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 ... 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 eom.springer.de R r077630.htm Springer Online Reference Works Rational Representation Category Representation theory of algebraic groups math stub ... 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
In number theory , a K rational point is a point on an algebraic variety where each coordinate of the point belongs to the field K . This means that, if the variety is given by a set of equations f sub i sub x sub 1 sub , ..., x sub n sub 0, j 1, ..., m then the K rational points are solutions x sub 1 sub , ..., x sub n sub K sup n sup of the equations. In the parlance of morphisms of schemes, a K rational point of a scheme X is just a morphism Spec K X . The set of K rational points is usually denoted X K . If a scheme mathematics scheme or variety X is defined over a field mathematics field k , a point x X is also called rational point if its residue field k x is isomorphic to k . Rational points of varieties constitute a major area of current research. For an abelian 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 fields. See also Arithmetic dynamics References unreferenced date July 2008 numtheory stub Category Algebraic geometry Category Diophantine geometry fr point rationnel ... more details
One source date February 2010 Infobox company name Rational Software logo Image RationalSoftware.png 180px Rational Software type genre fate Acquired by IBM predecessor successor foundation 1981 founder ... Rational Machines was founded by Paul Levy and Mike Devlin entrepreneur Mike Devlin in 1981 to provide ... 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 Environment was an integrated ... that could execute on a range of computing platforms. The Rational Environment was organized around ... recompilation times proportional to system size rather than size of change, the Rational Environment ..., which encouraged developer driven unit 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 in particular. The horizontally microprogrammed R1000 .... The company s name was later changed from Rational Machines to Rational to avoid emphasizing this proprietary hardware. When date February 2011 Rational provided code generators and the cross debuggers ... architecture descriptions PQCC . Organization Rational s field Practices underlying the later Rational ... base. Second generation products In 1990, Rational launched three parallel development efforts re implementation of the Rational Environment for Ada to run on Unix based workstations from Sun Microsystems Sun and IBM , development of a comparable Rational Environment for C to run on Unix ... that supported a graphical notation developed 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 ... developed and released by Rational s Bangalore team. Rose 1.0 was introduced at OOPSLA in 1992 ... Rational s late re entry into the market for object oriented modeling tools. Rose 2.0 ran on Windows ... 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 Cite book 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 number 3 date Summer 2001 pages 365 372 url http aca.metapress.com index HNUE6LUPB5GKDNYC.pdf 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 number 2 date June, 1982 ref The received orthodoxy of mental health professionals for more than a century views all suicides as irrational ... title Rational suicide? author DJ Mayo publisher Journal of Loss and Trauma year 1998 doi 10.1080 10811449808414441 ref See also Rational depression References Reflist Category Suicide Death 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
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
In mathematics , a rational function is any function mathematics function which can be written as the ratio ... by the function are necessarily rational. Definitions Image RationalDegree2byXedi.gif thumb right 250px Rational function of degree 2 br math y frac x 2 3x 2 x 2 4 math In the case of one variable, math x , math , a function is called a rational function if and only if it can be written in the form ... 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 irrational is not generally ... a rational expression . The math x math need not be a variable. In abstract algebra the math x math is called an indeterminate . A rational equation is an equation in which two rational expressions ... thumb right 250px Rational function of degree 3 br math y frac x 3 2x 2 x 2 5 math The rational ... pm sqrt 5 math . The rational function math f x frac x 2 2 x 2 1 math is defined for all real number ... 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 2 ... 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 is equal ... excepting division by the zero polynomial of two rational functions is itself a rational function ... unless care is taken. Taylor series The coefficients of a Taylor series of any rational ... the rational function equal to its Taylor series and collecting like terms . For example, math ... quad for k ge 2. math Conversely, any sequence that satisfies a linear recurrence determines a rational ..., since by using partial fractions partial fraction decomposition we can write any rational function ... In complex analysis , a rational function math f z frac P z Q z math is the ratio of two polynomials ... where Q z is 0 . The degree of a rational function is the maximum of the degree of a polynomial ... more details
Multiple issues unreferenced October 2010 notability October 2010 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 12 , 1984 unauthorised extended ... Studio albums Cold War Night Life album, 1982 CD release 1997 Rational Youth EP Rational Youth mini ..., 1999 Compilations Total Rational unauthorised Dutch compilation CD, 1994 All Our Saturdays 1981 1986 ... External links http www.rationalyouth.net Rational Youth s Official Web site http www.yulrecords.ca YUL Records. Rational Youth s label web site Category Musical groups established in 1981 Category Canadian New Wave musical groups Category Synthpop Canada band stub pl Rational Youth ... more details
In mathematics , a rational number is any number that can be expressed as the quotient or fraction a b ... is a rational number. The set mathematics set of all rational numbers is usually denoted by a boldface ... . The decimal expansion of a rational number always either terminates after finitely many numerical .... Moreover, any repeating or terminating decimal represents a rational number. These statements hold ... other integer radix base . A real number that is not rational is called irrational number irrational ... expansion of an irrational number continues forever without repeating. Since the set of rational numbers ... every real number is irrational. The rational numbers can be Formalism mathematics formally defined ... 0. In abstract algebra , the rational numbers together with certain operations of addition and multiplication ... is the field of algebraic number s. In mathematical analysis , the rational numbers form a dense set dense subset of the real numbers. The real numbers can be constructed from the rational numbers ... decimal s. Zero divided by any other integer equals zero, therefore zero is a rational number but division by zero is undefined . Terminology The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers. In mathematics, the adjective rational often means that the underlying field mathematics field considered is the field Q of rational numbers. For example, a rational integer is an algebraic integer which is also a rational number, which is to say, an ordinary integer, and a rational matrix is a matrix whose coefficients are rational numbers. Rational polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a polynomial over the rationals . However, rational function does not mean the underlying field is the rational numbers, and a algebraic curve rational algebraic curve is not an algebraic curve with rational coefficients HI Arithmetic see also Fraction mathematics Arithmetic with fractions ... 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
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 www.geocities.com rdsmarb rational fideism.html Domenic Marbaniang, Epistemics of Divine Reality , Internet Archive, http www.archive.org details EoistemicsOfDivineReality , 2009 ref faith must be corroborated with reason and experience in order to gain knowledge. See also Philosophical Theology Philosophy of Religion Fideism 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 , Internet Archive,2009 http www.archive.org details EpistemicsOfDivineReality Rational Fideism http www.scribd.com doc 48416873 Rational Fideism Category Epistemics Category Epistemological theories Category Rationalism philo stub ... more details
Economics sidebar NPOV language date October 2009 No footnotes date November 2009 Rational expectations ... formulation is that rational expectations are model consistent expectations , in that the agents ... The rational expectations assumption is used in many contemporary macroeconomic model s, game theory and other applications of rational choice theory . Since most macroeconomic models today study decisions ... made about expectations see Cobweb model . To assume rational expectations is to assume ... future income from that stock. Theory Rational expectations theory defines this kind of expectations ... from the market economic equilibrium equilibrium results. As a result, rational expectations ... is the equilibrium price in a simple market, determined by supply and demand . The theory of rational ... ex ante the actual price is equal to its rational expectation math P P epsilon math math E P P math where math P math is the rational expectation and math epsilon math is the random error term, which has an expected value of zero, and is independent of math P math . Rational expectations theories .... This may be regarded as unrealistic surely rational individuals would sooner or later realize the trend and take it into account in forming their expectations? The hypothesis of rational expectations ... from the expected values. The rational expectations hypothesis has been used to support some ... rate, not employment. This is a distinctly New Classical outcome. During the 1970s rational ... with the Lucas critique . However, rational expectations theory has been widely adopted throughout ... Fischer . Rational expectations theory is the basis for the efficient market hypothesis efficient ... time then the more interesting implications of the theory of rational expectations do not apply. In fact ... posited by rational expectations theorists. A further problem relates to the application of the rational ... holds true for rationality assumptions Even if all individuals have rational expectations, the representative ... 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
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. 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 Quotes 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 definition References references Category Aristotle Category Definition Category History of philosophy de Animal rationale philo stub ... more details
Rational depression is reason guided psychological depression . It can arise from adverse life circumstances. Such depression may not be responsive to traditional depression therapies. ref citation title Calling It Quits When Patients or Proxies Request to Withdraw or Withhold Life Sustaining Treatment After Spinal Cord Injury author KL Kirschner publisher Topics in Spinal Cord Injury Rehabilitation year 2007 url http thomasland.metapress.com index M4XW957794653825.pdf issn 1082 0744 doi 10.1310 sci1303 30 ref Traditionally, cognitive therapist s have asserted that there is no such thing as a rational depression since, while sadness can be a natural response to loss or disappointment, depression results from distorted thoughts. ref cite book title When bad things happen to rational people Cognitive therapy in adverse life circumstances author S Moorey publisher Frontiers of cognitive therapy year 1996 ref Christian leaders such as John Cassian , Isidore of Seville , Bede recognized the distinction between irrational depression which emerged frequently without any antecedent stressor and was therefore seemingly from God and rational depression. ref citation title Blue is the new black author Gordon Parker publisher World Journal of Biological Psychiatry year 2009 volume 10 issue 3 pages 177 179 doi 10.1080 15622970903083194 ref Research indicates that although disability is associated with particular socially mediated stressors, there is no theoretical or empirical evidence to indicate that depression and its role in the right to die is dynamically different, more natural, or more reasonable for disabled people than for non disabled people. ref citation title Depression in the Context of Disability and the Right to Die issn 1386 7415 volume 25 number 3 date May, 2004 doi 10.1023 B META.0000040058.24814.54 url http www.springerlink.com content q82301lp607104h6 publisher Theoretical Medicine and Bioethics ref Rational depression sometimes leads to rational suicide . ref ... more details
For other forms of egoism Egoism disambiguation Individualism sidebar In ethics ethical philosophy , rational selfishness also called rational egoism 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 ... discusses a theory that she called Rational egoism or more specifically Rational self interest . She ... 2006 , 2a. ref Thus, her view is a conjunction of both rational egoism in the standard sense and ethical ... ref Her book The Virtue of Selfishness 1964 explains in depth the concept of rational egoism. According to Rand s ethical egoism, every rational individual s own life should be his or her own highest ... more details
Rational addiction is the hypothesis that addictions to heroin, tobacco, television, etc. can be usefully modeled as specific kinds of rational, forward looking, optimal consumption plans. The canonical ..., G. and K. Murphy 1988 A theory of rational addiction . Journal of Political Economy , 96, 675 700 ..., religion, or food the article tried to reconcile addictions with the standard rational choice framework ... or co authored peer reviewed articles on rational addiction theory indicates that the researchers see ... a sequence of rational, welfare maximizing choices ref name Melberg 00 Melberg, H. O. and O. J. Rogeberg 2010 Rational Addiction Theory A Survey of Opinions . Journal of Drug Policy Analysis , Vol ... his future health. Rational choice amounts to comparing the benefit of smoking that cigarette to the total ... to tobacco have gone up. A sizeable econometric literature has developed on rational addiction, often reporting evidence in favor of rational addiction. For example, Gruber and Koszegi 2001 ref name gruber 00 Gruber, J. and B. Koszegi 2001 Is addiction rational? Theory and evidence . Quarterly Journal ... of Health Economics , 23, 1117 33. ref show, however, that the empirical version of the rational ... orphanides 00 Athanasios Orphanides Orphanides, Athanasios and Zervos, David 1995 Rational Addiction with Learning and Regret . Journal of Political Economy , 103 4 , 739 758. ref you make a rational ..., Enelberg J. and Feichtinger, Gustav 1993 Cyclical Consumption Patterns and Rational Addiction ... discounting ref name gruber 00 Gruber, J. and B. Koszegi 2001 Is addiction rational? Theory and evidence ... for and against drug prohibition drug liberal implications for policy. The fully rationalrational ... t impose externalities on others e.g., second hand smoke . Criticism Criticism of rational addiction ... and the Case of Rational Addiction Theories . Philosophy of Science , 71, 263 285. ref , and argues ... Category Rational choice theory ... more details
In mathematics , a rational variety is an algebraic variety , over a given field mathematics field K , which is birationally equivalent to projective space of some dimension over K . This is a question on its function field is it up to isomorphism math K x 1, dots , x n , math the field of all rational ... question asks whether a given field extension is rational , in the sense of being up to isomorphism the function field of a rational variety such field extensions are also described as purely ... L math isomorphic to a rational function field over math K math in the number of indeterminates ... concerns subfields L of K X , the rational functions in the single indeterminate X , for which the degree K X L is finite, and with K algebraically closed . Any such field is also rational, i.e. L K F for some rational function F . In geometrical terms this states that a non constant birational mapping ... variety is one covered by a rational variety, so that on the function field level it has a function ... s problem shows that for algebraic curves, rational and unirational are the same, and Castelnuovo s theorem implies that for complex surfaces unirational implies rational, because both are characterized ... examples Zariski surface s in characteristic p 0 that are unirational but not rational. harvtxt Clemens Griffiths 1972 showed that a cubic three fold is in general not a rational variety ... unirational 3 folds with non trivial torsion in their third cohomology group, which implies that they are not rational ... of many classical results, beginning with the case of cubic surface s which are rational varieties ... 2008 ref . See also Rational surface Severi Brauer variety Birational geometry Rationally connected ... which are not rational doi 10.1112 plms s3 25.1.75 id MathSciNet id 0321934 year 1972 journal ... last2 Smith first2 Karen E. last3 Corti first3 Alessio title Rational and nearly rational varieties ... BF01457099 . citation first R. G. last Swan title Invariant rational functions and a problem of Steenrod ... more details
Primary sources date October 2009 The concept known as Rational irrationality was introduced in The Myth of the Rational Voter . It was developed by economist Bryan Caplan to reconcile the widespread existence of irrationality irrational behavior with the assumption of rationality made by mainstream economics and game theory . The original purpose of the concept was to explain how allegedly detrimental policies could be implemented in a democracy, and unlike conventional public choice theory , Caplan posited that bad policies were selected by voters themselves. Essentially Rational Irrationality is a term used to describe a state in which a decision, of minimal immediate impact on a person, such as voting allows that person to make a decision which benefits him by adhering to his own rational adherence to personal belief without regard to the irrationality of the decisions cost benefit analysis in the big picture. Explanation A voter has very small chance of influencing policy. Therefore, the vote he casts is of little consequence to him. There is no incentive for a voter to invest time and energy analyzing the consequences of a policy. However, a voter derives some benefits from voting. It allows him to signal the kind of person he is, who he associates with. If there are no consequences in being mistaken about the effect of a policy, it becomes rational to hold the most pleasant opinion, or the one most conducive to socialization. Democracy as a Commons Problem Empty section date July 2010 Implications Empty section date July 2010 Potential Applications Category Economics Category Logic Category Voting systems Category Game theory ... more details
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
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 .... Two arbitrary rational maps math f colon V to W, g colon W to X math can not necessarily be composed ... that the range of any of them is dense, since open sets are dense in a variety . Since dominant rational maps can be composed, we can go further and talk about their inverses a dominant rational map 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 ... 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 or codomain, more categorically precisely 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 ... a rational map , in that there exist varieties which are birational but not isomorphic. The usual ... subset which does not change the field, a manifestation of the fact that a rational map depends ... a rational map, though tracing through the proof of the theorem it is possible to do so. See also Birational ... more details
Infobox company company name Rational AG company logo Image Rational AG Logo.svg 240px company type Aktiengesellschaft traded as FWB RAA foundation start date 1973 location Landsberg am Lech , Germany key people G nter Blaschke small chief executive officer CEO and chairman of the executive board small , Siegfried Meister small founder and chairman of the supervisory board small industry Manufacturing products Catering food steamer s and multi purpose oven s revenue 350.1 million small 2010 small ref name AR2010 cite web url http www.rational online.com fileadmin mediapool pdf Gesch C3 A4ftsberichte RAA GB2010 EN.pdf title Annual Report 2010 accessdate 3 April 2011 publisher Rational ref operating income 105.8 million small 2010 small ref name AR2010 net income 79.8 million small 2010 small ref name AR2010 assets 305.7 million small end 2010 small ref name AR2010 equity 230.3 million small end 2010 small ref name AR2010 num employees 1,090 small end 2010 small ref name AR2010 homepage http www.rational online.com GB en www.rational online.com intl yes Rational AG is a Germany German manufacturer of industrial food steamer s and oven s for catering purposes. The firm is headquartered in Landsberg am Lech, Germany. ref cite web url http investing.businessweek.com research stocks snapshot snapshot.asp?ric RAAG.DE title Snapshot of Rational AG RAAG accessdate 2009 03 29 ref Since 4 March 2009 it is a constituent of the MDAX index. History In 1973, the Rational GmbH was founded by Mr. Siegfried Meister. Three years later, the company started manufacturing combi steamer s for commercial kitchens. In the year 2000, the company went public and therefore is now named Rational Joint stock company AG . Besides the combi steamer , the company also sells the SelfCooking Center since 2004 and the VarioCooking Center since 2005, sold through FRIMA brand . Starting in 1991, the following ... Category Cooking appliance brands Category British Royal Warrant holders de Rational AG ... more details
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