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Real Analysis/Counting Numbers

Ordered Sets
An ordered set is a group of objects with an unambiguous sense of what is bigger. Instead of giving an abstract definition of ordering we will begin with some examples of ordered sets and explore what basic assumptions are necessary to flesh them out. This approach, though not as rigorous, should be more approachable. Our first and most important set is the natural numbers.

Natural Numbers
The basic set of analysis is the natural numbers $$\mathbb N=\{1,2,3,\ldots\}$$ (Some authors take $$\{0,1,2,\ldots\}$$ — when we wish to refer to this set, we use $$\mathbb{N}_0$$). The natural numbers are all you need for counting. This set is defined by its properties. The first property of the set of natural numbers $$\mathbb{N}$$ is that it has an equivalence relation $$=\ $$ meaning the following axioms are satisfied: These terse mathematical statements can be written in a less rigorous way. The first statement simply states that every natural number is equal to itself. The second states that the statement of equality is true regardless of the order you say it. The last statement says that when two natural numbers are equal and one of them equals something else they all three must be equal. These are the simple assumptions we make when we loosely talk about equality. This short list gives us a way to check if a proposed equality satisfies our notions of what it means for two things to be equal. Associated with this equivalence relation is an ordering meaning these additional axioms are satisfied: Trichotomy means that any two natural numbers are either equal or the bigger number can be uniquely chosen. Transitivity says that, if there is a third number that is bigger than the biggest of our first two, then it is bigger than the smaller one as well. With this we now have a concise definition of what it means for our numbers to have an order. Finally the natural numbers $$\mathbb{N}$$ have an associated operation called addition. The set $$\mathbb{N}$$ and the operation of addition $$+$$ satisfy the following axioms: Meaning if we add two natural numbers the result is a natural number. The order in which we add the numbers is not important and if I add two natural numbers the sum is bigger than either of the ones I started with. This is our notion of addition of positive numbers simplified to the basic assumptions. There is only one more assumption needed to bring the natural numbers as we know them into a well defined existence, the set of natural numbers is not empty. We can name the smallest element of $$\mathbb{N}$$ the number 1.
 * 1) Reflexivity
 * For all $$n\in\mathbb{N},\ n=n;$$
 * 1) Symmetry
 * For all $$n,m\in\mathbb{N},\ n=m$$ if and only if $$m=n$$;
 * 1) Transitivity
 * For all $$n,m,l\in\mathbb{N}$$ if $$n=m$$ and $$m=l$$, then $$n=l$$;
 * 1) Trichotomy
 * For all $$m,n\in\mathbb{N}$$, one and only one of the following hold:
 * $$mn$$
 * The notation $$m\le n$$ means $$mn$$ or $$m=n$$.
 * 1) Transitivity of < and >.
 * For all $$n,m,l\in\mathbb{N}$$, if $$n< m$$ and $$m\le l$$, then $$n< l$$.
 * For all $$n,m,l\in\mathbb{N}$$, if $$n> m$$ and $$m\ge l$$, then $$n> l$$.
 * 1) Closure
 * For all $$n,m\in\mathbb{N},n+m\in\mathbb{N}$$.
 * 1) Commutativity
 * For all $$n,m\in\mathbb{N},\ n+m=m+n$$.
 * 1) Associativity
 * For all $$n,m,l\in\mathbb{N},\ n+(m+l)=(n+m)+l$$.
 * Meaning we can unambiguously write $$n+m+l$$
 * 1) Compatibility with Ordering
 * For all $$n,m\in\mathbb{N},\ n<n+m$$
 * There exists a number $$1\in\mathbb{N}$$ so that $$1\le n$$ for all $$n\in\mathbb{N}$$

This statement simply says one exists and is less than or equal to any other natural number. With these assumptions we can go on and derive all the properties of $$\mathbb{N}$$.

Multiplication
On the natural numbers we can define a second operator, multiplication $$\times$$. The concept of multiplication on $$\mathbb{N}$$ is simply a shorthand for repeated addition. Here are the axioms of multiplication. Meaning natural numbers when multiplied give natural numbers. Any number times 1 gives itself. The order and the grouping is not important for multiplication. The last two give how multiplication behaves with respect to addition and the ordering on $$\mathbb{N}$$. Always writing $$n\times m$$ for multiplication is tedious so we abbreviate it as $$nm\ $$. Furthermore we abbreviate a repeated product of a particular number with a superscript counting the number of times the number is repeated. For example,
 * 1) Closure
 * For all $$ n,m\in\mathbb{N},\ n\times m\in\mathbb{N}$$.
 * 1) Identity
 * For all $$n\in\mathbb{N},\ n\times 1=n$$.
 * 1) Commutativity
 * For all $$n,m\in\mathbb{N},\ n\times m=m\times n$$.
 * 1) Associativity
 * For all $$n,m,l\in\mathbb{N},\ (n\times m)\times l=n\times (m\times l)$$,
 * meaning we may unambiguously write $$n\times m\times l$$.
 * 1) Distributivity
 * For all $$n,m,l\in\mathbb{N},\ n\times (m+l)=n\times m+n\times l$$.
 * 1) Compatibility with ordering
 * For all $$n,m\in\mathbb{N},\ m\le n\times m$$.
 * $$n\times n\equiv n^{2}.$$

Peano Axioms
An alternate derivation of the set of natural numbers can be characterized by a few axioms called the Peano or Dedekind–Peano axioms. A slight modification of the definitions of addition and multiplication in the Peano axioms would construct a different set where the element "0" (soon to be described) can actually be some natural number different from 0. These axioms can thus serve as the definition of the set of natural numbers.
 * 1) $$\exists0\in\mathbb{N}_0$$
 * There exists an element called "0" within the set of natural numbers.
 * 1) $$\forall x\in\mathbb{N}_0\ \exists S_x\in\mathbb{N}_0$$
 * Every natural number has a successor which is also a natural number.
 * 1) $$\forall x\in \mathbb{N}_0\ S_x\not=0$$
 * There is no natural number whose successor is 0.
 * 1) $$\forall x,y\in\mathbb{N}_0\ S_x=S_y\implies x=y$$
 * The successor function is one-to-one.
 * 1) $$\left(0\in A \land\left(x\in A\implies S_x\in A\right)\right)\implies\mathbb{N}_0\subset A$$
 * Mathematical induction: if 0 is within the set, and n being within the set implies that its successor is also within the set, then all natural numbers are within the set.

A natural number can be defined as an element of the set of natural numbers. These axioms can be used to prove very important basic theorems about basic operations and predicates, which are addition, multiplication, and order. Addition and multiplication are very important binary operators, and ordering is a very important binary predicate. They can be defined as follows:
 * Ordering
 * $$S_y > y\ $$
 * The successor of a number y is greater than the number y.
 * $$x >y \implies S_x>y$$
 * If a number x is greater than a number y then the successor of x is greater than y.
 * Addition
 * $$x+0=x\ $$
 * The sum of any number and zero returns the number.
 * $$x+S_y=S_{x+y}\ $$
 * The sum of a number x and the successor of a number y is the successor of the sum of x and y.
 * Multiplication
 * $$x\ 0=0$$
 * The product of any number and zero is zero.
 * $$x\ S_y=x+xy$$
 * The product of a number x and the successor of a number y is the sum of x and the product of x and y.

It is not necessary to know these definitions to know what addition, multiplication, and ordering mean. However, these definitions show how it is possible to construct addition, multiplication, and ordering from the Peano axioms.

Alternative constructions of addition and multiplication
The basic element does not necessarily have to be an identity for addition. Indeed, if the definitions of addition and multiplication were defined differently then the basic element is commonly written "1".

 * 1) $$\exists 1\in\mathbb{N}$$
 * 1 is an element of the set.
 * 1) $$\forall x\in\mathbb{N}\implies (x+1)\in\mathbb{N}\ $$
 * For any natural number x the sum of x and 1 is also a natural number.
 * 1) $$\forall x\in\mathbb{N}\ x+1\ne x$$
 * The sum of any natural number x and 1 is different from x.
 * 1) $$\forall x,y\in\mathbb{N}\ x+1=y+1\implies x=y$$
 * Adding one is a one-to-one function.
 * 1) $$(1\in A\land(x\in A\implies (x+1)\in A))\implies \mathbb{N}\subset A $$
 * Mathematical induction: If 1 is an element of a set A, where for any element x of A the sum of x and 1 is also in the set, then all the natural numbers are in the set A.

Zermelo Fraenkel Axioms
The natural numbers can also be constructed from sets. This is not a necessary step, and may be skipped, but it shows that set theory provides a sufficient basis for explaining the natural numbers. The Zermelo Fraenkel axioms provide sufficient conditions for a set which satisfies the Peano axioms.

 * 1) Let 0 be associated with the empty set $$\empty$$.
 * 2) Let the set $$S_n=n\cup\{n\}$$
 * The successor of a natural number n is the union of the set associated with n and the set containing n. Thus, each natural number is a set containing its predecessors.
 * For example, let 0 := $$\empty$$, then 1 := $$S_0=0\cup\{0\} = \empty \cup\{0\} $$ = {0}
 * $$ 2:= S_1=1\cup\{1\} $$ = {0}∪{1} = {0} ∪ = {0, {0}} = and hence {0,1}, 3 is {0,1,2}, and so on.
 * 1) $$\exists\mathbb{N}_0:\empty\in\mathbb{N}_0\land\left(\forall x\in\mathbb{N}_0\ x\cup\{x\}\in\mathbb{N}_0\right)$$
 * There is a set called $$\mathbb{N}_0$$ containing $$\empty$$ (the null set) and such that for any element x, the set $$x\cup\{x\}$$ is also within the set. This axiom is called the axiom of infinity and the set so constructed is identified with the natural numbers.

This construction of the natural numbers obviously satisfies the first three Peano axioms. The fact that Sx=Sy→x=y holds can be seen easily by the fact that $$x\cup\{x\}=y\cup\{y\}\implies x=y$$. The fifth Peano axiom holds because if $$\empty\in A\land\left(\forall x\in A\ x\cup\{x\}\in A\right)$$, then this set would be the natural numbers, and so the natural numbers would be a trivial subset of A.

Integers
If we extend $$\mathbb N$$ by adding $$0,$$ we can get the notion of an identity for addition, given by:
 * $$\exists 0:\forall n\in\mathbb{N},\ n+0=n,$$

stating that there is a number $$0$$, which, when added to a natural number, gives that number back again. Here we have a choice to make: where in our ordering does $$0$$ fit? The usual choice is $$0<1$$ and is the one we will make here, though it is worth pointing out the curious nature of this choice. Having defined zero, the possibility of an inverse of $$\mathbb{N}$$ arises. We denote the set of inverses as $$\mathbb{-N}$$, and the set satisfies the following axiom:
 * $$\forall n\in\mathbb{N},\ \exists (-n)\in\mathbb{-N}:n+(-n)=0.$$

Combining all three we obtain the integers
 * $$\mathbb{Z}=\{\mathbb{N}\}\cup\{0\}\cup\{\mathbb{-N}\}=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}$$.

The integers allow us to keep track of debts as well as count things, in other words to perform accounting. If you assume that the axioms for addition hold where the well ordering assumes takes the form
 * $$(\forall z\in\mathbb{Z})(\forall n\in\mathbb{N}),\ z<z+n,$$

and that the identity, closure and distributivity of multiplication hold for the integers $$\mathbb{Z},$$ then the multiplication operation can also be expanded to include all the integers $$\mathbb{Z}.$$

They can be constructed from the natural numbers easily. They can each be equivalence classes of ordered pairs (a,b) where a and b are both natural numbers. Then one can say that (a,b) and (c,d) are equal when a+d=b+c, the sum (a,b)+(c,d)=(a+c,b+d) and the product (a,b)(c,d)=(ac+bd,ad+bc) where the definitions of the sum and product of natural numbers are used. There is an additive identity element in the form (a,a) because (a,a)+(b,c)=(a+b,a+c) which is equivalent to (b,c) because a+b+c=a+b+c. All such elements of the form (a,a) are obviously equivalent. All elements (a,b) have the additive inverse (b,a) because (a,b)+(b,a)=(a+b,a+b). The best way to think of these ordered pairs (a,b) is to think of them as a-b. Thus, (a,a) can be thought of as "0."

Absolute Value
Typically, we think of the integers as extending to positive and negative infinity. Since this geometric notion is so fundamental to our understanding, we should like to talk about geometric properties. In particular, we need to know what is meant by the distance between two integers. To this end, we define the symbol $$|x|$$ as the function that gives the distance from $$x$$ to zero by mapping $$\mathbb{-N}$$ onto their respective inverses in $$\mathbb{N}$$ and mapping zero onto itself.
 * $$ |x| =\left\{\begin{matrix}x & \mbox{if }x \geq 0\\-x & \mbox{if }x < 0 \\\end{matrix}\right.$$

We can now define the distance between two integers, which we refer to as points in any geometric context, by taking the absolute value of their difference: $$d(x,y) = |x-y|\ $$. This distance function satisfies some nice, geometric properties: Pay special attention to the triangle inequality, as it will be used very often in later chapters.
 * 1) Positivity
 * $$d(x,y) \geq 0$$ and is equal to 0 iff $$x=y$$.
 * 1) Symmetry
 * $$d(x,y)=d(y,x).$$
 * 1) Triangle Inequality
 * $$d(x,y) + d(y,z) \geq d(x,z).$$

In general, any set with a distance function satisfying these properties is called a metric space. It is easy to show that the integers form a metric space under the metric d: Another fundamental property of the absolute value is that it is multiplicative:
 * 1) Positivity
 * 2) * If $$x \geq 0$$, then $$|x| = x \geq 0 $$.
 * 3) * If $$x < 0\ $$, then $$|x| = -x\ > 0 $$.
 * 4) Symmetry
 * 5) * If $$x - y \geq 0$$, then $$y - x \leq 0$$, so $$|x- y| = x - y = -(y-x) = |y-x|\ $$.
 * 6) * If $$x - y\ < 0$$, then $$y - x > 0\ $$, so $$|x- y| = -(x-y) = y-x = |y-x|\ $$.
 * 7) Triangle Inequality
 * 8) * If $$x \geq 0$$, then $$|x| = x \geq 0 > -|x|$$.
 * 9) * If $$x < 0\ $$, then $$|x| > 0 > x = -|x|\ $$.
 * 10) * This gives us $$-|x| \leq x \leq |x|$$ and $$-|y| \leq y \leq |y|$$.
 * 11) * Adding, we see that $$-(|x|+|y|) \leq x + y \leq |x| + |y|$$.
 * 12) * If $$x+y \geq 0$$, then $$|x + y| = x + y \leq |x| + |y|$$.
 * 13) * If $$x + y < 0\ $$, then $$|x + y| = -(x+y) \leq |x| + |y|$$.
 * 14) * Thus in all cases $$|x+y| \leq |x| + |y|$$.
 * 15) * Replacing $$x\ $$ with $$x-y\ $$ and $$y\ $$ with $$y-z\ $$ yields $$|x - y| + |y - z| \geq |x - z|$$.
 * $$|x||y| = |xy|.$$

The proof is left as an exercise. As above, it is simply a matter of checking all the cases.

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

title: Real Analysis/Counting Numbers


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Real Analysis/Introduction

Introduction
The subject of real analysis is concerned with studying the behavior and properties of functions, sequences, and sets on the Real number line, which we denote $$\mathbb{R}$$. Concepts that we wish to examine through real analysis include properties like Limits, Continuity, Derivatives (rates of change), Integration (amount of change over time). Many of these ideas that, on a conceptual or practical level, are dealt with at lower levels of mathematics, including a regular First-Year Calculus course, and so to the uninitiated reader the subject of Real Analysis may seem rather shallow and trivial. However, the depth and complexity (and arguably the beauty) of Real Analysis is that we wish to generalize these properties away from the "nice" functions and sets dealt with in everyday mathematics, and rigorously prove these properties for all objects in the universe of the Real numbers. Thus real analysis can to some degree be viewed as a development of a rigorous, well-proven framework to support spatial and conceptual ideas that we frequently take for granted.

Real Analysis is a very straightforward subject, in that it is simply a nearly linear development of the ideas mentioned above. However, as the object of Real Analysis is to make things that we may already "know" more rigorous and definite, we cannot begin our development on unproven assumptions. Thus the approach we take to set this matter straight is to define the real numbers axiomatically. In layman's terms, we set down the properties which we think define the real numbers. We then prove from these properties and these properties only that the real numbers behave in the way which we have come to understand that regular objects in space behave. Lastly, we build a number system, and show it satisfies these properties.

From that foundation, we will develop first a set of results about the real number line, and then move on to results on a two-dimensional plane of real numbers, and then eventually we will generalize many of our results into n-dimensions.

Note: A table of the math symbols used below and their definitions is available here.

title: Real Analysis/Introduction


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Quantum Mechanics/Complex Waves

In quantum mechanics we are interested in solutions to the Schrödinger equation that are renormalizable. One class of functions that does this is the complex exponentials, and we can write a solution to the Schrödinger equation as the sum of two complex exponentials $\Psi(x)= Ae^{ikx}+Be^{-ikx}$

which is the superposition of waves, one traveling from the left, the other from the right. k is the wave number which is in units of 1/m and is usually given by $$2*\pi /L$$ where L is determined by boundary conditions.

Let us consider a situation in which there is a wave traveling from the left to the right. On the interval (-$$\infty$$,0) V(x)E(x) and the solutions of the Schrödinger Equation are of the form $\Psi(x)= Ae^{i\sqrt{\frac{2m}{\hbar^2}(E(x)-V(x))}x}+Be^{-i \sqrt {\frac{2m}{\hbar^2}(E(x)-V(x))} x}$ on the first interval and $\Psi(x)= Ce^{\sqrt{\frac{2m}{\hbar^2}(E(x)-V(x))}x}+De^{- \sqrt {\frac{2m}{\hbar^2}(E(x)-V(x))}x}$ on the second interval

Even though at first glance our wave equation looks like a superposition of normal exponentials, it is still a complex wave though because the term inside of the square root is negative and this allows us to always have renormalizable solutions.

title: Quantum Mechanics/Complex Waves


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MATLAB Programming/Examples of MATLAB Command

Introduction
MATLAB is interesting in that it is dynamically compiled. In other words, when you're using it, you won't run all your code through a compiler, generate an executable, and then run the executable file to obtain a result. Instead, MATLAB simply goes line by line and performs the calculations without the need for an executable.

Partly because of this, it is possible to do calculations one line at a time at the command line using the same syntax as would be used in a file. It's even possible to write loops and branches at the command line if you want to. Of course this would often lead to a lot of wasted efforts, so doing anything beyond very simple calculations, testing to see if a certain function, syntax, etc. works, or calling a function you put into an .m file should be done within an .m file.

Examples
MATLAB can perform the functions of a simple calculator from the command line. We go thru some of the common Maths problem in the real life.

Here are few of mathematics problem that are demonstrated to be solved in MATLAB:

Painting Coverages
A house painter usually uses 10L of white paint to cover 120 square meter on average for single layer of coating.

Calculate how many 10L paint cans that house painter needed to buy to paint a room with size of 13m X 9m with height 5m from floor to ceiling.

Also the room have 2 windows with size of 1.5m*0.75m and 2m*1.25m and a door with size of 1.2m*3m

Earth to Sun Distance
Distances from sun to earth is equivalent to 150 million km (150,000,000 km)

If human launched a advance rocket capable of achieving constant speed of 7km/s (ignore all of the air frictions, gravitational pull), how many year(s) does the rocket to reach the sun from earth ?

Dice Roll for board games
You invite some friends come over to play a board game but somehow the dice that should be comes with board game is missing. Create a MATLAB programs to throw two dices (with 6 faces)

You need to use randi command to generate random numbers. More command can be learnt here: MATLAB Programming/Fundamentals of MATLAB/MATLAB operator

External Resources
ControlTheoryPro.com


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Cantonese/Lesson 9

Time on a clock

 * For telling time, 兩 is used instead of 二 for the hour, use 二 for the minute.
 * The Cantonese for "o'clock" is 點鐘. Study the following examples:

 * The Cantonese for "half past" is 點半 (and 半 means "half of"). Study the following examples:

 * The time not exactly "o'clock" can be said with the format "x點y分". Study the following examples:

title: Cantonese/Lesson 9


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User talk:Tommciver

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Thanks
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title: User talk:Tommciver


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Real Analysis/Rational Numbers

Before we can build up the notion of an ordered field, we first need some basic concepts from algebra.

Groups
Groups play an important role in mathematics. They describe the most basic structures in algebra. The subject of group theory studies the nature and structure of general groups. In this book we will mostly be concerned with groups you are already familiar with, so this section is just to set some standard terminology. We begin by defining a binary operation, which is often thought of as either multiplication or addition, depending on the context. To avoid confusion we will denote this operation by * while discussing general groups, but in specific cases we will generally use + or ·

Definition: A binary operation on a set S is a function from S×S→S

Definition: A group is a set G together with a binary operation on G that satisfies the following axioms.
 * G is closed under the binary operation. That is, for all x, y in G, x*y is in G.
 * The binary operation is associative. That is, for all x, y, and z in G,  x*(y*z)=(x*y)*z.
 * There exists an identity element, which we denote by e, that satisfies e*x=x*e=x for all x in G.
 * For all x in G there exists an inverse element, which we denote by x-1, so that x*x-1=x-1*x=e.

Examples
 * The integers $$\mathbb Z$$ together with the binary operation of addition are a group.
 * The rational numbers $$\mathbb Q$$ with the binary operation of addition are a group.
 * The non-zero rational numbers $$\mathbb{Q}\setminus\{0\}$$ with the binary operation of multiplication are a group.
 * The set $$\mathbb{Z}$$ together with the binary operation of multiplication is not a group.
 * The set $$\{-1,1\}$$ with the binary operation given by multiplication is a group.
 * The set $$\{e,o\}$$ with a binary relation given by $$e+e=e$$, $$e+o=o$$, $$o+e=o$$, and $$o+o=e$$ is a group. If one thinks of $$e$$ as a shorthand for even, and $$o$$ as a short hand for odd, these are the familiar rules from childhood "An even number plus an even number is again an even number", etc.

It is often useful to talk about when two groups are basically the same. It may happen that two groups have a different underlying set, and have a different binary operation, but behave exactly the same algebraically. When this happens the two groups are called isomorphic.

Definition The groups (G,*) and (H,⊗) are said to be isomorphic if there is a bijective function φ:G→H that satisfies the following two properties:
 * φ(eG)=eH, where eG is the identity element in G and eH is the identity element in H;
 * φ(x*y)=φ(x)⊗φ(y) for all x and y in G.

A Field
The set of integers $$\mathbb{Z}$$ and the operation of addition $$+\ $$ form a group, multiplication $$\times$$ lacks inverses. If we allow multiplication and addition to operate on $$\mathbb{Z}$$ we can define a set where every element except zero has a multiplicative inverse. This is the set of rational numbers.

Rational Numbers
The next standard extension adds the possibility of quotients or division, and gives us the rational numbers (or just rationals) $$\mathbb Q$$, which includes the multiplicative inverses of $$\mathbb{Z}\setminus\{0\}$$ of the form $$\frac{1}{z}$$ fractions such as $$\frac{1}{2}$$, as well as products of the two sets of the form $$\frac{z_1}{z_2}$$ such as $$\frac{64}{7}, \frac{17}{16\times 10^5}$$. The rationals allow us to use arbitrary precision, and they suffice for measurement.

The rational numbers can be constructed from the integers as equivalence classes of order pairs (a,b) of integers such that (a,b) and (c,d) are equivalent when ad=bc using the definition of multiplication of integers. These ordered pairs are, of course, commonly written $$\tfrac{a}{b}$$. One can define addition as (a,b)+(c,d)=(ad+bc,bd) and multiplication as (ac,bd) all using the definition of addition and multiplication of integers.

title: Real Analysis/Rational Numbers


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Real Analysis/Properties of Real Numbers

Some simple results
At this point there are a large number of very simple results we can deduce about these operations from the axioms. Some of these follow, and some of them have proofs. The remaining proofs should be considered exercises in manipulating axioms. The aim of these results is to allow us to perform any manipulations which we think are "obviously true" due to our experience with working with numbers. Unless otherwise quantified, the following should hold for all $$x,y\in\mathbb R$$.

 * $$0$$ is the only additive identity

 * Proof. Suppose $$x$$ is an additive identity, s.t. $$x \neq 0$$. Then, let $$y \in \mathbb{R}$$. Then, $$x + y = 0 + y$$, and this leads to $$x = 0$$. However, since we assumed that $$x \neq 0$$, this leads to a contradiction, therefore $$0$$ is the only additive identity. $$\Box$$

 * $$1$$ is the only multiplicative identity

 * Both additive and multiplicative inverses are unique. More formally: If both $$x+y=0$$ and $$x+z=0$$ then $$y=z$$; and if both $$xy=1$$ and $$xz=1$$ then $$y=z$$ (so that the notations $$-x$$ and $$x^{-1}$$ make sense).

 * Proof: For the case of addition: We have $$x+y=0$$ and $$x+z=0$$, so adding $$y$$ to the latter equation, we get $$(x+z)+y=0+y$$, but then by commutativity and associativity of addition we deduce that $$(x+y)+z=0+y$$, and by our other assumption $$0+z=0+y$$, and then by identity of addition $$z=y$$. $$\Box$$

 * $$-(-x)=x$$

 * $$\forall x\in\mathbb R\setminus\{0\}:(x^{-1})^{-1}=x$$

 * $$0\times x=0$$

 * $$0$$ has no multiplicative inverse (so division by $$0$$ can not make sense)

 * $$\forall n,m\in\mathbb Z:x^nx^m=x^{n+m}$$

 * $$\forall n,m\in\mathbb Z:(x^n)^m=x^{nm}$$

 * $$x>y\iff\neg x\leq y$$ (Here $$\neg$$ is logical negation, so $$\neg x\leq y$$ means "it is not the case that $$x\leq y$$".)

 * Proof: First we consider the implication $$\implies$$. Suppose $$x>y$$.  By definition, this means that $$x\not=y$$ and $$yy$$. $$\Box$$
 * Conversely, suppose $$\neg x\leq y$$. First, if we had $$x=y$$ then by reflexivity $$x\leq y$$, which is impossible, so in fact $$x\not=y$$.  Secondly, by totality we deduce that $$y\leq x$$.  These two conditions are exactly those required for $$x>y$$. $$\Box$$

 * $$x<y\iff\neg x\geq y$$

 * $$x$$ is non-positive if and only if $$x$$ is not positive

 * $$x$$ is non-negative if and only if $$x$$ is not negative

 * If $$x$$ is both non-positive and non-negative then $$x=0$$

 * $$x$$ is not both positive and negative

 * $$x\geq0\iff -x\leq0$$

 * Proof: Suppose $$x\geq0$$. By one of the axioms we get $$x+(-x)\geq0+(-x)$$.  By additive inverse this gives $$0\geq0+(-x)$$ and then by additive identity $$0\geq-x$$, as required.
 * The converse implication follows similarly. $$\Box$$
 * The converse implication follows similarly. $$\Box$$

 * $$(x\leq y\mbox{ and }z\leq0)\implies xz\geq yz$$

 * $$\forall x\in\mathbb R:x^2\geq 0$$

 * Proof: By totality of the order, we have either $$x\geq0$$ or $$x\leq0$$. In the first case we can apply the axiom linking the order to multiplication directly to $$0\leq x$$ and deduce $$0\leq x^2$$.  In the latter case we apply the last result in this list to $$0\leq x$$ and obtain $$x^2\geq0$$. $$\Box$$

 * $$1>0$$ and $$-1<0$$

Applications
Although it might be said that the entirety of this book is devoted to studying the applications of completeness, there are in particular some simple applications we can give easily which provide an indication as to how completeness solves the problem with the rationals described above.

Theorem (Square roots)
Let $$x\in\mathbb R$$ be non-negative. Then $$x$$ has a unique non-negative square root, denoted $$\sqrt{x}$$, which satisfies $$(\sqrt{x})^2=x$$.

Proof
We deal only with the case $$x\geq 1$$. The case $$x\in[0,1)$$ is left for the exercises.

First we note that when $$y,z\in\mathbb R$$ are non-negative, $$y<z\implies y^2x\geq 1$$, then $$y^2>y$$, so that $$y^2>x$$, and hence $$y\not\in S$$.

Putting these facts together, by the least upper bound axiom, we deduce that $$S$$ has a least upper bound, which we call $$s$$. We wish to show that $$s$$ is the square root of $$x$$ that we seek.

Certainly $$s$$ is positive, since $$1\in S$$ and so $$s\geq 1$$. In particular, we may divide by $$s$$.

To show that $$s^2=x$$, we eliminate the possibilities that $$s^2>x$$, and that $$s^2x$$. Let $$t=s-\frac{s^2-x}{2s}$$. Then:

$$t^2 = s^2 - (s^2-x) + \frac{(s^2-x)^2}{4s^2} = x + \frac{(s^2-x)^2}{4s^2} > x$$

So $$t$$ is in fact an upper bound for $$S$$, but this is impossible, since $$ts$$ and $$s$$ is an upper bound for $$S$$.

Thus we have concluded that $$s^2\geq x$$, and so $$s^2=x$$ as required.$$\Box$$

This argument may appear excessively complex (especially since some details are left for the exercises), and indeed there is a sense in which it is, and we shall be able to present a much neater argument later. Nevertheless, it suffices to show that we can find a square root of 2, and so avoid the immediate problem with the rationals posed at the beginning of this section. To show that no more elaborate construction will give rise to the same problem will have to wait until we reach the study of continuity.

Theorem (Archimedes axiom)
(Note that despite the name, this theorem is not an axiom to us, but a theorem we deduce from the other axioms.)

a) $$\forall x\in\mathbb R:\exists n\in\mathbb N:n>x$$

b) $$\forall x\in\mathbb R^*+:\exists n\in\mathbb N:\frac{1}{n}l-1$$. But then, $$n+1>l$$, and $$n+1\in\mathbb N$$ so we get the contradiction that $$l$$ is not an upper bound for $$\mathbb N$$ after all.

Thus, our supposition was false, and (a) holds.

b) Take $$x\in\mathbb R^+$$. Certainly $$x\not=0$$, so that we can invert $$x$$ to get $$x^{-1}\in\mathbb R^+$$.  Applying part (a) to $$x^{-1}$$, we can find $$n\in\mathbb N$$ with $$n>x^{-1}$$, and then inverting this inequality, we deduce $$\frac{1}{n}yn+2$$.

Now choose the least $$m\in\mathbb N$$ satisfying $$N-m xn$$

So, in summary, we have $$yn>N-m>xn$$, so $$y>\frac{N-m}{n}>x$$, and we have found the rational number we want.

To find an irrational number, we use what we have just deduced to first find a rational $$q\in(x+\sqrt{2},y+\sqrt{2})$$, so that $$q-\sqrt{2}\in(x,y)$$. Furthermore, $$q-\sqrt{2}$$ must be irrational, for if it were rational then we would also have $$q-(q-\sqrt{2})=\sqrt{2}$$ rational, and we know that it is not. $$\Box$$

Properties of Least Upper Bounds
We'll be doing a lot of work with least upper bounds, so it will be important to know how to use them efficiently in proofs. Here are some definitions and properties that are helpful in this respect:

Uniqueness of Least Upper Bounds
Every non-empty set that is bounded above has a unique least upper bound

Proof
Let $$a$$ and $$b$$ be two least upper bounds for a set $$S$$.

If $$a>b$$, then since $$b$$ is an upper bound for $$S$$, $$a$$ cannot be the least upper bound. Thus $$a \leq b$$. Similarly, $$a \geq b$$. Thus $$a = b$$, so $$S$$ can have only one least upper bound.

Existence of Greatest Lower Bounds
Every non-empty set S that is bounded below has a unique greatest lower bound, or infimum (denoted $$\inf S$$).

Proof
Let S be non-empty and bounded below. Let $$T := \{-x: x \in S\}$$.

Since S is non-empty, $$\exists x \in S$$. Thus $$-x \in T$$, so T is non-empty.

Since S is bounded below, $$ \exists M : \forall x \in S : x > M$$.

Then $$x \in T \implies -x \in S \implies -x > M \implies x < -M$$.

Thus T is bounded above by M, and therefore T has a least upper bound, $$\beta$$.

Since $$ x \in S \implies -x \in T \implies -x < \beta \implies x > -\beta$$, $$-\beta$$ is a lower bound for S.

Let $$\alpha$$ be a lower bound for S.

Then $$ x \in T \implies -x \in S \implies -x > \alpha \implies x < \alpha$$, so $$-\alpha$$ is an upper bound for T.

Since $$\beta$$ is the least upper bound for T, $$ -\alpha > \beta $$, and thus $$ \alpha < -\beta $$.

Thus all lower bounds for S are less than $$-\beta$$

In other words, $$-\beta$$ is a greatest lower bound for S.

Uniqueness follows similarly to the uniqueness of least upper bounds.

Theorem (Ordering of Sups and Infs)
If $$S \subseteq T$$, where S is non-empty and T is bounded above and below, then $$\inf T \leq \inf S \leq \sup S \leq \sup T$$

Proof
Since S is non-empty, it contains an element x. By definition, $$\inf S \leq x$$ and $$x \leq \sup S $$, so $$\inf S \leq \sup S$$.

Since T is bounded above, it has a least upper bound, $$\sup T$$.

Since t is in particular an upper bound for T, $$\forall x \in T: x \leq \sup T $$. Since $$S \subseteq T$$, $$x \in S \implies x \in T \implies x \leq \sup T$$.

Thus $$\sup T$$ is an upper bound for S, so $$\sup S$$ exists and by definition $$\sup S \leq \sup T$$.

Similarly, $$\inf S \geq \inf T$$.

Sum and Product Notation
We often need to take a sum or product of several real numbers at a time. Since "..." is given no meaning by our axioms, we can't just write "$$a_1 + a_2 + \dots + a_n $$". Thus we use the symbols $$\sum_{k=1}^{n} a_k$$ and $$\prod_{k=1}^{n} a_k $$ to denote the sum and product, respectively, over an arbitrary finite number of real numbers. We do this inductively, as follows:

 * $$\sum_{k=1}^{1} a_k = a_1$$ and $$\prod_{k=1}^{1} a_k = a_1$$
 * $$\sum_{k=1}^{n} a_k = a_{n} + \sum_{k=1}^{n-1} a_k$$ and $$\prod_{k=1}^{n} a_k = a_{n}\prod_{k=1}^{n-1}a_k$$

Now we can prove some properties of sums and products:

Properties

 * The order of summation can be changed arbitrarily. That is, if $$\{a_k: 1 \leq k \leq n\} = \{b_k: 1 \leq k \leq n\}$$, then $$\sum_{k=1}^{n} a_k = \sum_{k=1}^{n} b_k$$ and  $$\prod_{k=1}^{n} a_k = \prod_{k=1}^{n} b_k$$

Proof: This follows from commutativity and a rather nasty induction.

 * $$\sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k = \sum_{k=1}^{n} (a_k + b_k)$$ and $$\prod_{k=1}^{n} a_k \prod_{k=1}^{n} b_k = \prod_{k=1}^{n} (a_k b_k)$$

Proof: We proceed by induction. First, note that $$\sum_{k=1}^{1} a_k + \sum_{k=1}^{1} b_k = a_k + b_k = \sum_{k=1}^{1} (a_k + b_k)$$.

Now assume that $$\sum_{k=1}^{n-1} a_k + \sum_{k=1}^{n-1} b_k = \sum_{k=1}^{n-1} (a_k + b_k)$$. Then $$\sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k$$$$ =\sum_{k=1}^{n-1} a_k + a_n + \sum_{k=1}^{n-1} b_k + b_n$$$$ =\sum_{k=1}^{n-1} a_k + \sum_{k=1}^{n-1} b_k + a_n + b_n$$$$ = \sum_{k=1}^{n-1} (a_k + b_k) + (a_n + b_n)$$$$ = \sum_{k=1}^{n} (a_k + b_k)$$.

The statement for products follows similarly.

 * $$ c\sum_{k=1}^{n}a_k = \sum_{k=1}^{n}c a_k$$

Proof: Another induction. For $$n=1$$, $$c\sum_{k=1}^{1}a_k = c a_1 = \sum_{k=1}^{1}c a_k$$. Now assume the statement is true for n-1. Then $$c\sum_{k=1}^{n}a_k = c(\sum_{k=1}^{n-1}a_k + a_n) = \sum_{k=1}^{n-1}c a_k + c a_n = \sum_{k=1}^{n}c a_k$$.

a_k b_l$$
 * $$\sum_{k=1}^{n}(a_k) \sum_{l=1}^{m}(b_l) = \sum_{k=1}^{n} \sum_{l=1}^{m}

Proof: We induct on n. The previous property takes care of the case n=1. Assume the statement is true for n-1. Then $$\sum_{k=1}^{n}(a_k) \sum_{l=1}^{m}(b_k) $$$$= (\sum_{k=1}^{n-1}(a_k) + a_n)\sum_{l=1}^{m}(b_k) $$$$= \sum_{k=1}^{n-1}(a_k) \sum_{l=1}^{m}(b_k) + a_n \sum_{l=1}^{m}(b_k) $$$$= \sum_{k=1}^{n-1} \sum_{l=1}^{m}(a_k b_l) + \sum_{l=1}^{m}(a_n b_k) $$$$= \sum_{k=1}^{n} \sum_{l=1}^{m}(a_k b_l)$$

Most familiar properties of sums and products can be deduced by similar methods.

Análise real/Índice/Propriedades dos números reais


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Animal Care/Land hermit crab cleaning

Each new land hermit crab assessory should be cleaned before use. To avoid attracting insects, to avoid mold/fungus, and to avoid odors, the entire habitat must be cleaned regularly.

Water dishes should be emptied and refilled every 24 to 48 hours (bubblers can increase that time). Swap out the sponges on a weekly basis. Rinse the sponge well in dechlorinated water, squeeze it dry, let it sun dry completely, and then microwave for 45 seconds to sanitize it.

Feces or spilled food on the surface should be scooped up on a daily basis. On a monthly or bi-monthly basis, completely change the substrate (small habitats) or just remove the top 0.5 inch of substrate (larger habitats). In the latter case, add more substrate when the level gets too low or change the substrate at that time.

Do not clean the habitat or its assessories with anything stronger than vinegar and water (rinse well). Soaking in salt water can remove a lot of buildup from assessories and kill mold. Sea salt and baking soda can be used for scrubbing, but may scratch plexi-glass.

Depending on the materials contained in an assessory, baking, boiling, steaming, or microwaving can be used for sterilization. When microwaving wood, put it in a slightly ventilated, ziplock bag with some water to avoid fires. Diluted, pure vanilla essence has an antibacterial effect and will add a pleasant smell.

To reuse substrate, rinse, stir, and pour off the dirty water until the water is clear. To kill mold and fungus, completely dry the substrate in the sun or oven bake thin layers (0.5") at 150°C (300°F) until dry (20–30 minutes or more). Oven baking at this temperature is sufficient to kill any mites. Alternating between two sets of substrate simplifies this whole process.

CAUTION: Coral is fragile so just pour boiling water over it to sterilize. Some plastic items like exercise balls will melt if microwaved or boiled. Steam those items or allow boiling water to cool a few degrees before pouring it over them.

title: Animal Care/Land hermit crab cleaning


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User talk:Msmithma

Good luck! -- Herby talk thyme 09:22, 13 November 2006 (UTC)

title: User talk:Msmithma


text:
Real Analysis/Authors

Authors
Martín Smith-Martínez

SPat

A

Hypercube

SharingNotes

title: Real Analysis/Authors


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French/Lessons/Candide, (ou l'Optimisme)

Voici un extrait d'un conte de Voltaire, appelé Candide, tiré du texte que vous retrouverez dans Wikisource: Il y avait en Vestphalie, dans le château de monsieur le baron de Thunder-ten-tronckh, un jeune garçon à qui la nature avait donné les mœurs les plus douces. Sa physionomie annonçait son âme. Il avait le jugement assez droit, avec l'esprit le plus simple; c'est, je crois, pour cette raison qu'on le nommait Candide. Les anciens domestiques de la maison soupçonnaient qu'il était fils de la sœur de monsieur le baron et d'un bon et honnête gentilhomme du voisinage, que cette demoiselle ne voulut jamais épouser parce qu'il n'avait pu prouver que soixante et onze quartiers, et que le reste de son arbre généalogique avait été perdu par l'injure du temps.

Monsieur le baron était un des plus puissants seigneurs de la Vestphalie, car son château avait une porte et des fenêtres. Sa grande salle même était ornée d'une tapisserie. Tous les chiens de ses basses-cours composaient une meute dans le besoin; ses palefreniers étaient ses piqueurs; le vicaire du village était son grand-aumônier. Ils l'appelaient tous monseigneur, et ils riaient quand il faisait des contes.

Madame la baronne, qui pesait environ trois cent cinquante livres, s'attirait par là une très grande considération, et faisait les honneurs de la maison avec une dignité qui la rendait encore plus respectable. Sa fille Cunégonde, âgée de dix-sept ans, était haute en couleur, fraîche, grasse, appétissante. Le fils du baron paraissait en tout digne de son père. Le précepteur Pangloss était l'oracle de la maison, et le petit Candide écoutait ses leçons avec toute la bonne foi de son âge et de son caractère.

Rappel: L'imparfait
The above extract uses many verbs in the imperfect tense. This allows Voltaire to set the background to the story, in describing the castle of Thunder-ten-tronck (and its inhabitants) as it was. As well as être and avoir, we can find examples of verbs in the past tense such as « pesait, faisait, rendait, paraissait, appelaient ». Pangloss (enseigner) la métaphysico-théologo-cosmolonigologie. Il (prouver) admirablement qu'il n'y a point d'effet sans cause, et que, dans ce meilleur des mondes possibles, le château de monseigneur le baron (être) le plus beau des châteaux, et madame la meilleure des baronnes possibles
 * Voici deux extraits du même chapitre de Candide. Tous les verbes à l'imparfait sont remplacés par leurs infinitifs, entre parenthèses. Récrire ces extraits en conjuguant ces infinitifs à l'imparfait.

[...]

Candide (écouter) attentivement, et (croire) innocemment; car il (trouver) mademoiselle Cunégonde extrêmement belle, quoiqu'il ne prît jamais la hardiesse de le lui dire. Il (conclure) qu'après le bonheur d'être né baron de Thunder-ten-tronckh, le second degré de bonheur (être) d'être mademoiselle Cunégonde; le troisième, de la voir tous les jours; et le quatrième, d'entendre maître Pangloss, le plus grand philosophe de la province, et par conséquent de toute la terre.

Monsieur était...car...
Justifying your statements is crucial when writing, whether in French or in English. Consider this sentence from the first extract: Monsieur le baron était un des plus puissants seigneurs de la Vestphalie, car son château avait une porte et des fenêtres. The structure of this sentence is: Statement + car + statement. (Remember that car means 'as' or 'because'.) For example: Elle est intelligente, car elle a eu une bonne éducation.
 * Ecrivez cinq phrases en vous servant de ce modèle.


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User talk:203.111.237.44

Please do not replace Wikibooks pages with blank content. Blank pages are harmful to Wikibooks because they have a tendency to confuse readers. If it is a duplicate article, please redirect it to an appropriate existing page. If the page has been vandalised, please revert it to the last legitimate version. If you feel that the content of a page is inappropriate, please edit the page and replace it with appropriate content. If you believe there is no hope for the page, please see the deletion policy for how to proceed. If page is no longer useful, insert delete to its top. Thanks for contributing to Wikibooks! -- Herby talk thyme 09:31, 13 November 2006 (UTC)

title: User talk:203.111.237.44


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French/Lessons/Les Fleurs du Mal

L'albatros
Souvent, pour s’amuser, les hommes d’équipage Prennent des albatros, vastes oiseaux des mers, Qui suivent, indolents compagnons de voyage, Le navire glissant sur les gouffres amers.

À peine les ont-ils déposés sur les planches, Que ces rois de l’azur, maladroits et honteux, Laissent piteusement leurs grandes ailes blanches Comme des avirons traîner à côté d’eux.

Ce voyageur ailé, comme il est gauche et veule ! Lui, naguère si beau, qu’il est comique et laid ! L’un agace son bec avec un brûle-gueule, L’autre mime, en boitant, l’infirme qui volait !

Le Poète est semblable au prince des nuées Qui hante la tempête et se rit de l’archer ; Exilé sur le sol au milieu des huées, Ses ailes de géant l’empêchent de marcher.

Questions de compréhension
1. Pourquoi les hommes d'équipage prennent-ils des albatros? 2. Est-il facile pour les albatros de marcher? 3. Le Poète est comparé au « prince des nuées ». Qui est ce prince ? 4. Où est-il exilé ? 5. Qui l'empêche de marcher ?

Inversion: « À peine les ont-ils déposés sur les planches »
Hardly had they... Notice the curious inversion of subject and verb, in English as well as in French. Instead of the normal order of subject-direct object pronoun (DOP)-verb as we usually do in French, we now have DOP-verb-subject: instead of ils les ont déposés, we have les ont-ils déposés. Of course, this inversion occurs frequently in asking questions in French. However here we find the inversion in an affirmative sentence. There are certain words that, when used at the start of a sentence in French, require this inversion of subject and verb.
 * Quel temps fait-il?
 * Comment vas-tu?

title: French/Lessons/Les Fleurs du Mal


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Physical Activity/Skateboarding X-treme

Skateboarding is a great way to get around, show off to your friends, have fun - and get fit at the same time.

A skateboard is a short, sturdy board, with curved ends on both sides. Generally, one side is narrower than the other. The narrower side is the tail. The other, wider, side is the nose. It doesn't really matter which side is in front when you skate, but I prefer to have the nose in front.

Get a Board
If you want to become a skater, you first have to get a board.

Getting a good board can actually prove to be quite a challenge. From my personal experience, I would recommend getting a skateboard from a skate shop, not a supermarket or the internet. At a supermarket, the only boards they have are toys, and you DO NOT want a toy. At an internet skate shop, the boards might be really good, but you cannot really see them in person. Only get a board from an internet skate store if one of your friends recommends it AND your friend is a good skater. A good board usually cost more and they last longer.

Get the Feel of the Board
Once you have a board, it's time to get used to the feel of it.

Just stand on it at first, and swing back and forth. That will give you the idea of moving. Once you are comfortable with that, try going around your driveway, leaning on your toes to turn in, leaning on your heels to turn out.

Once you are comfortable with riding, notice: which foot you push with, and which foot is in front on the deck. Here is a list of what foot you are:

 * Goofy: you push your board with your left foot, and your right foot is in front
 * Regular: you shove your board along with your right foot, and your left foot is in front

Some people skate mongo, which is when they push with their front foot. I would not recommend that, but I am regular and not mongo. It's what ever you are most comfortable with; in other words, personal preference.

Tick-tack
The "Tick-tack" is sort of a trick, but is mostly a way to turn sharper then just leaning in or out.

This is how you do it:

Center your back foot on the tail (right foot if you are regular, left foot if you are goofy), and put your front foot about 3/4 up the deck. Once you are good at this, you might want to put your front foot farther down the board - maybe halfway down - to get an added spin and more speed.

Now apply slight pressure to your back foot and the nose of the skateboard will rise into the air. Next, move the nose either left or right by moving your body and stop the pressure on your back foot. Than should make the nose drop. Repeat this process, alternating which way you turn. For example, nose movement: left, right, left, right. That will help you gain speed, turn sharp, and say "Wow! I can do a 'trick!'"

title: Physical Activity/Skateboarding X-treme


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Juggling/Combat

Combat is a game played by jugglers.

Basic rules
The basic rules are quite simple:
 * At a signal everyone starts juggling.
 * The last person juggling a full set of balls wins.
 * Holding one ball while juggling the others in one hand is not permitted.

The combative aspect is that players intentionally juggle so as to interfere with each other, as long at there is no actual physical contact between players.

Combat is most often played with three balls per player. Very experienced jugglers may play combat with four or five balls (but the games are shorter), and brave jugglers may play with clubs.

Strategies
Basic strategies include:
 * Moving face-to-face with an opponent so the opponent gets confused about which balls to catch. (Only works on novice players.)
 * Throwing one ball high, and using the time to wave a hand in front of an opponent.
 * Grabbing an opponent's ball out of the air, and either dropping one of your own balls, or continuing with four balls.
 * Grabbing a ball from one opponent and throwing it at another opponent. (Not allowed when playing with clubs.)
 * If the game is played with clubs, then it is possible to hit an opponent's club out of the air with one of your own.
 * Throwing one ball high and far away, so it is possible to safely hold the two remaining balls while running away from an aggressive opponent.
 * And of course, turning away from all opponents and hoping that they will drop a ball before you do.

title: Juggling/Combat


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MediaWiki Administrator's Handbook/Introduction

Sysops, more commonly known on various projects as administrators or custodians are users that have access to restricted software tools used for maintenance. These tools are not available to all editors because the tools can be used to create a rather more serious level of disruption than the generally-available editing tools. It's important that a track record of good-faith editing be established first before granting adminship.

The different Wikimedia projects have different ways of vetting users who request the tools. On most projects this is done through an election process, but new projects (which do not have any bureaucrats, who have use of the "makesysop" tool) need to make the requests on Meta, and at least some projects (such as en.Wikiversity) use a mentoring system rather than open elections.

Getting to know how the tools work is fairly easy, as for the most part they're quite intuitive thanks to our thoughtful developers and years of trial and error. Knowing how and when they should be used is a different issue, since there are a lot of things that should not be done, and each wikimedia project will have different policies and guidelines regarding their use.

The golden rule is simple: never use an administrative tool unless you are sure it's the right thing to do. If in doubt, just leave it to a more experienced administrator, or just ask for advice.


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MediaWiki Administrator's Handbook/Special pages

Special pages are available to all Wikimedia users from a link on the sidebar. Administrators can access some special pages not available to non-administrators:

 * Block user
 * Import pages
 * Unwatched pages
 * View deleted pages


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MediaWiki Administrator's Handbook/Rollback

Rollback quickly reverts all consecutive edits by the most recent contributor to a particular page. The edit summary is automatically set to read "Reverted edit of A, changed back to last version by B", and the edit is marked as "minor".

The rollback tool should only be used to revert blatant vandalism. It should not be used to revert good-faith edits, to resolve content disputes, or to enforce a particular version of an article.

The rollback tool may remove constructive edits, and it may also leave earlier vandalism intact. As a result, after using the rollback tool, sysops should always check recent revisions and ensure that all disruptive edits have been reverted and all constructive edits restored.

Finally, after using the rollback tool, sysops should leave a note on the appropriate User talk: page and explain why rollback was used. Many "vandals" are simply new users who are unfamiliar with the ways of the wiki; by gently correcting them, a good sysop may lead them to contribute positively.

If you are an administrator you can use bot rollback, which will treat rollbacks as bot edits so they don't flood recent changes, good if a vandal bot comes and vandalizes lots of pages. To use this, you need to go to the user's Special:Contributions page and put &bot=1 at the end of the url, like this, http://en.wikipedia.org/w/wiki.phtml?title=Special:Contributions&target=SomePersistentVandal&bot=1. You can then click the rollback buttons on the user's contribution page so that the rollbacks don't flood recent changes.


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MediaWiki Administrator's Handbook/Protect

Protect prevents changes to a page by users below a certain user level.

 * Default: anyone can edit
 * Autoconfirmed: No IP editing, no editing by the newest accounts
 * Sysop: only editable by administrators

Pagemove permissions are "locked" by default to match the edit permissions, but can be unlocked to allow editing without allowing pagemoves, or in theory vice-versa (though there's rarely any reason to set it that way). This is often done to project namespace pages which should be editable by anyone, but should not be moved around.

Protecting the main page does not automatically protect the talk page.

What should be protected depends on the project, but generally:

 * Project and help namespace pages should generally be protected against pagemoves.
 * Policy pages may be protected
 * Widely used templates
 * Frequent vandal or spam targets (the different projects have vastly different rules governing when this should be done)
 * User namespace pages, on request of the user.
 * For a more general discussion, see Protected pages considered harmful (on meta).

Local Policies

 * en.wikipedia -- Wikipedia:Protection policy
 * en.wikibooks -- Semi-protection policy
 * en.wikiversity -- Wikiversity:Page protection templates
 * commons.wikimedia -- Commons:Protected page
 * en.wiktionary -- Wiktionary:Protected page guidelines
 * en.wikisource -- Wikisource:Protection policy
 * en.wikiquote -- Wikiquote:Protected page
 * en.wikinews -- Wikinews:Protection policy

Manual wiki/Anexo/Instalando el software


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User talk:62.232.224.14

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 11:17, 13 November 2006 (UTC)

Welcome to Wikibooks. We invite everyone to contribute constructively to our collection. Take a look at the welcome page if you would like to learn more about contributing. However, unconstructive edits are considered vandalism, and if you continue in this manner you may be blocked from editing without further warning. Please stop, and consider improving rather than damaging the hard work of others. Thanks. -- Herby talk thyme 11:52, 13 November 2006 (UTC)

title: User talk:62.232.224.14


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MediaWiki Administrator's Handbook/Block

Block prevents the user of a certain account or IP address from editing for a determined period of time.

When to block
Blocking policy varies from project to project, but it generally applies to the following categories of user:

 * Persistent vandals
 * Spammers
 * Trolls
 * Open or anonymous proxies
 * Threats or harassment
 * Persistently adding copyright violations
 * Inappropriate user names

Options

 * Expiry sets a time after which the block is automatically lifted by the MediaWiki software. The dropbox offers several frequently-used time periods, such as "24 hours", "1 week", and "1 month".  (Please see note below about block lengths.)
 * Block anonymous users only allows users from a blocked IP address to edit using a registered account, only restricting the edits of anonymous users. (This option is not available when blocking a registered username.)
 * Prevent account creation prevents the creation of new accounts from a blocked IP
 * Automatically block the last IP address used by this user, and any subsequent addresses they try to edit from blocks any IP used by this account

Unblocking
In order to unblock an account (to allow editing or increase the period of the block) go to Special:Ipblocklist where the option is available.

Length of block
The priority must be to minimise damage to the wiki. However when there is time or on review afterwards some or all of the following may be helpful in looking at the length of the block.
 * 1) User contributions.  Bear in mind that if a page is deleted it will not be in the contributions.  (This is often the case with spam bot type pages.)
 * 2) Block log history.  This may give guidance on previous blocks.
 * 3) WHOIS data
 * 4) User talk page on Wikipedia.  Given the level of traffic on WP compared to other Wiki's this may have useful history of both editing patterns and warnings.
 * 5) Block log on Wikipedia.  Again this may give guidance on level of nuisance and block period.
 * 6) Check out the spam database on DNS Stuff.  If one or more of the lines are red this is an indication of a lack of security of the IP and the likelihood spam may originate from it.
 * 7) Ask around!  It's always worth it.

Conflicting block lengths
If a particular user or IP is blocked more than once simultaneously, the MediaWiki software will honor the block that expires the soonest. (This may or may not be the shortest block.) In this case, the sysops who placed the blocks should contact each other and decide which length is most appropriate under the circumstances.

Indefinite blocks
The "indefinite" option in the Expiry field allows a sysop to block a particular user or IP without specifying an expiration. This block is effectively permanent, but it can be lifted at any time by any sysop. Indefinite blocks should be used sparingly on IP addresses because IPs tend to shift from user to user over time.

Custom block lengths
The "other" option in the Expiry field allows a sysop to define either a custom block length or a precise date and time when the user or IP will be able to edit again. Custom values must be entered in the GNU date input format.

Range blocks
The MediaWiki software allows sysops to block entire ranges of IP addresses. Using CIDR notation, enter an IP range between /16 and /31 (inclusive; MediaWiki does not support blocking larger ranges). If you are unfamiliar with CIDR and binary arithmetic, then you should not block ranges.

A properly-applied range block will remove editing privileges from all persons who connect from an IP in the affected range. As a result, it may unintentionally block valued contributors, so it may be worthwhile to consult a checkuser before applying a range block.

Range blocks are a drastic measure, and should only be used for brief periods as a last resort. Poorly-applied range blocks can shut out entire nations!

Projects

 * Commons -- Commons:Blocking policy
 * Meta --
 * Wikibooks -- Blocking policy
 * Wikinews -- Wikinews:Blocking policy
 * Wikipedia -- Wikipedia:Blocking policy
 * Wikiquote -- Wikiquote:Blocking policy
 * Wikisource -- Wikisource:Blocking policy
 * Wikispecies --
 * Wikiversity -- Wikiversity:Blocking policy
 * Wiktionary -- Wiktionary:Blocking policy


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User talk:213.48.45.42

Please do not replace Wikibooks pages with blank content. Blank pages are harmful to Wikibooks because they have a tendency to confuse readers. If it is a duplicate article, please redirect it to an appropriate existing page. If the page has been vandalised, please revert it to the last legitimate version. If you feel that the content of a page is inappropriate, please edit the page and replace it with appropriate content. If you believe there is no hope for the page, please see the deletion policy for how to proceed. If page is no longer useful, insert delete to its top. Thanks for contributing to Wikibooks! -- Herby talk thyme 12:17, 13 November 2006 (UTC)

title: User talk:213.48.45.42


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MediaWiki Administrator's Handbook/Editing in the MediaWiki Namespace

Editing pages in the MediaWiki namespace is one of the more complicated and serious abilities of an administrator, and must be undertaken with extreme caution because it can change the appearance of the entire website.

 * MediaWiki:Uploadtext – Text appearing on the upload page, including warnings about images without copyright information, etc.
 * MediaWiki:Sitenotice – Text that appears at the top of every page on the site, just below the tabs
 * MediaWiki:Copyrightwarning2 – The copyright warning that appears below the edit window.
 * MediaWiki:Common.css – Style sheet (CSS) defining site appearance, common to all skins
 * MediaWiki:Monobook.js – Contains scripts in JavaScript
 * MediaWiki:Edittools – Provides list of special characters appearing below edit box and save/preview buttons
 * MediaWiki:Sidebar – Defines 'navigation' box in top-left corner, and allows for the creation of other navigation boxes (such as the 'community' box on wikibooks)
 * MediaWiki:Revertpage – Text added to the edit summary when using the rollback tool.
 * MediaWiki:Monobook.css – Style sheet (CSS) defining site appearance in default skin.
 * MediaWiki:Licenses – List of available licenses for images/files appearing in a drop-down list on file upload page.
 * MediaWiki:Newmessagesdifflink
 * MediaWiki:Mainpage
 * MediaWiki:Sharedupload – Appears on the page of any image or file used in wikibooks but actually located on commons. Most wikis use a template linking to the commons page.
 * MediaWiki:Tagline – The text appearing just below a page title (on Wikibooks, this currently reads "From Wikibooks, open books for an open world").


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User talk:84.25.87.192

Please do not replace Wikibooks pages with blank content. Blank pages are harmful to Wikibooks because they have a tendency to confuse readers. If it is a duplicate article, please redirect it to an appropriate existing page. If the page has been vandalised, please revert it to the last legitimate version. If you feel that the content of a page is inappropriate, please edit the page and replace it with appropriate content. If you believe there is no hope for the page, please see the deletion policy for how to proceed. If page is no longer useful, insert delete to its top. Thanks for contributing to Wikibooks! -- Herby talk thyme 12:30, 13 November 2006 (UTC)

title: User talk:84.25.87.192


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Talk:MediaWiki Administrator's Handbook/Protect

Pagemove
Ok I can't see the toys but "Pagemove is linked by default, but can be unlinked." is not very clear to me as a reader (I'm assuming at least Johnny will be watching this!) -- Herby talk thyme 12:51, 13 November 2006 (UTC)

 * Ah, I'll fix that :). -- SB_Johnny | talk 13:40, 13 November 2006 (UTC)

 * Def better -- Herby talk thyme 13:47, 13 November 2006 (UTC)


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MediaWiki Administrator's Handbook/Asking for help

When uncertain about whether and how to use a tool, there are various venues both within and associated with each project where assistance can be found.

By project (with a link to the project admin list):

 * Wikibooks: Ask one of the other administrators on their talk page, or use the IRC channel (#wikibooks on freenode), or post a note to Reading room/Administrative Assistance
 * Commons
 * Meta:
 * Wikinews:
 * Wikipedia:
 * Wikiquote:
 * Wikisource:
 * Wikispecies:
 * Wikiversity:
 * Wiktionary:


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Talk:MediaWiki Administrator's Handbook/Block

Blocking
Something about checking IP addresses per Derbeth's comments here.

Also views on "appropriate" duration of blocks too would probably be helpful to people (any policies anywhere?) -- Herby talk thyme 12:54, 13 November 2006 (UTC)

No time at pres
Unblock needs dealing with here, I'll do it when I can if no one beats me to it! -- Herby talk thyme 20:00, 6 December 2006 (UTC)


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User talk:195.8.175.27

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 12:58, 13 November 2006 (UTC)

title: User talk:195.8.175.27


text:
User talk:194.70.74.210

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 13:05, 13 November 2006 (UTC)

title: User talk:194.70.74.210


text:
User:Pdavis68/monobook.js

// --- Begin Cacyle editor ---

// install User:Cacycle/diff text diff code document.write(' ');

// install User:Pilaf/Live_Preview page preview tool document.write(' ');

// install User:Cacycle/wikEd rich text editor document.write(' ');

// --- End Cacyle editor ---

// --- Begin Popup Preview ---

// User:Lupin/popups.js document.write(' ');

// --- End Popup Preview ---

title: User:Pdavis68/monobook.js


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MediaWiki Administrator's Handbook/How to Handle Copyright Violations

Copyright Violations -- usually referred to as "Copyvios" for short -- occur when copyrighted text or files are added to a wikimedia project. How this is dealt with depends on the type of material that was copied.

Copyright compatibility
Wikimedia projects are copyrighted using the GFDL. Some other copyrights are compatible with the GFDL for images and other files.

Copyrights that restrict the use of materials for any purpose are incompatible with the GFDL, and so including them in a GFDL document results in a violation of the material's copyright.

Image copyrights
All images uploaded to wikimedia projects must have their copyright status announced using one of the provided templates.

Public Domain, GFDL, and most Creative Commons licences are permitted.

"Fair Use" images are permitted on some projects, but not others.

Text copyrights
These can be harder to ferret out.

Text that is copied and pasted from other websites is the most common problem. In many cases the user who contributed the material will add a link to the source website. Most users are not familiar with how the copyright laws apply, so they should be dealt with kindly (but firmly).

Text from printed materials is less common, but much harder to catch.

Text copied from other wikimedia projects can be very difficult to catch.


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User talk:Securiger~enwikibooks

=Acetone peroxide synthesis= I have proposed the acetone peroxide synthesis for deletion. What do you think? Ewen 14:17, 13 November 2006 (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 Securiger. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Securiger~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 23:40, 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) 05:28, 19 April 2015 (UTC)

title: User talk:Securiger~enwikibooks


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MediaWiki Administrator's Handbook/Spam and Spammers

On wikimedia wikis, spamming is the addition of unwelcome commercial links and/or language, either added to already existing pages, or added as an entire page.

The various projects have vast differences in what is defined as spamming, and what to do about it.

Extension:ConfirmEdit
The most common way to protect your wiki from spam is to use ConfirmEdit extension. Extension:ConfirmEdit offers several variants of capchas:

SimpleCaptcha
SimpleCaptcha is the default captcha in ConfirmEdit. This captcha prompts user to solve a simple math question to prove that they are not spam bots:

SimpleCaptcha can be easily bypassed by simple script, so it is recommended to use more complex capchas.

FancyCaptcha
FancyCaptcha is a traditional graphic captcha, that prompts user to type letters that appear in the picture:

To create captcha's images you will need to run a python script. But you will not need python on your web-server to use this captcha, you may create images on your desktop, for example, and just copy them to your wiki web-server.

MathCaptcha
MathCaptcha is another captcha mechanism for ConfirmEdit. To run it you will require TeX support within your MediaWiki.

ConfirmEdit Notes
Note that ConfirmEdit is quite sensitive to version mismatch. The last version of ConfirmEdit should work well with the last version of MediaWiki. If you want to use ConfirmEdit with a legacy version of MediaWiki, you might spend a lot of time looking for a matching version of ConfirmEdit in the SVN repository. Some version matches are mentioned on the extension home page but not all of them.

Extension:ReCAPTCHA
ReCAPTCHA is a captcha extension for Mediawiki. It offers strong visual and audio capchas.

Using Abusefilters
MediaWiki's abuse filter extension can also be very useful in stopping spammers, in fact, it can outdo the other methods described above if configured properly while being less intrusive to the user. The main challenge is to determine how strict you want to be - you can be too strict but have virtually no spam - or you can be lenient and make do with some spam.

This site (en.wikibooks) mainly combats spam with the help of an abuse filter which blocks users with less than two posts from posting external links. This filter is particularly effective as spambots are focused on including external links and the abuse filter takes care of them, while the threshold is deliberately set low so that genuine users aren't affected.

If you use this method, expect false positives from time to time. You may wish to include a page where users can do that, in Wikibooks this page is used.


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Engineering Analysis/SISO Transformations

Let's say that we have a random variable X that is the input into a given system. The system output, Y is then also a random variable that is related to the input X by the response of the system. In other words, we can say that:

 * $$Y = g(X)$$

Where g is the mathematical relationship between the system input and the system output.

To discover information about Y, we can use the information we know about the r.v. X, and the relationship g:

 * $$f_Y(y) = \sum_i \frac{f_X(y)}{\left|\frac{dy}{dx}\right|}$$

Where xi are the roots of g.


text:
User talk:213.180.112.62

Please do not replace Wikibooks pages with blank content. Blank pages are harmful to Wikibooks because they have a tendency to confuse readers. If it is a duplicate article, please redirect it to an appropriate existing page. If the page has been vandalised, please revert it to the last legitimate version. If you feel that the content of a page is inappropriate, please edit the page and replace it with appropriate content. If you believe there is no hope for the page, please see the deletion policy for how to proceed. If page is no longer useful, insert delete to its top. Thanks for contributing to Wikibooks! -- Herby talk thyme 19:25, 13 November 2006 (UTC)

title: User talk:213.180.112.62


text:
Biblical Studies/Christianity/Seventh-day Adventist

Fundamental Beliefs
Seventh-day Adventists accept the Bible as their only creed and hold certain fundamental beliefs to be the teaching of the Holy Scriptures. These beliefs, as set forth here, constitute the church's understanding and expression of the teaching of Scripture. Revision of these statements may be expected at a General Conference session when the church is led by the Holy Spirit to a fuller understanding of Bible truth or finds better language in which to express the teachings of God's Holy Word.
 * 1) Holy Scriptures: The Holy Scriptures, Old and New Testaments, are the written Word of God, given by divine inspiration through holy men of God who spoke and wrote as they were moved by the Holy Spirit. In this Word, God has committed to man the knowledge necessary for salvation. The Holy Scriptures are the infallible revelation of His will. They are the standard of character, the test of experience, the authoritative revealer of doctrines, and the trustworthy record of God's acts in history. (2 Peter 1:20, 21; 2 Tim. 3:16, 17; Ps. 119:105; Prov. 30:5, 6; Isa. 8:20; John 17:17; 1 Thess. 2:13; Heb. 4:12.)
 * 2) Trinity: There is one God: Father, Son, and Holy Spirit, a unity of three co-eternal Persons. God is immortal, all-powerful, all-knowing, above all, and ever present. He is infinite and beyond human comprehension, yet known through His self-revelation. He is forever worthy of worship, adoration, and service by the whole creation. (Deut. 6:4; Matt. 28:19; 2 Cor. 13:14; Eph. 4:4-6; 1 Peter 1:2; 1 Tim. 1:17; Rev. 14:7.)
 * 3) Father:God the eternal Father is the Creator, Source, Sustainer, and Sovereign of all creation. He is just and holy, merciful and gracious, slow to anger, and abounding in steadfast love and faithfulness. The qualities and powers exhibited in the Son and the Holy Spirit are also revelations of the Father. (Gen. 1:1; Rev. 4:11; 1 Cor. 15:28; John 3:16; 1 John 4:8; 1 Tim. 1:17; Ex. 34:6, 7; John 14:9.)
 * 4) Son: God the eternal Son became incarnate in Jesus Christ. Through Him all things were created, the character of God is revealed, the salvation of humanity is accomplished, and the world is judged. Forever truly God, He became also truly man, Jesus the Christ. He was conceived of the Holy Spirit and born of the virgin Mary. He lived and experienced temptation as a human being, but perfectly exemplified the righteousness and love of God. By His miracles He manifested God's power and was attested as God's promised Messiah. He suffered and died voluntarily on the cross for our sins and in our place, was raised from the dead, and ascended to minister in the heavenly sanctuary in our behalf. He will come again in glory for the final deliverance of His people and the restoration of all things. (John 1:1-3, 14; Col. 1:15-19; John 10:30; 14:9; Rom. 6:23; 2 Cor. 5:17-19; John 5:22; Luke 1:35; Phil. 2:5-11; Heb. 2:9-18; 1 Cor. 15:3, 4; Heb. 8:1, 2; John 14:1-3.)
 * 5) Holy Spirit: God the eternal Spirit was active with the Father and the Son in Creation, incarnation, and redemption. He inspired the writers of Scripture. He filled Christ's life with power. He draws and convicts human beings; and those who respond He renews and transforms into the image of God. Sent by the Father and the Son to be always with His children, He extends spiritual gifts to the church, empowers it to bear witness to Christ, and in harmony with the Scriptures leads it into all truth. (Gen. 1:1, 2; Luke 1:35; 4:18; Acts 10:38; 2 Peter 1:21; 2 Cor. 3:18; Eph. 4:11, 12; Acts 1:8; John 14:16-18, 26; 15:26, 27; 16:7-13.)
 * 6) Creation: God is Creator of all things, and has revealed in Scripture the authentic account of His creative activity. In six days the Lord made "the heaven and the earth" and all living things upon the earth, and rested on the seventh day of that first week. Thus He established the Sabbath as a perpetual memorial of His completed creative work. The first man and woman were made in the image of God as the crowning work of Creation, given dominion over the world, and charged with responsibility to care for it. When the world was finished it was ``very good,'' declaring the glory of God. (Gen. 1; 2; Ex. 20:8-11; Ps. 19:1-6; 33:6, 9; 104; Heb. 11:3.)
 * 7) Nature of Man: Man and woman were made in the image of God with individuality, the power and freedom to think and to do. Though created free beings, each is an indivisible unity of body, mind, and spirit, dependent upon God for life and breath and all else. When our first parents disobeyed God, they denied their dependence upon Him and fell from their high position under God. The image of God in them was marred and they became subject to death. Their descendants share this fallen nature and its consequences. They are born with weaknesses and tendencies to evil. But God in Christ reconciled the world to Himself and by His Spirit restores in penitent mortals the image of their Maker. Created for the glory of God, they are called to love Him and one another, and to care for their environment. (Gen. 1:26-28; 2:7; Ps. 8:4-8; Acts 17:24-28; Gen. 3; Ps. 51:5; Rom. 5:12-17; 2 Cor. 5:19, 20; Ps. 51:10; 1 John 4:7, 8, 11, 20; Gen. 2:15.)
 * 8) Great Controversy: All humanity is now involved in a great controversy between Christ and Satan regarding the character of God, His law, and His sovereignty over the universe. This conflict originated in heaven when a created being, endowed with freedom of choice, in self-exaltation became Satan, God's adversary, and led into rebellion a portion of the angels. He introduced the spirit of rebellion into this world when he led Adam and Eve into sin. This human sin resulted in the distortion of the image of God in humanity, the disordering of the created world, and its eventual devastation at the time of the worldwide flood. Observed by the whole creation, this world became the arena of the universal conflict, out of which the God of love will ultimately be vindicated. To assist His people in this controversy, Christ sends the Holy Spirit and the loyal angels to guide, protect, and sustain them in the way of salvation. (Rev. 12:4-9; Isa. 14:12-14; Eze. 28:12-18; Gen. 3; Rom. 1:19-32; 5:12-21; 8:19-22; Gen. 6-8; 2 Peter 3:6; 1 Cor. 4:9; Heb. 1:14.)
 * 9) Life, Death, and Resurrection of Christ: In Christ's life of perfect obedience to God's will, His suffering, death, and resurrection, God provided the only means of atonement for human sin, so that those who by faith accept this atonement may have eternal life, and the whole creation may better understand the infinite and holy love of the Creator. This perfect atonement vindicates the righteousness of God's law and the graciousness of His character; for it both condemns our sin and provides for our forgiveness. The death of Christ is substitutionary and expiatory, reconciling and transforming. The resurrection of Christ proclaims God's triumph over the forces of evil, and for those who accept the atonement assures their final victory over sin and death. It declares the Lordship of Jesus Christ, before whom every knee in heaven and on earth will bow. (John 3:16; Isa. 53; 1 Peter 2:21, 22; 1 Cor. 15:3, 4, 20-22; 2 Cor. 5:14, 15, 19-21; Rom. 1:4; 3:25; 4:25; 8:3, 4; 1 John 2:2; 4:10; Col. 2:15; Phil. 2:6-11.)
 * 10) Experience of Salvation: In infinite love and mercy God made Christ, who knew no sin, to be sin for us, so that in Him we might be made the righteousness of God. Led by the Holy Spirit we sense our need, acknowledge our sinfulness, repent of our transgressions, and exercise faith in Jesus as Lord and Christ, as Substitute and Example. This faith which receives salvation comes through the divine power of the Word and is the gift of God's grace. Through Christ we are justified, adopted as God's sons and daughters, and delivered from the lordship of sin. Through the Spirit we are born again and sanctified; the Spirit renews our minds, writes God's law of love in our hearts, and we are given the power to live a holy life. Abiding in Him we become partakers of the divine nature and have the assurance of salvation now and in the judgment. (2 Cor. 5:17-21; John 3:16; Gal. 1:4; 4:4-7; Titus 3:3-7; John 16:8; Gal. 3:13, 14; 1 Peter 2:21, 22; Rom. 10:17; Luke 17:5; Mark 9:23, 24; Eph. 2:5-10; Rom. 3:21-26; Col. 1:13, 14; Rom. 8:14-17; Gal. 3:26; John 3:3-8; 1 Peter 1:23; Rom. 12:2; Heb. 8:7-12; Eze. 36:25-27; 2 Peter 1:3, 4; Rom. 8:1-4; 5:6-10.)
 * 11) Growing in Christ: By His death on the cross Jesus triumphed over the forces of evil. He who subjugated the demonic spirits during His earthly ministry has broken their power and made certain their ultimate doom. Jesus' victory gives us victory over the evil forces that still seek to control us, as we walk with Him in peace, joy, and assurance of His love. Now the Holy Spirit dwells within us and empowers us. Continually committed to Jesus as our Saviour and Lord, we are set free from the burden of our past deeds. No longer do we live in the darkness, fear of evil powers, ignorance, and meaninglessness of our former way of life. In this new freedom in Jesus, we are called to grow into the likeness of His character, communing with Him daily in prayer, feeding on His Word, meditating on it and on His providence, singing His praises, gathering together for worship, and participating in the mission of the Church. As we give ourselves in loving service to those around us and in witnessing to His salvation, His constant presence with us through the Spirit transforms every moment and every task into a spiritual experience. (Ps 1:1, 2; 23:4; 77:11, 12; Col 1:13, 14; 2:6, 14, 15; Luke 10:17-20; Eph 5:19, 20; 6:12-18; 1 Thess 5:23; 2 Peter 2:9; 3:18; 2 Cor. 3:17, 18; Phil 3:7-14; 1 Thess 5:16-18; Matt 20:25-28; John 20:21; Gal 5:22-25; Rom 8:38, 39; 1 John 4:4; Heb 10:25.)
 * 12) Church: The church is the community of believers who confess Jesus Christ as Lord and Saviour. In continuity with the people of God in Old Testament times, we are called out from the world; and we join together for worship, for fellowship, for instruction in the Word, for the celebration of the Lord's Supper, for service to all mankind, and for the worldwide proclamation of the gospel. The church derives its authority from Christ, who is the incarnate Word, and from the Scriptures, which are the written Word. The church is God's family; adopted by Him as children, its members live on the basis of the new covenant. The church is the body of Christ, a community of faith of which Christ Himself is the Head. The church is the bride for whom Christ died that He might sanctify and cleanse her. At His return in triumph, He will present her to Himself a glorious church, the faithful of all the ages, the purchase of His blood, not having spot or wrinkle, but holy and without blemish. (Gen. 12:3; Acts 7:38; Eph. 4:11-15; 3:8-11; Matt. 28:19, 20; 16:13-20; 18:18; Eph. 2:19-22; 1:22, 23; 5:23-27; Col. 1:17, 18.)
 * 13) Remnant and Its Mission: The universal church is composed of all who truly believe in Christ, but in the last days, a time of widespread apostasy, a remnant has been called out to keep the commandments of God and the faith of Jesus. This remnant announces the arrival of the judgment hour, proclaims salvation through Christ, and heralds the approach of His second advent. This proclamation is symbolized by the three angels of Revelation 14; it coincides with the work of judgment in heaven and results in a work of repentance and reform on earth. Every believer is called to have a personal part in this worldwide witness. (Rev. 12:17; 14:6-12; 18:1-4; 2 Cor. 5:10; Jude 3, 14; 1 Peter 1:16-19; 2 Peter 3:10-14; Rev. 21:1-14.)
 * 14) Unity in the Body of Christ: The church is one body with many members, called from every nation, kindred, tongue, and people. In Christ we are a new creation; distinctions of race, culture, learning, and nationality, and differences between high and low, rich and poor, male and female, must not be divisive among us. We are all equal in Christ, who by one Spirit has bonded us into one fellowship with Him and with one another; we are to serve and be served without partiality or reservation. Through the revelation of Jesus Christ in the Scriptures we share the same faith and hope, and reach out in one witness to all. This unity has its source in the oneness of the triune God, who has adopted us as His children. (Rom. 12:4, 5; 1 Cor. 12:12-14; Matt. 28:19, 20; Ps. 133:1; 2 Cor. 5:16, 17; Acts 17:26, 27; Gal. 3:27, 29; Col. 3:10-15; Eph. 4:14-16; 4:1-6; John 17:20-23.)
 * 15) Baptism: By baptism we confess our faith in the death and resurrection of Jesus Christ, and testify of our death to sin and of our purpose to walk in newness of life. Thus we acknowledge Christ as Lord and Saviour, become His people, and are received as members by His church. Baptism is a symbol of our union with Christ, the forgiveness of our sins, and our reception of the Holy Spirit. It is by immersion in water and is contingent on an affirmation of faith in Jesus and evidence of repentance of sin. It follows instruction in the Holy Scriptures and acceptance of their teachings. (Rom. 6:1-6; Col. 2:12, 13; Acts 16:30-33; 22:16; 2:38; Matt. 28:19, 20.)
 * 16) Lord's Supper: The Lord's Supper is a participation in the emblems of the body and blood of Jesus as an expression of faith in Him, our Lord and Saviour. In this experience of communion Christ is present to meet and strengthen His people. As we partake, we joyfully proclaim the Lord's death until He comes again. Preparation for the Supper includes self-examination, repentance, and confession. The Master ordained the service of foot washing to signify renewed cleansing, to express a willingness to serve one another in Christlike humility, and to unite our hearts in love. The communion service is open to all believing Christians. (1 Cor. 10:16, 17; 11:23-30; Matt. 26:17-30; Rev. 3:20; John 6:48-63; 13:1-17.)
 * 17) Spiritual Gifts and Ministries: God bestows upon all members of His church in every age spiritual gifts which each member is to employ in loving ministry for the common good of the church and of humanity. Given by the agency of the Holy Spirit, who apportions to each member as He wills, the gifts provide all abilities and ministries needed by the church to fulfill its divinely ordained functions. According to the Scriptures, these gifts include such ministries as faith, healing, prophecy, proclamation, teaching, administration, reconciliation, compassion, and self-sacrificing service and charity for the help and encouragement of people. Some members are called of God and endowed by the Spirit for functions recognized by the church in pastoral, evangelistic, apostolic, and teaching ministries particularly needed to equip the members for service, to build up the church to spiritual maturity, and to foster unity of the faith and knowledge of God. When members employ these spiritual gifts as faithful stewards of God's varied grace, the church is protected from the destructive influence of false doctrine, grows with a growth that is from God, and is built up in faith and love. (Rom. 12:4-8; 1 Cor. 12:9-11, 27, 28; Eph. 4:8, 11-16; Acts 6:1-7; 1 Tim. 3:1-13; 1 Peter 4:10, 11.)
 * 18) The Gift of Prophecy: One of the gifts of the Holy Spirit is prophecy. This gift is an identifying mark of the remnant church and was manifested in the ministry of Ellen. G. White . As the Lord's messenger, her writings are a continuing and authoritative source of truth which provide for the church comfort, guidance, instruction, and correction. They also make clear that the Bible is the standard by which all teaching and experience must be tested. (Joel 2:28, 29; Acts 2:14-21; Heb. 1:1-3; Rev. 12:17; 19:10.)
 * 19) Law of God: The great principles of God's law are embodied in the Ten Commandments and exemplified in the life of Christ. They express God's love, will, and purposes concerning human conduct and relationships and are binding upon all people in every age. These precepts are the basis of God's covenant with His people and the standard in God's judgment. Through the agency of the Holy Spirit they point out sin and awaken a sense of need for a Saviour. Salvation is all of grace and not of works, but its fruitage is obedience to the Commandments. This obedience develops Christian character and results in a sense of well-being. It is an evidence of our love for the Lord and our concern for our fellow men. The obedience of faith demonstrates the power of Christ to transform lives, and therefore strengthens Christian witness. (Ex. 20:1-17; Ps. 40:7, 8; Matt. 22:36-40; Deut. 28:1-14; Matt. 5:17-20; Heb. 8:8-10; John 15:7-10; Eph. 2:8-10; 1 John 5:3; Rom. 8:3, 4; Ps. 19:7-14.)
 * 20) Sabbath: The beneficent Creator, after the six days of Creation, rested on the seventh day and instituted the Sabbath for all people as a memorial of Creation. The fourth commandment of God's unchangeable law requires the observance of this seventh-day Sabbath as the day of rest, worship, and ministry in harmony with the teaching and practice of Jesus, the Lord of the Sabbath. The Sabbath is a day of delightful communion with God and one another. It is a symbol of our redemption in Christ, a sign of our sanctification, a token of our allegiance, and a foretaste of our eternal future in God's kingdom. The Sabbath is God's perpetual sign of His eternal covenant between Him and His people. Joyful observance of this holy time from evening to evening, sunset to sunset, is a celebration of God's creative and redemptive acts. (Gen. 2:1-3; Ex. 20:8-11; Luke 4:16; Isa. 56:5, 6; 58:13, 14; Matt. 12:1-12; Ex. 31:13-17; Eze. 20:12, 20; Deut. 5:12-15; Heb. 4:1-11; Lev. 23:32; Mark 1:32.)
 * 21) Stewardship: We are God's stewards, entrusted by Him with time and opportunities, abilities and possessions, and the blessings of the earth and its resources. We are responsible to Him for their proper use. We acknowledge God's ownership by faithful service to Him and our fellow men, and by returning tithes and giving offerings for the proclamation of His gospel and the support and growth of His church. Stewardship is a privilege given to us by God for nurture in love and the victory over selfishness and covetousness. The steward rejoices in the blessings that come to others as a result of his faithfulness. (Gen. 1:26-28; 2:15; 1 Chron. 29:14; Haggai 1:3-11; Mal. 3:8-12; 1 Cor. 9:9-14; Matt. 23:23; 2 Cor. 8:1-15; Rom. 15:26, 27.)
 * 22) Christian Behavior: We are called to be a godly people who think, feel, and act in harmony with the principles of heaven. For the Spirit to recreate in us the character of our Lord we involve ourselves only in those things which will produce Christlike purity, health, and joy in our lives. This means that our amusement and entertainment should meet the highest standards of Christian taste and beauty. While recognizing cultural differences, our dress is to be simple, modest, and neat, befitting those whose true beauty does not consist of outward adornment but in the imperishable ornament of a gentle and quiet spirit. It also means that because our bodies are the temples of the Holy Spirit, we are to care for them intelligently. Along with adequate exercise and rest, we are to adopt the most healthful diet possible and abstain from the unclean foods identified in the Scriptures. Since alcoholic beverages, tobacco, and the irresponsible use of drugs and narcotics are harmful to our bodies, we are to abstain from them as well. Instead, we are to engage in whatever brings our thoughts and bodies into the discipline of Christ, who desires our wholesomeness, joy, and goodness. (Rom. 12:1, 2; 1 John 2:6; Eph. 5:1-21; Phil. 4:8; 2 Cor. 10:5; 6:14-7:1; 1 Peter 3:1-4; 1 Cor. 6:19, 20; 10:31; Lev. 11:1-47; 3 John 2.)
 * 23) Marriage and the Family: Marriage was divinely established in Eden and affirmed by Jesus to be a lifelong union between a man and a woman in loving companionship. For the Christian a marriage commitment is to God as well as to the spouse, and should be entered into only between partners who share a common faith. Mutual love, honor, respect, and responsibility are the fabric of this relationship, which is to reflect the love, sanctity, closeness, and permanence of the relationship between Christ and His church. Regarding divorce, Jesus taught that the person who divorces a spouse, except for fornication, and marries another, commits adultery. Although some family relationships may fall short of the ideal, marriage partners who fully commit themselves to each other in Christ may achieve loving unity through the guidance of the Spirit and the nurture of the church. God blesses the family and intends that its members shall assist each other toward complete maturity. Parents are to bring up their children to love and obey the Lord. By their example and their words they are to teach them that Christ is a loving disciplinarian, ever tender and caring, who wants them to become members of His body, the family of God. Increasing family closeness is one of the earmarks of the final gospel message. (Gen. 2:18-25; Matt. 19:3-9; John 2:1-11; 2 Cor. 6:14; Eph. 5:21-33; Matt. 5:31, 32; Mark 10:11, 12; Luke 16:18; 1 Cor. 7:10, 11; Ex. 20:12; Eph. 6:1-4; Deut. 6:5-9; Prov. 22:6; Mal. 4:5, 6.)
 * 24) Christ's Ministry in the Heavenly Sanctuary: There is a sanctuary in heaven, the true tabernacle which the Lord set up and not man. In it Christ ministers on our behalf, making available to believers the benefits of His atoning sacrifice offered once for all on the cross. He was inaugurated as our great High Priest and began His intercessory ministry at the time of His ascension. In 1844, at the end of the prophetic period of 2300 days, He entered the second and last phase of His atoning ministry. It is a work of investigative judgment which is part of the ultimate disposition of all sin, typified by the cleansing of the ancient Hebrew sanctuary on the Day of Atonement. In that typical service the sanctuary was cleansed with the blood of animal sacrifices, but the heavenly things are purified with the perfect sacrifice of the blood of Jesus. The investigative judgment reveals to heavenly intelligences who among the dead are asleep in Christ and therefore, in Him, are deemed worthy to have part in the first resurrection. It also makes manifest who among the living are abiding in Christ, keeping the commandments of God and the faith of Jesus, and in Him, therefore, are ready for translation into His everlasting kingdom. This judgment vindicates the justice of God in saving those who believe in Jesus. It declares that those who have remained loyal to God shall receive the kingdom. The completion of this ministry of Christ will mark the close of human probation before the Second Advent. (Heb. 8:1-5; 4:14-16; 9:11-28; 10:19-22; 1:3; 2:16, 17; Dan. 7:9-27; 8:13, 14; 9:24-27; Num. 14:34; Eze. 4:6; Lev. 16; Rev. 14:6, 7; 20:12; 14:12; 22:12.)
 * 25) Second Coming of Christ: The second coming of Christ is the blessed hope of the church, the grand climax of the gospel. The Saviour's coming will be literal, personal, visible, and worldwide. When He returns, the righteous dead will be resurrected, and together with the righteous living will be glorified and taken to heaven, but the unrighteous will die. The almost complete fulfillment of most lines of prophecy, together with the present condition of the world, indicates that Christ's coming is imminent. The time of that event has not been revealed, and we are therefore exhorted to be ready at all times. (Titus 2:13; Heb. 9:28; John 14:1-3; Acts 1:9-11; Matt. 24:14; Rev. 1:7; Matt. 24:43, 44; 1 Thess. 4:13-18; 1 Cor. 15:51-54; 2 Thess. 1:7-10; 2:8; Rev. 14:14-20; 19:11-21; Matt. 24; Mark 13; Luke 21; 2 Tim. 3:1-5; 1 Thess. 5:1-6.)
 * 26) Death and Resurrection: The wages of sin is death. But God, who alone is immortal, will grant eternal life to His redeemed. Until that day death is an unconscious state for all people. When Christ, who is our life, appears, the resurrected righteous and the living righteous will be glorified and caught up to meet their Lord. The second resurrection, the resurrection of the unrighteous, will take place a thousand years later. (Rom. 6:23; 1 Tim. 6:15, 16; Eccl. 9:5, 6; Ps. 146:3, 4; John 11:11-14; Col. 3:4; 1 Cor. 15:51-54; 1 Thess. 4:13-17; John 5:28, 29; Rev. 20:1-10.)
 * 27) Millennium and the End of Sin: The millennium is the thousand-year reign of Christ with His saints in heaven between the first and second resurrections. During this time the wicked dead will be judged; the earth will be utterly desolate, without living human inhabitants, but occupied by Satan and his angels. At its close Christ with His saints and the Holy City will descend from heaven to earth. The unrighteous dead will then be resurrected, and with Satan and his angels will surround the city; but fire from God will consume them and cleanse the earth. The universe will thus be freed of sin and sinners forever. (Rev. 20; 1 Cor. 6:2, 3; Jer. 4:23-26; Rev. 21:1-5; Mal. 4:1; Eze. 28:18, 19.)
 * 28) New Earth: On the new earth, in which righteousness dwells, God will provide an eternal home for the redeemed and a perfect environment for everlasting life, love, joy, and learning in His presence. For here God Himself will dwell with His people, and suffering and death will have passed away. The great controversy will be ended, and sin will be no more. All things, animate and inanimate, will declare that God is love; and He shall reign forever. Amen. (2 Peter 3:13; Isa. 35; 65:17-25; Matt. 5:5; Rev. 21:1-7; 22:1-5; 11:15.)


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Real Analysis/Bibliography

This Wikibook is written by Wikibookians who were inspired by other books. The list is provided below.

 * 1) J. K. Hunter and B. Nachtergaele (2001) Applied Analysis World Scientific Publishing, Singapore
 * 2) W. Rudin (1953) Principles of mathematical analysis. 3ed. McGraw-Hill Inc. New York
 * 3) David A. Sprecher (1970) Elements of Real Analysis Dover Publications, Mineola, New York
 * 4) R.G. Bartle, D.R. Sherbert, (2000) Introduction to Real Analysis 3ed. John Wiley & Sons
 * 5) M. Longo, V. Valori, The Comparison Test — Not Just for Nonnegative Series, Mathematics Magazine, Volume 79, Number 3, June 2006, pp. 205-210(6)

title: Real Analysis/Bibliography


text:
User talk:128.117.32.11

Diesels do have throttles of course...

title: User talk:128.117.32.11