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Topology/Local Connectedness

A topological space is said to be locally connected at x (where x is a point of the space to be called X), if for each neighbourhood V of x, there is a connected neighbourhood U of x contained in V. If a space is locally connected at each of its points, we call the space locally connected. Surprisingly, local connectedness and connectedness are independent properties; a space may satisfy one of them without satisfying the other (see the examples below).

Examples
1. If a space has a basis consisting entirely of connected sets, then it is locally connected (to see this, let x be a point of the space. Then for each neighbourhood V of x, there is a basis element B containing x which is connected and contained in V. Then B is the desired connected neighbourhood of x contained in V).

2. From example 1, we can conclude that the set of real numbers R is locally connected and connected (since it is a linear continuum).

3. The set of rational numbers Q is neither locally connected nor connected. The same is true for R in the lower limit topology. In fact, a totally disconnected space cannot be locally connected unless it has the discrete topology.

4. The finite union of intervals is locally connected. In other words, elementary sets (see measure theory) in R are locally connected. In particular, the set
 * A = (-1,0) ∪ (0,1)

is locally connected but not connected.

5. The topologist's sine curve (if f(x) = sin (1/x), then the topologist's sine curve is just the closure of the image of (0,1] under f), is an example of a connected space which is not locally connected. The fact that the topologist's sine curve is connected follows from:
 * a) The set S = f((0,1]) is connected since it is the image of a connected space under a continuous map.
 * b) The closure of a connected space is connected.

The space is not locally connected at any point in the set B = [Closure (S)] – S. It is locally connected at every other point though.

Local path connectedness
A topological space, X, is locally path connected, if for each point x, and each neighborhood V of x, there is a path connected neighbourhood U of x contained in V. Similar examples to the previous ones, show that path connectedness and local path connectedness are independent properties. We will give a few more examples.

 * Examples

1. A locally path connected space is always locally connected. The proof uses the fact that every path connected space is connected.

2. An example of a path connected space that is not locally path connected is the comb space: if K = {1/n |n is a natural number}, then the comb space is defined by:

 * C = (K × [0,1]) ∪ ({0} × [0,1]) ∪ ([0,1] × {0}]

The comb space is path connected (this is trivial) but locally path connected at no point in the set A = {0} × (0,1]. It is however locally path connected at every other point.

Weakly Locally Connected
A weaker property that a topological space can satisfy at a point is known as ‘weakly locally connected’:

Definition

Let X be a topological space and x a point of X. If for every neighbourhood V of x, there is a connected subspace A of V containing a neighbourhood U of x, the topological space is said to be weakly locally connected at x. If the space is weakly locally connected at each of its points, it is said simply to be weakly locally connected.

We will give a few examples and there will be one important theorem on this concept later.

Examples

1. Every locally connected space is trivially weakly locally connected (let A = U in the definition).

2. Note that neither the comb space nor the topologist’s sine curve is weakly locally connected. It is easy to see that these two spaces are not weakly locally connected at each of the points that they aren't locally connected.

3. A totally disconnected space cannot be weakly locally connected unless it has the discrete topology (in other words, a totally disconnected space is weakly locally connected if and only if it has the discrete topology). An example is R in the lower limit topology which doesn't have the discrete topology and hence cannot be weakly locally connected at any point.

4. We will prove shortly that if a topological space is weakly locally connected (that is weakly locally connected at EVERY point), then it is locally connected at every point. However, the proof of this fact requires another notion related to local connectedness which is to be discussed in the next section.

5. The previous property may seem strange. If every weakly locally connected space is locally connected, then why call a space ‘weakly’ locally connected? It doesn't seem to be a weaker condition. However, if a space is weakly locally connected at just one point, then it need not be locally connected at that point. An example of such a space is the broom space. For this space to be locally connected at that point, it needs to be weakly locally connected in a NEIGHBOURHOOD of that point. Hence the definition of a weakly locally connected space does indeed make sense.

Components and path components
Define an equivalence relation ~ on a topological space X, so that we declare x~y if there is a connected subspace A of X containing both x and y. The collection of all equivalence classes are called the components of X (this is indeed an equivalence relation as the reader can check).

Define another equivalence relation ~2 (different from the first) to set x~2y if there is a path from x to y. Equivalently, x~2y if and only if there is a path connected subspace of X containing x and y. The collection of all equivalence classes with this equivalence relation are called the path components of the space (this is indeed an equivalence relation as the reader can check).

Being equivalence classes, the components are disjoint, nonempty subsets of the topological space X, whose union is X. The same is true for the path components.

 * Properties

1. Intuitively, the components are the largest possible connected subsets of X and the path components are the largest possible path connected subsets of X.

2. The components are always connected (to see this let C be a component (the equivalence class containing a certain point x of the space). Then C is nonempty since x~x. For each y in C, let Ay be a connected subset of X containing both y and x (since y~x). Take the union of all such Ay for y in C. This union is C. Since this is the union of connected spaces having the point x in common, this union is connected. Therefore, C is connected). A similar argument shows that the path components are always path connected and hence connected.

3. Every connected subset of X lies in a component of X. This justifies property 1. To see this, let A be connected (and nonempty or else the proof is trivial), and let x belong to A. Then if y belongs to A, A is a connected subset of X containing both y and x so that y~x. Since y was arbitrary, we have shown that A is a subset of the component containing x. A similar proof shows that every path connected subset of X is a subset of a path component of X.

4. The components are always closed. To see this we note that if C is a component, then C is connected, so that the closure of C is connected. Now, C is always a subset of the closure of C so that it is sufficient to verify that the closure of C is a subset of C. But we know that the closure of C is connected, and every connected subset of X is the subset of exactly one component. Therefore, C = Closure(C) and C is closed. This proof fails for the path components since the closure of a path connected space need not be path connected (for example, the topologist's sine curve).

5. If there are only finitely many components, then the components are also open. If C is a component, then its complement is the finite union of components and hence closed. Therefore, C is open. This proof fails for infinitely many components, as the set of rational numbers show.

 * Examples

1. The set of all real numbers has precisely one component; the whole space. In fact every connected space has only one component.

2. The set:

 * A = (-∞,0)∪(0,+∞)

has precisely two components; (-∞, 0) and (0, +∞). Note how these are the largest possible connected subsets of this space.

3. The set of rational numbers has countably many components. Each one point set is a component. In fact, for any totally disconnected space, the one point sets are the components of the space.

4. The path components of the spaces in examples 1,2 and 3 are exactly the same as the components. However, they are not the same in general. An example of this is the topologist's sine curve which has one component (since it is connected), but has two path components:

 * A = {0} × [-1,1]

 * B = f((0,1]) where f is the function defined by f(x) = sin(1/x)

Notice that A is closed; but not open. Also, B is open; but not closed. This shows that even though the components are always closed, the same thing does not hold for the path components.

5. In general, the path components are subsets of the components. To see this we note that the path components of a topological space are path connected and hence connected. Since connected subsets of X lie in a component of X, the result follows. We will prove later that the path components and components are equal provided that X is locally path connected.

6. The set I × I (where I = [0,1]) in the dictionary order topology has exactly one component (because it is connected) but has uncountably many path components. Indeed, any set of the form {a} × I is a path component for each a belonging to I.

7. Let f be a continuous map from R to Rℓ (R in the lower limit topology). Since R is connected, and the image of a connected space under a continuous map must be connected, the image of R under f must be connected. Therefore, the image of R under f must be a subset of a component of Rℓ. Since this image is nonempty, the only continuous maps from R to Rℓ, are the constant maps.

Quasicomponents
The concept of a quasicomponent is similar to that of a component and only brief attention will be given to it. We first make a definition:

Definition

Let X be a topological space. Define an equivalence relation on X to declare x~y if there is no separation of X into sets A and B such that x is an element of A and y is an element of B.

We note that the relation given in the definition is indeed an equivalence relation as the reader can readily check.

Properties

1. The components are always contained in the quasicomponents of X. Before proving this we will let x~1y if there is a connected subset of X containing both x and y (which is the equivalence relation used in defining components), and x~2y if x~y as in the definition given in this section. If x~1y, then there is a connected subset, C, of X containing x and y. If there was a separation of X into sets A and B such that x is an element of A and y is an element of B, then C would intersect both A and B. This implies in particular that ‘C intersection A’ and ‘C intersection B’ would form a separation on C contradicting the connectedness of C. Therefore, we must have that x~2y.

2. From the previous property, we may conclude that the path components of a space always lie in the quasicomponents of a space (since the path components always lie in the components).

Examples

1. There is only one quasicomponent of a connected space; the space itself. This follows from property 1.

2. In the next section, we will prove that if X is a locally connected topological space, then the quasicomponents of X are equal to the components of X.

3. If X is locally path connected, it is locally connected (as mentioned earlier) so that the components of X are equal to the quasicomponents of X. Since if X is locally path connected, the components of X and the path components of X are equal (which we shall prove in the next section), it follows that the quasicomponents, path components, and components are all equal if X is a locally path connected space.

4. Property 2 still holds if X is weakly locally connected (that is, weakly locally connected at each of its points). This is because a weakly locally connected space is always locally connected (which we shall prove imminently).

Theorems
In this section, we will prove theorems relevant to the material in this article.

 * Theorem 1

A topological space is locally connected if and only if the components of open sets are open.

 * Proof

Suppose X is locally connected (where X is the topological space in question). Let V be open in X and let C be a component of V. For each x in C, choose a connected neighbourhood U of x contained in V (by the local connectedness of X). Since U is connected, U is a subset of C. It follows that C is the union of open sets and is thus open.

Conversely, suppose that the components of open sets in a topological space, X, are open. Let x belong to X and let V be a neighbourhood of x. Let C be a component of V containing x. Then C is open by hypothesis. Therefore, C is a connected neighbourhood of X contained in V so that X is locally connected at x. Since x was arbitrary, X is locally connected.

 * Theorem 2

A topological space is locally path connected if the path components of open sets are open.

 * Proof

The proof is similar to theorem 1 and is omitted.

 * Theorem 3

The components and path components of a topological space, X, are equal if X is locally path connected.

 * Proof

Let P be a path component of X containing x and let C be a component of X containing x. We know that P is a subset of C by example 5 in the previous section. Suppose P is a proper subset of C. Since X is open, P is open in X by theorem 2. Take the union of all those path components of X disjoint from P and intersect them with C; call the union of all the resulting sets Q. Then, P and Q form a separation on C, since they are disjoint, nonempty (P is nonempty since x is in P; Q is nonempty, because we are assuming that P is a proper subset of X), and open in C (Q is open in C by theorem 2 and the definition of the subspace topology). This contradicts the connectedness of C.

 * Theorem 4

Let X be a locally path connected space. Then every open, connected subset of X is path connected.

 * Proof

Let U be an open connected subset of X and let x belong to U. Let A be the set of all points in U that can be joined by a path to X. Then A is open. If y is in A, then there is a path connected subset V of U containing y (by the local path connectedness of the space). Then we assert that V is a subset of A. If z is in V, then there is a path from z to y. Since there is a path from x to y (because y belongs to A), we can ‘paste’ these paths together to form a path from x to z. Therefore, z is in A. Therefore, V is a subset of A and A is open. Now if y is in U−A, then there is no path from y to x. If V is a path connected neighbourhood of y contained in U, then V is a subset of U−A. If z is in V, and if there is a path from z to x, then there would be a path from x to y since there is already a path from z to y. This cannot be since y is in U−A. Therefore, there is no path from z to x and z is in U−A. It follows that V is a subset of U−A and U−A is open. We know already that A is nonempty (x is in A). If U−A is nonempty, then A and U−A will form a separation on U contradicting the connectedness of U. Therefore, every point in U can be joined by a path to x. From this it follows that U is path connected.

Theorem 5

Let X be a weakly locally connected space. Then X is locally connected.

Proof

From theorem 1, it is sufficient to show that the components of open sets is open. Let U be open in X and let C be a component of U. Let x be an element of C. Then x is an element of U so that there is a connected subspace A of X contained in U and containing a neighbourhood V of x. Since A is connected and A contains x, A must be a subset of C (the component containing x). Therefore, the neighbourhood V of x is a subset of C. Since x was arbitrary, we have shown that each x in C has a neighbourhood V contained in C. This shows that C is open relative to U. Therefore, X is locally connected.

Theorem 6

Let X be a topological space. If X is locally connected, the quasicomponents of X are equal to the components of X.

Proof

We noted earlier in the section on quasicomponents that each component of the space is contained in a quasicomponent of the space. It suffices to show that the quasicomponents always lie in the components if the space is locally connected. From theorem 1, it follows that the components of X are open (since X is locally connected). Therefore, if x~1y doesn't hold (x is not equivalent to y under the equivalence relation defining components), then the component containing x is both open and closed (components are always closed; the component containing x is open because X is a locally connected topological space), so that there is a separation of X into sets A and B such that A contains x and B contains y (A is the component containing x, and B = X-A). Therefore, x cannot be equivalent to y under the equivalence relation defining quasicomponents. Therefore, we have proved that if y is not in the component containing x, y cannot be in the quasicomponent containing x. From elementary set theory, this implies that if y is in the quasicomponent containing x, it must be in the component containing x. Therefore, the quasicomponents of X are equal to the components of X.

 * Some applications of the theorems

1. From theorem 4, we can conclude that every open connected subset of R is path connected (since R is locally path connected). Also, every open, connected subset of R2 is path connected.

2. The set of rational numbers Q is not locally connected since the components of Q are not open in Q (see theorem 1).

3. The components and path components of an elementary subset of R are the same. Also, the elementary subsets of R are the finite union of intervals, since every elementary set is locally path connected.

4. Take the point, p = (0.5,0.5) in the plane. For each rational x in R, let Tx denote the line segment joining p to (x,0). Let T be the union of all such Tx. Then T is locally connected only at p. However, it isn't locally connected at any other point (if x is distinct from p and belongs to T, choose a neighbourhood of x (open in R^2) disjoint from p which doesn't intersect the x-axis. The intersection of this neighbourhood with T is homeomorphic with the countable union of open intervals in R (this we leave you to check in the exercises on homeomorphisms (see next section))and therefore cannot be connected. Therefore, no neighbourhood smaller than this neighbourhood can be connected). Therefore, T cannot be weakly locally connected by theorem 5.

5. See the infinite broom (or the broom space). This is an example of a space which is weakly locally connected at a particular point, but not locally connected at that point. The reader is directed to Wolfram MathWorld for a definition of the broom space. See also the section (in this article) on ‘weakly locally connected space’.

Exercises
Easy Questions

1. Prove that if X is locally path connected, then X is locally connected.

2. Prove that X has precisely one component if and only if X is connected. What is this component?

3. If f is a continuous, surjective map from a topological space X to a topological space Y, does it follow that Y is locally connected if X is locally connected? What conditions can you impose for this statement to be true, i.e. the continuous image of a locally connected space is locally connected (if your answer was yes to the first question, ignore the second; however, if your answer is yes you must prove it)?

4. a) Prove that the set I X I in the dictionary order topology is locally connected (Hint: What familiar connected subspace of R are basis elements for the topology on I X I homeomorphic to?) b*) Find the path components of this space (Hint: First show that this space is not path connected by obtaining a contradiction and using the intermediate value theorem (assume [0,1] is connected) c) Use your answer to b), to determine whether or not the space I X I is locally path connected.

5. Prove theorem 2

Normal Questions

'''In these questions, we will verify whether local connectedness is preserved under products and other such operations. The reader is expected to be familiar with the product topology.'''

6. a) Consider the set X = (-1,0) U (0,1) which is clearly not connected. Prove that this set is locally connected. Also find the components of this space (Hint: When proving that this space is locally connected, assume that (0,1) is connected. Then construct a homeomorphism from (0,1) to (-1,0) to show that (-1,0) is connected) b) Apply a theorem to show that the path components of this space are equal to the components of this space. Prove any assumptions that you make. c) Let E be an elementary subset of R (this term is derived from measure theory where E is said to be an elementary subset of R, if it is the finite union of intervals (not necessarily open), prove that E is locally connected by assuming that each interval is connected.

7. Suppose we take the finite product of X with itself. Is this new space locally connected? (Hint: Use the fact that the product of connected spaces is always connected)

8. Now, what if we take the infinite product of X with itself given the product topology; call this new space Y. We will check whether Y is locally connected or not in question 8. We leave the reader to have a think about this before he reads question 8.

See question 7 and think before reading question 8:

9. a) Are basis elements for the product topology on Y connected? (Hint: The projection map is continuous; therefore the projection of a connected space is always connected)

b) Possibly a) gives us a conjecture whether Y is locally connected or not. Depending on your answer to a), prove or disprove that Y is locally connected. (Hint: If your answer to a) was yes, find at least one connected neighbourhood of a point (note that a neighborhood of a point need NOT BE a basis element). If you answer to a) was no, proceed by contradiction:

If x is a point of Y and V is a neighborhood of x, suppose U is a connected neighborhood of x contained in V. Then U contains a basis element about X. Note that the projection map is continuous so that the image of U under each projection should be connected. Obtain a contradiction)

c) Is Y weakly locally connected? Justify your answer (Hint: The idea is similar to part b))

Research Questions

You will be asked to formulate hypothesis based on your knowledge:

10. Like components of a topological space are always closed, do you think path components satisfy a similar property?

We will analyse this question (see now question 9 and think before answering 10)

11. a) Is it possible to determine the path components of a connected space (without knowing what the space is)? Is it possible to determine the path components of a path connected space without knowing what the space is?

b) The topologist’s sine curve is not path connected. Assuming this determine its path components and analyze the properties these path components satisfy (determine whether they are closed or open or neither).

c) From 10.b), you should be able to answer question 9. As a question which we will leave the reader to think about:

12. Are components of a topological space necessarily locally connected?

13. Are path components of a topological space necessarily locally path connected? (Hint: This question is a little trickier than 7; you might have to construct your own space (a subspace of R^2))

14. a) Prove that homeomorphisms preserve path connectedness. b) Deduce that the topologist’s sine curve and the comb space are not homeomorphic.

The following question is a research type question:

15*. Like there are the components of a space, can you invent a similar equivalence relation that determines the ‘local components’ of a space. These ‘local components’ should be the largest locally connected subsets of the space.

Example Type Questions

16. Give an example where the path components of a space and the components of a space are not equal. Give a different example where the path components and quasicomponents of a space are not equal.

17. Find an example of a locally connected space that is not locally path connected.

18*. Give an example where the quasicomponents of a space and the components of the space are not equal.

19. If X is locally homeomorphic with Y and X is locally connected, need Y be locally connected? Justify your answer.

Questions on weakly locally connected spaces

20. Formulate a new notion called weak local path connectedness that uses the same idea as the definition of a weakly locally connected space.

a) Prove that every weakly locally path connected space is locally path connected

b) Give an example of a weakly locally path connected space that isn't locally path connected (at a particular point).

21. As in theorem 2, characterize weakly locally connected spaces in terms of components (Hint: Remember theorem 5)

Questions on Homeomorphisms

22. In the section on ‘applications of the theorems’, check the details of 4.

23. Prove that if X has ‘n’ components, and Y is homeomorphic to X, Y has ‘n’ components. Give an example where f is a continuous surjective map from X to Y, and X has ‘n’ components but Y doesn't.

24. a) Suppose p is a continuous, closed surjective map such that p^(-1) {y} is compact in X for each element y of Y (p: X->Y) (the reader who is not familiar with compactness is referred to Wikipedia). Prove that if Y is locally connected, so is X. b) Prove that if Y is connected so is X c) If X is locally connected, need Y be locally connected?

Difficult question

25. Is every connected Hausdorff space with more than one point uncountable (Hint: Every connected metric space with more than one point in uncountable; prove this assertion. See what difficulty arises with this proof when one considers a Hausdorff space. Remember that metric spaces are always first countable, normal, paracompact etc… whereas Hausdorff spaces aren’t)?

title: Topology/Local Connectedness


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OpenGL Programming/Intermediate/Normals

You can specify normals for faces you create. This is a vector that's perpendicular to the face. Specifying normals is important if you plan to use lighting. They tell OpenGL how the light is to be reflected. Although OpenGL is not a raytracer, it still influences how bright you see the face. OpenGL will not calculate the normals itself, you have to do it manually.

Specifying Normals
Normals are specified using the function glNormal3f and used before you make the vertices that belong to the face. Example:

// Create a square on the XY axis. The normal vector is then (0, 0, 1) glBegin(GL_QUADS); glNormal3f(0.0, 0.0, 1.0); glVertex3f(0.0, 0.0, 0.0); glVertex3f(0.0, 1.0, 0.0); glVertex3f(1.0, 1.0, 0.0); glVertex3f(1.0, 0.0, 0.0); glEnd;

Calculating Normals
There is a quite simple approach for calculating the normal vector for a triangle. Given p1, p2 and p3, the normal vector can be expressed as the cross product of (p2 - p1) and (p3 - p1), divided by the length of the result. In C code, using a struct type for the vectors:

struct vector a, b, n; double l; a.x = p2.x - p1.x; a.y = p2.y - p1.y; a.z = p2.z - p1.z; b.x = p3.x - p1.x; b.y = p3.y - p1.y; b.z = p3.z - p1.z; n.x = (a.y * b.z) - (a.z * b.y); n.y = (a.z * b.x) - (a.x * b.z); n.z = (a.x * b.y) - (a.y * b.x); // Normalize (divide by root of dot product) l = sqrt(n.x * n.x + n.y * n.y + n.z * n.z); n.x /= l; n.y /= l; n.z /= l; // Now pass n.x, n.y, n.z to glNormal3f

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

title: OpenGL Programming/Intermediate/Normals


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Template:Former featured book

Usage
This template should be placed on the book's main talk page.

title: Template:Former featured book


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User:Psychofarm/Dopamine and BiPolar Disorder

Dopamine
Dopamine pathways in the midbrain help to regulate motor activity, motivation and reward circuits. Behavioral changes of the manic phase such as increase in goal-directed behavior and excitement may involve these dopaminergic pathways. Mesolimbic dopaminergic pathways may be directly involved in the selection and coordination of goal-directed behavior, especially those elicted by reward stimuli. Homovanillic acid (HVA), a metabolite of dopamine, is reduced in the cerebral spinal fluid (CSF) of some depressed patients. Conversely, CSF levels of HVA are increased in some manic patients. Pharmacological agents which act on the dopamine pathways further implicate dopamine in the pathogenesis of bipolar symptoms. The dopamine precursor L-dopa can produce hypomania in bipolar patients. Amphetamine, which promotes dopamine release can cause hypomania in bipolar patients and can induce hypomanic symptoms in normal people. Bromocriptine and piribedil, which are direct dopamine agonists exhibit antidepressant properties and cause mania in some bipolar patients. Conversely, neuroleptics which block dopamine receptors have antimanic properties. Pramipexole, a dopamine agonist at the D2/D3 site, shows antidepressant efficacy as an augmentation strategy in Bipolar patients who are failing treatment with mood stabilizers.


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User:Psychofarm/footnotes and references text

Footnotes and References
Wikibooks has a really nice way to do footnotes and references. Much better than HTML, or most word processors! First put in a reference in the form:

"This hedgehog will live with us!"

(Click "edit" to see the code for this.) That makes a superscript number that your readers can click on. Then at the end of your file put in the note:

 * 1)  Chapayev, V.I., "Sesquipedalian Obfuscation in Late Early Middle Wikibooking," Journal of Very Specialized Research, 2005.

Remember the pound sign (#) to make the matching number appear.


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User talk:Gilblais

Welcome!
Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.--Whiteknight (Page) (Talk) 19:43, 26 December 2007 (UTC)

title: User talk:Gilblais


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User talk:Tophat~enwikibooks

Welcome!
Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.--Whiteknight (Page) (Talk) 21:23, 26 December 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Tophat. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Tophat~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:50, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 05:36, 19 April 2015 (UTC)

title: User talk:Tophat~enwikibooks


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Talk:Adventist Youth Honors Answer Book/Vocational/Radio Electronics

Shouldn't Ohms Law be written using capital-letters (e.g. V=I*R) ? Sure lower-case letters are sometimes used for alternating current and voltage, but resistance should still be the upper-case "R".

 * Yup. I changed it.  I had forgotten that lower case was generally for AC.  Thanks for reminding me. --Jomegat (talk) 01:29, 27 December 2007 (UTC)


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Talk:SL Psychology/Anxiety Disorders

I think that phobias is too broad of a section. Something like Social Phobia is often treated as seperate from your average phobias.

Shahadah - 26/12/07

title: Talk:SL Psychology/Anxiety Disorders


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Cookbook:Chervil

Chervil is a herb, used extensively in French cuisine.

Characteristics
A delicate herb, chervil has bright green feathery fern-like leaves and a slight resemblance to parsley. It has a flavour slightly reminiscent of liquorice.

Selection and storage
Chervil does not dry well and should be purchased fresh. Look for well-colored unwilted leaves.

Use
The herb is most common in French cooking, where it helps make up fines herbes and is used in a variety of dishes. Because the flavor is delicate, adding at the end of cooking is recommended. The leaves can also be used as a garnish.

title: Cookbook:Chervil


text:
Talk:Linux Guide/merge/Linux For Newbies/Command Line

Options vs Arguments
In the paragraph "Command Arguments", options like "-A" and "--symbolic" are refered to as arguments - along with "important notice.txt".

Surely it's more commanly to seperate between options like -A, -f, --symbolic, and arguments like something.txt, *.doc.


text:
User talk:Mariaparra

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Good luck! --Panic (talk) 23:58, 26 December 2007 (UTC)

title: User talk:Mariaparra


text:
User talk:Anemospilia

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Good luck! --Panic (talk) 23:58, 26 December 2007 (UTC)

title: User talk:Anemospilia


text:
User talk:Aaronsteers~enwikibooks

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Good luck! --Panic (talk) 23:58, 26 December 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Aaronsteers. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Aaronsteers~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

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Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 22:54, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 04:28, 19 April 2015 (UTC)

title: User talk:Aaronsteers~enwikibooks


text:
User talk:Rob Parkinson

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Good luck! --Panic (talk) 23:58, 26 December 2007 (UTC)

title: User talk:Rob Parkinson


text:
VHDL for FPGA Design/4-Bit ALU

4-Bit ALU VHDL Code
A combinatorial ALU with the following operations:

Generated Symbol
VHDL-kieli FPGA-suunnittelussa/4-bittinen ALU

title: VHDL for FPGA Design/4-Bit ALU


text:
User talk:Moistsam

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.

I wonder if you have seen the books Programmable Logic and Microprocessor Design? You might be interested in these books as well. --Whiteknight (Page) (Talk) 03:48, 27 December 2007 (UTC)

title: User talk:Moistsam


text:
User talk:Tkoskine

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Good luck! --Panic (talk) 05:45, 27 December 2007 (UTC)

title: User talk:Tkoskine


text:
User talk:Nesnad/sandbox/Emotions you have

Now it's time to find out what emotions are bottled up inside. Before we try to find out Mary's emotional state of mind, lets take a quick peak at some words we can use!

So Johnny, are you ready to find out if Mary is sad? She has been really moody recently, lets ask her!

Wow, get some sleep Mary! I wonder if she has a pillow. But wait a minute! What was that わたし thing? That isn't a name! If you didn't figure it out, it is the polite way to talk about yourself. Lets look at a few more.

Ahh, it looks like Mary got some sleep! Good for her. Lets find out how she's feeling today. Go ahead Johnny.

Phew, good to hear that Mary is back to her cheerful self. Don't want any frowns to ruin our communication! Lets try to request things we want!


text:
User talk:Punkouter

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Good luck! --Panic (talk) 06:34, 27 December 2007 (UTC)

title: User talk:Punkouter


text:
User talk:Regulars

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Good luck! --Panic (talk) 08:24, 27 December 2007 (UTC)

title: User talk:Regulars


text:
User talk:Oddalot

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Good luck! --Panic (talk) 08:24, 27 December 2007 (UTC)

title: User talk:Oddalot


text:
User talk:Grandad~enwikibooks

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Good luck! --Panic (talk) 08:24, 27 December 2007 (UTC)

Request for delete
Hello, I am going to request a deletion of a book that I believe you started, Simple Central Heating Using Easy Maths. If you want to work on it, I am fine with that but it currently has only chapter titles. NumberTheorist (talk) 01:58, 14 February 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Grandad. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Grandad~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:11, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 05:00, 19 April 2015 (UTC)

title: User talk:Grandad~enwikibooks


text:
User talk:Shelly bean

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Good luck! --Panic (talk) 08:24, 27 December 2007 (UTC)

title: User talk:Shelly bean


text:
User talk:Mikenucklesii

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Good luck! --Panic (talk) 08:24, 27 December 2007 (UTC)

title: User talk:Mikenucklesii


text:
User:Wakedream

I'm Wakedream, and I'm here--unless I'm dreaming.

Like many professional writers, I spend a lot of time looking for things to do other than work on what I'm supposed to be writing. Unlike many professional writers, those other things often involve writing. Perhaps this is why I'm not rich.

title: User:Wakedream


text:
Horticulture/Tradescantia

Spiderworts (Tradescantia) are a genus of an estimated 71 species of perennial plants in the family Commelinaceae, native to the New World from southern Canada south to northern Argentina. Though sometimes accounted a weed, spiderwort is cultivated for borders and also used in containers. Where it appears as a volunteer, it is often welcomed and allowed to stay. The genus takes its name from John Tradescant the elder, a 17th-century English plant collector and nurseryman.

The first species described, Virginia Spiderwort T. virginiana, is native to the eastern United States from Maine to Alabama, and Canada in southern Ontario. Virginia Spiderwort was introduced to Europe in 1629, where it is cultivated as a garden flower.

The Western Spiderwort T. occidentalis is listed as an endangered species in Canada, where the northernmost populations of the species are found at a few sites in southern Saskatchewan, Manitoba and Alberta; it is however more common further south in the United States south to Texas and Arizona.

The three species of Wandering Jew, one native to eastern Mexico, also belong to the tradescantia genus. Other names used for various species include Spider-lily, Cradle-lily, Oyster-plant and Flowering Inch Plant.

Description
They are weakly upright to scrambling plants, growing to 30-60 cm tall, and are commonly found individually or in clumps in wooded areas and fields. The leaves are long, thin and bladelike to lanceolate, from 3-45 cm long. The flowers are white, pink or purple but most commonly bright blue, with three petals and six yellow anthers. The sap is mucilaginous and clear. A number of the species flower in the morning and when the sun shines on the flowers in the afternoon they close up, but the flowers can remain open on cloudy days until evening.

Uses
The cells of the stamen hairs of some Tradescantia are colored blue, but when exposed to sources of ionizing radiation such as gamma rays, the cells mutate and change color to pink; they are one of the few tissues known to serve as an effective bioassay for ambient radiation levels.

title: Horticulture/Tradescantia


text:
Horticulture/Trifolium

Clover (Trifolium) is a genus of about 300 species of plants in the pea family Fabaceae. The genus has a cosmopolitan distribution; the highest diversity is found in the temperate Northern Hemisphere, but many species also occur in South America and Africa, including at high altitudes on mountains in the tropics.

They are small annual, biennial, or short-lived perennial herbaceous plants. The leaves are trifoliate (rarely 5- or 7-foliate), with stipules adnate to the leaf-stalk, and heads or dense spikes of small red, purple, white, or yellow flowers; the small, few-seeded pods are enclosed in the calyx.

Other closely related genera often called clovers include ../Melilotus/ (sweet clover) and ../Medicago/ (alfalfa or 'calvary clover'). The "shamrock" of popular iconography is sometimes considered to be young clover.

The scientific name derives from the Latin tres, "three", and folium, "leaf", so called from the characteristic form of the leaf, which has three leaflets (trifoliate); hence the popular name trefoil.

Shamrock, the traditional Irish symbol coined by Saint Patrick for the Holy Trinity, is commonly associated with clover, though also sometimes with ../Oxalis/ species, which also have trifoliate leaves. Clovers occasionally have leaves with four leaflets, instead of the usual three. These four-leaf clovers, like other rarities, are considered lucky.

Uses
Several species are extensively cultivated as fodder-plants. The most widely cultivated clovers are White clover ../Trifolium repens/ and Red clover ../Trifolium pratense/. Clover, either sown alone or in mixture with ryegrass, has for a long time formed a staple crop for soiling, for several reasons; it grows freely, shooting up again after repeated mowings; it produces an abundant crop; it is palatable to and nutritious for livestock; it grows in a great range of soils and climates; and it is appropriate either for pasturage or green composting.

"Clover sickness" in more recent times may also be linked to pollinator decline; clovers are most efficiently pollinated by bumblebees, which have declined as a result of agricultural intensification. Honeybees can also pollinate clover, and beekeepers are often in heavy demand from farmers with clover pastures. Farmers enjoy the benefits of increased reseeding that occurs with increased bee activity, which means that future clover yields remain abundant. Beekeepers benefit from the clover bloom as clover is one of the main nectar sources for honeybees.

T. repens, White or Dutch clover, is a perennial abundant in meadows and good pastures. The flowers are white or pinkish, becoming brown and deflexed as the corolla fades. T. hybridum, Alsike or Swedish clover, is a perennial which was introduced early in the 19th century and has now become naturalized in Britain. The flowers are white or rosy, and resemble those of the last species. T. medium, meadow or zigzag clover, a perennial with straggling flexuous stems and rose-purple flowers, is of little agricultural value.

Other British species are: T. arvense, Hare's-foot trefoil; found in fields and dry pastures, a soft hairy plant with minute white or pale pink flowers and feathery sepals; T. fragiferum, Strawberry clover, with densely-flowered, globose, rose-purple heads and swollen calyxes; T. procumbens, Hop trefoil, on dry pastures and roadsides, the heads of pale yellow flowers suggesting miniature hops; and the somewhat similar T. minus, common in pastures and roadsides, with smaller heads and small yellow flowers turning dark brown. The last named is often called Shamrock.

Clovers are a valuable survival food, as they are high in protein, widespread, and abundant. They are not easy to digest raw, but this can be easily fixed by juicing them or boiling them for 5-10 minutes. Dried flowerheads and seedpods can also be ground up into a nutritious flour and mixed with other foods. Dried flowerheads can also be steeped in hot water for a healthy, tasty tea.

Pests and diseases
In many areas, particularly on acidic soils, clover is short-lived due a combination of insect pests, diseases and nutrient balance; this is known as "clover-sickness". When crop rotations are managed so that clover does not recur at shorter intervals than eight years, it grows with much of its pristine vigor.

title: Horticulture/Trifolium


text:
Horticulture/Trifolium pratense

Trifolium pratense (Red Clover) s a species of clover, native to Europe, western Asia and northwest Africa, but planted and naturalised in many other regions. The plant was named Trifolium pratense by Carolus Linnaeus in 1753. Pratense is Latin for "found in meadows". It is widely grown as a fodder crop, valued for its nitrogen fixation which increases soil fertility. For these reasons it is used as a green manure crop. Several Cultivar Groups have been selected for agricultural use, mostly derived from var. sativum. It has become naturalised in many temperate areas, including the Americas and Australasia as an escape from cultivation.

It is the national flower of Denmark and the state flower of Vermont.

Description
It is a herbaceous perennial plant, very variable in size, growing to 20-80 cm tall. The leaves are alternate, trifoliate (with three leaflets), each leaflet 15-30 mm long and 8-15 mm broad, green with a characteristic pale crescent in the outer half of the leaf; the petiole is 1-4 cm long, with two basal stipules. The flowers are dark pink with a paler base, 12-15 mm long, produced in a dense inflorescence 2-3 cm diameter.

Varieties
There are seven varieties:
 * Trifolium pratense var. pratense Widespread.
 * Trifolium pratense var. americanum Southeastern Europe (despite the name).
 * Trifolium pratense var. frigidum Mountains of central and southern Europe (Pyrenees, Alps, Balkans).
 * Trifolium pratense var. maritimum Southern Baltic Sea coast.
 * Trifolium pratense var. parviflorum Europe.
 * Trifolium pratense var. sativum Mediterranean region. Robust-growing, with hairless or nearly hairless foliage.
 * Trifolium pratense var. villosum Alps. Densely hairy foliage.

Uses
The isoflavones and phytoestrogens from Red Clover have been used to treat the symptoms of menopause. Women who are pregnant and breastfeeding should avoid ingesting Red Clover. It has also been reported that red clover can be used for therapeutic purposes for coughs, bronchitis, eczema, sores, scrofula and can be gargled for mouth ulcers and sore throats. It is also an ingredient in eight-herb essiac tea.

Pests and Diseases
List of red clover diseases

title: Horticulture/Trifolium pratense