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User:TakuyaMurata/Sequences of numbers

The chapter begins with the discussion of definitions of real and complex numbers. The discussion, which is often lengthy, is shortened by tools from abstract algebra.

Real numbers
Let $$\mathbf{Q}$$ be the set of rational numbers. A sequence $$x_n$$ in $$\mathbf{Q}$$ is said to be Cauchy if $$|x_n - x_m| \to 0$$ as $$n, m \to \infty$$. Let $$I$$ be the set of all sequences $$x_n$$ that converges to $$0$$. $$I$$ is then an ideal of $$\mathbf{Q}$$. It then follows that $$I$$ is a maximal ideal and $$\mathbf{Q} / I$$ is a field, which we denote by $$\mathbf{R}$$.

The formal procedure we just performed is called field completion, and, clearly, a different choice of a valuation $$| \cdot |$$ could give rise to a different field. It turned out that every valuation for $$\mathbf{Q}$$ is either equivalent to the usual absolute or some p-adic absolute value, where $$p$$ is a prime. (Ostrowski's theorem) Define $$\mathbf{Q}_p = \mathbf{Q} / I$$ where $$I$$ is the maximal ideal of all sequences $$x_n$$ that converges to $$0$$ in $$| \cdot |_p$$.

A p-adic absolute value is defined as follows. Given a prime $$p$$, every rational number $$x$$ can be written as $$x = p^n a / b$$ where $$a, b$$ are integers not divisible by $$p$$. We then let $$|x|_p = p^{-n}$$. Most importantly, $$|\cdot|_p$$ is non-Archimedean; i.e.,
 * $$|x + y|_p \le \max \{ |x|_p, |y|_p \}$$

This is stronger than the triangular inequality since $$\max \{ |x|_p, |y|_p \} \le |x|_p + |y|_p$$

Example: $$|6|_2 = |18|_2 = {1 \over 2}$$

2 Theorem If $$|x_n - x|_p \to 0$$, then $$|x_n - x_m|_p$$.

Proof:
 * $$|x_n - x_m|_p \le |x_n - x|_p + |x - x_m|_p \to 0$$ as $$n, m \to \infty$$

Let $$s_n = 1 + p + p^2 + .. + p^{n-1}$$. Since $$(1 - p)s_n = 1 - p^n$$,
 * $$|s_n - {1 \over 1 - p}|_p = |{p^n \over 1 - p}| = p^{-n} \to 0$$

Thus, $$\lim_{n \to \infty} s_n = {1 \over 1 - p}$$.

Let $$\mathbf{Z}_p$$ be the closed unit ball of $$\mathbf{Q}_p$$. That $$\mathbf{Z}_p$$ is compact follows from the next theorem, which gives an algebraic characterization of p-adic integers.

2 Theorem ''
 * $$\mathbf{Z}_p \simeq \varprojlim \mathbf{Z} / p^n \mathbf{Z}$$

where the project maps $$f_n: \mathbf{Z} / p^{n+1} \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}$$ are such that $$f_n \circ \pi_{n+1} = \pi_n$$ with $$\pi_n$$ being the projection $$\pi_n: \mathbf{Z} \to \mathbf{Z} / p^n \mathbf{Z}$$.''

The theorem implies that $$ \mathbf{Z}_p$$ is a closed subspace of:
 * $$\prod_{n \ge 0} \mathbf{Z} / p^n \mathbf{Z}$$,

which is a product of compact spaces and thus is compact.

Sequences
We say a sequence of scalar-valued functions converges uniformly to another function $$f$$ on $$X$$ if $$\sup |f_n - f| \to 0$$

1 Theorem (iterated limit theorem) ''Let $$f_n$$ be a sequence of scalar-valued functions on $$X$$. If $$\lim_{y \to x} f_n(y)$$ exists and if $$f_n$$ is uniformly convergent, then for each fixed $$x \in X$$, we have:
 * $$\lim_{n \to \infty} \lim_{y \to x} f_n(y) = \lim_{y \to x} \lim_{n \to \infty} f_n(y)$$

Proof: Let $$a_n = \lim_{n \to \infty} f_n(x)$$, and $$f$$ be a uniform limit of $$f_n$$; i.e., $$\sup |f_n - f| \to 0$$. Let $$\epsilon > 0$$ be given. By uniform convergence, there exists $$N > 0$$ such that
 * $$2\sup |f - f_N| < \epsilon / 2$$

Then there is a neighborhood $$U_x$$ of $$x$$ such that:
 * $$ |f_N(y) - f_N(x)| < \epsilon / 2$$ whenever $$y \in U_x$$

Combine the two estimates:
 * $$|f(y) - f(x)| \le |f(y) - f_N(y)| + |f_N(y) - f_N(x)| + |f_N(x) - f(x)| < 2\sup |f - f_N| + \epsilon / 2 = \epsilon$$

Hence,
 * $$\lim_{y \to x} f(y) = f(x) = \lim_{n \to \infty} f_n(x)$$ $$\square$$

2 Corollary A uniform limit of a sequence of continuous functions is again continuous.

Real and complex numbers
In this chapter, we work with a number of subtleties like completeness, ordering, Archimedian property; those are usually never seriously concerned when Calculus was first conceived. I believe that the significance of those nuances becomes clear only when one starts writing proofs and learning examples that counter one's intuition. For this, I suggest a reader to simply skip materials that she does not find there is need to be concerned with; in particular, the construction of real numbers does not make much sense at first glance, in terms of argument and the need to do so. In short, one should seek rigor only when she sees the need for one. WIthout loss of continuity, one can proceed and hopefully she can find answers for those subtle questions when she search here. They are put here not for pedagogical consideration but mere for the later chapters logically relies on it.

We denote by $$\mathbb{N}$$ the set of natural numbers. The set $$\mathbb{N}$$ does not form a group, and the introduction of negative numbers fills this deficiency. We leave to the readers details of the construction of integers from natural numbers, as this is not central and menial; to do this, consider the pair of natural numbers and think about how arithmetical properties should be defined. The set of integers is denoted by $$\mathbb{Z}$$. It forms an integral domain; thus, we may define the quotient field $$\mathbb{Q}$$ of $$\mathbb{Z}$$, that is, the set of rational numbers, by
 * $$\mathbb{Q} = \left \{ {a \over b} : \mathit{b} \mbox{ are integers and }\mathit{b}\mbox{ is nonzero }\right \}$$.

As usual, we say for two rationals $$x$$ and $$y$$ $$x < y$$ if $$x - y$$ is positive.

We say $$E \subset \mathbb{Q}$$ is bounded above if there exists some $$x$$ in $$\mathbb{Q}$$ such that for any $$y \in E$$ $$y \le x$$, and is bounded below if the reversed relation holds. Notice $$E$$ is bounded above and below if and only if $$E$$ is bounded in the definition given in the chapter 1.

The reason for why we want to work on problems in analysis with $$\mathbb{R}$$ instead of is a quite simple one:

2 Theorem Fundamental axiom of analysis fails in $$\mathbb{Q}$$.

Proof: $$1, 1.4, 1.41, 1.414, ... \to 2^{1/2}$$. $$\square$$

How can we create the field satisfying this axiom? i.e., the construction of the real field. There are several ways. The quickest is to obtain the real field $$\mathbb{R}$$ by completing $$\mathbb{Q}$$.

We define the set of complex numbers $$\mathbb{C} = \mathbb{R}[i] / $$ where $$i$$ is just a symbol. That $$i^2 + 1$$ is irreducible says the ideal generated by it is maximal, and the field theory tells that $$C$$ is a field. Every complex number $$z$$ has a form:

 * $$z = a + bi + $$.

Though the square root of -1 does not exist, $$i^2$$ can be thought of as -1 since $$i^2 +  = -1 + 1 + i^2 +  = -1$$. Accordingly, the term $$$$is usually omitted.

2 Exercise Prove that there exists irrational numbers $$a, b$$ such that $$a^b$$ is rational.

Sequences
2 Theorem' ''Let $$s_n$$ be a sequence of numbers, be they real or complex. Then the following are equivalent:
 * (a) $$s_n$$ converges.
 * (b) There exists a cofinite subsequence in every open ball.

Theorem (Bolzano-Weierstrass) Every infinite bounded set has a non-isolated point.

Proof: Suppose $$S$$ is discrete. Then $$S$$ is closed; thus, $$S$$ is compact by Heine-Borel theorem. Since $$S$$ is discrete, there exists a collection of disjoint open balls $$S_x$$ containing $$x$$ for each $$x \in S$$. Since the collection is an open cover of $$S$$, there exists a finite subcover $$\{ S_{x_1}, S_{x_2}, ..., S_{x_n} \}$$. But
 * $$S \cap (\{ S_{x_1} \cup S_{x_2} \cup ... S_{x_n} \} = \{ x_1, x_2, ... x_n \} $$

This contradicts that $$S$$ is infinite. $$\square$$.

2 Corollary Every bounded sequence has a convergent subsequence. Proof:

The definition of convergence given in Ch 1 is general enough but is usually inconvenient in writing proofs. We thus give the next theorem, which is more convenient in showing the properties of the sequences, which we normally learn in Calculus courses.

In the very first chapter, we discussed sequences of sets and their limits. Now that we have created real numbers in the previous chapter, we are ready to study real and complex-valued sequences. Throughout the chapter, by numbers we always means real or complex numbers. (In fact, we never talk about non-real and non-complex numbers in the book, anyway)

Given a sequence $$a_n$$ of numbers, let $$E_j = \{ a_j, a_{j+1}, \cdot \}$$. Then we have:

The similar case holds for liminf as well.

Theorem ''Let $$s_j$$ be a sequence. The following are equivalent: Proof: From the triangular inequality it follows that:
 * (a) The sequence $$s_j$$ converges to $$s$$.
 * (b) The sequence $$s_j$$ is Cauchy; i.e., $$|s_n - s_m| \to 0$$ as $$n, m \to \infty$$.
 * (c) Every convergent subsequence of $$s_j$$ converges to $$s$$.
 * (d) $$\limsup_{j \to \infty} s_j = \liminf_{j \to \infty} s_j$$.
 * (e) For each $$\epsilon > 0$$, we can find some real number $$N$$ (i.e., N is a function of $$\epsilon$$) so that
 * $$|s_j - s| < \epsilon$$ for $$j \ge N$$.
 * $$|s_n - s_m| \le |s_n - n| + |s_m - m|$$.

Letting $$n, m \to \infty$$ gives that (a) implies (b). Suppose (b). The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence, say, $$s_k$$. Thus,

 * $$|s_n - s|$$
 * $$= |s_n - s_m + s_m - s| $$
 * $$\le |s_n - s_m| + |s_m - s| $$
 * $$< \epsilon \ 2 + \epsilon \ 2 = \epsilon$$
 * }. Therefore, $$s_j \to s$$. That (c) implies (d) is obvious by definition.
 * $$\le |s_n - s_m| + |s_m - s| $$
 * $$< \epsilon \ 2 + \epsilon \ 2 = \epsilon$$
 * }. Therefore, $$s_j \to s$$. That (c) implies (d) is obvious by definition.
 * $$< \epsilon \ 2 + \epsilon \ 2 = \epsilon$$
 * }. Therefore, $$s_j \to s$$. That (c) implies (d) is obvious by definition.

2 Theorem ''Let $$x_j$$ and $$y_j$$ converge to $$x$$ and $$y$$, respectively. Then we have:''
 * (a) $$\lim_{j \to \infty} (x_j + y_j) = x + y$$.
 * (b) $$\lim_{j \to \infty} (x_jy_j) = xy$$.

Proof: Let $$\epsilon > 0$$ be given. (a) From the triangular inequality it follows:
 * $$|(x_j + y_j) - (x + y)| = |x_j - x| + |y_j - y| < {\epsilon \over 2} + {\epsilon \over 2} = \epsilon$$

where the convergence of $$x_j$$ and $$y_j$$ tell that we can find some $$N$$ so that
 * $$|x_j - x| < {\epsilon \over 2}$$ and $$|y_j - y| < {\epsilon \over 2}$$ for all $$j > N$$.

(b) Again from the triangular inequality, it follows:

where we may suppose that $$|y_j - y| < 1$$. $$\square$$
 * $$|x_j y_j - xy|$$
 * $$= |(x_j - x)(y_j - y + y) + x(y_j - y)|$$
 * $$\le |x_j - x|(1 + |y|) + |x| |y_j - y|$$
 * $$\to 0$$ as $$j \to \infty$$
 * }
 * $$\le |x_j - x|(1 + |y|) + |x| |y_j - y|$$
 * $$\to 0$$ as $$j \to \infty$$
 * }
 * $$\to 0$$ as $$j \to \infty$$
 * }

Other similar cases follows from the theorem; for example, we can have by letting $$y_j = \alpha$$
 * $$\lim_{j \to \infty} (kx_j) = k \lim_{j \to \infty} x_j$$.

2.7 Theorem ''Given $$\sum a_n$$ in a normed space:

Proof: If (b) is true, then we can find a $$N > 0$$ and $$b$$ such that for all $$n \ge N$$:
 * (a) $$\sum \left \Vert a_n \right \Vert$$ converges.
 * (b) The sequence $$\left \Vert a_n \right \Vert^{1 \over n}$$ has the upper limit $$a^*< 1$$. (Archimedean property)
 * (c) $$\prod (a_n + 1)$$ converges.
 * $$\left\| a_n \right\|^{1 \over n} < b < 1$$. Thus (a) is true since:,
 * $$\sum_1^{\infty} \left \Vert a_n \right \Vert = a + \sum_N^{\infty} \left \Vert a_n \right \Vert \le a + \sum_0^\infty b^n = a + {1 \over 1 - b}$$ if $$a = \sum_1^{N-1}$$.

Continuity
Let $$f:\Omega \to \mathbb{R} \cup \{\infty, -\infty\}$$. We write $$\{ f > a \}$$ to mean the set $$\{ x : x \in \Omega, f(x) > a \}$$. In the same vein we also use the notations like $$\{f < a\}, \{f = a\}$$, etc.

We say, $$u$$ is upper semicontinuous if the set $$\{ f < c \}$$ is open for every $$c$$ and lower semicontinuous if $$-f$$ is upper semicontinuous.

2 Lemma The following are equivalent. Proof: Suppose $$u(z) < c$$. Then we find a $$c_2$$ such that $$u(z) < c_2 < c$$. If (i) is true, then we can find a $$\delta > 0$$ so that $$\sup_{|s-z|<\delta} u(s) \le c_2 < c$$. Thus, the converse being clear, (i) $$\iff$$ (ii). Assuming (ii), for each $$\epsilon > 0$$, we can find a $$\delta_\epsilon > 0$$ such that $$\sup_{|s-z|<{\delta_\epsilon}}u(s) < u(z) + \epsilon$$. Taking inf over all $$\epsilon$$ gives (ii) $$\Rightarrow$$ (iii), whose converse is clear. $$\square$$
 * (i) $$u$$ is upper semicontinuous.
 * (ii) If $$u(z) < c$$, then there is a $$\delta > 0$$ such that $$\sup_{|s-z|<\delta} f(s) < c$$.
 * (iii) $$\limsup_{s \to z} u(s) \le u(z)$$.

2 Theorem If $$u$$ is upper semicontinuous and $$< \infty$$ on a compact set $$K$$, then $$u$$ is bounded from above and attains its maximum.

Proof: Suppose $$u(x) < \sup_K u$$ for all $$x \in K$$. For each $$x \in K$$, we can find $$g(x)$$ such that $$u(x) < g(x) < \sup_K u$$. Since $$u$$ is upper semicontinuous, it follows that the sets $$\{u < g(x)\}$$, running over all $$x \in K$$, form an open cover of $$K$$. It admits a finite subcover since $$K$$ is compact. That is to say, there exist $$x_1, x_2, ..., x_n \in K$$ such that $$K \subset \{u < g(x_1)\} \cup \{u < g(x_1)\} \cup ... \{u < g(x_n)\}$$ and so:
 * $$\sup_K u \le \max \{ g(x_1), g(x_2), ..., g(x_n) \} < \sup_K u$$,

which is a contradiction. Hence, there must be some $$z \in K$$ such that $$u(z) = \sup_K u$$. $$\square$$

If $$f$$ is continuous, then both $$f^{-1} ( (\alpha, \infty) )$$ and $$f^{-1} ( (-\infty, \beta) )$$ for all $$\alpha, \beta \in \mathbb{R}$$ are open. Thus, the continuity of a real-valued function is the same as its semicontinuity from above and below.

2 Lemma ''A decreasing sequence of semicontinuous functions has the limit which is upper semicontinuous. Conversely, let $$f$$ be upper semicontinuous. Then there exists a decreasing sequence $$f_j$$ of Lipschitz continuous functions that converges to f.''

Proof: The first part is obvious. To show the second part, let $$f_j(x) = \inf_{y \in E} { f(x) + j^{-1} |y - x| }$$. Then $$f_j$$ has the desired properties since the identity $$|f_j(x) - f_j(y)| = j^{-1} |x - y|$$. $$\square$$

We say a function is uniform continuous on $$E$$ if for each $$\epsilon > 0$$ there exists a $$\delta > 0$$ such that for all $$x, y \in E$$ with $$|x - y| < \delta$$ we have $$|f(x) - f(y)|$$.

Example: The function $$f(x) = x$$ is uniformly continuous on $$\mathbb{R}$$ since
 * $$|f(x) - f(y)| = |x - y| < \delta = \epsilon$$

while the function $$f(x) = x^2$$ is not uniformly continuous on $$\mathbb{R}$$ since
 * $$|f(x) - f(y)| = |x^2 - y^2| = |x - y||x + y|$$

and $$|x + y|$$ can be made arbitrary large.

2 Theorem ''Let $$f: K \to \mathbb{R}$$ for $$K \subset \mathbb{R}$$ compact. If $$f$$ is continuous on $$K$$, then $$f$$ is uniformly continuous on $$K$$.''

Proof:

The theorem has this interpretation; an uniform continuity is a global property in contrast to continuity, which is a local property.

2 Corollary ''A function is uniform continuous on a bounded set $$E$$ if and only if it has a continuos extension to $$\overline{E}$$.

Proof:

2 Theorem if the sequence $$\{ f_j \}$$ of continuos functions converges uniformly on a compact set $$K$$, then the sequence $$\{ f_j \}$$ is equicontinuous.

Let $$\epsilon > 0$$ be given. Since $$f_j \to f$$ uniformly, we can find a $$N$$ so that:
 * $$|f_j(x) - f(x) | < \epsilon / 3$$ for any $$j > N$$ and any $$x \in K$$.

Also, since $$K$$ is compact, $$f$$ and each $$f_j$$ are uniformly continuous on $$K$$ and so, we find a finite sequence $$\{ \delta_0, \delta_1, ... \delta_N \}$$ so that:
 * $$|f(x) - f(y)| < \epsilon / 3$$ whenever $$|x - y| < \delta_0$$ and $$x, y \in E$$,

and for $$i = 1, 2, ... N$$,
 * $$|f_i(x) - f_i(y)| < \epsilon$$ whenever $$|x - y| < \delta_i$$ and $$x, y \in E$$.

Let $$\delta = \min \{ \delta_0, \delta_1, ... \delta_n \}$$, and let $$x, y \in K$$ be given. If $$j \le N$$, then
 * $$|f_j(x) - f_j(y)| < \epsilon$$ whenever $$|x - y| < \delta$$.

If $$j > N$$, then

whenever $$|x - y| < \delta$$. $$\square$$
 * $$|f_j(x) - f_j(y)|$$
 * $$\le |f_j(x) - f(x)| + |f(x) - f(y)| + |f(y) - f_j(y)|$$
 * $$< \epsilon$$
 * }
 * $$< \epsilon$$
 * }
 * }

2 Theorem A sequence $$f_j$$ of complex-valued functions converges uniformly on $$E$$ if and only if $$\sup_E | f_n - f_m | \to 0$$as $$n, m \to \infty$$

Proof: Let $$\| \cdot \| = \sup_E | \cdot |$$. The direct part follows holds since the limit exists by assumption. To show the converse, let $$f(x) = \lim_{j \to \infty} f_j(x)$$ for each $$x \in E$$, which exists since the completeness of $$\mathbb{C}$$. Moreover, let $$\epsilon > 0$$ be given. Since from the hypothesis it follows that the sequence $$\|f_j\|$$ of real numbers is Cauchy, we can find some $$N$$ so that:
 * $$\|f_n - f_m\| < \epsilon / 2$$ for all $$n, m > N$$.

The theorem follows. Indeed, suppose $$j > N$$. For each $$x$$, since the pointwise convergence, we can find some $$M$$ so that:
 * $$\|f_k(x) - f(x)\| < \epsilon / 2$$ for all $$k > M$$.

Thus, if $$n = \max \{ N, M \} + 1$$,
 * $$|f_j(x) - f(x)| \le |f_j(x) - f_n(x)| + |f_n(x) - f(x)| < \epsilon.$$ $$\square$$

2 Corollary A numerical sequence $$a_j$$ converges if and only if $$|a_n - a_m| \to 0$$ as $$n, m \to \infty$$.

Proof: Let $$f_j$$ in the theorem be constant functions. $$\square$$.

2 Theorem (Arzela) ''Let $$K$$ be compact and metric. If a family $$\mathcal{F}$$ of real-valued functions on K bestowed with $$\| \cdot \| = \sup_K | \cdot |$$ is pointwise bounded and equicontinuous on a compact set $$K$$, then: Proof: Let $$\epsilon > 0$$ be given. Then by equicontinuity we find a $$\delta > 0$$ so that: for any $$x, y \in K$$ with $$x \in B(\delta, y)$$
 * (i) $$\mathcal{F}$$ is uniformly bounded.
 * (ii) $$\mathcal{F}$$ is totally bounded.
 * $$|f(x) - f(y)| < \epsilon / 3$$

Since $$K$$ is compact, it admits a finite subset $$K_2$$ such that:
 * $$K \subset \bigcup_{x \in K_2} B(\delta, x)$$.

Since the finite union of totally and uniformly bounded sets is again totally and uniformly bounded, we may suppose that $$K_2$$ is a singleton $$\{ y \}$$. Since the pointwise boundedness we have:
 * $$|f(x)| \le |f(x) - f(y)| + |f(y)| \le \epsilon / 3 + |f(y)| < \infty$$ for any $$x \in K$$ and $$f \in \mathcal{F}$$,

showing that (i) holds. To show (ii) let $$A = \cup_{f \in \mathcal{F}} \{ f(y) \}$$. Then since $$\overline{A}$$ is compact, $$A$$ is totally bounded; hence, it admits a finite subset $$A_2$$ such that: for each $$f \in \mathcal{F}$$, we can find some $$g(y) \in A_2$$ so that:
 * $$|f(y) - g(y)| < \epsilon / 3$$.

Now suppose $$x \in K$$ and $$f \in \mathcal{F}$$. Finding some $$g(y)$$ in $$A_2$$ we have:
 * $$|f(x) - g(x)| \le |f(x) - f(y)| + |f(y) - g(y)| + |g(y) - g(x)| < \epsilon / 3$$.

Since there can be finitely many such $$g$$, this shows (ii). $$\square$$

2 Corollary (Ascoli's theorem) If a sequence $$f_j$$ of real-valued functions is equicontinuous and pointwise bounded on every compact subset of $$\Omega$$, then $$f_j$$ admits a subsequence converging normally on $$\Omega$$.

Proof: Let $$K \subset \Omega$$ be compact. By Arzela's theorem the sequence $$f_j$$ is totally bounded with $$\| \cdot \| = \sup_K | \cdot |$$. It follows that it has an accumulation point and has a subsequence converging to it on $$K$$. The application of Cantor's diagonal process on an exhaustion by compact subsets of $$\Omega$$ yields the desired subsequence. $$\square$$

2 Corollary ''If a sequence $$f_j$$ of real-valued $$\mathcal{C}^1$$ functions on $$\Omega$$ obeys: for some $$c \in \Omega$$,
 * $$\sup_j |f_j(c) | < \infty$$ and $$\sup_{x \in \Omega, j} |f_j'(x)| < \infty$$,

then $$f_j$$ has a uniformly convergent subsequence.

Proof: We want to show that the sequence satisfies the conditions in Ascoli's theorem. Let $$M$$ be the second sup in the condition. Since by the mean value theorem we have:
 * $$|f_j(x)| \le |f_j(x) - f_j(c)| + |f_j(c)| \le |x - c|M + |f_j(c)|$$,

and the hypothesis, the sup taken all over the sequence $$f_j$$ is finite. Using the mean value theorem again we also have: for any $$j$$ and $$x, y \in \Omega$$,
 * $$|f_j(x) - f_j(y)| \le |x - y| M$$

showing the equicontinuity. $$\square$$

First and second countability
2 Theorem (first countability) ''Let $$E$$ have a countable base at each point of $$E$$. Then we have the following: Proof: (i) Let $$\{B_j\}$$ be a base at $$x$$ such that $$B_{j + 1} \subset B_j$$. If $$x \in \overline{A}$$, then every $$B_j$$ intersects A. Let $$x_j \in B_j \cap E$$. It now follows: if $$x \in G$$, then we find some $$B_N \subset G$$. Then $$x_k \in B_k \subset B_N$$ for $$k \ge N$$. Hence, $$x_j \to x$$. Conversely, if $$x \not\in \overline{A}$$, then $$E \backslash \overline{E}$$ is an open set containing $$x$$ and no $$x_j \subset E$$. Hence, no sequence in $$A$$ converges to $$x$$. That (ii) is valid follows since $$f(x_j)$$ is the composition of continuous functions $$f$$ and $$x$$, which is again continuous. In other words, (ii) is essentially the same as (i). $$\square$$.
 * ''(i) For $$A \subset E$$, $$x \in \overline{A}$$ if and only if there is a sequence $$x_j \in A$$ such that $$x_j \to x$$.
 * ''(ii) A function is continuous on $$E$$ if and only if $$f(x_j) \to f(x)$$ whenever $$x_j \to x$$.

2. 2 Theorem $$\Omega$$ has a countable basis consisting of open sets with compact closure.

Proof: Suppose the collection of interior of closed balls $$G_{\alpha}$$ in $$\Omega$$ for a rational coordinate $$\alpha$$. It is countable since $$\mathbb{Q}$$ is. It is also a basis of $$\Omega$$; since if not, there exists an interior point that is isolated, and this is absurd.

2.5 Theorem In $$\mathbb{R}^k$$ there exists a set consisting of uncountably many components.

Usual properties we expect from calculus courses for numerical sequences to have are met like the uniqueness of limit.

2 Theorem (uncountability of the reals) The set of all real numbers is never a sequence.

Proof: See.

Example: Let f(x) = 0 if x is rational, f(x) = x^2 if x is irrational. Then f is continuous at 0 and nowhere else but f' exists at 0.

2 Theorem (continuous extension) ''Let $$f, g: \overline{E} \to \mathbb{C}$$ be continuous. If $$f = g$$ on $$E$$, then $$f = g on \overline{E}$$.

Proof: Let $$u = f - g$$. Then clearly $$u$$ is continuous on $$\overline{E}$$. Since $${0}$$ is closed in $$\mathbb{C}$$, $$u^{-1}(0)$$ is also closed. Hence, $$u = 0$$ on $$\overline{E}$$. $$\square$$

, and for each $$j$$, $$B_j = \overline{ \{ x_k : k \ge j \} }$$. Since for any subindex $$j(1), j(2), ..., j(n)$$, we have:
 * $$x_{j(n)} \in B_{j(1)} \cap B_{j(2)} ... B_{j(n)}$$.

That is to say the sequence $$B_j$$ has the finite intersection property. Since $$E$$ is coun

Since $$E$$ is first countable, $$E$$ is also sequentially countable. Hence, (a) $$\to$$ (b).

Suppose $$E$$ is not bounded, then there exists some $$\epsilon > 0$$ such that: for any finite set $$\{ x_1, x_2, ... x_n \} \subset E$$,
 * $$E \not \subset B(\epsilon, x_1) \cup ... B(\epsilon, x_n)$$.

Let recursively $$x_1 \in E$$ and $$x_j \in E \backslash \bigcup_1^{j-1} B(\epsilon, x_k)$$. Since $$d(x_n, x_m) > \epsilon$$ for any n, m, $$x_j$$ is not Cauchy. Hence, (a) implies that $$E$$ is totally bounded. Also,

3 Theorem the following are equivalent:
 * (a) $$\| x \| = 0$$ implies that $$x = 0$$.
 * (b) Two points can be separated by disjoint open sets.
 * (c) Every limit is unique.

3 Theorem The separation by neighborhoods implies that a compact set $$K$$ is closed in E.

Proof : If $$K = E$$, then $$K$$ is closed. If not, there exists a $$x \in E \backslash K$$. For each $$y \in K$$, the hypothesis says there exists two disjoint open sets $$A(y)$$ containing $$x$$ and $$B(y)$$ containing $$y$$. The collection $$\{ B(y) : y \in E \}$$ is an open cover of $$K$$. Since the compactness, there exists a finite subcover $$\{ B(y_1), B(y_2), ... B(y_n) \}$$. Let $$A(x)$$ = $$A(y_1) \cap A(y_2) ... A(y_n)$$. If $$z \in K$$, then $$z \in B(y_k)$$ for some $$k$$. Hence, $$z \not\in A(y_k) \subset A$$. Hence, $$A(x) \subset E \backslash K$$. Since the finite intersection of open sets is again open, $$A(x)$$ is open. Since the union of open sets is open,
 * $$E \backslash K = \bigcup_{x \not\in K} A(x)$$ is open. $$\square$$

Seminorm
Let $$G$$ be a linear space. The seminorm of an element in $$G$$ is a nonnegative number, denoted by $$\| \cdot \|$$, such that: for any $$x, y \in G$$,
 * 1) $$\|x\| \ge 0$$.
 * 2) $$\| \alpha x \| = |\alpha| \|x\|$$.
 * 3) $$\|x + y\| \le \|x\| + \|y\|$$. (triangular inequality)

Clearly, an absolute value is an example of a norm. Most of the times, the verification of the first two properties while whether or not the triangular inequality holds may not be so obvious. If every element in a linear space has the defined norm which is scalar in the space, then the space is said to be "normed linear space".

A liner space is a pure algebraic concept; defining norms, hence, induces topology with which we can work on problems in analysis. (In case you haven't noticed this is a book about analysis not linear algebra per se.) In fact, a normed linear space is one of the simplest and most important topological space. See the addendum for the remark on semi-norms.

An example of a normed linear space that we will be using quite often is a sequence space $$l^p$$, the set of sequences $$x_j$$ such that
 * For $$1 \le p < \infty$$, $$\sum_1^{\infty} |x_j|^p < \infty$$
 * For $$p = \infty$$, $$x_j$$ is a bounded sequence.

In particular, a subspace of $$l^p$$ is said to have dimension n if in every sequence the terms after nth are all zero, and if $$n < \infty$$, then the subspace is said to be an Euclidean space.

An open ball centered at $$a$$ of radius $$r$$ is the set $$\{ z : \| z - a \| < r \}$$. An open set is then the union of open balls.

Continuity and convergence has close and reveling connection.

3 Theorem ''Let $$L$$ be a seminormed space and $$f: \Omega \to L$$. The following are equivalent:''
 * (a) $$f$$ is continuous.
 * (b) Let $$x \in \Omega$$. If $$f(x) \in G$$, then there exists a $$\omega \subset \Omega$$ such that $$x \in \omega$$ and $$f(\omega) \subset G$$.
 * (c) Let $$x \in \Omega$$. For each $$\epsilon > 0$$, there exists a $$\delta > 0$$ such that
 * $$|f(y) - f(x)| < \epsilon$$ whenever $$y \in \Omega$$ and $$|y - x|< \delta$$

Proof: Suppose (c). Since continuity, there exists a $$\delta > 0$$ such that:
 * (e) Let
 * $$|f(x_j) - f(x)| < \epsilon$$ whenever $$|x_j - x| < \delta$$

The following special case is often useful; in particular, showing the sequence's failure to converge.

2 Theorem ''Let $$x_j \in \Omega$$ be a sequence that converges to $$x \in \Omega$$. Then a function $$f$$ is continuous at $$x$$ if and only if the sequence $$f(x_j)$$ converges to $$f(x)$$.''

Proof: The function $$x$$ of $$j$$ is continuous on $$\mathbb{N}$$. The theorem then follows from Theorem 1.4, which says that the composition of continuous functions is again continuous. $$\square$$.

2 Theorem Let $$u_j$$ be continuous and suppose $$u_j$$ converges uniformly to a function $$u$$ (i.e., the sequence $$\|u_j - u\| \to 0$$ as $$j \to \infty$$. Then $$u$$ is continuous.

Proof: In short, the theorem holds since
 * $$\lim_{y \to x}u(y) = \lim_{y \to x} \lim_{j \to \infty} u_j(y) = \lim_{j \to \infty} \lim_{y \to x} u_j(y) = \lim_{j \to \infty} u_j(x) = u(x)$$.

But more rigorously, let $$\epsilon > 0$$. Since the convergence is uniform, we find a $$N$$ so that
 * $$\|u_N(x) - u(x)\| \le \|u_N - u\| < \epsilon / 3$$ for any $$x$$.

Also, since $$u_N$$ is continuous, we find a $$\delta > 0$$ so that
 * $$\|u_N(y) - u_N(x)\| < \epsilon / 3$$ whenever $$|y - x| < \delta$$

It then follows that $$u$$ is continuous since:
 * $$\| u(y) - u(x) \| \le \| u(y) - u_N(y) \| + \| u_N(y) - u_N(x) \| + \| u_N(x) - u(x) \| < \epsilon$$

whenever $$|y - x| < \delta$$ $$\square$$

2 Theorem The set of all continuous linear operators from $$A$$ to $$B$$ is complete if and only if $$B$$ is complete.

Proof: Let $$u_j$$ be a Cauchy sequence of continuous linear operators from $$A$$ to $$B$$. That is, if $$x \in A$$ and $$\|x\| = 1$$, then
 * $$\|u_n(x) - u_m(x)\| = \| (u_n - u_m)(x) \| \to 0$$ as $$n, m \to \infty$$

Thus, $$u_j(x)$$ is a Cauchy sequence in $$B$$. Since B is complete, let
 * $$u(x) = \lim_{j \to \infty} u_j(x)$$ for each $$x \in A$$.

Then $$u$$ is linear since the limit operator is and also $$u$$ is continuous since the sequence of continuous functions converges, if it does, to a continuous function. (FIXME: the converse needs to be shown.) $$\square$$

Metric spaces
We say a topological space is metric or metrizable if every open set in it is the union of open balls; that is, the set of the form $$\{ x; d(x, y) < r \}$$ with radius $$r$$ and center $$y$$ where $$d$$, called metric, is a real-valued function satisfying the axioms: for all $$x, y, z$$
 * 1) $$d(x, y) = d(y, x)$$
 * 2) $$d(x, y) + d(y, z) \ge d(x, z)$$
 * 3) $$d(x, y) = 0$$ if and only if $$x = y$$. (Identity of indiscernibles)

It follows immediately that $$|d(x, y) - d(z, y)| \le d(x, y)$$ for every $$x, y, z$$. In particular, $$d$$ is never negative. While it is clear that every subset of a metric space is again a metric space with the metric restricted to the set, the converse is not necessarily true. For a counterexample, see the next chapter.

2 Theorem Let $$K, d)$$ be a compact metric space. If $$\Gamma$$ is an open cover of $$K$$, then there exists a $$\delta > 0$$ such that for any $$E \subset K$$ with $$\sup_{x, y \in E} d(x, y) < \delta$$ $$E \subset $$ some member of $$\Gamma$$

Proof: Let $$x, y$$ be in $$K$$, and suppose $$x$$ is in some member $$S$$ of $$\Gamma$$ and $$y$$ is in the complement of $$S$$. If we define a function $$\delta$$ by for every $$x$$
 * $$\delta(x) = \inf \{ d(x, z); z \in A \in \Gamma, x \ne A \}$$,

then
 * $$\delta(x) \le d(x, y)$$

The theorem thus follows if we show the inf of $$\delta$$ over $$K$$ is positive. $$\square$$

2 Lemma ''Let $$K$$ be a metric space. Then every open cover of $$K$$ admits a countable subcover if and only if $$K$$ is separable.''

Proof: To show the direct part, fix $$n = 1, 2, ...$$ and let $$\Gamma$$ be the collection of all open balls of radius $${1 \over n}$$. Then $$\Gamma$$ is an open cover of $$K$$ and admits a countable subcover $$\gamma$$. Let $$E_n$$ be the centers of the members of $$\gamma$$. Then $$\cup_1^\infty E_n$$ is a countable dense subset. Conversely, let $$\Gamma$$ be an open cover of $$K$$ and $$E$$ be a countable dense subset of $$K$$. Since open balls of radii $$1, {1 \over 2}, {1 \over 3}, ...$$, the centers lying in $$E$$, form a countable base for $$K$$, each member of $$\Gamma$$ is the union of some subsets of countably many open sets $$B_1, B_2, ...$$. Since we may suppose that for each $$n$$ $$B_n$$ is contained in some $$G_n \in \Gamma$$ (if not remove it from the sequence) the sequence $$G_1, ...$$ is a countable subcover of $$\Gamma$$ of $$K$$. $$\square$$

Remark: For more of this with relation to cardinality see.

2 Theorem ''Let $$K$$ be a metric space. Then the following are equivalent:'' Proof (from ): Throughout the proof we may suppose that $$K$$ is infinite. Lemma 1.somthing says that every sequence of a infinite compact set has a limit point. Thus, the first countability shows (i) $$\Rightarrow$$ (ii). Supposing that (iii) is false, let $$G_1, G_2, ...$$ be an open cover of $$K$$ such that for each $$n = 1, 2, ...$$ we can find a point $$x_n \in (\cup_1^n G_j)^c = \cap_1^n {G_j}^c$$. It follows that $$x_n$$ does not have a convergent subsequence, proving (ii) $$\Rightarrow$$ (iii). Indeed, suppose it does. Then the sequence $$x_n$$ has a limit point $$p$$. Thus, $$p \in \cap_1^\infty {G_j}^c$$, a contradiction. To show (iii) $$\Rightarrow$$ (i), in view of the preceding lemma, it suffices to show that $$K$$ is separable. But this follows since if $$K$$ is not separable, we can find a countable discrete subset of $$K$$, violating (iii). $$\square$$
 * (i) $$K$$ is compact.
 * (ii) Every sequence of $$K$$ admits a convergent subsequence. (sequentially compact)
 * (iii) Every countable open cover of $$K$$ admits a finite subcover. (countably compact)

Remark: The proof for (ii) $$\Rightarrow$$ (iii) does not use the fact that the space is metric.

2 Theorem A subset $$E$$ of a metric space is precompact (i.e., its completion is compact) if and only if there exists a $$\delta > 0$$ such that $$E$$ is contained in the union of finitely many open balls of radius $$\delta$$ with the centers lying in E.

Thus, the notion of relatively compactness and precompactness coincides for complete metric spaces.

2 Theorem (Heine-Borel) If $$E$$ is a subset of $$\mathbb{R}^n$$, then the following are equivalent.
 * (i) $$E$$ is closed and bounded.
 * (ii) $$E$$ is compact.
 * (iii) Every infinite subset of $$E$$ contains some of its limit points.

2 Theorem Every metric space is perfectly and fully normal.

2 Corollary Every metric space is normal and Hausdorff.

2 Theorem If $$x_j \to x$$ in a metric space $$X$$ implies $$f(x_n) \to f(x)$$, then $$f$$ is continuous.

Proof: Let $$E \subset X$$ and $$x \in $$ the closure of $$E$$. Then there exists a $$x_j \in E \to x$$ as $$j \to \infty$$ as $$j \to \infty$$. Thus, by assumption $$f(x_j) \to f(x)$$, and this means that $$f(x)$$ is in the closure of $$f(E)$$. We conclude: $$f(\overline{E}) \subset \overline{f(E)}$$. $$\square$$

2 Theorem ''Suppose that $$f: (X_1, d_1) \to (X_2, d_2)$$ is continuous and satisfies
 * $$d_2(f(x), f(y)) \ge d_1(x, y)$$ for all $$x, y$$

If $$F \subset X_2$$ and $$(F, d_2)$$ is a complete metric space, then $$f(F)$$ is closed.

Proof: Let $$x_n$$ be a sequence in $$F$$ such that $$\lim_{n, m \to \infty} d_2(f(x_n), f(x_m)) = 0$$. Then by hypothesis $$\lim_{n, m \to \infty} d_1(x_n, x_m) = 0$$. Since $$F$$ is complete in $$d_2$$, the limit $$y = \lim_{n \to \infty} x_n$$ exists. Finally, since $$f$$ is continuous, $$\lim_{n\to \infty} d_2(f(x_n), f(y)) = 0$$, completing the proof. $$\square$$

Measure

A measure, which we shall denote by $$\mu$$ throughout this section, is a function from a delta-ring $$\mathcal{F}$$ to the set of nonnegative real numbers and $$\infty$$ such that $$\mu$$ is countably additive; i.e., $$\mu (\coprod_1^\infty E_j) = \sum_1^{\infty} \mu (E_j)$$ for $$E_j$$ distinct. We shall define an integral in terms of a measure.

4.1 Theorem A measure $$\mu$$ has the following properties: given measurable sets $$A$$ and $$B$$ such that $$A \subset B$$, Proof: (1) $$\mu (B) - \mu (A) = \mu (B \backslash A \cup A) - \mu (A) = \mu (B \backslash A) + \mu (A) - \mu (A)$$. (2) $$\mu (\varnothing) = \mu (A \backslash A) = \mu (A) - \mu (A). (3) \mu (B) - \mu (A) = \mu (B \backslash A) \ge 0$$ since measures are always nonnegative. Also, $$\mu (B) - \mu (A) = 0$$ if $$A = B$$ since (2).
 * (1) $$\mu (B \backslash A) = \mu (B) - \mu (A)$$.
 * (2) $$\mu (\varnothing) = 0$$.
 * (3) $$\mu (A) \le \mu (B)$$ where the equality holds for $$A = B$$. (monotonicity)

One example would be a counting measure; i.e., $$\mu E$$ = the number of elements in a finite set $$E$$. Indeed, every finite set is measurable since the countable union and countable intersection of finite sets is again finite. To give another example, let $$x_0 \in \mathcal{F}$$ be fixed, and $$\mu(E) = 1$$ if $$x_0$$ is in $$E$$ and 0 otherwise. The measure $$\mu$$ is indeed countably additive since for the sequence $${E_j}$$ arbitrary, $$\mu(\coprod_0^{\infty} E_j) = 1 = \mu(E_j)$$ for some $$j$$ if $$x_0 \in \bigcup E_j$$ and 0 otherwise.

We now study the notion of geometric convexity.

2 Lemma
 * $${x+y \over 2} = a \left( a {x+y \over 2} + (1-a)y \right) + (1-a) \left( ax + (1-a){x+y \over 2} \right)$$ for any $$x, y$$.

2 Theorem ''Let $$D$$ be a convex subset of a vector space. If $$0 < a < 1$$ and $$f(ax + (1-a)y) \le af(x) + (1-a)f(y)$$ for $$x, y \in D$$, then
 * $$f({x+y \over 2}) \le {f(x) + f(y) \over 2}$$ for $$x, y \in D$$

Proof: From the lemma we have:
 * $$2a(1-a)f({x+y \over 2}) \le a(1-a) (f(x) + f(y))$$ $$\square$$

Let $$\mathcal{F}$$ be a collection of subsets of a set $$G$$. We say $$\mathcal{F}$$ is a delta-ring if $$\mathcal{F}$$ has the empty set, G and every countable union and countable intersection of members of $$\mathcal{F}$$. a member of $$\mathcal{F}$$ is called a measurable set and G a measure space, by analogy to a topological space and open sets in it.

2 Theorem (Hölder's inequality) ''For $$1 \le p \le \infty$$, if $$1/p + 1/q = 1$$, then
 * $$\int |fg|d\mu \le \left( \int |f|^p d\mu \right)^{1/p} \left( \int |g|^q d\mu \right)^{1/q}$$

Proof: By replacing $$f$$ and $$g$$ with $$|f|$$ and $$|g|$$, respectively, we may assume that $$f$$ and $$g$$ are non-negative. Let $$C = \int g^q d\mu$$. If $$C = 0$$, then the inequality is obvious. Suppose not, and let $$d\nu = {g^q d\mu \over C}$$. Then, since $$x^p$$ is increasing and convex for $$x \ge 0$$, by Jensen's inequality,
 * $$\int fg d\mu = \int fg^{(1-q)} d\nu C \le \left( \int f^p g^{-q} d\nu \right)^{(1/p)} C$$.

Since $$1/q = 1 - 1/p$$, this is the desired inequality. $$\square$$

4 Corollary (Minkowski's inequality) ''Let $$p \ge 1$$ and $$\| \cdot \|_p = \left( \int | \cdot |^p \right)^{1/p}$$. If $$f \ge 0$$ and $$\int \int f(x, y) dx dy < \infty$$, then
 * $$\| \int f(\cdot, y) dy \|_p \le \int \| f(\cdot, y) \|_p dy$$

Proof (from ): If $$p = 1$$, then the inequality is the same as Fubini's theorem. If $$p > 1$$,

By division we get the desired inequality, noting that $$(p - 1)q = p$$ and $$1 - {1 \over q} = {1 \over p}$$. $$\square$$
 * $$\int |\int f(x, y)dy|^p dx $$
 * $$= \int |\int f(x, y)dy|^{p-1} \int f(x, z)dz dx$$
 * align=right| (Fubini)
 * $$= \int \int |\int f(x, y)dy|^{p-1} f(x, z)dx dz$$
 * align=right| (Hölder)
 * $$\le \left( \int |\int f(x, y)dy|^{(p-1)q}dx \right)^{1/q} \int \| f(\cdot, z) \|_p dz$$
 * }
 * align=right| (Hölder)
 * $$\le \left( \int |\int f(x, y)dy|^{(p-1)q}dx \right)^{1/q} \int \| f(\cdot, z) \|_p dz$$
 * }

TODO: We can probably simplify the proof by appealing to Jensen's inequality instead of Hölder's.

Remark: by replacing $$\int dy$$ by a counting measure we get:
 * $$\|f+g\|_p \le \|f\|_p + \|g\|_p$$

under the same assumption and notation.

2 Lemma ''For $$1 \le p \le \infty$$ and $$a, b > 0$$, if $$1/p + 1/q = 1$$,
 * $$a^{1 \over p}b^{1 \over q} = \inf_{t > 0} \left( {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b \right)$$

Proof: Let $$f(t) = {1 \over p} t^{-{1 \over q}}a + {1 \over q} t^{1 \over p}b$$ for $$t > 0$$. Then the derivative
 * $$f'(t) = {1 \over pq} \left( t^{-{1 \over p}} a + t^{1 \over q} b \right)$$

becomes zero only when $$t = a/b$$. Thus, the minimum of $$f$$ is attained there and is equal to $$a^{1 \over p}b^{1 \over q}$$. $$\square$$

When $$t$$ is taken to 1, the inequality is known as Young's inequality.

2 Theorem (Hölder's inequality) ''For $$1 \le p \le \infty$$, if $$1/p + 1/q = 1$$, then
 * $$\int |fg| \le \left( \int |f|^p \right)^{1/p} \left( \int |f|^q \right)^{1/q}$$

Proof: If $$p = \infty$$, the inequality is clear. By the application of the preceding lemma we have: for any $$t > 0$$
 * $$\int |f|^{1 \over p} |g|^{1 \over q} \le {1 \over p} t^{-{1 \over q}}\int |f| + {1 \over q} t^{1 \over p} \int |g|$$

Taking the infimum we see that the right-hand side becomes:
 * $$\left( \int |f| \right)^{1/p} \left( \int |g| \right)^{1/q}$$ $$\square$$

The convex hull in $$\Omega$$ of a compact set K, denoted by $$\hat K$$ is:
 * $$\left \{ z \in \Omega : |f(z)| \le \sup_K|f|\mbox{ for }f\mbox{ analytic in }\Omega \right \}$$.

When $$f$$ is linear (besides being analytic), we say the $$\hat K$$ is geometrically convex hull. That $$K$$ is compact ensures that the definition is meaningful.

Theorem The closure of the convex hull of $$E$$ in $$G$$ is the intersection if all half-spaces containing $$E$$.

Proof: Let F = the collection of half-spaces containing $$E$$. Then $$\overline{co}(E) \subset \cap F$$ since each-half space in $$F$$ is closed and convex. Yet, if $$x \not\in \overline{co}(E)$$, then there exists a half-space $$H$$ containing $$E$$ which however does not contain x. Hence, $$\overline{co}(E) = \cap F$$. $$\square$$

Let $$\Omega \subset \mathbb{R}^n$$. A function $$f:\Omega \to \mathbb{R}^n$$ is said to be convex if
 * $$\sup_K |f - h| = \sup_{\mbox{b}K} | f - h |$$.

for some $$K \subset \Omega$$ compact and any $$h$$ harmonic on $$K$$ and continuous on $$\overline{K}$$.

Theorem ''The following are equivalent: Proof: Suppose (a). For each $$x \in [a, b]$$, there exists some $$\lambda \in [0, 1]$$ such that $$x = \lambda a + (1 - \lambda) b$$. Let $$A(x) = A(\lambda a + (1-\lambda) b) = \lambda f(a) + (1-\lambda) f(b)$$, and then since $$\mbox{b}[a, b] = \{a, b\}$$ and $$(f - A)(a) = 0 = (f - A)(b)$$,
 * (a) $$f$$ is convex on some $$K \subset \mathbb{R}$$ compact.
 * (b) if $$[a, b] \subset K$$, then
 * $$f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)$$ for $$\lambda \in [0, 1]$$.
 * (c) The difference quotient
 * $${f(x + h) - f(x) \over h}$$ increases as $$h$$ does.
 * (d) $$f$$ is measurable and we have:
 * $$f \left( {a + b \over 2} \right) \le {a + b \over 2}$$ for any $$[a, b] \subset K$$.
 * (e) The set $$\{ (x, y) : x \in [a, b], f(x) \le y \}$$ is convex for $$[a, b] \subset K$$.

Thus, (a) $$\Rightarrow$$ (b). Now suppose (b). Since $$\lambda = {x - a \over b - a}$$, for $$\lambda \in (0, 1)$$, (b) says:
 * $$|f(x)| = |f(x) - A(x) + A(x)|$$
 * $$\le \sup \{|f(a) - A(a)|, |f(b) - A(b)|\} + A(x)$$
 * $$=\lambda f(a) + (1-\lambda) f(b)$$
 * }
 * $$=\lambda f(a) + (1-\lambda) f(b)$$
 * }
 * }

Since $$a - x < b - x$$, we conclude (b) $$\Rightarrow$$ (c). Suppose (c). The continuity follows since we have:
 * $$(b - x)f(x) + (x - a)f(x)$$
 * $$\le (b -x)f(a) + (x - a)f(b)$$
 * $${f(a) - f(x) \over a - x}$$
 * $$\le {f(b) - f(x) \over b - x}$$
 * }
 * $$\le {f(b) - f(x) \over b - x}$$
 * }
 * $$\lim_{h > 0, h \to 0}f(x + h) = f(x) + \lim_{h > 0, h \to 0}h{f(x + h) - f(x) \over h}$$.

Also, let $$x = 2^{-1}(a + b)$$ such that $$a < b$$, for $$a, b \in K$$. Then we have:

Thus, (c) $$\Rightarrow$$ (d). Now suppose (d), and let $$E = \{ (x, y) : x \in [a, b], y \ge f(x) \}$$. First we want to show
 * $${f(a) - f(x) \over a - x}$$
 * $$\le {f(b) - f(x) \over b - x}$$
 * $$f(x) - f(a)$$
 * $$\le f(b) - f(x)$$
 * $$f \left( {a + b \over 2} \right)$$
 * $$\le {f(a) + f(b) \over 2}$$
 * }
 * $$f \left( {a + b \over 2} \right)$$
 * $$\le {f(a) + f(b) \over 2}$$
 * }
 * $$f\left({1 \over 2^n} \sum_1^{2^n} x_j \right) \le {1 \over 2^n} \sum_1^{2^n} f(x_j)$$.

If $$n = 0$$, then the inequality holds trivially. if the inequality holds for some $$n - 1$$, then

 * $$f \left( {1 \over 2^n} \sum_1^{2^n} x_j \right)$$
 * $$=f \left( {1 \over 2} \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j + {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right) \right)$$
 * $$\le {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_j \right) + {1 \over 2} f \left( {1 \over 2^{n - 1}} \sum_1^{2^{n - 1}} x_{2^{n - 1} + j} \right)$$
 * $$\le {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_j) + {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_{2^{n - 1} + j}) $$
 * $$= {1 \over 2^n} \sum_1^{2^n}f(x_j)$$
 * }
 * $$\le {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_j) + {1 \over 2^n} \sum_1^{2^{n - 1}} f(x_{2^{n - 1} + j}) $$
 * $$= {1 \over 2^n} \sum_1^{2^n}f(x_j)$$
 * }
 * $$= {1 \over 2^n} \sum_1^{2^n}f(x_j)$$
 * }
 * $$= {1 \over 2^n} \sum_1^{2^n}f(x_j)$$
 * }

Let $$x_1, x_2 \in [a, b]$$ and $$\lambda \in [0, 1]$$. There exists a sequence of rationals number such that:
 * $$\lim_{j \to \infty} {p_j \over 2q_j} = \lambda$$.

It then follows that:

Thus, (d) $$\Rightarrow$$ (e). Finally, suppose (e); that is, $$E$$ is convex. Also suppose $$K$$ is an interval for a moment. Then
 * $$f(\lambda x_1 + (1 - \lambda) x_2)$$
 * $$\le \lim_{j \to \infty} {p_j \over 2q_j} f(x_1) + \left( 1 - {p_j \over 2q_j} \right) f(x_2)$$
 * $$= \lambda f(x_1) + (1 - \lambda) f(x_2)$$
 * $$\le \lambda y_1 + (1 - \lambda) y_2$$.
 * }
 * $$= \lambda f(x_1) + (1 - \lambda) f(x_2)$$
 * $$\le \lambda y_1 + (1 - \lambda) y_2$$.
 * }
 * $$\le \lambda y_1 + (1 - \lambda) y_2$$.
 * }
 * $$f(\lambda a + (1 - \lambda) b) \le \lambda f(a) + (1 - \lambda) f(b)$$. $$\square$$

4. Corollary (inequality between geometric and arithmetic means)
 * $$\prod_1^n a_j^{\lambda_j} \le \sum_1^n \lambda_j a_j$$ if $$a_j \ge 0$$, $$\lambda_j \ge 0$$ and $$\sum_1^n \lambda_j = 1$$.

Proof: If some $$a_j = 0$$, then the inequality holds trivially; so, suppose $$a_j > 0$$. The function $$e^x$$ is convex since its second derivative, again $$e^x$$, is $$> 0$$. It thus follows:
 * $$\prod_1^n a_j^{\lambda_j} =e^{\sum_1^n \lambda_j \log(a_j)} \le \sum_1^n \lambda_j e^{\log(a_j)} = \sum_1^n \lambda_j a_j$$. $$\square$$

The convex hull of a finite set $$E$$ is said to be a convex polyhedron. Clearly, the set of the extremely points is the subset of $$E$$.

4 Theorem (general Hölder's inequality) ''If $$a_{jk} > 0$$ and $$p_j > 1$$ for $$j = 1, ... n_j$$ and $$k = 1, ... n_k$$ and $$\sum_1^{n_j} p_j = 1$$, then:''
 * $$\sum_{k=1}^{n_k} \prod_{j=1}^{n_j} a_{jk} \le \prod_{j=1}^{n_k} \left( \sum_{k=1}^{n_k} a_{jk}^{p_j} \right)^{1/p}$$ (Hörmander 11)

Proof: Let $$A_j = \left( \sum_{k} a_{jk}^{p_j} \right)^{1/p_j}$$.

4. Theorem A convex polyhedron is the intersection of a finite number of closed half-spaces.

Proof: Use induction. $$\square$$

4. Theorem The convex hull of a compact set is compact.

Proof: Let $$f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n) = \sum_1^n \lambda_j x_j$$. Then $$f$$ is continuous since it is the finite sum of continuous functions $$\lambda_j x_j$$. Since the intersection of compact sets is compact and
 * $$\mbox{co}(K) = \bigcap_{n = 1, 2, ...} f(\lambda_1, \lambda_2, ... \lambda_n, x_1, x_2, ... x_n)$$,

$$\mbox{co}(K)$$ is compact. $$\square$$

Example: Let $$p(z) = (z - s_1)(z - s_2) ... (z - s_n)$$. Then the derivative of $$p$$ has zeros in $$\mbox{co}(\{s_1, s_2, ... s_n\})$$.

Addendum

 * 1) Show the following are equivalent with assuming that $$K$$ is second countable.
 * 2) * (1) $$K$$ is compact.
 * 3) * (2) Every countable open cover of $$K$$ admits a finite subcover (countably compact)
 * 4) * (3) $$K$$ is sequentially compact.

Nets are, so to speak, generalized sequences.

 * 1) Show the following are equivalent with assuming Axiom of Choice:
 * 2) * (1) $$K$$ is compact; i.e, every open cover admits a finite subcover.
 * 3) * (2) In $$K$$ every net has a convergent subnet.
 * 4) * (3) In $$K$$ every ultrafilter has an accumulation point (Bourbaki compact)
 * 5) * (4) There is a subbase $$\tau$$ for $$K$$ such that every open cover that is a subcollection of $$\tau$$ admits a finite subcover. (subbase compact)
 * 6) * (5) Every nest of non-empty closed sets has a non-empty intersection (linearly compact) (Hint: use the contrapositive of this instead)
 * 7) * (6) Every infinite subset of $$K$$ has a complete accumulation point (Alexandroff-Urysohn compact)

title: User:TakuyaMurata/Sequences of numbers


text:
Cookbook:Falafel Salad

Thanks to dry falafel mix and the nature of simple salads, this recipe is pretty fast and easy. Look in the bulk bins at your local hippie grocery store for dry falafel mix.

Ingredients

 * 1 ⅔ cup falafel mix
 * 1 ¼ cup water
 * 1 head red leaf lettuce, chopped
 * 3 tomatoes, chopped in bite-size pieces
 * 1 cucumber, peeled and sliced
 * ½ cup Tahini Yogurt Sauce

Falafel mix

 * 1) In a large bowl, add water to falafel mix and stir well.
 * 2) Let stand 15 minutes to absorb the water.
 * 3) Form mix into ¾-inch balls.

Baking

 * 1) Place balls into mini-cupcake tins or onto a cookie sheet.
 * 2) Bake in oven at 375°F for 12 minutes.

Frying

 * 1) Heat about ½ inch vegetable oil to 375°F.
 * 2) Fry balls until brown and crisp (about 3 minutes).
 * 3) Drain and set aside.

Serving

 * 1) Toss vegetables with falafels and douse with tahini yogurt sauce. Falafel may be cold or hot for a delicious salad straight outta the PNW.

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

title: Cookbook:Falafel Salad


text:
Cookbook:Tahini Yogurt Sauce

Ingredients

 * 1 large lemon
 * 3 tbsp tahini
 * 8 oz (220 g) plain yogurt or soy-based substitute
 * 2 tbsp olive oil
 * 3 cloves garlic, minced

Procedure

 * 1) Squeeze juice from lemon into bowl or small food processor.
 * 2) Add remaining ingredients, and process well to combine.

title: Cookbook:Tahini Yogurt Sauce


text: User talk:Shashinmishra

title: User talk:Shashinmishra


text:
Talk:Mathematical Proof/Introduction/Notation

Quantifier or qualifier
What here is called a qualifier (for all, there is) is normally called a quantifier. See quantification. Otto 06:32, 10 April 2006 (UTC)


text:
User:Otto ter Haar

Just took a look here. I am since a while active at wikipedia.

title: User:Otto ter Haar


text:
Wikijunior talk:Languages/Greek

Aesop's Fables - verse version by Babrius
I've got a website full of Aesop's Fables.

Which one do we choose?

I'm thinking The Tortoise and the Hare or The Boy Who Cried Wolf.

There is a Perry translation alongside so think about this as we pick and choose.

--EuropracBHIT 07:36, 10 April 2006 (UTC).

 * Would be good as an example for Ancient Greek. Andreas 02:49, 27 October 2006 (UTC)

Phonetics of the Greek Letters
In both the "Words I can learn" and the song/story sections of this it doesn't have the phonetic pronunciations. I, as a person without any backround in Greek (like the people who would be reading this), have little clue how to pronounce the greek words nor how to say the fable. Prehapes a phonetic pronunciations should be given. N1person 14:01, 23 April 2006 (UTC)

Unsuitable text
The "simple story" is totally out of place and also misleading because it is written in a language that is completely antiquated. It differs from today's Greek more than Shakespearean English differs from today's English. See the Wikipedia article on Katharevousa for details. I replaced this text with a children's song. Andreas 01:49, 27 October 2006 (UTC)

title: Wikijunior talk:Languages/Greek


text:
User:Red Rooster

de:Benutzer:Red_Rooster

title: User:Red Rooster


text:
User:Kellen/Scratch

DynamicPageList test (chocolate desserts):
 category=Dessert recipes category=Chocolate recipes shownamespace=false order=descending

Pending todos list for Talk:Cookbook
Some really obvious things that need doing:
 * write Cookbook:Yeast, Cookbook:Broth
 * cleanup all our redirects
 * standardize a case and tense so we can get on with business and not have 10000x redirects
 * fix weird leading whitespace for categories included via recipesummary. Cookbook:Focaccia
 * cookbook disambiguation pages for multiple recipes
 * move Cookbook:Breakfast Recipes to Cookbook:Breakfast
 * sort Category:Confection recipes into Cookbook:Candy recipes or other appropriate cats
 * Deal with Category:Books to be merged; make recipe merge templates and add them appropriately. Take over Category:Duplicate recipes
 * put the cakes back in 'dessert' as they're different categorization schemes see Special:Contributions/Pfctdayelise

Category:Cookbook pages with recipe lists

Pages that need attention:
 * Cookbook:Basic foodstuffs
 * Cookbook:Basque_cuisine
 * Cookbook:Boiled_pasta
 * Cookbook:Boiling_rice
 * Cookbook:Asian Cuisine
 * Cookbook:Cereal Grain
 * Cookbook:Cereal
 * Cookbook:Chicken Soup
 * Cookbook:Crème Brûlée
 * Cookbook:FAQ
 * Cookbook:History_of_food_and_cooking or deletion.

Pages I was going to speedy delete, but Horning thought we should talk about:
 * Cookbook:Chicken Riggies
 * Cookbook:Chicken Rice Salad
 * Cookbook:Chettinad cuisine
 * Cookbook:Chana Masala
 * Cookbook:Carrots cooked kinpira style
 * Cookbook:Healthy_Vegan_Diet

Other things I think we should delete:
 * Cookbook:Churrasco
 * Cookbook:Creme_Anglaise/trial
 * Cookbook:Creamy_Delight_Smoothie
 * Cookbook:Easy_Orange_Smoothie
 * Cookbook:Ethics
 * Cookbook:Faggot
 * Cookbook:Ernie_Salad
 * Cookbook:Fantasy_and_Mythical_cuisines
 * Cookbook:Flourly_potatoes
 * Cookbook:Fruitmince
 * Cookbook:Harvard Healthy Eating Pyramid Recipes
 * Cookbook:Hot and Spicy Dip
 * Cookbook:Italian_Pasta
 * Cookbook:Jefferson Noodles
 * Cookbook:Low Fat Smoothie/Shake
 * Cookbook:Marrow
 * Cookbook:Menemen
 * Cookbook:Microwave Quesdilla
 * Cookbook:Nature
 * Cookbook:Pasta_alla_Carbonara
 * Cookbook:Fred_Steak
 * Category:Basic foodstuffs

This user's contribs:
 * Special:Contributions/TWODM256

Suggested categorization drives:
 * Category:Equipment
 * Category:Units
 * Category:Vegetables

Commercial Vegan Substitutions

 * † Available in refrigerated varieties.
 * ‡ Available in non-hydrogenated varieties.
 * 1 VeganRella's sister brands, TofuRella and AlmondRella contains casein. The Rella brands are all made on the same machinery, as noted in this mailing list post.
 * 2 Most of Galaxy Foods' (SoyCo) products are not vegan and are probably made on the same machinery.
 * 3 Yves also produces "Good Dogs" which contain butter.
 * 4 Lightlife also produces "Smart Franks" which contain eggs.
 * 5 A "ground" product which can be cut into patties.
 * 6 Most horchata (especially powder-based ones sold at Mexican restaurants) contains dried milk or casein.
 * 7 Most Boca products contain milk and eggs.
 * 7 Other Whole Foods 365 burgers contain milk and cheese.

title: User:Kellen/Scratch


text:
French/Texts/National anthems/Le cantique suisse

History of the Anthem
Originally written in German, the Swiss Psalm has been translated into all the official Swiss languages. The Swiss government has written a history of the Swiss anthem, known as the Swiss Psalm.

The often mentioned three-stanza English translation, exemplified below, is closer to the German lyrics arrangement than the French lyrics arrangement. However the versions by all languages are generally similar in spirit throughout with sentences rearranged or restructured within or beyond a stanza for example.

 * German 1st Stanza
 * Trittst im Morgenrot daher,
 * Seh' ich dich im Strahlenmeer,
 * Dich, du Hocherhabener, Herrlicher!
 * Wenn der Alpenfirn sich rötet,
 * Betet, freie Schweizer, betet!
 * Eure fromme Seele ahnt
 * Gott im hehren Vaterland,
 * Gott, den Herrn, im hehren Vaterland.

 * Common English translation of German 1st Stanza
 * When the morning skies grow red
 * And o'er us their radiance shed,
 * Thou, O Lord, appearest in their light.
 * When the Alps glow bright with splendour,
 * Pray to God, to Him surrender,
 * For you feel and understand,
 * That He dwelleth in this land.


text:
French/Texts/National anthems/La Brabançonne

About
La Brabançonne is the national anthem of Belgium.

Questions
1. What are the last 3 lines in English?

2. What is the blood spilt for Belgium described as in the third line?


text:
User:CR~enwikibooks

i was using this namespace to prototype for the incipient wikibook "longevity", but have since realized the whole project was more suited to a wikipedia "wikiproject", and am thus moving all the content there.

title: User:CR~enwikibooks


text:
Talk:NVIDIA

Being marked up for being a speedy delete
First of all, the note to be a speedy delete was added by an anonymous user, which IMHO is a form of vandalism. If they really had an issue here, they should have taken it up on this discussion page and tried to resolve the issue. Also, Wikibooks is about much more than just textbooks, and this does qualify as an instructional resource.

Writing a book can be a very painstaking process, and new users should not be expected to get everything down 100% to make it have the "appearance" of a book. Please, if there is a dispute with this book, bring it up here, especially during the initial development period.

I've taken down the VfD and invoked the Deletion policy on this content to give you a little bit of breathing room to create this content. Please take advantage of this time period. --Rob Horning 13:52, 10 April 2006 (UTC)

Availability of original content
Some of the content of the deleted Web page mentioned on the opening page here is available on the WayBack server of the Internet Archive at http://web.archive.org/web/20050102210051/www.gmpf.de/index.php/NVidia:Basic_Installation 216.7.19.20 12:28, 2 September 2007 (UTC)

There is already a german NVIDIA-Wiibook available
It's available at either http://wiki.mosnis.dyndns.org/NVIDIA in the Developmentversion for different Operatingsystems or in a Linux-Only Version at http://wiki.linux-club.de/NVIDIA licenced not under the GFDL but a CreativeCommons licence. Perhaps someone will help in translating? --87.78.254.74 (talk)

title: Talk:NVIDIA


text:
User:Barthax

Avid gamer, especially the older stuff.
 * Twin Galaxies Sinclair ZX Spectrum Referee.
 * System Leader at HARP for Sega Genesis/Megadrive & Bally Astrocade.
 * (Inactive) Confirmer at MARP (not that this is a big achievement!)
 * Accessible gaming achievements:
 * Twin Galaxies: A. Peter Mee
 * MAME Action Replay Page (MARP): Barthax
 * Home Action Replay Page (HARP): Barthax
 * CyberScore: Barthax
 * CivFanatics: Barthax
 * TMX Nations: Barthax

Various Wiki Guises:
 * Wikipedia
 * Meta Wiki
 * MediaWiki
 * GamerWiki
 * Wikibooks
 * Wiktionary
 * StrategyWiki
 * Atari-Forum Wiki

title: User:Barthax


text:
Colonizing Outer Space

Introduction

 * 1) Space colonization
 * 2) International space law
 * 3) Space advocacy
 * 4) Living in space
 * 5) Generalities about Off-Earth colonization
 * 6) Colonizing other celestial bodies

Resources

 * Astrodynamics
 * Space Transport and Engineering Methods

title: Colonizing Outer Space


text:
Colonizing Outer Space/Colonization/Law

International Space Law
International Space Law is a set of documents which govern activities in outer space by the governments of the Earth. While nobody actually lives permanently outside of Earth right now, to suggest there are no existing laws governing activities in space is not accurate either. Right now outer space is governed through various international treaties regarding who is responsible for the equipment that goes beyond Earth and what legal rights may be asserted by governments of many nations. Below are some of the major documents of space law and a brief overview of each document:

 * Outer Space Treaty - Basic governance of extraterrestrial bodies and relinquishment of sovereignty claims by Earth governments.
 * Moon Treaty - Follow up treaty to Outer Space Treaty, but has not been ratified by any major space-faring nation except for France.
 * Nuclear Test Ban Treaty - Prohibition of nuclear weapons in space and legal consequences of nuclear detonation in space. Also affects technologies like the Orion project and nuclear rockets.
 * Montevideo Convention - declaring requirements to be recognized as a "state" by the UN and other nations.
 * The Antarctic Treaty - The original international treaty for new territories post WWII, and the basis for much of the current space law even if it doesn't explicitly deal with celestial bodies.

United States of America
In April 2, 2012 the Competitive Enterprise Institute (CEI), a non-profit think tank with the purpose of advancing economic liberty and fighting over-regulation by big government released a new study named by Adjunct Scholar Rand Simberg. In it they argue that the U.S.A. should recognize transferable off-planet land claims. Arguing that the 1967 Outer Space Treaty only clearly prohibits declarations of national sovereignty, and noting that the U.S.A. is not a signatory of the 1979 Moon Treaty, which does outlaw private property claims in space. That a legal regime for real estate on the Moon, planets and asteroids could usher in a new era of space exploration at little or no cost to the U.S.A. government. But without off-planet property rights, investors have little incentive to fund space transportation or development.

title: Colonizing Outer Space/Colonization/Law


text:
Colonizing Outer Space/Colonization/Space/Generalities/Celestial Bodies

Colonizing Mars
Mars is a frequent topic of discussion regarding colonization possibilities. A project for the colonization of Mars provides an opportunity to examine how we live here on Earth and what aspects of our own planet and its effects on our lives we value and would wish to re-create on Mars. The Mars Project also provides a case-study for further planetary colonization.

Mars overall surface area is similar to the dry land surface Earth, it has large water reserves, and has carbon (locked as carbon dioxide in the atmosphere). It may have gone through similar geological and hydrological processes as Earth and contain valuable mineral ores, but this is debated.

Mars's atmosphere is very thin (averaging 800 Pa or about 0.8% of Earth sea-level atmospheric pressure) and the climate is colder. Its gravity is only around a third that of Earth. Mars is often the topic of discussion regarding terraforming to make the entire planet or at least large portions of it habitable.

As of 2011, according to the Journal of Cosmology, an interplanetary trip to Mars from Earth, could take as little as 10 months, and it would be possible for a privately-funded one-way mission to Mars to depart as soon as 20 years from now. But returning would be virtually impossible (due to payload and energy restrictions).

In regards to volunteers, in 2011 a special edition of the Journal of Cosmology, detailing and promoting a privately-funded one-way mission to Mars prompted more than 400 readers to volunteer as colonists. This leads to the conclusion that there are people willing to take on the risk, but it also creates the need to select those that should and are more capable of being successful.

Solar energy would be a problem in mars, there is not only night time to consider but a large amount of dust.

There is an ongoing debate on this topic on the Mars Civil-Cultural mailing list. One way that this topic can be explored is to look at the lessons learned from science fiction.

Life and fossils
There is increasing evidences that point to the, at least ancient, presence of life on Mars, this has created a strong scientific interest in colonizing Mars. A study from 2011 from NASA indicates that if life existed on at some point in Mars history, the best chances for it to have survived (or at least to have persisted longer) would have been below the surface. This hypothesis is based in geological studies performed by probes from NASA and the European Space Agency in more than 350 sites, along with what is known of mars today.

Water
Water is one of the most important substances for life as we know it. In the last three decades upon reviewing images of the surface of Mars taken by the Viking orbiters in the 70s, scientists became puzzled by the apparent sloping of rocky material around mountains and cliffs, these deposits, referred to as lobate debris apron, seems to extend sometimes for tens of miles. The debate around those formations generated the theory that they may have been formed by the lubrication of the rocky debris by a thin layer of ice, since similar features can be observed on Earth, for instance in Antarctica.

A new indication that water seems to exist on the planet came from data collected by the Shallow Subsurface Radar (SHARAD) instrument aboard the Mars Reconnaissance Orbiter probe, lunched in August 12, 2005 and operating at destination from 2006 onward. The radar results are consistent with massive deposits of water in middle latitudes, providing further support for the debris covered glacier hypothesis.

Equipment is available to extract in situ resources (water, air, etc.) from the Martian ground and atmosphere.

Agriculture
Martian soil contains several nutrients required to support to plant life. With fertilizer and environmental control, it could be used to grow plants.

Artifacts and equipment already available
There are several artifacts left on the surface of Mars from previous exploration and still working equipment, like monitoring satellites.

Colonies on the Moon
The Moon is Earth's only natural satellite, it is also known as Luna in Latin and other languages. Due to its proximity and relative familiarity, Earth's Moon is also frequently discussed as a target for colonization.

A colony on Luna has several important advantages over a colony on any other celestial body outside Earth. One big advantage is that it is close proximity to Earth. It would be comparatively easy to supply any colonization process, place and evacuate colonists, and use radio signals for reprogramming and remote control of robots and other devices on the Moon's surface. Another advantage is the low gravity, allowing for easier exchange of goods and services. But the topmost advantage is that we already have been there. Solar energy on the moon is a possibility by the moon has two week nights and similar to Mars dust will constitute a problem, not atmospheric dust but the surface regolith is an extreme problem, making nuclear power more attractive.

A major drawback of the Moon is its low abundance of volatiles necessary for life such as hydrogen and carbon. Water ice deposits was found in relative abundance, especially in some polar craters, that could serve as significant sources for these elements.

India's first unmanned lunar probe, launched in	22 October 2008, Chandrayaan-1 intended to last for 2 years in a compressive mapping of moon resources has generated data that leads NASA's scientists to estimate the existence of 600 million metric tons of water ice in craters around the Moon's north pole.

The existence of numerous craters near the poles of the Moon that have interiors that are in permanent sun shadow, makes them capable of maintaining very low temperatures. Without the effects from the Sun's radiation water can be indefinitely kept as ice. If this estimation is verified, mining this resource, could serve as a source of life support for future lunar bases as water and oxygen or as propellant for future spacecraft if converted to methane, dramatically reducing the cost of space exploration. NASA in April 2012 contracted Astrobotic Technology Inc. to determine whether its polar rover can deploy an ice-prospecting payload to the Moon, aiming for such verification.

There are also several magma caves that would facilitate the creation of protected habitats.

According to M.J. Fogg, "it is just possible to terraform the Earth's moon, so long as you are prepared to keep its imported volatiles topped up every few thousand years or so". http://www.users.globalnet.co.uk/~mfogg/gallery.htm
 * The lunar space elevator

Planed Moon Colonies
China

Mercury
There is a suggestion that Mercury could be colonized using the same technology, approach and equipment that is used in colonization of the Moon.

Venus
While the surface of Venus is far too hot and features atmospheric pressure at least 90 times that at sea level on Earth, its massive atmosphere offers an alternate location for colonization. At a height of approximately 50 km, the pressure is reduced to a few atmospheres, and the temperature would be between 40-100° C, depending on the height. This part of the atmosphere is probably within dense clouds which contain some sulphuric acid. Even these have a certain benefit to colonization, as they present a possible source for the extraction of water. Thus, the most likely colony on Venus would be a floating city in the clouds that stays above 50 km above the surface.

Europa
The Artemis Project designed a plan to colonize Europa, one of Jupiter's moons. Scientists were to inhabit igloos and drill down into the Europan ice crust, exploring any sub-surface ocean. It also discusses use of "air pockets" for human inhabitation.

Gas Giants
It may also be possible to colonize the three furthest gas giants with floating cities in their atmospheres. By heating hydrogen balloons large masses can be suspended underneath at roughly Earth gravity. Jupiter would be unsuitable for habitation due to its high gravity, escape velocity and radiation. Such colonies could export Helium-3 which would be used to fuel fusion reactors for energy.


text:
Talk:Blender 3D: Noob to Pro/Every Material Known to Man/Metal

This page looks like it could use some editing.

At the moment, it seems to serve no purpose.


text:
Talk:Brewing

Focus areas
I have created a page /Focus areas/ to get some corodination as to
 * What needs to be done
 * Who is all involved
 * Who needs help

This may not quite be the right way of doing it... but bold editing is what I am trying at the moment - bit of the blind leading from the front Andbrew.downes (talk) 09:13, 11 November 2009 (UTC)

This was the origninal text as an introduction: Brewing is the making of beer. Grain (most commonly barley) is allowed to sprout. Enzymes are produced within the grain which can convert starch into sugars. The sprouting is stopped by heating the grain (a process known as malting). Malted grain can be stored until needed. When the brewing process takes place, the malted grain is cracked or milled to allow access to the starchy interior and then soaked in warm water. The enzymes then turn the starch into sugar, and the resultant thin sweet liquid (known as wort) is drained off the spent grain. The wort is then boiled, usually with the addition of plant-based flavour modifiers. Boiling has the effect of sanitising the wort, and concentrating the sugars present because the volume is reduced as water is evaporated off. After cooling the wort, yeast is added and fermentation takes place: the yeast converts sugars to alcohol and carbon dioxide. The result is an alcoholic drink generically known as beer. Many subtleties and variations on the brewing process exist. I am planning on changing the text into a more 'introduction' style of text and drawing the reader in. Elfich 15:48, 10 April 2006 (UTC)

How would people feel about expanding this book to include home production of other fermented beverages such as wine, mead, and cider?

Re: Expanding to wine, mead, or cider
Wine mead and cider differ from beer in several aspects. I wouldn't think integrating their production with the beer making text would be useful, but seperate sections might be a good idea. Also, the production of these other beverages is not refered to as "brewing," so I don't think that this wikibook is obliged to discuss them based on the title. However, mentioning the basic processes couldn't hurt much, but each of them deserve their own book, IMHO. --Schusselig 02:51, 11 July 2007 (UTC)

Forwards
I've added a pair of forwards; to the student and to the instructor, with the hope of encouraging this book in the direction of becoming more of a text book than it currently is. I was trying to provide some guidance that a person wishing to learn how to brew (or teach) could follow through the book and become enamored with the process without reorganizing the book. I think it's important for a new brewer to start with an extract beer first, due to the smaller amount of initial investment required and also to the shorter period of time and in depth knowledge required to get that first batch consumed. This is probably due to the way I learned to brew myself; learning from my oldest brother who is still an extract only brewer and then later from reading books such as the Joy of Home Brewing whose organization begins with extract only, extract with grain adjuncts, and then finally with all-grain. While Mr. Papazian's books remain treasured parts of my library, I can't help but hate finding parts of the same material in three seperate parts of the book. I did not have anything to do with the initial organisation of this book, but I feel it can result in a more concise and usable book, provided a decent guide to encourage new comers. Perhaps a wikiversity class on brewing may be in order to fufill the need I see, but I believe that a simple guide would be sufficient while a class would better accomplish such goals live and in person. But I've digressed a bit...

So, I shall now state a goal for myself and any other author, with regard to this book; to write in a manner encouraging to potential brewers while still providing a complete work. Uh, that's kind of vague, but it's the best I have right now. Maybe I'll try again later..... --Schusselig 21:33, 26 July 2007 (UTC)

title: Talk:Brewing


text:
Brewing/Appendix

A

 * adjuncts Sources of fermentable sugar other than barley. In extract brewing, any grain added is sometimes referred to as "adjunct". Some brewers refer to coloured malts as adjuncts.
 * ale A style of beer that uses top fermenting yeast.
 * alpha acid An insoluble resin found in hops that provides the bitter characteristic of beer. The alpha acids are converted to more soluble iso alpha acids during the wort boiling in the brewhouse stage of the process.
 * amylase The enzyme produced in the barley during malting that is needed for the conversion of the malt starches in the mash to malt sugars.
 * amylose The technical term for large straight starch chains of glucose molecules.
 * amylopectin branched starch molecules.

B

 * back a brewhouse vessel, for example under-back or hop-back.
 * brewing Brewing is a generic term used to describe the entire beer production process, however it is technically correct only to refer to the time from beginning the boil until pitching yeast as brewing.
 * beta acid An bitter insoluble resin present in hops that is not considered to add noticeable bitterness.

D

 * dunkel German word for dark.

E

 * enzyme a biological catalyst that helps a chemical reaction to take place. An enzyme is a special type of protein and in brewing the enzymes are provided by the malt.
 * ester a flavour compound produced by the yeast during fermentation that results in a beer with a fruity flavour. Esters are promoted by higher fermentation temperatures and lower pitching rates. Examples of esters are: ethyl hexanoate (red apple flavour), iso-amyl acetate (banana flavour - sought after in a Bavarian weizen) ethyl acetate (solvent flavour - an unwanted ester).
 * extract the sugars (and to a lesser extent amino acids) recovered from the mash. There are various measures of extract, the most popular are degrees Plato (°P) or specific gravity (S.G.)

F

 * fermentation The process in which yeast converts fermentable sugars to alcohol, CO2, and flavouring compounds.
 * fermentable sugars The most common fermentable sugars in an all malt wort are glucose, maltose and maltotriose. Other commonly occurring fermentable sugars are sucrose and fructose.

G

 * grist Milled grain from which fermentable starches are obtained; usually barley malt and sometimes adjuncts.
 * gyle Unfermented wort set aside to provide sugars to be used in natural carbonation by krausening.

H

 * hops Flower buds of the hop plant used to bitter beer and provide flavor and aroma characteristics.

I

 * IPA or India Pale Ale A style of ale beer that originated from the liberal use of hops as a preservative in English beer destined for the India colony.

K

 * krauesen (noun) A layer of foam and dead yeast that forms on the surface of the fermenting beer during the primary fermentation stage.
 * krauesen (verb) A method by which unfermented gyle is added to fermented beer, originating from the same initial wort, to prime natural carbonation.

L

 * lager (noun) A style of beer that is produced using lower fermentaion temperatures and a special bottom fermenting yeast.
 * lager (verb) A method of conditioning beer by storing carbonated beer at near freezing temperatures.
 * lautering The process by which spent grist and/or hops are removed from wort.
 * lauter tun A vessel for producing clear wort from the mash. The slotted false bottom holds back the husk material which acts as filter material to produce clear wort.

M

 * malt Grain that has be allowed to germinate, and then dried.
 * mash A process by which fermentable sugars, starches, and other flavor characteristics are extracted from malted grains.
 * mash tun a vessel for the mixing of the malt grist and mashing liquor, and for the conversion of the malt starch to malt sugars. In the traditional British mash tun (or a two vessel brewhouse), the vessel is also used for wort separation.

P

 * pitch A term used to describe the indroduction of yeast to fermentable wort.

S

 * sparge A hot water rinse of grist (mash) to remove additional extract during the lautering process.
 * steep the immersion in water - barley is steeped before germination to increase the water content in the grain to get the germination process started.

T

 * tun a brewhouse vessel, for example Mash tun, Lauter tun

U

 * ullage The space between the surface of a contained liquid and the top of the container.
 * under-back A vessel between the lauter tun and the wort kettle used to hold the wort until the wort kettle is available. Normally only found in commercial breweries where multiple brews are produced per day.

W

 * wort (pronounced "wert") The product of the brewhouse process - a liquid rich in fermentable barley sugars with the hop bitterness components and nitrogen for yeast growth. The growth medium for the yeast.

Y

 * yeast Microorganisms that are applied to the task of converting sugars to alcohol, CO2, and flavouring compounds.

Z

 * zymurgy The science of using yeast for fermentation. Also a magazine published by the American Homebrewers Association.

Other
Edit this section to add terms that are not defined here, that you yourself cannot define.

title: Brewing/Appendix


text:
User:Dwilliams274

You rock!!!

title: User:Dwilliams274


text:
User:Eigendude

I am a PhD. student in electrical engineering.

title: User:Eigendude


text:
User talk:Sad1225

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally. Kellen T 19:18, 10 April 2006 (UTC)
 * Please also check out the Cookbook policy page and the Cookbook talk pages.

title: User talk:Sad1225


text:
Electronics/I/O

=Input/Output=

Input
For example: a computer disk can have a lot of information on it, and the user selects a particular disk for the task in hand. Then the user activates an input procedure to have the computer move a copy of the given required information from the disk, where the original remains, into a part of the computer's memory for further use. The original on the disk can, if desired, be deleted.

Output
For example: Some work has been done using a computer, and now the result is to be moved from the computer onto a disk, which is one of many possible output devices. Usually the computer's screen, which is also an output device, shows exactly what is being done by the user so as to ensure the information concerned ends up at the correct place, outside of the computer. Often a printer is used, which is another output device.

title: Electronics/I/O


text:
XML - Managing Data Exchange/Google earth

KML Introduction
(most content here is directly quoted from the KML wikipedia article)

KML (Keyhole Markup Language) is an XML-based Markup language for managing the display of three-dimensional geospatial data in the programs Google Earth, Google Maps, Google Mobile, ArcGIS Explorer and World Wind. (The word Keyhole is an earlier name for the software that became Google Earth; the software was produced in turn by Keyhole, Inc, which was acquired by Google in 2004. The term "Keyhole" actually honors the KH-11|KH reconnaissance satellites, the original eye-in-the-sky military reconnaissance system now some 30 years old.)

The KML file specifies a set of features (placemarks, images, polygons, 3D models, textual descriptions, etc.) for display in Google Earth, Maps and Mobile. Each place always has a longitude and a latitude. Other data can make the view more specific, such as tilt, heading, altitude, which together define a "camera view". KML shares some of the same structural grammar as Geography Markup Language|GML. Some KML information cannot be viewed in Google Maps or Mobile.

KML files are very often distributed as KMZ files, which are Data compression|zipped KML files with a .kmz extension. When a KMZ file is unzipped, a single "doc.kml" is found along with any overlay and icon images referenced in the KML.

Example KML document:

   New York City New York City  -74.006393,40.714172,0

The MIME type associated to KML is application/vnd.google-earth.kml+xml.

The MIME type associated to KMZ is application/vnd.google-earth.kmz.

Basic KML Document Types
For an XML document to recognize KML specific tags you must declare the KML namespace (listed below).

You will see this declaration in all the example files listed.

In order to see use the examples provided in this chapter you will need to copy and paste the text into any text editor. Next you will save the file as a .kml. This can be done by choosing "save as" and naming the file with a .kml extension (You might have to surround the name in quotes ie "test.kml").

Placemarks
Placemarks simply make a clickable pinpoint inside Google Earth at an exact location based on coordinates. This can be useful for marking a point of interest or a beginning and ending destination to a trip.

The example KML document in the introduction uses the Placemark tag. If you want to move the placemark to a different location all you would change are the coordinates.

Paths
Paths are a series of connected coordinates that can be edited with line styles for a more bold appearance inside Google Earth. The height and color can be adjusted for a more exaggerated appearance and better clarity.

The following example is a path from Atlanta, Georgia to Nashville, Tennessee. The code may look a bit complicated but it is mostly just styling/formatting tags with the 4 actual coordinates at near the end. So if you wanted to use the same style wall but just make a different path, all you would do is change the coordinates.

 Paths Path from Atlanta to Nashville

  7f00ffff 4        7f00ff00

<Placemark> Atlanta to Nashville Wall structured path <styleUrl>#yellowLineGreenPoly</styleUrl> <LineString> 1        1         <altitudeMode>absolute</altitudeMode> -84.40204442007513,33.75488573910702,83269 		-84.37837132006098,33.82567285375923,83269 		-84.79700041857893,35.30711817667424,83269 		-86.79210094043326,36.15389499208452,83269      </LineString>

</Placemark> </Document>

Overlays
Overlays are graphics that can be placed over an area in Google Earth marked by coordinates. These graphics can show how an area looked at a different point in time or during a special event (like a volcanic eruption).

This overlay example comes from Google's KML samples webpage and shows what Mt. Etna looked like during an actual eruption.

<?xml version="1.0" encoding="UTF-8"?> <kml xmlns="http://earth.google.com/kml/2.1"> <Folder> Ground Overlays Examples of ground overlays <GroundOverlay> Large-scale overlay on terrain Overlay shows Mount Etna erupting on July 13th, 2001. <Icon> http://code.google.com/apis/kml/documentation/etna.jpg </Icon> <LatLonBox> 37.91904192681665        37.46543388598137         15.35832653742206         14.60128369746704         -0.1556640799496235       </LatLonBox> </GroundOverlay> </Folder>

You can see from the code that it takes the image etna.jpg and places it over the coordinates listed.

Polygons
Polygons are a neat feature of Google Earth that allow 3-D shapes to be molded anywhere in Google Earth. These shapes can be useful for making neat presentations or just showing the world a structure actually looks.

This example is a polygon of Turner Field (The Atlanta Braves' home stadium) in Georgia. There is no styling on the polygon to keep the code simple.

<?xml version="1.0" encoding="UTF-8"?> <kml xmlns="http://earth.google.com/kml/2.1">

<Placemark> Turner Field <Polygon> 1      <altitudeMode>relativeToGround</altitudeMode> <outerBoundaryIs> <LinearRing> -84.39024224888713,33.73459764262901,28 		-84.38961532726215,33.73451197628319,28 		-84.38830478530726,33.7350571795205,28 		-84.38811742696677,33.73579651137399,28 		-84.38856034410841,33.73618350237595,28 		-84.38930790023139,33.73647497375488,28 		-84.38997872537549,33.73655338302832,28 		-84.39051294303495,33.73605785090994,28 		-84.39056804786146,33.73528763589146,28 		-84.39024224888713,33.73459764262901,28

</LinearRing> </outerBoundaryIs> </Polygon> </Placemark>

As you can see from the code, the polygon's 3-D shape is made up from the latitude and longitude coordinates with the height being determined by the 3rd column value inside the coordinates tag (28 in this case).

Google Earth
Google Earth is the name of Google's free software that is responsible for handling these KML documents. It is a virtual world created by a collage of satellite images where a user can manipulate the Earth in any way to see its landscapes, oceans, and cities.

You can find more information and download the software here.

Basic Interface Navigation
The user interface is fairly straight forward and is extremely easy for computer illiterate users to just jump right in and start exploring. You can completely ignore the toolbars and buttons if you want and just click and grab on the Earth to shake, spin, or roll it as you please. To zoom in, simply right-click and pull down or up depending on the speed which you prefer to "fall" toward the surface.

If you would like help finding a location you can type in the location (in the form of City, State) in the "Fly to.." search box. Google Earth will then spin around and zoom in to the location entered. If you want to take a virtual vacation, Google has some preset locations saved in the window below "Fly to..." labeled "Sightseeing." Simply click on one of these locations and be taken to the location where pictures, articles, and comments can be all be viewed.

Points of Interest
Points of interest that Google has already marked for users are marked with different color dots and icons. These can be clicked on for a variety of information ranging from a simple comment, to a panoramic photograph of that exact location. Advanced users, interested in the code make-up of such a feature, can right click on any of Google's marks and choose "Copy." This will copy all the code for that feature where you can just paste it into a text document to see all of the tags and references.

Other notes

 * Point of Interest
 * Geography Markup Language

Wikipedia in other languages

 * كيه إم إل (Arabian)
 * Keyhole Markup Language (German)
 * KML (Spanish)
 * Keyhole Markup Language (Italian)
 * Keyhole Markup Language (Hungarian)
 * Keyhole Markup Language (Dutch)
 * Keyhole Markup Language (Polish)
 * KML (Russian)

 * /Exercises/


text:
Talk:XML - Managing Data Exchange/Google earth

I am planning to go back and dress up my chapter some, I know right now it is not really easy on the eyes. I also want to add some more examples when I have more time to get familiar with the tutorial and add some exercises. Honestly right now I do not know what I could add for an exercise with this. I am just reading over it and trying to understand it myself.


text:
User:Johncub

Sorry, not much to say but didn't want to leave this blank. Feel free to leave me a message on my discussion page.

title: User:Johncub


text:
ActionScript Programming/AS In A Nutshell

Setting up an ActionScript application
To get started coding in ActionScript 2.0 (or AS 2.0 for short which is how I will be referring to it from now on), start up the Flash MX 2004 or later program. Then create a New Flash Document. When your new Document comes up, go to where the Timeline is and click the first frame of Layer 1. Now open up the Actions panel by either clicking on the Actions pane on the bottom of the screen or going to the File menu and clicking Window>Development Panels>Actions. Anything you type in here will be run first when you run or test the Flash movie. Now let’s create a simple program.

Put this code into the Actions panel:


text:
User:Redlentil/monobook.js

// User:Lupin/popups.js

mw.loader.load(            'https://en.wikipedia.org/w/index.php?title=User:Lupin/popups.js'             + '&action=raw&ctype=text/javascript&dontcountme=s');

title: User:Redlentil/monobook.js


text:
User talk:209.7.140.15

Please stop vandilising the RuneScape articles. Richard 20:28, 10 April 2006 (UTC)

title: User talk:209.7.140.15


text:
Umbraco

Table of Contents

 * /Installation/
 * /Templating/
 * Templates
 * FAQ (Frequently Asked Questions)
 * Reference
 * Plugins and Extensions
 * Samples and Articles
 * Help Out
 * Useful links
 * /Contribution guidelines/

title: Umbraco


text:
Umbraco/Installation

Requirements
Current system requirements can be seen on the Umbraco website.

Existing installation tutorials
An installation guide is available at the Umbraco website.

A hosted cloud version is also available from Umbraco, which eliminates the need for local installation.

Post install
You may wish to employ a method to disable the output of Debugging information when you install a production instance of Umbraco. See the Reference section for more details.

title: Umbraco/Installation


text:
Template:LaTeX/S-Green

This page uses material from Dr. Sheldon Green's Hypertext Help with LaTeX.

title: Template:LaTeX/S-Green


text:
Cookbook:Crêpe

A crêpe is a wide and thin pancake originating from France.

Batter
Crêpes should be very thin, so the batter for them must have certain characteristics. The batter should be thin and free of lumps in order to quickly and evenly coat the pan, but it needs to be strong enough to avoid tearing—the desired consistency is very similar to runny cream, and high-protein bread flour can contribute extra strength. Additionally, sieving the batter before use will eliminate any residual lumps. A bit of brandy or rum will add flavor to the batter, and a small amount of beer or sour milk will provide a subtle tang. Melted butter also adds flavor, but it can separate out of the batter if you choose to rest it overnight. If you do not want the batter to taste too "eggy", use fewer yolks than whites. If the batter is chilled overnight, it will thin slightly—be sure to account for this when preparing it. A basic ratio for the batter is 230 g eggs per 300 g flour, and then enough milk to make the correct consistency. Sugar and any other flavorings can then be added.

Cooking
Before you begin cooking, bring the batter to room temperature to avoid sticking. Choose a pan that you can lift and rotate easily so that you can quickly spread the batter around the pan. Ideally the pan should have sloping sides, and a non-stick coating is convenient. Even if you use a non-stick pan, be sure to brush the pan with a very thin coat of oil, which will help prevent sticking.

The pan should be heated over a fairly high heat before adding the batter. Once ready, add the minimum necessary batter to coat the bottom of the pan, and tilt to cover the entire base with a thin layer. The crêpe is ready to flip when the surface looks matte (i.e., no longer glistening with moisture). It usually takes 2–3 minutes for the first side to fully cook, followed by another fifteen seconds on the other side. Make sure to stir the remaining batter periodically while cooking to prevent the flour from sinking to the bottom of the bowl.

Instead of in a frying pan, crêpes can also be cooked on a specialized crêpe maker. These often afford better heat control and consistency, and the batter is spread using a special T-shaped wooden spreader. When cooking crêpes on one of these specialized plates, the batter can be slightly thicker since it is being manually spread out.

Troubleshooting

 * Brittle crepes: add an egg to the batter.

Equipment
Crêpes can be made in a normal skillet, but specialized hot plates exist as well. The cheaper plates have a very thin coated surface. Because the plate has almost no mass, these versions are very lightweight; however, the heat is not very consistent at the surface. These plates are cheaper because the materials and heating elements are durable, and they are okay if you use them only rarely. Additionally, making crepe on a coated plate is not that easy as it slips around on the surface and the coating can get scraped off.

More expensive plates have a solid cast-iron plate and a progressive thermostat where you can set the exact temperature the plate should have. Because of the massive cast iron they are very heavy, a plate with a diameter of 40 cm weighs around 23 kg. These devices are more durable and can last a lifetime. Before using, they must be seasoned with fat to create a nonstick surface.

Only buy a plate that is even and has no lip at the edge, as a lip makes the turning of the crepe more difficult.

Use
Crêpes are often filled and/or topped with other ingredients, such as chocolate-hazelnut spread, sugar, lemon, fruits, ice cream, and more. When eaten with savory toppings/fillings, they are usually called galettes. Crêpes can also be layered with cream or frosting to make what is called a crêpe cake.

title: Cookbook:Crêpe


text:
Talk:Guitar/Muting and Raking

I'm removing the gibberish that graced the discussion page earlier. Sameerkale 01:13, 29 January 2007 (UTC)

title: Talk:Guitar/Muting and Raking


text:
Talk:Quenya/Nouns

Untitled
This is a very interesting article. I have but one suggestion: perhaps you could include a pronunciation page? Pronunciation is one of the most important parts to learning a language; without pronunciation, we would not be able to rightly learn this language, or anyother. Just a thought... :( grandma

title: Talk:Quenya/Nouns


text:
Radiation Oncology/Mesothelioma

Malignant Pleural Mesothelioma

 * Staging
 * Official Guidelines
 * Overview
 * Mesothelioma randomized evidence''

title: Radiation Oncology/Mesothelioma