_1.jpg|P8-3]]

 * $$P_2(x_j)=\sum _{i=0}^{2}l_i(x_j)f(x_i)=l_0(x_j)f(x_0)+l_1(x_j)f(x_1)+l_2(x_j)f(x_2)$$

Also,
 * $$l_1(x_j)=\delta _{ij}=\left\{\begin{matrix}

1(i=j)\\ 0(i\neq j) \end{matrix}\right.$$ And
 * $$l_0(x_0)=1\,$$
 * $$l_0(x_1)=l_0(x_2)=0\,$$

 * $$l_1(x_1)=1\,$$
 * $$l_1(x_0)=l_1(x_2)=0\,$$

 * $$l_2(x_2)=1\,$$
 * $$l_2(x_0)=l_2(x_1)=0\,$$

Therefore,
 * $$P_2(x_0)=l_0(x_0)f(x_0)+l_1(x_0)f(x_1)+l_2(x_0)f(x_2)=f(x_0)\,$$
 * $$P_2(x_1)=l_0(x_1)f(x_0)+l_1(x_1)f(x_1)+l_2(x_1)f(x_2)=f(x_1)\,$$
 * $$P_2(x_2)=l_0(x_2)f(x_0)+l_1(x_2)f(x_1)+l_2(x_2)f(x_2)=f(x_2)\,$$

...

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\Rightarrow P_2(x_j)=\sum _{i=0}^{2}l_i(x_j)f(x_i)=f(x_j) $$ $$
 * $$\displaystyle
 * }
 * }

=Contributing members= EGM6341.s10.Team1.Kumanchik 17:07, 27 January 2010 (UTC) EGM6341.s10.Team1.Ya-Chiao Chang18:32, 27 January 2010 (UTC) Egm6341.s10.team1.lei 19:22, 27 January 2010 (UTC) Egm6341.s10.team1.toddmock 21:03, 27 January 2010 (UTC)

title: User:Egm6341.s2010.Team1/HW1


text:
User:Egm6341.s10/Report table/Best of HW1

Comments, corrections, best reports, best solutions


text:
Talk:Socialism/Socialist economics

wat is the best way to define socialist, capitolism, communist in a 5th grade vocab?

horrible description
the page was written by someone that has never taken an economics class. it should just be deleted until someone worthwhile wants to recreate it.

biased, if may i add
well, the description clearly is biased against socialist and marxian economic theory, and it clearly shows the author's ignorance about the subject matter. Jv22invierno 01:26, 23 January 2011 (UTC)

please delete this page (or change its title)
I totally agree with the previous guys. Who created this page doesn't have any knowledge of Marx's theory. Delete this page or change its title to "State capitalism economy" or "State planned economy". --94.160.131.19 23:26, 17 February 2011 (UTC)

title: Talk:Socialism/Socialist economics


text:
User:Jessicay

=Humanities I/The Commons=

title: User:Jessicay


text:
User talk:Leighblackall/2009

'''Hi there! I look forward to talking with you. Please add new topics at the top of my talk page. Thanks :)'''

User page structure
Re: your preference for new talk messages at the top (and more generally, searching for better ways ), I thought you might be interested in this user talk page which uses topic headings: m:user talk:Pathoschild. -- Jtneill - Talk - c 05:54, 10 November 2009 (UTC)
 * Very nice. Think I will adopt that style. Amazing to see such a huge talk log too. Nice for assessing learnings within wikiversity I would say! We should organise wikiversity days at uc hey? Leighblackall 09:11, 10 November 2009 (UTC)

Nomination for full custodianship
I've nominated you - Candidates for Custodianship/Leighblackall - you should try to be available during the next five days to answer any questions. -- Jtneill - Talk - c 23:25, 15 October 2009 (UTC)
 * Thanks JT, Will do my best. Am on the road with my android phone (still no Internet in my new home :( and will jump on any network computer I see free. Watching --Leighblackall 04:46, 17 October 2009 (UTC)

WYSIWIG editing
I know this is something you'd like to see: Proposal:WYSIWIG default editor -- Jtneill - Talk - c 02:13, 27 September 2009 (UTC)
 * It will make a big difference hey! --Leighblackall 02:39, 30 September 2009 (UTC)

Catalysts
Hi Leigh. I wonder if you could support the role of "catalyst" in some formalized way having to do with online facilitation and some of the other realms of virtual group dynamics into which you delve. I know you are "hip" to this thinking and I thought we (v:) could establish some formal methods to use here at home and abroad. See the discussion in a broader context CQ 11:59, 28 August 2009 (UTC)
 * Good suggestion CQ. I'll pass this message onto the course facilitator.. they need ideas for Facilitation of Wikis. --Leighblackall 02:42, 30 September 2009 (UTC)

OER Search and Discovery
Followup to the twitter note - there is a thread at Colloquium that I started, if you'd like to discuss OER search in more detail. --mikeu talk 10:03, 20 July 2009 (UTC)

Nomination for Probationary Custodianship
Hi Leigh,

I've nominated you here for probationary custodianship: Candidates_for_Custodianship. Could you indicate there whether you accept and who you would like to be mentored by? Sincerely, James - -- Jtneill - Talk - c 01:31, 6 July 2009 (UTC)
 * Hey Leigh, that's great news - check out the link above - just add a "yes please" to that page and hopefully we can get underway . Sincerely, James - -- Jtneill - Talk - c 04:05, 8 July 2009 (UTC)

✅ This begins the 4 week period of probationary custodianship. Good luck with the mop! --mikeu talk 12:45, 8 July 2009 (UTC)

 * If you have any questions or would like some advice you can drop by #wikiversity-en, where you can frequently find me and other custodians. Also, please add yourself to WV:STAFF as a probationary custodian with timezone and babel info.  --mikeu talk 12:59, 8 July 2009 (UTC)

 * BTW, babel refers to language fluency. EN is native speaker of English, and FR-1 is basic knowledge of French.  See User:Jade Knight's page for an example of how this is sometimes used.  It is a very handy system on multilingual wikis like  or for language learning project participants.  --mikeu talk 01:33, 9 July 2009 (UTC)

 * Hi Leigh, whilst by no means compulsory, also consider participating in, and contributing to, this learning project: How to be a MediaWiki sysop. -- Jtneill - Talk - c 23:55, 8 July 2009 (UTC)
 * I'm keen if you are to get you involved as a "participant" in this project and in so doing so, hopefully to improve it. What do you think? -- Jtneill - Talk - c 12:57, 2 August 2009 (UTC)

Hi Leigh, as discussed off-wiki, I've recommended an extension of your custodian probationary period to allow some more time to develop your use of Wikiversity. If you think it would be helpful we could brainstorm a list of things to tour and learn more about - we could do this on Skype if you like!? -- Jtneill - Talk - c 07:47, 5 August 2009 (UTC)

 * Thanks James, just at the moment (and for the last month or so) I've been a bit tied up with work to focus on Wikiversity based work. We are currently running the Facilitating online course - which has a number of participants, and soon to run Composing educational resources. AFter I am back from the Open Education conference August 25th, I will be able to focus a lot more on this. It is my intention to spend at least a year with Wikiversity is my primary vehicle for online educational development, and compare and contrast the experience with Wikieducator --Leighblackall 05:10, 12 August 2009 (UTC)
 * No worries, Leigh, when you're back around more, let us know, and we can discuss next steps . -- Jtneill - Talk - c 13:01, 12 August 2009 (UTC)
 * Hi Leigh, there seems to be support for an extended probationary period - could perhaps post to Candidates for Custodianship/Leighblackall with a proposed time-frame that could work for you? Basically IMHO we need sufficient time for you to try out / demonstrate use of the custodian-level tools and for you to demonstrate the kind of editing/projects/maintanence work you intend to be involved with. -- Jtneill - Talk - c 23:35, 27 August 2009 (UTC)
 * Hi James, I could use a Skype call (or Gizmo if you'd like to record it) to discuss these questions. Let's email a time and record a talk, adding the link here for others should anyone else be in this position some day. For me, its a case of not knowing what I don't know (and not making/having time to find out what I don't know). --Leighblackall 00:16, 28 August 2009 (UTC)
 * Whoops, I only just now saw your reply - sorry - will email/skype. -- Jtneill - Talk - c 23:29, 1 September 2009 (UTC)
 * What are you thinking about bringing your custodian probationary status to a close soon since we've run well over the typical period - would you like to be nominated for full custodian - do you feel ready? Is there anything else you'd like to "cover"? From my point of view, you seem to understand the tools and policy well enough to not abuse them and also know how to ask about something if you don't know. So, I don't think we need to string this out any further if you agree and are willing to make effective and good of use of the custodian-level functions if and when the need arises. As I mentioned in talking, for me custodianship, at least initially, was mostly just about being able to clean up my messes when I made them. -- Jtneill - Talk - c 22:34, 15 October 2009 (UTC)
 * Thanks JT. I think we may as well close it and see what others think. I am willing to take the responsibility, and answer any questions regarding the status assigned. As you say, it is mostly about the added flexibility it gives me in my own work. But if there are things that need to be done, such as helping others without the flexibility to manage their pages, I am more than willing to help. Generally speaking I have a no deletion policy personally, and would rather see things built on than deleted. That said, there have been several times I myself have created a page with teh wrong name or whatever, and wish for it to be deleted. That to me is pretty straight forward. What next then? --Leighblackall 22:57, 15 October 2009 (UTC)

Explicit content
Good questions/comments. I'm not sure, but I've asked about more about this here: Request custodian action. -- Jtneill - Talk - c 07:06, 5 July 2009 (UTC)

 * I have added that image to MediaWiki:Bad image list which prevents it from appearing in the page history . We don't get this sort of activity very often, but it has occured a few times recently.  As Jtneill mentioned we are discussing ways to prevent this in the future.  Wikiversity custodians don't have any direct control of what is hosted on commons.  However, anyone at wikiversity can go to commons and request that a file be deleted.  The official policy on such matters can be found at Commons:Project scope which in part states:

"'The counterpoint to that is that the statement “Commons is not censored” is not a valid argument for keeping a file that falls outside the permitted Commons scope, as set out above. Photographs of nudity and male and female genitalia are sometimes uploaded for non-educational motives, and such images are not exempt from the requirement to comply with the rules on scope. Often (but not always) such images add nothing educationally distinct to the stock of such images we hold already, and hence fail the test of being realistically useful for an educational purpose.'"

 * Given that this image is not used on a single wikimedia project and that Commons also has a policy about "Files apparently created and/or uploaded for the purpose of vandalism or attack" should make this is a good candidate for deletion.  Having said that, I have also seen debates on wiki where the justification for keeping such material is based on rather spurious arguments.  --mikeu talk 20:54, 5 July 2009 (UTC)

 * If an image is in MediaWiki:Bad image list that will prevent it from appearing on any current page or any history page. The image will simply not be visible anywhere on wikiversity.  However, this is not a manageable solution to the problem.  Trying to keep up with the new images that get uploaded to commons would be impossible.  And mostly pointless, since almost none of those images get used to vandalise a page here.  (There are only 5 entries on the bad image list, just to give you an idea of how rare it is.)  I see this problem as being a subset of a larger problem which is malicious editing to disrupt the site.  This might also be done by inserting vulgar language or by changing the layout of the page so that it is unreadable, for example.  In a few recent cases there were users who inserted these images in many pages.  The response we take is to block the user from editing and cleanup the mess.  There are some automated methods for preventing this.  But mostly we really on volunteers to quickly stop and revert this activity.  Given the small size of our community it is not always possible to handle this as rapidly as we would like.  I'd be interested in exploring other possibilities for handling this.  There have been many discussions on wikimedia sites about explicit images and vandalism.  I'll try to find some links to those pages.  --mikeu talk 22:47, 5 July 2009 (UTC)

 * Here is a page that summarizes some of the previous discussions on the topic. While much of it is specific to wikipedia, there are some points that are more general about wikimedia sites.  There is also a discussion on implementing a technical means to filter out explicit content.  BTW, I did tag the image that was on your page for speedy deletion, FWIW.  --mikeu talk 13:33, 6 July 2009 (UTC)

 * The file has been deleted from Commons for being "out of project scope."  --mikeu talk 22:16, 7 July 2009 (UTC)

Talk
FYI, I redesigned my talk page header a little - see what you think possibly as a template for others. -- Jtneill - Talk - c 00:56, 27 September 2008 (UTC)

The heck with the page header... This is about feedback. Anyone with a dog named LEGO has my vote! -pbr http://reiber.org

Lecture 2.0
Have you got some ideas for Lecture 2.0? How does Otago do it? -- Jtneill - Talk - c 01:54, 23 July 2008 (UTC)

RSS feed
FYI, unfortunately, the RSS extension is not currently available on WV. It is available on the sandbox server (which you can access), but I'm not sure how useful that is. Unfortunately, further, when I looked into it, it seemed unlikely to get approved by WMF because it would mean unable to directly control content on pages - e.g., if the feed is spammed. Sandbox server: http://www.sandboxserver.org/wiki/index.php?title=RSS_Reader -- Jtneill - Talk - c 06:17, 6 July 2009 (UTC)
 * This has been updated on the sandbox server; now see Extension:SimpleFeed -- Jtneill - Talk - c 10:34, 20 July 2009 (UTC)

Talk link in signature
Go to Special:Preferences. In the Signature field, add something like -- Jtneill - Talk - c, tick the box, and save. -- Jtneill - Talk - c 09:36, 8 July 2009 (UTC)

Greetings
It's nice to see you being active on the site... Sj 08:45, 1 August 2009 (UTC)

Course journals
I'm curious to follow up your ideas about emerging academics publishing coursework into journals. Perhaps we could add ideas about this to a page (course journals?) but also wanted to check what you'd call this? And we could add examples. I kind of tried in 2007 with social psychology (psychology) - e.g., see +  - but I'm keen to find other/better ways. -- Jtneill - Talk - c 07:42, 9 October 2009 (UTC)
 * Maybe course journal systems - goes nicely with open journal systems? -- Jtneill - Talk - c 07:43, 9 October 2009 (UTC)
 * Well, remembering that the first principle in the idea (from my perspective) was to dissolve the perceived relationship between student and teacher, and reinforce the relationship as Academic and Emerging Academic. These terms were inspired by a hidden, but locally vocal researcher in Otago named Russell Butson. He has some very advanced thinking about deschooling higher education. But one idea he has trialed has been emerging academics taking their existing essays forward into a journal - just as you say - a course journal. The only thing I bork at is that the word "course" reinforces that student/teacher paradigm.. or an idea that "this is not serious". I think it is as open journals by emerging academics, as in emerging artists, writers, sports people etc.--Leighblackall 02:32, 12 October 2009 (UTC)

Welcome aboard
Hi Leighblackall. You are now a "full custodian" (and sorry about the delay!).

Please make sure to update your status on Support staff, and put WV:RCA on your watchlist.

Congrats! --SB_Johnny talk 18:42, 24 October 2009 (UTC)

Yes! I'm very happy to see you've got the position. Best wishes and I'm sure you'll do a great job! --AFriedman 01:39, 25 October 2009 (UTC)

 * Yes, welcome aboard to adminship. As you know, this is serious business:

If you should stuff up, then a serious trout slapping and time in the village stocks may be applied. Hope we can continue to learn together. Sincerely, James -- Jtneill - Talk - c 23:25, 24 October 2009 (UTC)
 * James said it best :) Sj 00:21, 3 November 2009 (UTC)

title: User talk:Leighblackall/2009


text:
User:Santhosh babu

My name is santhosh babu now working as a teacher holds a law graduation

title: User:Santhosh babu


text:
User talk:Ayushi~enwikiversity

 Hello Ayushi, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 28 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 28 January 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Ayushi. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Ayushi~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:28, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:03, 19 April 2015 (UTC)

title: User talk:Ayushi~enwikiversity


text:
User talk:Jbennett

 Hello Jbennett, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 28 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 28 January 2010 (UTC)

title: User talk:Jbennett


text:
User talk:EmMain

 Hello EmMain, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 28 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 28 January 2010 (UTC)

title: User talk:EmMain


text:
User talk:Writer65~enwikiversity

 Hello Writer65, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Gbaor 08:33, 28 January 2010 (UTC)

 Woof! My name is Jack Russell. I am a dog and a Wikiversity mascot. I am pretty new around Wikiversity. Perhaps we can learn together! Tail wag...

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Writer65. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Writer65~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:48, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:25, 19 April 2015 (UTC)

title: User talk:Writer65~enwikiversity


text:
User talk:Pizaz24

 Hello Pizaz24, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Gbaor 08:33, 28 January 2010 (UTC)

 Woof! My name is Jack Russell. I am a dog and a Wikiversity mascot. I am pretty new around Wikiversity. Perhaps we can learn together! Tail wag...

title: User talk:Pizaz24


text:
User talk:Juha Suoranta

 Hello Juha Suoranta, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Gbaor 08:33, 28 January 2010 (UTC)

 Hi! My name is WikiversityJack. I am a pumpkin and a Wikiversity mascot. My aunt told me that fun + learning = Wikiversity. Let me know if I can help you out.

title: User talk:Juha Suoranta


text:
User talk:Chaetodipus

 Hello The New Mikemoral, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Gbaor 08:33, 28 January 2010 (UTC)

 Hi , I am Percy. Nice to meet you. I love the beach, swimming, fishing, and I am currently doing some research at Wikiversity on oceanography. I am also learning to play piano. What do you like? Can I be your mascot?

title: User talk:Chaetodipus


text:
User talk:Jessicay

 Hello Jessicay, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Gbaor 08:35, 28 January 2010 (UTC)

 Hi! My name is WikiversityJack. I am a pumpkin and a Wikiversity mascot. My aunt told me that fun + learning = Wikiversity. Let me know if I can help you out.

title: User talk:Jessicay


text:
User talk:Bci2

 Hello Bci2, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Gbaor 08:35, 28 January 2010 (UTC)

title: User talk:Bci2


text:
User talk:Santhosh babu

 Hello Santhosh babu, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Gbaor 08:35, 28 January 2010 (UTC)

 Hi , I am Percy. Nice to meet you. I love the beach, swimming, fishing, and I am currently doing some research at Wikiversity on oceanography. I am also learning to play piano. What do you like? Can I be your mascot?

title: User talk:Santhosh babu


text:
User:Waterfriend

STEAM PRESSURE CAUSES EARTHQUAKES – THE PRESSURE COOKER MODEL Posted on December 8, 2009 by waterfriend | Edit From my book, CHILDHOOD MEMOIRS Molecular Hydrology- The incredible journey of water molecules. Water molecules are like mischievous urchins: hyper active, restless and mercurial. They can grow wings and fly like butterflies into the air or metamorphose into earthworms and burrow into the earth eating up whatever comes in their way. The movement of these molecules may be invisible as in diffusion or osmosis; or gigantic like water falls and the mighty rivers in the continents or still mightier, like the rivers in the ocean called ocean currents. At present hydrology is restricted to the visible water in the surface of the continent. This is only a small fraction of the water contents of the continents. Huge quantities of snow and ice are deposited above the snowline of mountains; a small part of this melts and flows through rivers during summer. But the major part of it remains locked up in the valleys in what is known as glacier lakes. Because of the pressure of the snow and ice over the top of the mountains, the lower layer of ice melts and the water molecules are driven into the rock below, dissolving minerals and, through the pores in the rock the molecules travel down the minute channels which merge into bigger veins and traverse the whole length and breadth of the continents. The channels get broken, occasionally, near the surface of the earth and we get springs. The journey of the water into the Earth continues until the water molecules get heated up into steam. Thereafter the hot steam may once again travel to the higher strata of the contentment or spread even below the ocean bottom. On account of very high pressure steam pierces the bottom of the oceans and we have fresh water springs in the oceans, of which we know very little. I would like to call the study of this invisible water molecules, molecular hydrology. Obviously, this study will have to be conducted by a team of experts in various fields such as geography, geology, geophysics, oceanography, hydrology etc. which may yield surprising results. The Pressure Cooker Model A hydrological explanation for Earth Quakes, Volcanos, Tsunami et al The bottom of the oceans is subjected to very high pressure of the order of about 1000 atmospheres and over millions of years, the high pressure drives water molecules which may link up with subterranean water channels criss-crossing the continental rock system. Naturally this region receives geothermal energy from the interior of the earth, resulting in high pressure steam in all the region circling the entire globe up to the zones where basalt begins to melt. Whenever and wherever the pressure becomes too high, the steam is let off through holes in the bottom of the oceans or in the continental shelf itself. These springs come into contact with the water near zero temperature, resulting in two way movement of water; if the water holes are sufficiently large, the possibility of ocean water entering inside the steam zone cannot be ruled out. The instantaneous conversion of water into steam, thereafter, can create havoc in the whole region. The sudden withdrawal of the sea and its massive resurgence during Tsunami can be explained this way. The molten interior of the earth may be influenced by gravitational forces of the Sun and Moon, much as a balloon filled with water can be stretched, effecting the strain on the crust of the earth (Internal tides? Half of the earth in the region of Pacific Ocean is lesss in mass than the other half where the Himlayan mountains greatly adds to the mass of the earth. This results in a slight deviation of the center of gravity of the earth being slightly away from the geometric center. This deviation is further enhanced by the pull of the Sun and the Moon during new moon/full moon days. The Hindus believe that Amawas is inauspicious for all purposes. As children we were given holiday during Amawas days from the study of Vedas. A computer study of the accidents during the days near Amawasya and the rest of the days of the month may confirm this belief) It is generally of observed that earthquakes etc. occur during or near full Moon or new Moon days. The crust of the earth supporting continents and oceans is resting on a zone of steam, always vibrating, very much like the vent weight of a pressure cooker! Iceland is a model of the above .Hot springs and steam rising from inside is a common sight. The steam is used for working turbines producing electicity. There are earth quakes also round the year, but these are mild as steam pressure is released continuosly. We can utilize the steam inside the earth and end the menace of earth quakes ! Under water research station This can be designed after the manner of a space craft with strong outer casing whose volume can be increased or decreased at will, say, by operating the sliding dome, so that the craft may by allowed to sink down or go up in water or remain stationary, by providing fin-like flaps to be operated manually, this craft can be balanced. Propellers around the craft can be used to effect side-wise movement. Future possibilities If we are able to bore a well into the sea bottom, it may be possible to get unlimited quantities of steam out of the “pressure cooker” for operating thermal power station. Steam can also be used as fuel for cooking etc. Treasures may be fished out of the sea bottom.

title: User:Waterfriend


text:
User:Soau

Siksha O Anusandhan University
(Declared U/S 3 of the UGC Act, 1956)

Siksha O Anusandhan (SOA) University, is a premier University located at Bhubaneswar, the temple city of Orissa. The University offers undergraduate, post graduate, post graduate diploma and doctoral programmes in engineering / technology, medicine, dental, nursing, business administration, hospitality and tourism management, hospital administration, pharmacy, biotechnology and other new-age career courses. The University is proactive in its approach to understand and meet the changing requirements of students, employees and the community at large. The University aims at maintaining continuous learning, improvement and transformation in technical and professional education. The University emphasises on the quality of learning process in the perspective of the following vision and mission:

VISION

To build a state-of-the-art educational system, facilitating the process of imparting quality professional, and technical education in India and abroad.

MISSION

• To promote learning by students and faculty. • To promote quality teaching of global perspective.

• To acquire knowledge and skill through updated training and research.

• To achieve transformation of knowledge.

• To establish effective, efficient and transparent system of examination and evaluation.

• To promote the spirit of entrepreneurship among the students of the University.

• To foster close interaction with industries.

• To realize and adopt social responsibilities.

THE FACULTIES

Ø Institute of Technical Education & Research (ITER) - Faculty of Engineering and Technology

Ø Institute of Business & Computer Studies (IBCS) -Faculty of Management Sciences

Ø School of Hotel Management (SHM)-Faculty of Hotel Management

Ø Institute of Medical Sciences & SUM Hospital (IMS&SH) -Faculty of Medical Sciences

Ø Institute of Dental Sciences (IDS) - Faculty of Dental Sciences

Ø School of Pharmaceutical ScienceS (SPS)-Faculty of Pharmaceutical Sciences.

Ø SUM Nursing College (SNC)-Faculty of Nursing

Ø Center for Bio-Technology (CBT)-Faculty of Pharmaceutical Sciences.

title: User:Soau


text:
User:Cdpl94

I'm doing this Wiki thing because I really do use it, not having ever had a personal encyclopedia at hand, and because too many times I've looked for something and not found it, and because I enjoy editing other people's stuff almost more than I enjoy writing.

In reading the discussions about the internal soul searching of the Wiki world I found myself as one of the 99%, and in the under-represented categories. Being an activist, I'm going to do something about it by getting engaged.

You'll soon see what I'm interested in.

title: User:Cdpl94


text:
User talk:Cdpl94

 Hello Cdpl94, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. Jtneill - Talk - c 11:45, 28 January 2010 (UTC)

 Hi , I am Percy. Nice to meet you. I love the beach, swimming, fishing, and I am currently doing some research at Wikiversity on oceanography. I am also learning to play piano. What do you like? Can I be your mascot?

title: User talk:Cdpl94