In number theory , Li s criterion is a particular statement about the positivity of a certain sequence that is completely equivalent to the Riemann hypothesis . The criterion is named after Xian Jin Li, who presented it in 1997. Recently, Enrico Bombieri and Jeffrey C. Lagarias provided a generalization, showing that Li s positivity condition applies to any collection of points that lie on the Re s 1 2 axis. Definition The Riemann Xi function Riemann &xi function is given by math xi s frac 1 2 s s 1 pi s 2 Gamma left frac s 2 right zeta s math where is the Riemann zeta function . Consider the sequence math lambda n frac 1 n 1 left. frac d n ds n left s n 1 log xi s right right s 1 . math Li s criterion is then the statement that the Riemann hypothesis is completely equivalent to the statement that math lambda n 0 math for every positive integer n . The numbers math lambda n math may also be expressed in terms of the non trivial zeros of the Riemann zeta function math lambda n sum rho left 1 left 1 frac 1 rho right n right math where the sum extends over , the non trivial zeros of the zeta function. This conditionally convergent sum should be understood in the sense that is usually used in number theory, namely, that math sum rho lim N to infty sum Im rho le N . math A generalization Bombieri and Lagarias demonstrate that a similar criterion holds for any collection of complex numbers, and is thus not restricted to the Riemann hypothesis. More precisely, let R be any collection of complex numbers , not containing 1, which satisfies math sum rho ... the complex conjugate math overline rho math and math 1 rho math are in R , then Li s criterion ... 1 left 1 frac 1 rho right n right ge 0. math Bombieri and Lagarias also show that Li s criterion follows from Weil s criterion for the Riemann hypothesis. References cite journal author Enrico Bombieri ... doc bombieri.ps title Complements to Li s criterion for the Riemann hypothesis journal Journal of Number ... more details
Unreferenced date November 2006 Infobox mountain pass name Criterion Summit photo photo caption elevation ft 3360 elevation ref traversed U.S. Route 197 Oregon U.S. Route 197 location Wasco County, Oregon , USA range coordinates Coord 44.98272 120.99124 display inline,title type pass topo Criterion Summit el. 3360 ft. is a mountain pass in Oregon traversed by U.S. Route 197 Oregon U.S. Route 197 . Its location is south of Maupin, Oregon Maupin approximately 3 miles north of the intersection of US 197 with U.S. Route 97 Oregon U.S. Route 97 . It is marked with a sign, and a view point that has a map of all the mountain peaks you can see from its vantage point ranging from Mt. Bachelor to Mt. St. Helens Category Mountain passes of Oregon Category Transportation in Wasco County, Oregon Category Landforms of Wasco County, Oregon WascoOR geo stub ... more details
unref date January 2008 In statistics , an optimality criterion provides a measure of the fit of the data to a given hypothesis. The selection process is determined by the solution that optimizes the criteria used to evaluate the alternative hypotheses. The term has been used to identify the different criteria that are used to evaluate a phylogenetic tree and include maximum likelihood , Bayesian probability Bayesian , maximum parsimony , and minimum evolution . For example, in order to determine the best topology between two phylogenetic trees using the maximum likelihood optimality criterion, one would calculate the maximum likelihood score of each tree and choose the one that had the better score. However, different optimality criteria can select different hypotheses. In such circumstances caution should be exercised when making strong conclusions. Many other disciplines use similar criteria or have specific measures geared toward the objectives of the field. References Empty section date July 2010 See also Optimality in Pareto equilibrium Category Hypothesis testing ru ... more details
In quantum mechanics , in particular quantum information , the Range criterion is a necessary condition that a state must satisfy in order to be separable states separable . In other words, it is a separability criterion . The result Consider a quantum mechanical system composed of n subsystems. The state space H of such a system is the tensor product of those of the subsystems, i.e. math H H 1 otimes cdots otimes H n math . For simplicity we will assume throughout that all relevant state spaces are finite dimensional. The criterion reads as follows If is a separable mixed state acting on H , then the range of is spanned by a set of product vectors. Proof In general, if a matrix M is of the form math M sum i v i v i math , it is obvious that the range of M , Ran M , is contained in the linear span of math v i math . On the other hand, we can also show math v i math lies in Ran M , for all i . Assume without loss of generality i 1 . We can write math M v 1 v 1 T math , where T is Hermitian and positive semidefinite. There are two possibilities 1 span math v 1 subset math Ker T . Clearly, in this case, math v 1 in math Ran M . 2 Notice 1 is true if and only if Ker T math perp subset math span math v 1 perp math , where math perp math denotes orthogonal compliment. By Hermiticity of T , this is the same as Ran T math subset math span math v 1 perp math . So if 1 does not hold, the intersection Ran T math cap math span math v 1 math is nonempty, i.e. there exists some complex number such that math T w alpha v 1 math . So math M w langle w, v 1 rangle v 1 T w langle w, v 1 rangle alpha v 1. math Therefore math v 1 math lies in Ran M . Thus Ran M coincides with the linear span of math v i math . The range criterion is a special case of this fact. A density matrix acting ... that the range of is the linear span of these product states. This proves the criterion. References P. Horodecki, Separability Criterion and Inseparable Mixed States with Positive Partial Transposition ... more details
Unreferenced date December 2009 In quantum information theory , the reduction criterion is a necessary condition a mixed state must satisfy in order for it to be separable state separable . In other words, the reduction criterion is a separability criterion . Details Let H sub 1 sub and H sub 2 sub be Hilbert spaces of finite dimensions n and m respectively. L H sub i sub will denote the space of linear operators acting on H sub i sub . Consider a bipartite quantum system whose state space is the tensor product math H H 1 otimes H 2. math An un normalized mixed state is a positive linear operator density matrix acting on H . A linear map L H sub 2 sub L H sub 1 sub is said to be positive if it preserves the cone of positive elements, i.e. A is positive implied A is also. From the one to one correspondence between positive maps and entanglement witness es, we have that a state is entangled if and only if there exists a positive map such that math I otimes Phi rho math is not positive. Therefore, if is separable, then for all positive map , math I otimes Phi rho geq 0. math Thus every positive, but not completely positive map completely positive , map gives rise to a necessary condition for separability in this way. The reduction criterion is a particular example of this. Suppose H sub 1 sub H sub 2 sub . Define the positive map L H sub 2 sub L H sub 1 sub by math Phi A operatorname Tr A A. math It is known that is positive but not completely positive. So a mixed state being separable implies math I otimes Phi rho geq 0. math Direct calculation shows that the above expression is the same as math I otimes rho 1 rho geq 0 math where sub 1 sub is the partial trace of with respect to the second system. The dual relation math rho 2 otimes I rho geq 0 math is obtained in the analogous fashion. The reduction criterion consists of the above two inequalities. DEFAULTSORT Reduction Criterion Category Quantum information science ... more details
The Intuitive Criterion in Game Theory is a technique for equilibrium refinement in signaling games. In economics, signaling games are games in which a player with private information moves first. Private information generally refers to the player s hidden or unobservable type. Signaling games typically have lots of perfect Bayesian equilibria . Equilibrium refinement techniques are ways of reducing the set of equilibria. Most refinement techniques are broadly based on restricting beliefs off the equilibrium path. Off equilibrium actions or outcomes are those that are different from what is predicted in a Nash equilibrium. The intuitive criterion was presented by In Koo Cho and David M. Kreps in a 1987 article ref Cho, I K. & Kreps, D. M. 1987 Signaling games and stable equilibria. Quarterly Journal of Economics 102 179 221. ref . Their idea was to try to reduce the set of equilibria by requiring off equilibrium beliefs to be reasonable in some sense. This refinement of the solution concept allows the modeller to choose between multiple perfect Bayesian equilibria. Formally, we can eliminate a particular perfect Bayesian equilibrium by using the Intuitive Criterion if there is some type who has a deviation that is assured of yielding him a payoff above his equilibrium payoff as long as other players do not assign a positive probability to the deviation having been made by any type for whom this action is equilibrium dominated. Intuitively, we can eliminate a PBE if there is a type of player who wants to deviate even though he is not sure what the beliefs of other players are. The player is only sure that the other players will not think that he is a player who would find ... the Intuitive Criterion and suggested alternative refinements such as Universal Divinity . Example ... Criterion eliminates all pooling equilibria. In the game, there is also a continuum of separating equilibria , but the Intuitive Criterion eliminates all the separating equilibria except for the most ... more details
In mathematics, Cartan s criterion gives conditions for a Lie algebra in characteristic 0 to be solvable Lie algebra solvable , which implies a related criterion for the Lie algebra to be semisimple Lie algebra semisimple . It is based on the notion of the Killing form , a symmetric bilinear form on math mathfrak g math defined by the formula math K u,v operatorname tr operatorname ad u operatorname ad v , math where tr denotes the Trace linear algebra trace of a linear operator . The criterion was introduced by harvs txt authorlink lie Cartan first lie last Cartan year 1894 . Cartan s criterion for solvability Cartan s criterion for solvability states A Lie subalgebra math mathfrak g math of endomorphisms of a finite dimensional vector space over a field mathematics field of characteristic zero is solvable if and only if math Tr ab 0 math whenever math a in mathfrak g ,b in mathfrak g , mathfrak g . math The fact that math Tr ab 0 math in the solvable case follows immediately from Lie Kolchin theorem Lie s theorem that solvable Lie algebras in characteristic 0 can be put in upper triangular form. Applying Cartan s criterion to the adjoint representation gives A finite dimensional Lie algebra math mathfrak g math over a field mathematics field of characteristic zero is solvable if and only if math K mathfrak g , mathfrak g , mathfrak g 0 math where K is the Killing form . Cartan s criterion for semisimplicity Cartan s criterion for semisimplicity states A finite dimensional Lie algebra math mathfrak g math over a field mathematics field of characteristic zero is semisimple if and only if the Killing form is degenerate form non degenerate . harvtxt Dieudonn 1953 gave a very short proof that if a finite dimensional Lie algebra in any characteristic has a Quadratic Lie algebra non degenerate invariant bilinear form and no non zero abelian ideals, and in particular if its ... from Cartan s criterion for solvability that a semisimple algebra in characteristic 0 has a non ... more details
Economy Vol. 56, No. 1, pp. 23 34 ref A voting system satisfies the Condorcet criterion if it chooses the Condorcet winner when one exists. Any method conforming to the Condorcet criterion is known ... Antoine Nicolas Caritat, the Marquis de Condorcet . Relation to other criteria The Condorcet criterion implies the majority criterion that is, any system that satisfies the former will satisfy the latter ... criterion will not satisfy independence of irrelevant alternatives . The Condorcet criterion is also incompatible with the later no harm criterion , the participation criterion , and the consistency criterion . Compliance of methods Complying methods main Condorcet method The following methods comply with the Condorcet criterion Black Copeland s method Copeland Dodgson s method Kemeny Young ... criterion. This statement requires qualification in some cases see the individual subsections ... use, the Condorcet criterion may be violated. Consider an election in which 70 of the voters prefer ... criterion, which may not be accepted by all voting theorists. Other generalizations ... criterion in the following case. Consider an election consisting of five voters and three alternatives ... criterion. Consider, for example, the following vote count of preferences with three candidates ... preferred to both A and C. B must then win according to the Condorcet criterion. Using the rules ... preference, it is easy to construct an example which fails the Condorcet criterion. Consider ... clearly does not comply with the condorcet criterion for instance, if three voters vote for three candidates ... 1970 publisher Holden Day See also Condorcet loser criterion Condorcet method Ramon Llull 1232 1315 ..., was given credit for discovering the Borda count and Condorcet criterion Llull winner in the 13th ... more details
Other uses Refimprove date February 2008 Infobox Theatre name Criterion Theatre image CriterionTheatre.png caption The Criterion Theatre in September 2007 address Piccadilly Circus city City of Westminster ... 0.134056 architect Thomas Verity owner Criterion Theatre Trust lease capacity 588 on 3 levels type ... play The 39 Steps currentuse website www.criterion theatre.co.uk The Criterion Theatre is a West End ... the tiled staircases, were retained and can still be seen. The redesign placed the large Criterion ... of the book on The Criterion in 1903, Gilbert wrote I am sorry to say that in my mind is an absolute blank to the opening of The Criterion. I never saw Topseyturveydom . If you happen to have a copy ... actor Charles Wyndham became the manager and lessee in 1875 and under his management The Criterion ... off a considerable portion of the adjoining Criterion Restaurant. New corridors were built, with several ... 1884 states The Criterion Theatre, transformed from a stuffy band box to a convenient, handsome, and well ... and launched the writing career of Terence Rattigan . During World War II, the Criterion ... blitz and light entertainment programmes were both recorded and broadcast live. After the war, the Criterion ... the Criterion site was proposed for redevelopment, which caused protest as people feared the theatre ... of the arts, and his wife, theatre impresario Sally Greene . The couple set up the Criterion Theatre Trust, a registered charity created to protect the Criterion s future. From April 1989 to October ..., the Criterion retains a well preserved Victorian auditorium with an intimate atmosphere. Recent years ... 3 year 3Yoverview.html ref The Criterion s current production is The 39 Steps play The 39 Steps , adapted .... Peppa Pig s Party transferred to the Criterion Theatre for the 2010 Christmas period to play alongside The 39 Steps play The 39 Steps and will return for the 2011 Christmas period. Criterion Presents ... works premiered fr Criterion Theatre ... more details
Context date October 2009 A valid measure actually measures what it says it will measure. To define a measure as valid, one can assess different types of validity. The type of validity of measurement assessed depends on what the researcher wants to know. Criterion validity is one method for assessing the validity of a measure. In psychometrics , criterion validity is a measure of how well one Variable mathematics variable or set of variables predicts an outcome based on information from other variables, and will be achieved if a set of measures from a personality test relate to a behavioral criterion on which psychologist s agree. ref name pennington cite book last Pennington first Donald authorlink Donald Pennington title Essential Personality publisher Edward Arnold publisher Arnold date 2003 doi isbn 0340761180 page 37 ref A typical way to achieve this is in relation to the extent to which a score on a personality test can predict future performance or behavior . Another way involves Correlation correlating test scores with another established Psychological testing test that also measures the same Trait theory personality characteristic . ref name pennington Criterion or concrete validity is the extent to which the measures are demonstrably related to concrete criteria in the real world. This type of validity is often divided into concurrent and predictive subtypes. The term concurrent validity is reserved for demonstrations relating a measure to other concrete criteria assessed simultaneously. Predictive validity refers to the degree to which any measure can predict future or independent past events. These variables are often represented as intermediate and ultimate criteria. For example, let us say we are conducting a study on success in college. If we find out there is a high correlation between student grades in high school math classes and their success in college which can be measured by many possible variables , we would say there is high criterion related ... more details
In the field of epistemology , the problem of the criterion is an issue regarding the starting point of knowledge. This is a separate and more fundamental issue than the regress argument found in discussions on Epistemology Justification justification of knowledge. American philosopher Roderick Chisholm Roderick M. Chisholm in his Theory of Knowledge details the problem of the criterion with two sets of questions 1. What do we know? or What is the extent of our knowledge? 2. How do we know? or What is the criterion of knowing? An answer to either set of questions will allow us to devise a means of answering the other. Answering the former question set first is called particularism , whereas answering the latter set first is called methodism . A third solution, found untenable by many philosophers Citation needed date November 2011 for its inadequacy to give explanation is skepticism . A skeptic will proclaim that since one cannot have an answer to the first set of questions without first answering the second set, and one cannot hope to answer the second set of questions without first knowing the answers to the first set, we are, therefore, unable to answer either. This has the result of our being unable to justify any of our beliefs. Particularist theories organize things already known and attempt to use these particulars of knowledge to find a method of how we know, thus answering the second question set. Methodist theories propose an answer to question set two and proceed to use this to establish what we, in fact, know. Classical empiricism embraces the Methodist approach. References references http www.the philosopher.co.uk mille.htm Category Epistemology epistemology stub fi Kriteerin ongelma ... more details
In robust statistics , Peirce s criterion is a rule for eliminating outlier s from data sets, which was devised by Benjamin Peirce . Outliers removed by Peirce s criterion The problem of outliers Main outlier See also robust statistic In data set s containing real numbered measurements, the suspected outlier s are the measured values that appear to lie outside the cluster of most of the other data values. The outliers would greatly change the estimate of location if the arithmetic average were to be used as a summary statistic of location. The problem is that the arithmetic mean is very sensitive to the inclusion of any outliers in statistical terminology, the arithmetic mean is not robust statistic robust . In the presence of outliers, the statistician has two options. First, the statistician may remove the suspected outlier s from the data set and then use the arithmetic mean to estimate the location parameter. Second, the statistician may use a robust statistic, such as the median statistic. Peirce s criterion is a statistical procedure for eliminating outliers. Peirce s criterion The statistician and historian of statistics Stephen M. Stigler wrote the following about Benjamin Peirce ref name stig246 Stigler 1979 , page 246 ref blockquote In 1852 he published the first significance ... Stigler Stigler , 1973a . blockquote Peirce s criterion is derived from a statistical analysis of the Gaussian ... The method can be applied using a table Citation needed date March 2010 which lists criterion ... for the criterion. Peirce s criterion was used for decades at the United States Coast Survey . ref ... inclined statistical organization of the era. ref name stig246 blockquote Peirce s criterion was discussed ... type PDF HIGH Criterion for the Rejection of Doubtful Observations , Astronomical Journal II 45 1852 ... HIGH Errata to the original paper . cite journal title On Peirce s criterion authorlink Benjamin Peirce ... 140 160 isbn 0253372011 Ross, Stephen, Peirce s Criterion for the Elimination of Suspect Experimental ... more details
In mathematics , Eisenstein nowiki s nowiki criterion gives an easily checked necessary and sufficient ... the hypotheses of the criterion for p on the contrary it could make the criterion hold for some prime, even if it did not before the division . This criterion is certainly not applicable to all ... cases the criterion does not apply directly for any prime number , but it does apply after transformation ... s criterion to apply for a prime number p it must divide both non leading coefficients 15 and 10 ... into two quadratic factors could also be possible. Often Eisenstein s criterion does not apply ... 1 of x is not divisible by any prime, Eisenstein s criterion does not apply to H . But if one ... Eisenstein s criterion for the prime number 7. Since the substitution is an automorphism of the ring ... s criterion apply is a useful way to broaden its scope. Another possibility to transform a polynomial so as to satisfy the criterion, which may be combined with applying a shift, is reversing the order ... math 2 x sup 5 sup 4 x sup 2 sup 3 satisfies the criterion for math p 2 after reversing ... An important class of polynomials whose irreducibility can be established using Eisenstein s criterion ... example of math H , the coefficients 1 prevent Eisenstein s criterion from applying directly. However the polynomial will satisfy the criterion for p after substitution of math x 1 for x ... form math x sup p 1 sup ... x 1 . History The criterion is named after Gotthold Eisenstein . However, Theodor Sch nemann was the first to publish a version of the criterion, ref name Cox David A. Cox, Why Eisenstein proved the Eisenstein criterion and why Sch nemann discovered it first ... und angewandte Mathematik 32 , pp. 93 118. The criterion is formulated on p. 100 ref ... F x does not exceed math n , the criterion is correct however, and somewhat stronger than the formulation ... abh ngt , Journal f r die reine und angewandte Mathematik 39 , pp. 160 179. The criterion ... more details
Carath odory s criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carath odory . Its statement is as follows Let math lambda math denote the Lebesgue measure Lebesgue outer measure on math mathbb R n math , and let math E subseteq mathbb R n math . Then math E math is Lebesgue measurable if and only if math lambda A lambda A cap E lambda A setminus E math for every math A subseteq mathbb R n math . Notice that math A math is not required to be a measurable set. See also Carath odory s theorem Category Measure theory probability stub ... more details
cleanup date April 2010 Mean field theory gives sensible results as long as we are able to neglect fluctuations in the system under consideration. The Ginzburg Criterion tells us quantitatively when mean field theory is valid. It also gives the idea of an upper critical dimension, a dimensionality of the system above which mean field theory gives proper results, and the critical exponents predicted by mean field theory match exactly with those obtained by numerical methods. Example Ising Model If math phi math is the Phase transition Order parameters order parameter of the system, then mean field theory requires that the fluctuations in the order parameter are much smaller than the actual value of the order parameter near the critical point. Quantitatively, this means that br math displaystyle mathcal delta phi 2 quad ll quad phi 2 math Using this in the Landau theory , which is identical to the mean field theory for the Ising model, the value of the upper critical dimension comes out to be 4. If the dimension of the space is greater than 4, the mean field results are good and self consistent. But for dimensions less than 4, the predictions are less accurate. For instance, in one dimension, the mean field approximation predicts a phase transition at finite temperatures for the Ising model, whereas we know that there is none from the exact analytic solution that can be evaluated in one dimension except at T 0 or math T rightarrow infty math References Cite journal author V. L. Ginzburg year 1960 journal Soviet Physics Solid State volume 2 pages 1824 Cite journal author J. Als Nielsen and R. J. Birgeneau title Mean field theory, the Ginzburg criterion, and marginal dimensionality of phase transitions publisher AAPT year 1977 journal American Journal of Physics volume 45 number 6 pages 554 560 url http link.aip.org link ?AJP 45 554 1 doi 10.1119 1.11019 Category Statistical mechanics Category Fundamental physics concepts ... more details
climate calculators . Criterion has also served the U.S. Green Building Council as international ... links http www.crit.com Criterion Planners website Category Companies based in Portland, Oregon ... more details
Refimprove date May 2011 The participation criterion is a voting system criterion . It is also known ..., December 1994 date December 1994 ref as follows In a deterministic framework, the participation criterion ... framework, the participation criterion says that the addition of a ballot, where each candidate ... voting , range voting , and the Borda count all satisfy the participation criterion. Citation needed ... and Democracy accessdate 2011 05 12 ref fail. Voting systems that fail the participation criterion ... choice win. The participation criterion for voting systems is one example of a Participation constraint ... criterion is not in the use of particular voting systems, but in simple yes or no measures that place ... criterion, as a minority of voters preferring the no option could cause the measure to fail ... no s , by contrast, would pass the participation criterion. Examples Copeland Main Copeland s method This example shows that Copeland s method violates the Participation criterion. Assume four candidates ... criterion. Instant runoff voting Main Instant runoff voting This example shows that Instant runoff voting violates the Participation criterion. Assume three candidates A, B and C and 15 potential voters ... least preference. Thus, Instant runoff voting fails the Participation criterion. Kemeny Young method ... criterion. Assume four candidates A, B, C, D with 21 voters and the following preferences class ... to overcome the first one. Thus, Kemeny Young fails the Participation criterion. Majority Judgment Main Majority Judgment This example shows that Majority Judgment violates the Participation criterion ... criterion. Minimax Main Minimax Condorcet This example shows that the Minimax method violates the Participation criterion. Assume four candidates A, B, C, D with 18 potential voters and the following ... method fails the Participation criterion. Ranked pairs Main Ranked pairs This example shows that the Ranked pairs method violates the Participation criterion. Assume four candidates A, B, C and D ... more details
This article is about a voting system criterion. See monotonic function for a mathematical notion. The monotonicity criterion is a voting system criterion used to analyze both single and multiple winner voting system s. A voting system is monotonic if it satisfies one of the definitions of the monotonicity criterion, given below. Douglas R. Woodall , calling the criterion mono raise , defines it as A candidate x should not be harmed nowiki i.e., change from being a winner to a loser nowiki if x is raised on some ballots without changing the orders of the other candidates. ref D R Woodall, http www.votingmatters.org.uk ISSUE6 P4.HTM Monotonicity and Single Seat Election Rules , Voting matters , Issue 6, 1996 ref Note that the references to orders and relative positions concern the rankings of candidates other than X, on the set of ballots where X has been raised. So, if changing a set of ballots voting A B C to B C A causes B to lose, this does not constitute failure of Monotonicity, because in addition to raising B, we changed the relative positions of A and C. This criterion may be intuitively justified by reasoning that in any fair voting system, no vote for a candidate, or increase in the candidate s ranking, should instead hurt the candidate. It is a property considered in Arrow s impossibility theorem . Some political scientists, however, doubt the value of monotonicity as an evaluative measure of voting systems. David Austen Smith and Jeffrey Banks, for example, published an article in The American Political Science Review in which they argue that monotonicity in electoral ... the monotonicity criterion suffer an unusual form, where voters with enough information about other ... systems violate the mono raise criterion. Suppose a president were being elected among ... ref References references See also Voting system Voting system criterion DEFAULTSORT Monotonicity Criterion Category Voting system criteria ... more details
In number theory Euler s criterion is a formula for determining whether an integer is a quadratic residue modular arithmetic modulo a prime number prime . Precisely, Let p be an odd number odd prime and a an integer coprime to p . Then ref Gauss, DA, Art. 106 ref math a tfrac p 1 2 equiv begin cases ,1 pmod p & text if there is an integer x text such that a equiv x 2 pmod p 1 pmod p & text if there is no such integer. end cases math Euler s criterion can be concisely reformulated using the Legendre symbol ref Hardy & Wright, thm. 83 ref math left frac a p right equiv a p 1 2 pmod p. math The criterion first appeared in a 1748 paper by Leonhard Euler Euler . ref Lemmermeyer, p. 4 cites two papers, E134 and E262 in the Euler Archive ref Proof The proof uses fact that the residue classes modulo a prime number are a finite field field . See the article Characteristic algebra Case of fields prime field for more details. The fact that there are p 1 2 quadratic residues and the same number of nonresidues mod p is proved in the article quadratic residue . Fermat s little theorem says that math a p 1 equiv 1 pmod p. math This can be written as math a tfrac p 1 2 1 a tfrac p 1 2 1 equiv 0 pmod p. math Since the integers mod p form a field, one or the other of these factors must be congruent to zero. Now if a is a quadratic residue, a &equiv x sup 2 sup , math a tfrac p 1 2 equiv x 2 tfrac p 1 2 equiv x p 1 equiv1 pmod p. math So every quadratic residue mod p makes the first factor zero. Lagrange s theorem number theory Lagrange s theorem says that there can be no more than p 1 2 values of a that make the first factor zero. But it is known that there are p 1 2 ... the second factor zero. This is Euler s criterion. Examples Example 1 Finding primes for which ... 16 1 mod 17 , so it is not a quadratic residue. Euler s criterion is related to the Quadratic reciprocity ... euler index.html The Euler Archive DEFAULTSORT Euler s Criterion Category Modular arithmetic Category ... more details
Turf race infobox class Group 3 horse race Criterion Stakes image caption location Newmarket Racecourse July Course br Newmarket, Suffolk Newmarket , England inaugurated 1978 race type Flat racing Flat Thoroughbred website http www.newmarketracecourses.co.uk Newmarket distance 7f 1,408 metres track Turf, straight qualification Three years old and up small br excluding Group 1 winners br this year or last year small weight 8 Stone Imperial mass st 8 Pound mass lb 3yo br 9 st 3 lb 4yo small br Allowances br 3 lb for Filly fillies and mare s br Penalties br 8 lb for Group 2 winners br 5 lb for Group 3 winners br since August 31 last year small purse Pound sterling 50,000 2011 small br 1st 28,385 small bonuses The Criterion Stakes is a Conditions races Group 3 Flat racing flat Horse racing horse race in Great Britain open to thoroughbred s aged three years or older. It is run on the July Course at Newmarket Racecourse Newmarket over a distance of 7 furlongs 1,408 metres , and it is scheduled to take place each year in late June or early July. The present Criterion Stakes was established in 1978, and for a period it was classed at Listed level. It was promoted to Group 3 status in 1986. A previous race with the same name was an important event for two year olds. Its winners included Ormonde horse Ormonde 1885 , Flying Fox horse Flying Fox 1898 , Pretty Polly horse Pretty Polly 1903 and Gay Crusader 1916 . Records span style font size 90 Most successful horse no horse has won this race more than once Leading jockey 2 wins George Duffield Daring March 1978 , Noalcoholic 1982 Pat Eddery Cadeaux Genereux 1988 , Toussaud 1992 Walter Swinburn Zilzal 1989 , La Grange Music 1991 Richard ... 1 galopp sieger.de Criterion Stakes. http www.horseracingintfed.com racingDisplay.asp?section 8&racepid 52432 horseracingintfed.com International Federation of Horseracing Authorities Criterion Stakes ... pedigreequery.com Criterion Stakes Newmarket. cite book last Abelson first Edward coauthors John Tyrrel ... more details
In mathematics , Nevalinna s criterion in complex analysis , proved in 1920 by the Finnish mathematician Rolf Nevanlinna , characterizes holomorphic univalent functions on the unit disk which are star domain starlike . Nevanlinna used this criterion to prove the Bieberbach conjecture for starlike univalent functions Statement of criterion A univalent function h on the unit disk satisfying h 0 0 and h 0 1 is starlike, i.e. has image invariant under multilpication by real numbers in 0,1 , if and only if math z h prime z h z math has positive real part for z 1 and takes the value 1 at 0. Note that, by applying the result to a h rz , the criterion applies on any disc z r with only the requirement that f 0 0 and f 0 0. Proof of criterion Let h z be a starlike univalent function on z 1 with h 0 0 and h 0 1. For t 0, define ref harvnb Hayman 1994 p 14 ref math f t z h 1 e t h z , , math a semigroup of holomorphic mappinga of D into itself fixing 0. Moreover h is the Koenigs function for the semigroup f sub t sub . By the Schwarz lemma , f sub t sub z decreases as t increases. Hence math partial t f t z 2 le 0. math But, setting w f sub t sub z , math partial t f t z 2 2 Re , overline f t z partial t f t z 2 Re , overline w v w , math where math v w h w over h prime w . math Hence math Re , overline w h w over h prime w ge 0. math and so, dividing by w sup 2 sup , math Re , h w over w h prime w ge 0. math Taking reciprocals and letting t go to 0 gives math Re , z h prime z over h z ge 0 math for all z 1. Since the left hand side is a harmonic function , the maximum principle implies the inequality is strict. Conversely if math g z z h prime z over h z math has positive real part and g 0 1, then h can vanish only at 0, where it must have a simple zero. Now math partial theta arg h re i theta partial theta Im , log h z Im , partial ... 1921 proved that math a n le n. math In fact by Nevanlinna s criterion math g z z f prime z over ... more details
In nuclear fusion research, the Lawson criterion , first derived on fusion reactors initially classified by John D. Lawson scientist John D. Lawson in 1955 and published in 1957, ref J. D. Lawson, Some Criteria for a Power Producing Thermonuclear Reactor , Proceedings of the Physical Society B, Volume 70 1957 , p. 6 ref is an important general measure of a system that defines the conditions needed for a fusion reactor to reach ignition , that is, that the heating of the plasma by the products of the fusion reactions is sufficient to maintain the temperature of the plasma against all losses without external power input. As originally formulated the Lawson criterion gives a minimum required value for the product of the plasma electron density n sub e sub and the energy confinement time sub E sub . Later analyses suggested that a more useful figure of merit is the triple product of density, confinement time, and plasma temperature T . The triple product also has a minimum required value, and the name Lawson criterion often refers to this inequality. The product n sub e sub &tau sub E sub The confinement time math tau E math measures the rate at which a system loses energy to its environment. It is the energy content W divided by the power loss math P loss math rate of energy loss math tau E frac W P rm loss math For a fusion reactor to operate in steady state, as magnetic fusion ..., the Lawson criterion for the D T reaction will be derived here, but the same principle can be applied ... fusion ntau.svg right 250px thumb The Lawson criterion, or minimum value of electron density energy ... E sub minimizes near the temperature 25 electronvolt keV 300 million kelvins . The Lawson criterion ... math n e tau e math . This is the Lawson criterion. For the D T reaction, the physical value is at least .... Inertial confinement The Lawson criterion applies to inertial confinement fusion as well as to magnetic ... of the mass density R 1 g cm To satisfy this criterion at the density of solid D T 0.2 g cm ... more details
Unreferenced date December 2009 Tisserand s Criterion is used to determine whether or not an observed orbiting body, such as a comet or an asteroid , is the same as a previously observed orbiting body. While all the orbital parameters of an object orbiting the Sun during the close encounter with another massive body e.g. Jupiter can be changed dramatically, the value of a function of these parameters, called Tisserand s relation due to F lix Tisserand is approximately conserved, making possible to recognise the orbit after the encounter. Definition Tisserand s Criterion is computed in a circular restricted three body system. In a circular restricted three body system, one of the masses is assumed to be much smaller than the other two. The other two masses are assumed to be in a circular orbit about the system s center of mass. In addition, Tisserand s Criterion also relies on the assumptions that a one of the two larger masses is much smaller than the other large mass and b the comet or asteroid has not had a close approach to any other large mass. Two observed orbiting bodies are possibly the same if they satisfy or nearly satisfy Tisserand s Criterion math frac 1 2a 1 sqrt a 1 1 e 1 2 cos i 1 frac 1 2a 2 sqrt a 2 1 e 2 2 cos i 2 math where a is the semimajor axis , e is the eccentricity orbit eccentricity , and i is the inclination of the body s orbit. In other words, if a function of the orbital elements named Tisserand s parameter of the first observed body nearly equals the same function calculated with the orbital elements of the second observed body, the two bodies might be the same. Tisserand s relation The relation defines a function of orbital parameters, conserved approximately when the third body is far from the second perturbing mass. math frac 1 2a sqrt a 1 e 2 cos i approx rm const math The relation is derived from the Jacobi integral Jacobi constant selecting a suitable unit system and using some approximations. Traditionally, the units are chosen in order ... more details
The Cass Criterion is a central result in theory of overlapping generations model s in economics . It is named after David Cass . ref name Cass72 citation last1 Cass first1 David year 1972 title On capital overaccumulation in the aggregative neoclassical model of economic growth a complete characterization journal Journal of Economic Theory volume 4 pages 200 223 doi 10.1016 0022 0531 72 90149 4 issue 2 ref ref name BalaskoShell80 citation last1 Balasko first1 Yves authorlink Yves Balasko last2 Shell first2 Karl authorlink2 Karl Shell year 1980 title The overlapping generations model, I the case of pure exchange without money journal Journal of Economic Theory volume 23 pages 281 306 doi 10.1016 0022 0531 80 90013 7 issue 3 ref A major feature which sets overlapping generations models in economics apart from the standard model with a finite number of infinitely lived individuals is that the First welfare theorem might not hold, that is competitive equilibrium competitive equilibria may be not be Pareto optimal . If math p t math represents the vector of Arrow Debreu Good economics and accounting commodity prices prevailing in period math t math then a competitive equilibrium allocation is Pareto efficiency inefficient if and only if math sum t 0 infty frac 1 p t infty math . References reflist econ stub Category Economics models ... more details
In syntax , the theta criterion in its original form states that in a grammatical sentence, every theta role that a verb can assign must be realized by some verb argument argument , and each argument may bear only a single theta role. ref cite book title Lectures on Government and Binding The Pisa Lectures author Chomsky, Noam year 1981 1993 publisher Mouton de Gruyter See p. 36 ref So, for instance, for the verb send , which is associated with the theta roles of Agent, Goal and Theme, a well formed sentence might look like this Robert sent Mary a package. Here, the three theta roles are assigned to Robert , Mary and a package , respectively. The sentence would have been malformed if any of these arguments had been absent. Apparent violations It is often noted that sometimes arguments of the verb are not overtly present, and that sometimes the verb may take more arguments than specified in the lexicon Gladius Maximus died the death befitting a Roman soldier . John ate . Going to the store, be back later Linguists often explain sentences 2 and 3, where the verbs seem to be missing arguments, by positing a silent pronoun pro that occupies the phonologically empty argument positions. This can be justified by noting that one cannot get the meaning from sentence 2 that John ate something inedible, such as a rock, and that in sentence 3 there is a clear implied first person subject. In sentence 1, however, there is no clear theta position that the death befitting a Roman soldier is occupying it is unclear what licenses this cognate object , producing a pseudo transitive use of the verb died . One plausible explication is the interpretation of the verb die in this phrase as synonym of suffer , undergo or be impaired thus, the sentence may be interpreted as Gladius Maximus suffered the death befitting a Roman soldier . And then, the arguments are complete. References references Category Thematic roles ling stub nl Th ta criterium ... more details
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