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Advanced Mathematics for Engineers and Scientists/The Laplacian and Laplace's Equation

The Laplacian and Laplace's Equation
By now, you've most likely grown sick of the one dimensional transient diffusion PDE we've been playing with:

\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}\, $$

Make no mistake: we're not nearly done with this stupid thing; but for the sake of variety let's introduce a fresh new equation and, even though it's not strictly a separation of variables concept, a really cool quantity called the Laplacian. You'll like this chapter; it has many pretty pictures in it.

The Laplacian
The Laplacian is a linear operator in Euclidean n-space. There are other spaces with properties different from Euclidean space. Note also that operator here has a very specific meaning. As a function is sort of an operator on real numbers, our operator is an operator on functions, not on the real numbers. See here for a longer explanation.

We'll start with the 3D Cartesian "version". Let $$u = u(x, y, z, t)$$. The Laplacian of the function $$u$$ is defined and notated as:

\nabla^2 u = \Delta u = \operatorname{div} \ \operatorname{grad} \ u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\, $$

So the operator is taking the sum of the nonmixed second derivatives of $$u$$ with respect to the Cartesian space variables $$x, y, z$$. The "del squared" notation is preferred since the capital delta can be confused with increments and differences, and $$\mbox{div grad }u$$ is too long and doesn't involve pretty math symbols. The Laplacian is also known as the Laplace operator or Laplace's operator, not to be confused with the Laplace transform. Also, note that if we had only taken the first partial derivatives of the function $$u$$, and put them into a vector, that would have been the gradient of the function $$u$$. The Laplacian takes the second unmixed derivatives and adds them up.

In one dimension, recall that the second derivative measures concavity. Suppose $$y = f(x)$$; if $$f(x)$$ is positive, $$y$$ is concave up, and if $$f(x)$$ is negative, $$y$$ is concave down, see the graph below with the straight up or down arrows at various points of the curve. The Laplacian may be thought of as a generalization of the concavity concept to multivariate functions.

This idea is demonstrated at the right, in one dimension: $$u(x) = x^3 - x$$. To the left of $$x = 0$$, the Laplacian (simply the second derivative here) is negative, and the graph is concave down. At $$x = 0$$, the curve inflects and the Laplacian is $$0$$. To the right of $$x = 0$$, the Laplacian is positive and the graph is concave up.

Concavity may or may not do it for you. Thankfully, there's another very important view of the Laplacian, with deep implications for any equation it shows itself in: the Laplacian compares the value of $$u$$ at some point in space to the average of the values of $$u$$ in the neighborhood of the same point. The three cases are:

 * If $$u$$ is greater at some point than the average of its neighbors, $$\scriptstyle \nabla^2 u < 0$$.
 * If $$u$$ is at some point equal to the average of its neighbors, $$\scriptstyle \nabla^2 u = 0$$.
 * If $$u$$ is smaller at some point than the average of its neighbors, $$\scriptstyle \nabla^2 u > 0$$.

So the laplacian may be thought of as, at some point $$(x_0, y_0, z_0)$$:

\nabla^2 u \Big|_{x = x_0, y = y_0, Z = Z_0} \approx \mbox{average of u in the neighborhood of } (x_0, y_0, z_0) \ - \ u(x_0, y_0, z_0, t) \, $$

The neighborhood of some point is defined as the open set that lies within some Euclidean distance δ (delta) from the point. Referring to the picture at right (a 3D example), the neighborhood of the point $$(x_0, y_0, z_0)$$ is the shaded region which satisfies:

(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 < \delta^2 \, $$

Note that our one dimensional transient diffusion equation, our parallel plate flow, involves the Laplacian:

\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} = \alpha \nabla^2 u\, $$

With this mentality, let's examine the behavior of this very important PDE. On the left is the time derivative and on the right is the Laplacian. This equation is saying that:

 * The rate of change of $$u$$ at some point is proportional to the difference between the average value of $$u$$ around that point and the value of $$u$$ at that point.

For example, if there's at some position a "hot spot" where $$u$$ is on average greater then its neighbors, the Laplacian will be negative and thus the time derivative will be negative, this will cause $$u$$ to decrease at that position, "cooling" it down. This is illustrated below. The arrows reflect upon the magnitude of the Laplacian and, by grace of the time derivative, the direction the curve will move.

It's worth noting that in 3D, this equation fully describes the flow of heat in a homogeneous solid that's not generating it's own heat (like too much electricity through a narrow wire would).

Laplace's Equation
Laplace's equation describes a steady state condition, and this is what it looks like:

\nabla^2 u = 0\, $$

Solutions of this equation are called harmonic functions. Some things to note:

 * Time is absent. This equation describes a steady state condition.
 * The absence of time implies the absence of an IC, so we'll be dealing with BVPs rather then IBVPs.
 * In one dimension, this is the ODE of a straight line passing through the boundaries at their specified values.
 * All functions that satisfy this equation in some domain are analytic (informally, an analytic function is equal to its Taylor expansion) in that domain.
 * Despite appearances, solutions of Laplace's equation are generally not minimal surfaces.
 * Laplace's equation is linear.

Laplace's equation is separable in the Cartesian (and almost any other) coordinate system. So, we shouldn't have too much problem solving it if the BCs involved aren't too convoluted.

Laplace's Equation on a Square: Cartesian Coordinates

Imagine a 1 x 1 square plate that's insulated top and bottom and has constant temperatures applied at its uninsulated edges, visualized to the right. Heat is flowing in and out of this thing steadily through the edges only, and since it's "thin" and "insulated", the temperature may be given as $$u(x, y)$$. This is the first time we venture into two spatial coordinates, note the absence of time.

Let's make up a BVP, referring to the picture:

\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\, $$

\begin{align} u(0, y) = 0 \qquad \mbox{(Edge D)}\\ u(1, y) = 0 \qquad \mbox{(Edge B)}\\ u(x, 0) = 0 \qquad \mbox{(Edge A)}\\ u(x, 1) = 1 \qquad \mbox{(Edge C)} \end{align} $$

So we have one nonhomogeneous BC. Assume that $$u(x, y) = X(x) Y(y)$$:

\frac{\partial^2}{\partial x^2}(X(x) Y(y)) + \frac{\partial^2}{\partial y^2}(X(x) Y(y)) = 0\, $$

X(x)Y(y) + X(x)Y(y) = 0\, $$

-\frac{X(x)}{X(x)} = \frac{Y(y)}{Y(y)} = k^2\, $$

 * $$\Big\Downarrow$$

X''(x) = -k^2 X(x)\, $$

Y''(y) = k^2 Y(y)\, $$

As with before, calling the separation constant $$k^2$$ in favor of just $$k$$ (or something) happens to make the problem easier to solve. Note that the negative sign was kept for the $$X(x)$$ equation: again, these choices happen to make things simpler. Solving each equation and combining them back into $$u(x, y)$$:

X(x) = C_1 \cos(k x) + C_2 \sin(k x)\, $$

Y(y) = C_3 e^{k y} + C_4 e^{-k y}\, $$

 * $$\Big\Downarrow$$

u(x, y) = X(x) Y(y)\, $$

u(x, y) = \left(C_1 \cos(k x) + C_2 \sin(k x)\right)\left(C_3 e^{k y} + C_4 e^{-k y}\right)\, $$

At edge D:

u(0, y) = 0\, $$

\left(C_1 \cos(k \cdot 0) + C_2 \sin(k \cdot 0)\right)\left(C_3 e^{k y} + C_4 e^{-k y}\right) = 0\, $$

C_1 \left(C_3 e^{k y} + C_4 e^{-k y}\right) = 0 \Rightarrow C_1 = 0\, $$

 * $$\Big\Downarrow$$

u(x, y) = C_2 \sin(k x)\left(C_3 e^{k y} + C_4 e^{-k y}\right)\, $$

Note that the constants can be merged, but we won't do it so that a point can be made in a moment. At edge A:

u(x, 0) = 0\, $$

C_2 \sin(k x)\left(C_3 e^{k \cdot 0} + C_4 e^{-k \cdot 0}\right) = 0\, $$

Taking $$C_2$$ as $$0$$ would satisfy this particular BC, however this would yield a plane solution of $$u(x, y) = 0$$, which can't satisfy the temperature at edge C. This is why the constants weren't merged a few steps ago, to make it obvious that $$C_2$$ may not be $$0$$. So, we instead take $$C_3 = -C_4$$ to satisfy the above, and then combine the three constants into one, call it $$B$$:

u(x, y) = C_2 \sin(k x)\left(C_3 e^{k y} + C_4 e^{-k y}\right) \qquad ; \ C_3 = -C_4\, $$

 * $$\Big\Downarrow$$

u(x, y) = B \sin(k x)\left(e^{k y} - e^{-k y}\right)\, $$

Now look at edge B:

u(1, y) = 0\, $$

B \sin(k)\left(e^{k y} - e^{-k y}\right) = 0\, $$

It should go without saying by now that $$B$$ can't be zero, since this would yield $$u(x, y) = 0$$ which couldn't satisfy the nonzero BC. Instead, we can take $$k = n \pi$$:

u(x, y) = B \sin(n \pi x)\left(e^{n \pi y} - e^{-n \pi y}\right)\, $$

As of now, this solution will satisfy 3 of the 4 BCs. All that is left is edge C, the nonhomogeneous BC.

u(x, 1) = 1\, $$

B \sin(n \pi x)\left(e^{n \pi y} - e^{-n \pi y}\right) = 1\, $$

Neither $$B$$ nor $$n$$ can be contorted to fit this BC.

Since Laplace's equation is linear, a linear combination of solutions to the PDE is also a solution to the PDE. Another thing to note: since the BCs (so far) are homogeneous, we can add the solutions without worrying about nonzero boundaries adding up.

Though $$u(x, y)$$ as shown above will not solve this problem, we can try summing (based on $$n$$) solutions to form a linear combination which might solve the BVP as a whole:

u(x, y) = \sum_{n=1}^{\infty} u_n(x, y)\, $$

u_n(x, y) = B_n \sin(n \pi x)\left(e^{n \pi y} - e^{-n \pi y}\right)\, $$

 * $$\Big\Downarrow$$

u(x, y) = \sum_{n=1}^{\infty} B_n \sin(n \pi x)\left(e^{n \pi y} - e^{-n \pi y}\right)\, $$

Assuming this form is correct (review Parallel Plate Flow: Realistic IC for motivation), let's again try applying the last BC:

u(x, 1) = 1\, $$

\sum_{n=1}^{\infty} B_n \sin(n \pi x)\left(e^{n \pi \cdot 1} - e^{-n \pi \cdot 1}\right) = 1\, $$

It looks like it needs Fourier series methodology. Finding $$B_n$$ via orthogonality should solve this problem:

2 \sin(m \pi x) \cdot \sum_{n=1}^{\infty} B_n \sin(n \pi x)\left(e^{n \pi} - e^{-n \pi}\right) = 2 \sin(m \pi x) \cdot 1\, $$

\sum_{n=1}^{\infty} B_n \cdot 2 \sin(m \pi x) \sin(n \pi x)\left(e^{n \pi} - e^{-n \pi}\right) = 2 \sin(m \pi x) \, $$

\int_{0}^{1} \sum_{n=1}^{\infty} B_n \cdot 2 \sin(m \pi x) \sin(n \pi x) \left(e^{n \pi} - e^{-n \pi}\right) dx = \int_{0}^{1} 2 \sin(m \pi x) dx\, $$

\sum_{n=1}^{\infty} B_n \int_{0}^{1} 2 \sin(m \pi x) \sin(n \pi x) dx \left(e^{n \pi} - e^{-n \pi}\right) = 2 \frac{1 - \cos(m \pi)}{m \pi}\, $$

B_m \left(e^{m \pi} - e^{-m \pi}\right) = 2 \frac{1 - \cos(m \pi)}{m \pi}\, $$

B_n = 2 \frac{1 - (-1)^n}{n \pi \left(e^{n \pi} - e^{-n \pi}\right)}\, $$

$$m$$ was changed to $$n$$ in the last step. Also, for integer $$n$$, $$\cos(n \pi) = (-1)^n$$. Note that a Fourier sine expansion has been done. The solution to the BVP can finally be assembled:

u(x, y) = \sum_{n=1}^{\infty} B_n \sin(n \pi x)\left(e^{n \pi y} - e^{-n \pi y}\right)\, $$

u(x, y) = \sum_{n=1}^{\infty} 2 \frac{1 - (-1)^n}{n \pi \left(e^{n \pi} - e^{-n \pi}\right)} \sin(n \pi x)\left(e^{n \pi y} - e^{-n \pi y}\right)\, $$

That solves it!

It's finally time to mention that the BCs are discontinuous at the points $$(0, 1)$$ and $$(1, 1)$$. As a result, the series should converge slowly at those points. This is clear from the plot at right: it's a 25 term partial sum (note that half of the terms are $$0$$), and it looks perfect except at $$y = 1$$, especially near the discontinuities at $$x = 0$$ and $$x = 1$$.

Laplace's Equation on a Circle: Polar Coordinates
Now, we'll specify the value of $$u$$ on a circular boundary. A circle can be represented in Cartesian coordinates without too much trouble; however, it would result in nonlinear BCs which would render the approach useless. Instead, polar coordinates $$(r, \theta)$$ should be used, since in such a system the equation of a circle is very simple. In order for this to be realized, a polar representation of the Laplacian is necessary. Without going in to the details just yet, the Laplacian is given in (2D) polar coordinates:

\nabla^2 u = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2}\, $$

This result may be derived using differentials and the chain rule; it's not difficult but it's a little long. In these coordinates Laplace's equation reads:

\frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0\, $$

Note that in going from Cartesian to polar coordinates, a price was paid: though still linear, Laplace's equation now has variable coefficients. This implies that after separation at least one of the ODEs will have variable coefficients as well.

Let's make up the following BVP, letting $$u = u(r, \theta)$$:

\frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0\, $$

u(1, \theta) = \phi(\theta)\, $$

u(r, \theta) \ \mbox{is bounded inside the unit circle.}\, $$

This could represent a physical problem analogous to the previous one: replace the square plate with a disc. Note the apparent absence of sufficient BC to obtain a unique solution. The funny looking statement that u is bounded inside the domain of interest turns out to be the key to getting a unique solution, and it often shows itself in polar coordinates. It "makes up" for the "lack" of BCs. To separate, we as usual incorrectly assume that $$u(r, \theta) = R(r) \Theta(\theta)$$:

\frac{\partial^2}{\partial r^2}(R(r)\Theta(\theta)) + \frac{1}{r} \frac{\partial}{\partial r}(R(r)\Theta(\theta)) + \frac{1}{r^2} \frac{\partial^2}{\partial \theta^2}(R(r)\Theta(\theta)) = 0\, $$

R(r)\Theta(\theta) + \frac{1}{r} R'(r)\Theta(\theta) + \frac{1}{r^2} R(r)\Theta(\theta) = 0\, $$

r^2 R(r) + r R'(r) + R(r) \frac{\Theta(\theta)}{\Theta(\theta)} = 0\, $$

-\frac{\Theta(\theta)}{\Theta(\theta)} = \frac{r^2 R(r) + r R'(r)}{R(r)} = k^2\, $$

 * $$\Big\Downarrow$$

r^2 R''(r) + r R'(r) - k^2 R(r) = 0\, $$

\Theta''(\theta) + k^2 \Theta(\theta) = 0\, $$

Once again, the way the negative sign and the separation constant are arranged makes the solution easier later on. These decisions are made mostly by trial and error.

The $$R$$ equation is probably one you've never seen before, it's a special case of the Euler differential equation (not to be confused with the Euler-Lagrange differential equation). There are a couple of ways to solve it, the most general method would be to change the variables so that an equation with constant coefficients is obtained. An easier way would be to note the pattern in the order of the coefficients and the order of the derivatives, and from there guess a power solution. Either way, the general solution to this simple case of Euler's ODE is given as:

R(r) = C_1 r^k + C_2 r^{-k}\, $$

This is a very good example problem since it goes to show that PDE problems very often turn into obscure ODE problems; we got lucky this time since the solution for $$R$$ was rather simple though its ODE looked pretty bad at first sight. The solution to the $$\Theta(\theta)$$ equation is:

\Theta(\theta) = C_3 \cos(k \theta) + C_4 \sin(k \theta)\, $$

Combining:

u(r, \theta) = R(r)\Theta(\theta)\, $$

u(r, \theta) = \left(C_1 r^k + C_2 r^{-k}\right)(C_3 \cos(k \theta) + C_4 \sin(k \theta))\, $$

Now, this is where the English sentence condition stating that u must be bounded in the domain of interest may be invoked. As $$\scriptstyle r \to 0$$, the term involving $$r^{-k}$$ is unbounded. The only way to fix this is to take $$C_2 = 0$$. Note that if this problem were solved between two concentric circles, this term would be nonzero and very important. With that term gone, constants can be merged:

u(r, \theta) = r^k (A \cos(k \theta) + B \sin(k \theta))\, $$

Only one condition remains: $$u = \phi(\theta)$$ on $$r = 1$$, yet there are 3 constants. Let's say for now that:

\phi(\theta) = 2 \cos(4 \theta) + 3 \sin(4 \theta)\, $$

Then, it's a simple matter of equating coefficients to obtain:

u(r, \theta) = r^4 (2 \cos(4 \theta) + 3 \sin(4 \theta))\, $$

Now, let's make the frequencies differ:

\phi(\theta) = 4 \cos(3 \theta) + \sin(10 \theta)\, $$

Equating coefficients won't work. However, if the IC were broken up into individual terms, the sum of the solution to the terms just happens to solve the BVP as a whole:

\phi_3(\theta) = 4 \cos(3 \theta) \quad, \quad \phi_{10}(\theta) = \sin(10 \theta)\, $$

u_3(r, \theta) = r^3 4 \cos(3 \theta) \quad, \quad u_{10}(r, \theta) = r^{10} \sin(10 \theta)\, $$

u(r, \theta) = u_3(r, \theta) + u_{10}(r, \theta) = r^3 4 \cos(3 \pi \theta) + r^{10} \sin(10 \theta)\, $$

Verify that the solution above is really equal to the BC at $$r = 1$$:

u(1, \theta) = 1^3 4 \cos(3 \theta) + 1^{10} \sin(10 \theta)) = 4 \cos(3 \theta) + \sin(10 \theta) = \phi(\theta)\, $$

And, since Laplace's equation is linear, this must solve the PDE as well. What all of this implies is that, if some generic function $$\phi(\theta)$$ may be expressed as a sum of sinusoids with angular frequencies given by $$n$$, all that is needed is a linear combination of the appropriate sum. Notated:

u(r, \theta) = \sum_{n = 1}^{\infty} r^{n} (A_n \cos(n \theta) + B_n \sin(n \theta))\, $$

To identify the coefficients, substitute the BC:

u(1, \theta) = \sum_{n = 1}^{\infty} 1^{n} (A_n \cos(n \theta) + B_n \sin(n \theta))\, $$

\phi(\theta) = \sum_{n = 1}^{\infty} (A_n \cos(n \theta) + B_n \sin(n \theta))\, $$

The coefficients $$A_n$$ and $$B_n$$ may be determined by a (full) Fourier expansion on $$0 \leq \theta \leq 2 \pi$$. Note that it's implied that $$\phi(\theta)$$ must have period $$2 \pi$$ since we are solving this in a domain (a circle specifically) where $$0 \leq \theta \leq 2 \pi$$.

You probably don't like infinite series solutions. Well, it happens that through a variety of manipulations it's possible to express the full solution of this particular problem as:

u(r, \theta) = \frac{1}{2 \pi}\int_{0}^{2 \pi} \phi(\psi) \frac{1 - r^2}{1 - 2 r \cos(\theta - \psi) + r^2}\ d\psi\, $$

This is called Poisson's integral formula.

Derivation of the Laplacian in Polar Coordinates
Though not necessarily a PDEs concept, it is very important for anyone studying this kind of math to be comfortable with going from one coordinate system to the next. What follows is a long derivation of the Laplacian in 2D polar coordinates using the multivariable chain rule and the concept of differentials. Know, however, that there are really many ways to do this.

Three definitions are all we need to begin:

\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\, $$

x = r \cos(\theta)\, $$

y = r \sin(\theta)\, $$

If it's known that $$u = u(r, \theta) = u(r(x, y), \theta(x, y))$$, then the chain rule may be used to express derivatives in terms of $$r$$ and $$\theta$$ alone. Two applications will be necessary to obtain the second derivatives. Manipulating operators as if they meant something on their own:

\frac{\partial}{\partial x} = \frac{\partial}{\partial r}\frac{\partial r}{\partial x} + \frac{\partial}{\partial \theta}\frac{\partial \theta}{\partial x}\, $$

Applying this to itself, treating the underlined bit as a unit dependent on $$r$$ and $$\theta$$:

\frac{\partial^2}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}\right) = \frac{\partial}{\partial x}\left( \underline{ \frac{\partial}{\partial r}\frac{\partial r}{\partial x} + \frac{\partial}{\partial \theta}\frac{\partial \theta}{\partial x} }\right) \,$$

\frac{\partial^2}{\partial x^2} = \frac{\partial}{\partial r}\left( \underline{\frac{\partial}{\partial r}\frac{\partial r}{\partial x} + \frac{\partial}{\partial \theta}\frac{\partial \theta}{\partial x}} \right)\frac{\partial r}{\partial x} + \frac{\partial}{\partial \theta}\left( \underline{\frac{\partial}{\partial r}\frac{\partial r}{\partial x} + \frac{\partial}{\partial \theta}\frac{\partial \theta}{\partial x}} \right)\frac{\partial \theta}{\partial x} \,$$

\frac{\partial^2}{\partial x^2} = \left( \frac{\partial^2}{\partial r^2}\frac{\partial r}{\partial x} + \frac{\partial}{\partial r}\frac{\partial^2 r}{\partial r \partial x} + \frac{\partial^2}{\partial r \partial \theta}\frac{\partial \theta}{\partial x} + \frac{\partial}{\partial \theta}\frac{\partial^2 \theta}{\partial r \partial x} \right)\frac{\partial r}{\partial x} + \left( \frac{\partial^2}{\partial \theta \partial r}\frac{\partial r}{\partial x} + \frac{\partial}{\partial r}\frac{\partial^2 r}{\partial \theta \partial x} + \frac{\partial^2}{\partial \theta^2}\frac{\partial \theta}{\partial x} + \frac{\partial}{\partial \theta}\frac{\partial^2 \theta}{\partial \theta \partial x} \right)\frac{\partial \theta}{\partial x} \,$$

The above mess may be quickly simplified a little by manipulating the funny looking derivatives:

\frac{\partial^2 r}{\partial r \partial x} = \frac{\partial^2}{\partial x \partial r}(r) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial r}(r)\right) = \frac{\partial}{\partial x}(1) = 0 \,$$

\frac{\partial^2 \theta}{\partial \theta \partial x} = \frac{\partial^2}{\partial x \partial \theta}(\theta) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial \theta}(\theta)\right) = \frac{\partial}{\partial x}(1) = 0 \,$$

 * $$\Big\Downarrow$$

\frac{\partial^2}{\partial x^2} = \left( \frac{\partial^2}{\partial r^2}\frac{\partial r}{\partial x} + \frac{\partial^2}{\partial r \partial \theta}\frac{\partial \theta}{\partial x} + \frac{\partial}{\partial \theta}\frac{\partial^2 \theta}{\partial r \partial x} \right)\frac{\partial r}{\partial x} + \left( \frac{\partial^2}{\partial \theta \partial r}\frac{\partial r}{\partial x} + \frac{\partial}{\partial r}\frac{\partial^2 r}{\partial \theta \partial x} + \frac{\partial^2}{\partial \theta^2}\frac{\partial \theta}{\partial x} \right)\frac{\partial \theta}{\partial x} \,$$

This may be made slightly easier to work with if a few changes are made to the way some of the derivatives are written. Also, the $$y$$ variable follows analogously:

\frac{\partial^2}{\partial x^2} = \left( \frac{\partial^2}{\partial r^2}\frac{\partial r}{\partial x} + \frac{\partial^2}{\partial r \partial \theta}\frac{\partial \theta}{\partial x} + \frac{\partial}{\partial \theta}\frac{\partial}{\partial r}\left(\frac{\partial \theta}{\partial x}\right) \right)\frac{\partial r}{\partial x} + \left( \frac{\partial^2}{\partial \theta \partial r}\frac{\partial r}{\partial x} + \frac{\partial}{\partial r}\frac{\partial}{\partial \theta}\left(\frac{\partial r}{\partial x}\right) + \frac{\partial^2}{\partial \theta^2}\frac{\partial \theta}{\partial x} \right)\frac{\partial \theta}{\partial x} \,$$

\frac{\partial^2}{\partial y^2} = \left( \frac{\partial^2}{\partial r^2}\frac{\partial r}{\partial y} + \frac{\partial^2}{\partial r \partial \theta}\frac{\partial \theta}{\partial y} + \frac{\partial}{\partial \theta}\frac{\partial}{\partial r}\left(\frac{\partial \theta}{\partial y}\right) \right)\frac{\partial r}{\partial y} + \left( \frac{\partial^2}{\partial \theta \partial r}\frac{\partial r}{\partial y} + \frac{\partial}{\partial r}\frac{\partial}{\partial \theta}\left(\frac{\partial r}{\partial y}\right) + \frac{\partial^2}{\partial \theta^2}\frac{\partial \theta}{\partial y} \right)\frac{\partial \theta}{\partial y} \,$$

Now we need to obtain expressions for some of the derivatives appearing above. The most direct path would use the concept of differentials. If:

x = r \cos(\theta) \quad, \quad y = r \sin(\theta)\, $$

Then:

dx = dr \cos(\theta) - r \sin(\theta) d\theta \quad, \quad dy = dr \sin(\theta) + r \cos(\theta) d\theta\, $$

Solving by substitution for $$dr$$ and $$d \theta$$ gives:

dr = \cos(\theta) dx + \sin(\theta) dy \quad , \quad d\theta = -\frac{\sin(\theta)}{r} dx + \frac{\cos(\theta)}{r} dy \,$$

If $$z = z(x, y)$$, then the total differential is given as:

dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy \,$$

Note that the two previous equations are of this form (recall that $$r = r(x, y)$$ and $$\theta = \theta(x, y)$$, just like $$z$$ above), which means that:

dr = \frac{\partial r}{\partial x}dx + \frac{\partial r}{\partial y}dy \quad , \quad d\theta = \frac{\partial \theta}{\partial x}dx + \frac{\partial \theta}{\partial y}dy \,$$

Equating coefficients quickly yields a bunch of derivatives:

\frac{\partial r}{\partial x} = \cos(\theta) \quad, \quad \frac{\partial r}{\partial y} = \sin(\theta) \quad, \quad \frac{\partial \theta}{\partial x} = -\frac{\sin(\theta)}{r} \quad, \quad \frac{\partial \theta}{\partial y} = \frac{\cos(\theta)}{r} \,$$

There's an easier but more abstract way to obtain the derivatives above that may be overkill but is worth mentioning anyway. The Jacobian of the functions $$x(r, \theta)$$ and $$y(r, \theta)$$ is:

\frac{\partial (x, y)}{\partial (r, \theta)} = \begin{bmatrix} \cfrac{\partial x}{\partial r} & \cfrac{\partial x}{\partial \theta} \\ \cfrac{\partial y}{\partial r} & \cfrac{\partial y}{\partial \theta} \\ \end{bmatrix} = \begin{bmatrix} \cos(\theta) & -r \sin(\theta) \\ \sin(\theta) & r \cos(\theta) \\ \end{bmatrix} $$

Note that the Jacobian is a compact representation of the coefficients of the total derivative; using $$\mathbf{z} = \mathbf{z}(\mathbf{x})$$ as an example (bold indicating vectors):

d\mathbf{z} = \frac{\partial (z_1, z_2, ... \, z_n)}{\partial (x_1, x_2, ... \ , x_n)} \cdot d\mathbf{x} $$

So, it follows then that the derivatives that we're interested in may be obtained by inverting the Jacobian matrix:

\frac{\partial (r, \theta)}{\partial (x, y)} = \left(\frac{\partial (x, y)}{\partial (r, \theta)}\right)^{-1} $$

\begin{bmatrix} \cfrac{\partial r}{\partial x} & \cfrac{\partial r}{\partial y} \\ \cfrac{\partial \theta}{\partial x} & \cfrac{\partial \theta}{\partial y} \\ \end{bmatrix} = \begin{bmatrix} \cfrac{\partial x}{\partial r} & \cfrac{\partial x}{\partial \theta} \\ \cfrac{\partial y}{\partial r} & \cfrac{\partial y}{\partial \theta} \\ \end{bmatrix}^{-1} $$

\begin{bmatrix} \cfrac{\partial r}{\partial x} & \cfrac{\partial r}{\partial y} \\ \cfrac{\partial \theta}{\partial x} & \cfrac{\partial \theta}{\partial y} \\ \end{bmatrix} = \begin{bmatrix} \cos(\theta) & -r \sin(\theta) \\ \sin(\theta) & r \cos(\theta) \\ \end{bmatrix}^{-1} $$

\begin{bmatrix} \cfrac{\partial r}{\partial x} & \cfrac{\partial r}{\partial y} \\ \cfrac{\partial \theta}{\partial x} & \cfrac{\partial \theta}{\partial y} \\ \end{bmatrix} = \begin{bmatrix} \cos(\theta) & \sin(\theta) \\ -\cfrac{\sin(\theta)}{r} & \cfrac{\cos(\theta)}{r} \\ \end{bmatrix} $$

Though somewhat obscure, this is very convenient and it's just one of the many utilities of the Jacobian matrix. An interesting bit of insight is gained: coordinate changes are senseless unless the Jacobian is invertible everywhere except at isolated points, stated another way the determinant of the Jacobian matrix must be nonzero, otherwise the coordinate change is not one-to-one (note that the determinant will be zero at $$r = 0$$ in this example. An isolated point such as this is not problematic.).

Either path you take, there should now be enough information to evaluate the Cartesian second derivatives. Working on $$x$$:

\frac{\partial^2}{\partial x^2} = \left( \frac{\partial^2}{\partial r^2}\cos(\theta) - \frac{\partial^2}{\partial r \partial \theta}\frac{\sin(\theta)}{r} - \frac{\partial}{\partial \theta}\frac{\partial}{\partial r}\left(\frac{\sin(\theta)}{r}\right) \right)\cos(\theta) - \left( \frac{\partial^2}{\partial \theta \partial r}\cos(\theta) + \frac{\partial}{\partial r}\frac{\partial}{\partial \theta}\left(\cos(\theta)\right) - \frac{\partial^2}{\partial \theta^2}\frac{\sin(\theta)}{r} \right)\frac{\sin(\theta)}{r} \,$$

\frac{\partial^2}{\partial x^2} = \cos(\theta)^2\frac{\partial^2}{\partial r^2} - \frac{\sin(\theta)\cos(\theta)}{r}\frac{\partial^2}{\partial r \partial \theta} + \frac{\sin(\theta)\cos(\theta)}{r^2}\frac{\partial}{\partial \theta} - \frac{\sin(\theta)\cos(\theta)}{r}\frac{\partial^2}{\partial \theta \partial r} + \frac{\sin(\theta)^2}{r}\frac{\partial}{\partial r} + \frac{\sin(\theta)^2}{r^2}\frac{\partial^2}{\partial \theta^2} \,$$

\frac{\partial^2}{\partial x^2} = \cos(\theta)^2\frac{\partial^2}{\partial r^2} - \frac{2\sin(\theta)\cos(\theta)}{r}\frac{\partial^2}{\partial r \partial \theta} + \frac{\sin(\theta)\cos(\theta)}{r^2}\frac{\partial}{\partial \theta} + \frac{\sin(\theta)^2}{r}\frac{\partial}{\partial r} + \frac{\sin(\theta)^2}{r^2}\frac{\partial^2}{\partial \theta^2} \,$$

Proceeding similarly for $$y$$:

\frac{\partial^2}{\partial y^2} = \left( \frac{\partial^2}{\partial r^2}\sin(\theta) + \frac{\partial^2}{\partial r \partial \theta}\frac{\cos(\theta)}{r} + \frac{\partial}{\partial \theta}\frac{\partial}{\partial r}\left(\frac{\cos(\theta)}{r}\right) \right)\sin(\theta) + \left( \frac{\partial^2}{\partial \theta \partial r}\sin(\theta) + \frac{\partial}{\partial r}\frac{\partial}{\partial \theta}\left(\sin(\theta)\right) + \frac{\partial^2}{\partial \theta^2}\frac{\cos(\theta)}{r} \right)\frac{\cos(\theta)}{r} \,$$

\frac{\partial^2}{\partial y^2} = \sin(\theta)^2\frac{\partial^2}{\partial r^2} + \frac{\sin(\theta)\cos(\theta)}{r}\frac{\partial^2}{\partial r \partial \theta} - \frac{\sin(\theta)\cos(\theta)}{r^2}\frac{\partial}{\partial \theta} + \frac{\sin(\theta)\cos(\theta)}{r}\frac{\partial^2}{\partial \theta \partial r} + \frac{\cos(\theta)^2}{r}\frac{\partial}{\partial r} + \frac{\cos(\theta)^2}{r^2}\frac{\partial^2}{\partial \theta^2} \,$$

\frac{\partial^2}{\partial y^2} = \sin(\theta)^2\frac{\partial^2}{\partial r^2} + \frac{2\sin(\theta)\cos(\theta)}{r}\frac{\partial^2}{\partial r \partial \theta} - \frac{\sin(\theta)\cos(\theta)}{r^2}\frac{\partial}{\partial \theta} + \frac{\cos(\theta)^2}{r}\frac{\partial}{\partial r} + \frac{\cos(\theta)^2}{r^2}\frac{\partial^2}{\partial \theta^2} \,$$

Now, add these tirelessly hand crafted differential operators and watch the result collapse into just 3 nontrigonometric terms:

\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\, $$

\nabla^2 = \cos(\theta)^2\frac{\partial^2}{\partial r^2} - \frac{2\sin(\theta)\cos(\theta)}{r}\frac{\partial^2}{\partial r \partial \theta} + \frac{\sin(\theta)\cos(\theta)}{r^2}\frac{\partial}{\partial \theta} + \frac{\sin(\theta)^2}{r}\frac{\partial}{\partial r} + \frac{\sin(\theta)^2}{r^2}\frac{\partial^2}{\partial \theta^2} + \,$$

\qquad \sin(\theta)^2\frac{\partial^2}{\partial r^2} + \frac{2\sin(\theta)\cos(\theta)}{r}\frac{\partial^2}{\partial r \partial \theta} - \frac{\sin(\theta)\cos(\theta)}{r^2}\frac{\partial}{\partial \theta} + \frac{\cos(\theta)^2}{r}\frac{\partial}{\partial r} + \frac{\cos(\theta)^2}{r^2}\frac{\partial^2}{\partial \theta^2} \,$$

\nabla^2 = (\sin(\theta)^2 + \cos(\theta)^2)\frac{\partial^2}{\partial r^2} + \left(-\frac{2\sin(\theta)\cos(\theta)}{r} + \frac{2\sin(\theta)\cos(\theta)}{r}\right)\frac{\partial^2}{\partial r \partial \theta} + \, $$

\left(\frac{\sin(\theta)\cos(\theta)}{r^2} - \frac{\sin(\theta)\cos(\theta)}{r^2}\right)\frac{\partial}{\partial \theta} + \left(\frac{\sin(\theta)^2}{r} + \frac{\cos(\theta)^2}{r}\right)\frac{\partial}{\partial r} + \left(\frac{\sin(\theta)^2}{r^2} + \frac{\cos(\theta)^2}{r^2}\right)\frac{\partial^2}{\partial \theta^2} \,$$

\nabla^2 = \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + \frac{1}{r^2}\frac{\partial^2}{\partial \theta^2}\, $$

That was a lot of work. To save trouble, here is the Laplacian in other two other popular coordinate systems:

Cylindrical:

\nabla^2 u = \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2} + \frac{\partial^2 u}{\partial z^2} \,$$

Spherical:

\nabla^2 u = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial u}{\partial r}\right) + \frac{1}{r^2 \sin(\theta)}\frac{\partial}{\partial \theta}\left(\sin(\theta) \frac{\partial u}{\partial \theta}\right) + \frac{1}{r^2 \sin(\theta)^2}\frac{\partial^2 u}{\partial \phi^2} \,$$

Derivatives have been combined wherever possible (not done previously).

Concluding Remarks
This was a long, involved chapter. It should be clear that the solutions derived work only for very simple geometries, other geometries may be worked with by grace of conformal mappings.

The Laplacian (and variations of it) is a very important quantity and its behaviour is worth knowing like the back of your hand. A sampling of important equations that involve the Laplacian:

 * The Navier Stokes equations.
 * The diffusion equation.
 * Laplace's equation.
 * Poisson's equation.
 * The Helmholtz equation.
 * The Schrödinger equation.
 * The wave equation.

There's a couple of other operators that are similar to (though less important than) the Laplacian, which deserve mention:

 * Biharmonic operator, in three Cartesian dimensions:

\nabla^4 = \frac{\partial^4 u}{\partial x^4} + \frac{\partial^4 u}{\partial y^4} + \frac{\partial^4 u}{\partial z^4} + 2\frac{\partial^4 u}{\partial x^2 \partial y^2} + 2\frac{\partial^4 u}{\partial y^2 \partial z^2} + 2\frac{\partial^4 u}{\partial x^2 \partial z^2} \,$$

The biharmonic equation is useful in linear elastic theory, for example it can describe "creeping" fluid flow:

\nabla^4 \psi = 0 \,$$

 * d'Alembertian:

\Box^2 = \nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \,$$

The wave equation may be expressed using the d'Alembertian:

\Box^2 u = 0 \,$$

Though expressed with the Laplacian is more popular:

\nabla^2 u = \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} \,$$


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ERP Internals/Modules/Investment Manager

Every enterprise has a reserve of funds. As the enterprise grows larger, the total reserve becomes larger and larger. Eventually, it is large enough that it is prudent for the enterprise to invest part of it to gain a higher rate of return than would be possible in a checking account. These investments include GICs, mutual funds, stocks, bonds, and other investment options. Investment Manager keeps track of these investments. Only large ERP packages have an Investment Manager module.

 * Functions
 * Reports

Tables
The first table is the Investment table which contains one record for every investment and the second table is the Investment Transaction table which contains one record for every investment transaction. There is at least one transaction record for every investment and there can be many.

Data Exchange
Investment Manager sends transactions to General Ledger.

Transactions
Investment Manager

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


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ERP Internals/Modules/Pricing

The ERP provides six different prices for each item. The Pricing module calculates these prices based on pricing rules. Every ERP package has some form of Pricing module.

 * Functions
 * Reports

Data Exchange
Pricing does not send or receive transactions to/from other modules.

Transactions
Pricing

title: ERP Internals/Modules/Pricing


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ERP Internals/Modules/Receiving

Receiving receives the physical goods from suppliers, verifies their quantity and quality and delivers them to their destination within the enterprise. It interacts with Purchasing by flagging Purchase Order lines as received and, after authorization from Purchasing, with Inventory by reducing on-order and increasing on-hand quantities. See Purchasing for a description of invoice matching. Normally, Shipping and Receiving occur in the same physical location. All ERP packages have a receiving function although it might be included in Inventory rather than made a separate module.

 * Functions
 * Reports

Data Exchange
Receiving sends transactions to Purchasing and Inventory.

Transactions
Receiving

title: ERP Internals/Modules/Receiving


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ERP Internals/Modules/Shipping

Shipping receives orders from Warehouse Manager (picking) if there is no manufacturing involved or from Manufacturing if there is. It packs the goods, selects transportation to satisfy the customer's due date, and generates shipping documents such as labels, packing slips, and bills of lading. Customers can be notified electronically when the goods are shipped and Order Entry/Invoicing is notified so the order can be invoiced. Normally, Shipping and Receiving occur in the same physical location. Only large ERP packages have a separate Shipping module.

 * Functions
 * Reports

Data Exchange
Shipping does not send or receive transactions to/from other modules.

Transactions
Shipping

title: ERP Internals/Modules/Shipping


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ERP Internals/Glossary/Bond

A bond is a debt owned by the enterprise to the bondholder. Commercial bonds are normally issued in units of $1,000. Bondholders receive regular interest on their investment.

title: ERP Internals/Glossary/Bond


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ERP Internals/Glossary/Bondholder

A bondholder is an individual or a enterprise that owns a bond.

title: ERP Internals/Glossary/Bondholder


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ERP Internals/Glossary/ABC

ABC has a meaning in Inventory and is also an acronym for Activity-based Costing.

In Inventory, "A" items are high value items that are closely controlled, "B" items are medium value items that have moderate control, and "C" items are low value items that are loosely controlled. See Cycle Counting.

title: ERP Internals/Glossary/ABC


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ERP Internals/Glossary/Physical Count

Once a year, every item is counted to correct errors that accumulated during the year. Contrast with Cycle Counting.

title: ERP Internals/Glossary/Physical Count


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ERP Internals/Glossary/Cycle Counting

In Inventory, a method in which some items are counted every week (or even day in the case of large businesses) based on the ABC classification. Cycle Counting avoids the major disruption of Physical Counting.

title: ERP Internals/Glossary/Cycle Counting


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Talk:Animal Behavior/Operant Conditioning

i propose to include the content of this page in the section on Learning Robert Huber 00:22, 24 January 2007 (UTC)

content transfer
i am not sure whether i did this right. i have inserted the content of this page into the section structure of the book - the section on Learning. i then made some major changes to it there but i left this original page as it was. is there a way to copy the original edit history along with the text? thanks, Robert Huber 16:24, 25 January 2007 (UTC)


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ERP Internals/Glossary/Back Order

A customer orders a certain quantity of an item. If the business does not have enough to fill the order, some of the items are placed on back order and shipped to the customer at a later date when they are received from the manufacturer or created by Manufacturing.

title: ERP Internals/Glossary/Back Order


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ERP Internals/Glossary/Employee

A person who works for the enterprise and is paid a salary.

title: ERP Internals/Glossary/Employee


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ERP Internals/Glossary/Cash on Delivery

Payment is required when the order is delivered to a customer.

title: ERP Internals/Glossary/Cash on Delivery


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ERP Internals/Glossary/Customer

A person or enterprise that purchases the enterprise's goods and/or services. In the case of not-for-profits, a customer is referred to as a "donor".

title: ERP Internals/Glossary/Customer


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ERP Internals/Glossary/Customer Relationship Marketing

Presenting a unified version of all the customer information and interactions. CRM is an add-on to many of the earlier ERP packages.


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ERP Internals/Glossary/Supplier Relationship Marketing

Presenting a unified version of all the supplier information and interactions. SRM is an add-on to many of the earlier ERP packages.


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ERP Internals/Glossary/Deposit

Typically, an enterprise makes a bank deposit once a day. This consists of the checks the enterprise has received plus any cash or credit/debit card receipts.

Alternatively, deposit means a payment made by a customer to an enterprise before work starts on a job or project. The deposit is typically a percentage of the job price.

title: ERP Internals/Glossary/Deposit


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ERP Internals/Glossary/Donor

Someone that contributes to a not-for-profit enterprise. In a fuzzy kind of way, "donors" can be considered "customers" in that they are a source of revenue.

title: ERP Internals/Glossary/Donor


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ERP Internals/Glossary/Discount Date

Invoices that allow credit normally allow a discount if the invoice is paid within a certain number of days. This is the discount date. It is calculated using the invoice date and the terms.

title: ERP Internals/Glossary/Discount Date


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ERP Internals/Glossary/Due Date

Invoices that are not COD normally require that the invoice be paid within a certain number of days. This is the Due Date. It is calculated using the invoice date and the terms.

title: ERP Internals/Glossary/Due Date


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ERP Internals/Glossary/Economic Order Quantity

Based on the cost of the goods and the cost of storage, an optimum order quantity can be calculated. I'll add the formula later.


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ERP Internals/Glossary/Estimate

A document presented to a potential customer that provides prices but does not guarantee them. Contrast with Quotation.

title: ERP Internals/Glossary/Estimate


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ERP Internals/Glossary/Quotation

A document presented to a potential customer that guarantees prices. Contrast with Estimate.

title: ERP Internals/Glossary/Quotation


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ERP Internals/Glossary/Fixed Asset

Something the enterprise has purchased that has a useful life of more than one year and is not for resale

title: ERP Internals/Glossary/Fixed Asset


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ERP Internals/Glossary/First In First Out

In Inventory, the concept that the first item received is the first item sold. This is the most common practise. Contrast with LIFO and Average.


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ERP Internals/Glossary/Last In First Out

In Inventory, the concept that the last item received is the first item sold. This is an unusual practise. Contrast with FIFO and Average.


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ERP Internals/Glossary/Interest

Money paid on a debt. This can be interest on a bond or interest on an overdue invoice.

title: ERP Internals/Glossary/Interest


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ERP Internals/Glossary/Invoice

A record of an order by a customer of goods and/or services. Invoices have a unique #, a date, terms and details of the goods and services ordered.

title: ERP Internals/Glossary/Invoice


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ERP Internals/Glossary/Invoice Date

The date the invoice was issued. This is used in the calculation of the Due Date and the Discount Date.

title: ERP Internals/Glossary/Invoice Date


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ERP Internals/Glossary/Electronic Data Interchange

A method in which data such as invoices, purchase order, and order acknowledgements can be exchanged between companies electronically. This is being replaced by XML.


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ERP Internals/Glossary/Password

A unique sequence of letters and numbers that identifies a user.

title: ERP Internals/Glossary/Password


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ERP Internals/Glossary/Mutual Fund

A company that buys and sells stocks, bonds and other investments in hopes of providing a good return to its owners.

title: ERP Internals/Glossary/Mutual Fund


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ERP Internals/Glossary/National Account

A customer that has multiple locations but is treated as a single entity.

title: ERP Internals/Glossary/National Account


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ERP Internals/Glossary/Outstanding

Goods or services are often provided on credit. The invoice have a Due Date. If they are unpaid by their Due Date, they are outstanding.

title: ERP Internals/Glossary/Outstanding


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ERP Internals/Glossary/Purchase Order

A Purchase Order authorizes a supplier to ship products to or perform services for the enterprise and invoice them for it.

title: ERP Internals/Glossary/Purchase Order


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ERP Internals/Glossary/Receiving

The department of the enterprise that receives goods from suppliers. Receiving and Shipping normally happen at the same physical location.

title: ERP Internals/Glossary/Receiving


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ERP Internals/Glossary/Shipping

The department of the enterprise that sends goods from customers. Receiving and Shipping normally happen at the same physical location.

title: ERP Internals/Glossary/Shipping


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ERP Internals/Glossary/Request for Proposal

A document sent to potential suppliers requesting a document specifying how they will meet complex requirements. Contrast with RFQ.


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ERP Internals/Glossary/Request for Quotation

A document sent to potential suppliers requesting a document specifying how they will meet relatively simple requirements. Contrast with RFP.


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Jeep Liberty/Factory Options

Over the years, the different factory options available on the Liberty have changed. This page documents the options by year available.

Drivetrain

Options

title: Jeep Liberty/Factory Options


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Roulette/Mech Sys

Back | Other Statistical Systems

Mechanical systems are systems which rely on the inherent flaws of a roulette wheel as a mechanical device or contiguous number spans across the wheel layout. This is a fundamental difference in strategy between the way that a table game and a computer generates random numbers.

The basic principle underlying mechanical betting systems is exploiting the flaw in the way that the machine generates the numbers. As opposed to an electronic random number generator where any number is theoretically independent of any other number, a wheel is constructed with numbers situated permanently adjacent to one another. Therefore the numbers are not completely independent to one another because if the ball is right on the edge of falling from one cup to another, the number those cups represents are fixed. This can be especially useful when using Table clocking to determine a biased wheel, but as technology is making finding a biased wheel very difficult, the underlying theory is still interesting to consider. These systems are generally reserved for the American tables because it’s next to impossible to create contiguous sets on the European layout. However the casinos have solved this problems and already have special announce bets that can make this process easier.

If the ball is predestined to fall a certain region of the wheel (the 0 pocket for example), and you cover the four numbers to each side of the 0 {12,35,3,26,0,32,15,19,4}, as the ball jostles and bounces it is more likely to land in the pockets immediately next to the number than the other side of the wheel. However covering these numbers, with the same relative return, and without dropping a ton of chips is the challenge. Finding a betting system that most efficiently covers the numbers on the region of the board is the challenge to the mechanical system.

In this author’s opinion, this is the ONLY system worth exploring with real money in a casino – or buy a wheel and try it at home, but as of the time of this writing (2008) good professional 30” wheels cost around $4000, and a 27” around $2000. When one considers some systems can be explored with a bankroll of less than $100, it may be worth hitting a casino first.

What makes this most interesting is that mathematically the odds don’t change. Covering 24 numbers no matter where they are has the same probability as placing a bet which covers the numbers in sequence around the wheel. Perhaps it’s just a visceral reaction, but there have been many times where the ball seems to land in the pocket for the number which was bet only to have it slide one pocket over. I don’t know physics well enough to say if there is any reason why covering a region of numbers on a roulette wheel is more advantageous than not, but it just feels right. Besides the math has proved ineffectual.

There is only one book that I have been able to find which covers this in any great detail; Beating the Casinos at Their Own Game by Peter Svoboda ISBN 0-7570-0005-3. I make direct references to this source only because this particular section of this book is available on Google Books. This book also covers several other games in interesting ways and I highly recommend buying a copy or getting it from the interlibrary loan from your local library.

Mr. Svoboda goes into several systems and makes several interesting points, but one thing which he does not stress is that it is VITAL to find a roulette wheel with the exact same layout order as the one he uses in his book. It is possible that a wheel could look very similar but have a different order to their numbering which would require the player to analyze that particular arrangement and betting strategy. For instance I have seen some antique tables which have a reversed layout where the 2 is on the left of the 0 as opposed to the right, however everything else was in the same order so this may be a moot point.

The thing that I really like is that Mr. Svoboda also makes all of his systems mathematically related to winning percentages, and his strategies cover sectors of the wheel rather than sections of the layout. Additionally they fit within the constraints of the table limits imposed by most US casinos.

I’ll go into this more later.

Standing Systems
Standing systems (for lack of a better term) are those where the same bet is played on the same locations irregardless of previous wins or losses. This is opposed to progressive systems as described in the next section.

System 1
On the American layout, the best known contiguous bet combination is based on the center column. That placement covers the span from 23 to 5 (clockwise) with five gaps {0, 7, 9, 23, and 30}, and it only includes two numbers {8 and 29} which are outside the span. This covers a span width of 15 numbers which is almost exactly two fifths of the wheel.

This bet can be played two different ways.
 * Place 4 chips on the center column and one chip on each of the numbers in set {0, 7, 9, 28, and 30}
 * Return: If the numbers {2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35} hit, then the return is: (4*2)-5=3. If the numbers {0, 7, 9, 28, 30}hit, then the return is: (1*35)-(4+4)=27.  If the numbers {00, 1, 3, 4, 6, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 31, 33, 34, 36} hit, then the return is -9.

 * Place 1 chip each on two street bets {7, 8, 9} and {28, 29, 30}, and one chip on 0, and two chips on the center column.
 * Return: If the numbers {2, 5, 11, 14, 17, 20, 23, 26, 32, 35} hit, then the return is: (2*2)-3=1. If the number {0} hits, then the return is (1*35)-4=31.  If the numbers {7, 9}or {28, 30} hit, then the return is: (1*11)-4=7.  If the numbers {8 or 29} hit, then the return is ((2*2)+(1*11))-2=13.  If the numbers {00, 1, 3, 4, 6, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 31, 33, 34, 36} hit, then the return is -5.

System 2
This combines two spans to cover 23 of the 38 numbers on the wheel.

 * Place 1 chip on each of the 4 street bets {1, 2, 3}, {13, 14, 15}, {22, 23, 24}, and {34, 35, 36}. This is the first span.  Place 1 chip on each of the two street bets {4, 5, 6} and {31, 32, 33}, and place 1 chip on each of the two splits {16, 19} and {18, 21}.
 * Return: If the numbers {1, 2, 3, 4, 5, 6, 13, 14, 15, 22, 23, 24, 31, 32, 33, 34, 35, 36} hit, then the return is: (1*11)-7=4. If the number {16, 19, 18, 21} hit, then the return is: (1*17)-7=10.  If the numbers {0, 00, 7, 8, 9, 10, 11, 12, 17, 20, 25, 26, 27, 28, 29, 30} hit, the return is -8.

Progressive Systems
Progressive systems change depending on the previous win/loss history. These systems are based upon the mechanical layout as opposed to statistical systems as described in this chapter.

System 1
Next | Math

title: Roulette/Mech Sys


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Radiation Oncology/Hyperthermia

Hyperthermia

• Hyperthermia uses higher temperatures than diathermy, which is the deep heating of body tissue for relaxation or physical therapy

• When combined with radiation therapy, it is called thermoradiotherapy

Hyperthermic intraperitoneal chemotherapy

• Intraperitoneal hyperthermic chemoperfusion (HIPEC or IPHC) is a type of hyperthermia therapy used in combination with surgery in the treatment of advanced abdominal cancers

• In this procedure, warmed anti-cancer medications are infused and circulated in the peritoneal cavity (abdomen) for a short period of time

• The chemotherapeutic agents generally infused during IPHC are mitomycin-C and cisplatin

• Santiago González-Moreno, et al. World J Gastrointest Oncol. 2010 Feb 15; 2(2): 68–75. Hyperthermic intraperitoneal chemotherapy: Rationale and technique ( 21160924)

Whole-body hyperthermia
 * H Wehner et al. Whole-body hyperthermia with water-filtered infrared radiation: technical-physical aspects and clinical experiences. Int J Hyperthermia. 2001 Jan-Feb;17(1):19-30. ( 11212877)
 * Eighty treatments of patients in an advanced stage of cancer (40 male, 40 female, April 1998-April 1999) were performed with a WBH device with water-filtered infrared radiation, IRATHERM 2000.
 * The mean duration of the rising phase (37.5-41.5 degrees C) was 87 +/- 19 min.
 * WBH with water-filtered infrared radiation in combination with induced hyperglycaemia and relative hyperoxaemia (procedure of systemic Cancer Multistep Therapy) is a safe and practicable procedure with good tolerability in the case of experienced application.

 * Subpage: Level I evidence
 * Breast cancer
 * Melanoma
 * Cervical cancer

RT +/- Hyperthermia

 * Duke (1994-2001)
 * Randomized. 109 patients with superficial tumors (<= 3cm depth); breast 65%, H&N 13%, melanoma 10%. Tumors had to be "heatable" on initial HT. Arm 1) RT alone vs. Arm 2) RT + HT. Hyperthermia to >43C x1 hour, 2x/week. RT dose 30-60 Gy if previous RT, otherwise 60-70 Gy in 1.8-2.0 Gy/fx
 * 2005 PMID 15860867 -- "Randomized trial of hyperthermia and radiation for superficial tumors." (Jones EL, J Clin Oncol. 2005 May 1;23(13):3079-85.)
 * Outcome: CR RT 42% vs. RT+HT 66% (SS). Patients with prior RT had most benefit (68% vs. 23%, SS)
 * Toxicity: Well tolerated, 1 Grade III thermal burn
 * Conclusion: Significant local control benefit for HT

 * International Collaborative Hyperthermia Group (1988-1991) -- combined results from 5 PIII trials
 * Randomized. 5 trials merged due to slow accrual, closed early after benefit. 306 patients. Advanced primary or recurrent BCA; 50% active disease outside treatment area. Could have had prior RT. Trials: Dutch Hyperthermia Group (DHG), Medical Research Council (MRC BrI and BrR), European Society of Hyperthermic Oncology (ESHO), and Princes Margaret (PMH). Target hyperthermia 43C. RT given in various fractions.
 * 1996 PMID 8690639 -- "Radiotherapy with or without hyperthermia in the treatment of superficial localized breast cancer: results from five randomized controlled trials. International Collaborative Hyperthermia Group." (Vernon CC, Int J Radiat Oncol Biol Phys. 1996 Jul 1;35(4):731-44.)
 * Outcome: CR RT 41% vs. RT+HT 59% (SS); greatest effect in recurrent lesions in previous RT, where re-irradiation dose was low. 2-year OS ~40% (NS), 74% patients progressed outside HT area during follow-up
 * Conclusion: Combined result demonstrated efficacy of hyperthermia as adjunct to RT for treatment of recurrence

 * RTOG, 1991 PMID 1903023 -- "Randomized phase III study comparing irradiation and hyperthermia with irradiation alone in superficial measurable tumors. Final report by the Radiation Therapy Oncology Group." (Perez CA, Am J Clin Oncol 1991 Apr;14(2):133-41.)
 * Randomized. 307 patients with superficial tumors. Treated with RT alone or RT + HT (following, to 42.5C for 45-60 min)
 * Response: CR RT 30% vs. RT+HT 32%; improved LC in tumors <3cm
 * Toxicity: comparable, but 30% thermal blisters in heated tumors

Timing of Hyperthermia

 * Jefferson, 1993 (1983-1990) PMID 8515137 -- "Randomized trial of one versus two adjuvant hyperthermia treatments per week in patients with superficial tumours." (Engin K, Int J Hyperthermia. 1993 May-Jun;9(3):327-40.)
 * Randomized. 41 patients with 44 superficial tumors. Treated with QW vs. BIW HT + RT (mean dose 45 Gy)
 * 2-month CR: 59% QW HT vs. 55% BIW HT
 * Predictive factors for CR: RT dose, tumor volume; Predictive factors for duration of response: minimum tumor temperature, tumor volume
 * Toxicity: No difference.
 * Conclusion: No difference in response or duration between once weekly and twice weekly hyperthermia

Duration of Hyperthermia

 * Temperature should be converted to 43 celsius-equivalent minutes
 * Each 1C above 43C requires 1/2 time for equivalent thermal damage (e.g. 1 min @ 45C = 4 min @ 43C)
 * Each 1C below 43C requires 4x increase in time for equivalent thermal damage (e.g. 1 min @ 42C = 0.25 min @ 43C)
 * t2/t1 = 2^(Temp1-Temp2)

Hyperthermic Intraperitoneal Chemotherapy (HIPEC)

 * S J Valle et al. Hyperthermic Intraperitoneal Chemotherapy (HIPEC) Methodology, Drugs and Bidirectional Chemotherapy. Indian J Surg Oncol. 2016 Jun;7(2):152-9. ( 27065705)
 * Wim Ceelen et al. Hyperthermic Intraperitoneal Chemotherapy: A Critical Review. Cancers (Basel). 2021 Jun 22;13(13):3114. ( 34206563)

title: Radiation Oncology/Hyperthermia


text:
Roulette/Mechanics/Table

Back | Other Statistical Systems

The roulette table has a layout printed onto a felt cover. There are generally only two types of layouts:

American Layout

The American layout reflects the 38 possible numbers on the American wheel. The main difference between this layout and the European layout is that at the top of the numbers there are a 0 and a 00. This layout also has a unique betting option where one can play the 0, 00, 1, 2, and 3 however this only pays out 6:1 making it the only bet on the table where the house edge is greater than 5.263%.

European Layout

The European (aka French) layout reflects the 37 possible numbers on the European wheel. The main difference besides the lack of the 00 is that European tables will generally allow announced bets. In a casino you would tell the croupier the bet you would like and s/he would take your chips and place them on a special marker area to indicate your bet. This marker area can vary widely among table makers.

The house edge is 2.70% on all bets, except in Europe on even money bets (even/odd, red/black, 1-18/19-36). There, if you place an even money bet and the ball lands on zero, you will not forfeit the bet. Instead, in some casinos you will get half your bet back, while in others your bet will be "imprisoned", meaning your bet will be unchanged but it will have to play the next spin. Either way, the house edge for even money bets is 1.35%, the lowest house edge in any kind of roulette.

This method of betting is especially used if you want to bet on segments of the roulette wheel (see Mechanical Systems). These are the Announce Bets and the chip increments needed:

images to come

work in progress

Next | Wheel Layout

title: Roulette/Mechanics/Table


text:
The Linux Kernel/Modules

A kernel module is a code that can be loaded into the kernel image at will, without requiring users to rebuild the kernel or reboot their computer. Modular design ensures that you do not have to make a monolithic kernel that contains all code necessary for hardware and situations.

Common kernel modules are device drivers, which directly access computer and peripheral hardware.

Kernel modules have a .ko extension as of kernel 3.0.

Using modules
To determine loaded modules run the command lsmod, which reads the file /proc/modules. An example output of lsmod is the following:

Module                 Size  Used by ctr                    13049  1 ccm                   17773  1 snd_hda_codec_hdmi    46368  1 snd_hda_codec_cirrus   18855  1 btusb                 32412  0 hid_generic           12548  0 arc4                  12608  2 b43                  387371  0 mac80211             630669  1 b43 hid_apple             13386  0 bcm5974               17589  0 joydev                17381  0 cfg80211             484040  2 b43,mac80211 ssb                   62379  1 b43 hid_holtek_mouse      12625  0

Please note that the output printed by lsmod is usually very long and contains several kernel modules listed. This example has been shortened for obvious reasons.

In order to add a module to the kernel you can use the modprobe command. A kernel module daemon kmod, often executes modprobe in order to load a module. Modprobe is passed a string argument that may be an actual module name or an alias to module. Modprode search /proc/modprobe.conf to relate an module alias to an actual module. In order to determine module dependencies, modprobe searches /lib/module/version/modules.dep and determines if other modules must be loaded before the requested module can be. The modules.dep file is created using the depmod -a command, which determines if a particular module calls functions or variables (symbols) that are defined by other modules.

If there are dependencies, insmod loads these before it loads the requested module. While insmod requires detailed instructions about the path and file names of requested modules, modprobe does not. Here is an example for "spidev.ko":

sudo modprobe spidev

Modprobe, insmod and depmod are found in the module-init-tools or kmod packages.

Loaded modules are listed in /proc/modules. Use modinfo modulename, for file modulename.ko, to determine module information. Here is an example:

modinfo spidev

Note that you cannot use X, such as an xterm, as the kernel only prints directory to consoles or log files.

A simple module
The simplest kernel module:

Kernel modules must always contain these two functions, init_module and cleanup_module. Insmod calls init_module when it loads the modules into the kernel and rmsmod calls cleanup_module when it removes the module. printk is a logging macro used by the kernel and is assigned a priority of <1>. Kernel priorities, of which there are eight, are defined in kernel.h. Each kernel priority has a particular intended meaning and a related definition:

When using printk when syslogd and klogd are running, the output of printk will be appended to /var/log/messages. printk if called with low priority will only be appended to /var/log/messages. To ensure that it prints to the console use KERN_ALERT priority.

Compiling the simple module
Kernel modules are not compiled in the same manner as userspace code. Here is a sample makefile:

obj-m += hello.o

all: modules

.DEFAULT: $(MAKE) -C $(KDIR) M=$$PWD $@

Evoke the command make to compile your simple module.

Using the module_init and module_exit macros you can change the name of the init_module and cleanup_module(void).

The __init macro frees the memory of initialization function in the module after the built-in driver, for example, has been loaded. This does not occur, however, with modprobe.

The __exit macro prevents the cleanup_module from being loaded during initialization, as it will not be needed in the kernel while the kernel is in use. These macros can be found in linux/init.h, and are useful for managing memory. __initdata is analogous to __init but for variables.

The MODULE_LICENSE macro allows you to determine the license of code, and define it for your module.

MODULE_PARM allows you to pass command line parameters when insmod initializes the module.

In order to span modules across several files, the make file must be altered.

obj-m += hello.o obj-m += startstop.o startstop-objs := start.o stop.o

default: modules

.DEFAULT: $(MAKE) -C $(KDIR) M=$$PWD $@

Testing your simple module
% make make[1]: Entering directory `/usr/src/linux-2.6.10' CC [M] /home/ldd3/src/misc-modules/hello.o Building modules, stage 2. MODPOST CC /home/ldd3/src/misc-modules/hello.mod.o LD [M] /home/ldd3/src/misc-modules/hello.ko make[1]: Leaving directory `/usr/src/linux-2.6.10' % su root# insmod ./hello.ko Hello, world root# rmmod hello Goodbye cruel world root#

Writing modules
The symbols that are available to modules can be found in /proc/kallsyms.

Modules can be written to replace the kernel's system calls.

To see what system calls are executed by a userspace process evoke the process using strace.

Kernel space and userspace
A CPU has different modes, 80386 has four. For a description of rings, see the Wikipedia rings entry.

References

 * The Linux Kernel Module Programming Guide
 * http://lwn.net/images/pdf/LDD3/ch02.pdf

title: The Linux Kernel/Modules


text:
General Astronomy/Atomic Emission and Absorption

Most of the information about stars is obtained from studying electromagnetic radiation. Interstellar dust can be observed to get information, as well.

Electromagnetic radiation includes UV (ultraviolet), radio waves, and X-rays. There are two kinds of waves: longitudinal (such as the way sound travels), and transverse (the way light and other electromagnetic radiation travels). Transverse waves are measured by their wavelength and frequency. Wavelength is represented by the Greek letter $$\lambda$$ (lambda). Longer wavelengths make shorter frequency.

The formula for frequency is

$$f=\frac{c}{\lambda}$$

where $$c$$ is the speed of EM radiation, $$3*10^8$$ meters per second.

Astronomers can learn a lot about distant objects by analyzing spectra. The 3 types of spectra are continuous, emission, and absorption. A continuous spectrum comes from a high pressure blackbody, or thermal source (such as a light bulb). An emission spectrum has bright lines. It is caused by cool, low-pressure gas. An absorption spectrum has dark lines it. It appears when a blackbody's light passes through a cool, low-pressure gas. For example, our sun's surface emits a continuous spectrum, but it becomes an absorption spectrum once it reaches us, because it has passed through the sun's atmosphere. Because of this, all normal stars have absorption spectra.

Since each element has a unique spectrum, astronomers can determine the chemical composition of stars by analyzing their spectra. They can also determine the pressure of the object by the type of spectrum: continuous (high pressure), or emission (low pressure).


text:
User talk:MattSanford

Hello and welcome to Wikibooks!

Here are some tips to help you get started:
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You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page. For site news, see the Bulletin board. It might be an idea if you add this page to your "watchlist" so that you can see when any new information is posted there.

Good luck! -- Herby talk thyme 08:17, 24 January 2007 (UTC)

title: User talk:MattSanford


text:
User talk:80.24.190.14

Thanks for experimenting with Wikibooks. Your test worked, and has now been removed. Please use Sandbox for any other tests you want to do, since testing material in modules will normally be removed quickly. Please see the welcome page if you would like to learn more about contributing to our project. Thanks. -- Herby talk thyme 08:21, 24 January 2007 (UTC)

title: User talk:80.24.190.14


text:
Category:Nepali recipes

Recipes from Nepal.

title: Category:Nepali recipes


text:
Haskell/Web programming

An example web application, using the HAppS framework, is hpaste, the Haskell paste bin. Built around the core Haskell web framework, HAppS, with HaXmL for page generation, and binary/zlib for state serialisation.

The HTTP and Browser modules exist, and might be useful.

/Not in book

title: Haskell/Web programming


text:
Template:Miskito/Review head

Review heading template for Miskito language course
 * parameter 1: instructions (or leave a space)

' Review'''

title: Template:Miskito/Review head


text:
Template:Miskito/Review

Review template for Miskito language course
 * parameter 1: input
 * parameter 2: output

title: Template:Miskito/Review


text:
User talk:82.148.97.69

Please do not add inappropriate external links to Wikibooks. Wikibooks is intended to be read without the need for external reference and requires content rather than external links particularly those which are for advertising or promotion. Inappropriate links may include (but are not limited to) links to personal web sites, links to web sites with which you are affiliated, and links that exist to attract visitors to a web site or promote a product. If you feel the link should be added to the article, then please discuss it on the article's talk page rather than re-adding it. See the welcome page to learn more about Wikibooks. Thank you. -- Herby talk thyme 12:38, 24 January 2007 (UTC)

title: User talk:82.148.97.69


text:
Template:AY patch unavailable

/Honors with Unavailable Insignia

/Templates

title: Template:AY patch unavailable


text:
User talk:Cafetrottoir

Please do not add inappropriate external links to Wikibooks. Wikibooks is intended to be read without the need for external reference and requires content rather than external links particularly those which are for advertising or promotion. Inappropriate links may include (but are not limited to) links to personal web sites, links to web sites with which you are affiliated, and links that exist to attract visitors to a web site or promote a product. If you feel the link should be added to the article, then please discuss it on the article's talk page rather than re-adding it. See the welcome page to learn more about Wikibooks. Thank you. -- Herby talk thyme 12:49, 24 January 2007 (UTC)

title: User talk:Cafetrottoir


text:
User talk:Lonjers~enwikibooks

Hello and welcome to Wikibooks!

Here are some tips to help you get started:
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 * Remember to conduct any editing experiments in the sandbox.
 * If you're a Wikipedian, see Wikibooks for Wikipedians for a primer on how things work here (it's a little different).
 * If you want to base your work here on materials from Wikipedia, please use WB:RFI to bring the material over in compliance with the GFDL.
 * If you're an instructor and plan on using wikibooks for a class project, see Guidelines for class projects
 * Please say hello at the Staff Lounge with any questions or ideas.
 * Eventually, you might want to read the Manual of Style and Policies and Guidelines.
 * Please take a look at Naming policy before starting a new book.
 * Remember to maintain a Neutral point of view.
 * Explore, be bold in editing pages, and have fun!

You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page. For site news, see the Bulletin board. It might be an idea if you add this page to your "watchlist" so that you can see when any new information is posted there.

Good luck! -- Herby talk thyme 12:58, 24 January 2007 (UTC)

Talk:Demystifying Depression
Noticing your comments here (& reading them and the previous ones as an "outsider") it looks as though the "book" can be improved and that no one is very active on it at present. I am not knowledgeable in the subject area nor a particularly active editor at present but this is a Wiki - feel free to work on improving it, it is what Wikis are about! It does look as tho the talk page could do with tidying - some of the older comments look redundant now - and I would help with that if you want me too - regards -- Herby talk thyme 12:58, 24 January 2007 (UTC)

Your account will be renamed
Hello,

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

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

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

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:26, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 05:12, 19 April 2015 (UTC)

title: User talk:Lonjers~enwikibooks


text:
User talk:Blackmelas

Hello and welcome to Wikibooks!

Here are some tips to help you get started:
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 * Remember to conduct any editing experiments in the sandbox.
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 * If you're an instructor and plan on using wikibooks for a class project, see Guidelines for class projects
 * Please say hello at the Staff Lounge with any questions or ideas.
 * Eventually, you might want to read the Manual of Style and Policies and Guidelines.
 * Please take a look at Naming policy before starting a new book.
 * Remember to maintain a Neutral point of view.
 * Explore, be bold in editing pages, and have fun!

You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page. For site news, see the Bulletin board. It might be an idea if you add this page to your "watchlist" so that you can see when any new information is posted there.

Good luck! -- Herby talk thyme 13:16, 24 January 2007 (UTC)

Cookbook:Pancake
G'day, thanks for fixing this up, it was bugging me (and others, obviously!) that it was presenting the instructions off a packet mix as a recipe. cheers, Webaware talk 13:37, 24 January 2007 (UTC)
 * The user above may be from foreign parts but they speak for me too! A good fix -- Herby  talk thyme 13:42, 24 January 2007 (UTC)

title: User talk:Blackmelas


text:
Template:Specific book stub

This template adds a page to Category:/Stubs.

 * Note that your stub category should probably link to Help:Stub

title: Template:Specific book stub


text:
Adventist Youth Honors Answer Book/Vocational/Computer

| /Honors with Divisional Variants


text: Template:Haskell/Stub

title: Template:Haskell/Stub


text:
Category:Book:Haskell/Stubs

See also Haskell/To do and Haskell/Notes for contributors.

title: Category:Book:Haskell/Stubs


text:
Adventist Youth Honors Answer Book/Vocational/Computer (North American Division)

1. Describe the function of and point out the following components of a personal computer

 * a. CPU:

 * b. Memory (RAM):

The mother board is located inside the main unit and contains the core circuitry of the computer, including the CPU. It also contains "slots" that other circuit cards such can be plugged into to add extra functions to the computer, such as a modem, networking, video acceleration, or a television tuner. The functions provided by these add-on cards (or daughter cards) are often absorbed into the motherboard as new technology is introduced. Modems, networking, sound, graphics, hard drive controllers, and USB adapters are examples of technology introduced as add-on cards, but now commonly found on the motherboard.
 * c. Mother board:
 * d. Hard Drive:

An external drive is a storage device that is connected to a computer via a USB (or other) port. The drive itself remains outside the computer. The advantage of an external drive is that it is easier to install and it can be moved from one computer to another. This feature can be used for physically transporting large amounts of data between computers, or securing the data in an off-site location. An internal drive is a storage device that is installed inside the computer. The advantage of an internal drive is that it takes up less space and is generally more reliable than an external drive. USB stands for Universal Serial Bus. It is a convenient method of interfacing many different devices to a computer. USB devices include cameras, printers, flash memory drives, external hard drives, music players, and many others. USB ports are located on the back, front, or both sides of a computer, and sometimes even on computer monitors. A USB device is simply plugged into the port, and the computer recognizes the device and prepares it for use. An optical drive is a device which reads and optionally writes data to optical storage media such as CD-ROMS and DVDs. An input device is any device through which data can be provided to a computer. Input devices include, the mouse, keyboard, network card, storage devices, cameras, and microphones.
 * e. External Drive:
 * f. Internal Drive:
 * g. USB:
 * h. Optical Drive:
 * i. Input devices:
 * j. Monitor:

 * k. Keyboard:

A printer is a device for capturing the output of a computer on paper. A mouse is a device that is used by human operators to control a computer. It can be used to move a cursor on a screen, highlight data on the screen, "drag and drop" files from one folder to another, control the action in a computer game, and manipulate data in many other ways. Modem stands for Modulator/Demodulator, and it a hardware device which allows computers to communicate with one another over a telephone line (or some other audio channel). A network card serves the same function except that it connects to a computer network (such as ethernet) and is much, much faster than a modem. Modems are used for "dial-up" internet connections, and network cards are used for "broadband" internet connections. A digital camera is a device which digitally captures and stores an image in the form of a photograph. The photograph can then be transferred to a computer where it can be stored, transmitted to other computers, or digitally manipulated. A scanner is a device which is used to digitize a two-dimensional image, such as a printed photograph, a page from a book, or a store receipt.
 * l. Printer:
 * m. Mouse:
 * n. Modem / Network Card:
 * o. Digital Camera:
 * p. Scanner:

2. Describe the proper handling and storage techniques of disks, CDs, CDR’s, DVDs, Flash/USB drives, and other equivalent optical media devices.
CD's, CDR's, and DVD's are all forms of optical media, and as such, must be protected from scratches. They are fairly impervious to water, and in fact can be cleaned with it (just be sure to completely dry them off before trying to use them). Store them in protective cases, envelopes, or some other container designed to store them.

Flash/USB drives are incredibly durable and can take a lot of punishment. But that doesn't mean they should be abused. Though they are tough, they can still be crushed or snapped. Do not open the casing, as this will expose the internal circuitry to static electricity which can damage them. Unlike the optical media described above, flash drives can be damaged by liquids.

3. Explain the difference between read-only, write once, and write-rewrite media. What are examples of each?

 * Read-only: Read-only media is prepared at the factory and cannot be altered by the consumer. The most prevalent examples of read-only memory today are the CD-ROM and the DVD.
 * Write-once: Write-once media is blank when purchased and is written by the consumer. Once it is written to, the data on it cannot be changed.  An example of write-once media is the CDR (or CD-R).
 * Write-rewrite: Write-rewrite media can be written, erased, and rewritten by the end user. An example is the CD-RW.  This type if media is generally more expensive than the CD, the DVD, or the CDR.

Windows
Click the "Start" button. Then click "My Computer" with the right mouse button, and select "Properties". This will bring up a window. Select the "General" tab, and the computer will display the processor speed, memory capacity, as well as a number of other bits of information.

To find the storage capacity, click the "Start" button and then use the left mouse button to select "My Computer". This will open a new window which will show all the drives on the computer. Generally, the "C:" drive is the hard drive, so right-click on that and select "Properties" - this will show you the drive capacity as well as how much of the drive is in use.

Be sure to pay attention to the use of the left mouse button verses the right mouse button.

Mac
Click on the "Apple" at the top left corner of the screen. Now click on "About This Mac". This will bring up a window that contains the processor type and speed and how much memory the computer has. To find the storage capacity on the hard drive, click the "More Info..." button. Now click on "ATA" under "Hardware". There should be two "ATA Bus" sections. Click on the bottom section or the one that doesn't contain anything about a CD drive. The hard drive capacity is under "Capacity".

Linux
At the command prompt, type the following command:

cat /proc/cpuinfo

This will tell you the processor speed as well as the memory capacity. For the storage capacity, type:

df

This will yield an output similar to this: Filesystem          1K-blocks      Used Available Use% Mounted on /dev/hda2             20315844   7085116  12182088  37% / /dev/hda1              101086     11451     84416  12% /boot /dev/hda5            16674136  12303988   3523124  78% /home

This output above indicates that the computer has a single hard drive with three partitions. The first of these is the root partition which has a total capacity of 20 Gigabytes, the second is 101 Megabytes, and the third is 16.7 Gigabytes.

5. What are the advantages of increasing processor speed, hard drive storage capacity and RAM on a computer?
Increasing the processor speed of a computer will allow it to make more computations per second, thus improving its performance. Increasing the amount of RAM in a computer will also increase its performance, but for a different reason. When a computer program needs RAM, it requests it from the operating system (OS). If free memory is available, the OS assigns it to the program and all is well. If there is not enough memory to satisfy the request, the OS will copy the contents of some other program's RAM to the hard drive, and then give that to the requesting program. When the other program needs that memory again, the OS restores it from the hard drive (after writing the other program's version of the memory to the hard drive). This process is called "swapping," and it can really slow down a computer. However, all the programs continue to work as designed, and the computer makes maximum use of the resources available to it. Increasing RAM capacity will eliminate or reduce the amount of swapping that goes on.

Increasing the storage capacity of a computer will not make it run any faster (as increasing the processor speed and memory capacity will), but it will allow more files and programs to be stored on the computer.

a. Backup of personal files
An important fact about modern computing is that computers eventually fail. If the hard drive on a computer fails, it may be difficult or impossible to recover the data that is stored on it. Forward-thinking computer users keep all of their important personal files backed up on external media so that when the computer fails, they still have their data.

b. Whole system image backup
It is not nearly as important to back up program files, because the program can be re-installed when the computer is repaired or replaced. However, re-installing every program that a computer user needs can be a time-consuming and frustrating process. If the whole system is backed up, it can be restored in short order.

c. Surge protection
The A/C power that feeds most businesses and households is subject to all kinds of disturbances that can disrupt or damage a computer. Lightning strikes can induce large currents in powrlines, and these translate into "surge" voltages. The increased voltage can easily damage a computer. Most often, this affects its power supply module. A surge protector is designed to absorb these voltage surges, preventing damage to the computer.

d. Internet safety hardware/software
An unfortunate fact of modern life is that the world is filled with evil (or immature) people who think it great fun to destroy data on a stranger's computer. Some of these people are capable of "hacking" their way into computer systems and altering files - even taking complete control of the computer. The computer can then be used to launch attacks against other computers. Sometimes this is done in order to install malicious software designed to analyze keystrokes. When the program detects that someone is engaged in online banking activities or ordering merchandise with a credit card, it springs into action, gathering account numbers and passwords which are then passed to the programmer. Once hacking was confined to a small community of immature, thoughtless programmers looking for a thrill, but it is becoming more and more an activity conducted by organized crime.

Firewalls and antivirus software are designed to combat these operations, but in order to be effective, they must be installed and constantly updated.

==8. Write a 250-word essay or give a three-minute oral report about the history of computers. Include prominent events and personalities that are significant to the development of the computer, both hardware and software. This report should focus on the development of the personal computer, not the internet or other accessory functions related to computing.== This research should be performed by the person earning the honor, but we will provide a few names for plugging into a search engine:

Steve Wozniak and Steve Jobs are probably the most important people in the early history of personal computer hardware, as together, they formed Apple Computer which produced the first commercially viable home computer. Bill Gates and the company he founded (Microsoft) have had more influence on computer software than anyone else. Other people to know about include Brian Kernighan, Dennis Ritchie, Ken Thompson, Richard Stallman, Vint Cerf, and Linus Torvalds.

==9. Spend one week charting the time you spend on a computer. List what time was spent on schoolwork, gaming, online, etc. At the end of the week, evaluate with your counselor, family, or group leader how your management of computer time relates to the Bible’s instructions on stewardship of our time and resources (Romans 14:12; Psalms 31:15; Ecclesiastes 3:1-8; Ephesians 5:15-16).== So then, each of us will give an account of himself to God. - Romans 14:12, NIV

My times are in your hands; deliver me from my enemies and from those who pursue me. - Psalm 14:12, NIV

Be very careful, then, how you live—not as unwise but as wise, making the most of every opportunity, because the days are evil. - Ephesians 5:15-16, NIV.

We are responsible for how we spend our time, and God expects us to use it in His service. Ecclesiastes 3:1-8 starts out

There is a time for everything, and a season for every activity under heaven: a time to be born and a time to die, a time to plant and a time to uproot - Ecclesiastes 3:1,2, NIV.

Likewise, there is a time to compute and a time to not compute. It is very easy to become addicted to computing, and as with all things, moderation is the key to temperance.

==10. Dialogue with a long-term computer user about the advantages/disadvantages of Macs and PCs. Some questions you should ask include:a. What operating system does each use? What are some advantages of that OS?b. What compatibility issues do these two types of computers have in relation to data sharing and program installation?c. What type of industries / careers tend to use each type of computer system?== With the importance of computing in today's world, it is highly likely that you know someone who could be described as "a long-term computer user." Perhaps a staff member in your Pathfinder club fits this description. Invite this person to attend a meeting to have this discussion. It would be a good idea to provide this person with the questions ahead of time to provide ample opportunity for preparations.

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Adventist Youth Honors Answer Book/Vocational/Computer - Advanced

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Category:Lithuanian recipes

Recipes from Lithuania.

title: Category:Lithuanian recipes