importantly, the law applies ( as the name indicates ) only when a large number of observations are considered. there is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be " balanced " by the others ( see the gambler's fallacy ). the lln only applies to the average. therefore, while other formulas that look similar are not verified, such as the raw deviation from " theoretical results " : not only does it not converge toward zero as n increases, but it tends to increase in absolute value as n increases.


squaring both sides yields 2b2 = a2. since the expression on the left is an integer multiple of 2, the right expression is by definition divisible by 2. that is, a2 is even, which implies that a must also be even, as seen in the proposition above ( in # proof by contraposition ).


the theory of linear programming dictates that under mild assumptions ( if the linear program has an optimal solution, and if the feasible region does not contain a line ), one can always find an extreme point or a corner point that is optimal. the obtained optimum is tested for being an integer solution. if it is not, there is guaranteed to exist a linear inequality that separates the optimum from the convex hull of the true feasible set.


in mathematical logic and computer science, the lambda - mu calculus is an extension of the lambda calculus introduced by m. parigot. it introduces two new operators : the μ operator ( which is completely different both from the μ operator found in computability theory and from the μ operator of modal μ - calculus ) and the bracket operator. proof - theoretically, it provides a well - behaved formulation of classical natural deduction. one of the main goals of this extended calculus is to be able to describe expressions corresponding to theorems in classical logic.


in most non - strict languages the non - strictness extends to data constructors. this allows conceptually infinite data structures ( such as the list of all prime numbers ) to be manipulated in the same way as ordinary finite data structures. it also allows for the use of very large but finite data structures such as the complete game tree of chess. non - strictness has several disadvantages which have prevented widespread adoption : because of the uncertainty regarding if and when expressions will be evaluated, non - strict languages generally must be purely functional to be useful.


another constraint is adjacency. this constraint applies to the case when the " cake " is a disputed territory that has to be divided among neighboring countries. in this case, it may required that the piece allocated to each country is adjacent to its current territory ; this constraint is handled by hill's land division problem. in land division there are often two - dimensional geometric constraints, e. g., each piece should be a square or ( more generally ) a fat object.


define the constant φ { \ displaystyle \ phi } as φ = min s ⊂ x, π ( s ) ≤ 1 2 q ( s × s c ) π ( s ). { \ displaystyle \ phi = \ min _ { s \ subset x, \ pi ( s ) \ leq { \ frac { 1 } { 2 } } } { \ frac { q ( s \ times s ^ { c } ) } { \ pi ( s ) } }. } the operator k, { \ displaystyle k, } acting on the space of functions from | x | { \ displaystyle | x | } to | x | { \ displaystyle | x | }, defined by ( k ) ( x ) = y k ( x, y ) ( y ) { \ displaystyle ( k \ phi ) ( x ) = \ sum _ { y } k ( x, y ) \ phi ( y ) } has eigenvalues λ 1 ≥ λ 2 ≥ ≥ λ n { \ displaystyle \ lambda _ { 1 } \ geq \ lambda _ { 2 } \ geq \ cdots \ geq \ lambda _ { n } }.


in some sense the 0 - 1 indicator function is the most natural loss function for classification. it takes the value 0 if the predicted output is the same as the actual output, and it takes the value 1 if the predicted output is different from the actual output. for binary classification with y = { − 1, 1 } { \ displaystyle y = \ { - 1, 1 \ } }, this is : v ( f ( x → ), y ) = θ ( − y f ( x → ) ) { \ displaystyle v ( f ( { \ vec { x } } ), y ) = \ theta ( - yf ( { \ vec { x } } ) ) } where θ { \ displaystyle \ theta } is the heaviside step function.


in metadata, an identification scheme is used to identify unique records in a set. if a data element is used to identify a record within a data set, the data element uses the identifier representation term. an identification scheme should be contrasted with a classification scheme. classification schemes are used to classify individual records into categories. many records in a data set may be in a single category.


in telecommunications, a remote digital terminal ( rdt ) typically accepts e1, t1 or oc - 3 digital lines to communicate with a telephone access network ( an ) or telephone exchange ( local digital switch, lds ) on one side, and forms a local exchange ( le ) on the other, which is connected to " plain old telephone service " ( pots ) lines.


in statistics and machine learning, lasso ( least absolute shrinkage and selection operator ; also lasso or lasso ) is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the resulting statistical model. it was originally introduced in geophysics, and later by robert tibshirani, who coined the term. lasso was originally formulated for linear regression models. this simple case reveals a substantial amount about the estimator.


in the following table are the decimal values of the kaktovik digits up to three places to the left and to the right of the units'place.


in relational algebra, a rename is a unary operation written as ρ a / b ( r ) { \ displaystyle \ rho _ { a / b } ( r ) } where : r is a relation a and b are attribute names b is an attribute of rthe result is identical to r except that the b attribute in all tuples is renamed to a. for an example, consider the following invocation of ρ on an employee relation and the result of that invocation : formally, the semantics of the rename operator is defined as follows : ρ a / b ( r ) = { t : t ∈ r }, { \ displaystyle \ rho _ { a / b } ( r ) = \ { \ t : t \ in r \ \ }, } where t { \ displaystyle t } is defined as the tuple t, with the b attribute renamed to a, so that : t = { ( c, v ) | ( c, v ) ∈ t, c = b } ∪ { ( a, t ( b ) ) }. { \ displaystyle t = \ { \ ( c, v ) \ | \ ( c, v ) \ in t, \ c \ neq b \ \ } \ cup \ { \ ( a, \ t ( b ) ) \ \ }. } = = references = =


operation itself is triggered by writing data to a triggering operand of an operation. thus, an operation is executed as a side effect of the triggering data transport. therefore, executing an addition operation in tta requires three data transport definitions, also called moves.


in probability theory, a joint distribution can be factorized into a product between conditional and marginal distributions q ( x, y ) = p ( x y ) π ( y ) { \ displaystyle q ( x, y ) = p ( x \ mid y ) \ pi ( y ) } the analog of this rule in the kernel embedding framework states that c x y π, { \ displaystyle { \ mathcal { c } } _ { xy } ^ { \ pi }, } the joint embedding of q ( x, y ), { \ displaystyle q ( x, y ), } can be factorized as a composition of conditional embedding operator with the auto - covariance operator associated with π ( y ) { \ displaystyle \ pi ( y ) } c x y π = c x y c y y π { \ displaystyle { \ mathcal { c } } _ { xy } ^ { \ pi } = { \ mathcal { c } } _ { x \ mid y } { \ mathcal { c } } _ { yy } ^ { \ pi } } where c x y π = e, { \ displaystyle { \ mathcal { c } } _ { xy } ^ { \ pi } = \ mathbb { e }, } c y y π = e. { \ displaystyle { \ mathcal { c } } _ { yy } ^ { \ pi } = \ mathbb { e }. } in practical implementations, the kernel chain rule takes the following form c ^ x y π = c ^ x y c ^ y y π = υ ( g + λ i ) − 1 g ~ diag ( α ) φ ~ t { \ displaystyle { \ widehat { \ mathcal { c } } } _ { xy } ^ { \ pi } = { \ widehat { \ mathcal { c } } } _ { x \ mid y } { \ widehat { \ mathcal { c } } } _ { yy } ^ { \ pi } = { \ boldsymbol { \ upsilon } } ( \ mathbf { g } + \ lambda \ mathbf { i } ) ^ { - 1 } { \ widetilde { \ mathbf { g } } } \ operatorname { diag } ( { \ boldsymbol { \ alpha } } ) { \ boldsymbol { \ widetilde


{ \ phi } } } ^ { t } }


to 4 : 00 p. m. eastern time. index etfs are also sometimes weighted by revenue rather than market capitalization.


in proof theory and constructive mathematics, the principle of independence of premise states that if φ and θ are sentences in a formal theory and φ → θ is provable, then ( φ → θ ) is provable. here x cannot be a free variable of φ, while θ can be a predicate depending on it. the main application of the principle is in the study of intuitionistic logic, where the principle is not generally valid. the principle is valid in classical logic.


in project management, communication management must address the following questions : what information needs to flow in and out of the project? who needs what information? when is the information needed?


in mathematics, and more specifically in abstract algebra, a * - algebra ( or involutive algebra ) is a mathematical structure consisting of two involutive rings r and a, where r is commutative and a has the structure of an associative algebra over r. involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a hilbert space and hermitian adjoints. however, it may happen that an algebra admits no involution.


in mathematics, a real - valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. equivalently, a function is convex if its epigraph ( the set of points on or above the graph of the function ) is a convex set. a twice - differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. well - known examples of convex functions of a single variable include a linear function f ( x ) = c x { \ displaystyle f ( x ) = cx } ( where c { \ displaystyle c } is a real number ), a quadratic function c x 2 { \ displaystyle cx ^ { 2 } } ( c { \ displaystyle c } as a nonnegative real number ) and a exponential function c e x { \ displaystyle ce ^ { x } } ( c { \ displaystyle c } as a nonnegative real number ).


in taxonomy, an undescribed taxon is a taxon ( for example, a species ) that has been discovered, but not yet formally described and named. the various nomenclature codes specify the requirements for a new taxon to be validly described and named. until such a description has been published, the taxon has no formal or official name, although a temporary, informal name is often used. a published scientific name may not fulfil the requirements of the codes for various reasons.


in probability theory, especially as that field is used in statistics, a group family of probability distributions is a family obtained by subjecting a random variable with a fixed distribution to a suitable family of transformations such as a location - scale family, or otherwise a family of probability distributions acted upon by a group. consideration of a particular family of distributions as a group family can, in statistical theory, lead to the identification of an ancillary statistic.


in order for programs and interrupt handlers to work without interference and share the same hardware memory and access to the i / o system, in a multitasking operating systems running on a digital system with a single cpu / mcu it is required to have some sort of software and hardware facilities to keep track of an executing processes data ( memory page addresses, registers etc. ) and to save and recover them back to the state they were in before they were suspended. this is achieved by a context switching. : 3. 3 the running programs are often assigned a process context identifiers ( pcid ). in linux - based operating systems, a set of data stored in registers is usually saved into a process descriptor in memory to implement switching of context. pcids are also used.


however, if this would result in the same output level as the previous bit time, a 0 level is output instead. initially, the encoder output present state is assumed at 0 level when the first bit arrives at the encoder input. the new line - coding scheme violates the encoding rule of nrz - l when a sequence of 1s or 0s arrives and hence, it overcomes some of their deficiencies. during the violation period for a run of 1s or 0s, it operates on the same encoding rule of the polar rz but with pulse occupancy of full period.


in model theory, a mathematical discipline, a β - model ( from the french " bon ordre ", well - ordering ) is a model which is correct about statements of the form " x is well - ordered ". the term was introduced by mostowski ( 1959 ) as a strengthening of the notion of ω - model. in contrast to the notation for set - theoretic properties named by ordinals, such as ξ { \ displaystyle \ xi } - indescribability, the letter β here is only denotational.


in multi - channel systems, channels are used for separate purposes.


in the area of modern algebra known as group theory, a tarski monster group, named for alfred tarski, is an infinite group g, such that every proper subgroup h of g, other than the identity subgroup, is a cyclic group of order a fixed prime number p. a tarski monster group is necessarily simple. it was shown by alexander yu. olshanskii in 1979 that tarski groups exist, and that there is a tarski p - group for every prime p > 1075. they are a source of counterexamples to conjectures in group theory, most importantly to burnside's problem and the von neumann conjecture.


in number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer n as the sum of k squares, where representations that differ only in the order of the summands or in the signs of the numbers being squared are counted as different, and is denoted by rk ( n ).


the tsallis distributions have been applied to problems in the fields of statistical mechanics, geology, anatomy, astronomy, economics, finance, and machine learning. the distributions are often used for their heavy tails. note that tsallis distributions are obtained as box – cox transformation over usual distributions, with deformation parameter λ = 1 − q { \ displaystyle \ lambda = 1 - q }. this deformation transforms exponentials into q - exponentials.


in october 2006, html inventor and w3c chair tim berners - lee, introducing a major w3c effort to develop a new html specification, posted in his blog that, " the attempt to get the world to switch to xml... all at once didn't work. the large html - generating public did not move... some large communities did shift and are enjoying the fruits of well - formed systems... the plan is to charter a completely new html group. " the current html5 working draft says " special attention has been given to defining clear conformance criteria for user agents in an effort to improve interoperability... while at the same time updating the html specifications to address issues raised in the past few years. " ian hickson, editor of the html5 specification criticizing the improper use of xhtml in 2002, is a member of the group developing this specification and is listed as one of the co - editors of the current working draft. simon pieters researched the xml - compliance of mobile browsers and concluded " the claim that xhtml would be needed for mobile devices is simply a myth ".


nor can there be an infinite number of middle terms between the first principle and the conclusion. in all demonstration, the first principles, the conclusion, and all the intermediate propositions, must be necessary, general and eternal truths. of things that happen by chance, or contingently, or which can change, or of individual things, there is no demonstration.


we call it β 2, { \ displaystyle \ beta _ { 2 }, } and define a new sequence ⟨ β 2, i ⟩ { \ displaystyle \ langle \ beta _ { 2, i } \ rangle } similar to the previous sequence. we can repeat this process, getting a sequence of sequences ⟨ β j, i ⟩ { \ displaystyle \ langle \ beta _ { j, i } \ rangle } where each element of a sequence is greater than every member of the previous sequences. then for each i < α, { \ displaystyle i < \ alpha, } ⟨ β j, i ⟩ { \ displaystyle \ langle \ beta _ { j, i } \ rangle } is an increasing sequence contained in c i, { \ displaystyle c _ { i }, } and all these sequences have the same limit ( the limit of ⟨ β j, i ⟩ { \ displaystyle \ langle \ beta _ { j, i } \ rangle } ).


in the japanese kantenji braille, the standard 8 - dot braille patterns 356, 1356, 3456, and 13456 are the patterns related to braille pattern dots - 245, since the two additional dots of kantenji patterns 0245, 2457, and 02457 are placed above the base 6 - dot cell, instead of below, as in standard 8 - dot braille.


in the european union, the proposed artificial intelligence act includes requirements to disclose copyrighted material used to train generative ai systems, and to label any ai - generated output as such. in the united states, a group of companies including openai, alphabet, and meta signed a voluntary agreement with the white house in july 2023 to watermark ai - generated content. in china, the interim measures for the management of generative ai services introduced by the cyberspace administration of china regulates any public - facing generative ai. it includes requirements to watermark generated images or videos, regulations on training data and label quality, restrictions on personal data collection, and a guideline that generative ai must " adhere to socialist core values ".


for detecting performance regressions, software performance tests are run on a regular basis, to monitor the response time and resource usage metrics of the software after subsequent changes. unlike functional regression tests, the results of performance tests are subject to variance - that is, results can differ between tests due to variance in performance measurements ; as a result, a decision must be made on whether a change in performance numbers constitutes a regression, based on experience and end - user demands. approaches such as statistical significance testing and change point detection are sometimes used to aid in this decision.


in the finnish language, there are three classes of vowels – front, back, and neutral, where each front vowel has a back vowel pairing. grammatical endings such as case and derivational endings – but not enclitics – have only archiphonemic vowels u, o, a, which are realized as either back or front inside a single word. from vowel harmony it follows that the initial syllable of each single ( non - compound ) word controls the frontness or backness of the entire word. non - initially, the neutral vowels are transparent to and unaffected by vowel harmony.


in mathematics, a noncototient is a positive integer n that cannot be expressed as the difference between a positive integer m and the number of coprime integers below it. that is, m − φ ( m ) = n, where φ stands for euler's totient function, has no solution for m. the cototient of n is defined as n − φ ( n ), so a noncototient is a number that is never a cototient. it is conjectured that all noncototients are even. this follows from a modified form of the slightly stronger version of the goldbach conjecture : if the even number n can be represented as a sum of two distinct primes p and q, then p q − φ ( p q ) = p q − ( p − 1 ) ( q − 1 ) = p + q − 1 = n − 1.


in number theory, selberg's identity is an approximate identity involving logarithms of primes named after atle selberg. the identity, discovered jointly by selberg and paul erdos, was used in the first elementary proof for the prime number theorem.


in probability theory, coupling is a proof technique that allows one to compare two unrelated random variables ( distributions ) x and y by creating a random vector w whose marginal distributions correspond to x and y respectively. the choice of w is generally not unique, and the whole idea of " coupling " is about making such a choice so that x and y can be related in a particularly desirable way.


in the dynamics of the sandpile automaton defined above, some stable configurations ( 0 ≤ z ( v ) < 4 { \ displaystyle 0 \ leq z ( v ) < 4 } for all v ∈ g { s } { \ displaystyle v \ in g \ setminus \ { s \ } } ) appear infinitely often, while others can only appear a finite number of times ( if at all ). the former are referred to as recurrent configurations, while the latter are referred to as transient configurations. the recurrent configurations thereby consist of all stable non - negative configurations which can be reached from any other stable configuration by repeatedly adding grains of sand to vertices and toppling. it is easy to see that the minimally stable configuration z m { \ displaystyle z _ { m } }, where every vertex carries z m ( v ) = d e g ( v ) − 1 { \ displaystyle z _ { m } ( v ) = deg ( v ) - 1 } grains of sand, is reachable from any other stable configuration ( add d e g ( v ) − z ( v ) − 1 ≥ 0 { \ displaystyle deg ( v ) - z ( v ) - 1 \ geq 0 } grains to every vertex ).


kenkichi iwasawa ( 1941 ) proved that a p - group g is an iwasawa group if and only if one of the following cases happens : g is a dedekind group, or g contains an abelian normal subgroup n such that the quotient group g / n is a cyclic group and if q denotes a generator of g / n, then for all n ∈ n, q−1nq = n1 + ps where s ≥ 1 in general, but s ≥ 2 for p = 2. in berkovich & janko ( 2008, p. 257 ), iwasawa's proof was deemed to have essential gaps, which were filled by franco napolitani and zvonimir janko. roland schmidt ( 1994 ) has provided an alternative proof along different lines in his textbook.


in practice, there are a wide rarity of methods that are utilized, most of which are reconstruct 3 - d information ( volume ) from 2 - d signals ( image ). typically used methods are ct, mri, pet and spect. and the filtered back projection based on the principles introduced above are commonly applied.


in relational database management systems, a user - defined function provides a mechanism for extending the functionality of the database server by adding a function, that can be evaluated in standard query language ( usually sql ) statements. the sql standard distinguishes between scalar and table functions. a scalar function returns only a single value ( or null ), whereas a table function returns a ( relational ) table comprising zero or more rows, each row with one or more columns. user - defined functions in sql are declared using the create function statement.


for example, for aes with a 256 - bit key, k { \ displaystyle k } is a 256 - bit number and f { \ displaystyle f } is a 128 - bit number. encrypting block p { \ displaystyle p } with logical index ( tweak ) i { \ displaystyle i } uses the following formula : x = f ⊗ i, c = e k ( p ⊕ x ) ⊕ x. { \ displaystyle { \ begin { aligned } x & = f \ otimes i, \ \ c & = e _ { k } ( p \ oplus x ) \ oplus x. \ end { aligned } } } here multiplication ⊗ { \ displaystyle \ otimes } and addition ⊕ { \ displaystyle \ oplus } are performed in the finite field ( gf ( 2 128 ) { \ displaystyle { \ text { gf } } \ left ( 2 ^ { 128 } \ right ) } for aes ).


the solution corresponding to the original constrained optimization is always a saddle point of the lagrangian function, which can be identified among the stationary points from the definiteness of the bordered hessian matrix. the great advantage of this method is that it allows the optimization to be solved without explicit parameterization in terms of the constraints. as a result, the method of lagrange multipliers is widely used to solve challenging constrained optimization problems. further, the method of lagrange multipliers is generalized by the karush – kuhn – tucker conditions, which can also take into account inequality constraints of the form h ( x ) ≤ c { \ displaystyle h ( \ mathbf { x } ) \ leq c } for a given constant c { \ displaystyle c ~ }.


in synchronous duplex mode of operation hardware coupling is provided between two processors which execute same set of instructions and compare the results continuously. if mismatch occurs then the faulty processor is identified and taken out of service within a few milliseconds. when system is operating normally, the two processors have same data in memories at all times and simultaneously receive information from exchange environment.


a property that is invariant under such deformations is a topological property. basic examples of topological properties are : the dimension, which allows distinguishing between a line and a surface ; compactness, which allows distinguishing between a line and a circle ; connectedness, which allows distinguishing a circle from two non - intersecting circles.


this is especially useful when dealing with the semantics of different side effects, such as nontermination, mutable state or nondeterminism. instead of giving two variants of the semantics, one for the call - by - name evaluation order and one for the call - by - value one, one can simply give a semantics for the cbpv term language ; one gets two semantics for lambda - calculus by composing this cbpv semantics with the same cbv and cbn translations from lambda - calculus. = = references = =


one way to construct a family of sets with these parameters, each two sharing an element, is to choose a single element to belong to all the subsets, and then form all of the subsets that contain the chosen element. the erdos – ko – rado theorem states that when n { \ displaystyle n } is large enough for the problem to be nontrivial ( n ≥ 2 r { \ displaystyle n \ geq 2r } ) this construction produces the largest possible intersecting families.


in number theory, many arithmetic functions are integer - valued.


in mathematics, a lucas – carmichael number is a positive composite integer n such that if p is a prime factor of n, then p + 1 is a factor of n + 1 ; n is odd and square - free. the first condition resembles the korselt's criterion for carmichael numbers, where - 1 is replaced with + 1. the second condition eliminates from consideration some trivial cases like cubes of prime numbers, such as 8 or 27, which otherwise would be lucas – carmichael numbers ( since n3 + 1 = ( n + 1 ) ( n2 − n + 1 ) is always divisible by n + 1 ). they are named after edouard lucas and robert carmichael.


in psychology, data collection often relies on samples drawn from western, educated, industrialized, rich and democratic ( weird ) populations, which impedes the overall generalizability of findings. crowdsourcing the data collection process by relying on multi - lab data collection and online crowdsourcing platforms ( e. g., amazon mechanical turk, prolific ) makes it easier to reach a wider audience of participants from different cultural backgrounds and non - weird populations. when the research question makes it possible to rely on internet samples, it is also an easy way to recruit larger samples of participants with minimal financial input and within short amounts of time. most of the time, members of the general public are recruited to undergo studies as research participants but they can also be recruited to collect data and observations.


in the character sets developed for computing, each upper - and lower - case letter is encoded as a separate character. in order to enable case folding and case conversion, the software needs to link together the two characters representing the case variants of a letter. ( some old character - encoding systems, such as the baudot code, are restricted to one set of letters, usually represented by the upper - case variants. ) case - insensitive operations can be said to fold case, from the idea of folding the character code table so that upper - and lower - case letters coincide.


in the following example, there is a common swedish surname astrom written in the two alternative methods, the first one with a precomposed a ( u + 00c5 ) and o ( u + 00f6 ), and the second one using a decomposed base letter a ( u + 0041 ) with a combining ring above ( u + 030a ) and an o ( u + 006f ) with a combining diaeresis ( u + 0308 ). astrom ( u + 00c5 u + 0073 u + 0074 u + 0072 u + 00f6 u + 006d ) astrom ( u + 0041 u + 030a u + 0073 u + 0074 u + 0072 u + 006f u + 0308 u + 006d ) except for the different colors, the two solutions are equivalent and should render identically. in practice, however, some unicode implementations still have difficulties with decomposed characters. in the worst case, combining diacritics may be disregarded or rendered as unrecognized characters after their base letters, as they are not included in all fonts.


jackson and mr c. b. thompson. proper time studies are based on repeated observation, so that motions performed on the same part differently by one or many workers can be recorded, to determine those values that are truly repetitive and measurable.


this discrete problem may be ill - conditioned, depending on the original problem and the chosen quadrature rule. since the linear equations require o ( n 3 ) { \ displaystyle o ( n ^ { 3 } ) } operations to solve, high - order quadrature rules perform better because low - order quadrature rules require large n { \ displaystyle n } for a given accuracy. gaussian quadrature is normally a good choice for smooth, non - singular problems.


6. evaluation and visualization finally, the clustering models can be assessed by various metrics. and it is sometimes helpful to visualize the results by plotting the clusters into low ( two ) dimensional space. see multidimensional scaling as a possible approach.


in the field of artificial intelligence, the most difficult problems are informally known as ai - complete or ai - hard, implying that the difficulty of these computational problems, assuming intelligence is computational, is equivalent to that of solving the central artificial intelligence problem — making computers as intelligent as people, or strong ai. to call a problem ai - complete reflects an attitude that it would not be solved by a simple specific algorithm. ai - complete problems are hypothesised to include computer vision, natural language understanding, and dealing with unexpected circumstances while solving any real - world problem. currently, ai - complete problems cannot be solved with modern computer technology alone, but would also require human computation. this property could be useful, for example, to test for the presence of humans as captchas aim to do, and for computer security to circumvent brute - force attacks.


that is to say, when one or more values are missing for a case, most statistical packages default to discarding any case that has a missing value, which may introduce bias or affect the representativeness of the results. imputation preserves all cases by replacing missing data with an estimated value based on other available information.


in the case where r x = 3 { \ displaystyle r _ { x } = 3 } and r y = 7 { \ displaystyle r _ { y } = 7 }, the length of an input variable y { \ displaystyle \ mathbf { y } } from source y { \ displaystyle y } is 7 bits, therefore it can be sent lossless with 7 bits independent of any other bits. based on the knowledge that x { \ displaystyle \ mathbf { x } } and y { \ displaystyle \ mathbf { y } } have hamming distance at most one, for input x { \ displaystyle \ mathbf { x } } from source x { \ displaystyle x }, since the receiver already has y { \ displaystyle \ mathbf { y } }, the only possible x { \ displaystyle \ mathbf { x } } are those with at most 1 distance from y { \ displaystyle \ mathbf { y } }. if we model the correlation between two sources as a virtual channel, which has input x { \ displaystyle \ mathbf { x } } and output y { \ displaystyle \ mathbf { y } }, as long as we get y { \ displaystyle \ mathbf { y } }, all we need to successfully " decode " x { \ displaystyle \ mathbf { x } } is " parity bits " with particular error correction ability, taking the difference between x { \ displaystyle \ mathbf { x } } and y { \ displaystyle \ mathbf { y } } as channel error. we can also model the problem with cosets partition.


in symmetric pairing - based cryptography the group g { \ displaystyle g } is equipped with a pairing e : g × g → t { \ displaystyle e : g \ times g \ to t } which is bilinear. this map gives an efficient algorithm to solve the decisional diffie - hellman problem. given input ( g, g a, g b, h ) { \ displaystyle ( g, \, g ^ { a }, \, g ^ { b }, \, h ) }, it is easy to check if h { \ displaystyle h } is equal to g a b { \ displaystyle g ^ { ab } }. this follows by using the pairing : note that e ( g a, g b ) = e ( g, g ) a b = e ( g, g a b ).


the goal may be to visualize such data by points in euclidean space whose distance matrix approximates a given dissimilarity matrix as well as possible — this is known as multidimensional scaling. alternatively, given two sets of data already represented by points in euclidean space, one may ask how similar they are in shape, that is, how closely can they be related by a distance - preserving transformation — this is procrustes analysis. some of the distances may also be missing or come unlabelled ( as an unordered set or multiset instead of a matrix ), leading to more complex algorithmic tasks, such as the graph realization problem or the turnpike problem ( for points on a line ).


in the field of linguistics, the term passive is applied to a wide range of grammatical structures. linguists therefore find it difficult to define the term in a way that makes sense across all human languages. the canonical passive in european languages has the following properties : the subject is not an agent. there is a change in : word order ; or in nominal morphology — the form of the nouns in the sentence.


in mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or laplacian matrix. the adjacency matrix of a simple undirected graph is a real symmetric matrix and is therefore orthogonally diagonalizable ; its eigenvalues are real algebraic integers. while the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, although not a complete one. spectral graph theory is also concerned with graph parameters that are defined via multiplicities of eigenvalues of matrices associated to the graph, such as the colin de verdiere number.


in dirac representation, the four contravariant gamma matrices are γ 0 = ( 1 0 0 0 0 1 0 0 0 0 − 1 0 0 0 0 − 1 ), γ 1 = ( 0 0 0 1 0 0 1 0 0 − 1 0 0 − 1 0 0 0 ), γ 2 = ( 0 0 0 − i 0 0 i 0 0 i 0 0 − i 0 0 0 ), γ 3 = ( 0 0 1 0 0 0 0 − 1 − 1 0 0 0 0 1 0 0 ). { \ displaystyle { \ begin { aligned } \ gamma ^ { 0 } & = { \ begin { pmatrix } 1 & 0 & 0 & 0 \ \ 0 & 1 & 0 & 0 \ \ 0 & 0 & - 1 & 0 \ \ 0 & 0 & 0 & - 1 \ end { pmatrix } }, & \ gamma ^ { 1 } & = { \ begin { pmatrix } 0 & 0 & 0 & 1 \ \ 0 & 0 & 1 & 0 \ \ 0 & - 1 & 0 & 0 \ \ - 1 & 0 & 0 & 0 \ end { pmatrix } }, \ \ \ \ \ gamma ^ { 2 } & = { \ begin { pmatrix } 0 & 0 & 0 & - i \ \ 0 & 0 & i & 0 \ \ 0 & i & 0 & 0 \ \ - i & 0 & 0 & 0 \ end { pmatrix } }, & \ gamma ^ { 3 } & = { \ begin { pmatrix } 0 & 0 & 1 & 0 \ \ 0 & 0 & 0 & - 1 \ \ - 1 & 0 & 0 & 0 \ \ 0 & 1 & 0 & 0 \ end { pmatrix } } ~. \ end { aligned } } } γ 0 { \ displaystyle \ gamma ^ { 0 } } is the time - like, hermitian matrix. the other three are space - like, anti - hermitian matrices.


in software engineering and mathematics, numerical error is the error in the numerical computations.


as a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers ; see cardinality of the continuum for details. the theorem is named for german mathematician georg cantor, who first stated and proved it at the end of the 19th century.


in mathematics, the reflexive closure of a binary relation r { \ displaystyle r } on a set x { \ displaystyle x } is the smallest reflexive relation on x { \ displaystyle x } that contains r. { \ displaystyle r. } a relation is called reflexive if it relates every element of x { \ displaystyle x } to itself. for example, if x { \ displaystyle x } is a set of distinct numbers and x r y { \ displaystyle xry } means " x { \ displaystyle x } is less than y { \ displaystyle y } ", then the reflexive closure of r { \ displaystyle r } is the relation " x { \ displaystyle x } is less than or equal to y { \ displaystyle y } ".


note that the bit b i { \ displaystyle b _ { i } } is what decides which basis a i { \ displaystyle a _ { i } } is encoded in ( either in the computational basis or the hadamard basis ). the qubits are now in states which are not mutually orthogonal, and thus it is impossible to distinguish all of them with certainty without knowing b { \ displaystyle b }.


as the names of these two tests implies, destructive testing will destroy the part that is being tested while nondestructive testing enables the test piece to be used afterwards. there are several methods available in each of these types. this section outlines some requirements of testing plastic welds as well as the different types of destructive and non - destructive methods that are applicable to plastic welding and go over some of the advantages and disadvantages.


in addition : rule a2 does not perform any reduction on its own. however, it is still useful, because of its " shuffling " effect that can create new opportunities for applying the other rules ; rule a1 is not used in practice, because it may increase proof size ; rules b1, b2, b2'and b3 are directly responsible for the reduction, as they produce a transformed root clause stronger than the original one ; the application of a b rule may lead to an illegal proof ( see the example below ), as some literals missing in the transformed root clause may be involved in another resolution step along the path to the proof root. therefore, the algorithm also has to " reconstruct " a legal proof when this happen. the following example shows a situation where the proof becomes illegal after the application of b2'rule : applying rule b2'to the highlighted context : the proof is now illegal because the literal o { \ displaystyle o } is missing from the transformed root clause.


in some early dsp and risc processors, the documentation advises programmers to avoid such dependencies in adjacent and nearly adjacent instructions ( called delay slots ), or declares that the second instruction uses an old value rather than the desired value ( in the example above, the processor might counter - intuitively copy the unincremented value ), or declares that the value it uses is undefined. the programmer may have unrelated work that the processor can do in the meantime ; or, to ensure correct results, the programmer may insert nops into the code, partly negating the advantages of pipelining.


further, the single pass aspect slowed or prevented getting the compiler bootstrapped out of z80 assembly language and onto pascal itself. since the compiler was monolithic, the conversion to pascal could not be done one section at a time, but had to proceed as a wholesale replacement.


if one also assumes the axiom of choice, then all sets can be enumerated so that it coincides up to relabeling with the most general form of enumerations. since set theorists work with infinite sets of arbitrarily large cardinalities, the default definition among this group of mathematicians of an enumeration of a set tends to be any arbitrary α - sequence exactly listing all of its elements. indeed, in jech's book, which is a common reference for set theorists, an enumeration is defined to be exactly this. therefore, in order to avoid ambiguity, one may use the term finitely enumerable or denumerable to denote one of the corresponding types of distinguished countable enumerations.


in spring 2008 nokia introduced a built in sip voip client for the very first time to the mass market device ( nokia 6300i ) running series 40 operating system. later that year ( nokia 6260 slide was introduced introducing slightly updated sip voip client. nokia maintains a list of all phones that have an integrated voip client in forum nokia. aircell's battle with some companies allowing voip calls on flights is another example of the growing conflict of interest between incumbent operators and new voip operators.


in queueing models, a discipline within the mathematical theory of probability, the quasi - birth – death process describes a generalisation of the birth – death process. : 118 as with the birth - death process it moves up and down between levels one at a time, but the time between these transitions can have a more complicated distribution encoded in the blocks.


in real - life data, particularly for large datasets, the notions of rough approximations were found to be excessively restrictive. therefore, an extension of drsa, based on stochastic model ( stochastic drsa ), which allows inconsistencies to some degree, has been introduced. having stated the probabilistic model for ordinal classification problems with monotonicity constraints, the concepts of lower approximations are extended to the stochastic case. the method is based on estimating the conditional probabilities using the nonparametric maximum likelihood method which leads to the problem of isotonic regression. stochastic dominance - based rough sets can also be regarded as a sort of variable - consistency model.


in statistics, a scan statistic or window statistic is a problem relating to the clustering of randomly positioned points. an example of a typical problem is the maximum size of a cluster of points on a line or the longest series of successes recorded by a moving window of fixed length. joseph naus first published on the problem in the 1960s, and has been called the " father of the scan statistic " in honour of his early contributions. the results can be applied in epidemiology, public health and astronomy to find unusual clusters of events. it was extended by martin kulldorff to multidimensional settings and varying window sizes in a 1997 paper, which is ( as of 11 october 2015 ) the most cited article in its journal, communications in statistics – theory and methods. this work lead to the creation of the software satscan, a program trademarked by martin kulldorff that applies his methods to data.


in software development, formal methods are mathematical approaches to solving software ( and hardware ) problems at the requirements, specification, and design levels. formal methods are most likely to be applied to safety - critical or security - critical software and systems, such as avionics software. software safety assurance standards, such as do - 178c allows the usage of formal methods through supplementation, and common criteria mandates formal methods at the highest levels of categorization. for sequential software, examples of formal methods include the b - method, the specification languages used in automated theorem proving, raise, and the z notation.


in the case of coronals, the symbols ⟨ t, d ⟩ are normally used for the stop portion of the affricate regardless of place. for example, is commonly seen for. the exemplar languages are ones that have been reported to have these sounds, but in several cases, they may need confirmation.


in mathematical logic, lob's theorem states that in peano arithmetic ( pa ) ( or any formal system including pa ), for any formula p, if it is provable in pa that " if p is provable in pa then p is true ", then p is provable in pa. if prov ( p ) means that the formula p is provable, we may express this more formally as if p a p r o v ( p ) → p { \ displaystyle { \ mathit { pa } } \ vdash { \ mathrm { prov } ( p ) \ rightarrow p } } then p a p { \ displaystyle { \ mathit { pa } } \ vdash p }. an immediate corollary ( the contrapositive ) of lob's theorem is that, if p is not provable in pa, then " if p is provable in pa, then p is true " is not provable in pa. for example, " if 1 + 1 = 3 { \ displaystyle 1 + 1 = 3 } is provable in pa, then 1 + 1 = 3 { \ displaystyle 1 + 1 = 3 } " is not provable in pa. lob's theorem is named for martin hugo lob, who formulated it in 1955. it is related to curry's paradox.


such groups are elementarily equivalent to the integers ( z, +, < ) { \ displaystyle ( \ mathbb { z }, +, < ) }. z - groups are an alternative presentation of presburger arithmetic. occasionally, ( z ) - group is used to mean a zassenhaus group, a special type of permutation group.


the pixel x is predicted by the loco - i predictor according to the following guesses : x = { min ( a, b ) if c ≥ max ( a, b ) max ( a, b ) if c ≤ min ( a, b ) a + b − c otherwise. { \ displaystyle x = \ left \ { { \ begin { aligned } & \ min ( a, b ) \ quad \, { \ mbox { if } } \, c \ geq \ max ( a, b ) \ \ & \ max ( a, b ) \ quad { \ mbox { if } } \, c \ leq \ min ( a, b ) \ \ & a + b - c \ quad \, { \ mbox { otherwise } }. \ \ \ end { aligned } } \ right. } the three simple predictors are selected according to the following conditions : ( 1 ) it tends to pick b in cases where a vertical edge exists left of the x, ( 2 ) a in cases of an horizontal edge above x, or ( 3 ) a + b – c if no edge is detected.


in the era of rapid technology evolution, it is not the biggest that survives, but the fastest. the sooner the end product is delivered without major defects, the sooner feedback can be received, and incorporated into the next iteration. the shorter the iterations, the better the learning and communication within the team. with speed, decisions can be delayed.


this problem was solved, with an algorithm, in the same original paper by gale and shapley, in which the stable marriage problem was solved. the hospitals / residents problem with couples allows the set of residents to include couples who must be assigned together, either to the same hospital or to a specific pair of hospitals chosen by the couple ( e. g., a married couple want to ensure that they will stay together and not be stuck in programs that are far away from each other ). the addition of couples to the hospitals / residents problem renders the problem np - complete. the assignment problem seeks to find a matching in a weighted bipartite graph that has maximum weight.


in relational databases, a table can be a ( mathematical ) set or a multiset, depending on the presence of unicity constraints on some columns ( which turns it into a candidate key ). sql allows the selection of rows from a relational table : this operation will in general yield a multiset, unless the keyword distinct is used to force the rows to be all different, or the selection includes the primary ( or a candidate ) key. in ansi sql the multiset keyword can be used to transform a subquery into a collection expression : is a general select that can be used as subquery expression of another more general query, while transforms the subquery into a collection expression that can be used in another query, or in assignment to a column of appropriate collection type.


in statistics, cluster sampling is a sampling plan used when mutually homogeneous yet internally heterogeneous groupings are evident in a statistical population. it is often used in marketing research. in this sampling plan, the total population is divided into these groups ( known as clusters ) and a simple random sample of the groups is selected. the elements in each cluster are then sampled.


the symbol " ≈ " is also used for this purpose. in physics and astronomy, a tilde can be used between two expressions ( e. g. h ~ 10−34 j s ) to state that the two are of the same order of magnitude. in statistics and probability theory, the tilde means " is distributed as " ; see random variable ( e. g. x ~ b ( n, p ) for a binomial distribution ). a tilde can also be used to represent geometric similarity ( e. g. ∆abc ~ ∆def, meaning triangle abc is similar to def ).


in mathematics, the surgery structure set s ( x ) { \ displaystyle { \ mathcal { s } } ( x ) } is the basic object in the study of manifolds which are homotopy equivalent to a closed manifold x. it is a concept which helps to answer the question whether two homotopy equivalent manifolds are diffeomorphic ( or pl - homeomorphic or homeomorphic ). there are different versions of the structure set depending on the category ( diff, pl or top ) and whether whitehead torsion is taken into account or not.


if two of these symbols are used, one on the left and the mirror image of it on the right, it almost always indicates a set, as in { a, b, c } { \ displaystyle \ { a, b, c \ } }, the set containing three members, a { \ displaystyle a }, b { \ displaystyle b }, and c { \ displaystyle c }. but if it is used only on the left, it groups two or more simultaneous equations. there are other symbols of grouping.


in the 80386 microprocessor and later, virtual 8086 mode ( also called virtual real mode, v86 - mode, or vm86 ) allows the execution of real mode applications that are incapable of running directly in protected mode while the processor is running a protected mode operating system. it is a hardware virtualization technique that allowed multiple 8086 processors to be emulated by the 386 chip. it emerged from the painful experiences with the 80286 protected mode, which by itself was not suitable to run concurrent real - mode applications well. john crawford developed the virtual mode bit at the register set, paving the way to this environment. vm86 mode uses a segmentation scheme identical to that of real mode ( for compatibility reasons ), which creates 20 - bit linear addresses in the same manner as 20 - bit physical addresses are created in real mode, but are subject to protected mode's memory paging mechanism.


in order to prevent intel x86 - specific code from slipping into the operating system, due to developers being used to developing on x86 chips, windows nt 3. 1 was initially developed using non - x86 development systems and then ported to the x86 architecture. this work was initially based on the intel i860 - based dazzle system and, later, the mips r4000 - based jazz platform. both systems were designed internally at microsoft. windows nt 3. 1 was released for intel x86 pc compatible, pc - 98, dec alpha, and arc - compliant mips platforms.


in n - dimensional euclidean space over the real numbers, r n { \ displaystyle \ mathbb { r } ^ { n } }, the standard basis is denoted e1, e2, e3,... en. each basis vector ei points along the positive xi axis, with the basis being orthonormal. component j of ei is given by the kronecker delta : a vector in r n { \ displaystyle \ mathbb { r } ^ { n } } takes the form : similarly for the order - 2 tensor above, for each vector a and b in r n { \ displaystyle \ mathbb { r } ^ { n } } : or more generally :


mathematical theorems are known about combinations of relation properties, such as " a transitive relation is irreflexive if, and only if, it is asymmetric ". of particular importance are relations that satisfy certain combinations of properties. a partial order is a relation that is reflexive, antisymmetric, and transitive, an equivalence relation is a relation that is reflexive, symmetric, and transitive, a function is a relation that is right - unique and left - total ( see below ). since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations. the above concept of relation has been generalized to admit relations between members of two different sets ( heterogeneous relation, like " lies on " between the set of all points and that of all lines in geometry ), relations between three or more sets ( finitary relation, like " person x lives in town y at time z " ), and relations between classes ( like " is an element of " on the class of all sets, see binary relation § sets versus classes ).


in the early 1960s peter swinnerton - dyer used the edsac - 2 computer at the university of cambridge computer laboratory to calculate the number of points modulo p ( denoted by np ) for a large number of primes p on elliptic curves whose rank was known. from these numerical results birch & swinnerton - dyer ( 1965 ) conjectured that np for a curve e with rank r obeys an asymptotic law p ≤ x n p p ≈ c log ( x ) r as x → ∞ { \ displaystyle \ prod _ { p \ leq x } { \ frac { n _ { p } } { p } } \ approx c \ log ( x ) ^ { r } { \ mbox { as } } x \ rightarrow \ infty } where c is a constant. initially this was based on somewhat tenuous trends in graphical plots ; this induced a measure of skepticism in j. w. s. cassels ( birch's ph. d.


it is easy to find an initial solution that visits all the cities but will likely be very poor compared to the optimal solution. the algorithm starts with such a solution and makes small improvements to it, such as switching the order in which two cities are visited. eventually, a much shorter route is likely to be obtained.


in mid - 1822 hamilton began a systematic study of laplace's mecanique celeste. in november and december 1822 he completed his first three original mathematical papers. on his first visit to dunsink observatory, he showed two of them to brinkley, who asked for a more developed form.


in relation to the 2016 united states presidential election, individuals associated with online message boards, such as 4chan, noted a similarity between kek and the character pepe the frog. this was later paired with images of pepe, resulting in a resurgence of interest in the ancient deity. believers in kek say that repeating integers, often called “ dubs ”, are the prima materia of reality, and that their occurrence invoke the deity. elon musk has made numerous references to pepe and even to kek, among others within the perceived illuminati such as donald trump, who tweeted himself as a version of the frog. believers have cited this as evidence of memetic synchronicity.


the darmon – granville theorem uses faltings's theorem to show that for every specific choice of exponents ( x, y, z ), there are at most finitely many coprime solutions for ( a, b, c ). : p.


together with k. - m. chung, d. schuch, w. ulmer and b. zeiger a unified understanding of molecular interactions was developed based on a nonlinear schrodinger equation. hartmann thereby pioneered the discovery of one self - interacting field as the foundation of chemistry. h. hartman emerited in 1982 and died two years later. h. hartmann was honoured as member of the deutsche akademie der naturforscher leopoldina, the gesellschaft osterreichischer chemiker, the accademia nazionale die lincei, the royal danish academy of sciences and letters, the comitato premio of fondazione balzan, the international academy of quantum molecular science, and the akademie der wissenschaften und der literatur zu mainz.


in most theories up to the eighteenth century, light was pictured as being made up of particles. since particle models cannot easily account for the refraction, diffraction and birefringence of light, wave theories of light were proposed by rene descartes ( 1637 ), robert hooke ( 1665 ), and christiaan huygens ( 1678 ) ; however, particle models remained dominant, chiefly due to the influence of isaac newton. in the early 19th century, thomas young and august fresnel clearly demonstrated the interference and diffraction of light, and by 1850 wave models were generally accepted. james clerk maxwell's 1865 prediction that light was an electromagnetic wave – which was confirmed experimentally in 1888 by heinrich hertz's detection of radio waves – seemed to be the final blow to particle models of light.


in a linear normalization, the weight t i { \ displaystyle t _ { i } } of a descriptor is mapped to a size scale of 1 through f, where t min { \ displaystyle t _ { \ min } } and t max { \ displaystyle t _ { \ max } } are specifying the range of available weights. s i = f max ⋅ ( t i − t min ) t max − t min { \ displaystyle s _ { i } = \ left \ lceil { \ frac { f _ { \ max } \ cdot ( t _ { i } - t _ { \ min } ) } { t _ { \ max } - t _ { \ min } } } \ right \ rceil } for t i > t min { \ displaystyle t _ { i } > t _ { \ min } } ; else s i = 1 { \ displaystyle s _ { i } = 1 } s i { \ displaystyle s _ { i } } : display fontsize f max { \ displaystyle f _ { \ max } } : max. fontsize t i { \ displaystyle t _ { i } } : count t min { \ displaystyle t _ { \ min } } : min.


in number theory, a factorion in a given number base b { \ displaystyle b } is a natural number that equals the sum of the factorials of its digits. the name factorion was coined by the author clifford a. pickover.


consider a class that contains the following procedure to print a string on standard output after a new line : the following snippet, assumed to be in the same class, uses print _ on _ new _ line to demonstrate the mixing of open arguments and open targets in agents used as arguments to the same routine. this example uses the procedure do _ all for linear structures, which executes the routine modeled by an agent for each item in the structure. the sequence of three instructions prints the strings in my _ list, converts the strings to lowercase, and then prints them again. procedure do _ all iterates across the structure executing the routine substituting the current item for either the open argument ( in the case of the agents based on print _ on _ new _ line ), or the open target ( in the case of the agent based on to _ lower ). open and closed arguments and targets also allow the use of routines that call for more arguments than are required by closing all but the necessary number of arguments : the eiffel agent mechanism is detailed in the eiffel iso / ecma standard document.


it was introduced by gottlob frege, developed by moses schonfinkel, and further developed by haskell curry. uncurrying is the dual transformation to currying, and can be seen as a form of defunctionalization. it takes a function f { \ displaystyle f } whose return value is another function g { \ displaystyle g }, and yields a new function f ′ { \ displaystyle f'} that takes as parameters the arguments for both f { \ displaystyle f } and g { \ displaystyle g }, and returns, as a result, the application of f { \ displaystyle f } and subsequently, g { \ displaystyle g }, to those arguments. the process can be iterated.


in queueing theory, an adversarial queueing network is a model where the traffic to the network is supplied by an opponent rather than as the result of a stochastic process. the model has seen use in describing the impact of packet injections on the performance of communication networks. the model was first introduced in 1996. the stability of an adversarial queueing network can be determined by considering a fluid limit. = = references = =


each base station is adapted to communicate over a radio link with at least one user's mobile station located within the radio range of said base station. the advantage is that the equipment for wifi, 5g and other protocols can be centralized in one place, with remote antennas attached via fiber optic serving all protocols. it greatly reduces the equipment and maintenance cost of the network. rof technology enables convergence of fixed and mobile networks.


in the plan 9 operating system from bell labs ( mid - 1980s onward ), union mounting is a central concept, replacing several older unix conventions with union directories ; for example, several directories containing executables, unioned together at a single / bin directory, replace the path variable for command lookup in the shell. plan 9 union semantics are greatly simplified compared to the implementations for posix - style operating systems : the union of two directories is simply the concatenation of their contents, so a directory listing of the union may display duplicate names. also, no effort is made to recursively merge subdirectories, leading to an extremely simple implementation. directories are unioned in a controllable order ; u / name, where u is a union directory, denotes the file called name in the first constituent directory that contains such a file.


in mathematics, a monogenic semigroup is a semigroup generated by a single element. monogenic semigroups are also called cyclic semigroups.


in recent years, sentence embedding has seen a growing level of interest due to its applications in natural language queryable knowledge bases through the usage of vector indexing for semantic search. langchain for instance utilizes sentence transformers for purposes of indexing documents. in particular, an indexing is generated by generating embeddings for chunks of documents and storing ( document chunk, embedding ) tuples.


in mathematics, a fact is a statement ( called a theorem ) that can be proven by logical argument from certain axioms and definitions.


unlike stochastic independence and uncorrelatedness, mean independence is not symmetric : it is possible for y { \ displaystyle y } to be mean - independent of x { \ displaystyle x } even though x { \ displaystyle x } is mean - dependent on y { \ displaystyle y }. the concept of mean independence is often used in econometrics to have a middle ground between the strong assumption of independent random variables ( x 1 x 2 { \ displaystyle x _ { 1 } \ perp x _ { 2 } } ) and the weak assumption of uncorrelated random variables ( cov ( x 1, x 2 ) = 0 ). { \ displaystyle ( \ operatorname { cov } ( x _ { 1 }, x _ { 2 } ) = 0 ). }


in mathematics, many sequences of numbers or of polynomials are indexed by nonnegative integers, for example, the bernoulli numbers and the bell numbers. in both mechanics and statistics, the zeroth moment is defined, representing total mass in the case of physical density, or total probability, i. e. one, for a probability distribution. the zeroth law of thermodynamics was formulated after the first, second, and third laws, but considered more fundamental, thus its name. in biology, an organism is said to have zero - order intentionality if it shows " no intention of anything at all ". this would include a situation where the organism's genetically predetermined phenotype results in a fitness benefit to itself, because it did not " intend " to express its genes. in the similar sense, a computer may be considered from this perspective a zero - order intentional entity, as it does not " intend " to express the code of the programs it runs. in biological or medical experiments, initial measurements made before any experimental time has passed are said to be on the 0 day of the experiment. in genomics, both 0 - based and 1 - based systems are used for genome coordinates. patient zero ( or index case ) is the initial patient in the population sample of an epidemiological investigation.


in statistical mechanics, the softargmax function is known as the boltzmann distribution ( or gibbs distribution ) : : 7 the index set 1, …, k { \ displaystyle { 1, \, \ dots, \, k } } are the microstates of the system ; the inputs z i { \ displaystyle z _ { i } } are the energies of that state ; the denominator is known as the partition function, often denoted by z ; and the factor β is called the coldness ( or thermodynamic beta, or inverse temperature ).


in mathematics, particularly in abstract algebra, a semigroup with involution or a * - semigroup is a semigroup equipped with an involutive anti - automorphism, which — roughly speaking — brings it closer to a group because this involution, considered as unary operator, exhibits certain fundamental properties of the operation of taking the inverse in a group : uniqueness, double application " cancelling itself out ", and the same interaction law with the binary operation as in the case of the group inverse. it is thus not a surprise that any group is a semigroup with involution. however, there are significant natural examples of semigroups with involution that are not groups.


in mathematics, newton's identities, also known as the girard – newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. evaluated at the roots of a monic polynomial p in one variable, they allow expressing the sums of the k - th powers of all roots of p ( counted with their multiplicity ) in terms of the coefficients of p, without actually finding those roots. these identities were found by isaac newton around 1666, apparently in ignorance of earlier work ( 1629 ) by albert girard. they have applications in many areas of mathematics, including galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity.


in order to speak to larger groups of people, a need arose to increase the volume of the human voice. the earliest devices used to achieve this were acoustic megaphones. some of the first examples, from fifth - century - bc greece, were theater masks with horn - shaped mouth openings that acoustically amplified the voice of actors in amphitheaters. in 1665, the english physicist robert hooke was the first to experiment with a medium other than air with the invention of the " lovers'telephone " made of stretched wire with a cup attached at each end. in 1856, italian inventor antonio meucci developed a dynamic microphone based on the generation of electric current by moving a coil of wire to various depths in a magnetic field.


on x86 - 64 processors in native long mode, the address translation scheme uses pae but adds a fourth table, the 512 - entry page - map level 4 table, and extends the page directory pointer table to 512 entries instead of the original 4 entries it has in protected mode. this means that 48 bits of virtual page number are translated, giving a virtual address space of up to 256 tb. for some processors, a mode can be enabled with a fifth table, the 512 - entry page - map level 5 table ; this means that 57 bits of virtual page number are translated, giving a virtual address space of up to 128 pb. : 141 – 153 in the page table entries, in the original specification, 40 bits of physical page number are implemented. page table structures


in mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. today, these dualities are usually collected under the label stone duality, since they form a natural generalization of stone's representation theorem for boolean algebras. these concepts are named in honor of marshall stone. stone - type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics. this article gives pointers to special cases of stone duality and explains a very general instance thereof in detail.


except for certain logical issues, in this case cbd specializes to traditional treatments of contextuality in quantum physics. in particular, for consistently connected cyclic systems the noncontextuality criterion above reduces to d ( c n ) ≤ n − 2, { \ displaystyle d \ left ( { \ mathcal { c } } _ { n } \ right ) \ leq n - 2, } which includes the bell / chsh inequality ( n = 4 { \ displaystyle n = 4 } ), kcbs inequality ( n = 5 { \ displaystyle n = 5 } ), and other famous inequalities. that nonlocality is a special case of contextuality follows in cbd from the fact that being jointly distributed for random variables is equivalent to being measurable functions of one and the same random variable ( this generalizes arthur fine's analysis of bell's theorem ).


in number theory, all powers of three are perfect totient numbers. the sums of distinct powers of three form a stanley sequence, the lexicographically smallest sequence that does not contain an arithmetic progression of three elements. a conjecture of paul erdos states that this sequence contains no powers of two other than 1, 4, and 256.


the multicast dissemination protocol ( mdp ) was an initial attempt to ensure reliability and to address the problem of ack implosions through the use of nacks. mdp used selective negative acknowledgement ( nack ) to support reliability. additionally, mdp implemented probabilistic techniques to suppress redundant nacks in the multicast group ( and thereby avoid nack implosions ).


the scp responds with a geographic number, e. g. 0121 xxx xxxx, and the call is actually routed to a phone. by this architecture : 08xxx ( non - geographic numbers ) can be set up in a few scp nodes rather than having to be set up in every telephone exchange in the country. geographic numbers can be hidden revenue can be generated by non - telecoms companies from people making telephone calls to services – e. g. telephone voting


one of the first system calls made by the c standard i / o library is in an isatty ( ) call used to determine if the program is being run interactively by a human ( in which case isatty ( ) will succeed and the library will write its output a line at a time so the user sees a regular flow of text ) or as part of a pipeline ( in which case it writes a block at a time for efficiency ). if a library routine fails for some reason unrelated to a system call ( for example, because a user name wasn't found in the password file ) and a naive programmer blindly calls the normal error reporting routine perror ( ) on every failure, the leftover enotty will result in an utterly inappropriate " not a typewriter " ( or " not a teletype ", or " inappropriate ioctl for device " ) being delivered to the user. for many years the unix mail program sendmail contained this bug : when mail was delivered from another system, the mail program was being run non - interactively. if the destination address was local, but referred to a user name not found in the local password file, the message sent back to the originator of the email was the announcement that the person they were attempting to communicate with was not a typewriter.


in mathematics, finite field arithmetic is arithmetic in a finite field ( a field containing a finite number of elements ) contrary to arithmetic in a field with an infinite number of elements, like the field of rational numbers. there are infinitely many different finite fields. their number of elements is necessarily of the form pn where p is a prime number and n is a positive integer, and two finite fields of the same size are isomorphic. the prime p is called the characteristic of the field, and the positive integer n is called the dimension of the field over its prime field. finite fields are used in a variety of applications, including in classical coding theory in linear block codes such as bch codes and reed – solomon error correction, in cryptography algorithms such as the rijndael ( aes ) encryption algorithm, in tournament scheduling, and in the design of experiments.


in mathematical optimization, the method of lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints ( i. e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables ). it is named after the mathematician joseph - louis lagrange. the basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. the relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the lagrangian function. the method can be summarized as follows : in order to find the maximum or minimum of a function f ( x ) { \ displaystyle f ( x ) } subjected to the equality constraint g ( x ) = 0 { \ displaystyle g ( x ) = 0 }, form the lagrangian function, l ( x, λ ) ≡ f ( x ) + λ ⋅ g ( x ) { \ displaystyle { \ mathcal { l } } ( x, \ lambda ) \ equiv f ( x ) + \ lambda \ cdot g ( x ) } and find the stationary points of l { \ displaystyle { \ mathcal { l } } } considered as a function of x { \ displaystyle x } and the lagrange multiplier λ { \ displaystyle \ lambda ~ }.


in order to allow the system to work even with the high inter - unit latencies, each processor used an 8 - deep instruction pipeline. branches used a variable delay slot, the end of which was signaled by a bit in the next instruction. the bit indicated that the results of the branch had to be re - merged at this point, stalling the processor until this took place.


from the beginning, descriptively near sets have proved to be useful in applications of topology, and visual pattern recognition, spanning a broad spectrum of applications that include camouflage detection, micropaleontology, handwriting forgery detection, biomedical image analysis, content - based image retrieval, population dynamics, quotient topology, textile design, visual merchandising, and topological psychology. as an illustration of the degree of descriptive nearness between two sets, consider an example of the henry colour model for varying degrees of nearness between sets of picture elements in pictures ( see, e. g., § 4. 3 ). the two pairs of ovals in fig. 1 and fig. 2 contain coloured segments. each segment in the figures corresponds to an equivalence class where all pixels in the class have similar descriptions, i. e., picture elements with similar colours. the ovals in fig. 1 are closer to each other descriptively than the ovals in fig. 2.


for example, the hamming distance measures the minimal number of substitutions needed, while the levenshtein distance measures the minimal number of deletions, insertions, and substitutions ; both of these can be thought of as distances in an appropriate graph. graph edit distance is a measure of dissimilarity between two graphs, defined as the minimal number of graph edit operations required to transform one graph into another. wasserstein metrics measure the distance between two measures on the same metric space.


in ontologies designed to serve natural language processing ( nlp ) and natural language understanding ( nlu ) systems, ontology concepts are usually connected and symbolized by terms. this kind of connection represents a linguistic realization. terms are words or a combination of words ( multi - word units ), in different languages, used to describe in natural language an element from reality, and hence connected to that formal ontology concept that frames this element in reality. the lexicon, the collection of terms and their inflections assigned to the concepts and relationships in an ontology, forms the ‘ ontology interface to natural language ’, the channel through which the ontology can be accessed from a natural language input.


in model theory, a forking extension of a type is an extension of that type that is not free whereas a non - forking extension is an extension that is as free as possible. this can be used to extend the notions of linear or algebraic independence to stable theories. these concepts were introduced by s. shelah.


like ordinary categories, they contain objects ( the 0 - simplices of the simplicial set ) and morphisms between these objects ( 1 - simplices ). but unlike categories, the composition of two morphisms need not be uniquely defined. all the morphisms that can serve as composition of two given morphisms are related to each other by higher order invertible morphisms ( 2 - simplices thought of as " homotopies " ).


in structural proof theory, the nested sequent calculus is a reformulation of the sequent calculus to allow deep inference. = = references = =


in some cases it may be desirable to add logic redundancy. one of those cases is to avoid race conditions whereby an output can fluctuate because different terms are " racing " to turn off and on. to explain this in more concrete terms the karnaugh map to the right shows the minterms for the following function : f ( a, b, c, d ) = e ( 6, 8, 9, 10, 11, 12, 13, 14 ). { \ displaystyle f ( a, b, c, d ) = e ( 6, 8, 9, 10, 11, 12, 13, 14 ). \ } the boxes represent the minimal and / or terms needed to implement this function : f = a c + a b + b c d.


in proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. the number of cases sometimes can become very large. for example, the first proof of the four color theorem was a proof by exhaustion with 1, 936 cases. this proof was controversial because the majority of the cases were checked by a computer program, not by hand. the shortest known proof of the four color theorem as of 2011 still has over 600 cases.


in statistics, kernel - independent component analysis ( kernel ica ) is an efficient algorithm for independent component analysis which estimates source components by optimizing a generalized variance contrast function, which is based on representations in a reproducing kernel hilbert space. those contrast functions use the notion of mutual information as a measure of statistical independence.


in terms of actual computer processing, the principal steps are to 1 ) digitize the performed, analog music, 2 ) do successive short - term, fast fourier transform ( ffts ) to obtain the time - varying spectra, 3 ) identify the peaks in each spectrum, 4 ) analyze the spectral peaks to get pitch candidates, 5 ) connect the strongest individual pitch candidates to get the most likely time - varying, pitch contour, 6 ) map this physical data into the closest music - notation terms. these fundamental steps, originated by piszczalski in the 1970s, became the foundation of automatic music transcription. the most controversial and difficult step in this process is detecting pitch. the most successful pitch methods operate in the frequency domain, not the time domain.


in mathematics, fenchel's duality theorem is a result in the theory of convex functions named after werner fenchel. let ƒ be a proper convex function on rn and let g be a proper concave function on rn. then, if regularity conditions are satisfied, inf x ( f ( x ) − g ( x ) ) = sup p ( g ∗ ( p ) − f ∗ ( p ) ). { \ displaystyle \ inf _ { x } ( f ( x ) - g ( x ) ) = \ sup _ { p } ( g _ { * } ( p ) - f ^ { * } ( p ) ). } where ƒ * is the convex conjugate of ƒ ( also referred to as the fenchel – legendre transform ) and g * is the concave conjugate of g. that is, f ∗ ( x ∗ ) : = sup { ⟨ x ∗, x ⟩ − f ( x ) | x ∈ r n } { \ displaystyle f ^ { * } \ left ( x ^ { * } \ right ) : = \ sup \ left \ { \ left. \ left \ langle x ^ { * }, x \ right \ rangle - f \ left ( x \ right ) \ right | x \ in \ mathbb { r } ^ { n } \ right \ } } g ∗ ( x ∗ ) : = inf { ⟨ x ∗, x ⟩ − g ( x ) | x ∈ r n } { \ displaystyle g _ { * } \ left ( x ^ { * } \ right ) : = \ inf \ left \ { \ left. \ left \ langle x ^ { * }, x \ right \ rangle - g \ left ( x \ right ) \ right | x \ in \ mathbb { r } ^ { n } \ right \ } }


in the context of digital privacy, individual privacy is the notion that individuals have a right to exist freely on the internet, in that they can choose what type of information they are exposed to, and more importantly, that unwanted information should not interrupt them. an example of a digital breach of individual privacy would be an internet user receiving unwanted ads and emails / spam, or a computer virus that forces the user to take actions, which otherwise they would not. in such cases, the individual does not exist digitally without interruption from unwanted information ; thus their individual privacy has been infringed upon.


in order to define a model structure on the category of simplicial sets, one has to define fibrations, cofibrations and weak equivalences. one can define fibrations to be kan fibrations. a map of simplicial sets is defined to be a weak equivalence if its geometric realization is a weak homotopy equivalence of spaces. a map of simplicial sets is defined to be a cofibration if it is a monomorphism of simplicial sets.


in mathematics, a subring of r is a subset of a ring that is itself a ring when binary operations of addition and multiplication on r are restricted to the subset, and which shares the same multiplicative identity as r. for those who define rings without requiring the existence of a multiplicative identity, a subring of r is just a subset of r that is a ring for the operations of r ( this does imply it contains the additive identity of r ). the latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings ( and they may have a multiplicative identity that differs from the one of r ). with definition requiring a multiplicative identity ( which is used in this article ), the only ideal of r that is a subring of r is r itself.


in the early 1990s, engineers at japan's hitachi found ways to compress the reduced instruction sets so they fit in even smaller memory systems than ciscs. such compression schemes were used for the instruction set of their superh series of microprocessors, introduced in 1992. the superh instruction set was later adapted for arm architecture's thumb instruction set.


these three people are the most commonly cited fastest typists in online typing communities. all of their records were set on the qwerty keyboard layout.


bf6 f3 40. gxf3 exf3 41. bxg5 bxh3 42. bf4 + 1 – 0


he identifies commitment as the distinguishing factor between desire and intention, noting that it leads to ( 1 ) temporal persistence in plans and ( 2 ) further plans being made on the basis of those to which it is already committed. the bdi software model partially addresses these issues. temporal persistence, in the sense of explicit reference to time, is not explored.


this is to be contrasted with principal component analysis which seeks to minimize the mean square error of all residuals. before the advent of high - speed computers, considerable effort was devoted to finding approximate solutions to the problem, particularly in estimating the communalities by other means, which then simplifies the problem considerably by yielding a known reduced correlation matrix. this was then used to estimate the factors and the loadings.


in mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1 / b. a number is said to be normal in base b if, for every positive integer n, all possible strings n digits long have density b−n. intuitively, a number being simply normal means that no digit occurs more frequently than any other. if a number is normal, no finite combination of digits of a given length occurs more frequently than any other combination of the same length.


in mathematics, a semigroup with no elements ( the empty semigroup ) is a semigroup in which the underlying set is the empty set. many authors do not admit the existence of such a semigroup. for them a semigroup is by definition a non - empty set together with an associative binary operation. however not all authors insist on the underlying set of a semigroup being non - empty.


in the 1980s, rump made an example. he made a complicated function and tried to obtain its value. single precision, double precision, extended precision results seemed to be correct, but its plus - minus sign was different from the true value.


the model is formulated in such a way that given x i { \ displaystyle x _ { i } }, y i { \ displaystyle y _ { i } } are independent ( conditional independence of the observable variables given the markov random field ). in the vast majority of the related literature, the number of possible latent states is considered a user - defined constant. however, ideas from nonparametric bayesian statistics, which allow for data - driven inference of the number of states, have been also recently investigated with success, e. g.


other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups. an operation of arity two that involves several sets is sometimes also called a binary operation. for example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar. such binary operations may be called simply binary functions. binary operations are the keystone of most algebraic structures that are studied in algebra, in particular in semigroups, monoids, groups, rings, fields, and vector spaces.


in the mercury system, two series of rotors were used. the first series, dubbed the control maze, had four rotors, and stepped cyclometrically as in typex. five outputs from the control maze were used to determine the stepping of five rotors in the second series of rotors, the message maze, the latter used to encrypt and decrypt the plaintext and ciphertext. a sixth rotor in the message maze was controlled independently and stepped in the opposite direction to the others.


in order to improve the speed of floating - point division calculations on the pentium chip over the 486dx, intel opted to replace the shift - and - subtract division algorithm with the sweeney, robertson, and tocher ( srt ) algorithm. the srt algorithm can generate two bits of the division result per clock cycle, whereas the 486's algorithm could only generate one. it is implemented using a programmable logic array with 2, 048 cells, of which 1, 066 cells should have been populated with one of five values : −2, −1, 0, + 1, + 2. when the original array for the pentium was compiled, five values were not correctly downloaded into the equipment that etches the arrays into the chips – thus five of the array cells contained zero when they should have contained + 2. as a result, calculations that rely on these five cells acquire errors ; these errors can accumulate repeatedly owing to the recursive nature of the srt algorithm.


many browser games became free to play in order to attract more visits. at the early age of smartphones, mobile games were paid to download because there was usually no interface for a smartphone to install a physical copy. standardization and the ubiquity of mobile platforms that allowed for easy purchases by customers, brought on initially by the iphone app store and followed closely by the android marketplace and other competitors, resulted in a wide spread move towards microtransactions and indirect monetization.


in order to grow their collections, music librarians purchase books and music reference materials, subscribe to serials, order music scores and sound recordings, and buy or license electronic resources. this involves not only contact with publishers and other agencies that provide music materials, but also budgetary management of library funds.


to retain compatibility with the 7 - bit representation, the behaviour of bytes 0xa0 and 0xff was originally left undefined. the first c1 control code set to be registered for use with iso 2022 was din 31626, a specialised set for bibliographic use which was registered in 1979. the general - use iso / iec 6429 set was registered in 1983, although the ecma - 48 specification upon which it was based had been first published in 1976. further editions of the standards altered the provisions to an extent. for instance, a further revision to ecma - 35 and iso 2022 in 1985 introduced the concept of a 96 - code graphical character set.


selective retention – refers to the process of categorizing and interpreting information in a way that favors one category or interpretation over another. furthermore, they just simply forget the unsympathetic material. groups and group norms work as mediators.


however, the database system does not guarantee any ordering of the rows unless an order by clause is specified in the select statement that queries the table. an equally valid representation of a relation is as an n - dimensional chart, where n is the number of attributes ( a table's columns ). for example, a relation with two attributes and three values can be represented as a table with two columns and three rows, or as a two - dimensional graph with three points. the table and graph representations are only equivalent if the ordering of rows is not significant, and the table has no duplicate rows.


in queueing theory, a discipline within the mathematical theory of probability, a fluid queue ( fluid model, fluid flow model or stochastic fluid model ) is a mathematical model used to describe the fluid level in a reservoir subject to randomly determined periods of filling and emptying. the term dam theory was used in earlier literature for these models. the model has been used to approximate discrete models, model the spread of wildfires, in ruin theory and to model high speed data networks. the model applies the leaky bucket algorithm to a stochastic source.


the hardware tracing modules ( etm and etb ) are compatible, but updated, versions of those used in the arm9. in particular, trace semantics were updated to address parallel instruction execution and data transfers. arm makes an effort to promote recommended verilog coding styles and techniques.


in most systems a stack frame has a field to contain the previous value of the frame pointer register, the value it had while the caller was executing. for example, the stack frame of drawline would have a memory location holding the frame pointer value that drawsquare uses ( not shown in the diagram above ). the value is saved upon entry to the subroutine. having such a field in a known location in the stack frame enables code to access each frame successively underneath the currently executing routine's frame, and also allows the routine to easily restore the frame pointer to the caller's frame, just before it returns.


in phonetics, a bilabial consonant is a labial consonant articulated with both lips.


in order to avoid o ( n log n ) complexity, the lisp 2 algorithm uses three different passes over the heap. in addition, heap objects must have a separate forwarding pointer slot that is not used outside of garbage collection. after standard marking, the algorithm proceeds in the following three passes : compute the forwarding location for live objects. keep track of a free and live pointer and initialize both to the start of heap.


an object is classified by a plurality vote of its neighbors, with the object being assigned to the class most common among its k nearest neighbors ( k is a positive integer, typically small ). if k = 1, then the object is simply assigned to the class of that single nearest neighbor. in k - nn regression, the output is the property value for the object.


what is the format of the information? who will be responsible for transmitting and providing the information? moreover, this only serves as guidelines and must take into account other factors like cost and access to information. as defined by the project management institute ( 1996 ), project communications management includes processes required to ensure timely and appropriate generation, collection, dissemination, storage, and ultimate disposition of project information. the following communication processes are involved in project communications management, to wit : communication planning – in this phase, the problems, needs and future plans are being identified to ensure attainment of goals and objectives.


in mathematical logic and computer science, homotopy type theory ( hott ) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of ( abstract ) homotopy theory applies. this includes, among other lines of work, the construction of homotopical and higher - categorical models for such type theories ; the use of type theory as a logic ( or internal language ) for abstract homotopy theory and higher category theory ; the development of mathematics within a type - theoretic foundation ( including both previously existing mathematics and new mathematics that homotopical types make possible ) ; and the formalization of each of these in computer proof assistants. there is a large overlap between the work referred to as homotopy type theory, and as the univalent foundations project.


in mathematics and abstract algebra, group theory studies the algebraic structures known as groups. the concept of a group is central to abstract algebra : other well - known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. linear algebraic groups and lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.


in mathematics, particularly in abstract algebra, a ring r is said to be stably finite ( or weakly finite ) if, for all square matrices a and b of the same size with entries in r, ab = 1 implies ba = 1. this is a stronger property for a ring than having the invariant basis number ( ibn ) property. namely, any nontrivial stably finite ring has ibn. commutative rings, noetherian rings and artinian rings are stably finite. subrings of stably finite rings and matrix rings over stably finite rings are stably finite. a ring satisfying klein's nilpotence condition is stably finite.


it applies to predicates of the form e x p r ∈ { e x p r 1, …, e x p r n } { \ displaystyle expr \ in \ { expr _ { 1 }, \ dots, expr _ { n } \ } }. in this case, it generates n test classes such that a predicate of the form e x p r = e x p r i { \ displaystyle expr = expr _ { i } } is added to each of them. mandatory test set ( mts ).


in mathematics, a formal group law is ( roughly speaking ) a formal power series behaving as if it were the product of a lie group. they were introduced by s. bochner ( 1946 ). the term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. formal groups are intermediate between lie groups ( or algebraic groups ) and lie algebras. they are used in algebraic number theory and algebraic topology.


this allows for rapid prototyping. lazy evaluation is often combined with memoization, as described in jon bentley's writing efficient programs. after a function's value is computed for that parameter or set of parameters, the result is stored in a lookup table that is indexed by the values of those parameters ; the next time the function is called, the table is consulted to determine whether the result for that combination of parameter values is already available. if so, the stored result is simply returned.


their correct definition is rather technical. they are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. there are also maps ( or morphisms ) from one sheaf to another ; sheaves ( of a specific type, such as sheaves of abelian groups ) with their morphisms on a fixed topological space form a category.


in telephony, a zip tone ( also known as a whisper tone or call waiting tone ) is a call - progress tone which indicates a new incoming call is either connecting or waiting depending on the application. unlike a ringtone, which alerts those near a telephone to answer it, a zip tone alerts someone already on the line — for example a telephone operator, call center agent, or telephone subscriber with call waiting service — that action is needed for an incoming call such as pressing a button or reciting a phrase ( e. g. " may i help you? " ). this way of offering an incoming call to an available call center agent is also referred to as'forced calling '.


, s n { \ displaystyle s _ { 1 },..., s _ { n } } are the n { \ displaystyle n } sentences in the corpus, but n { \ displaystyle n } is the number of words in the corpus. this normalizes the perplexity by the length of the text, allowing for more meaningful comparisons between different texts or models. suppose the average sentence xi in the corpus has a probability of 2 − 190 { \ displaystyle 2 ^ { - 190 } } according to the language model.


a closely related vector is the angular wave vector ( or angular wavevector ), with a typical unit being radian per metre. the wave vector and angular wave vector are related by a fixed constant of proportionality, 2π radians per cycle. it is common in several fields of physics to refer to the angular wave vector simply as the wave vector, in contrast to, for example, crystallography. it is also common to use the symbol k for whichever is in use. in the context of special relativity, wave vector can refer to a four - vector, in which the ( angular ) wave vector and ( angular ) frequency are combined.


in statistics, the k - nearest neighbors algorithm ( k - nn ) is a non - parametric supervised learning method first developed by evelyn fix and joseph hodges in 1951, and later expanded by thomas cover. it is used for classification and regression. in both cases, the input consists of the k closest training examples in a data set. the output depends on whether k - nn is used for classification or regression : in k - nn classification, the output is a class membership.


additive smoothing is a type of shrinkage estimator, as the resulting estimate will be between the empirical probability ( relative frequency ) x i / n { \ textstyle \ textstyle { x _ { i } / n } }, and the uniform probability 1 / d { \ textstyle \ textstyle { 1 / d } }. invoking laplace's rule of succession, some authors have argued that α should be 1 ( in which case the term add - one smoothing is also used ), though in practice a smaller value is typically chosen. from a bayesian point of view, this corresponds to the expected value of the posterior distribution, using a symmetric dirichlet distribution with parameter α as a prior distribution. in the special case where the number of categories is 2, this is equivalent to using a beta distribution as the conjugate prior for the parameters of the binomial distribution.


in the us and some other countries, the industry standard today for physical properties and raw materials constituents is astm c 1364, the standard specification for architectural cast stone. membership in astm international ( founded in 1898 as the american chapter of the international association for testing and materials and most recently known as the american society for testing and materials ) exceeds 30, 000 technical experts from more than 100 countries who comprise a worldwide standards forum. the astm method of developing standards has been based on consensus of both users and producers of all kinds of materials.