we may assume by probability amplification that both circuit families have success probability at least 5 / 6. then we create a composite algorithm where the circuits for l 1 { \ displaystyle l _ { 1 } } and l 2 { \ displaystyle l _ { 2 } } are run independently, and we set p to the conjunction of p 1 { \ displaystyle p _ { 1 } } and p 2 { \ displaystyle p _ { 2 } }, and q to the conjunction of q 1 { \ displaystyle q _ { 1 } } and q 2 { \ displaystyle q _ { 2 } }. it is not hard to see by a union bound that this composite algorithm correctly decides membership in l 1 ∩ l 2 { \ displaystyle l _ { 1 } \ cap l _ { 2 } } with ( conditional ) probability at least 2 / 3. more generally, combinations of these ideas show that postbqp is closed under union and bqp truth - table reductions.


in number theory, tunnell's theorem gives a partial resolution to the congruent number problem, and under the birch and swinnerton - dyer conjecture, a full resolution.


in the 1992 article " extending and formalizing the framework for information systems architecture " john f. sowa and john zachman present the framework and its recent extensions and show how it can be formalized in the notation of conceptual graphs. also in 1992 : john zachman's co - author john sowa proposed the additions of the scope perspective of the ‘ planner ’ ( bounding lists common to the enterprise and its environment ) and the detailed representation perspective of the ‘ sub - contractor ’ ( being the out - of - context vendor solution components ). the who, when and why columns were brought into public view, the notion of the four levels of metaframeworks and a depiction of integration associations across the perspectives were all outlined in the paper. keri anderson healey assisted by creating a model of the models ( the framework metamodel ) which was also included in the article. later during the 1990s methodologists like clive finkelstein refocused on the top two framework rows which he labeled enterprise engineering and has one of the most successful methods for converging the business needs with information technology engineering implementation, and determining a logical build sequence of the pieces.


the piotrowski law is a case of the so - called logistic model ( cf. logistic equation ). it was shown that it covers also language acquisition processes ( cf. language acquisition law ). text block law : linguistic units ( e. g. words, letters, syntactic functions and constructions ) show a specific frequency distribution in equally large text blocks.


a symplectomorphism is an isomorphism of symplectic manifolds. a permutation is an automorphism of a set. in geometry, isomorphisms and automorphisms are often called transformations, for example rigid transformations, affine transformations, projective transformations. category theory, which can be viewed as a formalization of the concept of mapping between structures, provides a language that may be used to unify the approach to these different aspects of the basic idea.


in number theory, the prime omega functions ω ( n ) { \ displaystyle \ omega ( n ) } and ω ( n ) { \ displaystyle \ omega ( n ) } count the number of prime factors of a natural number n. { \ displaystyle n. } thereby ω ( n ) { \ displaystyle \ omega ( n ) } ( little omega ) counts each distinct prime factor, whereas the related function ω ( n ) { \ displaystyle \ omega ( n ) } ( big omega ) counts the total number of prime factors of n, { \ displaystyle n, } honoring their multiplicity ( see arithmetic function ). that is, if we have a prime factorization of n { \ displaystyle n } of the form n = p 1 α 1 p 2 α 2 p k α k { \ displaystyle n = p _ { 1 } ^ { \ alpha _ { 1 } } p _ { 2 } ^ { \ alpha _ { 2 } } \ cdots p _ { k } ^ { \ alpha _ { k } } } for distinct primes p i { \ displaystyle p _ { i } } ( 1 ≤ i ≤ k { \ displaystyle 1 \ leq i \ leq k } ), then the respective prime omega functions are given by ω ( n ) = k { \ displaystyle \ omega ( n ) = k } and ω ( n ) = α 1 + α 2 + + α k { \ displaystyle \ omega ( n ) = \ alpha _ { 1 } + \ alpha _ { 2 } + \ cdots + \ alpha _ { k } }. these prime factor counting functions have many important number theoretic relations.


in software development, a sanity test ( a form of software testing which offers " quick, broad, and shallow testing " ) evaluates the result of a subset of application functionality to determine whether it is possible and reasonable to proceed with further testing of the entire application. sanity tests may sometimes be used interchangeably with smoke tests insofar as both terms denote tests which determine whether it is possible and reasonable to continue testing further. on the other hand, a distinction is sometimes made that a smoke test is a non - exhaustive test that ascertains whether the most crucial functions of a programme work before proceeding with further testing whereas a sanity test refers to whether specific functionality such as a particular bug fix works as expected without testing the wider functionality of the software. in other words, a sanity test determines whether the intended result of a code change works correctly while a smoke test ensures that nothing else important was broken in the process.


in probability theory, regular conditional probability is a concept that formalizes the notion of conditioning on the outcome of a random variable. the resulting conditional probability distribution is a parametrized family of probability measures called a markov kernel.


in model theory, a branch of mathematical logic, and in algebra, the reduced product is a construction that generalizes both direct product and ultraproduct. let { si | i ∈ i } be a family of structures of the same signature σ indexed by a set i, and let u be a filter on i. the domain of the reduced product is the quotient of the cartesian product i ∈ i s i { \ displaystyle \ prod _ { i \ in i } s _ { i } } by a certain equivalence relation ~ : two elements ( ai ) and ( bi ) of the cartesian product are equivalent if { i ∈ i : a i = b i } ∈ u { \ displaystyle \ left \ { i \ in i : a _ { i } = b _ { i } \ right \ } \ in u } if u only contains i as an element, the equivalence relation is trivial, and the reduced product is just the original cartesian product. if u is an ultrafilter, the reduced product is an ultraproduct. operations from σ are interpreted on the reduced product by applying the operation pointwise.


in queueing theory, bartlett's theorem gives the distribution of the number of customers in a given part of a system at a fixed time.


the euclidean algorithm for computing the greatest common divisor of two integers is one example. given two integers a { \ displaystyle a } and b { \ displaystyle b }, the algorithm performs o ( log a + log b ) { \ displaystyle o ( \ log a + \ log b ) } arithmetic operations on numbers with at most o ( log a + log b ) { \ displaystyle o ( \ log a + \ log b ) } bits. at the same time, the number of arithmetic operations cannot be bounded by the number of integers in the input ( which is constant in this case, there are always only two integers in the input ).


the computation of the uspca and cspca solutions is demanding when the data matrix is large. with projection spca ( pspca ) approximate cspca solutions can computed much more efficiently by simply projecting the current first pc onto subsets of the variables. this means that the solutions can be computed with efficient linear regression routines. pspca is fast and can be shown to explain a proportion of the variance of the dataset comparable with that explained by cspca.


cash prizes of varying size, up to us $ 200, 000 ( and prizes up to $ 20, 000 awarded ), were offered for factorization of some of them. the smallest rsa number was factored in a few days. most of the numbers have still not been factored and many of them are expected to remain unfactored for many years to come.


in mathematics, hooley's delta function ( δ ( n ) { \ displaystyle \ delta ( n ) } ), also called erdos - - hooley delta - function, defines the maximum number of divisors of n { \ displaystyle n } in { \ displaystyle } for all u { \ displaystyle u }, where e { \ displaystyle e } is the euler's number. the first few terms of this sequence are 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 4 { \ displaystyle 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 4 } ( sequence a226898 in the oeis ).


hans van der laan : modern primitive. the sequence was described by ian stewart in his scientific american column mathematical recreations in june 1996.


in pseudocode the algorithm can be expressed as ( 0 - based array ) :


specialized hobbyist websites ( like geocaching. com ) faq and its benchmark hunting forum can provide more information. some of these marks have precise " adjusted " coordinates ( latitude and longitude ). the adjusted coordinates are precise to sub - centimeter accuracy, while others, typically true elevation bench marks, have only coordinates scaled from a map.


c variables are used directly while register names are quoted as string literals. _ _ asm in microsoft visual c + + ( msvc ), borland / embarcadero c compiler, and descendents. this syntax is not based on iso rules at all ; programmers simply write asm inside a block without needing to conform to c syntax.


the dispatch has all these three services combined into one dispatch for the best multi - coordinated response to an incident or an emergency. and also facilitates in information management, emergency communication and care of citizens. these services are the main structure for a response to an emergency. it can happen that, for a specific emergency, the co - operation with another service is needed, for instance the ministry of defence, water board ( s ) or rijkswaterstaat. the safety region can integrate these other services into their structure by adding them to specific conferences on operational or administrative level. all regions operate according to the coordinated regional incident management system.


the statement that two sets a and b are equinumerous is usually denoted a ≈ b { \ displaystyle a \ approx b \, } or a b { \ displaystyle a \ sim b }, or | a | = | b |. { \ displaystyle | a | = | b |. } the definition of equinumerosity using bijections can be applied to both finite and infinite sets, and allows one to state whether two sets have the same size even if they are infinite.


it is possible and sometimes desirable for the condition to always evaluate to be true. this creates an infinite loop. when an infinite loop is created intentionally there is usually another control structure that allows termination of the loop.


note that the four points x { \ displaystyle x }, y { \ displaystyle y }, x ∧ y { \ displaystyle x \ wedge y }, and x ∨ y { \ displaystyle x \ vee y } form a rectangle in euclidean space ( in the sense that x ∧ y − x = y − x ∨ y { \ displaystyle x \ wedge y - x = y - x \ vee y }, x − x ∨ y = x ∧ y − y { \ displaystyle x - x \ vee y = x \ wedge y - y }, and x ∧ y − x { \ displaystyle x \ wedge y - x } and x − x ∨ y { \ displaystyle x - x \ vee y } are orthogonal ). on the other hand, supermodularity and concavity together guarantee that u ( x ∨ y − λ v ) − u ( y ) ≥ u ( x ) − u ( x ∧ y + λ v ). { \ displaystyle u ( x \ vee y - \ lambda v ) - u ( y ) \ geq u ( x ) - u ( x \ wedge y + \ lambda v ). } for any λ ∈ { \ displaystyle \ lambda \ in }, where v = y − x ∧ y = x ∨ y − x { \ displaystyle v = y - x \ wedge y = x \ vee y - x }. in this case, crucially, the four points x { \ displaystyle x }, y { \ displaystyle y }, x ∨ y − λ v { \ displaystyle x \ vee y - \ lambda v }, and x ∧ y + λ v { \ displaystyle x \ wedge y + \ lambda v } form a backward - leaning parallelogram in euclidean space.


the standard specifies that the correct encoding of a code point uses only the minimum number of bytes required to hold the significant bits of the code point. longer encodings are called overlong and are not valid utf - 8 representations of the code point. this rule maintains a one - to - one correspondence between code points and their valid encodings, so that there is a unique valid encoding for each code point. this ensures that string comparisons and searches are well - defined.


in morphology, a null morpheme or zero morpheme is a morpheme that has no phonetic form. in simpler terms, a null morpheme is an " invisible " affix. it is a concept useful for analysis, by contrasting null morphemes with alternatives that do have some phonetic realization.


that cannot be your task. there is no such thing as valid jury nullification. you would violate your oath and the law if you willfully brought a verdict contrary to the law given to you in this case. " the ninth circuit upheld the first three sentences of the jury's instruction and overruled the remainder but deemed that instruction a harmless error and affirmed the conviction.


again, it depends on the context how exactly the word " filtration " is to be understood. descending filtrations are not to be confused with the dual notion of cofiltrations ( which consist of quotient objects rather than subobjects ). filtrations are widely used in abstract algebra, homological algebra ( where they are related in an important way to spectral sequences ), and in measure theory and probability theory for nested sequences of σ - algebras. in functional analysis and numerical analysis, other terminology is usually used, such as scale of spaces or nested spaces.


in bnpf encoding, a single byte ( 8 bits ) would be represented by a highly redundant character framing sequence starting with a single ascii " b ", eight ascii characters where a " 0 " would be represented by a " n " and a " 1 " would be represented by a " p ", followed by an ending ascii " f ". these ten - character ascii sequences were separated by one or more whitespace characters, therefore using at least eleven ascii characters for each byte stored ( 9 % efficiency ). the ascii " n " and " p " characters differed in four bit positions, providing excellent protection from single punch errors. alternative schemes were also available where " l " and " h " or " 0 " and " 1 " were also available to represent data bits, but in both of these encoding schemes, the two data - bearing ascii characters differ in only one bit position, providing very poor single punch error detection.


in the following definitions, x t { \ displaystyle \ mathbf { x } ^ { \ textsf { t } } } is the transpose of x { \ displaystyle \ mathbf { x } }, x ∗ { \ displaystyle \ mathbf { x } ^ { * } } is the conjugate transpose of x { \ displaystyle \ mathbf { x } } and 0 { \ displaystyle \ mathbf { 0 } } denotes the n - dimensional zero - vector.


alice sends | ψ ⟩ { \ displaystyle | \ psi \ rangle } over a public and authenticated quantum channel e { \ displaystyle { \ mathcal { e } } } to bob. bob receives a state e ( ρ ) = e ( | ψ ⟩ ⟨ ψ | ) { \ displaystyle { \ mathcal { e } } ( \ rho ) = { \ mathcal { e } } ( | \ psi \ rangle \ langle \ psi | ) }, where e { \ displaystyle { \ mathcal { e } } } represents both the effects of noise in the channel and eavesdropping by a third party we'll call eve. after bob receives the string of qubits, both bob and eve have their own states.


in the early days of computing the total capacity of hdds was specified in seven to nine decimal digits frequently truncated with the idiom millions. by the 1970s, the total capacity of hdds was given by manufacturers using si decimal prefixes such as megabytes ( 1 mb = 1, 000, 000 bytes ), gigabytes ( 1 gb = 1, 000, 000, 000 bytes ) and terabytes ( 1 tb = 1, 000, 000, 000, 000 bytes ). however, capacities of memory are usually quoted using a binary interpretation of the prefixes, i. e. using powers of 1024 instead of 1000.


the formula is used in calculating the frequency of each note in the piece. the values are then added together and divided by the number of notes. this is the average frequency of those notes. it is said that such techniques were used by classical composers, especially those who involved mathematics heavily in their music.


the general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the size of n. since these numbers are smaller, they are more likely to be smooth than the numbers inspected in previous algorithms. this is the key to the efficiency of the number field sieve. in order to achieve this speed - up, the number field sieve has to perform computations and factorizations in number fields. this results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve. the size of the input to the algorithm is log2 n or the number of bits in the binary representation of n. any element of the order nc for a constant c is exponential in log n. the running time of the number field sieve is super - polynomial but sub - exponential in the size of the input.


in modular arithmetic, any unit fraction can be converted into an equivalent whole number using the extended euclidean algorithm. this conversion can be used to perform modular division : dividing by a number x { \ displaystyle x }, modulo y { \ displaystyle y }, can be performed by converting the unit fraction 1 / x { \ displaystyle 1 / x } into an equivalent whole number modulo y { \ displaystyle y }, and then multiplying by that number. in more detail, suppose that x { \ displaystyle x } is relatively prime to y { \ displaystyle y } ( otherwise, division by x { \ displaystyle x } is not defined modulo y { \ displaystyle y } ). the extended euclidean algorithm for the greatest common divisor can be used to find integers a { \ displaystyle a } and b { \ displaystyle b } such that bezout's identity is satisfied : in modulo - y { \ displaystyle y } arithmetic, the term b y { \ displaystyle by } can be eliminated as it is zero modulo y { \ displaystyle y }. this leaves that is, a { \ displaystyle a } is the modular inverse of x { \ displaystyle x }, the number that when multiplied by x { \ displaystyle x } produces one. equivalently, thus division by x { \ displaystyle x } ( modulo y { \ displaystyle y } ) can instead be performed by multiplying by the integer a { \ displaystyle a }.


in the 1760s, johann heinrich lambert was the first to prove that the number π is irrational, meaning it cannot be expressed as a fraction a / b { \ displaystyle a / b }, where a { \ displaystyle a } and b { \ displaystyle b } are both integers. in the 19th century, charles hermite found a proof that requires no prerequisite knowledge beyond basic calculus. three simplifications of hermite's proof are due to mary cartwright, ivan niven, and nicolas bourbaki.


with some luck it will be the target of the search. if the target is not found by the time the search pattern has reached maximum convenient radius, the centre point may be shifted and another search started. this can be repeated as often as necessary, but the positions of the centre points must be chosen to allow the full search area to be covered.


( where φ ( β / α ) { \ displaystyle \ varphi ( \ beta / \ alpha ) } denotes the formula formed by substituting all free occurrences of the variable α { \ displaystyle \ alpha } in φ { \ displaystyle \ varphi } by β { \ displaystyle \ beta }. ) likewise, finding a counterexample disproves ( proves the negation of ) a universal conclusion. this is used in a proof by contradiction.


even the " system specific " code needs modification only slightly from implementation to implementation. the result is great amounts of implementation work saved while porting the system. finally, it was an error to enter into a second round of optimization before bootstrapping the compiler.


in mathematics, a differential invariant is an invariant for the action of a lie group on a space that involves the derivatives of graphs of functions in the space. differential invariants are fundamental in projective differential geometry, and the curvature is often studied from this point of view. differential invariants were introduced in special cases by sophus lie in the early 1880s and studied by georges henri halphen at the same time. lie ( 1884 ) was the first general work on differential invariants, and established the relationship between differential invariants, invariant differential equations, and invariant differential operators.


in order to understand how sounds are made, experimental procedures are often adopted. palatography is one of the oldest instrumental phonetic techniques used to record data regarding articulators. in traditional, static palatography, a speaker's palate is coated with a dark powder. the speaker then produces a word, usually with a single consonant.


in the 1960s, the term double dabble was also used for a different mental algorithm, used by programmers to convert a binary number to decimal. it is performed by reading the binary number from left to right, doubling if the next bit is zero, and doubling and adding one if the next bit is one. in the example above, 11110011, the thought process would be : " one, three, seven, fifteen, thirty, sixty, one hundred twenty - one, two hundred forty - three ", the same result as that obtained above.


using matrices, this formula can be written x o l d = a x n e w, { \ displaystyle \ mathbf { x } _ { \ mathrm { old } } = a \, \ mathbf { x } _ { \ mathrm { new } }, } where " old " and " new " refer respectively to the firstly defined basis and the other basis, x o l d { \ displaystyle \ mathbf { x } _ { \ mathrm { old } } } and x n e w { \ displaystyle \ mathbf { x } _ { \ mathrm { new } } } are the column vectors of the coordinates of the same vector on the two bases, and a { \ displaystyle a } is the change - of - basis matrix ( also called transition matrix ), which is the matrix whose columns are the coordinate vectors of the new basis vectors on the old basis. this article deals mainly with finite - dimensional vector spaces. however, many of the principles are also valid for infinite - dimensional vector spaces.


in mathematical analysis, agmon's inequalities, named after shmuel agmon, consist of two closely related interpolation inequalities between the lebesgue space l ∞ { \ displaystyle l ^ { \ infty } } and the sobolev spaces h s { \ displaystyle h ^ { s } }. it is useful in the study of partial differential equations. let u ∈ h 2 ( ω ) ∩ h 0 1 ( ω ) { \ displaystyle u \ in h ^ { 2 } ( \ omega ) \ cap h _ { 0 } ^ { 1 } ( \ omega ) } where ω ⊂ r 3 { \ displaystyle \ omega \ subset \ mathbb { r } ^ { 3 } }. then agmon's inequalities in 3d state that there exists a constant c { \ displaystyle c } such that ‖ u ‖ l ∞ ( ω ) ≤ c ‖ u ‖ h 1 ( ω ) 1 / 2 ‖ u ‖ h 2 ( ω ) 1 / 2, { \ displaystyle \ displaystyle \ | u \ | _ { l ^ { \ infty } ( \ omega ) } \ leq c \ | u \ | _ { h ^ { 1 } ( \ omega ) } ^ { 1 / 2 } \ | u \ | _ { h ^ { 2 } ( \ omega ) } ^ { 1 / 2 }, } and ‖ u ‖ l ∞ ( ω ) ≤ c ‖ u ‖ l 2 ( ω ) 1 / 4 ‖ u ‖ h 2 ( ω ) 3 / 4.


in number theory, the numbers of the form x2 + xy + y2 for integer x, y are called the loschian numbers ( or loeschian numbers ). these numbers are named after august losch. they are the norms of the eisenstein integers. they are a set of whole numbers, including zero, and having prime factorization in which all primes congruent to 2 mod 3 have even powers ( there is no restriction of primes congruent to 0 or 1 mod 3 ).


heyting implication, y ⇒ z { \ displaystyle y \ rightarrow z }, is an alternative notation for z y { \ displaystyle z ^ { y } }. the above adjunction results translate to implication ( ⇒ : h × h → h { \ displaystyle \ rightarrow : h \ times h \ to h } ) being right adjoint to meet ( ∧ : h × h → h { \ displaystyle \ wedge : h \ times h \ to h } ). this adjunction can be written as ( − ∧ y ) ( y ⇒ − ) { \ displaystyle ( - \ wedge y ) \ dashv ( y \ rightarrow - ) }, or more fully as : in the category of topological spaces, the exponential object z y { \ displaystyle z ^ { y } } exists provided that y { \ displaystyle y } is a locally compact hausdorff space.


in the european union, rrm data analysis showed that authentic information was " de - contextualized ", " manipulated " and " distorted ", then used by " questionable " writers on " untrustworthy " sites to " seed conversations ", which were then " framed using a divisive and inflammatory narrative. " these would reference the original authentic source. in a " coordinated fashion " the information would be " amplified " with the most " susceptible communities " targeted.


in elliptic geometry, an elliptic rectangle is a figure in the elliptic plane whose four edges are elliptic arcs which meet at equal angles greater than 90°. opposite arcs are equal in length. in hyperbolic geometry, a hyperbolic rectangle is a figure in the hyperbolic plane whose four edges are hyperbolic arcs which meet at equal angles less than 90°. opposite arcs are equal in length.


in telecommunications, transmission is the process of sending or propagating an analog or digital signal via a medium that is wired, wireless, or fiber - optic. transmission system technologies typically refer to physical layer protocol duties such as modulation, demodulation, line coding, equalization, error control, bit synchronization and multiplexing, but it may also involve higher - layer protocol duties, for example, digitizing an analog signal, and data compression. transmission of a digital message, or of a digitized analog signal, is known as data transmission. examples of transmission are the sending of signals with limited duration, for example, a block or packet of data, a phone call, or an email.


in mathematics, the binary octahedral group, name as 2o or ⟨ 2, 3, 4 ⟩ is a certain nonabelian group of order 48. it is an extension of the chiral octahedral group o or ( 2, 3, 4 ) of order 24 by a cyclic group of order 2, and is the preimage of the octahedral group under the 2 : 1 covering homomorphism spin ( 3 ) → so ( 3 ) { \ displaystyle \ operatorname { spin } ( 3 ) \ to \ operatorname { so } ( 3 ) } of the special orthogonal group by the spin group. it follows that the binary octahedral group is a discrete subgroup of spin ( 3 ) of order 48. the binary octahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism spin ( 3 ) sp ( 1 ) { \ displaystyle \ operatorname { spin } ( 3 ) \ cong \ operatorname { sp } ( 1 ) } where sp ( 1 ) is the multiplicative group of unit quaternions. ( for a description of this homomorphism see the article on quaternions and spatial rotations. )


the optional selector channel also enabled the use of other high speed devices such as the 1200 lpm 0776 printer or the 2000 lpm 0770 printer. the machine had either 4k or 16k memory chips, and typical machines had between 128 and 512 kib memory. it ran an os called os / 3, and could run up to 7 jobs at one time, not counting various os extensions such as the print spooler and telecommunications access ( icam ).


lastly, the difference between the two highest values of fev1 should also be within 150 ml. the highest fvc and fev1 may be used from each different test. until the results of three tests meet the criteria of reproducibility, the test can be repeated up to eight times. if it is still not possible to get accurate results, the best three tests are used.


for example, right triangles, isosceles triangles and equilateral triangles are non - generic and non - degenerate. in fact, degenerate cases often correspond to singularities, either in the object or in some configuration space. for example, a conic section is degenerate if and only if it has singular points ( e. g., point, line, intersecting lines ).


these incorrect responses often matched the incorrect response of the majority group ( i. e., actors ). overall, 75 % of participants gave at least one incorrect answer out of the 12 critical trials. in his opinion regarding the study results, asch put it this way : " that intelligent, well - meaning, young people are willing to call white black is a matter of concern. "


397 - 398 ). cook and reckhow ( 1973 ) say it most succinctly : the indirect instructions are necessary in order for a fixed program to access an unbounded number of registers as the inputs vary. " ( p. 73 ) unbounded capacities of registers versus bounded capacities of state - machine instructions : the so - called finite state part of the machine is supposed to be – by the normal definition of algorithm – very finite both in the number of " states " ( instructions ) and the instructions'sizes ( their capacity to hold symbols / signs ).


in statistics, the generalized dirichlet distribution ( gd ) is a generalization of the dirichlet distribution with a more general covariance structure and almost twice the number of parameters. random vectors with a gd distribution are completely neutral. the density function of p 1, …, p k − 1 { \ displaystyle p _ { 1 }, \ ldots, p _ { k - 1 } } is − 1 p k b k − 1 − 1 i = 1 k − 1 { \ displaystyle \ left ^ { - 1 } p _ { k } ^ { b _ { k - 1 } - 1 } \ prod _ { i = 1 } ^ { k - 1 } \ left } where we define p k = 1 − i = 1 k − 1 p i { \ textstyle p _ { k } = 1 - \ sum _ { i = 1 } ^ { k - 1 } p _ { i } }. here b ( x, y ) { \ displaystyle b ( x, y ) } denotes the beta function. this reduces to the standard dirichlet distribution if b i − 1 = a i + b i { \ displaystyle b _ { i - 1 } = a _ { i } + b _ { i } } for 2 i k − 1 { \ displaystyle 2 \ leqslant i \ leqslant k - 1 } ( b 0 { \ displaystyle b _ { 0 } } is arbitrary ).


in mathematics, two sets are said to be disjoint sets if they have no element in common. equivalently, two disjoint sets are sets whose intersection is the empty set. for example, { 1, 2, 3 } and { 4, 5, 6 } are disjoint sets, while { 1, 2, 3 } and { 3, 4, 5 } are not disjoint. a collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint.


the child element is within the parent element, such as in a venn diagram. this structure is most effective in representing simple hierarchical relationships. for example, when directing someone to open a file on a computer desktop, one may first direct them towards the main folder, then the subfolders within the main folder.


in statistics, modes of variation are a continuously indexed set of vectors or functions that are centered at a mean and are used to depict the variation in a population or sample. typically, variation patterns in the data can be decomposed in descending order of eigenvalues with the directions represented by the corresponding eigenvectors or eigenfunctions. modes of variation provide a visualization of this decomposition and an efficient description of variation around the mean. both in principal component analysis ( pca ) and in functional principal component analysis ( fpca ), modes of variation play an important role in visualizing and describing the variation in the data contributed by each eigencomponent. in real - world applications, the eigencomponents and associated modes of variation aid to interpret complex data, especially in exploratory data analysis ( eda ).


in the case of the leaky bucket algorithm as a queue, the only defined limit for this algorithm is the bandwidth of its output. the bandwidth limit for the connection may be specified in a traffic contract. a bandwidth limit may be specified as a packet or frame rate, a byte or bit rate, or as an emission interval between the packets.


in modern mathematics, one often studies spaces whose points are themselves mathematical objects. a distance function on such a space generally aims to measure the dissimilarity between two objects. here are some examples : functions to a metric space. if x is any set and m is a metric space, then the set of all bounded functions f : x → m { \ displaystyle f \ colon x \ to m } ( i. e. those functions whose image is a bounded subset of m { \ displaystyle m } ) can be turned into a metric space by defining the distance between two bounded functions f and g to be this metric is called the uniform metric or supremum metric.


in the united kingdom, the official charts company compiles a weekly compilation albums chart, limited to various artists compilations and soundtrack compilations.


in probability theory, the law ( or formula ) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. it expresses the total probability of an outcome which can be realized via several distinct events, hence the name.


here, the word " broadcast " is used in the sense of conveying the state to two or more recipients. for multiple recipients to each receive the state, there must be, in some sense, a way of duplicating the state. the no - broadcast theorem generalizes the no - cloning theorem for mixed states.


later releases added the ability to write unformatted dumps, called at that time core image dumps. in modern operating systems, a process address space may contain gaps, and it may share pages with other processes or files, so more elaborate representations are used ; they may also include other information about the state of the program at the time of the dump. in unix - like systems, core dumps generally use the standard executable image - format : a. out in older versions of unix, elf in modern linux, system v, solaris, and bsd systems, mach - o in macos, etc.


various multi - compartment models can be used in the areas of pharmacokinetics and pharmacology, in the support of efforts in drug discovery, and in environmental science. in humans and related organisms, there are five major body compartments : the blood plasma, interstitial fluids, fat tissues, intracellular fluids, and transcellular fluids, the latter of which includes fluids in the pleural ( peritoneal ) cavity. the relative percents of body mass of these are included in the following table.


in statistics, intra - rater reliability is the degree of agreement among repeated administrations of a diagnostic test performed by a single rater. intra - rater reliability and inter - rater reliability are aspects of test validity.


it not only breaks the isolation that operating systems should provide between programs and hardware, raising both stability and security concerns, but also could leave the graphics hardware in an inconsistent state if two or more user space programs try to do the mode - setting at the same time. to avoid these conflicts, the x server became in practice the only user space program that performed mode - setting operations ; the remainder user space programs relied on the x server to set the appropriate mode and to handle any other operation involving mode - setting. initially the mode - setting was performed exclusively during the x server startup process, but later the x server gained the ability to do it while running.


{ \ displaystyle { \ frac { \ sum _ { v } d ( v ) ^ { 2 } } { 2 | e | } } = { \ frac { | v | } { 2 | e | } } ( \ mu ^ { 2 } + \ sigma ^ { 2 } ) = { \ frac { \ mu ^ { 2 } + \ sigma ^ { 2 } } { \ mu } } = \ mu + { \ frac { \ sigma ^ { 2 } } { \ mu } }. } for a graph that has vertices of varying degrees ( as is typical for social networks ), σ 2 { \ displaystyle { \ sigma } ^ { 2 } } is strictly positive, which implies that the average degree of a friend is strictly greater than the average degree of a random node.


the consumer has income i { \ displaystyle i }, and hence a budget set of affordable packages b ( p, i ) = { x : p ⋅ x ≤ i }, { \ displaystyle b ( p, i ) = \ { x : p \ cdot x \ leq i \ }, } where p ⋅ x = i l p i x i { \ displaystyle p \ cdot x = \ sum _ { i } ^ { l } p _ { i } x _ { i } } is the dot product of the price and quantity vectors. the consumer has a utility function u : r + l → r. { \ displaystyle u : \ mathbb { r } _ { + } ^ { l } \ rightarrow \ mathbb { r }. } the consumer's marshallian demand correspondence is defined to be x ∗ ( p, i ) = argmax x ∈ b ( p, i ) u ( x ) { \ displaystyle x ^ { * } ( p, i ) = \ operatorname { argmax } _ { x \ in b ( p, i ) } u ( x ) }


in business telephony, a telephone extension may refer to a phone on an internal telephone line attached to a private branch exchange ( pbx ) or centrex system. the pbx operates much as a community switchboard does for a geographic telephone numbering plan and allows multiple lines inside the office to connect without each phone requiring a separate outside line. in these systems, one usually has to dial a number ( typically 9 in north america, 0 in europe ) to tell the pbx to connect with an outside landline ( also called ddco, or direct dial central office ) to dial an external number.


in order to use a hurricane tracking chart, one needs access to latitude / longitude pairs of the cyclone's center and maximum sustained wind information in order to know which symbol to depict. new tropical cyclone information is available at least every twelve hours in the southern hemisphere and at least every six hours in the northern hemisphere from regional specialized meteorological centers and tropical cyclone warning centers. in decades past, newspaper, television, and radio ( including weather radio ) were primary sources for this information. local television stations within threatened markets would advertise tropical cyclone positions within the morning, evening, and nightly news during their weather segments. the weather channel includes the information within their tropical updates every hour during the atlantic and pacific hurricane seasons. starting in the mid 1990s, the world wide web allowed for the development of ftp and web sites by the bureau of meteorology in australia, canadian hurricane centre, central pacific hurricane center, the nadi tropical cyclone centre / fiji meteorological service, japan meteorological agency, joint typhoon warning center, meteo - france la reunion, national hurricane center, and the philippine atmospheric, geophysical and astronomical services administration which allows the end user to get their information from their official products.


in modern usage, metric is used almost exclusively in commercial transactions. these units are mostly historical, although they are still used in some limited contexts and in maltese idioms and set phrases. many of these terms are directly related to arabic units and some to sicilian units. the weights and measures ordinance of 1921 established uniformity in the conversion of such weights and measures. all these measures were defined as simple multiples of the imperial units then in use in britain.


in mathematics, the chung – fuchs theorem, named after chung kai - lai and wolfgang heinrich johannes fuchs, states that for a particle undergoing a random walk in m - dimensions, it is certain to come back infinitely often to any neighborhood of the origin on a one - dimensional line ( m = 1 ) or two - dimensional plane ( m = 2 ), but in three or more dimensional spaces it will leave to infinity. specifically, if a position of the particle is described by the vector x n { \ displaystyle x _ { n } } : where z 1, z 2, …, z n { \ displaystyle z _ { 1 }, z _ { 2 }, \ dots, z _ { n } } are independent m - dimensional vectors with a given multivariate distribution, then if m = 1 { \ displaystyle m = 1 }, e ( | z i | ) < ∞ { \ displaystyle e ( | z _ { i } | ) < \ infty } and e ( z i ) = 0 { \ displaystyle e ( z _ { i } ) = 0 }, or if m = 2 { \ displaystyle m = 2 } e ( | z i 2 | ) < ∞ { \ displaystyle e ( | z _ { i } ^ { 2 } | ) < \ infty } and e ( z i ) = 0 { \ displaystyle e ( z _ { i } ) = 0 }, the following holds : however, for m ≥ 3 { \ displaystyle m \ geq 3 },


that means we can compute and store f { \ displaystyle f } on all the possible values in t = ( log log q ) m { \ displaystyle t = ( \ log \ log q ) ^ { m } } time and space. if we take d = log q { \ displaystyle d = \ log q }, we get m = log n log log q { \ displaystyle m = { \ tfrac { \ log n } { \ log \ log q } } }, so the time / space requirement is just n log log q log log log q. { \ displaystyle n ^ { \ frac { \ log \ log q } { \ log \ log \ log q } }. } kedlaya and umans further show how to combine this preprocessing with fast ( fft ) multipoint evaluation. this allows optimal algorithms for many important algebraic problems, such as polynomial modular composition.


bot is a natural type for the " null pointer " value ( a pointer which does not point to any object ) of languages like java : in java, the null type is the universal subtype of reference types. null is the only value of the null type ; and it can be cast to any reference type. however, the null type is not a bottom type as described above, it is not a subtype of int and other primitive types. a type system including both top and bot seems to be a natural target for type inference, allowing the constraints on an omitted type parameter to be captured by a pair of bounds : we write s < : x < : t to mean " the value of x must lie somewhere between s and t. " in such a scheme, a completely unconstrained parameter is bounded below by bot and above by top.


in the remainder of this article, base ten is assumed. the single - digit final state reached in the process of calculating an integer's additive persistence is its digital root. put another way, a number's additive persistence counts how many times we must sum its digits to arrive at its digital root.


in the board game industry, playtesting applies both to feedback gathered during the early design process as well as late stage exposure to the target audience by a game's publisher. major types of boardgame testing include local testing — where a designer, developer, or publisher representative moderates the test in person, and remote testing, where groups receive copies of the game or files to assemble their own version. wizards of the coast ran a public playtest of their new dungeon command miniatures game. in this case, they used the feedback generated on the rules to improve the game but also used feedback on the playtest itself to improve logistics on the d & d next playtest. steve jackson games uses munchkin players from the area around their offices to test new cards and expansions, as well as distributing playtest packages at conventions. according to the sjg website, this is done " so we can observe carefully which cards work well, which jokes aren't as funny as we thought, and so on. "


for instance, the xml document and the ascii tree have the same structure. xml trees do not show the content in an instance document, only the structure of the document. in this example product is the root element of the tree and the two child nodes of product are name and details. details contains two child nodes, description and price. the tree command in windows and * nix also produce a similar tree structure and path.


in order for a table to be in sixth normal form, it has to be in fifth normal form first and then it requires that each table satisfies only trivial join dependencies. let ’ s take a simple example with a table already in 5nf : here, in the users table, every attribute is non null and the primary key is the username : users _ table this table is in 5nf because each join dependency is implied by the unique candidate key of the table ( username ). more specifically, the only possible join dependencies are : { username, status }, { username, department }. the 6nf version would look like this : users users _ dept so, from one table in 5nf, 6nf produces two tables.


in mathematics, the lexicographic or lexicographical order ( also known as lexical order, or dictionary order ) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set. there are several variants and generalizations of the lexicographical ordering. one variant applies to sequences of different lengths by comparing the lengths of the sequences before considering their elements. another variant, widely used in combinatorics, orders subsets of a given finite set by assigning a total order to the finite set, and converting subsets into increasing sequences, to which the lexicographical order is applied. a generalization defines an order on a cartesian product of partially ordered sets ; this order is a total order if and only if all factors of the cartesian product are totally ordered.


by 1995, the internet was fully commercialized in the u. s. when the nsfnet was decommissioned, removing the last restrictions on use of the internet to carry commercial traffic. as technology advanced and commercial opportunities fueled reciprocal growth, the volume of internet traffic started experiencing similar characteristics as that of the scaling of mos transistors, exemplified by moore's law, doubling every 18 months.


in standard truth - functional propositional logic, distribution in logical proofs uses two valid rules of replacement to expand individual occurrences of certain logical connectives, within some formula, into separate applications of those connectives across subformulas of the given formula. the rules are where " { \ displaystyle \ leftrightarrow } ", also written ≡, { \ displaystyle \, \ equiv, \, } is a metalogical symbol representing " can be replaced in a proof with " or " is logically equivalent to ".


in mathematics, in the area of quantum information geometry, the bures metric ( named after donald bures ) or helstrom metric ( named after carl w. helstrom ) defines an infinitesimal distance between density matrix operators defining quantum states. it is a quantum generalization of the fisher information metric, and is identical to the fubini – study metric when restricted to the pure states alone.


the cylinder was rotated every 24 hours, providing an approximate time for a given quake. luigi palmieri, influenced by mallet's 1848 paper, invented a seismometer in 1856 that could record the time of an earthquake. this device used metallic pendulums which closed an electric circuit with vibration, which then powered an electromagnet to stop a clock.


( e. g. in a claim bob instance of human, bob is the subject and an instance, while the object, human, is an ordinary class ; but a further claim that human instance of animal species makes " animal species " a metaclass because it has a member, " human ", that is also a class ). owl 2 dl supports metaclasses by a feature called punning, in which one entity is interpreted as two different types of thing — a class and an individual — depending on its syntactic context. for example, through punning, an ontology could have a concept hierarchy such as harry the eagle instance of golden eagle, golden eagle subclass of bird, and golden eagle instance of species.


in mathematics and its applications, the root mean square of a set of numbers x i { \ displaystyle x _ { i } } ( abbreviated as rms, rms or rms and denoted in formulas as either x r m s { \ displaystyle x _ { \ mathrm { rms } } } or r m s x { \ displaystyle \ mathrm { rms } _ { x } } ) is defined as the square root of the mean square ( the arithmetic mean of the squares ) of the set. the rms is also known as the quadratic mean ( denoted m 2 { \ displaystyle m _ { 2 } } ) and is a particular case of the generalized mean. the rms of a continuously varying function ( denoted f r m s { \ displaystyle f _ { \ mathrm { rms } } } ) can be defined in terms of an integral of the squares of the instantaneous values during a cycle. for alternating electric current, rms is equal to the value of the constant direct current that would produce the same power dissipation in a resistive load. in estimation theory, the root - mean - square deviation of an estimator is a measure of the imperfection of the fit of the estimator to the data.


this could be confused with the greek letter theta on a badly focused display, but in practice there was no confusion because theta was not ( then ) a displayable character and very little used anyway. an alternative, the slashed zero ( looking similar to the letter o except for the slash ), was primarily used in hand - written coding sheets before transcription to punched cards or tape, and is also used in old - style ascii graphic sets descended from the default typewheel on the teletype model 33 asr. this form is similar to the symbol ∅ { \ displaystyle \ emptyset }, or " ∅ " ( unicode character u + 2205 ), representing the empty set, as well as to the letter ø used in several scandinavian languages.


in processor design, there are two ways to increase on - chip parallelism with fewer resource requirements : one is superscalar technique which tries to exploit instruction - level parallelism ( ilp ) ; the other is multithreading approach exploiting thread - level parallelism ( tlp ). superscalar means executing multiple instructions at the same time while thread - level parallelism ( tlp ) executes instructions from multiple threads within one processor chip at the same time. there are many ways to support more than one thread within a chip, namely : interleaved multithreading : interleaved issue of multiple instructions from different threads, also referred to as temporal multithreading. it can be further divided into fine - grained multithreading or coarse - grained multithreading depending on the frequency of interleaved issues.


consequently, an irreducible ideal of a noetherian ring is primary. various methods of generalizing primary ideals to noncommutative rings exist, but the topic is most often studied for commutative rings. therefore, the rings in this article are assumed to be commutative rings with identity.


in mathematics, a partition of a set is a grouping of its elements into non - empty subsets, in such a way that every element is included in exactly one subset. every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. a set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory.


in statistics, a sampling frame is the source material or device from which a sample is drawn. it is a list of all those within a population who can be sampled, and may include individuals, households or institutions. importance of the sampling frame is stressed by jessen and salant and dillman. in many practical situations the frame is a matter of choice to the survey planner, and sometimes a critical one. some very worthwhile investigations are not undertaken at all because of the lack of an apparent frame ; others, because of faulty frames, have ended in a disaster or in cloud of doubt.


in the kochen – specker article the possibility is discussed that the value attribution v ( a ) { \ displaystyle v ( \ mathbf { a } ) } may be context - dependent, i. e. observables corresponding to equal vectors in different columns of the table need not have equal values because different columns correspond to different measurement arrangements. since subquantum reality ( as described by the hidden - variable theory ) may be dependent on the measurement context, it is possible that relations between quantum - mechanical observables and hidden variables are just homomorphic rather than isomorphic. this would make obsolete the requirement of a context - independent value attribution. hence, the ks theorem only excludes noncontextual hidden - variable theories. the possibility of contextuality has given rise to the so - called modal interpretations of quantum mechanics.


in mathematics, finite - dimensional distributions are a tool in the study of measures and stochastic processes. a lot of information can be gained by studying the " projection " of a measure ( or process ) onto a finite - dimensional vector space ( or finite collection of times ).


in the subbytes step, each byte a i, j { \ displaystyle a _ { i, j } } in the state array is replaced with a subbyte s ( a i, j ) { \ displaystyle s ( a _ { i, j } ) } using an 8 - bit substitution box. note that before round 0, the state array is simply the plaintext / input. this operation provides the non - linearity in the cipher.


in number theory, zero - sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. concretely, given a finite abelian group g and a positive integer n, one asks for the smallest value of k such that every sequence of elements of g of size k contains n terms that sum to 0. the classic result in this area is the 1961 theorem of paul erdos, abraham ginzburg, and abraham ziv. they proved that for the group z / n z { \ displaystyle \ mathbb { z } / n \ mathbb { z } } of integers modulo n, explicitly this says that any multiset of 2n − 1 integers has a subset of size n the sum of whose elements is a multiple of n, but that the same is not true of multisets of size 2n − 2.


in physics, averroes did not adopt the inductive method that was being developed by al - biruni in the islamic world and is closer to today's physics. rather, he was — in the words of historian of science ruth glasner — a " exegetical " scientist who produced new theses about nature through discussions of previous texts, especially the writings of aristotle. because of this approach, he was often depicted as an unimaginative follower of aristotle, but glasner argues that averroes's work introduced highly original theories of physics, especially his elaboration of aristotle's minima naturalia and on motion as forma fluens, which were taken up in the west and are important to the overall development of physics. averroes also proposed a definition of force as " the rate at which work is done in changing the kinetic condition of a material body " — a definition close to that of power in today's physics.


for example, a typical server runs at 425 w and vmware estimates a hardware reduction ratio of up to 15 : 1. a virtual machine ( vm ) can be more easily controlled and inspected from a remote site than a physical machine, and the configuration of a vm is more flexible. this is very useful in kernel development and for teaching operating system courses, including running legacy operating systems that do not support modern hardware.


in multi - stage job scheduling problems, there are other options for the machine environments : o : open - shop problem. every job j { \ displaystyle j } consists of m { \ displaystyle m } operations o i j { \ displaystyle o _ { ij } } for i = 1, …, m { \ displaystyle i = 1, \ ldots, m }.


in response, the field of integrity management has been born to help clients conduct business, even in the most challenging of markets, without compromising their ethics. with the help of expert advice, companies can go beyond abiding by the law to taking a voluntary, proactive approach to ensuring a company's activities promote behaving responsibly, with fairness, sustainability, and cultural sensitivity in the communities in which they operate. it is very different in this respect from the more reactive field of risk management, although some risk management companies have attempted to embrace ethical risk as an area of specialism. many risk management consultancies are adjuncts to private security companies.


{ \ displaystyle ~ - 2 \ ln ( ) = - 2 \ ; \ sum _ { i = 1 } ^ { k } x _ { i } \ ln \ left ( { \ frac { \ pi _ { i } } { p _ { i } } } \ right ) ~. } ( the factor − 2 { \ displaystyle ~ - 2 ~ } is chosen to make the statistic asymptotically chi - squared distributed, for convenient comparison to a familiar statistic commonly used for the same application. ) if the null hypothesis is true, then as n { \ displaystyle ~ n ~ } increases, the distribution of − 2 ln ( ) { \ displaystyle ~ - 2 \ ln ( ) ~ } converges to that of chi - squared with k − 1 { \ displaystyle ~ k - 1 ~ } degrees of freedom.


legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing zeros of the factorials. daniel bernoulli and leonhard euler interpolated the factorial function to a continuous function of complex numbers, except at the negative integers, the ( offset ) gamma function. many other notable functions and number sequences are closely related to the factorials, including the binomial coefficients, double factorials, falling factorials, primorials, and subfactorials. implementations of the factorial function are commonly used as an example of different computer programming styles, and are included in scientific calculators and scientific computing software libraries. although directly computing large factorials using the product formula or recurrence is not efficient, faster algorithms are known, matching to within a constant factor the time for fast multiplication algorithms for numbers with the same number of digits.


in mathematics, the euclidean distance between two points in euclidean space is the length of a line segment between the two points. it can be calculated from the cartesian coordinates of the points using the pythagorean theorem, therefore occasionally being called the pythagorean distance. these names come from the ancient greek mathematicians euclid and pythagoras, although euclid did not represent distances as numbers, and the connection from the pythagorean theorem to distance calculation was not made until the 18th century.


geometrically, it is the product of the euclidean magnitudes of the two vectors and the cosine of the angle between them. these definitions are equivalent when using cartesian coordinates. in modern geometry, euclidean spaces are often defined by using vector spaces. in this case, the dot product is used for defining lengths ( the length of a vector is the square root of the dot product of the vector by itself ) and angles ( the cosine of the angle between two vectors is the quotient of their dot product by the product of their lengths ). the name " dot product " is derived from the centered dot " · " that is often used to designate this operation ; the alternative name " scalar product " emphasizes that the result is a scalar, rather than a vector ( as with the vector product in three - dimensional space ).


such dogmatic assertions can't be proved. the statements are not based on possible experience. in section ii, the discipline of pure reason in polemics, kant argues strongly against the polemical use of pure reason.


if one requires a more differentiated hierarchy of animals – to differentiate, say, those who provide milk from those who provide nothing except meat at the end of their lives – that is an intermediary level of abstraction, probably dairyanimal ( cows, goats ) who would eat foods suitable to giving good milk, and meatanimal ( pigs, steers ) who would eat foods to give the best meat - quality. such an abstraction could remove the need for the application coder to specify the type of food, so they could concentrate instead on the feeding schedule. the two classes could be related using inheritance or stand alone, and the programmer could define varying degrees of polymorphism between the two types.


in the u. s., sms is often charged both at the sender and at the destination, but, unlike phone calls, it cannot be rejected or dismissed. the reasons for lower uptake than other countries are varied. many users have unlimited " mobile - to - mobile " minutes, high monthly minute allotments, or unlimited service.


in mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. ternary relations may also be referred to as 3 - adic, 3 - ary, 3 - dimensional, or 3 - place. just as a binary relation is formally defined as a set of pairs, i. e. a subset of the cartesian product a × b of some sets a and b, so a ternary relation is a set of triples, forming a subset of the cartesian product a × b × c of three sets a, b and c. an example of a ternary relation in elementary geometry can be given on triples of points, where a triple is in the relation if the three points are collinear. another geometric example can be obtained by considering triples consisting of two points and a line, where a triple is in the ternary relation if the two points determine ( are incident with ) the line.


in spatial statistics and stochastic geometry, to measure the statistical correlation relationship between points of a point process, the pair correlation function of a point process n { \ displaystyle { n } } is defined as : ρ ( x 1, x 2 ) = μ ( 2 ) ( x 1, x 2 ) μ ( 1 ) ( x 1 ) μ ( 1 ) ( x 2 ), { \ displaystyle \ rho ( x _ { 1 }, x _ { 2 } ) = { \ frac { \ mu ^ { ( 2 ) } ( x _ { 1 }, x _ { 2 } ) } { \ mu ^ { ( 1 ) } ( x _ { 1 } ) \ mu ^ { ( 1 ) } ( x _ { 2 } ) } }, } where the points x 1, x 2 ∈ r d { \ displaystyle x _ { 1 }, x _ { 2 } \ in r ^ { d } }. in general, ρ ( x 1, x 2 ) ≥ 0 { \ displaystyle \ rho ( x _ { 1 }, x _ { 2 } ) \ geq 0 } whereas ρ ( x 1, x 2 ) = 1 { \ displaystyle \ rho ( x _ { 1 }, x _ { 2 } ) = 1 } corresponds to no correlation ( between points ) in the typical statistical sense.


in the 1990s, the object - oriented programming paradigm was applied to database technology, creating a new database model known as object databases. this aims to avoid the object – relational impedance mismatch – the overhead of converting information between its representation in the database ( for example as rows in tables ) and its representation in the application program ( typically as objects ). even further, the type system used in a particular application can be defined directly in the database, allowing the database to enforce the same data integrity invariants. object databases also introduce the key ideas of object programming, such as encapsulation and polymorphism, into the world of databases.


the second and third columns would give, ( 1, 1 ), ( 2, 1 ), ( 2, 2 ) and ( 1, 2 ) ; again, all possible ordered pairs each appearing once. the same statement would hold had the first and second columns been used. this is thus an orthogonal array of strength two.


in the united states the majority of smartphones use ipv6, but only a small percent of computers and tablets use ipv6. as of april 2022, 46. 2 % of google users in the us use ipv6.


in programming languages that support generics ( a. k. a. parametric polymorphism ), the programmer can extend the type system with new constructors. for example, a c # interface like ilist makes it possible to construct new types like ilist or ilist. the question then arises what the variance of these type constructors should be.


at each single observation in the sequence, probabilities to be used for calculations at the next observation are computed. the smoothing step can be calculated simultaneously during the backward pass. this step allows the algorithm to take into account any past observations of output for computing more accurate results. the forward – backward algorithm can be used to find the most likely state for any point in time. it cannot, however, be used to find the most likely sequence of states ( see viterbi algorithm ).


in mathematics, an extraneous solution ( or spurious solution ) is a solution, such as that to an equation, that emerges from the process of solving the problem but is not a valid solution to the problem. a missing solution is a solution that is a valid solution to the problem, but disappeared during the process of solving the problem. both are frequently the consequence of performing operations that are not invertible for some or all values of the variables, which prevents the chain of logical implications in the proof from being bidirectional.


in the distributed approach, each developer works directly with their own local repository, and changes are shared between repositories as a separate step.


in mathematical logic, a ground term of a formal system is a term that does not contain any variables. similarly, a ground formula is a formula that does not contain any variables. in first - order logic with identity with constant symbols a { \ displaystyle a } and b { \ displaystyle b }, the sentence q ( a ) ∨ p ( b ) { \ displaystyle q ( a ) \ lor p ( b ) } is a ground formula. a ground expression is a ground term or ground formula.


{ \ displaystyle \ ln ( m _ { i }! ) } is a constant, it will not give any additional information regarding the position of the maximum, so let's consider α ( m | e ) = i k { \ displaystyle \ alpha ( m \ vert e ) = \ sum _ { i } ^ { k } \ left } where α { \ displaystyle \ alpha } is something that shares the same maximum position as p ( m | e ) { \ displaystyle p ( m \ vert e ) }.


all diagonal elements m i i { \ displaystyle m _ { ii } } in a tuned filter are equal to zero because a susceptance vanishes at the resonant frequency. important merit of the matrix m { \ displaystyle \ mathbf { m } } is the fact that it allows to directly compute the frequency response of the equivalent network having the inductively coupled resonant circuits,. therefore it is convenient to use this matrix when designing the cross - coupled filters. the coupling matrices m { \ displaystyle \ mathbf { m } }, in particular, are used as coarse models of filters. utilization of a coarse model allows to quicken filter optimization manyfold because of computation of the frequency response for the coarse model does not consume cpu time with respect to computation for the real filter.


in mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to " prove " them. these techniques were introduced by john blissard and are sometimes called blissard's symbolic method. they are often attributed to edouard lucas ( or james joseph sylvester ), who used the technique extensively.


in the case of indivisible items, we have the following strong versions of the two welfare theorems : any competitive equilibrium maximizes the social welfare ( the sum of utilities ), not only over all realistic assignments of items, but also over all fractional assignments of items. i. e., even if we could assign fractions of an item to different people, we couldn't do better than a competitive equilibrium in which only whole items are assigned. if there is an integral assignment ( with no fractional assignments ) that maximizes the social welfare, then there is a competitive equilibrium with that assignment.


in mathematics and computer science, a matroid oracle is a subroutine through which an algorithm may access a matroid, an abstract combinatorial structure that can be used to describe the linear dependencies between vectors in a vector space or the spanning trees of a graph, among other applications. the most commonly used oracle of this type is an independence oracle, a subroutine for testing whether a set of matroid elements is independent. several other types of oracle have also been used ; some of them have been shown to be weaker than independence oracles, some stronger, and some equivalent in computational power. many algorithms that perform computations on matroids have been designed to take an oracle as input, allowing them to run efficiently without change on many different kinds of matroids, and without additional assumptions about what kind of matroid they are using. for instance, given an independence oracle for any matroid, it is possible to find the minimum weight basis of the matroid by applying a greedy algorithm that adds elements to the basis in sorted order by weight, using the independence oracle to test whether each element can be added. in computational complexity theory, the oracle model has led to unconditional lower bounds proving that certain matroid problems cannot be solved in polynomial time, without invoking unproved assumptions such as the assumption that p = np. problems that have been shown to be hard in this way include testing whether a matroid is binary or uniform, or testing whether it contains certain fixed minors.


priority inversion can also reduce the perceived performance of the system. low - priority tasks usually have a low priority because it is not important for them to finish promptly ( for example, they might be a batch job or another non - interactive activity ).


in mathematics, the chvatal – sankoff constants are mathematical constants that describe the lengths of longest common subsequences of random strings. although the existence of these constants has been proven, their exact values are unknown. they are named after vaclav chvatal and david sankoff, who began investigating them in the mid - 1970s. there is one chvatal – sankoff constant γ k { \ displaystyle \ gamma _ { k } } for each positive integer k, where k is the number of characters in the alphabet from which the random strings are drawn. the sequence of these numbers grows inversely proportionally to the square root of k. however, some authors write " the chvatal – sankoff constant " to refer to γ 2 { \ displaystyle \ gamma _ { 2 } }, the constant defined in this way for the binary alphabet.


in bifurcating ( fully resolved ) trees, each internal branch induces a quartet whose leaves are either subtrees of the original tree or actual leaves of the original tree. if the topology of a quartet extracted from the reference species tree is embedded in the gene tree, the quartet is compatible with the gene tree. conversely, incompatible strongly supported quartets indicate potential hgt events. quartet mapping methods are much more computationally efficient and naturally handle heterogeneous representation of taxa among gene families, making them a good basis for developing large - scale scans for hgt, looking for highways of gene sharing in databases of hundreds of complete genomes.


in order to assess whether financial management for it services has been successfully deployed, the following key performance indicators may be examined : do the predicted budgets match the actual expenditure? has user behavior changed to follow the corporate it goals? are charges seen by users and customers to be simple, fair and in line with organisational goals?


in numerical analysis, an incomplete cholesky factorization of a symmetric positive definite matrix is a sparse approximation of the cholesky factorization. an incomplete cholesky factorization is often used as a preconditioner for algorithms like the conjugate gradient method. the cholesky factorization of a positive definite matrix a is a = ll * where l is a lower triangular matrix.


in a specific example, the goal of verschelde, et al. was the integration of several different ontology libraries into a larger one that contained more definitions of different subspecialties ( medical, molecular biological, etc. ) and was able to distinguish between ambiguous tags ; the result was a data - warehouse like effect, with easy access to multiple databases through the use of ontologies. in a separate project, bertens, et al. constructed a lattice work of three ontologies ( for anatomy and development of model organisms ) on a novel framework ontology of generic organs. for example, results from a search of ‘ heart ’ in this ontology would return the heart plans for each of the vertebrate species whose ontologies were included. the stated goal of the project is to facilitate comparative and evolutionary studies.


in the early years, acar was mainly used to investigate the physics of the electron - positron annihilation process. in the 1930s several annihilation mechanism were discussed. otto klemperer could show with his angular correlation setup that the electron - positron pairs annihilate mainly into two gamma quanta which are emitted anti - parallel. in the 1950s, it was realized that by measuring the deviation from collinearity of the annihilation radiation information about the electronic structure of a solid can be obtained. during this time mainly setups with'long slit geometry'were used.


the same heuristic renumbering algorithm that reduce the bandwidth are also used to reduce the skyline. the basic and one of the earliest algorithms to do that is reverse cuthill – mckee algorithm. however, skyline storage is not as popular for very large systems ( many millions of equations ) because skyline cholesky is not so easily adapted for massively parallel computing, and general sparse methods, which store only the nonzero entries of the matrix, become more efficient for very large problems due to much less fill - in.


richard cole and uzi vishkin show that there is a distributed algorithm that reduces the number of colors from n to o ( log n ) in one synchronous communication step. by iterating the same procedure, it is possible to obtain a 3 - coloring of an n - cycle in o ( log * n ) communication steps ( assuming that we have unique node identifiers ). the function log *, iterated logarithm, is an extremely slowly growing function, " almost constant ".


in numerical optimization, the broyden – fletcher – goldfarb – shanno ( bfgs ) algorithm is an iterative method for solving unconstrained nonlinear optimization problems. like the related davidon – fletcher – powell method, bfgs determines the descent direction by preconditioning the gradient with curvature information. it does so by gradually improving an approximation to the hessian matrix of the loss function, obtained only from gradient evaluations ( or approximate gradient evaluations ) via a generalized secant method. since the updates of the bfgs curvature matrix do not require matrix inversion, its computational complexity is only o ( n 2 ) { \ displaystyle { \ mathcal { o } } ( n ^ { 2 } ) }, compared to o ( n 3 ) { \ displaystyle { \ mathcal { o } } ( n ^ { 3 } ) } in newton's method. also in common use is l - bfgs, which is a limited - memory version of bfgs that is particularly suited to problems with very large numbers of variables ( e. g., > 1000 ). the bfgs - b variant handles simple box constraints. the algorithm is named after charles george broyden, roger fletcher, donald goldfarb and david shanno.


in morpheme - based morphology, word forms are analyzed as arrangements of morphemes. a morpheme is defined as the minimal meaningful unit of a language. in a word such as independently, the morphemes are said to be in -, de -, pend, - ent, and - ly ; pend is the ( bound ) root and the other morphemes are, in this case, derivational affixes. in words such as dogs, dog is the root and the - s is an inflectional morpheme.


in mathematics, an isomorphism is a structure - preserving mapping between two structures of the same type that can be reversed by an inverse mapping. two mathematical structures are isomorphic if an isomorphism exists between them. the word isomorphism is derived from the ancient greek : ισος isos " equal ", and μορφη morphe " form " or " shape ". the interest in isomorphisms lies in the fact that two isomorphic objects have the same properties ( excluding further information such as additional structure or names of objects ).


hardware execution of java bytecode, such as that offered by arm's jazelle, was also explored to offer significant performance improvements. the performance of a java bytecode compiled java program depends on how optimally its given tasks are managed by the host java virtual machine ( jvm ), and how well the jvm exploits the features of the computer hardware and operating system ( os ) in doing so. thus, any java performance test or comparison has to always report the version, vendor, os and hardware architecture of the used jvm. in a similar manner, the performance of the equivalent natively compiled program will depend on the quality of its generated machine code, so the test or comparison also has to report the name, version and vendor of the used compiler, and its activated compiler optimization directives.


in some cases, an evaluation is done to determine the fit of a development proposal within its environment. it also allows comparison of alternatives. another purpose is to seek the best use of resources to maximize economic vitality.


in number theory, sophie germain's theorem is a statement about the divisibility of solutions to the equation x p + y p = z p { \ displaystyle x ^ { p } + y ^ { p } = z ^ { p } } of fermat's last theorem for odd prime p { \ displaystyle p }.


in mathematics, a sastry automorphism, is an automorphism of a field of characteristic 2 satisfying some rather complicated conditions related to the problem of embedding ree groups of type 2f4 into chevalley groups of type f4. they were introduced by sastry ( 1995 ), and named and classified by bombieri ( 2002 ) who showed that there are 22 families of sastry automorphisms, together with 22 exceptional ones over some finite fields of orders up to 210.


{ \ displaystyle ( \ neg a ) \ wedge ( \ neg b ). } if either a or b were true, then the disjunction of a and b would be true, making its negation false. presented in english, this follows the logic that " since two things are both false, it is also false that either of them is true ".


each returning traveler must have enough supplies on the way back. the problem asks for the minimum number of accompanying travelers needed to reach the other base.


consider a function f ∈ k ( c ) ∗ { \ displaystyle f \ in { \ overline { k } } ( c ) ^ { * } }, then we can look at the formal sum div ( f ) = p ∈ c o r d p ( f ) { \ textstyle ( f ) = \ sum _ { p \ in c } { \ mathrm { ord } _ { p } ( f ) } }. here o r d p ( f ) { \ displaystyle \ mathrm { ord } _ { p } ( f ) } denotes the order of f { \ displaystyle f } at p { \ displaystyle p }. we have that ord p ( f ) < 0 { \ displaystyle _ { p } ( f ) < 0 } if f { \ displaystyle f } has a pole of order − o r d p ( f ) { \ displaystyle - \ mathrm { ord } _ { p } ( f ) } at p { \ displaystyle p }, o r d p ( f ) = 0 { \ displaystyle \ mathrm { ord } _ { p } ( f ) = 0 } if f { \ displaystyle f } is defined and non - zero at p { \ displaystyle p } and o r d p ( f ) > 0 { \ displaystyle \ mathrm { ord } _ { p } ( f ) > 0 } if f { \ displaystyle f } has a zero of order o r d p ( f ) { \ displaystyle \ mathrm { ord } _ { p } ( f ) } at p { \ displaystyle p }.


in recursion theory, e { \ displaystyle \ phi _ { e } } denotes the computable function with index ( program ) e in some standard numbering of computable functions, and e b { \ displaystyle \ phi _ { e } ^ { b } } denotes the eth computable function using a set b of natural numbers as an oracle. a set a of natural numbers is turing reducible to a set b if there is a computable function that, given an oracle for set b, computes the characteristic function χa of the set a. that is, there is an e such that χ a = e b { \ displaystyle \ chi _ { a } = \ phi _ { e } ^ { b } }. this relationship is denoted a ≤t b ; the relation ≤t is a preorder. two sets of natural numbers are turing equivalent if each is turing reducible to the other.


in mathematics, a primorial prime is a prime number of the form pn # ± 1, where pn # is the primorial of pn ( i. e. the product of the first n primes ). primality tests show that pn # − 1 is prime for n = 2, 3, 5, 6, 13, 24,... ( sequence a057704 in the oeis ) pn # + 1 is prime for n = 0, 1, 2, 3, 4, 5, 11,... ( sequence a014545 in the oeis ) the first term of the second sequence is 0 because p0 # = 1 is the empty product, and thus p0 # + 1 = 2, which is prime. similarly, the first term of the first sequence is not 1, because p1 # = 2, and 2 − 1 = 1 is not prime. the first few primorial primes are 2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 ( sequence a228486 in the oeis ) as of october 2021, the largest known primorial prime ( of the form pn # − 1 ) is 3267113 # − 1 ( n = 234, 725 ) with 1, 418, 398 digits, found by the primegrid project. as of 2022, the largest known prime of the form pn # + 1 is 392113 # + 1 ( n = 33, 237 ) with 169, 966 digits, found in 2001 by daniel heuer. euclid's proof of the infinitude of the prime numbers is commonly misinterpreted as defining the primorial primes, in the following manner : assume that the first n consecutive


in mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects a, b, c { \ displaystyle a, b, c }, the canonical map : a × b + a × c → a × ( b + c ) { \ displaystyle : a \! \ times \! b \, + a \! \ times \! c \ to a \! \ times \! ( b + c ) } is an isomorphism, and for all objects a { \ displaystyle a }, the canonical map 0 → a × 0 { \ displaystyle 0 \ to a \ times 0 } is an isomorphism ( where 0 denotes the initial object ). equivalently, if for every object a { \ displaystyle a } the endofunctor a × − { \ displaystyle a \ times - } defined by b ↦ a × b { \ displaystyle b \ mapsto a \ times b } preserves coproducts up to isomorphisms f { \ displaystyle f }. it follows that f { \ displaystyle f } and aforementioned canonical maps are equal for each choice of objects. in particular, if the functor a × − { \ displaystyle a \ times - } has a right adjoint ( i. e., if the category is cartesian closed ), it necessarily preserves all colimits, and thus any cartesian closed category with finite coproducts ( i. e., any bicartesian closed category ) is distributive.


therefore, an informed user is one who can protect and achieve the best security out of the system they use. because of the importance of end - user security and the impact it can have on organisations the uk government set out a guidance for the public sector, to help civil servants learn how to be more security aware when using government networks and computers. while this is targeted to a certain sector, this type of educational effort can be informative to any type of user. this helps developers meet security norms and end users be aware of the risks involved. reimers and andersson have conducted a number of studies on end user security habits and found that the same type of repeated education / training in security " best practices " can have a marked effect on the perception of compliance with good end user network security habits, especially concerning malware and ransomware.


alternatively, one can visualize infinitely many 3 - dimensional planes that go through n = 2 { \ displaystyle n = 2 } fixed points. more generally, to estimate a least squares model with k { \ displaystyle k } distinct parameters, one must have n ≥ k { \ displaystyle n \ geq k } distinct data points. if n > k { \ displaystyle n > k }, then there does not generally exist a set of parameters that will perfectly fit the data.


in the early 1980s, researchers at uc berkeley and ibm both discovered that most computer language compilers and interpreters used only a small subset of the instructions of complex instruction set computing ( cisc ). much of the power of the cpu was being ignored in real - world use. they realized that by making the computer simpler and less orthogonal, they could make it faster and less costly at the same time. at the same time, cpu calculation became faster in relation to the time for needed memory accesses.


in operating systems, write barrier is a mechanism for enforcing a particular ordering in a sequence of writes to a storage system in a computer system. for example, a write barrier in a file system is a mechanism ( program logic ) that ensures that in - memory file system state is written out to persistent storage in the correct order.


this ensures semantically rigorous designs, preserving identical semantics throughout the chip design flow, which included extensive use of formal verification techniques. without such attention, integrating an arm11 with third - party designs could risk exposing hard - to - find latent bugs. due to arm cores being integrated into many different designs, using a variety of logic synthesis tools and chip manufacturing processes, the impact of its register - transfer level ( rtl ) quality is magnified many times. the arm11 generation focused more on synthesis than previous generations, making such concerns more of an issue.


timing advance is significant for privacy and communications security, as its combination with other variables can allow gsm localization to find the device's position and track the mobile phone user. ta is also used to adjust transmission power in space - division multiple access systems. this limited the original range of a gsm cell site to 35 km as mandated by the duration of the standard timeslots defined in the gsm specification.


dect phones use allocated spectrum outside the ism bands that differs in europe and north america. ultra - wideband lans require more spectrum than the ism bands can provide, so the relevant standards such as ieee 802. 15. 4a are designed to make use of spectrum outside the ism bands. despite the fact that these additional bands are outside the official itu - r ism bands, because they are used for the same types of low power personal communications, they are sometimes incorrectly referred to as ism bands as well.


in the analysis of algorithms, the input to breadth - first search is assumed to be a finite graph, represented as an adjacency list, adjacency matrix, or similar representation. however, in the application of graph traversal methods in artificial intelligence the input may be an implicit representation of an infinite graph. in this context, a search method is described as being complete if it is guaranteed to find a goal state if one exists. breadth - first search is complete, but depth - first search is not. when applied to infinite graphs represented implicitly, breadth - first search will eventually find the goal state, but depth first search may get lost in parts of the graph that have no goal state and never return.


in the category of sets, the morphisms between sets x and y are the functions from x to y. it results that the set of the functions from x to y that is denoted y x { \ displaystyle y ^ { x } } in the preceding section can also be denoted hom ( x, y ). { \ displaystyle \ hom ( x, y ). } the isomorphism ( s t ) u s t × u { \ displaystyle ( s ^ { t } ) ^ { u } \ cong s ^ { t \ times u } } can be rewritten hom ( u, s t ) hom ( t × u, s ). { \ displaystyle \ hom ( u, s ^ { t } ) \ cong \ hom ( t \ times u, s ). }


these are called supplementary services and are commonly invoked by a vertical service code. examples include " call waiting ", " call forward on busy " etc. call control software, because of its central place in the operation of the telephone network, is marked by both complexity and reliability. call control systems will typically require many thousands of person years in development.


in mathematics, a complete set of invariants for a classification problem is a collection of maps f i : x → y i { \ displaystyle f _ { i } : x \ to y _ { i } } ( where x { \ displaystyle x } is the collection of objects being classified, up to some equivalence relation { \ displaystyle \ sim }, and the y i { \ displaystyle y _ { i } } are some sets ), such that x x ′ { \ displaystyle x \ sim x'} if and only if f i ( x ) = f i ( x ′ ) { \ displaystyle f _ { i } ( x ) = f _ { i } ( x') } for all i { \ displaystyle i }. in words, such that two objects are equivalent if and only if all invariants are equal. symbolically, a complete set of invariants is a collection of maps such that ( f i ) : ( x / ) → ( y i ) { \ displaystyle \ left ( \ prod f _ { i } \ right ) : ( x / \ sim ) \ to \ left ( \ prod y _ { i } \ right ) } is injective. as invariants are, by definition, equal on equivalent objects, equality of invariants is a necessary condition for equivalence ; a complete set of invariants is a set such that equality of these is also sufficient for equivalence. in the context of a group action, this may be stated as : invariants are functions of coinvariants ( equivalence classes, orbits ), and a complete set of invariants characterizes the coinvariants ( is a set of defining equations for the coinvariants ).


the unit of measurement for the log - odds scale is called a logit, from logistic unit, hence the alternative names. see § background and § definition for formal mathematics, and § example for a worked example. binary variables are widely used in statistics to model the probability of a certain class or event taking place, such as the probability of a team winning, of a patient being healthy, etc. ( see § applications ), and the logistic model has been the most commonly used model for binary regression since about 1970.


in the 1840s, mathematicians were generally unprepared to understand grassmann's ideas. in the 1860s and 1870s various mathematicians came to ideas similar to that of grassmann's, but grassmann himself was not interested in mathematics anymore. : 46 adhemar jean claude barre de saint - venant developed a vector calculus similar to that of grassmann, which he published in 1845. he then entered into a dispute with grassmann about which of the two had thought of the ideas first.


in mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set - theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. it also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations. any set of sets closed under the set - theoretic operations forms a boolean algebra with the join operator being union, the meet operator being intersection, the complement operator being set complement, the bottom being ∅ { \ displaystyle \ varnothing } and the top being the universe set under consideration.


in probability theory and statistics, a covariance matrix ( also known as auto - covariance matrix, dispersion matrix, variance matrix, or variance – covariance matrix ) is a square matrix giving the covariance between each pair of elements of a given random vector. intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. as an example, the variation in a collection of random points in two - dimensional space cannot be characterized fully by a single number, nor would the variances in the x { \ displaystyle x } and y { \ displaystyle y } directions contain all of the necessary information ; a 2 × 2 { \ displaystyle 2 \ times 2 } matrix would be necessary to fully characterize the two - dimensional variation. any covariance matrix is symmetric and positive semi - definite and its main diagonal contains variances ( i. e., the covariance of each element with itself ). the covariance matrix of a random vector x { \ displaystyle \ mathbf { x } } is typically denoted by k x x { \ displaystyle \ operatorname { k } _ { \ mathbf { x } \ mathbf { x } } }, σ { \ displaystyle \ sigma } or s { \ displaystyle s }.


go tos were largely replaced by the perform statement and procedures, which promoted modular programming and gave easy access to powerful looping facilities. however, perform could be used only with procedures so loop bodies were not located where they were used, making programs harder to understand. cobol programs were infamous for being monolithic and lacking modularization. cobol code could be modularized only through procedures, which were found to be inadequate for large systems.


in software engineering, a version control system is used to keep track of versions of a set of files, usually to allow multiple developers to collaborate on a project. the repository keeps track of the files in the project, which is represented as a graph. a distributed version control system is made up of central and branch repositories. a central repository exists on the server. to make changes to it, a developer first works on a branch repository, and proceeds to commit the change to the former.


in number theory, a full reptend prime, full repetend prime, proper prime : 166 or long prime in base b is an odd prime number p such that the fermat quotient q p ( b ) = b p − 1 − 1 p { \ displaystyle q _ { p } ( b ) = { \ frac { b ^ { p - 1 } - 1 } { p } } } ( where p does not divide b ) gives a cyclic number. therefore, the base b expansion of 1 / p { \ displaystyle 1 / p } repeats the digits of the corresponding cyclic number infinitely, as does that of a / p { \ displaystyle a / p } with rotation of the digits for any a between 1 and p − 1. the cyclic number corresponding to prime p will possess p − 1 digits if and only if p is a full reptend prime. that is, the multiplicative order ordp b = p − 1, which is equivalent to b being a primitive root modulo p. the term " long prime " was used by john conway and richard guy in their book of numbers. confusingly, sloane's oeis refers to these primes as " cyclic numbers ".


in some programming languages, any expression can be evaluated in a context that expects a boolean data type. typically ( though this varies by programming language ) expressions like the number zero, the empty string, empty lists, and null evaluate to false, and strings with content ( like " abc " ), other numbers, and objects evaluate to true. sometimes these classes of expressions are called " truthy " and " falsy " / " false ".


in probability theory and information theory, the variation of information or shared information distance is a measure of the distance between two clusterings ( partitions of elements ). it is closely related to mutual information ; indeed, it is a simple linear expression involving the mutual information. unlike the mutual information, however, the variation of information is a true metric, in that it obeys the triangle inequality.


in statistics, prediction is a part of statistical inference. one particular approach to such inference is known as predictive inference, but the prediction can be undertaken within any of the several approaches to statistical inference. indeed, one possible description of statistics is that it provides a means of transferring knowledge about a sample of a population to the whole population, and to other related populations, which is not necessarily the same as prediction over time.


in order to facilitate the development of applications for mobile devices, and consistency thereof, various approaches have been taken. most companies that ship a product ( e. g. apple, ipod / iphone / ipad ) provide an official software development kit ( sdk ). they may also opt to provide some form of testing and / or quality assurance ( qa ). in exchange for being provided the sdk or other tools, it may be necessary for a prospective developer to sign a some form of non - disclosure agreement, or nda, which restricts the sharing of privileged information.


in spectral graph theory, the alon – boppana bound provides a lower bound on the second - largest eigenvalue of the adjacency matrix of a d { \ displaystyle d } - regular graph, meaning a graph in which every vertex has degree d { \ displaystyle d }. the reason for the interest in the second - largest eigenvalue is that the largest eigenvalue is guaranteed to be d { \ displaystyle d } due to d { \ displaystyle d } - regularity, with the all - ones vector being the associated eigenvector. the graphs that come close to meeting this bound are ramanujan graphs, which are examples of the best possible expander graphs. its discoverers are noga alon and ravi boppana.


in sound and music, sampling is the reuse of a portion ( or sample ) of a sound recording in another recording. samples may comprise elements such as rhythm, melody, speech, sound effects or longer portions of music, and may be layered, equalized, sped up or slowed down, repitched, looped, or otherwise manipulated. they are usually integrated using electronic music instruments ( samplers ) or software such as digital audio workstations. a process similar to sampling originated in the 1940s with musique concrete, experimental music created by splicing and looping tape.


k is then the gaussian curvature of the space at the time when a ( t ) = 1. r is sometimes called the reduced circumference because it is equal to the measured circumference of a circle ( at that value of r ), centered at the origin, divided by 2π ( like the r of schwarzschild coordinates ). where appropriate, a ( t ) is often chosen to equal 1 in the present cosmological era, so that d σ { \ displaystyle \ mathrm { d } \ mathbf { \ sigma } } measures comoving distance. alternatively, k may be taken to belong to the set { −1, 0, + 1 } ( for negative, zero, and positive curvature respectively ).


in modern grammar, a particle is a function word that must be associated with another word or phrase to impart meaning, i. e., it does not have its own lexical definition. according to this definition, particles are a separate part of speech and are distinct from other classes of function words, such as articles, prepositions, conjunctions and adverbs. languages vary widely in how much they use particles, some using them extensively and others more commonly using alternative devices such as prefixes / suffixes, inflection, auxiliary verbs and word order. particles are typically words that encode grammatical categories ( such as negation, mood, tense, or case ), clitics, fillers or ( oral ) discourse markers such as well, um, etc. particles are never inflected.


in the united kingdom, examiners usually follow guidelines issued by the association of chief police officers ( acpo ) for the authentication and integrity of evidence. they were updated to version 5 in october 2011 when computer based evidence was replaced with digital evidence reflecting the development of investigating information security incidents in a wider context. the guidelines consist of four principles : principle 1 : no action taken by law enforcement agencies, persons employed within those agencies or their agents should change data which may subsequently be relied upon in court. principle 2 : in circumstances where a person finds it necessary to access original data, that person must be competent to do so and be able to give evidence explaining the relevance and the implications of their actions.


finally, the dissemination of foreign intelligence information needed for an approved purpose..


in phonetics and linguistics, a phone is any distinct speech sound or gesture, regardless of whether the exact sound is critical to the meanings of words. in contrast, a phoneme is a speech sound in a given language that, if swapped with another phoneme, could change one word to another. phones are absolute and are not specific to any language, but phonemes can be discussed only in reference to specific languages.


in the context of dikw, information meets the definition for knowledge by description ( " information is contained in descriptions " ), and is differentiated from data in that it is " useful ". " information is inferred from data ", in the process of answering interrogative questions ( e. g., " who ", " what ", " where ", " how many ", " when " ), thereby making the data useful for " decisions and / or action ". " classically, " states a 2007 text, " information is defined as data that are endowed with meaning and purpose. "