Wiktionarypar rationalRational may refer to Rationality , a concept of reason Rational number , a number that can be expressed as a ratio of two integers Rational function , a mathematical function which can be written as the ratio of two polynomial functions Rational Software , formerly Rational Software Corporation, a software company now owned by IBM a central paradigm of Rational Emotive Behavior Therapy The Rational temperament in the Keirsey Temperament Sorter Rational choice theory , a theoretical paradigm in microeconomics Rational AG , a German manufacturer of food processors See also Rationalism disambiguation Rationale , a liturgical vestment worn by clergy a common misspelling Rationality disambig cs Racion ln de Rational fr Rationnel pt Racional es Racional ... more details
Rational approximation may refer to Diophantine approximation , the approximation of real numbers by rational numbers. Pad approximation , the approximation of functions by rational functions. See also Simple rational approximation Mathdab ... more details
In mathematics , in the representation theory of algebraic group s, a linear representation of an algebraic group is said to be rational if, viewed as a map from the group to the general linear group, it is a rational map of algebraic varieties. Finite direct sums and products of rational representations are rational. A rational math G math module is a module that can be expressed as a sum not necessarily direct of rational representations. see Group representation References http www.jstor.org view 00029327 di994362 99p00143 Extensions of Representations of Algebraic Linear Groups http www.encyclopediaofmath.org index.php Rational representation Springer Online Reference Works Rational Representation Category Representation theory of algebraic groups algebra stub ... more details
One source date February 2010 Infobox company name Rational Software logo File RationalSoftware.png 180px Rational Software type genre fate Acquired by IBM predecessor successor foundation Start date 1981 ... rational footnotes intl Rational Machines was founded by Paul Levy and Mike Devlin entrepreneur ..., particularly explicit modular architecture and iterative development. Rational was sold for US 2.1 billion to IBM on February 20, 2003. Rational Environment First released in 1985, the R1000 Rational ... platforms. The Rational Environment was organized around a persistent intermediate representation ... to system size rather than size of change, the Rational Environment supported the definition of subsystems ... testing. The Rational Environment ran on custom hardware, the Rational R1000 , which implemented a high level architecture optimized for execution of Ada programs in general and the Rational Environment ... Rational Machines to Rational to avoid emphasizing this proprietary hardware. When date February 2011 Rational provided code generators and the cross debuggers for then popular instruction set architectures ... Rational s field Practices underlying the later Rational Unified Process RUP iterative development ... In 1990, Rational launched three parallel development efforts re implementation of the Rational Environment ... of a comparable Rational Environment for C to run on Unix based workstations from Sun and IBM, and development ... by Grady Booch . Apex, the Rational Environment for Ada, was launched on Sun and IBM Unix platforms in 1993, and the Rational Environment for C followed on the same platforms a year later. A version of Apex that ran on Microsoft Windows NT was successfully developed and released by Rational s Bangalore ... as built navigable class diagrams, helped overcome Rational s late re entry into the market .... UML and RUP In 1994, Rational merged with Verdix , a public company that produced a wide array of Ada compilers targeted to many architecture OS combinations. The resulting entity was named Rational ... more details
Inline citations date January 2012 In number theory , a rational point is a point in space each of whose coordinate s are rational that is, the coordinates of the point are elements of the Field mathematics field of rational number s, as well as being elements of larger fields that contain the rational numbers, such as the real numbers and the complex numbers . For example, math 3, &minus 67 4 is a rational point in 2 dimensional space, since 3 and &minus 67 4 are rational numbers. A special case of rational points are integer point s, that is, when all of the coordinates are integers, e.g., math ... rational point is a point in a space where each coordinate of the point belongs to the field K , as well as being elements of larger fields containing the field K . This is analogous to rational points ... case of K rational points are those that belong to a Ring mathematics ring of algebraic integers existing inside the field K . Rational or K rational points on algebraic varieties See Diophantine geometry If a K rational point P exists on that is, lies on an algebraic variety, and the variety is given ... of all of the equations simultaneously. Rational as well as K rational points that lie on an algebraic ... variety A , the K rational points form a Group mathematics group . The Mordell Weil theorem states that the group of rational points of an abelian variety over K is finitely generated if K is a number field . The Weil conjectures concern the distribution of rational points on varieties over finite field s. Example 1 The point math 3, &minus 67 4 is one of the infinite set of rational points on the straight line given by the equation math y 67 4 2 x &minus 3 . This set of rational points forms ... out that there are also algebraic curves which contain just finitely many rational points e.g. a line math y a x contains only one rational point math 0, 0 for any irrational number a or no rational ... x sup 2 sup &minus 2 y sup 0 . Although P is not a rational point, since the coordinate 2 is not rational ... more details
Orphan date February 2009 Unreferenced date May 2010 In computer science rational languages are a category of formal language s. They are defined as the set mathematics set of strings for which some rational series assigns non zero values also known as that series support mathematics support . When the semiring of the rational series is boolean logic boolean , the associated rational languages are simply the regular language s. Category Formal languages Formalmethods stub ... more details
Rational Tau is a Unified Modeling Language UML and SysML modeling tool supporting automated code generation and model verification. Tau is today maintained by IBM . It was part of the IBM acquisition of Telelogic 2008. External links http ibm.com software awdtools tau IBM Rational Tau Category UML tools uml stub ... more details
In economics , game theory , decision theory , and artificial intelligence , a rational agent is an agent which has clear preferences , models uncertainty via expected values , and always chooses to perform the action that results in the Optimization mathematics optimal outcome for itself from among all feasible actions. Rational agents are also studied in the fields of cognitive science , ethics , and philosophy, including the philosophy of practical reason . A rational agent can be anything that makes decisions, typically a person, firm, machine, or software. The action a rational agent takes depends on the preferences of the agent the agent s information of its environment, which may come from past experiences the actions, duties and obligations available to the agent the estimated or actual benefits and the chances of success of the actions. In game theory and classical economics , it is often assumed that the actor s, people, and firms are rational. However, the extent that people and firms behave rationally is subject to debate. Economists often assume the models of rational choice theory and bounded rationality to formalize and predict the behavior of individuals and firms. Rational agents sometimes behave in manners that are counter intuitive to many people, as in the Traveler s dilemma . Artificial intelligence has borrowed the term rational agents from economics to describe autonomous programs that are capable of goal directed behavior. Today there is a considerable overlap between AI research, game theory and decision theory. Rational agents in AI are closely related to intelligent agent s , autonomous software programs that display intelligence. NOTOC References Economics and game theory Citation last Osborne first Martin last2 Rubinstein first2 Ariel year 2001 title A Course in Game Theory publisher MIT Press location Cambridge, Mass. page 4 isbn 0262650401 ... rationality Rational choice theory Game theory Software Intelligent agent Software agent Category ... more details
Rational fideism is the philosophy philosophical view that considers faith to be precursor for any reliable knowledge . Whether one considers rationalism or empiricism , either of them ultimately tends to belief in reason or experience respectively as the absolute basis for their methods. Thus, faith is basic to knowability. However, as Domenic Marbaniang shows, ref Rational Fideism. http books.google.com books?id fL dAqxShiwC&printsec frontcover Domenic Marbaniang, Epistemics of Divine Reality , http books.google.com books?id fL dAqxShiwC&printsec frontcover , 2009 ref faith must be corroborated with reason and experience in order to gain knowledge. See also Fideism Philosophical theology Philosophy of religion References Reflist External links On Kierkegaard s Fideism http www.blackwell synergy.com doi pdf 10.1111 1468 2265.00084 Stanford on Fideism http www.seop.leeds.ac.uk entries fideism Marbaniang, Domenic. Epistemics of Divine Reality ,ISBN 978 1 105 16077 6 http books.google.com books?id fL dAqxShiwC&printsec frontcover Rational Fideism http www.scribd.com doc 48416873 Rational Fideism Four Principles of Rational Fideism http marbaniang.wordpress.com 2011 09 30 four principles of rational fideism Category Epistemics Category Epistemological theories Category Rationalism epistemology stub ... more details
On isolated rational singularities of surfaces id MathSciNet id 0199191 year 1966 journal American ... University Press jstor 2373050 Citation last1 Lipman first1 Joseph title Rational singularities, with applications ... more details
Aristotelianism Rational animal is a classical definition of man. ref http plato.stanford.edu entries cognition animal ref Though it is often attributed to first appearing as a definition in Aristotle s Metaphysics Aristotle Metaphysics , Aristotle does not define it here. In the Nicomachean Ethics I.13, Aristotle states that the human being has a rational principle. The definition of man as a rational animal was common in scholastical philosophy . ref name CathEnc Catholic Encyclopedia inline 1 wstitle Man ref Catholic Encyclopedia states that this definition means that in the system of classification and definition shown in the Arbor Porphyriana , man is a Substance philosophy substance , corporeal , life living , sentient , and Rationality rational . ref name CathEnc Influence and legacy Descartes In Meditation II of Meditations on First Philosophy , Descartes arrives at his famous cogito ergo sum I am, I exist claim. He then goes on to wonder What am I? He considers and rejects, rational animal blockquote Shall I say a rational animal ? No for then I should have to inquire what an animal is, what rationality is, and in this one question would lead me down the slope to other harder ones. ref The Philosophical Writings of Descartes Volume II . Translated by John Cottingham, Robert Stoothoff, Dugald Murdoch. Cambridge University Press. 1984. ref blockquote Later influences Neo kantian philosopher Ernst Cassirer , in his work An Essay on Man 1944 , altered Aristotle definition to label man as a symbolic animal . This definition has been influential in the field of philosophical anthropology , where it has been reprised by Gilbert Durand . Quotes Copy section to Wikiquote Man is a rational animal who always loses his temper when called upon to act in accordance with the dictates of reason. Oscar Wilde It has been said that man is a rational animal. All my life I have been searching for evidence which could support this. Bertrand Russell See also Genus differentia ... more details
distinguish Intelligent design In chemical biology and biomolecular engineering , rational design is the strategy of creating new molecules with a certain functionality, based upon the ability to predict how the molecule s structure will affect its behavior through physical models. This can be done either from scratch or by making calculated variations on a known structure, and is usually contrasted with directed evolution . Applications of rational design include Protein design Nucleic acid design Drug design nano tech stub biology stub Category Nanotechnology ... more details
In mathematics , a rational function is any function mathematics function which can be written as the ratio ... values taken by the function are necessarily rational number s. Definitions In the case of one Variable mathematics variable , math x , math , a function is called a rational function if and only ... degree. Every polynomial function is a rational function with math Q x 1 math . A function that cannot be written in this form for example, math f x sin x math is not a rational function but the adjective ... math frac P x Q x math is called a rational expression . The math x math need not be a variable. In abstract algebra the math x math is called an Indeterminate variable indeterminate . A rational equation is an equation in which two rational expressions are set equal to each other. These expressions ... by zero is rejected. Examples multiple image header Examples of rational functions align right direction vertical width 300 image1 RationalDegree2byXedi.svg alt1 Rational function of degree 2 caption1 Rational function of degree 2 math y frac x 2 3x 2 x 2 4 math image2 RationalDegree3.svg alt2 Rational function of degree 3 caption2 Rational function of degree 3 math y frac x 3 2x 2 x 2 5 math The rational function math f x frac x 3 2x 2 x 2 5 math is not defined at math x 2 5 Leftrightarrow x pm sqrt 5 math . The rational function math f x frac x 2 2 x 2 1 math is defined for all real number ... 2 2 i 2 1 frac 1 2 1 1 frac 1 0 math , which is undefined. The rational function math f x frac x 3 2x ... such as f x is a rational function since constants are polynomials. Note that the function itself is rational, even though f x is irrational for all x . The rational function math f x frac x x math ... excepting division by the zero polynomial of two rational functions is itself a rational function ... rational function satisfy a Recurrence relation linear recurrence relation , which can be found by setting the rational function equal to its Taylor series and collecting like terms . For example, math ... more details
Infobox Software name IBM Rational Rhapsody screenshot caption developer IBM latest release version 7.6.1 ... 01.ibm.com software awdtools rhapsody IBM Rational Rhapsody Now called IBM Rational Rhapsody , a modeling ... systems and software. Rational Rhapsody uses graphical models to generate software applications ... language Java and C Sharp programming language C . Rational Rhapsody helps diverse teams ... 3 6 2006 ref Rhapsody became an Rational Software IBM Rational product after the acquisition of Telelogic ... room publisher IBM date Apr 3, 2008 accessdate 2009 10 20 ref Since the rebranding, Rational Rhapsody has been integrated with IBM Rational Quality Manager and IBM Rational Publishing Engine . Editions Engineers and developers might select one of the following Rational Rhapsody editions for their specific needs http www.ibm.com software rational products rhapsody developer Rational Rhapsody Developer ..., model, and debugging operations http www.ibm.com software rational products rhapsody designer Rational ... with Rational Publishing Engine http www.ibm.com software rational products rhapsody swarchitect index.html Rational Rhapsody Architect for Software provides a UML and SysML based software development ... automatically Develop automotive applications using AUTOSAR http www.ibm.com software rational products rhapsody sysarchitect Rational Rhapsody Architect for Systems Engineers helps systems engineers ... tedious manual tasks with systems engineering toolkit XML Metadata Interchange XMI and Rational ... with the Jazz based Rational Team Concert solution through Eclipse Develop systems of systems with additional DoDAF, MODAF or UPDM support http www 01.ibm.com software rational products rhapsody designmanager Rational Rhapsody Design Manager enables you to share, collaborate and manage ... Rational Jazz technology See also List of UML tools External links http www.ibm.com software awdtools rhapsody Rational Rhapsody Website http www.ibm.com developerworks rational products rhapsody Rational ... more details
suicide merge right to die date December 2010 Rational suicide is the reason guided decision to end one s life, specifically, when these reasons are the result of an indifferent or temperate weighing of pros and cons. It has been argued that a suicide can be Rationality rational and still nonetheless be a mistake. ref citation title What is Wrong with Rational Suicide author A Pilpel, L Amsel url http www.avitalpilpel.com uploads What is wrong with rational suicide April 2010.pdf ref Rational suicide is sometimes viewed as a Subjectivity subjective concept Jerome Motto writes, What may be an inconvenience, a source of discomfort, or an embarrassment to one person represents unbearable agony, excruciating pain, or intolerable humiliation to another. ref Citation title Rational suicide? implications for mental health professionals author James L. Werth ref One study of mental health counselor s found that 80 of respondents were moderately supportive of the idea that people can make well reasoned decisions that death is their best option. ref citation title Rational Suicide An Empirical Investigation of Counselor Attitudes publisher Journal of Counseling & Development volume 79 date Summer 2001 pages 365 372 url http aca.metapress.com index HNUE6LUPB5GKDNYC.pdf issue 3 ref It is believed that in coming years clinic ians will increasingly be confronted by patients declaring their intention to exercise their Right to die right to suicide or to commit what they describe as a rational suicide. ref citation title Rational suicide Considerations for the clinician publisher Psychiatric Quarterly issn 1573 6709 doi 10.1007 BF01064749 volume 54 date June, 1982 issue 2 last1 Siegel first1 Karolynn journal Psychiatric Quarterly pages 77 84 pmid 7146216 ref The received orthodoxy of mental ... persons should always be prevented from ending their own lives. ref citation title Rational suicide ... journal Journal of Loss and Trauma volume 3 issue 2 pages 193 203 ref See also Rational depression References ... more details
For other forms of egoism Egoism disambiguation Individualism sidebar In ethics ethical philosophy , rational egoism also called rational selfishness or egotism is the principle that an action is rational ... egoism is about motivation and ethical egoism is about morality , rational egoism is a view about ... from amoralism . Philosophy Rational egoism is discussed by the nineteenth century English philosopher Henry Sidgwick in The Methods of Ethics . ref Sidgwick 1907 ref A method of ethics is any rational ..., namely, rational egoism, dogmatic intuitionism and utilitarianism . Rational egoism is the view that, if rational, an agent regards quantity of consequent pleasure and pain to himself alone ... for preferring rational egoism over utilitarianism . Although utilitarianism can be provided with a rational basis and reconciled with the morality of common sense, rational egoism appears to be an equally ... Two objections to rational egoism are given by the English philosopher Derek Parfit , who discusses ... the rational egoist point of view, it is rational to contribute to a pension scheme now, even though ... and future self. Social sciences Rational egoism is the standard behavioral postulate of neoclassical economics and of rational actor theory in political science . The economic assumption of rational ..., in political science, the assumption of rational egoism gives rise to the paradox of voting given ..., voting in elections is not rational unless the number of electors is very small. ref Mueller 1989 . ref Supporters of rational egoism, however, might point out that this argument is not sound because ... also discusses a theory that she called Rational egoism or more specifically Rational self interest ... Moseley 2006 , 2a. ref Thus, her view is a conjunction of both rational egoism in the standard sense ... explains in depth the concept of rational egoism. According to Rand s ethical egoism, every rational ... . Peter Singer ed. , Blackwell Oxford. Brink, D. 1992, Sidgwick and the Rationale for Rational ... more details
Primary sources date October 2009 The concept known as rational irrationality was popularized by economist ... economics bcaplan eejformat.doc title Caplan s original paper on rational irrationality publisher ... www.gmu.edu departments economics bcaplan ratirnew.doc title Caplan s paper on rational irrationality versus rational ignorance publisher Bryan Caplan date accessdate 2012 02 13 ref The theory, along with its implications for democracy, was expanded upon by Caplan in his book The Myth of the Rational ... url http econfaculty.gmu.edu bcaplan larrydeb.htm title Debate between Caplan and Iannaccone on rational ... one s actual goals, given one s actual beliefs. Rational irrationality describes a situation where it is instrumentally rational to be epistemically irrational. Caplan argues that rational irrationality ... for irrationality is high. Rational irrationality versus doublethink Rational irrationality ... to evidence that may contract their beliefs. Sources of preferences over beliefs For rational ... these rarely impose real world costs. Thus, it may be instrumentally rational to be epistemically ... may tend to relax their intellectual standards. ref name CaplanReligionDebate Politics Rational irrationality in individual political beliefs Politics is a situation where rational irrationality ... a situation where voters may be rationally irrational. Rational irrationality and systemic biases For rational irrationality at an individual level to have an effect on political outcomes, it is necessary ... of systemic biases in beliefs about economics in his book The Myth of the Rational Voter ... externality for other voters. Caplan believes that the rational irrationality of voters is one of the reasons ... trade versus protectionism . Philosopher Michael Huemer , in a TEDx talk on rational irrationality in politics, cited the war on terror and protectionism as two examples of rational irrationality in politics ... of office, and the platforms of candidates of major political parties. Like rational irrationality ... more details
Inappropriate tone date December 2007 Refimprove date September 2007 Rational ignorance occurs when the cost of educating oneself on an issue exceeds the potential benefit that the knowledge would provide. Ignorance about an issue is said to be rational when the cost of educating oneself about the issue sufficiently to make an informed decision can outweigh any potential benefit one could reasonably expect to gain from that decision, and so it would be irrational to waste time doing so. This has consequences for the quality of decisions made by large numbers of people, such as general election ... theorized that voters anomalous beliefs cannot be attributed solely to rational ignorance he states that irrational systemic bias is also at work, and he has developed a theory of rational irrationality to explain this. ref citation doi 10.1007 978 0 306 47828 4 171 chapter Rational Irrationality title ... the positive and negative factors in nutrition. Applications Marketers can take advantage of rational ... between them, then it is more rational for a consumer to just take his chances on whichever of the two ... of issues that a person needs to consider in order to make a rational decision between candidates ... of rational ignorance was drawn from studies of voter apathy , which reached particularly ... with the importance of political choices. Additionally, rational ignorance is scrutinized for its ... teams ced econ 8 3No29.pdf The Theory of Rational Ignorance http www.gmu.edu departments economics bcaplan ratirnew.doc Rational Ignorance vs. Rational Irrationality http www.jstor.org view 00129682 di952648 95p0129t 0 Rational Behavior Under Complete Ignorance account needed http bbs.cenet.org.cn uploadImages 200332715474972903.pdf Rational Irrationality Some Economics of Self Management http dx.doi.org doi 10.1016 j.jet.2005.02.005 Would Rational Voters Acquire Costly Information? http dx.doi.org 10.1007 s00182 006 0051 4 Rational Ignorance and Voting Behavior DEFAULTSORT Rational Ignorance ... more details
Refimprove date July 2007 Rational Recovery and Rational Recovery Systems, Inc. is a commercial vendor ... designed as a direct counterpoint to Alcoholics Anonymous AA and twelve step program s. Rational ... drug addictions. Rational Recovery is a commercial trade mark, along with the Addictive Voice Recognition ..., and lectures. The Rational Recovery program is based on the premise that the addict both desires and is capable of permanent, planned abstinence . However, the Rational Recovery program recognizes ... as greatly enhanced, by the substance. ref name Beck See Beck, et al. ref This ambivalence is the Rational ..., and determination to remain abstinent by constantly reminding themselves of the rational basis ... themselves and their rational minds from a bodily impulse that has no regard for responsibility ... voice, not the other way around. In his book, Rational Recovery , Trimpey calls the addict s addictive ... easier to say no to the addictive voice, than to give in. Moreover, this separation of the rational ..., Australia and the UK. These facilities base their programs on the success of Rational emotive ..., as well as for substance abuse. While Rational Recovery and AA promote abstinence, the programs use radically different strategies. Rational Recovery repeatedly claims that there is no better time ... is contradictory to never using again. That is, Rational Recovery says, AA proposes that if you ... 2011 . Rational Recovery does not regard alcoholism as a disease , but rather a voluntary behavior. Rational Recovery discourages adoption of the forever recovering drunk persona. There are no Rational ... , and Moderation Management p. 22 . Trimpey asserts that some have accused Rational Recovery of being anti religious. The Rational Recovery FAQ states Quotation Rational Recovery has voiced the conscientious ... What is Rational Recovery? archiveurl http web.archive.org web 20080107024602 http www.rational.org faq.html Bot retrieved archive archivedate 2008 01 07 ref Rational Recovery claims to remain neutral ... more details
Unreferenced date December 2009 Orphan date December 2009 Infobox Album Name Total Rational Type Compilation Artist Rational Youth Cover Released 1994 Recorded 1981 82 Genre Synthpop Length Label Rams Horn Producer Pat Deserio Reviews Last album Malade br 1985 This album Total Rational br 1996 Next album All Our Saturdays 1981 1986 br 1996 Total Rational is a Rational Youth compilation album, released without authorisation from the band by the Netherlands Dutch label Rams Horn, who had previously released the Cold War Night Life album and some singles on license from YUL records. Total Rational contains the entire Cold War Night Life album, single tracks and extended mixes, including the unauthorised extended remix of Dancing On The Berlin Wall . Unfortunately, the running order is much altered from the original album also, it appears to have been mastered from vinyl copies. Track listing Saturdays in Silesia 4 03 Close to Nature 4 35 Ring the Bells 5 10 Power Zone 3 36 Dancing on the Berlin Wall 4 33 Beware the Fly 4 03 City of Night 3 48 Just a Sound in the Night 4 49 Pile ou Face 5 37 Le Meilleur des Mondes 2 54 Cite Phosphore 4 00 Saturdays in Silesia extended version 7 14 Dancing on the Berlin Wall Dutch 1984 extended edit 5 43 City of Night danse mix 7 08 Personnel Tracy Howe vocals, synthesizers Bill Vorn synthesizers, programming, vocoder Kevin Komoda synthesizers 1990s pop album stub Category 1994 compilation albums ... more details
Rational Synergy is a software tool that provides software configuration management SCM capabilities for all artifacts related to software development including source code , documents and images as well as the final built software executables and libraries. Rational Synergy also provides the repository for the http www 01.ibm.com software awdtools change Rational change management tool known as Telelogic change Rational Change . Together these two tools form an integrated configuration management and change management environment that is used in software development organizations that need controlled SCM processes and an understanding of what is in a build of their software. The name Synergy refers to its database level integration with Change Management that provides views into what is in a build in terms of defects History Synergy began life in 1988 as a research project for computer aided software engineering by software developer Pete Orelup at Computers West of Irvine California. Computers West was supporting itself through contract software development and an application for finance and insurance at automobile dealerships on the Pick OS, and probably had fewer than 10 employees. In 1989, the company decided to pursue development of a software configuration management and version control product, renamed itself CaseWare, and hired three more developers, Alan Wright, Kris Meissner, and Greg Holmberg. The system was re imagined as a platform for building SCM systems running ... 2008 08 27. ref Under Telelogic, Continuus was renamed to Synergy. In 2008 IBM Rational Software Rational announced that it had purchased Telelogic. ref cite web url http www 01.ibm.com software rational ... 08 27. ref Rational Synergy is now part of the IBM Rational family of SCM tools. Notes reflist 2 External links http www.ibm.com support docview.wss?uid swg27016565 Rational Synergy Information Centers Category Configuration management de Rational Synergy fr Rational Synergy ... more details
merge to cyclotomic field date January 2012 discuss Talk cyclotomic field Gaussian rational In mathematics , a Gaussian rational number is a complex number of the form p qi , where p and q are both rational number s. The set of all Gaussian rationals forms the Gaussian rational field mathematics field , denoted Q i , obtained by adjoining the imaginary number i to the field of rationals. It thus provides an example of an algebraic number field , which is both a quadratic field and a cyclotomic field since i is a 4th root of unity . Like all quadratic fields it is a Galois extension of Q with Galois group cyclic group cyclic of order two, in this case generated by complex conjugation , and is thus an abelian extension of Q , with conductor algebraic number theory conductor 4. The field of Gaussian rationals is neither ordered field ordered nor topologically complete space complete . The Gaussian integer s Z i form the ring of integers of Q i . References Ian Stewart mathematician Ian Stewart , David O. Tall , Algebraic Number Theory , Chapman and Hall , 1979, ISBN 0412138409. Chap.3. Category Cyclotomic fields Numtheory stub fr Rationnel de Gauss it Intero di Gauss Campo dei quozienti ... more details
Rational Youth is a Canada Canadian new wave music New Wave band, originally formed in 1981 in Montreal , Quebec , by synthesizer wizards Tracy Howe, former singer and drummer for Montreal punk band The Normals , and Bill Vorn, both of whom idolized the German synthpop pioneers Kraftwerk . Among the band s notable singles were City of Night , Saturdays in Silesia and Dancing on the Berlin Wall, from their debut album, Cold War Night Life 1982 and No More and No Less from Heredity 1985 . In 1999, Rational Youth, with a new lineup of original frontman Howe and new keyboard players Jean Claude Cutz and Dave Rout, released its first album after fourteen years, To the Goddess Electricity . Rational Youth toured throughout Scandinavia over the next two years, playing its final concert on November 3, 2001, at the Tinitus Festival in Stockholm, Sweden. Original members Tracy Howe and Bill Vorn came together again in 2009, recording a new version of their 1982 hit Dancing On The Berlin Wall in honour of the twentieth anniversary of the fall of the Wall. Personnel Tracy Howe vocals, synthesizers, guitar Bill Vorn synthesizers, vocoder Kevin Komoda Discography Singles I Want to See the Light & Coboloid Race I Want to See the Light Coboloid Race 12 , 1981 Cite Phosphore 7 , 1982 City of Night 12 , 1982 Saturdays in Silesia 7 12 , 1982 In Your Eyes 7 12 , 1983 Dancing on the Berlin Wall Dutch ... And Blood CD single, 1999 Albums Studio albums Cold War Night Life album, 1982 CD release 1997 Rational Youth EP Rational Youth mini album, 1983 Heredity album Heredity album, 1985 never released on CD To the Goddess Electricity album, 1999 Compilations Total Rational unauthorised Dutch compilation ... www.yulrecords.ca YUL Records Rational Youth s label website Category Musical groups established in 1981 Category Canadian New Wave musical groups Category Synthpop groups Canada band stub pl Rational Youth sv Rational Youth ... more details
In algebraic geometry , a branch of mathematics , a rational surface is a surface birational geometry birationally equivalent to the projective plane , or in other words a rational variety of dimension two. Rational surfaces are the simplest of the 10 or so classes of surface in the Enriques Kodaira classification of complex surfaces, and were the first surfaces to be investigated. Structure Every non singular rational surface can be obtained by repeatedly blowing up a minimal rational surface. The minimal rational surfaces are the projective plane and the Hirzebruch surfaces &Sigma sub r sub for r 0 or r &ge 2. Invariants The plurigenera are all 0 and the fundamental group is trivial. Hodge diamond table border 0 cellpadding 2 cellspacing 0 tr th th th th th 1 th tr tr th th th 0 th th th th 0 th tr tr th 0 th th th th 1 n th th th th 0 th tr tr th th th 0 th th th th 0 th tr tr th th th th th 1 th tr table where n is 0 for the projective plane, and 1 for Hirzebruch surface s and greater than 1 for other rational surfaces. The Picard group is the odd unimodular lattice I sub 1, n sub , except for the Hirzebruch surface s &Sigma sub 2 m sub when it is the even unimodular lattice II ... sub 2 sub the irregularity and second plurigenus both vanish is rational. This is used in the Enriques Kodaira classification to identify the rational surfaces. harvtxt Zariski 1958 proved that Castelnuovo ... that any unirational complex surface is rational, because if a complex surface is unirational then its irregularity and plurigenera are bounded by those of a rational surface and are therefore all 0, so the surface is rational. Most unirational complex varieties of dimension 3 or larger are not rational ... surface s that are not rational. At one time it was unclear whether a complex surface such that q and P sub 1 sub both vanish is rational, but a counterexample an Enriques surface was found by Federigo Enriques . Examples of rational surfaces Bordiga surface s A degree 6 embedding of the projective ... more details
In mathematics , in particular the subfield of algebraic geometry , a rational map is a kind of partial function between algebraic varieties . This article uses the convention that varieties are irreducible. Definition Formal definition Formally, a rational map math f colon V to W math between two varieties is an equivalence class of pairs math f U, U math in which math f U math is a morphism of varieties from an open set math U subset V math to math W math , and two such pairs math f U, U math and math f U , U math are considered equivalent if math f U math and math f U math coincide on the intersection math U cap U math this is, in particular, vacuous truth vacuously true if the intersection is empty, but since math V math is assumed irreducible, this is impossible . The proof that this defines an equivalence relation relies on the following lemma If two morphisms of varieties are equal on any open set, then they are equal. math f math is said to be birational if there exists a rational map math g colon W to V math which is its inverse, where the composition is taken in the above sense. The importance of rational maps to algebraic geometry is in the connection between such maps and maps ... math . Even a cursory examination of the definitions reveals a similarity between that of rational map and that of rational function in fact, a rational function is just a rational map whose range is the projective line. Composition of functions then allows us to pull back rational functions along a rational map, so that a single rational map math f colon V to W math induces a homomorphism of fields ... of projective varieties with dominant rational maps over a fixed base field, for example math mathbb ... of varieties which requires a globally defined morphism to witness the isomorphism, not merely a rational ... which does not change the field, a manifestation of the fact that a rational map depends only ... , isomorphic to that of math mathbb P 2 k math . Note that at no time did we actually produce a rational ... more details
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