about an algorithm Israeli food corporation Osem company In mathematical optimization , the orderedsubsetexpectationmaximization OSEM method NOT algorithm is an iterative method that is used in computed tomography . In applications in medical imaging , the OSEM method is used for positron emission tomography , for single photon emission computed tomography , and for Computed tomography X ray computed tomography . The OSEM method is related to the expectationmaximization EM method of statistics. The OSEM method is also related to methods of filtered back projection . References Hudson, H.M., Larkin, R.S. 1994 Accelerated image reconstruction using ordered subsets of projection data , IEEE Trans. Medical Imaging , 13 4 , 601&ndash 609 doi 10.1109 42.363108 Hutton, Brian F., Hudson, H. Malcolm, Beekman, Freek J. 1997 A clinical perspective of accelerated statistical reconstruction , European Journal of Nuclear Medicine and Molecular Imaging , 24 7 , 797&ndash 808 doi 10.1007 BF00879671 Ing Tsung Hsiao and Hsuan Ming Huang 2010 An accelerated ordered subsets reconstruction algorithm using an accelerating power factor for emission tomography , Physics in Medicine and Biology 55 599. doi 10.1088 0031 9155 55 3 003 External links http osem.s pla.net Home page with IEEE paper and more information http osem.s pla.net Software Category Optimization algorithms and methods Category Medical statistics Category Medical imaging statistics stub ... more details
of a portfolio. The EM algorithm and its faster variant Orderedsubsetexpectationmaximization ...In statistics , an expectationmaximization EM algorithm is an iterative method for finding maximum likelihood ... This idea is further extended in generalized expectationmaximization GEM algorithm, in which one ... download?doi 10.1.1.9.9735&rep rep1&type pdf The ExpectationMaximization Algorithm , by Frank Dellaert, gives an easier explanation of EM algorithm in terms of lowerbound maximization. http www.seanborman.com publications EM algorithm.pdf The ExpectationMaximization Algorithm A short tutorial ... Class hierarchy in C GPL including Gaussian Mixtures DEFAULTSORT Expectationmaximization ... depends on unobserved latent variable s. The EM iteration alternates between performing an expectation E step, which computes the expectation of the Likelihood function Log likelihood log likelihood evaluated using the current estimate for the parameters, and a maximization M step, which computes ... seeks to find the MLE of the marginal likelihood by iteratively applying the following two steps Expectation ... theta mathbf X , mathbf Z right , math Maximization step M step Find the parameter that maximizes ... values are what is computed in the M step. Properties Speaking of an expectation E step is a bit ... circumstances, it is convenient to view the EM algorithm as two alternating maximization steps. ref ... be viewed as Expectation step Choose q to maximize F math q t operatorname arg ,max q F q, theta t math Maximization step Choose to maximize F math theta t 1 operatorname arg ,max theta F q t , theta ... mr 1452026 ref Additionally EM can be utilised with constrained estimation techniques. Expectation conditional maximization ECM replaces each M step with a sequence of conditional maximization CM steps ... likelihood ratio and the divergence. Obtaining this Q function is a generalized E step. Its maximization ... first Yasuo title The EM algorithm Surrogate likelihood maximization using logarithmic information ... more details
Maximization or maximisation can refer to Exaggeration Maximization in the sense of exaggeration Entropy maximizationMaximization economics Profit maximization Utility maximization problem Budget maximizing model Shareholder value maximization Optimization mathematics Expectationmaximization algorithm statistics See also Minimization disambiguation disambig ... more details
wiktionary expectationExpectation may refer to Expectation epistemic Expected value , in mathematical probability theory Expectation value quantum mechanics Expectationmaximization algorithm , in statistics Great Expectations , a 1860 61 novel serial by Charles Dickens Expectations album , a 1971 album by Keith Jarrett disambig Short pages monitor This long comment was added to the page to prevent it from being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Long comment. Please do not remove the monitor template without removing the comment as well. de Erwartung nl Verwachting tr Beklenti ... more details
Expert subject Mathematics date November 2008 An entropy maximization problem is a convex optimization problem of the form maximize math f 0 vec x sum i 1 n x i log x i math subject to math A vec x leq b, quad mathbf 1 T vec x 1 math where math vec x in mathbb R n math is the optimization variable, math A in mathbb R m times n math and math b in mathbb R m math are problem parameters, and math mathbf 1 math denotes a vector whose components are all 1. See also Principle of maximum entropy External links cite book last Boyd first Stephen coauthors Lieven Vandenberghe title Convex Optimization publisher Cambridge University Press date 2004 pages p. 362 isbn 0521833787 url http www.stanford.edu boyd cvxbook bv cvxbook.pdf accessdate 2008 08 24 Category Mathematical optimization Category Convex optimization mathapplied stub ... more details
refimprove date October 2006 In economics , profit maximization is the short run or long run process by which a firm determines the price and output economics output level that returns the greatest Profit economics profit . There are several approaches to this problem. The total revenue&ndash total cost perspective relies on the fact that profit equals revenue minus cost and focuses on minimizing this difference, and the marginal revenue &ndash marginal cost perspective is based on the fact that total profit reaches its maximum point where marginal revenue equals marginal cost. Basic definitions Any costs incurred by a corporation firm may be classed into two groups fixed costs and variable costs . Fixed costs, which occur only in the short run, are incurred by the business at any level of output, including zero output. These may include equipment maintenance, rent, wages of employees whose numbers cannot be increased or decreased in the short run, and general upkeep. Variable costs change with the level of output, increasing as more product is generated. Materials consumed during production often have the largest impact on this category, which also includes the wages of employees who ... total cost perspective Image profit max total small.svg right thumb 250px Profit Maximization The Totals ... right thumb 250px Profit maximization using the marginal approach An alternative perspective relies ... maximization solutions involve the use of game theory . Case in which maximizing revenue is equivalent ... by maximizing revenues. Changes in total costs and profit maximization A firm maximizes profit by operating ..., which is equivalent to picking a preferred quantity to produce and sell . The profit maximization ... maximization The general rule is that firm maximizes profit by producing that quantity of output where marginal revenue equals marginal costs. The profit maximization issue can also be approached from ... ProfitMaximizationInPerfectCompetition Profit Maximization in Perfect Competition ... more details
In probability theory , a conditional expectation also known as conditional expected value or conditional ... distribution . The concept of conditional expectation is extremely important in Andrey Kolmogorov Kolmogorov ... of conditional probability itself is actually defined in terms of conditional expectation. Introduction Let X and Y be discrete random variable s, then the conditional expectation of X given the event ... for individual values of y since both sides are zero , it should hold for any measurable subset ... to define both conditional expectation and conditional probability. Formal definition Let math ... B subseteq mathcal A math . Then a conditional expectation of X given math scriptstyle mathcal ... E X mathcal B math is simply the name of the conditional expectation function. Discussion A couple ... the required property that a conditional expectation must satisfy. The required property has the same form as the last expression in the Introduction section. Existence of a conditional expectation ... of the same conditional expectation will only differ on a null set set of probability zero . The ... expectation math scriptstyle E X mathcal B math over a finer grained algebra math scriptstyle ... the conditional expectation using the pre image of with respect to Y math mathcal B sigma Y Y 1 left Sigma right Y 1 S S in Sigma math This suffices to ensure that the conditional expectation is &sigma Y measurable. Although conditional expectation is defined to condition on events in the underlying ... with respect to the Borel algebra Borel algebra on 0,1 . Note that the expectation of this random ... operatorname P A mathcal B math is the conditional expectation of the indicator function for A math ... for all &omega &Omega . An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation. For the trivial sigma algebra math mathcal ... of conditional expectation that we provided above, the fact that Y is a real random variable is irrelevant ... more details
For the 2006 South Korean television drama Great Inheritance Infobox Korean film name The Greatest Expectation image Greatest Expectation movie poster.jpg caption Theatrical poster hangul hanja linktext linktext rr Widaehan yusan mr Widaehan yusan director Oh Sang hoon producer writer Lee Won hyeong br Lee Won jae br Oh Sang hoon starring Lim Chang jung br Kim Seon ah br Kim Soo mi music Jo Seong woo cinematography editing Kyeong Min ho distributor CJ Entertainment released 24 October 2003 runtime 115 min. country South Korea language Korean budget The Greatest Expectation is a 2003 South Korea n romantic comedy film. External links http www.hancinema.net korean movie The Greatest Expectation.php?ac 1 The Greatest Expectation at HanCinema Cinema of Korea DEFAULTSORT Greatest Expectation, The Category 2003 films Category Korean language films Category South Korean films Category South Korean romantic comedy films SouthKorea film stub id The Greatest Expectation ... more details
Distinguish Expected utility Unreferenced date December 2009 Context date October 2008 A qualification introduced by Jeremy Bentham , to distinguish between two different types of utility utilities , or, rather, sources of utility for utility, being identical to hedonism pleasure , remains always qualitatively the same . Expectation utilities are future regarding, and thus imply desires and beliefs natural utilities are not. Expectation utilities allow for long term projects, which provide a higher proportion of utility than the natural utilities favored by those agents that can only pursue their immediate interest. DEFAULTSORT Expectation Utilities Category Utility Econ theory stub ... more details
Unreferenced date December 2009 In English law , the concept of legitimate expectation arises from administrative law , a branch of public law . In proceedings for judicial review , it applies the principles of fairness and reasonableness to the situation where a person has an expectation or interest in a public body retaining a long standing practice or keeping a promise. Discussion The traditional constraint on a public body has been the test of irrationality , also known as Wednesbury unreasonableness ..., when faced with claims of a legitimate expectation, the courts have begun to require public officials ... expectation to protect the public interest in consistency and certainty through a test of proportionality ... a decision is taken if he or she has a legitimate expectation of being heard. But the fact that a person ... expectation unless fairly specific representations as to policies affecting future conduct ... any change is finally decided and, if the citizen s expectation is real, the courts might require the public body to identify an overriding public interest to trump the particular expectation. This supplements ... or irrational. The idea of legitimate expectation has received sufficient mention both academically and in case law so as to effectively merit being referred to as the doctrine of legitimate expectation. Legitimate expectation has been said to be a new category of fairness, with the dicta of Bingham ... for a public authority to do so, with Legitimate expectation thus being the public law equivalent ... 2000 showed that legitimate expectation is recognized in cases where the relevant authority had made ... with contract and the doctrine of estoppel. The clearest mention of the doctrine of legitimate expectation ... on its own. Reference to legitimate expectation is also made in the early and famous GCHQ case ... expectation as being Legitimate or reasonable expectation may arise either from an express promise ... See also Legitimate expectation in Singapore Category English administrative law de Vertrauensschutz ... more details
About the concept of expectation as a thought or belief expectation in the context of probability theory and statistics expected value In the case of uncertainty , expectation is what is considered the most likely to happen. An expectation, which is a belief that is centered on the future , may or may not be realistic. A less advantageous result gives rise to the emotion of disappointment . If something happens that is not at all expected it is a surprise emotion surprise . An expectation about the behavior or performance of another person, expressed to that person, may have the nature of a strong request, or an order. Expectations of well being Richard Lazarus asserts that people become accustomed to positive or negative life experiences which lead to favorable or unfavorable expectations of their present and near future circumstances. Lazarus notes the widely accepted philosophical principle that happiness depends on the background psychological status of the person...and cannot be well predicted without reference to one s expectations. ref name Lazarus Lazarus, Richard. Emotion and Adaptation . Oxford University Press, 1991. ref With regard to happiness or unhappiness, Lazarus notes that people whose objective conditions of life are those of hardship and deprivation often make a positive assessment of their well being, while people who are objectively well off...often make a negative assessment of their well being. Lazarus argues that the most sensible explanation of this apparent ... that a person s expectation is directly linked to the Self fulfilling prophecy Self fulfilling Prophecy . Whether or not such an expectation was truthful or not has little or no effect on the outcome ... will see the expectation to its inevitable conclusion. There is an inherent danger in this kind ... of in terms of causality or cause and effect, the higher a person s expectation and the lower ... expectation div style column count 3 moz column count 3 webkit column count 3 Collective belief Delusion ... more details
In music cognition melodic expectation is the tendency for a person listening to a melody to have a feeling or Expectation epistemic expectation for what might come next in the melody. For example, if the ascending musical partial octave do re mi fa sol la ... is heard, listeners familiar with Western music will have a strong expectation to hear one more note, ti , to complete the octave. Leonard Meyer Leonard Meyer s Emotion and Meaning in Music 1956 is the classic text in music expectation. Meyer s starting point is the belief that the experience of music as a listener is derived from one s emotions and feelings about the music, which themselves are a function of relationships within the music itself. Meyer writes that listeners bring with them a vast body of musical experiences that, as one listens to a piece, conditions one s response to that piece as it unfolds. Meyer argued that music s evocative power derives from its capacity to generate, suspend, prolongate, or violate these expectations. Meyer models listener expectation in two levels. On a perceptual level, Meyer draws on Gestalt psychology to explain how listeners build mental representations of auditory phenomena. Above this raw perceptual level, Meyer argues that learning shapes and re shapes one s expectations over time. Implication Realization Model main Implication Realization Narmour s 1992 Implication Realization I R Model is a detailed formalization based on Meyer s work on expectation. The theory focuses on how implicative intervals set up expectations for certain realizations to follow. Extensions to the Implication ... an expectation , Margulis attempts to model intervalic local expectation as well as more deeply schematic global expectation. For this, Margulis relies on Lerdahl s and Jackendoff s GTTM 1983 ... is explicit and realizable, and yields quantitative output. The output melodic expectation at each ... Press. ISBN 0 19 505834 8. Margulis, E. H. 2005 . A Model of Melodic Expectation . Music Perception ... more details
, one cannot have a reasonable expectation of privacy in things held out to the public. A well known ... car. While a person may have a subjective expectation of privacy in his her car , it is not always ... do not have an objective expectation of privacy. ref See, e.g., Oliver v. United States . ref Privacy and search The expectation of privacy is crucial to distinguishing a legitimate, reasonable ... expectation of privacy 2 and that expectation of privacy must be reasonable, in the sense ... expectation that the evidence obtained would not be available to the public. In other words, the person ... , then no subjective expectation of privacy is assumed. ref Mark Tunick 1998 Practices and Principles ... Objectivity philosophy objectively would society at large deem a person s expectation of privacy to be reasonable ... reasonable expectation of privacy and thus no search when officers hovering ... a reasonable expectation of privacy in the contents of their e mail in United States v. Warshak , although ... more details
Infobox Album See Wikipedia WikiProject Albums Name Experiments In Expectation Album Experiments in Expectation Cover Type Album Artist Kill Sadie Released 2001 Genre Post hardcore Label Dim Mak Records This album Experiments In Expectations br 2001 Album ratings rev1 AbsolutePunk.net rev1Score 91 ref http www.absolutepunk.net showthread.php?t 334751 AbsolutePunk review ref rev2 Allmusic rev2Score rating 3 5 ref Allmusic class album id r655102 ref Experiments in Expectation was the final full length release album by Seattle based band Kill Sadie . It was released on Dim Mak Records in 2001. Band Photography by Shawn Brackbill . Track listing The Ivy League Donor Prescription Epidemic 6 21 The Laugh Track for Contemporary Music 2 46 Rebirth Through Adaptation 2 07 Erf The Place You Live 4 49 The Quieting Function of Mouth 3 50 The Surgeon s Muse 3 31 Untitled Number Three Hundred and Three 2 07 The Cocktail Party Effect 3 35 A Ride in the Contrifuge 3 38 An Antiquated Bluff 8 33 References Reflist 2000s punk album stub Category 2001 albums Category Kill Sadie albums ... more details
Infobox Album See Wikipedia WikiProject Albums Name Annihilation of Expectation Type ep Longtype Split album Artist The Knife Trade Through the Eyes of the Dead Cover Annihilation of Expectation.jpeg Released 2005 Recorded November 2004 Genre Metalcore Deathcore Length 36 17 Label Lovelost Records Producer Jamie King Reviews Chronology Through the Eyes of the Dead Last album The Scars of Ages EP br 2004 This album Annihilation of Expectation br 2005 Next album Bloodlust album Bloodlust br 2005 Annihilation of Expectation was a split recorded by melodic death metal band The Knife Trade who contributed the first five tracks and deathcore act Through the Eyes of the Dead who contributed the final five tracks . Track listing tracklist title1 Intro length1 1 21 title2 The Silent Life length2 3 47 title3 Mortification Under Scarlet Skies length3 3 53 title4 Resurrection length4 3 23 title5 Off With Their Heads length5 4 51 title6 Intro length6 0 53 title7 Abyssal Creation length7 4 10 title8 Reconstruction Of The Female Anatomy length8 3 50 title9 Erratic Perception length9 4 11 title10 Truest Shade Of Crimson length10 6 02 total length 36 17 Credits The Knife Trade Joey Stiletto Vocals Ryan Knight Guitar Ryan McCullough Guitar Daniel Shroyer Bass Nate Lumbra Drums Through The Eyes Of The Dead Anthony Gunnells Vocals Justin Longshore Guitar Chris Anderson Guitar Jeff Springs Bass Dayton Cantley Drums Trivia Ryan Knight, guitarist for The Knife Trade, would go on to join Arsis , and later go on to join The Black Dahlia Murder . This album is Through The Eyes of The Dead s last album released on Lovelost Records. DEFAULTSORT Annihilation Of Expectation Category Through the Eyes of the Dead albums Category Split EPs Category 2005 EPs 2000s metalcore album stub 2000s death metal album stub ... more details
Expectation propagation EP is a technique in Bayesian machine learning , developed by Thomas Minka . EP finds approximations to a probability distribution . It uses an iterative approach that leverages the factorization structure of the target distribution. It differs from other Bayesian approximation approaches such as Variational Bayesian methods . References Minka, T. 2001 Expectation Propagation for Approximate Bayesian Inference http research.microsoft.com en us um people minka papers ep minka ep uai.pdf Corrected pdf In Jack S. Breese, Daphne Koller Eds. UAI 01 Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence, University of Washington, Seattle, Washington, USA, August 2 5, 2001. pages 362&ndash 369 External links http research.microsoft.com minka papers ep Minka s EP papers http research.microsoft.com en us um people minka papers ep roadmap.html List of papers using EP . Category Machine learning Category Bayesian statistics compsci stub ... more details
The Expectations Hypothesis of the term structure of interest rates also known as Term structure Yield Curve is the proposition that the long term rate is determined by the market s expectation for the short term rate plus a constant risk premium Sarno et al., 2007 . It has been found that the expectation hypothesis has been tested and rejected using a wide variety of interest rates, over a variety of time periods and monetary policy regimes Guidolin et al., 2008 . This analysis is supported in a study conducted by Sarno, where it is concluded that while conventional bivariate procedure provides mixed results, the more powerful testing procedures, for example expanded Vector Autoregression test, suggest rejection of the EH throughout the maturity spectrum examined. A common reason given for the failure of the EH is that the risk premium is not constant as the expectation hypothesis requires, but is time varying. However, research by Guidolin and Thornton 2008 suggest otherwise. It is postulated that the expectation hypothesis fails because short term interest rates are not predictable to any significant degree. While traditional term structure tests mostly indicate that expected future interest rates are ex post inefficient forecasts, Froot 1989 has an alternative take on it. At short maturities, the expectation hypothesis fails. At long maturities, however, changes in the yield curve reflect changes in expected future rates one for one. 1 i lt n 1 i st year 1 1 i st year 2 cdots 1 i st year n This hypothesis assumes that the various maturities are perfect substitutes and suggests that the shape of the yield curve depends on market participants expectations of future interest rates. These expected rates, along with an assumption that arbitrage opportunities will be minimal, is enough information to construct a complete yield curve. For example, if investors have an expectation of what 1 year interest rates will be next year, the 2 year interest rate can be calculate ... more details
Unreferenced stub auto yes date December 2009 Contract law Expectation damages are damages recoverable from a breach of contract breach of contract by the non breaching party. It originates from an injured party s interest in realizing the value of the expectancy that was created by the promise of the other party. The purpose of expectation damages is to put the non breaching plaintiff party in the position he would have occupied had the contract been fulfilled. Expectation damages can be contrasted to reliance damages and restitution restitution damages , which are remedies that address other types of interests of parties involved in contract enforceable promises . The default for expectation damages are monetary damages which are subject limitations or exceptions see below Expectation damages are measured by the diminution in value, coupled with consequential damages consequential and incidental damages incidental damages. Measuring Expectation Damages In expectation damages, the measure of damages is the difference between what was given and what was promised, along with consequential and incidental expenses MINUS any payments received from the breaching party and any costs saved ... of the bargain. However, it is important to note that expectation damages are not punitive its theoretical ... anticipated profit. NOTE Expectation damage calculations can be different in contracts with the United ..., AND POLICY ARGUMENTS FOR EXPECTATION DAMAGES IN CONTRACTS TERMINATED FOR THE CONVENIENCE OF THE GOVERNMENT ... t use any of Wisda s Landscaping s services, expectation damages paid to Wisda s Landscaping would be 500 ... Townsend is forced to hire another service for 60 an hour, expectation direct damages paid to Neal ... contract . edit Potential Exceptions or Limitations to General Rule on Measure of Damages expectation ... See also Smith v. Bolles , 132 U.S. 125 1889 , limiting expectation damages from fraudulent conduct ... Code UCC DEFAULTSORT Expectation Damages Category Contract law Law stub ... more details
Unreferenced date August 2010 Expert subject date August 2010 An ordered set is in order theory in mathematics is an ambiguous term referring to a Set mathematics set that is either a partially ordered set or a totally ordered set . A set with a binary relation R on its elements that is reflexive relation reflexive for all a in the set, a R a , antisymmetric relation antisymmetric if a R b and b R a , then a b and transitive relation transitive if a R b and b R c , then a R c is described as a partially ordered set or poset . If the binary relation is antisymmetric, transitive and also total for all a and b in the set, a R b or b R a , then the set is a totally ordered set . If every non empty subset has a least element, then the set is a Well order well ordered set . In information theory , an ordered set is a non data carrying set of bits as used in 8b 10b encoding . See also Array data structure computing ordered set not mentioned on blue link List computing ordered set not mentioned on blue link Category Order theory is R un mengjafr i mk nn Velordna mengd sr uk ... more details
Pythagorean expectation is a formula invented by Bill James to estimate how many games a baseball team should have won based on the number of run baseball runs they scored and allowed. Comparing a team s actual and Pythagorean winning percentage can be used to evaluate how lucky that team was by examining the variation between the two winning percentages . The term is derived from the formula s resemblance to the Pythagorean theorem . The basic formula is math mathrm Win frac text runs scored 2 text runs scored 2 text runs allowed 2 frac 1 1 text runs allowed text runs scored 2 math where Win is the winning ratio generated by the formula. The expected number of wins would be the expected winning ratio multiplied by the number of games played. Empirical origin Empirically, this formula correlates fairly well with how baseball teams actually perform. However, statisticians since the invention of this formula found it to have a fairly routine error, generally about 3 games off. For example, in 2002, the New York Yankees scored scored 897 runs, allowing 697 runs. According to James original formula, the Yankees should have won 62.35 of their games. math mathrm Win frac text 897 2 text 897 2 text 697 2 0.623525865 math Based on a 161 game season, the Yankees should have won 100.38 games. The 2002 Yankees actually went 103 58. ref http www.baseball reference.com teams NYY 2002.shtml Baseball reference.com 2002 NY Yankees Statistics Bot generated title ref In efforts to fix this error, statisticians have performed numerous searches to find the ideal exponent. If using a single number exponent, 1.83 is the most accurate, and the one used by baseball reference.com, the premier website for baseball statistics across teams and time. ref http www.baseball reference.com about faq.shtml ... to adapt James Pythagorean expectation to professional basketball while a researcher at STATS, Inc ... Baseball Pythagorean expectation. http www.pro football reference.com blog ?p 337 Adjusting football ... more details
In mathematics , an ordered field is a field mathematics field together with a total order ing of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered ... theory of ordered fields and formally real field s. An ordered field necessarily has characteristic .... This implies that an ordered field necessarily contains an infinite number of elements. Finite field s cannot be ordered. Every subfield of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is isomorphism isomorphic to the rational number s. Any Dedekind complete ordered field is isomorphic to the real numbers. Squares are necessarily non negative in an ordered field. This implies that the complex number s cannot be ordered since the square of the imaginary unit i is 1. Every ordered field is a formally real field. Definition There are two equivalent definitions of an ordered field. Def 1 appeared first historically .... Def 1 A total order on F A Field mathematics field F , , together with a total order on F is an ordered ... then 0 a b Def 2 A positive cone of F A prepositive cone of a field F is a subset P F that has the following ... . The element &minus 1 is not in P . If in addition, the subset F is the union of P and &minus ... . An ordered field is a field F together with a positive cone P . Equivalence of the two definitions ... of ordered fields If x y and y z , then x z . transitive property transitivity If x y and z ..., which is a contradiction An ordered field has characteristic algebra characteristic 0. Since 1 ... be ordered. Squares are non negative. 0 a for all a in F . Follows by a similar argument to 1 0 Every subfield of an ordered field is also an ordered field inheriting the induced ordering . The smallest .... If every element of an ordered field lies between two elements of its rational subfield, then the field ... ordered field and contains infinitesimal s. For example, the real number s form an Archimedean ... more details
In mathematics , an ordered pair a , b is a pair of mathematical object s. In the ordered pair a , b ..., the objects are called the first and second coordinates , or the left and right projections of the ordered pair. The order in which the objects appear in the pair is significant the ordered pair a , b is different from the ordered pair b , a unless a b . Ordered pairs are also called tuple 2 tuples ... of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n tuple n tuple s ordered lists of n objects . For example, the ordered triple a , b , c can be defined as a , b ... function mathematics functions are defined in terms of ordered pairs. Generalities Let math a 1, b 1 math and math a 2, b 2 math be ordered pairs. Then the characteristic or defining property of the ordered pair is math a 1, b 1 a 2, b 2 quad text if and only if quad a 1 a 2 text and b 1 b 2. math The Set mathematics set of all ordered pairs whose first entry is in some set X and whose ... . A binary relation between sets X and Y is a subset of X × Y . If one wishes to employ the math ... the ordered pair may be denoted by the variant notation math left langle a,b right rangle. math Defining the ordered pair using set theory The above characteristic property of ordered pairs is all that is required to understand the role of ordered pairs in mathematics. Hence the ordered pair can be taken ... sets of some sort. Hence if the ordered pair is not taken as primitive, it must be defined as a set. ref Quine has argued that the set theoretical implementations of the concept of the ordered ... about theoretical entities . ref Several set theoretic definitions of the ordered pair are given below. Wiener s definition Norbert Wiener proposed the first set theoretical definition of the ordered ... pbk. . van Heijenoort states the simplification this way By giving a definition of the ordered pair ... 1967 224. van Heijenoort observes that the resulting set that represents the ordered pair has ... more details
In mathematics , an ordered semigroup is a semigroup S , together with a partial order that is compatible with the semigroup operation, meaning that x y implies z x z y and x z y z for all x , y , z in S . If S is a Group mathematics group and it is ordered as a semigroup, one obtains the notion of ordered group , and similarly if S is a monoid it may be called ordered monoid . References T.S. Blyth, Lattices and Ordered Algebraic Structures , Springer, 2005, ISBN 1 85233 905 5, chap. 11. Category Ordered algebraic structures Category Semigroup theory Abstract algebra stub ... more details
In abstract algebra , an ordered ring is a commutative ring math R math with a total order such that for all a , b , and c in R if a b then a c b c . if 0 a and 0 b then 0 ab . Ordered rings are familiar from arithmetic . Examples include the real number s. The rationals and reals in fact form ordered field s. The complex number s do not form an ordered ring or ordered field . In analogy with real numbers, we call an element c 0, of an ordered ring positive if 0 c and negative number negative if c 0. The set of positive or, in some cases, nonnegative elements in the ring R is often denoted by R sub sub . If a is an element of an ordered ring R , then the absolute value of a , denoted a , is defined thus math a begin cases a, & mbox if 0 leq a, a, & mbox otherwise , end cases math where a is the additive inverse of a and 0 is the additive identity element . A discrete ordered ring or discretely ordered ring is an ordered ring in which there is no element between 0 and 1. The integers are a discrete ordered ring, but the rational numbers are not. Basic properties For all a , b and c in R If a b and 0 c , then ac bc . ref OrdRing ZF 1 L9 ref This property is sometimes used to define ordered rings instead of the second property in the definition above. ab a b . ref OrdRing ZF 2 L5 ref An ordered ring that is not trivial ring trivial is infinite. ref ord ring infinite ref Exactly one of the following is true a is positive, a is positive, or a 0. ref OrdRing ZF 3 L2, see also OrdGroup decomp ref This property follows from the fact that ordered rings are abelian group abelian , linearly ordered group s with respect to addition. An ordered ring R has no zero divisor s if and only if the positive ring elements are Closure mathematics closed under multiplication i.e. if a and b are positive, then so is ab . ref OrdRing ZF 3 L3 ref In an ordered ring, no negative element is a square ... Ordered groups Category Real algebraic geometry ca Anell ordenat es Anillo ordenado pl Pier cie ... more details
The term ordered list can refer to an ordered list HTML a mathematical sequence disambig Short pages monitor This long comment was added to the page to prevent it being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Longcomment. Please do not remove the monitor template without removing the comment as well. ... more details
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