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Three sigma rule

Prediction interval (on the y-axis) given from the standard score (on the x-axis). The y-axis is logarithmically scaled (but the values on it are not modified).

In statistics, the 68–95–99.7 rule is a shorthand used to remember the percentage of values that lie within a band around the mean in a normal distribution with a width of two, four and six standard deviations, respectively; more accurately, 68.27%, 95.45% and 99.73% of the values lie within one, two and three standard deviations of the mean, respectively. In mathematical notation, these facts can be expressed as follows, where X is an observation from a normally distributed random variable, μ is the mean of the distribution, and σ is its standard deviation:

In the empirical sciences the so-called three-sigma rule of thumb expresses a conventional heuristic that "nearly all" values are taken to lie within three standard deviations of the mean, i.e. that it is empirically useful to treat 99.7% probability as "near certainty".[1] The usefulness of this heuristic of course depends significantly on the question under consideration, and there are other conventions, e.g. in the social sciences a result may be considered "significant" if its confidence level is of the order of a two-sigma effect (95%), while in particle physics, there is a convention of a five-sigma effect (99.99994% confidence) being required to qualify as a "discovery".

The "three sigma rule of thumb" is related to a result also known as the three-sigma rule, which states that even for non-normally distributed variables, at least 88.8% of cases should fall within properly-calculated three-sigma intervals. It follows from Chebyshev's Inequality. For unimodal distributions the probability of being within the interval is at least 95%. There may be certain assumptions for a distribution that force this probability to be at least 98%.[2]

Cumulative distribution function
Diagram showing the cumulative distribution function for the normal distribution with mean (µ) 0 and variance (σ2 ) 1.

These numerical values "68%, 95%, 99.7%" come from the cumulative distribution function of the normal distribution.

The prediction interval for any standard score z corresponds numerically to (1−(1−Φ(z))·2).

For example, Φ(2) ≈ 0.9772, or Pr(X ≤ μ + 2σ) ≈ 0.9772, corresponding to a prediction interval of (1 − (1 − 0.97725)·2) = 0.9545 = 95.45%. Note that this is not a symmetrical interval – this is merely the probability that an observation is less than μ + 2σ. To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding):

This is related to confidence interval as used in statistics: X ¯ ± 2 σ n {\displaystyle {\bar {X}}\pm 2{\frac {\sigma }{\sqrt {n}}}} is approximately a 95% confidence interval when X ¯ {\displaystyle {\bar {X}}} is the average of a sample of size n {\displaystyle n} .

Normality tests

The "68–95–99.7 rule" is often used to quickly get a rough probability estimate of something, given its standard deviation, if the population is assumed to be normal. It is also as a simple test for outliers if the population is assumed normal, and as a normality test if the population is potentially not normal.

To pass from a sample to a number of standard deviations, one first computes the deviation, either the error or residual depending on whether one knows the population mean or only estimates it. The next step is standardizing (dividing by the population standard deviation), if the population parameters are known, or studentizing (dividing by an estimate of the standard deviation), if the parameters are unknown and only estimated.

To use as a test for outliers or a normality test, one computes the size of deviations in terms of standard deviations, and compares this to expected frequency. Given a sample set, one can compute the studentized residuals and compare these to the expected frequency: points that fall more than 3 standard deviations from the norm are likely outliers (unless the sample size is significantly large, by which point one expects a sample this extreme), and if there are many points more than 3 standard deviations from the norm, one likely has reason to question the assumed normality of the distribution. This holds ever more strongly for moves of 4 or more standard deviations.

One can compute more precisely, approximating the number of extreme moves of a given magnitude or greater by a Poisson distribution, but simply, if one has multiple 4 standard deviation moves in a sample of size 1,000, one has strong reason to consider these outliers or question the assumed normality of the distribution.

For example, a 6σ event corresponds to a chance of about two parts per billion. For illustration, if events are taken to occur daily, this would correspond to an event expected every 1.4 million years. This gives a simple normality test: if one witnesses a 6σ in daily data and significantly fewer than 1 million years have passed, then a normal distribution most likely does not provide a good model for the magnitude or frequency of large deviations in this respect.

In The Black Swan, Nassim Nicholas Taleb gives the example of risk models according to which the Black Monday crash would correspond to a 36-σ event: the occurrence of such an event should instantly suggest that the model is flawed, i.e. that the process under consideration is not satisfactorily modelled by a normal distribution. Refined models should then be considered, e.g. by the introduction of stochastic volatility. In such discussions it is important to be aware of problem of the gambler's fallacy, which states that a single observation of a rare event does not contradict that the event is in fact rare. It is the observation of a plurality of purportedly rare events that increasingly undermines the hypothesis that they are rare, i.e. the validity of the assumed model. A proper modelling of this process of gradual loss of confidence in a hypothesis would involve the designation of prior probability not just to the hypothesis itself but to all possible alternative hypotheses. For this reason, statistical hypothesis testing works not so much by confirming a hypothesis considered to be likely, but by refuting hypotheses considered unlikely.

Table of numerical values

Because of the exponential tails of the normal distribution, odds of higher deviations decrease very quickly. From the rules for normally distributed data for a daily event:

Range Expected fraction of population inside range Approximate expected frequency outside range Approximate frequency for daily event
μ ± 0.5σ 0.382924922548026 2 in 3 Four times a week
μ ± σ 0.682689492137086 1 in 3 Twice a week
μ ± 1.5σ 0.866385597462284 1 in 7 Weekly
μ ± 2σ 0.954499736103642 1 in 22 Every three weeks
μ ± 2.5σ 0.987580669348448 1 in 81 Quarterly
μ ± 3σ 0.997300203936740 1 in 370 Yearly
μ ± 3.5σ 0.999534741841929 1 in 2149 Every six years
μ ± 4σ 0.999936657516334 1 in 15787 Every 43 years (twice in a lifetime)
μ ± 4.5σ 0.999993204653751 1 in 147160 Every 403 years (once in the modern era)
μ ± 0.999999426696856 1 in 1744278 Every 4776 years (once in recorded history)
μ ± 5.5σ 0.999999962020875 1 in 26330254 Every 72090 years (thrice in history of modern humankind)
μ ± 6σ 0.999999998026825 1 in 506797346 Every 1.38 million years (twice in history of humankind)
μ ± 6.5σ 0.999999999919680 1 in 12450197393 Every 34 million years (halfway since the extinction of dinosaurs)
μ ± 7σ 0.999999999997440 1 in 390682215445 Every 1.07 billion years (a quarter of Earth's history)
μ ± xσ erf ( x 2 ) {\displaystyle \operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)} 1 in 1 1 erf ( x 2 ) {\displaystyle {\tfrac {1}{1-\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)}}} Every 1 1 erf ( x 2 ) {\displaystyle {\tfrac {1}{1-\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)}}} days
See also
  1. this usage of "three-sigma rule" entered common usage in the 2000s, e.g. cited in Schaum's Outline of Business Statistics. McGraw Hill Professional. 2003. p. 359, and in Grafarend, Erik W. (2006). Linear and Nonlinear Models: Fixed Effects, Random Effects, and Mixed Models. Walter de Gruyter. p. 553.
  2. See:
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Phi Sigma Epsilon


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Phi Sigma Kappa


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Snake Eyes (also released as Snake-Eyes) is a fictional character from the G.I. Joe: A Real American Hero toyline, comic books, and cartoon series. He is one of the original and most popular members of the G.I. Joe Team , and is most known for his relationships with Scarlett and Storm Shadow . Snake Eyes is one of the most prominent characters in the G.I. Joe: A Real American Hero franchise, having appeared in every series of the franchise since its inception. He is portrayed by Ray Park in the 2009 live-action film G.I. Joe: The Rise of Cobra , and the 2013 sequel G.I. Joe: Retaliation . Profile Snake Eyes is the code name of a member of the G.I. Joe Team. He is the team's original commando , and much of his history and information, including his real name, place of birth and service number , have remained "CLASSIFIED" throughout all depictions of his origin. All that is known for certain is his rank/grade (originally U.S. Army Sergeant/E-5 , eventually reaching Sergeant First Class/E-7 before it too was ma ...more...

Slater–Condon rules


Within computational chemistry , the Slater–Condon rules express integrals of one- and two-body operators over wavefunctions constructed as Slater determinants of orthonormal orbitals in terms of the individual orbitals. In doing so, the original integrals involving N-electron wavefunctions are reduced to sums over integrals involving at most two molecular orbitals, or in other words, the original 3N dimensional integral is expressed in terms of many three- and six-dimensional integrals. The rules are used in deriving the working equations for all methods of approximately solving the Schrödinger equation that employ wavefunctions constructed from Slater determinants. These include Hartree–Fock theory , where the wavefunction is a single determinant, and all those methods which use Hartree–Fock theory as a reference such as Møller–Plesset perturbation theory , and Coupled cluster and Configuration interaction theories. In 1929 John C. Slater derived expressions for diagonal matrix elements of an approximate Ha ...more...

List of Mega Man X characters


This is a list of recurring characters appearing in the Mega Man X series of video games developed and published by Capcom . Unless otherwise stated, each of these characters is a reploid ; an artificially intelligent android . Names are organized in order of appearance, and characters who only appear in a single game are covered in the article for the corresponding game. Maverick Hunters The Maverick Hunters (Irregular Hunters イレギュラーハンター in Japan) are a group of Reploids who protect humans and other Reploids from Mavericks and the heroes of the Mega Man X series, and the protagonists of each game are prominent Maverick Hunters. When they are introduced in Mega Man X , they have already existed for quite a while, having been previously founded by Dr. Cain (who has since retired). From Mega Man X on, they battle Sigma and the other Mavericks . X Voiced by (English): Michael Donovan ( Ruby-Spears cartoon), Peter Von Gomm (2003), Mark Gatha (2004-2005), Ted Sroka (2017) Voiced by (Japanese): Megumi Ogata (1993) ...more...

Ninja Gaiden (2004 video game)


Ninja Gaiden is an action-adventure hack and slash video game developed by Team Ninja for the Xbox video game console. It went through five years of development before its release by Tecmo in 2004, and had a number of expansion packs and two remakes , Ninja Gaiden Black and Ninja Gaiden Sigma . The game follows the fictional story of Ryu Hayabusa , a master ninja , in his quest to recover a stolen sword and avenge the slaughter of his clan. Tecmo specifically targeted Ninja Gaiden at a western audience, and despite difficulties in obtaining content ratings due to the game's graphic depictions of violence, it was generally well received, and 362,441 copies were sold in North America in the first month after its release. Nevertheless, the game had to be censored for release in some regions, and Japanese sales were poor, with only 60,000 in the four months following its release. Making use of the Xbox's internet connectivity , Ninja Gaiden was the focus of a series of online contests across North America, Europe ...more...

Context-free grammar


In formal language theory, a context-free grammar ( CFG ) is a certain type of formal grammar : a set of production rules that describe all possible strings in a given formal language. Production rules are simple replacements. For example, the rule A   →   α {\displaystyle A\ \to \ \alpha } replaces A {\displaystyle A} with α {\displaystyle \alpha } . There can be multiple replacement rules for any given value. For example, A   →   α {\displaystyle A\ \to \ \alpha } A   →   β {\displaystyle A\ \to \ \beta } means that A {\displaystyle A} can be replaced with either α {\displaystyle \alpha } or β {\displaystyle \beta } . In context-free grammars, all rules are one-to-one, one-to-many, or one-to-none. These rules can be applied regardless of context. The left-hand side of the production rule is always a nonterminal symbol. This means that the symbol does not appear in the resulting formal language. So in our case, our language contains the letters α {\displaystyle \alpha } and β {\displaystyle \beta } but not A ...more...

Sigma Pi (literary society)


Sigma Pi ( ΣΠ ) is one of the four male literary societies of Illinois College . It is the oldest literary society at Illinois College and one of the oldest literary societies in the United States, having been founded on Saturday, June 24, 1843. Sigma Pi resides in the oldest college building in Illinois, Beecher Hall. William Jennings Bryan , a three time presidential nominee, is one of its most distinguished members. Origin "To Samuel Willard and Henry Wing, the idea first occurred of founding a society which would live while their Alma Mater sat in proud eminence. Accordingly, on Saturday, June 24th, 1843 the first regular meeting of the Sigma Pi Society was held in room 32, Old College Building." Thus was born the first permanent literary society at Illinois College. Sigma Pi was nameless until Barbour Lewis suggested the appropriateness of keeping "Union and Progress" and Samuel Willard selected the corresponding Greek Words, Sustasis Kai Prokape along with Henry Wing's suggestion that Sigma Pi adopt th ...more...

Chebyshev's inequality


In probability theory , Chebyshev's inequality (also spelled as Tchebysheff's inequality , Russian : Нера́венство Чебышёва , also called Bienaymé-Chebyshev inequality ) guarantees that, for a wide class of probability distributions , no more than a certain fraction of values can be more than a certain distance from the mean . Specifically, no more than 1/k of the distribution's values can be more than k standard deviations away from the mean (or equivalently, at least 1−1/k of the distribution's values are within k standard deviations of the mean). The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers . In practical usage, in contrast to the 68–95–99.7 rule , which applies to normal distributions , Chebyshev's inequality is weaker, stating that a ...more...

Leibniz formula for determinants


In algebra , the Leibniz formula , named in honor of Gottfried Leibniz , expresses the determinant of a square matrix in terms of permutations of the matrix elements. If A is an n×n matrix, where a is the entry in the ith row and jth column of A, the formula is where sgn is the sign function of permutations in the permutation group S, which returns +1 and −1 for even and odd permutations , respectively. Another common notation used for the formula is in terms of the Levi-Civita symbol and makes use of the Einstein summation notation , where it becomes which may be more familiar to physicists. Directly evaluating the Leibniz formula from the definition requires Ω ( n ! ⋅ n ) {\displaystyle \Omega (n!\cdot n)} operations in general—that is, a number of operations asymptotically proportional to n factorial —because n! is the number of order-n permutations. This is impractically difficult for large n. Instead, the determinant can be evaluated in O(n ) operations by forming the LU decomposition A = L U {\displayst ...more...

Pareto distribution


The Pareto distribution , named after the Italian civil engineer , economist , and sociologist Vilfredo Pareto , is a power law probability distribution that is used in description of social , scientific , geophysical , actuarial , and many other types of observable phenomena. Definition If X is a random variable with a Pareto (Type I) distribution, then the probability that X is greater than some number x, i.e. the survival function (also called tail function), is given by where x is the (necessarily positive) minimum possible value of X, and α is a positive parameter. The Pareto Type I distribution is characterized by a scale parameter x and a shape parameter α, which is known as the tail index. When this distribution is used to model the distribution of wealth, then the parameter α is called the Pareto index . Properties Cumulative distribution function From the definition, the cumulative distribution function of a Pareto random variable with parameters α and x is Probability density function It follows ( ...more...

Foveon X3 sensor


The Foveon X3 sensor is an image sensor for digital cameras, designed by Foveon, Inc. (now part of Sigma Corporation ) and manufactured by Dongbu Electronics. It uses an array of photosites, each of which consists of three vertically stacked photodiodes , organized in a two-dimensional grid. Each of the three stacked photodiodes responds to different wavelengths of light; that is, each has a different spectral sensitivity curve. This difference is because different wavelengths of light penetrate silicon to different depths. The signals from the three photodiodes are then processed, resulting in data that provides the amounts of three additive primary colors , red, green, and blue. The sensor was first deployed in 2002 in the Sigma SD9 DSLR camera, and most recently in Sigma SD14 which was released in 2007 and Sigma DP2 from 2012. The development of the Foveon X3 technology is the subject of the 2005 book The Silicon Eye by George Gilder . Operation Color absorption in silicon and the Foveon X3 sensor. See t ...more...

Orbital hybridisation


In chemistry , orbital hybridisation (or hybridization ) is the concept of mixing atomic orbitals into new hybrid orbitals (with different energies, shapes, etc., than the component atomic orbitals) suitable for the pairing of electrons to form chemical bonds in valence bond theory . Hybrid orbitals are very useful in the explanation of molecular geometry and atomic bonding properties. Although sometimes taught together with the valence shell electron-pair repulsion (VSEPR) theory , valence bond and hybridisation are in fact not related to the VSEPR model. History Chemist Linus Pauling first developed the hybridisation theory in 1931 in order to explain the structure of simple molecules such as methane (CH) using atomic orbitals . Pauling pointed out that a carbon atom forms four bonds by using one s and three p orbitals, so that "it might be inferred" that a carbon atom would form three bonds at right angles (using p orbitals) and a fourth weaker bond using the s orbital in some arbitrary direction. In real ...more...

Gamma matrices


In mathematical physics , the gamma matrices , { γ 0 , γ 1 , γ 2 , γ 3 } {\displaystyle \{\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3}\}} , also known as the Dirac matrices , are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cℓ( R ). It is also possible to define higher-dimensional gamma matrices . When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space , the column vectors on which the matrices act become a space of spinors , on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts . Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin-½ particles. In Dirac representation , the four contravariant gamma matrices are γ 0 {\displaystyle \gamma ^{0}} is the time-like matrix and ...more...

Frontier molecular orbital theory


In chemistry , frontier molecular orbital theory is an application of MO theory describing HOMO / LUMO interactions. History In 1952, Kenichi Fukui published a paper in the Journal of Chemical Physics titled "A molecular theory of reactivity in aromatic hydrocarbons." Though widely criticized at the time, he later shared the Nobel Prize in Chemistry with Roald Hoffmann for his work on reaction mechanisms. Hoffman's work focused on creating a set of four pericyclic reactions in organic chemistry, based on orbital symmetry, which he coauthored with Robert Burns Woodward , entitled "The Conservation of Orbital Symmetry." Fukui's own work looked at the frontier orbitals, and in particular the effects of the Highest Occupied Molecular Orbital ( HOMO ) and the Lowest Unoccupied Molecular Orbital ( LUMO ) on reaction mechanisms, which led to it being called Frontier Molecular Orbital Theory (FMO Theory). He used these interactions to better understand the conclusions of the Woodward–Hoffmann rules . Theory Fukui re ...more...

Linear discriminant analysis


Ronald Fisher Linear discriminant analysis ( LDA ) is a generalization of Fisher's linear discriminant , a method used in statistics , pattern recognition and machine learning to find a linear combination of features that characterizes or separates two or more classes of objects or events. The resulting combination may be used as a linear classifier , or, more commonly, for dimensionality reduction before later classification . LDA is closely related to analysis of variance (ANOVA) and regression analysis , which also attempt to express one dependent variable as a linear combination of other features or measurements. However, ANOVA uses categorical independent variables and a continuous dependent variable , whereas discriminant analysis has continuous independent variables and a categorical dependent variable (i.e. the class label). Logistic regression and probit regression are more similar to LDA than ANOVA is, as they also explain a categorical variable by the values of continuous independent variables. T ...more...

Impossible Creatures


Impossible Creatures is a real-time strategy game released in 2003 and developed by Relic Entertainment in conjunction with Microsoft Studios . Its unique feature is that the armies used are all created by the player. The armies consist of 9 creatures; each one is a combination of any two animals from a list of 76 (51 with no downloads). Many animals possess inherent abilities to add more strategic depth to the game. There is an extensive single-player campaign as well as online multiplayer functionality with different game modes, add-ons, custom maps, mods, and scenarios. Impossible Creatures was followed up later by a free downloadable expansion entitled Insect Invasion which added new creatures and abilities to the game. The last official add on for Impossible Creatures was released in 2004. Relic has stated that they have no further plans for the Impossible Creatures universe. On November 12, 2015, Impossible Creatures was released on Steam as Impossible Creatures: Steam Edition , by THQ Nordic . Relic En ...more...

Volatility (finance)


In finance , volatility (symbol σ) is the degree of variation of a trading price series over time as measured by the standard deviation of logarithmic returns . Historic volatility is derived from time series of past market prices. An implied volatility is derived from the market price of a market traded derivative (in particular an option) Volatility terminology Volatility as described here refers to the actual volatility , more specifically: actual current volatility of a financial instrument for a specified period (for example 30 days or 90 days), based on historical prices over the specified period with the last observation the most recent price. actual historical volatility which refers to the volatility of a financial instrument over a specified period but with the last observation on a date in the past near synonymous is realized volatility , the square root of the realized variance , in turn calculated using the sum of squared returns divided by the number of observations. actual future volatility whi ...more...

Control chart


Control charts , also known as Shewhart charts (after Walter A. Shewhart ) or process-behavior charts , are a statistical process control tool used to determine if a manufacturing or business process is in a state of control . Overview If analysis of the control chart indicates that the process is currently under control (i.e., is stable, with variation only coming from sources common to the process), then no corrections or changes to process control parameters are needed or desired. In addition, data from the process can be used to predict the future performance of the process. If the chart indicates that the monitored process is not in control, analysis of the chart can help determine the sources of variation, as this will result in degraded process performance. A process that is stable but operating outside desired (specification) limits (e.g., scrap rates may be in statistical control but above desired limits) needs to be improved through a deliberate effort to understand the causes of current performanc ...more...

University of Alabama


The University of Alabama ( Alabama or UA ) is a public research university located in Tuscaloosa , Alabama , United States, and the flagship of the University of Alabama System . Founded in 1820, UA is the oldest and largest of the public universities in Alabama . UA offers programs of study in 13 academic divisions leading to bachelor's, master's, Education Specialist , and doctoral degrees. The only publicly supported law school in the state is at UA. Other academic programs unavailable elsewhere in Alabama include doctoral programs in anthropology , communication and information sciences, metallurgical engineering, music, Romance languages , and social work . As one of the first public universities established in the early 19th century southwestern frontier of the United States, the University of Alabama has left a vast cultural imprint on the state, region and nation over the past two centuries. The school was a center of activity during the American Civil War and the Civil Rights Movement . The Univers ...more...

G.I. Joe: Sigma 6


G.I. Joe: Sigma 6 is a line of military-themed action figures and toys produced by Hasbro , re-imagining the characters of the 1980s toyline, G.I. Joe: A Real American Hero . The Sigma 6 toy line served several purposes for Hasbro. First, it allowed them to depart from the classic 3 ¾-inch format of the A Real American Hero series of the 1980s; most Sigma 6 action figures stand at approximately 8 inches (200 mm) and have more articulation. Second, the new series offered them the chance to streamline the story and characters, stripping away old continuity and "rebooting" the franchise with younger versions of the cast - similar to but different from their 1980s namesakes - rendered in the popular anime style. The line proved to be polarizing with G.I. Joe fans and ultimately unsuccessful at retail. Sigma 6 was cancelled after two years leaving many showcased sets and characters unproduced. In 2007 Hasbro would later go back to the classic 3¾”- 4” scale for the 25th anniversary line and beyond. History In 2005, ...more...

Laboratory quality control


Laboratory quality control is designed to detect, reduce, and correct deficiencies in a laboratory's internal analytical process prior to the release of patient results, in order to improve the quality of the results reported by the laboratory. Quality control is a measure of precision, or how well the measurement system reproduces the same result over time and under varying operating conditions. Laboratory quality control material is usually run at the beginning of each shift, after an instrument is serviced, when reagent lots are changed, after calibration, and whenever patient results seem inappropriate. Quality control material should approximate the same matrix as patient specimens, taking into account properties such as viscosity, turbidity, composition, and color. It should be simple to use, with minimal vial to vial variability, because variability could be misinterpreted as systematic error in the method or instrument. It should be stable for long periods of time, and available in large enough quant ...more...

Sequent calculus


Sequent calculus is, in essence, a style of formal logical argumentation where every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen ) instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference , giving a better approximation to the style of natural deduction used by mathematicians than David Hilbert's earlier style of formal logic where every line was an unconditional tautology. There may be more subtle distinctions to be made; for example, there may be non-logical axioms upon which all propositions are implicitly dependent. Then sequents signify conditional theorems in a first-order language rather than conditional tautologies. Sequent calculus is one of several extant styles of proof calculus for expressing line-by-line logical arguments. Hilbert style . Every line is an unconditional tautology (or theorem). Gentzen style. Every line is a ...more...

Mitsubishi Galant


The Mitsubishi Galant is an automobile that was manufactured by Mitsubishi from 1969 to 2012. The name was derived from the French word galant, meaning "chivalrous". There have been nine distinct generations; cumulative sales exceed five million. It began as a compact sedan, but over the course of its life evolved into a mid-size car . Initial production was based in Japan, but since 1994 the American market was served by vehicles assembled at the former Diamond-Star Motors (DSM) facility in Normal, Illinois . First generation (A50; 1969–1973) The first generation of the car, initially known as the Colt Galant, was released in December 1969 at a new Mitsubishi Japanese dealership called Galant Shop . The design was dubbed "Dynawedge" by Mitsubishi, referring to the influence of aerodynamics on the silhouette. Three models were available, powered by the new 'Saturn' engine in 1.3- (AI model) or 1.5-liter (AII and AIII) configurations. 1.4- and 1.6-liter versions (14L and 16L) replaced these in September 197 ...more...

Relational algebra


Relational algebra , first created by Edgar F. Codd while at IBM, is a family of algebras with a well-founded semantics used for modelling the data stored in relational databases, and defining queries on it. The main application of relational algebra is providing a theoretical foundation for relational databases , particularly query languages for such databases, chief among which is SQL . Introduction Relational algebra received little attention outside of pure mathematics until the publication of E.F. Codd 's relational model of data in 1970. Codd proposed such an algebra as a basis for database query languages. (See section Implementations .) Five primitive operators of Codd's algebra are the selection , the projection , the Cartesian product (also called the cross product or cross join), the set union , and the set difference . Set operators The relational algebra uses set union , set difference , and Cartesian product from set theory , but adds additional constraints to these operators. For set union and ...more...

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