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Tectonic hazards/Shake-table

Shake-table in earthquake engineering is a very effective experimental equipment to verify seismic performance associated with the direct damage to individual buildings or their models under specified ground shaking. The best way to do it is to put a building structure on a shake-table that simulates the earth shaking and to watch what may happen next (if you have no time to stand out in the field and wait for a real earthquake to strike). The earliest shake-table experiments were performed more than a century ago

Modern shake-tables typically consist of a rectangular platform that is driven in up to six degrees of freedom (DOF) by servo-hydraulic or other types of actuators. Test specimens are fixed to the platform and shaken, often to the point of failure.

Concurrent performance testing
Concurrent testing on a shake-table is mostly used when some new building technology has to be compared against the old one; see, e.g., the images below:

Those testings might include both real (physical) and virtual (computer) experiments.

title: Tectonic hazards/Shake-table


text:
User:Egm6321.f10.team4.Yoon/Mtg10

 Mtg 10: Tue, 14 Sep 10

[[media: 2010_09_14_15_00_52.djvu | Page 10-1]]
On the lecture note [[media: 2010_09_09_14_48_41.djvu | (2)p.9-4 ]] we have,

Apply 2nd exactness condition to find h:

For (1) to be exact:

Solving (5) for integrating factor h(x,y) is usually not easy. So, we will consider 2 particular cases:

case 1: Suppose $$\displaystyle h_y = 0 $$; then (5) becomes:

[[media: 2010_09_14_15_00_52.djvu | Page 10-2]]
case 2: Assume $$\displaystyle h_x=0 \Rightarrow h $$ is a function of y only.

Read F09 lecture notes and find the expression for for h(y) for case2

Application: General non-homogenous L1_ODE_VC

If $$\displaystyle P(x) \neq 0; \ \ \ \ \ \forall x $$

[[media: 2010_09_14_15_00_52.djvu | Page 10-3]]
Eqn.(1) [[media: 2010_09_14_15_00_52.djvu | p.10-2:]]

Eqn.(5) [[media: 2010_09_14_15_00_52.djvu | p.10-2:]]

Recall,

Application: Non-homog L1_ODE_VC

Read F09 $$\displaystyle h(x)=x$$ and thus $$\displaystyle y(x) = \frac{x^3}{4}+\frac{k}{x}$$

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

title: User:Egm6321.f10.team4.Yoon/Mtg10


text:
User:Egm6321.f10.team4.Yoon/Mtg14

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

SC-L1-ODE-VC and SC-L1-ODE-CC
 Mtg 14: Thu, 23 Sep 10

[[media: 2010_09_23_13_57_17.djvu | Page 14-1]]
Linear Time-Invariant system $$\displaystyle \equiv$$ SC-L1-ODE-CC Linear Time-Variant system $$\displaystyle \equiv$$ SC-L1-ODE-VC

Particular case: When n=1 (L1-ODE-VC)

Open loop: u(t) is prescribed.

L1-ODE-VC: Varying coefficients

SC-L1-ODE-CC: Constant coefficients

[[media: 2010_09_23_13_57_17.djvu | Page 14-2]]
Recall:

Bryson&Ho 1975 p.450

Compare [[media: 2010_09_23_13_57_17.djvu | Eqn.(3) to Eqn.(4) p.14-1]] (Constant Coefficients)

[[media: 2010_09_23_13_57_17.djvu | Page 14-3]]
$$\displaystyle \mathbf{\Phi} $$ is related to Integrating Factor

Prop. of $$\displaystyle \color{blue}{\mathbf{\Phi}(t, t_0)}$$: (State transition matrix)

Recall SC-L1-ODE-CC

title: User:Egm6321.f10.team4.Yoon/Mtg14


text:
User:Egm6321.f10.team4.Yoon/Mtg16

= EGM6321 - Principles of Engineering Analysis 1, Fall 2010 =

Mtg 16: Tue, 28 Sep 10

[[media: 2010_09_28_15_08_25.djvu | Page 16-1]]
[[media: 2010_09_23_14_52_54.djvu |Eqn.(1) and Eqn.(2) p.15-4:]]

If $$\displaystyle h=k_1 $$ (constant) $$\displaystyle \ \ \Rightarrow h_x=h_y = 0 $$ 1st integral $$\displaystyle \underbrace{\Phi(x,y,p)}_{\color{blue}{k_1 + xyp}} = k_2$$ k3 and k4 are two constants of integration(2nd order equation.)

End Application

Note: Recall N1-ODE $$\displaystyle F(x,y,y')=0$$ is exact if $$\displaystyle \exists \underbrace{\Phi(x,y)}_{=k}$$ such that $$\displaystyle F= \frac{d\Phi}{dx}(x,y)=0$$

[[media: 2010_09_28_15_08_25.djvu | Page 16-2]]
2nd exactness condition: $$\displaystyle \underbrace{N_x}_{\color{blue}{f_x}}=\underbrace{M_y}_{\color{blue}{g_y}}$$

cf.[[media: 2010_09_23_14_52_54.djvu | Eqn.(1) and Eqn.(2) p.15-3:]]

Clairaut 1739-40 proved $$\displaystyle \Phi_{xy}=\Phi_{yx}$$.

End Note

N2-ODE: 2nd exactness condition
Back to N2-ODE: 2nd relation of exactness condition 2, i.e., [[media: 2010_09_23_14_52_54.djvu |Eqn.(2) p.15-3:]] is easy to prove:

=> Just find $$\displaystyle g_{pp}$$ using definition of g in [[media: 2010_09_23_14_52_54.djvu | Eqn.(3) p.15-2:]]. see F09

1st relation of exactness condition 2: [[media: 2010_09_23_14_52_54.djvu | Eqn.(1) p.15-3:]] $$\displaystyle \Phi(x,y,p)$$ - 3 arguments

Recall $$\displaystyle f=\Phi_p $$[[media: 2010_09_23_14_52_54.djvu |Eqn.(4) p.15-2]], $$\displaystyle g=\Phi_x + \Phi_y p $$[[media: 2010_09_23_14_52_54.djvu |Eqn.(3) p.15-2]]

[[media: 2010_09_28_15_08_25.djvu | Page 16-4]]
F09: Use $$\displaystyle \Phi_{xy} = \Phi_{yx} $$, and $$\displaystyle \underbrace{_{\color{blue}{Eqn.(2)p.16-3}}}_{\color{blue}{\Phi_x}} $$ and $$\displaystyle \underbrace{\color{blue}{_{Eqn.(1)}}}_{\color{blue}{\Phi_y}}$$, to obtain the 1st relation for exactness condition 2, i.e., [[media: 2010_09_28_15_08_25.djvu |Eqn.(1) p.15-3]].

title: User:Egm6321.f10.team4.Yoon/Mtg16


text:
User:Egm6321.f10.team4.Yoon/Mtg18

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_09_30_13_56_04.djvu | Mtg 18:]] Thu, 30 Sep 10

= Connection of linear systems with Reynolds Transport Theorem =

[[media: 2010_09_30_13_56_04.djvu | Page 18-1]]
Note: HW [[media: 2010_09_23_14_52_54.djvu | p.15-1]]

Dimensional Analysis

Connection to Reynolds Transp. Theorem(RTT) Material description (Lagrangian)

[[media: 2010_09_30_13_56_04.djvu | Page 18-2]]
End Note

[[media: 2010_09_29_07_17_06.djvu | p.17-3]], Ex2 continued: 2 Choices: 1) Assume

[[media: 2010_09_30_13_56_04.djvu | Page 18-3]]
2) Assume

title: User:Egm6321.f10.team4.Yoon/Mtg18


text:
User:Egm6321.f10.team4.Yoon/Mtg20

No class; exam 1

title: User:Egm6321.f10.team4.Yoon/Mtg20


text:
User:Egm6341.s11.team2.S

$$ \int $$

University of Florida

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group page

=Useful Links=

Template

Group Page

Lecture Plan

[http://en.wikipedia.org/wiki/Wikipedia:How_to_edit_a_page How to Edit a Page? ]

Help:Displaying_a_formula

wolframalpha

Manual

 * [CST]

HW2 Problem 2
Note that for $$ f(x) = {e^x} $$

$$ {f^{(n)}}({x_0}) = {f^{(n)}}(0) = {e^0} = 1 $$

So,

$$ {p_n}(x) = f(0) + \frac{f^{(1)}}(0) + \frac{f^{(2)}}(0) + ... + \frac{f^{(n)}}(0) $$

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$$  \displaystyle {p_n}(x) = 1 + \sum\limits_{i = 1}^\infty {\frac} $$     (2-1)
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and

$$ {R_{n + 1}}(x) = \frac{f^{(n + 1)}}(\xi ) = \frac{e^\xi } $$

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$$  \displaystyle {R_{n + 1}}(x) = \frac{e^\xi } $$     (2-2)
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$$  \displaystyle {e^x} = {p_n}(x) + {R_{n + 1}}(x) = 1 + \sum\limits_{i = 1}^\infty {\frac}  + \frac{e^\xi } $$     (2-3)
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From Eq(2-3)

$$ {e^x} - 1 = \sum\limits_{i = 1}^\infty {\frac}  + \frac{e^\xi }$$

so,

$$ f(x) = \frac{x} = \frac{x} = \sum\limits_{i = 1}^\infty {\frac}  + \frac{e^\xi } $$

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

$$  \displaystyle f(x) = {p_n}(x) + {R_{n + 1}}(x) = \frac{x} = \sum\limits_{i = 1}^\infty {\frac}  + \frac{e^\xi } $$     (2-4)
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So for $$ f(x) = \frac{x} $$

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

$$  \displaystyle {p_n}(x)= \sum\limits_{i = 1}^\infty {\frac} $$
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$$  \displaystyle {R_{n + 1}}(x)=\frac{e^\xi } $$

where $$ \xi \in [{x_0},x] $$ (2-5)
 *
 * }

= Problem 2.12 Numerically calculating integral of a function using Lagrange method =

Ref: Egm6341.s10.team3.sa/HW2

Given
$$ f(x)=\frac{e^{x}-1}{x} $$ on $$ x\in[0,1] (a) $$ and $$ x\in[0,1] (b) $$

Find
A) Construct $$ f_{n}(x)= \sum_{i=0}^{n}l_{i,n}(x)f(x_{i}) $$ for $$\displaystyle n = 1, 2, 4, 8, 16 $$ for both intervals.

B) Plot $$\displaystyle f(x) $$ and $$\displaystyle f_{n}(x) $$ for n = 1, 2, 4, 8, 16.

C) Compute $$ I_{n}=\int_{b}^{b}f_{n}(x)dx $$ for n = 1, 2, 4, 8 and compare to $$\displaystyle I $$.

D) For n = 5, plot $$\displaystyle l_{0}, l_{1}, l_{2} $$.

Solution
A) Considering the following equations:

$$  \displaystyle l_{i,n}(x) = \prod_{j=0,i\neq j}^{n}\frac{x-x_{j}}{x_{i}-x_{j}} $$ and $$  \displaystyle

f_{n}=\sum_{i=0}^{n}l_{i,n}(x)f(x_{i}) $$

for different n values and $$ x\in[0,1] $$, the followings can be calculated:

When n = 1

$$ \displaystyle x_{0} = 0, x_{1} = 1 $$.

$$\displaystyle l_{0,1}=\prod_{j=0,j\neq 0}^{1}\frac{x-x_{j}}{x_{0}-x_{j}}=\frac{x-1}{0-1}, \quad f(x_{0})=f(0)=1, $$

$$\displaystyle l_{1,1}=\prod_{j=0,j\neq 1}^{1}\frac{x-x_{j}}{x_{1}-x_{j}}=\frac{x-0}{1-0}, \quad f(x_{1})=f(1)=e-1. $$

When n = 2

$$ x_{0} = 0, x_{1} = \frac{1}{2}, x_{2} = 1 $$.

$$\displaystyle l_{0,2}=\prod_{j=0,j\neq 0}^{2}\frac{x-x_{j}}{x_{0}-x_{j}}=\frac{(x-\frac{1}{2})(x-1)}{(0-\frac{1}{2}(0-1)}, \quad f(x_{0})=f(0)=1, $$

$$\displaystyle l_{1,2}=\prod_{j=0,j\neq 1}^{2}\frac{x-x_{j}}{x_{1}-x_{j}}=\frac{(x-0)(x-1)}{(\frac{1}{2}-0)(\frac{1}{2}-1)}, \quad f(x_{1})=f(\frac{1}{2})=2\left(e^{\frac{1}{2}}-1\right), $$

$$\displaystyle l_{2,2}=\prod_{j=0,j\neq 2}^{2}\frac{x-x_{j}}{x_{2}-x_{j}}=\frac{(x-0)(x-\frac{1}{2})}{(1-0)(1-\frac{1}{2})}, \quad f(x_{2})=f(1)=e-1. $$

When n = 4

$$ x_{0} = 0, x_{1} = \frac{1}{4}, x_{2} = \frac{2}{4}, x_{3} = \frac{3}{4}, x_{4} = 1 $$.

$$\displaystyle l_{0,4}=\prod_{j=0,j\neq 0}^{4}\frac{x-x_{j}}{x_{0}-x_{j}}=\frac{(x-\frac{1}{4})(x-\frac{2}{4})(x-\frac{3}{4})(x-1)}{(0-\frac{1}{4})(0-\frac{2}{4})(0-\frac{3}{4})(0-1)}, \quad f(x_{0})=f(0)=1, $$

$$\displaystyle l_{1,4}=\prod_{j=0,j\neq 1}^{4}\frac{x-x_{j}}{x_{1}-x_{j}}=\frac{(x-0)(x-\frac{2}{4})(x-\frac{3}{4})(x-1)}{(\frac{1}{4}-0)(\frac{1}{4}-\frac{2}{4})(\frac{1}{4}-\frac{3}{4})(\frac{1}{4}-1)}, \quad f(x_{1})=f(\frac{1}{4})=4\left(e^{\frac{1}{4}}-1\right), $$ $$\displaystyle l_{2,4}=\prod_{j=0,j\neq 2}^{4}\frac{x-x_{j}}{x_{2}-x_{j}}=\frac{(x-0)(x-\frac{1}{4})(x-\frac{3}{4})(x-1)}{(\frac{2}{4}-0)(\frac{2}{4}-\frac{1}{4})(\frac{2}{4}-\frac{3}{4})(\frac{2}{4}-1)}, \quad f(x_{2})=f(\frac{2}{4})=2\left(e^{\frac{2}{4}}-1\right), $$

$$\displaystyle l_{3,4}=\prod_{j=0,j\neq 3}^{4}\frac{x-x_{j}}{x_{3}-x_{j}}=\frac{(x-0)(x-\frac{1}{4})(x-\frac{2}{4})(x-1)}{(\frac{3}{4}-0)(\frac{3}{4}-\frac{1}{4})(\frac{3}{4}-\frac{2}{4})(\frac{3}{4}-1)}, \quad f(x_{3})=f(\frac{3}{4})=\frac{4}{3}\left(e^{\frac{3}{4}}-1\right), $$

$$\displaystyle l_{4,4}=\prod_{j=0,j\neq 4}^{4}\frac{x-x_{j}}{x_{4}-x_{j}}=\frac{(x-0)(x-\frac{1}{4})(x-\frac{2}{4})(x-\frac{3}{4})}{(1-0)(1-\frac{1}{4})(1-\frac{2}{4})(1-\frac{3}{4})}, \quad f(x_{4})=f(1)=e-1, $$

When n = 8

$$ x_{0} = 0, x_{1} = \frac{1}{8}, x_{2} = \frac{2}{8}, x_{3} = \frac{3}{8}, x_{4} = \frac{4}{8}, x_{5} = \frac{5}{8}, x_{6} = \frac{6}{8}, x_{7} = \frac{7}{8}, x_{8} = 1 $$.

$$\displaystyle l_{0,8}=\prod_{j=0,j\neq 0}^{8}\frac{x-x_{j}}{x_{0}-x_{j}}=\frac{(x-\frac{1}{8})(x-\frac{2}{8})(x-\frac{3}{8})(x-\frac{4}{8})(x-\frac{5}{8})(x-\frac{6}{8})(x-\frac{7}{8})(x-1)}{(0-\frac{1}{8})(0-\frac{2}{8})(0-\frac{3}{8})(0-\frac{4}{8})(0-\frac{5}{8})(0-\frac{6}{8})(0-\frac{7}{8})(0-1)}, \quad f(x_{0})=f(0)=1, $$

$$\displaystyle l_{1,8}=\prod_{j=0,j\neq 1}^{8}\frac{x-x_{j}}{x_{1}-x_{j}}=\frac{(x-0)(x-\frac{2}{8})(x-\frac{3}{8})(x-\frac{4}{8})(x-\frac{5}{8})(x-\frac{6}{8})(x-\frac{7}{8})(x-1)}{(\frac{1}{8}-0)(\frac{1}{8}-\frac{2}{8})(\frac{1}{8}-\frac{3}{8})(\frac{1}{8}-\frac{4}{8})(\frac{1}{8}-\frac{5}{8})(\frac{1}{8}-\frac{6}{8})(\frac{1}{8}-\frac{7}{8})(\frac{1}{8}-1)}, \quad f(x_{1})=f(\frac{1}{8})=8\left(e^{\frac{1}{8}}-1\right), $$

$$\displaystyle l_{2,8}=\prod_{j=0,j\neq 2}^{8}\frac{x-x_{j}}{x_{2}-x_{j}}=\frac{(x-0)(x-\frac{1}{8})(x-\frac{3}{8})(x-\frac{4}{8})(x-\frac{5}{8})(x-\frac{6}{8})(x-\frac{7}{8})(x-1)}{(\frac{2}{8}-0)(\frac{2}{8}-\frac{1}{8})(\frac{2}{8}-\frac{3}{8})(\frac{2}{8}-\frac{4}{8})(\frac{2}{8}-\frac{5}{8})(\frac{2}{8}-\frac{6}{8})(\frac{2}{8}-\frac{7}{8})(\frac{2}{8}-1)}, \quad f(x_{2})=f(\frac{2}{8})=\frac{8}{2}\left(e^{\frac{2}{8}}-1\right), $$

$$\displaystyle l_{3,8}=\prod_{j=0,j\neq 3}^{8}\frac{x-x_{j}}{x_{3}-x_{j}}=\frac{(x-0)(x-\frac{1}{8})(x-\frac{2}{8})(x-\frac{4}{8})(x-\frac{5}{8})(x-\frac{6}{8})(x-\frac{7}{8})(x-1)}{(\frac{3}{8}-0)(\frac{3}{8}-\frac{1}{8})(\frac{3}{8}-\frac{2}{8})(\frac{3}{8}-\frac{4}{8})(\frac{3}{8}-\frac{5}{8})(\frac{3}{8}-\frac{6}{8})(\frac{3}{8}-\frac{7}{8})(\frac{3}{8}-1)}, \quad f(x_{3})=f(\frac{3}{8})=\frac{8}{3}\left(e^{\frac{3}{8}}-1\right), $$

$$\displaystyle l_{4,8}=\prod_{j=0,j\neq 4}^{8}\frac{x-x_{j}}{x_{4}-x_{j}}=\frac{(x-0)(x-\frac{1}{8})(x-\frac{2}{8})(x-\frac{3}{8})(x-\frac{5}{8})(x-\frac{6}{8})(x-\frac{7}{8})(x-1)}{(\frac{4}{8}-0)(\frac{4}{8}-\frac{1}{8})(\frac{4}{8}-\frac{2}{8})(\frac{4}{8}-\frac{3}{8})(\frac{4}{8}-\frac{5}{8})(\frac{4}{8}-\frac{6}{8})(\frac{4}{8}-\frac{7}{8})(\frac{4}{8}-1)}, \quad f(x_{4})=f(\frac{4}{8})=\frac{8}{4}\left(e^{\frac{4}{8}}-1\right), $$

$$\displaystyle l_{5,8}=\prod_{j=0,j\neq 5}^{8}\frac{x-x_{j}}{x_{5}-x_{j}}=\frac{(x-0)(x-\frac{1}{8})(x-\frac{2}{8})(x-\frac{3}{8})(x-\frac{4}{8})(x-\frac{6}{8})(x-\frac{7}{8})(x-1)}{(\frac{5}{8}-0)(\frac{5}{8}-\frac{1}{8})(\frac{5}{8}-\frac{2}{8})(\frac{5}{8}-\frac{3}{8})(\frac{5}{8}-\frac{4}{8})(\frac{5}{8}-\frac{6}{8})(\frac{5}{8}-\frac{7}{8})(\frac{5}{8}-1)}, \quad f(x_{5})=f(\frac{5}{8})=\frac{8}{5}\left(e^{\frac{5}{8}}-1\right), $$

$$\displaystyle l_{6,8}=\prod_{j=0,j\neq 6}^{8}\frac{x-x_{j}}{x_{6}-x_{j}}=\frac{(x-0)(x-\frac{1}{8})(x-\frac{2}{8})(x-\frac{3}{8})(x-\frac{4}{8})(x-\frac{5}{8})(x-\frac{7}{8})(x-1)}{(\frac{6}{8}-0)(\frac{6}{8}-\frac{1}{8})(\frac{6}{8}-\frac{2}{8})(\frac{6}{8}-\frac{3}{8})(\frac{6}{8}-\frac{4}{8})(\frac{6}{8}-\frac{5}{8})(\frac{6}{8}-\frac{7}{8})(\frac{6}{8}-1)}, \quad f(x_{6})=f(\frac{6}{8})=\frac{8}{6}\left(e^{\frac{6}{8}}-1\right), $$

$$\displaystyle l_{7,8}=\prod_{j=0,j\neq 7}^{8}\frac{x-x_{j}}{x_{7}-x_{j}}=\frac{(x-0)(x-\frac{1}{8})(x-\frac{2}{8})(x-\frac{3}{8})(x-\frac{4}{8})(x-\frac{5}{8})(x-\frac{6}{8})(x-1)}{(\frac{7}{8}-0)(\frac{7}{8}-\frac{1}{8})(\frac{7}{8}-\frac{2}{8})(\frac{7}{8}-\frac{3}{8})(\frac{7}{8}-\frac{4}{8})(\frac{7}{8}-\frac{5}{8})(\frac{7}{8}-\frac{6}{8})(\frac{7}{8}-1)}, \quad f(x_{7})=f(\frac{7}{8})=\frac{8}{7}\left(e^{\frac{7}{8}}-1\right), $$ $$\displaystyle l_{8,8}=\prod_{j=0,j\neq 8}^{8}\frac{x-x_{j}}{x_{8}-x_{j}}=\frac{(x-0)(x-\frac{1}{8})(x-\frac{2}{8})(x-\frac{3}{8})(x-\frac{4}{8})(x-\frac{5}{8})(x-\frac{6}{8})(x-\frac{7}{8})}{(1-0)(1-\frac{1}{8})(1-\frac{2}{8})(1-\frac{3}{8})(1-\frac{4}{8})(1-\frac{5}{8})(1-\frac{6}{8})(1-\frac{7}{8})}, \quad f(x_{8})=f(1)=e-1. $$

When n = 16 $$ x_{0} = 0, x_{1} = \frac{1}{16}, x_{2} = \frac{2}{16}, x_{3} = \frac{3}{16}, x_{4} = \frac{4}{16}, x_{5} = \frac{5}{16}, x_{6} = \frac{6}{16}, x_{7} = \frac{7}{16}, x_{8} = \frac{8}{16}, x_{9} = \frac{9}{16}, x_{10} = \frac{10}{16}, x_{11} = \frac{11}{16}, x_{12} = \frac{12}{16}, x_{13} = \frac{13}{16}, x_{14} = \frac{14}{16}, x_{15} = \frac{15}{16}, x_{16} = 1 $$.

$$\displaystyle l_{0,16}=\prod_{j=0,j\neq 0}^{16}\frac{x-x_{j}}{x_{0}-x_{j}}=\frac{(x-\frac{1}{16})(x-\frac{2}{16})(x-\frac{3}{16})(x-\frac{4}{16})(x-\frac{5}{16})(x-\frac{6}{16})(x-\frac{7}{16})(x-\frac{8}{16})(x-\frac{9}{16})(x-\frac{10}{16})(x-\frac{11}{16})(x-\frac{12}{16})(x-\frac{13}{16})(x-\frac{14}{16})(x-\frac{15}{16})(x-1)}{(0-\frac{1}{16})(0-\frac{2}{16})(0-\frac{3}{16})(0-\frac{4}{16})(0-\frac{5}{16})(0-\frac{6}{16})(0-\frac{7}{16})(0-\frac{8}{16})(0-\frac{9}{16})(0-\frac{10}{16})(0-\frac{11}{16})(0-\frac{12}{16})(0-\frac{13}{16})(0-\frac{14}{16})(0-\frac{15}{16})(0-1)}, \quad f(x_{0})=f(0)=1, $$

$$\displaystyle l_{1,16}=\prod_{j=0,j\neq 1}^{16}\frac{x-x_{j}}{x_{1}-x_{j}}=\frac{(x-0)(x-\frac{2}{16})(x-\frac{3}{16})(x-\frac{4}{16})(x-\frac{5}{16})(x-\frac{6}{16})(x-\frac{7}{16})(x-\frac{8}{16})(x-\frac{9}{16})(x-\frac{10}{16})(x-\frac{11}{16})(x-\frac{12}{16})(x-\frac{13}{16})(x-\frac{14}{16})(x-\frac{15}{16})(x-1)}{(\frac{1}{16}-0)(\frac{1}{16}-\frac{2}{16})(\frac{1}{16}-\frac{3}{16})(\frac{1}{16}-\frac{4}{16})(\frac{1}{16}-\frac{5}{16})(\frac{1}{16}-\frac{6}{16})(\frac{1}{16}-\frac{7}{16})(\frac{1}{16}-\frac{8}{16})(\frac{1}{16}-\frac{9}{16})(\frac{1}{16}-\frac{10}{16})(\frac{1}{16}-\frac{11}{16})(\frac{1}{16}-\frac{12}{16})(\frac{1}{16}-\frac{13}{16})(\frac{1}{16}-\frac{14}{16})(\frac{1}{16}-\frac{15}{16})(\frac{1}{16}-1)}, \quad f(x_{1})=f(\frac{1}{16})=16\left(e^{\frac{1}{16}}-1\right), $$

and it can be calculate up to $$ l_{16,16} $$ in the same manner.

$$\displaystyle l_{16,16}=\prod_{j=0,j\neq 16}^{16}\frac{x-x_{j}}{x_{16}-x_{j}}=\frac{(x-0)(x-\frac{1}{16})(x-\frac{2}{16})(x-\frac{3}{16})(x-\frac{4}{16})(x-\frac{5}{16})(x-\frac{6}{16})(x-\frac{7}{16})(x-\frac{8}{16})(x-\frac{9}{16})(x-\frac{10}{16})(x-\frac{11}{16})(x-\frac{12}{16})(x-\frac{13}{16})(x-\frac{14}{16})(x-\frac{15}{16})}{(1-0)(1-\frac{1}{16})(1-\frac{2}{16})(1-\frac{3}{16})(1-\frac{4}{16})(1-\frac{5}{16})(1-\frac{6}{16})(1-\frac{7}{16})(1-\frac{8}{16})(1-\frac{9}{16})(1-\frac{10}{16})(1-\frac{11}{16})(1-\frac{12}{16})(1-\frac{13}{16})(1-\frac{14}{16})(1-\frac{15}{16})}, \quad f(x_{16})=f(1)=e-1, $$

And for different n values and $$ x\in[-1,1] $$, the followings can be calculated:

When n = 1

$$ \displaystyle x_{0} = -1, x_{1} = 1 $$.

$$\displaystyle l_{0,1} = \prod\limits_{j = 0,j \ne 0}^1 {\frac} = \frac,\quad f({x_0}) = f( - 1) =  - ({e^{ - 1}} - 1) $$

$$ {l_{1,1}} = \prod\limits_{j = 0,j \ne 1}^1 {\frac} = \frac,\quad f({x_1}) = f(1) = e - 1. $$

When n = 2

$$ \displaystyle x_{0} = -1, x_{1} = 0, x_{2} = 1 $$.

$${l_{0,2}} = \prod\limits_{j = 0,j \ne 0}^2 {\frac} = \frac,\quad f({x_0}) = f( - 1) =  - ({e^{ - 1}} - 1), $$

$${l_{1,2}} = \prod\limits_{j = 0,j \ne 1}^2 {\frac} = \frac,\quad f({x_1}) = f(0) = 1 $$

$${l_{2,2}} = \prod\limits_{j = 0,j \ne 2}^2 {\frac} = \frac,\quad f({x_2}) = f(1) = e - 1. $$

When n = 4

$$ \displaystyle x_{0} = -1, x_{1} = -\frac{1}{2}, x_{2} = 0, x_{3} = \frac{1}{2}, x_{4} = 1 $$.

$$ {l_{0,4}} = \prod\limits_{j = 0,j \ne 0}^4 {\frac} = \frac,\quad f({x_0}) = f( - 1) =  - ({e^{ - 1}} - 1)$$

$${l_{1,4}} = \prod\limits_{j = 0,j \ne 1}^4 {\frac} = \frac,\quad f({x_1}) = f( - \frac{1}{2}) =  - 2\left( {{e^{ - \frac{1}{2}}} - 1} \right), $$

$${l_{2,4}} = \prod\limits_{j = 0,j \ne 2}^4 {\frac} = \frac,\quad f({x_2}) = f(0) = 1, $$

$${l_{3,4}} = \prod\limits_{j = 0,j \ne 3}^4 {\frac} = \frac,\quad f({x_3}) = f(\frac{1}{2}) = 2\left( {{e^{\frac{1}{2}}} - 1} \right), $$

$$ {l_{4,4}} = \prod\limits_{j = 0,j \ne 4}^4 {\frac} = \frac,\quad f({x_4}) = f(1) = e - 1, $$

When n = 8

$$ \displaystyle x_{0} = -1, x_{1} = -\frac{3}{4}, x_{2} =- \frac{2}{4}, x_{3} = -\frac{1}{4}, x_{4} = 0, x_{5} = \frac{1}{4}, x_{6} = \frac{2}{4}, x_{7} = \frac{3}{4}, x_{8} = 1 $$.

$$ l_{0,8} = \prod\limits_{j = 0,j \ne 0}^8 {\frac} = \frac,\quad f({x_0}) = f( - 1) =  - ({e^{ - 1}} - 1), $$

$${l_{1,8}} = \prod\limits_{j = 0,j \ne 1}^8 {\frac} = \frac,\quad f({x_1}) = f( - \frac{3}{4}) =  - \frac{4}{3}\left( {{e^{ - \frac{3}{4}}} - 1} \right), $$

$$ {l_{2,8}} = \prod\limits_{j = 0,j \ne 2}^8 {\frac}  = \frac,\quad f({x_2}) = f( - \frac{2}{4}) =  - 2\left( {{e^{ - \frac{2}{4}}} - 1} \right), $$

$$ {l_{3,8}} = \prod\limits_{j = 0,j \ne 3}^8 {\frac}  = \frac,\quad f({x_3}) = f( - \frac{1}{4}) =  - 4\left( {{e^{ - \frac{1}{4}}} - 1} \right)$$

$$ {l_{4,8}} = \prod\limits_{j = 0,j \ne 4}^8 {\frac}  = \frac,\quad f({x_4}) = f(0) = 1, $$

$$ {l_{5,8}} = \prod\limits_{j = 0,j \ne 5}^8 {\frac} = \frac,\quad f({x_5}) = f(\frac{1}{4}) = 4\left( {{e^{\frac{1}{4}}} - 1} \right),$$

$$ {l_{6,8}} = \prod\limits_{j = 0,j \ne 6}^8 {\frac}  = \frac,\quad f({x_6}) = f(\frac{2}{4}) = 2\left( {{e^{\frac{2}{4}}} - 1} \right), $$

$$ {l_{7,8}} = \prod\limits_{j = 0,j \ne 7}^8 {\frac} = \frac,\quad f({x_7}) = f(\frac{3}{4}) = \frac{4}{3}\left( {{e^{\frac{3}{4}}} - 1} \right),$$

$${l_{8,8}} = \prod\limits_{j = 0,j \ne 8}^8 {\frac} = \frac,\quad f({x_8}) = f(1) = e - 1. $$

When n = 16 $$ \displaystyle x_{0} = -1, x_{1} = -\frac{7}{8}, x_{2} = -\frac{6}{8}, x_{3} = -\frac{5}{8}, x_{4} = -\frac{4}{8}, x_{5} = -\frac{3}{8}, x_{6} = -\frac{2}{8}, x_{7} = \frac{1}{8}, x_{8} = 0, x_{9} = \frac{1}{8}, x_{10} = \frac{2}{8}, x_{11} = \frac{3}{8}, x_{12} = \frac{4}{8}, x_{13} = \frac{5}{8}, x_{14} = \frac{6}{8}, x_{15} = \frac{7}{8}, x_{16} = 1 $$.

$$ l_{0,16} = \prod\limits_{j = 0,j \ne 0}^{16} {\frac} = \frac,\quad f({x_0}) = f( - 1) =  - ({e^{ - 1}} - 1), $$

$${l_{1,16}} = \prod\limits_{j = 0,j \ne 1}^{16} {\frac} = \frac,\quad f({x_1}) = f( - \frac{7}{8}) =  - \frac{8}{7}\left( {{e^{ - \frac{7}{8}}} - 1} \right),$$

and it can be calculate up to $$ l_{16,16} $$ in the same manner.

$$ {l_{16,16}} = \prod\limits_{j = 0,j \ne 16}^{16} {\frac}  = \frac,\quad f({x_{16}}) = f(1) = e - 1, $$

B) Considering the following equation, the approximated integral can be calculated:

$$ {I_n} = \int\limits_a^b {\sum\limits_{i = 0}^n {{l_{i,n}}({x_i}).f({x_i})dx} } $$

note that $$ {E_n} = \left| {I - {I_n}} \right| $$ is the difference between the exact integral $$\displaystyle I $$ and approximated integral $$\displaystyle  I_n $$ The exact integral was calculated using wolframalpha

C)

 * [[File:HW2 P12 A.png|1000px|center|thumb|Figure 12.1: $$\displaystyle f(x), f_1(x), f_2(x), f_4(x), f_8(x) for x\in[0,1] $$]]
 * [[File:HW2 P12 B.png|1000px|center|thumb|Figure 12.2: $$\displaystyle f(x), f_1(x), f_2(x), f_4(x), f_8(x) for x\in[-1,1] $$]]

D)
 * [[File:HW2 P12 C.png|1000px|center|thumb|Figure 12.3: $$\displaystyle f(x), l_{0,5},l_{1,5},l_{2,5} for x\in[0,1] $$]]

= Problem 2.2 Taylor series expansion of sin(x) =

Ref: Egm6341.s10.team3.sa/HW1

Given
$$ f(x)=sin x, x \in [0,\pi] $$

Find
Construct a Taylor series of $$f(x)$$ around $$x_0 = \frac{\pi}{4}$$ for $$\displaystyle n=0,1,2......,10.$$

Construct a Taylor series of $$f(x)$$ around $$x_0 = \frac{3\pi}{8}$$ for $$\displaystyle n=0,1,2......,10.$$

Estimate the maximum $$\displaystyle R(x)$$ at $$ x=\frac{3\pi}{4}$$

Plot the series for each $$\displaystyle n $$

Solution
Considering

$$ {p_n}(x) = f({x_0}) + \frac{f^{(1)}}({x_0}) + \frac{f^{(2)}}({x_0}) + ... + \frac{f^{(n)}}({x_0}) $$

for $$ \displaystyle f({x}) = \sin ({x}) $$ around the point $${x_0} = \frac{\pi }{4} $$ the follwoing equations can be calculated for $$ \displaystyle n= 0,1,2,...,10.$$ Note that $$f({x_0}) = \sin (\frac{\pi }{4}) = \frac{2} $$

$${p_0}(x) = \frac{2}$$

$${p_1}(x) = \frac{2} + \frac\cos (\frac{\pi }{4}) $$

$${p_2}(x) = \frac{2} + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) $$

$${p_3}(x) = \frac{2} + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) - \frac\cos (\frac{\pi }{4}) $$

$${p_4}(x) = \frac{2} + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) - \frac\cos (\frac{\pi }{4}) + \frac\sin (\frac{\pi }{4}) $$

$${p_5}(x) = \frac{2} + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) - \frac\cos (\frac{\pi }{4}) + \frac\sin (\frac{\pi }{4}) + \frac\cos (\frac{\pi }{4}) $$

$$ {p_6}(x) = \frac{2} + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) - \frac\cos (\frac{\pi }{4}) + \frac\sin (\frac{\pi }{4}) + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) $$

$${p_7}(x) = \frac{2} + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) - \frac\cos (\frac{\pi }{4}) + \frac\sin (\frac{\pi }{4}) + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) - \frac\cos (\frac{\pi }{4})$$

$$ {p_8}(x) = \frac{2} + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) - \frac\cos (\frac{\pi }{4}) + \frac\sin (\frac{\pi }{4}) + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) - \frac\cos (\frac{\pi }{4}) + \frac\sin (\frac{\pi }{4}) $$

$${p_9}(x) = \frac{2} + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) - \frac\cos (\frac{\pi }{4}) + \frac\sin (\frac{\pi }{4}) + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) - \frac\cos (\frac{\pi }{4}) + \frac\sin (\frac{\pi }{4}) + \frac\cos (\frac{\pi }{4}) $$

$${p_{10}}(x) = \frac{2} + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) - \frac\cos (\frac{\pi }{4}) + \frac\sin (\frac{\pi }{4}) + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) - \frac\cos (\frac{\pi }{4}) + \frac\sin (\frac{\pi }{4}) + \frac\cos (\frac{\pi }{4}) - \frac\sin (\frac{\pi }{4}) $$

which can be simplified to the following equations:

Considering

$$ {p_n}(x) = f({x_0}) + \frac{f^{(1)}}({x_0}) + \frac{f^{(2)}}({x_0}) + ... + \frac{f^{(n)}}({x_0}) $$

for $$ \displaystyle f({x}) = \sin ({x}) $$ around the point $${x_0} = \frac{3\pi }{8} $$ the follwoing equations can be calculated for $$ \displaystyle n= 0,1,2,...,10.$$

$${p_0}(x) = \sin (\frac{8}) $$

$${p_1}(x) = \sin (\frac{8}) + \frac\cos (\frac{8})$$ $${p_2}(x) = \sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8}) $$

$${p_3}(x) = \sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8}) - \frac\cos (\frac{8}) $$

$${p_4}x) = \sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8}) - \frac\cos (\frac{8}) + \frac\sin (\frac{8}) $$

$${p_5}(x) = \sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8}) - \frac\cos (\frac{8}) + \frac\sin (\frac{8}) + \frac\cos (\frac{8}) $$

$${p_6}(x) = \sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8}) - \frac\cos (\frac{8}) + \frac\sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8})$$

$${p_7}(x) = \sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8}) - \frac\cos (\frac{8}) + \frac\sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8}) - \frac\cos (\frac{8}) $$

$${p_8}(x) = \sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8}) - \frac\cos (\frac{8}) + \frac\sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8}) - \frac\cos (\frac{8}) + \frac\sin (\frac{8}) $$

$${p_9}(x) = \sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8}) - \frac\cos (\frac{8}) + \frac\sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8}) - \frac\cos (\frac{8}) + \frac\sin (\frac{8}) + \frac\cos (\frac{8}) $$

$${p_{10}}(x) = \sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8}) - \frac\cos (\frac{8}) + \frac\sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8}) - \frac\cos (\frac{8}) + \frac\sin (\frac{8}) + \frac\cos (\frac{8}) - \frac\sin (\frac{8})$$

which can be simplified to the following equations:

Matlab Code and Graphs
And here, the graphs show the results. Note that the Matlab code is the same for part A and B and in order to run the Matlab code for part B, the value for $$ x_0 $$ needs to be changed to $$ \frac{8} $$

 * HW2 P2 A.png
 * HW2 P2 B.png

=HW3-P6=

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

$$\displaystyle {q_{n + 1}}(x) = \prod\limits_{j = 0}^n {(x - {x_j})} = (x - {x_0})(x - {x_1})...(x - {x_{n - 1}})(x - {x_n}) $$    (6.1)
 *
 * }

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

q_{n + 1}^{n + 1}(x) = (n + 1)! $$     (6.2)
 * $$\displaystyle
 * $$\displaystyle
 *
 * }

We know that $$ \displaystyle{q_{n + 1}}(x) \in {p_{n + 1}}$$, so (6.1) can be written as:

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

{q_{n + 1}}(x) = \underbrace _{f(x)} + \underbrace {{a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ... + {a_1}{x^1} + {a_0}{x^0}}_{{p_n}(x)} $$     (6.3)
 * $$\displaystyle
 * $$\displaystyle
 *
 * }

note that in (6.3), $$ {a_{n + 1}} = 1\displaystyle $$.

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

q_{n + 1}^{n + 1}(x) = {f^{n + 1}}(x) + p_n^{n + 1}(x) $$     (6.4)
 * $$\displaystyle
 * $$\displaystyle
 *
 * }

where $$\displaystyle p_n^{n + 1}(x) = 0 $$ and $$\displaystyle {f^{n + 1}}(x) $$ can be calculated as follows:

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

f(x) = {x^{n + 1}} $$
 * $$\displaystyle
 * $$\displaystyle
 *
 * }

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

{f^1}(x) = (n + 1){x^n} $$
 * $$\displaystyle
 * $$\displaystyle
 *
 * }

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

{f^2}(x) = (n + 1)(n){x^{n - 1}} $$
 * $$\displaystyle
 * $$\displaystyle
 *
 * }

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

{f^3}(x) = (n + 1)(n)(n - 1){x^{n - 2}} $$
 * $$\displaystyle
 * $$\displaystyle
 *
 * }

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

{f^{n - 1}}(x) = (n + 1)(n)(n - 1)......(2){x^{n - (n - 1)}} $$
 * $$\displaystyle
 * $$\displaystyle
 *
 * }

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

{f^n}(x) = (n + 1)(n)(n - 1)......(2)(1){x^{n - n}} $$
 * $$\displaystyle
 * $$\displaystyle
 *
 * }

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

{f^{n + 1}}(x) = (n + 1)! $$
 * $$\displaystyle
 * $$\displaystyle
 *
 * }

and consequently

= Summary=

Licensing
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 * style="background: #ccf;" |Multi-licensed with any Creative Commons Attribution Share-Alike License
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 * colspan="3" |I agree to multi-license my text contributions, unless otherwise stated, under Wikipedia's copyright terms and the Creative Commons license Attribution Share-Alike version 3.0 and the Creative Commons license Attribution Non-Commercial Share-Alike version 3.0. Please be aware that other contributors might not do the same, so if you want to use my contributions under the Creative Commons terms, please check the CC dual-license and Multi-licensing guides.
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title: User:Egm6341.s11.team2.S


text:
User:Egm6321.f10.team4.Yoon/Mtg12

 Mtg 12: Thu, 16 Sep 10

[[media: 2010_09_16_14_59_27.djvu | Page 12-1]]
Remark: [[media: 2010_09_16_13_50_21.djvu | Eqn.(6) p.11-3]] is more general than [[media:2010_09_14_15_00_52.djvu | Eqn.(5) p.10-2]] (K.p.512) since the latter needs the condition $$\bar b(x) \neq 0 \ \forall x$$ End Remark

Question: [[media: 2010_09_16_13_50_21.djvu | p.11-4]] continued. Yes

= Class of N1-ODE either exact or can be made exact by integrating factor method =

(Euler, Celestial mechanics)

where $$\displaystyle a(\cdot), b(\cdot), c(\cdot)$$ are arbitrary functions.

Application:

[[media: 2010_09_16_14_59_27.djvu | Page 12-2]]
F09: Show Eqn.(1) is either exact or can be made exact

Derive Eqn.(1) - Eqn.(3) [[media: 2010_09_16_14_59_27.djvu | p.12-1]] consider

[[media: 2010_09_16_14_59_27.djvu | Page 12-3]]
Eqn.(6) [[media: 2010_09_16_14_59_27.djvu | Page 12-2]]:

Use Eqn.(1) and Eqn.(3) in [[media: 2010_09_09_13_54_49.djvu | Eqn.(3) p.8-2]](1st condition of exactness) => [[media: 2010_09_16_14_59_27.djvu | Eqn.(1) - Eqn.(3) p.12-1]]

= L2-ODE-VC with missing dependant variable =

[[media: 2010_09_16_14_59_27.djvu | Page 12-4]]
Let $$\displaystyle p(x) := y'(x)$$ [[media: 2010_09_16_14_59_27.djvu | Eqn.(4) p.12-3]]: $$\displaystyle \underbrace{a_2(x)p'+a_1(x)p = b(x)}_{\color{blue}{\text{L1-ODE-VC}}}$$

title: User:Egm6321.f10.team4.Yoon/Mtg12


text:
Counseling/Jungian personality types

Jungian personality types
Personality traits are described in Jung's personality code as letters are biopolar in that they represent opposing traits along a linear measure. They measure the distance along a pole an individual tends to that trait, and away from the opposing trait. Jung described introversion and extraversion as the key polar traits and refered to them as attitudes; they comprise the first letter of the code. The next two letters in the code measure sensing versus intuition, and feeling versus thinking; Jung referred to them as functions. The final pole, perceiving versus judging, comes from the Myers-Briggs typographic model, and is also a function.

Poles
For Jung, an attitude "plays the principal role in an individual's adaptation or orientation to life" (1921, para. 1). The two factors for this pole, extraversion and introversion, describe how a person draws mental energy. The extravert draws mental energy from the surrounding environment, or as Dyce explains, "from a crowd" (J. Dyce, personal communication, n.d.). The introvert draws mental energy internally from the ideas that develop within his mind. The mental energy, what Dyce also calls "psychic energy," is called libido in Jung's model. An individual's approach to life, or attitude, is determined by his prefered way to obtain libido; either externally from the surrounding environment, or interally from personal thoughts (Jung, 1976).

Libido: Nutrition for the mind
Jung describes the need for libido as a driving force, or what BF Skinner might think of as a hunger; libido is effectively nutrition for the mind. Jung describes the extravert as absorbing llibido from the surrounding environment by attaching to it as an "object." To Jung, the extravert is, hence, objective, as the psychic energy is based on surrounding reality. The introvert, conversely, develops this nutrition within his mind from the "subject" of his thoughts; the introvert is subjective to Jung, and his psychic metabolism (to extend the nutrition analogy) is synthesis.

Bi-poles as a "bent universe"
Jung is emphatic that mental health hinges on a person's adaptability along the poles of his bipolar model (J. Dyce, personal communication, n.d.). Thinking problems may result for extraverts when they are overwhelmed by the information that they pull in, whereas introverts may attempt to "coerce facts" to resemble a preconceived image that they have created (Jung, 1921, para. 87). When individuals tend to extremes along either pole, and neglect the qualities of the other end of the pole, they will very likely compensate for the trait of the pole they have neglected (Jung, 1976, p. 3) and move towards the counter-pole (p. 291). His three pole model can be thought of as a "bent universe" because those at polar extremes will ultimately compensate in some way. In his "bent universe," the extreme extrovert cannot help but return to the subject of his thoughts, or his "soul" (p. 293). In the opposite polar direction, the introvert cannot help but crash into reality, or the "object" that defines surrounding reality. If the extravert is intent on his attachment to the object (as the participant self-reported that he has experienced at times), then he creates a mythology that serves as a substitute for introverted thought, or the subject. However, if he is not careful, the compensating restoration of the "soul" may ultimately destroy the material benefits that extroversion may have provided (p. 340). In a similar sense, the extreme introvert creates a facsimile of reality that Jung describes as a "dream" (p. 169).

Jung's two other bipolar measures describe how an individual functions with respect to his role. They are "function-types" (1976, p. 330), and describe how an individual collects information and makes decisions based on how he "attends" (J. Dyce, personal communication, n.d.). Information can be collected through the normal senses, which is sensing, or through a "sixth sense" (J. Dyce, personal communication, n.d.) of intuition. Intuition is how the unconscious perceives. Decisions can be made logically and in a detached way, or in a personal value-oriented way, creating the thinking/feeling bipole.

The fourth pole from the Myers-Briggs model adds a third functional dimension that describes prefered lifestyle in terms of either judgement, which is rigidly well-ordered and highly-organized, or perception, which is flexible as it allows for spontaneity, providing an ability to shift between tasks (J. Dyce, personal communication, n.d.).

Plato and Aristotle: Jung's approach to subjective- and objectiveness
To help illustrate the fundamental importance of the introvert/extravert pole, and to show that Jung's theory was not entirely his own invention, but was based on others' supporting ideas, including the classics, Jung inserted a passage by Heine at the very beginning of the book describing Plato in introverted terms, and Aristotle in extraverted terms. Plato, in Heine's passage, is mystical, and hence subjective in Jung's model; and Aristotle is orderly and practical (presumably with respect to information organization). Aristotle is openly attached to the "object" of his surrounding environment, and hence an extravert and objective. This assessment contradicts the common perception of both Plato and Aristotle being objective, but in different ways. Plato's Forms created a basis for orderly science, and Aristotle's disciplined observational approach contributed to the scientific method. Both of them are further perceived as highly objective because of their resistance to sentimentality. The Similarminds questionaire makes a similar distinction with questions that define a rational/sentimental bipole ("Personality Test," n.d.). Jung further contradicts the common perception of Plato as objective by attaching the concept of "empathy," and hence sentimentality, to objectivity (p. 48), and by showing that empathy's antithesis, abstraction, is used by the introvert to synthesize a version of the world within his mind--as he may be afraid of the real world (p. 505). As Jung's ideas are the basis for the instrument of this assessment, this conclusion by Jung underscores the necessity to assess the individual in the context of his personality and experiences, and not in the context of contemporary society with its many preconceptions.

Sense/intuition and thinking/feeling
Jung's two functional bipoles, sense/intuition and thinking/feeling, also obey the rules of his "bent universe" that forces compensation at the extremes. Dyce describes intuition as an added, or sixth sense (J. Dyce, personal communcation, n.d.). This implies that unintuitive individuals who are sense-oriented rely on superficial cues from the surrounding environment, and hence may not be able to interpret the meanings of surrounding phenomena, and therefore are distanced, and possibly afraid (p. 505). "Sense," therefore, might be counter-intuitively interpreted as an introverted pole and "intuition" an extroverted, and also empathic, pole. Jung self-debates the extra- and introverted nature of the sensing/intuition bipole, but agrees that intuition is important, along with empathy, for understanding others (p. 473). Thinking and feeling are even easier to explain; thinking is the domain of the introvert. Jung describes it as thinking of subjects, and, as thinking is the polar opposite to feeling, the thinking/feeling bipole further attaches Plato to subjectivity.

In Psychological Types, Jung describes a dichotomous bipolar model of opposing traits that not only explains personality phenomena along the established poles used by the Myers-Briggs, but introduces many related descriptors that can create a flexible matrix that is applicable to diverse therapeutic scenarios (Jung, 1976). Jung built his theory by critically examining other theories being developed during the early 20th Century, and further supports it with classical anecdotes. Jung's writing is self-critical in a way that creates possibilities for alternate analyses, which allowed the participant to modify his interpretation of the intuitive/sensing bipole to improve his self-assessment. Psychological Types gives advanced insights into the relationships between many psychological concepts. The most important of these are empathy, abstraction, and conceptualization (Jung, 1976, p 48), as he reported seeing these concepts being implemented to define emotional intelligence.

title: Counseling/Jungian personality types


text:
Psycholinguistics/Development of Speech Production

Introduction
Speech production is an important part of the way we communicate. We indicate intonation through stress and pitch while communicating our thoughts, ideas, requests or demands, and while maintaining grammatically correct sentences. However, we rarely consider how this ability develops. We know infants often begin producing one-word utterances, such as "mama," eventually move to two-word utterances, such as "gimme toy" and finally sound like an adult. However, the process itself involves development not only of the vocal sounds (phonology), but also semantics (meaning of words), morphology and syntax (rules and structure). How do children learn to this complex ability? Considering that an infant goes from an inability to speak to two-word utterances within 2 years, the accelerated development pattern is incredible and deserves some attention. When we ponder children's speech production development more closely, we begin to ask more questions. How does a child who says "tree" for "three" eventually learn to correct him/herself? How does a child know "nana" (banana) is the yellow,boat-shaped fruit he/she enjoys eating? Why does a child call all four-legged animals "horsie"? Why does this child say "I goed to the kitchen"? What causes a child to learn words such as "doggie" before "hand"? This chapter will address these questions and focus on the four areas of speech development mentioned: phonology, semantics, and morphology and syntax.

Prelinguistic Speech Development
Throughout infancy, vocalizations develop from automatic, reflexive vocalizations with no linguistic meaning to articulated words with meaning and intonation. In this section, we will examine the various stages an infant goes through while developing speech. In general, researchers seem to agree that as infants develop they increase their speech-like vocalizations and decrease their non-speech vocalizations (Nathani, Ertmer, & Stark). Many researchers (Oller, ; Stark, as cited in Nathani, Ertmer, & Stark, 2006). Many researchers (Oller; Stark, as cited in Nathani, Ertmer, & Stark,) have documented this development and suggest growth through the following five stages: reflexive vocalizations, cooing and laughing, vocal play (expansion stage), canonical babbling and finally, the integration stage.

Stage 1: Reflexive Vocalization
As newborns, infants make noises in responses to their environment and current needs. These reflexive vocalizations may consist of crying or vegetative sounds such as grunting, burping, sneezing, and coughing (Oller). Although it is often thought that infants of this age do not show evidence of linguistic abilities, a recent study has found that newborns’ cries follow the melody of their surrounding language input (Mampe, Friederici, Christophe, & Wermke). They discovered that the French newborns’ pattern was a rising contour, where the melody of the cry rose slowly and then quickly decreased. In comparison, the German newborns’ cry pattern rose quickly and slowly decreased. These patterns matched the intonation patterns that are found in each of the respective spoken languages. Their finding suggest that perhaps infants vocalizations are not exclusively reflexive and may contain patterns of their native language.

Stage 2: Gooing, Cooing and Laughing
Between 2 and 4 months, infants begin to produce “cooing” and “gooing” to demonstrate their comfort states. These sounds may often take the form of vowel-like sounds such as “aah” or “oooh.” This stage is often associated with a happy infant as laughing and giggling begin and crying is reduced. Infants will also engage in more face-to-face interactions with their caregivers, smiling and attempting to make eye contact (Oller).

Stage 3: Vocal Play
From 4 to 6 months, and infants will attempt to vary the sounds they can produce using their developing vocal apparatus. They show a desire to explore and develop new sounds which may include yells, squeals, growls and whispers(Oller). Face-to-face interactions are still important at this stage as it promotes development of conversation abililities. Beebe, Alson, Jaffe et al. found that even at this young age, infants’ vocal expression show a “dialogic structure” - meaning that, during interactions with caregivers, infants were able to take turns vocalizing.

Stage 4: Canonical babbling
After 6 months, infants begin to make and combine sounds that are found in their native language, sometimes known as “well-formed syllables,” which are often replicated in their first words(Oller). During this stage, infants combine consonants and vowels and replicate them over and over - they are thus called reduplicated babble. For example, an infant may produce ‘ga-ga’ over and over. Eventually, infants will begin to string together multiple varied syllables, such as ‘gabamaga’, called variegated babbles. Other times, infants will move right into the variegated babbles stage without evidence of the reduplicated babbles (Oller). Early in this stage, infants do not produce these sounds for communicative purposes. As they move closer to pronouncing their first words, they may begin to use use sounds for rudimentary communicative purposes(Oller).

Stage 5: Integration

In the final stage of prelinguistic speech, 10 month-old infants use intonation and stress patterns in their babbling syllables, imitating adult-like speech. This stage is sometimes known as conversational babble or gibberish because infants may also use gestures and eye movements which resemble conversations(Oller). Interestingly, they also seem to have acoustic differences in their vocalizations depending on the purpose of their communication. Papaeliou and Trevarthen found that when they were communicating for social purposes they used a higher pitch and were more expressive in their vocalizations and gestures than when exploring and investigating their surroundings. The transition from gibberish to real words is not obvious(Oller) as this stage often overlaps with the acquisition of an infant’s first words. These words begin when an infant understands that the sounds produced are associated with an object .During this stage, infants develop vocal motor schemes, the consistent production of certain consonants in a certain period of time. Keren-Portnoy and Marjorano’s study showed that these vocal motor schemes play a significant part in the development of first words as children who children who mastered them earlier, produced words earlier. These consistent consonants were used in babble and vocal motor schemes, and would also be present in a child’s first words. Evidence that a child may understand the connection between context and sounds is shown when they make consistent sound patterns in certain contexts (Oller). For example, a child may begin to call his favorite toy “mub.” These phonetically-consistent sound patterns, known as protowords or quasi-words, do not always reflect real words, but they are an important step towards achieving adult-like speech(Otomo ; Oller). Infants may also use their proto-words to represent an entire sentence (Vetter). For example, the child may say “mub” but may be expressing “I want my toy”, “Give me back my toy” “Where is my toy?”, etc.

Phonological Development
When a child explicitly pronounces their first word they have understood the association between sounds and their meaning Yet, their pronunciation may be poor, they produce phonetic errors, and have yet to produce all the sound combinations in their language. Researchers have come up with many theories about the patterns and rules children and infants use while developing their language. In this section, we will examine some frequent error patterns and basic rules children use to articulate words. We will also look how phonological development can be enhanced.

Patterns of Speech
Depending on their personalities and individual development, infants develop their speech production slightly differently. Some children,productive learners, attempt any word regardless of proper pronunciation (Rabagaliati, Marcus, & Pylkkänen). Conservative learners (Rabagaliati, Marcus, & Pylkkänen), are hesitant until they are confident in their pronunciation. Other differences include preference to use nouns and name things versus use of language in a more social context. (Bates et al., as cited in Smits-Bandstra). Although infants vary in their first words and the development of their phonology, by examining the sound patterns found in their early language, researchers have extracted many similar patterns. For example, McIntosh and Dodd examined these patterns in 2 year olds and found that they were able to produce multiple phonemes but were lacking. They were also able to produce complex syllables. Vowel errors also occurred, although consonant errors are much more prevalent. The development of phonemes continues throughout childhood and many are not completely developed until age 8 (Vetter).

 * Emergence and Mastery of Speech Sounds

Phonological Errors
As a child pronounces new words and phonemes, he/she may produce various errors that follow patterns. However, all errors will reduce with age (McIntosh & Dodd). Although each child does not necessarily produce the same errors, errors can typically be categorized into various groups. For example, they are multiple kinds of consonant errors. A cluster reduction involves reducing a multiple consonants in a row (ie: skate). Most often, a child will skip the first consonant (thus skate becomes kate), or they may leave out the second stop consonant (consonant deletion - Wyllie-Smith, McLeod, & Ball) (thus skate becomes sate). This type of error has been found by McIntosh and Dodd. For words that have multiple syllables, a child may skip the unstressed syllable at the beginning of the sentence (ie: potato becomes tato) or in the middle of a sentence (ie: telephone becomes tephone) (Ganger & Brent). This omission may simply be due to the properties of unstressed syllables as they are more difficult to perceive and thus a child may simply lack attention to it. As a child grows more aware of the unstressed syllable, he/she may chose to insert a dummy syllable in place of the unstressed syllable to attempt to lengthen the utterance (Aoyama, Peters, & Winchester ). For example, a child may say [ə hat] (‘ə hot’) (Clark, as cited in Smits-Bandstra). Replacement shows that the child understands that there should be some sound there, but the child has inserted the wrong one. Another common phonological error pattern is assimilation. A child may pronounce a word such that a phoneme within that word sounds more like another phoneme near it (McIntosh & Dodd). For example, a child may say “”gug” instead of “bug”. This kind of error may also be seen for with vowels and is common in 2 year-olds, but decreases with age (Newton).

Definition of Error Patterns

 * Definition of error pattern.JPG

Factors affecting development of phonology
As adequate phonology is an important aspect in effective communication, researchers are interested in factors that can enhance it. In a study done by Goldstein and Schwade, it was found that interactions with caregivers provided opportunities for8-10 month old infants to increase their babbling of language sounds (consonant-vowel syllables and vowels). This study also found that infants were not simply imitations their caregivers vocalizations as they were producing various phonological patterns and had longer vocalizations! Thus, it would seem that social feedback from caregivers advances infants phonological development. On the other hand, factors such as hearing impairment, can negatively affect phonological development (Nicolaidis ). A Greek population with hearing impairments was compared to a control group and it was found that they have a different pattern of pronunciation of phonemes. Their pattern displayed substitutions (ie:[x] for target /k/), distortions (ie: place of articulation)and epenthesis/cluster production (ie:[ʃtʃ] or [jθ] for /s/) of words.

Semantic Development
When children purposefully use words they are trying to express a desire, refusal, a label or for social communication (Ninio & Snow ). As a child begins to understand that each word has a specific purpose, they will inevitably need to learn meaning of multiple words. Their vocabulary will rapidly expand as they experience various social contexts, sing songs, practice routines and through direct instruction at school (Smits-Bandstra, 2006). In this section, we will examine children’s first words, their vocabulary spurt, and what their semantic errors are like.

First Words
Many studies have analyzed the types of words found in early speech. Overall, children’s first words are usually shorter in syllabic length, easier to pronounce, and occur frequently in everyday speech (Storkel, 2004 ). Whether early vocabularies have a noun-bias or not tends to divide researchers. Some researchers argue that the noun bias, or children’s tendency to produce names for objects, people and animals, is sufficient evidence of this bias (Gllette et al.). However, this bias may not be entirely accurate. Recently, Tardif studied first words cross-culturally between English, Cantonese and Mandarin 8-16 month old infants and found interesting differences. Although all children used terms for people, there was much variation between languages for animals and objects. This suggests that there may be some language differences in which types of words children acquire first.

Vocabulary Spurt
Around the age of 18 months, many infants will undergo a vocabulary spurt, or vocabulary explosion, where they learn new words at an increasingly rapid rate (Smits-Bandstra) ; Mitchell & McMurray,2009. Before onset of this spurt, the first 50 words a child learned as usually acquired at a gradual rate (Plunkett, as cited in Smits-Bandstra) .Afterward the spurt, some studies have found upwards of 20 words learned per week( Mitchell and McMurray). There has been a lot of speculation about the process underlying the vocabulary spurt and there are three main theories. First, it has been suggested that the vocabulary spurt results from the naming insight (Reznick and Gldfield). The naming insight is a process where children begin to understand that referents can be labeled, either out of context or in place of the object. Second, this period seems to coincide with Piaget’s sensorimotor stage in which children are expanding their understanding of categorizing concepts and objects. Thus, children would necessarily need to expand their vocabulary to label categories (Gopnik). Finally, it has been suggested that leveraged learning may facilitate the vocabulary explosion (Mitchell & McMurray). Learning any word begins slowly - one word is learned, which acts as a ‘leverage’ to learn the next word, then those two words can each facilitates learning a new word, and so on. Learning therefore becomes easier. It is possible that not all children experience a vocabulary spurt, however. Some researchers have tested to determine whether there truly is an accelerated learning process. Interestingly, Ganger and Brent used a mathematical model and found that only a minority of the infants studied fit the criteria of a growth spurt. Thus the growth spurt may not be as common as once believed.

Semantic Errors
Even after a child has developed a large vocabulary; errors are made in selecting words to convey the desired meaning. One type of improper word selection is when children invent a word (called lexical innovation). This is usually because they have not yet learned a word associated with the meaning they are trying to express, or they simply cannot retrieve it properly. Although made-up words are not real words, it is fairly easy to figure out what a child means, and sometimes easier to remember than the traditional words (Clarke, as cited in Swan). For example, a child may say “pourer” for “cup” (Clarke, as cited in Swan) .These lexical innovations show that the child is able to understand derivational morphology and use it creatively and productively (Swan).

Sometimes children may use a word in an inappropriate context either extending or restricting use of the word. For example, a child says “doggie” while pointing to any four-legged animal - this is known as overextension and is most common in 1-2 year olds (McGregor, et al. Bloomquist; Bowerman; Jerger & Damian). Other times, children may use a word only in one specific context, this is called underextension (McGregor, et al. Bloomquist; Bowerman; Jerger & Damian). For example, they may only say “baba” for their bottle and not another infant’s bottle. Semantic errors manifest themselves in naming tasks and provide an opportunity to examine how children might organize semantic representations. In McGregor et al.’s naming pictures task for 3-5 year olds, errors were most often related to functional or physical properties (ie: saying chair for saddle). Why are such errors produced? McGregor et al. proposed three reasons for these errors:
 * 1. they are attempting to fill the lexical gap because they do not have a current label for that particular item.
 * 2. They don’t know the target word sufficiently and thus have difficulty determining which label the target should receive.
 * 3. They temporarily cannot retrieve the word.

Grammatical and Morphological Development
As children develop larger lexicons, they begin to combine words into sentences that become progressively long and complex, demonstrating their syntactic development. Longer utterances provide evidence that children are reaching an important milestone in beginning the development of morphosyntax (Aoyama et al.). Brown developed a method that would measure syntactic growth called mean length of the utterance (MLU). It is determined by recording or listening to a 30-minute sample of a child’s speech, counting the number of meaningful morphemes (semantic roles – see chart below) and dividing it by the number of utterances. Meaningful morphemes can be function words (ie: “of” ), content words (ie: “cat”) or grammatical inflections (ie: -s). Utterances will include each separate thought conveyed thus repetitions, filler words, recitations, or titles and compound words would be counted as one utterance. Brown ended up with 5 different stages to describe syntactical development: Stage I (MLU 1.0-2.0), Stage II (MLU 2.0-2.5), Stage III (MLU 2.5-3.0), Stage IV (MLU 3.0-3.5) Stage V (MLU 3.5-4.0).

What is this child's MLU?

 * {| class="wikitable sortable"

! Sample of speech: !! # of utterances: !! # of morphemes: !! MLU:
 * “Mommy, want cookie. No dinner! Drink juice.” ||
 * “Mommy, want cookie. No dinner! Drink juice.” ||

3

1-	Mommy want cookie. 2-	No dinner 3-	Drink juice 1-	Mommy 2-	want 3-	Cookie 4-	no 5-	dinner 6-	drink 7-	juice
 * 7
 * 7/3= 2.33

 * }

Two-word utterances
Around the age of 18 months, children’s utterances are usually in two-word forms such as “want that, mommy do, doll fall, etc.” (Vetter . In English, these forms are dominated by content words such as nouns, verbs and adjectives and are restricted to concepts that the child is learning based on their sensorimotor stage as suggested by Piaget (Brown) . Thus, they will express relations between objects, actions and people. This type of speech is called telegraphic speech. During this development stage, children are combining words to convey various meanings. They are also displaying evidence of grammatical structure with consistent word orders and inflections.(Behrens & Gut; Vetter) . Once the child moves from Stage 1, simple sentences begin to form and the child begins to use inflections and function words (Aoyama et al.) . At this time, the child develops grammatical morphemes (Brown) which are classified into 14 different categories organized by acquisition (See chart below).These morphemes modify the meaning of the utterance such as tense, plurality, possession, etc. There are two theories for why this particular order takes place. The frequency hypothesis suggests that children acquire the morphemes they hear most frequently in adult speech. Brown argued against this theory by analyzing adult speech where articles were the most common word form, yet children did not acquire articles quickly. He suggested that linguistic complexity may account for the order of acquisition where the less complex morphemes were acquired first. Complexity of the morphemes was determined based on semantics (meaning) and/or syntax (rules) of the morpheme. In other words, a morpheme with only one meaning such as plurality (-s) is easier to learn than the copula “is” (which encodes number and time the action occurs). Brown also suggested that for a child to have successfully mastered a grammatical morpheme, they must use it properly 90% of the time.
 * {| class="wikitable"

! Order !! Morpheme !! Example
 * 1 || Present progressive (-ing) || running
 * 2-3|| in, on || sit on chair
 * 4 || Plural (-s) || cookies
 * 5 || Past irregular || ran,drew
 * 6 || Possessive (-'s) || Daddy's toy
 * 7|| Uncontractible copula (is) || That is my cookie.
 * 8|| Articles (the, a)|| The cat ; "a" dog
 * 9|| Past regular (-ed)|| jumped
 * 10 || Third person regular || he cooks
 * 11 || Third person irregular || he has my toy
 * 12 || Uncontractible auxiliary (was, do)|| do you have one?
 * 13 || Contractible copula (-'m, are || You are here
 * 14 || Contractible auxiliary (-s)|| He's coming!
 * }
 * 9|| Past regular (-ed)|| jumped
 * 10 || Third person regular || he cooks
 * 11 || Third person irregular || he has my toy
 * 12 || Uncontractible auxiliary (was, do)|| do you have one?
 * 13 || Contractible copula (-'m, are || You are here
 * 14 || Contractible auxiliary (-s)|| He's coming!
 * }
 * 12 || Uncontractible auxiliary (was, do)|| do you have one?
 * 13 || Contractible copula (-'m, are || You are here
 * 14 || Contractible auxiliary (-s)|| He's coming!
 * }
 * 14 || Contractible auxiliary (-s)|| He's coming!
 * }

Syntactic Errors
As children begin to develop more complex sentences, they must learn to use to grammar rules appropriately too. This is difficult in English because of the prevalence of irregular rules. For example, a child may say, “I buyed my toy from the store.” This is known as an overregularization error. The child has understood that there are syntactic patterns and rules to follow, but overuses them, failing to realize that there are exceptions to rules. In the previous example, the child applied a regular part tense rule (-ed) to an irregular verb. Why do these errors occur? It may be that the child does not have a complete understanding of the word meaning and thus incorrectly selects it (Pinker, et al.). Brooks et al. suggested that these errors may be categorization errors. For example, intransitive or transitive verbs appear in different contexts and thus the child is required to learn that certain verbs appear only in certain contextes. (Brooks). Interestingly, Hartshorne and Ullman found a gender difference for overregularization errors. Girls were more than three times more likely than boys to produce overregularizations. They concluded that girls were more likely to overgeneralize associatively, whereas boys overgeneralized only through rule-governed methods. In other words, girls, who remember regular forms, better than boys, quickly associated their rule forms to similar sounding words (ie: fold-folded, mold-molded, but they would say hold becomes holded). Boys, on the other hand, will use the regular rule when they have difficulty retrieving the irregular form (ie: past tense form - ed added to irregular form run becomes runed) (Hartshorne & Ullman).

Another common error committed by children is omission of words from an utterance. These errors are especially prevalent in their early speech production, which frequently lack function words (Gerken, Landau, & Remez). For example, a child may say “dog eat bone” forgetting function words “the” and “a”.This type of error has been frequently studied and researchers have proposed three main theories to account for omissions. First, it may be that children may focus on words that have referents (Brown). For example, a child may focus on “car” or “ball”, rather than “jump” or “happy.” The second theory suggests children simply recognize the content words which have greater stress and emphasis (Brown). The final theory, suggested by Gerken, involves an immature production system. In their study, children could perceive function words and classify them into various syntactic categories, yet still omitted them from their speech production.

Summary
In this chapter, the development of speech production was examined in the areas of prelinguistics, phonology, semantics, syntax and morphology. As an infant develops, their vocalizations will undergo a transition from reflexive vocalizations to speech-like sounds and finally words. However, their linguistic development does not end there. Infants underdeveloped speech apparatus restricts them from producing all phonemes properly and thus they produce errors such as consonant cluster reduction, omissions of syllables and assimilation. At 18 months, many children seem to undergo a vocabulary spurt. Even with a larger vocabulary, children may also overextend (calling a horse a doggie) or underextend (not calling the neighbors’ dog, doggie) their words. When a child begins to combine words, they are developing syntax and morphology. Syntactic development is measured using mean length of the utterance (MLU) which is categorized into 5 stages (Brown). After stage II, children begin to use grammatical morphemes (ie: -ed, -s, is) which encode tense, plurality, etc. As with other areas of linguistic development, children also produce errors such as overregularization (ie: “I buyed it”) or omissions (ie: “dog eat bone”). In spite of children’s early errors patterns, children will eventually develop adult-like speech with few errors. Understanding and studying child language development is an important area of research as it may give us insight into underlying processes of language as well as how we might be able to facilitate it or treat individuals with language difficulties.

Learning Exercise
1. Watch the video clips of a young boy CC provided below.

Video 1 Video 2 Video 3 Video 4 Video 5

 * (a)For each clip, indicate at which stage of prelinguistic development (Stage 1 -5) CC is and how you arrived at this conclusion.
 * (b) An infant says "lalu" whenever she is given her bottle or wants her bottle. What is this called and during which stage of prelinguistic development does it occur?

2. The following is a transcription of conversations between a mother (*MOT) and a child (*CHI) from Brown's (1970) corpus. You can ignore the # symbol as it represents unintelligible utterances. Use the charts found in the section on "Grammatical and Morphological Development" to help answer this question.
 * (a) Find 4 examples of different semantic roles used by the child and identify the type of semantic role.
 * example: 'Mommy telephone' - agent + object
 * (b) Find examples that would suggest that the child is not yet producing the following grammatical morphemes:
 * Possessive morphemes ('s)
 * in, on
 * Present progressive (-ing)
 * (*hint: Compare the mother's repetition to the child's utterance or the child's repetition to the mother's utterance.Do they match?)
 * (c) Based on this data sample, does this child appear to have mastered the grammatical morpheme category of articles (the, a)? Why or not why?

3. Below are examples of children's speech. These children are displaying some characteristics of terms of we have covered in this chapter. The specfic terms found in each video are provided. Find examples of these terms within their associated video. Indicate which type of development (phonological, semantic, syntactic) is associated with each of these term.
 * MOT:	let me see.
 * MOT:	over here +...
 * MOT:	you have tapioca on your finger.
 * CHI:	tapioca finger.
 * MOT:	here you go.
 * CHI:	more cookie.
 * MOT:	you have another cookie right on the table.
 * CHI:	Mommy fix.
 * MOT:	want me to fix it ?
 * MOT:	alright.
 * MOT:	bring it here.
 * CHI:	bring it.
 * CHI:	bring it.
 * MOT:	bring it here.
 * CHI:	that Kathy.
 * CHI:	that Kathy.
 * MOT:	yes # that's Kathy.
 * CHI:	op(en).
 * CHI:	op(en).
 * MOT:	no # we'll leave the door shut.
 * CHI:	why ?
 * MOT:	because I want it shut.
 * CHI:	Mommy.
 * MOT:	I'll fix it once more and that's all.
 * CHI:	Mommy telephone.
 * MOT:	well # go and get your telephone.
 * MOT:	yes # he gave you your telephone.
 * MOT:	who are you calling # Eve ?
 * CHI:	my telephone.
 * CHI:	Kathy cry.
 * MOT:	yes # Kathy was crying.
 * MOT:	Kathy was unhappy.
 * MOT:	what is that ?
 * CHI:	letter.
 * MOT:	Eve's letter.
 * CHI:	Mommy letter.
 * MOT:	there's Mommy's letter.
 * CHI:	Eve letter.
 * MOT:	what is that ?
 * CHI:	a fly.
 * MOT:	yes # a fly.
 * MOT:	why don't you go in the room and kill a fly ?
 * MOT:	you go in the room and kill a fly.
 * CHI:	fly.
 * MOT:	yes # you get a fly.
 * MOT:	oh # what's that ?
 * CHI:	hat.
 * MOT:	I'm going to go in the basement # Eve.

4.
 * (a) 24 month old Alex has learned that objects have names. He labels most things or asks "what that?". He is also speaking consistently in telegraphic speech (two-word utterances). It would seem that Alex has undergone which developmental milestone? Explain the theories on how this developmental milestone is reached.
 * (b) Some of Alex's first words were “birdy, milk, mama, sissy, ball, doggie”. This shows evidence of which bias?
 * (c) Alex uses "birdy" when referring to the family pet parakeet, but also when he sees butterflies, airplanes, or kites. What is his error?

5.The following are examples of children’s speech errors. Name the error and the type of development it is associated with (phonological, syntactic, morphological, or semantic). Can you explain why such an error occurs?
 * (a) juice holder (cup)
 * (b) apple (while eating an orange)
 * (c) he “goed” away
 * (d) sool (for school)
 * (e) limiting “kitty” to child’s pet cat
 * (f) ephant (elephant)
 * (g) kuk (buck)
 * (h) “want shoe” (I want the shoe) or “plate green” (the plate is green)

Learning Exercise Answers
Click here!


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User talk:Phanitnan

 Hello Phanitnan, 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.

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 * 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://twiter.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! --Ottava Rima (talk) 02:06, 16 January 2011 (UTC)

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User:Techexplo/Case Study 1

=Tech Explo Case Study 1= Due January 22 by 23:59

Instructions

 * Each group select one of the following cases. Use technology in m-Commerce to solve the problem
 * Your group has 30 minutes to collect data from the web
 * Present in class
 * Each group post results of your case study below

Case 1
Master students need to keep a journal, making 4 entries a day. They have to watch a vocabulary video and complete the homework

Case 2
An insurance company needs to quickly have a field agent document an accident but reaching the scene, recording a conversation with the customer.

 * 1) Numbered list item

Group 1
Supasin

Methinee

''The first recommendation, buying items on phone, how does this relate to the case? Download application is fine but please what application in this case that help with English? I like messaging, but again how does this help with learning English? Please relate your answers to how it solves the problem. Grade 4.6/5''--Techexplo 01:21, 23 January 2011 (UTC)

 * 1) Purchase items on your mobile phone. Most modern mobile phones have the capability to connect to the Internet via a GPRS or 3G connection. These connections can be used to access websites and online shopping is one type of M-commerce.
 * 2) Download application that useful with your study. In this case we use LiveJournal for example.
 * 3) Use the messaging services on your phone. Communicating with business colleagues using a mobile phone is technically classified as M-commerce. Use mobile technology to advertise your products and services directly to the consumer.
 * 4) For vocabulary video, iphone already has youtube application and you can search in anytime, everywhere that you want.
 * 5) M-commerce can make our life is easy because you can take your mobile everywhere that you want. When you want to send some message you can do it by accessing  to e-mail.

Group 2
Anand

Charyanaan

''You give a good survey of cell phone technology, and I'd like to see more focus on want you to select what features, options, or particular technology that would solve the problem in this case. For example, the muli-card phones are interesting but how does this help a student study English. The information sounds too much like it is copied from the source. I want to know what you think. Grade:3.5/5''--Techexplo 01:15, 23 January 2011 (UTC)

Mobiles or the Cellular Phones are widely being used almost in very country around the globe. The use defers user to user. It is being used to in trading shares, watching breaking news and entrainment, connecting with friends and family and education and so on.

There are different types of cell phones having different features:

Multi-card hybrid phones

A hybrid mobile phone can take more than one SIM card, even of different types. The SIM and RUIM cards can be mixed together, and some phones also support three or four SIMsThey are popular in India and Indonesia, attributed to lower on-net call rates.

3D mobile phone

Spice Mobile has launched Spice View M-67 3D Dual-SIM Mobile Phone. India's first 3D mobile phone allowing users to view 3D pictures and videos along with phone's user interface without 3D glasses. The phone is equipped with 2 megapixel camera, but only takes 2D.

3G

International Mobile Telecommunications-2000 (IMT — 2000), better known as 3G or 3rd Generation, is a generation of standards for mobile phones and mobile telecommunications services fulfilling specifications by the International Telecommunication Union. Application services include wide-area wireless voice telephone, mobile Internet access, video calls and mobile TV, all in a mobile environment. Compared to the older 2G and 2.5G standards, a 3G system must allow simultaneous use of speech and data services, and provide peak data rates of at least 200 kbit/s according to the IMT-2000 specification. Recent 3G releases, often denoted 3.5G and 3.75G, also provide mobile broadband access of several Mbit/s to laptop computers and smartphones. The following standards are typically branded 3G:

4G

The recently released 4th generation, also known as Beyond 3G, aims to provide broadband wireless access with nominal data rates of 100 Mbit/s to fast moving devices, and 1 Gbit/s to stationary devices defined by the ITU-R 4G systems may be based on the 3GPP LTE (Long Term Evolution) cellular standard, offering peak bit rates of 326.4 Mbit/s. It may perhaps also be based on WiMax or Flash-OFDM wireless metropolitan area network technologies that promise broadband wireless access with speeds that reaches 233 Mbit/s for mobile users. The radio interface in these systems is based on all-IP packet switching, MIMO diversity, multi-carrier modulation schemes, Dynamic Channel Assignment (DCA) and channel-dependent scheduling. A 4G system should be a complete replacement for current network infrastructure and is expected to be able to provide a comprehensive and secure IP solution where voice, data, and streamed multimedia can be given to users on a "Anytime, Anywhere" basis, and at much higher data rates than previous generations. Sprint in the US has claimed its WiMax network to be "4G network" which most cellular telecoms standardization experts dispute repeatedly around the world. Sprint's 4G is seen as a marketing gimmick as WiMax itself is part of the 3G air interface. The officially accepted, ITU ratified standards-based 4G networks are not expected to be commercially launched until 2011.

Students get a lot of help from Mobile and its applications such as: ibook and YouTube in 4G. Applications on Mobile phone allow us to access them anywhere and anytime. It is Universal! Everything on the palm:

Mobile internet browser allows users to access webs including audio, video, and texts, including multi languages dictionary. Especially for students this instrument is great gift to enhance the knowledge. Since the mobiles are very tiny piece of machine with negligible weight and no codes need to access. It can be great time save indeed. Students travel at least couple of hours to attend the universities. And if they use public transportation, may not use time in preparing for class or studying or reading a book on the bus or subway. Thing around will divert the attention so easily, the objects are surrounded in such situation. But, Mobile phone comes over all such barriers. User can get 80 to 90 percent concentrate even on public transportations. For example, using YouTube, there are number of videos with clear quality of voice and explanation in details. Just using headphone user can feel in the different place means the atmosphere will not be barrier for user and can make good use of the time.

Share and connect easily:

While doing homework at the master level the student needs to do his best, best of group is better than best of each. So sharing information and doing it together makes work precious. And however the students don’t stay together. In like such situations mobile phone allows to share information and connect each other with very lower cost.

Give birth to the death moments:

Since the world of IT has been revolutionized human nature change to large extend, previous generation was able to concentrate for longer time than the present generation. The students can focus to lecturers for first 15 to 20 minutes and if it is last class of the day may for 5 minutes and topic are enormously important at this level. The mobile again help to record the whole lecture and can repeat it whenever needed. It will be even more useful when the language is less understood.

Your world with you!

The modern mobile phones like iphone 4G are like your mobile computers in terms of internet browser. Their limitless access to websites can gather information from the desired webs as if it can be done on computers.

Games:

There are same games applications which improve language skill, the language based games and other games like business games and Sudoku.

By Anand

Group 3
Piyanut

Usanee

Case 2
An insurance company needs to quickly have a field agent document an accident but reaching the scene, recording a conversation with the customer.

''You do a good job of explaining how m-Commerce will solve the insurance company, but you explain m-Commerce in a very general way. I would like to see details of how to apply the technology like there is in Group 4 but with your explaination of how this solves the company that you wrote here. Grade:4.4/5''--Techexplo 00:55, 23 January 2011 (UTC)

Use technology in M-Commerce to solve the problem
 * 1) Providing wider reach - access on your demand from anywhere anytime
 * 2)  In this case, An insurance company needs to quickly have a field agent document an accident but reaching the scene, recording a conversation with the customer.So the insurance company can send data about an accident to customer immediately via wifi on mobile-phone such as e-mail, Facebook,Twitter,sms, mms or else.
 * 3) Purely personal
 * 4)     It's security and privacy between company and customer.
 * 5) easy-to-use
 * 6)     It's easy and fast to provide and sending the information between company and customer.
 * 7) secure
 * 8)     It's very security bacause  it must use personal username and password when you need to access the data information.
 * 9) location and time independent
 * 10)     You can connect with other anytime anywhere that you want and have internet signal.
 * 11) Reducing transaction cost
 * 12)     It can reduce cost of transportation or transaction.
 * 13) Streamline business processes
 * 14)     As fast as the best when you do business on the processing. It makes the impression to your customer.
 * 15) Competitive pricing and efficiency
 * 16)   It's very cheap and fast to using mobile commerce.
 * 17) Reducing time to order
 * 18)    It's very fast and save time by transaction via mobile commerse without transportations.

Group 4
Nannapad

Apawan

................... Use Technology in M-commerce to Solve the Problem in following case;

Master students need to keep a journal, making 4 entries a day. They have to watch a vocabulary video and complete the homework

''These are excellent recommendations. I'd like to see you select a few, explain them well and make sure to explain how they solve the problem that was given. Look at Group 3 for a good example. Grade:4.5/5''--Techexplo 00:48, 23 January 2011 (UTC)
 * 1) Work on the Internet via Mobile-phone with connecting to the Internet.
 * 2) Searching Vocabulary Video from the Internet such as Google,  yahoo. Etc.
 * 3) Submit the Journal via SMS, MMS, E-mail, by Text or Voice etc.
 * 4) Update the data from the Mobile-phone with connecting to the Internet.
 * 5) Gathering information  that is happens in time-critical  and reports it by message  or capturing  photographs, and then we can transmit it without needing any bulky equipment, it is on hands.
 * 6) M-commerce will make them easy to complete the homework because it is mobility, convenience and Reach ability in anytime and anywhere.
 * 7) We can use the Microsoft Office Applications on Mobile phone that is available on the Mobile Phone today; this is an important tools to create a good documents and edit pages.
 * 8) Currently, Mobile-phone will collect a lots of data, messages and photos that we can put its on the sim card or a memory card. So, the student can collect the data as much as they need.

title: User:Techexplo/Case Study 1


text:
User:Vickyessar

Hi Vicky !!!

title: User:Vickyessar


text:
User:Egm6321.f10.team4.Yoon/Mtg22

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_10_08_17_07_22.djvu | Mtg 22:]] Fri, 8 Oct 10

= Non-exact L2-ODE-VC, integrating factor (cont'd) =

[[media: 2010_10_08_17_07_22.djvu | Page 22-1]]
where m,n $$\displaystyle \in \mathbb{R}$$to be determined.

Application
For application on [[media: 2010_09_30_15_51_32.djvu | Eq (1), p.19-1]]:

F09: solve L1-ODE-VC Eqn. (3) for y(x)(IFM)

= Mathematical Structure of first integral for a class of "exact" L2-ODE-VC =

First integral $$\displaystyle \phi$$

[[media: 2010_10_08_17_07_22.djvu | Page 22-2]]
Consider the L2-ODE-VC:

Answer:

See [[media: Egm6321.f09.mtg13.djvu |F09 p.13-1]] for the homework problem statement on derivation of Eqn.(3).

Application: Select $$\displaystyle \phi$$ satisfying Eqn.(3)

[[media: 2010_10_08_17_07_22.djvu | Page 22-3]]
= Exact Nn-ODE's (Non-linear nth order)=

Exactness Condition 1:

[[media: 2010_10_08_17_07_22.djvu | Page 22-4]]
Exactness Condition 2:

Case n=1: N1-ODE

[[media: 2010_10_08_17_07_22.djvu | Page 22-5]]
See [[media: egm6321.f09.mtg13.djvu | F09 Mtg 13]] for more details.

title: User:Egm6321.f10.team4.Yoon/Mtg22


text:
User:Egm6321.f10.team4.Yoon/Mtg24

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_10_14_14_02_27.djvu | Mtg 24:]] Thu, 14 Oct 10

= Are the Legendre and Bessel L2-ODEs-VC exact ? =

[[media: 2010_10_14_14_02_27.djvu | Page 24-1]]
F09: Legendre Equation L2-ODE-VC

Ref: K.2003 p.31; See [[media: 2010_09_02_13_55_50.djvu | Eqn(1)p.5-4]]

1)Verify exactness of (1) using 2 methods 1a) [[media: 2010_09_23_14_52_54.djvu | Eqn.(1) & Eqn.(2) p.15-3]] 1b) [[media: 2010_10_08_17_07_22.djvu | Eqn.(3) p.22-4]] 2) If Eqn.(1) is not exact, see whether if it can be made exact using IFM with $$\displaystyle h(x,y) = x^m y^n$$ ; see [[media: 2010_09_30_15_51_32.djvu | p.19-1]] & [[media: 2010_10_08_17_07_22.djvu | p22-1]]

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HW: Do the same for Bessel equation $$\displaystyle F=(1-x^2)y'' - 2xy' + (x^2- \nu^2)y = 0; \qquad \upsilon \in \mathbb{R}$$ Ref.: K. 2003p.58
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Next we look at vibration, which is an important engineering application.

[[media: 2010_10_14_14_02_27.djvu | Page 24-2]]
= Vibration: Chladni patterns =

The above figure provides a conceptual explanation of the experimental setup to display the Chaldni patterns.

Heat equations:

Note the plus sign in the diffusion ($$\displaystyle {\rm div \ grad}$$) term.

1-D case:

=Equation of motion for Euler-Bernoulli beam=

Note the minus sign for the internal force term $$EI \frac{\partial^4 u}{\partial x^4}$$.

Free vibration
$$\displaystyle f=0 $$

Note:

Finite element discretization of the heat equation:

End Note

title: User:Egm6321.f10.team4.Yoon/Mtg24


text:
User:Egm6321.f10.team4.Yoon/Mtg26

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_10_19_14_56_03.djvu | Mtg 26:]] Tue, 19 Oct 10

= Note on particle motion with air resistance =

[[media: 2010_09_16_14_59_27.djvu | Page 26-1]]
Note:

Particle motion with air resistance:

End Note

= Application: Euler L2-ODE-VC =

[[media: Egm6321.f09.mtg16.djvu | F09 Mtg 16]]: Higher-order derivative:

[[media: 2010_10_19_14_56_03.djvu | Page 26-2]]
Back to [[media: 2010_10_14_14_56_10.djvu | Eqn.(4)p.25-3]] (Euler L2-ODE-VC): Use [[media: 2010_10_14_14_56_10.djvu | Eqn.(1)&(2) p.25-4]] in [[media: 2010_10_14_14_56_10.djvu | Eqn.(4) p.25-3]] to obtain

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$$  \displaystyle y_{tt}-3y_t+2y=0 $$     (2)
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Stage2: Trial Solution $$\displaystyle e^{rt} $$

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$$  \displaystyle y = e^{rt} $$      (3)
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Use (3) in (2) $$\displaystyle \Rightarrow $$ characteristic equation for r(=root)

[[media: 2010_10_19_14_56_03.djvu | Page 26-3]]
Now we consider a special class of 3rd order ODEs.

= Application: Euler L3-ODE-VC =

Consider the following L3-ODE-VC:

Method 2:

title: User:Egm6321.f10.team4.Yoon/Mtg26


text:
User:Egm6321.f10.team4.Yoon/Mtg28

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_10_21_14_48_38.djvu | Mtg 28:]] Thu, 21 Oct 10

= Application: Nonhomogeneous L2-ODE-CC cont'd =

[[media: 2010_10_21_14_48_38.djvu | Page 28-1]]
Recall:

Undergraduate; solve [[media: 2010_10_21_14_05_59.djvu | Eqn(2) p.27-2]] 1) Find $$\displaystyle y_H$$ (Homogeneous Solutions) 2) Find $$\displaystyle y_p$$ (Particular Solutions) Use look-up table (F09 Mtgs [[media: Egm6321.f09.mtg20.djvu | 20]], [[media: Egm6321.f09.mtg22.djvu | 22]]) with a few Particular cases

HW 5.5: Special IFM to solve for general f(t) in [[media: 2010_10_21_14_05_59.djvu | Eqn(2) p.27-2]].

End Recall

{| style="width:100%" border="0" align="left" HW 5.5: cont'd
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2) Trial Solution for integrating factor

which is similar to trial solution for Euler L2-ODE-CC (a homogeneous Ln-ODE-CC), where $$\displaystyle \alpha $$ is unknown to be determined.

[[media: 2010_10_21_14_48_38.djvu | Page 28-2]]
Assume $$\displaystyle \bar a_2 = 0$$ to reduce the order of the resulting ODE so that

2.1) Find $$\displaystyle \bar a_1 $$ and $$\displaystyle \bar a_0 $$in terms of $$\displaystyle a_2, a_1, a_0$$.

Hint: Differentiate the r.h.s. of (1) and compare to integrand on l.h.s. of (1); see also [[media: 2010_10_21_14_48_38.djvu | Eqn(2) p.28-1]].

2.2) Find quadratic equation for $$\displaystyle \alpha $$

Hint: Use $$\displaystyle (2)_2$$ & $$\displaystyle (2)_3$$

2.3) Reduced-order equation: [[media: 2010_10_21_14_48_38.djvu | Eqn(1),(2)p.28-1]] lead to

which is an L1-ODE-CC easily solvable by the IFM.

[[media: 2010_10_21_14_48_38.djvu | Page 28-3]]
2.4) Use the IFM.

[[media: 2010_09_14_15_00_52.djvu | Eqn(1)p.10-3:]]

$$\displaystyle h(t) = \exp [ \int \underbrace{\frac{\bar a_0}{\bar a_1}}_{\displaystyle \beta} dt ] = e^{\beta t} $$

Find y (for general excitation f(t))

2.5) Show that $$\displaystyle \alpha \beta = \frac{a_0}{a_2}$$ and $$\displaystyle \alpha + \beta = \frac{a_1}{a_2}$$, thus $$\displaystyle (\alpha, \beta)$$ are roots of the quadratic equation:

$$\displaystyle (\lambda - \alpha)(\lambda - \beta) = \lambda^2 - \underbrace{(\alpha + \beta)}_{\displaystyle \frac{a_1}{a_2}}\lambda + \underbrace{\alpha \beta}_{\displaystyle\frac{a_0}{a_2}} = 0$$

which is the same as [[media: 2010_10_21_14_48_38.djvu | Eqn(3) p.28-2.]]

2.6) Deduce the expression for the particular Solution $$\displaystyle y_P $$ for a general excitation f(t).

2.7) Verify with table of particular solutions consider $$\displaystyle f(t)=t^2 $$

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Next the method of variation of parameters.

= Variation of parameters: Reduction-of-order method 2 =

[[media: 2010_10_21_14_48_38.djvu | Page 28-4]]
Variation of parameters: a reduction-of-order method History: (Battin 1999 Astrodynamics) Euler 1748, 1752: Mutual perturbations of Jupiter & Saturn Variation of orbital elements variation of constants Lagrange 1782: General variation of parameters perturbations comets on elliptic orbits Dramatic success: Adams(1819 - 1892), LeVerrier (1811-1877) Discovery of Neptune based on irregular motion of Uranus. Neptune: observed (after prediction) by Galle 23 Sep 1846

Important homogeneous L2-ODE-VC
Homogeneous L2-ODE-VC (e.g. Legendre equation, Bessel equation, ...) Given one homogeneous solution $$\displaystyle u_1(x)$$ Find 2nd homogeneous solution $$\displaystyle u_2(x)$$ such that

Assume full homogeneous solution of the form $$\displaystyle y(x) = U(x) u_1(x)$$, where $$\displaystyle U(x)$$ is the unknown function to be found.

title: User:Egm6321.f10.team4.Yoon/Mtg28


text:
User:Egm6321.f10.team4.Yoon/Mtg30

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_10_28_14_00_19.djvu | Mtg 30:]] Thu, 28 Oct 10

= Variation of parameters: Reduction-of-order method 2 (cont'd) =

[[media: 2010_10_28_14_00_19.djvu | Page 30-1]]
cf. K.p.6, Eq.(1.3) 2nd homogeneous solution.

Particular Solution

Comparison with the method in King 2003 p.8
Now we look at some applications.

[[media: 2010_10_28_14_00_19.djvu | Page 30-2]]
= Application: Legendre equation with n=1 =

Legendre equation with n=1

[[media: 2010_10_28_14_00_19.djvu | Page 30-3]]
= Application: King 2003 p.28 (Pb.1.1 ab) =

Consider

Trial Solution:

title: User:Egm6321.f10.team4.Yoon/Mtg30


text:
User:Egm6321.f10.team4.Yoon/Mtg32

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_11_02_14_58_42.djvu | Mtg 32:]] Thu, 16 Sep 10

= Summary =

[[media: 2010_11_02_14_58_42.djvu | Page 32-1]]

 * IFM solves all L1_ODE_VC
 * Not so for L2-ODE-VC

If 1 Homogeneous Solution => Solve using variation of parameters
 * L2-ODE-VC

= Polynomial series as trial solution: Frobenius method =

[[media: 2010_10_28_14_54_20.djvu | Eqn(2) p.31-1 ]]=> use more general trial solution with c and r constants to be determined

Use Wolfram Alpha to find (c,r) $$\displaystyle _{\Rightarrow} $$ c=-1, r = $$\displaystyle _{\pm} $$ i; See [[media: 2010_10_28_14_54_20.djvu | p.31-2]].

[[media: 2010_11_02_14_58_42.djvu | Page 32-2]]
[[media: 2010_10_28_14_00_19.djvu | Eqn(2) p.30-3 ]] => Wolfram Alpha => one Valid solution

same as [[media: 2010_10_28_14_00_19.djvu | Eqn(2) p.30-3 ]] Note: [[media: 2010_11_02_14_58_42.djvu | Eqn (1)p.32-1]] is similar to starting pt for method of Frobenius(see K.2003)

L2-ODE-VC Trial solution: to be determined => 1 homogeneous solution as a series with factor d0(1 integer constant)

e.g., Legendre polynomials (finite series) as first homogeneous solutions of Legendre equation.

1st homogeneous solutions = $$\displaystyle [P_0(x), P_1(x), P_2(x), ...] $$ 2nd homogeneous solutions = $$\displaystyle [Q_0(x), Q_1(x), Q_2(x), ...] $$ Given a 1st homogeneous solution $$\displaystyle P_i(x), i=0,1,2,...,$$ corresponding 2nd homogeneous solution $$\displaystyle Q_i(x) $$ is obtained by variation parameters.

= Legendre functions =

[[media: 2010_11_02_14_58_42.djvu | Page 32-3]]
Specifically:

[[media:2010_09_02_14_58_46.djvu | Eqn(2) P.6-1]] $$\displaystyle \equiv P_1(x) $$

[[media:2010_09_02_14_58_46.djvu | Eqn(3) P.6-1]] $$\displaystyle \equiv Q_1(x) $$

[[media: 2010_10_28_14_00_19.djvu | p.30-2 ]], Application: Given $$\displaystyle P_1(x), $$ find $$\displaystyle Q_1(x)$$ by variation of parameters

Now...

= Examples: Non-homogeneous L2-ODE-VC =

title: User:Egm6321.f10.team4.Yoon/Mtg32


text:
User:Egm6321.f10.team4.Yoon/Mtg34

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_11_04_14_51_31.djvu | Mtg 34:]] Thu, 4 Nov 10

= Heat conduction on a sphere (cont'd) =

[[media: 2010_11_04_14_51_31.djvu | Page 34-1]]
Cartesian coordinates in terms of spherical coordinates:

Infinitesimal length in spherical coordinates
Now general curvilinear coordinates.

Laplace operator in spherical coordinates
For spherical coordinates:

$$\displaystyle h_1h_2h_3 = r^2\cos \theta $$

Axisymetric problems: No dependence on $$\displaystyle \varphi$$ (longitude).

[[media: 2010_11_04_14_51_31.djvu | Page 34-3]]
cf. King 2003 p.45 (in which the mathematical physics convention is used).

Unit sphere

Problem: Find $$\displaystyle \Psi(r,\theta)$$ such that $$\displaystyle \Delta \Psi = 0$$ and

Seperation of variables
(See [[media: 2010_09_02_13_55_50.djvu | Mtg25]]: Vibration of beam, Euler equation with constant coefficient $$\displaystyle \equiv $$ Euler L4-ODE-CC)

Ansatz:

[[media: 2010_11_04_14_51_31.djvu | Page 34-4]]
Use[[media: 2010_11_04_14_51_31.djvu | Eqn(4) p.34-3 in Eqn(1) p.34-3]]

Simplify $$\displaystyle \frac{1}{r^2} $$ and divide by $$\displaystyle R\Theta $$

Separated equations
2 seperated equations:

The L2-ODE-VC (2) is an Euler (homogeneous) L2-ODE-VC, see [[media: 2010_10_14_14_56_10.djvu | Eqn(2) p.25-3]].

title: User:Egm6321.f10.team4.Yoon/Mtg34


text:
User:Egm6321.f10.team4.Yoon/Mtg36

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_11_09_15_00_14.djvu | Mtg 36:]] Tue, 9 Nov 10

[[media: 2010_11_09_15_00_14.djvu | Page 36-1]]
Note:

Axisymmetry problem [[media: 2010_11_04_14_51_31.djvu | p.34-2]].

Q: what if axis of symmetry is arbitrarily oriented (i.e. no longer the South-to-North axis)?

A: Change the coordinate system such the new S-to-N axis coincides with new axis of symmetry => Same equation

End Note

= Heat conduction on a sphere (cont'd) =

[[media: 2010_11_05_07_46_02.djvu | Eqn(0) p.35-2, Eqn(3) p.35-3 ]]

[[media: Egm6321.f09.mtg31.djvu | F09 p.31-2]]: solve quadratic equation (1) to obtain (2) and (3).

[[media: 2010_11_09_15_00_14.djvu | Page 36-2]]
[[media: 2010_11_05_07_46_02.djvu | Eqn(1) p.35-1 ]]

Superposition: Now 2nd seperated equation [[media: 2010_11_05_07_46_02.djvu | Eqn(3) p.34-4 ]](Legendre equation if k=n(n+1)) Family of 1st homogeneous Solutions = Legndre Polynomials

Some problems to solve...

[[media: 2010_11_09_15_00_14.djvu | Page 36-3]]
Back to heat equation on a sphere...

Family of 2nd homogeneous solutions $$\displaystyle \left\{ Q_0, Q_1, Q_2, ...\right\}$$(see [[media: 2010_11_02_14_58_42.djvu | p.32-2 ]])

For each n: with $$\displaystyle \mu := \sin \theta $$ from [[media: 2010_11_05_07_46_02.djvu | Eqn(1) p.35-2 ]].

where $$\displaystyle R_n(r)$$ is given in (1) p.36-2, and $$\displaystyle \Theta_n(\theta)$$ is given in (1) above.

title: User:Egm6321.f10.team4.Yoon/Mtg36