Conservapedia - User contributions [en]

Neoconservatism

2007-03-15T21:52:58Z

Pixologic:

A '''neoconservative''' (colloquially, ''neocon'') is a former liberal turned conservative. Neoconservatives believe that democracy can and should be installed around the world, even in Muslim countries such as Iraq, [[Iran]], and [[Saudi Arabia]]. Neoconservatives were responsible for persuading President [[George W. Bush]] to invade Iraq, and they have lobbied for military action against [[Iran]] also.

==Origin==
The root of neoconservatism lie in a group of largely Jewish intellectuals who attended City College of New York in the mid- to late 1930's and early 1940's, a group that included [[Irving Kristol]], [[Daniel Bell]], [[Irving Howe]], [[Nathan Glazer]] and [[Daniel Patrick Moynihan]].

== Publications ==

The leading publications of neoconservatives are Commentary and The [[Weekly Standard]]. Many Washington, D.C. think tanks, such as the American Enterprise Institute (AEI) and the Project for the New American Century (PNAC), are now dominated by neoconservatives.

== Social Issues ==

Neoconservatives often take [[libertarian]] positions on social issues, and are unlikely to agree with religious conservatives on issues like abortion, prayer in school and same-sex marriage. Neoconservatives differ from libertarians in that neoconservatives tend to support [[Big government]] policies to further their objectives, and to support erosion of American liberty by [[Patriot act]] and other measures.

In anticipation of vacancies on the [[U.S. Supreme Court]], the neoconservatives urged the selection of [[Michael McConnell]], a [[libertarian]]-leaning jurist, and [[J. Michael Luttig]], who declared ''[[Roe v. Wade]]'' to be "super-stare decisis"<ref>http://althouse.blogspot.com/2005/07/arlen-specter-makes-up-term.html</ref> and later left the judiciary to become general counsel of Boeing.<ref>http://www.washingtonpost.com/wp-dyn/content/article/2006/05/10/AR2006051000929.html</ref> Both were passed over in filling the vacancies with religious conservatives.

Neoconservatives are small in number but dominate television talk shows, Washington-based think tanks, and advisory positions in Republican Administrations. An example of a neoconservative is [[Bill Kristol|William Kristol]], founder of the [[Weekly Standard]].

== References ==

<references/>

[[Category:Politics]]

Field (mathematics)

2007-03-15T21:18:53Z

Pixologic:

A '''field''' is a commutative [[Ring (mathematics)|ring]] which contains a non-zero multiplicative identity and all non-zero elements have multiplicative inverses. Everyday examples of fields include the [[real numbers|real numbers]], [[complex numbers]] and the [[rationals]]. Besides the trivial field, the set of rationals is the smallest number field, so any other number field is an extension field of the rationals.

[[Category:Mathematics]]

Cauchy sequences

2007-03-15T21:09:27Z

Pixologic: Cauchy sequences moved to Cauchy sequence

#REDIRECT [[Cauchy sequence]]

Complete metric space

2007-03-15T21:08:51Z

Pixologic:

A complete metric space is a [[metric space]] in which every [[Cauchy sequence]] converges.
[[Category:Mathematics]]

Complete metric space

2007-03-15T21:08:32Z

Pixologic:

A complete metric space is a [[metric space]] in which every [[Cauchy sequence]] converges.

Hilbert Space

2007-03-15T21:08:05Z

Pixologic:

Hilbert space is a [[normed vector space]] that is also a [[complete metric space]]. A Hilbert space is always a [[Banach space]], but the converse need not hold.
[[Category:Mathematics]]

Banach space

2007-03-15T21:06:33Z

Pixologic:

A Banach space is a complete, [[normed vector space]].
[[Category:Mathematics]]

Banach space

2007-03-15T21:06:00Z

Pixologic:

A Banach space is a complete, [[normed vector space]].

Riemannian geometry

2007-03-15T21:03:55Z

Pixologic:

Riemmanian geometry is a non-euclidean geometric theory developed by mathematicians [[Bernhard Riemann]] and [[Carl Friedrich Gauss]], and was later used by [[Albert Einstein]] in formulating his General Theory of Relativity.
[[Category:Mathematics]]

Hilbert Space

2007-03-15T21:03:43Z

Pixologic:

Hilbert space is a [[normed vector space]] that is also a complete [[metric space]]. A Hilbert space is always a [[Banach space]], but the converse need not hold.
[[Category:Mathematics]]

Hilbert Space

2007-03-15T21:00:32Z

Pixologic:

Hilbert space is a [[normed vector space]] that is also a complete [[metric space]]. A Hilbert space is always a [[Banach space]], but the converse need not hold.

Quantum mechanics

2007-03-15T20:59:07Z

Pixologic:

Quantum mechanics consists of the breakthrough in physics in the 1920s in understanding how particles behave inside atoms. Classical mechanics, as discovered mostly by [[Isaac Newton]], cannot explain atomic behavior.

Classical mechanics would predict that an electron orbits a proton just as planets orbit the sun. Classical electromagnetism would predict that the orbiting electron would emit a time-varying electrical field just as a radio station does. But the electron would lose energy as it emits this radiation, and would orbit closer and closer to the proton, until it collapses into the proton! Such a model cannot be correct.

Quantum mechanics posits that an electron behaves as both a wave and a particle. As a result of the wave nature of the electron, the position of the electron is never precisely known. Whenever it is attempted to be measured, knowledge of the electron's velocity is lost. Hence, there is an inherent uncertainty that prevents precisely measuring both the position and the momentum simultaneously. This is known as the [[Werner Heisenberg|Heisenberg]] uncertainty principle.

Quantum mechanics forms the basis for our understanding of chemical reactions, as well as all computers and electronic devices today.

An important aspect of Quantum Mechanics is the predictions it makes about the radioactive decay of isotopes. This decay is random, and the randomness is inherently a quantum mechanical effect. As a result, the decay follows "1st-order kinetics" - typically resulting in an exponential decrease in the amount of isotope present as a function of time. The characteristic time required for 1/2 of the original amount of isotope to decay is known as the "half-life" and can vary from billionths of a second to billions of years.

==Mathematics==

The mathematics of Quantum mechanics is formulated using the language of infinite dimensional [[Hilbert Space]].

For an excellent discussion of quantum mechanics, see:
http://www.chemistry.ohio-state.edu/betha/qm/

See also: [[Momentum (operator)]]

[[Category:Physics]]

Quantum mechanics

2007-03-15T20:56:26Z

Pixologic:

Quantum mechanics consists of the breakthrough in physics in the 1920s in understanding how particles behave inside atoms. Classical mechanics, as discovered mostly by [[Isaac Newton]], cannot explain atomic behavior.

Classical mechanics would predict that an electron orbits a proton just as planets orbit the sun. Classical electromagnetism would predict that the orbiting electron would emit a time-varying electrical field just as a radio station does. But the electron would lose energy as it emits this radiation, and would orbit closer and closer to the proton, until it collapses into the proton! Such a model cannot be correct.

Quantum mechanics posits that an electron behaves as both a wave and a particle. As a result of the wave nature of the electron, the position of the electron is never precisely known. Whenever it is attempted to be measured, knowledge of the electron's velocity is lost. Hence, there is an inherent uncertainty that prevents precisely measuring both the position and the momentum simultaneously. This is known as the [[Werner Heisenberg|Heisenberg]] uncertainty principle.

Quantum mechanics forms the basis for our understanding of chemical reactions, as well as all computers and electronic devices today.

An important aspect of Quantum Mechanics is the predictions it makes about the radioactive decay of isotopes. This decay is random, and the randomness is inherently a quantum mechanical effect. As a result, the decay follows "1st-order kinetics" - typically resulting in an exponential decrease in the amount of isotope present as a function of time. The characteristic time required for 1/2 of the original amount of isotope to decay is known as the "half-life" and can vary from billionths of a second to billions of years.

==Mathematics==

The mathematics of Quantum mechanics is formulated using the language of infinite dimensional [[Hilbert Spaces]].

For an excellent discussion of quantum mechanics, see:
http://www.chemistry.ohio-state.edu/betha/qm/

See also: [[Momentum (operator)]]

[[Category:Physics]]

Karl Popper

2007-03-15T20:35:54Z

Pixologic:

Sir Karl Popper (1902-1994) was an insightful Austrian-born philosopher who had an influence on science. Popper claimed that science should only include a theory that is "[[falsifiable]]", which means capable of being proven false if it is false. Popper wrote that "the scientific status of a theory is its [[falsifiability]], or refutability, or testability."<ref>http://www.stephenjaygould.org/ctrl/popper_falsification.html</ref>

Popper proposed applying his theory of falsification to proposals in ethics and politics. Popper's approach even has merit in debunking claims that "if only ..." something happened, then everything would be better. "If only I were a wealthy man, then ...." There is no way to falsify such claims even if they were false, so there is no point in wasting any time thinking about them.

'''Sources:'''

<references/>

Riemannian geometry

2007-03-15T20:31:53Z

Pixologic:

Riemannian geometry

2007-03-15T20:31:37Z

Pixologic:

Riemmanian geometry is a non-euclidean geometric theory developed by mathematicians [[Bernhard Riemann]] and [[Carl Friedrich Gauss]], and was used by [[Albert Einstein]] in formulating his General Theory of Relativity.

String theory

2007-03-15T20:30:33Z

Pixologic:

String theory, (or super-string theory when coupled with supersymmetry), is a class of models in [[theoretical physics]] which replace zero-dimensional points (particles) in four-dimensional spacetime with one-dimensional strings in an eleven-dimensional spacetime as the fundamental building block of the universe. The elusive goal is to develop a set of equations that unify all known natural forces (gravitational, electromagnetic, weak, and strong). A leading proponent of string theory has been Edward Witten of the [[Institute for Advanced Study]].

There is much debate on whether string theory is truly scientific. Under the [[positivist]] philosophy, models must be falsifiable (i.e. it must be possible to define an experiment which will prove or disprove the theory) for them to qualify as science, and then they must survive these tests. In more general terms, a model is not scientific unless we can use it to make predictions about the world we observe, and it is consistent with those predictions. String theory has yet to yield a particular definitive test, or to produce any models that resemble any known particles or forces.

Some string theorists, such as Leonard Susskind, say that string theory may not be testable but may still provide insights on multiple universes. His recent book is ''The Cosmic Landscape: String Theory and the Illusion of Intelligent Design''.
[[Category:Physics]]

Riemannian geometry

2007-03-15T20:29:46Z

Pixologic:

Riemmanian geometry is a non-euclidean geometric theory developed by mathematicians [[Bernhard Riemann]] and [[Carl Friedrich Gauss]].

Riemannian geometry

2007-03-15T20:29:26Z

Pixologic:

Riemmanian geometry is a non-euclidean theory of geometry developed by mathematicians [[Bernhard Riemann]] and [[Carl Friedrich Gauss]].

Albert Einstein

2007-03-15T20:25:31Z

Pixologic:

Albert Einstein (1879-1955) was a theoretical physicist who contributed to the theory of a [[Special Relativity]] and formulated the theory of [[General Relativity]]. He also contributed to the study of [[Photoelectric Effect]], [[Brownian Motion]] and [[Bose-Einstein Statistics]]. He was critical of [[quantum mechanics]] and has proposed the [[EPR-Paradox]] in attempt to disprove it. The last fifteen years of his life were spent in spent unsuccessfully searching for a [[Unified Field Theory]]. Born in Germany, he won the [[Nobel Prize]] for his work on the [[Photoelectric Effect]] in 1921.<ref>http://nobelprize.org/nobel_prizes/physics/laureates/1921/</ref> He immigrated to the United States in the late 1930s where, upon the urging of a colleague in physics, recommended that President Franklin D. Roosevelt develop an [[atomic bomb]] project.<ref>[http://www.dannen.com/ae-fdr.html]</ref>

Einstein's best-known scientific work is his 1905 papers on [[Special Relativity]]. The theory of relativity had already been published by [[Hendrik Lorentz]] and [[Henri Poincaré]]. Einstein's contribution was to give a description of the Lorentz-Poincare theory without reference to the luminiferous aether, and to clarify some of the physical consequences of the theory.

Einstein then wrote a series of papers applying relativity to gravity using the mathematics of [[Riemannian Geometry]], resulting in [[General Relativity]]. Much of the actual mathematics of general relativity was worked out by Marcel Grossman, David Hilbert, and others. Einstein's main contribution was to give an alternate derivation of the field equations, and to apply them to explain an anomaly in the advance of the perihelion in Mercury's orbit.

Sir [[Arthur Eddington]] popularized General Relativity by publishing an explanation of it and by attempting to prove the theory experimentally in a dramatic test. Eddington embarked on a highly publicized trip into the Atlantic Ocean off Africa in order to observe the bending of [[star]]light passing nearby the [[Sun]] during the total [[solar eclipse]] of May 29, 1919. Eddington returned with a declaration that he had proven Einstein's theory of [[General Relativity]], making Einstein extremely famous, when in fact he had selectively used data to attain this result. Eddington was passed over for the Nobel Prize.

==Cosmological Constant==

Einstein revised his work on [[General Relativity]] to include a '[[fudge-factor]]' which he termed the [[Cosmological Constant]]. He was trying to account for the apparent steady-state nature of the universe.

[[Edwin Hubble]], of whom [[NASA]]'s famous [[Hubble Space Telescope]] is named in honor, later convinced him otherwise. At one point he had Einstein actually making direct [[observation]]s alongside him, and Hubble's [[finding]]s showed conclusively that the [[universe]] was indeed [[expand]]ing, as predicted by certain solutions of GR's field equations in the absence of a cosmological constant. Einstein recanted his original [[philosophy]], and accepted as true the notion of a changing universe having a definite [[begin]]ning.

He later said of his Cosmological Constant, "It was the biggest blunder of my life."

An interesting end-note on this term: Principle investigators from [[Princeton]] [[University]] first published findings in 1998 apparently revealing an [[accelerating]] expansion for the universe<ref>http://map.gsfc.nasa.gov/m_mm.html</ref>. Because this initially seemed like it might be a sort of 'anti-gravity' effect (similar in concept to Einstein's original notion), it briefly became quite popular for scientists to dub the [[phenomenon]] the 'Cosmological Constant' as well. For more information, see: [[Dark Energy]].

==References==

"The Nobel Prize in Physics 1921"[http://nobelprize.org/nobel_prizes/physics/laureates/1921/]

"On The Electrodynamics Of Moving Bodies"[http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf]

"Does The Inertia Of A Body Depend Upon Its Energy-Content?"[http://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf]

[[Category:Biographies]]
[[Category:Physics]]

Electromagnetic wave

2007-03-15T19:28:34Z

Pixologic:

A transverse wave composed of an oscillating electrical field and a magnetic field that oscillates perpendicular to the electric field.<ref>Wile, Dr. Jay L. ''Exploring Creation With Physical Science''. Apologia Educational Ministries, Inc. 1999, 2000</ref>

Electromagnetic waves was predicted by the classical laws of electricity and magnetism, known as Maxwell's equations:

::<math>\nabla \cdot \mathbf{E} = 0 \qquad \qquad \qquad \ \ (1)</math>
::<math>\nabla \times \mathbf{E} = -\frac{\partial}{\partial t} \mathbf{B} \qquad \qquad (2)</math>
::<math>\nabla \cdot \mathbf{B} = 0 \qquad \qquad \qquad \ \ (3)</math>
::<math>\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial}{\partial t} \mathbf{E} \qquad \ \ \ (4)</math>
:where <math>\nabla</math> is a vector differential operator
Maxwell's equations is an example of a system of [[partial differential equations]].

[[Category:Physics]]

Electromagnetic wave

2007-03-15T19:28:20Z

Pixologic:

A transverse wave composed of an oscillating electrical field and a magnetic field that oscillates perpendicular to the electric field.<ref>Wile, Dr. Jay L. ''Exploring Creation With Physical Science''. Apologia Educational Ministries, Inc. 1999, 2000</ref>

Electromagnetic waves was predicted by the classical laws of electricity and magnetism, known as Maxwell's equations:

::<math>\nabla \cdot \mathbf{E} = 0 \qquad \qquad \qquad \ \ (1)</math>
::<math>\nabla \times \mathbf{E} = -\frac{\partial}{\partial t} \mathbf{B} \qquad \qquad (2)</math>
::<math>\nabla \cdot \mathbf{B} = 0 \qquad \qquad \qquad \ \ (3)</math>
::<math>\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial}{\partial t} \mathbf{E} \qquad \ \ \ (4)</math>
:where
::<math>\nabla</math> is a vector differential operator
Maxwell's equations is an example of a system of [[partial differential equations]].

[[Category:Physics]]

F-theory

2007-03-15T19:21:24Z

Pixologic:

'''F-theory''' is a modified version of [[String theory]] in which a second [[time]] variable is introduced. It allows string theorists to construct new, more realistic vacua in the form of F-theory compactified on elliptically fibered [[Calabi-Yau]] four-folds. The letter "F" stands for "Father", like the "M" in [[M-theory]] is often taken to stand for "Mother".
[[Category:Physics]]

M-theory

2007-03-15T19:21:10Z

Pixologic:

M-theory is a theory conjectured by [[Edward Witten]] of the Institute for Advanced Study, that would unify the five different superstring theories ([[Type IIA string theory]], [[Type IIB string theory]], [[Heterotic string theory]], [[SO(32) string theory]], and heterotic ''E''8×''E''8 string theory). [[Edward Witten]] has used M-theory to explain a number of observed dualities, such as the T-duality, U-duality and S-duality.
[[Category:Physics]]

String theory

2007-03-15T19:20:43Z

Pixologic:

String theory, (or super-string theory when coupled with super-symmetry), is a class of models in [[theoretical physics]] which replace zero-dimensional points (particles) in four-dimensional spacetime with one-dimensional strings in an eleven-dimensional spacetime as the fundamental building block of the universe. The elusive goal is to develop a set of equations that unify all known natural forces (gravitational, electromagnetic, weak, and strong). A leading proponent of string theory has been Edward Witten of the [[Institute for Advanced Study]].

There is much debate on whether string theory is truly scientific. Under the [[positivist]] philosophy, models must be falsifiable (i.e. it must be possible to define an experiment which will prove or disprove the theory) for them to qualify as science, and then they must survive these tests. In more general terms, a model is not scientific unless we can use it to make predictions about the world we observe, and it is consistent with those predictions. String theory has yet to yield a particular definitive test, or to produce any models that resemble any known particles or forces.

Some string theorists, such as Leonard Susskind, say that string theory may not be testable but may still provide insights on multiple universes. His recent book is ''The Cosmic Landscape: String Theory and the Illusion of Intelligent Design''.
[[Category:Physics]]

M-theory

2007-03-15T19:20:08Z

Pixologic:

M-theory

2007-03-15T19:19:53Z

Pixologic:

M-theory is a theory conjectured by [[Edward Witten]] of the Institute for Advanced Study, that would unify the five different superstring theories ([[Type IIA string theory]], [[Type IIB string theory]], [[Heterotic string theory]], [[SO(32) string theory]], and [[heterotic ''E''8×''E''8 string theory]]). [[Edward Witten]] has used M-theory to explain a number of observed dualities, such as the T-duality, U-duality and S-duality.

F-theory

2007-03-15T19:13:16Z

Pixologic:

F-theory

2007-03-15T19:12:07Z

Pixologic:

'''F-theory''' is a modified version of [[String theory]] in which a second [[time]] variable is added. It allows string theorists to construct new, more realistic vacua in the form of F-theory compactified on elliptically fibered [[Calabi-Yau]] four-folds. The letter "F" stands for "Father", like the "M" in [[M-theory]] is often taken to stand for "Mother".

F-theory

2007-03-15T19:12:01Z

Pixologic:

'''F-theory''' is a modified version of [[String Theory]] in which a second [[time]] variable is added. It allows string theorists to construct new, more realistic vacua in the form of F-theory compactified on elliptically fibered [[Calabi-Yau]] four-folds. The letter "F" stands for "Father", like the "M" in [[M-theory]] is often taken to stand for "Mother".

F-theory

2007-03-15T19:11:48Z

Pixologic:

'''F-theory''' is a modified version of [[string Theory]] in which a second [[time]] variable is added. It allows string theorists to construct new, more realistic vacua in the form of F-theory compactified on elliptically fibered [[Calabi-Yau]] four-folds. The letter "F" stands for "Father", like the "M" in [[M-theory]] is often taken to stand for "Mother".

Hausdorff space

2007-03-15T19:06:07Z

Pixologic:

Hausdorff space is a [[topological space]] in which, for any pair of distinct points x and y, there exist disjoint open sets U and V, such that x is in U and y is in V. Almost all spaces studied in analysis are Hausdorff.

An important property of Hausdorff spaces is that sequences and nets converge to a unique point.
[[Category:Mathematics]]

Regular space

2007-03-15T18:59:10Z

Pixologic:

Regular space is a Hausdorff [[topological space]] in which, given any closed set F and any point x that does not belong to F, there exist an open set U of x and an open set V of F that are disjoint.
[[Category:Mathematics]]

Normal space

2007-03-15T18:56:35Z

Pixologic:

Normal space is a Hausdorff [[topological space]] in which, given any pair of disjoint closed sets E and F, there exist neighbourhoods U of E and V of F that are disjoint.
[[Category:Mathematics]]

Paracompact space

2007-03-15T18:53:23Z

Pixologic:

A paracompact space is a Hausdorff [[topological space]] in which every [[open cover]] admits an open, locally finite refinement. Example of paracompact space include [[metric space]], locally compact second-countable space, and regular Lindelöf space.
[[Category:Mathematics]]

Paracompact space

2007-03-15T18:51:24Z

Pixologic:

A paracompact space is a Hausdorff [[topological space]] in which every [[open cover]] admits an open, locally finite refinement.
[[Category:Mathematics]]

Paracompact space

2007-03-15T18:51:10Z

Pixologic:

A paracompact space is a Hausdorff [[topological space]] in which every [[open cover]] admits an open locally finite refinement.
[[Category:Mathematics]]

Paranormal space

2007-03-15T18:49:17Z

Pixologic:

A [[topological space]] is said to be '''paranormal''' if every countable discrete collection of [[closed set]]s has a [[locally finite]] open expansion.
[[Category:Mathematics]]

2-category

2007-03-15T18:44:56Z

Pixologic:

2-category is a [[category]] with morphisms in between morphisms. It is defined as a category enriched over the category of categories and functors, with the monoidal structure induced by the composition.

[[Category:Mathematics]]

Stack

2007-03-15T18:40:47Z

Pixologic:

'''stack''' is a concept introduced in [[algebraic geometry]] to generalize [[algebraic varieties]] and [[schemes]]. They were originally proposed by [[David Mumford]] on the compactifications of moduli of curves; such stack is now called a [[Deligne-Mumford stack]]. They were later generalized by [[Michael Artin]] to [[Artin stack]]s, or sometimes also called[[algebraic stack]]s. More generally a ''stack'' is a [[category]] acting like a [[moduli space]] with a ''universal family'' parametrizing a family of mathematical objects such as schemes or [[topological space]]s.

==External links==
* ''[http://www.ams.org/notices/200304/what-is.pdf What is... a Stack?]''

[[Category:Mathematics]]

Stack

2007-03-15T18:40:18Z

Pixologic:

'''stack''' is a concept introduced in [[algebraic geometry]] to generalize [[algebraic varieties]] and [[schemes]]. They were originally proposed by [[David Mumford]] on the compactifications of moduli of curves; such stack is now called a [[Deligne-Mumford stack]]. They were later generalized by [[Michael Artin]] to [[Artin stack]], or sometimes also called[[algebraic stack]]. More generally a ''stack'' is a [[category]] acting like a [[moduli space]] with a ''universal family'' parametrizing a family of mathematical objects such as schemes or [[topological space]]s.

==External links==
* ''[http://www.ams.org/notices/200304/what-is.pdf What is... a Stack?]''

[[Category:Mathematics]]

Twistor theory

2007-03-15T18:21:52Z

Pixologic:

Twistor theory was originally developed by Sir Roger Penrose in 1967, it is a mathematical theory that seeks to describe the geometry of four dimensional real space-time via the geometry of 4-dimensional complex space with metric signature (2,2).

Twistor space has also been used by Edward Witten to efficiently calculate the scattering amplitude of [[Feynman diagrams]].

==External links==
* [http://users.ox.ac.uk/~tweb/00001/index.shtml Roger Penrose - ''On the Origins of Twistor Theory'']
* [http://users.ox.ac.uk/~tweb/00002/index.shtml Roger Penrose - ''The Central Programme of Twistor Theory'']
* [http://users.ox.ac.uk/~tweb/00003/index.shtml Richard Jozsa - ''Applications of Sheaf Cohomology in Twistor Theory'']
* [http://users.ox.ac.uk/~tweb/00004/ Fedja Hadrovich - ''Twistor primer'']
[[Category:Mathematics]]

Real numbers

2007-03-15T18:20:45Z

Pixologic: /* Formal definition */

Real numbers can be thought of as numbers which can be represented by some infinite or finite decimal representation, such as 0.707106781187...

In classical physics, measurements of things that can vary smoothly and continuously, like speed or temperature, are treated as real numbers.

In a number line representation, the real numbers correspond to all the points on a geometric line. The distance between any two points on a line is a real number.

In computer programming, some computer languages such as FORTRAN include a ''real'' data type that is intended to represent real numbers.<ref>In reality, the actual values the computer uses are very-high-precision fractions which can equal or approximate real numbers</ref>

The real numbers include within them all of these other kinds of numbers:

*The "natural numbers" or positive integers, 1, 2, 3, ...
*Zero and the negative integers
*Fractions, 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 &sqrt;10 = 3.162277660168... π = 3.1415926525..., 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 cuts]] or [[Cauchy Sequences]], i.e. it is the compeletion 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). The reals contain ''an infinite number of finite numbers.''

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, 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."

==Notes and references==
<references/>

[[Category:Mathematics]]

Real numbers

2007-03-15T18:20:24Z

Pixologic:

Real numbers can be thought of as numbers which can be represented by some infinite or finite decimal representation, such as 0.707106781187...

In classical physics, measurements of things that can vary smoothly and continuously, like speed or temperature, are treated as real numbers.

In a number line representation, the real numbers correspond to all the points on a geometric line. The distance between any two points on a line is a real number.

In computer programming, some computer languages such as FORTRAN include a ''real'' data type that is intended to represent real numbers.<ref>In reality, the actual values the computer uses are very-high-precision fractions which can equal or approximate real numbers</ref>

The real numbers include within them all of these other kinds of numbers:

*The "natural numbers" or positive integers, 1, 2, 3, ...
*Zero and the negative integers
*Fractions, 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 &sqrt;10 = 3.162277660168... π = 3.1415926525..., 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 cuts]] or [[Cauchy Sequences]], i.e. it is the compeletion of the rational numbers.

==Infinity==

The real numbers ''do not'' include <math>\infty</math> or <math>-\infty</math> (infinity and minus infinity). The reals contain ''an infinite number of finite numbers.''

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, 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."

==Notes and references==
<references/>

[[Category:Mathematics]]

Functor

2007-03-15T18:19:17Z

Pixologic:

In [[category theory]], a '''functor''' is a [[morphism]] between categories.

Functors were first proposed in [[algebraic topology]], where algebraic objects such as the [[fundamental group]] are associated to [[topological space]]s, and algebraic [[homomorphism]]s are associated to continuous functions. Functors are now used to relate various categories.
[[Category:Mathematics]]

Prime Number Theorem

2007-03-15T18:16:06Z

Pixologic:

The Prime Number Theorem is one of the most famous theorem in mathematics. It states that the number of primes not exceeding n is asymptotic to <math>\frac{n}{\log(n)}</math>, where log(n) is the logarithm of (n) to the base e.

The number of primes not exceeding n is commonly written as <math>\pi(n)</math>, and an asymptotic relationship between a(n) and b(n) is commonly designated as a(n)~b(n). (This does not mean that a(n)-b(n) is small as n increases. It means the ratio of a(n) to b(n) approaches one as n increases.)

The Prime Number Theorem thus states that <math> \pi(n) </math>~<math> n/ \log(n)</math> .

In other words, the limit (as n approaches infinity) of the ratio of pi(n) to n/log(n) is one. Put a third way, n/log(n) is a good approximation for <math>\pi(n)</math>.

== Equivalent Statements ==

[[Carl Friedrich Gauss]] conjectured the equivalent statement that <math>\pi(x)</math> was asymptotic to <math>\mbox{Li}(x)</math> defined as:

<math> \mbox{Li}(x) = \int_2^x \frac{dt}{\ln t}</math>.

In fact, for large x this turns out to be a better approximation than <math>\pi(x)</math>.
[[Category:Mathematics]]

Functor

2007-03-15T18:15:56Z

Pixologic:

In [[category theory]], a '''functor''' is a [[morphism]] between categories.

Functors were first proposed in [[algebraic topology]], where algebraic objects such as the [[fundamental group]] are associated to [[topological space]]s, and algebraic [[homomorphism]]s are associated to continuous functions. Recently, functors are used to relate various categories.
[[Category:Mathematics]]

Category theory

2007-03-15T18:15:48Z

Pixologic:

Category theory is a branch of mathematics that study abstraction of mathematical structures and the relationships between them. Many concepts in mathematics, computer science and mathematical physics can be phrased in the language of categories and morphisms between categories. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942 as a tool in the study of algebraic topology. Category theory has been jokingly referred to, by some mathematicians, as "abstract nonsense" refers, due to its high level of abstraction.
[[Category:Mathematics]]

Morphism

2007-03-15T18:15:41Z

Pixologic:

In [[category theory]], a morphism is a mapping between two categories. For example, morphisms between sets are functions; morphisms between groups are group homomorphisms; morphisms between topological spaces are continuous functions; morphisms between category of functors are natural transformations.
[[Category:Mathematics]]

Algebraic topology

2007-03-15T18:15:32Z

Pixologic:

'''Algebraic topology''' is a branch of mathematics that uses [[abstract algebra]] to understand [[topological space]]s.
[[Category:Mathematics]]