text:
User:Eml5526.s11.team5.srv/hw5

Home Work 5.2
Solving the differential euation using weak form with appropriate basis functions which satisfy constraint breaking solution

P.D.E

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

$$ \frac{\partial}{\partial x} \left[ (2+3x) \frac{\partial u}{\partial x} \right] + 5x = 0 \ \ \forall x \in \left] 0,1 \right[ $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle                                                                          (Eq. 5.2.1)
 * }

Boundary Conditions

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

$$u \ at \ {x=1} \ is \ 4 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle                                                                          (Eq. 5.2.2)
 * }

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

$$  \frac{\partial u}{\partial x} \  at \ x = 0 \ is \ -6 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle                                                                          (Eq. 5.2.3)
 * }

Find
 Find approximate solution using weak form and compare it to exact one

 * Consider basis function satisfying constraint basis solution $$ \boldsymbol{F_{I}}, \ where \ I = p, f , e $$

 * Compute $$ u_n^h (x = 0.5) \quad $$ for $$ n = 2, 4, 6..... \quad $$ until convergence of $$ u^h (0.5)\ to \ 0(10^{-6}) $$

 * error

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

$$ e_n(0.5) = u_n(0.5) - u_n^h(0.5) \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle                                                                       (Eq. 5.2.4)
 * }

 * Plot $$ e_n(0.5) \quad $$ vs. $$ n \quad $$

Solution
Strong form

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

$$ \frac{\partial}{\partial x} \left[ (2+3x) \frac{\partial u}{\partial x} \right] + 5x = 0 \ \ \forall x \in \left] 0,1 \right[ $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle                                                                       (Eq. 5.2.5)
 * }

Boundary conditions

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

$$u \ at \ {x=1} \ is \ 4 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle                                                                          (Eq. 5.2.6)
 * }

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

$$  \frac{\partial u}{\partial x} \  at \ x = 0 \ is \ -6 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle                                                                          (Eq. 5.2.7)
 * }

The exact solution of the displacement can be found as follows.

Differentiating eqn 5.2.5, we get,

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

$$ (2+3x)\frac{\partial u}{\partial x} + \frac{5x^2}{2} = c_1 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle                                                                          (Eq. 5.2.8)
 * }

Substituting the natural B.C in the above equation, we get,

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

$$ c_1 = -12 \quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.9)
 * }

Therefore eqn 5.2.8 becomes,

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

$$ (2+3x)\frac{\partial u}{\partial x} + \frac{5x^2}{2} = -12 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.10)
 * }

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

$$\frac{\partial u}{\partial x} = \frac{-12-\frac{5x^2}{2}}{(2+3x)} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.11)
 * }

Integrating the above equation, we get,

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$$u = \frac{1}{108}(-45x^2 + 60x - 472 log (2+3x) +60 ) + c_2 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.12)
 * }

Substituting the Essential B.C in the above equation, we get,

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

$$c_2 = 10.3394 \quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.13)
 * }

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

$$u = \frac{1}{108}(-45x^2 + 60x - 472 log (2+3x) +60 ) + 10.3394 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.14)
 * }

Weak Form

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$$  \sum_{i} c_i \left [ \sum_{j} \left \{ \tilde{K_{ij}} \tilde{d_j} \right \} - \tilde{F_i} \right ] = 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.15)
 * }

where,

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$$\tilde{K_{ij}} = \int_{0}^{1} b_i' \ a_2 \ b_j ' \ dx $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.16)
 * }

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

$$ \tilde{F_i} = b_i (\alpha )h + \int_{0}^{1} b_i f dx $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.17)
 * }

Using Polynomial Basis Functions:

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$$b_j = (x-1)^j \ where \ j = 0,1,2,... $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.18)
 * }

Let us try for n = 2;

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

$$ b_j = [ 1 \quad (x-1) \quad (x-1)^2]$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.19)
 * }

We know that,

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$$ u^h (x) = \sum_{0}^{2} d_j b_j(x) = d_0 + (x-1)d_1 + (x-1)^2 d_2 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.20)
 * }

Substituting the essential B.C;

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$$ d_0 = 4 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.21)
 * }

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

$$ b_j' = [ 0 \quad 1 \quad 2(x-1)] $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.22)
 * }

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

$$K_{01} = K_{10} = K_{02} = K_{20} = 0 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.23)
 * }

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

$$ K_{11} = \int_{0}^{1}(1)(2+3x)(1)dx = 3.5 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.24)
 * }

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

$$ K_{12} = K_{21} = \int_{0}^{1}(1)(2+3x)2(x-1)(1)dx = 2\int_{0}^{1}(3x^2-x-2)dx = -3 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.25)
 * }

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

$$K_{22} = \int_{0}^{1}2(x-1)(2+3x)2(x-1)dx = 4\int_{0}^{1}(2+3x)(x-1)dx = 3.6667 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.26)
 * }

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

$$ F_0 = b_0(\alpha)(12) + \int_{0}^{1}b_0 (5x)dx = (1)(12) + \int_{0}^{1} 5xdx = 14.5 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.27)
 * }

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

$$F_1 = b_1(\alpha)(12) + \int_{0}^{1}b_1 (5x)dx = (-1)(12) + \int_{0}^{1} 5x(x-1)dx = -12.833 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.28)
 * }

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

$$ F_2 = b_2(\alpha)(12) + \int_{0}^{1}b_2 (5x)dx = (12) + \int_{0}^{1} 5x(x-1)^2dx = 12.4166 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.29)
 * }

Hence we obtain 3 linearly independent equations, (i.e) one from the essential boundary condition and the other two from the weak form. These three equations can be written in matrix form as follows.

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$$ K.d = F \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle
 * }

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

$$ \begin{bmatrix} 1 & 0 & 0\\ 0& 3.5 & -3\\  0&  -3 & 3.6667  \end{bmatrix} \begin{Bmatrix} d_0\\ d_1 \\ d_2 \end{Bmatrix} = \begin{Bmatrix} 4\\ -12.833 \\ 12.4166 \end{Bmatrix}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.30)
 * }

Solving for $$ d_0 \quad $$, $$ d_1 \quad$$ and $$ d_2 \quad$$ , we get,

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

$$ d_0 = 4; \qquad d_1 = -2.55768; \qquad d_2 = 1.2937$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.31)
 * }

Therefore the displacement equation is given by ,

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

$$ u^h(x) = 4-2.55768(x-1) + 1.2937(x-1)^2 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.32)
 * }

At $$ x = 0.5 \quad $$ ,

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

$$ u^h(0.5) = 4-2.55768(0.5-1) + 1.2937(0.5-1)^2 = 5.602265 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.33)
 * }

The exact solution can be found from the eqn. 5.2.14. It is given by ,

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$$ u(0.5) = 5.59352829 \quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle
 * }

Hence the error we get for $$ n = 2 $$ is ,

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

$$u^h(0.5) - u(0.5) = 8.74117 \times 10^{-3} \quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.2.34)
 * }

The outputs are as follow:

$$ \int_\omega

\begin{bmatrix} cos(\phi)  \\ cos(x+\phi) \\ cos(2x+\phi) \end{bmatrix}

[2(0 - d_1cos(x+\phi) - 4d_2cos(2x+\phi))] dx := $$ 0   0.3818   -1.0137         0    0.2919    0.7126         0    0.1781    2.3464

$$ \int_\omega

\begin{bmatrix} cos(\phi)  \\ cos(x+\phi) \\ cos(2x+\phi) \end{bmatrix} -3 dx \quad := $$

2.1213   0.8099   -0.5376

P.D.E

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

$$ \frac{\partial}{\partial x} \left[ (2+3x) \frac{\partial u}{\partial x} \right] + 5x = 0 \ \ \forall x \in \left] 0,1 \right[ $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.1)
 * }

Boundary Conditions

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

$$u \ at \ {x=0} \ is \ 4 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.2)
 * }

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

$$  \frac{\partial u}{\partial x} \  at \ x = 1 \ is \ \frac{12}{5} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.3)
 * }

Finding the approximate solution using weak form and its comparison to exact solution

 * Consider basis function satisfying constraint basis solution $$ \boldsymbol{F_{I}}, \ where \ I = p, f , e $$

 * Compute $$ u_n^h (x = 0.5) \quad $$ for $$ n = 2, 4, 6..... \quad $$ until convergence of $$ u^h (0.5)\ to \ 0(10^{-6}) $$

 * error

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

$$ e_n(0.5) = u_n(0.5) - u_n^h(0.5) \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                       (Eq. 5.1.4)
 * }

 * Plot $$ e_n(0.5) \quad $$ vs. $$ n \quad $$

Solution
Strong form

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

$$ \frac{\partial}{\partial x} \left[ (2+3x) \frac{\partial u}{\partial x} \right] + 5x = 0 \ \ \forall x \in \left] 0,1 \right[ $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                       (Eq. 5.1.5)
 * }

Boundary conditions

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

$$u \ at \ {x=0} \ is \ 4 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.6)
 * }

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

$$  \frac{\partial u}{\partial x} \  at \ x = 1 \ is \ \frac{12}{5} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.7)
 * }

Finding of the Exact Solution
The exact solution of the displacement can be found as follows.

Integrating eqn 5.2.5, we get,

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

$$ (2+3x)\frac{\partial u}{\partial x} + \frac{5x^2}{2} = c_1 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.8)
 * }

Substituting the natural B.C in the above equation, we get,

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

$$ c_1 = 14.5 \quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.9)
 * }

Therefore eqn 5.1.8 becomes,

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

$$ (2+3x)\frac{\partial u}{\partial x} + \frac{5x^2}{2} = 14.5 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.10)
 * }

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

$$\frac{\partial u}{\partial x} = \frac{14.5-\frac{5x^2}{2}}{(2+3x)} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.11)
 * }

Integrating the above equation, we get,Reference : www.wolframalpha.com

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

$$u = \frac{1}{108}(-45x^2 + 60x + 482 log (2+3x) +60 ) + c_2 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.12)
 * }

Substituting the Essential B.C in the above equation, we get,

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

$$c_2 = 0.3509542497 \quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.13)
 * }

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

$$u = \frac{1}{108}(-45x^2 + 60x + 482 log (2+3x) +60 ) + 0.3509542497 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.14)
 * }

Finding of the Approximate Solution using Weak form
Weak Form

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

$$  \sum_{i} c_i \left [ \sum_{j} \left \{ \tilde{K_{ij}} \tilde{d_j} \right \} - \tilde{F_i} \right ] = 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.15)
 * }

where,

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

$$\tilde{K_{ij}} = \int_{0}^{1} b_i' \ a_2 \ b_j ' \ dx $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.16)
 * }

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

$$ \tilde{F_i} = b_i (\alpha )h + \int_{0}^{1} b_i f dx $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.17)
 * }

Finding of the Approximate Solution using Polynomial Basis Function satisfying Constraint Breaking Solution in Weak form

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

$$b_j = x^j \ where \ j = 0,1,2,... $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.18)
 * }

Let us try for n = 2;

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

$$ b_j = [ 1 \quad (x) \quad (x)^2]$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.19)
 * }

We know that,

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

$$ u^h (x) = \sum_{0}^{2} d_j b_j(x) = d_0 + (x)d_1 + (x)^2 d_2 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.20)
 * }

Substituting the essential B.C;

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

$$ d_0 = 4 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.21)
 * }

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

$$ b_j' = [ 0 \quad 1 \quad 2(x)] $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.22)
 * }

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

$$K_{01} = K_{10} = K_{02} = K_{20} = 0 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.23)
 * }

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

$$ K_{11} = \int_{0}^{1}(1)(2+3x)(1)dx = 3.5 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.24)
 * }

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

$$ K_{12} = K_{21} = \int_{0}^{1}(1)(2+3x)2(x)(1)dx = 2\int_{0}^{1}(3x^2-x-2)dx = 4 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.25)
 * }

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

$$K_{22} = \int_{0}^{1}2(x)(2+3x)2(x)dx = 4\int_{0}^{1}(2+3x)(x)dx = 5.6667 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.26)
 * }

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

$$ F_0 = b_0(\alpha)(12) + \int_{0}^{1}b_0 (5x)dx = (1)(12) + \int_{0}^{1} 5xdx = 14.5 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.27)
 * }

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

$$F_1 = b_1(\alpha)(12) + \int_{0}^{1}b_1 (5x)dx = (1)(12) + \int_{0}^{1} 5x(x)dx = 13.6667 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.28)
 * }

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

$$ F_2 = b_2(\alpha)(12) + \int_{0}^{1}b_2 (5x)dx = (12) + \int_{0}^{1} 5x(x)^2dx = 13.25 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.29)
 * }

Hence we obtain 3 linearly independent equations, (i.e) one from the essential boundary condition and the other two from the weak form. These three equations can be written in matrix form as follows.

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

$$ K.d = F \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle
 * }

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

$$ \begin{bmatrix} 1 & 0 & 0\\ 0& 3.5 & 4\\  0&  4 & 5.6667  \end{bmatrix} \begin{Bmatrix} d_0\\ d_1 \\ d_2 \end{Bmatrix} = \begin{Bmatrix} 4\\ 13.6667 \\ 13.25 \end{Bmatrix}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.30)
 * }

Solving for $$ d_0 \quad $$, $$ d_1 \quad$$ and $$ d_2 \quad$$ , we get,

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

$$ d_0 = 4; \qquad d_1 = 6.3769; \qquad d_2 = -2.1631$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.31)
 * }

Therefore the displacement equation is given by ,

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

$$ u^h(x) = 4+6.3769(x) -2.1631(x)^2 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.32)
 * }

At $$ x = 0.5 \quad $$ ,

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

$$ u^h(0.5) = 4+6.3769(0.5) -2.1631(0.5)^2 = 6.647675 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.33)
 * }

The exact solution can be found from the eqn. 5.1.14. It is given by ,

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

$$ u(0.5) = 6.671155 \quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle
 * }

Hence the error we get for $$ n = 2 $$ is ,

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

$$u^h(0.5) - u(0.5) = -0.023480 \quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle                                                                          (Eq. 5.1.34)
 * }

MATLAB code with output
The K matrix and F matrix which we get for the convergence upto 10^-6 is as follows:

K =

1        0         0         0         0         0         0         0         0         0    3.5000    4.0000    4.2500    4.4000    4.5000    4.5714    4.6250    4.6667         0    4.0000    5.6667    6.6000    7.2000    7.6190    7.9286    8.1667    8.3556         0    4.2500    6.6000    8.1000    9.1429    9.9107   10.5000   10.9667   11.3455         0    4.4000    7.2000    9.1429   10.5714   11.6667   12.5333   13.2364   13.8182         0    4.5000    7.6190    9.9107   11.6667   13.0556   14.1818   15.1136   15.8974         0    4.5714    7.9286   10.5000   12.5333   14.1818   15.5455   16.6923   17.6703         0    4.6250    8.1667   10.9667   13.2364   15.1136   16.6923   18.0385   19.2000         0    4.6667    8.3556   11.3455   13.8182   15.8974   17.6703   19.2000   20.5333

F =

[4  13.6667   13.2500   13.0000   12.8333   12.7143   12.6250   12.5556   12.5000]^T

d =

[4.0000   7.2498   -5.4295    4.9208   -5.0302    4.5192   -3.0193    1.2522   -0.2348]^T

The outputs are as follow:

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=2 = 6.647645e+000

Error at n=2 is 2.351072e-002

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=3 = 6.678544e+000

Error at n=3 is 7.388478e-003

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=4 = 6.671260e+000

Error at n=4 is 1.047460e-004

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=5 = 6.670884e+000

Error at n=5 is 2.712998e-004

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=6 = 6.671152e+000

Error at n=6 is 3.583888e-006

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=7 = 6.671166e+000

Error at n=7 is 1.009645e-005

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=8 = 6.671156e+000

Error at n=8 is 9.634049e-008

Finding of the Approximate Solution using Fourier Basis Function satisfying Constraint Breaking Solution in Weak form
For $$ n = 1 \quad $$ Referencewww.wolframalpha.com

Referencewww.wolframalpha.com

Referencewww.wolframalpha.com

Referencewww.wolframalpha.com

Referencewww.wolframalpha.com

MATLAB code with output
The K matrix and F matrix which we get for the convergence upto 10^-6 is as follows:

K =

1        0         0         0         0         0         0         0         0         0    1.1444   -1.3612    3.2564   -0.5408    3.7008    2.3208    1.2335    4.7249         0   -1.3612    2.3556   -4.1866    2.4001   -5.8462   -0.0674   -4.7299   -2.9460         0    3.2564   -4.1866    9.5121   -2.3503   11.6194    5.4649    5.7892   12.8497         0   -0.5408    2.4001   -2.3503    4.4879   -5.4725    5.3501   -8.9290    3.8999         0    3.7008   -5.8462   11.6194   -5.4725   16.8127    2.2054   14.2443   12.2128         0    2.3208   -0.0674    5.4649    5.3501    2.2054   14.6873   -9.6620   19.6948         0    1.2335   -4.7299    5.7892   -8.9290   14.2443   -9.6620   23.4828   -3.0985         0    4.7249   -2.9460   12.8497    3.8999   12.2128   19.6948   -3.0985   32.5172

F =

[4  -6.1075   11.6035  -18.9907   13.0886  -27.2503    3.4218  -23.8065   -8.5011]^T

d =

[4.0000   47.4091   48.5755   -3.4993  -30.6212   -4.1185    7.2053    0.9062   -0.4248]^T

The outputs are as follow:

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=1 = 6.619837e+000

Error at n=1 is 5.131875e-002

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=2 = 6.671392e+000

Error at n=2 is 2.367707e-004

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=3 = 6.671145e+000

Error at n=3 is 1.055939e-005

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=4 = 6.671156e+000

Error at n=4 is 3.459119e-007

Finding of the Approximate Solution using Exponential Basis Function satisfying Constraint Breaking Solution in Weak form
For $$ n = 2 \quad $$ Referencewww.wolframalpha.com

Referencewww.wolframalpha.com

Referencewww.wolframalpha.com

Referencewww.wolframalpha.com

Referencewww.wolframalpha.com

MATLAB code with output
The K matrix and F matrix which we get for the convergence upto 10^-6 is as follows:

K =

1.0e+008 *

1        0         0         0         0         0         0         0         0         0    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0001    0.0003         0    0.0000    0.0000    0.0000    0.0000    0.0001    0.0002    0.0006    0.0017         0    0.0000    0.0000    0.0000    0.0001    0.0003    0.0008    0.0022    0.0062         0    0.0000    0.0000    0.0001    0.0003    0.0008    0.0025    0.0072    0.0206         0    0.0000    0.0001    0.0003    0.0008    0.0026    0.0077    0.0225    0.0649         0    0.0000    0.0002    0.0008    0.0025    0.0077    0.0232    0.0682    0.1973         0    0.0001    0.0006    0.0022    0.0072    0.0225    0.0682    0.2014    0.5858         0    0.0003    0.0017    0.0062    0.0206    0.0649    0.1973    0.5858    1.7106

F =

1.0e+004 * [4  0.0023    0.0085    0.0249    0.0692    0.1885    0.5107    1.3817    3.7387]^T

d =

[4.0000  171.4273 -268.6688  254.1458 -153.5759   59.9076  -14.6354    2.0398   -0.1239]^T

The outputs are as follow:

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=2 = 6.328507e+000

Error at n=2 is 3.426491e-001

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=3 = 6.661733e+000

Error at n=3 is 9.422741e-003

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=4 = 6.681880e+000

Error at n=4 is 1.072431e-002

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=5 = 6.670738e+000

Error at n=5 is 4.179779e-004

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=6 = 6.669725e+000

Error at n=6 is 1.431086e-003

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=7 = 6.670941e+000

Error at n=7 is 2.143177e-004

@x=0.5 Exact U=6.671156e+000

U(weak form) for n=8 = 6.671171e+000

Error at n=8 is 1.566386e-005

>>

title: User:Eml5526.s11.team5.srv/hw5


text:
User:Eml5526.s11.team5.srv/hw4

Given

 * that is ,

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

$$F_{p} = \left \{x^{j}; \quad j= 0,1,2,... \right \} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 4.5.2)
 * }

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

$$F_{c} = \left \{cos (jx);\quad j= 0,1,2,... \right \} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 4.5.3)
 * }

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

$$F_{s} = \left \{1, sin (jx); \quad j= 1,2,... \right \} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 4.5.4)
 * }

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

$$F_{f} = \left \{F_{p} = \left \{cos(jx); \quad j= 0,1,2,...                \qquad sin (kx); \quad j = 1,2,... \right \}\right \} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 4.5.5)
 * }

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

$$F_{p} = \left \{e^{jx}; \quad j = 0,1,2,... \right \}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 4.5.6)
 * }

Solution
1.

Consider

................

................

................

Consider

................

................

................

Consider

................

................

................

Consider

................

................

................

Consider

................

................

................

2.

The plot showing the phase shift in the basis function according to the constraint breaking solution is given below.

(i) Polynomial basis function

(ii) Cosine basis function

(iii) Sine basis function

(iv) Fourier basis function

(v) Exponential basis function

The Matlab code for plotting the graphs shown above is given below.

3.

The Gram Matrix is defined as:

For this case,

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

$$ b_1(x)=1 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle                                                       (Eq. 4.5.15)
 * }

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

$$ b_2(x)= \ e^{x} \quad $$ $$
 * style="width:90%" |
 * style="width:90%" |
 *  $$ \displaystyle                                                       (Eq. 4.5.16)
 * }

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

$$  b_3(x)= \ e^{2x} \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle                                                       (Eq. 4.5.17)
 * }

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

Now we need to calculate each element in the $$ \Gamma $$ Matrix: $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle\!
 * }

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

Since $$ \int_\Omega b_i(x)b_j(x)\,dx = \int_\Omega b_j(x)b_i(x)\,dx$$,  $$\Gamma_{ij}$$ will be equivalent to $$\Gamma_{ji}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle                                                       (Eq. 4.5.18)
 * }

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

$$ \Gamma_{1,1} = \int_{-2}^{4} 1*1*dx = \left [ x\right ]_{0}^{1} = 1 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle\!
 * }

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

$$ \Gamma_{1,2} = \Gamma_{2,1} = \int_{-2}^{4} 1*e^{x}*dx = \left [ e^{x}\right ]_{-2}^{4} = 54.463 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle\!
 * }

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

$$ \Gamma_{1,3} = \Gamma_{3,1} = \int_{-2}^{4} 1*e^{2x}*dx = \left [ e^{2x}/2 \right ]_{-2}^{4} = 1490.4699 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle\!
 * }

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

$$ \Gamma_{2,2} = \int_{-2}^{4} e^{x}*e^{x}*dx = \left [ e^{2x}/2\right ]_{-2}^{4} = 1490.4699 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle\!
 * }

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

$$ \Gamma_{2,3} = \Gamma_{3,2} = \int_{-2}^{4} e^{x}*e^{2x}*dx = \left [ e^{3x}/3\right ]_{-2}^{4} = 54251.6 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle\!
 * }

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

$$ \Gamma_{3,3} = \int_{-2}^{4} e^{2x}*e^{2x}*dx = \left [ e^{4x}/4 \right ]_{-2}^{4} = 2221527.63 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle\!
 * }

For the value of the determinant, refer http://www.wolframalpha.com

The determinant of the Gram Matrix is not equal to zero.

Therefore,

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

title: User:Eml5526.s11.team5.srv/hw4


text:
User:Eml5526.s11.team5.srv/hw3

Problem
 Show that the equivalent stiffness of a  spring aligned in the x direction for the bar of thickness t with a centered square hole shown in the figure is :

 where E is the Young's Modulus and t is the width of the bar

Solution
To find the equivalent stiffness of the bar, we have to subdivide the bar into 4 parts and each part behaves like an individual spring with stiffness K1, K2, K3 and K4 respectively. It can be shown from the figure below.

The Equivalent spring - mass system can be shown from the figure below.

From the Hooke's Law,

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

$$\sigma = E \varepsilon$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 3.5.2)
 * }

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

$$\frac{F}{A} = E \frac{\delta }{L}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 3.5.3 )
 * }

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

$$\frac{F}{\delta} = E \frac{A}{L}$$ $$ From the definition of Stiffness(K),
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 3.5.4 )
 * }

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

$$K = \frac{F}{\delta}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle\!
 * }

Therefore $$ \displaystyle (Eq. 3.5.4 ) $$ becomes

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

$$K = E \frac{A}{L}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 3.5.5   )
 * }

Hence the stiffness of the springs K1, K2, K3 and K4 can be determined as,

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

$$K_{1} = \frac{EA_{1}}{L_{1}} = \frac{Eat}{\frac{l}{10}}$$                   (From figure 2) $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle\!
 * }

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

$$ K_{1}= \frac{10Eat}{l}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 3.5.6  )
 * }

Similarly

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

$$K_{4}= \frac{10Eat}{l}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq.3.5.7   )
 * }

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

$$K_{2} = \frac{EA_{2}}{L_{2}} = \frac{E. 2b. t}{\frac{4l}{5}}$$                  (From figure 2) $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle\!
 * }

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

$$K_{2}= \frac{5Ebt}{2l}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 3.5.8  )
 * }

Similarly

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

$$K_{3}= \frac{5Ebt}{2l}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 3.5.9  )
 * }

From Figure 2 and 3, The total stiffness of the bar (K) can be derived as follows,

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

$$\frac{1}{K} = \frac{1}{K_{1}} + \frac{1}{(K_{2} + K_{3})} + \frac{1}{K_{4}}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 3.5.10)
 * }

Due to symmetry,$$K_{1} = K_{4}\quad$$ and $$K_{2} = K_{3},\quad$$ therefore,

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

$$\frac{1}{K} = \frac{2}{K_{1}} + \frac{1}{2K_{2}}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 3.5.11)
 * }

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

$$K = \frac{2K_{1}K_{2}}{K_{1}+4K_{2}} = \frac{2(\frac{10Eat}{l})(\frac{5Ebt}{2l})}{(\frac{10Eat}{l} + \frac{20Ebt}{2l})}\quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle\!
 * }

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

$$K = \frac{(\frac{10Eat}{l})(\frac{5Ebt}{l})}{\frac{10Eat}{l}(a+b)}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle\!
 * }

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

$$K = \frac{5Etab}{(a+b)l}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 3.5.12)
 * }

title: User:Eml5526.s11.team5.srv/hw3


text:
User:Eml5526.s11.team5.srv/hw2

Home Work 2.5
'''Show that $$\overset{b _{i}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$  where i = 1, 2,. . ., n is the same as $$\overset{W}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$   such that     $$\overset{W}{\rightarrow}=\sum\limits_{i=1}^{n}$$'''

Solution:
For i = 1, 2,. . ., n

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

$$\overset{b _{i}}{\rightarrow}\centerdot\overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 2.5.1)
 * }

Multiplying the arbitrary constant αi   on both sides ;

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

$${{\alpha }_{i}}(\overset{b _{i}}{\rightarrow}\centerdot\overset{P}{\rightarrow}\overset{(V)}{\rightarrow})=0$$ $$ From the properties of Dot product of a vector, $$c\left( \vec{u}\cdot \vec{v} \right)=c(\vec{u})\cdot \vec{v}=\vec{u}\cdot c(\vec{v})$$, where 'c' is an arbitrary constant.
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 2.5.2)
 * }

Therefore, equation (2.5.2) becomes,

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

$${{\alpha }_{i}}\overset{b _{i}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle\!$$
 * }

For i = 1;

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

$${{\alpha }_{1}}\overset{b _{1}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq.1)
 * }

For i = 2;

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

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

$${{\alpha }_{2}}\overset{b _{2}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 *  $$ \displaystyle (Eq. 2)
 * }

Similarly, For all i = n;

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

$${{\alpha }_{n}}\overset{b _{n}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. n)
 * }

Adding the equations Eq. 1, Eq. 2 .... upto Eq. n, we get,

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

$${{\alpha }_{1}}\overset{b _{1}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}$$ + $${{\alpha }_{2}}\overset{b _{2}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}$$ +. . . . . . + $${{\alpha }_{n}}\overset{b _{n}}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 2.5.3)
 * }

Using the Distributive property of dot Product of a vector, Eqn. 2.5.3 can be written as,

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

$$({{\alpha }_{1}}\overset{b _{1}}{\rightarrow} + {{\alpha }_{2}}\overset{b _{2}}{\rightarrow} + . . . . . . + {{\alpha }_{n}}\overset{b _{n}}{\rightarrow})\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. 2.5.4)
 * }

The above equation can be written as,

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

$$\sum\limits_{i=1}^{n}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow} = 0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 2.5.5)
 * }

Since $$\sum\limits_{i=1}^{n} = \overset{W}{\rightarrow},$$ Eqn(2.5.5) becomes

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

$$\overset{W}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 2.5.6)
 * }

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

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

$$\overset{W}{\rightarrow}\centerdot \overset{P}{\rightarrow}\overset{(V)}{\rightarrow}=0$$ $$
 *
 $$ \displaystyle\!
 * }

Vignesh Solai Rameshbabu--Eml5526.s11.team5.srv 21:09, 30 January 2011 (UTC)

Home Work 2.8
Show that $$\int_{\Omega}w^{h}(x)P(u^{h}(x))dx = 0 \;\; \forall\; w^{h}(x)$$ where $$\;\;w^{h}(x) = \sum_{i=1}^{n}c_{i}b_{i}(x)$$is the same as  $$\int_{\Omega}b_{i}(x)P(u^{h}(x))dx = 0 \;\; for\;\; i = 1,2,...,n$$   where i = 1 to n

$$\int_{\Omega}w^{h}(x)P(u^{h}(x))dx = 0  \;\Leftrightarrow\;  \int_{\Omega}b_{i}(x)P(u^{h}(x))dx = 0 $$

Solution
'Let '

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

$$\int_{\Omega}w^{h}(x)P(u^{h}(x))dx = 0 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 2.8.1)
 * }

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

$$\int_{\Omega}b_{i}(x)P(u^{h}(x))dx = 0 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 2.8.2)
 * }

1)To proove
 * {| style="width:100%" border="0"

$$\int_{\Omega}w^{h}(x)P(u^{h}(x))dx = 0  \;\Rightarrow\;  \int_{\Omega}b_{i}(x)P(u^{h}(x))dx = 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle\!
 * }

Choice 1:
 * $$\;\; \big[c_{1}, c_{2}, c_{3},.........., c_{n}\big] = \big[ 1,0,0,...,0 \big] $$

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

$$\therefore\;\; w^{h}(x) = b_{1}(x) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle\!
 * }

Therefore Eq. 2.8.1 becomes,

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

$$\displaystyle \;\;\int_{\Omega}b_{1}(x)P(u^{h}(x))dx = 0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 2.8.3)
 * }

Choice 2:
 * $$\;\; \big[c_{1}, c_{2}, c_{3},............, c_{n}\big] = \big[ 0,1,0,...,0 \big] $$

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

$$\therefore\;\; w^{h}(x) = b_{2}(x) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle\!
 * }

Therefore Eq. 2.8.1 becomes,

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

$$\displaystyle \;\;\int_{\Omega}b_{2}(x)P(u^{h}(x))dx = 0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 2.8.4)
 * }

Choice n:
 * $$\;\; \big[c_{1}, c_{2}, c_{3},............, c_{n}\big] = \big[ 0,0,0,...,1 \big] $$

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

$$\therefore\;\; w^{h}(x) = b_{n}(x) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle\!
 * }

Therefore Eq. 2.8.1 becomes,

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

$$\displaystyle \;\;\int_{\Omega}b_{n}(x)P(u^{h}(x))dx = 0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 2.8.5)
 * }

In General, Eq. 2.8.3, Eq. 2.8.4, Eq. 2.8.5 can be written as,

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

|$$\displaystyle \int_{\Omega}b_{i}(x)P(u^{h}(x))dx = 0 \;\; where\;\; i = 1,2,...,n  \;( which \;is \;the\; same \;as\; Eq. 2.8.2) $$ $$
 *
$$\displaystyle\!

 * }
 * }

2)To proove
 * {| style="width:100%" border="0"

$$  \int_{\Omega}b_{i}(x)P(u^{h}(x))dx = 0     \;\Rightarrow\;  \int_{\Omega}w^{h}(x)P(u^{h}(x))dx = 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle\!
 * }

For i = 1, 2,. . ., n

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

$$\int_{\Omega}b_{i}(x)P(u^{h}(x))dx = 0 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 2.8.6)
 * }

Multiplying the arbitrary constant $$\;\;c_{i}\;\;$$   on both sides ;

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

$$\;\int_{\Omega}c_{i}b_{i}(x)P(u^{h}(x))dx = 0 \;\; for\;\; i = 1,2,...,n$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 2.8.7)
 * }

For i = 1;

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

$$\;\int_{\Omega}c_{1}b_{1}(x)P(u^{h}(x))dx = 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq.1)
 * }

For i = 2;

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

$$\;\int_{\Omega}c_{2}b_{2}(x)P(u^{h}(x))dx = 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq.2)
 * }

Similarly, For all i = n;

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

$$\;\int_{\Omega}c_{n}b_{n}(x)P(u^{h}(x))dx = 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq.n)
 * }

Adding the equations Eq. 1, Eq. 2 .... upto Eq. n, we get,

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

$$\int_{\Omega}c_{1}b_{1}(x)P(u^{h}(x))dx + \int_{\Omega}c_{2}b_{2}(x)P(u^{h}(x))dx + .\; .\;. + \int_{\Omega}c_{n}b_{n}(x)P(u^{h}(x))dx = 0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 2.8.8)
 * }

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

$$\;\;\int_{\Omega}\left [ c_{1}b_{1}(x) + c_{2}b_{2}(x)+ .\; .\;. + c_{n}b_{n}(x)\right ]P(u^{h}(x))dx = 0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 2.8.9)
 * }

The above equation can be written as,

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

$$\displaystyle \Rightarrow\;\;\int_{\Omega}\sum_{i=1}^{n}c_{i}b_{i}(x)P(u^{h}(x))dx = 0$$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 2.8.10)
 * }

Since $$\sum\limits_{i=1}^{n} = \;w^{h}(x)$$ Eqn(2.8.10) becomes

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

$$\int_{\Omega}w^{h}(x)P(u^{h}(x))dx = 0 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *
 $$ \displaystyle (Eq. 2.8.11)
 * }

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

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

$$\int_{\Omega}w^{h}(x)P(u^{h}(x))dx = 0 \;\; \forall\; w^{h}(x)\;\;\; (which \;is\; the\; same\; as\; Eq. 2.8.1)$$ $$
 *
 $$ \displaystyle\!
 * }

Vignesh Solai Rameshbabu--Eml5526.s11.team5.srv 21:09, 30 January 2011 (UTC)

title: User:Eml5526.s11.team5.srv/hw2


text:
User:Eml5526.s11.team5.srv/hw1

Rectangular Area
Vignesh Solai Rameshbabu--Eml5526.s11.team5.srv 02:06, 23 January 2011 (UTC)

title: User:Eml5526.s11.team5.srv/hw1


text:
Wikiversity talk:List of banned users

In an edit to WV:Bans, it was asserted that User:Wikademia was banned.

Wikiversity:Community Review/User:Wikademia was cited. That Review did not attract adequate participation to secure a community ban. Rather, admins involved in the block of Wikademia did open up a CR, but for whatever reason, perhaps it wasn't noticed, hardly anyone commented, and the discussions were closed by a sysop who had been a blocking admin, essentially confirming his own block. That's fine for an ad hoc decision -- this wasn't improper -- but is not adequate to determine a ban. There was no determination of a ban there, merely a block was defacto sustained through lack of opposition. This is sometimes called, on Wikipedia, a "community ban," but more commonly, the term is used there for a ban that was actually determined by the community, with a neutral close.

The difference between a block and a ban is important if it's interpreted that:
 * The talk page of a banned user will be blanked or even deleted, replaced with a ban template.
 * No custodian is allowed to unblock because it was a "community ban," not merely the decision of an admin or a few.

A true community ban will have a close by a neutral custodian (WV:Bans is unclear on close; it implies that custodians have no role to play in bans, but that is a misunderstanding. I'd, personally, require a neutral 'crat closure for a ban. Every ban should have someone who becomes the enforcer or arbiter of it, someone who can undo it if new conditions arise, without a new community discussion. That aspect of how wikis work (by creating ad hoc neutral "supervisors" of something that requires admin tools, such as page deletion or blocking) is often not understood, but when it is, the wiki works with far less fuss, on Wikipedia, DRV is avoided, etc. I'll explain if anyone wants the explanation...). --Abd 23:41, 9 April 2011 (UTC)

title: Wikiversity talk:List of banned users


text:
User talk:Geoff Plourde/1

Trevor MacInnis 04:49, 25 August 2006 (UTC)

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

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

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

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --mikeu talk 23:00, 5 October 2008 (UTC)

The old template is a bit out of date... --mikeu talk 23:00, 5 October 2008 (UTC)

Completion status
Try 25%done. It's pretty gross, but you're well underway so I'd use 25%. -- Jtneill - Talk - c 05:34, 6 October 2008 (UTC)

American History
Looks like a significant improvement—way more attractive. I also noticed you made a tasks page, which is appreciated; I've added it to the School tasks page for you. If a browser is not at a high enough resolution, however, the History page displays somewhat strangely. Not sure how to fix this, but thought I'd note it.

One thing to consider is what exactly a "minor" would mean to the Department of American History, given the fact that it would not be (for the foreseeable future) accredited at all. You may want to create a page for the "minor" itself, listing coursework (etc.) and explaining this. Another thing to consider is what exactly the function of course numbering serves for the Department of American History. Finally, in conducting your course, make sure you keep in mind that many of Wikiversity's current visitors are very sporadic, and not highly motivated; do not expect to find students who will attend an online course for no credit regularly. As you design your course, keep this sort of thing in mind. At the same time, I believe that if an instructor were to be aggressive in keeping contact with visiting students, posting messages and discussion on student talk pages, it may be possible to get students to become more active and regular than the History School has managed so far. Whatever approach you try, I wish you the best of luck.

Finally: It would be much appreciated if you could visit the History talk page and post your comments on some of the discussions there. I'm trying to develop some school-wide policies, goals, and practices, but it's been very difficult to get feedback on proposals.

If there's anything I can do for you, just let me know. The Jade Knight (d'viser) 14:44, 6 October 2008 (UTC)
 * Not sure what resolution; I use IE windowed. I'm guessing 1024x768 was close to the size, though maybe not exactly.  Some people out there are still using 800x600, though.  The Jade Knight (d'viser) 02:17, 7 October 2008 (UTC)

Re: page deletion. Drop a note to User:Emesee since it looks like he just created that page today. He can also delete if anything is in the way of the move. --mikeu talk 18:21, 8 October 2008 (UTC)

Hello, I'm interested in your course. Keep in mind I'm not an acedemic and it has been 10 years since I was in a college setting. I look forward to start your course.--Jolie 19:44, 8 October 2008 (UTC)

just curious
I noticed that you struck out this edit but left the other part in place. I was just wondering if you consider this to be an "open issue" that still needs to be addressed? --mikeu talk 15:50, 9 October 2008 (UTC)
 * No longer open Geo.plrd 18:17, 10 October 2008 (UTC)

JWSchmidt and IRC
Forgive me if I am asking the wrong person, but I noticed that JWSchmidt is still banned on IRC. Do you still believe this is necessary? I understand if you do, just wanted to double check that it is still necessary. Geo.plrd 06:43, 11 October 2008 (UTC)

 * Well, there are people concerned on both sides of the issue, so I think it needs to be discussed first. --SB_Johnny talk 14:18, 11 October 2008 (UTC)

North American History
Best to talk about it on the appropriate talk pages, so everyone can see the discussion and give input. Emesee appears to have objected over at the Canadian History page. The Jade Knight (d'viser) 05:15, 13 October 2008 (UTC)
 * I don't think anyone will object, either. Just make sure you include all the content that is there now somewhere, even if only at a sub-page or archive of some sort.  The Jade Knight (d'viser) 05:30, 13 October 2008 (UTC)

Emesee's decided to act unilaterally on this one.
Please see Community_Review, as it relates to consensus-building at the School of History. The Jade Knight (d'viser) 06:23, 13 October 2008 (UTC)

US history course
thought I'd leave my chat here as an easier way to get ahold of you.

I apologize- but I started a rewrite of your lecture notes!

I now see the textbook. ok. so in the talk page of the lecture notes, I asked whether meso-american history is really US history? I just want to be sure because it doesn't make a lot of sense to me. (and I didn't see it on the cliff notes version of US history).

but thats ok if it is, I'll begin looking into it.

 * ok heres the link to the college board and its specified content (required for an AP history course)
 * your right on!! that's exactly the topic that given in the textbook.
 * Clearly the college board wants the Inca/Aztek/maya, hohokum/anasazi, and the mound builders mentioned.
 * no specific indication about the clovis. --Jolie 20:13, 13 October 2008 (UTC)

take a look at the rough draft rewrite of your notes. comments? Deciding what is relevant is actually much harder than getting the story right and the narrative succinct and I welcome any assistance you could give me. The cliff notes have been rather helpful.

 * One stewing thought. clearly the lecture notes were meant to prove a point to the average reader.  however a textbook should be more disciplined... there's no POINT to make exactly. so mentioning the indian removal of 1830 is definitely out of place.  A disciplined historical textbook, keeps to the epoch. and we are a Long ways away from 1830.--Jolie 20:17, 13 October 2008 (UTC)

 * actually, no I read about the indian removal act of 1830 in the wikipedia article about the 5 civilized tribes.--Jolie 11:47, 14 October 2008 (UTC)

So in the next couple days I hope to get this material into the textbook.

I wanted to add a little superfluous comment that I love this course! I love the idea we get to redo the book. this is great scholarship! thanks for the course. --Jolie 17:12, 13 October 2008 (UTC) more work on the textbook. a history of southwestern indian tribes has been posted. it is as rough as the desart sandstone landscape itself. but it is there. I have already completed some reseaches on the mound buildersr.--Jolie 17:08, 15 October 2008 (UTC)

The moundbuilders are done. Unfortunately, as the writing continued I felt less and less inspired. What really stumped me was a good summary of their accomplishments. The southwestern indians had a much better summary. I also thought I got stuck onto dimensions and shallow description- but the overall scale of the missippian accomplishement is incredible!

I need a bit of a mind break. Perhaps I can make things better in a few hours.

by the way thanks for the compliment.--Jolie 17:11, 16 October 2008 (UTC)

I appreciate your thoughts on the justification on the new textbook. Jade knight is asking us to account for this action. Wikimedia is an interesting and unfamiliar forum for me and I don't always understand what is 'ethical'.

Is it more ethical to redo a wikibook or alter the writing of the current one ,with conflicting material? Couldn't the latter be considered a form of vandalism. And yet, myself I would probably WELCOME a contribution to my material rather than the rewriting the topic altogether. The fact that the book seems abandoned, needs considered, too. if the author is not around to work with; how do you exactly build of his work???

That's what the day is about SO far. I didn't want to get stuck in a philosophical dilemma about ethics (So much of WV IS now days)- I wanted to learn. but to edit, means to make choices that some might see as unethical. I think this issue is our a test of what we consider to be ethical editing.

Even once this project has done, It will remain an ethical dilemma. even for WB. Do they boot out the current book from their 'featured textbook', will they think we wer right to redo the textbook, rather than add to the current book? If our goal is the Authoritative textbook of US history. We need to be aware that this might seem controversial to people in WB.--Jolie 15:55, 20 October 2008 (UTC)

Whew! thats done, the Mayan, Incans and Azteks seemed hard to do. On to the vikings.--Jolie 16:43, 22 October 2008 (UTC)

please provide a little bit of organization in the book. I believe the next topic is "3.	American Indian cultures of North America at the time of European contact " this should be rather straitforward. We've already covered the major missippian culture that dominated the south and midwest. The Northeastern indian cultures were less unified.--Jolie 16:54, 22 October 2008 (UTC)

 * sorry, geo I think we should give. I expect mounting opposition to our efforts.  on the other hand, we've gotten unpredented permission to severely edit "US history", so.. why not?  Please understand I have the highest regard for your assistance, guidance, and judgement.--Jolie 18:43, 22 October 2008 (UTC)

 * Ok I saw your post on wikibooks, which gave me some thought. While I understand the appeal of altering AP US history (which is basically so imcomplete and stubbed that it would be a blank slate), I think you should ponder one more thought.

 * What good does it do for there to be a FEATURED book in wikibooks, where the bering land bridge doesn't exist and Mayans study asteroids to redict the end of the world. have you seen the vikings chapter?  I don't think its good for wikimedia.  I may not be an expert but I can write more NPOV and closer to the college board content than that.

 * Now peruse some later information. The civil war chapter is a little disorganized and unsiccinct but its not euro-centric. When the text coverges, the job becomes easier. Just like we wouldn't want to keep stuff like this in wikipedia to confuse and detract from serious study; I don't like the idea of keeping it in wikibooks. (The fact that it is featured will direct many students to it)

 * Wikibooks doesn't need to have a euro-centric textbook that claims to meet college board requirements and doesn't.

 * IMHO, we would do wikimedia a favor to replace it. again your opinion is everything. I will continue fleshing out the next subject "First European contact with the Native Americans."

 * If you want to work on AP US history, it would be a fairly staitforward way of getting consensus on what we wanted to do in the first place and I would gladly comply. Feel free to delete anything you don't want left lying around this is meant to be semi-private to you--Jolie 15:07, 23 October 2008 (UTC)

 * Well I could do rework US history (and still think this is the best idea as opposed to changing the stub AP US history). I could do it!  the first chapter will Look amazingly like "United we stand"  but then AP US history would look amazingly like "United we stand"

Give me the go away and I will whip out my chainsaw.

fundementally I am stuck on ONE thing, US history is a 'featured book'! if we're going to basically wipe out content and put our own thing in there... it is US history that it should go. I will pose one more 'are you sure post to mike' on wikibooks check his answer carefully.--Jolie 17:18, 24 October 2008 (UTC)

well, Mike is taking a wiki-break. really I have the feeling I'm not doing what I ought to do. If I can PUT our material in US history, you were willing to do that right? well, I gotta get back to life; but on Monday. I will take the crowbar to US history. and If I break it- I can apologize profusely to Mike.lifeguard to get all my changes unmade.

maybe your right and US history is just 'built' wrong for the AP. but I'm not going to know it until I try to change it. so have a good weekend.--Jolie 17:52, 24 October 2008 (UTC)

Believe it or NOt I am still making project.

In addition to the text, I think it is important to be transparent about what I think of the text (note things that seem weak, Justify what theories I include,etc) I hope that such a conversation will help other people collaborate with me on future changes,  Which I most assuredly welcome.

This will pre-empt the effort to support the text with citations. (NOTE: General convention is NOT to clutter the text of a textbook with lots of citations). I'm not sure where citations belong. Also If I have a fact, should I state where that fact comes from in the talk page? I didn't see any references in the US History book (at all).

 * Are you one of several IP addresses that have been tweaking the chapter headings?(205.215.88.74; 205.215.119.85; 205.215.119.108) One of them left a short message in french that I believes say "One more thing to change, I'll leave the rest alone". I tried to reply in french, back.  Sadly my french is very poor and rusty. Perhaps there are actually other people interested in the book and if so, I hope they introduce themselves.--Jolie 16:04, 29 October 2008 (UTC)

US history course2
ok well someone on IP addresses 205 is out to make our job hard. I've already complained about it at wikibooks. I've assumed in the past that he was doing a best effort at (unknowingly) improving our book. However the european explorers section contained ,honest, a reference to the marco polo game. that seems very vandalism. its kind of a shame for if its vandalism its a very subtle vandalism with occasional well written stuff.

but since we want college board material, I am at wits end to figure out how to get through to him (and if he IS a vandal, he's loving my weak gestures at communicating)

at any rate. we have moved to about 1500, and the first european map of the new world. Next in land; the spanish explorations of florida; (which were prompted by a bizarre search for a 'fountain of youth'). A brief mention of Southwestern spanish settlements in CA. and then on to the french and English. English settelment is more researched than the French.

 * Actually I would welcome your help a little on french settlement of the new world. How deeply should I delve into canadian history??  While the french contributions to the new world were significant, they seemed to handle it much different than the conquistadors (who basically destroyed powerful meso-american culture and greedily seized whatever they wanted from the native americans).

 * I'm not even sure who began 'New France' as it was called. If the Spanish had columbus and England had Cabot, who was the man that developed new france?  Most of the History I could dig up had to do with a Northwest passage, and that was a french dream that didn't seem to pan out.  (I suspect that the sea passage, which does exist, isn't reliably ice free enough to allow European Trade ships.)

 * And another mystery is way the spanish seem to falter when the french really solidify their position in North America. The spanish are destroying kingsdoms siezing vast amounts of gold, competiting with brazil to supply an increasing hunger for slavery (in the 16th cetury) and then a few hundred years, France has taken control of the american south (ie New orleans) and expanding throughout Canada.

 * The war between french and english influence in North America, is of course not really over until the French and Indian War deep in the colonial period. But what of the fight between Spain and France?  I suspect a conflict in Europe, but I know nothing about it.  A few clues and I'll take it to wikipedia.  You might want to mention something more about france and spain in your lecture notes (in the interest of closely following college board requirements).--Jolie 14:13, 11 November 2008 (UTC)

 * Some internet searching does yield some answers. France's 'christopher columbus' was undoubtably Jacques Cartier who much like christopher, explored, made a settlement and resupplied the settlement at New france though several voyages.

 * Another force for french exploration was the french 'hugenaught' (spelling?), an early protestant movement that garnered sympathy in france and sought asylum from dictorial decrees. They would found a small colony in florida (that got demolished by spanish conquistadors) in the mid 16th century.

 * anyway, despite the gold spain had seized Native America, The growing superpower in European affairs would be france.  By the reign of the 'sun king' france would be considered epitomy of european culture.  Spain would lose much during the disastrous 'war of spanish succession' where the possibility of a combined spanish and french monarchy caused war throughout the world.

 * those are some of the answers I have been finding. already I have gotten immersed in the details and as normal I will have to simplify what I have to make progress in the next several sections of the textbook.  I'm still interested in what you think might be useful to mention.  Since some issues span large chunks of the colonial period ('war of spanish succession' particularly) the description of france and spanish empire might be tricky.--Jolie 16:45, 11 November 2008 (UTC)

 * Looking through the 'age of discovery' as it is called. I've come to the conclusion that the story is more multifaceted than merely columbus, cabot, amerigo and a famous map.  There is also Verrazzano, cartier, hudson, desoto, balboa and lots of others.  In a section that quickly goes throught the spanish, french and english settlements in short order, Its not succinct writing include all these people.  I'm sure you could devote whole books to just one of these topics.

 * I think then I need to focus on people that play pivotal parts of American history. I'm still undecided on the might and savagery of spanish actions in south america and mexico.  Meso-American history never made sence in a US history book (Whatever the college board says); its unclear if the conquistador's attack on them is a revelant.  I think some of that history is already been told.

 * anyway look for changes and new expansion in the textbook- I want to get through english settlement.--Jolie 17:03, 11 November 2008 (UTC)

A week-long business trip and holiday preparations as well as some extra work at work has kept me inactive from this project for several weeks. I came back to it some today and worked to establish some clarity with Englands action in the 16th century. England seems to gain determination during the very early 17th century and the whole story kind of burst out of me. I am no so sure I like what I wrote but I gained a few decades and studied throught the anglo-spanish war of 1585 today. I hope to make progress in the days to come. Wikiversity seems pretty quiet these days and I sense I am not the only one preoccupied with other things.--JoliePA 16:27, 16 December 2008 (UTC)

SAR
Thanks for contributing to SAR mission 2008 01. You are welcome to contribute more. This is an real case, unsolved, that is being redeveloped as a training mission. Some of the contributors are people in the local community who have some experience as SAR field team members, who are interested in managing SAR missions. They need coaching to learn to wear the ICS "hats" aka "thinking caps" of planning, logistics, and operations. Actual work on the ground in the local community may result from this Wikiversity exercise. --Una Smith 16:37, 14 October 2008 (UTC)

 * Well I noticed some things that most certainly should be done and added them. Another thing to do is to retrace his steps on the day he disappeared. Then start asking everyone around the area if they have seen him or the tricycle. This might end up with a lead. Another thing is to search caves and any unused buildings. I remember reading a case study where someone disappeared and was found in a old chicken coop. Geo.plrd 00:36, 15 October 2008 (UTC)

 * Yup, that's the idea. Many of those tasks have already been done.  For various practical and pedogogic reasons, this exercise involves first working out what should be done, then considering what has been done already.  There is an important step that should happen before tasks are developed, which is to examine scenarios and draw up a list of mission objectives to deal with those scenarios.  If that step is skipped, often the tasks done will focus on just one scenario, ignoring others.  --Una Smith 14:47, 15 October 2008 (UTC)

Bills of Rights and Social Contracts
Hi Geoff - I agree to an extent that we need something like these things - but only insofar as other more developed wikis (like Wikipedia) have developed policies to mitigate against specific types of incivility, or potentially damaging or unproductive activity. I personally wouldn't go as far as Wikipedia in creating a proliferation of policies - I always advocate keeping things as simple as possible - structurally, at least. ;-) So, I'd probably advocate something along the lines of a set of key pillars - perhaps civility, scholarliness, and appropriate means and ends - that all work in cohort with each other. I'm going to post something for discussion on the Colloquium soon. Any ideas you have would be great. :-) Cormaggio talk 09:24, 15 October 2008 (UTC)

History Project Box
This may be silly, but I currently have a strong dislike of the History project box for the reason that it files projects directly into the Main history cat. McCormack was working with me to fix this when he took off on wikibreak. The Jade Knight (d'viser) 12:20, 16 October 2008 (UTC)
 * Not sure what you mean; the History project box is a template McCormack created; you added it to Third Reich. The Jade Knight (d'viser) 23:34, 16 October 2008 (UTC)
 * Thanks for doing that. The Jade Knight (d'viser) 07:21, 17 October 2008 (UTC)

Change username
I can do the rename of "Geo.plrd" to "Geoff Plourde" but I want to check to make sure you are ready for unified login before I do it. Have you set the password and email address on all the accounts so that they are the same? This will make the unification easier if it is done first. There is also help at meta:Help:SUL if you need it. --mikeu talk 12:55, 28 October 2008 (UTC)
 * Please see b:WB:RENAME - I've requested confirmation that you own the account on English Wikipedia. – Mike.lifeguard | @en.wb 21:14, 28 October 2008 (UTC)
 * Confirmation is here. The account has been renamed.  --mikeu talk 04:00, 29 October 2008 (UTC)

Tx & Categories
Welcome back and tx 4 contributing to the community review and bureaucrat nomination as well as new content. Just a tip - try to include at least a category on each page e.g., Category:Religious studies. -- Jtneill - Talk - c 02:11, 21 March 2010 (UTC)

Gospel of John
I was pleasantly surprised to see you working on the New Testament course. It looks good and comprehensive, and I can't wait to see it when its completed. We need more high quality theology courses here! I just wrote a course on the Gospel of John, as a sort of biblical foundation for the Christian leadership program I've been working on. Feel free to look at it, or let me know if you need any help. God bless, Geoff Plourde 18:59, 19 April 2010 (UTC)

 * Geoff: Thanks for your comments on my talk page. The course is intended to be a general introduction and overview of the New Testament for those who have little or no familiarity with it (its title is "Introduction to the New Testament"). I think it serves its purpose -- that of being an entry point -- quite well. What is needed now is the next level -- courses which focus more in depth on specific books, sections, or topics. Your course on the Gospel of John does exactly that. I've suggested it as a next step in the "In Conclusion..." section at the end of the course, and also listed it under "Other Resources." It would be appropriate for people who already have some exposure to the NT to skip the introductory course altogether and go to the more specialized courses, such as yours, so I've changed the comments under the heading a little to make it clear that the course is aimed at newcomers, not at those who are ready for the next level. It would be nice to have in-depth courses covering all of the NT books eventually. I'm also hoping that someone's going to do something on the OT, as well. As of now, there's nothing at all. --gdm 16:31, 22 April 2010 (UTC)

Probationary custodianship
Welcome aboard, Geoff - you are now a probationary custodian. Please add yourself to Staff and review WV:Policies. You may also like to participate in How to be a Wikimedia sysop. If I can do anything to help, please let me know. Sincerely, James -- Jtneill - Talk - c 07:29, 2 August 2010 (UTC)

Subpage
Hi, I recommend to found a subpage that our discussion will not mix with the discussion in here.--Juan de Vojníkov 08:50, 2 August 2010 (UTC)
 * No problem, see /mentoring 2010 Geoff Plourde 16:00, 2 August 2010 (UTC)

Decline of RSD
Perhaps I should have noted that Category:20th Century Philosophy exists, that is why I thought the empty Category:20th-century philosophy could be deleted. Thenub314 19:20, 2 August 2010 (UTC)
 * Thanks for the heads up, went back and deleted the duplicate Geoff Plourde 22:44, 3 August 2010 (UTC)

You forgot
You forgot to make a diff stating that Juan will be your mentor and that he has the right to reverse your actions, desysop you, etc. Please do so either on the Candidates page or on his talk page and place a diff on the Candidates page of that statement. Ottava Rima (talk) 18:11, 3 August 2010 (UTC)
 * Is that required? Adambro 18:13, 3 August 2010 (UTC)
 * It is tradition, at least. Otherwise, JWS will say something about Geoff never having accepted the mentorship or whatever. Ottava Rima (talk) 18:33, 3 August 2010 (UTC)
 * I've accepted Juan as my mentor on the candidate page. Geoff Plourde 19:27, 3 August 2010 (UTC)
 * But not the ramifications of the mentorship. Normally, there needs to be a diff to link so any questions can just be directed to that. A binding agreement that allows him to act in the fullest extent. Ottava Rima (talk) 19:31, 3 August 2010 (UTC)
 * I never made such an agreement. Perhaps this should be discussed more generally because current policy doesn't seem to suggest that the mentor should be given the right to reverse any actions by the candidate. I think the recent situation with Abd has shown that perhaps isn't a good idea. Mentors should guide probationary custodians, not force them to act as they demand. Probationary custodians should feel able to act as they consider appropriate without worrying that they may be quickly overruled without a proper discussion. Adambro 20:44, 3 August 2010 (UTC)
 * WMF policy does not allow for administrators on any project except through community election or appointed as part of official function of the WMF. Therefore, no one has the right to act with the admin tools unless they meet those conditions. It isn't about "guiding" at all. As I stated before, the temporary custodianship only gives a mentee the ability to use the mentor's admin right. They are not trusted enough to act on their own or "without worrying". Ottava Rima (talk) 20:52, 3 August 2010 (UTC)
 * I'm not aware of the WMF policy regarding administrators, perhaps you could link to it, but our policy with regards to custodianship talks about your mentor being "your first person to contact" with any questions. I see nothing in our policy to suggest your custodian rights are just an extension of your mentor's nor that they will be able to overrule anything you may do as a probationary custodian. As I've suggested, I'm not comfortable with that idea but perhaps you should raise it for discussion at Wikiversity talk:Custodianship or similar, since we've high-jacked Geoff's talk page, with a view to seeing what others think about clarifying the policy. Adambro 21:32, 3 August 2010 (UTC)
 * I never agreed to having my actions reversed either. However I do think a mentor could be said to be responsible for the actions of the mentee, and the current policy allows for desysop after 48 hours if the mentor no longer wants the responsibility. -- dark lama  22:28, 3 August 2010 (UTC)
 * I think the whims of one vocal individual is not a good argument for tradition or for making decisions. -- dark lama  22:42, 3 August 2010 (UTC)

Well, I am wondering what is going on!--Juan de Vojníkov 20:49, 3 August 2010 (UTC)

 * Perhaps it has something to do with the special promise I made in my candidacy. I made that up, and Diego Grez followed it. Policy does not provide for the rights that I gave to Ottava with my probationary custodianship. A mentor may require that or not. I have argued that in theory the custodial rights of the probationer are a kind of extension of those of the mentor, but they are also independent, as shown by the 48 hour period the probationer has to find another mentor if the original mentor withdraws. The probationer has no mentor during that period. Essentially, a custodian should not mentor someone unless s/he is willing to supervise and take responsibility for the probationer's actions. For another precedent, SB_Johnny mentored User:Salmon of Doubt and requested and required an agreement for immediate desysop at the discretion of SBJ. I didn't offer that much, but I did offer a right to reverse actions, and a right to prohibit actions, with prior agreement to immediately accept desysop by Ottava (and, in fact, any other supporting custodian) if I violated such a prohibition. Emergency desysop is always available, probably should be requested by a 'crat, but it could in theory be anyone who files a report. If it doesn't look very clear, it will probably be dismissed quickly. Ottava did go for immediate desysop, but without following the provided process, and there was no emergency and he did not claim there was one, he claimed it was routine. It wasn't. Mess. Here, Juan, it's completely up to you. You consented unconditionally, but you could change that if you want, or leave it alone. --Abd 02:21, 6 August 2010 (UTC)

Transcluded page that was blanked.
Re Topic:Neural Symbolic Learning and Reasoning/Topics, I didn't think to check for "What links here." I was on an i-Phone. There would have been little harm if it had been deleted, but if I get tools back, now I know to check for transclusion for a blanked page. I did put a note on the attached talk page about the transclusion, but maybe that's overkill. Thanks. --Abd 02:10, 6 August 2010 (UTC)
 * No problem. I almost deleted the whole thing right after he blanked it, but decided to contact him instead. Geoff Plourde 02:18, 6 August 2010 (UTC)

Copyright
As part of your mentorship, you were asked about copyright. I deleted this as being a clear copyvio. Try to see if you can put together any evidence to verify and then post it here to show the steps you used to verify it. Ottava Rima (talk) 12:58, 8 August 2010 (UTC)
 * How we know that the contributer was not an original autor to that work? Why the original author havent recieved the chance to get/offer permission or revrite it?--Juan de Vojníkov 13:24, 8 August 2010 (UTC)
 * It was posted by an IP, who lack the ability to have solid proof of who they are and their authorship. They would also have to prove that they were the poster of the original statement on the original copyrighted website, and that they, and not the webmaster, has the original copyright. On some websites, there is an author's name, and they conflict. Such shady authorship and problems make it a mess that Wiki is best not to deal with, especially with the spammish nature of multiple websites. This is stuff that Geoff should pay attention to. Ottava Rima (talk) 23:02, 8 August 2010 (UTC)
 * Evidence can be found at here and at several other websites and newsgroups. I placed portions of the text into Google to locate the website, then used Ctrl+F to perform a standard text to text direct comparison and confirm. Geoff Plourde 19:41, 8 August 2010 (UTC)

Response to comment about "dinging."
diff, permanent link to discussion.

The term "ding" was used there as a colloquial expression for making a comment critical of an action.

Yes, we need particularly to "ding" users or, especially, custodians who do something we think contrary to policy or the welfare of the community. However, the original "ding" comment was about something I did not do, not something I did.

I was just "dinged" today on my Talk page for doing what I believe is proper, but a custodian thinks not. Criticism of actions by teachers, staff or students is, in fact, essential to a functional and productive academic environment, and if you imagine it will just happen spontaneously, well, you are a long-time participant here. How has it been working? If you look at the comments I made in that discussion, they were civil and collaborative. Now, two weeks later, you want to dig it up? To criticize a word I used, or the criticism? If the latter, shouldn't you practice what you preach? Which is more important, wheel-warring -- or the appearance of wheel-warring -- or using a mild word like "ding?" There is real wheel-warring going on, from time to time; this particular incident was just a mistake, nobody ended up looking bad. I'd thought!

Thanks for your work improving Wikiversity. --Abd 18:20, 10 August 2010 (UTC)

A missed opportunity to help a newbie?
Geoff, I came across a problem on Requests for deletion. You can see it now at the RfD undeletion request I section-titled and supported. When I was looking for the edit that had placed the request there, I came across this archiving from that section. This would have been an opportunity to notice the request when it was still fairly fresh. It would have stood out as an unsigned section, exposed by what you removed, as you can see in the diff. Obviously, we all overlook things sometimes, but I thought I should let you know. Thanks. --Abd 23:02, 10 September 2010 (UTC)

Activity
So no interest in wv any more?--Juan de Vojníkov 05:56, 28 September 2010 (UTC)

Full custodian
Congratulations, you are now a full custodian. Thanks for your contributions. -- Jtneill - Talk - c 09:00, 29 November 2010 (UTC)

title: User talk:Geoff Plourde/1


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Survey research and design in psychology/Tutorials/Multiple linear regression/Instructor notes

This tutorial takes ~100-110 minutes:
 * 1) Explain the general steps and assumptions involved in MLR as background to the tutorial exercises and as summary of the lecture/reading content (approx. 15-20 mins)
 * 2)  Allen and Bennett exercises:
 * 3) Demonstrate step-by-step, with participants following the steps to perform standard and hierarchical MLRs
 * 4) Francis exercises
 * 5) Demonstrate step-by-step, with participants following the steps to perform standard, hierarchical, and step-wise MLRs
 * 6) Point to and briefly explain key points from the example MLR write-up.doc
 * 7) Approximately one paragraph each for descriptives, model/assumptions, R squared, coefficients, conclusion/summary.
 * 8) What tables are needed? (Descriptives (could be combined with EFA), Correlations and Coefficients (could be combined))
 * 9) Allow for any questions


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Openness and collaboration

Welcome to the Wikiversity learning project about openness and collaboration. This is a "Learning by doing" project where Wikiversity community members explore how to promote openness and collaboration at Wikiversity.

This learning project was started in response to the recent call for Openness by the the Wikimedia Foundation Board of Trustees.

Methods
The methods used in this learning project will include action research aimed at removing existing barriers to openness, participation and collaboration at Wikiversity.

Questions

 * 1) Why is the content of this learning project being removed, repeatedly?
 * 2) Why was this learning project nominated for deletion?
 * 3) How are "openness", "collaboration", and "participants" defined by the Wikimedia Foundation?
 * 4) What do "openness" and "collaboration" mean to current and potential participants of Wikiversity?
 * 5) What barriers keep people from collaboration and participation at Wikiversity?
 * 6) What barriers keep Wikiversity from becoming more open?
 * 7) What are the barriers to collaboration in general?
 * 8) What reasons have past participants decided to stop collaborating and participating at Wikiversity?
 * 9) How do potential participants want to collaborate?
 * 10) Why do current participants continue to collaborate and participate at Wikiversity?
 * 11) What things do current and potential participants want Wikiversity to be more open about?
 * 12) How can Wikiversity include personal opinions that promote open and education-based debates?
 * 13) How can Wikiversity encourage participants to collaborate when a person's methods discourage collaboration?
 * 14) How can Wikiversity encourage more participation in community decisions and open discussions?
 * 15) What can Wikiversity do to resolve conflicts quicker and for longer in ways that encourage retention and increases participation in learning?
 * 16) What can Wikiversity do to make policies, practices, and the collaborative culture friendlier to increase participation in learning?
 * 17) Where do new participants first learn about Wikiversity?
 * 18) When and how do people decide to collaborate?
 * 19) When and how do people decide to participate?
 * 20) What attracts people to Wikiversity?
 * 21) What qualities promote openness?
 * 22) What qualities promote teamwork and collaboration?
 * 23) What qualities promote closeness?
 * 24) What qualities promote conflict?

Proposals for change

 * Policy development

Related resources

 * Edit using your real name
 * strategy.wikimedia.org, Mailing List: related discussions.
 * Wikipedia article about openness
 * Cultural evolution and individual development of openness and conservatism by Alberto Acerbi, Magnus Enquist and Stefano Ghirlanda.
 * Do Wikimedians have few or many opportunities to learn about what disrupts Wikimedia wiki projects? A model by Pontus Strimling, Magnus Enquist and Kimmo Eriksson suggests that cultural evolution can only be adaptive when there are rich learning opportunities.
 * Citizendium, a wiki branching off Wikipedia started by Wikipedia co-founder Larry Sanger restricts the wiki primarily to editing by experts only.
 * Why academics do and do not participate on wikis

title: Openness and collaboration


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Instructional design/LMS interactivity

Introduction | Lesson 1 | Lesson 2| Practice| References Next

Introduction
Have you ever had to take a self-paced eLearning course and been bored to tears? Many people will admit to 'multi-tasking' while an instructional video remains playing in a browser window, paying closer attention to their alternative task than the original task. However, those same people are also able to describe, in detail, a course where they have felt completely engrossed with its contents, a course where they have payed close attention to each word spoken or written by classmates and instructors, or a course where they have gone above and beyond to research and read non-assigned course materials. What has been the difference in these seemingly opposite, yet similar environments? What causes a course to captivate learners on multiple levels; cognitive, affective, and effective and what holds that same course back?

The courses that captivate learners are engaging, meaning they involve the learner, and are also interactive, requiring the learner to participate with the material in some manner. One of the key ingredients to creating effective instruction and a positive student learning experience is interactivity.

Overview of Lesson
Learners will be presented with research that describes the necessity of creating self-paced online instruction that is interactive. A video will be viewable to compare self-paced online courses with and without interactivity. The video will refer to the benefits of interaction and the holes in the learning that can be caused by not having interactions. From the videos, a list will be presented to offer characteristics of engaging interactions that can be included in self-paced online courses. Learners will also be provided with an opportunity to contribute to the list.

Learners will then be presented with a review of the non-interactive section of a course and the objectives for that course that was presented in the first section of the lesson. An improved version of the course will follow outlining the differences between the courses. The steps that were taken to create the “new” course will be identified and verified against the list of characteristics.

Finally, the learners will be shown a second example of a non-interactive course. The learners will be able to include their ideas of activities that they think would add interactivity to the course.

Lesson Goal
Corporate trainers and course designers will use this course to explore creating online courses with interactivity.

Key Learning Objectives
Learners will be able to
 * identify characteristics of interaction within a self-paced eLearning course.
 * identify steps for improving interaction in self-paced eLearning courses.
 * create at least one interactive activity for the course.

To learn more about creating interactions go to the Lesson  section of this course.

title: Instructional design/LMS interactivity


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Instructional design/LMS interactivity/Lesson 1

Introduction | Lesson 1 | Lesson 2| Practice| References Next

Introduction to Creating Interactions in Online Self-paced Courses
Interactivity should be seen not as a single type of instruction but one that is on a continuum. Where the interation for any particular course lies depends on the conditions of the learning environment and can take on a multitude of formats.

For this lesson’s purposes interactivity in eLearning self-paced courses includes

 * user participation
 * ability to control the content

Interactivity is a vital component to online instruction and studies have shown that when students are actively involved in self-study learning retention can increase by as much as 20%. Interactive instruction provides opportunities for learners to interact with peers, teachers, and content in a manner that will enhance their educational experience, taking them beyond the limitations of a traditional classroom where learning is teacher centered. Learners are able to control content, navigation, time, and sequence. Students are provided with opportunities to apply and develop critical thinking skills while being an active participant. In recent years, the technology of the computer has been used to enhance curriculum, which can dramatically affect attention and engagement of the learner. Using technology such as the computer allows learners to constantly be participants rather than spectators in their instruction and with this technology feedback is able to come immediately after instruction has taken place. Something that psychologists agree is best

Examples of Courses with Interactivity
The following are examples of courses that have been created with interactivity that is descried in the list above.
 * http://www.cisco.com/web/learning/netacad/peterpacket2/NewBuild_forweb/default.htm
 * http://www.oxfamamerica.org/multimedia/flash/a-seat-at-the-table/
 * http://www.suddenlysmart.com/examples/Patient_Management/player.html

Examples of Courses with NO Interactivity
The next set of courses for you to view show courses that do not contain a high level of interactivity as defined by this lesson.
 * http://www.commoncraft.com/rss_plain_english
 * http://www.youtube.com/user/YaleCourses#p/c/BCB3059E45654BCE/0/qd3MJPHaotQ
 * http://openlearn.open.ac.uk/mod/oucontent/view.php?id=398572&section=1.1.1

List of Interactivity Characteristics
The above courses illustrate interactivity in the following ways:

 * Animations
 * Simulations
 * Navigation
 * Hot Spots
 * Self-selection of actions
 * World Problems
 * Branching
 * Practice
 * Immediate Feedback
 * Assessment
 * Practice with Feedback
 * Quizz

Other examples of interactivity in online courses include
 * asynchronous discussion
 * synchronous discussion
 * discussion boards
 * debates
 * forum postings
 * blogs

Now that you have learned how interaction can help, proceed to Lesson 2 of this course.


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Instructional design/LMS interactivity/Lesson 2

Introduction | Lesson 1 | Lesson 2| Practice| References Next

Non-interactive Course
The following video is an example of a course with no interactivity. While viewing the video your attention to be brought to where there is no interactivty. Depending on your internet connection the loading of the course could take a few moments.

Course with no interactivity

Open the course in a new window by holding down your shift key and right clicking the link above.

After you have viewed the above course, continue to the next section to view two sections of the PowerPoint course that could be added for interaction.

Course with Interactivity
Below is a link to a new version of the open source course we showed in the video above.

Course with interactivity

Open the course in a new window by holding down your shift key and right clicking the link above.

Identify steps that we taken in creating new version of course
To create the new course we first must follow the following steps.

 * 1) Identify areas that are lacking in the current version of the course.
 * 2) Decide what type of interaction to include.
 * 3) Gather all the elements of the interaction.
 * 4) Create that interaction.
 * 5) Publish a new course.

For the above example, here were our specific steps.

 * 1) Identified that we wanted to create interaction for:
 * 2) Entering notes.
 * 3) Adding a new slide.
 * 4) Adding text and an image to a slide.
 * 5) Created slides in PowerPoint and imported them into Cativate.
 * 6) In Captivate we added interactive elements to the imported slides such as text entry boxes and hotspots with popup windows to provide feedback.
 * 7) We then added it to this page for you to view.

Now you can help create interaction in the Practice section of this course.

Introduction | Lesson 1 | Lesson 2| Practice| References Next


text:
Multiple linear regression/Quiz

Multiple linear regression practice quiz

MLR I
 {Multiple linear regression (MLR) is a __________ type of statistical analysis. - univariate - bivariate + multivariate
 * type=""}

{The following types of data can be used in MLR (choose all that apply) + Interval or higher dependent variable (DV) + Interval or higher independent variables (IVs) + Dichotomous IVs
 * type="[]"}

{A linear regression (LR) analysis produces the equation Y = 0.4X + 3. This indicates that: - When Y = 0.4, X = 3 - When Y = 0, X = 3 - When X = 3, Y = 0.4 + When X = 0, Y = 3
 * type=""}

{A LR analysis produces the equation Y = -3.2X + 7. This indicates that: + A 1 unit increase in X results in a 3.2 unit decrease in Y. - A 1 unit decrease in X results in a 3.2 unit decrease in Y. - A 1 unit increase in X results in a 3.2 unit increase in Y. - An X value of 0 would would increase Y by 7.
 * type=""}

{The main purpose(s) of (LR) is/are (choose all that apply): + Predicting one variable on the basis of another + Explaining one variable in terms of another - Describing the relationship between one variable and another - Exploring the relationship between one variable and another
 * type="[]"}

{When writing regression formulae, which of the following refers to the predicted value on the dependent variable (DV)? - Y + Y (hat) - X - X (hat) - a
 * type=""}

{The major conceptual limitation of all regression techniques is that one can only ascertain relationships, but never be sure about underlying causal mechanism. + True - False
 * type=""}

{In MLR, the square of the multiple correlation coefficient or R2 is called the + Coefficient of determination - Variance - Covariance - Cross-product - Big R
 * type=""}

MLR II
 {In MLR, a residual is the difference between the predicted Y and actual Y values. + True - False
 * type=""}

{Shared and unique variance among multiple variables can be represented by a diagram that includes overlapping circles. This is referred to as a: - Homogeneity diagram - 3-way scatterplot + Venn diagram (2 circles) or Ballantine diagram (3 or more circles) - Pie chart - Path diagram
 * type=""}

{In an MLR, the r between the two IVs is 1. Therefore, R will equal the r between one of the IVs and the DV. (Hint: Draw a Venn Diagram.) + True - False
 * type=""}

{In a MLR, if the two IVs are correlated with the DV and the two IVs are correlated with one another, the rps (partial correlations) will be _______ in magnitude than the rs (Hint: Draw a Venn Diagram.) - Equal + Smaller - Larger - Impossible to tell
 * type=""}

{In MLR, the unique variance in the DV explained by a particular IV is estimated by its: - Zero-order correlation squared (r2) - Multiple correlation coefficient squared (R2) + Semi-partial correlation squared (sr2)
 * type=""}

{Interaction effects can be tested in MLR by using IVs that represent: - Cross-products between the IVs and DV + Cross-products of IVs - Semi-partial correlations squared (sr2)
 * type=""}

{A researcher wants to assess the extent to which social support from group members can explain changes in participants' mental health (MH) which is measured at the beginning and end of an intervention program. What MLR design could be used? + Hierarchical with pre-MH in Step 1 - Hierarchical with cross-products of IVs in Step 2
 * type=""}

title: Multiple linear regression/Quiz


title: Literature/1923