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Real Analysis/The real numbers

Why we need the real numbers
This is a good juncture to justify the subject of real analysis, which essentially reduces to justifying the necessity of studying $$\R$$. So, what is missing? Why do we need anything beyond the rationals?

The first sign of trouble is square roots. Famously, $$\sqrt2$$ is not rational – in other words, there is no rational number which squares to $$2$$ (see the exercises). This fact has a curious consequence – consider the following function:

$$f:\Q\to\Q\ ;\ x\mapsto\begin{cases}0&:x^2<2\\1&:x^2>2\\\end{cases}$$

Clearly this function has a dramatic jump in it around the rational $$1.4$$, where it suddenly changes from being equal to zero and starts being equal to one. However, it's difficult (or even impossible) to pin down exactly where this jump happens. Any specific rational number is safely on one side or the other, and, indeed, in the standard Topology on $$\Q$$, this function is continuous (don't worry if that makes no sense to you).

It is this flaw which the real numbers are designed to repair. We will define the real numbers $$\R$$ so that no matter how clever we try to be, if a function has a 'jump' in the way that $$f$$ does, then we will always be able to find a specific number at which it jumps.

The following sections describe the properties of $$\R$$ which make this possible.

Different perspectives
In order to prove anything about the real numbers, we need to know what their properties are. There are two different approaches to describing these properties – axiomatic and constructive.

An axiomatic approach
When we take an axiomatic approach, we simply make a series of assertions regarding $$\mathbb R$$, and assume that they hold.

The assertions that we make are called axioms – in a mathematical context this term means roughly 'basic assumption'.

The advantage of this approach is that it is then clear exactly what has been assumed, before proceeding to deduce results which rely only on those assumptions.

The disadvantage of this approach is that it might not be immediately clear that any object satisfying the properties we desire even exists!

A constructive approach
With a constructive approach, we are not happy simply to assume exactly what we want, but rather we try to construct $$\R$$ from something simpler, and then prove that it has the properties we want. In this way, what could have been axioms become theorems. There are several different ways to do this, starting from $$\Q$$ and using some method to 'fill up the gaps between the rationals'.

All of these methods are fairly complex and will be put off until the next section.

The axioms
So, what are these axioms which we will need? The short version is to say that $$\R$$ is a complete ordered field. This is in fact saying a great many things:
 * That $$\R$$ is a totally ordered field.
 * That $$\R$$ is complete in this ordering (Note that the meaning of completeness here is not quite the same as the common meaning in the study of partially ordered sets).
 * That the algebraic operations (addition and multiplication) described by the field axioms interact with the ordering in the expected manner.

In more detail, we assert the following:
 * 1) $$\R$$ is a field. For this, we require binary operations addition (denoted $$+$$) and multiplication (denoted $$\times$$) defined on $$\R$$, and distinct elements $$0$$ and $$1$$ satisfying:
 * 2) $$(\R,+,0)$$ is a commutative group, meaning:
 * 3) $$\forall x,y,z\in\R:(x+y)+z=x+(y+z)$$ (associativity)
 * 4) $$\forall x,y\in\R: x+y=y+x$$ (commutativity)
 * 5) $$\forall x\in\R: x+0=x$$ (identity)
 * 6) $$\forall x\in\R:\exists y\in\R:x+y=0$$ (inverse)
 * 7) $$(\R\setminus\{0\},\times,1)$$ is a commutative group, meaning:
 * 8) $$\forall x,y,z\in\R\setminus\{0\}:(x\times y)\times z=x\times (y\times z)$$ (associativity)
 * 9) $$\forall x,y\in\R\setminus\{0\}:x\times y=y\times x$$ (commutativity)
 * 10) $$\forall x\in\R\setminus\{0\}:x\times1=x$$ (identity)
 * 11) $$\forall x\in\R\setminus\{0\}:\exists y\in\R\setminus\{0\}:x\times y=1$$ (inverse)
 * 12) $$\forall x,y,z\in\R:x\times(y+z)=(x\times y)+(x\times z)$$ (distributivity)
 * 13) $$\R$$ is a totally ordered set. For this we require a relation (denoted by $$\le$$) satisfying:
 * 14) $$\forall x\in\R: x\le x$$ (reflexivity)
 * 15) $$\forall x,y,z\in\R:(x\le y\text{ and }y\le z)\implies x\le z$$ (transitivity)
 * 16) $$\forall x,y\in\R:(x\le y\text{ and }y\le x)\implies x=y$$ (anti-symmetry)
 * 17) $$\forall x,y\in\R:\text{either }x\le y\text{ or }y\le x$$ (totality)
 * 18) $$\R$$ is complete in this order (see below for details).
 * 19) The field operations and order interact in the expected manner, meaning:
 * 20) $$\forall x,y,z\in\R:x\le y\implies(x+z)\le(y+z)$$
 * 21) $$\forall x,y,z\in\R:(x\le y\text{ and }0\le z)\implies(x\times z)\le(y\times z)$$

This is a substantial list, and if you are not used to axiomatic mathematics (or even if you are!) it may seem somewhat daunting, especially since we have yet to give details of what completeness means. This is amongst the longest list of axioms in any region of mathematics, but if you examine each in turn, you will find that they all state things which you have probably taken for granted as 'the way numbers behave' without a second thought.

These axioms are so exacting that there is a sense in which they specify the real numbers precisely. In other words $$\R$$ is the only complete ordered field.

Further notation
Having defined these operations and relations on $$\R$$, we need to introduce more notation to aid in talking about them. Hopefully all these conventions should be familiar to you, but it is important to formally present them all to avoid confusion following from misunderstanding of notation:

 * Rather than writing $$\times$$ for multiplication, we may simply denote it by juxtaposition. In other words, we write $$xy$$ to denote $$x\times y$$.
 * Since both multiplication and addition are associative, we omit unnecessary bracketing when several numbers are added or multiplied. In other words, rather than writing $$(x+y)+z$$ or $$x+(y+z)$$, which are equal, we simply write $$x+y+z$$ to denote their common value.
 * To further save writing of brackets, by convention, multiplication has a higher precedence than addition. So, for example, the expression $$x+yz$$ should be interpreted as $$x+(yz)$$, not as $$(x+y)z$$.
 * The number $$x+y$$ is called the sum of $$x$$ and $$y$$.
 * The number $$xy$$ is called the product of $$x$$ and $$y$$.
 * The additive inverse of $$x$$ is written $$-x$$, and called the negative or negation of $$x$$. So, $$x+(-x)=0$$.
 * The multiplicative inverse of $$x$$ is written $$x^{-1}$$, and called the reciprocal, or simply the inverse of $$x$$. So, $$x(x^{-1})=1$$.
 * We define the binary operation of subtraction as follows: For $$x,y\in\mathbb R$$, we set $$x-y=x+(-y)$$. The number $$x-y$$ is called the difference of $$x$$ and $$y$$.
 * Subtraction has the same precedence as addition (less than that of multiplication), and when the two operations are mixed without bracketing, left-associativity is implied. For example, $$a+b-c-d+e$$ should be interpreted as $$(((a+b)-c)-d)+e$$.
 * We define the binary operation of division as follows: For $$x,y\in\mathbb R$$, with $$y\not=0$$, we set $$x/y=x(y^{-1})$$. The number $$x/y$$ is called the quotient of $$x$$ and $$y$$, and is also denoted $$\frac{x}{y}$$.
 * Division has a higher precedence than that of addition or subtraction, but there is no simple convention as to how to handle mixed multiplication and division. Using the $$\frac{x}{y}$$ notation, rather than the $$x/y$$ notation helps to avoid confusion.
 * We define the binary operation of exponentation as follows: For $$x\in\mathbb R$$ and $$n\in\mathbb N_0$$ we define $$x^n$$ recursively by $$x^0=1$$ and $$x^{n+1}=(x^n)x$$.  Then for $$n\in\mathbb Z$$, with $$n<0$$, we define $$x^n=(x^{-1})^{-n}$$.
 * Exponentation has a higher precedence than any of division, multiplication, addition and subtraction. For example, $$ab^2+d^3$$ should be interpreted as $$(a(b^2))+(d^3)$$.
 * We write $$x\geq y$$ to mean $$y\leq x$$.
 * We write $$xy$$ to mean $$y0$$.
 * To say $$x$$ is negative means $$x<0$$.
 * To say $$x$$ is non-positive means $$x\leq 0$$.
 * To say $$x$$ is non-negative means $$x\geq 0$$.
 * We also introduce notation for several common varieties of subsets of $$\mathbb R$$. All of these subsets are called intervals:
 * $$[a,b]=\{x\in\mathbb R:a\leq x\leq b\}$$ (called the closed interval from $$a$$ to $$b$$)
 * $$(a,b)=\{x\in\mathbb R:a<x<b\}$$ (called the open interval from $$a$$ to $$b$$)
 * $$[a,b)=\{x\in\mathbb R:a\leq x<b\}$$
 * $$(a,b]=\{x\in\mathbb R:a<x\leq b\}$$
 * In all these cases, $$a$$ is called the lower limit of the interval, and $$b$$ is called the upper limit.
 * An excluded lower limit (as in the second and fourth cases) may be replaced by $$-\infty$$ to indicate that there is no lower restriction. For example $$(-\infty,b]=\{x\in\mathbb R:x\leq b\}$$.
 * Similarly, an excluded upper limit (as in the second and third cases) may be replaced by $$\infty$$. For example, $$(-\infty,\infty)=\mathbb R$$.
 * Some specific intervals which appear frequently are the closed unit interval, or just unit interval, which is $$[0,1]$$, and $${\mathbb R}^+=(0,\infty)$$, the positive real numbers.

Completeness
The rational numbers $$\mathbb Q$$ satisfy all of the axioms above which have been explained in detail, and so if we are to escape the problem which we described above then we clearly need something more. This 'something more' is completeness. There are several equivalent ways of describing completeness, but most of them require us to know about Sequences, which we do not introduce until the next chapter, so for the moment we can only give one definition.

Upper bounds
Let $$A\subseteq\mathbb R$$. We say $$b\in\mathbb R$$ is an upper bound for $$A$$ if
 * $$\forall s\in A:s\leq b$$

For example, $$3$$ is an upper bound for $$[0,1]$$, as is $$1$$, but $$\frac{1}{2}$$ is not, because $$1\in[0,1]$$ and $$1>\frac{1}{2}$$. A set with an upper bound $$b$$ is said to be bounded above by $$b$$.

Least Upper Bound
We say $$s$$ is a least upper bound or supremum for $$A$$ if $$s$$ is an upper bound for $$A$$, and $$b$$ is any upper bound for $$A$$ then $$s\leq b$$. More formally:
 * $$(\forall a\in A:a\leq s)\mbox{ and }(\forall b\in\mathbb R:((\forall a\in A:a\leq b)\implies(s\leq b)))$$

Similarly, we say $$b\in\mathbb R$$ is a lower bound for $$A$$ if
 * $$\forall a\in A:a\geq b$$

and we say $$i$$ is a greatest lower bound or infimum for $$A$$ if:
 * $$(\forall a\in A:a\geq i)\mbox{ and }(\forall b\in\mathbb R:((\forall a\in A:a\geq b)\implies(i\geq b)))$$

The supremum and infimum of a set $$A$$ are denoted $$\sup A$$ and $$\inf A$$ respectively.

The Least upper bound axiom
Now we are finally ready to state the last axiom: This is the axiom of the real numbers that finally satisfies what was lacking in the rationals: completeness. It is worth noting at this point, to avoid possible confusion, that in the study of general partially ordered sets, the definition of completeness is that every subset has a least upper bound, and there is no condition that they be non-empty or bounded above. Nevertheless, we really do wish to impose these two conditions in this case.
 * If $$S\subseteq\mathbb R$$ is non-empty and has an upper bound, then $$S\ $$ has a least upper bound in $$\mathbb R$$.

Other completeness axioms
There are other equivalent ways to state the completeness axiom, but they involve sequences, so we shall delay them until after the discussion of that topic. Because of the existence of these other forms, this axiom is sometimes called the least upper bound axiom.

authors: יהודה שמחה ולדמן

title: Real Analysis/The real numbers


text:
Real Analysis/Sequences

Definition
Sequences occur frequently in analysis, and they appear in many contexts. While we are all familiar with sequences, it is useful to have a formal definition.

 * Definition A sequence of real numbers is any function a : N→R.

Often sequences such as these are called real sequences, sequences of real numbers or sequences in R to make it clear that the elements of the sequence are real numbers. Analogous definitions can be given for sequences of natural numbers, integers, etc.

Given a sequence (xn), a subsequence, notated as $$(x_{n_j})_{j=1}^\infty$$, is a sequence where (nj) is strictly increasing sequence of natural numbers.

For example, taking nj=2j would the subsequence consisting of every other element of the original sequence, that is (x2, x4, x6, …).

Sequence Notation
However, we usually write an for the image of n under a, rather than a(n). The values an are often called the elements of the sequence. To make a distinction between a sequence and one of its values it is often useful to denote the entire sequence by $$(a_n)_{n=1}^\infty$$, or just (an). Some employ set notation and denote it as {an} When specifying a particular sequence, it may be written in the form (a1, a2, a3, …), when the sequence is infinite, or (a1, a2, …, an) when the sequence is finite. We tend only to discretely write down enough elements is so that the pattern is clear, which is typically 3 times.

Examples of Sequences
(1, 2, 3, 4, …), (1, -2, 3, -4, …), and (1, π, π2, π3, π4, …) are all examples of sequences. Note, however, that there need not be any particular pattern to the elements of the sequence. For example, we may specify an to be the n-th digit of π. Often sequences are defined recursively. That is, to specify some initial values of the sequence, and then to specify how to get the next element of the sequence from the previous elements. For example, consider the sequence x1=1, x2=1, and xn = xn−1 + xn−2 for n ≥ 3. This sequences is known as the Fibonacci sequence, and its first few terms are given by (1, 1, 2, 3, 5, 8, 13, …). Another familiar example of a recursive sequence is Newton's method. With an initial guess x0 for the zero of a function, Newton's method tells you how to construct the next guess. In this way you generate a sequence which (hopefully) converges to the zero of the function.

Operations on Sequences
We can also perform algebraic operations on sequences. In other words, we can add, subtract, multiply, divide sequences. These operations are simply performed element by element, for completeness we give the definitions.

 * Definition Given two sequences (xn) and (yn) and a real number c, we define the following operations:

Classification of Sequences
Some properties of sequence are so important that they are given special names.

Some of these terms are prefixed with strictly because the term increasing is used in some contexts with meaning either that of strictly increasing or of non-decreasing, and similarly decreasing can mean the same as either strictly decreasing, or non-increasing. As a result, these ambiguous terms are usually prefixed with and strictly. We will try adhere to using this unambiguous term.

From here, we will also describe properties of sequences based on boundedness, a word which we will define for sequences below.

Convergence and Limits
A further important property of sequences (arguably the most important property from the perspective of analysis) is the property of convergence. This property can be easily described by extending the epsilon-delta definition. However, because sequences are relative to counting numbers, there exists an additional way to imagine convergence. Both methods are described below.

 * Definition Let (xn) be a sequence of real numbers. The sequence (xn) is said to converge to a real number a.
 * if for all ε>0, there exists N in N such that |xn-a|<ε for all n≥N.

If (xn) converges to a then we say a is the limit of (xn) and write

 * $$\lim_{n\to\infty}x_n=a$$

or

 * $$x_n\to a$$ as $$n\to\infty$$.

This is read xn approaches a as n approaches ∞. If it is clear which variable is playing the role of n then this may be abbreviated to simply xn→a or lim xn=a.

If a sequence converges, then it is called convergent.

It is also useful to extend this concept and allow sequences whose limits are either ∞ or −∞

 * Definition We say xn→∞ as n→∞ if for every M in R there is a natural number N so that xn≥ M for all n≥N.  We say xn→−∞ as n→∞ if for every M in R there is a natural number N so that xn≤ M for all n≥N.

Despite this, we do not refer to sequences such as these as convergent. They are instead called divergent.

Although convergence can be proven using the epsilon-delta definition as proof, another method to prove convergences of sequences is through mathematical induction, since sequences are referenced using counting numbers. Through this method, some theorems are easier to prove. However, proof using mathematical induction cannot generalize to real numbers like a proof using epsilon-delta can.

The following theorems will prove that variations of a convergent sequence, expressed either through inductive notation, limit notation, or Cauchy notation, converges to exactly one number. This may seem intuitively clear, but remember that intuition often fails us when it comes to limits. It is also in proper mathematical style to rigorously prove every mathematical notion presented to us.

Theorem (Uniqueness of limits)
A sequence can have at most one limit. In other words: if xn → a and xn → b then a = b.

Proof
Suppose the sequence has two distinct limits, so a≠b. Let ε=|a−b|/3.

Certainly ε>0, using the definition of convergence twice we can find natural numbers Na and Nb so that

 * $$|x_n-a|\leq\epsilon$$ for all n > Na.

and

 * $$|x_n-b|\leq\epsilon$$ for all n > Nb.

Taking k=max(Na,Nb) then both of these conditions hold for xk. Hence we deduce that |xk−a|≤ε and |xk−b|≤ε. Applying the triangle inequality, we see
 * $$|a-b|=|(x_k-b)-(x_k-a)|\leq 2\epsilon=(2/3)|a-b|,$$

which is a contradiction. Thus, any sequence has at most one limit. $$\Box$$

Theorem (Convergent Sequences Bounded)
If the subsequence $$(x_n)_{n=1}^{\infty}$$ is a convergent sequence, then it is bounded.

Proof
Let $$a=\lim_{n\to\infty}x_n$$, and let ε = 1.

From the definition of convergence there exists a natural number N such that

 * $$|x_n-a|\leq 1$$ for all n ≥ N.

The sequence $$(x_n)_{n=N+1}^{\infty}$$ is bounded above by a+1 and below by a−1. Let M = max(|x1|,|x2|,|x3|,…,|xN|, |a|+1). It follows that −M ≤ xn ≤ M for all n in N. Hence the sequence is bounded. $$\Box$$

Theorem (Boundedness of Cauchy Sequences)
If $$(x_n)_{n=1}^{\infty}$$ is a Cauchy sequence, then it is bounded.

Proof
Let (xn) be a Cauchy sequence. By the definition of a Cauchy sequence, there is a natural number N such that |xn−xm|<1 for all n,m > N. In particular, |xN+1−xm|<1 for all m > N. It follows by the reverse triangle inequality that |xm| < |xN+1| + 1. If we take M=max(|x1|, |x2|, …, |xN|, |xN+1| + 1), then |xn| ≤ M for all n in N. $$\Box$$

The following theorem tells us that algebraic operations on sequences commute with the taking limits. This simple theorem is a useful tool in computing limits.

Properties of Sequences
Given our new definition of convergence, it should be essential that we can use the values we get from them algebraically and whether or not we can apply algebraic intuition in regards to converging sequences as well.

Algebraic Operations
If (xn) and (yn) are convergent sequences and a ∈ R, the following properties hold:
 * 1) $$\lim_{n\to\infty}(x_n+y_n)=\lim_{n\to\infty}(x_n)+\lim_{n\to\infty}(y_n)$$.
 * 2) $$\lim_{n\to\infty}(x_ny_n)=\left(\lim_{n\to\infty}x_n\right)\left(\lim_{n\to\infty}y_n\right)$$.
 * 3) $$\lim_{n\to\infty}(ax_n)=a\lim_{n\to\infty}(x_n)$$.
 * 4) $$\lim_{n\to\infty}\left(\frac{x_n}{y_n}\right)=\frac{\displaystyle \lim_{n\to\infty}x_n}{\displaystyle \lim_{n\to\infty}y_n}$$ (assuming yn ≠ 0 for all n in N and lim y_n ≠ 0).
 * 5) If xn ≤ yn for every n in N, then $$\lim_{n\to\infty}x_n \leq \lim_{n\to\infty}y_n$$.

Proof
1. Let x=lim xn and y=lim yn. We need to show that for any ε>0 there is natural number N so that if n≥ N, then |(xn + yn) − (x + y)|≤ε. Given any ε>0 we have ε/3>0 so from the definition of convergence there is a natural number Nx so that |xn−x|≤ε/3 for all n>Nx, similarly we can choose Ny |yn−y|≤ε/3 for all n>Ny.

Let N=max(Nx ,Ny). If n>N, then by the triangle inequality we have
 * $$|(x_n+y_n)-(x+y)|=|(x_n-x)+(y_n-y)| \leq \epsilon /3 + \epsilon /3<\epsilon,$$

which is what we needed to show. $$\Box$$

2. Let x=lim xn and y=lim yn. Since these sequences are convergent they are bounded. Let Mx be a bound for (xn) and let My be a bound for (yn). By increasing these quantities of necessary we may also assume Mx > x and My > y. Given ε>0, there exists some Nx and Ny such that
 * $$|x_n - x| < {\epsilon \over 2M_y}$$ for n > Nx and
 * $$|y_n - y| < {\epsilon \over 2M_x}$$ for n > Ny.

Then for every n > max(Nx, Ny),

$$\Box$$
 * $$|x_n y_n - xy|\,$$
 * $$= |(x_n - x)y_n + x(y_n - y)|\,$$
 * $$\leq |x_n - x|M_y + M_x |y_n - y|$$
 * $$\leq \frac{\epsilon}{2} + \frac{\epsilon}{2}\leq \epsilon.$$
 * }
 * $$\leq |x_n - x|M_y + M_x |y_n - y|$$
 * $$\leq \frac{\epsilon}{2} + \frac{\epsilon}{2}\leq \epsilon.$$
 * }
 * $$\leq \frac{\epsilon}{2} + \frac{\epsilon}{2}\leq \epsilon.$$
 * }

3. Let yn = a for all n in N. The statement now follows from 2. $$\Box$$

4. We can reduce this to showing that lim (1/yn) exists and equals 1/(lim yn). Then it follows by 2 that we have:
 * $$\lim_{n\to\infty}\left(\frac{x_n}{y_n}\right)= \left( \lim_{n \to \infty} {1 \over y_n} \right) \left( \lim_{n \to \infty} {x_n} \right)=\frac{\displaystyle \lim_{n\to\infty} x_n}{\displaystyle \lim_{n\to\infty} y_n}.$$

Let y=lim yn. By the exercises, since y and yn are not 0, we can find δ > 0 so that |y_n| > δ and |y| > δ. It follows that 1/|yny|<1/δ2. Given ε > 0 choose n in N so that |yn − y| < δ2ε. We have
 * $$\left| {1 \over y_n} - {1 \over y} \right| = |y - y_n|{1 \over |y_n y|} \le \frac{|y - y_n|}{\delta^2} < \epsilon$$.

Hence, $$\lim_{n \to \infty} {1 \over y_n} = {1 \over y}.$$ $$\Box$$

5. We first can reduce to the case when one sequence is identically 0. To see this let zn = xn − yn. Then zn < 0 for all n in N. Let z = lim zn. Suppose that z > 0 then we can then find a natural number N so that
 * $$|z - z_N| < z\,$$.

Since zN ≤ 0 < z, the absolute value equals z − zN. Subtracting z we find that −zN < 0. Hence zN is positive. Contradiction. Therefore we must have that z ≤ 0. Which means that by 1 we get:
 * $$\lim_{n \to \infty} x_n - \lim_{n \to \infty} y_n = \lim_{n \to \infty} z_n = z \le 0.$$

Therefore lim xn ≤ lim yn $$\square$$

Theorem (Squeeze/Sandwich Limit Theorem)
This is the important squeeze theorem that is a cornerstone of limits. Since converging sequences can also be thought of through limit notions and notations, it should also be wise if this important theorem applies to converging sequences as well.

Given sequences (xn), (yn), and (wn), if (xn) and (yn) converge to a and xn ≤ wn ≤ yn, then wn converges to a.

Proof
Fix ε > 0. We need to find an N such that |wn − a| < ε if n > N. Since (xn) → a and (yn) → a the definition of convergence ensures that there exists integers Nx and Ny so that |xn − a| < ε for n > Nx and |yn − a| < ε for n > Ny.

Let N=max(Nx, Ny). Then, for all n > N we have −ε < xn − a and yn − a < ε. Since xn < wn < yn, it follows that xn − a < wn − a < yn − a.

Thus if n ≥ N, then −ε < xn − a < wn − a < yn − a < ε. In other words, |wn − a| < ε.$$\Box$$

Completeness
The following results are closely related to the completeness of the real numbers.

Theorem (Convergence of Monotone sequences)
Any monotone, bounded sequence converges. If the sequence is non-decreasing, then the sequence converges to the least upper bound of the elements of the sequence. If the sequence is non-increasing, then the sequence converges to the greatest lower bound of the elements of the sequence

Proof
Let (xn) be any monotone sequence that is bounded by a real number M. Without loss of generality, assume (xn) is non-decreasing. Since (xn) is bounded above, it has a least upper bound by the least upper bound axiom. Let x = sup {xn | n ∈ N}. We will now show that (xn) → x.

Fix ε > 0. As was shown in the exercises, if s = sup(A), then for any ε > 0 there is an element a in A  so that  s − ε < a < s. Hence, it follows that there exists an N in N so that x − ε < xN < x.

For any n > N, since xn is non-decreasing, we have that
 * $$x- \epsilon < x_N \leq x_n < x$$.

Thus |x − xn| < ε and by the definition of convergence, (xn) converges to x. $$\Box$$

Theorem (Nested intervals property)
If there exists a sequence of closed intervals In = [an, bn] = {x | an ≤ x ≤ bn} such that In+1 ⊆ In for all n, then ∩In is nonempty.

Proof
Since In+1 ⊆ In it follows that an ≤ an+1 and bn+1 ≤ bn.

Since (an) and (bn) are monotonic sequences they converge by the previous theorem. Furthermore, since an < bn for all n, it follows that lim an ≤ lim bn.

By the monotonicity of (an) and (bn) we have for every n
 * $$a_n \leq \lim_{n\to\infty}a_n \leq \lim_{n\to\infty}b_n \leq b_n.$$

Therefore lim an ∈ [an, bn] for every n, which implies that
 * $$\lim_{n\to\infty}a_n \in \bigcap _{n=1}^{\infty} I_n.$$

Thus the intersection is nonempty. $$\Box$$

Theorem (Bolzano—Weierstrass)
Every bounded sequence of real numbers contains a convergent subsequence.

Proof
Let (xn) be a sequence of real numbers bounded by a real number M, that is |xn| < M for all n. We define the set A by A = {r | |r| ≤ M and r < xn for infinitely many n}. We note that A is non-empty since it contains −M and A is bounded above by M. Let x = sup A.

We claim that, for any ε > 0, there must be infinitely many points of xn in the interval (x − ε, x + ε). Suppose not and fix an ε > 0 so that there are only finitely many values of xn in the interval (x − ε, x + ε). Either x ≤ xn for infinitely many n or x ≤ xn for at most only finitely many n (possibly no n at all). Suppose x< xn for infinitely many n. Clearly in this case x ≠ M. If necessary restrict ε so that x + ε ≤ M. Set r = x + ε/2 we have that r < xn for infinitely many n because there are only finitely many xn in the set [x,r] and x must be less than infinitely many xn, furthermore |r| < M. Thus r is in A, which contradicts that x is an upper bound for A. Now suppose x< xn for at most finitely many n. Set y = x − ε/2. Then there are at most only finitely man n so that xn ≥ y. Thus, if r < xn for infinitely many n, we have that r ≤ y. This means that y is an upper bound for A that is less than x, contradicting that x wast the least upper bound of A. In either case we arrive at a contradiction, thus we must have that for any ε > 0, there must be infinitely many points of xn in the interval (x − ε, x + ε). Now we show there is a subsequence that converges to x. We define the subsequence inductively, choose any xn 1 from the interval (x − 1, x + 1). Assuming we have chosen xn 1, …, xn k−1 , choose xn k to be an element in the interval (x − 1/k, x + 1/k) so that nk∉{n1, …, nk−1}, this is possible as there are infinitely many elements of (xn) in the interval. Notice that for this choice of xn k we have that |x − xn k |<1/k. Hence for any ε>0, if we take any k > 1/ε, then |xn k -x| < ε. That is the subsequence (xn k ) → x. $$\Box$$

Theorem (Cauchy criterion)
A sequence converges if and only if it is Cauchy. Although this seems like a weaker property than convergence, it is actually equivalent, as the following theorem shows:

Proof
First we show that if (xn) → x then xN is Cauchy. Now suppose that for a given ε > 0 we wish to find an N so that |xn − xm| < ε for all n, m > N. We will choose N so that for all n ≥ N we have that |xn − x| < ε/2. By the triangle inequality, for any n, m > N we have:

 * $$|x_n - x_m| \leq |x_n - x| + |x_m - x| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$$.

Thus (xn) is a Cauchy sequence.

Now we show that if (xn) is a Cauchy sequence, then it converges to some x. Let (xn) be a Cauchy sequence, and let ε > 0. By the definition of a Cauchy Sequence, there exits a natural number L so that |xn − xm| < ε/2 whenever n, m > L. Since (xn) is a Cauchy sequence it is bounded. By the Bolzano—Weierstrass theorem, it has a convergent subsequence (xn k ) that converges to some point x. Now we will show that the whole sequence converges to x

Because (xn k ) converges, we can choose a natural number M so that if nk > M, then |xn k − x| < ε/2. Let N = max(L, M), and fix any nk > N. For n > N we have that

 * $$|x_n - x| \leq |x_n - x_{n_k}| + |x_{n_k} - x| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$$.

Thus by definition of convergence (xn) → x. $$\Box$$

These theorems all describe different aspects of the completeness of the real numbers. The reader will notice that the least upper bound property was used heavily in this section, and it is the axiom that separates the real numbers from the rational numbers. While these theorems would be false for the rational numbers, not all of them can substitute for the least upper bound property. The Cauchy criterion and the nested intervals property are not strong enough to imply the least upper bound property without additional assumptions, while the Convergence of Monotone sequences theorem and the Bolzano—Weierstrass property do imply the least upper bound property.

Limit superior and limit inferior
Limits turn out to be a very useful tool in analysis, their primary draw back is that they may not always exist. Occasionally it is useful to have some notion of limit that makes sense for any sequence. To this end we introduce the limit superior (often just called the "lim sup") and the limit inferor (often called the "lim inf").

Definition For a sequence (xn) we define the limit superior, denoted lim sup by:
 * $$\limsup_{n\to\infty} x_n=\inf_{k}\sup_{n\geq k} x_n.$$

Similarly we define the limit inferior, denoted by lim inf by:
 * $$\liminf_{n\to\infty} x_n=\sup_{k}\inf_{n\geq k} x_n.$$

If (xn) is not bounded above, we say that lim sup xn = ∞. If (xn) is not bounded we say that lim inf xn = −∞.

Notice that for bounded sequences the lim sup and the lim inf always exist. As we know general bounded sequence the limit doesn't always exist. But in the case when the lim sup and lim inf are equal, life is nicer as the next theorem shows.

Theorem (Limit Superior and Inferior)
Let (xn) be a bounded sequence. Then (xn) → x if and only if lim sup xn = x = lim inf xn.

Proof
First suppose (xn) → x. Fix an ε > 0 choose a natural number N so that x − ε < xn < x + ε for any n > N. Hence for any k > N we have that
 * $$x-\epsilon\leq \sup_{n>k}x_n\leq x+\epsilon$$

and hence x − ε < lim sup xn < x + ε. Since ε was arbitrary, this can only happen if lim sup xn = x. A similar argument shows that lim inf xn = x.

Now suppose lim inf xn = x = lim sup xn, and we wish to show that lim xn = x.

First recall that the x=lim sup xn is defined as:
 * $$x=\inf_k\sup_{n>k} x_n$$

Given an ε > 0, since we can get arbitrarily close to the infimum, we can choose we will choose Nls so that
 * $$\sup_{n>N_{ls}} x_n - x < \epsilon$$

Similarly recall that the x=lim inf xn is defined as:
 * $$x=\sup_k\inf_{n>k} x_n$$

Since we can get arbitrarily close to the supremum, we can choose we will choose Nli so that
 * $$x-\sup_{n>N_{li}} x_n< \epsilon$$

Let N = max(Nls, Nli). Now if n > N, then
 * $$\inf_{n> N_{li}}\leq \inf_{n\geq N}\leq x \leq \sup_{n\geq N}\leq \sup_{n> N_{ls}}$$

Hence for any n > N
 * $$x-\sup_{n> N_{ls}} x_n \leq x-x_n\leq x-\inf_{n>N_{li}} x_n.$$

By our choice of Nls and Nli this implies for any n > N

 * $$-\epsilon \leq x-x_n\leq \epsilon.\quad\Box$$

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

title: Real Analysis/Sequences


text:
XML - Managing Data Exchange/The one-to-many relationship

Introduction
In a one-to-many relationship, one object can reference several instances of another. A model is mapped into a schema whereby each data model entity becomes a complex element type. Each data model attribute becomes a simple element type, and the one-to-many relationship is recorded as a sequence. Exhibit 1:Data model for 1:m relationship

In the previous chapter, we introduced a simple XML schema, XML document, and an XML stylesheet for a single entity data model. We now include more features of each of the key aspects of XML.

Implementing a one-to-many relationship
There are three different techniques for implementing a one-to-many relationship:

Containment relationship: A structure is defined where one element is contained within another. The "contained" element ceases to exist when the "container" element is removed. For instance, where a city has many hotels, the hotels are "contained" in the city. Intra-document relationships: In a case where you have one city with many hotels, rather than a city containing hotels, a hotel will have a "location in" relationship to a city. A city id is used as a reference on the hotel element. Therefore, rather than the hotels being contained in the city, they now just reference the city's id via the cityRef attribute. This is very similar to a foreign key in a relational database. Inter-document relationships: The inter-document relationship is much like the intra-document relationship. It also uses the id and idRef attributes to assign an attribute to a parent attribute. The difference is that the inter-document relationship is used when tables, such as the city and hotel tables, might live in different filesystems or tablespaces.

Exhibit 2:Checklist for deciding what technique to use:

XML schema
Some of the built-in data types for an XML schema were introduced in the previous chapter, but still, there are more that are very useful, such as anyURI, date, time, year, and month. In addition to the built-in data types, a custom data type can be defined by the schema designer to accept specific data input. As we have learned, data are defined in XML documents using markup tags defined in an XML schema. However, some elements might not have values. An empty element tag can be used to address this situation. An empty element tag (and any custom markup tag) can contain attributes that add additional information about the tag without adding extra text to the element. An example will be shown in the chapter, using attributes in an empty element tag.

Empty elements with attributes in XML document
Elements can have different content types depending on how each element is defined in the XML schema. The different types are element content, mixed content, simple content, and empty content. An XML element consists of everything from the start of the element tag to the close of that element tag.

 * An element with element content is the root element - everything in between the opening and closing tags consists of elements only.
 * A mixed content element is one that has text and as well as other elements between its opening and closing tags.
 * A simple content element is one that contains only text between its opening and closing tags.
 * An empty content element, which is an empty element, is one that does not contain anything between its opening and closing tags (or the element tag is opened and ended with a single tag, by using / before the closing of the opening tag.

An empty element is useful when there is no need to specify its content or that the information describing the element is fixed. Two examples illustrated this concept. First, a picture element that references the source of an image with its attributes, but has no need in specifying text content. Second, the owner’s name is fixed for a company, thus it can specify the related information inside the owner tag using attributes. An attribute is meta-information, information that describes the content of the element.

XML schema data types
Some of the commonly used data types, such as string, decimal, integer, and boolean, are introduced in chapter 2. The following are a few more data types that are useful.

Exhibit 3:Other data types:

More data types
Besides the built-in data types, custom data types can be created as required. A custom data type can be a simple type or complex type. For simplicity, we create a custom data type that is a simple type, which means that the element does not contain other elements or attributes. It contains text only. The creation of a custom simple type starts from using a built-in simple type and applying it with restrictions, or facets, to limit the acceptable values of the tag. A custom simple type can be nameless or named. If the custom simple type is to be used only once, then it makes sense to not name it; thus, that custom type will only be used in where it is defined. Since a named custom type can be referenced (by its name), that custom type can be used wherever necessary.

A pattern can be used to specify exactly how the content of the element should look. For example, one might want to specify the format of a telephone number, a postal code, or a product code. By having a defined pattern for certain elements, the data exchanged will be uniform and the values will be consistent when stored in a database. A useful way to set patterns is through Regex, which will be discussed in later chapters.

Schema examples
The following is a schema that extends the schema introduced in the previous chapter to include a one-to-many relationship of city to hotels with two examples of custom data types.

Exhibit 1:Data model for 1:m relationship
 * [[Image:Xmldmcityhotel.png|1:m relationship - City Hotel]]

Important, this is a continuing example, so new code is added to the last chapter's example!

Inter-document example
Refers to Chapter 2 - A single entity for steps in using NetBeans to create the above XML schema.

XML document
Attributes

 * The valid element naming structure applies to attribute names as well
 * In a given element, all attributes’ names must be unique
 * An attribute may not contain the symbol ‘<’ The character string ‘&lt;’ can be used to represent it
 * Each attribute must have a name and a value. (i.e. , filename is the name and pan_pacific.jpg is the value)
 * If the assigned value itself contains a quoted string, the type of quotation marks must differ from those used to enclose the entire value. (For instance, if double quotes are used to enclose the whole value then use single quotes for the string: John Smith )

Table 3-2: XML Document for a one-to-many relationship – city_hotel.xml

Refers to Chapter 2 - A single entity for steps in using NetBeans to create the above XML document.

Exercises
In order to learn more about the one-to-many relationship, exercises are provided.

Answers
In order to learn more about the one-to-many relationship, answers are provided to go with the exercises above.


text:
Computers for Beginners/More Basics

Background Information
Any information that is stored on the computer for later use is saved in a file. If there was music on your computer, it might be saved in an mp3 or ogg vorbis file. If one acquired a picture of a monkey and wanted to save if for later use, they might save it as a jpg or png file.

File Naming
When there is a file on the computer, it is saved on the storage drive with a name. This name can be whatever you may desire, but it will have an extension at the end. For example, the picture of a monkey might be called monkey.png. There may be spaces, dashes, and underscores in file names. Using any other special characters such as parenthesis and @ symbols is generally a bad idea. Capital Letters can also be used. If you had a band with the name of Green Pickles and a song called Soup, the name for this song on the computer could be GreenPickles-Soup.mp3, Green Pickles - Soup.mp3, green_pickles-soup.mp3, or whatever you want.

There will also be times when files don't have an extension. Those are mostly text files or executable files which will be explained.

Organizing and Creating Files
Hard drives are very big these days, and people have a whole bunch of files. To organize all of these files, a method that can be thought of as a brief case can be used. To look at files, humans use a file manager. Clicking on My Computer from the desktop or start menu will do.

Try this:

 * 1) Click on My Computer on the Desktop or Windows Explorer (Start -> Accessories -> Windows Explorer).
 * 2) Go to Tools -> Folder Options... and click on the File Types tab.
 * 3) If "Hide extensions for known file types" is checked, uncheck it by clicking in the checkbox.
 * 4) * The previous two steps are not necessary but very useful.
 * 5) Go to the c: drive (pronounced "c drive") by double clicking on it.
 * 6) * You have just dived into your hard drive. This is where most of your files are stored.
 * 7) * a is the floppy drive, d is usually the cd-rom or dvd-rom drive, and other peripherals like usb drives are the next letters.
 * 8) Scroll up and down. You should see quite a few folders. One will be called "Program Files" and another one will be called "Windows." There will also be more.
 * 9) The first thing you need is a place to put all of your personal documents. Luckily, Windows has these folders already pre-made. There are folders called My Documents in Windows.
 * 10) When you are in the c drive, right click in empty space and click New -> Folder. You can also do this by going to File -> New -> Folder. Give the folder your name by typing it, and press Enter on the keyboard when your finished.
 * 11) Now, go into that folder by double clicking on it.
 * 12) If you want, make a few more folders inside of that one for more organization. For example, have one for pictures, one for school/work, one for music, and whatever you can think of. You can also make those folders as you go along.
 * 13) Close the window when finished.

Now we are going to make a simple file.
 * 1) Open Notepad.
 * 2) Type "Hello, I am practicing my computer skills" or whatever you want to type.
 * 3) Click on File -> Save As
 * 4) Navigate to C:\YourName\ and call the text file practice.txt.
 * 5) Open My Computer or Windows Explorer.
 * 6) Go to C:\YourName\
 * 7) You will see that there is a file called practice.txt.
 * 8) Double click on it.
 * 9) * practice.txt was just opened by Notepad!
 * 10) Close Notepad.

Moving Files with the Cut, Copy, and Paste
You will now learn how to move files.
 * 1) Open up My Computer or Windows Explorer
 * 2) Make a new folder in your folder called computer-practice. For example, it could be C:\YourName\computer-practice\.
 * 3) Go to C:\YourName\, right click on practice.txt, and press copy.
 * 4) Go to C:\YourName\computer-practice\
 * 5) Right click in empty space and choose paste.
 * 6) Go to C:\YourName\, right click on practice.txt, and press delete.
 * 7) * When the file is deleted, it is not gone forever. To remove it more, read about the  .
 * 8) * To save time, cut could have been used instead of copy. The difference is that when you paste after pressing cut, the file is deleted from its original location. You shouldn't use this too much just in case something goes wrong.

Moving one file at a time is fun, but sometimes we need to move a whole bunch at a time.
 * 1) Go to C:\YourName\computer-practice\
 * 2) Copy practice.txt
 * 3) Press paste lots of times. (Hold ctrl and keep pressing v)
 * 4) * The new files will be renamed to something like "copy of practice.txt"
 * 5) There are two ways to select all of the files in this folder.
 * 6) Press Ctrl-A or Edit -> Select All
 * 7) Click the one on the top, hold shift, and click the one on the bottom.
 * 8) When they are all highlighted, right click on one of them and press copy.
 * 9) * As usual, the same thing can be done with Edit -> Copy or Ctrl-v
 * 10) Go to C:\YourName\
 * 11) Paste the files
 * 12) Delete all the text files that you just made in C:\YourName\

Sometimes, you only want to select some of the files in a folder
 * 1) Go to C:\YourName\computer-practice\
 * 2) Click on the first file.
 * 3) Hold shift down and click on another one towards the middle.
 * 4) * The files you clicked on and everything in between has been selected.
 * 5) Click in empty space so everything is deselected.
 * 6) Click on the first file again.
 * 7) Hold ctrl and click another file.
 * 8) * The files you clicked are the only ones that are selected.
 * 9) Hold ctrl and click yet another file.

Searching For Files
Knowing the name of a file, but not the location, can be very aggravating. Who wants to spend hours searching through an endless amount of folders? The answer is not very many people and that is what file searching is for.

Let's say that someone has a file called important.zip. This person lost it, so they are now going to find it quickly. Here's what the person would do.
 * 1) Right click Start -> Search.
 * 2) Type in the file name, select where the file may be located [i.e. My Documents, My Computer.]

There are other things to be done with the find utility. One feature is using *. The * means that anything can be there. For example, searching for *.ogg would find any file that had a .ogg at the end. Searching for *report* would find anything that had the word report in the file name.

Shortcuts
A shortcut is a file that leads to another file or a website. There main use is for the desktop (The screen behind all the windows. Icons are on the desktop).

Here is one way to use a shortcut.
 * 1) Right click on the desktop.
 * 2) Click New -> Shortcut
 * 3) A wizard will come up.
 * 4) Make the shortcut name "Notepad" and the link location C:\WINDOWS\system32\notepad.exe.

With this shortcut, you can open Notepad in the click of a button. You may now delete this shortcut if you think you'll never use it by right clicking on it.

Here is one way to use an Internet shortcut.
 * 1) Open up My Computer or Windows Explorer
 * 2) Go to C:\YourName\computer-practice\
 * 3) Do the same thing that you would do to make a folder, but click shortcut instead.
 * 4) A wizard will come up.
 * 5) Make the shortcut name "Never Used A Computer" and the link location http://en.wikibooks.org/wiki/Never_Used_A_Computer/.

With this shortcut, you can go to this book's website when you are in the computer-practice folder. You may now delete this shortcut because you'll probably never use it.

Compression
When a file or more than one file is too big, compression can be used. Compression is used on the internet a lot because it is better to download smaller files. If you've ever seen a file with a .zip extension, then you have witnessed the most common type of compression in Windows. There are also compression formats with a .gz, .bz2, or .rar extension.

Extracting
"I got a .zip file from the Internet, now what?" is a common question. The answer is actually quite simple starting with Windows XP. All you have to do is right click on it in My Computer or Windows Explorer and choose "Extract All..." A wizard will come up and it is pretty straightforward. When you click next, it will ask you where to put the files. The default folder is usually good, but you can change it if necessary. Also, if the file has a password, click the Password... button and type it in. When you click Next, the files will be extracted to the directory, and one more thing will come up. It'll be a check box that says "Show extracted files". It is a good idea to make this unchecked because it'll open up a new window if you don't, which is annoying.

If you have an older version of Windows or want to be able to extract files other than zip files, a great option is 7-zip. It works basically the same way as the first method, or you can use their 7-zip File Manager which should be in the Start Menu.

Compressing
Just like files can be uncompressed, it is also possible to make your own .zip files. The easiest way to do this is to do these steps.
 * 1) Open up My Computer or Windows Explorer
 * 2) Right click on a file or folder that you want to be compressed.
 * 3) Choose Send To -> Compressed (zipped) Folder
 * 4) * A .zip file with the same name as the original file for folder will be created in the same directory as the original file or folder.
 * 5) * For example, if there was a file called C:\Ronald\Bacon\BaconBenefits.doc, and you compressed it, there would now be a file called C:\Ronald\Bacon\BaconBenefits.zip. BaconBenefits.doc is still there.
 * 6) If you want to add to the archive, right click the .zip file and click Open.
 * 7) You can add more files by copying and pasting them in the new window that opens up.

The same can be done in other applications such as 7-zip, WinZip, and WinRar.

Types of Applications
There are a few types of applications
 * Open Source - Free software where the source code, the language programmers use to make computers do their jobs, is available to everyone. Anyone can change the code and see how it works.
 * Freeware - This is free software that you may acquire legally without any fees.
 * Shareware - This is free software that you may acquire legally without any fees. However, there may be some restrictions (e.g. more advanced features may require a payment.)
 * Commercial - This software is normally sold in boxed packages.
 * Free-Trial - This software allows a period of time where the software can be used without cost, but extended use requires a small payment.

Installing Applications in Linux
Most Linux-based operating systems come with a package manager, a tool that simplifies the process of installing and removing programs, and automatically updates installed software. A list of how to install application in major distributions follows (application= name of application you want to install):

Arch/Chakra Linux
pacman -S application

Fedora/Red Hat Linux
dnf install application

Ubuntu
apt install application

Installing Applications in Windows
Your computer probably came with some software already on it, but what if you need to do something fancy? For almost every task you could think of doing with a computer, there is a software program to let your computer do it.

There are a few ways to install applications

 * Install from the internet
 * Install a program from the Microsoft store application. This simplifies the install, and handles updates for you.
 * You download a file, extract it if it's compressed (.zip, .bz2, .gz, etc.), click on the file with a .exe extension, and follow instructions.
 * On a very rare occasion you have to compile a program from source. This can be tough, but the program will probably have a file called INSTALL that should be read.
 * Install from a CD or DVD - You put the disc in the drive and follow instructions. The defaults will usually work fine.

Learning is fun, so here's some more useful info. An executable file has either no extension at all or a .exe extension. When a file is executable, you can run it and it will do something. For example, the executable for Notepad is C:\WINDOWS\system32\notepad.exe. When you click on Start -> Accessories -> Notepad, it is really the same thing as going to notepad.exe in Windows Explorer and double-clicking it.

Windows
Sometimes a application is installed, but you don't want it anymore. Removing unwanted applications is like eating pie, easy.
 * 1) Go to Start -> Control Panel -> Add or Remove
 * 2) Highlight what you want to remove by clicking on it.
 * 3) Press Remove or Change/Remove
 * 4) Follow the instructions. They will most likely be pretty straight forward.

Customizing
This chapter will not explain every secret and tweak in Windows. What it will do is describe things that everyone should know in terms of customizing their computing experience to suite their needs.

Contents
Applications usually put themselves in the Start Menu automatically, but you may find that there is something that needs to be added or tweaked.

There are two ways to edit what is inside of the start menu. Try the first one, and if that doesn't work, the second method will work.

Auto-Hide the Taskbar
Have you ever seen someone's desktop and noticed that there is no taskbar, but then when they put their mouse at the bottom of the screen, it appears? That is called auto-hiding the taskbar. This is the kind of thing where some people love it and some people hate it. Here's how to do it. To make it normal again, do the same thing but this time uncheck the box.
 * 1) Right click on the taskbar.
 * 2) Click Properties.
 * 3) Check Auto-hide the taskbar.
 * 4) Press OK or Apply. (Apply won't close the window)

Show Quick Launch
To deactivate it, do the same thing but this time uncheck the box.
 * 1) Do the first two steps as Auto-Hide the Taskbar.
 * 2) Right click on the taskbar.
 * 3) Click Properties.
 * 4) Check Show Quick Launch.

Screen Size (Resolution)
There are two factors in determining the size of an image on the monitor. The first one is obvious, the physical size of the monitor. These would be values like 22", 24", 28", etc. Of course, this can't be changed without buying a new monitor.

The other thing that can determine the size of images on the screen is the resolution. To understand it, you have to know a little bit about how monitors work. It's not much.

A pixel is dot on the screen that has exactly one color. When all of those dots are combined, they make up what you see on the monitor. The resolution is how many dots are on the screen. When people describe resolution, they are talking about how many rows and columns of pixels there are, like a chess board. For example, a resolution of 1920x1080 means that there are 1920 pixels going from the left to the right of the screen in one row, and there are 1080 pixels going from the top to the bottom of the screen in one column.

Changing the resolution is very simple in Windows.
 * 1) First right click on the desktop, the area in the screen where the icons are, and click properties. You can also do the same thing by going in Start -> Control Panel -> Display.
 * 2) After that, go to the Settings tab.
 * 3) Click and hold the slider under Screen Resolution. Let go when you are at a suitable resolution.
 * 4) * 1920x1080 is a good choice. If your monitor can go higher, then do that if the text isn't too small.
 * 5) Press Apply.
 * 6) * If the screen looks really weird and gets all funky, don't panic. If you don't press Ok to the dialog that comes up, it will automatically change back to where it was in a few seconds.

title: Computers for Beginners/More Basics


text: Japanese/Kanji/Lesson 4

title: Japanese/Kanji/Lesson 4


text:
Cookbook:Boeuf Bourguignon

| France | Meat | Stews

Boeuf bourguignon (French for Burgundy beef) is a well known, traditional French stew prepared with beef braised in red wine (originally Burgundy wine) and beef broth, flavored with garlic, onions, carrots, a bouquet garni and garnished with mushrooms. It is one of many examples of peasant dishes slowly evolving into haute cuisine. Most likely, the method of slowly simmering beef in wine originated as a means of tenderizing cuts of meat that were too tough to cook any other way. Also, simmering these two ingredients together helps to create a unique and pleasant flavor.

Formerly, chefs larded the meat with lardons, but modern beef is so tender and well marbled that this time-consuming technique is rarely necessary.

French culinary expert Auguste Escoffier first published the boeuf bourguignon recipe once the dish became a standard of French cuisine. However, over time it has undergone subtle alterations, owing to changes in cooking equipment and available food supplies.

Ingredients

 * 1 kg beef (blade steak or stewing shoulder cut)
 * 3 large onions, chopped
 * 1–2 cloves of garlic
 * 300 g of chopped carrot
 * 300 g of button mushrooms
 * Bouquet of herbs: basil, rosemary and thyme tied with string
 * 2–4 cups of red wine
 * Salt
 * Pepper
 * Olive oil
 * Flour

Preparation

 * 1) Cut beef into large, 5 cm chunks. Season with salt, pepper and olive oil.
 * 2) Cook beef a small amount at a time in a dutch oven or large saucepan until it's golden brown on the outside. Remove each portion as it's done and set aside.
 * 3) Add onions and garlic to the same pot and cook till golden brown.
 * 4) Sprinkle with 2 tablespoons of flour. Cook a few minutes longer.
 * 5) Add 1–2 cups of red wine.
 * 6) Bring wine to a boil then simmer for 5 minutes.
 * 7) Add beef, carrots, herbs, and mushrooms.
 * 8) Add another couple cups of red wine and water (enough to cover the meat and vegetables).
 * 9) Cook for 2–3 hours, stirring occasionally. Remove surface scum as required.

Notes, tips, and variations

 * Use leftovers to make meat pies or puff pastries.
 * Add a small amount of dried Trompettes de la mort (Craterellus cornucopioides). For a dish serving 4, 4 trompettes can be used depending on taste. Cut off parts that may contain sand, rinse, cut in small pieces, soak for 5 minutes in warm water and add to the dish after adding the wine. Can be used instead of garlic.
 * Use a low-fat cut of beef and serve with boiled potatoes or steamed rice on the side to lower the fat and calories of this dish.

title: Cookbook:Boeuf Bourguignon


text:
Discrete Mathematics/Axiomatic set theory

See also: Topology/Set Theory


text: User:SchuhFits

title: User:SchuhFits


text:
Talk:Climate Change

I would strongly suggest that this entire wikibook be deleted as a "wikibook" and it become a subchapter of a parental topic such as atmospheric science, meteorology, and climatology. It should also be written from the point of neutrality - which I would suggest means that as long as their is a controversial discussion about "climate change" that it not be represented from a single point of view (i.e. man-made, and/or natural).

Per the person who made this statement - "the vast majority of the scientific community now believe that we must cut our emissions or else face some truly terrible consequences." You should perhaps read this: http://en.wikipedia.org/wiki/Milankovitch_cycles.

Hmmm... I have to say this book is attracting more interest from Climate Change cynics than I had suspected. My own understanding of the situation is that - yes - the scientific community was unsure of the causal link between man-made pollution and climate change. But things have moved on significantly in the past few years and it's now almost certain that humans are responsible for the recent increases in temperatures that we have witnessed.

For example: http://www.nature.com/news/2005/050131/full/050131-14.html

For those of you with access to the academic literature, I urge you to run some searches on "climate change". The vast majority of the scientific community now believe that we must cut our emissions or else face some truly terrible consequences. People (not to mention countless animals and plants) are already dying because of our actions. Please look at the up to date facts before posting any cynical content on this book.

Please. For the good of our planet. - Dan AKA Jack 7:10am 9th Feb 2005 GMT

Follow the money.

You need to understand academic literature and research before you can properly judge this. Consider three equally prominent (or obscure) researchers with equally sound (or unsound) science. One researcher says that we are causing our doom by our actions, one says that we are doomed no matter what, and one says that everything will be OK. Which of the three will get funding?

Going against the flow of money is a dumb career move, at least in the short term. Consider the cases of stomach ulcers (caused by bacteria) and cancer growths (which emit chemicals that cause blood vessel growth). Money went to the researchers denouncing these now-proven theories. In the field of computer science, from about 1985 to 1995 all OS researchers had to accept the idea that microkernels were superior. Those who pointed out the idiocy of microkernels were denied funding. Linus Torvalds, lead developer for Linux, was even taunted/confronted/teased/insulted by one of the big-name OS researchers for going against the fad... yet today we see that Linux is a solid high-performing OS while not even one true microkernel (no, MacOS X only pays it lip service) is into the big leagues. That big-name OS researcher had a profitable textbook to promote; you can see the bias if you follow the money. For many, tenure is at stake, and so they must support those who will be voting to grant tenure or not.

Climate change research is certainly not immune to these political problems. Researchers must defend themselves and their profitable theories. It's all "gloom and doom, but send more money and maybe we can figure something out or see how bad it is".

AlbertCahalan 02:15, 20 Feb 2005 (UTC)

I have just contributed some words on anthropogenic climate change. Although this is not my field, I don't want to see it parched and lifeless, if I may be allowed to freely mix metaphors:-)

My comment on Pascal's wager may be of interest regarding the arguments for and against the validity of climate change evidence.

Patrick, aka:

Logicmanalf 19:49, 31 May 2005 (BST)

When we refer to 'emissions' we may need to distinguish between the different sorts of emissions: carbon dioxide contributes around 20% of the total greenhouse effect and methane, nitrous oxide and a number of other gases contribute another 20%. Keith Thomas 12:00am GMT 14 February 2005

Sources from wikipedia
(William M. Connolley 23:09, 19 Feb 2005 (UTC)) There are lots of source pages within wikipedia for content here: the most obvious are: http://en.wikipedia.org/wiki/global_warming, http://en.wikipedia.org/wiki/Historical_temperature_record, http://en.wikipedia.org/wiki/Attribution_of_recent_climate_change and links therein.

 * (William M. Connolley 21:42, 20 Feb 2005 (UTC)) Hmmm... well, the wikipedia climate change pages are fairly mature. I can't quite see the point in fighting the same old wars over here.

some scientifically oriented content
I've been editing this page a bit. I'm trying to insert some sound science into the description of climate change. I think the organization of the page needs some work too, and if I have a chance I'll try to arrange things in a little more coherent order. Skymath 07:52, 6 December 2005 (UTC)

Orphaned pages
I found some orphaned pages related to this book:
 * Climate change-Contained "combustion"

--Derbeth talk 13:57, 11 January 2006 (UTC)

Great Global Warming Swindle
A programme by the same name as this was screened in the UK in March 2007. It caused great debate because it provided a number of reasonably highly profiled academics who repudiated many of the claims of the climate change scientists. Upon closer scrutiny, however, and after watching the online debate, it would seem that the contrarian view (one often taken by this particular documentary film maker) was somewhat misinformed and misrepresented some of the facts. This massaging was sufficient to call into doubt the integrity of the programme. Others discontent with the programme, in various discussion venues on the internet cited a webpage called exxonsecrets.com which indicated cash flows from Exxon, to key government departments and research individuals, who presumably have been on the payroll to promote a degree of disinformation. It would seem that the only thing absolutely certain about climate change is that it produces spin.

 * I think that climate changes is evident, even to the proverbial blind person. The reason can be varied but thinking that humans have no impact is the same as stating that acid rain never existed, that we are not about to reach the planet record population of 9 billions (and all the related implications) and the it is becoming increasingly rare to find an ecosystem without stress due to human actions.
 * The issue is that there are geopolitic interests behind the disinformation. Coal producers for instance, and at the other end of the spectrum solar cell producers or heolic turbine manufactures. This of course in due to economic motives and in great part of the world, economics dictates policies and shapes the future. For me some of the greatest errors coming from this are the "fake solutions" like carbon trading or similar concepts, like most of the carbon capturing proposal, all these invested interests are creating a general mess on the ability to scientifically address the issues so we can at least stabilize the situation.   --Panic (talk) 21:43, 8 December 2010 (UTC)

GLOBAL DIMMING
The article fails to mention anything about global dimming. This effect, under it's current understanding, is a very large factor in the progress of the climate. Someone should add it in. LealandA (talk) 13:09, 17 November 2008 (UTC)

title: Talk:Climate Change


text:
Talk:Computer Programming/Archive1

namespaces
I have moved things back into the proper namespace, articles on programming should be in the Programming: namespace (IMO the new naming scheme wasn't very workable and was a bit cumbersome in some times). If you wish to update the older articles, update the articles.

Hope I'm not treading on many feet here. I like the new systematic handling of this textbook, it just needs to be named better. Dysprosia 08:40, 20 Aug 2003 (UTC)

Basic pascal and ... are functional languages? They are PROCEDURAL languages. Functional languages are very different languages e..g Haskell.

Shouldn't these things be in seperate text books? I don't see why they all should be considered one book. Do you ever go to the bookstore and buy a book that tells you how to program 20 languages? LittleDan 19:43, 8 Oct 2003 (UTC)
 * I agree. Noldoaran 17:00, 8 Dec 2003 (UTC)

 * Is there one language that people could agree on to write one textbook in? Once we have one textbook, most of it can likely be ported to other languages.  Another option is to write one textbook completely language neutral (maybe pseudo-code) and have separate textbooks that teach languages specifically without the accompanying concepts (presumably with the concept book as a prerequisite).  Any thoughts?  Xygorn 5:49, 9 Dec 2003 (UTC)
 * Nope, doesn't work like that. Each language has its own nuances and particulars that need to be taken into consideration - take C for example. In order to teach people how to write good C, you have to instruct them as to what the Standard says on things, about bounds checking, and all that lovely C-jazz, which you don't need to in other languages. Other languages may have their own special specifics. Dysprosia 06:03, 9 Dec 2003 (UTC)

I thought that Computer programming is like the gateway page to all the Programming: languages? Each language can be considered to be in its own volume? I don't know, just my 2c Dysprosia 09:36, 23 Oct 2003 (UTC)

The way Computer programming is positioned on the main page, I interperted it to be a shelf of it's own with the various texts on languages therin. Though Computer programming does need to have it's own book however for the programming patterns that are not language dependent, like looping in general or how to build a recursive function/method to keep all of the book from having to be Language101. As it stands, I see Computer programming as a very good introduction page. Dbroadwell 03:21, 22 Dec 2003 (UTC)

Should beginners care about speed?
I'm going to remove a lot of discussion on speed differences in languages, which I think is largely irrelevant these days, especially for beginners in programming.

Even if you don't agree about speed of execution, at least you would agree about speed of compiling right? -- Kowey 13:15, 20 Dec 2003 (UTC)

 * Beginners, no. Compilation speed, yes. But what languages are that slow to be noticeable? Dysprosia 13:39, 20 Dec 2003 (UTC)

I think there should at least be a mention of Forth, though I don't know what category it belongs in.

To do
The Abstraction Leaks sub paragraph needs to be filled in. Kowey 16:49, 22 Nov 2003 (UTC)

I have changed the reference about Visual Basic as being easily learned to read simply as Basic because. I have also added a paragraph about availability and suitability of a language choice, and the need to learn to read legacy source code so it can be maintained or updated.
 * Visual Basic is NOT an easily learned language and was never designed as a teaching language. Microsoft seem to have developed it as a common automation technique for various Microsoft Office applications and it has been so useful or necessary that a lot of people now use it as a programming language.
 * Basic was originally developed as a teaching language.

Feel free to adjust/reverse if any of this is not correct 210.86.34.179 07:26, 7 Jan 2004 (UTC)

 * If that's the case, we might as well just get rid of Basic altogether (not an idelogical statement) because nobody actually teaches with it. VB is easy not because of the language but because of the visual GUI editor.  -- Kowey 09:23, 9 Jan 2004 (UTC)

BTW: Has anyone heard about Structured Programming - which is a language independent technique and can be done in any language the programmer might choose, including English. Rather than launching straight into a discussion about what language to choose, one could discuss the benefits of structuring and planning a programming task so that it can be organised into modules and sections before coding commences. The use of Structured Programming techniques can do as much for the speed of a programming application as the choice of language as they create efficient source code. 210.86.34.179 07:26, 7 Jan 2004 (UTC)

 * Of course. Wish there were more people capable of doing it. Wish I had time to write a book about it. But doing it right takes time and effort; I'm not sure if anybody here has that kind of time to spend. --BenTels 06:05, 25 Jan 2004 (PST)

I think the term Scripting has been abused. Scripting languages are used to script behaviours in executables. Lua and JavaScript are used exclusively for scripting, hence they would be Scripting Languages, however Python and Perl are only rarely used for embedding in other executables, so I wouldn't call them Scripting Languages. Interpreted would be a better name.

 * Lisp and perl are not interpreted, they are compiled just in time. CommonLisp runs as fast as C. The proper classification should include static and dynamic; where dynamic languages pushes as much choices to run-time and static languages try to do everything in compile-time.

 * For ease of learning I would sugest dropping Basic (which will teach you very bad programming) and opt for PLT Scheme as a good place to start. A small compact language with a good book to learn if from.

Hi, I just stumbled above the wikibooks site. I think the programming book is, err.. a bit incomplete :-) I mean, it is only talking about coosing a programming language! What about
 * Algorithms
 * complexity
 * sorting
 * searching
 * Data structures
 * procedural programming
 * structured programming
 * object oriented programming
 * functional programming
 * how to design a program (bottom up, top down, ...)

I could contribute to any of these subjects from time to time. I think it is important to create a good structure of these topics first, and THEN start writing some content. Not important stuff should be left out, who cares about all the tiny bits of advantages/disadvantages one programming language has over another? Just choose one that allows a simple start, and get going. IMHO Ruby would be a very good choice. It's both extremely simple to learn and all of the above programming concepts are possible with it.

We should switch to a Computer Science namespace.
Nobody gets a degree in computer programming; programming is just one aspect of computer science. Theoretical courses like discrete math might never involve writing programs. I suggest we use a computer science namespace for stuff like data structures and algorithms, and programming should be a category inside computer science that describes the nuances of various different languages.--Waxmop 00:23, 20 Jul 2004 (UTC)

 * I think things are fine as they stand. It would be unfortunate to have Computer science:Programming:Some Language as the title for an article. There doesn't need to be any highly visible overlap - we can create new wikibooks under the Computer science namespace or whatever if need be. Dysprosia 08:41, 20 Jul 2004 (UTC)

 * Why not use the category system instead of namespaces? A book on algorithms could be part of the Category:Computer science category. – [[User:Mxn|Minh Nguyễn (talk, blog) ]] 16:17, 13 Aug 2004 (UTC)

 * I agree with Minh Nguyễn. The category system is much more appropriate than a hierarchial namespace. --DavidCary 03:36, 15 Aug 2004 (UTC)

 * I don't think a "computer science" namespace is a good idea. Also, few or none of the other books (i.e. I couldn't find any) do that. Instead, they create a namespace for each book, and each chapter is at Bookname:Chapter name. I think this is a good idea. Categories are good too. The current Programming namespace is not good. --Yath 21:16, 22 Sep 2004 (UTC)

Exercises and solutions
I have not seen a consistent handling of exercises and their solutions in Wikibooks so far (someone correct me if I've missed it). I think exercises can reasonably be embedded in a page with explanatory text, or collected on separate exercise pages. I prefer to do the former. I plan to have separate 'solutions' pages though, which will have the same name as the page with the exercises, and '-solutions' appended. See Programming:Python Creating Python programs. --Yath 21:52, 24 Sep 2004 (UTC)

Setting the PATH in Windows
Several of these books have places (e.g. Programming:Python Getting Python) where a guide to setting the PATH in Windows would be helpful. I do not have a Windows system to experiment on. It would be great if someone could write this up. --Yath 01:04, 25 Sep 2004 (UTC)

Too much stuff!
Ugh. This module has become rather unwieldy. How do you think we should trim it? -- Kowey 09:08, 4 Oct 2004 (UTC)

Modules? Comments? Versioning?
 * Sequence
 * Selection
 * Repetition
 * Input/Output

title: Talk:Computer Programming/Archive1


text:
A-level Mathematics/OCR/C1/Differentiation

Motivation
Finding the gradient of a straight line is simple. For a line $$y=mx + c$$, the gradient is m. But how do you find the gradient of a curve at a particular point? Suppose we want to find the gradient of the tangent line to $$y=x^2$$ at the point (3,9). Immediately, this question poses a problem. How are we supposed to find the gradient of a line when only one point can be known? We can use a sequence of chords:

The gradient of the tangent line at (3, 9) is 6. As you can see, this method is rather tedious. Luckily for us we have the derivative, which can accurately and quickly determine any rate of change.

Derivative
The derivative is the rate of change of a function. It is found using the gradient of a line formula when the distance between the first and second point approaches 0: $$\lim_{h \to 0}\frac{f\left(x+h\right)-f\left(x\right)}{h} = f'(x)$$. A function is differentiable at the point a only if the limit exists at the point a. The notation for derivative is $$\frac{dy}{dx}\ or\ f' \left ( x \right )$$.

Differentiation Rules

 * 1) Derivative of a constant function:

$$\frac{d}{dx} \left (c \right) = 0$$
 * 1) The Power Rule:

$$\frac{d}{dx} \left (x^n \right) = nx^{n - 1}$$
 * 1) The Constant Multiple Rule:

If c is a constant and f(x) is a differentiable function: $$\frac{d}{dx} c f \left ( x \right ) = c \frac{d}{dx} f \left ( x \right )$$
 * 1) The Sum Rule:

$$\frac{d}{dx} \begin{bmatrix} f \left ( x \right ) + g \left ( x \right ) \end{bmatrix} = \frac{d}{dx} f \left ( x \right ) +  \frac{d}{dx} g \left ( x \right )$$
 * 1) The Difference Rule:

$$\frac{d}{dx} \begin{bmatrix} f \left ( x \right ) - g \left ( x \right ) \end{bmatrix} = \frac{d}{dx} f \left ( x \right ) -  \frac{d}{dx} g \left ( x \right )$$

Gradient of a Line
The derivative is used to find the gradient of a line at the point a. Then the point a is substituted into the derivative. The resultant will give us the gradient at the point a. The first derivative of a distance function is velocity and the second derivative is acceleration.

The distance that a car has travelled is given by the function $$s = 1.25t^2 -5$$t. Where s is the distance travelled in metres and t is the time elapsed. What is the velocity of the car when 12 seconds have elapsed?

1) Using the power, the constant multiple rule and the difference derivative rules we get v = 2.5t - 5:

2) Now we substitute the time into the function: v = 2.5x12 - 5, we get that v = 25 metres/second.

So after 12 seconds the car will be travelling at 25 metres per second.

Tangent Lines
Since we can find the gradient of a function at any point we can also find the equation of the tangent line to the function at any point:

What is the equation of the tangent line for the curve $$y = x^2 + 3x -16$$ at the point x = -4?

First we need to find the derivative of the equation.

$$\frac{dy}{dx} = 2x + 3$$

Then we find the gradient at the point x = -4.

$$2\left ( -4 \right) + 3 = -5$$

Using the original function we find the corresponding y value to the x value at the point of tangency.

$$y = -4^2 + 3 \left (-4 \right) -16 = -12$$

Then we use the point-gradient equation to find the equation.

$$y + 12 = -5 \left ( x + 4 \right )$$ so the equation of the tangent line is y = -5x - 32....

Normal Lines
A Normal line is the line perpendicular to the Tangent Line. Therefore when we determine the equation of the normal line the procedure is very similar to when we determining the equation of the tangent line. Example:

What is the equation of the normal line for the curve $$y = x^2 + 3x -16$$ at the point when x=-4?

First we need to find the derivative of the equation.

$$\frac{dy}{dx} = 2x + 3$$

Then we need to solve for the given x value.

$$2\left ( -4 \right) + 3 = -5$$

Then we use the perpendicular line equation to find the gradient of the normal line.

$$m_1 \times m_2=-1$$ so $$-5 \times m_2=-1$$ so the gradient is $$m_2=\frac{-1}{-5}$$ which is $$\frac{1}{5}$$.

Then we find the y value corresponding to x value

$$y = -4^2 + 3 \left (-4 \right) -16 = -12$$

Then we use the point-gradient equation to find the equation.

$$y + 12 = .2 \left ( x + 4 \right )$$ so the equation of the normal line is y = .2x - 11.2

Higher Derivatives
By C2, you will be expected to have a brief understanding of the applications of the second (and third, in some cases) derivative of a function. The second derivative is exactly what it sounds like, it is the derivative of a derivative. The third derivative is the derivative of the second derivative, and so on.The notation for the second derivative is, assuming we are talking in terms of y with respect to x: $$\frac{d^2y}{dx^2}$$ And subsequent derivatives simply have larger numbers in their indices.

Application of Derivatives to Graphs
Derivatives also help us to graph functions, by locating minimum, maximum, and inflection points. They can also determine the interval on which a function is convex or concave.

Convex - The graph is below the tangent lines. Example $$-x^2$$, think of a frown.

Concave - The graph is above the tangent lines. Example $$x^2$$, think of a smile.

Inflection Point - The point when a function changes from convex to concave or vice versa. Example $$x^3$$ at x = 0.

Maximum Point - The highest point of a function on a convex interval.

Minimum Point - The lowest point of a function on a concave interval.

Stationary Point - A point where f'(c) = 0

Rules of Stationary Points

 * If $$f' \left ( c \right ) = 0$$ and $$f'' \left ( c \right ) <0$$, then c is a local maximum point of f(x). The graph of f(x) will be concave on the interval.
 * If $$f' \left ( c \right ) = 0$$ and $$f'' \left ( c \right ) >0$$, then c is a local minimum point of f(x). The graph of f(x) will be convex on the interval.
 * If $$f' \left ( c \right ) = 0$$ and $$f \left ( c \right ) = 0$$ and $$f' \left ( c \right ) \ne 0$$, then c is a local inflection point of f(x).

Locating And Evaluating Stationary Points
The Function

The original function is used to determine when the function crosses the x and y axis's.

The First Derivative Test

The first derivative is used to find all the minimum, maximum or inflection points. At these points f'(x) = 0. Also if we make a sign chart we can see at which intervals the function is increasing and decreasing, but the second derivative is a better test of this.

The Second Derivative Test

The second derivative is used to find the points when a function is concave or when it is convex at these points f''(x) = 0. Then you need to make a sign chart. The interval(s) that the sign of the second derivative is positive the function is concave and if the sign is negative the function is convex. When f'(a) is zero and on that interval f(x) is negative then the point a'' is a maximum point, before this point the function will be increasing afterwards the function will be decreasing. When f'(a) is zero and on that interval f(x) is positive then the point a'' is a minimum point, before this the function will be decreasing afterwards the function will be increasing.

The Third Derivative Test

The third derivative test determines if a point is an inflection point. If f(c) = 0 and $$f'(c) \ne 0$$, then f(c) is an inflection point.

An Example
Sketch the graph of $$f(x) = x^3 -4x^2$$

1) The function will tell us that:
 * $$f(x) = x^2(x - 4)$$
 * $$0 = x^2(x - 4)$$ x = 0 and x=4 are the x-intercepts.
 * $$y = 0^2(0 - 4)$$ y = 0 is the y-intercept.

2) Then we use the first derivative test.

 * $$f'(x) = 3x^2 - 8x$$
 * $$0 = x(3x - 8)$$ x = 0 and x=2.66 are the minimum or maximum or inflection points.

3) Then we use the second derivative test.
 * f''(x) = 6x - 8
 * 0 = 6x -8 so x = $$\frac{4}{3}$$.
 * We now make a number line __f  (0) = -8______ 1.333____f  (2) = 4.
 * Before 1.333 the function will be convex after 1.333 the function will be concave.

4) Then we use the third derivative test.

 * f'''(x) = 6
 * $$f'''(\frac{4}{3}) = 6$$ The point is an inflection point.

5) Now we gather all the information into a table. Also we need to find the corresponding y values to all the x values that we obtained from steps 1 to 3 using f(x). 6)Now we can draw our graph:


text:
The Written Word

Table of contents

 * /Introduction/
 * /What Is A Word?/
 * /Historical Background/
 * /The Book/
 * /What does the word do for you?/

title: The Written Word


text:
Radiation Oncology/Simulation devices

Patient immobilization devices

 * BOHAP - Blue OverHead Arm Positioner

 * Karolinska Hospital (Sweden)(1998-2001) -- head mask vs head-and-neck mask setup
 * Randomized. 241/260 patients. H&N cancers excluding NPC. Arm 1) Head mask vs. Arm 2) Head-and-shoulder mask. Both Posicast thermoplastic masks. Assessed reproducibility (port films and table positions), patient tolerability (fit, pain, movement within, claustrophobia), and skin damage
 * 2005 PMID 15629618 -- "Randomized trial on two types of thermoplastic masks for patient immobilization during radiation therapy for head-and-neck cancer." (Sharp L, Int J Radiat Oncol Biol Phys. 2005 Jan 1;61(1):250-6.)
 * Outcome: No difference in reproducibility of setup
 * Toxicity: Claustrophobia head mask 45% vs. head-and-shoulder masks 58% (SS), otherwise no difference in tolerability. Skin toxicity Grade 3 head mask 21% vs. head-and-shoulder mask 39% (SS)
 * Conclusion: Smaller head masks reduced claustrophobial and skin reactions, without compromising reproducibility of the setup

title: Radiation Oncology/Simulation devices


text:
Cookbook:Soda

Soda is an ambiguous term that can refer to:

 * baking soda
 * soda pop: fountain drinks, carbonated beverages, etc.

title: Cookbook:Soda


text:
Radiation Oncology/Head & Neck/General

General Information About H&N Cancers

HPV Status

 * Please also see the Oropharynx page

 * DAHANCA 5 (1986-1990)
 * Randomized. 414 pts w/ pharynx or supraglottic larynx ca. RT median 66 Gy. Randomized: 1) +/- Nimorazole. Pts with low hgb level underwent 2nd randomization: 2) +/- transfusion.
 * HPV Response; 2009 PMID 19289615 -- "Effect of HPV-associated p16INK4A expression on response to radiotherapy and survival in squamous cell carcinoma of the head and neck." (Lassen P, J Clin Oncol. 2009 Apr 20;27(12):1992-8. Epub 2009 Mar 16.)
 * Subset analysis. p16/HPV+ in 22%
 * Outcome: 5-year LRC p16+ 58% vs p16- 28% (SS), DSS 72% vs 34% (SS), OS 62% vs 26% (SS). p16 remained strong predictor on MVA
 * Conclusion: Expression of p16 has a major impact on treatment response and survival
 * HPV & Hypoxia; 2010 PMID 19910068 -- "HPV-associated p16-expression and response to hypoxic modification of radiotherapy in head and neck cancer." (Lassen P, Radiother Oncol. 2010 Jan;94(1):30-5. Epub 2009 Nov 10.)
 * Subset analysis. 331 patients with pretreatment tumor blocks, stained for p16. p16+ in 25% (oropharynx 37%, SGL 21%, others 10%)
 * Outcome: Nimorazole improved LRC overall (HR 0.7, SS). p16- improved LRC overall (HR 0.4, SS). If p16-, nimorazole improved LRC (HR 0.7, SS), but if p16+, nimorazole did not improved LRC (HR 0.9, NS)
 * Conclusion: Hypoxic modification improved outcome in HPV/p16 negative patients, but had no impact in HPV/p16 positive patients

Lymphatic Risk

 * MD Anderson; 1972 (1948-1965) PMID 5031238 -- "Distribution of cervical lymph node metastases from squamous cell carcinoma of the upper respiratory and digestive tracts." (Lindberg R, Cancer. 1972 Jun;29(6):1446-9.)
 * Retrospective. 2044 patients, previously untreated. Incidence and topographic distribution of neck LNs

 * University of Florida; 1995 PMID 7782203 -- "Retropharyngeal adenopathy as a predictor of outcome in squamous cell carcinoma of the head and neck." (McLaughlin MP, Head Neck. 1995 May-Jun;17(3):190-8.)
 * Retrospective. 619 patients. Review of pretreatment CT and/or MRI to determine presence of retropharyngeal LNs
 * Outcome: Highest incidence in NPC (74%) and pharyngeal wall (19%). Neck relapse and DM significantly higher, RFS and OS significantly worse if present
 * Conclusion: Retropharyngeal adenopathy predictor of poor outcome

 * Washington University; 2002 (1997-2000) PMID 12128118 -- "Determination and delineation of nodal target volumes for head-and-neck cancer based on patterns of failure in patients receiving definitive and postoperative IMRT." (Chao KS, Int J Radiat Oncol Biol Phys. 2002 Aug 1;53(5):1174-84.)
 * Retrospective. 126 patients treated with IMRT. System for nodal target volumes used. Median F/U 2.2 years. Patterns of failure analyzed.
 * Outcome: persistent/recurrent nodal disease in 12% of definitive IMRT patients and 9% of postop IMRT patients
 * Conclusion: Development of guidelines for nodal target volumes

Incidence of contralateral and bilateral lymph nodes:
 * >30% cN+ bilateral - pharyngeal wall (50%), pyriform sinus (49%), supraglottis (39%)
 * cN- but pN+ bilateral - pyriform sinus (59%), BOT (55%), phar. wall (37%), oral tongue (33%), supraglottis (26%), FOM (21%), glottic larynx (15%)
 * Adopted from Chao 2002 (PMID 12128118)

Higher dose may "make up for" prolonged treatment time

 * MSKCC, 1990 - PMID 2325418 — "Impact of the time interval between surgery and postoperative radiation therapy on locoregional control in advanced head and neck cancer." Schiff PB et al. J Surg Oncol. 1990 Apr;43(4):203-8.
 * Patients receiving < 60 Gy had locoregional recurrence rate of 7% when SRI < 6 weeks vs 27% when greater than 6 weeks. However when doses > 60 Gy were given, failure rates were 15% and 12%, respectively.

Recursive Partitioning Analysis

 * RTOG RPA, 1996 - PMID 8646692 — "Recursive partitioning analysis of 2105 patients treated in Radiation Therapy Oncology Group studies of head and neck cancer." Cooper JS et al. Cancer. 1996 May 1;77(9):1905-11.
 * 2105 pts. For survival, most predictive factor was T stage. For T1-T2, next most important was tumor location, whereas for T3-T4, it was KPS. For LRC, N stage was most important; for N0, T stage was next most important, whereas for N+ number of treatment fractions was.
 * Survival: Group 1) T2 or less, glottic, age < 75; 2) Group 1 but age >= 75; 3) T2 or less, not glottic, N0-2, KPS >= 80, age < 75; or T3-4 N0-2 site:NP,OP,SGL,sinus KPS 90-100; 4) T2 or less, not glottic, N3, KPS >=80; or T3-4, N0-2, KPS 90-100, oral cavity, hypopharynx, or glottis; or T3-4 N3 or Nx KPS < 90; 5) T2 or less, not glottic, KPS <80; or N3, T3-4, KPS 90-100, N3; or T3-4, N1-3, KPS < 90.
 * Local control:

 * PMID 11443750 - "Validation of the RTOG recursive partitioning classification for head and neck tumors." Cooper et al. - Tested validity using a separate database from RTOG 85-27.

 * PMID 15672358, 2005 - "Comparison of the Radiation Therapy Oncology Group recursive partitioning classification and Union Internationale Contre le Cancer TNM classification for patients with head and neck carcinoma."
 * 2166 pts classified both by RPA and TNM stage and compared overall survival and loco-regional DFS. No differences between the systems in terms of survival, but for locoregional control, RPA system depended on treatment and was not generalizable.

Surgery + RT vs. Chemo-RT

 * Singapore (1996-2000)
 * Randomized. Stopped early due to slow accrual. 199 patients, resectable Stage III/IV SCHNC excluding NPC and salivary glands (larynx 32% (supraglottis 23%), oral cavity 27%, oropharynx 21%, hypopharynx 12%). T4 56%. Arm 1) surgery + adjuvant RT 60/30 vs. Arm 2) RT 66/33 + concurrent cisplatin 20 mg/m2 + 5-FU 1000 mg/m2 x2 cycles. 90% received at least 1 cycle of chemo
 * 2005 PMID 16012523 -- "Surgery and adjuvant radiotherapy vs concurrent chemoradiotherapy in stage III/IV nonmetastatic squamous cell head and neck cancer: a randomised comparison." (Soo KC, Br J Cancer. 2005 Aug 8;93(3):279-86.) Median F/U 6 years
 * Outcome: 3-year DFS: S+RT 50% vs. chemo-RT 40% (NS). Organ preservation (larynx/hypopharynx 68%, oropharynx 55%, oral cavity 21%). Chemo-RT group had poor surgical salvage of 47%, with no long-term survivors (possibly due to larger proportion of T4 and oral cavity cancers)
 * Conclusion: Chemo-RT not superior to surgery+RT, but can be attempted for organ preservation in larynx, hypopharynx, and oropharynx. Poor organ preservation (and salvage) in oral cavity

Reviews

 * PMID 3278390, 1988 - "Radiation Therapy Oncology Group (RTOG) studies in head and neck cancer."

Prediction of Response

 * MSKCC, 2007 PMID 17416856 -- "Identification of angiogenesis/metastases genes predicting chemoradiotherapy response in patients with laryngopharyngeal carcinoma." (Ganly I, J Clin Oncol. 2007 Apr 10;25(11):1369-76.)
 * Gene arrays. Correlation of 277 genes (angiogenesis and/or mets) to locoregional control
 * MDM2 and erbB2 are predictors of locoregional failure in patients treated with chemo-RT

Patterns of failure

 * M.D.Anderson, Hong, 1985 - PMID 4027864 — "Patterns of relapse in locally advanced head and neck cancer patients who achieved complete remission after combined modality therapy." Hong WK et al. Cancer. 1985 Sep 15;56(6):1242-5.
 * 103 pts treated with induction chemotherapy followed by surgery and/or RT. 71 pts were free of disease. 5-year recurrence rate was 51% (39% local and 26% distant failure). Relapse patterns were affected by: site (oral cavity more likely to fail locally, hypopharynx more likely to have DM); type of treatment (surgery + RT had lower local failure); TN stage (T3-4N3 had higher risk of local and distant failure); oropharynx (higher local + distant failure).

Planned Neck Dissection

 * General consensus suggests that adjuvant neck dissection is not necessary for patients with N1 neck and CR after chemo-RT
 * Historically, ipsilateral neck recurrence was lower for N2/N3 disease after primary RT + adjuvant neck dissection than either modality alone
 * The necessity of adjuvant neck dissection after chemo-RT is controversial
 * ~25% of patients with clinical/radiographic CR who undergo neck dissection have residual disease
 * 30-40% of patients with clinical/radiographic detectable disease who undergo neck dissection have no residual disease
 * Thus, overall accuracy of neck response by clinical/radiographic evaluation is ~60%
 * PET shows promise, and appears more accurate than clinical, CT, or MRI neck evaluation
 * It may be reasonable to observe patients with clinical/radiographic CR, and negative PET 12 weeks after completing chemo-RT
 * It is not clear what negative PET means in setting of clinical/radiographic detectable disease
 * TROG 98.02 subset analysis suggests that after chemo-RT, ipsilateral failure is low in patients with clinical/radiographic CR

 * RTOG 98.02
 * Subset analysis. TROG 98.02 is Phase II randomized trial of RT 70/35 with Arm 1) concurrent cisplatin/tirapazamine vs. Arm 2) concurrent cisplatin/5-FU. Subset patients with initial N2-3 disease, who achieved complete clinical/radiological CR at 12 weeks (N2 63%, N3 40%) and no planned neck dissection was performed
 * 2008 PMID 18286488 -- "N2-N3 neck nodal control without planned neck dissection for clinical/radiologic complete responders-Results of Trans Tasman Radiation Oncology Group Study 98.02." (Corry J, Head Neck. 2008 Feb 19 [Epub ahead of print]). Median F/U 4.3 years
 * Outcome: First failure: local 4%, locoregional 2%, distant 28%, locoregional + distant 6%. No patients with neck-only failure
 * Conclusion: Patients with CR do not need planned neck disection

Radiation Injury
Please see Radiation Oncology/Toxicity/Head & Neck

Other histologic types
Sarcomatoid carcinoma - also called spindle cell carcinoma
 * MDACC; 1998 PMID 9591559 -- "Radiation therapy for early stage (T1-T2) sarcomatoid carcinoma of true vocal cords: outcomes and patterns of failure." (Ballo MT, Laryngoscope. 1998 May;108(5):760-3.)
 * 5-yr LC 94% for T1 and 54% for T2. 10-year DSS 92%, OS 63%
 * Conclusion: similar control rates to typical squamous cell carcinoma. The histologic diagnosis of sarcomatoid carcinoma by itself should not influence the decision to treat a patient with early stage glottic disease with irradiation.

Other Resources

 * eContouring Webinar - Head and Neck with Kenneth Hu, M.D.
 * ASTRO/ARRO Journal Club Webinar 2012 - "RT + Cetuximab for Locoregionally Advanced Head and Neck Cancer: 5 Year Update" with Alexander Lin, M.D.

To Add
add: hypopharynx - randomized surgery vs chemo/rt

title: Radiation Oncology/Head & Neck/General


text:
Radiation Oncology/Treatment time

This page discusses how the time from surgery, overall treatment time, and time to deliver a fraction have an impact on the success of treatment.
 * See also: the appropriate Radiobiology concepts.

The concept of surgery to radiotherapy interval (SRI):
 * effect of boost dose
 * sequencing of chemotherapy and radiotherapy (in particular for breast cancer)
 * Radiation Oncology:H&N - see discussion for head and neck tumors
 * Radiation Oncology:Breast overview - see discussion for breast cancer

The concept of overall treatment time:
 * cervical cancer

Head & Neck

 * Aarhus, 2007 (Denmark) PMID 17493700 -- "Tumor progression in waiting time for radiotherapy in head and neck cancer." (Jensen AR, Radiother Oncol. 2007 Jul;84(1):5-10.)
 * Retrospective. 61 patients with SCCHN, with diagnostic and treatment-planning CT scan. Median interval between scans 28 days (5-95)
 * Outcome: 62% patients measurable increase in tumor volume; median increase 46% (6-495%). 20% new LN mets, 16% progression in T-stage
 * Doubling time: median 99 days, but median for faster half was 30 days.
 * Conclusion: Negative impact of waiting time; majority showed progression within 4 week waiting time
 * Editorial PMID 17574695 -- "Killing time: The consequences of delays in radiotherapy." (Mackillop WJ, Radiother Oncol. 2007 Jul;84(1):1-4.)

 * Wisconsin; 2007 PMID 17305251 -- "Is there an optimum overall time for head and neck radiotherapy? A review, with new modelling." (Fowler JF, Clin Oncol (R Coll Radiol). 2007 Feb;19(1):8-22.)
 * LQ modelling for various schedules, calculated tumor log cell kill, with acute and late complications
 * Conclusion: Schedules of 4-6 weeks give 11 log10 cell kill; most require two fractions/day

 * Ontario, 1996 PMID 12118558 -- "The effect of delay in treatment on local control by radiotherapy." (Mackillop WJ, Int J Radiat Oncol Biol Phys. 1996 Jan 1;34(1):243-50.)
 * Monte Carlo simulation, applied to tonsillar CA
 * Outcome: at some point for every patient, probability of local control decreases sharply over a relatively short time period. LC rate decreases by ~10% per month for tonsillar CA
 * Conclusion: Waiting times for RT should be As Short As Reasonably Achievable (ASARA)

NSCLC

 * Netherlands; 2003 PMID 14676792 -- "Accelerated regrowth of non-small-cell lung tumours after induction chemotherapy." (El Sharouni SY, Br J Cancer. 2003 Dec 15;89(12):2184-9.)
 * Retrospective. 23 patients with NSCLC. Interval times between end of induction of start of RT evaluated. Mean time to RT 80 days
 * Outcome: 41% potentially curable patients became incurable; tumor median doubling 29 days (8.3-171 days)
 * Conclusion: Accelerated tumor cell proliferation, much shorter doubling time than for untreated tumors. Gain from chemo (tumor decrease) was lost with waiting for RT

title: Radiation Oncology/Treatment time