text:
Ordinary Differential Equations/Separable 1

First Order Differential Equations

This section deals with a technique of solving differential equation known as Separation of Variables. Before we begin discussing separation of variables it is very helpful to recall the theorem behind integration by substitution from calculus.

Which states that if $$f(x)$$ is a continuous function and $$y$$ has a derivative that is a continuous function - hence $$y$$ being a function itself, then integrating such chained functions yields
 * $$\int f\big(y(x)\big) y'\,dx = \int f(y)\,dy$$

In other words, our integrand $$f(y(x))y'(x)$$ and our antiderivate will be the same if we took the antiderivative of $$f$$ and plugged it into $$y(x)$$. This provides a very useful tool to solve many ODE's.

To take an example let's consider the differential equation:
 * $$y'=ay$$

We comment the constant function $$y(x)=0$$ is a solution. We will focus on finding solutions that are not zero. As commented before it is a consequence of the uniqueness of solutions that two solutions can never intersect. For this problem it means that solutions are always positive or always negative because they cannot cross $$y(x)=0$$. For easier understanding, picture a normal Cartesian coordinate system. The solutions can only be above or below the x-axis.

For this reason we are justified for the remainder in assuming that $$y(x)\neq 0$$. Thus we can divide by $$y(x)$$ as division by zero is undefined, to find that:

 * $$\frac{1}{y}y'=a$$

Now integrating both sides with respect to $$x$$ we have that

 * $$\int \frac{1}{y}y'\,dx=\int a\,dx$$

Using the theorem about integration by substitution theorem:

 * $$\int \frac{1}{y}\,dy=\int a\,dx$$

Or

 * $$\ln |y|=ax+C$$.
 * $$y=\pm e^C e^{ax}=Ce^{ax}$$.

Finally we comment that, $$e^c>0$$ and $$e^c$$ collapsing to $$C$$, so technically speaking the above derivation only shows that we get a solution if $$C>0$$ or if $$C<0$$, taking into account $$\pm$$. Notice on the other hand that if we allow $$C=0$$ then we can recover the solution $$y(x)=0$$. Thus we can represent all solutions as
 * $$y(x)=Ce^{ax}$$

where $$C$$ is a real number.

This method is known as separation of variables because after we have divided we have one side that is entirely in terms of $$y$$ and the other side entirely in terms of $$x$$.

Comments about notation and heuristics
While the change of variables theorem is the mathematical justification, using Leibniz' notation for derivatives allows for excellent mnemonics for remembering how separation of variables works.

We present two notationally different but equivalent derivations here:

Beginning from the point where we have divided by $$y(x)$$ and using Leibniz notation we have that:
 * $$\frac{1}{y}\frac{dy}{dx}=a.$$

One might not view this as fully separated because there is still a "dx" on the left hand side. But one can fully separate the equation by multiplying by "dx", to get:
 * $$\frac{1}{y}\,dy=a\,dx$$

Then placing an integral in front of each term one gets:
 * $$\int\frac{1}{y}\,dy=\int a\,dx$$.

This arrives at the correct place, but treating "dx" as a number is purely heuristic.

A second mnemonic which is fairly similar uses Leibniz notation to remind us how the change of variables theorem should work. Beginning from the step in the original example where we integrated both sides with respect to x, and using Leibniz notation we have:
 * $$\int\frac{1}{y}\frac{dy}{dx}\,dx=\int a\,dx$$.

Now we can "cancel the dx" to get that $$\int\frac{1}{y}\frac{dy}{\cancel{dx}}\,\cancel{dx}=\int\frac{1}{y}\,dy$$. Thus
 * $$\int\frac{1}{y}\,dy=\int a\,dx$$.

Again, "dx" is not a number, so it cannot really be canceled. But the heuristic does correctly give the change of variables formula.

The reader should use which ever of the above methods seems most clear to them individually. The book will undoubtedly use anyone of the three at different points in its exposition.

What is a Separable Equation?
An ordinary differential equation is called separable if it can be rewritten in the form:

 * $$M(x)+N(y)y'=0$$

Integrating both sides with respect to $$x$$

 * $$\int M(x)\,dx + \int N(y)y'\,dx = C$$

Now by one of the methods we discussed above we can change and simplfy the second integral to

 * $$\int M(x)\,dx + \int N(y)\,dy = C$$

An equation where you can factor out both P and Q into separate functions of x and y

$$P_1(x)P_2(y)dx+Q_1(x)Q_2(y)dy=0$$

is called separable because the equation can become an equation with separated variables.

 * $$\frac {P_1(x)}{Q_1(x)}dx + \frac {Q_2(y)}{P_2(y)}dy = 0$$

 * $$\frac {P_1(x)}{Q_1(x)}dx = - \frac {Q_2(y)}{P_2(y)}dy$$

However, attention must be paid to this process of division. The cases when $$Q_1(x)=0$$ and $$P_2(y)=0$$ will be additional possible integral solutions that will not be found through the integral because of this division.

Trivial cases
There are two special cases of a separable equation that make the solution almost trivial. These are the cases where either

 * $$N(y)=k \,$$

or

 * $$M(x)=k, \,$$

where k is a constant. In either case, the equation changes from being in terms of 3 variables and reduces down to 2. When we can throw out one of the two variables, the solution can be achieved by a simple integral.

No y term: N(y) = k
If N(y)=k, we can treat k as part of M(x), and turn N(y) into 1. That is, if we divide both sides by k and let f(x) = M(x)/k. Then we're left with

 * $$\frac{dy}{dx}=M(x)/k=f(x)$$.

This is now basic calculus - take the integral of both sides. The general solution is

 * $$y=\int f(x)dx+C,$$

where C is a constant.

Let's take a look at a few examples.

Example 1

 * $$\frac{dy}{dx}=x^2+7$$

Here, we have a derivative equaling a function of x. The original function is the antiderivative of this, that is, we have to integrate to find it. Now we integrate both sides of the equation, w.r.t.x (with regard to 'x'):

 * $$\int \frac{dy}{dx}\,dx=\int(x^2+7)\,dx$$

 * $$y=\frac{1}{3}x^3+7x+C$$

This is the general solution. A function that solves a differential equation is said to satisfy it, so

 * $$y=\frac{1}{3}x^3+7x+C$$

satisfies the differential equation

 * $$\frac{dy}{dx}=x^2+7$$.

Example 2

 * $$\frac{dy}{dx}= \sin x+e^x$$

This works the same as before - integrate both sides with respect to x.

 * $$\int \frac{dy}{dx}\, dx= \int\sin x+e^x\, dx$$

 * $$y=- \cos x+e^x+C \,$$

Note that even in this simple case, it may be impossible to write down a formula for the solution because f(x) cannot be integrated in elementary terms. In this case, writing the integral of y, which is the function we are looking for, symbolically with an integral symbol is deemed to be a solution.

No x term: M(x)=k
Another special case is when

 * $$M(x)=k.$$

The only difference is how we make it into an integrable form. Our equation is

 * $$N(y)\frac{dy}{dx}= -k.$$

This type of equation is called an autonomous differential equation. The differential equation has no explicit dependence on the independent variable x except through the function y. We will have more to say about this type of equation later, but for the moment we note that this type of equation is always separable.

Integrating both sides with respect to x we get

 * $$\int N(y)\frac{dy}{dx}\,dx= \int k\,dx.$$

We now have two integrals, each with respect to the functions contained in them. The solution is of the form

 * $$\int N(y)\,dy=kx+C.$$

You can now try to solve for y. It is possible that the left hand side doesn't have an antiderivative that can be expressed as a nice formula, and in that case, we leave it in a as clean a form as possible, that is with the integral symbol unresolved. Another important concept emerges here.

Example 4

 * $$\frac{dy}{dx}=\frac{k}{y}$$

Here we multiply through by y, and multiply through by dx. The rest of the math works as before.

 * $$ydy=kdx \,$$

 * $$\int ydy=\int kdx \,$$

 * $$\frac{1}{2}y^2=kx+C$$

 * $$y^2=2kx+C \,$$

In this case, its more convenient to solve for y2 than y. If we take the square root to solve for y, we change the equation (since the square root of a number is always positive, we would lose half of each solution curve). In these cases its just fine to leave it in its current form.

You may have noticed that the right hand side of the equation, before solving for y, is the same for both examples. When f(x)=k, the right hand side will always be kx+C where k is whatever constant is in the problem. It only changes when f(x) is not constant.

As before, it may be impossible to do the integral. Even if it is possible, it may then not be possible to invert the equation to give y explicitly as a function of x, for example if we get tan(y) + 1/y = x. Again, this does not matter; the result is regarded as a valid solution.

General Separable Variable Equations
In the most general case, neither M(x) or N(y) is a constant. In this case, we use the same method as before (although the variables might have to be separated first). The only difference is that both sides result in non-trivial integrals, so we have a little more work to do. This can produce some untidy solutions, and ODEs are known for this. This untidiness often means that we might want to check our result at the end.

Example 5

 * $$\frac{dy}{dx}=y^3x$$

Dividing by y3 and multiplying by dx gives

 * $$\frac{dy}{y^3}=xdx$$

 * $$\int\frac{dy}{y^3}=\int xdx$$

 * $$-\frac{1}{2y^2}=\frac{1}{2}x^2+C$$

 * $$y^2=-\frac{1}{x^2+C}$$

Example 6

 * $$\frac{dy}{dx}=\frac{y}{x}$$

We multiply by $$\frac{dx}{y}$$:

 * $$\frac{dy}{y}=\frac{dx}{x}$$

 * $$\int\frac{dy}{y}=\int\frac{dx}{x} \,$$

 * $$\ln(y)=\ln(x)+C \,$$

 * $$y=e^{\ln(x)+C}=e^Ce^{\ln(x)}=Cx \,$$

Again $$e^c$$ collapsing into $$C$$.

Example 7

 * $$\frac{dy}{dx}=xy+y$$

This doesn't look separable at first glance. A little factoring can be applied though:

 * $$\frac{dy}{dx}=(x+1)y$$

This is separable.

 * $$\frac{dy}{y}=(x+1)dx$$

 * $$\int\frac{dy}{y}=\int(x+1)dx$$

 * $$\ln(y)=\frac{1}{2}x^2+x+C$$

 * $$y=Ce^{\frac{1}{2}x^2+x}$$

Problems with Initial Conditions
Solving an initial value problem is fairly easy. Solve for y as we did above. Then once you have the equation, plug in b for y and a for x. Then, solve for the constant.

Example 8

 * $$\frac{dy}{dx}=ky, \quad y(0)=10$$

As we saw in example 3, the general solution to this is y=Cekx. Let's plug in our boundary conditions:

 * $$10=Ce^{0k} \,$$
 * $$10=Ce^{0} \,$$
 * $$C=10 \,$$

 * $$y=10e^{kx} \,$$

This is the particular solution.

Unfortunately in some cases care must be used when solving for the constant.

Example 9

 * $$2yy'=2x+2\qquad y(0)=-2$$

This equation is already separated, so integrating both sides
 * $$\int 2yy'\,dx=\int 2x+2\,dx$$
 * $$y^2=x^2+2x+C$$

By plugging in we find that:
 * $$C=(-2)^2=4$$

So now we have that:
 * $$y^2=x^2+2x+4.$$

But if we continue to solve for y we must be careful. One may naively write: $$y=\pm\sqrt{x^2+2x+4}.$$

But there is a problem, we are looking for a function y(x), which means the function must be single valued (see Single-valued function). And we should be careful what we mean by $$\pm$$. Do we mean that for each different x there is a different possibility of a sign?

Fortunately since the function has to be differentiable (and hence continuous), the function y(x) cannot jump back and forth between positive and negative values. So we must mean there is one choice of sign:
 * $$y=\sqrt{x^2+2x+4};$$     Or
 * $$y=-\sqrt{x^2+2x+4}.$$

But which one? It is a nice consequence of uniqueness, that only one of the two functions could solve the problem. Both solve the differential equation, so we can double check the initial condition. In the first instance y(0)=2, so that is not our solution, but for the second function y(0)=-2 as desired. Thus the solution is given by:
 * $$y=-\sqrt{x^2+2x+4}.$$

Example 10
Another interesting example that illustrates problems that may arise is:
 * $$y'=6xy^{2/3}\qquad y(0)=0.$$

In order to solve this initial value problem we begin by separating variables.

 * $$\int y^{-2/3}y'\,dx=\int 6x\,dx$$
 * $$3y^{1/3}=3x^2+C$$
 * $$y=(x^2+C)^3$$

Now to plug in the initial conditions we have that:
 * $$0=(0^2+C)^3$$, Thus
 * $$C=0$$

So we have a solution $$y(x)=x^6$$. But by inspection we also have the solution $$y(x)=0$$. So there are at least two solutions to this problem. Perhaps there are more, at the moment we have no way to be sure. Worse, there is nothing that comes up while separating variables to indicate we are in a situation where the situations are not unique. For this reason we will return to issue and discuss the general mathematical theorems that guarantee the existence of unique solutions.

This may be important for many reasons. For example, often one believes a real life process is described by a function which satisfies an ODE. Usually using the ODE, and actual measurements of the process you'd like to deduce what function is governing the process so you can make predictions about future behaviour. But if the solutions to the ODE are not unique, it is difficult to tell when you have found the one solution which is really giving the behaviour of the real world process.

Example 11
In the example above we saw that sometimes a differential equation is not well behaved and admits multiple solutions. In this example we see that sometimes the problem is perfectly well behaved but if we are not careful we may think we find multiple solutions.

 * $$y'=4y^{1/2}x\qquad y(0)=4.$$

 * $$\int y^{-1/2}y'\,dx=\int 4x\,dx$$
 * $$2y^{1/2}=2x^2+C$$
 * $$y=(x^2+C)^2$$

Now by plugging in the initial condition we find that
 * $$4=(0^2+C)^2=C^2$$

At this point it is very tempting to claim that:
 * $$C=\pm 2$$

And we appear get two solutions
 * $$y=(x^2+2)^2$$
 * $$y=(x^2-2)^2$$

But if we are careful we can see that the second equation is not really a solution. To see this let's plug it into the equation.

If
 * $$y=(x^2-2)^2$$, then
 * $$y'=4x(x^2-2)$$, and
 * $$y^{1/2}=|x^2-2|$$.

Thus when we plug into $$y'=4y^{1/2}x$$ we are trying to equate
 * $$4x(x^2-2)$$ and $$4x|x^2-2|$$,

but notice the first equation is negative for $$00$$ we can drop the absolute values in the second expression.

So what went wrong? We introduced the ambiguity when we were solving for $$y$$ by squaring both sides of the equation. We could have avoided this issue if we solved for $$C$$ immediately after integrating, in which case we would have been working with the equation $$2y^{1/2}=2x^2+C$$ and:
 * $$2(4)^{1/2}=2(0)^2+C$$, so
 * $$C=4$$

Plugging this back in and solving for y we get:
 * $$2y^{1/2}=2x^2+4$$
 * $$y^{1/2}=x^2+2$$
 * $$y=(x^2+2)^2$$

Proceed to the next part of the lesson: Examples


text:
Talk:French/Lessons/Transportation

Untitled
I came across this when recording audio for this lesson; would "quel jour est-il" be more common than "quel jour sommes-nous aujourd'hui"? Please comment. I will record the audio file later when this has been cleared up. Jonojet
 * You could be bold and add a second section oof how to say it the normal way, and leave the way we have it here as well. And hey, could we sacrifice alphabetical order and have the things on audio file first and the things that aren't can be after that?Lotusduck 17:37, 26 December 2005 (UTC)

Subject-past participle agreement
This section says: ''When conjugating with être, the past participles of the above verbs must agree with the the subject of a sentence in gender and plurality. Note that there is no agreement if these verbs are conjugated with avoir.''

While this explanation is correct for the type of sentences shown, there is an exception for verbs conjugated with avoir if the participle is preceded by a direct object. In that case, the participle agrees with the gender of the preceding direct object (according to Glanville Price "A Comprehensive French Grammar" p354). Perhaps this is more complication than we need to cover in this lesson, though. Recent Runes (talk) 19:03, 29 October 2010 (UTC)

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

title: Talk:French/Lessons/Transportation


text:
User talk:Jonojet

French Textbook - adding audio, perhaps some solutions to exercises, vocab lists?

Hey Jonojet, I'd be really greatful if you'd consider commenting on my proposed restructure for the French Wikibook at User:Serge/French. As one of the main contributors to this book I value your opinion. Thanks, Serge 06:15, September 8, 2005 (UTC)

title: User talk:Jonojet


text:
User:Crosstimer

 * See my main profile in Wikipedia: Crosstimer's Profile

I am a native Hong Kong student living in Tseung Kwan O.

Works

 * Written Chinese

title: User:Crosstimer


text:
Talk:Editing Wikitext

Double
Can I ask -- in what way is this book intended to be different from the Editing Help page? - Lynx7725 6 July 2005 11:15 (UTC)
 * This is supposed to be more of a textbook than a manual. Messedrocker 01:56, 27 November 2005 (UTC)

Merge
I suggest merging the entire MediaWiki User Guide into this Editing Wikitext book. Both books seem to have exactly the same topic and audience -- people who edit the content at some MediaWiki instance. --DavidCary (talk) 04:08, 31 May 2009 (UTC)
 * Please do not do it.  The result is most likely the demise of both.   By all means improve this book by adding sections that have been forgotten;   subscripts and superscripts e.c.t., with a view to improving it further.   Mathematical notation needs a step by step instruction within this set in my opinion.   Perhaps somebody would care to have a whack at it?  Anonymouse. (0800 GMT, 7-Aug-2009)
 * I'd propose not to merge "Editing Wikitext" with "Mediawiki User Guide". While the actual subject matter seems to be the same, the books have a very different style. The wikibook "Editing Wikitext" has a misleading title, as its subject matter is "Editing Wikitext with Mediawiki". "Editing Wikitext" is chatty, and has colorful backgrounds and typographic niceties, while "Mediawiki User Guide" is terse in words, and plain in typography. --Dan Polansky (talk) 09:45, 20 August 2009 (UTC)
 * I'd propose not to merge "Editing Wikitext" with "Mediawiki User Guide". While the actual subject matter seems to be the same, the books have a very different style. The wikibook "Editing Wikitext" has a misleading title, as its subject matter is "Editing Wikitext with Mediawiki". "Editing Wikitext" is chatty, and has colorful backgrounds and typographic niceties, while "Mediawiki User Guide" is terse in words, and plain in typography. --Dan Polansky (talk) 09:45, 20 August 2009 (UTC)

 * I find the book in it's current form very usefull :) I'm very happy i found it! The user guide should probley be the "reference guide" and this the "user guide" Pearts (talk) 23:00, 31 July 2010 (UTC)

Dead link
On the Further Reading section, the link to " Mayflower Image Search Engine" lands on a 404 error.
 * ✅ This tool hasn't been replaced on the new server. JackPotte (discuss • contribs) 14:55, 13 April 2018 (UTC)

title: Talk:Editing Wikitext


text:
Portuguese/Present tense irregular verbs

O verbo reflexivo
In Brazil

In Portugal


text:
Talk:Portuguese/Present tense irregular verbs

Untitled
this was taken from the German Portuguese wikibook


text:
User:Quadell

I mainly hang out on Wikipedia. This link goes to my user page there.

So far I've only written Cookbook:Pot Stickers.

title: User:Quadell


text:
User:Sparkit~enwikibooks

Sparkit's wiki home-base on wikipedia.

title: User:Sparkit~enwikibooks


text:
User:Jessemerriman~enwikibooks

Ello ello, I'm Jesse Merriman. Until I feel like putting any more info up here, just see my website.

title: User:Jessemerriman~enwikibooks


text:
MATLAB Programming/Psychtoolbox/ptb ind general setup.m

%ptb_ind_general_setup.m %use to set up some values to make code portable between different %versions of psychtoolbox %This does not refer to any screen functions. % % Robert Kohr - July 6, 2005 - created file

v = version; v = str2num(v(1)); osx = 0; oldmac = 0; win = 0;

if(strcmp(computer, 'MAC')|strcmp(computer, 'MAC2')) if(v<6) %classic mode comp_type = 'oldmac'; oldmac=1; else comp_type = 'osx'; osx =1; end else comp_type = 'win'; win=1; end left_arrow_name = 'left'; right_arrow_name = 'right'; up_arrow_name = 'up'; down_arrow_name = 'down';

if(oldmac) dir_sep = ':'; left_arrow_name = 'left'; right_arrow_name = 'right'; up_arrow_name = 'up'; down_arrow_name = 'down'; elseif(win) dir_sep = '/'; else %os x   left_arrow_name = 'LeftArrow'; right_arrow_name = 'RightArrow'; up_arrow_name = 'RightArrow'; down_arrow_name = 'RightArrow'; dir_sep = '/'; end


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Talk:World History/Ancient Civilizations

"People began to do things in similar ways (it is a property of human nature to want to belong)."

I'm not sure about this - aren't people more likely to do things the way other people in their civilization do because they've grown up used to that idea and they don't know any other way? For example, people here usually build rectangular doors not because they want to have the same door as everyone else, but because they don't think about any other kind of door. Humans may want to belong, but I don't think that's a main reason behind homogenized cultures. But I didn't edit that because I have no background in psychology or anthropology -- can anyone explain or comment?--Lady~Macbeth 6 July 2005 20:26 (UTC)

Persians
I found that the section on 'Persians' is under development. I wish to remark here that I'm currently wiriting a book on Iranian History. You are free to borrow content from my book. Thanks :-) -Ravichandar84 (talk) 03:28, 8 February 2008 (UTC)

Source for dates?
According to "Western Civilisations" Lerner,Meacham,Burns 12th ed. [1993] vol. 1 p29 Sargon the Great conquered all of Sumer in 2320BCE and that Sumer gained it's independence in 2130BCE. The date for the conquering in the wikibook is 2125BCE. There seems to be a discrepency here! does anyone know the source for the date and others mentioned in the wikibook?

Zfishwiki (talk) 18:28, 28 April 2010 (UTC)

Criticism of content priorities
In my belief too much information is given regarding Sargon the Great and his descendents and their respective campaigns. This is supposed to be a world history, not a political history of Sumer. The time would be better spent on describing what life was like for the inhabitants of Sumer and on important technical innnovations such as the wheel, the lunar calendar and writing. And whilst we are on the subject of writing: what is all that guff about the Georgian Asomtavruli alphabet. It surely does not deserve a place in a brief History of Mesopotamia

also see my critique of the structure in the talk page of World History

Zfishwiki (talk) 18:38, 28 April 2010 (UTC)

Sources?
Could someone provide sources for the following?

From an anthropological viewpoint Sumerians belonged to the Caucasian, Mediterranean, Balkan European race.

For instance, the ancient Asomtavruli alphabet of the modern Georgian language has ethno-cultural contacts with the Sumerian world.

But Sumerian did not confide the Semites with the Majuscule alphabet, the secret spiritual alphabet that has a lot of similarity to ancient Georgian Asomtavruli alphabet

the Semites inhabiting the town of Akkad conquered Sumer and made the Sumerian scientists create an alphabet for them

Zfishwiki (talk) 12:26, 29 April 2010 (UTC)

my big question is isn't the Abyssinian kingdom an ancient civilization i have read a lot about abyssinian(ethiopian) kingdom when i wrote my paper & i have read that ethiopian civilization dates many years back before christ birth. evidences for this are;1, Diodorus Siculus bkii, says that egypt was a colony of ethiopia & the lows & script of both lands were the same but the hieroglyphic script was universal in ethiopia but confined in priestly clasess alone in egypt,2; Herodotuss called ethiopians "wise men occuping upper nile, men of long like, whose manners & customs pertain to the golden age those virtous mortals whose feasts & banquets ownerd by jupiter him self" in greek times egyptians depicted ethiopia as an ideal state 3; the puranas,the ancient historical books of india speaks of the civilization of ethiopia as being older than that of egypt. This Sanskirt books mention the names of old kushite kings that were worshiped in india & who were adopted & changed to suit the fancy of the later people of greece & rome.

Other Civ texts

 * Ancient River Valley
 * Saylor.org's_Ancient_Civilizations_of_the_World

Some of the sentences read confusingly to me
I feel I am of least average intelligence with decent comprehension. Yet I feel some of the sentences are incomprehensible. Too many facts crammed into one sentence perhaps. For example - " The first known mention of Akkad is in an inscription of Enshakushanna of Uruk, where he claims to have defeated Agade—indicating that it was in existence well before the days of Sargon of Akkad, who the Sumerian kinglist claims to have built it."


text:
Biblical Studies/Christianity/Christian Theology/Transcendence

Transcendence addresses the nature of God to act within history, while being above human history. The Christian faith in Jesus illuminates this idea. On the one hand, the presence of Jesus (for the Christian) reflects God's intervention in the world to provide hope and salvation. On the other hand, Christians also believe that Jesus arose from the dead, and as such is observing human history without directly manipulating it. So for the Christian, Jesus is considered transcendental. Prayer reflects the hope that God will once again intervene in the course of human affairs, and influence a favorable outcome.

In developing a deeper understanding of transcendence, it is important for the theist to reconcile additional views of God's intervention in human history with the established idea of humanity's "free will."


text:
User:Aya/Wikibooks

This page was originially created in an effort to change this page into a bona fide Wikibook about Wikibooks itself, including how to create them, and how to work with them (linking to other pages in the Wikibooks namespace as appropriate). The idea is to try to create a normative index to all these pages, while merging any duplicate content. It should include information on common conventions and practises. This may already exist somewhere else. Add it to the lists and categories of Wikibooks, and try to get it linked from the main page, since the main page doesn't really tell you what the site is all about, and the other pages are somewhat disorganised.

Subpages

 * An analysis of wiki-based communication
 * A critique of Wikibooks
 * Lists of pages
 * List of books

Misc info

 * Seems to have come from http://textbooks.wikipedia.org/
 * Up-to-date DB dumps available from http://download.wikimedia.org/wikibooks/en/
 * Sidebar at MediaWiki:Sidebar
 * IRC-relay at http://www.tiredbrain.com/wikinews/irc/irc.cgi
 * Mailing lists at http://mail.wikipedia.org/mailman/listinfo
 * MediaWiki CVS tree at http://cvs.defau.lt/
 * Wikimania vids at http://www.knams.wikimedia.org/wikimania/
 * Site notice at MediaWiki:Sitenotice

 * Namespaces

 * -2 - Media
 * -1 - Special
 * 0 - (Main)
 * 1 - Talk
 * 2 - User
 * 3 - User talk
 * 4 - Wikibooks
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 * 12 - Help
 * 13 - Help talk
 * 14 - Category
 * 15 - Category talk

title: User:Aya/Wikibooks


text:
User:Lady~Macbeth~enwikibooks

Lady~Macbeth is a student in New Jersey and is too boring to be able to write a good page about herself.

title: User:Lady~Macbeth~enwikibooks


text:
Software Engineers Handbook/Supporting Processes/Configuration Management

Here is the Wikipedia description of configuration management.

Assessing the size of the development team and the scope of the development are two things that need to be identified very early in the project. The larger the team, the more important it is to develop a configuration management strategy that keeps the developers moving without stepping on each others' toes (having a tool that supports a local developer sandbox, integration, and release areas automatically as opposed to "making them by hand", like we've always done). The more complicated the development, the more complicated the configuration management strategy. Specifically, is support of multiple releases of the same product necessary? If so, a heavier weight configuration management strategy (one that supports branching) is more desirable.

Being able to automatically re-create software from the baseline repository is a must, independent of size of project or team. It provides product predictability expected of manufacturing processes.

Some popular tools to support configuration management include:
 * Makefile and make: Originally these were designed to allow a programmer to tweak one source file and then type "make" to regenerate all the executables, without having to remember the exact name of each and every source file or the exact compile-time options for each one. Today, many programmers also include regression tests in the Makefile.
 * distributed version control systems. A programmer uses a VCS to pull the latest version (tip) from the repository into a sandbox (also called a working directory), test it out, make some changes, and then commit any improvements back into the repository.


text:
Talk:Software Engineers Handbook/Supporting Processes/Configuration Management

Thanks to Bill Clare for the original info!


text:
Talk:Introduction to Sociology/Health and Medicine

Foccault
I hate postmodernism, I loathe every sentence of it. Nevertheless since it's so prevelant amongst sociologists a nod in the direction of Foccaults views of medicine might be neccessary.

Additional Topics Might Include

 * STDs
 * Euthanasia
 * theory:
 * Structural-Functional – defines healthy
 * Sick people are exempt from normal responsibilities
 * Sick people need to want to get well
 * Sick people need to seek help
 * Symbolic Interaction
 * Illness is socially constructed
 * Social-Conflict
 * Health care is a commodity under capitalism
 * Profit Motive
 * Politics of medicine

Additional Suggested Reading

 * Barnes P, Powell-Griner E, McFann K, Nahin R. CDC Advance Data Report #343. Complementary and Alternative Medicine Use Among Adults: United States, 2002. May 27, 2004. Online
 * Benedetti,F., Maggi,G., Lopiano, L. Open Versus Hidden Medical Treatments: The Patient's Knowledge About a Therapy Affects the Therapy Outcome. Prevention & Treatment, Volume 6, Article 1, posted June 23, 2003. APA online
 * Berga, S.L., Marcus, M.D., Loucks, T.L. 2003. Recovery of ovarian activity in women with functional hypothalamic amenorrhea who were treated with cognitive behavior therapy. Fertility and Sterility, Volume 80, Issue 4, Pages 976-981. Abstract
 * Bond J. & Bond S. (1994) Sociology and Healthcare Churchill Livingstone
 * Campos, Paul. The Obesity Myth. Gotham Books, May 2004. ISBN 1592400663.
 * Critser, Greg. Fat Land: how Americans became the fattest people in the World. Houghton Miffin Jan 2003. ISBN 0618164723.
 * Devereaux PJ, Heels-Ansdell D, Lacchetti C, Haines T, Burns KEA, Cook DJ, et al. Payments for care at private for-profit and private not-for-profit hospitals: a systematic review and meta-analysis. CMAJ 2004;170(12):1817-24.
 * Devereaux PJ, Choi PT, Lacchetti C, Weaver B, Schunemann HJ, Haines T, et al. A systematic review and meta-analysis of studies comparing mortality rates of private for-profit and private not-for-profit hospitals. CMAJ 2002;166(11):1399-406.
 * Downing, A.M., Hunter, D.G. Validating clinical reasoning: a question of perspective, but whose perspective? Man Ther. 2003 May;8(2):117-9. Review. PMID 12890440 Manual Therapy Online
 * Eisenberg, D.M. 1997. Advising patients who seek alternative medical therapies. Ann Intern Med. 127:61-69. PMID 9214254
 * Eisenberg, D.M., Davis, R.B., Ettner, S.L. 1998. Trends in alternative medicine use in the United States, 1990-1997. JAMA. 280:1569-1575. PMID 9820257
 * Ernst, E. 2003. Obstacles to research in complementary and alternative medicine. Medical Journal of Australia. Sep 15;179(6):279-80. PMID 12964907 MJA online
 * Ewing, John A. Detecting Alcoholism: The CAGE Questionaire. JAMA. 252: 1905-1907, 1984
 * Fenton MV, Morris DL. The integration of holistic nursing practices and complementary and alternative modalities into curricula of schools of nursing. Altern Ther Health Med. 2003 Jul-Aug;9(4):62-7. PMID 12868254
 * Flanagin, A., Carey, L.A., Fontanarosa, P.B. Prevalence of articles with honorary authors and ghost authors in peer-reviewed medical journals. JAMA. 1998 Jul 15;280(3):222-4. Full text
 * Flier JS. Obesity wars: molecular progress confronts an expanding epidemic. Cell 2004;116:337-50. PMID 14744442.
 * Goldberg, J., Hayes, W., and Huntley, J. "Understanding Health Disparities." Health Policy Institute of Ohio (November 2004).
 * Gonsalkorale, W.M., Miller, V., Afzal, A., Whorwell, P.J. 2003. Long term benefits of hypnotherapy for irritable bowel syndrome. Gut. Nov;52(11):1623-9. PMID 14570733
 * Gunn, I.P. 1998. A critique of Michael L. Millenson's book, Demanding medical excellence: doctors and accountability in the information age, and its relevance to CRNAs and nursing. AANA J. Dec;66(6):575-82. Review. PMID 10488264
 * Janssen I, Katzmarzyk PT, Ross R. Waist circumference and not body mass index explains obesity-related health risk. Am J Clin Nutr 2004;79:379-84 PMID 14985210
 * Kleijnen, J.; Knipschild, P.; ter Riet, G. 1991. Clinical trials of homoeopathy. BMJ. Feb 9;302(6772):316-23.
 * Larkin, M. Whose article is it anyway? Lancet. 1999 Jul 10;354(9173):136. Editorial
 * Linde, K., Clausius, N., Ramirez, G. 1998. Are the clinical effects of homeopathy placebo effects? A meta-analysis of placebo-controlled trials. Lancet. 1997 Sep 20;350(9081):834-43. Erratum in: Lancet 1998 Jan 17;351(9097):220. PMID 9310601
 * McDonough, J., Gibbs, B., Scott-Harris, J., Kronebusch, K., Navarro, A., and Taylor, K. A. "State Policy Agenda to Eliminate Racial and Ethnic Health Disparities," Commonwealth Fund (June 2004).
 * Michalsen, A., Ludtke, R., Buhring, M. 2003. Thermal hydrotherapy improves quality of life and hemodynamic function in patients with chronic heart failure. Am Heart J. Oct;146(4):E11. PMID 14564334
 * Mokdad AH, Marks JS, Stroup DF, Gerberding JL. Actual causes of death in the United States, 2000. JAMA 2004;291:1238-45. PMID 15010446
 * Saxon DW, Tunnicliff G, Brokaw JJ, Raess BU. Status of complementary and alternative medicine in the osteopathic medical school curriculum. J Am Osteopath Assoc. 2004 Mar;104(3):121-6. PMID 15083987
 * Schwan, R.; Albuisson, E.; Malet, L.; Loiseaux, M.N.; Reynaud, M.; Schellenberg, F.; Brousse, G.; Llorca, P.M. 2004. The use of biological laboratory markers in the diagnosis of alcohol misuse: an evidence-based approach. Drug and Alcohol Dependency. June 11; 74(3):273-9.
 * Smedley, B., Stith, A., and Nelson, A. "Unequal Treatment: Confronting Racial and Ethnic Disparities in Health Care." Institute of Medicine (2002).
 * Snow V, Barry P, Fitterman N, Qaseem A, Weiss K; Clinical Efficacy Assessment Subcommittee of the American College of Physicians. Pharmacologic and surgical management of obesity in primary care: a clinical practice guideline from the American College of Physicians. Ann Intern Med 2005;142:525-31. Fulltext. PMID 15809464.
 * Snyderman, R. Weil AT. Integrative medicine: bringing medicine back to its roots. Arch Intern Med. 2002;162:395-397. PMID: 11863470
 * Tonelli, M.R. 2001. The limits of evidence-based medicine. Respir Care. Dec;46(12):1435-40; discussion 1440-1. Review. PMID 11728302
 * Tonnesen, H.; Hejberg, L.; Frobenius, S.; Andersen, J.R. 1986. Erythrocyte mean cell volume--correlation to drinking pattern in heavy alcoholics. Acta Med Scand. 219(5):515-8.
 * Vangipuram SD, Sheele J, Atkinson RL, Holland TC, Dhurandhar NV. A human adenovirus enhances preadipocyte differentiation. Obes Res 2004;12:770-7. PMID 15166297.
 * Vickers, A. Alternative Cancer Cures: "Unproven" or "Disproven"? CA Cancer J Clin 2004 54: 110-118. Online
 * Wetzel MS, Eisenberg DM, Kaptchuk TJ. Courses involving complementary and alternative medicine at US medical schools. JAMA. 1998;280(9):784 -787. PMID 9729989
 * Whitmer RA, Gunderson EP, Barrett-Connor E, Quesenberry CP Jr, Yaffe K. Obesity in middle age and future risk of dementia: a 27 year longitudinal population based study. BMJ 2005;330:1360. PMID 15863436.
 * Woolhandler S, Himmelstein DU. When Money is the Mission — The High Costs of Investor-Owned Care. N Engl J Med 1999;341:444-6.
 * Zagorsky JL. Is Obesity as Dangerous to Your Wealth as to Your Health? Res Aging 2004;26:130-152. PDF fulltext.
 * Zalewski, Z. 1999. Importance of Philosophy of Science to the History of Medical Thinking. CMJ 40: 8-13. CMJ online


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Talk:Portuguese Grammar

Please be aware that there has already been extensive work on the Portuguese Language Wikibook on grammar. Maybe instead of creating a second book we could combine the two?? - Serge

 * it doens't look like a grammar book, this is a love poem in portuguese

title: Talk:Portuguese Grammar


text:
Multibody Mechanics

This web text is a reference for all topics in the mechanics of multibody systems. A format similar to that of an encyclopedia has been adopted to clearly take advantage of web style referencing and encourage other authors to add new topics or complete existing ones where they are lacking.

A topic is deemed complete when it includes the following elements:
 * an overview of the method or operation outlining its application.
 * a complete analytical description (and derivation where appropriate).
 * example of computational implementation (in a MATLAB and Octave compatible m-file).

Table of Contents

 * Introduction [[File:00%.svg]]
 * Notational Conventions [[File:00%.svg]]
 * Topics
 * Kinematics
 * Angular Velocity [[File:00%.svg]]
 * Basis Vectors[[File:00%.svg]]
 * Euler Angles (see Rotation Angles)
 * Euler Parameters [[File:00%.svg]]
 * Quaternions (see Euler Parameters)
 * Reference Frame [[File:00%.svg]]
 * Rotation Angles [[File:00%.svg]]
 * Vector and Tensor Operations [[File:00%.svg]]

title: Multibody Mechanics


text:
World Stamp Catalogue/Oceania

Stamp issuing Australasian and Pacific countries and dependent territories

 * Australia - AUS
 * Christmas Island (Indian Ocean) - CXR
 * Cocos Island - CCK
 * Fiji - FJI
 * New Zealand - NZL
 * Niue - NIU
 * Norfolk Island - NFK
 * Pitcairn - PCN
 * Western or British Samoa - WSM
 * Tonga - TON

title: World Stamp Catalogue/Oceania


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World Stamp Catalogue/New Zealand

=Stamps of New Zealand=

Postage stamps and postal history of New Zealand

1873, 1874 & 1882 Queen Victoria "Sidefaces"
1873:

1/2d Pink/Rose (Newspaper Rate) 1874 First Sidefaces:

1d Lilac 2d Red 3d Brown 4d Maroon 6d Blue 1/ Green 2/ Red 5/ Grey (The 2/ and 5/ are very hard to find.) 1882 Second Sidefaces:

1/2d Black 1d Red 2d Purple 21/2d Blue 3d Yellow 4d Green 5d Grey 6d Brown 8d Blue 1/ Chestnut

1898 Pictorials
1/2d Mt Cook (black) 1d Taupo 2d Pembroke (red) 21/2d Wakatipu 3d Huias 4d Terraces 5d Otira 6d Kiwi (green) 8d Canoe 9d Terraces 1/ Kaka 2/ Milford 5/ Mt Cook Notes: Four stamps from this set can also be found overprinted OFFICIAL: 3d Huias, 1/ Kaka, 2/ Milford and 5/ Mt Cook. The 21/2d Wakatipu can also be found with the spelling variation "Wakitipu", These are not hard to find and are most commonly seen unused. The 5/ Mt Cook is not scarce but it is difficult to find in good condition with a regular cancellation. Fiscally used embossed, circular "crown" cancels, or pen cancels are common and worth much less than catalogue value, One leading NZ dealer sells fiscally used copies for $35NZ which is less than 1/20th of the catalogue price. Also beware of "mint no gum" of this stamp as most are fiscally used with the pen cancel cleaned off.

1900 Pictorials
1/2d Mt Cook (green), 1d Terraces, 11/2d Boer War, 2d Pembroke, 4d Taupo, 6d Kiwi. Notes: This set contains mostly reprints of designs from the 1898 pictorials, in different colors and values. 3 stamps from this set can also be found overprinted OFFICIAL: 1/2 Mt Cook, 2d Pembroke, 6d Kiwi.

1901 Universal.
1d Zealandia (Can be found with or without official overprint)

1906 Christchurch Exhibition
1/2d Canoe 1d Carving 3d Landing 6d Annexation

1907 Redrawn Pictorials
1d Zealandia (diagonal lines on globe) 3d Huias 6d Kiwi (red) 1/ Kaka

Notes: Same designs as earlier issues - but these are smaller (except for the zealandia which can be identified by the lines on the globe.) The Zealandia and 6d Kiwi can also be found overprinted Official.

1909 King Edward VII
1/2d Green 1d Zealandia Dominion 2d Mauve 3d Brown 4d Orange 4d Yellow 5d Brown 6d Red 8d Blue 1/ Orange

Note: 1/2d Green, 1d Dominion (Same design as zealandia with words "Dominion Of" at top of stamp) 3d Brown, 6d Red, 8d Blue & 1/ Orange can also be found with Official overprints.

1913 Auckland Exhibition
1/2 Green 1d Dominion 3d Brown 6d Red

Note: These are stamps from the King Edward issue of 1909, overprinted for the exhibition.

1915 King George V
1/2d Green 11/2d Grey 11/2d Orange-Brown 2d Violet/Purple 2d Yellow 3d Brown 4d Yellow 4d Violet/Purple 41/2d Green 5d Blue 6d Red 71/2d Brown 8d Blue 8d Brown 9d Olive 1/ Orange

These exist in a variety of different printings and some can be found overprinted Official. The 1/2d Green can also be found overprinted "War Stamp".

1920 Victory
1/2d Green 1d Red 11/2d Brown 2d Two Pence overprint on 1/2d Green 3d Brown 6d Purple 1/ Orange

Note: The 6d and 1/ are commonly found with heavy parcel cancels, these are worth considerably less than catalogue values.

1923 Map Stamp
1d Red

1925 Dunedin Exhibition
1/2d Green 1d Red 4d Mauve/Purple

1926 Admirals
1d Red 2/ Blue 3/ Purple

Note: Heavy parcel cancels & pen or punched/perfin cancels are common on the 3/ and worth a lot less than catalogue values, one of the leading New Zealand dealers supplies parcel cancels of these at about 1/4th of their regular used price. 1d Red and 2/ Blue can also be found overprinted Official.

1931 Airmails
3d Chocolate/Brown 4d Purple 5d Five Pence overprint on 3d green 7d Orange/Brown 7d Blue with Trans Tasman overprint.

Note: The 5d and 7d Blue exist only with the overprints.

1935 Air Mail
1d Red 3d Violet 6d Blue

1935 Silver Jubilee
1/2d Green 1d Red 6d Orange

1935 Pictorials
xx xx
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1938 George VI
xx xx
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1940 Centennial
xx xx
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1953-4 Queen Elizabeth
xx xx
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 * Number of stamps issued:

1960 Pictorials
xx xx
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 * Number of stamps issued:

1967 Decimal Currency
xx xx
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1970 Pictorials
xx xx
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1975 Pictorials
xx xx
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1980 - 1989 Issues
xx xx
 * Number of issues:
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1990 - 1994 Issues
xx xx
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1995 - 1998 Issues
xx xx
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1999 - 2000 Issues
xx xx
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2001 - 2002 Issues
xx xx
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2003 - 2004 Issues
xx xx
 * Number of issues:
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2005 - 2006 Issues
xx xx
 * Number of issues:
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New Zealand |New Zealand

title: World Stamp Catalogue/New Zealand


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Ordinary Differential Equations/Separable 2

First-Order Differential Equations

=Separable Variables: Real-World Examples=

This page gives some examples of where simple separable variable DEs are found in the world around us.

Acceleration, velocity, and position
The classic real world example of differential equations is the relationship between acceleration, velocity, and position.

 * $$a(t)=v'(t)=x''(t) \,$$

So if you're given an equation for acceleration, you can figure out both velocity and position.

Example 1 - Constant Acceleration
Lets say that acceleration is a constant, g (the acceleration due to gravity, or about 10 m s-2.  The initial velocity at t=0 is v0.  The initial position is x0. Solve for v and x.

First, you need to solve for v. I.w.r.t.x.

 * $$v'=a=g \,$$

 * $$\int v' dv=\int g dt \,$$

 * $$v=gt+C \,$$

Now plug in to find C.

 * $$v_0=g \times 0+C \,$$

 * $$C=v_0 \,$$

 * $$v(t)=gt+v_0 \,$$

Now we solve for x.

 * $$x'=v=gt+v_0 \,$$

 * $$\int x' dt=\int (gt+v_0)dt$$

 * $$x=\frac{1}{2}gt^2+v_0t+C$$

Again, now we solve for C.

 * $$x_0=\frac{1}{2}g \times 0^2+v_0 \times 0+C$$
 * $$x_0=C \,$$

 * $$x(t)=-\frac{1}{2}gt^2+v_0t+x_0$$

Anyone who has studied physics will recognize this as the basic equation for position for an object undergoing a constant force in one dimension.

Example 2 - Resistive Medium
Lets say we're traveling through a medium that resists movement. In this medium, $$a=-v^2$$. Solve for v and x, given that the initial velocity was 10 m/s and the initial displacement was 0 m.

 * $$a=v'=-v^2 \,$$

 * $$\frac{dv}{dt}=-v^2 $$

 * $$\int \frac{1}{v^2}dv=\int -1 dt$$

 * $$-\frac{1}{v}=-t+C$$

 * $$v=\frac{1}{t+C}$$

Notice that as t increases, velocity decreases. This is what you'd expect if the medium was resisting your movement and slowing you down over time. Now substitute in the initial velocity at t=0

 * $$10=\frac{1}{0+C}$$
 * $$C=\frac{1}{10}$$

 * $$v=\frac{1}{t+0.1}$$

Now to solve for x.

 * $$v=x'=\frac{1}{t+0.1}$$

 * $$\frac{dx}{dt}=\frac{1}{t+0.1}$$

 * $$\frac{dx}{1}=\frac{dt}{t+0.1}$$

 * $$\int 1 dx=\int \frac{1}{t+0.1} dt$$

 * $$x=ln(t+0.1)+D \,$$

The position increases throughout, but it increases ever more slowly as time goes on. This is again what you'd expect for a medium resisting motion. Since velocity is never less than 0, we never stop going forward, but we go exponentially less far over time. Putting in our boundary conditions,

 * $$0=ln(0+0.1)+D \,$$
 * $$D=-ln(0.1) \,$$

 * $$x=ln(t+0.1)-ln(0.1)=ln(10t+1) \,$$

Exponential Growth and Decay
One of the most common differential equations in science is

 * $$y'=ky \,$$.

The solution to this is

 * $$y=Ce^{kt} \,$$.

If k is positive, this is called exponential growth. If k is negative, its exponential decay. Both are used in science, for very different reasons.

Population Growth
Lets say we have a group of animals in the wild. We want to know how many animals there will be in t years. We know how many there are now. We also know the birth rate and death rate. Can we solve this problem?

Of course we can. First, we need to figure out the rate of growth. If the birth rate is B, and the death rate is D, the total rate of change is (B-D). Since this is the rate, we need to multiply it by the current population to get the population growth. The final equation looks like

 * $$\Delta P=(B-D)P \,$$

where P is the population. That looks like the equation for exponential growth, doesn't it? As a matter of fact, change the 'delta' to a differential, and it is. The growth factor is (B-D).

Example 3

 * In a certain population of rabbits, the birth rate is 10%. The death rate is 15%.  The initial population is 100.  How many rabbits are there after 10 years?  Will we always have rabbits?

From our solution to the exponential equation:

 * $$P=Ce^{(B-D)t} \,$$

 * $$100=Ce^{(B-D)\times 0}=Ce^0 \,$$

 * $$C=100 \,$$

 * $$P=100e^{-0.05t} \,$$

 * $$P(10)=100e^{-0.5}\approx 61$$

Unfortunately, we will not always have rabbits. Since the growth rate is negative, they will eventually go extinct. (Note: we will never actually hit 0, but in real life you can't have less than 1 rabbit.  If we were measuring a continuous property instead of a discrete one, we would always have something, it would just get very small).

Radioactive Decay
Another situation for exponential growth is radioactive isotopes. If you have a sample of radioactive material, individual atoms will randomly decay or not decay. While you can't know exactly how many atoms decay and when, you do know the average rate of decay. Every λ years, half of the atoms left will decay. This period of time, λ, is called a half life. The activity of the sample (how many decays per second) is called A. Mathematically, this looks like

 * $$\frac{dA}{dt}=\lambda A$$

These problems look just like the problem above. Just like rabbits, we will eventually run out of radioactive atoms as well.


text:
Ordinary Differential Equations/Separable 4

Existence problems
1) f(x,y) has no discontinuities, so a solution exists. $$\frac{\partial {f} }{\partial {y} }$$ has no discontinuities, so the solution is unique.

2) f(x,y) is not defined for the point (-1,10) because ln(x) is not defined. So no solution exists.

3) f(x,y) has discontinuities at y=1 and -1, but not at 0 so a solution exists. $$\frac{\partial {f} }{\partial {y} }$$ has no discontinuities at (0,16) so the solution is unique.

4) f(x,y) has discontinuities at y<0, but not at 1 so a solution exists. $$\frac{\partial {f} }{\partial {y} }$$ is discontinuous at 1, so the solution is not unique

5) f(x,y) has discontinuities at -3 and -4, but not at 0 so a solution exists. $$\frac{\partial {f} }{\partial {y} }$$ has no discontinuities at (5,9) so the solution is unique.

6) f(x,y) has a discontinuity at x=5, so no solution exists.

Separable equations
7) $$y'=y^3sec^2(x)$$

$$\frac{dy}{y^3}=sec^2(x)dx$$

$$\int \frac{dy}{y^3}=\int sec^2(x)dx$$

$$-\frac{1}{2y^2}=tan(x)+C$$

$$y=-\frac{1}{\sqrt{(2tan(x)+C)}}$$

8) $$y'=\frac{5y^2+6}{y}$$

$$\frac{ydy}{5y^2+6}=dx$$

$$\int \frac{ydy}{5y^2+6}=\int dx$$

$$\frac{1}{10}ln(5y^2+6)=x+C$$

$$y=\pm\sqrt{Ce^{10x}-\frac{6}{5}}$$

9) $$y'=x^3/y^3$$

$$y^3dy=x^3dx$$

$$\int y^3dy=\int x^3dx$$

$$\frac{1}{4}y^4=\frac{1}{4}x^4+C$$

$$y=(x^4+C)^{\frac{1}{4}}$$

10) $$y'=x^2+3x-9$$

$$dy=(x^2+3x-9)dx$$

$$\int dy=\int (x^2+3x-9)dx$$

$$y=\frac{1}{3}x^3+\frac{3}{2}x^2-9x+C$$

11) $$y'=cos(y)/sin(y)$$

$$\frac{sin(y)dy}{cos(y)}=dx$$

$$\int \frac{sin(y)dy}{cos(y)}=\int dx$$

$$-ln(cos(y))=x+C$$

$$y=arccos(Ce^x)$$

12) $$y'=\frac{cos(x)}{sin(y)}$$

$$sin(y)dy=cos(x)dx$$

$$\int sin(y)dy=\int cos(x)dx$$

$$-cos(y)=sin(x)+C$$

$$y=arccos(-sin(x)+C)$$

Initial value problems
13) $$y'=cos(x)+sin(x),y(0)=1$$

$$dy=(cos(x)+sin(x))dx$$

$$\int dy=\int (cos(x)+sin(x))dx$$

$$y=sin(x)-cos(x)+C$$

$$1=sin(0)-cos(0)+C=0-1+C=C-1$$

$$C=2$$

$$y=sin(x)-cos(x)+2$$

14) $$y'=7y^2,y(5)=9$$

$$\frac{dy}{y^2}=7dx$$

$$\int \frac{dy}{y^2}=\int 7dx$$

$$-\frac{1}{y}=7x+C$$

$$y=\frac{1}{-7x+C}$$

$$9=\frac{1}{-7*5+C}$$

$$C=\frac{316}{9}$$

$$y=\frac{1}{-7x+\frac{316}{9}}$$


text:
Ordinary Differential Equations/Separable 3

Existence problems
Tell whether the following initial value problems have a solution or not, and if its solution is unique.

1) $$y'=(12x^2+5x)(y+9y^3),y(7)=11$$

2) $$y'=ln(7x),y(-1)=10$$

3) $$y'=\frac{x+7x^2-6x^3}{y^2-1},y(0)=16$$

4) $$y'=xln(y-1),y(1)=1$$

5) $$y'=\frac{x^3+5x}{y^2+7y+12},y(5)=9$$

6) $$y'=\frac{y+7y^2}{x-5},y(5)=4$$

Separable equations
7) $$y'=y^3sec^2(x),$$

8) $$y'=\frac{5y^2+6}{y}$$

9) $$y'=x^3/y^3$$

10) $$y'=x^2+3x-9$$

11) $$y'=cos(y)/sin(y)$$

12) $$y'=\frac{cos(x)}{sin(y)}$$

Initial value problems
13) $$y'=cos(x)+sin(x),y(0)=1$$

14) $$y'=7y^2,y(5)=9$$


text:
Talk:Sun Certified Web Services Developer Certification

Maby you should shorten the titles a bit, the toc is very messy now. --August 7 July 2005 09:14 (UTC)

move?
move to Sun Certified Developer For Java Web Services for descriptiveness maybe? --Remi (talk) 07:17, 25 February 2008 (UTC)


text:
Wi-Fi/Setting up a hotspot

This page provides general information about configuring a wireless hotspot.

Components
Wireless networks require hardware, and to a lesser degree, software.

Hardware
At a minimum, the following are needed:

 * A router — This typically connects to a high-speed internet connection and manages the network's communication with the Internet. If you're dealing with a network not connected to the Internet (Rare), a switch or hub may be used. If you're only connecting two computers to each other (An "Ad Hoc" network), only adapters are needed.
 * Adapters for each machine on the network — These will be expansion cards for desktop computer and PCMCIA cards for laptops.

If you have a large area to cover, separate antennas and/or access points may be needed.

Software
All computers on the network need to have an operating system version (Windows, Mac OS, Linux, etc) that can deal with wireless networking.

 * All Windows from 98 on have some degree of ability in this field. 98, ME and 2000 require additional software and drivers. XP with the latest updates from Microsoft will work automatically with many network cards, Vista will work natively.
 * Mac OS 9 has some wireless networking ability, 10 will work natively.
 * Most recent Linux kernels have built-in wireless networking capability.

Hardware setup
Assuming a basic router-and-clients setup, the following hardware will be needed:

 * 1) At least one router
 * 2) A wireless adapter for each machine that doesn't already have one (Some laptops have built in antennas)
 * 3) A cable and adapter for each machine that will be wired into the network
 * 4) Depending on the layout of your house and devices, you may need some repeaters. The following section will explain this.

Location of equipment
Most likely your router will have to connect to your modem (If you don't have an Internet connection, you can disregard this). If this is in a central location for your planned uses, then you will likely not need anything beyond the router and adapters. If you're installing the modem and are planning to have wireless networking, consider installing it in a central location.

If not, you can run more wiring in your house to get it to the router, or accept a weaker signal in the parts of the house farther from the router. If this is an issue, you may want to invest in a repeater that can bring your wireless signal to the far corners of your house.

As discussed previously, wireless signal strength is of significant importance. Some points of information about signals and permeability:

 * Wireless signals can pass through drywall relatively easily, but if it must pass through a significant number of walls the signal will degrade rapidly.
 * Wireless signals do not travel well through cinder block walls.
 * Wireless signals can be disrupted by ducts of a climate control system.
 * Water can greatly distort wireless signals. Take into account aquariums, sinks, and water pipes.  Remember, the human body is 66% water, so locations with a high density of people can distort wireless signals. This also includes rain or humidity, so consider bath/shower rooms or other regularly wet areas.

If you want coverage outdoors (which can be very nice during good weather with a laptop!), consider locating the router or a repeater near a window facing the direction you want to cover. An upper floor can be an excellent place for a router or signal propagator for coverage around the outside of the building, and possibly inside as well.

Wiring up the router
Most residential routers are fairly simple to set up and have standardized markings. The usual setup is this:

 * 1) Plug the Ethernet (data) cable from your cable or DSL modem into the "WAN" port (If your modem has USB and Ethernet ports, use the Ethernet port and don't connect anything to the USB port).
 * 2) Plug any computers you wish to hard-wire to your router to one of the "LAN" ports. If every computer is to connect wirelessly, skip this step.
 * 3) If possible, disconnect the antenna, or shield it from transmission.
 * 4) Shut off/unplug your modem for about 15 seconds
 * 5) Restore power to your modem
 * 6) Plug in your router

The above steps may or may not cause the router to obtain an internet connection from the modem. These are VERY generic steps, which will sometimes work if the user's Internet Service Provider uses DHCP. If not, the router must be manually configured, and only the physical connections listed here will be of use to the user.

After a few seconds (At most), the router should start up with the default settings. Leave the antenna disconnected/blocked until you've had a chance to change the wireless security settings, as router defaults are notoriously insecure.

Router configuration
The vast majority of residential routers are configured by a "web-based interface" (A webpage which you can view with Firefox or IE or whatever you use to surf the web).

To access this webpage, a special address (called an IP address) is used. This is normally detailed in the users manual or "quick start guide". The username and password to be used are also listed. If a "quick start wizard" comes up, you can follow it through or dismiss it. The rest of this guide assumes that it didn't appear or was dismissed.

Once logged in to the router, the first steps should be to strengthen the security. See below...

Expansion cards should be installed with the computer turned off. PCMCIA cards for laptops can be inserted at any time. If the operating system doesn't automatically detect and set up the new networking hardware, try using the disc that came with it.

title: Wi-Fi/Setting up a hotspot


text:
Talk:Systems Theory/Open/Closed System Structure

Purpose of this module
What is the purpose of this article?

Is this article about system behavior and the major types of systems or is it specifically about IT systems?

Is this supposed to be a monlogue against the Great Satan, Microsoft, or is it supposed to be about systems theory?

There appears to be too much emphasis on IT systems with quite a bit of half baked philosophy thrown in.


text:
User:Daveamz~enwikibooks

I discovered this enciclopedia by mistake in a simply Google Search and I think is great because you(we) can be involved directly in this "Mass Culture Project" and by part of it by editing pages and keep'em up to date.

title: User:Daveamz~enwikibooks


text:
User talk:Oniony~enwikibooks

 * Thanks for all your contributions to the book - it's really coming along. add your name to London:Regular contributors and anyone else who you know is working on the book! Charlie123 7 July 2005 14:41 (UTC)

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally. (Donovan|Geocachernemesis|Interact) 00:21, 15 July 2005 (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 Oniony. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Oniony~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 23:34, 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) 05:19, 19 April 2015 (UTC)

title: User talk:Oniony~enwikibooks


text:
User talk:169.71.50.11

Why not get yourself a username? Just click on "Create an account" top right. Thanks for great work on the London book - see London:Regular contributors and add your name. Charlie123 7 July 2005 14:48 (UTC)

title: User talk:169.71.50.11


text:
EBay

This book will explain techniques you can use to find and win the item of your dreams, while advising the reader on how to avoid fraud and make the most of their experience throughout.

The first half of this book will concentrate on how to buy on eBay. Once you are experienced, you can move on to the second part of the book on selling.

Section I: Buying
Anticipated contents:

Introduction to Buying

Examining Listings and Searching

Understanding Item Descriptions and Pre-Purchase Considerations

Bid to Win: Tips and Techniques

Paying for Items Safely

If there is a problem

Leaving Feedback

Top 10 Tips

Section II: Selling
Anticipated contents:

Introduction to Selling

What will and won't sell

Writing Good Descriptions

Considering Payment Methods

Considering Shipping Methods

The Tools of the Trade

Putting it all together and Making a Listing

Managing Questions

Receiving Payment

Sending the Item

If there is a problem

Leaving Feedback

Top 10 Tips

title: EBay


text:
User:Nintenoid~enwikibooks

Who am I?!
Well, I'm an 18 year old from the South West of the United Kingdom.

My qualifications include 8 A* GCSEs and 6 grade A AS Levels.

Next year I am starting work as a software engineer.

Books
I have just started work on Understanding eBay. I intend to write a book to help AQA AS Computing students, as all the books on this subject are, frankly, very poor.

I'm happy to help out with anything else you want me to do, so feel free to contact me.

title: User:Nintenoid~enwikibooks


text:
Talk:Spanish/Exercises/Comparisons

Untitled
Should #5 read "Los niños juegan..."? My Spanish/English dictionary says the third person present tense form of jugar should be juegan, not jugan, but I left it unedited because I don't speak Spanish.

title: Talk:Spanish/Exercises/Comparisons


text:
Robotics/Computer Control/Control Architectures

Robots need some form of artificial intelligence in order to be able to navigate around a room or to solve a problem. Such AI can range from simple, as in following particular rules when a specific input is received, through more complicated, as in fuzzy logic rules or simple neural nets, to complicated, as in state of the art machine learning methods.

How complicated you can make your AI depends on a few factors:
 * Processing power: How fast is your microprocessor, is it 8-, 16-, or 32-Bit, How many microprocessors do you use?
 * Available memory: Both temporary and "permanent" memory. Limited amounts of memory restrict how much data can be held at a time, e.g. a map of the surroundings, sensor history, movement history. Systems with much permanent storage and little ram are slower than systems with large amounts of ram.
 * Your knowledge: AI is a very broad subject. Parts of it are very math intensive and can be tricky to understand.


text:
How to Solve the Rubik's Cube

Notation
There are several notations; please refer to this notation guide.

Briefly:

 * There are six sides to the cube, presented as Front, Back, Left, Right, Up and Down. They are usually referred to by their one-letter abbreviations.
 * In the isometric diagrams below, where a corner points out at you, you see the F, R and U sides. The F faces to the left.
 * Movements are presented as one quarter rotation (90 degrees) of an external face per movement. This means that the center tile colors are not changed. In our diagrams, F is blue, R is red and U is yellow. The other three colors are typically orange opposite red, green opposite blue and white opposite yellow
 * Quarter-rotations of that face's layer default to clockwise. Counter-clockwise rotations are often referred to as "inverted" and indicated by ′, for instance, R′. (The ′ is commonly read as "prime", "apostrophe", "tick mark", "anti-clockwise", "anti" or "i" for inverted). Half-rotations (180 degrees) are indicated by the digit "2", for instance, R2 (meaning 2 quarter-rotations following the one-letter abbreviation).
 * To see what is happening on the sides of the other three colors, rotate the cube as a whole, described as rotating along the x, y, z space axis, all pointing out of the page. x is R, y is U and z is F, but since this sort of move also changes the colors of the center-tiles, it is used sparingly.

Example solve
As an example, let's consider a complete solve. 25 move scrambles are used to mix up the cube. Our sample scramble is:

 * U B′ R2 D′ U′ R U2 B R′ B2 L2 R F2 R2 U2 R B U2 F2 L2 F2 D R B2 R2

The solve is:

 * R′ B R D2 F2 L U′ F U R′ D R F D F′ F′ D′ F U2 R′ D′ R U2 U′ F′ D′ F U U B′ D′ B U′ y2 F D2 F2 R F R′ B′ D F D′ B D F′ F2 D M D2 M′ D F2 (54 moves)

Methods
There are a large variety of methods to solve the Rubik's Cube. Here we list methods that are described in some detail in this Wikibook, before briefly reviewing other methods in the following sections.
 * Beginner
 * Beginner (alternative)
 * Fridrich Method

Classical Methods
The first widely publicized solutions to the Rubik's Cube appeared in the early 1980s, when quite a number of solutions where published in books and articles. For example, see Philip Marshall's comparison of various classical methods. Here we mention two solutions from around 1981 by David Singmaster , and James Nourse.

One of the key observations was that the solution can be broken down into several steps. Most of the "standard" classical approaches solve the cube layer by layer. For example, solve 1. all edges in the top layer, 2. the corners in the top layer, 3. the edges in the middle or horizontal layer, 4. the edges in the bottom layer, and 5. the corners in the bottom layer, which completes the solution. There are a number of relevant variations, for example step 2 and 3 can be combined (see the Fridrich method below), or step 5 is often split into first putting the pieces in the right place and then fixing their orientation. The bottom line is that such steps simplify the solution process because there exist algorithms (sequences of face turns) that can handle the individual steps efficiently. Solution methods differ not only in the steps, but also in the set of algorithms suggested for the individual steps.

As Marshall discusses, many of the more recent methods have their roots in these early solutions. This is also apparent in several of the methods that are listed below, which however include a number of improvements. The solution steps have been modified, and the sets of algorithms have been improved and optimized for different solutions steps. Furthermore, some of the early methods were not fully explained (or had gaps), and many improvements in presentation were made.

In the following, we comment on beginner methods, where the goal is simplicity (usually at the cost of efficiency), and on advanced methods, which give faster and/or shorter solutions (usually at the cost of increased complexity).

Beginner Methods
The term beginner method is used differently by different authors. A beginner method should be simple, but what is considered simple depends on the person, and changes quickly as experience is gained. As Marshall discusses, some of the early simple methods required 10 to 20 algorithms and needed 100 to 150 moves to solve the cube. For example, he reports an average of 110 moves for Nourse's method with 12 algorithms, which comes down to 100 moves adding shortcuts involving 20 algorithms in total.

A modern method could be called simple if it only needs 5 algorithms or less. If you are used to advanced methods with 50 plus algorithms, then 10 or less is also simple. Also, the algorithms should not be too long and complicated themselves. Some of the recent beginner methods with 5 or less algorithms are surprisingly efficient, requiring less than 100 and in some cases only about 70 moves. This is quite nice compared to some advanced speedcubing methods that require 40 to 60 moves or so on average but use 50 plus algorithms.

Recent beginner methods
Solution guide at Rubiks.com: The guide at the official Rubik's site seems to be one of the classic layer-by-layer methods. It lists 14 algorithms.

Heise's beginner method: This is a representative example of an optimized, classical layer-by-layer method. The original strategy is credited to David Singmaster. It requires four algorithms. (Some of the easy steps are labeled intuitive and not counted as algorithms, but this is common practice.) The solution is presented with the help of animations.

Wikibook Beginner method: Presented on these wikibook pages, this layer-by-layer method uses 5 to 8 algorithms, depending on how you count them. Although perhaps not optimal in the number of algorithms, it demonstrates a rather successful idea. Only 3 of 4 corner pieces and 3 of 4 edge pieces are solved in the first two layers. Leaving those pieces as free workspace can simplify some of the later steps. See the Petrus method below for the classical example of a block building method that avoids solving certain pieces too early.

8355 method: Another layer-by-layer method using two pieces as workspace. Requires 3 algorithms. In this case the workspace allows a simplification compared to, say, Heise's beginner method.

Philip Marshall's method: An edges-first method that relies on only 2 algorithms. Reported to require only 65 moves on average by Marshall. Notice that the above layer-by-layer with workspace methods can be turned into edges-first methods if solving the three corners of the first layers is delayed until all edges have been solved. The Marshall method achieves its simplicity by using one particular 4-turn algorithm to do all the edges, and one 8-turn algorithm for the corners. Some elementary setup turns are needed as well.

Single algorithm methods: The 8355 method and Marshall method can be reduced to single algorithm methods, see the Single Alg Cube Solution and the Y-move method. Basically, there are two elementary 4-turn commutators, the "S-move" and the "Y-move", which are used in these and other methods to solve the edges. It turns out that these commutators can also be applied repeatedly to replace the corner algorithms in the 8355 and Marshall method. There is a small loss of efficiency, but in this way single algorithm methods are possible. Another single algorithm method was originally developed by Camilo Vladimir de Lima Amaral, who called it "Less is more method" or "Amaral Method".

Are there zero algorithm methods? The answer is yes since single face turns are not counted as algorithms, and the cube can certainly be solved with those. However, the idea of algorithms is to group single face turns into something manageable by humans. Relying only on "intuition" and single face turns has not led to a beginners method yet.

In conclusion, there are several recent beginner methods that improve on the classical layer-by-layer beginner's methods, although simple can mean different things. Simple methods with just 1 to 4 algorithms are possible, where the main point is that this is less than 10 or 20. Still, a beginner may prefer a printed list of 5 or 10 algorithms over a less explicit method using only a 1 or 2 algorithms. On the other hand, a method with fewer algorithms is easier to memorize and easier to understand than a more complex method.

"Algorithms of last layer":Some algorithms of fixing last layer-

1. Making cross-(F R U R' U' F' if you have a horizontal bar & F U R U' R' F' if you have a back-left hook)

2. Matching edge colours-(R U R' U R U2 R')

3. Fixing corners-(U R U' L' U R' U' L)

4. Matching corners-(D R' D' R)

Faster methods
While the above methods may be good for a beginner, they are too slow to be used in speedcubing. The most popular method for speedcubers is very similar to the Wikibook Beginner method above, except steps 2 and 3 are combined, and the last layer is solved in two steps instead of three. The inventor of this common method is Jessica Fridrich. With this method, speedcubers with good dexterity and memory can average under 20 seconds after a few months of hard practice. However, to learn the method you must learn 78 algorithms. There are methods just as fast that require far fewer algorithms to be memorized. Here is a brief synopsis of several popular speedcubing methods:

Layer by Layer methods
Fridrich Method: A very fast First 2 Layers (or F2L) method, start by solving a cross on one face, then proceeding to solve the First 2 Layers pairing up edge and corner combinations and putting them into their slot. This is followed by solving the Last Layer in two steps, first orienting all pieces (one color on the last layer), then permuting them (solving the ring around the last layer). The basic method has 78 algorithms (without the inverse of them), and is recognized as one of the fastest methods currently in use.

F2L Alternatives: Methods that follow the same principle as Fridrich's method, but using different algorithms. Many of the algorithms are shared but there are a few differences, so there should be one to suit your fingers:
 * Bob Burton:
 * Shotaro 'Macky' Makisumi:
 * Speedcubing.com Collection: speedcubinglovers.com

ZB method: This method was developed independently by Ron van Bruchem and Zbigniew Zborowski in 2003. After solving the cross and three c/e pairs, the final F2L pair is solved while orienting LL edges. This is known as ZBF2L or ZBLS. The last layer can then be solved in one algorithm, known as ZBLL. The ultimate method requires several hundred algorithms. Lars Vandenbergh's site has ZBF2L algorithms, used in his VH system. There is also a more recent Google spreadsheet at. ZBLL algorithms can be found on Doug Li's webpage.

ZZ method: This method was created in 2006 by Zbigniew Zborowski, the co-creator of the ZB method. It has three basic steps: EOLine, F2L, and LL. EOLine stands for Edge Orientation Line. The orientation of edges is defined as either good or bad. Good meaning the edge can be placed into the correct position with a combination of R, L, U, D, F2, or B2, moves. Bad meaning it would require an F, F′, B, or B′ move to be moved into its correct position. Any F, F′, B, or B′ move will cause the four edges on that slice to change from its current state, good or bad, to the opposite state. The Line portion of EOLine is forming a line on the bottom of the cube that consists of the DB edge and the DF edge in their correct positions. The next step is F2L, First 2 Layers. It uses block building techniques to solve the two remaining 1x2x3 blocks of the F2L using only R, U, and L moves. This allows for very quick solving of F2L as it does not require cube rotation. The final step of the ZZ method is LL, Last Layer, and it can be broken into multiple steps or maintained as one depending on the algorithms used. There are two main approaches to this method OLL and PLL, Orientation of LL and Permutation of LL, and COLL  and EPLL , Corner OLL and Edge PLL. The first, OLL and PLL, is to use one of 7 algorithms to solve the top layer (OLL) and then permute the edge and corners into their correct positions (PLL), this requires 21 algorithms. The total algorithms required for the first approach of solving LL is 28. The second approach to solving LL is to solve the top and the corners in one algorithm (COLL) and then solve the edges (EPLL). COLL requires 40 algorithms and EPLL requires 4, making the total 44 algorithms. The second approach is faster due to the ease of recognition and speed of execution of EPLL.

VH method: Created by Lars Vandenbergh and Dan Harris, as a stepping stone from Fridrich to ZB. First, F2L without one c/e-pair is solved with Fridrich or some other method. Then the last pair is paired up, but not inserted. Then it's inserted to F2L and LL edges are oriented in one go. Then, using COLL, corners of LL are solved while preserving edge orientation. Then edges are permuted.

Block methods
Petrus System: Created by Lars Petrus. One of the shortest methods in terms of face turns per solve, the Petrus method is often used in fewest moves contests. Petrus reasoned that as you construct layers, further organization of the cube's remaining pieces is restricted by what you have already done. For a layer-based solution to continue after constructing the first layer, the solved portion of the cube would have to be temporarily disassembled while the desired moves were made, then reassembled afterwards. Petrus sought to get around this quagmire by solving the cube outwards from one corner, leaving him with unrestricted movement on several sides of the cube as he progressed. There are not as many algorithms to learn compared to the other F2L methods, but it takes a lot of dedication to master. The basis of the method is to create a 2 × 2 × 3 block on the cube, then proceed to solve a 3 × 3 × 2 block, but also flipping the edges on the Last Layer. Then the Last Layer is solved in two steps, first corners and then edges.

Heise method: Created by Ryan Heise. First, one inner square and three outer squares are built intuitively. Then they are placed correctly while orienting remaining edges. After that you create two c/e-pairs, and solve the remaining edges. The last 3 corners are solved using a commutator.

Gilles Roux Method: Another unique method, but works in blocks like the Petrus method. You start by solving a 1 × 2 × 3 block and then solve another 1 × 2 × 3 block on the other side of the cube. Next you solve the last 4 corners and finally the edges and centers. Has only 24 algorithms to learn.

Corners first methods
Waterman Method: Created by Mark Waterman. Advanced corners first method, with about 90 algorithms to learn. Solve a face on L, do the corners on R and then solve the edges. An extremely fast method.

Jelinek Method: Created by Josef Jelinek. This method is very similar to Waterman's.

Create a solved 2 × 2 × 2 cube on one corner and rotate the remaining blocks (it may take a while but you will eventually solve it).

Catta Method
From its creator the Italian Giuseppe Catta, unique and different from all, based on structural pieces (in the case of the 3x3x3 cube of only 2 types, edges and medians), from the edges towards the center and hinged on the two opposite faces. Very important note as these 3 principles with the Catta method subsequently guide the resolution of Rubik's Cubes of any size NxNxN, except for implementing additional algorithms consistent respectively with two additional structural pieces with the 4x4x4 (4 "centres" instead of 1 "diagonal internal crowns", not piece, of the 3x3x3; and 2 lateral medians instead of 1 of the 3x3x3) and a further structural piece with the 6x6x6 ("symmetrical median internal crowns").

This method - discovered and codified not by chance by an architect - is extremely didactic, simpler, more orderly, understandable and easier to memorize than the others, and its approach is paradigmatic of a way of dealing with things as a whole and as a whole, to system, from general to particular. It teaches us to see the whole whole at once, as a system of parts rather than as a sum or juxtaposition of parts.

Three "Levels of Difficulty"
A procedure has been developed whereby a complete beginner can learn and master the Cube by moving up through three self-contained "Levels of Difficulty".

The lowest level deliberately maintains configurations in which every face exhibits horizontal and vertical symmetry, so it also enables a number of "pretty patterns" to be constructed - such as Chequers, Crosses, Stripes and Central 'Dots'. Level Two involves solving cubes which have been scrambled using only 180-degree turns. Techniques acquired in those earlier stages, remain useful when continuing up to the next level.

How to solve the 2x2x2 (Mini/Pocket Cube)
If you know how to solve the 3x3x3 cube and 4x4x4 (see above), solving the 2x2x2 can be achieved by treating the cube as a 3x3x3 in which the centre and edge cubes are in a permanent solved state regardless of what moves you make. In other words, the solution consists of only the corner-solving steps of the 3x3x3 solution. The important thing is simply to remember which side is which (since you can't see the middle layers, because they aren't actually there), although this shouldn't be too difficult.

Other solution pages
Here are some of the more popular solution pages listed. All are different, although they mostly use a similar layer by layer method. Usually you will need Java to see the animations used.

Using animations:
 * Rubik's Cube solution for Beginners (rubiksplace.com)
 * Beginners Solution By Michiel van der Blonk
 * Beginners solution By Christophe Goudey.

Using pictures:
 * Rubik's Cube solution with animations (rubiksplace.com)
 * Simple solution by Rick Rayner
 * Beginners' solution by Alan Chang
 * Beginner's Guide (unknown author)
 * Beginners' solution (translated into multiple languages) by Jasmine Lee
 * Beginners' solution (unknown author, requires purchase for the later steps)
 * Simplest method / Beginners' solution (How to Solve Rubix Cube)

Using video:
 * Video tutorial by Tyson Mao.
 * Blogspot Site
 * How to solve a Rubik's Cube at the official Rubik's Cube website

Text only:
 * Beginners Solution (text) by Mark Jeays

Alternative methods

 * Solving the Rubik's Cube for Speed, a block method by Lars Petrus
 * Solving the Rubik's Cube A corners first method by Matthew Monroe
 * Ultimate solution to the Rubik's Cube An edges-first method by Philip Marshall, requiring the memorization of only 2 algorithms and requiring an average of only 65 moves to solve.
 * The Single Alg Cube Solution and the Y-Move Method are beginner methods that each rely on only a single, four-turn algorithm.

Solving Programs

 * Rubik's Cube Solver App
 * Automatic cube solver
 * Cube Explorer 4.10 – a fast program for finding optimal or near optimal solutions to the cube (less than 20 moves total!).

Background on the mathematics

 * Wikipedia article: Optimal solutions for Rubik's Cube

Related puzzles

 * Rubik Puzzles describes other puzzles designed by Erno Rubik

title: How to Solve the Rubik's Cube


title: User:Ral315


text:
Back Pain

Millions of people suffer back pain. This book is a self-help guide to dealing with back pain, written by people who have experienced back pain themselves and by doctors who specialise in the subject. Particular emphasis is placed on the experience people have had with different treatment options.

Authors
Dr Gordon Cameron: Dr Cameron is a specialist in joint pain and back pain. He is based in Edinburgh, Scotland.

Table of Contents

 * /Introduction/
 * /Causes/
 * /Treatment/
 * First Aid Treatment for Back Pain
 * Behavior Based Treatment
 * Drug Based Treatment
 * Surgery Based Treatment

title: Back Pain


text:
Back Pain/Treatment-Behavior

Home Exercises and Stretching
A highly recommended book to start with is Robin McKenzie's "7 Steps to a Pain-Free Life: How to Rapidly Relieve Back and Neck Pain". These exercises don't help everybody, but for a sizable portion of the population they do a great deal of good.

Also, consult Pete Egoscue's "Pain Free: A Revolutionary Method for Stopping Chronic Pain". You can also receive expert advice from Postural Restoration Therapists and Posture Alignment Specialists (PAS). See egoscueuniversity.com and posturalrestoration.com

Other Behavior Modifications
Many people with lower back pain find that bad posture while sitting contributes to their pain. A good chair can help alleviate this problem, but a cheaper and more universal solution is a lumbar support roll. These come in many forms, from the home-made to professionally designed solutions sold in most back-pain stores.

A lumbar roll is placed just above the belt line, while sitting up straight in a normal chair. It causes your spine to bend forward in the same way it should naturally, but is often lost with age.

Two easy home alternatives to a lumbar roll is a roll of paper towels (Small sizes work best) or a rolled up towel.

title: Back Pain/Treatment-Behavior


text:
Cookbook:Mexican Seasoning

Mexican Seasoning is sometimes just chili powder, though many variations on this theme exist. Make your own mixture ahead of time and store in a small shaker for easy use.

Ingredients

 * ½ cup chili powder
 * 1 tablespoon salt
 * 1–2 teaspoons cumin (seed or powder)
 * 1 teaspoon or more garlic
 * 1 teaspoon or more of the following:
 * Mexican oregano
 * Basil
 * Coriander
 * Onion (powder or granulated)
 * Cinnamon
 * Crushed red pepper
 * Black pepper

Procedure

 * 1) Combine all ingredients.
 * 2) Store in a cool, dry place.

title: Cookbook:Mexican Seasoning


text: User:Ctrl build

title: User:Ctrl build


text: User:Ctrl build/Buttons

title: User:Ctrl build/Buttons


text: Template:User page

title: Template:User page


text:
Talk:Scouting/BSA/Family Life Merit Badge

What does this have to do with Family Life? --216.57.68.190 8 July 2005 00:32 (UTC)


text: User talk:Ctrl build

title: User talk:Ctrl build


text: User:Ctrl build/sandbox

title: User:Ctrl build/sandbox


text:
Cookbook talk:Outdoor Cooking

You know, a section on lightweight backpacking cooking would be nice. --216.57.68.190 8 July 2005 00:48 (UTC)

Safety Rules source
The section on Food Safety Rules uses text adapted from the public domain U.S. Department of Agriculture article Food Safety While Hiking, Camping & Boating.

title: Cookbook talk:Outdoor Cooking