Conservapedia - User contributions [en]

Talk:Quaternion

2009-01-19T01:39:15Z

Pyfgcr:

This article is full of nonsense. Quaternions and quaternionic integers are different things (analogous to the difference between the real numbers and the integers). Multiplication of quaternions is not commutative as stated, but multiplicative inverses nonetheless exist and _are_ unique. Moreover, the quaternions are not denoted Q (that's the rationals), but rather H, in honor of Hamilton. Allan Quartermain, whom the article claims is the namesake of the quaternions, is apparently a fictional character, and certainly was not Hamilton's mentor and had nothing whatsoever to do with the quaternions. I'll edit the article to reflect these changes, but this article is in need of serious work.

I would recommend at least temporary deletion of this article until a proper version is written. I suspect the original article was a parody to test the editing standards of Conservapedia.

-- Pyfgcr

Talk:Quaternion

2009-01-19T01:31:55Z

Pyfgcr:

Talk:Quaternion

2009-01-19T01:31:42Z

Pyfgcr:

Quaternion

2009-01-19T01:26:15Z

Pyfgcr: This article was full of nonsense and I suspect it was a parody. Allan Quatermain was a fictional character, not Hamilton's mentor. There is no longer anything false in the article.

In [[mathematics]], a '''quaternion''' is a four-dimensional [[object]] important in [[group theory]] and [[geometry]]. As with the complex numbers, the quaternions can be viewed as an extension of the [[real number]] line. Unlike the complex numbers, however, the quaternions are not a [[field]], since multiplication is not commutative. Instead, the quaternions are a '''skew field'''

Quaternions were invented by [[Irish]] [[mathematician]] William Rider Hamilton in the 1840s. Their unusual appearance prompted him to give them the pseudo-[[Latin]]ate name "quaternion integer". Quaternions have proved useful in describing the mechanics of [[rotation]].

==Operations==
The quaternions obey all the usual arithmetic operations. Quaternions may be [[addition|added]], [[subtraction|subtracted]], and [[multiplication|multiplied]]. Addition is are [[associative]] and [[commutative]], while multiplication is only associative. Moreover, addition distributes over multiplication, and so the quarternions are termed a skew field.

[[Category:Mathematics]]

Talk:Quaternion

2009-01-19T01:18:24Z

Pyfgcr: New page: This article is full of nonsense. Quaternions and quaternionic integers are different things (analogous to the difference between the real numbers and the integers). Multiplication of qu...

Richard Stallman

2009-01-19T01:05:16Z

Pyfgcr: Remark on RMS's politics

'''Richard M. Stallman''' ('''RMS''') is the founder of the [[GNU]] project and the [[Free Software Foundation]]. He is the brains behind the [[GNU]] General Public License (GPL). The GNU GPL is used in the [[Linux]] operating system, and in other popular software. Stallman was part of MIT's [[Artificial Intelligence Laboratory]] in the 1970s, where he did a lot of creative computer programming (which he and his friends called "[[hacking]]").

He conceived of a notion of "[[free software]]" where the software is free as in "free speech", as opposed to "free beer", as he explains it. It is formally licensed as an alternative to the [[public domain]] and to other free licenses. He as actively promoted forming a worldwide community of programmers and software users who can all equally share the benefits of new and improved software donated to the community. Stallman is also an [[atheist]], and his website espouses a number of radical liberal political viewpoints, leading some to suspect that he is a [[communist]]. He has also dealt with the communist governments of Cuba and Venezuela, seeking to convince them to utilize open source software.

== External links ==
[http://www.stallman.org Stallman's website]

{{DEFAULTSORT:Stallman, Richard}}

[[Category:Computer Science]]
[[category:Atheists]]

P-adic values

2009-01-19T01:01:14Z

Pyfgcr: Statement of Ostrowski's theorem is incorrect: we have to get an absolute value from the p-adic valuation described -- the valuation itself is not an absolute value.

Given a [[prime number]] p the '''p-adic value''' is the function, denoted <math>v_p</math> which takes as its argument a natural number n and returns the power of p appearing in the [[prime factorization]] of that number (equivalently, the highest power of p which divides n):
<math>v_p(x)=\max\{n:p^n\mid x\}</math>. For example, the p-adic values of 60 for p=2,3,5,7,11,13... are 2,1,1,0,0,0,.... One can associate with the p-adic valuation an absolute value <math>|n|_p=p^{-v_P(n)}</math>.

By convention, <math>v_p(0)=\infty</math> for all primes p.

Here are some important properties of p-adic values:

* p-adic values convert multiplication into addition (akin to the logarithm function): <math>v_p(xy) = v_p(x) + v_p(y)</math>.
* p-adic values satisfy the [[Archimedes|archimedean]] inequality: <math>v_p(x+y) \le \min\{v_p(x),v_p(y)\}</math>.
* Equality holds in the above so long as <math>v_p(x)\ne v_p(y)</math>.
* The [[Fundamental Theorem of Arithmetic|fundamental theorem of arithmetic]] can be restated compactly using p-adic values: For all natural numbers n, <math>n=\prod_pp^{v_p(n)}</math> where p ranges over all primes.
* p-adic values can be extended to the rational numbers by defining <math>v_p(x/y)=v_p(x)-v_p(y)</math> for all integers x,y.
* Ostrowski's theorem states that the only absolute values on the field of rational numbers are the real [[absolute value]] (which some mathematicians view as the "prime at infinity") and the p-adic absolute values described above.

p-adic values are used most commonly in [[number theory]] and [[algebra]], especially in the theory of [[commutative]] [[ring]]s.

Completing the field of rational numbers with respect to p-adic values yiels the field of [[p-adic numbers]].

[[Category:Mathematics]]
[[Category:Algebra]]

Talk:Field extension

2009-01-19T00:54:04Z

Pyfgcr: New page: This definition's a bit awkward. Really the extension field has not only the same identity element, but also the same addition and multiplication on when restricted to the smaller field. ...

This definition's a bit awkward. Really the extension field has not only the same identity element, but also the same addition and multiplication on when restricted to the smaller field. This is required but stronger than the second sentence here. Has anyone thought about things like degree of an algebraic extension here and maybe adding some articles about Galois theory?

Cohomology

2009-01-19T00:47:42Z

Pyfgcr: It's called "elliptic cohomology", not "elliptic curve cohomology". There is a connected to elliptic curves, so leaving the link.

In [[category theory]], the '''cohomology''' of an [[exact sequence]] studies the relationship between the [[cokernel]]s and [[coimage]]s, when it is possible to take their [[quotient]]. Cohomology is the category-theoretic [[dual]] of [[homology]].

Well-known examples include [[de Rham cohomology]] in [[differential geometry]], [[elliptic curve|elliptic]] cohomology, and [[Hyperbola|hyperbolic]] cohomology of infinite [[Abelian]] [[Group (mathematics)|groups]].

[[Category:Algebra]]

Talk:Bilinear function

2009-01-19T00:09:28Z

Pyfgcr: New page: It seems like more explanation is needed here. In what two variables is a Mobius transformation linear? We can regard it as a bilinear function in the projective coordinates on CP^1, but...

It seems like more explanation is needed here. In what two variables is a Mobius transformation linear? We can regard it as a bilinear function in the projective coordinates on CP^1, but does the author have something else in mind?

-- Pyfgcr

Entire function

2009-01-18T23:12:20Z

Pyfgcr: Added brief discussion of Liouville's theorem.

In [[complex analysis]], an '''entire function''' is a [[function]] that is [[analytic]] on the whole [[complex plane]].
[[category:Complex analysis]].

The main result governing the behavior of entire functions is Liouville's theorem, which states that a bounded entire function is constant. Here an entire function <math>f</math> is said to be bounded if there exists a constant <math>M</math> such that for all <math>z \in \mathbb C</math> the bound <math>f(z)<M</math> holds. Liouville's theorem yields a simple proof of the fundamental theorem of algabra: if <math>p(z)</math> were a polynomial with no roots in the complex plane, then one can prove that <math>1/p(z)</math> would be a bounded entire function, and thus constant.

Imaginary number

2009-01-18T23:05:51Z

Pyfgcr: Made more precise.

An '''imaginary number''' in mathematics is any number that is a multiple the imaginary unit, defined as, <math>i^{2} = -1</math>

An imaginary number is of the form, <math>k i</math>, where ''k'' is a [[real number]].

For example <math>\sqrt{-1}</math> has imaginary representation of

:<math>\sqrt{-1}= \pm i</math>.

When a imaginary number is added to real number, they form a [[complex number]]. Imaginary numbers are mathematically useful because they fabricate solutions to every polynomial [[equation]] - for example, the equation <math>x^2+1=0</math> has no real solution.

The analysis of imaginary numbers forms the basis for the field of [[mathematics]] known as
[[complex analysis]].

[[category:mathematics]]
[[category:complex analysis]]

Real number

2009-01-18T19:35:05Z

Pyfgcr: Somebody got a digit of pi wrong... tsk tsk.

'''Real numbers''' include within them all of these other kinds of [[number]]s:

*The positive [[integer]]s, 1, 2, 3, ...
*[[Zero]] and the negative integers
*[[Fraction]]s, like 355/113
*Any decimal representation which terminates (comes to an end), like 6.023, because this is just a way of writing a fraction (in this case, 6023/1000)
*Any decimal representation which repeats or recurs, like 1.86292929292929..., because these can be shown to be fractions
*[[Irrational numbers]], like <math>\sqrt{10} = 3.162277660168...</math>π = 3.1415926535..., whose decimal representations never repeat or terminate.

==Formal definition==

Formally, real numbers are defined as the unique [[Field (mathematics)|field]] which is [[ordered]], [[Complete (mathematics)|complete]], and [[Archimedean]]. The reals can be constructed from the [[rationals]] by means of [[Dedekind cut]]s or [[Cauchy sequence]]s, i.e. it is the completion of the [[metric space]] of rational numbers.

==Infinity==

The real numbers ''do not'' include <math>\infty</math> or <math>-\infty</math> (infinity and minus infinity). However, there are non-standard models of real numbers which include <math>\infty</math> or include both <math>\infty</math> and <math>-\infty</math>.

There is no largest real number, because you can always make a real number larger by adding 1 (or 137.035 or 6.023·1023) to it, and no smallest real number, because you can always make a real number smaller by subtracting from it.

Every real number is finite. One way to see this is to observe that you cannot subtract infinity from itself—the result is indeterminate—but, for any real number '''''x,''''' then '''''x - x = 0''''', exactly.

It is sometimes convenient to have a set of numbers that ''does'' include infinity. For example, in computer programming, "real arithmetic" is often done by a specific system defined by standard IEEE 754-1985; this system is built in to modern processor chips. It provides for values which print out as INF and -INF and which participate in arithmetic as if they were numbers. Thus, division by zero, which was often an error that stopped calculation on older machines, can be a legal operation which simply produces a +INF or -INF result. The system of numbers implemented in IEEE 754 is known in mathematics as the "affinely extended real numbers."

==Real line==

The real numbers can be thought of as a [[line]], called the '''real line'''. Each real number represents a point on the real line. However, it is a mistake to think of the real line as a row of individual points, like beads. There is no real number “just to the right” of a given real number. This is because the real numbers, like the rational numbers, are a [[dense set]], so points accumulate around each other.<ref>http://abstractmath.org/MM/MMRealNumbers.htm</ref>

The real line is useful as a [[coordinate system]] for [[graph]]ing [[functions]]. Thus, the [[x-axis]] and [[y-axis]] are both instances of the real line. The real line is the basis for geometric [[measurement]]s, and more generally for ideas in [[metric topology]].

==Notes and references==
{{reflist}}

[[Category:Mathematics]]

Topological group

2009-01-18T18:30:15Z

Pyfgcr: Definition corrected.

A topological group is a [[topological space]] that is also a [[Group (mathematics)|group]] such that the [[binary operation]] defined on the group is a [[continuous function|continuous map]], as is the operation of inversion.

Every group G can be turned into a topological group by giving G the [[discrete topology]]. Every topological group is a [[completely regular space]].

[[category:Topology]]
[[Category:Algebra]]

Bob Jones University

2009-01-18T18:22:22Z

Pyfgcr: "Its" now also noted for the proper use of possessives on "it's" conservapedia page

{{University
|name=Bob Jones University
|image=
|text=rgb(211,222,250)
|background=rgb(55,113,153)
|type=Private
|city= Greenville, South Carolina
|sports=
|colors=dark blue, light blue, whtie
|mascot=
|website=http://www.bju.edu/
}}
'''Bob Jones University''' is a [[conservative]], [[Christian]] liberal arts college in Greenville, [[South Carolina]]. It welcomes [[homeschoolers]]: 29% of the 2007-2008 student body was homeschooled.{{fact}}

Its Mission Statement is: Within the cultural and academic soil of liberal arts education, Bob Jones University exists to grow Christlike character that is Scripturally disciplined; others-serving; God-loving; Christ-proclaiming; and focused Above.

The Rev. Dr. [[Ian Paisley]], MP, leader of the [[Democratic Unionist Party]] and First Minister of the [[Northern Ireland]] Executive, holds an honorary doctorate from Bob Jones University. [[Billy Graham]] also attended Bob Jones University, although he ended up graduating from Wheaton College in Illinois. [[John D. Ashcroft]] accepted an honorary degree from BJU in May 1999. Republican senators [[Jesse Helms]] (N.C.) and [[Strom Thurmond]] (S.C.), along with Republican Representatives [[Lindsey Graham]] (S.C.) and [[Asa Hutchinson]] (Ark.), also received honorary degrees. <ref name="wash">Washington Post: Bob Jones: A Magnet School for Controversy, University's Policies Haunt GOP Hopefuls, By Juliet Eilperin and Hanna Rosin, Friday, February 25, 2000; Page A06 [http://www.washingtonpost.com/wp-srv/WPcap/2000-02/25/045r-022500-idx.html]</ref>

[[Asa Hutchinson]] and his brother, [[Tim Hutchinson]], graduated from BJU in the early '70s. <ref name="wash" />

For years, Bob Jones University was a routine campaign stop for conservative Republicans visiting South Carolina. [[Ronald Reagan]], [[Dan Quayle]], [[Pat Buchanan]] and [[Robert J. Dole]] campaigned at BJU. <ref name="wash" /> Presidential candidate [[Alan Keyes]] and Democratic South Carolina Gov. Jim Hodges also campaigned there. <ref>Salon.com: Jonesing for votes, George W. Bush's speech at a college that bans interracial dating raises questions about his compassion, by Jake Tapper [http://archive.salon.com/politics2000/feature/2000/02/03/bob_jones/index.html]</ref>

==Accreditation==

Bob Jones, Sr., the founder of BJU, was fearful of obtaining academic accreditation for the school. Over the years, as pressure mounted on the institution to give its students the benefits of accredited degrees, the university moved towards membership in the Transnational Association of Christian Colleges and Schools. Accreditation was not obtained until 2005.<ref>bju.edu: Bob Jones University Granted National Accreditation, Greenville, S.C., November 8, 2006[http://www.bju.edu/accreditation.html]</ref>

==College Life==

Bob Jones University is well known for its stringent rules it applies to its students, and many of its rules are based on a strict interpretation of the [[Bible]]. Pornography, homosexuality and sexual activity outside of marriage are all strictly banned. Students are not allowed to have posters in their dorms or watch films or listen to music (including Christian-influenced music). Strict dress codes also apply. Conservative and modest dress styles are required and tattoos, dyed hair and body piercings (with the exception of ear piercings for female students) are all banned.

==Political Controversy==

In the primary race in 2000, [[George W. Bush]] campaigned at BJU, where he stated "I look forward to publicly defending our conservative philosophy." <ref>At Bob Jones U., A Disturbing Lesson About The Real George W., by Derrick Jackson[http://www.commondreams.org/views/020900-101.htm]</ref> His Republican rival, [[John McCain]], used the visit against Bush in a telephone calling campaign that turned out [[Catholic]] voters for the Senator. The McCain ad stated "Governor George Bush has campaigned against Sen. John McCain by seeking the support of southern fundamentalists who express anti-Catholic views." McCain said the calls didn't accuse Bush of bigotry, only the people Bush was turning to for support. <ref>CBS News: Bush Regrets Bob Jones U. Calls Visit There "A Missed Opportunity" NEW YORK, Feb. 27, 2000[http://www.cbsnews.com/stories/2000/02/27/politics/main165529.shtml]</ref>

Bush reacted to the political distortions by criticizing the attempts at guilt by association. He wrote a letter affirming that he is not a religious bigot and sent it to Cardinal John O'Connor of New York.<ref>The letter contained this sentence, which is often quoted out of context to imply that Bush was criticizing evangelicals, when he was not: "On reflection, I should have been more clear in disassociating myself from anti-Catholic sentiments and racial prejudice. It was a missed opportunity, causing needless offense, which I deeply regret." CNN: George W. Bush's Visit to Bob Jones University Continues to Stir up Controversy, aired February 27, 2000 - 8:09 p.m. ET [http://edition.cnn.com/TRANSCRIPTS/0002/27/wv.12.html]</ref>

Such spats are common in primaries, and McCain subsequently endorsed Bush for president in 2000 and Bush has endorsed McCain for president in 2008.{{fact}}

== Racial Controversy ==
Bob Jones University did not enroll black students until 1971. Afterwards it had a policy against interracial dating and marriage, for which the Internal Revenue Service revoked the the university's 501(c)(3) tax exempt status in 1976. The University challenged this action, but lost when the Supreme Court upheld the ruling in 1983. <ref>This ruling was upheld, over the objection of the U.S. Solicitor General, in 1983 by the ruling of the Supreme Court of the United States in Bob Jones University v. United States. BOB JONES UNIVERSITY v. UNITED STATES, 461 U.S. 574 (1983) [http://caselaw.lp.findlaw.com/cgi-bin/getcase.pl?court=US&vol=461&invol=574].</ref>

The university announced in 2000 that its policy against interracial dating was no longer being enforced and was officially discarded. <ref>CNN: Bob Jones University ends ban on interracial dating, March 4, 2000 [http://archives.cnn.com/2000/US/03/04/bob.jones/]</ref>

In late 2008, the university issued a statement <ref>BJU: ''Statement about Race at Bob Jones University'' [http://www.bju.edu/about/race.html]</ref>, apologizing for its earlier policies.

== References ==
<references/>

== External Links ==

*[http://www.bju.edu Official Website]
*[http://www.bju.edu/prospective/expect/rhall.html Residence Hall Life]
*[http://www.bju.edu/accreditation.html Accreditation Press Release]

{{Nb_US_universities}}

GNU/Linux

2009-01-18T18:08:07Z

Pyfgcr: Balmer -> Ballmer

[[Image:Tuxie.jpg|thumb|right|"Tux", the Linux mascot|90px]]
The '''GNU/Linux''' operating system is [[free software]] created to replace the [[Unix]] operating system.
The GNU project was started by ace programmer [[Richard Stallman]] and was the impetus behind the creation of the [[Free Software Foundation]]. Except for the small but essential [[kernel]] (called Linux after its creator [[Linus Torvalds]]), most of the operation system code was created by Stallman and other FSF contributors. There's a dispute over whether Stallman or Torvalds deserves more credit. The trade press refer to the overall system as "Linux", which annoys Stallman no end.

The Linux [[kernel]] that communicates with the hardware, supporting many other software components (such as the graphical user interface, file system and media players), but those who are unaware of or not concerned about the history of the project commonly used "Linux" to refer to the whole operating system. The Linux kernel was developed by Finnish grad student [[Linus Torvalds]].

Linux distinguishes itself from other operating systems such as UNIX and [[Microsoft Windows]] in that the [[source code]] is distributed freely under various [[open source]] licenses. In essence, this means that anybody can modify the code to their needs, and that the development of Linux happens in an open community, rather than in a closed commercial environment. Any improvements to the code will be contributed to the community, and any software that is based upon viral licenses will be, in turn, licenses under these. The Linux kernel itself is licensed under the GNU [[General Public License]] (GPL).

Linux is also different to the closed source operating system vendors in that the software is distributed by many different companies. Major Linux distributions include [http://www.redhat.com/ Red Hat], [http://www.novell.com/linux/ SUSE], [http://www.debian.org/ Debian] and [http://www.ubuntu.com/ Ubuntu]. There are literally hundreds of Linux distributions as can be seen on [http://distrowatch.com DistroWatch.com].

Numerous sources<ref>[http://blogs.zdnet.com/open-source/index.php?p=210 Intellectual Property - Left?]</ref> <ref>[http://www.flickr.com/photos/daviderickson/718933691/]</ref>, including Steve Ballmer<ref>[http://www.theregister.co.uk/2000/07/31/ms_ballmer_linux_is_communism/ MS' Ballmer: Linux is communism]</ref>, one of the driving minds behind the success of Microsoft, have claimed that the Open Source movement is inherently Communist. Both Open Source and Communism shun the idea of personal property, instead favoring a communal ownership where no single entity has control or authority.

In 2005 [http://www.forbes.com/ Forbes.com] posted [http://www.forbes.com/2005/03/15/cz_dl_0315linux_print.html an article] estimating Linux ran 60% of the world's top supercomputers at that time. In 2003 the [[IBM]] Linux Technology Center [http://www.ibm.com/developerworks/library/l-rel/?ca=dgr-lnxw01LTP concluded] that Linux has enterprise class reliability. Linux servers can run without reboot for years as can usually be seen at the [http://uptime.netcraft.com/up/today/top.avg.html Longest uptimes] URL on [http://www.netcraft.com/ Netcraft.com]. Another location to check on Linux uptime statistics is the [http://counter.li.org/reports/uptimestats.php Machine uptimes] page at [http://counter.li.org/ Linux Counter].

Owing to the nature of open source software, many variants of a Linux distribution may be created by using the original code and making changes to it to suit a particular need. For example, there is also a [[Ubuntu]] [[Christian]] Edition.<ref>[http://www.whatwouldjesusdownload.com/christianubuntu/2006/07/about-ubuntu-christian-edition.html Ubuntu Christian Edition]</ref>

[[Image:Gnomesword large.png|right|thumb|250px|Screenshot of Ubuntu Christian Edition.]]

It is likely that Linux is becoming one of the most commonly adopted operating systems in the world. However, this is difficult to quantify with hard evidence since most Linux distributions are given away for "free" and there are few sales records or marketing numbers to review. While personal computers in the United States and other "first world nations" still overwhelmingly use [[Microsoft]] operating systems such as Windows XP, Linux is a common choice for web servers, file servers and embedded platforms, thanks to its perceived reliability, low/no cost, and the fact that modifications to the source code can readily be made by anyone.

As an example, in March of 2007 the server hosting the the Conservapedia web site was running the Linux operating system<ref>[http://toolbar.netcraft.com/site_report?url=http://www.conservapedia.com NetCraft site report for Conservapedia.com]</ref>

==Linux Pre-Installed==
From its inception, and with exception of the server market, it was difficult to find new computers available with Linux pre-installed. Users typically have to download the Linux distribution of their choice and install it on a computer themselves, slowing the increase in computers using a desktop version of Linux, as many home computer users find installing an operating system a difficult task. This situation drastically changed in 2007 when [[Dell]] started selling laptop and desktop computers to the general public with Linux pre-installed<ref>[http://www.dell.com/content/topics/global.aspx/alliances/en/linux?c=us&cs=555&l=en&s=biz Dell and Linux]</ref>.

During 2008, a new type of low cost laptop computer, the "netbook" was introduced by most of the major manufacturers. To keep costs down, Linux was offered on most of the lines as an alternative to Windows XP (Windows Vista being unable to run on the low powered computers), bringing Linux into the mainstream for the first time.

== References ==
<references />

==External links==
*[http://www.kernel.org/ The Linux Kernel Archives]
*[http://www.linux-foundation.org/en/Main_Page The Linux Foundation]

[[Category:Information technology]]

Residue calculus

2009-01-18T17:54:07Z

Pyfgcr: Purged nonsense

The '''residue calculus''' is a method of definite [[integration]] which relies heavily on [[Cauchy's residue theorem]]. The idea is to rewrite a definite integral on the [[real]] line as limit of integrals in the [[complex plane]] which are, in some sense, easier to compute.

==Example==
Here is a simple example. Suppose we want to find the [[integral]] of <math>\log(x)/(1+x^2)</math> from 0 to <math>\infty</math>. We begin by choosing an [[analytic]] branch of the [[logarithm]], defined everywhere in <math>\mathbb{C}</math> except for the line consisting of negative purely imaginary numbers (that is, <math>\log(z)=\log|z|+i\arg(z)</math> where <math>\arg(z)</math> is specified to take values in <math>(-\pi/2,3\pi/2)</math>. Now, choose real numbers <math>r,R</math> where <math>0<r<1</math> and <math>R>1</math>. Then, let <math>\Gamma(r,R)</math> be the positively oriented [[contour]] consisting of the clockwise upper semicircular [[arc]] from <math>-r</math> to <math>r</math>, the directed [[line segment]] from <math>r</math> to <math>R</math>, the anticlockwise semicircular arc from <math>R</math> to <math>-R</math>, and the directed line segment from <math>-R</math> to <math>-r</math> (this looks like a rainbow in the complex plane).

The reason we chose this contour is that it necessarily avoids the inevitable singularity of the logarithm at 0, and it includes inside of it, the pole of the function <math>f(z)=log(z)/(1+z^2)</math>. By Cauchy's residue theorem, we have <math>I:=\int_{\Gamma(r,R)}f(z)\,dz</math>= <math>2\pi i</math> times the residue of the [[pole]] at <math>z=i</math>. This is easily calculated, and it is equal to <math>\pi/4</math>. Thus, the integral evaluates to <math>\pi^2 i/2</math>.

The idea is to now take <math>r\rightarrow 0^+</math> and <math>R\rightarrow\infty</math>. Some easy calculations (left to the reader) show that the integrals along the semicircular arcs vanish, leaving us with only the integrals along the [[real axes]]. For <math>(0,\infty)</math> we get back the integral we are trying to calculate <math>I</math>, and on <math>(-\infty,0)</math> we get (since arg here is equal to <math>\pi i</math>): <math>I+\pi i\int_{-\infty}^0 dz/(1+z^2)=I+\pi^2 i/2</math>. Thus, plugging everything back in, <math>2I+\pi^2 i/2=\pi^2 i/2</math>, so <math>I=0</math>.

[[Category:Calculus]]
[[Category:integration]]

Fractal

2009-01-18T17:50:53Z

Pyfgcr: Added a remark on biology of fractals

[[Image:SsFb.gif|A computer generated image of a leaf.|right|thumb]]
A '''fractal''' is a [[design]] which contains in detail a shape which looks like the whole. This can be repeated ''ad infinitum'', meaning that fractals do not cease in their [[similarity]].

The fractal nature of some plants has convinced many people that God must have designed the natural world (see [[fern]]s). The leaves of a fern are shaped very much like the frond. Evolutionary biologists have yet to propose a satisfactory explanation for the spontaneous appearance of complex patterns in living organisms.

*Fractals have been studied by mathematicians for over a century. The name was coined by [[Benoit Mandelbrot]], a mathematician at [[Yale University]]. He calls fractals the "geometry of nature." [http://www.wsu.edu/DrUniverse/fractal.html]

Damien Jones wrote:
*A fractal is a shape that, when you look at a small part of it, has a similar (but not necessarily identical) appearance to the full shape. Take, for example, a rocky mountain. From a distance, you can see how rocky it is; up close, the surface is very similar. Little rocks have a similar bumpy surface to big rocks and to the overall mountain. [http://www.fractalus.com/info/layman.htm]

==External links==
*[http://www.wsu.edu/DrUniverse/fractal.html Ask Dr. Universe] - Washington State University
*[http://www.creativitysoftware.com/fractals-images/pages-hs/fern-fronds.htm Fractal fronds]
*[http://www.fractalus.com/info/layman.htm Fractals, in Layman's Terms]
[[category:mathematics]]

Fractal

2009-01-18T17:49:03Z

Pyfgcr: "ad infinitum" was misspelled

[[Image:SsFb.gif|A computer generated image of a leaf.|right|thumb]]
A '''fractal''' is a [[design]] which contains in detail a shape which looks like the whole. This can be repeated ''ad infinitum'', meaning that fractals do not cease in their [[similarity]].

The fractal nature of some plants has convinced many people that God must have designed the natural world (see [[fern]]s). The leaves of a fern are shaped very much like the frond.

*Fractals have been studied by mathematicians for over a century. The name was coined by [[Benoit Mandelbrot]], a mathematician at [[Yale University]]. He calls fractals the "geometry of nature." [http://www.wsu.edu/DrUniverse/fractal.html]

Damien Jones wrote:
*A fractal is a shape that, when you look at a small part of it, has a similar (but not necessarily identical) appearance to the full shape. Take, for example, a rocky mountain. From a distance, you can see how rocky it is; up close, the surface is very similar. Little rocks have a similar bumpy surface to big rocks and to the overall mountain. [http://www.fractalus.com/info/layman.htm]

==External links==
*[http://www.wsu.edu/DrUniverse/fractal.html Ask Dr. Universe] - Washington State University
*[http://www.creativitysoftware.com/fractals-images/pages-hs/fern-fronds.htm Fractal fronds]
*[http://www.fractalus.com/info/layman.htm Fractals, in Layman's Terms]
[[category:mathematics]]

Talk:Space

2009-01-18T17:47:48Z

Pyfgcr:

Did a 12-year-old write this article?

The second line is a quote from Hitchhiker's Guide to the Galaxy. [[User:JoshuaZ|JoshuaZ]] 15:46, 30 June 2007 (EDT)

==Recent==

Please be careful about presenting modern theories as fact. Also, your understanding of the Big Bang theory has some errors. Current thought is that the universe will expand forever, not go through a contraction. The contraction belief was popular before it was discovered, barring some future discoveries, that there simply isn't enough matter to cause a contraction. [[User:Learn together|Learn together]] 00:33, 12 July 2007 (EDT)

== Wrong name ==

This article should be renamed "Outer Space". '''---[[user:DLerner]]---''' 07:26, 19 May 2008 (EDT)

Is there any reason not to delete this article? It has no meaningful content.

Space

2009-01-18T17:47:12Z

Pyfgcr: "It's" is not a possessive.

For the [[astronomy]] term, see [[Outer space]]

'''Space''' is, in its simplest terms, the [[geometry]] of three [[dimension]]s.

[[Category:Geometry]]

Mark Foley

2009-01-18T16:20:01Z

Pyfgcr:

[[Image:1 63 foley mark media.jpg|thumb|150px|right|Mark Foley]]
'''Mark Foley''' (born on September 8, 1954) is a liberal [[Republican]] who served as a member of the [[House of Representatives]] from 1995 to 2006. Before serving in the House, Foley was a member of the Lake Worth City Council, Florida House of Representatives, and Florida State Senate. As a Congressman, Foley served on the [[House Ways and Means Committee]] as well as chairing the House Caucus on Missing and Exploited Children. He also played a role in the recount of the [[Florida]] ballots in the 2000 presidential election, challenging that the media's work would only "undermine the legitimacy of the presidency." <ref>http://archive.democrats.com/view.cfm?id=838</ref>

Foley resigned from the House of Representatives on September 26, 2006 after allegations that he had sent sexually suggestive emails and IM conversations to underaged male Senate Pages. As of April 2007 he remains "under investigation" regarding his communications with the pages, according to Florida and [[FBI]] officials, but no charges filed.<ref>http://www.newsobserver.com/114/story/558510.html</ref> After Foley's resignation, his lawyer said that Foley had been abused by a [[priest]] as a boy, and that his client was entering a rehabilitation center for [[alcoholism]]. <ref>http://www.harpers.org/sb-republicans-1160492797.html</ref>

==References==
<References/>

{{DEFAULTSORT:Foley, Mark}}

[[Category:Former United States Representatives]]
[[Category:Republican Party]]

Complex residue

2009-01-18T16:07:38Z

Pyfgcr:

The '''complex residue''' or simply '''residue''' of a meromorphic function ''f''(''z'') is the coefficient of the <math>1 \over z</math> term in ''f''(''z'')'s [[Laurent series]] expansion.
[[category:complex analysis]]

Algebraic topology

2009-01-18T15:39:47Z

Pyfgcr:

'''Algebraic topology''' is a branch of mathematics that uses [[abstract algebra]] to understand [[topological space]]s.
 
One of the most useful tools in algebraic topology is the [[fundamental group]], <math>\pi_{1}(X)</math> of a topological space <math>X</math>. The definition is as follows: Let <math>p</math> be a fixed point in <math>X</math>. Consider the space of all [[curve]]s <math>\gamma:[0,1]\rightarrow X</math> which begin and end at <math>p</math>. We consider two such curves to be ([[homotopy|homotopically]]) equivalent if we can [[continuous]]ly deform the first curve into the second. More precisely, we consider <math>\gamma_1</math> to be equivalent to <math>\gamma_2</math>
if there exists a [[mapping]] <math>H:[0,1]\times[0,1]\rightarrow X</math> with <math>H(t,0) = \gamma_1(t)</math> and <math>H(t,1) = \gamma_2(t)</math>. We can define an group structure on the resulting [[equivalence class]] of curves by declaring <math>\gamma_1\cdot\gamma_2</math> to be the curve <math>\gamma_2</math> followed by <math>\gamma_1</math>, and rescaled so that the resulting curve still has domain <math>[0,1]</math>. This group is called the fundamental group of <math>X</math>
 
Example: The most important example is the topological space <math>S^1</math>. It can be shown that the fundamental group of <math>S^1</math> is isomorphic to the additive group of [[integer]]s <math>\mathbf{Z}</math>.

In addition to aiding in the classification of topological spaces, methods in algebraic topology considerably simplify some proofs in [[abstract algebra]]. For example, the proof that there is an injection from the free group on three generators to the free group on two generators is most easily expressed in terms of the topological concept of covering spaces.

[[Category:Topology]]
[[Category:Algebra]]

Homotopy group

2009-01-18T15:08:52Z

Pyfgcr: General expansion

'''Homotopy groups''' are tools used in [[algebraic topology]] to classify [[topological space]]s. The different ways to map an '''n-[[sphere]]''' continuously into a given topological space are divided into [[equivalence class]]es, called '''homotopy classes'''. The set of homotopy classes of maps of the n-sphere into a space may be endowed with a group structure by a means analogous to the concatenation operation used to construct the [[fundamental group]]; this group is usually denoted <math>\pi_n</math>. However, as long as <math>n \geq 2</math>, the homotopy groups <math>\pi_n(X)</math> are [[Abelian group|abelian groups]].

Homotopy groups are notoriously difficult to compute, in contrast with homology and cohomology groups, where are generally computable: even the higher homotopy groups of spheres are not fully understood. For example, the group <math>\pi_3(S^2)</math> is isomorphic to the group of integers, generated by the Hopf fibration. Spectral sequences are an important tool in the computation of higher homotopy groups.

[[category: Topology]]

{{stub2}}

One-point compactification

2009-01-18T14:56:11Z

Pyfgcr:

If Y is a compact, [[Hausdorff space]]; X is a topological space such that Y-X contains one element and the closure of X equals Y; then Y is the one-point compactification of X. The One-point compactification is the minimal compactification one can perform on X. A space X has a one-point compactification if and only if it is itself locally compact and Hausdorff.

[[Category:Topology]]

Trivial topology

2009-01-18T14:54:30Z

Pyfgcr:

The '''trivial topology''' (sometimes called the '''indiscrete topology''') on a set X only makes the [[empty set]] and the entire set X an open subset of X. This is the topology on X with the smallest [[cardinality]].

[[category: Topology]]

Continuous function

2009-01-18T03:29:26Z

Pyfgcr: Topological definition wasn't quite right... fixed.

{{merge|calculus}}
In [[calculus]], a [[function]] ''f(x)'' is said to be '''continuous''' at point ''c'' if ''f(c)'' equals the limit of ''f(x)'' as x approaches c from both the positive and negative directions.

Another way of understanding this is by recognizing that a discontinuous function over a specific interval is one that has a gap in the interval, or one having different limits at a particular point depending on whether it is approached from the positive or negative directions.

A simple example of a continuous function would be Y = 2X + 5.

An example of a discontinuous function is Y = 1/X, which has no value for X = 0; also the limits of the function as X approaches zero from each side are different.

A [[differentiable function]] is always continuous, but a continuous function is not always differentiable.

A function f: X -> Y mapping elements in a [[topological space]] X to a topological space Y is continuous if for every [[open set]] U in Y, the inverse image of U under f is an open subset of X.

A continuous function maps a convergent [[sequence]], [[net]], or [[filter]] to a convergent sequence, net, or filter, respectively.

A continuous function maps a [[compact space]] to a [[compact space]].

[[category: mathematics]]
[[category: Topology]]

Algebraic topology

2009-01-18T03:26:38Z

Pyfgcr: It makes more sense to say the fundamental group is isomorphic to Z as a group. It's awkward to say a group is isomorphic to a ring; really, it's isomorphic to the additive group of that ring.