text:
University of Florida/Egm6321/f09.team1.gzc/Mtg6

Mtg 6: Sun, 16 Jan 11

page6-1

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

 * style="width:95%" |
 * style="width:95%" |

Proof of Taylor series continued

Since

$$  \displaystyle \int_{x_0}^{x} {f}^{1}(t)dt=[ {f}^{0}(t)]_{t= {x}_{0}}^{t=x} $$     (1)
 *
 * }

$$  \displaystyle \Rightarrow \color{red}(e) \color{blue} p.5-3 $$

Int. by parts (1)

$$  \displaystyle \color{red}(2) \begin{cases} & \ \color{black} \int_{x_0}^{x} \color{blue}  \underset{u^{'}}{ \underbrace{1}} \underset{v}{ \underbrace{ \color{black}{{f}^{(1)}(t)} }} \color{black}dt= {[uv]}^{x}_{x_0}- \int_{}^{}{uv}^{'}   \\ & \ \color{black}=  {[tf^{(1)}{t}]}^{t=x}_{t=x_0}-  \color{blue}  \underset{ \alpha}{ \underbrace{ \color{black}{ \int_{x_0}^{x}tf^{(2)}dt} }} \\ & \  \color{black}= xf^{1}(x)-x_{0}f^{(1)}(x)0)- \color{blue} \alpha  \\ & \  \color{red}+ \color{blue} xf^{(1)}(x_0) \color{red}- \color{blue}xf^{(1)}(x_{0})\\ & \     \color{black} =\color{blue} \underset{ \int_{x_{0}}^{x}xf^{(2)}(t)dt=: \beta}{  \underbrace{\color{black}[xf^(1)(x)-xf^(1)(x_{0})] }} \color{black}+(x-x_{0}){f}^{(1)}(x_{0}) - \color{blue}   \alpha \end{cases}

$$

page6-2

Combine [ + β - α ] into a single int. use (2) p.6-1 in (2) p.5-3 :

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

 * style="width:95%" |
 * style="width:95%" |

$$  \displaystyle f(x)=f(x_{0})+ \frac{ {(x-x_{0})}^{ \color{blue} 1}}{ \color{blue}1! } {f}^{(1)}(x_{0})+ \color{blue} \underset{ \beta-\alpha}{ \underbrace{ \color{black} \int_{x_{0}}^{x}(x-t)f^{(2)}(t)dt }} $$

(1)
 *
 * }

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

 * style="width:95%" |
 * style="width:95%" |

HW*2.1: 1) Do integration by parts on last term (integration) of (1) to reveal 3 more terms in Taylor series, i.e. ,

$$  \displaystyle \frac{ {x- {x}_{0}}^{2}}{2!} {f}^{(2)}( {x}_{0})+ \frac{ {x- {x}_{0}}^{3}}{3!} {f}^{(3)}( {x}_{0})+ \frac{ {x- {x}_{0}}^{4}}{4!} {f}^{(4)}( {x}_{0}) $$
 * }

plus remainder2)Use IMVT to expression remainder in terms of f(s)(ξ) for s belong[x0,x]3) Assume(3)&(4)p.3-3 correct, do intergration by parts once more

page6-3

to verify(3)&(4) p.3-3 for (n+1) expansionwith R(n+2)(x)
4) UseIMVT on(4) p.3-3to show(5)p.3-3

HW*2.2:$$f(x)=sinx, x\in[0, \pi]$$ Constrast Taylor Series of f(.) around

$$  \displaystyle \begin{cases} & \ {x}_{0}= \frac{ \pi}{4} \ \color{blue} S10 \\ & \ {x}_{0}= \frac{3 \pi}{4} \ \color{blue} S11 \end{cases} $$ for n = 0,1,2,...,10

Plot these series (for each n) Find (estimate) max $$ \left|{R}_{n+1}(x=\frac{3 \pi}{4}) \right|$$

$${\color{red}(5)} {\color{blue}p.3-3} \Rightarrow \left|{R}_{n+1}(x=\frac{3 \pi}{4}) \right| \leqq \frac{{x-{x}_{0}}^{n+1}}{(n+1)!}{\color{blue}(\alpha)}$$

page6-4

$${\color{blue} \alpha}=max \left|{f}^{(n+1)}(t) \right| $$

$$\color{blue} \underset{\leqq 1 (f(x)=sinx)}{ \underbrace{ \color{black} t \in [{x}_{0},x]}}$$

Note:Motivation for pf of Taylor series expansion. (similar technique will be used)

 * higher order analysis of Trap. rule (not in A.)
 * Richardson extrap.
 * clenshaw-Cwetis quadrature
 * chebyshew poly (orthog.) Recent devel. using chebyshew poly to solveL2_ODE_VC (Linear 2nd order ODE with varying coefficient ) combine of symbolics + numericsReference : Trefethen's chebfun.

page6-5


text:
University of Florida/Egm6321/f09.team1.gzc/Mtg8

Mtg 8: Wed, 19 Jan 11

page8-1

Legendre poly. Pn(x):

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

 * style="width:95%" |
 * style="width:95%" |

$$ {P}_{n}(x)= \sum_{i=0}^{{\color{red}[{\color{black}n/2}]} } {(-1)}^{i} \frac{(2n-2i)! {x}^{n-2i}}{ {2}^{n}i!(n-i)!(n-2i)!}$$

(1)
 *
 * }

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

 * style="width:95%" |
 * style="width:95%" |

$$ {\color{red}[ {\color{black} n/2}]} = integer \ part \ of \ n/2$$

(2)
 *
 * }

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

 * style="width:95%" |
 * style="width:95%" |

e.g., m = 5, n/2 = 2.5 , [2.5] = 2

$$ {\color{red}(3)} \ {P}_{0}=1 \color{blue} \in {P}_{0} $$

$$ {\color{red}(4)} \ {P}_{1}=x \color{blue} \in {P}_{1} $$

$$ {\color{red}(5)} \ {P}_{2}= \frac{1}{2}(3{x}_{2}-1)  \color{blue} \in {P}_{1} $$

$$ {\color{red}(6)} \ {P}_{3}= \frac{1}{2}(5{x}_{3}-3x)  \color{blue} \in {P}_{1} $$

$$ {\color{red}(7)} \ {P}_{4}= \frac{35}{8}{x}^{4}-\frac{15}{4}{x}^{2}+\frac{3}{8}  \color{blue} \in {P}_{1} $$

$${\color{blue}{P}_{n}} \ = \ set \ of \ poly. \ of \ degree \ \leqslant \ n$$

HW 2.6:Verify(3)-(7) using(1)-(2)

$${\color{red}(5)} \Rightarrow \ {P}^{2}=0 \Rightarrow {x}_{1,2} = {\color{red}-} \frac{1}{\sqrt{3}} \ or {\color{red}+} \frac{1}{\sqrt{3}}$$

page8-2

Weights wi, i = 1,...,n ((1)p.7-5)

 * }

$$ \underline { \color{blue} {NOTE:}} \ {E}_{n}(f)=0 \ \forall f \in {P}_{2n-1} \ since \ f^{\color{red}(2n)}(x)=0 \. \ Only \ need \ to \ use \ n \ int. \ pts\ \left\{x_{i}, \ i=1,...n\right\} \ to \ int. \ exactly \ any \ poly \ in \ P_{2n-1} \ i.e., \ of \ degree \ \leqq \ 2n-1$$

$${\color{blue} {Newton-cotes \ method:}} \ I(f)$$

$$History \ Newton \ cotes \ \color{blue}  \rightarrow \ Lecture \ plan \ suli+Meyers(2003)$$

page8-3

$${\color{blue}1)} \ Approx. \ f(.) \ using \ {\color{blue} { \underline { {\color{black} Lagrange}}}} \ interp. \  funcs \  \Rightarrow \  {f}^{ {\color{red} L}}_{n}(x) $$

$$ Int. \ exactly \ {f}^{ {\color{red} L}}_{n}(x) \ \Rightarrow \ \color{blue}  \underbrace{ \color{black} I( {f}^_{n})}  $$

$$I_{n}(f) \ := I( {f}^_{n})=\int {f}^_{n}(x)dx \ \color{red} (1) $$

$${f}^_{n}(x)=P_{n}(x)= \sum_{i=0}^{\infty} {\color{blue} \underset{Lagrange \ interp. \ func}{ {\color{black} \underbrace{l_{i,n}(x)}}}} \ f(x_{i}) \ \color{red} (2)$$

NOTE: Demonstrated Wolfram Alpha(WA) e.g., (debt usa)/(gdp usa) integrate x from 0 to 1 link WA comp. results in HW reportsAvoid plagiarism

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


text:
University of Florida/Egm6321/f09.team1.gzc/Mtg10

Mtg 10: Sat, 22 Jan 11

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

 * style="width:95%" |
 * style="width:95%" |

page10-1

HW2.9:Use (1) p.8-1 to generate P5(x), and matlabl command"roots"to comp. the roots of P5(x)to check values in table on p. 7-5. Plot the roots on [-1,+1] using
matlab "plot" command (plot dots "."

with coordinator (xi,yi), i = 1,...,5 : use "markersinge" 15)

xi: roots of P5(x)

yi: yi = 0

Plot (x, y, '.', 'markersige', 15)

Repeat the above for P10(x)

observe the location of the roots near end points -1 and +1

(prepare for Runge phenomenon)

NOTE:

Lagrange interp. cont'd p.9-4

page10-2

Simpson's rule (simple)

$$[a,b] \ x_{0}=a, \ x_{1}=\frac{a+b}{2}, \ x_{2}=b \ \color{red} (1)$$

$$ {\color{blue} { \underline {Method 1:}}} \ f_{2 }(x)=P_{2}(x) \overset{=}c_{2}x^{2}+c_{1}x+c_{0}$$

c0, c1,  c2   unknowns

p2=(xi) = f(xi)   i= 0, 1, 2               (3)

3 equations for 3 unknowns {ci}

Method 2:Use lagrangeinterp.(2) p.8-3(1) p.9-2

$$ {\color{red} {(4)}} = \sum_{i=0}^  {\color{blue} \underset{l_{i}(x)}{ \underbrace}}f(x_{i})$$

Equiv. of meth1 and meth 2:           (3)

$$p_{2}(x_{j})= \sum_{i=0}^{2} \color{blue} \underset{ \delta_{ij}}{ \underbrace}f(x_{i})=f{x_j} \ \color{red}(5)$$

page10-3

$$l_{0{\color{blue},2}}=l_{0{\color{blue},2}}= \prod_{j=0 {\color{red}j \neq i}}^{n=2} \frac{x-x_{j}}{x_{0}-x_{j}}= \frac{(x-x_{1})(x-x_{2})}{(x_{0}-x_{1})(x_{0}-x_{2})} \ {\color{blue} \in P_{2}}$$

It can be verified that

l0(x0)=1, l0(x1) = l0(x2) = 0

li(xj) = δij                 i,j = 0. 1. 2

$$  {\color{blue} \underset{ \underbrace{\color{black}l_{1{\color{blue}(i=1),2(n=2)}}(x)}}}= \prod_{j=0, {\color{red}{j \neq 1}}}^{2 {\color{blue}(n=2)}}\frac{x-x_{j}}{x_{i{\color{blue}(i=1)}}-x_{j}}=\frac{(x-x_{0})(x-x_{2})} \ {\color{blue} \in P_{2}}$$

$$l_{1}(x_{1})=1 \, \ l_1(x_{0})=l_{1}(x_{2})=0$$

page10-4

$${l}^{''}_{1}(x_{1})<0$$

HW*2.10: Use(2) & (3)p.10-2 to find

expression for {ci} in terms (xi, f(xi)) i=0,1,2.

HW*2.11: Use (4) p.10-2 to

derive simple Simpson's rule

HW*2.11:f(x)-ex1/x on [o,1] S10and on [-1,1]S11

x0=-1, xn=+1

Consider n=1(Trap), 2(Simp), 4, 8, 16

page10-5

Constrast fn(x) as in (2) p.8-3.

Plot f, fn , n= 1, 2, 4, 8, 16

Compare
$$ {I}_{n}= \int_{a}^{b}f_{n}(x)dx $$ n=1, 2, 4, 8

and compare to I (use WA with more digits)

For n=5 plot l0, l1, l2 How would l3, l4, l5 look like?

HW*2.13:show

$$ \underset{Simple \ Trap.}{ \underbrace} \Rightarrow  \underset{Compare \ Trap.}{ \underbrace}$$

$$\underset{Simple \ Trap.}{ \underbrace} \Rightarrow  \underset{Compare \ Trap.}{ \underbrace} $$
 * }


text:
University of Florida/Egm6321/f09.team1.gzc/Mtg12

Mtg 12: Wed, 26 Jan 11

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

 * style="width:95%" |
 * style="width:95%" |

[[media: Nm1.s11.mtg12.djvu| page12-1]]

Appl. of LIET p.11-2

$$Let \ {f}^{L}_{n}(t) \ be \ Lagrange \ interpolation \ of \ order \ n \ to \ f(t). \ \Rightarrow \ {f}^{L}_{n}(.) \ \in \ \color{blue}  \underset{set \ of \ poly \ of \ degree \  \leqslant \ n}{ \underbrace{ {\color{black}P_{n} }}}$$

$${\color{blue} n=1} \ {f}^{L}_{1}(.) \ interpolation \ exactly$$

$${\color{blue} P_{1}(x) \rightarrow}straight \ lines \ (linear \ functions) \ i.e., \ e_{1}(P_{1},t) \ \equiv \ 0 \ \forall \ t$$

 * nm1.s11.Mtg12.pg1.fig1.svg

$${\color{blue}n=2} \ {f}^{L}_{2}(.) \ interpolation \ exactly \ parabolas \ (polynomial \ of \ {\color{blue}\underset{degree}{ \underbrace{\color{black}order}}}) \ 2$$

$${\color{blue}n=3} \ cubic \ polynomial$$

$$\color{blue} \vdots$$

$${\color{blue}n} \ {f}^{L}_{n}(.) \ interpolation \ exatcly \ \color{blue}\overset{ \in P_{n}}{ \overbrace}$$

$$i.e., \ e_(p_;t) \ =0 \ \forall t$$

[[media: Nm1.s11.mtg12.djvu| page12-2]]

LIET: (2) p.11-3

$$IF \ f \ = \ p \in \ P_{n} \ \Rightarrow \ f^{(n+1)} \ \equiv \ 0$$

$${\color{red} (2)} \ {\color{blue} \ p.11-3} \ \Rightarrow \ e_(p,t)=0 \ \forall t$$

$$i.e., \ {f}^{L}_{n}(.) \ interpolation \ {\color{red}exactly} \ p\in P_{n}$$

$$\color{blue} Method \ to \ comp. \ \left\{w_{i,n}; \ i=0,..,n \right\} \ given \ \left\{x_{i}; \ i=0,..,n \right\} \ \underline {not} \ necessarily \ equidistant.$$

$${\color{blue}\underline {j=0}} \ consider \ f=p_{\color{blue}j}=p_{0}\in P_{0} \ {\color{red}constant}$$

$$e_{\color{blue}n} \left( \underset{{\color{blue} \underbrace{{\color{black}p_{\color{red}0}}}};t} \right) \ = {\color{blue}\underset{1}{ \underbrace}}- {f}^{L}_{{\color{blue}n}(t)}={\color{blue}1}-{f}^{L}_{n}(t)=0$$

$$\Rightarrow {\color{blue} \underset{ \sum_{i=0}^{n}l_{i,n}(t){\color{red} \underset{1}{ \underbrace}}}{ \underbrace}}=1 \ \Rightarrow \ {\color{red} \int_{a}^{b}}{\color{blue} \left( \underset{{f}^{L}_{n}(t)}{ \underbrace}\right)}dt \ = \ {\color{blue} \underset{b-a}{ \underbrace}}$$

[[media: Nm1.s11.mtg12.djvu| page12-3]]

$$\Rightarrow \ {\color{blue} \underset{w_{i,n}}{ \underbrace{ \left( {\color{red} \int_{a}^{b}}l_{i,n}(t)dt\right)}}.{\color{red}1}}=b-a$$

$$\Rightarrow$$

$${\color{blue} \underline {j=1}} \ let \ f=p_=p_{1}\in P_{1}$$

$$choose \ f(x)=p_{1}(x)= \ {\color{red}x} \in P_{1}$$

$$e_{\color{blue} \left( \underset{ \underbrace}{\color{black};t }\right)} \ = \ {\color{blue} \underset{ \underbrace}}-{f}^{L}_{\color{blue}n}(t) ={\color{blue}t}-f_{n}(t)=0$$

$$\Rightarrow \underset{ {\color{blue}\underbrace{\color{black}{f}^{L}_{n}(t)}}}={\color{red}t} \ \Rightarrow \ {\color{red} \int_{a}^{b}}{\color{blue} \underset{{f}^{L}_{n}(t)}{ \underbrace{ \left( {\color{black} \sum_{i=0}^{n}l_{i,n}(t).}{\color{red}x_{i}}\right)}dt}}={\color{blue} \underset{(b^{2}-a^{2})/2}{ \underbrace}}$$

$$\Rightarrow$$

 * }


text:
University of Florida/Egm6321/f09.team1.gzc/Mtg20

Mtg 20: Wed, 16 Feb 11

 * nm1.s11.Mtg20.pg1.fig1.svg

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

 * style="width:95%" |
 * style="width:95%" |

page20-1

Pf of SSET p.19-3 cont'd

$${\color{red}(1)}{\color{blue}p.19-3 \ Rolle's \ thm} \ \Rightarrow \ \exists \  \varsigma_{1} \in \  ] \ 0,1 \ [ \ st \ G^{\color{blue}(1)( \varsigma_{1})}  \overset{=}0$$

$$Now \ G^{(1)}(0) \ \overset{=}0 \ {\color{red}why ?}$$

$${\color{red}(1)} \ {\color{blue}p.19-1:} \ G^{(1)}(t)\overset{=}e^{(1)}(t)-5t^{4}e(1)$$

$${\color{red}(5)} {\color{blue}p.18-3:} \ e(t)=A(t)-{A}^{L}_{2}(t) \ {\color{red}(4)}$$

$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ e^{(1)}(t)=A^{(1)}-{A}^{L{\color{red}(1)}}_{2}(t) \ {\color{red}(5)}$$

page20-2

$$A(t)= \int_{+t}^{-t} -=\int_{k}^{-t} -+\int_{t}^{k}-$$

$$A^{(1)} \overset{=}F(-t)+F(t)$$

$$k\in{\color{red}]}-t,t{\color{red}[} \ ({\color{red}k is constant})$$

(4)p.20-1& (5)p.18-3:

$$A^{L{\color{red}(1)}}_{2}(t)\overset{=} \frac{1}{3}[F(-t)+4F(0)+F(t)]+\frac{t}{3}[F^{(1)}(-t)+F^{(1)}(t)$$

(5) p.20-1:

$$e^{(1)}(0)=A^{(1)}(0)-A^{L(1)}_{2}(0)=2F(0)-[2F(0)+0]=0 \ {\color{red}(3)}$$

$${\color{red}(3) \ and \ (3)}{\color{blue}p.20-1} \Rightarrow{\color{red}(2)}{\color{blue}p.20-1}$$

Recall:

$$G^{(1)}( \xi_{1})=0 \ {\color{red}(1)}{\color{blue}p.20-1}$$

$$G^{(1)}( 0)=0 \ {\color{red}(2)}{\color{blue}p.20-1}$$
 * }

 * nm1.s11.Mtg20.pg2.fig1.svg

page20-3

$$Rolle's theorem \Rightarrow \ \forall\xi_{2}\in]0,\xi_{1}[ st \ \ G^{(2)}(\xi_{2})=0 \ {\color{red}(1)}$$

$$Again, \ G^{(2)}(0)\overset{=}0 \ {\color{blue}HW^{*}4.3}$$

$${\color{red}(1) \ and \ (2)} \ Rolle's theorem \ \Rightarrow \ \forall\xi_{3}\in]0,\xi_{2}[ \ G^{(2)}(\xi_{2})=0 \ st \ \ G^{(3)}(\xi_{3})\overset{=}0$$

$${\color{red}(1)}{\color{blue}p.19-1:} \ G^{(3)}(t)\oversete^{(3)}(t)-{\color{red} \underset{(\xi)(4)(3)}{ \underbrace}}e(1)$$

$$e^{(3)}(t) \underset{=}-\frac{t}{3}[F^{(3)}(t)-F^{(3)}(-t)] \ {\color{red}(5)}$$

$$G^{(3)}(\xi_{3})=-\frac{\xi_{3}}{3}{\color{green} \overset{Apply \ DMVT}{ \overbrace}}-60(\xi_{3})^{2}e(1) \underset{ \overset{=}}0 $$

$$ \underset{ \overset{=}}-\frac{\xi_{3}}{3}[{\color{blue} \underset{\xi_{3}-(-\xi_{3})}{ \underbrace{\color{black}2\xi_{3}}}}F^{(4)}(\xi_{4})]-60(\xi_{3})^{2}e(1) \ \xi_{4}\in \ ]-\xi_{3},\xi_{3}[$$


text:
University of Florida/Egm6321/f09.team1.gzc/Mtg22

Mtg 30: Mon, 14 Mar 11

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

 * style="width:95%" |
 * style="width:95%" |

page22-1

Comp. Trap. rule error: $$ h={\color{red}\frac{n}} {\color{blue}\leftarrow}{\color{red}(3)}{\color{blue}p.7-4}$$

 * Mtg22.pg1.fig1.svg

$${E}^_{n} \underset{:=} \ I-{I}^{T}_{n}= \int_{a}^{b}f(x)dx-h[{\color{red}\frac{1}{2}}f(x_)+f(x_{1})+ \cdots+f(x_{n-1}){\color{red}\frac{1}{2}}f(x_)] \ {\color{red}(1)}$$

$$= \sum_{i=1}^{n}{\color{blue} \left\{{\color{black} \int_{x_{i-1}}^{x_{i}}f(x)dx-\frac{h}{2}[f(x_{i-1})+f(x_{i})]} \right\}}$$

(1)p.17-2:

$$ \left|{E}^{T}_{n} \right| \leqslant \frac{h^{3}}{12} \underset{ {\color{blue}\underset{ \overset{=: \overline{M_{2}}}}}}{\sum_{i=1}^{n}(max{\color{red} \left|{\color{black}f^(\xi)} \right|})}{\color{red} \ (2)}$$

(4)p.17-1:

$$ \overline{M_{2}} \leqslant M_{2} \ {\color{red}(4)}$$

$${\color{red}(5)}$$
 * }

page22-2

HW*4.8: Comp. Simpson error

$$ \left|{E}^_ \right| \leqslant \frac{(b-a)^{5}}{2880n^{4}}{\color{blue} \underset{{\color{red}(4)}p.17-1}{ \underbrace}}=\frac{(b-a)h^}{2880}M_{4} \ \color{red}(1)$$

HW*4.8: See HW*2.4 p.7-3

1)Use error estimate for Taylor Series, Compare Trap., Compare Simpson, to findn

Q(10-6), and compare to number results.

2)Numerically find the power ofhin error : Plot

logerror vslogh, and meas. slop with least square

(lin. regression).

 * nm1.s11.Mtg22.pg2.fig1.svg$$E=ah^+Q(h^{k+1}), \ $$   $$ \ logE=loga+{\color{red}k}logh$$

page22-3

HW*4.9: pf. of SSET, G(.) in (1) p.19-1

A)Redo the pf for 2 cases

$${\color{blue}1)} \ G(t):= \ e(t)-t^{\color{red}4}e(1) \ {\color{red}(1)}$$

$${\color{blue}2)} \ G(t):= \ e(t)-t^{\color{red}6}e(1) \ {\color{red}(2)}$$

Print out where pf breaks down.

B)For G(t) as in(1)p.19-1(w/t5), find

G(3)(0)and follow same steps in pf to

see what happen.

HW*4.10:1)Don'tUse matlabtrapzforcompare Trap.in

your code to produce Table 5.1 n A.p.255(p.22-4)

 $${\color{red}(3)}I= \int_{0}^{\pi}e^{x}{\color{blue}sinx}dx$$(use WAto find I)


text:
University of Florida/Egm6321/f09.team1.gzc/Mtg30

Mtg 30: Mon, 14 Mar 11

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

 * style="width:95%" |
 * style="width:95%" |

page30-1

$${\color{blue} \underline {Appl \ of \ HOTR \  p.29-2:}} \ {\color{red} \underline {{\color{blue}Corrected \ Trap. \ rule}s}}$$

$${\color{blue}CT_(n)}={\color{blue}\underset{CT_(n)}{\underbrace}}+ {a}^_h^+Q(h^)$$

$$I=CT_(n)+Q(h^)$$

$${\color{blue}CT_(n)}={\color{blue}CT_(n)}+{a}^_h^+Q(h^)$$

$$I=CT_(n)+Q(h^)$$

HW*5.6:See HW*2.4 p.7-3, HW*5.4, p.29-6

Compare In using CTk (n),, for n=2,4,

8,16,... until error I-In=Q(10-6).

HW*5.7:Discuss ppros and cons of intergration methods

1)Taylor series3)Compare Simpson5)CTk(n)

2)Compare Trap.4)Romberg(Richardson)

page30-2

Pf of HOTRE p.29-2:

$${E}^{T}_{n}=I-I_{n}= \int_{a}^{b}f(x)dx-T_(n)$$

$$=\sum_{k=0}^{n-1}[\int_{x_{k}}^{x_{k+1}}f(x)dx-\frac{h}{2}{\color{blue} \left\{{\color{black}f(x_{k})+f(x_{kH})} \right\}}] \ {\color{red}(1)}$$

$$x_{k}:=a+kh \, \ h:=(b-a)/n \ {\color{red}(2)}$$

$$Transfer \int_{x_{k}}^{x_{kH}}-{\color{blue}dx} \ to \ \int_{-1}^{+1}-{\color{blue}dt} $$

$$x(t):=\frac{x_{k}+x_{k+1}}{2}+t\frac{h}{2}, \ t\in [-1,+1] \ {\color{red}(3)}$$

$$ {\color{blue}\begin{cases} & \ {\color{black}x(-1)=x_{k}} \\ & \ {\color{black}x(0)=(x_{k}+x_{k+1})/2} \ {\color{red}(4)} \\ & \ {\color{black}x(+1)=x_{k+1}} \end{cases}}$$

$${E}^{T}_{n} = \frac{h}{2} \sum_{k=0}^{n-1}[\int_{-1}^{+1}g_{k}(t)dt-{\color{blue} \left\{{\color{black}{g}^{(-1)}_{k}+ {g}^{(+1)}_{k}} \right\}}] \ {\color{red}(5)}$$

$${g}^{(t)}_{k}:=f(x(t)) \ st \ x\in[x_{k},x_{k+1}] \ {\color{red}(6)}$$

page30-3

$$ \int_{-1}^{+1} {\color{blue}\underset{P_{1}(t)}{ \underbrace}}g^(t)dt \overset{=}{\color{blue} \underset{:=E}{ \underbrace}} \ {\color{red}(1)}$$

Step2a: Some motivation

HW*5.8:

$${\color{blue} \underset{\frac{d^{i}}{dt^{i}}{g}^{(t)}_{k}}{ \underbrace}}=(\frac{h}{2})^{i}f^{\color{blue}(i)}(x(t)) \ {\color{red}(2)}$$

$$x\in[x_{k},x_{k+1}]$$

$$To \ obtain \ {\color{blue} \underline {\color{black}higher \ powers \ of \ h}} \ {\color{blue}\Rightarrow} \ Take \ higher$$

$$derivation \ of \ g_{k}(t) \ {\color{blue}\Rightarrow} \ {\color{blue} \underline {\color{black}successive \ int. \ by \ parts}}$$

$$(recall \ pf \ of \ Taylor \ series \ {\color{blue}Mtg5 \ and \ Mtg6})$$

$${\color{red}(1)\Rightarrow}E=\int_{-1}^{+1} {\color{blue} \underset{P_{1}(t)}{ \underbrace}}{\color{black}g^}dt$$

$$P_{2}(t) \overset{:=}c_{1}\frac{t^{2}}{2!}+c_{3} \, \ c_{2}\overset{=}0$$

page30-4

Step2b:Intergration by parts

$$E=[P_(t)g^{(1)}(t)]^{+1}_{-1}-{\color{blue}\overset{A \overset{2b}{=}}{ \overbrace}}+\int_{-1}^{+1}P_g^(t)dt \ {\color{red}(3)}$$

$$P_\overset{=} \int P_(t)dt=-\frac{t^{3}}{6}+ \alpha t +\beta {\color{blue}(new \ intergration \ constant)}=c_{1}\frac{t^{3}}{3!}+C_{3}t+C_{4} \ (C_{2}\overset{=}0)$$

Pb:Find(α, β)=(C3, C4) to cancel terms withg(2) (evenorder derivation of g, i.e., oddpower of h ; here h3)

$$i.e., \ make $$

$$\Rightarrow$$

(recall P1)

$$Consider \ P_ \ st \ P_=0 \, \ P_(+1 \ and \ -1)=0 \ {\color{red}(6)}$$

page30-5

 * nm1.s11.Mtg30.pg5.fig1.svg

$$P_{3}(0)=0\Rightarrow C_{4}=0=\beta \ {\color{red}(1)}$$

$$P_{3}(-1)=0=-\frac{(-1)^{3}}{6}+\alpha(-1)\Rightarrow C_{3}=\alpha=\frac{1}{6} \ {\color{red}(2)}$$

$$P_{3}(+1) \overset{=} -P_{3}(-1)=0 \ {\color{red}(3)}$$

Summary, Steps2ab:

step3a: (3)&(5) p.30-4 ; intergration by parts

$$E=[P_(t)g^{(1)}(t)]^{+1}_{-1}+\int_{-1}^{+1}P_(t)g^(t)dt$$

$$=[P_(t)g^{(1)}(t)]^{+1}_{-1}+[P_(t)g^{(3)}(t)]^{+1}_{-1}-\int_{-1}^{+1}P_(t)g^(t)dt \ {\color{red}(5)}$$

page30-6

$$P_(t)=\int P_(t)dt=-\frac{t^{4}}{4!}+\frac{t^{2}}{12}+\alpha$$

$$=C_{1}\frac{t^{4}}{4!}+C_{3}\frac{t^{2}}{2!}+ \underset{C_{5}} \ {\color{red}(1)}$$

$$C_{1} \ and \ C_{3} \ in \ {\color{red}(4)}{\color{blue}p.30-5}$$

step3b:Intergration by parts

$${\color{red}(5)}{\color{blue}p.30-5}\Rightarrow \ E=[P_g^+P_g^]^{+1}_{-1}$$

$$[P_g^]^{+1}_{-1}{\color{red}+}\int_{-1}^{+1}P_g^dt \ {\color{red}(2)}$$

$$P_(t)=\int P_(t)dt=\frac{t^{5}}{120}+\frac{t^{3}}{36}+\alpha{\color{black}t+}{\color{blue} \underset{C_{6}}{ \underbrace}} \ {\color{red}(3)}$$

$$Want \ eliminate \ g^ \ (thus \ h^);$$

$$So \ make $$

$$\Rightarrow$$

(recall P1, P3 oddfuncs)

page30-7

Similar to (6)p.30-4:

$$Consider \ P_(t) \ st \ P_(0)=0, \ P_(+1 \ and \ -1)=0 \ {\color{red}(1)}$$

$$\Rightarrow \ C_{6}=0 \, \ C_{5}=-\frac{7}{360} \ {\color{blue}HW^{*}5.{\color{red}8}} \ {\color{red}(2)}$$

Summar, Steps3ab:

$${\color{red}(3)}$$

$${\color{red}(4)}$$

Compare to first part in(5)p.30-5

$$E=[P_(t)g^(t)]^{+1}_{-1} \ {\color{red}+} \ P_(t)g^(t)$$
 * }


text:
University of Florida/Egm6321/f09.team1.gzc/Mtg34

Mtg 34: Wed, 23 Mar 11

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

 * style="width:95%" |
 * style="width:95%" |

page34-1

1) Supersonic intercaptor: Find angle of attack and thrust to reach a given altitude inmin time

Reference: Bryson & Denham 1962

2) Bunt maneuver of aircraft: min altitude to reach target with a bunt maneuver (inverted loop)

page34-2

$${\color{blue}V} \ = \ velocity \ of \ P \ (airplane \ = \ pt)$$

$${\color{blue}T} \ = \ thrust \ // \ airplace \ axis$$

$$\alpha \ = \ \angle \ ({\color{blue}V,} \ {\color{red}T} ) \ = \ angle \ of \ attack$$

$$\gamma \ = \ \angle \ (x, \ {color{blue}V}) \ = \ angle \ between \ honz \ x \ axis \ and \ veloctity$$

$$D \ = \ axial \ aero \ force \ // \ {\color{red}T}$$

$$(parall. \ airplane \ axis)$$

page34-3

$$L \ = \ Tramsv. \ aero. \ force \ \bot \ {\color{red}T}$$

$$W \ = \ mg \ = \ airplane \ weight.$$

$$Equation \ of \ motion \ (EOM)$$

$${color{blue}Control \ input:} \ {\color{blue} \underset{time \ dependent}{ \underbrace{{\color{red}T}(t)}\, \ \alpha(t)} }$$

$${\color{blue}state \ var,:} \ x(t), \ y(t) \ v(t), \ r(t)$$

$${\color{blue} \begin{cases} & \ {\color{black}D=\frac{L}{2} \ C_{d} \ \rho \ V^{2} \ S_{reft}}  \\ & \ {\color{black}L=\frac{L}{2} \ C_{l} \ \rho \ V^{2} \ S_{reft}} \end{cases}}$$ $$C_{d}=drg \ coefficient \ {\color{green}|} \ \rho=air \ density$$ $$C_{l}=lift \ coefficient \ {\color{green}|} \ \S_{ref}=ref. \ area \ of \ airplane$$ page34-4

$$C_{d}=A_{1}\alpha^{2}+A_{2}\alpha+A_{3} \ {\color{red}(1)}$$

$$(A_{1}, \ A_{2}, \ A_{3})=curve \ fitting \ coefficient$$

$$C_{l}=B_{1}\alpha+B_{2} \ {\color{red}(2)}$$

$$(B_{1}, \ B_{2})=curve \ fitting \ coefficient$$

$$\rho=C_{1}h^{2}+C_{2}h+C_{3} \ {\color{red}(3)}$$

$$(C_{1}, \ C_{2}, \ C_{3})=curve \ fitting \ coefficient$$

Reference: S & Z 2007

$${\underline { \color{blue} {Kinematics: }}} \ \frac{dx}{dt}=V_{x}=Vcosr \ {\color{red}(4)}$$

$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{dy}{dt}=V_{y}=Vsinr \ {\color{red}(5)}$$

$${\underline { \color{blue} {Kinematics: }}}{\color{blue}Euler \ equations:} \ \frac{d}{dt} {\underline {P}}= \sum_ {\underline {F}}_ \ {\color{red}(6)}$$

page34-5

Note: Parameteriztion of ellipse

$${\color{blue} \underset{cos^{2}t}{ \underbrace}}+{\color{blue} \underset{sin^{2}t}{ \underbrace}}=1 \ {\color{blue}ellipse}$$

<{\color{blue}dl={\color{red}[{\color{blue}dx^{2}+dy^{2}}]^{1/2}}}/math>

$$Eccentricity: \ e= \left(1-\frac{b^{2}}{a^{2}} \right)^{1/2}$$

$$C=\int dl=a \int_{t=0}^{2\pi}[1-e^{2}cos^{2}t]^{1/2}dt$$

$$=41\int_{\alpha=0}^{\frac{\pi}{2}}[1-e^{2}sin^{2}\alpha]^{\frac{1}{2}}d\alpha=4aE(e)$$
 * }


text:
Book Reviews/Cradle to Cradle

Cradle to Cradle is book that was coauthored by William McDonough and Michael Braungart and self-Published in 2002. Bearing the slogan, "Remaking the way we make things," Cradle to Cradle sets out to address the concept of waste by design, and offer us a radically different yet familiar solution.

Introduction: This book is not a tree
IIn the introduction, William and Michael are introduced as an architect and a chemist respectively, and their storied reasons for starting the c2c movement are outlined. Main Points:
 * Bill's quote, "design is a signal of intention."
 * what is our intention?

 * Can we design products that "[love] all the children, of all species, for all time?"[1]
 * Can we, the human species, leave an ecological footprint on this planet that we can "delight in, not lament?" [2]

Chapter One: A Question of Design
NoahBonn 19:35, 14 August 2011 (UTC)

title: Book Reviews/Cradle to Cradle


text:
User:EGM6341.s11.team1.Chiu/S10 Mtg41

Mtg 41: Tue, 13 Apr 10

[[media: Egm6341.s10.mtg41.djvu| page41-1]]

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

 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

HW:  [[media: Egm6341.s10.mtg40.djvu| page40-3]] continued

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

 * style="width:95%" |
 * style="width:95%" |

Initial condition $$ \mathbf{Z}(t_{0})= \mathbf{Z}_{0} \ $$  S+Z 2007 page 316

 * }

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

 * style="width:95%" |
 * style="width:95%" |

Controls: $$ \mathbf{u}(t)= \left \lfloor T(t) \alpha(t) \right \rfloor \ $$

 * }

 * nm1.s11.Mtg41.pg1.fig1.svg

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

 * style="width:95%" |
 * style="width:95%" |

Solve for $$ \mathbf{Z}(t), \ t \in \left [ 0,40 \right ] \ $$  at   $$ \left \| \underset{ \color{blue} \underset{2 \ successive}{\uparrow}}{ \mathbf{Z}^{(k+1)}}- \underset{ \color{blue} \underset{numerical \ solutions}{\uparrow}}{ \mathbf{Z}^{(k)}} \right \|_{ \infty} \leq \theta(10^{-6}) \ $$

 * }

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

 * style="width:95%" |
 * style="width:95%" |

with solution  $$ \color{blue} \mathbf{Z}^{(k+1)} \ $$   obtained with double number of nodes compose to   $$  \color{blue} \mathbf{Z}^{(k)}. \ $$

Run also matlab ode45.

 * }

 * }

[[media: Egm6341.s10.mtg41.djvu| page41-2]]

Question: Why did Simpson's rule pop out in HS algorithms? (Why not some other rules?)

Answer: Recall Simpson can integrate exactly polynomial of  $$ degree \leq 3. \ $$ (See [[media: Egm6341.s10.mtg41.djvu| page14-1]] for the theorem, and [[media: Egm6341.s10.mtg13.djvu| page13-3]] for the geometric interpretation of the area cancellation)

HS algorithm: collocation (enforcing compliance with differential equation at 2 end points and mid point); use cubic Hermitian interpretation.

Upon enforcing compliance with differential equation at mid point (closing gap     $$ \Delta \ $$), then Simpson's rule naturally pops out, since it can integrate exactly a cubic polynomial with 3 points. [[media: Egm6341.s10.mtg39.djvu| page34-3]]

End Question & Answer

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

 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

HW: Solve logistic equation using inconsistent trapezoid-Simpson algorithm: [[media: Egm6341.s10.mtg39.djvu| (3) page36-3]]

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

 * style="width:95%" |
 * style="width:95%" |

$$ Z_{i+1}=Z_{i}+ \frac{ \frac{h}{2}}{3} \left [ f_{i}+4f_{i+ \frac{1}{2}}+f_{i+1} \right ] \ $$

 * }

[[media: Egm6341.s10.mtg41.djvu| page41-3]]

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

 * style="width:95%" |
 * style="width:95%" |

with     $$ Z_{i+ \frac{1}{2}}= \frac{1}{2}(Z_{i}+Z_{i+1}) \ $$      (linear interpretation) Trapezoid

 * }

Compare with HS results using same values of     $$ h \ $$

 * }

title: User:EGM6341.s11.team1.Chiu/S10 Mtg41


text:
User:Opriter/PhD

I will use this page for my onPhD

title: User:Opriter/PhD


text:
User:EGM6341.s11.team1.Chiu/S10 Mtg42

Mtg 42: Tue, 13 Apr 10

[[media: Egm6341.s10.mtg42.djvu| page42-1]]

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

 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

HW:

 * nm1.s11.Mtg42.pg1.fig1.svg

show: $$ \overline{CD}= \overline{AB}+hot \ $$     for $$ d \gamma \ $$     small

 * }

Parameterization of ellipse:

 * nm1.s11.Mtg42.pg1.fig2.svg

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

 * style="width:95%" |
 * style="width:95%" |

Eliminate t     $$ \Rightarrow \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 \ $$      HW

 * }

[[media: Egm6341.s10.mtg42.djvu| page42-2]]

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

 * style="width:95%" |
 * style="width:95%" |

Arc length:     $$ dl= \left [ dx^{2}+dy^{2} \right ]^{ \frac{1}{2}} \ $$      HW

 * }

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

 * style="width:95%" |
 * style="width:95%" |

Eccentricity    $$ e= \left [ 1- \frac{b^{2}}{a^{2}} \right ]^{ \frac{1}{2}} \ $$

 * }

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

 * style="width:95%" |
 * style="width:95%" |

$$ \Rightarrow c = \int dl = a \int^{2 \pi}_{t=0} \left [ 1-e^{2}cos^{2}t \right ]^{ \frac{1}{2}}dt \ $$     HW

 * }

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

 * style="width:95%" |
 * style="width:95%" |

$$ = 4a \underbrace{ \int^{ \frac{ \pi}{2}}_{ \alpha=0} \left [ 1-e^{2}sin^{2} \alpha \right ]^{ \frac{1}{2}}d \alpha}_{ \color{blue} elliptic \ integration \ 2nd \ kind} \ $$     HW

 * }

End Parameterization of ellipse

Clenshaw-Curtis:

*Periodicity (trapezoidal error)

*Runge phenomenon

Change of variables: $$ x= \underset{ \color{blue} \underset{chebyshev}{ \uparrow}}{cos \theta} \ $$

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

 * style="width:95%" |
 * style="width:95%" |

$$ I= \int^{+1}_{-1}f(x)dx= \int^{ \pi}_{0}f(cos \theta) \underbrace{sin \theta d \theta}_{ \color{blue} -dx} \ $$     HW

 * }

Integrate simpler if cosine series of     $$ f(cos \theta) \ $$ known.

[[media: Egm6341.s10.mtg42.djvu| page42-3]]

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

 * style="width:95%" |
 * style="width:95%" |

$$ f(cos \theta)= \underbrace{ \frac{a_{0}}{2}+ \sum^{ \infty}_{k=1}a_{k}cos(k \theta)}_{\color{blue} cosine \ series} \ $$

 * }

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

 * style="width:95%" |
 * style="width:95%" |

$$ a_{k}= \frac{2}{ \pi} \int^{ \pi}_{0}f(cos \theta)cos(k \theta)d \theta \ $$     HW

 * }

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

 * style="width:95%" |
 * style="width:95%" |

$$ I=a_{0}+ \sum^{ \infty}_{k { \color{blue}\to j}=1} \frac{2a_{2j}}{1-(2j^{2})} \ $$     HW

 * }

$$ a_{k} \ $$     evaluated number using trapezoidal rule which corresponds to the Discrete Cosine Transfer (DCT). (which can be computed efficiently using FFT: Fast Fourier Transfer)

title: User:EGM6341.s11.team1.Chiu/S10 Mtg42


title: File:Nm1.s11.Mtg41.pg1.fig1.svg


title: File:Nm1.s11.Mtg42.pg1.fig1.svg


title: File:Nm1.s11.Mtg42.pg1.fig2.svg


text:
User talk:NoahBonn

 Hello NoahBonn, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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.
 * Explore Wikiversity with the links to your left.

 * 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.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Abd 02:48, 15 August 2011 (UTC)

title: User talk:NoahBonn


text:
User talk:Egm6321.f09.team1.gzc

 Hello Egm6321.f09.team1.gzc, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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.
 * Explore Wikiversity with the links to your left.

 * 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.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Abd 02:50, 15 August 2011 (UTC)

title: User talk:Egm6321.f09.team1.gzc


text:
User talk:Elton

 Hello Striker, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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.
 * Explore Wikiversity with the links to your left.

 * 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.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Abd 02:51, 15 August 2011 (UTC)

title: User talk:Elton


text:
User talk:Patriciablackwell

 Hello Patriciablackwell, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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.
 * Explore Wikiversity with the links to your left.

 * 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.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Abd 02:55, 15 August 2011 (UTC)

Links on your user page
Welcome to Wikiversity. Although we encourage participants to express themselves on their user page, please keep in mind our policies on external links. --Draicone (talk) 11:11, 15 August 2011 (UTC)
 * I saw your edit and this removal mentioned above. If you can tell me what you are interested in, what educational resources you might like to develop or work on or use, I may be able to help you, and appropriate links from a resource may be much more allowable, plus, when you are open about who you are, as many of us are, limited links to, say, our own web sites may be allowed. Let me know, you can respond here, I should be watching this page, or you may be able to pop a note on my Talk page. You may also, if you enable email, send me an email to get my attention. Again, welcome to Wikiversity, I hope the removal wasn't too disconcerting, Draicone is following general policy. --Abd 19:22, 16 August 2011 (UTC)

title: User talk:Patriciablackwell


text:
User talk:BenjaminWyatt

Blocked
Linkspam --Abd 03:09, 15 August 2011 (UTC)

title: User talk:BenjaminWyatt


text:
User talk:Simon Gardner

 Hello Simon Gardner, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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.
 * Explore Wikiversity with the links to your left.

 * 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.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Abd 03:12, 15 August 2011 (UTC)

title: User talk:Simon Gardner


text:
Deskbound: A Love Letter to Your Body

Deskbound: A Love Letter to Your Body is a lecture presentation by Kelly Starrett for Google. The video is available for free viewing on YouTube via the following link:

http://www.youtube.com/watch?v=kfg_e6YG37U

1. "Locked at 90°" The first point that Kelly makes is that the average white-collar working american literally spends the majority of his/her waking hours in a seated position. The implications of this are the gradual development of limitations in the mobility of one's ankles, knees, and especially hips. Kelly likens the practice to keeping ones elbow locked at 90° such as in a cast all day. In either such a situation, we will see disability arise when the individual wants to engage in activities that require additional range of motion such as playing sports or even tying ones shoes.

NoahBonn 05:18, 15 August 2011 (UTC)

title: Deskbound: A Love Letter to Your Body


title: User:Northamerica1000


text:
User talk:Christine42

 Hello Christine42, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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.
 * Explore Wikiversity with the links to your left.

 * 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.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Abd 15:39, 15 August 2011 (UTC)

Account history
I created the page Naturality using the accout of Jivasu. I was instructed to do so from him Christine42 14:35, 15 August 2011 (UTC)
 * Thanks, see User talk:Jivasu. --Abd 15:39, 15 August 2011 (UTC)
 * And I see that you are the webmaster, apparently, for jivasu.com. If you aren't, I'm sure that someone will tell us! Given what you have posted, I don't think there is any harm, and some good, in being explicit about this. If there is, please let me know and I'll delete this revision. But I actually recommend you put this on your user page. Many WV users do disclose real life identity, because it matters in academia. --Abd 16:22, 15 August 2011 (UTC)

The future of the Naturality page

 * It is possible that some conflict will arise if someone thinks that the Naturality resource is biased. I'm not addressing this now, but if there is some "criticism of Naturality" it can be handled by creating a top level page that is completely neutral, linking to subpages that are from differing points of view. Wikiversity handles conflict, while satisfying overall WMF neutrality policy, by forking. The subpages would have a warning that they represent a point of view. But that's not needed yet, I'm just letting you know how I understand to handle such things in case I'm not around if it comes up. Remember, this is a collaborative community, and we should always seek maximized consensus.
 * You may expand on sections by creating subpages. Place a link where you want to refer to the subpage. For example, you could place a link /Innate intelligence/ in the section header where you have that, instead of the plain text. This would create a link to the subpage, initially a redlink, it will turn blue when you create the content there. Notice that I did not capitalize "intelligence." That's standard wikiversity style, sentence case is used for resource names, and generally in section headers as well, unless it's a proper noun. I'll fix this if you don't, later.
 * Often new users will create new top-level pages, causing "topic sprawl." While your subpages may refer to general concepts, pages on general concepts should not be written just from the Naturality point of view, but you may, of course, refer to those pages and you might also refer to your pages from them.
 * Wikibooks would have *only* subpages for most of your content. The main page would be a title page with links to the content.
 * I hope you are having fun, I certainly am. Fun is something I don't see mentioned in Naturality. Where does it fit in this system? --Abd 16:42, 15 August 2011 (UTC)

title: User talk:Christine42


text:
User:!proxy

This account was created by Abd to reserve the account name as meaning "No proxy." --!proxy 15:24, 15 August 2011 (UTC)
 * Confirmed: Abd 15:25, 15 August 2011 (UTC)

title: User:!proxy


text: File:Austerity (PDG).Country 1.pdf

title: File:Austerity (PDG).Country 1.pdf


text: File:Austerity (PDG).Country 2.pdf

title: File:Austerity (PDG).Country 2.pdf


text: File:Austerity (PDG).Country 3.pdf

title: File:Austerity (PDG).Country 3.pdf


text: File:Austerity (PDG).Information.pdf

title: File:Austerity (PDG).Information.pdf


text: File:Austerity (PDG).Instructor.pdf

title: File:Austerity (PDG).Instructor.pdf


text: File:Austerity (PDG).Observer.pdf

title: File:Austerity (PDG).Observer.pdf


text:
User:Egm6321.f11/Licensing info

I agree to multi-license all my contributions, including my user pages, as described below (MultiLicenseWithCC-BySA-Any):

title: User:Egm6321.f11/Licensing info


text:
User:Egm6321.f11/Interesting news

Free digital texts begin to challenge costly college textbooks in California, By Gale Holland, Los Angeles Times Staff Writer August 18, 2008. Would-be reformers are trying to beat the high cost -- and, they say, the dumbing down -- of college materials by writing or promoting open-source, no-cost online texts.

IMPORTANT: The perils of plagiarizing wikipedia; see Falling exam passes blamed on Wikipedia 'littered with inaccuracies', 21 June 2008, by Martyn Mclaughlin, News.Scotsman.com. "WIKIPEDIA and other online research sources were yesterday blamed for Scotland's falling exam pass rates."

title: User:Egm6321.f11/Interesting news


text:
Talk:Naturality

Clean up references from section headers
Section headers shouldn't have references in them. They are mostly used for building the Table of Contents (TOC) for a page, and then the TOC headers are used in page history for edits to a section. References complicate that, and the reference numbers will shift as references shift, so History links to sections will not work any more. I'm going to remove them, I don't have time to do more than that today, but they may be replaced with some text inside the section as appropriate. The first reference header I went to, the reference was strange: to a 13-page article in the Oxford Concise English Dictionary? . Something is off here.
 * The removal edits are just before the time of this note, and all the edit summaries are "rm ref". I'm doing this as separate edits to make it easier to work with, I'm fixing the automatic edit summary to the new form of the section header. --Abd 16:17, 15 August 2011 (UTC)

title: Talk:Naturality


text:
University of Florida/Egm6321/f09.team1.gzc/Mtg36

Mtg 34: Wed, 23 Mar 11

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

 * style="width:95%" |
 * style="width:95%" |

page36-1

Abstract formulation cont'd p.35-4

$${\underline{Init. \ cond.}} \ {\underline{Z}}(t_{0})={\underline{Z}}_{0} \ {\color{blue}given}$$

$${\color{blue}\underline{Opt. \ control \ pb.:}} \ Find \ {\underline{u}} \ st. \ min \ J({\underline{z}}, \ {\underline{u}})$$

$$st. \ {\underline{ \overset{ \bullet}{X}}}={\underline{f}}({underline{z}}, \ {underline{u}}, \ t)$$

$${\color{blue}Init. \ cond.:} \ {\underline{z}}(t_{0})={\underline{z}}_{0}$$

$${\color{blue}Ineq. \ constr.:} \ {\underline{g}}({\underline{z}}, \ t) \leqslant {\underline{0}}$$

$${\color{blue}Equal. \ constr.:} \ {\underline{h}}({\underline{z}}, \ t)={\underline{0}}$$

J(z,u)=Obj. funcion or Performance Index

Ex1: supersonic interceptor

$$J:= \ \int_{t_{0}}^{t_{f}}dt= {\color{blue}\underset{unkown \ to \ be \ determinded}{ \underbrace}}-t_{0} \, \ t_{f}=t_{f}({underline{z}}, \ {underline{u}})$$

page36-2

Ex2: Bunt manuuver

$$J=\int_{t_{0}}^{t_{f}}{\color{blue} \underset{altitude}{ \underbrace}}(t)dt \ {\color{blue}fig.p.34-1:}Area \ under \ curve$$

$${\color{blue}Ineq. \ constr.:} \ {\underline{g}}({\underline{z}}, \ t) \leqslant {\underline{0}}$$

e.g., \ T_{min} \leqslant T(t) \leqslant T_{max}

$$ \Leftrightarrow \ T(t) \leqslant T_{max} \ \forall t$$

$$ \ \ {\color{red}-}\leqslant \ {\color{red}-}T_ \ \forall t$$

$${\color{blue}g_{1}:=} \ T(t)-T_{max}\leqslant 0$$

$${\color{blue}g_{2}:=} \ {\color{red}-}T(t){\color{red}+}T{min}\leqslant 0$$

$$\Leftrightarrow \ {\color{blue} \underset{ \underbrace}} \ {\color{blue} \underset{ \leqslant}{ {\color{black}\leqslant}}} \ {\color{blue} \underset} $$

page36-3

$${\color{blue}Another \ ineq. \ constr. \ :}h(t)\leqslant h_{max} \ {\color{blue}S \ and \ Z \ 2007}$$

$${\color{blue}Solution \ form \ of \ opt. \ contr. \ : \ Direct \ transcription}$$

$$convert \ \ into$$

$${\color{red}\underline}$$

$${\color{blue}\Rightarrow \ Discretize \ abs. \ form. \ (OESs) \ in \ time} \ {\color{blue} \underset{scalar}{ \underbrace}}=f(z,t)$$

$$f_{n}:=f(z_{n},t_{n})$$

$${\color{blue}Hermitian \ interp.:} \ z(t) \cong \ P_(t)= \sum_{i=0}^ c_{i}t^{i}$$

page36-4

$${\color{blue}dof \ =}degree. \ of \ freedom$$

$${\color{red}(1) \begin{cases} & \ {\color{black}d_{1}=z_{n}, \ d_{3}=z_{n+1}}\\ & \ {\color{black}d_{2}= \overset{ \bullet}{z}_{n}, \ d_{4}= \overset{ \bullet}{z}_{n+1}} \end{cases}}$$ $$P_(t)= \sum_{i=0}^c_{i}t^{i}=\sum_{i={\color{red}1}}^{\color{blue} \underset{basis \ function}{ \underbrace}} \ {\color{red}(2)} $$

Hermite-Simpson algo:

$$[t_{i}, \ t_{i+1}] \ st \ t(s)=(1-s)t_{i}+st_{i+1} \ {\color{red}(3)}$$

$$s=0 \ \Rightarrow \ t(0)=t_{i}$$

$$s=1 \ \Rightarrow \ t(1)=t_{i+1}$$

$$z(s) \overset \overset{=} \sum_{i=0}^c_{i}s^{i}{\color{blue} \begin{cases} & at \ t_{i} \ and t_{i+1}, \ enforce\\ & compliance \ with \ ODE \end{cases}}$$ $${\color{blue} \begin{cases} & {\color{black} \overset{ \bullet}{z}_{i} \overset{=}f_{i} := \ f(z_{i}, \ t_{i})} \\ & {\color{black} \overset{ \bullet}{z}_{i+1} \overset{=}f_{i+1} := \ f(z_{i+1}, \ t_{i+1})} \end{cases}}$$ page36-5

$${\color{blue}In \ general,} \ \forall \ t\in \ ]t_{i}, t_{i+1}[, \ i.e., \ t \neq t_{i} \ and \ t \neq t_{i+1}, \ \overset{ \bullet}{z}_{t} \neq f_{t} \cdot$$
 * }


text:
User:EGM6341.s11.team1.Chiu/Mtg36

Mtg 34: Wed, 23 Mar 11

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

 * style="width:95%" |
 * style="width:95%" |

page36-1

Abstract formulation cont'd p.35-4

$${\underline{Init. \ cond.}} \ {\underline{Z}}(t_{0})={\underline{Z}}_{0} \ {\color{blue}given}$$

$${\color{blue}\underline{Opt. \ control \ pb.:}} \ Find \ {\underline{u}} \ st. \ min \ J({\underline{z}}, \ {\underline{u}})$$

$$st. \ {\underline{ \overset{ \bullet}{X}}}={\underline{f}}({underline{z}}, \ {underline{u}}, \ t)$$

$${\color{blue}Init. \ cond.:} \ {\underline{z}}(t_{0})={\underline{z}}_{0}$$

$${\color{blue}Ineq. \ constr.:} \ {\underline{g}}({\underline{z}}, \ t) \leqslant {\underline{0}}$$

$${\color{blue}Equal. \ constr.:} \ {\underline{h}}({\underline{z}}, \ t)={\underline{0}}$$

J(z,u)=Obj. funcion or Performance Index

Ex1: supersonic interceptor

$$J:= \ \int_{t_{0}}^{t_{f}}dt= {\color{blue}\underset{unkown \ to \ be \ determinded}{ \underbrace}}-t_{0} \, \ t_{f}=t_{f}({underline{z}}, \ {underline{u}})$$

page36-2

Ex2: Bunt manuuver

$$J=\int_{t_{0}}^{t_{f}}{\color{blue} \underset{altitude}{ \underbrace}}(t)dt \ {\color{blue}fig.p.34-1:}Area \ under \ curve$$

$${\color{blue}Ineq. \ constr.:} \ {\underline{g}}({\underline{z}}, \ t) \leqslant {\underline{0}}$$

e.g., \ T_{min} \leqslant T(t) \leqslant T_{max}

$$ \Leftrightarrow \ T(t) \leqslant T_{max} \ \forall t$$

$$ \ \ {\color{red}-}\leqslant \ {\color{red}-}T_ \ \forall t$$

$${\color{blue}g_{1}:=} \ T(t)-T_{max}\leqslant 0$$

$${\color{blue}g_{2}:=} \ {\color{red}-}T(t){\color{red}+}T{min}\leqslant 0$$

$$\Leftrightarrow \ {\color{blue} \underset{ \underbrace}} \ {\color{blue} \underset{ \leqslant}{ {\color{black}\leqslant}}} \ {\color{blue} \underset} $$

page36-3

$${\color{blue}Another \ ineq. \ constr. \ :}h(t)\leqslant h_{max} \ {\color{blue}S \ and \ Z \ 2007}$$

$${\color{blue}Solution \ form \ of \ opt. \ contr. \ : \ Direct \ transcription}$$

$$convert \ \ into$$

$${\color{red}\underline}$$

$${\color{blue}\Rightarrow \ Discretize \ abs. \ form. \ (OESs) \ in \ time} \ {\color{blue} \underset{scalar}{ \underbrace}}=f(z,t)$$

 * nm1.s11.Mtg36.pg3.fig1.svg

$$f_{n}:=f(z_{n},t_{n}) \ $$

$${\color{blue}Hermitian \ interp.:} \ z(t) \cong \ P_(t)= \sum_{i=0}^ c_{i}t^{i}$$

page36-4

$${\color{blue}dof \ =}degree. \ of \ freedom$$

$${\color{red}(1) \begin{cases} & \ {\color{black}d_{1}=z_{n}, \ d_{3}=z_{n+1}}\\ & \ {\color{black}d_{2}= \overset{ \bullet}{z}_{n}, \ d_{4}= \overset{ \bullet}{z}_{n+1}} \end{cases}}$$ $$P_(t)= \sum_{i=0}^c_{i}t^{i}=\sum_{i={\color{red}1}}^{\color{blue} \underset{basis \ function}{ \underbrace}} \ {\color{red}(2)} $$

Hermite-Simpson algo:

$$[t_{i}, \ t_{i+1}] \ st \ t(s)=(1-s)t_{i}+st_{i+1} \ {\color{red}(3)}$$

$$s=0 \ \Rightarrow \ t(0)=t_{i}$$

$$s=1 \ \Rightarrow \ t(1)=t_{i+1}$$

$$z(s) \overset \overset{=} \sum_{i=0}^c_{i}s^{i}{\color{blue} \begin{cases} & at \ t_{i} \ and t_{i+1}, \ enforce\\ & compliance \ with \ ODE \end{cases}}$$ $${\color{blue} \begin{cases} & {\color{black} \overset{ \bullet}{z}_{i} \overset{=}f_{i} := \ f(z_{i}, \ t_{i})} \\ & {\color{black} \overset{ \bullet}{z}_{i+1} \overset{=}f_{i+1} := \ f(z_{i+1}, \ t_{i+1})} \end{cases}}$$ page36-5

$${\color{blue}In \ general,} \ \forall \ t\in \ ]t_{i}, t_{i+1}[, \ i.e., \ t \neq t_{i} \ and \ t \neq t_{i+1}, \ \overset{ \bullet}{z}_{t} \neq f_{t} \cdot$$

title: User:EGM6341.s11.team1.Chiu/Mtg36


text:
User:EGM6341.s11.team1.Chiu/Mtg34

Mtg 34: Wed, 23 Mar 11

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

 * style="width:95%" |
 * style="width:95%" |

page34-1

1) Supersonic intercaptor: Find angle of attack and thrust to reach a given altitude inmin time

 * Nm1.s11.Mtg34.pg1.fig1.svg

Reference: Bryson & Denham 1962

2) Bunt maneuver of aircraft: min altitude to reach target with a bunt maneuver (inverted loop)

 * Nm1.s11.Mtg34.pg1.fig2.svg

page34-2

$${\color{blue}V} \ = \ velocity \ of \ P \ (airplane \ = \ pt)$$

$${\color{blue}T} \ = \ thrust \ // \ airplace \ axis$$

$$\alpha \ = \ \angle \ ({\color{blue}V,} \ {\color{red}T} ) \ = \ angle \ of \ attack$$

$$\gamma \ = \ \angle \ (x, \ {color{blue}V}) \ = \ angle \ between \ honz \ x \ axis \ and \ veloctity$$

$$D \ = \ axial \ aero \ force \ // \ {\color{red}T}$$

$$(parall. \ airplane \ axis)$$

page34-3

$$L \ = \ Tramsv. \ aero. \ force \ \bot \ {\color{red}T}$$

$$W \ = \ mg \ = \ airplane \ weight.$$

$$Equation \ of \ motion \ (EOM)$$

$${color{blue}Control \ input:} \ {\color{blue} \underset{time \ dependent}{ \underbrace{{\color{red}T}(t)}\, \ \alpha(t)} }$$

$${\color{blue}state \ var,:} \ x(t), \ y(t) \ v(t), \ r(t)$$

$${\color{blue} \begin{cases} & \ {\color{black}D=\frac{L}{2} \ C_{d} \ \rho \ V^{2} \ S_{reft}}  \\ & \ {\color{black}L=\frac{L}{2} \ C_{l} \ \rho \ V^{2} \ S_{reft}} \end{cases}}$$

$$C_{d}=drg \ coefficient \ {\color{green}|} \ \rho=air \ density$$

$$C_{l}=lift \ coefficient \ {\color{green}|} \ \S_{ref}=ref. \ area \ of \ airplane$$

page34-4

$$C_{d}=A_{1}\alpha^{2}+A_{2}\alpha+A_{3} \ {\color{red}(1)}$$

$$(A_{1}, \ A_{2}, \ A_{3})=curve \ fitting \ coefficient$$

$$C_{l}=B_{1}\alpha+B_{2} \ {\color{red}(2)}$$

$$(B_{1}, \ B_{2})=curve \ fitting \ coefficient$$

$$\rho=C_{1}h^{2}+C_{2}h+C_{3} \ {\color{red}(3)}$$

$$(C_{1}, \ C_{2}, \ C_{3})=curve \ fitting \ coefficient$$

Reference: S & Z 2007

$${\underline { \color{blue} {Kinematics: }}} \ \frac{dx}{dt}=V_{x}=Vcosr \ {\color{red}(4)}$$

$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{dy}{dt}=V_{y}=Vsinr \ {\color{red}(5)}$$

$${\underline { \color{blue} {Kinematics: }}}{\color{blue}Euler \ equations:} \ \frac{d}{dt} {\underline {P}}= \sum_ {\underline {F}}_ \ {\color{red}(6)}$$

page34-5

Note: Parameteriztion of ellipse

$${\color{blue} \underset{cos^{2}t}{ \underbrace}}+{\color{blue} \underset{sin^{2}t}{ \underbrace}}=1 \ {\color{blue}ellipse}$$

<{\color{blue}dl={\color{red}[{\color{blue}dx^{2}+dy^{2}}]^{1/2}}}/math>

$$Eccentricity: \ e= \left(1-\frac{b^{2}}{a^{2}} \right)^{1/2}$$

$$C=\int dl=a \int_{t=0}^{2\pi}[1-e^{2}cos^{2}t]^{1/2}dt$$

$$=41\int_{\alpha=0}^{\frac{\pi}{2}}[1-e^{2}sin^{2}\alpha]^{\frac{1}{2}}d\alpha=4aE(e)$$

title: User:EGM6341.s11.team1.Chiu/Mtg34


text:
User:EGM6341.s11.team1.Chiu/Mtg30

Mtg 30: Mon, 14 Mar 11

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

 * style="width:95%" |
 * style="width:95%" |

page30-1

$${\color{blue} \underline {Appl \ of \ HOTR \  p.29-2:}} \ {\color{red} \underline {{\color{blue}Corrected \ Trap. \ rule}s}}$$

$${\color{blue}CT_(n)}={\color{blue}\underset{CT_(n)}{\underbrace}}+ {a}^_h^+Q(h^)$$

$$I=CT_(n)+Q(h^)$$

$${\color{blue}CT_(n)}={\color{blue}CT_(n)}+{a}^_h^+Q(h^)$$

$$I=CT_(n)+Q(h^)$$

HW*5.6:See HW*2.4 p.7-3, HW*5.4, p.29-6

Compare In using CTk (n),, for n=2,4,

8,16,... until error I-In=Q(10-6).

HW*5.7:Discuss ppros and cons of intergration methods

1)Taylor series3)Compare Simpson5)CTk(n)

2)Compare Trap.4)Romberg(Richardson)

page30-2

Pf of HOTRE p.29-2:

$${E}^{T}_{n}=I-I_{n}= \int_{a}^{b}f(x)dx-T_(n)$$

$$=\sum_{k=0}^{n-1}[\int_{x_{k}}^{x_{k+1}}f(x)dx-\frac{h}{2}{\color{blue} \left\{{\color{black}f(x_{k})+f(x_{kH})} \right\}}] \ {\color{red}(1)}$$

$$x_{k}:=a+kh \, \ h:=(b-a)/n \ {\color{red}(2)}$$

$$Transfer \int_{x_{k}}^{x_{kH}}-{\color{blue}dx} \ to \ \int_{-1}^{+1}-{\color{blue}dt} $$

$$x(t):=\frac{x_{k}+x_{k+1}}{2}+t\frac{h}{2}, \ t\in [-1,+1] \ {\color{red}(3)}$$

$$ {\color{blue}\begin{cases} & \ {\color{black}x(-1)=x_{k}} \\ & \ {\color{black}x(0)=(x_{k}+x_{k+1})/2} \ {\color{red}(4)} \\ & \ {\color{black}x(+1)=x_{k+1}} \end{cases}}$$

$${E}^{T}_{n} = \frac{h}{2} \sum_{k=0}^{n-1}[\int_{-1}^{+1}g_{k}(t)dt-{\color{blue} \left\{{\color{black}{g}^{(-1)}_{k}+ {g}^{(+1)}_{k}} \right\}}] \ {\color{red}(5)}$$

$${g}^{(t)}_{k}:=f(x(t)) \ st \ x\in[x_{k},x_{k+1}] \ {\color{red}(6)}$$

page30-3

$$ \int_{-1}^{+1} {\color{blue}\underset{P_{1}(t)}{ \underbrace}}g^(t)dt \overset{=}{\color{blue} \underset{:=E}{ \underbrace}} \ {\color{red}(1)}$$

Step2a: Some motivation

HW*5.8:

$${\color{blue} \underset{\frac{d^{i}}{dt^{i}}{g}^{(t)}_{k}}{ \underbrace}}=(\frac{h}{2})^{i}f^{\color{blue}(i)}(x(t)) \ {\color{red}(2)}$$

$$x\in[x_{k},x_{k+1}]$$

$$To \ obtain \ {\color{blue} \underline {\color{black}higher \ powers \ of \ h}} \ {\color{blue}\Rightarrow} \ Take \ higher$$

$$derivation \ of \ g_{k}(t) \ {\color{blue}\Rightarrow} \ {\color{blue} \underline {\color{black}successive \ int. \ by \ parts}}$$

$$(recall \ pf \ of \ Taylor \ series \ {\color{blue}Mtg5 \ and \ Mtg6})$$

$${\color{red}(1)\Rightarrow}E=\int_{-1}^{+1} {\color{blue} \underset{P_{1}(t)}{ \underbrace}}{\color{black}g^}dt$$

$$P_{2}(t) \overset{:=}c_{1}\frac{t^{2}}{2!}+c_{3} \, \ c_{2}\overset{=}0$$

page30-4

Step2b:Intergration by parts

$$E=[P_(t)g^{(1)}(t)]^{+1}_{-1}-{\color{blue}\overset{A \overset{2b}{=}}{ \overbrace}}+\int_{-1}^{+1}P_g^(t)dt \ {\color{red}(3)}$$

$$P_\overset{=} \int P_(t)dt=-\frac{t^{3}}{6}+ \alpha t +\beta {\color{blue}(new \ intergration \ constant)}=c_{1}\frac{t^{3}}{3!}+C_{3}t+C_{4} \ (C_{2}\overset{=}0)$$

Pb:Find(α, β)=(C3, C4) to cancel terms withg(2) (evenorder derivation of g, i.e., oddpower of h ; here h3)

$$i.e., \ make $$

$$\Rightarrow$$

(recall P1)

$$Consider \ P_ \ st \ P_=0 \, \ P_(+1 \ and \ -1)=0 \ {\color{red}(6)}$$

page30-5

 * nm1.s11.Mtg30.pg5.fig1.svg

$$P_{3}(0)=0\Rightarrow C_{4}=0=\beta \ {\color{red}(1)}$$

$$P_{3}(-1)=0=-\frac{(-1)^{3}}{6}+\alpha(-1)\Rightarrow C_{3}=\alpha=\frac{1}{6} \ {\color{red}(2)}$$

$$P_{3}(+1) \overset{=} -P_{3}(-1)=0 \ {\color{red}(3)}$$

Summary, Steps2ab:

step3a: (3)&(5) p.30-4 ; intergration by parts

$$E=[P_(t)g^{(1)}(t)]^{+1}_{-1}+\int_{-1}^{+1}P_(t)g^(t)dt$$

$$=[P_(t)g^{(1)}(t)]^{+1}_{-1}+[P_(t)g^{(3)}(t)]^{+1}_{-1}-\int_{-1}^{+1}P_(t)g^(t)dt \ {\color{red}(5)}$$

page30-6

$$P_(t)=\int P_(t)dt=-\frac{t^{4}}{4!}+\frac{t^{2}}{12}+\alpha$$

$$=C_{1}\frac{t^{4}}{4!}+C_{3}\frac{t^{2}}{2!}+ \underset{C_{5}} \ {\color{red}(1)}$$

$$C_{1} \ and \ C_{3} \ in \ {\color{red}(4)}{\color{blue}p.30-5}$$

step3b:Intergration by parts

$${\color{red}(5)}{\color{blue}p.30-5}\Rightarrow \ E=[P_g^+P_g^]^{+1}_{-1}$$

$$[P_g^]^{+1}_{-1}{\color{red}+}\int_{-1}^{+1}P_g^dt \ {\color{red}(2)}$$

$$P_(t)=\int P_(t)dt=\frac{t^{5}}{120}+\frac{t^{3}}{36}+\alpha{\color{black}t+}{\color{blue} \underset{C_{6}}{ \underbrace}} \ {\color{red}(3)}$$

$$Want \ eliminate \ g^ \ (thus \ h^);$$

$$So \ make $$

$$\Rightarrow$$

(recall P1, P3 oddfuncs)

page30-7

Similar to (6)p.30-4:

$$Consider \ P_(t) \ st \ P_(0)=0, \ P_(+1 \ and \ -1)=0 \ {\color{red}(1)}$$

$$\Rightarrow \ C_{6}=0 \, \ C_{5}=-\frac{7}{360} \ {\color{blue}HW^{*}5.{\color{red}8}} \ {\color{red}(2)}$$

Summar, Steps3ab:

$${\color{red}(3)}$$

$${\color{red}(4)}$$

Compare to first part in(5)p.30-5

$$E=[P_(t)g^(t)]^{+1}_{-1} \ {\color{red}+} \ P_(t)g^(t)$$

title: User:EGM6341.s11.team1.Chiu/Mtg30


text:
User:EGM6341.s11.team1.Chiu/Mtg22

Mtg 30: Mon, 14 Mar 11

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

 * style="width:95%" |
 * style="width:95%" |

page22-1

Comp. Trap. rule error: $$ h={\color{red}\frac{n}} {\color{blue}\leftarrow}{\color{red}(3)}{\color{blue}p.7-4}$$

 * Mtg22.pg1.fig1.svg

$${E}^_{n} \underset{:=} \ I-{I}^{T}_{n}= \int_{a}^{b}f(x)dx-h[{\color{red}\frac{1}{2}}f(x_)+f(x_{1})+ \cdots+f(x_{n-1}){\color{red}\frac{1}{2}}f(x_)] \ {\color{red}(1)}$$

$$= \sum_{i=1}^{n}{\color{blue} \left\{{\color{black} \int_{x_{i-1}}^{x_{i}}f(x)dx-\frac{h}{2}[f(x_{i-1})+f(x_{i})]} \right\}}$$

(1)p.17-2:

$$ \left|{E}^{T}_{n} \right| \leqslant \frac{h^{3}}{12} \underset{ {\color{blue}\underset{ \overset{=: \overline{M_{2}}}}}}{\sum_{i=1}^{n}(max{\color{red} \left|{\color{black}f^(\xi)} \right|})}{\color{red} \ (2)}$$

(4)p.17-1:

$$ \overline{M_{2}} \leqslant M_{2} \ {\color{red}(4)}$$

$${\color{red}(5)}$$

page22-2

HW*4.8: Comp. Simpson error

$$ \left|{E}^_ \right| \leqslant \frac{(b-a)^{5}}{2880n^{4}}{\color{blue} \underset{{\color{red}(4)}p.17-1}{ \underbrace}}=\frac{(b-a)h^}{2880}M_{4} \ \color{red}(1)$$

HW*4.8: See HW*2.4 p.7-3

1)Use error estimate for Taylor Series, Compare Trap., Compare Simpson, to findn

Q(10-6), and compare to number results.

2)Numerically find the power ofhin error : Plot

logerror vslogh, and meas. slop with least square

(lin. regression).

 * nm1.s11.Mtg22.pg2.fig1.svg$$E=ah^+Q(h^{k+1}), \ $$   $$ \ logE=loga+{\color{red}k}logh$$

page22-3

HW*4.9: pf. of SSET, G(.) in (1) p.19-1

A)Redo the pf for 2 cases

$${\color{blue}1)} \ G(t):= \ e(t)-t^{\color{red}4}e(1) \ {\color{red}(1)}$$

$${\color{blue}2)} \ G(t):= \ e(t)-t^{\color{red}6}e(1) \ {\color{red}(2)}$$

Print out where pf breaks down.

B)For G(t) as in(1)p.19-1(w/t5), find

G(3)(0)and follow same steps in pf to

see what happen.

HW*4.10:1)Don'tUse matlabtrapzforcompare Trap.in

your code to produce Table 5.1 n A.p.255(p.22-4)

 $${\color{red}(3)}I= \int_{0}^{\pi}e^{x}{\color{blue}sinx}dx$$(use WAto find I)

title: User:EGM6341.s11.team1.Chiu/Mtg22


text:
User:EGM6341.s11.team1.Chiu/Mtg20

Mtg 20: Wed, 16 Feb 11

 * nm1.s11.Mtg20.pg1.fig1.svg

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

 * style="width:95%" |
 * style="width:95%" |

page20-1

Pf of SSET p.19-3 cont'd

$${\color{red}(1)}{\color{blue}p.19-3 \ Rolle's \ thm} \ \Rightarrow \ \exists \  \varsigma_{1} \in \  ] \ 0,1 \ [ \ st \ G^{\color{blue}(1)( \varsigma_{1})}  \overset{=}0$$

$$Now \ G^{(1)}(0) \ \overset{=}0 \ {\color{red}why ?}$$

$${\color{red}(1)} \ {\color{blue}p.19-1:} \ G^{(1)}(t)\overset{=}e^{(1)}(t)-5t^{4}e(1)$$

$${\color{red}(5)} {\color{blue}p.18-3:} \ e(t)=A(t)-{A}^{L}_{2}(t) \ {\color{red}(4)}$$

$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ e^{(1)}(t)=A^{(1)}-{A}^{L{\color{red}(1)}}_{2}(t) \ {\color{red}(5)}$$

page20-2

$$A(t)= \int_{+t}^{-t} -=\int_{k}^{-t} -+\int_{t}^{k}-$$

$$A^{(1)} \overset{=}F(-t)+F(t)$$

$$k\in{\color{red}]}-t,t{\color{red}[} \ ({\color{red}k is constant})$$

(4)p.20-1& (5)p.18-3:

$$A^{L{\color{red}(1)}}_{2}(t)\overset{=} \frac{1}{3}[F(-t)+4F(0)+F(t)]+\frac{t}{3}[F^{(1)}(-t)+F^{(1)}(t)$$

(5) p.20-1:

$$e^{(1)}(0)=A^{(1)}(0)-A^{L(1)}_{2}(0)=2F(0)-[2F(0)+0]=0 \ {\color{red}(3)}$$

$${\color{red}(3) \ and \ (3)}{\color{blue}p.20-1} \Rightarrow{\color{red}(2)}{\color{blue}p.20-1}$$

Recall:

$$G^{(1)}( \xi_{1})=0 \ {\color{red}(1)}{\color{blue}p.20-1}$$

$$G^{(1)}( 0)=0 \ {\color{red}(2)}{\color{blue}p.20-1}$$

 * nm1.s11.Mtg20.pg2.fig1.svg

page20-3

$$Rolle's theorem \Rightarrow \ \forall\xi_{2}\in]0,\xi_{1}[ st \ \ G^{(2)}(\xi_{2})=0 \ {\color{red}(1)}$$

$$Again, \ G^{(2)}(0)\overset{=}0 \ {\color{blue}HW^{*}4.3}$$

$${\color{red}(1) \ and \ (2)} \ Rolle's theorem \ \Rightarrow \ \forall\xi_{3}\in]0,\xi_{2}[ \ G^{(2)}(\xi_{2})=0 \ st \ \ G^{(3)}(\xi_{3})\overset{=}0$$

$${\color{red}(1)}{\color{blue}p.19-1:} \ G^{(3)}(t)\oversete^{(3)}(t)-{\color{red} \underset{(\xi)(4)(3)}{ \underbrace}}e(1)$$

$$e^{(3)}(t) \underset{=}-\frac{t}{3}[F^{(3)}(t)-F^{(3)}(-t)] \ {\color{red}(5)}$$

$$G^{(3)}(\xi_{3})=-\frac{\xi_{3}}{3}{\color{green} \overset{Apply \ DMVT}{ \overbrace}}-60(\xi_{3})^{2}e(1) \underset{ \overset{=}}0 $$

$$ \underset{ \overset{=}}-\frac{\xi_{3}}{3}[{\color{blue} \underset{\xi_{3}-(-\xi_{3})}{ \underbrace{\color{black}2\xi_{3}}}}F^{(4)}(\xi_{4})]-60(\xi_{3})^{2}e(1) \ \xi_{4}\in \ ]-\xi_{3},\xi_{3}[$$

title: User:EGM6341.s11.team1.Chiu/Mtg20


text:
User:EGM6341.s11.team1.Chiu/Mtg12

Mtg 12: Wed, 26 Jan 11

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

 * style="width:95%" |
 * style="width:95%" |

[[media: Nm1.s11.mtg12.djvu| page12-1]]

Appl. of LIET p.11-2

$$Let \ {f}^{L}_{n}(t) \ be \ Lagrange \ interpolation \ of \ order \ n \ to \ f(t). \ \Rightarrow \ {f}^{L}_{n}(.) \ \in \ \color{blue}  \underset{set \ of \ poly \ of \ degree \  \leqslant \ n}{ \underbrace{ {\color{black}P_{n} }}}$$

$${\color{blue} n=1} \ {f}^{L}_{1}(.) \ interpolation \ exactly$$

$${\color{blue} P_{1}(x) \rightarrow}straight \ lines \ (linear \ functions) \ i.e., \ e_{1}(P_{1},t) \ \equiv \ 0 \ \forall \ t$$

 * nm1.s11.Mtg12.pg1.fig1.svg

$${\color{blue}n=2} \ {f}^{L}_{2}(.) \ interpolation \ exactly \ parabolas \ (polynomial \ of \ {\color{blue}\underset{degree}{ \underbrace{\color{black}order}}}) \ 2$$

$${\color{blue}n=3} \ cubic \ polynomial$$

$$\color{blue} \vdots$$

$${\color{blue}n} \ {f}^{L}_{n}(.) \ interpolation \ exatcly \ \color{blue}\overset{ \in P_{n}}{ \overbrace}$$

$$i.e., \ e_(p_;t) \ =0 \ \forall t$$

[[media: Nm1.s11.mtg12.djvu| page12-2]]

LIET: (2) p.11-3

$$IF \ f \ = \ p \in \ P_{n} \ \Rightarrow \ f^{(n+1)} \ \equiv \ 0$$

$${\color{red} (2)} \ {\color{blue} \ p.11-3} \ \Rightarrow \ e_(p,t)=0 \ \forall t$$

$$i.e., \ {f}^{L}_{n}(.) \ interpolation \ {\color{red}exactly} \ p\in P_{n}$$

$$\color{blue} Method \ to \ comp. \ \left\{w_{i,n}; \ i=0,..,n \right\} \ given \ \left\{x_{i}; \ i=0,..,n \right\} \ \underline {not} \ necessarily \ equidistant.$$

$${\color{blue}\underline {j=0}} \ consider \ f=p_{\color{blue}j}=p_{0}\in P_{0} \ {\color{red}constant}$$

$$e_{\color{blue}n} \left( \underset{{\color{blue} \underbrace{{\color{black}p_{\color{red}0}}}};t} \right) \ = {\color{blue}\underset{1}{ \underbrace}}- {f}^{L}_{{\color{blue}n}(t)}={\color{blue}1}-{f}^{L}_{n}(t)=0$$

$$\Rightarrow {\color{blue} \underset{ \sum_{i=0}^{n}l_{i,n}(t){\color{red} \underset{1}{ \underbrace}}}{ \underbrace}}=1 \ \Rightarrow \ {\color{red} \int_{a}^{b}}{\color{blue} \left( \underset{{f}^{L}_{n}(t)}{ \underbrace}\right)}dt \ = \ {\color{blue} \underset{b-a}{ \underbrace}}$$

[[media: Nm1.s11.mtg12.djvu| page12-3]]

$$\Rightarrow \ {\color{blue} \underset{w_{i,n}}{ \underbrace{ \left( {\color{red} \int_{a}^{b}}l_{i,n}(t)dt\right)}}.{\color{red}1}}=b-a$$

$$\Rightarrow$$

$${\color{blue} \underline {j=1}} \ let \ f=p_=p_{1}\in P_{1}$$

$$choose \ f(x)=p_{1}(x)= \ {\color{red}x} \in P_{1}$$

$$e_{\color{blue} \left( \underset{ \underbrace}{\color{black};t }\right)} \ = \ {\color{blue} \underset{ \underbrace}}-{f}^{L}_{\color{blue}n}(t) ={\color{blue}t}-f_{n}(t)=0$$

$$\Rightarrow \underset{ {\color{blue}\underbrace{\color{black}{f}^{L}_{n}(t)}}}={\color{red}t} \ \Rightarrow \ {\color{red} \int_{a}^{b}}{\color{blue} \underset{{f}^{L}_{n}(t)}{ \underbrace{ \left( {\color{black} \sum_{i=0}^{n}l_{i,n}(t).}{\color{red}x_{i}}\right)}dt}}={\color{blue} \underset{(b^{2}-a^{2})/2}{ \underbrace}}$$

$$\Rightarrow$$

title: User:EGM6341.s11.team1.Chiu/Mtg12


text:
User:EGM6341.s11.team1.Chiu/Mtg10

Mtg 10: Sat, 22 Jan 11

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

 * style="width:95%" |
 * style="width:95%" |

page10-1

HW2.9:Use (1) p.8-1 to generate P5(x), and matlabl command"roots"to comp. the roots of P5(x)to check values in table on p. 7-5. Plot the roots on [-1,+1] using
matlab "plot" command (plot dots "."

with coordinator (xi,yi), i = 1,...,5 : use "markersinge" 15)

xi: roots of P5(x)

yi: yi = 0

Plot (x, y, '.', 'markersige', 15)

Repeat the above for P10(x)

observe the location of the roots near end points -1 and +1

(prepare for Runge phenomenon)

NOTE:

Lagrange interp. cont'd p.9-4

page10-2

Simpson's rule (simple)

$$[a,b] \ x_{0}=a, \ x_{1}=\frac{a+b}{2}, \ x_{2}=b \ \color{red} (1)$$

$$ {\color{blue} { \underline {Method 1:}}} \ f_{2 }(x)=P_{2}(x) \overset{=}c_{2}x^{2}+c_{1}x+c_{0}$$

c0, c1,  c2   unknowns

p2=(xi) = f(xi)   i= 0, 1, 2               (3)

3 equations for 3 unknowns {ci}

Method 2:Use lagrangeinterp.(2) p.8-3(1) p.9-2

$$ {\color{red} {(4)}} = \sum_{i=0}^  {\color{blue} \underset{l_{i}(x)}{ \underbrace}}f(x_{i})$$

Equiv. of meth1 and meth 2:           (3)

$$p_{2}(x_{j})= \sum_{i=0}^{2} \color{blue} \underset{ \delta_{ij}}{ \underbrace}f(x_{i})=f{x_j} \ \color{red}(5)$$

page10-3

$$l_{0{\color{blue},2}}=l_{0{\color{blue},2}}= \prod_{j=0 {\color{red}j \neq i}}^{n=2} \frac{x-x_{j}}{x_{0}-x_{j}}= \frac{(x-x_{1})(x-x_{2})}{(x_{0}-x_{1})(x_{0}-x_{2})} \ {\color{blue} \in P_{2}}$$

It can be verified that

l0(x0)=1, l0(x1) = l0(x2) = 0

li(xj) = δij                 i,j = 0. 1. 2

$$  {\color{blue} \underset{ \underbrace{\color{black}l_{1{\color{blue}(i=1),2(n=2)}}(x)}}}= \prod_{j=0, {\color{red}{j \neq 1}}}^{2 {\color{blue}(n=2)}}\frac{x-x_{j}}{x_{i{\color{blue}(i=1)}}-x_{j}}=\frac{(x-x_{0})(x-x_{2})} \ {\color{blue} \in P_{2}}$$

$$l_{1}(x_{1})=1 \, \ l_1(x_{0})=l_{1}(x_{2})=0$$

page10-4

$${l}^{''}_{1}(x_{1})<0$$

HW*2.10: Use(2) & (3)p.10-2 to find

expression for {ci} in terms (xi, f(xi)) i=0,1,2.

HW*2.11: Use (4) p.10-2 to

derive simple Simpson's rule

HW*2.11:f(x)-ex1/x on [o,1] S10and on [-1,1]S11

x0=-1, xn=+1

Consider n=1(Trap), 2(Simp), 4, 8, 16

page10-5

Constrast fn(x) as in (2) p.8-3.

Plot f, fn , n= 1, 2, 4, 8, 16

Compare
$$ {I}_{n}= \int_{a}^{b}f_{n}(x)dx $$ n=1, 2, 4, 8

and compare to I (use WA with more digits)

For n=5 plot l0, l1, l2 How would l3, l4, l5 look like?

HW*2.13:show

$$ \underset{Simple \ Trap.}{ \underbrace} \Rightarrow  \underset{Compare \ Trap.}{ \underbrace}$$

$$\underset{Simple \ Trap.}{ \underbrace} \Rightarrow  \underset{Compare \ Trap.}{ \underbrace} $$

title: User:EGM6341.s11.team1.Chiu/Mtg10


text:
User:EGM6341.s11.team1.Chiu/Mtg8

Mtg 8: Wed, 19 Jan 11

page8-1

Legendre poly. Pn(x):

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

 * style="width:95%" |
 * style="width:95%" |

$$ {P}_{n}(x)= \sum_{i=0}^{{\color{red}[{\color{black}n/2}]} } {(-1)}^{i} \frac{(2n-2i)! {x}^{n-2i}}{ {2}^{n}i!(n-i)!(n-2i)!}$$

(1)
 *
 * }

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

 * style="width:95%" |
 * style="width:95%" |

$$ {\color{red}[ {\color{black} n/2}]} = integer \ part \ of \ n/2$$

(2)
 *
 * }

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

 * style="width:95%" |
 * style="width:95%" |

e.g., m = 5, n/2 = 2.5 , [2.5] = 2

$$ {\color{red}(3)} \ {P}_{0}=1 \color{blue} \in {P}_{0} $$

$$ {\color{red}(4)} \ {P}_{1}=x \color{blue} \in {P}_{1} $$

$$ {\color{red}(5)} \ {P}_{2}= \frac{1}{2}(3{x}_{2}-1)  \color{blue} \in {P}_{1} $$

$$ {\color{red}(6)} \ {P}_{3}= \frac{1}{2}(5{x}_{3}-3x)  \color{blue} \in {P}_{1} $$

$$ {\color{red}(7)} \ {P}_{4}= \frac{35}{8}{x}^{4}-\frac{15}{4}{x}^{2}+\frac{3}{8}  \color{blue} \in {P}_{1} $$

$${\color{blue}{P}_{n}} \ = \ set \ of \ poly. \ of \ degree \ \leqslant \ n$$

HW 2.6:Verify(3)-(7) using(1)-(2)

$${\color{red}(5)} \Rightarrow \ {P}^{2}=0 \Rightarrow {x}_{1,2} = {\color{red}-} \frac{1}{\sqrt{3}} \ or {\color{red}+} \frac{1}{\sqrt{3}}$$

page8-2

Weights wi, i = 1,...,n ((1)p.7-5)

$$ \underline { \color{blue} {NOTE:}} \ {E}_{n}(f)=0 \ \forall f \in {P}_{2n-1} \ since \ f^{\color{red}(2n)}(x)=0 \. \ Only \ need \ to \ use \ n \ int. \ pts\ \left\{x_{i}, \ i=1,...n\right\} \ to \ int. \ exactly \ any \ poly \ in \ P_{2n-1} \ i.e., \ of \ degree \ \leqq \ 2n-1$$

$${\color{blue} {Newton-cotes \ method:}} \ I(f)$$

$$History \ Newton \ cotes \ \color{blue}  \rightarrow \ Lecture \ plan \ suli+Meyers(2003)$$

page8-3

$${\color{blue}1)} \ Approx. \ f(.) \ using \ {\color{blue} { \underline { {\color{black} Lagrange}}}} \ interp. \  funcs \  \Rightarrow \  {f}^{ {\color{red} L}}_{n}(x) $$

$$ Int. \ exactly \ {f}^{ {\color{red} L}}_{n}(x) \ \Rightarrow \ \color{blue}  \underbrace{ \color{black} I( {f}^_{n})}  $$

$$I_{n}(f) \ := I( {f}^_{n})=\int {f}^_{n}(x)dx \ \color{red} (1) $$

$${f}^_{n}(x)=P_{n}(x)= \sum_{i=0}^{\infty} {\color{blue} \underset{Lagrange \ interp. \ func}{ {\color{black} \underbrace{l_{i,n}(x)}}}} \ f(x_{i}) \ \color{red} (2)$$

NOTE: Demonstrated Wolfram Alpha(WA) e.g., (debt usa)/(gdp usa) integrate x from 0 to 1 link WA comp. results in HW reportsAvoid plagiarism

title: User:EGM6341.s11.team1.Chiu/Mtg8


text:
User:EGM6341.s11.team1.Chiu/Mtg6

Mtg 6: Sun, 16 Jan 11

page6-1

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

 * style="width:95%" |
 * style="width:95%" |

Proof of Taylor series continued

Since

$$  \displaystyle \int_{x_0}^{x} {f}^{1}(t)dt=[ {f}^{0}(t)]_{t= {x}_{0}}^{t=x} $$     (1)
 *
 * }

$$  \displaystyle \Rightarrow \color{red}(e) \color{blue} p.5-3 $$

Int. by parts (1)

$$  \displaystyle \color{red}(2) \begin{cases} & \ \color{black} \int_{x_0}^{x} \color{blue}  \underset{u^{'}}{ \underbrace{1}} \underset{v}{ \underbrace{ \color{black}{{f}^{(1)}(t)} }} \color{black}dt= {[uv]}^{x}_{x_0}- \int_{}^{}{uv}^{'}   \\ & \ \color{black}=  {[tf^{(1)}{t}]}^{t=x}_{t=x_0}-  \color{blue}  \underset{ \alpha}{ \underbrace{ \color{black}{ \int_{x_0}^{x}tf^{(2)}dt} }} \\ & \  \color{black}= xf^{1}(x)-x_{0}f^{(1)}(x)0)- \color{blue} \alpha  \\ & \  \color{red}+ \color{blue} xf^{(1)}(x_0) \color{red}- \color{blue}xf^{(1)}(x_{0})\\ & \     \color{black} =\color{blue} \underset{ \int_{x_{0}}^{x}xf^{(2)}(t)dt=: \beta}{  \underbrace{\color{black}[xf^(1)(x)-xf^(1)(x_{0})] }} \color{black}+(x-x_{0}){f}^{(1)}(x_{0}) - \color{blue}   \alpha \end{cases}

$$

page6-2

Combine [ + β - α ] into a single int. use (2) p.6-1 in (2) p.5-3 :

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

 * style="width:95%" |
 * style="width:95%" |

$$  \displaystyle f(x)=f(x_{0})+ \frac{ {(x-x_{0})}^{ \color{blue} 1}}{ \color{blue}1! } {f}^{(1)}(x_{0})+ \color{blue} \underset{ \beta-\alpha}{ \underbrace{ \color{black} \int_{x_{0}}^{x}(x-t)f^{(2)}(t)dt }} $$

(1)
 *
 * }

HW*2.1: 1) Do integration by parts on last term (integration) of (1) to reveal 3 more terms in Taylor series, i.e. ,

$$  \displaystyle \frac{ {x- {x}_{0}}^{2}}{2!} {f}^{(2)}( {x}_{0})+ \frac{ {x- {x}_{0}}^{3}}{3!} {f}^{(3)}( {x}_{0})+ \frac{ {x- {x}_{0}}^{4}}{4!} {f}^{(4)}( {x}_{0}) $$

plus remainder2)Use IMVT to expression remainder in terms of f(s)(ξ) for s belong[x0,x]3) Assume(3)&(4)p.3-3 correct, do intergration by parts once more

page6-3

to verify(3)&(4) p.3-3 for (n+1) expansionwith R(n+2)(x)
4) UseIMVT on(4) p.3-3to show(5)p.3-3

HW*2.2:$$f(x)=sinx, x\in[0, \pi]$$ Constrast Taylor Series of f(.) around

$$  \displaystyle \begin{cases} & \ {x}_{0}= \frac{ \pi}{4} \ \color{blue} S10 \\ & \ {x}_{0}= \frac{3 \pi}{4} \ \color{blue} S11 \end{cases} $$ for n = 0,1,2,...,10

Plot these series (for each n) Find (estimate) max $$ \left|{R}_{n+1}(x=\frac{3 \pi}{4}) \right|$$

$${\color{red}(5)} {\color{blue}p.3-3} \Rightarrow \left|{R}_{n+1}(x=\frac{3 \pi}{4}) \right| \leqq \frac{{x-{x}_{0}}^{n+1}}{(n+1)!}{\color{blue}(\alpha)}$$

page6-4

$${\color{blue} \alpha}=max \left|{f}^{(n+1)}(t) \right| $$

$$\color{blue} \underset{\leqq 1 (f(x)=sinx)}{ \underbrace{ \color{black} t \in [{x}_{0},x]}}$$

Note:Motivation for pf of Taylor series expansion. (similar technique will be used)

 * higher order analysis of Trap. rule (not in A.)
 * Richardson extrap.
 * clenshaw-Cwetis quadrature
 * chebyshew poly (orthog.) Recent devel. using chebyshew poly to solveL2_ODE_VC (Linear 2nd order ODE with varying coefficient ) combine of symbolics + numericsReference : Trefethen's chebfun.

page6-5

$${\color{blue}\underline{Note:}} \ $$

$${\color{red} \underset{Quadrilateral}{ \underbrace{\color{blue}rature=}}{\color{black} Num. \ int.}}$$

$${\underline{Greeks}}:\ Meas. \ area$$

$${\color{red} \underset{Cube}{ \underbrace{\color{blue}ture}}}$$

$$Vol. \ = \sum \ Cubes$$

Num. Int. Using Taylor Series cont'd

$$I_{n}= \int_{0}^{1}f_{n}(x)dx=\int_{0}^{1}$$

{\color{blue} \underset{f_{n}(x)=P_{n}(x)}{ \underbrace{\color{black}dx}}}

$$ \overset{=} \sum_{j=1}^{n}\frac{1}{j!j}$$

$${\color{red}(2)} \underset{ \xi \in [0,x]}{f(x)-f_{n}(x)}=R_{n+1}(x)=\frac{(x-0)^{n+1}}{x(n+1)!}exp(\xi)$$

title: User:EGM6341.s11.team1.Chiu/Mtg6


text:
User:Spadkins

My name is Stephen Adkins.
 * http://www.shareduniverse.net/home/stephen-adkins

title: User:Spadkins


text:
User:Hailong.Chen

Hi, I'm a newbie...

What You See Is What You Get.

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

$$\displaystyle
 * style="width:92%; padding:10px; border:2px solid #000000" |
 * style="width:92%; padding:10px; border:2px solid #000000" |

\frac{\partial \color {red} w}{\partial x}+\frac{\partial \color {red} w}{\partial y}=2xy

$$
 * }

$$\frac{\partial^2 y}{\partial x^2}$$

$$\left( {\begin{array}{*{20}{c}} && \\   && \\   && \end{array}} \right)$$

$$\mu :={{\left( \frac{{{\upsilon }^{3}}}{\varepsilon } \right)}^{1/4}}$$

title: User:Hailong.Chen


text:
User talk:Jemmasanderson

 Hello Jemmasanderson, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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.
 * Explore Wikiversity with the links to your left.

 * 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.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Abd 02:51, 16 August 2011 (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.

Motivation and emotion
Hi. Welcome to the unit and to Wikiversity! I look forward to learning with you and I hope you find the topic and the unit rewarding. Please feel free to let me know if you have any suggestions or would like help at any point along the way. Sincerely, James Neill, -- Jtneill - Talk - c 06:29, 16 August 2011 (UTC)

 Hi Jemma, just thought I would drop past our group's individual pages after our first tute. T'was great catching up again and I look forward to seeing your page develop, it seems like a very interesting article. Do you think Tamagotchis would have the same effect or at least a by product? As I was just discussing on Michelle's page, no I haven't bought the app yet! Will speak soon. Blondlox 14:21, 28 August 2011 (UTC)

Structure & Tutorial 2
Hi, I apologise in advance - I will not be able to make the second tutorial due to work commitments. However, I have placed a rough draft of my chapter structure on the chapter’s discussion page for all to review and comment. I will be more than happy to return the favour and will keep an eye out for your structures and ideas on your pages. Thanks! Blondlox 14:11, 7 September 2011 (UTC)

title: User talk:Jemmasanderson


text:
User talk:TabithaJ

 Hello TabithaJ, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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.
 * Explore Wikiversity with the links to your left.

 * 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.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Abd 03:03, 16 August 2011 (UTC)

 Hello , you can call me Jacob Frank. I'd like to be your mascot and welcome you to Wikiversity. Even though I was a plain and untutored man back in 18th-century Europe, that still didn't stop me from contributing my ideas to the world. Neither should it stop you. As a religious leader who developed the doctrine of "purification through transgression," my advice to you is to be bold and to ignore all rules—yes, these are two of Wikiversity's official policies! Happy editing.

Motivation and emotion
Hi. Welcome to the unit and to Wikiversity! I look forward to learning with you and I hope you find the topic and the unit rewarding. Please feel free to let me know if you have any suggestions or would like help at any point along the way. Sincerely, James Neill, -- Jtneill - Talk - c 06:30, 16 August 2011 (UTC)

Return Greetings
Hi James! Thank you for the welcome to the Unit. I'm looking forward to working wtih you. Kindest Regards, TabithaJ 10:28, 24 August 2011 (UTC)
 * Well done on working how to use/reply to messages on your user talk page. -- Jtneill - Talk - c 04:09, 17 October 2011 (UTC)

Peer review comments
Thanks for commenting on several chapter talk pages - much appreciated. It can feel a bit artificial/awkward at first, but it definitely helps in getting the collaboration going. -- Jtneill - Talk - c 04:09, 17 October 2011 (UTC)

title: User talk:TabithaJ


title: File:Nm1.s11.Mtg34.pg1.fig1.svg


title: File:Nm1.s11.Mtg34.pg1.fig2.svg


title: File:Nm1.s11.Mtg34.pg5.fig1.svg


title: File:Nm1.s11.Mtg36.pg3.fig1.svg


text:
User:U112052

I am a mature aged student, working towards a third career either as a counsellor or psychotherapist.

In terms of Motivation and Emotion, I am fascinated by anti-social personality disorder/ sociopathy/psychopathy and how these creatures operate in the workforce. Looking forward to adding a chapter to our book on the subject.

title: User:U112052


text:
User:TabithaJ

Greetings :-) I'm in my final year of the Graduate Diploma of Psychological Science. I'm really looking forward to formally studying Motivation & Emotion, as opposed to making my own "quasi" observations about the motivations and emotions of myself and others!

Many of the areas within the subject are my personal areas of interest. Specifically, I'm quite interested in obesity and appetite regulation, personality disorders, achievement orientations, empathy, power dynamics, coping and intimacy. I'd like to do my text book chapter on either obesity or personality disorders, as I believe the impact of these issues on society and within personal relationships is chronic and warrants further education on my part. I'd like to wish you all the best for your personal learning journey for the remainder of 2011 :-)

TabithaJ 04:49, 16 August 2011 (UTC)

title: User:TabithaJ


text:
User:Jay-bird

Welcome to my home page. I am a Psychology student at University of Canberra, currently enrolled in a fascinating unit called Motivation and Emotion. Check out the book we are co-writing, over at Motivation and Emotion 2011.

My chapter is about how people respond to adversity and suffering in various ways. Of special interest is the way people sometimes respond to traumatic events and situations by creating a new sense of meaning in their lives. These ideas are based in the existential philosophy of Viktor Frankl, and the area of positive psychology. It explores the ways that growth occurs in people's lives. I hope that it will add something of value to this area of research.

Jay-bird 21:40, 20 October 2011 (UTC)

title: User:Jay-bird


text:
User talk:HarryMMM

 Hello HarryMMM, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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.
 * Explore Wikiversity with the links to your left.

 * 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.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! Jtneill - Talk - c 06:27, 16 August 2011 (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.

Welcome to Motivation and emotion
Hi. Welcome to the unit and to Wikiversity! I look forward to learning with you and I hope that you find the topic and the unit rewarding. Please feel free to let me know if you have any suggestions or would like help at any point along the way. Sincerely, James Neill, -- Jtneill - Talk - c 06:27, 16 August 2011 (UTC)

title: User talk:HarryMMM


title: User:U990911


text:
User:AlexMC

I can type text.

I can type big.

I can also jump puddles, read all sorts of books, listen to music (except that Rap rubbish and opera) and play guitar (debatable). I am taking a double degree, Psych and Arts (Politics & International Relations and Literary Studies) and I'm trying to find my way around Wiki!

title: User:AlexMC


text: User:Jemmasanderson

title: User:Jemmasanderson


text:
User talk:U112052

 Hello U112052, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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.
 * Explore Wikiversity with the links to your left.

 * 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.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! Jtneill - Talk - c 06:31, 16 August 2011 (UTC)

 Hello , you can call me Jacob Frank. I'd like to be your mascot and welcome you to Wikiversity. Even though I was a plain and untutored man back in 18th-century Europe, that still didn't stop me from contributing my ideas to the world. Neither should it stop you. As a religious leader who developed the doctrine of "purification through transgression," my advice to you is to be bold and to ignore all rules—yes, these are two of Wikiversity's official policies! Happy editing.

Welcome to Motivation and emotion
Hi. Welcome to the unit and to Wikiversity! I look forward to learning with you and I hope that you find the topic and the unit rewarding. Please feel free to let me know if you have any suggestions or would like help at any point along the way. Sincerely, James Neill, -- Jtneill - Talk - c 06:31, 16 August 2011 (UTC)

Comment on emotional intelligence subtitle
FYI, I've commented on your book chapter - see Talk:Motivation and emotion/Book/Emotional intelligence. Sincerely, James -- Jtneill - Talk - c 03:49, 14 October 2011 (UTC) Many thanks JamesU112052 09:39, 14 October 2011 (UTC)

Emotional intelligence
I've changed the colour of those boxes to LightGrey - see where I changed. See also Named colors. -- Jtneill - Talk - c 21:47, 19 October 2011 (UTC)

External links should look OK now. -- Jtneill - Talk - c 03:20, 4 November 2011 (UTC)

I have added some feedback here: Talk:Motivation and emotion/Book/Emotional intelligence. -- Jtneill - Talk - c 04:55, 4 November 2011 (UTC)

title: User talk:U112052


text:
User talk:Jay-bird

 Hello Jay-bird, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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.
 * Explore Wikiversity with the links to your left.

 * 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.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! Jtneill - Talk - c 06:32, 16 August 2011 (UTC)

 Hi! Our name is Vicky-Verity. We are twins and are Wikiversity mascots. We think Wikiversity is a great place to learn and make friends (when we aren't sleeping). Leave us a message!

Welcome to Motivation and emotion
Hi. Welcome to the unit and to Wikiversity! I look forward to learning with you and I hope that you find the topic and the unit rewarding. Please feel free to let me know if you have any suggestions or would like help at any point along the way. Sincerely, James Neill, -- Jtneill - Talk - c 06:32, 16 August 2011 (UTC)

hi James, I am experimenting with adding images to my page. I thought I followed your instuctions for downloading a media file. But the images are not showing on mypage, just the link address. What might I be doing wrong? I have clicked the icon "Embedded file". Is that the right icon ?

Thanks, Jay-bird 00:32, 2 September 2011 (UTC)

Peer review comments
Thanks for commenting on several chapter talk pages - much appreciated. It can feel a bit artificial/awkward at first, but it definitely helps in getting the collaboration going. -- Jtneill - Talk - c 04:08, 17 October 2011 (UTC)

Motivation and emotion/Book/Meaning and logotherapy
Hi Jude - Thanks for asking. I've fixed those things on the meaning chapter page and added an example table. Tables are a little tricky to work with - if you know how many rows and columns you want, let me know, and I can tweak it some more for you. Sincerely, James. -- Jtneill - Talk - c 09:29, 21 October 2011 (UTC)

Thank you!
Hi Jude

Thanks for your comments regarding my multimedia presentation - it does feel a little awkward listening to your own voice for that long so I'm relieved to hear some positive feedback from you. I hope you are managing yours ok too - I'm sure it will be great!

In other news, check out our awesome mark for the Learning presentation - WHAT A TEAM!!!

See you next week for our final week for the year, and thank you also for always being a friendly face with a positive attitude! ShaunaB 07:36, 12 November 2011 (UTC)

title: User talk:Jay-bird


text: User:LanaJ

title: User:LanaJ


text:
User:Aimesch

aimesch

title: User:Aimesch


text:
User talk:LanaJ

 Hello LanaJ, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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.
 * Explore Wikiversity with the links to your left.

 * 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.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! Jtneill - Talk - c 06:33, 16 August 2011 (UTC)

 Hi! Our name is Vicky-Verity. We are twins and are Wikiversity mascots. We think Wikiversity is a great place to learn and make friends (when we aren't sleeping). Leave us a message!

Welcome to Motivation and emotion
Hi. Welcome to the unit and to Wikiversity! I look forward to learning with you and I hope that you find the topic and the unit rewarding. Please feel free to let me know if you have any suggestions or would like help at any point along the way. Sincerely, James Neill, -- Jtneill - Talk - c 06:33, 16 August 2011 (UTC)

title: User talk:LanaJ


text:
User talk:Aimesch

 Hello Aimesch, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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.
 * Explore Wikiversity with the links to your left.

 * 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.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! Jtneill - Talk - c 06:35, 16 August 2011 (UTC)

 Hello , you can call me Jacob Frank. I'd like to be your mascot and welcome you to Wikiversity. Even though I was a plain and untutored man back in 18th-century Europe, that still didn't stop me from contributing my ideas to the world. Neither should it stop you. As a religious leader who developed the doctrine of "purification through transgression," my advice to you is to be bold and to ignore all rules—yes, these are two of Wikiversity's official policies! Happy editing.

Welcome to Motivation and emotion
Hi. Welcome to the unit and to Wikiversity! I look forward to learning with you and I hope that you find the topic and the unit rewarding. Please feel free to let me know if you have any suggestions or would like help at any point along the way. Sincerely, James Neill, -- Jtneill - Talk - c 06:35, 16 August 2011 (UTC)

title: User talk:Aimesch


text:
User:Crazydaisy

Articles
Hi. I have some articles on gambling in Uni. students, but if you have more I'd really appreciate it if you send me them or the links (or whatever). If you have among them any that discuss 'Personality' -which, if they are like the ones I already have, they probably wont  :(  - I'd be esp. interested.

Thanks AlexMC 01:04, 27 October 2011 (UTC)

title: User:Crazydaisy