Conservapedia - User contributions [en]

Federal

2008-06-30T23:49:36Z

Henry8th: Replacing page with 'Call the FBI!! (oh. already on this page)'

Call the FBI!! (oh. already on this page)

Prayer

2008-06-30T23:49:14Z

Henry8th: Replacing page with 'Some people are against prayer in schools! Call the FBI!!'

Some people are against prayer in schools! Call the FBI!!

George H. W. Bush

2008-06-30T23:49:07Z

Henry8th: Replacing page with 'Call the FBI!!!'

Call the FBI!!!

Richard Lenski

2008-06-30T23:49:05Z

Henry8th: Replacing page with 'Call the FBI!!!'

Call the FBI!!!

Autism

2008-06-30T23:49:03Z

Henry8th: Replacing page with 'Autism cover up! Call the FBI!!'

Autism cover up! Call the FBI!!

Talk:Richard Lenski

2008-06-30T23:48:54Z

Henry8th:

==Formatting==
I borrowed a template from another wiki which might make the data section a bit easier to comprehend. I will include it here pending approval of senior sysops. [[User talk:MargeryCampbell|Marge]] 12:03, 28 June 2008 (EDT)

Looks great. Please install in the entry as desired.--[[User:Aschlafly|Aschlafly]] 12:55, 28 June 2008 (EDT)
:Okay, I've moved it. I've also cut it from this page for economy's sake. [[User talk:MargeryCampbell|Marge]] 13:07, 28 June 2008 (EDT)
::Actually it makes the article look amaturish. For starters you have written in capitals and you have not explained why the missing data is significant which is the point of the side-by-side template to do a full rebuttle. [[User:DanielB|DanielB]] 19:44, 28 June 2008 (EDT)

:::I agree. A discussion specifying what one expects from the 'missing' data and what usefulness such information would likely shed is necessary.--[[User:Argon|Argon]] 12:57, 30 June 2008 (EDT)
::::I agree that it is necessary to establish exactly what is being asked for in the terse comments on the right hand side, sadly such a discussion could get the majority of the contributors banned for the 90/10 rule. I myself have not contributed much to this question recently as a result of this rule. Could the powers that be explicitly suspend this rule on this issue so that the question may be freely discussed? --[[User:British_cons|British_cons]] [[User_talk:British_cons|(talk)]] 13:12, 30 June 2008 (EDT)

== Listing fields ==

I'm not a parodist, though I did make a mistake on what a CV is.

== Suggestion: Disagreements with E Coli paper should go to a new section? ==

Shouldn't this be a brief bio of Lenski, and a link to the controversy over the paper, including the discussions and the emails, be part of that page instead of here? It looks pretty bad putting it on the main page. 18:11, 30 June 2008 (EDT)
On the other hand, after doing some research, I don't think it's standard for faculty to put degree fields in a list of their degrees.

I googled "MSU biology", then went over the "Faculty List A-G". Here is what I found:

No list of degrees: 2
List with fields: 3
List without fields: 12

Finally, Oberlin is a liberal college, and they do focus mostly on liberal arts. [[User:Drochld|Drochld]] 14:26, 29 June 2008 (EDT)

==Template / formatting==
Out of interest, I help to administer a private wiki where I work, and think this template is quite nice. Might I ask where you found it, and what (if any) associated copyright restrictions there are on it? [[User:Henry8th|Henry8th]] 15:49, 29 June 2008 (EDT)

: Really, "Henry8th"??? That's an amazing coincidence you describe: you just happen to administer a "private wiki" at "work". Before wasting people's time with your silliness, how about convincing us a bit more of the earnestness of comment, as in providing the wiki version number, the type of your employer and the location of your "work". I'm sure the wiki developers who freely donate their efforts for use by all would be thrilled to learn of your profitable use.--[[User:Aschlafly|Aschlafly]] 16:33, 29 June 2008 (EDT)
:: We run Mediawiki 1.6.7 (yes, I know we're incredibly out of date, but we can't upgrade to a newer one as that would require PHP5, whereas we need that server running PHP4 to support other older software). I make it a habit not to disclose too much personal information on the internet, I'm afraid, but I'm a PhD student at a UK university and I help to administer a wiki run by my research group. By "private" I mean "accessible only to people within my research group"; we find that a wiki is an efficient way for us to edit documents collaboratively. [[User:Henry8th|Henry8th]] 16:47, 29 June 2008 (EDT)
:: The copyright policy here says "Conservapedia grants a non-exclusive license to you to use any of the content (other than images) on this site with or without attribution".[[User:JPohl|JPohl]] 11:53, 30 June 2008 (EDT)
:::However, since it was admitted above that the template is not CP's creation, it should first be established whether or not the template was used in a correct way (and if any attributions are needed) before others borrow it from here. [[User:ZTak|ZTak]] 12:06, 30 June 2008 (EDT)
::::I would ask the same question as ZTak above. Firstly, from a legal point of view I would want to be absolutely sure that any code I might upload to a server at my university was properly licensed (we have regular software audits, of course). Secondly, from an ethical point of view I would like to make sure that I can properly credit the original author of the template. As someone whose work involves writing computer programs, I certainly wouldn't appreciate it if someone took software I'd written and didn't attribute authorship correctly. [[User:Henry8th|Henry8th]] 14:36, 30 June 2008 (EDT)

== Fact checking could use some work on the omissions section ==

I strongly encourage editors to confirm that the topics on this entry's 'omissions' section page really refer to "data not provided" and do not reflect the lack of understanding of the research. At a bare minimum it is necessary to completely read the actual article. 'Skimming' it quickly is not fair to the goals of this wiki or Lenksi's lab. To at least be diligent, I recommend that the authors of the "omissions" section go to Lenksi's lab web page here (https://www.msu.edu/~lenski/), and peruse (read, not skim) the many papers related to the Long-Term Evolution project that are referenced in the current paper.--[[User:Argon|Argon]] 11:57, 30 June 2008 (EDT)

== I do not understand why my last edit was reversed ==

It seems to be common practice (and therefore not "odd") to not list what fields faculty majored in, and Oberlin is very liberal. [[User:Drochld|Drochld]] 12:34, 30 June 2008 (EDT)

I'm not certain what 'liberal' has to do with anything and "odd" has pejorative overtones. Rather than worry about 'oddness' why not investigate the degrees? In his graduate program Lenski studied insects and evolution and around 1983, worked with Bruce Levin, moving into the study of microbial evolution.--[[User:Argon|Argon]] 12:51, 30 June 2008 (EDT)

== This man is nothing but a farce ==

Why must we, the children of God, even acknowledge this liberal atheist (as if there is any other form of an atheist) on our glorious site? We are only feeding his overinflated ego by doing so. --[[User:Dexter111344|Dexter111344]] 12:52, 30 June 2008 (EDT)

== A problem with: "Lenski is best known for his claim to have observed [[evolution]] of [[E. coli]] in a long-term laboratory study," ==

This recent study is but one of many papers by Lenksi ''et al'' regarding the evolution of bacterial populations in continuous culture. He is best known professionally for his ''in vitro'' experiments studying the population dynamics of genetic adaptations in bacteria and other studies in microbial evolution.--[[User:Argon|Argon]] 13:14, 30 June 2008 (EDT)

== don't clutter this entry with meaningless edits???? ==

The topic of the page is 'Richard Lenski'. I added additional bibliographic information about Lenski's professional awards and they are reversed as "meaningless edits"? And the 'odd' comment remains despite the fact that it's common to leave the subjects of the degrees out of web pages?--[[User:Argon|Argon]] 13:43, 30 June 2008 (EDT)
:Yes, various links to things such as [[data]] as well. I thought data was central to the question at hand? Why not link to it (such as it is)? And a mention of his work with virtual organisms seems relevant - especially as it's mentioned again at the end.--[[User:British_cons|British_cons]] [[User_talk:British_cons|(talk)]] 13:47, 30 June 2008 (EDT)

And now I'm a "Lenski supporter"? Anyone who actually read the paper and referenced the pertinent section of the article that clearly specified the glucose concentration used is a supporter? I think that addition of 'Lenksi supporter' is needlessly inflammatory. Furthermore, with regard to the question: WHERE IS THE DATA MEASURING OR CONFIRMING THE 25 MG/LITER GLUCOSE? DID THAT CONCENTRATION EVER CHANGE?", I also wrote: "A weighed amount of glucose is mixed into the medium. The paper states that the concentration listed was used in the media throughout the continuous subculturing experiment." and that got removed. Resubmitted.--[[User:Argon|Argon]] 18:14, 30 June 2008 (EDT)

== shouldn't we move the E Coli paper discussion to a separate page? ==

It looks really cheezy to have this right on Lenski's biography, very unprofessional.
Just have a link and move all that discussion to something like Lenski_E_Coli_controversy,
don't erase a thing. [[User:Encinocathy|Cathy]] 18:11, 30 June 2008 (EDT)

== Wrong format for a debate ==

Ignoring all the other issues with this "article", the side-by-side table is really not made for a discussion. Maybe move that part to a debate page? It's quite silly to see an encyclopedia article doing a question-and-answer session with anonymous participants. --[[User:KevinM|KevinM]] 18:18, 30 June 2008 (EDT)
:Agreed. It's silly to post the page without doing the basic research. Better to corral the innuendos into debate pages.--[[User:Argon|Argon]] 18:20, 30 June 2008 (EDT)
::Also, we have yet to discovere the proper attribution for the side-by-side table, and as such it should be taken down or it might constitute a violation of copyright. [[User:ZTak|ZTak]] 18:25, 30 June 2008 (EDT)
:::Given that [[User:MargeryCampbell|Marge]] said she took the template from elsewhere, it would presumably be straightforward for her to tell us where she found it. Given that information I'd happily aid the project by checking out this other wiki's copyright policy before reporting back here. [[User:Henry8th|Henry8th]] 18:45, 30 June 2008 (EDT)
:My issue is mostly about using a left/right table when the left side is basically just the intro section while the right side is suddenly filled with a weird pseudo-debate that goes back and forth. It would be better to do it like this:
==Visual Inspection==
===Lenski's Paper===
(quote)
===Discussion===
(back and forth; not as debate, but in more encyclopedic "A claims B, to which C points out D." style)

==Study Omissions==
(quote)
:And so on. It would make the article more readable. And it would have the added benefit of NOT USING SO MANY ALL-CAPS SENTENCES, BECAUSE DESPITE WHAT SOME SITES SAY, CAPSLOCK IS NOT THE CRUISE CONTROL FOR COOL. --[[User:KevinM|KevinM]] 19:12, 30 June 2008 (EDT)

Copyright should be respected. Call the FBI!!

Talk:Richard Lenski

2008-06-30T22:45:21Z

Henry8th: /* Wrong format for a debate */

Hermitian matrix

2008-06-30T18:47:41Z

Henry8th: A start.

A '''Hermitian matrix''' is one that satisfies <math>M=M^\dagger</math>, where <math>M^\dagger</math> is the Hermitian conjugate of <math>M</math> (i.e., the matrix formed by transposing <math>M</math> and taking the complex conjugate of each element). As an example, the most general 2x2 Hermitian [[matrix]] has the form

<math>
\begin{pmatrix}
a & b \\
b^* & c
\end{pmatrix}
</math>

for arbitrary complex numbers <math>a,b,c</math>. In the case where all elements of the matrix are real, a Hermitian matrix becomes symmetric (as Hermitian conjugation then becomes equivalent to transposition).

==Properties of Hermitian matrices==
* The [[eigenvalue|eigenvalues]] are all real.
* The [[eigenvector|eigenvectors]] corresponding to different eigenvalues are orthogonal.

Because of these properties, Hermitian matrices have important applications in [[quantum mechanics]].

Matrix

2008-06-30T18:46:34Z

Henry8th:

''For the 1999 film, see [[The Matrix]].''

A '''matrix''' (pl.: "matrices," [[Latin]] origin) is a complex ordering, in deliberate fashion, of [[numeral|numerals]]. In [[mathematics]], a "matrix" is a regular grid of numbers, which may be manipulated and solved through intermediate-level [[algebra]]. Matrix algebra is usually taught in [[sophomore]] [[high school]] level mathematics.

More formally, a matrix is an example of a rank-2 [[tensor]].

Alternately, a '''matrix''' may also be a complex ordering of a group of equivalent objects, especially where the order is imposed to gain incidental benefit from the [[synergy]] of the networked objects.

==Mathematics==
In mathematics, matrices can be manipulated in a variety of ways, including [[addition]] and [[multiplication]].

===Addition of matrices===
For example, to add two matrices, one would add their respective elements, thus:

<math>\begin{bmatrix}
x & y & z \\
1 & 3 & 5 \\
0 & 2 & 0
\end{bmatrix} + \begin{bmatrix}
0 & 3 & 1 \\
4 & 3 & {x+2} \\
0 & 4 & v
\end{bmatrix} </math>

would equal

<math>\begin{bmatrix}
{x+0} & {y+3} & {z+1} \\
{1+4} & {3+3} & {5+(x+2)} \\
{0+0} & {2+4} & {0+v}
\end{bmatrix} = \begin{bmatrix}
x & {y+3} & {z+1} \\
5 & 6 & {x+7} \\
0 & 6 & v
\end{bmatrix} </math>

===Multiplication of matrices===
To multiply two matrices, one uses the rule "go along the rows and down the columns". This is best illustrated by a specific example: a matrix times a vector:

<math>\begin{bmatrix}
a & b \\
c & d
\end{bmatrix} \begin{bmatrix}
e \\ f
\end{bmatrix}=\begin{bmatrix}
ae+bf \\
ce+df \end{bmatrix}
</math>

It is important to note that matrix multiplication is not commutative: in general, <math>AB \neq BA</math> for two matrices <math>A</math> and <math>B</math>. This has important consequences in [[quantum mechanics]].

To see why matrix multiplication works the way it does, we will use [[suffix notation]]. Consider first forming the product of two matrices, <math>AB</math>, which is itself a matrix. Then form the product <math>ABx</math>. Matrix multiplication is associative, so we can consider this as either <math>(AB)x</math> or <math>A(Bx)</math>. In suffix notation,

<math>
\sum_j(AB)_{ij}x_j = \sum_k A_{ik}(Bx)_k=\sum_{j,k}A_{ik}B_{kj}x_j
</math>
The vector <math>x</math> is arbitrary, so we can therefore deduce the rule for finding the product of two matrices:

<math>
(AB)_{ij}=\sum_k A_{ik}B_{kj}
</math>

===Matrix concepts===

====Basic concepts====

*[[Adjoint]]
*[[Determinant]]
*[[Diagonal matrix]]
*[[Identity matrix]]
*[[Inverse matrix]]
*[[Null, column and row space]]
*[[Scalar]]
*[[Trace]]
*[[Transpose matrix]]
*[[Vector]]
*[[Zero matrix]]

====Advanced concepts====

*[[Basis]]
*[[Eigenspace]]
*[[Eigenvalue]]
*[[Eigenvector]]
*[[Gram-Schmidt process]]
*[[Hermitian matrix]]
*[[Jordan canonical form]]
*[[Laplacian]]
*[[Linear independence]]
*[[Matrix diagonalization]]
*[[Matrix reformation]]
*[[Matrix transformation]]
*[[Orthogonal matrix]]
*[[Orthonormal matrix]]
*[[Resultant]]
*[[Span]]
*[[Systems of linear equations]]
*[[Transcriptor]]
*[[Wronskian]]

[[Category:Linear algebra]]
[[Category:Computers]]

Talk:Richard Lenski

2008-06-30T18:36:30Z

Henry8th: /* Template / formatting */

==Formatting==
I borrowed a template from another wiki which might make the data section a bit easier to comprehend. I will include it here pending approval of senior sysops. [[User talk:MargeryCampbell|Marge]] 12:03, 28 June 2008 (EDT)

Looks great. Please install in the entry as desired.--[[User:Aschlafly|Aschlafly]] 12:55, 28 June 2008 (EDT)
:Okay, I've moved it. I've also cut it from this page for economy's sake. [[User talk:MargeryCampbell|Marge]] 13:07, 28 June 2008 (EDT)
::Actually it makes the article look amaturish. For starters you have written in capitals and you have not explained why the missing data is significant which is the point of the side-by-side template to do a full rebuttle. [[User:DanielB|DanielB]] 19:44, 28 June 2008 (EDT)

:::I agree. A discussion specifying what one expects from the 'missing' data and what usefulness such information would likely shed is necessary.--[[User:Argon|Argon]] 12:57, 30 June 2008 (EDT)
::::I agree that it is necessary to establish exactly what is being asked for in the terse comments on the right hand side, sadly such a discussion could get the majority of the contributors banned for the 90/10 rule. I myself have not contributed much to this question recently as a result of this rule. Could the powers that be explicitly suspend this rule on this issue so that the question may be freely discussed? --[[User:British_cons|British_cons]] [[User_talk:British_cons|(talk)]] 13:12, 30 June 2008 (EDT)

== Listing fields ==

I'm not a parodist, though I did make a mistake on what a CV is.

On the other hand, after doing some research, I don't think it's standard for faculty to put degree fields in a list of their degrees.

I googled "MSU biology", then went over the "Faculty List A-G". Here is what I found:

No list of degrees: 2
List with fields: 3
List without fields: 12

Finally, Oberlin is a liberal college, and they do focus mostly on liberal arts. [[User:Drochld|Drochld]] 14:26, 29 June 2008 (EDT)

==Template / formatting==
Out of interest, I help to administer a private wiki where I work, and think this template is quite nice. Might I ask where you found it, and what (if any) associated copyright restrictions there are on it? [[User:Henry8th|Henry8th]] 15:49, 29 June 2008 (EDT)

: Really, "Henry8th"??? That's an amazing coincidence you describe: you just happen to administer a "private wiki" at "work". Before wasting people's time with your silliness, how about convincing us a bit more of the earnestness of comment, as in providing the wiki version number, the type of your employer and the location of your "work". I'm sure the wiki developers who freely donate their efforts for use by all would be thrilled to learn of your profitable use.--[[User:Aschlafly|Aschlafly]] 16:33, 29 June 2008 (EDT)
:: We run Mediawiki 1.6.7 (yes, I know we're incredibly out of date, but we can't upgrade to a newer one as that would require PHP5, whereas we need that server running PHP4 to support other older software). I make it a habit not to disclose too much personal information on the internet, I'm afraid, but I'm a PhD student at a UK university and I help to administer a wiki run by my research group. By "private" I mean "accessible only to people within my research group"; we find that a wiki is an efficient way for us to edit documents collaboratively. [[User:Henry8th|Henry8th]] 16:47, 29 June 2008 (EDT)
:: The copyright policy here says "Conservapedia grants a non-exclusive license to you to use any of the content (other than images) on this site with or without attribution".[[User:JPohl|JPohl]] 11:53, 30 June 2008 (EDT)
:::However, since it was admitted above that the template is not CP's creation, it should first be established whether or not the template was used in a correct way (and if any attributions are needed) before others borrow it from here. [[User:ZTak|ZTak]] 12:06, 30 June 2008 (EDT)
::::I would ask the same question as ZTak above. Firstly, from a legal point of view I would want to be absolutely sure that any code I might upload to a server at my university was properly licensed (we have regular software audits, of course). Secondly, from an ethical point of view I would like to make sure that I can properly credit the original author of the template. As someone whose work involves writing computer programs, I certainly wouldn't appreciate it if someone took software I'd written and didn't attribute authorship correctly. [[User:Henry8th|Henry8th]] 14:36, 30 June 2008 (EDT)

== Fact checking could use some work on the omissions section ==

I strongly encourage editors to confirm that the topics on this entry's 'omissions' section page really refer to "data not provided" and do not reflect the lack of understanding of the research. At a bare minimum it is necessary to completely read the actual article. 'Skimming' it quickly is not fair to the goals of this wiki or Lenksi's lab. To at least be diligent, I recommend that the authors of the "omissions" section go to Lenksi's lab web page here (https://www.msu.edu/~lenski/), and peruse (read, not skim) the many papers related to the Long-Term Evolution project that are referenced in the current paper.--[[User:Argon|Argon]] 11:57, 30 June 2008 (EDT)

== I do not understand why my last edit was reversed ==

It seems to be common practice (and therefore not "odd") to not list what fields faculty majored in, and Oberlin is very liberal. [[User:Drochld|Drochld]] 12:34, 30 June 2008 (EDT)

I'm not certain what 'liberal' has to do with anything and "odd" has pejorative overtones. Rather than worry about 'oddness' why not investigate the degrees? In his graduate program Lenski studied insects and evolution and around 1983, worked with Bruce Levin, moving into the study of microbial evolution.--[[User:Argon|Argon]] 12:51, 30 June 2008 (EDT)

== This man is nothing but a farce ==

Why must we, the children of God, even acknowledge this liberal atheist (as if there is any other form of an atheist) on our glorious site? We are only feeding his overinflated ego by doing so. --[[User:Dexter111344|Dexter111344]] 12:52, 30 June 2008 (EDT)

== A problem with: "Lenski is best known for his claim to have observed [[evolution]] of [[E. coli]] in a long-term laboratory study," ==

This recent study is but one of many papers by Lenksi ''et al'' regarding the evolution of bacterial populations in continuous culture. He is best known professionally for his ''in vitro'' experiments studying the population dynamics of genetic adaptations in bacteria and other studies in microbial evolution.--[[User:Argon|Argon]] 13:14, 30 June 2008 (EDT)

== don't clutter this entry with meaningless edits???? ==

The topic of the page is 'Richard Lenski'. I added additional bibliographic information about Lenski's professional awards and they are reversed as "meaningless edits"? And the 'odd' comment remains despite the fact that it's common to leave the subjects of the degrees out of web pages?--[[User:Argon|Argon]] 13:43, 30 June 2008 (EDT)
:Yes, various links to things such as [[data]] as well. I thought data was central to the question at hand? Why not link to it (such as it is)? And a mention of his work with virtual organisms seems relevant - especially as it's mentioned again at the end.--[[User:British_cons|British_cons]] [[User_talk:British_cons|(talk)]] 13:47, 30 June 2008 (EDT)

Talk:Richard Lenski

2008-06-29T20:47:53Z

Henry8th: /* Template / formatting */

Talk:Richard Lenski

2008-06-29T19:49:25Z

Henry8th:

Curl

2008-06-28T13:37:48Z

Henry8th:

The '''curl''' is a way of expressing a certain type of [[derivative]] of a [[vector field]]. It is defined for fields of 3-dimensional vectors on 3-dimensional space. The curl of a vector field is another vector field.

The curl is written as though it were the [[cross product]] of the special symbol "<math>\nabla</math>" (which is commonly called "del" or "nabla"), with the given vector field, like this: <math>\nabla \times \vec V</math>. This is usually pronounced "curl V" or "del cross V".

In ordinary [[Cartesian coordinates]], the curl is calculated as:

:<math>\nabla \times \vec V = (\ \ \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z},\ \ \ \ \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x},\ \ \ \ \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}\ \ )</math>

or, using a notation like the [[determinant]] notation for cross product,
:<math>\nabla \times \vec V = \begin{vmatrix}
\hat x & \hat y & \hat z \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
V_x & V_y & V_z
\end{vmatrix}</math>
where <math>\hat x</math>, <math>\hat y</math>, and <math>\hat z</math> are the unit basis vectors.

If one thinks of <math>\nabla</math> as being a fictional vector field with components <math>(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})</math>, one can sort of see that the cross product notation makes sense. This is also useful for remembering how to calculate a curl.

The curl is a true vector field operation—the result is independent of the coordinate system that is used. The proof of that, and its ramifications, are beyond the scope of this page.

The curl operation has an intrinsic "handedness" to it. Any physical phenomenon described by the curl operation (for example, magnetic fields), involves some kind of "right-hand rule".

The curl is an extremely important operation in physics, mathematics, and engineering. It is perhaps most famous for its appearance in [[Maxwell's Equations]].

Intuitively, the curl measures the degree to which the vector field rotates around a given point. If you were to measure the curl of the vector field of wind speed in the vicinity of a meteorological low pressure area, then, keeping in mind that the [[Coriolis force]] makes the wind move in a counterclockwise vortex in the Northern hemisphere, it would be a vector pointing upward. (The ''reason'' for the Coriolis force is not important here, we're just talking about the ''observation'' that the air moves in a counterclockwise vortex.) The way to see that the curl points upward is to visualize a giant right hand with the fingers curled counterclockwise—the thumb would point upward.

Vector fields with a curl of zero are called ''irrotational''.

[[Category:Mathematics]]
[[Category:Physics]]

Taylor series

2008-06-24T22:32:57Z

Henry8th: \frac{df(x_0)}{dx} != \frac{df(x)}{dx} evaluated at x_0; text already states that derivatives are to be evaluated at x_0

The '''Taylor series''' of a function is useful for approximating a [[function|mathematical function]] near to some particular point. For a function <math>f(x)</math>, the Taylor series about the point <math>x_0</math> is

<math>f(x)=f(x_0)+(x-x_0)\frac{df(x)}{dx}+\frac{(x-x_0)^2}{2!}\frac{d^2f(x)}{dx^2}+\ldots+\frac{(x-x_0)^N}{N!}\frac{d^Nf(x)}{dx^N}</math>

where each of the derivatives is to be evaluated at <math>x=x_0</math>. If as <math>N\rightarrow\infty</math> the [[series (mathematics)|series]] converges, then it is exact. Otherwise, it can be used as an approximation. Often, Taylor series are performed around <math>x_0=0</math>, in which case they are sometimes also known as a [[Maclaurin series]].

===Examples of common Taylor series===
<math>e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots</math>

<math>\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots</math>

<math>\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots</math>

====Extensions of the Exponential Function====

Consider the exponential of [[imaginary number]] <math>yi</math>,

<math>e^{yi}=1+yi+\frac{(yi)^2}{2!}+\frac{(yi)^3}{3!}+\frac{(yi)^4}{4!}+\frac{(yi)^5}{5!}+\ldots</math>
:<math>=1+yi-\frac{y^2}{2!}-i\frac{y^{3}}{3!}+\frac{y^4}{4!}+i\frac{y^5}{5!}-\ldots</math>

:<math>=(1-\frac{y^2}{2!}+\frac{y^4}{4!}-\ldots)+i(y-\frac{y^{3}}{3!}+\frac{y^5}{5!}-\ldots)</math>

:<math>=\cos y+i\sin y</math>

By the power laws then all [[complex numbers]] have an exponential,

<math>e^{x+yi}=e^{x}(\cos y+i\sin y)</math>.

[[category:mathematics]]

Entropy

2008-05-12T22:05:18Z

Henry8th: Added some definitions

Entropy is a measure of disorder or information content in a system, first postulated by [[Lazare Carnot]] in 1803.

The second law of thermodynamics states that entropy will always increase over time within a closed system, defining a closed system as one in which neither matter nor energy may enter or leave.

==Definitions==
===Thermodynamic definition===
In classical [[thermodynamics]], if a small amount of energy dQ is supplied to a system from a reservoir held at temperature T, the change in entropy is given by

<math>
dS=\frac{dQ}{T}
</math>

===Statistical mechanics definition 1===
If a system can be arranged in W different ways, the entropy is

<math> S= k_B \log W </math>

where <math>k_B</math> is Boltzmann's constant.

===Statistical mechanics definition 2===
Label the different states a thermodynamic system can be in by <math>i=1,2,3\ldots N</math>. If the probability of finding the system in state i is <math>p_i</math>, the entropy is

<math>
S=-k_B \sum_i^N p_i \log p_i
</math>

This definition is closely related to ideas in [[information theory]], where the definition of information content is very similar to the definition of entropy.

==See also==

* [[history of entropy|History of the development of entropy]]

==References==
<references/>
[[category:physics]]

Abortion industry

2008-05-12T20:04:03Z

Henry8th: typo

The '''abortion industry''' is a billion-dollar industry that profits from performing [[abortions]]. It receives millions of dollars in taxpayer money in some states.

Along with its allies in the media and other professions, the '''abortion industry''' is the single most powerful industry in exerting political influence over the [[Democratic Party]].
[[category:politics]]
[[category:abortion]]

Fourier transformation

2008-05-12T20:02:56Z

Henry8th:

The Fourier transformation (often called the "Fourier transform" or "Fourier integral") is an invertible integral transformation that decomposes a square integrable, [[piecewise continuous functions ]] on a [[topological group]] into a linear combination of basis elements. It can be thought of as the ultimate extension of the [[Fourier series]], in which the interval of periodicity becomes infinitely long and the "coefficients" infinitely close together, becoming a function instead of an infinite series.

Often, functions which are difficult to analyze in one topological group become much easier to analyze when transformed to another topological group.

The formulas usually favored by mathematicians are the "normalized" form. Given a function f(k) defined on the entire real line, its Fourier transform g(k) is given by:

:<math>g(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-ixk}\, dx</math>

The inverse transform, that recovers the original function, is:

:<math>f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} g(k) e^{ixk}\, dk</math>

The constants in front of the integrals are arbitrary, so long as their product is <math>1/2\pi</math>. In order to make the forward and inverse transforms as similar as possible, an oft-used convention is to set them both equal to <math>1/\sqrt{2\pi}</math>.

==Discrete Fourier transformation==
Discrete Fourier transformations are defined on discrete topological groups, and the integral is replaced by summation.

[[category: mathematics]]

Determinant

2008-05-12T19:38:28Z

Henry8th: Just starting: definition, and some useful properties

The '''determinant''' of a [[matrix]] (written |A|) is a single number that depends on the elements of the matrix A. Determinants only exist for square matrices (i.e., ones where the number of rows equals the number of columns).

==How to calculate==
We need to introduce two notions: the ''minor'' and the ''cofactor'' of a matrix element. Also, the determinant of a 1x1 matrix equals the sole element of that matrix.

;Minor: The minor <math>m_{ij}</math> of the element <math>a_{ij}</math> of an NxN matrix is the determinant of the (N-1)x(N-1) matrix formed by removing the ith row and jth column from M.
;Cofactor: The cofactor <math>C_{ij}</math> equals the minor <math>m_{ij}</math> multiplied by <math>(-1)^{i+j}</math>

The determinant is then defined to be the sum of the products of the elements of any one row or column with their corresponding cofactors.

===2x2 case===
For the 2x2 matrix

<math>\begin{pmatrix} a & b \\ c & d\end{pmatrix}</math>

the determinant is simply ad-bc (for example, using the above rule on the first row).

===3x3 case===
For a general 3x3 matrix

<math>\begin{pmatrix}
A_{11} & A_{12} & A_{13} \\
A_{21} & A_{22} & A_{23} \\
A_{31} & A_{32} & A_{33} \\
\end{pmatrix}</math>

we can expand along the first row to find

<math>
|A|=A_{11}\begin{vmatrix}A_{22} & A_{23} \\ A_{32} & A_{33} \end{vmatrix}-
A_{12}\begin{vmatrix}A_{21} & A_{23} \\ A_{31} & A_{33} \end{vmatrix}+
A_{13}\begin{vmatrix}A_{21} & A_{22} \\ A_{31} & A_{32} \end{vmatrix}
</math>

where each of the 2x2 determinants is given above.

==Properties of determinants==
The following are some useful properties of determinants. Some are useful computational aids for simplifying the algebra needed to calculate a determinant. The first property is that <math>|M|=|M^T|</math> where the superscript "T" denotes transposition. Thus, although the following rules refer to the rows of a matrix they apply equally well to the columns.

* The determinant is unchanged by adding a multiple of one row to any other row.
* If two rows are interchanged the sign of the determinant will change
* If a common factor <math>\alpha</math> is factored out from each element of a single row, the determinant is multiplied by that same factor.
* If all the elements of a single row are zero (or can be made to be zero using the above rules) then the determinant is zero.
* <math>|AB|=|A||B|</math>

In practice, one of the most efficient ways of finding the determinant of a large matrix is to add multiples of rows and/or columns until the matrix is in triangular form such that all the elements above or below the diagonal are zero, for example

<math>\begin{pmatrix}
A_{11} & A_{12} & A_{13} \\
0 & A_{22} & A_{23} \\
0 & 0 & A_{33} \\
\end{pmatrix}</math>.

The determinant of such a matrix is simply the product of the diagonal elements (use the cofactor expansion discussed above and expand down the first column).

[[category:mathematics]]

Curl

2008-05-12T19:07:48Z

Henry8th:

The '''curl''' is a way of expressing a certain type of [[derivative]] of a [[vector field]]. It is defined for fields of 3-dimensional vectors on 3-dimensional space. The curl of a vector field is another vector field.

The curl is written as though it were the [[cross product]] of the special symbol "<math>\nabla</math>" (which is commonly called "del" or "nabla"), with the given vector field, like this: <math>\nabla \times \vec V</math>. This is usually pronounced "curl V" or "del cross V".

In ordinary [[Cartesian coordinates]], the curl is calculated as:

:<math>\nabla \times \vec V = (\ \ \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z},\ \ \ \ \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x},\ \ \ \ \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}\ \ )</math>

or, using a somewhat notation like the [[determinant]] notation for cross product,
:<math>\nabla \times \vec V = \begin{vmatrix}
\hat x & \hat y & \hat z \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
V_x & V_y & V_z
\end{vmatrix}</math>
where <math>\hat x</math>, <math>\hat y</math>, and <math>\hat z</math> are the unit basis vectors.

If one thinks of <math>\nabla</math> as being a fictional vector field with components <math>(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})</math>, one can sort of see that the cross product notation makes sense. This is also useful for remembering how to calculate a curl.

The curl is a true vector field operation—the result is independent of the coordinate system that is used. The proof of that, and its ramifications, are beyond the scope of this page.

The curl operation has an intrinsic "handedness" to it. Any physical phenomenon described by the curl operation (for example, magnetic fields), involves some kind of "right-hand rule".

The curl is an extremely important operation in physics, mathematics, and engineering. It is perhaps most famous for its appearance in [[Maxwell's Equations]].

Intuitively, the curl measures the degree to which the vector field rotates around a given point. If you were to measure the curl of the vector field of wind speed in the vicinity of a meteorological low pressure area, then, keeping in mind that the [[Coriolis force]] makes the wind move in a counterclockwise vortex in the Northern hemisphere, it would be a vector pointing upward. (The ''reason'' for the Coriolis force is not important here, we're just talking about the ''observation'' that the air moves in a counterclockwise vortex.) The way to see that the curl points upward is to visualize a giant right hand with the fingers curled counterclockwise—the thumb would point upward.

Vector fields with a curl of zero are called ''irrotational''.

[[Category:Mathematics]]
[[Category:Physics]]

Bernoulli's principle

2008-05-12T19:07:00Z

Henry8th: wikilink added

'''Bernoulli's Principle''' relates the [[pressure]], [[velocity]], and relative height of a point in the flow of a [[fluid]]. It is one of the most used relationships in aerodynamics.

The integrated form of Bernoulli's equation for a point is:
:<math> P + {1 \over 2} \rho V^2 + \rho g h = constant </math>
where
:P is pressure
:ρ is [[fluid]] [[density]]
:V is the [[speed]] of the flow
:g is the [[acceleration]] due to [[gravity]] and
:h is the height of the flow above the reference, or datum, point.
This equation applies to all points along a streamline of a flow, unless the flow is [[curl|irrotational]], in which case it applies everywhere in the flow.

In aerodynamics, flows are often fast enough where ρgh term (the gravitation potential term) is very small in relation to the other terms and can be neglected. In this case, Bernoulli's equation says that points of higher speed in the flow have lower pressure. This conclusion is fundamental to the principles of flight. When air flows over an airfoil it travels faster over the top of the wing because of its shape. From Bernoulli's equation, the pressure on the top of the wing is lower than the pressure on the bottom because of this difference in speed. This pressure difference is where lift comes from, which is essential for flight.

[[Category:physics]]

Information entropy

2008-05-12T18:44:17Z

Henry8th: Math formatting & couple of small errors

'''Information theory''' is the study of "information" or data, in a statistical manner.

== Overview ==

The well known entity in the theory of information, seemingly founded by [[Claude Shannon]], is the [[information entropy]] defined as

<math> H = -\sum_i p(i) \log_2 p(i) </math>

where the index i runs through all positive integers up to n, for instance (i = 1, 2, …, n). Note that the logarithm is taken to base 2. This entropy may calculated for any statistical frequency function, such that <math>\sum p(i) = 1</math>. The maximum in H (= log(n)) is obtained when all p(i) are equal. Mathematically, H is also equivalent to entropy and disorder in physics.

H may also be interpreted as an [[average information]] based on a certain axiom that when an event occurs with probability p(i), then we get the [[self-information]] - log [p(i) ]. This information is only about probability and has no mental meaning.

Shannon derived the formula from axioms, which in a natural way may be interpreted as information. When we toss a coin we may have two different messages, each with probability 0.5 which may be denoted 0 and 1. If the coin is tossed two times in a sequence we may have the following four possible messages ‘00’, ‘01’, ‘10’ and ‘11’, each with probability 0.25, which may be seen as the information from a pair of tosses.

A reasonable postulate is that the sum of information from any two different messages should equal the total information in the joined messages. And one way to accomplish this is to let the self-information equal the negative logarithm of the probability. Thus, the information in a pair of tosses becomes

<math> -\log(0.5) - \log(0.5) = -\log(0.25) </math> or <math>\log(2) + \log(2) = \log(4) </math>

Another reasonable postulate is that when a message increases in length, then its information increases. For instance, a triple of tosses gives messages with probability = 0.125 and a quadruple with probability = 0.0625. and

<math> -\log(0.0625) > -\log(0.125). </math>

And an event occurring with probability = 1 is a certain event, which will carry no information. Consequently -log(1) = 0.

If all these properties should be maintained for all possible divisions of messages written in different alphabets, then the negative logarithm of the probability is a very good candidate for being a measure of information.

More generally - the word “we” may be followed by “are”, but hardly ever by “am” – some part Y of a message may depend on another part X, making the average information H(Y|X) less than H(XY), i. e. the average information when X and Y are statistically independent. We have

<math> H(Y|X)=H( XY ) - H(X)</math>

and the only functions satisfying this formula are those proportional to the [[entropy]] H, Åslund, 1961. Then, entropy may be seen as [[average information]].

== The continuous case ==

In the continuous case the discrete frequency function is replaced by a continuous probability density function, (p. d. f.) , f(x), and H may be written

<math> H = \int f(x) \log f(x) dx .</math>

A problem here is that H becomes uncertain because of the infinitesimally small dx. A way to evade the problem is to define H in such way that H equals the logarithm of the volume – in analogy to the cardinality n in the discrete case - covered by a uniform p. d. f. over the volume. Thus H may be defined as

<math> H = \int f(x) \log f(x) dx .</math>

The “volume” covered by a Gaussian p. d. f. is proportional to the volume of the ellipsoid of concentration and equal to <math>\sqrt{2\pi e^n |M|} </math>, where M is the moment matrix of the Gaussian.

The [[average information]] may be maximized by a certain genetic algorithm, [[Gaussian adaptation]], keeping the [[mean fitness]] – i. e. the average probability of becoming a parent to new individuals in the progeny – constant (and without any need for any knowledge about average information as a criterion function). This is illustrated by the figure below, showing Gaussian adaptation climbing a mountain crest in a phenotypic landscape.

The lines in the figure are part of a contour line enclosing a region of acceptability in the landscape. At the start the cluster of red points represents a very homogeneous population with small variances in the phenotypes. Evidently, even small environmental changes in the landscape, may cause the process to become extinct.

http://web.telia.com/~u91131915/mountain_crest.GIF

After a sufficiently large number of generations, the increase in [[average information]] may result in the green cluster. Actually, the mean fitness is the same for both red and green cluster (about 65%). The effect of this adaptation is not very salient in a 2-dimensional case, but in a high-dimensional case, the efficiency of the search process may be increased by many orders of magnitude. Besides, a Gaussian distribution has the highest [[average information]] as compared to other distributions having the same second order moment matrix, Middleton, 1960.

== History of applications ==

A comprehensive outline of applications such as telecommunication, channel coding, computer science, physics, neurobiology, electrical engineering and so on may be found elsewhere on the Internet.

Less well known is the use of information theory in simulated evolution by random search. For instance, the average speed of a random walk with Gaussian distributed steps in a hypercube or simplex is proportional to -P log( P ), where P is the average probability that a step will lead to a new point inside the hypercube, Kjellström, 1969. This may be interpreted as the [[self-information]] –log(P) divided by the work or time – proportional to 1/P – needed to get the information on the average.

The function -P log( P ) is positive in the P-interval, 0 <= P <= 1, equals zero at the end points and attains its maximum at P = 1/e = 37%.

A more general theorem of efficiency based on information theory may be found in a paper by Kjellström, 1991.

== See also ==
* [[Information]]

== References ==

* Brooks, D. R. & Wiley, E. O. Evolution as Entropy, Towards a unified theory of Biology. The University of Chicago Press, 1986.
* Kjellström, G. Network Optimization by Random Variation of component values. Ericsson Technics, vol. 25, no. 3, pp. 133-151, 1969.
* Kjellström, G. On the Efficiency of Gaussian Adaptation. Journal of Optimization Theory and Applications, vol. 71, no. 3, Dec. 1991.
* Shannon, C.E. A Mathematical Theory of Communication, Bell Syst. Techn. J., Vol. 27, pp 379-423, (Part I), 1948.
* Åslund, N. The fundamental theorems of information theory (Swedish). Nordisk Matematisk Tidskrift, Band 9, Oslo 1961.

[[Category:Mathematics]]
[[Category:Theories]]

Cross product

2008-05-12T18:21:43Z

Henry8th:

The cross product (or vector product) of two [[vector|vectors]] is itself a vector, and is written <math>a\times b</math>. The magnitude of the resulting vector is

<math>
|a\times b|=|a| |b| \sin\theta
</math>

where <math>\theta</math> is the angle between the two vectors. The direction of the cross product is defined by the right hand rule: with your right hand, point your fingers along the direction of the first vector and curl them towards the second vector. The direction your thumb points gives the direction of the cross product.

===See also===
[[Dot product]]

[[category:mathematics]]

Dot product

2008-05-12T18:20:09Z

Henry8th:

The '''dot product''' is defined for two [[Vector|vectors]] X and Y to be:

: <math>X \cdot Y = |X||Y|\cos\theta</math>

where |x| is the norm and theta is the angle between the vectors.

Unlike the [[cross product]], the dot product is a [[scalar]], not a vector, and has no direction.

It follows from the above definition that the dot product of X and Y is 0 if X is perpendicular to Y.

Alternatively, one can describe the dot product as the length of the geometric projection of X onto Y times the length of Y, when the tails of the two vectors are placed at the same point. It must be remembered though, that the dot product is positive if the angle between the two vectors is less than 90 degrees, negative if the angle is between 90 and 180 degrees.

===See also===
[[Cross product]]
[[category:mathematics]]
[[category:physics]]

User:Henry8th

2008-05-11T23:51:22Z

Henry8th:

Things I like:

* [[Physics]]
* [[Mathematics|Maths]]
* [[Computers]]

Conditional probability

2008-05-11T22:38:56Z

Henry8th: New page: A '''conditional probability''' is the probability that some event A occurs given that a different event B has already occurred. This is written P(A|B) (often read as "the probability ...

A '''conditional probability''' is the [[probability]] that some event A occurs given that a different event B has already occurred. This is written P(A|B) (often read as "the probability of A given B"). The conditional probability satisfies the equation

<math>
P(A|B)=P(AB)/P(B)
</math>

where P(AB) is the probability of both events A and B occurring and P(B) is the probability of event B occurring.

===Example===

A bag has 5 red balls and 3 green balls inside it. You pick out a ball and do not replace it in the bag afterwards.

Q. What is the probability of removing a red ball?

A. 5/8

Q. What is the probability of removing a red ball, ''given that you have already removed a red ball''?

A. There are now 4 red balls and 3 greens, so: 4/7

In the second case, the question requires a conditional probability: P(red|red).

===See also===
[[Bayes'_theorem]]

[[Category:Probability and Statistics]]

Cambridge

2008-05-11T22:18:54Z

Henry8th:

:''For the US city, see [[Cambridge, Massachusetts]].''
'''Cambridge''' is a famous [[England|English]] university city in the county of [[Cambridgeshire]]. It is located north of [[London]], and is surrounded by a number of smaller towns and villages. It is unusual in England in lying slightly below sea level, in the middle of the Fens region. People who attend the university are known as Cantabrigians and, having graduated, can put 'Cantab.' after their name and degree.

Cambridge is best known for the [[University of Cambridge]].
[[Category:Towns in Cambridgeshire]]

Noether's Theorem

2008-05-11T22:07:20Z

Henry8th:

'''Noether's theorem''' is a mathematical theorem that relates [[symmetry|symmetries]] of a [[Hamiltonian]] or [[Lagrangian Dynamics|Lagrangian]] system to conserved quantities.
[[Category:Mathematics]]

Lagrangian Dynamics

2008-05-11T22:02:59Z

Henry8th:

'''Lagrangian dynamics''' are an alternative to [[Newtonian mechanics]]. Lagrangian dynamics are particularly useful in solving for the equations of a motion of a system operating under constraints. Often it is possible, using Lagrangian dynamics, to solve for the equations of motion for a system without needing to solve for the constraint forces, which is why they are often preferred over Newtonian mechanics for these types of problems. It is important to note that both methods will yield the same end result, the only thing that changes is the method of arriving at the end result.

One form of the Lagrangian equation is as follows:

<math>\frac{\partial}{\partial t}({\partial L }/ {\partial \dot q}) - {\partial L} /{\partial q} = Q </math>

Where:
L, the Lagrangian function, is defined as L = T - V, where T is the total kinetic energy of the system and V is the total potential energy of the system, q is the generalized coordinate, and Q is the generalized force. <math>{\partial L }/ {\partial \dot q}</math> is known as the generalized momentum.

===Example===
Consider a mass m attached a spring with spring constant k. The system has a single degree of freedom: x, the displacement of the mass from its equilibrium position. Then,

<math>T=\frac{1}{2}m\dot x^2</math>

<math>V=\frac{1}{2}kx^2</math>

Thus,

<math>
{\partial L }/ {\partial \dot x} = m\dot x
</math>

<math>
{\partial L}/{\partial x} = -kx
</math>

(Note that the generalized momentum is the same as the "normal" Newtonian momentum of mass times velocity in this problem.) <math>Q=0</math> for this simple problem, and so the equation of motion is

<math> m\ddot x = -kx </math>

which is the same as the result one arrives at by just considering the forces acting on the mass in Newtonian mechanics.

[[category:physics]]

Cross product

2008-05-11T21:53:14Z

Henry8th: New page: The cross product (or vector product) of two vectors is itself a vector, and is written <math>a\times b</math>. The magnitude of the resulting vector is <math> |a\times b|=|a| ...

User:Henry8th

2008-05-11T21:41:33Z

Henry8th: New page: Things I like: * Physics * Mathematics * Computers

Things I like:

* [[Physics]]
* [[Mathematics]]
* [[Computers]]

Tensor

2008-05-11T21:39:21Z

Henry8th: New page: Vectors and matrices are examples of more general objects called tensors. Tensors are defined via their transformation properties: suppose we have a set of numbers <m...

[[vector|Vectors]] and [[matrix|matrices]] are examples of more general objects called tensors. Tensors are defined via their transformation properties: suppose we have a set of numbers <math>v_i</math>, and we want to know how their values change under rotation of Cartesian axes. If the values in the new co-ordinate system <math>v'_i</math> can be written

<math>
v'_i=L_{ij}v_j
</math>

where <math>L_{ij}</math> are the elements of a rotation matrix then the <math>v_i</math> are said to be the components of a rank one tensor. Similarly, the components of a rank two tensor satisfy

<math>
a'_{ij}=L_{im}L_{jn}a_{mn}
</math>

and for higher order tensors, we just keep adding more of the <math>L_{ij}</math> rotation matrices. Scalars, vectors and matrices are rank zero, rank one and rank two tensors respectively.

==See also==
[[Suffix notation]]

[[category:mathematics]]

Tensor index notation

2008-05-11T21:33:13Z

Henry8th: New page: Suffix notation is a method of notation which is of use when dealing with tensors. Particular examples of tensors include vectors and matrices, and suffix ...

Suffix notation is a method of notation which is of use when dealing with [[tensor|tensors]]. Particular examples of tensors include [[vector|vectors]] and [[matrix|matrices]], and suffix notation can greatly simplify algebraic manipulations involving these types of mathematical object.

The components of a vector (with respect to some co-ordinate system) might be written <math>\boldsymbol{x}=(x_1,x_2,x_3)</math>. More concisely, we could write <math>x_i</math> for the components of the vector, where <math>i=1,2,3</math>. To motivate this notation, we will consider the equation <math>Ax=b</math> for some matrix <math>A</math> and vectors <math>x,b</math>. We will use the convention that if <math>A</math> is a matrix, then <math>(A)_{ij}=a_{ij}</math> is the element of that matrix in the ith row and jth column. One way of thinking of vector equations is as a shorthand for a set of simultaneous equations - each component of the vectors gives an equation. Explicitly, consider the set of three equations for the three unknowns <math>x_1, x_2, x_3</math>:

<math>
a_{11} x_1+a_{12} x_2+a_{13} x_3 = b_1
</math>

<math>
a_{21}x_1+a_{22}x_2+a_{23}x_3 = b_2
</math>

<math>
a_{31}x_1+a_{32}x_2+a_{33}x_3 = b_3
</math>

This can be rewritten in a matrix/vector form as equation <math>Ax=b</math>:

<math>
\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
=
\begin{pmatrix}
b_1 \\
b_2 \\
b_3
\end{pmatrix}
</math>

Comparison of these two forms should convince you that the "go along the column and down the rows" rule for multiplying a matrix and a vector is sensible. We can also write the above system of equations more succinctly in suffix notation. We notice that in any of the three equations, the first index on the <math>a_{ij}</math> elements is fixed whilst the second varies from 1 to 3. Thus:

<math>
\sum_{j=1}^3 a_{1j}x_j = b_1
</math>

<math>
\sum_{j=1}^3 a_{2j}x_j = b_2
</math>

<math>
\sum_{j=1}^3 a_{3j}x_j = b_3
</math>

Even more succinctly, we can write this as the single expression

<math>
\sum_{j=1}^3 a_{ij}x_j=b_i
</math>

When you see such an equation, remember that it is a shorthand notation for writing three equations at once, for <math>i=1,2,3</math> (in 3D).

[[Category:Mathematics]]

Matrix

2008-05-11T21:09:14Z

Henry8th:

A '''matrix''' (pl.: "matrices," [[Latin]] origin) is a complex ordering, in deliberate fashion, of [[numeral|numerals]]. In [[mathematics]], a "matrix" is a regular grid of numbers, which may be manipulated and solved through intermediate-level [[algebra]]. Matrix algebra is usually taught in [[sophomore]] [[high school]] level mathematics.

More formally, a matrix is an example of a rank-2 [[tensor]].

Alternately, a '''matrix''' may also be a complex ordering of a group of equivalent objects, especially where the order is imposed to gain incidental benefit from the [[synergy]] of the networked objects.

==Mathematics==
In mathematics, matrices can be manipulated in a variety of ways, including [[addition]] and multiplication.
===Addition of matrices===
For example, to add two matrices, one would add their respective elements, thus:

<math>\begin{bmatrix}
x & y & z \\
1 & 3 & 5 \\
0 & 2 & 0
\end{bmatrix} + \begin{bmatrix}
0 & 3 & 1 \\
4 & 3 & {x+2} \\
0 & 4 & v
\end{bmatrix} </math>

would equal

<math>\begin{bmatrix}
{x+0} & {y+3} & {z+1} \\
{1+4} & {3+3} & {5+(x+2)} \\
{0+0} & {2+4} & {0+v}
\end{bmatrix} = \begin{bmatrix}
x & {y+3} & {z+1} \\
5 & 6 & {x+7} \\
0 & 6 & v
\end{bmatrix} </math>

===Multiplication of matrices===
To multiply two matrices, one uses the rule "go along the rows and down the columns". This is best illustrated by a specific example: a matrix times a vector:

<math>\begin{bmatrix}
a & b \\
c & d
\end{bmatrix} \begin{bmatrix}
e \\ f
\end{bmatrix}=\begin{bmatrix}
ae+bf \\
ce+df \end{bmatrix}
</math>

It is important to note that matrix multiplication is not commutative: in general, <math>AB \neq BA</math> for two matrices <math>A</math> and <math>B</math>. This has important consequences in [[quantum mechanics]].

To see why matrix multiplication works the way it does, we will use [[suffix notation]]. Consider first forming the product of two matrices, <math>AB</math>, which is itself a matrix. Then form the product <math>ABx</math>. Matrix multiplication is associative, so we can consider this as either <math>(AB)x</math> or <math>A(Bx)</math>. In suffix notation,

<math>
\sum_j(AB)_{ij}x_j = \sum_k A_{ik}(Bx)_k=\sum_{j,k}A_{ik}B_{kj}x_j
</math>
The vector <math>x</math> is arbitrary, so we can therefore deduce the rule for finding the product of two matrices:

<math>
(AB)_{ij}=\sum_k A_{ik}B_{kj}
</math>

[[Category:Mathematics]]
[[Category:Computers]]

Taylor series

2008-05-11T18:28:05Z

Henry8th: added category

The Taylor series of a function is useful for approximating a mathematical function near to some particular point. For a function <math>f(x)</math>, the Taylor series about the point <math>x_0</math> is

<math>f(x-x_0)=f(x_0)+x_0\frac{df}{dx}+\frac{x_0^2}{2!}\frac{d^2f}{dx^2}+\ldots+\frac{x_0^N}{N!}\frac{d^Nf}{dx^N}</math>

where each of the derivatives is to be evaluated at <math>x=x_0</math>. If <math>N\rightarrow\infty</math> the series is exact, otherwise it is an approximation. Often, Taylor series are performed around <math>x_0=0</math>, in which case they are sometimes also known as a Maclaurin series.

===Examples of common Taylor series===
<math>e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots</math>

<math>\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots</math>

<math>\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots</math>

[[category:mathematics]]

Taylor series

2008-05-11T18:22:01Z

Henry8th: formatting change

Taylor series

2008-05-11T18:20:22Z

Henry8th: New page: The Taylor series of a function is useful for approximating a mathematical function near to some particular point. For a function <math>f(x)</math>, the Taylor series about the point <math...

The Taylor series of a function is useful for approximating a mathematical function near to some particular point. For a function <math>f(x)</math>, the Taylor series about the point <math>x_0</math> is
<math>f(x-x_0)=f(x_0)+x_0\frac{df}{dx}+\frac{x_0^2}{2!}\frac{d^2f}{dx^2}+\ldots+\frac{x_0^N}{N!}\frac{d^Nf}{dx^N}</math>
where each of the derivatives is to be evaluated at <math>x=x_0</math>. If <math>N\rightarrow\infty</math> the series is exact, otherwise it is an approximation. Often, Taylor series are performed around <math>x_0=0</math>, in which case they are sometimes also known as a Maclaurin series.

===Examples of common Taylor series===
<math>e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots</math>

<math>\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots</math>

<math>\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots</math>

Complex number

2008-05-11T18:07:38Z

Henry8th:

A '''complex number''' is a [[number]] composed of two parts - a [[Real number|real]] component and an [[Imaginary number|imaginary]] component, of the form <math>a + bi</math>, where ''a'' and ''b'' are real numbers and <math>i^2 = -1</math>.

Whereas the real numbers can be represented as all the possible points on an infinitely extended [[number line]], to represent all the complex numbers requires the use of a two dimensional coordinate system, usually with the real components on the horizontal axis (the ''abscissa'') and the imaginary components on the vertical axis (the ''ordinate''). This representation is known as the Argand diagram.

The complex numbers form an [[algebraic closure|algebraically closed]] [[field (mathematics)|field]] but do not permit a non-trivial ordering that is preserved under operations. They are the algebraic closure of the [[real numbers]].

Many functions used in real analysis can be extended in to complex numbers using [[Taylor series]]. This is the subject of [[complex analysis]].

===Polar notation===
The complex number <math>a+bi</math> can also be written in the form <math>\rho e^{i\theta}</math>, where
: <math>\rho^2=a^2+b^2</math> is the magnitude squared
: <math>\theta=\arctan\frac{b}{a}</math> is the phase
If a line is drawn on the Argand diagram from the origin to a given complex number, the length of that line will be <math>\rho</math> and the angle it makes to the real (horizontal) axis will be <math>\theta</math>. This leads to a straight-forward geometric interpretation for multiplication by a complex number: multiplying a complex number by <math>e^{i\theta}</math> is equivalent to an anticlockwise rotation through an angle <math>\theta</math> in the Argand diagram.

===In popular culture===
In [[Yvgeny Zamyatin]]'s satirical novel ''[[We (novel)|We]]'', the narrator's psychological distress at contemplating the concept of complex numbers becomes a metaphor for the limitations of totalitarian systems of thought.

[[Category:Mathematics]]
[[category:complex analysis]]

Talk:Mystery:Why is Phonics Taught in the UK but not the US?

2008-05-11T12:14:27Z

Henry8th:

==Illiterate democrat votes==
Can I see the source that states that the Dems rely heavily on illiterate votes?

Even Fox News will tell you that Obama gets most of his support from the well-educated
and younger, upwardly mobile, voters.
[[User:AdenJ|AdenJ]] 01:11, 11 May 2008 (EDT)

== UK ==
"The Labour Party is not struggling and does not fear losing power". Do you have any idea about UK politics? Did you see the recent election results where Labour came in third? Have you read any UK newspapers or watched any UK news reports? All of them would contradict the notion that the Labour Party is not struggling. [[User:Henry8th|Henry8th]] 08:14, 11 May 2008 (EDT)