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User:TakuyaMurata/Calculus

Module and linear space
An additive group $$G$$ is said to be a module over $$R$$, or R-module for short, if the scalars, the members of a ring $$R$$, satisfy the following properties: if $$x, y \in G$$ and $$\alpha, \beta \in R$$
 * (i) Both $$\alpha x$$ and $$x + y$$ are in $$G$$
 * (ii) $$(\alpha \beta) x = \alpha (\beta x)$$ (associativity)
 * (iii) $$\alpha (x + y) = \alpha x + \alpha y$$ and $$(\alpha + \beta) x = \alpha x + \beta x$$ (distribution law)
 * (iv) $$1_R x = x$$

By definition, every abelian group itself is a module over $$\mathbb{Z}$$, since $$x + x + x + ... = nx$$ and $$n$$ is a scalar. Finally, a linear space is a module over a field. Defining the notion of dimension is a bit tricky. However, we can safely say a $$\mathcal{K}$$-vector space is finite-dimensional if it has a finite basis; that is, we can find linear independent vectors $$e_1, e_2, ..., e_n$$ so that $$\mathcal{V} = \{ a_1 e_1 + a_2 e_2 + ... + a_n e_n; a_j \in \mathcal{F} \}$$. Such a basis need not be unique.

3 Theorem ''Let $$\mathcal{V}$$ be a finite-dimensional $$\mathcal{K}$$-vector space. Then $$\mathcal{V}^*$$ has the same dimension as $$\mathcal{V}$$ does; that is, every basis for $$\mathcal{V}$$ has the same cardinality as every basis for $$\mathcal{V}^*$$ does.''

It can be shown that the map $$\mathcal{V} \to \mathcal{V}^*$$ cannot be defined constructively. (TODO: need to detail this matter)

1 Theorem If $$\mathcal{X}$$ is a TVS and every finite subset of $$\mathcal{X}$$ is closed, it then follows that $$\mathcal{X}$$ is a Hausdorff space.

Proof: Let $$x, y \in X$$ with $$x \ne y$$ be given. Moreover, let $$\Omega$$ be the complement of the singleton $$\{y\}$$, which is open by hypothesis. Since the function $$f(z) = x + z$$ is continuous at $$0$$ and $$f(0) = x$$ is in $$\Omega$$, we can find an $$\omega$$ open and such that $$\{x\} + \omega \subset \Omega$$. Here, we used, and would do so henceforward, the notation $$A + B = $$ the union of $$\{ x + y \}$$ taken all over $$x \in A$$ and $$y \in B$$. Furthermore, since the function $$g(x) = -x$$ is continuous and so is its inverse, namely $$g$$, we may assume that $$\omega = -\omega$$ by replacing $$\omega$$ by the intersection of $$\omega$$ and $$-\omega$$. By repeating the same construction for each $$x + z$$ where $$z \in \omega$$, we find $$\omega$$ so that $$\{x\} + \omega + \omega \subset \Omega$$. It then follows that $$\{x\} + \omega$$ and $$\{y\} + \omega$$ are disjoint. Indeed, if we write $$x + z = y + w$$ for some $$z, w \in \omega$$, then $$y = x + z - w \in \Omega$$, a contradiction. $$\square$$

Normed spaces
A vector space is said to be normed if it is a metric space and its metric $$d$$ has the form:
 * $$d(x, y) = \|x - y\|$$

Here, the function $$\|\cdot\|$$, called a norm, has the property (in addition to that it induces the metric) that $$\| \lambda x \| = |\lambda| \| x \|$$ for any scalar $$\lambda$$. We note that:
 * $$\|x+y\| = d(x, -y) \le d(x, 0) + d(0, -y) = \|x\| + \|y\|$$

and
 * $$d(x+z, y+z) = \|x - y\| = d(x,y)$$ for any $$x, y, z$$.

It may go without saying but a vector space is infinite-dimensional if it is not finite-dimensional.

3 Theorem ''Let $$\mathcal{X}$$, $$\mathcal{Y}$$ be normed spaces. If $$\mathcal{X}$$ is an infinite-dimensional and if $$\mathcal{Y}$$ is nonzero, there exists a linear operator $$f:\mathcal{X} \to \mathcal{Y}$$ that is not continuous.

Baire's theorem
A normed space is said to be complete when every Cauchy sequence in it converges in it.

3 Theorem ''Let $$E$$ is a subspace of a Banach space $$G$$ carrying the same norm. Then the following are equivalent:''
 * (a) $$E$$ is complete.
 * (b) $$E$$ is closed in $$G$$.
 * (c) $$\sum \|x_k\| < \infty$$ implies $$\sum x_k$$.

Proof: (i) Show (a) $$\iff$$ (b). If $$E$$ is complete, then every Cauchy sequence in $$E$$ has the limit in $$E$$; thus, $$E$$ is closed. Conversely, if $$E$$ is closed, then every Cauchy sequence converges in $$E$$ since $$G$$ is complete. Hence, $$E$$ is complete. (ii) Show (a) $$\iff$$ (c). Let $$x_j \in G$$ be a Cauchy sequence. Then
 * $$\left \| \sum_0^n x_k - \sum_0^m x_k \right \| = \left \| \sum_m^n x_k \right \| \le \sum_m^n \| x_k \| \to 0$$ as $$n, m \to \infty$$.

Thus, $$\sum x_k$$ is Cauchy, and converges in $$G$$ since the completeness. Conversely, since a Cauchy sequence is convergent, we can find its subsequence $$x_k$$ such that $$\| x_{k+1} - x_k \| < 2^{-k}$$. Then
 * $$\sum \| x_{k+1} - x_k \| < \infty$$.

If the summation condition holds, then it follows that $$\sum x_{k + 1} - x_k$$ converges in $$G$$. Hence, $$x_j$$ converges in $$G$$ as well. $$\square$$

3 Corollary $$Q$$ is incomplete but dense in $$\mathbb{R}$$.

Proof: $$\mathbb{Q}$$ is not closed in $$\mathbb{R}$$. Since $$\mathbb{R} \backslash \mathbb{Q}$$ has empty interior, $$\overline{\mathbb{Q}} = \overline{\mathbb{R}}$$. $$\square$$

We say a set has dense complement if its closure has empty interior.

The next is the theorem whose importance is not what it says literally but that of consequences. Though the theorem can be proved more generally for a pseudometric space; e.g., F-space, this classical formulation suffices for the remainder of the book.

3 Theorem A complete normed space $$G$$ which is nonempty is never the union of a sequence of subsets of $$G$$ with dense complement.

Proof: Let $$E_n \subset G$$ be a sequence of subsets of $$G$$ with dense complement. Since $$\overline{E_1}$$ has empty interior and $$G$$ has nonempty interior, there exists an nonempty open ball $$S_1 \subset (G \backslash \overline{E_1})$$ with the radius $$\le 2^{-1}$$. Since $$\overline{E^2}$$ has empty interior and $$S_1$$ has empty interior, again there exists an nonempty open ball $$S_2 \subset (S_1 \backslash \overline{E_1})$$ with the radius $$\le 2^{-2}$$. Iterating the construction ad infinitum we get the decreasing sequence $$S_n$$. Now let $$x_n$$ be the sequence of the centers of $$S_n$$. Then $$x_n$$ is Cauchy since: for some $$N \le n, m$$
 * $$\|x_n - x_m\| < 2^{-N} + 2^{-N} \to 0$$ as $$N \to \infty$$.

It then follows $$x_n$$ converges in $$G \backslash \bigcup^{\infty} E_n$$ from the compleness of $$G$$. $$\square$$

3 Corollary (open mapping theorem) ''If $$A$$ and $$B$$ are Banach spaces, then a continuous linear surjection $$f: A \to B$$ maps an open set in $$A$$ to an open set in $$B$$.

Proof: Left as an exercise.

The following gives an nice example of the consequences of Baire's theorem.

3 Corollary (Lipschitz continuity) ''Let $$S_n$$ = the set of functions $$u \in \mathcal{C}^0 ([0, 1])$$ such that there exists some $$x \in [0, 1]$$ such that:
 * $$|u(x + h) - u(x)| \le n |h|$$ for all $$x + h \in [0, 1]$$.

Then (i) $$\mathcal{C}^0 ([0, 1])$$ is complete, (ii) $$S_n$$ is closed and has dense complement, and (iii) there exists a $$u \in \mathcal{C}^0 ([0, 1])$$ that is not in any $$S_n$$; i.e., one that is differentiable nowhere.

Proof: (i) $$[0, 1]$$ is complete; thus, $$\mathcal{C}^0$$ is a Banach space by some early theorem. (ii) Let $$u_j \in S_n$$ be a sequence, and suppose $$u_j \to u$$. Then we have:

Thus, $$u \in S_n$$; i.e., $$S_n$$ is closed. Stone-Weierstrass theorem says that every continuous function can be uniformly approximated by some infinitely differentiable function; thus, we find a $$g \in \mathcal{C}^{\infty}([0, 1])$$ such that:
 * $$|u(x + h) - u(x)|$$
 * $$\le |u(x+h) - u_j(x+h)| + |u_j(x + h) - u_j(x)| + |u_j(x) - u(x)|$$
 * $$\to n|h|$$ as $$j \to \infty$$
 * }
 * $$\to n|h|$$ as $$j \to \infty$$
 * }
 * }
 * $$\| u - g \|$$.

If we let $$v = g + {\epsilon \over 2} \sin Nx$$, then
 * $$v \in \mathcal{C}^0 ([0, 1]) \backslash S_n$$

Hence, $$S_n$$ has dense complement. Finally, (iii) follows from Baire's theorem since (i) and (ii). $$\square$$

More concisely, the theorem says that not every continuity is Lipschitz because of Baire's theorem.

3 Lemma In a topological space $$X$$, the following are equivalent: Proof: The lemma holds since an open set is dense if and only if its complement has empty interior. $$\square$$
 * (i) Every countable union of closed sets with empty interior has empty interior.
 * (ii) Every countable intersection of open dense sets is dense.

When the above equivalent conditions are true, we say $$X$$ is a Baire space.

3 Theorem If a Banach space $$G$$ has a Schauder basis, a unique sequence of scalars such that
 * $$\| x - \sum_1^n \alpha_k x_k \| \to 0$$ as $$n \to \infty$$,

then $$G$$ is separable.''

Proof:

The validity of the converse had been known as a Basis Problem for long time. It was, however, proven to be false in 19-something by someone.

Duality
The kernel of a linear operator $$f$$, denoted by $$\ker(f)$$, is the set of all zero divisors for $$f$$. A kernel of a linear operator is a linear space since $$f(x) = 0 implies that f(\alpha x) = 0$$ and $$f(x) = 0 = f(y)$$ implies $$0 = f(x + y)$$. Moreover, a linear operator has zero kernel if and only it is injective.

3 Theorem ''Let $$f$$ be a linear functional. Then $$f$$ is continuous if and only if $$\ker(f)$$ is closed.''

Proof: If $$f$$ is continuous, then $$\ker(f) = f^{-1} \mid_{\{0\}}$$ is closed since a finite set is closed. Conversely, suppose $$f$$ is not continuous. Then there exists a sequence $$x_j \to x$$ such that
 * $$\lim_{j \to \infty} f(x_j) - f(x) = \lim_{j \to \infty} f(x_j - x) \ne 0$$

In other words, $$\ker(f)$$ is not closed. $$\square$$

3 Theorem ''If $$f$$ is a linear functional on $$l^p$$, then
 * $$f(x_1, x_2, x_3, ...) = \sum_1^{\infty} x_k y_k$$''

Proof: Let $$y_k = f(\delta(1, k), \delta(2, k), \delta(3, k), ...)$$ where $$\delta(j, k) = 1$$ if $$j = k$$ else $$0$$. $$\square$$

The dual of a linear space $$G$$, denoted by $$G^*$$, is the set of all of linear operators from $$G$$ to $$\mathbb{F}$$ (i.e., either $$\mathbb{C}$$ or $$\mathbb{R}$$). Every dual of a linear space becomes again a linear space over the same field as the original one since the set of linear spaces forms an additive group.

Theorem ''Let G be a normed linear space. Then''
 * $$\|x\| = \sup_{\|y\| = 1} ||$$ and $$\|y\| = \sup_{\|x\| = 1} ||$$.

The duality between a Banach space and its dual gives rise to.

Example: For $$p$$ finite, the dual of $$l^p$$ is $$l^q$$ where $$1/p + 1/q = 1$$.

3 Theorem (Krein-Milman) The unit ball of the dual of a real normed linear space has an extreme point.

Proof: (TODO: to be written)

The theorem is equivalent to the AC.

The Hahn-Banach theorem
3 Theorem (Hahn-Banach) ''Let $$\mathcal{X}, \mathcal{Y}$$ be normed vector spaces over real numbers. Then the following are equivalent.''
 * (i) Every collection of mutually intersecting closed balls of $$\mathcal{Y}$$ has nonempty intersection. (binary intersection property)
 * (ii) If $$\mathcal{M} \subset \mathcal{X}$$ is a subspace and $$f: \mathcal{M} \to \mathcal{Y}$$ is a continuous linear operator, then $$f$$ can be extend to a $$F$$ on $$\mathcal{X}$$ such that $$\|f\| = \|F\|$$. (dominated version)
 * (iii) If the linear variety $${x} + \mathcal{M}$$ does not meet a non-empty open convex subset $$G$$ of $$\mathcal{X}$$, then there exists a closed hyper-plane $$H$$ containing $${x} + \mathcal{M}$$ that does not meet $$G$$ either. (geometric form)

3 Corollary If the equivalent conditions hold in the theorem, $$\mathcal{Y}$$ is complete.

Proof: Consider the identity map extended to the completion of $$\mathcal{Y}$$. $$\square$$

3 Corollary ''Let $$f$$ be a linear operator from a Banach space $$\mathcal{X}$$ to a Banach space $$\mathcal{Y}$$. If there exists a set $$\Gamma$$ and operators $$f_1:\mathcal{X} \to l^\infty (\Gamma)$$ and $$f_2:l^\infty (\Gamma) \to \mathcal{X}$$ such that $$f_2 \circ f_1$$ and $$\|f_2\| = \|f_1\|$$, then $$f$$ can be extended to a Banach space containing $$\mathcal{X}$$ without increase in norm.

Hilbert spaces
A linear space $$\mathcal{X}$$ is called a pre-Hilbert space if for each ordered pair of $$(x, y)$$ there is a unique complex number called ''an inner product of $$x$$ and $$y$$ and denoted by $$\langle x, y \rangle_\mathcal{X}$$ satisfying the following properties:
 * (i) $$\langle x, y \rangle_\mathcal{X}$$ is a linear operator of $$x$$ when $$y$$ is fixed.
 * (ii) $$\langle x, y \rangle_\mathcal{X} = \overline {\langle y, x \rangle_\mathcal{X}}$$ (where the bar means the complex conjugation).
 * (iii) $$\langle x, x \rangle \ge 0$$ with equality only when $$x = 0$$.

When only one pre-Hilbert space is being considered we usually omit the subscript $$\mathcal{X}$$.

We define $$\| x \| = \langle x, x \rangle^{1/2}$$ and indeed this is a norm. Indeed, it is clear that $$\| \alpha x \| = | \alpha | \| x \|$$ and (iii) is the reason that $$\| x \| = 0$$ implies that $$x = 0$$. Finally, the triangular inequality follows from the next lemma.

3 Lemma (Schwarz's inequality) $$|\langle x, y \rangle| \le \|x\|\|y\|$$ where the equality holds if and only if we can write $$x = \lambda y$$ for some scalar $$\lambda$$.

If we assume the lemma, then since $$\operatorname{Re}(\alpha) \le | \alpha |$$ for any complex number $$\alpha$$ it follows:

 * $$\| x + y \|^2$$
 * $$= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2$$
 * $$\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 $$
 * $$\le (\| x \| + \| y \|)^2$$
 * }
 * $$\le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 $$
 * $$\le (\| x \| + \| y \|)^2$$
 * }
 * $$\le (\| x \| + \| y \|)^2$$
 * }

Proof of Lemma: The lemma is just a special case of the next theorem:

3 Theorem Let $$\mathcal{H}$$ be a pre-Hilbert and $$S \subset \mathcal{H}$$ be an orthonormal set (i.e., for $$u, v \in E$$ $$\langle u, v \rangle = 1$$ iff $$u = v$$ iff $$\langle u, v \rangle$$ is nonzero.) Proof: (TODO)
 * (i) $$\sum_{u \in S} \langle x, u \rangle \le |x|$$ for any $$x \in \mathcal{H}$$.
 * (ii) The equality holds in (i) if and only if $$S$$ is maximal in the collection of all orthonormal subsets of $$\mathcal{H}$$ ordered by $$\subset$$.

3 Theorem ''Let $$u_j$$ be a sequence in a pre-Hilbert space with $$\|u_j\| = 1$$. If $$\Gamma = \sum_{j \ne k} | \langle u_j, u_k \rangle |^2 < \infty$$, then
 * $$(1 - \Gamma) \sum_{j=m}^n | \alpha_j |^2 \le \| \sum_{j=m}^n \alpha_j u_j \|^2 \le (1 + \Gamma) \sum_{j=m}^n | \alpha_j |^2$$ for any sequence $$\alpha_j$$ of scalars.

Proof: Let $$I$$ be a set of all pairs $$(i, j)$$ such that $$m \le i \le n$$, $$m \le j \le n$$ and $$i \ne j$$. By Hölder's inequality we get:
 * $$\sum_{(j, k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle | \le \sum_{j=m}^n | \alpha_j |^2 \Gamma$$.

Since
 * $$\| \sum_{j=1}^\infty \alpha_j u_j \|^2 \le \sum_{j=m}^n |\alpha_j|^2 + \sum_{(j,k) \in I} | \langle \alpha_j u_j, \alpha_k u_k \rangle |$$,

we get the second inequality. Moreover,
 * $$\sum_{j=m}^n | \alpha_j |^2 \le \sum_{j=m}^n \langle \alpha_j u_j, \alpha_j u_j \rangle + \sum_{(j, k) \in I} \langle \alpha_j u_j, \alpha_k u_k \rangle + | \langle \alpha_j u_j, \alpha_k u_k \rangle |$$

and this gives the first inequality. $$\square$$

3 Theorem (Bessel's inequality) ''Let $$U$$ be an orthonormal subset of a pre-Hilbert space. Then for each $$x$$ in the space,
 * $$\sum_{u \in U} |\langle x, u \rangle|^2 \le \|x\|^2$$

where the sum can be obtained over some countable subset of $$U$$ and the equality holds if and only if $$U$$ is maximal; i.e., $$U$$ is contained in no other orthogonal sets.

Proof: First suppose $$U$$ is finite; i.e., $$U = \{ u_1, u_2, ... u_n \}$$. Let $$\alpha_j = \langle x, u_j \rangle$$. Since for each $$k$$, $$\langle x - \sum_{j=1}^n \alpha_j u_j, u_k \rangle = \langle x, u_k \rangle - \alpha_k \langle u_k, u_k \rangle = 0$$, by the preceding theorem or by direct computation,

Now suppose that $$U$$ is maximal. Let $$y = \sum_{j=1}^n \langle x, u_j \rangle u_j$$. Then by the same reasoning above, $$x - y$$ is orthogonal to every $$u_j$$. But since the assumed maximality $$x = y$$. Hence,
 * $$\|x\|^2$$
 * $$=\| x - \sum_{j=1}^n \alpha_j u_j \| + \| \sum_{j=1}^n \alpha_j u_j \|^2 $$
 * $$\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2$$
 * }
 * $$\ge \| \sum_{j=1}^n \alpha_j u_j \|^2 = \sum_{j=1}^n |\alpha_j|^2$$
 * }
 * }
 * $$\sum_{j=1}^n |\langle x, u_j \rangle|^2 = \| \sum_{j=1}^n |\langle x, u_j \rangle u_j |^2 = \|y\|^2 = \|x\|^2$$. Conversely, suppose that $$U$$ is not maximal. Then there exists some nonzero $$x$$ such that $$\langle x, u \rangle = 0$$ for every $$u \in U$$. Thus,
 * $$\sum_{j=1}^n | \langle x, u_j \rangle |^2 = 0 < \|x\|^2$$.

The general case follows from the application of Egorov's theorem. $$\square$$

3 Corollary In view of Zorn's Lemma, it can be shown that a set satisfying the condition in (ii) exists. (TODO: need elaboration)

3 Lemma The function $$f(x) = \langle x, y \rangle$$ is continuous each time $$y$$ is fixed.

Proof: If $$f(x) = \langle x, y \rangle$$, from Schwarz's inequality it follows:
 * $$| f(z) - f(x) | = | \langle z - x, y \rangle | \le \| z - x \| \| y \| \to 0$$ as $$z \to x$$. $$\square$$

Given a linear subspace $$\mathcal{M}$$ of $$\mathcal{H}$$, we define: $$\mathcal{M}^\bot = \{ y \in \mathcal{H}; \langle x, y \rangle = 0, x \in \mathcal{M} \}$$. In other words, $$\mathcal{M}^\bot$$ is the intersection of the kernels of the continuous functionals $$f(x) = \langle x, y$$, which are closed; hence, $$M^\bot$$ is closed. (TODO: we can also show that $$\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot$$)

3 Lemma ''Let $$\mathcal{M}$$ be a linear subspace of a pre-Hilbert space. Then $$z \in \mathcal{M}^\bot$$ if and only if $$\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}$$.''

Proof: The Schwarz inequality says the inequality
 * $$| \langle z, z + w \rangle | \le \|z\| \|z + w \|$$

is actually equality if and only if $$z$$ and $$z + w$$ are linear dependent. $$\square$$

3 Theorem (Riesz) ''Let $$\mathcal{X}$$ be a pre-Hilbert space and $$\mathcal{M}$$ be its subspace. The following are equivalent:'' Proof: If $$\overline{\mathcal{M}} = \mathcal{H}$$ and $$z \in \mathcal{M}^\bot$$, then $$z \in \overline{\mathcal{M}} \cap \mathcal{M}^\bot = {0}$$. (Note: completeness was not needed.) Conversely, if $$\overline{\mathcal{M}}$$ is not dense, then it can be shown (TODO: using completeness) that there is $$y \in \overline{\mathcal{M}}$$ such that
 * (i) $$\mathcal{X}$$ is a complete.
 * (ii) $$\mathcal{M}$$ is dense if and only if $$\mathcal{M}^\bot = \{ 0 \}$$.
 * (iii) Every continuous linear functional on $$\mathcal{X}^*$$ has the form $$f(x) = \langle x, y \rangle$$ where y is uniquely determined by $$f$$.
 * $$\|x - y\| = \inf \{ \|x - w\|; w \in \mathcal{M} \}$$.

That is, $$0 \ne x - y \in \overline{M}^\bot$$. In sum, (i) implies (ii). To show (iii), we may suppose that $$f$$ is not identically zero, and in view of (ii), there exists a $$z \in \ker(f)^\bot$$ with $$\|z\| = 1$$. Since $$f(xf(z) - f(x)z) = 0$$,
 * $$0 = \langle xf(z) - f(x)z, z \rangle = \langle x, \overline f(z) z \rangle - f(x)$$.

The uniqueness holds since $$\langle x, y \rangle = \langle x, y_2 \rangle$$ for all $$x$$ implies that $$y = y_2$$. Finally, (iii) implies reflexivility which implies (i). $$\square$$

A complete pre-Hilbert space is called a Hilbert space.

3 Corollary ''Let $$\mathcal{M}$$ be a a closed linear subspace of a Hilbert space Proof: (i) Let $$x \in \mathcal{H}$$ be given. Define $$f(w) = \langle w, x \rangle$$ for each $$w \in \mathcal{M}$$. Since $$f$$ is continuous and linear on $$\mathcal{M}$$, which is a Hilbert space, there is $$y \in \mathcal{M}$$ such that $$f(w) = \langle w, y \rangle$$. It follows that $$\langle w, x - y \rangle = 0$$ for any $$w \in \mathcal{M}$$; that is, $$x - y \in \mathcal{M}^\bot$$. The uniqueness holds since if $$y_2 \in \mathcal{M}$$ and $$x - y_2 \in \mathcal{M}^\bot$$, then $$f(w) = \langle x, y_2 \rangle$$ and the representation is unique. (ii) If $$x \in \mathcal{M}$$, then since $$x$$ is orthogonal to $$\mathcal{M}^{\bot \bot}$$. Thus, $$\mathcal{M} \subset \mathcal{M}^{\bot\bot}$$ and taking closure on both sides we get: $$\overline{\mathcal{M}} \subset \overline {M^{\bot\bot}} \subset M^{\bot\bot}$$. Also, if $$x \in \overline{M}^{\bot\bot}$$, then we write: $$x = y + z$$ where $$y \in \overline{M}$$ and $$z \in \overline{M}^\bot$$ and $$\|z\|^2 = \langle x - y, z \rangle = \langle x, z \rangle = 0$$. Thus, $$x = y \in \overline{\mathcal{M}}$$. Since $$\mathcal{M} \subset \overline{\mathcal{M}}$$ implies that $$\overline{\mathcal{M}} \subset \mathcal{M}^\bot$$ and $$\mathcal{M}^{\bot\bot} \subset \overline{\mathcal{M}}^{\bot\bot}$$, the corollary follows. $$\square$$
 * ''(i) For any $$x \in \mathcal{H}$$ we can write $$x = y + z$$ where $$y \in \mathcal{M}$$ and $$z \in \mathcal{M}^\bot$$ and $$y, z$$ are uniquely determined by $$x$$.
 * (ii) then $$\mathcal{M}^{\bot\bot} = \bar \mathcal{M}$$.

Integration
3 Theorem (Fundamental Theorem of Calculus) The following are equivalent. Proof: Suppose (ii). Since we have:
 * (i) The derivative of $$\int_a^x f(t) dt$$ at $$x$$ is $$f(x)$$.
 * (ii) $$f$$ is absolutely continuous.
 * $$\inf_{x \le t \le y} f(t) \le (x-y)^{-1} \int_x^y f(t) \le \sup_{x \le t \le y} f(t)$$,

for any $$a$$,
 * $$\lim_{y \to x} (x-y)^{-1} (\int_{a}^y f(t)dt - \int_{a}^x f(t)dt) = f(x)$$. $$\square$$

Differentiation
Differentiation of $$f$$ at $$x$$ is to take the limit of the quotient by letting $$h \rightarrow 0$$:
 * $${f(x + h) - f(x) \over h}$$.

When the limit of the quotient indeed exists, we say $$f$$ is differentiable at $$x$$. The derivative of $$f$$, denoted by $$\dot f$$, is defined by $$f(x)$$ = the limit of the quotient at $$x$$.

3.8. Theorem The power series:
 * $$u = \sum_0^{\infty} a_j z^j$$

is analytic inside the radius of convergence.

Proof: The normal convergence of $$u$$ implies the theorem.

To show that every analytic function can be represented by a power series, we will, though not necessarily, wait for Cauchy's integral formula.

We define the norm in $$\mathbb{R}^n$$, thereby inducing topology;
 * $$\| x \| = \| \sum_1^n x_j^2 \|^{1/2}$$.

The topology in this way is often called a natural topology of $$\mathit{R}^2$$, since so to speak we don't artificially induce a topology by defining $$\sigma$$.

3. Theorem (Euler's formula)
 * $$z = |z|e^{i \theta} = |z|(cos \theta + sin \theta)$$ If $$z \in \mathbb{C}$$.

Proof:

3. Theorem (Cauchy-Riemann equations) ''Suppose $$u \in \mathcal{C}^1(\Omega)$$. We have:
 * $${\partial u \over \partial \bar z} = 0$$ on $$\Omega$$ if and only if $${\partial u \over \partial x} - {1 \over i} {\partial u \over \partial y} = 0$$ on $$\Omega$$.

Proof:

3. Corollary ''Let $$u, v$$ are non-constant and analytic in $$\Omega$$. If $$\mbox{Re }u = \mbox{Re }v$$, then $$u = v$$.

Proof: Let $$g = u - v$$. Then $$0 = \mbox{Re }g = g + \bar g$$. Thus, $$\mbox{Im }g = 0$$, and hence g = 0. $$\square$$

This furnishes examples of functions that are not analytic. For example, $$u(x + iy) = x + iy$$ is analytic everywhere and that means $$v(x + iy) = x + icy$$ cannot be analytic unless $$c = 1$$.

A operator $$f$$ is bounded if there exists a constant $$C > 0$$ such that for every $$x$$:
 * $$\| f(x) \| \le C \| x \|$$.

3.1 Theorem Given a bounded operator $$f$$, if
 * $$\alpha = \inf \{ C : \| f(x) \le C \| x \| \}$$, $$\beta = \sup_{\| x \| \le 1} \| f(x) \|$$ and $$\gamma = \sup_{\| x \| = 1} \| f(x) \|$$,

then $$\alpha = \beta = \gamma$$.

Proof: Since $$\beta \le \gamma$$ can be verified (FIXME) and $$\gamma$$ is inf,
 * $$\|f(x)\| = \left\|f\left({x \over \|x\|}\right)\right\| \|x\| \le \alpha \|x\| \le \beta \|x\| \le \gamma \|x\|$$.

Thus,
 * $$\alpha \le \beta \le \gamma$$.

But if $$\alpha \|x\| < \gamma \|x\|$$ in the above, then this is absurd since $$\gamma$$ is sup; hence the theorem is proven.

We denote by $$\| f \|$$ either of the above values, and call it the norm of $$f$$

3.2 Corollary A operator $$f$$ is bounded if and only if is continuous.

Proof: If $$f$$ is bounded, then we find $$\| f \|$$ and since the identity: for every $$x$$ and $$h$$
 * $$\| f(x + h) - f(x) \| \le \| f \| \| h \|$$,

$$f$$ is continuous everywhere. Conversely, every continuous operator maps a open ball centered at 0 of radius 1 to some bounded set; thus, we find the norm of $$f$$, $$\| f \|$$, and the theorem follows after the preceding theorem. $$\square$$

3. Theorem If F is a linear space of dimension $$n$$, then it has exactly $$n$$ subspaces including F and excluding {0}.

Proof: F has a basis of n elements.

Theorem ''If H is complete, then $$H^n = \{ \sum_1^n x_j e_j : x_j \in E \}$$ (i.e., a cartesian space of E) is complete '' Proof: Let $$z_j \in H$$ be a Cauchy sequence. Then we have:
 * $$|z_n - z_m| = \left| \sum_1^n (z|e_j)e_j - \sum_1^m (z|e_j)e_j \right| \to 0$$ as $$n, m \to \infty$$.

Since orthogonality, we have:
 * $$|(x_n - x_m) e_1 + (y_n - y_m) e_2| = |x_n - x_m| + |y_n - y_m|$$,

and both $$x_j$$ and $$y_j$$ are also Cauchy sequences. Since completeness, the respective limits $$x$$ and $$y$$ are in $$E$$; thus, the limit $$z = x e_1 + y e_2$$ is in E_2. $$\square$$

The theorem shows in particular that $$\mathbb {R}, \mathbb {R^n}, \mathbb{C}, \mathbb{C^n}$$ are complete.

3. Theorem (Hamel basis) The Axiom of Choice implies that every linear space has a basis

Proof: We may suppose the space is infinite-dimensional, otherwise the theorem holds trivially.

FIXEME: Adopt. 3. Theorem (Fixed Point Theorem) ''Suppose a function f maps a closed subset $$F$$ of a Banach space to itself, and further suppose that there exists some $$c < 1$$ such that $$\|f(x) - f(y)\| \le c \| x - y \|$$ for any $$x$$ and $$y$$. Then $$f$$ has a unique fixed point.''

Proof: Let $$s_n$$ be a sequence: $$x, f(x), f(f(x)), f(f(f(x))), ... $$. For any $$n$$ for some $$x \in F$$. Then we have:
 * $$\| s_{n + 1} - s_n \| = \| f(s_n) - f(s_{n - 1}) \| \le c \| s_n - s_{n - 1} \|$$.

By induction it follows:
 * $$\| s_{n + 1} - s_n \| = c^n \| s_1 - s_0 \|$$.

Thus, $$s_n$$ is a Cauchy sequence since:

That $$F$$ is closed puts the limit of $$s_n$$ in $$F$$. Finally, the uniqueness follows since if $$f(x) = x$$ and $$f(y) = y$$, then
 * $$\| s_{n+1} + ... s_{n+k} - s_n \| \le \sum_0^k \|s_{n + j}\| \le \| s_1 - s_0 \| c^n \sum_0^k c^j$$
 * $$= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})$$.
 * }
 * $$= \| s_1 - s_0 \| c^n(c - 1)^{-1}(1 - c^{k+1})$$.
 * }
 * $$\|f(x) - f(y)\| = \|x - y\| \le c \| x - y \|$$ or $$1 \le c$$unless $$x = y$$. $$\square$$

3. Corollary (mean value inequality) ''Let $$f : \mathbb{R}^n \to \mathbb{R}^m$$ be differentiable. Then there exists some $$z = (1 - t)x + ty$$ for some $$t \in [0, 1]$$ such that
 * $$\|f(x) - f(y)\| \le \| f' (z) \| \| x - y \|$$''

where the equality holds if $$n = m = 1$$ (mean value theorem).

Proof:

Theorem ''Let $$f: E \to \mathbb{R}$$ where $$E \subset \mathbb{R}^n$$ and is open. If $$D_1 f, D_2 f, ... D_n f$$ are bounded in $$E$$, then $$f$$ is continuous.''

Proof: Let $$\epsilon > 0$$ and $$x \in E$$ be given. Using the assumption, we find a constant $$M$$ so that:
 * $$\sup_E | D_i f | < M$$ for $$i = 1, 2, ... n$$.

Let $$\delta = \epsilon (nM)^{-1}$$. Suppose $$|h| < \delta$$ and $$x + h \in E$$. Let
 * $$\phi_k(t) = f \left( x + \sum_1^k(h \cdot e_j)e_j + t(h \cdot e_{k+1})e_{k+1} \right)$$.

Then by the mean value theorem, we have: for some $$c \in (0, 1)$$,

It thus follows: since $$\phi_k(0) = \phi_{k-1}(1)$$,
 * $$| \phi_k(1) - \phi_k(0) |$$
 * $$= |h| \left| D_k f(x + \sum_1^k (h \cdot e_j)e_j + c(h \cdot e_{k+1})e_{k+1}) \right|$$
 * $$< |h| M$$.
 * }
 * $$< |h| M$$.
 * }
 * }

 * $$| f(x+h) - f(x) |$$
 * $$= |\phi_n(1) - \phi_1(0) | = \left| \sum_1^n \phi_k(1) - \phi_k(0) \right|$$
 * $$< |h| nM < \epsilon$$ $$\square$$
 * }
 * $$< |h| nM < \epsilon$$ $$\square$$
 * }
 * }

Theorem (differentiation rules)' Given $$f, g: \mathbb{R} \to \mathbb{R}$$ differentiable, Proof: (b) and (c) follows after we apply (a) to them with $$log$$, $$h(x) = x^{-1}$$ and the implicit function theorem. $$\square$$.
 * (a) (Chain Rule) $$D(g \circ f) = (D(g) \circ f)D(f)$$.
 * (b) (Product Rule) $$D(fg) = D(f)g + fD(g)$$.
 * (b) (Quotient Rule) $$D(f / g) = g^{-2} (D(f)g - fD(g))$$.

Theorem (Cauchy-Riemann equations) ''Let $$\Omega \subset \mathbb{C}$$ and $$u:\Omega \to \mathbb{C}$$. Then $$u$$ is differentiable if and only if $${\partial \over \partial x}u$$ and $${\partial \over \partial y}u$$ are continuous on $$\Omega$$ and $${\partial \over \partial z} u = 0$$ on $$\Omega$$.''

Proof: Suppose $$u$$ is differentiable. Let $$z \in \Omega$$ and $$x = \mbox{Re}z$$ and $$y = \mbox{Im}z$$.

Since $$x = {z + \overline{z} \over 2}$$ and $$y = {z - \bar z \over 2i}$$, the Chain Rule gives:
 * $$u'(z)$$
 * $$= \lim_{h \in \mathbb{R} \to 0} {u(x+h, y) - u(x,y) \over h} = {\partial \over \partial x}u(z)$$
 * $$= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)$$
 * }
 * $$= \lim_{h \in \mathbb{R} \to 0} {u(x, y+h) - u(x,y) \over ih} = {1 \over i} {\partial \over \partial y}u(z)$$
 * }
 * }

Conversely, let $$z \in \Omega$$. It suffices to show that $$u'(z) = {\partial \over \partial x} u(z)$$. Let $$\epsilon > 0$$ be given and $$x = \Re z$$ and $$y = \Im z$$. Since the continuity of the partial derivatives and that $$\Omega$$ is open, we can find a $$\delta > 0$$ so that: $$B(\delta, z) \subset \Omega$$ and for $$s \in B(\delta, z)$$ it holds:
 * $${\partial \over \partial \bar z} u$$
 * $$= \left( {\partial x \over \partial \bar z} {\partial \over \partial x} + {\partial y \over \partial \bar z} {\partial \over \partial y} \right) u$$
 * $$={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u$$
 * $$=0$$.
 * }
 * $$={1 \over 2} \left( {\partial \over \partial x} - {1 \over i} {\partial \over \partial y} \right) u$$
 * $$=0$$.
 * }
 * $$=0$$.
 * }
 * $$\left| {\partial \over \partial x}(u(s) - u(z)) \right| < \epsilon / 2$$ and $$\left| {\partial \over \partial y}(u(s) - u(z)) \right| < \epsilon / 2$$.

Let $$h \in B(\delta, 0)$$ be given and $$h_1= \Re h$$ and $$h_2 = \Im h$$. Using the mean value theorem we have: for some $$s_1, s_2 \in B(\delta, z)$$,

where $${\partial \over \partial y} u = i {\partial \over \partial x} u$$ by assumption. Finally it now follows:
 * $$u(x+h_1, y+h_2) - u(x, y)$$
 * $$= u(x+h_1, y+h_2) - u(x, y+h_2) + u(x, y+h_2) - u(x, y)$$
 * $$= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)$$
 * }
 * $$= h_1 {\partial \over \partial x} u(s1) + h_2 {\partial \over \partial y} u(s2)$$
 * }
 * }

 * $$\left| {u(z + h) - u(z) \over h} - {\partial \over \partial x} u(z) \right|$$
 * $$= \left| {h_1 \over h} \right| \left| {\partial \over \partial x} (u(s_1) - u(z)) \right| + \left| {h_2 \over h} \right| \left| {\partial \over \partial x} (u(s_2) - u(z)) \right|$$
 * $$< \epsilon$$ $$\square$$
 * }
 * $$< \epsilon$$ $$\square$$
 * }
 * }

3 Corollary ''Let $$u \in \mathcal{A}(\Omega)$$ and suppose $$\Omega$$ is connected. Then the following are equivalent:'' Proof: That (a) $$\Rightarrow$$ (b) is obvious. Suppose (b). Since we have some constant $$M$$ so that for all $$z \in \Omega$$,
 * (a) $$u$$ is constant.
 * (b) $$\mbox{Re}u$$ is constant.
 * (c) $$|u|$$ is constant.
 * $$|e^u| = |e^{\Re u} e^{i \Im u}| = |e^{\Re u}| = e^M$$,

clearly it holds that $$|u| = |\log e^u| = M$$. Thus, (b) $$\Rightarrow$$ (c). Suppose (c). Then $$M^2 = |u|^2 = u \overline{u}$$. Differentiating both sides we get:
 * $$0 = {\partial \over \partial z} u\overline{u} = u {\partial \over \partial z} \overline{u} + \overline{u} {\partial \over \partial z} u$$.

Since $$u \in \mathcal{A}(\Omega)$$, it follows that $${\partial \over \partial z} \overline{u} = 0$$ and $$\overline{u} {\partial \over \partial z}u = 0$$. If $$\overline{u} = 0$$, then $$u = 0$$. If $${\partial \over \partial z}u = 0$$, then $$u$$ is constant since $$\Omega$$ is connected. Thus, (c) $$\Rightarrow$$ (a). $$\square$$

We say a function has the open mapping property if it maps open sets to open sets. The maximum principle states that equivalently
 * if a function has a local maximum, then the function is constant.

3 Theorem ''Let $$u: \Omega \to \mathbb{C}$$. The following are equivalent:''
 * (a) $$u$$ is harmonic.
 * (b) $$u$$ has the mean value property.

3 Theorem ''Let $$u: \Omega \to \mathbb{C}$$. If $$u$$ has the open mapping property, then the maximum principle holds.''

Proof: Suppose $$u \in \mathcal{A}(\Omega)$$ and $$\Omega$$ is open and connected. Let $$\omega = \{ z \in \Omega : |u(z)| = \sup_\Omega |u| \}$$. If $$u$$ has a local maximum, then $$\omega$$ is nonempty. Also, $$\Omega$$ is closed in $$\Omega$$ since $$\Omega = u^{-1}({\omega})$$. Let $$a \in \omega$$. Since $$\Omega$$ is open, we can find a $$r > 0$$ so that: $$B = B(r, a) \subset \Omega$$. Since $$u(\omega)$$ is open by the open mapping property, we can find a $$\epsilon > 0$$ so that $$B(\epsilon, u(a)) \subset u(\omega)$$. This is to say that $$u(a) < \epsilon + u(z)$$ for some $$z \subset B(r, a)$$. This is absurd since $$a \in \omega$$ and $$u(z) \le u(a)$$ for all $$z \in \Omega$$. Thus, $$u = \sup_\Omega |u|$$ identically on $$B(r, a)$$ and it thus holds that $$B(r, a) \subset \omega$$ and $$\omega$$ is open in $$\Omega$$. Since $$\Omega$$ is connected, $$\Omega = \omega$$. Therefore, $$u = \sup_\Omega |u|$$ on $$\Omega$$. $$\square$$

Addendum
Exercise ''Let $$f \in \mathcal{A}(\mathbb{C})0$$. Then $$f$$ is a polynomial of degree $$ > n$$ if and only if there are constants $$A$$ and $$B$$ such that $$|f(z)| \le A + B|z|^n$$ for all $$z \in \mathbb{C}$$.''

Exercise 2 ''Let $$f: A \to A$$ be linear. Further suppose $$A$$ has dimension $$n < \infty$$. Then the following are equivalent:''
 * 1) $$f^{-1}$$ exists
 * 2) $$\det(f) \ne 0$$ where $$\det(f) = \sum_1^n sgn(\sigma) x_{\sigma(i)j}$$
 * 3) The set
 * $$\left\{ f \begin{bmatrix} 1 \\ \vdots \\ 0 \end{bmatrix}, f \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} ... f \begin{bmatrix} 0 \\ \vdots \\ 1 \end{bmatrix} \right\}$$ has dimension $$n$$.

title: User:TakuyaMurata/Calculus


text:
Cookbook:Homemade Corn Tortillas

Tortillas are a type of flatbread popular in Mexico and Central America, commonly made from wheat flour or masa. This recipe is for the masa (corn) variety; see Flour Tortilla for the wheat flour version.

Ingredients

 * 2 cups masa (masa harina, masa lista)
 * 2 cups water
 * 1 teaspoon salt

Procedure

 * 1) In a large bowl, stir together the masa and salt.
 * 2) Add the water, and mix together well to form a dough.
 * 3) Turn out onto a lightly floured board and knead for 5 minutes.
 * 4) Divide the dough into ¼ cup (3 oz / 90 g) portions, and form them into balls.
 * 5) Roll each ball into a flat round about 6 inches (10 cm) in diameter and ⅛ inches (0.25 cm) thick.
 * 6) Heat a large heavy skillet over medium high heat.
 * 7) Place the tortillas one at a time into the dry hot skillet; cook until brown on one side, then turn and brown the other side.
 * 8) Remove from the skillet and keep warm in cloth towel.

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

title: Cookbook:Homemade Corn Tortillas


text:
User:Shadowsong~enwikibooks

Hey, I'm Shadowsong, I recently joined Wikipedia because I like the idea of an open-source encyclopedia and I got tired of my posts being signed with a bunch of numbers. I probably won't be doing a huge amount of editing since my knowledge in most areas isn't advanced enough to really help out with articles. Most of my contributions will be in the RuneScape series as I've been playing the game for several months now and I know quite a bit about it. I also like to go around correcting spelling errors because they annoy me.

My other interests include reading and tae kwon do. My all-time favourite book is His Dark Materials by Philip Pullman. I also try to read Dutch books ocassionally since that's my native language and I'm trying to reteach myself spelling. I've been living in America for a couple of years and the English influence is starting to show. Other than that my life basically consists of going to school and doing homework, I'm in 10th grade right now and school is getting to be very time-consuming.

title: User:Shadowsong~enwikibooks


text:
User:Tawker/monobook.js

// w: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:Tawker/monobook.js


text:
User:Ande B.~enwikibooks

Hello! I am Ande B. Now that I've said that, my name won't be showing up as a red link on the wiki pages. You can leave messages on my talk page or send me a wiki email. Ande B. 07:37, 8 April 2006 (UTC)

title: User:Ande B.~enwikibooks


text:
Cognitive Psychology and Cognitive Neuroscience

Contents

 * 1) /Cognitive Psychology and the Brain/
 * 2) /Problem Solving from an Evolutionary Perspective/
 * 3) /Evolutionary Perspective on Social Cognitions/
 * 4) /Behavioural and Neuroscience Methods/
 * 5) /Motivation and Emotion/
 * 6) /Memory/
 * 7) /Memory and Language/
 * 8) /Imagery/
 * 9) /Comprehension/
 * 10) /Neuroscience of Text Comprehension/
 * 11) /Situation Models and Inferencing/
 * 12) Knowledge Representation and Hemispheric Specialisation
 * 13) /Reasoning and Decision Making/
 * 14) /Present and Future of Research/
 * 15) /Index/

Stuff for students

 * 1) /Hints for good writing/
 * 2) /Some useful syntax/
 * 3) /Links for possible sources/
 * 4) /Copyright/

Organization

 * 1) /Cross References/
 * 2) /OverallStructure/


text:
User:Aschoeke

Hello,

A pic of mine

I am Andrej. I am from Germany and study Cognitive Science in Osnabrueck. I work mainly on the CPCN-Project.

Tests

 * User:Aschoeke/Template:CVUnav_test
 * User:Aschoeke/Template:User CVU2-en_test

title: User:Aschoeke


text:
Icelandic/Alphabet and Pronunciation

Stafróf og framburður
 * Alphabet and pronunciation

The Icelandic alphabet consists of 32 letters. There are also three letters only used for foreign words, and one deleted letter (which is sometimes still used only for foreign words). The Icelandic language uses the latin alphabet, which is the same as the English alphabet and most Western European languages. There are some letters that are not found in English, and even some letters that only Icelandic uses.

Íslenska stafrófið

Letters C, Q, and W are not part of the Icelandic alphabet, but are used in foreign words:

The final letter, Z, no longer appears in Icelandic words as of 1973, and therefore is no longer used in the Icelandic language either. However, it is still technically used only in very rare cases preserved in historic names of structures, organizations, and the like, such as Verzló (a school in Reykjavík). And words that once contained the letter Z nowadays contain the letter S.

With the exception of letters C, Q and W, these letters do appear in the Icelandic language in foreign words, unlike the deleted letter Z, which only appears in very rare cases in foreign words and therefore are also used more often than the deleted letter is.

Until 1980, the Icelandic alphabet used to consist of 36 letters and also included the 3 letters that are only used for foreign words and the deleted letter (which is still technically only used for foreign words but very rarely) (and computers still order this way):

A Á B C D Ð E É F G H I Í J K L M N O Ó P Q R S T U Ú V W X Y Ý Z Þ Æ Ö

How the letters are pronounced
Notes:

 * Icelandic words never begin with Ð, and no words end with Þ.
 * I and Y share the same pronunciation, as do Í and Ý.
 * HV is pronounced as KV, but is sometimes pronounced as Scots WH.
 * J, L, M, N, and R are voiceless before H and in most areas before K, P, and T (no English equivalent)
 * L and R are voiceless at the end of a word
 * There are no silent letters in Icelandic., though in spoken language some letters might produce a different sound than usual.
 * Double BB, DD, GG, RR, and SS are pronounced for longer than their monograph equivalents.
 * Double FF is pronounced as English F.
 * Double LL is pronounced something like tl.
 * Double MM and NN are often pronounced as pm and tn.
 * Double KK, PP, and TT are pronounced with an H to their left and pronounced for longer than their monograph equivalents.
 * If a K is followed by a t, it is pronounced similarly to a Spanish j (e.g. lukt – lantern).
 * Likewise, a P followed by a t changes into an f sound (e.g. Að skipta – to shift).
 * F in the middle of a word is often pronounced as a v (e.g. Að skafa – to shave).
 * If you are not able to type in Icelandic letters, you can substitute Ð with DH, Þ with TH, Æ with AE, and Á, É, Í, Ó, Ö, Ú, Ý with AA, EE, II, OO, OE, UU, YY.

Stress
Stress in Icelandic always falls on the first syllable.

Dialects of Icelandic
All dialects of Icelandic have assimilated into the standard spoken language, but people from Reykjavík tend to speak a little differently from people from Akureyri, Egilsstaðir, Ísafjörður and other countryside towns and villages. For example, the word for hot dog in Icelandic is pylsa; in Akureyri, they would say pil-sah but in Reykjavík you often will hear pulsa. Another example is the word for to want, langar: in Ísafjörður (the northwestern part of Iceland), you often will hear lahng-ar but in Reykjavík you will hear lángar.

title: Icelandic/Alphabet and Pronunciation


text:
Talk:Cognitive Psychology and Cognitive Neuroscience

The old discussion page has gone to Talk:Cognitive_Psychology_and_Cognitive_Neuroscience/Archive.

About

 * Started: 8 April 2006
 * Size: 89,200 words. (Sep 2008)

How to go on
Considering the big amount of work and the little amount of time we had, our course made a quiet good effort. I want to encourage anybody who wants to, to contribute to the exisiting frame. Here are some suggestions:

ToDo

 * 1) References not only at the bottom, but also in the text ("As we can see in the text of Zwaan (1)...")
 * 2) Correct the referencing section
 * 3) Finding better suitable pictures
 * 4) If there are publications with good graphics, kindly ask the publisher to license it under a free license.
 * 5) Extend anything of which you have a good knowledge
 * 6) As we are not native speaker, it is likely *cough* that there are problems with the language. ;)
 * 7) If you are a student, spread the word about this project and especially this book :)
 * 8) Create a PDF-Version

Please pay attention to

 * 1) Good referencing. This is serious scientific content!
 * 2) Structure.

A little note
Guys, I just want to say: this is great!! Reaally appreciate!! MethoxyRoxy 8 sept. 2006 13:29

Perceptual systems
Why is there nothing on perceptual systems? They are most often considered a part of cognitive neuroscience. -- Heida Maria

Something is repeated
Some parts of this books are repeated at two different places in this book. I have to look into this later. --Dirk Hünniger (discuss • contribs) 21:20, 18 March 2013 (UTC)

 * Thanks Dirk I have noted that as I was going through this books'' Quiin D (discuss • contribs) 18:51, 11 June 2024 (UTC)

 * Hi Quiin D,
 * Could you correct it in the printable version of the books and leave me a message on my talk page when you are done. I will regenerate the pdf in turn. Otherwise you can also do this yourself https://mediawiki2latex.wmcloud.org/ Yours --Dirk Hünniger (discuss • contribs) 15:24, 30 June 2024 (UTC)
 * Could you correct it in the printable version of the books and leave me a message on my talk page when you are done. I will regenerate the pdf in turn. Otherwise you can also do this yourself https://mediawiki2latex.wmcloud.org/ Yours --Dirk Hünniger (discuss • contribs) 15:24, 30 June 2024 (UTC)


text:
Wikibooks:WikiProject Editorial board

The Editorial Board is a project that helps to proofread, edit, and revise books, so that they can become higher-quality resources. Any book can be nominated for review by an editorial board, although featured books, and generally good books are given priority.

Goals
Volunteer members of this project will:
 * Find and mark books that are "good"
 * Proofread good books, edit, revise, and generally improve those books.

The project should use the guideline Editorial board guidelines.

Nominations
Books that are in need of review from an editorial board can be listed below.

Resources

 * Editorial board guidelines
 * meta:Help:Displaying a formula
 * Help:Contents
 * Volunteer editors

Members
People who are interested in helping this project are encouraged (but not required) to list their usernames below. All wikibookians are allowed to participate in this project, with or without identifying themselves as "members" beforehand. Project members are encouraged to add this page to their watchlists. Project members are also encouraged to add their names to Volunteer editors by qualification.

 * xixtas 00:24, 11 January 2007 (UTC)
 * Dragontamer 02:50, 11 January 2007 (UTC)
 * RobinH 11:45, 11 January 2007 (UTC)
 * Whiteknight 18:03, 11 January 2007 (UTC)
 * within focus 16:25, 14 January 2007 (UTC)

Also see: WikiProject Volunteer editors if you are prepared to act as a member of an editorial board.

title: Wikibooks:WikiProject Editorial board


text:
Wikibooks talk:Editorial board

Turn this into a proposal
There is some talk about turning this page from a wikiproject into a policy proposal. Implicit in this talk is the idea of making the editorial board more then simply a casual group project into a standing entity with some "authority" over specific matters here on wikibooks. An editorial board constructed in this matter would either be: The second option would open up the idea of decentralized authority, which many people might agree with. Barring the details, should this be a policy proposal or a wikiproject? --Whiteknight (talk) (projects) 03:10, 30 January 2007 (UTC)
 * 1) A local analog of the wikipedia Arbcom, or
 * 2) A local counterpoint of our own local Arbcom (which is also in a proposal stage).

 * I'm all for making it a policy proposal. We can have an elected group of several talented book-writers who are well-versed with book production and are also experts in one or more fields. We won't have full coverage of all subjects possibly, but I think such a group of perhaps five or seven individuals will form a very nice board. This way we can keep behavior issues separate from book production, a thing that could certainly slow down progress here. -within focus 22:31, 30 January 2007 (UTC)

 * I'm not really sure what is intended here, what does the editorial board have to do with arbcom? I think that it might be a better idea to keep it as a wikiproject so that it doesn't encourage people to go to arbcom with editorial disputes. I think instead the editorial board should probably be more involved with books in which the editors are not directly involved so that a more NPOV decision on what's a good book can be reached, and more appropriate decisions can be made. Of course, the actual authors of the book would have a say on why a book could deserve to be recognized and what should be improved, just the "authority" on whether something qualifies as a good book should be left to a more neutral party IMO. If this became part of the policy, of course, I would probably support it. Mattb112885 22:43, 30 January 2007 (UTC)

 * The idea is that content disputes (disputes between one or more editors concerning the content of a page) is not the same as a behavior dispute (although there is plenty of potential for one to turn into the other). I think it would be a bad idea to make the arbcom, which is essentially a disciplinary committee, handle content disputes as well. We would invoke the editorial board to handle disputes that did concern the authors of the book: if those authors are stalemated by a content dispute, we need to have a "higher authority" on matters of content to turn to. Beyond this "official duty" of the editorial board, there would also be a number of unofficial duties, such as reviewing, editing, and publishing good wikibooks. An editorial board would likely be kept small, and be able to call upon "experts" in various fields to help answer questions on particular subjects. --Whiteknight (talk) (projects) 22:56, 30 January 2007 (UTC)

Changes, Moving forward
I have made a few small changes to the text of this proposal, some of which are relatively small. In particular, in response to some criticism, I have removed mention of the editorial board being able to resolve disputes. The task of content dispute resolution will be left to Decision making, and Dispute resolution, and this board will not be an official part of that process. That said, however, I have added text that the board can lend guidance to wikibookians if asked, which means that the board could potentially be asked to act as a mediator in a content dispute. I leave it up to the discretion of the board as to whether or not to participate in such activities.

I really want to move forward on this issue, even if the current text of this policy is not satisfactory. I do want to have some kind of editorial-board-ish entity here, and I do want to work towards editing and eventually publishing (depending on a local definition of "publish") some of our better books. Changes to this proposal are fine, but in the interest of progress, I would ask that all criticisms be constructive. --Whiteknight (talk) (projects) 15:07, 7 February 2007 (UTC)

 * The new text about guidance seems more appropriate to me. I like your most recent change as far as having a centralized point of contact because, indeed, though there are many contributors to the project anyone who would be interested in using our books for classes and such, and they'd need someone to contact for more information (the staff lounge seems to be the place where this kind of thing occurs at the moment). Question: Would you want this to be a liaison to publishing companies too? Though I guess this could be considered a part of the "etc", but IMO this would a different ball of wax than a liaison to consumers because with publishers you have to make sure they follow the GFDL, which is fundamentally different from traditional publishing methods, and this could require the assistance of a lawyer and the courts and all of that if they decide they want to try and break the law and claim exclusive rights or something... maybe not a direct issue for the editorial board but probably is one for the publishing board, unless the books are to be published independently or something. Regards. Mattb112885 01:58, 12 February 2007 (UTC)

 * My vision basically includes printed versions, and versions of our books on CD being distributed to schools and libraries. I don't think we should follow the traditional "publication" model, where books get an ISBN number, and are sold for profit. Our books are already freely available from our website, So i dont think we need to worry about getting our books to appear on Google Books, or Amazon, or anything like that. We can call PDFs of our book the "finished product", and then distribute those books directly to consumers electronically. Once the teachers/libraries had our books in PDF form, they could print them for the students, or distribute them in some other manner. As to the legal issues, I want to get the WMF involved in this process. Jimbo has expressed some interest and support in the past, and that's something that I think we need to exploit. --Whiteknight (talk) (projects) 02:09, 12 February 2007 (UTC)

 * Sounds good to me, the PDFs are available here but we could distribute them on CD for convenience and such, and I agree with your wish to get WMF involved, they have more legal weight than a couple of volunteers (I know they usually are the ones that want to know about major copyright stuff). And the money we make from the CDs would simply be used to help keep the project running, kind of like a fundraiser. Regards. Mattb112885 02:32, 12 February 2007 (UTC)

 * I'm not particularly interested in making any money off our books, and I think that making any kind of profit off our books would probably just create more legal problems. Of course, we can work on these details later. --Whiteknight (talk) (projects) 03:10, 12 February 2007 (UTC)

 * Ah, but what if an individual author (say, he is the primary author of an excellent book) wants to make profit? I would say, he can go and publish it on his own, with the "wikibooks" version being edited by the board. He can request it to be taken off and we should be kind enough to comply. I doubt it has to be written down, but I think it would be optimal for the members of the board to keep something like that in mind. (it seems like common sense to me) --Dragontamer 04:11, 16 February 2007 (UTC)
 * I dont think that's an issue. Any person can sell a GFDL textbook for profit, whether or not they are the author of the book. There are no restrictions on the ability to profit from GFDL texts, although since they are freely available here, it is unlikely that anybody is going to pay much for them. Since all text contributions here have been expressly released to Wikibooks for use under the GFDL, a request from an author to remove those materials from our site isn't binding and cannot be enforced. --Whiteknight (talk) (projects) 19:32, 16 February 2007 (UTC)

title: Wikibooks talk:Editorial board


text:
Wikibooks:Editing disputes policy

Resolving editing disputes
An edit war occurs when text is reverted repeatedly to a previous form or when repeated changes or minor additions are made that amount to a repeated reversion of text. If reversions occur more than three times any contributor or an administrator may create a disputed edit page, containing the disputed text, and revert the original text to the form that was present before the dispute arose. The live page may not be changed whilst the dispute is in progress. The disputed edit page may not be referenced on any of the pages of the 'live' book although it may be referenced in Talk/Discussion pages. The edit dispute must be resolved on the disputed edit page before integration back into the book. Resolution will have occurred if all parties agree, or failing this, and after a cooling off period of two weeks, if a majority of parties including a neutral administrator agree. PDF Files may not be created whilst a book has a disputed edit page and will be deleted if produced.

If an edit war is resolved the original contributors may not restart the same dispute. Contributors who do so may be suspended or banned.

See further
For additional guidelines on editing modules:
 * Editorial policies and guidelines
 * Be bold in updating pages
 * What Wikibooks is not

For additional guidelines on editing and refactoring talk pages:
 * Talk page

title: Wikibooks:Editing disputes policy


text:
Template:Learning the vi Editor/key

<>

title: Template:Learning the vi Editor/key


text: Template:Learning the vi Editor/Ex

title: Template:Learning the vi Editor/Ex


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D Programming/Garbage collector

This page should be a documentation of the internals of the actual GC implementation, the interface, restrictions ...

Introduction
The actual Implementation has following characteristics:
 * conservative (vs. precise and internal pointers)
 * stop-the-world (vs. incremental vs. concurrent)
 * mark and sweep (vs. copying)
 * non-moving (vs. moving)

see also Garbage collection

File structure

 * std.thread:Support the GC with the actual position of the stackpointer.
 * /internal/gc/gc.d: the GC interface, a wrapper around the implementation.
 * /internal/gc/gcx.d: the GC implementation
 * /internal/gc/gcstub.d: stub for the gc, to prevent linking into application

Threads
To stop the world, the gc calls Thread.pauseAll. The stop mechanism on Linux works with the Signals SIGUSR1 and SIGUSR2. In case of pausing the Thread, the signal handler stores the current stack pointer in stackTop, so this pointer is always uptodate.

How does it work
The GC is only called when memory should be allocated and currently none is available. If the gc is not able to free enough memory, new memory is requested from the system.

Entry point is gcx.d fullCollect

 * 1) mark phase
 * 2) scan stacks of all threads
 * 3) scan roots (global variables)
 * 4) scan ranges (non gc managed memory which can also contain references to gc-objects)
 * 5) finalize all collected objects
 * 6) sweep the unused memory

Further information
/Thoughts about better GC implementations/

title: D Programming/Garbage collector


text:
D Programming/RTAI

Why use RTAI for real-time applications?
Real-time application need

 * efficient executing code
 * deterministic behaviour

Actually development for real-time applications is mainly made in embedded systems in the language C.

More and more is done in C++ and on PC platforms. With D there is an alternative to C++. But there are non-deterministic features in D (like in C++ also)

 * dynamic memory allocation
 * dynamic casts
 * garbage collection
 * exception handling

In a real-time environment not all has to be deterministic. So it is in the hand of the developer not to use these features in the real-time task.

 * /Installing/
 * /RTAI Bindings/
 * /LXRT/
 * /Accessing PCI hardware/
 * /User Space Interrupts/
 * /Examples/

title: D Programming/RTAI


text:
D Programming/RTAI/Accessing PCI hardware

With kernel 2.6 all information about devices is hold in the /sys directory. You can find your device under /sys/bus/pci/devices/. Each PCI device directory is named with the a combination of "bus:slot.function". In the device directory you find files named:
 * vendor: Vendor-ID is a unique number for each vendor. It is assign from the pci-sig and requires a membership.
 * device: Device-ID is a vendor unique number, chosen by the vendor.
 * irq: the assigned IRQ IRQ Handling in RTAI
 * resources: The memory images of the device with their flags
 * resource0: A virtual file for the config space of the device
 * resource1-5: Virtual files with the memory images.

Format of .../resource
A file can look like this: 0x00000000feafe000 0x00000000feafefff 0x0000000000000200 0x00000000feaf0000 0x00000000feaf7fff 0x0000000000000200 0x00000000feafd000 0x00000000feafdfff 0x0000000000000200 0x00000000feafc000 0x00000000feafcfff 0x0000000000000200 0x00000000feac0000 0x00000000feadffff 0x0000000000000200 0x00000000fea40000 0x00000000fea7ffff 0x0000000000000200 0x0000000000000000 0x0000000000000000 0x0000000000000000

The lines, are the images 0..6, 0 is config space, 1-5 are device dependent, 6 is an optional ROM. The first column is the start addresses of the images, the second column contains the end address (inclusive). The third column contains the flags for the image. The constants for the flags can be found in /usr/src/linux/include/linux/ioport.h.

 * IORESOURCE_IO: The image can only be access via IO-ports (outb, inb,.. macros)
 * IORESOURCE_MEM: The image is accessible via normal memory access
 * IORESOURCE_IRQ: ???
 * IORESOURCE_DMA: ???
 * IORESOURCE_PREFETCH: access to the image can be cached (no side effects)
 * IORESOURCE_READONLY: the memory is read only
 * IORESOURCE_CACHEABLE: memory writes can be cached
 * IORESOURCE_RANGELENGTH: ???
 * IORESOURCE_SHADOWABLE: ???
 * IORESOURCE_BUS_HAS_VGA: ???
 * IORESOURCE_DISABLED: ???
 * IORESOURCE_UNSET: ???
 * IORESOURCE_AUTO: ???
 * IORESOURCE_BUSY: Driver has marked this resource busy

mmap an image
The image file .../resource1-5 are ready to use. So you have only to do the two steps, open and mmap: uint aLength = 4096 uint mFileDescriptor = open( "/sys/bus/pci/devices/0000:02:09.0/resource2", O_RDWR ); uint* ptr = cast(uint*) mmap(       null,        aLength,        PROT_READ|PROT_WRITE,        MAP_SHARED,        mFileDescriptor,        0); // If ptr == -1, than the user was probably not root assert( cast(uint)ptr != uint.max ); uint[] mMemory = ptr[0 .. aLength/4];

If you are done, don't forget to close the resources: munmap( mMemory.ptr, mMemory.length ); close( mFileDescriptor );


text:
D Programming/RTAI/User Space Interrupts

RTAI has user space interrupts called RTAI USI.

Let Linux assign a IRQ number for you PCI device
But before one can use them, the hardware interrupt has to be mapped to interrupt in Linux. Therefore it is necessary to build a kernel module for the pci device, that registers the interrupt. With this action, the device gets a new interrupt number, which can be used in the usi.

Writing an interrupt handler
Create a realtime thread:

In the thread make a loop, looking like that: public void run{ const int irqNum = 5; rt_request_irq_task( irqNum, null, RT_IRQ_TASK, 1 ); rt_startup_irq( irqNum ); rt_enable_irq( irqNum ); while( true ){ int overrun = rt_irq_wait( irqNum ); if( overrun == RT_IRQ_TASK_ERR ) break; // if program should quit, break here // Do the irq handling rt_ack_irq; }  rt_release_irq_task( irqNum ); }


text:
User:Maebert

 * -] That's me. (Not to scale.)

title: User:Maebert


text:
User:Pehrenbr

BEST HEADLINE IN THE WORLD
= EVEN BETTER HEADLINE =

 * ABC
 * a
 * b

title: User:Pehrenbr


text:
D Programming/RTAI/LXRT

To make the task timing with RTAI you have two possibilities You can read about this in An overview of RTAI schedulers. To show the differences:
 * periodic timer mode
 * oneshot timer mode
 * periodic mode: The 8254 chip is used. It runs with a fixed frequency of 1193180 Hz. If you want to make a cyclic event for 1 kHz, it can set the timer period to 100 µs. In the cyclic task always call an rt_sleep_until with the next time. The result is 1 kHz task with jitter of 100 µs.
 * oneshot mode: The internal clock goes with the cpu clock. For each timer event, RTAI has to reprogram the timer for the next event. This is an overhead, but has the advantage of highest time resolution.

To make a cyclic task with LXRT, first create a new thread as usual. threadHandle = rt_thread_create( &threadFunc )

then implement the thread like this void threadFunc {    RT_TASK *handler; handler = rt_task_init( 0, 0, 0, 0); if (handler==null){ // error handling }        rt_allow_nonroot_hrt; mlockall(MCL_CURRENT | MCL_FUTURE); //rt_set_periodic_mode; rt_set_oneshot_mode; RTIME period = nano2count( 1_000_000_000l ); start_rt_timer(period); rt_make_hard_real_time; RTIME currentTime = rt_get_time_ns; while( stopCondition ) {      currentTime += period; rt_sleep_until( currentTime ); // the work to do ... }    stop_rt_timer; rt_make_soft_real_time; rt_task_delete(handler); } With such setup, you can achieve jitter lessen than 10 µs.

title: D Programming/RTAI/LXRT


text:
D Programming/RTAI/Examples

 * /Parallel port pinout/

title: D Programming/RTAI/Examples


text:
D Programming/RTAI/Examples/Parallel port pinout

25 pin Parallel Port Pinout Details


text:
User:Iroewer

=Hallö= Ich bin eine Biene...

title: User:Iroewer


text:
Wikibooks talk:Editing disputes policy

Add a comment or vote:

Comment
Is there a similar policy at English Wikipedia? How does it work? --Derbeth talk 19:33, 28 April 2006 (UTC)

 * Wikipedia contains numerous topical articles from the status of a particular pop group to the affairs of the British deputy prime minister; an edit dispute policy that takes disputes to one side, out of the public eye, would not be acceptable in such an environment.

 * At Wikipedia there is a 3 reversions rule where no more than 3 reversions may be made but this is often ignored and just makes the argument into a slanging match on the talk page eventually resulting in appeals to arbitration committees etc. It also allows editors to wait until a few changes have been made and then sneak their interpretation into an article. An editable version of the disputed page, out of the eye of readers, would be a way of focussing on the problem of achieving a workable text rather than concentrating on any perceived idiocies of opponents. RobinH 09:16, 30 April 2006 (UTC)

 * I've made some proposed alterations to the policy text, inserted below beneath a header. Originally I just put everything in the active voice, so the policy was clearer to me, but then I began seeing things which I thought we should try to work out. Existing PDF files should be allowed to stay, especially if the book was already in complete form, for example.
 * Even as I've written it, though, I can see other difficulties. What happens when an edit war spills out of one page or chapter into the whole book?    Do we need further clarification of the difference between vandalism and an edit war?  Do the authors and editors of a completed book have any right to keep their book's particular take on a subject more or less even? (For example, if someone decides the Ancient History textbook needs more alien invasions and cites Zechariah Stitchin in the bibliography, do those changes stay over the objections of ancient historians? Why shouldn't the new contributors be asked to go out and produce their own book, Aliens in the Ancient World? Andrew Watt 16:01, 5 June 2006 (UTC)

Resolving Editing Disputes

 * An edit war occurs when a user or a small group of users (3 or fewer) repeatedly revert text to a previous form, or when such a user or group make a series of minor changes or additions that amount to a repeated reversion of text. Edit wars are not Vandalism; they are differences of opinion regarding the substance of a given book. Ideallly, edit wars should be resolved with research and references to primary sources, not bickering and personal attacks.

 * If a user or group perform reversions more than three times, any contributor or an administrator may remove the text from the main page, and move it to a disputed edit page which contains both versions of the disputed text. The main page should be reverted to original text present before the dispute began. The live page may not be changed while the dispute is in progress.

 * A banner and single link to the disputed edit page may be placed on the main page section of the book where the dispute has occurred. Otherwise it may not be referenced on any of the pages of the 'live' book although it may be referenced in Talk/Discussion pages. Interested users -- readers and contributors -- must resolve disputed edit page before integration back into the book. Resolution will have occurred if all parties agree; or failing this, and after a cooling-off period of two weeks, if a 2/3 majority of involved parties (defined as all users involved in the debate, plus active users who contributed to the book but were not involved in the disputed edit debate), plus one neutral administrator agree; the goal should be to have a large number of project participants happy with the results of the editorial procedure, so that they remain productive and participating users.

 * Users may not attach PDF files to a book while a book has a disputed edit page. They should be deleted when attached, though existing PDF file versions of the book may remain.

 * Once an edit war is resolved, the user or group may not restart the same dispute. However, In the interest of NPOV, a user or group of users should be encouraged to develop their own book projects on the same subject, so that would-be readers can make use of everyone's expertise, and have multiple resources for their own efforts.

 * Possible solutions to edit wars can include but are not limited to:
 * Contrasting paragraphs in which roughly equal POVs are given roughly equal space;
 * Contrasting paragraphs in which a distinctly minority viewpoint receives some but not equal space;
 * Research by both parties, and substantial links to high-quality primary and secondary sources; or
 * a new Wikibook project in which either new or old contributors develop a parallel but distinct text with different editorial assumptions.

 * As always, good faith on the part of all users until proven otherwise is assumed. Readers and contributors alike All contributors should seek to present high-quality products on Wikibooks. Contributors who continue an edit war after resolution are formally warned, then suspended for two weeks, and finally banned.

 * Once resolved the dispute may not be re-opened by any editor without the agreement of a majority of editors, or in the case of a tied vote, the agreement of a majority of editors and a neutral administrator. Disputes that have been resolved may only be re-opened through editor's communicating on Talk Pages, not through pre-emptive changes of text in the book.

 * Editors who do not adhere to these rules will be warned and, if warnings are ignored, banned.

---

 * That makes no sense. How do we re-open a dispute and get agreement from other editors, unless we raise the topic again? we have to mention the dispute, and argue the point if we want to get a majority of editors to agree with us. --Whiteknight (talk) (projects) 18:53, 5 June 2006 (UTC)

 * The editors would contact each other on the Talk Page or the editor's Personal Talk Pages and debate the idea of re-opening the dispute - perhaps this would help them to resolve it without the need to re-open it. RobinH 18:58, 5 June 2006 (UTC) - I have added a line to the text above to include this. RobinH 19:02, 5 June 2006 (UTC)

 * I crossed out some more sections of my policy edit to remove some bureaucracy, and I don't much like what I see, even though I wrote some of it. I think we do need some sort of way of resolving editorial disputes within books, that allows them to work towards a higher vision — but books also need to be completed, and shutting down a whole book or section for days or weeks while uncompromising editors on both sides wrangle...  well, that's not what I'm here to do.  Andrew Watt 19:38, 5 June 2006 (UTC)

Yes, this should be an enforced policy
1. RobinH 18:59, 10 April 2006 (UTC)

I agree with this for now, but the policy needs to be cleaned up. --Whiteknight (talk) (current) 17:42, 28 April 2006 (UTC)
 * This policy needs far too much work right now for me to cast my vote towards it. --Whiteknight (talk) (projects) 18:05, 5 June 2006 (UTC)

2. The content could be more robust and informative, but the basics are there. -Matt 02:02, 30 April 2006 (UTC)

3. Gentgeen 10:07, 3 May 2006 (UTC). One question, who has the authority to ban users who revisit a dispute

4. Good enough. Je suis 16:34, 5 June 2006 (UTC)

No, this policy should be rejected
1. I appreciate the good intentions, but this is far too prescriptive. Generally speaking, the more red tape, the more it is exploited by those who troll and act in bad faith. WP has seen this happen time and time again. In my view, if a disruptive user adds something entirely unhelpful - such as a number of whole paragraphs into a GCSE Chemistry textbook that is unconnected with the GCSE syllabus - it should be deleted and if the user persists in re-adding it, despite having pointed out to them why it is being deleted, that disruptive user should be blocked. Under this policy, the good faith contributor seeking to remove the paragraphs will fail - they will be the first to do 3 reverts, they will be the one having to set up an alternative version of the page, and the disruptive editor will have succeeded in preventing a PDF version of the page being created. If the dispute does get resolved the right way, and a later disruptive editor (perhaps a sockpuppet of the previous one) re-adds the disputed material, the original editor would end up being suspended or banned if they tried to stop it - or, indeed, even mentioned it, as that would rekindle the old dispute. In short, this proposal prevents admins from dealing with trolls the way they should be dealt with (ie being quickly chucked off the site), and instead requires good users to give equal (or more weight if we add in the sockpuppets) to those who seek to disrupt, Jguk 16:54, 5 June 2006 (UTC)

 * Which admin would do the blocking and on what authority? Would an admin be able to judge on a technical issue?

 * It seems to me that your main objection to the policy is the way that it could lead to an unfair ban. You mentioned the problem of a sock puppet re-instating an erroneous change to trigger a ban of another editor - clearly the policy needs to address this. The 3 revert victory for the non-good faith contributor would not occur because the book is reverted back to before the edit dispute (ie: before this contributor objected).

 * The principle idea of the policy is to get serious edit disputes out of the mainstream of Wikibooks. Taking the "edit war" offline is the first step. This would prevent editors from ruining the book during the disagreement. There would also be no personal benefit from the argument because it would be out of the public eye.

 * What is needed is a way forward that answers your concerns about bans but still protects books against serious disputes.

 * I would suggest changing the text about banning if the dispute is reopened to:

 * "The dispute may not be re-opened by any editor without the agreement of a majority of editors, or in the case of a tied vote, the agreement of a majority of editors and a neutral administrator. Editors who do not adhere to these rules will be warned and, if warnings are ignored, banned."

 * (Changing the text in the book in such a way as to produce a reversion is reopening the dispute).

 * This adresses your concern that a non-good faith editor might try to trap another editor. There is no '3 revert victory' because the text is re-instated to the form it had before the first objection during the phase of compromise on an "disputed edit" page. RobinH 18:39, 5 June 2006 (UTC)

 * I have integrated this change into Andrew's text above. RobinH 18:48, 5 June 2006 (UTC)

2.
 * I have reconsidered this policy, and removed my yes vote from above, because I feel that this policy is too wordy, too demanding, and too preachy. We don't need to be told precisely how many reverts constitute an edit war, how long a person must be banned for such a violation, who may or may not discuss the topic, etc. A discussion should never be closed: we don't need a majority of editors to continue a discussion on which complete concensus was not reached. I feel that if an edit war or a similar dispute is raging in which page content is changing quickly and in bad faith, administrators can choose to protect the page from edits for an arbitrary "cool-off" time, and to redirect attention to the appropriate discussion venue. Edit conflicts are allowed to happen, and users are permitted to disagree ad infinitem: people with differing beliefs on the content of a particular page should never be banned, reprimanded, or have their content arbitrarily rolled-back to an older version simply because they care. --Whiteknight (talk) (projects) 19:00, 5 June 2006 (UTC)

 * But what about a biology text book for 15 year olds. Suppose some cult disagrees with evolution and continually reverts the text, what sort of study guide would that provide if the text were always changing? What if the admin were a member of a faith group with strong feelings on the issue? We need some sort of constitutional protection before such things happen. RobinH 19:09, 5 June 2006 (UTC)

 * All this policy is attempting to do is to provide a stable text even if the subject matter is found to be contentious by a particular editor or small group in society. It seeks to do this by allowing full discussion and consensus building on the "disputed edit" page. RobinH 19:13, 5 June 2006 (UTC)

 * I don't think stability is worth the price of attempting to script everybody's actions in the case of a dispute. If one admin is clearly biased, and uses those biases to insert controversial text into a given module, then that admin can be disciplined accordingly. What we should not do is take away the power of judgement and freewill from our admins. Even if we do have this policy in place with all it's rules and penalties, biased admins can still abuse their powers if they choose to. Adding in all sorts of thou-shalt-nots is not a good idea. --Whiteknight (talk) (projects) 23:29, 5 June 2006 (UTC)

 * Are we trying to make stable texts? No, we're trying to make free texts.  Stable texts are produced for the profit of their copyright holders and publishers.  Free texts reflect the needs of their users.  Bernard of Clairvaux in the 12th century sent a letter in which he complains, "I sent you my book to read, but you have kept it and copied it and sent it to the world, and now there are so many words I cannot take back, and men have put words in my mouth that they think I should have said."  At least we have the satisfaction of a history, so that what we said, and what was added to us, is recorded.  That's about the best we can hope for here.  I vote No. Andrew Watt 19:43, 5 June 2006 (UTC)

 * I agree with you about rigid texts. Entirely. But in the Editing Disputes Policy all we are trying to achieve is a relative stability in the special circumstances of a dispute that has got out of hand. Although you are voting "No" to rigidly stable texts should that "No" also apply to a process that allows disputing editors to move to a consensus rather than fighting an edit war on the pages of a book? I would propose to adopt what is largely your text for this special case of edit warring (see above). RobinH 20:42, 5 June 2006 (UTC)

 * Here's the text of this proposed policy again, revised to active voice, but without my additions (which were a bad idea... for which I apologize).

 * An edit war occurs when a user reverts text to a previous form repeatedly, or when a user or group of users make repeated changes or minor additions that amount to a repeated reversion of text. If they make reversions more than three times, any contributor or an administrator may create a disputed edit page, containing the disputed text, and revert the original text to the form that was present before the dispute arose. No one may edit the live page while the dispute is in progress, nor reference the dispute page on any other page of the live book, though it may be discussed on Talk/Discussion pages. Interested parties must resolve difficulties on the disputed edit page before the text is integrated back into the book. All interested parties must find a compromise; if they cannot, after two weeks, a majority of parties plus a neutral administrator may agree on a text.   No one may attach PDF Files while a book has a disputed edit page and administrators will delete them when uploaded.
 * If an edit war is resolved the original contributors may not restart the same dispute. Contributors who do so may be suspended or banned.

3. I see this policy as punitive rather than editorial. It creates an opportunity for bad-faith contributors to manipulate policy, and it sets no guidelines for moving toward a better text: "Revert a text too often and we'll smack you one" vs. "Find ways to integrate many visions into a text." Some books would become filled with disputed edit pages because rival editorial groups would be more interested in bringing this Editing disputes policy into action than in finding a way to a better text. I say yes to a process that mediates between dissenting and argumentative editors, but I'm not sure that this is it. How about a more filled-out version of, disputed edits should be settled through research, citation of primary sources, direct communication and frank exchanges of ideas between users, with some arbitration by a neutral administrator when necessary? -Andrew Watt 19:42, 6 June 2006 (UTC)

4. Oppose: this policy isn't a policy, more like a description. I'd suggest putting teeth into it, modeling after w:WP:3RR, and using other toofs such as w:WP:RFC in order to keep it "democratized". At least there should be a clear way for the admins to wiegh the merits of each side aside from personal opinion. -- SB_Johnny 15:07, 19 July 2006 (UTC)

5. rejected: A clear Book Community policy should be created first and any reversions should be policed by its members contributors/authors that have invested time providing content to a work.--Panic 00:47, 10 October 2006 (UTC)

Comments

 * Editing disputes boil down to one of three situations:
 * Two or more editors, all here with the aim of improving Wikibooks, disagreeing over something.
 * Two or more editors, where there is at least one Wikibookian acting in good faith with the aim of improving Wikibooks, and at least one user who isn't
 * Two or more editors, all of whom are not editing here with good faith
 * Situation 1 may get heated, but really is very likely to be resolved somehow. We can point out ideas to help them resolve their differences and remind people that talking about differences and aims is most likely to lead to resolution. But really, this sort of situation will be resolved.
 * In situation 2, we should warn and then, if necessary, block the disruptive editor (at least until there is a commitment to behave) and let the good faith editor continue improving wikibooks uninterrupted.
 * Situation 3, once identified, is relatively easy. An admin comes in and blocks everyone until they agree to behave.
 * The only problem then is how to distinguish between the three situations. This calls for some judgment - and this judgment call should be based on admins' knowledge of the users, including user contribution history, how the user accesses wikibooks (if known) (eg open proxies are out), whether likely sockpuppets are involved, whether there is evidence of typical troll behaviour, whether it appears that the user is genuinely trying to help improve the wikibook or wikibooks or is only concerned with the particular area in dispute, evidence of disruptive behaviour on another Wikimedia project, and so forth. Admins, aided by comments from other users, ought to be free to use their experience and judgment on this one - trying to give hard and fast rules is, quite simply, unlikely to work, Jguk 21:03, 6 June 2006 (UTC)

Move to rejected
I propose that this policy be moved to "rejected" status in 2 weeks time. RobinH 11:40, 8 August 2006 (UTC)

title: Wikibooks talk:Editing disputes policy


text:
D Programming/wxD

wxD is a D language graphical user interface wrapping (or binding) of the wxWidgets cross-platform graphical toolkit

wxD is licensed under the "wxWidgets License" which is LGPL, but with a special exception to allow distribution of statically-linked binaries on the user's own terms. This allows wxWidgets to be used in proprietary products without having to dynamically link.

Its main website is at http://wxd.sourceforge.net/

title: D Programming/wxD


text:
User:Niwde

=About= A high school student in MSEC, CK, "Taiwan".

Likely joining the Chemistry Group in MSEC.

 * Not a native english speaker, so expressing problems do accur.
 * Just got interests in Chemistry and wiki projects.
 * Wiki Newbie.
 * Hoping to learn whatsoever by having me editings being corrected.

=Books I will most likely edit= Organic_Chemistry

General_Chemistry

=Blahz= well...

title: User:Niwde


text:
Talk:Novial

Deletion of older modules
Deletion policy allows the speedy deletion of:

 * "A page that has been nominated for deletion due to a general reorganization of the Wikibook by the contributors. In this situation, please note the location of the relevant discussion that occured regarding the page cleanup."

Recently Nov ialiste, the main contributor, reorganised this book and listed some modules for deletion on the front Novial page. There was some discussion at User talk:Kernigh. Thus I proceed to delete the nominated modules. --Kernigh 16:26, 8 April 2006 (UTC)

Thanks.Novialiste 13:10, 11 April 2006 (UTC)

pdf?
This is an excellent Wikibook and I'd like to thank Novialiste for taking the time and effort to further this fascinating language. When will you be doing a pdf version? And when will you finish the translation of AIL? Delano 03:51, 15 March 2007 (UTC)

 * The translation of AIL will proceed little by little. I didn't think of making a pdf but I will look into it. Thanks for your encouragement. Novialiste 17:15, 18 October 2007 (UTC)

Novial clubs and information?
I know that Novial is an easier, cleaner and more international language design than Esperanto or even Ido, and I would love to see it succeed and develop a community of millions of speakers. However, it seems like the Novial community is vanishingly small. Nov_ialiste, are you around? What other Novial resources can you point me to?

By the way, I would like to comment that this book is too brief for its own good, and it is written in a fairly confusing way. The book seems abbreviated excessively,

 * Using phrases like "D F R" instead "German/French/Russian" or at least standard language codes like DE/FR/RU
 * Failing to quote words or otherwise separate them from the sentences that discuss them, e.g. <> instead of <<'Besono' is from FR 'besoin', not from 'besogne'>>.
 * Failing to provide translations for many/most Novial phrases: <>
 * Showing equivalence by mere juxtaposition, <> instead of, say, <>

Also, I can't believe Mr. Jesperson doesn't bother to introduce any of the crucial words in Novial such as 'li', 'e', 'es', 'in', etc., before using them. In addition, he seems to assume we are all familiar with multiple European languages, so he thinks there are some points he does not need to explain. Finally, his English is very complex and of an archaic style, so non-native English speakers may have difficulty with it. It seems to me this book could use a revised edition written more clearly and less compactly. ...sigh... Maybe a third translation is in order, English-to-English with simpler phrasing and some clarifying notes added. --Qwertie (talk) 00:39, 10 December 2010 (UTC)

Tutorial group not attended (?)
PokestarFan: I don't doubt that your wording is linguistically better, as I'm not a native English speaker, but I'm not sure whether the current is correct. The user engage/invested in this Wikibook hasn't made an edit (under his/her regular account, Nov_ialiste, on novwp) since 2010, and the number of people knowing Novial well, could be around say 20 in total in the world, perhaps. Therefore, it's very possible that it's rather a matter of years than of days before one could expect an answer, and in this way I think your wording is misleading. Regards, 786r (discuss • contribs) 17:24, 20 August 2017 (UTC)

title: Talk:Novial


text:
User:Apape

Hi, my name is Anna and I study Cognitive Science at the  University of Osnabrueck, Germany. I was one of the writers of the wikibook Cognitive Psychology and Cognitive Neuroscience.

title: User:Apape


text:
User:Effeietsanders

Hello, I'm effeietsanders, a user from the Dutch Wikipedia. Please check my userpage there or on the English Wikipedia for more information.

nl:user:effeietsanders

title: User:Effeietsanders


text:
Muggles' Guide to Harry Potter/Magic/Incendio

Overview
Incendio causes the victim or object to burst into flame.