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User:Egm6322.s09.three.liu/HW reports

=General Nonlinear PDE=

Variable
If $$  \left \{ {x}_{i} \right \}=   \left \{ {x}_{1},...,{x}_{n}   \right \} $$,

then $$\left \{ {x}_{i} \right \}$$has n independent variables.

Example 1: 1-D Case With Time Variable (i.e. 1-D time dependent problem)

If $$(x, t)=({x}_{1},{x}_{2})$$,then

 * $${x}_{1}=x$$
 * $$ {x}_{2}=t$$

$${x}_{1}$$ is the spatial variable; $${x}_{2}$$ is the temporal variable.

Example 2: 3-D Time Dependent Problems

$$ (x,y,z,t)=({x}_{1},{x}_{2},{x}_{3},{x}_{4})$$

 * $$ x=x_{1}$$
 * $$ y=x_{2}$$
 * $$ z=x_{3}$$
 * $$ t=x_{4}$$

Components of PDE
The unknown functions are expressed as $${u}_{1}$$,$${u}_{2}$$,...

e.g.

Navier-Stokes Equation in 3D is a typical PDE where (u1,u2,u3) represents the velocity field along (x,y,z) coordinates and (x1,x2,x3,x4) represent the independent variables (x,y,z,t).

$${u}_{i}(\left \{ {x}_{j} \right \});$$ $$i=1,2,3$$ $$j=1,2,3,4$$

In this case, the PDE has one unknown function u ;

n independent variable $$\left \{ {x}_{i} \right \}$$, i=1,2,...,n;

m th partial derivative: $$\frac{\partial^m u}{\partial x_{i_1}\,...,\partial x_{i_m}}$$

i1,...,im=1,...,n

Nonlinear PDEs
$$F( \left \{ {x}_{i} \right \}, \left \{ \frac{\partial u}{\partial x_i} \right \},\frac{\partial^2 u}{\partial x_i\,\partial x_j},...)=0$$, i=1,2,...,n

This general form of PDE has n independent variables: $$\left \{ {x}_{i} \right \}$$ and 1 unknown function u.

$$\left \{ {x}_{i} \right \}$$ has n arguments

$$\left \{ \frac{\partial u}{\partial x_i} \right \}$$ is computation of gradient of u, and has n arguments

$$\frac{\partial^2 u}{\partial x_i\,\partial x_j}$$ is the Hessian derivative, and has $$\frac{n(n+1)}{2}$$ arguments

$${H}_{ij}$$ is the Hessian Matrix, and $${H}_{ij}:=\frac{\partial^2 u}{\partial x_i\,\partial x_j}$$, (":=" means "equal by definition")

$$H^T=H$$

The PDE F[(x,y),u,(ux,uy),(uxx,uxy,uyy)]=0 has 2 independent variables and 1 unknown function.

e.g. $$2{u}_{xx}+3{u}_{yy}=ax^2+bx$$

Laplace Equation: $${u}_{xx}+{u}_{yy}=0$$

$$5({u}_{xx})^2+7({u}_{yy})^2=0$$ is a nonlinear PDE since the second derivatives uxx and uyy are squared.

Linearity of PDE
 u  is an unknown function; and  L  is an operator.

 L(u)  is linear with respect to  u  if

$$L(\alpha\,\!u+\beta\,\!u)=\alpha\,\!L(u)+\beta\,\!L(v)$$

$$F((x,y),u,({u}_{x},{u}_{y}),({u}_{xx}{u}_{xy}{u}_{yy}))=0$$

$$2{u}_{xx}+3{u}_{yy}=ax^2+bx$$

$${u}_{xx}+{u}_{yy}=0$$

If  L  is a operator, and $$L(u)=2{u}_{xx}+3{u}_{yy}-7x^2-x$$

$$ L(\alpha u+ \beta v)= \alpha \left (2 {u}_{xx}+3{u}_{yy} \right ) +\beta \left ( 2 {v}_{xx} +3 {v}_{yy} \right ) +f(x) $$.

$$ \alpha L(u)+ \beta L(v)= \alpha \left [2 {u}_{xx} +3 {u}_{yy} +f(x) \right ] +\beta \left [2 {v}_{xx}+3 {v}_{yy} + f(x) \right ] $$

$$ L(\alpha u+ \beta v) \ne \alpha L(u) +\beta L(v) $$

and then, the operator  L  is nonlinear.

=Linearity for N-Dimensions=

Example of Linear map for $$R^n$$ (n-dimensional case)
 R n $$\rightarrow$$ $$\mathbb{R}$$n is a set-to-set map.

If x $$\in$$  R  n; y $$\in$$ $$\mathbb{R}$$n,

then x $$\rightarrow$$ y=Ax is a point-to-point map.

A $$\in$$ $$\mathbb{R}$$n×n, and is a n×m matrix.

D (domain) =  R n

R (range) = $$\mathbb{R}$$n

General Case
When $$\mathbb{R}$$m $$\rightarrow$$ $$\mathbb{R}$$n; m $$\neq$$ n

If x $$\in$$ $$\mathbb{R}$$n; y $$\in$$ $$\mathbb{R}$$n; and y=Ax,

then A is a n×m matrix, A $$\in$$ $$\mathbb{R}$$n×m

Now consider: y=Ax+b;

y is a n×1 matrix; A is a n×m matrix; x is a m×1 matrix; b is a n×1 matrix.

M($$\cdot$$): $$\mathbb{R}$$m $$\rightarrow$$ $$\mathbb{R}$$n

If x $$\in$$ $$\mathbb{R}$$m; y $$\in$$ $$\mathbb{R}$$n;

then x $$\longmapsto$$ y=Ax+b

So clearly,M( 0 )= b $$\neq$$ 0 $$\Rightarrow$$ M($$\cdot$$) is not a linear map, and M($$\cdot$$) is not homogeneous, either.

 Example of Rotation

In the case y=Rx+b

$$\begin{Bmatrix} y_1 \\ y_2 \end{Bmatrix}$$ = $$\begin{bmatrix} R_{11} &R_{12}    \\ R_{21} &R_{22} \end{bmatrix}$$$$\begin{Bmatrix} x_1 \\ x_2 \end{Bmatrix}$$ + $$\begin{Bmatrix} b_1 \\ b_2 \end{Bmatrix}$$ For more: Link title

 Note:  If A $$\Leftrightarrow$$ B, then A is equivalent to B.

If A $$\Rightarrow$$ B, then A is sufficient condition for B;

If A $$\Leftarrow$$ B, then B is sufficient condition for A, or A is necessary condition for B.

[A $$\Rightarrow$$ B]$$\Leftrightarrow$$ [negation of B $$\Rightarrow$$ negation of A]

=Transformation of Coordinates=

Assume $$R^m$$ and $$R^n$$ are expressions of vectors (tensors). Divergent maps of vector field (vector valued function) into a scalar function, in other words, domain and range of div(·) are functions spaces.

 Note:  In P9.1, it can't be transformed to coordinates of Eqn(5) by 2nd order linear PDEs (e.g. Eqn(2))

In coordinates:

$$x$$=$$\Phi$$($$\bar{x}$$,$$\bar{y}$$)

$$y$$=$$\psi$$($$\bar{x}$$,$$\bar{y}$$)

in coordinate transformation of Eqn(5) in P9-1

$$u$$(x,y)=$$u$$($$\Phi$$($$\bar{x}$$,$$\bar{y}$$),$$\psi$$($$\bar{x}$$,$$\bar{y}$$))

=$$u$$($$\bar{x}$$,$$\bar{y}$$)           (abuse of notation by using "u")

=$$\bar{u}$$($$\bar{x}$$,$$\bar{y}$$)      (more rigorous notation)

 Example：

Let $$u(x)=ax+b$$

$$ x=x(\bar{x})

=sin\bar{x} =\Phi(\bar{x}$$)

$$u(x)=u(\Phi(\bar{x}))

=asin(\bar{x})+b

=u(\bar{x})

=\bar{u}(\bar{x})$$

$$u(\bar{x})$$is an abuse of notation; $$\bar{u}(\bar{x})$$ is more rigorous.

$$u_x(x,y)=\frac{\partial u}{\partial x}(x,y)=u_x(\Phi(\bar{x},\bar{y}),\Psi(\bar{x},\bar{y})) =\frac{\partial}{\partial x}\bar{u}(\bar{x},\bar{y}) =\frac{\partial \bar{u}}{\partial \bar{x}}\frac{\partial \bar{x}}{\partial x}+\frac{\partial \bar{u}}{\partial \bar{y}}\frac{\partial \bar{y}}{\partial x}$$

 Define:  $$\bar{x}=\bar{x}(x,y)$$=$$\bar{\Phi}(x,y)$$; $$\bar{y}=\bar{y}(x,y)$$=$$\bar{\Psi}(x,y)$$

then $$u_y(x,y)=\frac{\partial}{\partial y}\bar{u}(\bar{x},\bar{y})=\bar{u}_{\bar{x}}\frac{\partial \bar{x}}{\partial y}+\bar{u}_{\bar{y}}\frac{\partial \bar{y}}{\partial y}$$

 Matrix form:

$$\partial_xu=[\frac{\partial \bar{x}}{\partial x},\frac{\partial \bar{y}}{\partial x}]\begin{Bmatrix} \partial_\bar{x} \\ \partial_\bar{y} \end{Bmatrix}(\bar{u})$$

$$ \begin{Bmatrix} \partial_x \\ \partial_y \end{Bmatrix}=\begin{bmatrix} \frac{\partial \bar{x}}{\partial x} & \frac{\partial \bar{y}}{\partial x}     \\ \frac{\partial \bar{x}}{\partial y} & \frac{\partial \bar{y}}{\partial y} \end{bmatrix} \begin{Bmatrix} \partial_\bar{x} \\ \partial_\bar{y} \end{Bmatrix}$$

 Jacobian Matrix:  $$\begin{bmatrix} \frac{\partial \bar{x}}{\partial x} & \frac{\partial \bar{y}}{\partial x}     \\ \frac{\partial \bar{x}}{\partial y} & \frac{\partial \bar{y}}{\partial y} \end{bmatrix}$$; (Sometimes it is defined as transformation of Jacobian matrix)

Note: An easier and more general expression:

$$(x_1,...,x_n)\longmapsto(\bar{x}_1,...,\bar{x}_n)$$

Indicial notation:

$$\bar{x}_i=\bar{x}_i(x_1,...,x_n)$$

$$\underline{J}$$=$${[\frac{\partial \bar{x}_i}{\partial x_j}]}$$n×n

i: row index; j: column index.

Example of Nonlinear Coordinate Transformation
One example of nonlinear coordinate transformation is from Cartesian Coordinate to polar coordinate:

e.g.

(x,y): Cartesian coordinate;

$$(r,\theta)$$: polar coordinate;

The transformation between the two coordinates is:

$$x=r\cos\theta=\bar{x}\cos\bar{y}$$

$$y=r\sin\theta=\bar{x}\sin\bar{y}$$

Homework: Transform the Laplace Equation into polar coordinates, and compare to published results

Find the Jacobian Matrix $$\bar J$$ and transform from Cartesian Coordinate (x,y) to polar coordinate (r,$$\theta$$)

i.e. a particular case for Eq(1) in P13-1.

Further, particular Eq(2) (P8) to Laplace Equation in Cartesian Coordinate.

Recall Laplace Equation in tensor (coordinate free)

notation: $$ div(grad\ u)=0$$

Transform Laplace Equation into polar coordinate, and compute result to published result.

Homework: Compare and translate notations between those in LP and ours Compare the expansion of Equation 1 on page 12-3 to LP (Lapiduce & Binder, 1982), found on page 14 of their book. Translate theirs into our notation.

1st partial derivative under transform of coordinates: Eq(1) in P12-3. (using chain rule)

In LP, P14: 2nd partial derivative (using chain rule), translate it into new coordinate (ours).

i.e. ours: $$(\bar x_1, \bar x_2)=(\bar x, \bar y)$$

$$\bar {x_i}=\bar{x_i} (x_1,x_2) $$    (i=1,2,...)

LP: $$\xi=\phi(x,y)$$

$$\eta=\psi(x,y)$$

LP (1.2.5 ab): 1st partial derivative

LP (1.2.6 $$\theta$$): 2nd partial derivative

Obtain the complete form:

$$u_{xx}=u_{\xi\xi}(\phi_{x})^2+2 u_{\xi \eta} \phi_{x} \psi_{x}+u_{\eta \eta}+(\psi_{x})^2+...$$

Recall Jacobian Matrix    (P12-3)

$$\bar x_{i}=\bar x_{i}(x_1,...,x_n)$$

$$\bar J=\begin{bmatrix} \partial {\bar x_i}     \\ \partial {\bar x_j} \end{bmatrix} $$

2-Dimensions

$$\bar J=\begin{bmatrix} J_{11} & J_{12} \\ J_{21} & J_{22} \end{bmatrix}=\begin{bmatrix} \phi_{x} & \phi_{y} \\ \psi_{x} & \psi_{y} \end{bmatrix}$$

(1.2.6) in LP is:

$$u_{xx}=u_{\bar x \bar x}(J_{11})^2+2 u_{\bar x \bar y} J_{11} J_{21}+u_{\bar y \bar y}^2$$

Derive above equations: (1.2.6~7~8) in LP at once by matrix operator form: Eq(1) P12-3

$$\begin{Bmatrix} \partial x \\ \partial y \end{Bmatrix} \left \lfloor \partial_x \; \partial_y \right \rfloor =\begin{bmatrix} \partial {xx}     & \partial {xy}      \\ \partial {yx}     & \partial {yy} \end{bmatrix}=\bar J^T \begin{Bmatrix} \partial {\bar x} \\ \partial {\bar y} \end{Bmatrix} \left \lfloor \partial_x \; \partial_y \right \rfloor \bar J $$

Note:

$$\left \lfloor \partial_x \; \partial_y \right \rfloor= \left \lfloor \partial_{\bar x} \; \partial_{\bar y} \right \rfloor \bar J$$

$$(\bar A \bar B)^T={\bar B}^T{\bar A}^T$$

$$\alpha={\bar J}^T \begin{bmatrix} \partial_{\bar x \bar x}     & \partial_{\bar x \bar y}      \\ \partial_{\bar y \bar x}     & \partial_{\bar y \bar y}     \end{bmatrix} \bar J $$

Find $$\partial_{xx}$$

$$\begin{bmatrix} J_{11}     & J_{21}      \\ J_{12}     & J_{22} \end{bmatrix} \begin{bmatrix} \partial_{\bar x \bar x}     & \partial_{\bar x \bar y}      \\ \partial_{\bar y \bar x}     & \partial_{\bar y \bar y}  \end{bmatrix} \begin{bmatrix} J_{11}     & J_{12}      \\ J_{21}     & J_{22} \end{bmatrix} = \begin{bmatrix} a    & ? \\ ?      & ? \end{bmatrix} $$

$$a=(\partial_{\bar x \bar x}(J_{11})^2)+\partial {\bar x \bar y}J_{11}J_{21}+\partial {\bar y \bar x}J_{11}J{21}+\partial {\bar y \bar x}J_{11}J_{21}+\partial {\bar y \bar y}(J_{21})^2 $$

=lecture 19=

In the rotation and translation of coordinates,

$$\alpha=\beta+\gamma$$

and in this equation,

$$\alpha = \underline{J}^{T}\begin{Bmatrix} \partial_{\underline{x}}\\ \partial_{\underline{y}} \end{Bmatrix} \left \lfloor \partial_{\overline{x}} \; \partial_{\overline{y}} \right \rfloor \underline{J}+ \begin{Bmatrix} \partial_{x}\\ \partial_{y} \end{Bmatrix} \left \lfloor \partial_{\overline{x}} \; \partial_{\overline{y}} \right \rfloor \underline{J}$$

$$ \ \beta=\underline{J}^{T}\begin{Bmatrix} \partial_{\underline{x}}\\ \partial_{\underline{y}} \end{Bmatrix} \left \lfloor \partial_{\overline{x}} \; \partial_{\overline{y}} \right \rfloor \underline{J}$$

$$ \gamma= \begin{Bmatrix} \partial_{x}\\ \partial_{y} \end{Bmatrix} \left \lfloor \partial_{\overline{x}} \; \partial_{\overline{y}} \right \rfloor \underline{J}$$

$$\displaystyle {\rm div}( {\rm grad} \, u )=u_{xx}+u_{yy}=u_{rr}+\frac {1}{r^2}u_{\theta \theta}+\left \lfloor \frac {1}{r} \; \theta \right \rfloor \begin{Bmatrix} \partial_r u \\ \partial_{\theta} u \end{Bmatrix} $$

In this case,

$$\alpha=\displaystyle {\rm div}( {\rm grad} \, u )$$

$$\beta=u_{rr}+\frac {1}{r^2}u_{\theta \theta}$$

$$\gamma=\left \lfloor \frac {1}{r} \; \theta \right \rfloor \begin{Bmatrix} \partial_r u \\ \partial_{\theta} u \end{Bmatrix}$$

Classifications
In the general form of PDE:

$$au_{xx}+2bu_{xy}+cu_{yy}+du_{x}+eu_{y}+fu+g=0$$

it can be classified by the value of $$ac-b^2$$.

$$ac-b^2=det \underline A$$

$$ac-b^2<0 \Rightarrow$$ PDE is hyperbolic.

$$ac-b^2=0\Rightarrow$$ PDE is parabolic.

$$ac-b^2>0\Rightarrow$$ PDE is elliptic.

For example:

In the equation:

$$u_{xx}+u_{yy}=0 $$...(1)

$$\Rightarrow a=1,b=0,c=1$$

$$\Rightarrow ac-b^2=1>0$$

$$\Rightarrow$$ This PDE is elliptic

Question: what would be the classification of diffusion operator $$\displaystyle {\rm div}( {\rm grad} \, u )$$ in polar coordinate?

Another way to find the type of equation (1):

$$\underline \bar A:=\underline J \underline A \underline J^T =\begin{bmatrix} A     & B      \\ B & C \end{bmatrix} =\begin{bmatrix} \bar a     & \bar b      \\ \bar b & \bar c \end{bmatrix} \to$$ preferred form

$$det \underline \bar A=\bar a \bar c-\bar b^2$$

$$\bar a=1; \bar b=0; \bar c=\frac {1}{r^2} (r\ne0)$$

$$\Rightarrow$$The equation (1) is an elliptic PDE.

Relationship Between Classifications and Transformations
Observations from the verification of PDE classifications:

1. Diffusion operator remains elliptic in polar coordinates.

Question:How about a different transformation of coordinate? Would classification remain the same?

2. Does classification make sense if it changes under transformation of coordinate?

The answer is no, because physics (e.g. distribution of temperature as a result of solution of heat equation) must remain the same regardless of how heat equation was solved (under different coordinate system).

Therefore, classification better remains the same under different coordinate system for it to make sense.

=mtg29= Homework: Plot $$J_n(z)$$ and $$Y_n(z)$$ for $$n=0,1,2 ...$$$$z \in \Re$$ Use Matlab and include code.

Superposition: 2 separation of variables, constant $$\rho$$, $$\lambda$$

$$\sum_{\lambda}\sum_{\rho}[A(\lambda)exp(-\lambda t)][B(\rho)sin(\sqrt{\rho}\theta)+C(\rho)cos(\sqrt{\rho}\theta)] [D(\rho,\lambda)J_{\sqrt{\rho}}(\sqrt{\lambda} r)+E(\rho,\lambda)Y_{\sqrt{\rho}}(\sqrt{\lambda} r)]$$

Up to now, no restriction on $$\rho$$ and $$\lambda$$, i.e. they can be continuous real numbers (as opposed to discrete integral numbers as in Fourier Series: think of Fourier integral or Fourier Transformation and Laplace Transformation)

$$\Rightarrow$$ Replace discrete sum $$\sum$$ by continuous sum of $$\int$$.

Define:

$$\bar{B}(\rho,\lambda):=A(\lambda)B(\rho)$$

$$\bar{C}(\rho,\lambda):=A(\lambda)C(\rho)$$

General solution:

$$u(r,\theta,t)=\int d\lambda \int d\rho[exp(-\lambda t)][\bar{B}(\rho,\lambda)sin(\sqrt{\rho}\theta)+\bar{C}(\rho,\lambda)cos(\sqrt{\rho}\theta)][D(\rho,\lambda)J_{\sqrt{\rho}}(\sqrt{\lambda} r)+E(\rho,\lambda)Y_{\sqrt{\lambda}r}] $$

The Primitive of Functions
The primitive for function F is:

$$Q(x,\xi):=\int F(x,\xi)d\xi$$

e.g. The primitive for x is:

$$\int xd\xi=\frac {1}{2}x^2+k$$

The definition for the integral of x is:

$$\int_{x=a}^{x=b}x dx=\frac {1}{2}(b^2-a^2)$$

HW: Check the definition of indefinite integral The indefinite integral of a function $$f(x)$$ can be denoted by

$$\int f(x)dx$$

For example, if $$n \ne -1$$, we have

$$\int x^n dx=\frac {x^{n+1}}{n+1}+C$$

In this case, $$C$$ denotes an arbitrary constant. This is because the function $$f(x)$$ has many indefinite integrals, but any two differ by a constant. Consequently, if one antiderivative of a function $$f(x)$$ is found and added by an arbitrary constant, every indefinite integral for the function is found.

Generally, the indefinite integral for $$F(x,\xi)$$ is

$$ \int F(x,\xi)d\xi=G(x,\xi=B(x))-G(x,\xi=A(x))$$

$$\frac {d}{dx}F(x,\xi)

=[\frac {\partial G(x,B(x))}{\partial x}+\frac {\partial G(x,B(x))}{\partial \xi}]-[\frac {\partial G(x,A(x))}{\partial x}+\frac {\partial G(x,A(x))}{\partial \xi}]$$

$$=[\int_{\xi=A(x)}^{\xi=B(x)}\frac {\partial}{\partial x}F(x,\xi) d\xi+F(x,B(x))\frac {dB(x)}{dx}]-[\int_{\xi=\xi_0}^{\xi=A(x)}\frac {\partial}{\partial x}F(x,\xi) d\xi+F(x,A(x))\frac {dA(x)}{dx}]$$

$$\frac {\partial}{\partial x}F(x,\xi)=\int_{\xi=A(x)}^{\xi=B(x)}\frac {\partial}{\partial x}F(x,\xi) d\xi+F(x,B(x))\frac {dB(x)}{dx}-F(x,A(x))\frac {dA(x)}{dx}$$

Divergence operator in polar coordinate

Basic vector in polar coordinates:

$$\bar{\underline{e_i}}=\frac {\partial P}{\partial \bar{x_i}}; i=1,2,...$$

$$\bar{\underline{e_1}}=\underline{e_r}=\frac {\partial P}{\partial r}=\frac {\partial (\overrightarrow{OP})}{\partial r}; (\bar{x_1}=r)$$

$$\overrightarrow{OP}=x\overrightarrow{i}+y\overrightarrow{j}=(rcos\theta)\underline{i}+(rsin\theta)\underline{j}$$

$$\bar{\underline{e_2}}=\underline{e_\theta}=\frac {\partial P}{\partial \theta}; (\bar{x_x}=\theta)$$

$$ \underline{e_r}=\frac {\partial P}{\partial r}=cos\theta\underline{i}+sin\theta\underline{j}$$

$$\underline{e_\theta}=\frac{\partial P}{\partial \theta}=-rsin \theta \underline{i}+rcos\theta \underline{j}$$

$$\begin{Vmatrix} \underline{e_r} \end{Vmatrix}=1$$

$$\begin{Vmatrix} \underline{e_\theta} \end{Vmatrix}\neq1=r$$

=mtg 39=

Homework: Get the Canonical form for 2nd order linear Elliptic PDE x

Homework: Get the Canonical form for 2nd order linear Hyperbolic PDE For hyperbolic PDE: $$\lambda_1 u_{xx}-\lambda_2 u_{yy}=g$$

$$\begin{Bmatrix} \partial_{x} \\ \partial_{y} \end{Bmatrix} =J_B \begin{Bmatrix} \partial_{\xi} \\ \partial_{\eta} \end{Bmatrix}$$  ... (1)

The Jacobian matrix is:

$$ J_B= \begin{bmatrix} \frac {1}{\sqrt {\lambda_1}} & 0 \\ 0 & \frac {1}{\sqrt {\lambda_2}} \end{bmatrix}$$

so the hyperbolic PDE can be written as:

$$\left \lfloor \partial_x \partial_y \right \rfloor \begin{bmatrix} u \lambda_1 & 0 \\ 0 & -u \lambda_2 \end{bmatrix} \begin{bmatrix} \partial_x \\ \partial_y \end{bmatrix}=g$$

substituting (1) into the above equation gives:

$$\left \lfloor \partial_{\xi} \partial_{\eta} \right \rfloor \begin{bmatrix} \frac {1}{\sqrt {\lambda_1}} & 0 \\ 0 & \frac {1}{\sqrt {\lambda_2}} \end{bmatrix} \begin{bmatrix} u \lambda_1 & 0 \\ 0 & -u \lambda_2 \end{bmatrix} \begin{bmatrix} \frac {1}{\sqrt {\lambda_1}} & 0 \\ 0 & \frac {1}{\sqrt {\lambda_2}} \end{bmatrix} \begin{bmatrix} \partial_{\xi} \\ \partial_{\eta} \end{bmatrix}=g$$

So the canonical form for this hyperbolic PDE is

$$u_{\xi \xi}-u_{\eta \eta}=g$$

Homework: Get the Canonical form for 2nd order linear Parabolic PDE x

Homework: General classification of 2nd order linear PDEs to 2nd order nonlinear PDEs x

Homework: Verify (2) in Solution of (1). Derivation of d'Alembert solution for 1-D wave equation x

Consider: g=0 in Equation $$U_{\xi\xi}-U_{\eta}=0$$

$$U_{\xi\eta}=0$$

$$U{\xi,\eta}=F(\xi)+G(\eta)$$

where $$F(\cdot)$$ and $$G(\cdot)$$ are any function that is differentiable to first order.

$$c^2w_{xx}=w_{tt}$$

$$\tau w_{xx}+P=mw_{tt}$$

Classification of conics and 2nd order linear PDEs.

Transformation of coordinate via eigenvalue problem.

$$\bar a=a(\phi_x)^2+b\phi_x \phi_y+c(\phi_y)^2$$

$$\bar b=...$$

$$\bar c=a(\psi_x)^2+b \psi_x \psi_y+c(\psi_y)^2$$

$$\bar{\underline A}=\underline J \underline A \underline J^T= \begin{vmatrix} \bar a & \bar b \\ \bar b & \bar c \end{vmatrix}$$

To obtain form 2 of canonical form

Eq (3) P 39-1 set $$\bar a=\bar c=0$$

title: User:Egm6322.s09.three.liu/HW reports


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User:Egm6322.s09.three.liu/final report

=Wave Equation=

The Derivation of Wave Equation
Consider an infinite small segment of an elastic string shown in the picture, a small amplitude vibration of this string conform to the wave equation.

In the picture,

$$u(x,t)$$ is the vertical displacement of the string from the $$x$$ axis.

$$\theta(x,t$$) is the angle between the string and a horizontal line.

$$T(x,t)$$ is the tension in the string.

$$\rho(x)$$ is the mass density of the string.

$$F(x,t)$$ is the external force on the string.

Applying Newton's law in the vertical direction:

$$\rho(x) \sqrt{\Delta x^2+\Delta u^2} \frac {\partial^2 u}{\partial t^2}(x,t)=T(x+\Delta x,t)sin\theta(x+\Delta x,t)-T(x,t)sin\theta(x,t)+F(x,t)\Delta x$$

Divide $$\Delta x$$ on both RHS and LHS and take the limit as $$\Delta x \rightarrow 0$$ gives:

$$\rho(x)\sqrt{1+(\frac {\partial u}{\partial x})^2}\frac {\partial^2 u}{\partial t^2}(x,t)$$

$$=\frac {\partial}{\partial x}[T(x,t)sin\theta(x,t)]+F(x,t)$$

$$=\frac {\partial T}{\partial x}(x,t)sin\theta(x,t)+T(x,t)cos\theta(x,t)\frac{\partial \theta}{\partial x}{x,t}+F(x,t)$$ ...(1)

From the geometric relation in the picture,

$$sin\theta(x,t)=\frac{\frac{\partial u}{\partial x}(x,t)}{\sqrt{1+(\frac{\partial u}{\partial x}(x,t))^2}}$$

$$cos\theta(x,t)=\frac{1}{\sqrt{1+(\frac{\partial u}{\partial x}(x,t))^2}}$$

$$\theta(x,t)=tan^{-1}\frac{\partial u}{\partial x}(x,t)$$

$$ \frac{\partial \theta}{\partial x}(x,t)=\frac{\frac{\partial^2 u}{\partial x^2}(x,t)}{1+(\frac{\partial u}{\partial x}(x,t))^2}$$

Because $$\left | \theta(x,t) \right \vert<<1$$,

$$\sqrt{1+(\frac{\partial u}{\partial x})^2}\approx 1$$

$$ sin\theta(x,t)\approx\frac{\partial u}{\partial x}(x,t)$$

$$cos\theta(x,t)\approx1$$

$$\frac{\partial \theta}{\partial x}(x,t)\approx\frac{\partial^2 u}{\partial x^2}(x,t)$$

Substituting these into equation (1) gives

$$\rho(x) \frac{\partial^2 u}{\partial t^2}(x,t)=\frac{\partial T}{\partial x}(x,t)\frac{\partial u}{\partial x}(x,t)+T(x,t)\frac{\partial^2 u}{\partial x^2}+F(x,t)$$

Assume that the string only has transverse vibration, which means the horizontal force on it is zero.

So $$T(x+\Delta x,t)cos\theta(x+\Delta x,t)-T(x,t)cos\theta(x,t)=0$$

As $$\Delta x \rightarrow 0$$

$$\frac{\partial}{\partial x}[T(x,t)cos\theta(x,t)]=0$$

For small vertical vibration,$$cos\theta \rightarrow 1$$, and$$\frac{\partial T}{\partial x}(x,t)\rightarrow 0$$

Therefore, Equation (1) can be further simplified to:

$$\rho(x)\frac{\partial^2 u}{\partial t^2}(x,t)=T(t)\frac{\partial^2 u}{\partial x^2}(x,t)+F(x,t)$$

Assume the density is constant and no external forces, the second order 1-D wave equation finally goes to

$$\frac{\partial^2 u}{\partial t^2}(x,t)=c^2 \frac{\partial^2 u}{\partial x^2}(x,t), c=\sqrt{\frac{T}{\rho}}$$

In water waves, $$c$$ is the phase speed.

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

title: User:Egm6322.s09.three.liu/final report


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User:CarrieBerg/CarrieBerg Recipe

By: Mrs. Chaucer

Ingredients:

2 cups Couplets that Rhyme

1 cup chopped Older Tales

1 dash Middle English

3 tsp Mixed-Up Romance (can be substituted with 3 tsp Bitter Revenge if Romance is too sweet)

1 lb Different Perspectives

2 tbsp Intergroup Conflict/Rivalry

1 Mediator

2 Marinated Fools

2 Sober Sages

A few Feminine Touches

A Smattering of Realism

A Touch of Irony

1 can Intelligent Frosting

1 bottle Religious Sprinkles

Preheat oven to 375 degrees. Sift Couplets well then pour them into a bowl and mix with Older Tales, Middle English, Different Perspectives, Feminine Touches, and Realism. In a separate bowl mix Intergroup Conflict, Mediator, Marinated Fools, Sober Sages, and Irony. Pour together and stir well. Mix on high for 3 minutes while slowly adding in Romance (or Revenge). Pour into a 12x12 baking pan and bake for 30 minutes. Test to see if done. Now you should see it rise to form a nice firm story with only a few holes. Frost with a veneer of Intelligence and sprinkle liberally with Religion. Cut into 24 uneven pieces. Serve with a tall glass of alcohol to enjoy to the fullest.

(This was written in about 2 minutes in class as a joke. My teacher for my class on Chaucer had written "Ingredients to make a Canterbury Tale" on the board, and I, well, took him literally. When he asked for ideas, I read him this. The entire class applauded me. It is one of my fondest college memories, and so I'm sharing it with all of you.)

title: User:CarrieBerg/CarrieBerg Recipe


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User:CarrieBerg/A Matter of Perspective

Once, when I was younger, I wondered why anyone would become a villain. Now that time has taught me the truth, I find myself wistful for the days when I did not know the answer. But you can’t change what you know, even if you can travel back in time.

You see, once I had been the most powerful sorceress in the world. I had mastered all the known spells, from Wendor’s Flying Oscillator to the Seven Mystical Ilms of Dontara. Most of them learned from long harsh study under ancient sorcerers. I even went on to create two or three new spells, thus assuring the world that I was indeed more powerful than my tutors. You might have heard of Flidarine’s Mysterious Cube. I consider it to be my best spell. Though I find the Fifth Level of Torr is usually my most well known creation.

Spells were not the only measure of how powerful a hero I was. I had also traveled to all the known countries, and a few unknown for good measure. I battled scores of evil beasts, some too terrible to name. And I had saved numerous people and places from certain doom. Legends were created of my deeds, and tales were sung about me from the Purple Sandseas of Yllor to the mountains of Kr’gga

In short, I was young and proud and powerful.

But as the centuries went by, I found that travel began to get tiresome. For a long time I wondered why. It had never bothered me before. But one night, while camping in the forest because the superstitious farmers didn’t want me in their village, it came to me. I was miserable traveling because I wished to settle down. To have a place of my own, like those farmers, where I was accepted for who I was.

It happens to the best adventurers. We have seen the world twice over. We have spent our entire lives traveling hither and yon, fixing all the world’s problems. Someday we wake up and come to find we would much rather stay in a bed we know will be clean and warm and dry. We would rather have the time and space to make potions in an actual laboratory instead of on the fly over a fire in a makeshift cauldron. And the only thing we wish to fix is a warm cup of coffee in the morning, a sandwich at lunch, and a spicy stew for dinner.

So it was with me.

But a retired hero is never as welcome as an active one. Or so I quickly found out. For while the villagers are pleased enough when we stroll through their towns able and willing to fix their problems, they are just as pleased, if not more so, when we leave. Soon finding a place to settle down became quite ridiculous. At each city I traveled through, I would politely hint that I would like to live there. I’d say something like this, “I am going to stop traveling soon and build a tower somewhere, perhaps with a garden.” The words were no sooner out of my mouth when someone would say,

“I hear Ablivan is lovely this time of year.”

“Aye, and they have lots of uninhabited land.”

“I’ll get you a map, and you can be one your way.”

It was like this in every town, city, village, and even crofter’s halls. The place mentioned always changed, but the message was quite clear. I could settle down, just not by them. For each place they mentioned was always at least a kingdom and a half away.

Now, I know you must be thinking that this is when I became a villain. I lost my calm, razed a township in some remote area and built myself a tower. And the humble garden I had wanted began as a battlefield watered with blood. Well, I’m sorry to disappoint you, but you would be wrong.

I have never killed. I was not about to start then. And even though I know both Eulucid’s Mantle of Death and Grembator’s Mighty Rock; to this day I have never used them. The beasts I fought in my youth I did not defeat with death, but with compromise, a show of force, or my personal favorite: a riddle contest. The last was always the most fun. They would always agree, believing they could out riddle me. But they always lost. And the price for playing was always what harm they were causing. Lose and they must stop killing, razing fields, or eating the souls of the unwary. Since my magic sealed the bargain there was no way out. Some left to raze elsewhere, thinking it was just in that region that they agreed too stop. Some saw the wisdom in change and lived reformed lives. Most cursed my name, and swore that I too would suffer someday as they did. I had never believed them.

Until now.

For with each rejection I grew more and more depressed. I, the Greatest Sorceress of this Century, could not even find a place to grow herbs in peace! I was pathetic. To come so far and do so much for the people only to be thwarted from achieving the only goal I ever wanted for myself. It was then, at my lowest point, that I met him.

It was not, as the tales tell, on a dark and stormy night, in the Forest of Dreams. He was not wearing black, blending in with the shadows. He is not even evil. At least, not what I consider evil. Not anymore. And yet he is considered the worst of the worst. The most wanted villain in all two-hundred and seven kingdoms, city-states and principalities. All because he believes that heroes should not die alone in the wilderness.

But I will say this, the tales did get one thing right: he did offer me my heart’s desire.

I was in Zudoran, trying to lose my cares in one of their many fine bars. Wishing I could get drunk again. But I couldn’t. I couldn’t because of a minor side affect from casting Vityavar’s Ever-Flowing Waters. The spell created the waters of life, which stopped aging. It was necessary for any self-respecting sorcerer or sorceress to cast since it takes a century alone just to learn all the spells there are. But no one ever mentions the side affect - no drink would every again taste as sweet or get you as drunk as that glorious elixir. And it could only be cast once in a lifetime. And the lifetime of a sorceress was very long indeed without a way to dull the pain.

At first I did not notice him. I was too busy trying to create a spell that would allow, even if just for a moment, the beer to cause drunkenness. I was failing miserably at that as well, which certainly wasn’t helping my mood. When my drink was done, he bought me another. I have to admit, I did not think much of him. His hair badly needed to be cut, his cloak was so old it had been bleached by the weather to a faded grey, his boots were scuffed and worn. He seemed like just another rogue trying to play up to the damsel in distress. Not that many people had tried before. Usually my reputation preceded me. And, to tell you the truth, I didn’t look like much either.

I have never been your classic sorceress with flashing eyes the color of thunder, hair like a shadowed fog and skin as smooth and pale as a princess’s. Life traveling had tanned my skin a uniform brown. My hair had always been halfway between blond and brown, halfway between curly and straight. I grew sick of wasting spells to keep it mysterious and flowing and smooth. Now it hung down my back, a tangled thicket who’s only mystery was if its color was closer to dirty blond or to sun-kissed brown. My eyes do not flash, but they do shift. Sometimes it is that way with sorcerers. A rare few of us have this unique feature. Our eyes shift from seeming as common as a villager’s to as dangerous as an elemental’s. Mine shift from homey brown to dazzling gold. They can shift at will or by my command. Most of the time I don’t bother to shift them anymore. So I never know exactly what color they are to other people. But in my pain that evening, I am sure they were a dangerous glinting gold. A subtle signal to back off.

So I turned to warn him off, eyes flashing. But when I looked into his eyes, as faded gray as his cloak, they held too much sympathy and understanding. I still remember the conversation we had next as clearly as if it happened yesterday, and not two centuries ago.

“Are you the Sorceress Flidarine?”

“So what if I am.” I was in no mood to be grateful for his courtesy.

“I’ve heard you’re looking to build a home.”

“Yes. But I haven’t had much luck.”

“Have you tried the Hills of Breven?”

Here we go again, I thought. He was trying to get me out of the city. Sending me halfway to nowhere out in some forsaken wilderness that hadn’t seen civilization for two thousand years. The Hills of Breven were known for three things. One: they were the site were the Sorcerer Durran had last been seen, which was little over two thousand years ago. And he was still at large as the most wanted villain. He was at large because two: the Sorcerer Durran was believed to be the reason the city of L’hta walled itself up and supposedly disappeared. It was the only city located in the hills. And three: no one had dared to live in the hills after L’hta had vanished. Rumors were that anyone who tried also vanished. He seemed to sense my resentment and said,

“It’s not as bad as you think. L’hta is the chosen retirement city for heroes. I should know. I made it so.”

At his words I looked up and I found I couldn’t speak. For behind the look of the worn traveler were the eyes of a sorcerer. The color of flashing storm clouds. How could I have missed it? His eyes shifted like mine. It was a few minutes before he spoke again. His voice edged with the knowledge and sorrow of ages.

“It is, I find, a matter of perspective.”

Intrigued at meeting such a notorious figure, my resentment vanished. Durran shifted into a more comfortable position before continuing.

“When we heroes are traveling, there is no problem. Heroes are wild and free, and traveling is wild and free. We spread the wildness around, so to speak. Keep it in check. So when we are on the road, all is well. The world is in balance. But when a hero settles down, all that wildness inside him has a nasty tendency of coming out.

“Sure, he may say he is retired, but then old enemies show up. New pupils arrive to be trained. That mundane garden somehow grows a rare herb that healers and alchemists from around to world wish to study or purchase. Kings send messengers, rivals set up a home nearby, lovers past and present make a nuisance of themselves, and the careful balance of the world is upset. At least, that is how those who are not heroes see it.

“Heroes have a special place in the world, but never for long. We are expected to do great deeds, create wonderful tales, and then go out in a burst of glory. Nice and tidy. There to solve problems when you need them, and neatly packed away when you don’t. We are not supposed to fade away over the centuries, tending gardens and taking up hobbies we couldn’t do while saving the world. The new heroes can’t have the old heroes hanging around. Never mind that the old heroes, like us, teach the young ones all they know. Never mind that we no longer go around saving this village here, or that damsel there.

“We’ve had our time. Now we want our reward: peace and quiet without the weight of the world on our shoulders. And most of all, we want someplace to call home. Someplace were we can grow our gardens, make our meals, read our books in peace. Someplace were we don’t have to envy the people we protect - those people who have nice homes yet never realize how nice they are. Someplace were we won’t be run out of town for being different, magical, or wild. That someplace is L’hta.

“I was in the same situation as you are now. I found that I was weary of travel, that I wanted a warm fire in a cozy fireplace and bookshelves filled with all the books I had collected over the years, but never found time to read. I wanted a place where I could sit down and just relax. Perhaps create a new spell or two if I had the silence to think about it. I found that each city I went to turned me away with feeble excuses. Some agreed - but only if I gave up my magic and all heroic deeds I had done. Unwilling to give up who I was, I finally took one of the earlier suggestions. I went to the Hills of Breven and I built L’hta.

“At first it was just my house. But other heroes soon heard of what I had done and came to see if it was true - that I was now retired. I was, and some found they rather liked the retired life. It beats getting killed in battle. Soon L’hta was a collection of buildings. From towers made of glass to simple farmsteads. Not everyone there is a sorcerer. Great sword-slingers sometimes wish to retire, as do all manner of other heroes. Since I knew how hard it was to get accepted into a city, I let them stay.

“Sadly, some left. I did not try to stop them. It was their choice, after all, even if I thought it was the wrong choice. They changed their names and settled into normal cities. But by giving up their names, they gave up that part of their lives. They were no longer heroes. They are all dead now, their true names forgotten, their heroic deeds lost with time. It was sad to see them leave, but it was not surprising. They craved to settle in a real city, with common folk. It was painfully obvious that while we could live and build our homes in L’hta, it would never be common.

“For the villagers were right. With all those heroes living together, wildness poured out into the countryside. Strange occurrences began to happen in the hills. To protect the land, I shifted the city, altering it to be just outside this world, this plane, this time. We heroes of L’hta are a people apart. Ironically, it was then that I became branded as a villain. For saving the heroes from the world, and the world from the heroes, I was now the ultimate evil. The villagers and kings, the courtiers and merchants, they all believed that I had killed the heroes. To them it was a perfect ending to a rather messy matter - here was their needed blaze of glory! No longer were the useless heroes hanging around. And a new villain was created.

“I realized then that heroes and villains are not so different. It is, as a said, a matter of perspective. What saves the world for one person is proclaimed as evil by the next. So I sent out invitations to the villains to come and live with us. Most believed it was a plot to kill them. I did not blame them, if I had been invited to live in a city by my ancient enemy, I probably wouldn’t go either. They have an easier time of it out here as well. They do not need acceptance for where they live. They are quite willing to raze a city or two to get the perfect space to build a fortress. Still, there are a few who joined us. Those who grew tired of endlessly tormenting minions. Of being feared wherever they went. We do not judge their pasts, for they too just want a place of peace and quiet where they can relax.

“But the damage was done. Word spread that not only had I destroyed L’hta, now I was building up an army of villains in the hills. So we are all villains now. All the greatest heroes of Millidor. Now the Dark Army of Durran. How amusing it would be if they could see this ‘great and terrible army’ now! Sadly they are all too aware that I am still ‘recruiting’ after all these years. Perhaps someday they will understand.”

With that he fell silent. His eyes filled with painful memories. I understood. Sorcerers do not easily forget anything. We are trained to remember all the words of all the spells. Perhaps we are trained too well. For memories never fade with time for us. Pain is never dulled. And alcohol does nothing. Sometimes it is terrible to be a sorcerer. Or sorceress.

It should be obvious what happened next. He had won me over with his tale. For he offered me what I most desired: a place to call home. A welcome place where I could be accepted, even if it was a pale imitation of an actual city. I did not mind. At the time I would have gone to anywhere I would be accepted. His offer was just the first. So when he asked if I would join the other heroes, I agreed. I would become the latest member of what is still called the Dark Army of Durran. Never mind that we have never attacked anything, or done any harm. Never mind that we seldom leave L’hta, now that we live there in peace.

But we can leave. It is as Durran said; L’hta is a pale imitation. Sometimes we have to come back and walk the lands once more to remember how a real city feels. Or to attend a family member’s funeral, one who was not a hero. Or the funeral of a hero who stayed behind, giving up their name and so giving up their agelessness. We are all wanted now, so we must be careful not to reveal ourselves. We do not regret our retirement. It is nice to relax with a cup of tea and a new book, able to read without interruptions. To engage our minds in a friendly argument with someone we once fought against.

I wasn’t the last to join. Indeed, in the past two centuries more and more heroes have been arriving. Our city expands to hold us all. If the outside world had any idea as to the number of us, they would have quailed in their boots and fled to the farthest corners of the world. It is probably a good thing that we are not evil. At least, not in the way they believe.

Is it really so wrong to want a little peace? To settle down and grow a garden filled with periwinkles and Mipp’s Wondrous Snigberries? Are we evil for wanting to live out the rest of our lives without having to run off and save something or someone?

Some would say yes. That a hero's sole purpose is to save things. But you see, as Durran said, it is all a matter of perspective. While you may believe we are villains now, we are willing to wait in L’hta until a time when heroes are not so plentiful. Until you realize that perhaps chasing us away was not the best course of action. When that day comes, we will return, seeking out our ancestral homelands, visiting places openly where we once walked, and making our homes among you.

This time you will not turn us away, for you will find that life without adventure is a horribly dull way to live. And on that day, once again we will be heroes.

title: User:CarrieBerg/A Matter of Perspective


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User:CarrieBerg/Feathers

“Lady.”

Jalline turned around. A face was pressed again the bars of a cell. It belonged to an old woman, her eyes seemed to glow fever bright. Jalline pitied her. To be ill and locked away was far worse than to be healthy. At least Jalline still had her health.

“I am no great lady.”

“But you are. You just do not know it yet.” The old woman’s eyes grew brighter still. “I am sorry, but you are mistaken.” Jalline turned to go. The guards tugged at her bonds forcing her to take a quick step.

“Wait! Wait... there is something I must tell you!”

Jalline knew it was foolish, but the old woman sounded so worried she couldn’t help herself, without thinking she stopped and spread out her wings. They slipped through the bars of the cells on either side of her, preventing her from moving, at least temporarily. She turned back to the old woman. “I’m listening.”

“You will have to make a choice.”

“What choice?” Jalline longed to make the woman more comfortable, even though a slow chill was creeping up her spine.

“Hurry it up, would you? It’s not like I have all day to watch you!” the guard called, after a few short tugs on the rope. Seeing that there was no way to move Jalline while her wings were laced between the bars, he settled back, grimacing at his newest prisoner.

“A difficult choice, but it must be yours! When the time comes, you must be willing to give up your wings!”

Jalline went cold. “My wings?” she whispered. Her wings were her pride and joy. They were of the prophecy, or so the priests said. She was a child of earth and sky, and the unusual coloring of her wings proved it. They were not white or silvery gray, like the other sylphii. No, her’s were the soft blues of the sky at dawn, the pale greens of tender shoots of grass. Her’s were where the earth met the sky. To give up her wings would be... Jalline couldn’t even imagine.

It wasn’t just her vanity at the loss of their colors. No. It was without them, she would lose her entire way of life. She was a sylphii, and what would a sylphii be without her wings? To forever be denied the rush of air in flight? Their tender softness as they cocooned her at night? Their subtle weight balancing on her back? To lose her wings would be to lose her life.

“I...”

“Hush child. You need not chose now. But remember. When the time comes, give them freely.”

“I will never give up my wings!” Jalline whispered fiercely.

The old woman’s eyes seemed almost normal now. They shone with a sympathy that was so deep it hurt Jalline to look into them for long. Jalline turned her head away. The woman spoke, “But they will take them just the same.”

Jalline looked back at the old woman’s cell. The woman had turned around, her back facing Jalline. Jalline stared in horror at the scars on her back. They will take them just the same...  Jalline looked around at all the other prisoners. Those that were turned away also had telltale scars. Some sat staring out windows blankly, no life behind their dull eyes. It had fled when they lost their wings. Their wings. Others crowded close to their doors, watching her pass with hungry eyes. Some reached out to touch her wings, a gentle caress as if remembering what they had lost. For they were all sylphii. And they had all lost their wings.

When Jalline reached her cell, she sat and stared out the window for a long time, alone with her thoughts. For while the hopelessness of the situation had not touched her at first, now she truly knew fear. She drew herself into a tiny ball. They would not take her wings. She wasn’t going to stay here for long. The priests would come and get her. And as the hours grew longer, her thoughts circled around. Always coming back to the phrase, they would not take her wings.

 .:.:.:.:.:.

Days became months. Months dragged on into what could have been a year. Jalline didn’t know. She didn’t keep track. She found that counting only made things worse. Some of the other prisoners counted. They kept track on their walls. Brown stains of dried blood ticking off how long they had been here. Where ever here was. There was plenty of time for reflection. Plenty of time for thinking. For hearing.

Jalline still could not get used to the screams. The raw sounds of agony. The sounds that came from losing your wings. She knew what they did. She had been here long enough to meet with the doctors. To be shown their great lab. To be told that she would serve science wonderfully. The tests were not terrible. Not in the way you would expect in a prison. No one was tortured. No, the scientists and doctors had said, that would interfere with their understanding of how the wings worked.

How they worked. How the muscles pulled, how the wings functioned. How could there be four wings? Why, birds needed just two, so why did the sylphii have four? Jalline had watched their cold, cold eyes look over her wings. She watched as they poked and prodded. As they took notes. They were amazed. Never before had they seen wings like hers. It was an honor to study them. So they said. All the while Jalline had the words of the old woman running through her head. They will take them just the same... It was only a matter of time. And she knew it.

Still, she did not give up hope. The priests were bound to notice she was gone now. And any day they would come in and take her back. They would tell her captors it had been an honest mistake. She was in the wrong place at the wrong time. Had seen more than she should have. She wouldn’t tell anyone. She would be sequestered away at the High Temple. They would keep an eye on her. But even as Jalline thought this, she knew it wasn’t true.

She had seen far more than a little. And it hadn’t been an accident. The High Priests of the Temple of Air had sent her to find out what had happened at the Court of Uromal. There had been whispers on the winds. That the King of Uromal was dead. Murdered. Murdered by the sylphii that had been at court. When Jalline went to find out what happened, what she had found was even worse.

The rumors were true. The king had been murdered. As had his entire court. There were no members of the royal family left alive. Not only that, but the sylphii had done it. The young sylphii priests, so sure that they could rule better than these short-sighted humans, had killed the court and were setting up a court of their own. Jalline was appalled. The ancient laws stated that the sylphii were not to rule. They were to be the caretakers, and watch over the land and all the races that lived. But it was becoming obvious that there was a new sect of power rising. One that believed the sylphii should rightly be the rulers. They were more majestic, longer-lived, and had more powers than these puny humans. So why should they continue to serve?

Jalline had brought her report back to the High Priests. They had been just as shocked and appalled. But after a week Jalline knew something had gone wrong. News came in that the dissident priests were not punished, simply recalled to the floating islands of the sylphii. Then there came rumors of some of the other priests disappearing. Not the dissident priests. The priests who still followed the old ways. It was not long after Jalline had gotten that news that she was captured.

She didn’t remember much. The old priests had known what she risked to bring back her information, so she had been kept in the inner Temple. But someone must have tipped the dissident priests off. The last thing Jalline remembered was seeing the inner courtyard. And then her world went black. She hadn’t returned to consciousness until she arrived at the prison. So she had no idea where she was. All she knew was that the prison was located somewhere high. Probably not one of the sylphii’s floating islands. No. There were humans here. Sylphii would never bring humans to their islands. So it had to be on the mainland. In the mountains, perhaps. For what made this prison all the more cruel was the constant whistling wind that reminded every inhabitant of what they once had, but now could never have again.

.:.:.:.:.:.

Things continued on for a while. The cycle of eating, sleeping, resting, waiting, testing continued. Until the day the old woman was taken away. Jalline had never forgotten her words. As the old woman was dragged out of her cell, she called to Jalline, her fever bright eyes fixed on Jalline’s face,

“Remember! If a sylphii loses her wings willingly, wishes can be made on the feathers of hope!”

The next day, there was no news of the old woman. Instead a new prisoner arrived to take her cell. Jalline knew then that she was dead. A heaviness settled into her heart. She had never even learned her name. Dread had been gripping Jalline more all the time. She rarely slept anymore. She flinched away from people and walls. She allowed nothing to touch her wings. She didn’t even open them to catch a strong breeze when one managed to blow through the cells. Shadows formed under her eyes. If she had a mirror, she could have seen how sunken her cheeks had become, how sallow her skin.

Her eyes started to look like all the other’s. Filled with a seeping despair that leeched the life out of everything. There was a kind of fear that comes with waiting too long. A fear that the waiting will end. Fear that it will go on. Sometimes she seemed alert, on a knife’s edge. She would jump at the sound of feet. Not even sighing with relief when the guards came for someone else. For that would just mean that she had to wait longer. Other times she was lethargic. Sick of waiting. Sick of living. But the scientists would not allow the prisoners to die. No, they were fed twice a day, and bathed several times a week. Jalline tried to be alert then. She didn’t want anyone to touch her wings.

Day and night began to blur together. Now there was just a twilit cell. And the waiting. It seemed to almost have a presence of its own in her cell. A tangible feel. But that had changed with the old woman’s last words. They were able to drive back the waiting. Just a little, but it was enough. A little spark of life came back to Jalline’s eyes.

For there was a tale. An ancient sylphii tale about the power of gifts freely given. The greatest gift, of course, was your wings. It was a simple learning tale, not often told once you grew past the age of seven or eight, since that was when most sylphii stopped believing in the fairy tales. It started with a girl, Anelli, who wanted to fly to the moon. Now, she couldn’t fly to the moon, and she knew that, but how she dreamed! Each night, she would stare into the sky, trying to bridge the distance with her eyes. As she grew older, she would practice flying at the high altitudes for longer each day, and she would practice breathing exercises. For she knew if she was to fly to the moon she would have to go a very long way and the air would get very thin.

Now, about the time she was sixteen, she had tried to fly to the moon ever night for two years with no success. Not one to give up easily, she vowed to continue trying. Most other sylphii laughed at her impossible dream, but one young man was impressed by her determination. So impressed was Lanar that he asked Anelli to marry him. At first she said no, for he could never be first in her heart, her heart was given to the moon. However, he was persistent, and each night he flew with her and asked her to give it a try. If she really could never love him, then they would part ways. So finally, she gave in to his proposal and they were married.

Now, for a while, nothing had really changed. She continued to try to fly to the moon each night, and he continued to fly with her. Then one beautiful clear night they were out flying when a storm blew up out of nowhere. Thinking this was the boost she needed, Anelli flew into the heart of the storm and rose higher and higher in the sky. Lanar tried to fly after her, but he lost her in the quickly darkening sky. Undaunted by the storm, he flew on in the direction that he had last seen her. Pride shone in his eyes when he finally broke through the storm and saw her sailing high above him. But the light dimmed just a quickly when Lanar watched as Anelli was hit by an errant gust of wind and thrown like a ragdoll back down into the storm.

Lanar dove after her, headless to the dangers that swirled around him. He caught Anelli, but not before permanently damaging himself. The strong winds had torn two of his wings partially off. His blood dripped around as he held Anelli close. He crash landed in the mountains, using his body to shelter Anelli. She was unconscious for almost a day. The lack of air had affected her more than she had thought it would. Lanar watched over her, though it was clear he would no last that much longer.

When Anelli awoke, she saw the concern in Lanar’s eyes. She knew that he would never replace her dreams of flying to the moon, but now she saw that he didn’t have to. Lanar held her close, and Anelli saw the terrible wounds on his back.

“I have a gift for you,” he said in a weak voice. Anelli shook her head. This couldn’t be happening. Her dreams had caused him to lose his life.

“I won’t take no for an answer.” Lanar held Anelli away and looked deep into her eyes. “You must accept it.” He turned his back to her and spoke, “I want you to take a feather.”  He waited for Anelli to take one. When he felt no tug from Anelli he continued. “You must take a feather. My wings are destroyed and I will not make it back to our home. My time is not long. You must take a feather. A feather given freely from wings given to love can grant you one wish. When I die, you must make your wish. Then you can finally make it to the moon. Promise me... Promise me you will take a feather.”

It broke Anelli’s heart to hear him plead with her, and so she plucked a feather. As she did, Lanar sighed and slipped into unconsciousness. Anelli, with tears streaming down her face, gathered Lanar up in her arms and flexed her wings. She launched herself into the air with her precious burden. Up and up she flew until she was higher than she had ever flown before. Once she was up higher than even the storm had taken her, she looked down on Lanar’s peaceful face and let drop the feather. And so she made her wish.

Now, the story never did say what Anelli wished for. Some say she brought him back to life with her wish. Other’s say she wished for all sylphii to be granted the same power, and then plunged to her death with Lanar still in her arms. Still others say she used it to wish to travel to the moon. But one thing is true. On nights of the full moon, you can see Anelli and Lanar locked in their final embrace if you look at closely at the moon. That is why one side is blood red and the other is silvery white.

Jalline hadn’t even realized she had told the tale aloud. But when she looked up after telling it, the eyes of all the prisoners were fixed on her. She felt she finally understood the old woman’s message. Even if the feathers granted nothing, they would give hope. Or was it that if her wings were freely given, her feathers would be able to grant miracles? It wouldn’t hurt to try. And so Jalline stretched out her wings for the first time in months and attempted to smile.

“Please, take one” she said in a soft voice. She spread her wings to the sides of her cell, their blue and green quite brilliant against the gray of the cell. As they stretched to their fullest, they slipped between the bars of the cells on either side, allowing the prisoners to pluck a feather. While she could not reach everyone from within her cell, when she was let out she found ways to make sure each prisoner could touch her wings and gather a single feather. There were not a lot of them, so it was no great loss. Even if she was allowed to fly, the loss of a double handful of feathers would not hinder her flight. But she knew she would never leave this prison. That she would never fly again. She knew now that no priests would come for her. For during those long weeks where she was lost in thought she had remembered whose face she had seen as she entered the prison. The face of Land Priest Jankan. And if a priest had known she was here all this time, then they would not be sending anyone to get her out.

.:.:.:.:.:.

The gift of feathers seemed to give life back to the prisoners. No longer did some stare blankly out their windows. Some would talk softly to each other. To tell of what they would wish for. Some said nothing. If asked they would reply it wouldn’t work this time either. And so Jalline learned that the old woman had done the same, and had convinced a few others to do the same, but that when the time came for their wings to be removed, not one sylphii could give them freely, and so the wishes counted for nothing.

Jalline knew it would be hard. Hard to give them up. But she also knew her time was running out. She was the only prisoner left with wings. She had to be next. She knew she would die without her wings, but if by giving them freely she could help the others, then perhaps it was for the best. It wasn’t likely that she would ever fly or be free again anyway, so maybe it would be worth it. But it was still hard to image them gone. Each night she would touch her wings. Savor the feel of the downy feathers, of the stiff primaries and secondaries. Soak in their color. She had plucked two to keep. One green and one blue, so that she would remember there colors. Not that she planned on living long after they were gone, but she thought maybe to looking at the feathers would help her forget that they were gone.

It seemed she had been right. A week after giving her feathers away, they finally came for her. Jalline thought she was ready. But how could anyone be ready for this? Fear gripped her heart, but she tried to smile. She couldn’t look at the other prisoners. Their eyes said too much. Too much hope, too much fear, too much sympathy. As soon as she saw the lab door, her smile faded and Jalline paled.

There was nothing unusual about the lab. Nothing that would lead anyone to believe what was about to happen would. It was still pristine, white, clean. There not a lot of doctors there. Only two. And only one scientist to record observations. Usually they were all their. All poking and prodding. Perhaps they had done this so many times one more wasn’t anything special and so they had gone off to do better things. Jalline didn’t care either way. She was just grateful there were so few. She shut her eyes before stepping into the room. She was struggling so hard to stay composed.

She was half-dragged, half-carried across the room and placed on the usual stool. She sat limply, waiting for the words that would end her world. But they didn’t come. Instead, the doctors continued to get things ready and the scientist started on a new page of notes. In time one of the doctors asked her to strip and then lay on a table, face down. She nodded her understanding and moved. It was usually like this. Except for the lying on the table part. She hoped that she would be wrong. And then she took a look at the tools next to the table.

A scalpel.

The room spun once and Jalline struggled not to loose control. She had to hang on just a little bit longer. Fear wrapped around her spine. She forced herself not to move. If she was going to do this, she would do it right. It was so eerie. No one spoke of what was to happen, but it was obvious. There really was no other reason she could be there. One of the doctors was busy stretching out her wings and clamping them down. Jalline tried not to whimper. The clamps hurt. But she knew it was nothing compared to what was to come. She knew she had to speak soon, or she would lose her nerve entirely. Then next time one of the doctors came close, she spoke,

“Please” Jalline whispered, her mouth horrible dry, she felt like she was going to throw up. But the others were out there hoping, so she found strength to continue, “I give you my wings freely.”

“How odd.” she heard a scientist say. He wrote her words down in his notes. “Did you know that is the first time anyone said that? Usually they are pleading with us by now. Begging for release. Begging to lose an arm or a leg. Anything but their wings. Why, you’d almost think she wanted this.”

Jalline wished she could block out his voice, but she could not. She tried hard not to think at all, but her mind refused to go blank. Little sounds kept intruding. She heard the footsteps of the doctors as they finished preparations. She heard the scribbling of the scientists pen against his paper. She felt the cold bite of the scalpel and the burn that traced its path, catching her by surprise. A gasp escaped her. The cold was accompanied by rolling waves of burning pain. They rolled over her body, radiating out from her upper right wing. It was over quickly. The final slice was made. The sickening sound as the wing was pull from her muscles.

Through it all Jalline stayed awake. Locked in a haze of pain. She could swear she was flexing her wing, trying to get the pain to stop. But she couldn’t be. Her wing was gone. And finally, with that thought the pain overwhelmed her and she slipped into peaceful unconsciousness.

.:.:.:.:.:.

Jalline awoke later to the sounds of sirens. She opened her eyes. The room swam into focus. Or the floor did. She was on her stomach. Something wasn’t quite right. She rolled to her side and the knowledge came crashing down on her. She had no wings. She curled herself up in a ball and just sat staring at the wall for sometime. And then smoke started to curl into the room. Jalline blinked and looked at the door. It was open. There was no one in the room. She didn’t even know what room it was. She had never been here before. And she was dressed again.

She warred with herself. Part of her just wanted to curl up and die. She had lost her wings. What hope did she have of living? But a larger part of her had decided to live. Or at least live long enough to look out into the hall. And so she sat up. There shouldn’t be smoke here. There shouldn’t be... but there was. Jalline moved to the door. She fell. It wasn’t easy to walk. Her sense of balance seemed to be off. Then she remembered the weight of her wings was missing. She didn’t know her body was trying to overcompensate for their loss. But she caught on pretty quickly that walking would be out of the question for now. She sat down hard on the floor. The fall jarred her back. It seemed sore, and when she tentatively touched it, it was dry. Her wounds hadn’t reopened. At least the doctors had done a good job part of her mind seemed to say. She couldn’t even see what they had done. Crawling, Jalline stuck her head out into the hall. The sirens were louder there. And there was more smoke. Was the lab on fire?

Jalline crawled out into the hall and started moving away from the fire. There was no one around. Where was everyone? Had the prisoners escaped? Had their wishes worked? Jalline couldn’t stop the flow of questions through her mind. But they stopped soon enough. Crawling was harder than she remembered. Her knees hurt. And she didn’t seem to be getting anywhere very fast. Perhaps she could just lay down here for a few minutes and rest...

No!

She would not rest now. Not until she got outside. And so Jalline started to look for a door. After what seemed like forever she found one. With great care, she pulled herself up and turned the doorknob. Then she fell forward using her fall to push on the door. It was opening! She tumbled gracelessly from the burning building.

The sunlight seared her eyes. It had been so long since she had seen it. A breeze tugged playfully at her hair. Jalline shut her eyes and tried to hold back the welling of feeling deep inside her. She was free. She had never thought she would be free again. But freedom meant little when you couldn’t walk or fly. The loss of what had happened hit her. She curled herself into a ball and cried. She did not know how long she stayed their, wrapped in a ball, oblivious to the world. The wind kept tugging at her, as if to draw her away from the building. But Jalline wouldn’t be moved.

When the sun dropped below a mountain peak, Jalline finally stop crying. Through tear washed eyes she looked around. She had been right. The lab was on the mainland. And it was in the mountains. It seemed to be located on near a ledge. The prison windows must have faced over the cliff. That was why the wind seemed so strong. Jalline slowly uncurled her battered form. Her knees were bruised and stiff. Her arms were in agony. It was better not to think about her back. Slowly, every so slowly, she sat up. She was not by the door anymore. It seemed the wind had moved her. It was obvious why. There was not much left of the building. Flames raged out all the doors and windows, fueling the inferno within. Jalline couldn’t imagine anyone surviving that. She hoped the other prisoners had gotten what they wished for.

Before her eyes could tear up again, Jalline turned and looked towards the cliff edge. It didn’t seem too far away. But she knew she could no longer move fast. So she began the agonizing process of crawling to the edge. It was harder to crawl out here than inside. There were no smooth, polished floors. While the grass was pleasant, rocks frequently bit into her hands and legs, causing more pain and bruises. Jalline had to take several breaks. She was very weak, and not much accustomed to moving. But she was determined. And so, by the time the moon hung fully overhead, Jalline had made it to the edge of the cliff. She leaned over the top and looked. Darkness obscured everything below. She smiled a desperate smile.

Jalline stood up slowly. She didn’t want to fall. As she tottered on the edge, she searched her ragged dress for the feathers she had kept. Somewhere she had lost one. She pulled the other out. It was impossible to tell what color it had been by the light of the moon. Jalline looked up at the moon and started laughing, tears streaming down her face. The moon was full. She could see Anelli and Lanar locked in their embrace. “I’m coming to join you,” she whispered as she jumped off the edge. Almost as an afterthought, she let her feather fall. Perhaps wishes do come true.

title: User:CarrieBerg/Feathers


text:
User:CarrieBerg/Recipe For A Chaucerian Comedy

Recipe By: Mrs. Chaucer

Ingredients:

2 cups Couplets that Rhyme

1 cup chopped Older Tales

1 dash Middle English

3 tsp Mixed-Up Romance (can be substituted with 3 tsp Bitter Revenge if Romance is too sweet)

1 lb Different Perspectives

2 tbsp Intergroup Conflict/Rivalry

1 Mediator

2 Marinated Fools

2 Sober Sages

A few Feminine Touches

A Smattering of Realism

A Touch of Irony

1 can Intelligent Frosting

1 bottle Religious Sprinkles

Preheat oven to 375 degrees. Sift Couplets well then pour them into a bowl and mix with Older Tales, Middle English, Different Perspectives, Feminine Touches, and Realism. In a separate bowl mix Intergroup Conflict, Mediator, Marinated Fools, Sober Sages, and Irony. Pour together and stir well. Mix on high for 3 minutes while slowly adding in Romance (or Revenge). Pour into a 12x12 baking pan and bake for 30 minutes. Test to see if done. Now you should see it rise to form a nice firm story with only a few holes. Frost with a veneer of Intelligence and sprinkle liberally with Religion. Cut into 24 uneven pieces. Serve with a tall glass of alcohol to enjoy to the fullest.

(This was written in about 2 minutes in class as a joke. My teacher for my class on Chaucer had written "Ingredients to make a Canterbury Tale" on the board, and I, well, took him literally. When he asked for ideas, I read him this. The entire class applauded me. It is one of my fondest college memories, and so I'm sharing it with all of you.)


text: File:Sodium-potassium pump new.jpg

title: File:Sodium-potassium pump new.jpg


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User:CarrieBerg/Hummingbird Visions

I thought I saw a hummingbird

Hovering in the air

But no sooner than I saw it

It was no longer there

I couldn't see its wings

It hovered in midflight

Right beside my window

Much to my delight

I saw it again today

I'm sure this one is the same

It rested on a dangling vine

By my window again

I could see it clearly

As it sat down to rest

Its wings, no longer hovering,

Were as green as its breast

But as soon as I got my camera

It upped and flew away!

So all I have to prove I saw it is

The poem I wrote today

title: User:CarrieBerg/Hummingbird Visions


text: File:Sodium potassium pump best.jpg

title: File:Sodium potassium pump best.jpg


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User:CarrieBerg/The Wanderer's Dance

Round and round the fire light

Twirling oh so fast

Glowing orbs pass us by

Their beauty unsurpassed

A host of lesser beings join

Into the splendid dance

Exotic bodies twinkling

They hold our minds entranced

A simple melody is played

But upon each refrain

A subtle depth is added

Forming a feeling most arcane

And in our darkest hour

We find the sky all aglow

From the planetai and stars

That make up this midnight show

title: User:CarrieBerg/The Wanderer's Dance


text:
User:CarrieBerg/Flowers Of Spring

Petals fall down

Like soft, sweet-smelling snow

Whirling and twirling

As the winds blow

As they turn and dance

Through the air

They fall upon

Textbooks, tables and hair

Sometimes they clog windshields

Some people complain

But their delicate beauty

Does not long remain

Once they fall to the ground

They are soon gone

Easily squished by a boot

Only their scent lingers on

But oh! how they look

Gently raining down

Such wonders of nature

Spring brings around

title: User:CarrieBerg/Flowers Of Spring


text:
BSD

BSD stands for Berkeley Software Distribution and is an operating system. Most of its common variations are considered to be Free/Libre and Open Source Software (FLOSS). This resource is similar to its counterpart Linux and one should explore things like Basic commands in Linux and Directories in Linux. Most work for the BSDs also. FreeBSD is what we'll be looking at primarily but we will also cover NetBSD, OpenBSD and others.

See the main article at Wikipedia

Getting FreeBSD
Getting FreeBSD is easy. This course is based on the FreeBSD 7.2-RELEASE which came from here. If you are new to installing operating systems, you might consider getting only the 7.2-RELEASE-i386-livefs.iso to look at a Live CD version. PC-BSD and FreeSBIE are also available. You can also learn FreeBSD if you have an Amazon AWS account, as there is free version available on AWS Marketplace, and probably some other cloud hosting providers offers too FreeBSD installations.

see also Creating a bootable disk

Learning BSD
We chose FreeBSD for the i386 family. Look again at the Getting FreeBSD page to see if it matches your test machine. If you don't have a test machine, there are many BSD servers online where you can get a shell account.

Basic Commands
user@example.com# sysctl hw.model hw.machine hw.ncpu hw.model: Intel(R) Xeon(R) CPU E3-1230 V2 @ 3.30GHz hw.machine: amd64 hw.ncpu: 8
 * CPU Information:

 * Memory installed:

Package Management (FreeBSD_Ports):
 * List installed packages:
 * Install a package
 * Delete a package

Installing FreeBSD
Our CD set:
 * 7.x-RELEASE-i386-bootonly.iso
 * 7.x-RELEASE-i386-disc1.iso
 * 7.x-RELEASE-i386-disc2.iso
 * 7.x-RELEASE-i386-disc3.iso
 * 7.x-RELEASE-i386-docs.iso

Our Hardware: Our test machine will be a Dell PowerEdge 2400 server with one i386 class Pentium IV (even though there is a socket for a second).

''This course is under construction... see the talk page 03:00, 29 April 2009 (UTC)''

Updating FreeBSD
FreeBSD updates can be done using  command.

Managing Service
Services are managed using  command and the ,  , and   options.

Personalize command prompt
You can personalize your command prompt by editing  file. Change set prompt line to your desire value, for example:

title: BSD


text:
Tiny Core Linux

Tiny Core Linux is probably the smallest Linux distribution available. It is being developed by Robert Shingledecker (of Damn Small Linux fame) as "a nomadic ultra small desktop capable of booting from cdrom, pendrive, or frugally from a hard drive".

This is a Learning by doing module.

Get Tiny
Go to http://distro.ibiblio.org/pub/linux/distributions/tinycorelinux/release/ and download either bootfloppy.dd or preferably tinycore_1.4.iso. You'll need to copy these as bootable images.

NOTE: If you are new to Linux and working from a Windows® machine you should learn about RaWrite.

Run core
After you burn it you can learn it! TCL doesn't need your hard drive to run. It runs from RAM like a live CD. Just leave the disc in the drive and reboot your computer. You may have to go into the BIOS and change the "boot order".

Learn Linux
When TCL boots (loads itself into RAM), it quickly produces a very simple Graphical User Interface via a window manager called Tiny X. Three icons appear at the bottom of the window:

Terminal
Open the terminal application. At the command prompt type: ls /bin

This will show you all the commands (utilities) contained in BusyBox, a stripped down collection of the GNU Core Utilities.

Compare these with Basic commands in Linux.

Basic commands
Notice that your command prompt looks something like this:

tc@box:~$

The tilde ( ~ ) indicates that you are in your /home directory. Earlier you used the ls command to list the contents of the /bin or binaries directory. Now use the cd command to change directories to /bin.

Now your command prompt looks like this:

tc@box:/bin$

Now type ls with no arguments (just hit "Enter"). The resulting list is the same. Type cd with no arguments and you'll be taken back to your /home directory.

See Directories in Linux

Control panel
Clicking on the middle icon on the desktop opens a set of buttons that allow you to setup and configure common functions and run various utilities. See a brief summary, Control Panel and a screenshot from wiki.tinycorelinux.com.

App-browser
To download apps using the app-browser. Open app-browser and click apps -> cloud -> browse. This will load a library of app you can download, from there you can also search for apps. once you have found an app you want change the "Onboot" option to "download+load". let it download and done. :)

title: Tiny Core Linux


text:
Talk:Simplified DIY Rooftop Wind Turbine

Three Bladed Professional Kit
This durable efficiend rooftop wind turbine kit is a mini scaled version of the proven large utility sized wind turbines. It's stainless steel parts can withstand over 130mph wind! http://windenergy7.com/turbines/?p=41

 * Someone added this entry, yet it does not belong here, only efficient designs are discussed (this is a mere regular and probably inefficient design). It is kit-based, but that that allone does not mean its cost-efficient (what's the energy output ? )


text:
DIY airborne wind turbine

This article is about building efficient and simple airborne wind turbines (AWT) EnergyKiteSystems. There are several types of do-it-yourself (DIY) known designs of airborne wind turbines. Some are balloon lifted, some are kytoon lifted, some are kite lifted, some are the kytoon itself with lofted generator. Two general types by virture of position of the generator occur: 1. generator is lofted in the air; 2. generator is in a groundstation.

AWT LTA (lighter than air) Hydrogen-based with generator on ground
Magnus-effect

AWT LTA hydrogen-based with generator lofted in air
Magnus-effect kytoon type: The wind turbine would be similar to the Magenn Wind turbine, yet with some special distinctions:
 * The balloon would -in contrast to the Magenn Wind turbine- be filled with hydrogen. This gas (aldough inflammable) is non-polluting (eliminating danger of accidental release) and cheaper. It is also producable on-site (by hydrogen generator). Possible release of the gas from the balloon over time can thus be easily adressed. Perhaps a generator may be foreseen on the balloon for this purpose.
 * Low-cost building of the turbine itself
 * An airborne wind turbine is a design concept for a wind turbine that is supported in the air without a tower.[1] When the generator is on the ground,2 then the tethered aircraft (kite, kytoon) need not carry the generator mass or have a conductive tether. When the generator is aloft, then a conductive tether would be used to transmit energy to the ground, either mechanically or through electrical conductors. Some systems would have the advantage of tapping an almost constant wind and doing so without a set of slip rings or yaw mechanism, without the expense of tower construction. Kites and 'helicopters' come down when there is insufficient wind; kytoons and blimps resolve the matter. Also, bad weather such as lightning or thunderstorms, could temporarily suspend use of the machines, probably requiring them to be brought back down to the ground and covered. Some schemes require a long power cable and, if the turbine is high enough, an aircraft exclusion zone. When the generator is ground-based, the tether need not be conductive. As of 2008, no commercial airborne wind turbines are in regular operation

DIY AWT HTA kite system
This turbine (although more expensive than the rooftop wind turbine) would be quicker to deploy and could be a more cost-efficient turbine in areas with little wind at low-altitude or function as first turbine to determine the place where to most wind can be caught in a specific plot of land. Needs cooperative work dough. Feel free to add info.

title: DIY airborne wind turbine


text:
Cell biology/Introduction

Why Bother?
Cells are at an intermediate level of the structural hierarchy of living things. All organisms except viruses have cells as fundamental components, and even viruses depend on the machinery of a cell to reproduce. Cells are composed of molecules and multicellular organisms such as humans are composed of many types of cells that cooperate to form tissues and organs. A student learning about cell biology should also consider studying the chemistry of living organisms, although cell biology can be understood without deep analysis of chemical details. You can use the Wikiversity Biochemistry resources to provide the basic information you should have about the chemistry of living organisms. For more extensive online information about Biochemistry, there is an online textbook available through the PubMed system.

What is Cell Biology?
A large part of Cell Biology is the study of how biological molecules (mainly proteins and nucleic acids like DNA) account for the structure and function of cells. Cell Biologists also work to account for the structure and function of multicellular organisms in terms of their cellular components. The human body has many specialized cell types that cooperate to provide all of the functions of the human body. Cell biologists are very much concerned with the hormonal and other chemical signals that pass between cells and coordinate the activities of the trillions of cells in a human body.

Basic Cell Structure
Cell Biology is concerned with the structure and function of cells. "Structure" and "function" are two different ways of looking at the same thing; structures exist to accomplish certain functions and we account for biological function in terms of the structural components of living organisms. Microscopy is an important technique in Cell Biology that reveals cell structure and guides us towards an understanding of how cells function. Most cells are too small to be seen by the naked eye and the spatial distributions of molecular components in cells can only be revealed by microscopy. (Cell Microscopy Art Gallery.)

The structural integrity of cells depends on proteins that form the cytoskeleton. One of the major types of cytoskeletal proteins is keratin. Panel B (Figure 1) shows the cellular location of a type of keratin protein inside these dog kidney epithelial cells. In panels A and B, the green color shows the location of DNA inside the cell nuclei.

Basic Cell Function
Microscopy of dog (A and B) or mouse (C) cells. In Panels A and B, green shows the location of DNA in cell nuclei. Red shows the position of a cell surface protein (A) or a cytoskeletal protein (B). Panel C shows a mitotic cell with condensed chromosomes.]] The functions of cells are mostly made possible by the protein molecules of cells. The images shown to the right (Figure 1) illustrate microscopy of the cellular locations of two specific proteins. The human genome project has found that humans have genes that provide instructions for about 20,000 different proteins. In Panel A, the red color shows the location of a protein called beta-catenin. Much of the beta-catenin of cells is located at the cell surface where it functions to promote normal cell adhesion. Most cells of the body are part of solid tissues. Many cells of solid tissues have special cell surface proteins that allow adjacent cells to adhere tightly to each other.

DNA is the genetic molecule of cells that carries the molecular instructions used by cells to make protein molecules. Cells, as the basic functional components of living organisms, have the capacity to reproduce and divide into daughter cells. During cell reproduction, the cell nucleus disintegrates and the DNA molecules are packaged as condensed mitotic chromosomes. Panel C (Figure 1) shows two sets of mouse chromosomes ready to be packaged into the two daughter cells of a dividing parental cell.

Cell Biology and Medicine
Microscopy of dog (A and B) or mouse (C) cells. In Panels A and B, green shows the location of DNA in cell nuclei. Red shows the position of a cell surface protein (A) or a cytoskeletal protein (B). Panel C shows a mitotic cell with condensed chromosomes.]] Many diseases can be understood and treated at the cellular level. For example, cancers are diseases caused by abnormal cell growth or other forms of abnormal "antisocial" behavior by cells. The protein beta-catenin (Figure 1A) is a proto-oncogene that when mutated or abnormally regulated inside cells can contribute to cancers such as colon cancer. Mutated keratin (Figure 1B) genes can cause skin blistering diseases. Abnormal keratins and abnormal autoimmune attacks on proteins that cooperate with keratins to hold skin cells together can cause diseases in humans such as Pemphigus.

Many common medical diagnosis methods such as the Pap smear use microscopy of cells to identify cancerous and pre-cancerous cells.

An area of medical research that has been in the news lately (2005) is the study of stem cells. The cells shown in Figure 1C are embryonic stem cells that have the capacity to differentiate into neurons. There is much interest in the idea that diseases such as Parkinson's disease can be treated by transplantation of stem cells.

Biology Background Information
The online Cell Biology textbook has great background information that would be covered in an introductory Biology class. Make sure you are familiar with this information before diving into more advanced lectures.

Next Lesson

Return Home

title: Cell biology/Introduction


text:
Chemistry Basics

 * 1) The International System of Units, International System of Units
 * 2) Nomenclature of organic chemistry,Naming in Organic Chemistry,
 * 3) Logarithms
 * 4) Nuclides
 * 5) Radioactive Decay
 * 6) Gases Laws, Gases and Gas Laws
 * 7) Partial Pressure and Mole Fraction
 * 8) Osmosis
 * 9) Diffusion
 * 10) pKa
 * 11) pH
 * 12) Buffers

Return Home

title: Chemistry Basics


text:
Biochemistry/Biomolecules

 * Nucleic Acids
 * Amino Acids
 * Proteins
 * Carbohydrates
 * Lipids

Return Home

title: Biochemistry/Biomolecules


text:
Protein References

 * DNA Replication
 * Laboratory Nucleic Acid Synthesis
 * Nucleic Acid Degradation, Pyrimidine Metabolism and Purine Metabolism
 * Peptide Bond
 * Laboratory Protein Synthesis
 * Micronutrients and Vitamins
 * Enzymes
 * Regulation of metabolic pathways
 * Associated: Hemoglobin, Gas Transportation

Return Home

title: Protein References


text:
Biochemistry/Amino acids References

 * Decarboxylation
 * Transamination
 * Amino Acid Synthesis
 * Deanimation
 * General amino acid catabolism
 * Special amino acid metabolism
 * Amino acid interconversion
 * Urea Cycle, Amino Acid Catabolism
 * Gamma-glutamyl cycle
 * Associated: Gout, Maple Syrup Disease

Return Home

title: Biochemistry/Amino acids References


text:
Carbohydrate References

 * Glycolysis
 * Krebs' cycle
 * Pyruvate decarboxylation
 * The ATP cycle
 * Oxidative phosphorilation
 * C4 carbon fixation
 * Photosynthesis
 * Calvin Cycle
 * Dickens-Horecker cycle
 * Hexose interconversions
 * Neoglucogenesis
 * Glycogenesis
 * Glycogenolysis
 * Cori cycle
 * Carbohydrate regulation mechanism: glucagon and insulin
 * Associated: ...

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title: Carbohydrate References


text:
Lipid References

 * fatty acids
 * Essential fatty acids, item, poly-unsaturated fatty acids (PUFA); eicosanoids: prostaglandins, thromboxanes, prostacyclin and leukotrienes.
 * The lipid bilayer, item, and transport across it
 * Lipid peroxidation
 * Acylglycerols
 * Triacylglycerols
 * Sn system
 * Carnitine and membrane transport of fatty acids
 * Fatty acid oxidations
 * Beta oxidation
 * Ketogenesis
 * Lipid synthesis (lipogenesis)
 * Arachidonic acid cascade and the eicosanoids (above)
 * Associated: ...
 * Lipoproteins and cholesterol
 * The serum lipoprotein transport system
 * Cholesterol metabolism, regulation and transport
 * Lipoprotein metabolism
 * Chylomicra and VLDL
 * LDL and HDL
 * Regulation of lipoprotein metabolism

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title: Lipid References


text:
User:Jagdish~enwikiversity

 Reforms required in the Indian Democracy

General elections 2009 have proved that the Voter turnout has dwindled. The reasons stated may be long enough, however taking in to consideration the fact that huge expenditure has been incurred to create awareness amongst the voters, the turnout is dismal. Earlier the candidates used to arrange vehicles for transporting the voters to the booth and back. The candidates used to do so in their own interest. However the firm approach taken by the Election Commission in implementation of the model code of conduct has proved that the earlier instances of higher voter turn out were far from genuine. however still the robustness of Indian Democracy has survived many a onslaught of useless politicians. The neighbouring state has exhibited the classic case of failure due to lack of preparedness to accept idea of democracy. Now in order to further strengthen the Indian Democracy, it is felt that certain initiatives need to be taken :

Ban the entry of illiterates : no person who is illiterate and cannot read and write independently should be allowed to contest elections. Minimum requirement should be Higher Secondary education.

Verification of Character and Antecedents : All those contest election should be subjected to verification of Character and Antecedents. Even today candidates are submitting false declarations and are not questionned for the same. An agency should be formed for verification of the claim of the candidates. The candidates should be made to put out the details in public instead of keeping it in safe custody of Election Commission.

Candidates should be brought under the RTI Act 2005 : At present only the civil servants are subjected to RTI Act 2005 and the candidates are left out. In order to bring in 'transparency and accountability' the candidates should be brought under the Right to Information Act 2005.

Balance sheet of constituency or the M.P./M.L.A. should be made public: the income and expenditure of every M.P./M.L.A.should be made public. They should be made to submit details of Income and Expenditure tot he general public. Similarly the Balance Sheet of the constituency should be declared to give idea about the investment done, work done.

Right to Call Back: The general public/ voter should not be made to suffer full term for the inefficiency of the elected representative. The Public should be made Master in true sense of the Public Servant. It should be possible to cal;l back an Elected Representative through representation of atleast 50% voters in the area.

Voting to be made compulsory: The national database of the voters should be updated after every election and entries be made about those who failed to caste their vote. Various benefits should be linked to Voting and hence those who fail to caste vote should be deprived of certain privileges like filing petitions and complaints etc.

Right to caste Negative vote : the public should be given right to reject all the candidates. Candidates failing to secure minimum 40% vote of total voters should be declared unsuccessful and in case more than one candidate securing more than 40% votes, the candidate securing highest votes should be declared elected.

Catch them Young: Democratic values need to be inculcated at a very young age. for the purpose, it must be made mandatory to have student council in each and every school and elections conducted every year to instill the faith of democracy.

Research and development: Democratic values need to be developed through thorough research using various disciplines like sociology, psychology, management etc.

Amendments in law: Law should be amended suitably to take away draconian provisions to impose presidential rule or emergency in any state by the centre.

title: User:Jagdish~enwikiversity


text:
User:CarrieBerg/A Lonely City

I step outside and I’m blinded

By the light of the sun

I don’t know if it’s the Light of Truth

Or just the lies we tell everyone

I turn around and I can see again

Or is it a Shadow of a Doubt

Darkness surrounds me

Even though it’s still light out

Please don’t shout

I can still hear

Though I can’t see

Can you see me?

I walk away and I am lost

I seem to be in a haze

Wandering around aimlessly

Like I have been for days

As I travel on my solitary way

I pass by others on the street

We nod politely as we pass

But no one ever seems to speak

Who did I meet?

I can still see

Though I can’t say

Can you talk today?

There is nothing wrong it seems

There are people all around

The only difference today

Is that they make no sound

No one seems to stop and chat

They stoically ignore all in their way

As if they are the only people

To walk the Earth today

What did you say?

I can still talk

Though I can’t hear

Are you near?

I remember when the streets

Were full of pedestrian noise

People walked and talked

Instead of listening to toys

People isolate themselves

Music becomes a protective frost

Cut off reality and we are

A population of the lost

What is the cost?

Do you know?

The price we pay

Adds up each day

Each day I watch the people

Draw back a little more

Alone, but not yet lonely

They are quickly opening that door

Absorbed with their gadgets

They forget about this fact:

Technology cannot replace

The joys of human contact

Is that what you lack?

Can you see?

Are you talking on your phone?

Do you hear you are alone?

title: User:CarrieBerg/A Lonely City


text:
User:CarrieBerg/The Journey

I want to take you fishing in the streams of time

I want to take you sailing ‘cross the dreams in you mind

But, unfortunately you happen to be blind

To all that I want to show you

I want to steal a kiss from the jaws of death

I want to take you swimming in the waters of Lethe

But, unfortunately it will take your dying breath

That river can’t be reached in life

I want to open up the eye of the storm

I want to take you back to when the world was born

But, unfortunately you would always mourn

If I stole you away from here

I want to take you dancing when the world will burn

I want to watch of wheels of the sky turn

But, unfortunately you would never learn

From all that I want to show you

I will have to travel alone to where I would go

I will have to be alone with all that I would know

But, unfortunately I would miss you if you didn’t follow

So I’ll stay and tell you of my journeys

title: User:CarrieBerg/The Journey


text:
User:CarrieBerg/The Weather Debate

A crash in the sky!

A wondrous light!

Thunder and Lightning!

A storm’s here tonight!

Clouds hurry in,

They don’t want to be late,

As the sky opens up

For the thunderous debate.

Lightning flashes insults

Thunder yells back,

Soon the Wind joins in

With a lashing attack.

Rain angrily drizzles

Then openly pours,

As Thunder responds badly

And Lightning scores.

Looks like the argument

Could last all night,

When Thunder finally sees

Lightning’s bright light.

Points of view change

Both sides agree,

Clouds break up quickly

There’s no more to see.

As Clouds are parting

Wind sighs and leaves,

They know another debate will come

Next time Thunder and Lightning disagree.

title: User:CarrieBerg/The Weather Debate


text:
User:CarrieBerg/Daydreaming

Grass above me

Clouds below me

As I drift

Through endless sky

Birds swim through air

As I stroll along

No restraints

Just floating free

High above

You are watching

Envious

Of what you see

How can you

Sit there watching?

Wouldn’t you rather

Float with me?

So lie back

Let your mind wander

Come along

And dream with me

title: User:CarrieBerg/Daydreaming


text:
Coast

The learning activity has three parts:

Earth, water planet
{What share of Earth surface is covered by the sea? - Half. + Aproximately 70%. - Less than one tenth.
 * type="[]"}

Lithosphere, hydrosphere and atmosphere
{Left to right, how are they ordered? - Below the sea, above the stone, and at the top of the air. - The lithosphere surounding outside the hydrosphere, and both bellow the atmosphere. + Water, that is more heavy, below the air, and the ground below both.
 * type="[]"}

title: Coast


text:
User:CarrieBerg/Archaeology

Through eyes unblinded by day

And seeing even in darkest night

The illusions that are lost in the mist

Are illumed with phosphorescent light

The ancient truths long untold

Are re-etched in unfeeling stone

And voices rusted with disuse

These forgotten prophecies intone

Oh! Would that we can remember

What has once gone before

But the collective memory of the present

Destroys these artifacts of yore

Left with words unwritten

Deciphering the blanks like a code

Searching for answers in shadows

Reaching in vain for all that is old

How can the past be forgotten?

How can the present be blind?

What can we learn of the future?

What marvelous things will we find?

But the dust has long been collecting

The words are coated in grime

Soon the voices will forever be silenced

There wisdom lost again in time

Shall we ever learn our lesson?

Can the past shine on today?

Or will the lights be broken

Shattered as we hurry on our way?

title: User:CarrieBerg/Archaeology


text:
User:CarrieBerg/Last Waltz

A dance in the darkness

One last little song

To remember the good times

To forget the pain

The night grows ever longer

No one notices the end

When the dancers are taken

To be born again

And the night still goes on

The dance is never-ending

And the night still goes on

The dawn is long to show

The dancers change partners

The songs change their tunes

The stars change positions

During the night

Just one more dance

That’s what they all say

With a couple more drinks

They’re almost at daylight

And the dance still goes on

The dawn is far from showing

And the dance still goes on

The night has long to go

A few more steps

A couple of twirls

A bow and a curtsey

And they’re away

Gossip and whispers

Fade into the background

As the dance continues

And the music does play

And the night still goes on

The dance is never-ending

And the night still goes on

The dawn is long to show

Finally the dancers

Tired from waltzing

Smiling and nodding

Try hard to leave

But it’s too late

They cannot go now

For it's past dawn

And it’s death they receive

And the dance still goes on

The dawn is long past now

And the dance still goes on

For all eternity

title: User:CarrieBerg/Last Waltz


text:
User:CarrieBerg/Peppermint Dreams

Peppermint Dreams

Nothing’s as it seems

Rose tinted glasses cover our eyes,

Clouds turn out to be

Cotton candy,

What a surprise...

Peppermint Dreams

Nothing’s as it seems

The world twirls around,

Watch the time fly

As it streams by,

Never again to be found...

Peppermint Dreams

Nothing’s as it seems

All the world’s a sensation,

Touch, sound and sight

Hearing and tasting’s alright,

Of course, it’s just part of our imagination...

Peppermint Dreams

Nothing’s as it seems

What a great illusion,

It seems so real

Just think of what we can feel,

But that’s my conclusion...

Peppermint Dreams......

title: User:CarrieBerg/Peppermint Dreams


text:
User:CarrieBerg/Intertwining Trees

Two trees lovingly intertwine,

Around each other they do wind,

Forever together their barks do bind,

Into a wooden knot.

In spring one tree flowers white,

The other, deep pink, is not as bright,

The colors compliment each other in the light,

As the flowers bloom.

Summer turns to autumn, you know,

And on the trees, fruit starts to show,

Apples and crab apples together grow,

Until they fall and rot.

The leaves and fruit have gone away,

Cold winds blow as children play,

Still the dark and light bark does stay,

Intertwined in a snowy tomb.

title: User:CarrieBerg/Intertwining Trees