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 Édouard 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 Chvátal–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 Václav Chvátal and David Sankoff, who began investigating them in the mid-1970s.There is one Chvátal–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 Chvátal–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: ἴσος isos "equal", and μορφή 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, ϕ e {\displaystyle \phi _{e}} denotes the computable function with index (program) e in some standard numbering of computable functions, and ϕ 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 = ϕ 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 Adhémar Jean Claude Barré 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 concrète, 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."


(See example below.) After their introduction in the 1968 ALGOL 68 "Final Report", W-grammars were widely considered as too powerful and unconstrained to be practical.This was partly a consequence of the way in which they had been applied; the 1973 ALGOL 68 "Revised Report" contains a much more readable grammar, without modifying the W-grammar formalism itself. Meanwhile, it became clear that W-grammars, when used in their full generality, are indeed too powerful for such practical purposes as serving as the input for a parser generator.


In military acquisition, full operating capability or full operational capability (FOC) is the completion of a development effort. This is usually preceded by an initial operating capability or initial operational capability (IOC) phase. For the United States Department of Defense military acquisition FOC is defined as "in general attained when all units and/or organizations in the force structure scheduled to receive a system have received it and have the ability to employ and maintain it. The specifics for any particular system FOC are defined in that system’s Capability Development Document (CDD) and Capability Production Document (CPD).


Attention to the location of items in the search can be used to guide visual searches. This was demonstrated by valid cues improving the identification of targets relative to the invalid and neutral conditions. A visual search display can also influence how fast an observer responds to a spatial probe.


For vitamin C, vitamin D, vitamin E, vitamin K, calcium, phosphorus, magnesium, and manganese, the current highest RDAs are up to 50% higher than the older Daily Values used in labeling, whereas for other nutrients the recommended needs have gone down. A side-by-side table of the old and new adult Daily Values is provided at Reference Daily Intake. As of October 2010, the only micronutrients that are required to be included on all labels are vitamin A, vitamin C, calcium, and iron.


ASHRAE Guideline 4-2014 provides performance indices criteria for model calibration. The performance indices used are normalized mean bias error (NMBE), coefficient of variation (CV) of the root mean square error (RMSE), and R2 (coefficient of determination). ASHRAE recommends a R2 greater than 0.75 for calibrated models. The criteria for NMBE and CV RMSE depends on if measured data is available at a monthly or hourly timescale.


In the event of an attack from an aggressor, a state would massively retaliate by using a force disproportionate to the size of the attack. The aim of massive retaliation is to deter another state from attacking first. For such a strategy to work, it must be made public knowledge to all possible aggressors. The aggressor also must believe that the state announcing the policy has the ability to maintain second-strike capability in the event of an attack.


The MIPS design team that designed the R4300i started the company SandCraft, which designed the R5432 for NEC and later produced the SR71000, one of the first out-of-order execution processors for the embedded market. The original DEC StrongARM team eventually split into two MIPS-based start-ups: SiByte which produced the SB-1250, one of the first high-performance MIPS-based systems-on-a-chip (SOC); while Alchemy Semiconductor (later acquired by AMD) produced the Au-1000 SoC for low-power uses. Lexra used a MIPS-like architecture and added DSP extensions for the audio chip market and multithreading support for the networking market.


If these decisions change later in the project then the cone will widen. Original research for engineering and construction in the chemical industry demonstrated that actual final costs often exceeded the earliest "base" estimate by as much as 100% (or underran by as much as 50%). Research in the software industry on the Cone of Uncertainty stated that in the beginning of the project life cycle (i.e. before gathering of requirements) estimates have in general an uncertainty of factor 4 on both the high side and the low side. This means that the actual effort or scope can be 4 times or 1/4 of the first estimates. This uncertainty tends to decrease over the course of a project, although that decrease is not guaranteed.


In some networks, routing is complicated by the fact that no single entity is responsible for selecting paths; instead, multiple entities are involved in selecting paths or even parts of a single path. Complications or inefficiency can result if these entities choose paths to optimize their own objectives, which may conflict with the objectives of other participants. A classic example involves traffic in a road system, in which each driver picks a path that minimizes their travel time. With such routing, the equilibrium routes can be longer than optimal for all drivers.


In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors.


Later work by Kummer and others always divided the problem into first and second cases. Sophie Germain primes and safe primes have applications in public key cryptography and primality testing. It has been conjectured that there are infinitely many Sophie Germain primes, but this remains unproven.


In more recent times, the term bit slicing was reused by Matthew Kwan to refer to the technique of using a general-purpose CPU to implement multiple parallel simple virtual machines using general logic instructions to perform single-instruction multiple-data (SIMD) operations. This technique is also known as SIMD within a register (SWAR). This was initially in reference to Eli Biham's 1997 article A Fast New DES Implementation in Software, which achieved significant gains in performance of DES by using this method.


In the following query, four different types of data are retrieved from a single table (status) and for a single user ("me"): This query can run by querying the Facebook graph endpoint /fql with the parameters set to q=


For the 1991 and 1992 seasons, this meant potentially severe angles for short field goal attempts, since the hashmark width remained at 53 ft 4 in (16.26 m). In 1993, the NCAA narrowed the distance between the hashmarks to 40 ft (12.19 m), matching what was the width of hashmarks in the NFL from 1945 through 1971; as mentioned above, the NFL narrowed the hashmarks in 1972 to goalpost width at 18.5 feet (5.64 m). Canadian hash marks in amateur play are 51 feet (16 m) apart, 24 yards from each sideline.


In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. For a given natural number k, a number n is called k-perfect (or k-fold perfect) if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number. As of 2014, k-perfect numbers are known for each value of k up to 11.It is unknown whether there are any odd multiply perfect numbers other than 1. The first few multiply perfect numbers are: 1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, ... (sequence A007691 in the OEIS).


In scientific inquiry and academic research, data fabrication is the intentional misrepresentation of research results. As with other forms of scientific misconduct, it is the intent to deceive that marks fabrication as unethical, and thus different from scientists deceiving themselves. There are many ways data can be fabricated. Experimental data can be fabricated by reporting experiments that were never conducted, and accurate data can be manipulated or misrepresented to suit a desired outcome.


This rule not only replaces the representation of a monophthong with that of a diphthong, but it also destroys the reversibility of any capitalization process in digital environments, as the combination of uppercase letter and uppercase iota would normally be converted back to lowercase letter and lowercase iota. It is therefore strongly recommended, both for the integrity of text and for the practical compatibility with digital environments, that lowercase letter and iota subscript should be capitalized in all situations and contexts as uppercase letter and iota adscript. A future revision of the above-mentioned Unicode stipulation is linguistically stipulated and digitally inevitable, as its application is both destructive to the text and impractical in digital applications. In the ASCII-based encoding standard Beta Code, the iota subscript is represented by the pipe character "|" placed after the letter.


In mathematical logic, computational complexity theory, and computer science, the existential theory of the reals is the set of all true sentences of the form where the variables X i {\displaystyle X_{i}} are interpreted as having real number values, and where F ( X 1 , … X n ) {\displaystyle F(X_{1},\dots X_{n})} is a quantifier-free formula involving equalities and inequalities of real polynomials. A sentence of this form is true if it is possible to find values for all of the variables that, when substituted into formula F {\displaystyle F} , make it become true.The decision problem for the existential theory of the reals is the problem of finding an algorithm that decides, for each such sentence, whether it is true or false. Equivalently, it is the problem of testing whether a given semialgebraic set is non-empty. This decision problem is NP-hard and lies in PSPACE, giving it significantly lower complexity than Alfred Tarski's quantifier elimination procedure for deciding statements in the first-order theory of the reals without the restriction to existential quantifiers.


Thus in the sentence Here is the big house, both house and big house are N-bars, while the big house is a noun phrase. In the sentence I like big houses, both houses and big houses are N-bars, but big houses also functions as a noun phrase (in this case without an explicit determiner). In some modern theories of syntax, however, what are called "noun phrases" above are no longer considered to be headed by a noun, but by the determiner (which may be null), and they are thus called determiner phrases (DP) instead of noun phrases.


Given a finite group G = ⟨S⟩ and g ∈ G, find a straight-line program computing g over S. The constructive membership problem is often studied in the setting of black box groups. The elements are encoded by bit strings of a fixed length. Three oracles are provided for the group-theoretic functions of multiplication, inversion, and checking for equality with the identity.


In reverse mathematics, Brouwer's theorem can be proved in the system WKL0, and conversely over the base system RCA0 Brouwer's theorem for a square implies the weak Kőnig's lemma, so this gives a precise description of the strength of Brouwer's theorem.


In practice, a strict adherence to the basic three-structure template of structured programming yields highly nested code, due to inability to exit a structured unit prematurely, and a combinatorial explosion with quite complex program state data to handle all possible conditions. Two solutions have been generally adopted: a way to exit a structured unit prematurely, and more generally exceptions – in both cases these go up the structure, returning control to enclosing blocks or functions, but do not jump to arbitrary code locations. These are analogous to the use of a return statement in non-terminal position – not strictly structured, due to early exit, but a mild relaxation of the strictures of structured programming. In C, break and continue allow one to terminate a loop or continue to the next iteration, without requiring an extra while or if statement.


If τ1,...,τm,σ1,...,σn are ramified types then as in simple type theory there is a type (τ1,...,τm,σ1,...,σn) of "predicative" propositional functions of τ1,...,τm,σ1,...,σn. However, there are also ramified types (τ1,...,τm|σ1,...,σn) that can be thought of as the classes of propositional functions of τ1,...τm obtained from propositional functions of type (τ1,...,τm,σ1,...,σn) by quantifying over σ1,...,σn. When n=0 (so there are no σs) these propositional functions are called predicative functions or matrices.


In such a cyberattack the attacker places malware within the networks through a variety of methods (malware-laden website, targeted email, infected USB stick, socially engineered access, etc.) and then the malware propagates within the network. Most of the time existing cyber defenses clear the attacker tools from standard serves and IT workstations (IT endpoints) but the cyber defense software cannot access the embedded processors within medical devices. Most of the embedded operating systems within medical devices are running on Microsoft Windows 7 and Windows XP. The security in these operating systems is no longer supported.


Additionally, unlike bounds on the total time, it is a suitable form of analysis even for algorithms that produce an infinite sequence of outputs. The notion of polynomial delay was first introduced by David S. Johnson, Mihalis Yannakakis and Christos Papadimitriou. == References ==


In telecommunication, communications survivability is the ability of communications systems to continue to operate effectively under adverse conditions, though portions of the system may be damaged or destroyed. Various methods may be used to maintain communications services, such as using alternate routing, different transmission media or methods, redundant equipment, and sites and equipment that are radiation hardened.


An even nontotient may be one more than a prime number, but never one less, since all numbers below a prime number are, by definition, coprime to it. To put it algebraically, for p prime: φ(p) = p − 1. Also, a pronic number n(n − 1) is certainly not a nontotient if n is prime since φ(p2) = p(p − 1). If a natural number n is a totient, it can be shown that n · 2k is a totient for all natural number k. There are infinitely many even nontotient numbers: indeed, there are infinitely many distinct primes p (such as 78557 and 271129, see Sierpinski number) such that all numbers of the form 2ap are nontotient, and every odd number has an even multiple which is a nontotient.


In statistics and psychometrics, reliability is the overall consistency of a measure. A measure is said to have a high reliability if it produces similar results under consistent conditions:"It is the characteristic of a set of test scores that relates to the amount of random error from the measurement process that might be embedded in the scores. Scores that are highly reliable are precise, reproducible, and consistent from one testing occasion to another.


The theorem also includes a converse: if two quantum states do commute, there is a method for broadcasting them: they must have a common basis of eigenstates diagonalizing them simultaneously, and the map that clones every state of this basis is a legitimate quantum operation, requiring only physical resources independent of the input state to implement—a completely positive map. A corollary is that there is a physical process capable of broadcasting every state in some set of quantum states if, and only if, every pair of states in the set commutes. This broadcasting map, which works in the commuting case, produces an overall state in which the two copies are perfectly correlated in their eigenbasis. Remarkably, the theorem does not hold if more than one copy of the initial state is provided: for example, broadcasting six copies starting from four copies of the original state is allowed, even if the states are drawn from a non-commuting set. The purity of the state can even be increased in the process, a phenomenon known as superbroadcasting.


In the United States, in the context of stockholder lawsuits, typically relating to the sale or merger of a public company, the Delaware Court of Chancery has required sufficient disclosures be made to a board of directors and shareholders to “provide a balanced, truthful account of all matters” and said “When a document ventures into certain subjects, it must do so in a manner that is materially complete and unbiased by the omission of material facts.” In a Memorandum Opinion in the CheckFree/Fiserv merger Chancellor Chandler underlined that the earlier In re Pure Resources Court had established the proper frame of analysis for disclosure of financial data: “tockholders are entitled to a fair summary of the substantive work performed by the investment bankers upon whose advice the recommendations of their board as to how to vote on a merger or tender rely.” According to the certification hypothesis fairness opinions may also serve the interest of the shareholders by mitigating informational asymmetries in corporate transactions. == References ==


In mathematics, a generating set Γ of a module M over a ring R is a subset of M such that the smallest submodule of M containing Γ is M itself (the smallest submodule containing a subset is the intersection of all submodules containing the set). The set Γ is then said to generate M. For example, the ring R is generated by the identity element 1 as a left R-module over itself. If there is a finite generating set, then a module is said to be finitely generated. This applies to ideals, which are the submodules of the ring itself.


The pseudo code stores the prefix sum in variable x {\displaystyle x} and the sum over all nodes in a sub cube in variable σ {\displaystyle \sigma } . This makes it possible for all nodes in 1-sub cube to receive the sum over the 0-sub cube in every step. This results in a factor of log  p {\displaystyle \log p} for T start {\displaystyle T_{\text{start}}} and a factor of n log  p {\displaystyle n\log p} for T byte {\displaystyle T_{\text{byte}}}: T ( n , p ) = ( T start + n T byte ) log  p {\displaystyle T(n,p)=(T_{\text{start}}+nT_{\text{byte}})\log p} .


RMO and PowerPC allow reordering of reads to the same location. These models violate sequential order in examples A and B. An additional relaxation allowed in these models is that memory operations following a read operation can be overlapped and reordered with respect to the read. Alpha and RMO allow a read to return the value of another processor's early write. From a programmer's perspective these models must maintain the illusion of write atomicity even though they allow the processor to read its own write early.


Dialling 123 from a few mobile services, such as O2, also obtains a speaking clock service. The Giffgaff network uses the same service as O2. The service is not available on the 3 mobile telephone network, as they use 123 as the number for their voicemail services.


In mathematics, the Pocklington–Lehmer primality test is a primality test devised by Henry Cabourn Pocklington and Derrick Henry Lehmer. The test uses a partial factorization of N − 1 {\displaystyle N-1} to prove that an integer N {\displaystyle N} is prime. It produces a primality certificate to be found with less effort than the Lucas primality test, which requires the full factorization of N − 1 {\displaystyle N-1} .


In this way, the first path will be drawn as an x-monotone polyline, a type of curve that is automatically non-self-crossing, and the second path will similarly be drawn as a y-monotone polyline. This type of drawing places the vertices in an integer lattice of dimensions linear in the graph sizes. Similarly defined layouts also work, with larger but still linear grid sizes, when both graphs are caterpillars or when both are cycle graphs.


After returning from these recursive calls, vertex 2 is added to X and removed from P. The iteration of the inner loop of the algorithm for v = 4 makes a recursive call to the algorithm with R = {4}, P = {3,5,6}, and X = Ø (although vertex 2 belongs to the set X in the outer call to the algorithm, it is not a neighbor of v and is excluded from the subset of X passed to the recursive call). This recursive call will end up making three second-level recursive calls to the algorithm that report the three cliques {3,4}, {4,5}, and {4,6}. Then, vertex 4 is added to X and removed from P. In the third and final iteration of the inner loop of the algorithm, for v = 6, there is a recursive call to the algorithm with R = {6}, P = Ø, and X = {4}.


There is also generally an expectation that there be a close resemblance between the real world and the features of the model in an ontology.In theory, an ontology is a "formal, explicit specification of a shared conceptualisation". An ontology renders shared vocabulary and taxonomy which models a domain with the definition of objects and/or concepts and their properties and relations.Ontologies are the structural frameworks for organizing information and are used in artificial intelligence, the Semantic Web, systems engineering, software engineering, biomedical informatics, library science, enterprise bookmarking, and information architecture as a form of knowledge representation about the world or some part of it. The creation of domain ontologies is also essential to the definition and use of an enterprise architecture framework.


In the United States, housing segregation is the practice of denying African Americans and other minority groups equal access to housing through the process of misinformation, denial of realty and financing services, and racial steering. Housing policy in the United States has influenced housing segregation trends throughout history. Key legislation include the National Housing Act of 1934, the G.I. Bill, and the Fair Housing Act.


This has enabled download speeds in excess of 100 Mbit/s, over distances up to 35 miles (56 km) from the transmission site. In the early days of MMDS, it was known as "wireless cable" and was used in a variety of investment scams that still surface today. Frequent solicitations of wireless cable fraud schemes were often heard on talk radio shows like The Sonny Bloch Show in the mid-1990s.Several US telephone companies attempted television services via this system in the mid-1990s – the Tele-TV venture of Bell Atlantic, NYNEX and Pacific Bell; and the rival Americast consortium of Ameritech, BellSouth, SBC, SNET and GTE.


In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance. A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.


An IBM MPIC was also used in the RS/6000 7046 Model B50.The Apple Hydra Mac I/O (MIO) chip (from the 1990s classic Mac OS era) implemented a MPIC alongside a SCSI controller, ADB controller, GeoPort controller, and timers. The Apple implementation of "Open PIC" (as the Apple documentation of this era spells it) in their first MIO chip for the Common Hardware Reference Platform was based on version 1.2 of the register specification and supported up to two processors and up to 20 interrupt sources. A MPIC was also incorporated in the newer K2 I/O controller used in the Power Mac G5s.Freescale also uses a MPIC ("compatible with the Open PIC") on all its PowerQUICC and QorIQ processors. The Linux Kernel-based Virtual Machine (KVM) supports a virtualized MPIC with up to 256 interrupts, based on the Freescale variants.


In programming language theory, subtyping (also subtype polymorphism or inclusion polymorphism) is a form of type polymorphism in which a subtype is a datatype that is related to another datatype (the supertype) by some notion of substitutability, meaning that program elements, typically subroutines or functions, written to operate on elements of the supertype can also operate on elements of the subtype. If S is a subtype of T, the subtyping relation (written as S <: T, S ⊑ T, or S ≤: T ) means that any term of type S can safely be used in any context where a term of type T is expected. The precise semantics of subtyping here crucially depends on the particulars of how "safely be used" and "any context" are defined by a given type formalism or programming language. The type system of a programming language essentially defines its own subtyping relation, which may well be trivial, should the language support no (or very little) conversion mechanisms.


Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countable sets, satisfies the first-order sentence that intuitively states "there are uncountable sets". A mathematical explanation of the paradox, showing that it is not a contradiction in mathematics, was given by Skolem (1922). Skolem's work was harshly received by Ernst Zermelo, who argued against the limitations of first-order logic, but the result quickly came to be accepted by the mathematical community.


There are also several methods to approximate a shortest addition chain, and which often require fewer multiplications than binary exponentiation; binary exponentiation itself is a suboptimal addition-chain algorithm. The optimal algorithm choice depends on the context (such as the relative cost of the multiplication and the number of times a given exponent is re-used).The problem of finding the shortest addition chain cannot be solved by dynamic programming, because it does not satisfy the assumption of optimal substructure. That is, it is not sufficient to decompose the power into smaller powers, each of which is computed minimally, since the addition chains for the smaller powers may be related (to share computations). For example, in the shortest addition chain for a15 above, the subproblem for a6 must be computed as (a3)2 since a3 is re-used (as opposed to, say, a6 = a2(a2)2, which also requires three multiplies).


The category of pointed sets and pointed maps has both products and coproducts, but it is not a distributive category. It is also an example of a category where 0 × A {\displaystyle 0\times A} is not isomorphic to 0 {\displaystyle 0} .Many algebraic structures are pointed sets in a rather trivial way. For example, groups are pointed sets by choosing the identity element as the basepoint, so that group homomorphisms are point-preserving maps. : 24 This observation can be restated in category theoretic terms as the existence of a forgetful functor from groups to pointed sets. : 582 A pointed set may be seen as a pointed space under the discrete topology or as a vector space over the field with one element.As "rooted set" the notion naturally appears in the study of antimatroids and transportation polytopes.


The range of colors, sizes and shapes is far wider than in those grown or sold in Africa, Europe or the Americas. Southeast Asian languages do not make the distinction between "bananas" and "plantains" that is made in English (and Spanish). Thus both Cavendish cultivars, the classic yellow dessert bananas, and Saba cultivars, used mainly for cooking, are called pisang in Malaysia and Indonesia, kluai in Thailand and chuối in Vietnam.


The treaty stipulated that the use of cryptography with short key-lengths (56-bit for symmetric encryption, 512-bit for RSA) would no longer be export-controlled. Cryptography exports from the US became less strictly regulated as a consequence of a major relaxation in 2000; there are no longer very many restrictions on key sizes in US-exported mass-market software.


Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space. Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.


For how long? When do they regress to already seen words?


While the algorithmic complexity implies a deterministic description of an object (it measures the information content of an individual sequence), the statistical complexity, like forecasting complexity, implies a statistical description, and refers to an ensemble of sequences generated by a certain source. Formally, the statistical complexity reconstructs a minimal model comprising the collection of all histories sharing a similar probabilistic future, and measures the entropy of the probability distribution of the states within this model. It is a computable and observer-independent measure based only on the internal dynamics of the system, and has been used in studies of emergence and self-organization.


In music using the twelve tone technique, combinatoriality is a quality shared by twelve-tone tone rows whereby each section of a row and a proportionate number of its transformations combine to form aggregates (all twelve tones). Much as the pitches of an aggregate created by a tone row do not need to occur simultaneously, the pitches of a combinatorially created aggregate need not occur simultaneously. Arnold Schoenberg, creator of the twelve-tone technique, often combined P-0/I-5 to create "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively. "Combinatoriality is a side effect of derived rows, where the initial segment or set may be combined with its transformations (T,R,I,RI) to create an entire row.


In the compact disc system, cross-interleaved Reed–Solomon code (CIRC) provides error detection and error correction. CIRC adds to every three data bytes one redundant parity byte.


In the Japanese kantenji braille, the standard 8-dot Braille patterns 2567, 12567, 24567, and 124567 are the patterns related to Braille pattern dots-1345, since the two additional dots of kantenji patterns 01345, 13457, and 013457 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


{\displaystyle \sum _{v}\left({\frac {d(v)}{|E|}}{\frac {1}{2}}\right)d(v)={\frac {\sum _{v}d(v)^{2}}{2|E|}}.} We know from the definition of variance that ∑ v d ( v ) 2 | V | = μ 2 + σ 2 , {\displaystyle {\frac {\sum _{v}d(v)^{2}}{|V|}}=\mu ^{2}+\sigma ^{2},} where σ 2 {\displaystyle \sigma ^{2}} is the variance of the degrees in the graph. This allows us to compute the desired expected value as ∑ v d ( v ) 2 2 | E | = | V | 2 | E | ( μ 2 + σ 2 ) = μ 2 + σ 2 μ = μ + σ 2 μ .


In most programming languages, the simplest method to convert a floating point number to an integer does not do floor or ceiling, but truncation. The reason for this is historical, as the first machines used ones' complement and truncation was simpler to implement (floor is simpler in two's complement). FORTRAN was defined to require this behavior and thus almost all processors implement conversion this way. Some consider this to be an unfortunate historical design decision that has led to bugs handling negative offsets and graphics on the negative side of the origin.A bit-wise right-shift of a signed integer x {\displaystyle x} by n {\displaystyle n} is the same as ⌊ x 2 n ⌋ {\displaystyle \left\lfloor {\frac {x}{2^{n}}}\right\rfloor } .


Moreover, time-sharing was new ground. Many of the concepts involved, such as virtual memory, remained unproven. For example: At the time, nobody could explain why the troubled Manchester/Ferranti Atlas virtual memory "didn't work".


In the context of economics, for example, this is usually economic cost or regret. In classification, it is the penalty for an incorrect classification of an example. In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramér in the 1920s. In optimal control, the loss is the penalty for failing to achieve a desired value. In financial risk management, the function is mapped to a monetary loss.


In mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as a + b c d , {\displaystyle {a+b{\sqrt {c}} \over d},} for integers a, b, c, d; with b, c and d non-zero, and with c square-free. When c is positive, we get real quadratic irrational numbers, while a negative c gives complex quadratic irrational numbers which are not real numbers.


These often arise when discretising one-dimensional problems.Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.


An early version was demonstrated to a special committee of the society, the Recording Group, and this was later presented to the AIA. This was favourably received and the database has been in development ever since. In 2002 the society received an award from the AIA.


{\displaystyle 480=2!\cdot 2!\cdot 5!} . Every tree has a number of symmetries that is a Jordan–Pólya number, and every Jordan–Pólya number arises in this way as the order of an automorphism group of a tree. These numbers are named after Camille Jordan and George Pólya, who both wrote about them in the context of symmetries of trees.These numbers grow more quickly than polynomials but more slowly than exponentials. As well as in the symmetries of trees, they arise as the numbers of transitive orientations of comparability graphs and in the problem of finding factorials that can be represented as products of smaller factorials.


The paper is also darkened in proportion to its exposure to light, so a second reversal results which restores light and dark to their normal order.Negatives were once commonly made on a thin sheet of glass rather than a plastic film, and some of the earliest negatives were made on paper.Transparent positive prints can be made by printing a negative onto special positive film, as is done to make traditional motion picture film prints for use in theaters. Some films used in cameras are designed to be developed by reversal processing, which produces the final positive, instead of a negative, on the original film. Positives on film or glass are known as transparencies or diapositives, and if mounted in small frames designed for use in a slide projector or magnifying viewer they are commonly called slides.


The image on the right shows the permutohedron of order 4, which is the truncated octahedron. Its vertices are the 24 permutations of (1, 2, 3, 4). Parallel edges have the same edge color. The 6 edge colors correspond to the 6 possible transpositions of 4 elements, i.e. they indicate in which two places the connected permutations differ. (E.g. red edges connect permutations that differ in the last two places.)


Once researchers determine their preferred statistical model, different forms of regression analysis provide tools to estimate the parameters β {\displaystyle \beta } . For example, least squares (including its most common variant, ordinary least squares) finds the value of β {\displaystyle \beta } that minimizes the sum of squared errors ∑ i ( Y i − f ( X i , β ) ) 2 {\displaystyle \sum _{i}(Y_{i}-f(X_{i},\beta ))^{2}} . A given regression method will ultimately provide an estimate of β {\displaystyle \beta } , usually denoted β ^ {\displaystyle {\hat {\beta }}} to distinguish the estimate from the true (unknown) parameter value that generated the data.


Another category of bug is called a race condition that may occur when programs have multiple components executing at the same time. If the components interact in a different order than the developer intended, they could interfere with each other and stop the program from completing its tasks. These bugs may be difficult to detect or anticipate, since they may not occur during every execution of a program.


In the fair cake-cutting problem, classic allocation rules such as divide and choose are not RM. Several rules are known to be RM: When the pieces may be disconnected, any allocation rule maximizing a concave welfare function of the absolute (not normalized) utilities is RM. In particular, the Nash-optimal rule, absolute-leximin rule and absolute-utilitarian rule are all RM. However, if the maximization uses the relative utilities (utilities divided by total cake value) then most of these rules are not RM; the only one that remains RM is the Nash-optimal rule.


The most common and challenging criticism to metalinguistic description theories was put forth by Kripke himself: they seem to be an ad hoc explanation of a single linguistic phenomenon. Why should there be a metalinguistic theory for proper nouns (like names) but not for common nouns, count nouns, verbs, predicates, indexicals and other parts of speech. Another recent approach is two-dimensional semantics. The motivations for this approach are rather different from those that inspired other forms of descriptivism, however. Two-dimensional approaches are usually motivated by a sense of dissatisfaction with the causal theorist explanation of how it is that a single proposition can be both necessary and a posteriori or contingent and a priori.


The log-logistic distribution is the probability distribution of a random variable whose logarithm has a logistic distribution. It is similar in shape to the log-normal distribution but has heavier tails. Unlike the log-normal, its cumulative distribution function can be written in closed form.


Since there can be various levels of data classification and user clearances, this implies a quantified scale for robustness. For example, more robustness is indicated for system environments containing classified Top Secret information and uncleared users than for one with Secret information and users cleared to at least Confidential. To promote consistency and eliminate subjectivity in degrees of robustness, an extensive scientific analysis and risk assessment of the topic produced a landmark benchmark standardization quantifying security robustness capabilities of systems and mapping them to the degrees of trust warranted for various security environments.


In the case where all the matrices are Hermitian positive definite and all the eigenvalues are distinct, λ i = λ 0 i + x 0 i ⊤ ( δ K − λ 0 i δ M ) x 0 i x i = x 0 i ( 1 − 1 2 x 0 i ⊤ δ M x 0 i ) + ∑ j = 1 j ≠ i N x 0 j ⊤ ( δ K − λ 0 i δ M ) x 0 i λ 0 i − λ 0 j x 0 j {\displaystyle {\begin{aligned}\lambda _{i}&=\lambda _{0i}+\mathbf {x} _{0i}^{\top }\left(\delta \mathbf {K} -\lambda _{0i}\delta \mathbf {M} \right)\mathbf {x} _{0i}\\\mathbf {x} _{i}&=\mathbf {x} _{0i}\left(1-{\tfrac {1}{2}}\mathbf {x} _{0i}^{\top }\delta \mathbf {M} \mathbf {x} _{0i}\right)+\sum _{j=1 \atop j\neq i}^{N}{\frac {\mathbf {x} _{0j}^{\top }\left(\delta \mathbf {K} -\lambda _{0i}\delta \mathbf {M} \right)\mathbf {x} _{0i}}{\lambda _{0i}-\lambda _{0j}}}\mathbf {x} _{0j}\end{aligned}}} for infinitesimal δ K {\displaystyle \delta \mathbf {K} } and δ M {\displaystyle \delta \mathbf {M} } (the higher order terms in (3) being neglected). So far, we have not proved that these higher order terms may be neglected. This point may be derived using the implicit function theorem; in next section, we summarize the use of this theorem in order to obtain a first order expansion.


In probability a quasi-stationary distribution is a random process that admits one or several absorbing states that are reached almost surely, but is initially distributed such that it can evolve for a long time without reaching it. The most common example is the evolution of a population: the only equilibrium is when there is no one left, but if we model the number of people it is likely to remain stable for a long period of time before it eventually collapses.


In statistical research, it is an aim to measure the magnitudes of phenomena. For this purpose, phenomena have to be grouped and categorized, so that distinct and discrete counting units can be defined. It must be possible to allocate all observations to mutually exclusive categories, so that they are properly quantifiable.


In statistics, the mean absolute scaled error (MASE) is a measure of the accuracy of forecasts. It is the mean absolute error of the forecast values, divided by the mean absolute error of the in-sample one-step naive forecast. It was proposed in 2005 by statistician Rob J. Hyndman and Professor of Decision Sciences Anne B. Koehler, who described it as a "generally applicable measurement of forecast accuracy without the problems seen in the other measurements." The mean absolute scaled error has favorable properties when compared to other methods for calculating forecast errors, such as root-mean-square-deviation, and is therefore recommended for determining comparative accuracy of forecasts.


This can be viewed as a special case of the result relating two minimal Stinespring representations. Alternatively, there is an isometry scalar matrix {uij}ij ∈ Cnm × nm such that A i = ∑ j = 1 u i j B j . {\displaystyle A_{i}=\sum _{j=1}u_{ij}B_{j}.} This follows from the fact that for two square matrices M and N, M M* = N N* if and only if M = N U for some unitary U.


A measurement system can be accurate but not precise, precise but not accurate, neither, or both. For example, if an experiment contains a systematic error, then increasing the sample size generally increases precision but does not improve accuracy. The result would be a consistent yet inaccurate string of results from the flawed experiment.


In most of these systems, the hardware is not open to run general purpose applications. Bare machine computing originated with the application object (AO) concept invented by Karne at Towson University. It evolved over the years into dispersed operating systems (DOSC), and eventually into the BMC paradigm.


In category theory, a map may refer to a morphism. The term transformation can be used interchangeably, but transformation often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory.


In the foundations of mathematics, formalism is associated with a certain rigorous mathematical method: see formal system. In common usage, a formalism means the out-turn of the effort towards formalisation of a given limited area. In other words, matters can be formally discussed once captured in a formal system, or commonly enough within something formalisable with claims to be one. Complete formalisation is in the domain of computer science. Formalism also more precisely refers to a certain school in the philosophy of mathematics, stressing axiomatic proofs through theorems, specifically associated with David Hilbert. In the philosophy of mathematics, therefore, a formalist is a person who belongs to the school of formalism, which is a certain mathematical-philosophical doctrine descending from Hilbert.


The original, strong, form of the conjecture with exponent 1/2 has never been disproved, although it is no longer believed to be true and the term Hall's conjecture now generally means the version with the ε in it. For example, in 1998, Noam Elkies found the example 4478849284284020423079182 - 58538865167812233 = -1641843, for which compatibility with Hall's conjecture would require C to be less than .0214 ≈ 1/50, so roughly 10 times smaller than the original choice of 1/5 that Hall suggested. The weak form of Hall's conjecture would follow from the ABC conjecture.


In mathematics, for given real numbers a and b, the logarithm logb a is a number x such that bx = a. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logb a is an integer k such that bk = a. In number theory, the more commonly used term is index: we can write x = indr a (mod m) (read "the index of a to the base r modulo m") for rx ≡ a (mod m) if r is a primitive root of m and gcd(a,m) = 1. Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general. Several important algorithms in public-key cryptography, such as ElGamal, base their security on the assumption that the discrete logarithm problem (DLP) over carefully chosen groups has no efficient solution.


In relationally synthesized Odia words, base morphemes (root words) join with bound morphemes to express grammatical function. The Odia language has a tendency for commonly used words to have a 2:1 morpheme-word ratio i.e. on average; there are two morphemes in a single word. Because of this tendency, Odia is said to "possess morphology" since almost each used word has an internal compositional structure in terms of morphemes. In the Odia language, generally, separate words are used to express syntactic relationships which imparts an isolating tendency, while using inflectional morphology could have made the language more synthetic.


In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number a is equal to itself (reflexive).


In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, see Relative concreteness below). This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the category of topological spaces and the category of groups, and trivially also the category of sets itself. On the other hand, the homotopy category of topological spaces is not concretizable, i.e. it does not admit a faithful functor to the category of sets. A concrete category, when defined without reference to the notion of a category, consists of a class of objects, each equipped with an underlying set; and for any two objects A and B a set of functions, called morphisms, from the underlying set of A to the underlying set of B. Furthermore, for every object A, the identity function on the underlying set of A must be a morphism from A to A, and the composition of a morphism from A to B followed by a morphism from B to C must be a morphism from A to C.


In object-oriented programming, users can inherit the properties and behaviour of a superclass in subclasses. A subclass can override methods of its superclass, substituting its own implementation of the method for the superclass's implementation. Sometimes the overriding method will completely replace the corresponding functionality in the superclass, while in other cases the superclass's method must still be called from the overriding method. Therefore, most programming languages require that an overriding method must explicitly call the overridden method on the superclass for it to be executed.


In the absence of intervention in the foreign exchange market by national government authorities, the exchange rate of a country's currency is determined, in general, by market forces of supply and demand at a point in time. Government authorities may intervene in the market from time to time to achieve specific policy objectives, such as maintaining its balance of trade or to give its exporters a competitive advantage in international trade.


{\displaystyle \mathbf {y} =\mathbf {X} \mathbf {\beta } +\mathbf {\varepsilon } ,\qquad \operatorname {E} =0,\ \operatorname {Cov} =\mathbf {\Omega } .} Here β ∈ R k {\displaystyle \beta \in \mathbb {R} ^{k}} is a vector of unknown constants (known as “regression coefficients”) that must be estimated from the data. Suppose b {\displaystyle \mathbf {b} } is a candidate estimate for β {\displaystyle \mathbf {\beta } } .


In statically typed languages (such as C++ and Java), the term generic functions refers to a mechanism for compile-time polymorphism (static dispatch), specifically parametric polymorphism. These are functions defined with TypeParameters, intended to be resolved with compile time type information. The compiler uses these types to instantiate suitable versions, resolving any function overloading appropriately.


In queueing theory, a discipline within the mathematical theory of probability, a fork–join queue is a queue where incoming jobs are split on arrival for service by numerous servers and joined before departure. The model is often used for parallel computations or systems where products need to be obtained simultaneously from different suppliers (in a warehouse or manufacturing setting). : 78–80 The key quantity of interest in this model is usually the time taken to service a complete job. The model has been described as a "key model for the performance analysis of parallel and distributed systems." Few analytical results exist for fork–join queues, but various approximations are known. The situation where jobs arrive according to a Poisson process and service times are exponentially distributed is sometimes referred to as a Flatto–Hahn–Wright model or FHW model.


In mathematics and computer science, a shortest-path tree rooted at a vertex v of a connected, undirected graph G is a spanning tree T of G, such that the path distance from root v to any other vertex u in T is the shortest path distance from v to u in G. In connected graphs where shortest paths are well-defined (i.e. where there are no negative-length cycles), we may construct a shortest-path tree using the following algorithm: Compute dist(u), the shortest-path distance from root v to vertex u in G using Dijkstra's algorithm or Bellman–Ford algorithm. For all non-root vertices u, we can assign to u a parent vertex pu such that pu is connected to u, and that dist(pu) + edge_dist(pu,u) = dist(u). In case multiple choices for pu exist, choose pu for which there exists a shortest path from v to pu with as few edges as possible; this tie-breaking rule is needed to prevent loops when there exist zero-length cycles.


These include, among others: the "is greater than", "is equal to", and "divides" relations in arithmetic; the "is congruent to" relation in geometry; the "is adjacent to" relation in graph theory; the "is orthogonal to" relation in linear algebra.A function may be defined as a special kind of binary relation. Binary relations are also heavily used in computer science.


In order to identify which transitions must be made, the protocol detects sharing using a special bus line named CopiesExist. All other caches snoop all memory operations and raise the CopiesExist(C) if they detect a "snoop hit", i.e. if they have a copy of the data in their own cache. An arrow that goes from nowhere to a state represents a newly loaded block.


The weights used are usually based on the aggregate input shares either factor earns. This ratio is often quoted as: 33% return to capital and 67% return to labor (in Western nations).In a growing economy, capital is accumulated faster than people are born, so the denominator in the growth function under the MFP calculation is growing faster than in the ALP calculation. Hence, MFP growth is almost always lower than ALP growth. (Therefore, measuring in ALP terms increases the apparent capital deepening effect.) MFP is measured by the "Solow residual", not ALP.


The problem is to approximate the integral of a function f as the average of the function evaluated at a set of points x1, ..., xN: ∫  s f ( u ) d u ≈ 1 N ∑ i = 1 N f ( x i ) . {\displaystyle \int _{^{s}}f(u)\,{\rm {d}}u\approx {\frac {1}{N}}\,\sum _{i=1}^{N}f(x_{i}).} Since we are integrating over the s-dimensional unit cube, each xi is a vector of s elements.


Values also have types, which can be checked and queried at runtime. Typing of variables also allows polymorphisms to be resolved at compile time. However, this is different from the polymorphism used in object-oriented function calls (referred to as virtual functions in C++) which resolves the call based on the value type as opposed to the supertypes the variable is allowed to have.


In the following soft clustering example, the reference vector Y {\displaystyle Y\,} contains sample categories and the joint probability p ( X , Y ) {\displaystyle p(X,Y)\,} is assumed known. A soft cluster c k {\displaystyle c_{k}\,} is defined by its probability distribution over the data samples x i: p ( c k | x i ) {\displaystyle x_{i}:\,\,\,p(c_{k}|x_{i})} . Tishby et al. presented the following iterative set of equations to determine the clusters which are ultimately a generalization of the Blahut-Arimoto algorithm, developed in rate distortion theory. The application of this type of algorithm in neural networks appears to originate in entropy arguments arising in the application of Gibbs Distributions in deterministic annealing.


So we must sum over all these possibilities: Pr ( Median = v ) = ∑ i = 0 n ∑ k = 0 n N ! i ! ( N − i − k ) !


Most computers organize files into hierarchies using folders, directories, or catalogs. The concept is the same irrespective of the terminology used.


In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively. In mathematical notation, these facts can be expressed as follows, where Pr() is the probability function, Χ is an observation from a normally distributed random variable, μ (mu) is the mean of the distribution, and σ (sigma) is its standard deviation: The usefulness of this heuristic especially depends on the question under consideration. In the empirical sciences, the so-called three-sigma rule of thumb (or 3σ rule) expresses a conventional heuristic that nearly all values are taken to lie within three standard deviations of the mean, and thus it is empirically useful to treat 99.7% probability as near certainty.In the social sciences, a result may be considered "significant" if its confidence level is of the order of a two-sigma effect (95%), while in particle physics, there is a convention of a five-sigma effect (99.99994% confidence) being required to qualify as a discovery.


In the first version of the above-cited paper, the authors proved the asymptotic time complexity of the algorithm to be O ~ ( log  ( n ) 12 ) {\displaystyle {\tilde {O}}(\log(n)^{12})} (using Õ from big O notation)—the twelfth power of the number of digits in n times a factor that is polylogarithmic in the number of digits. However, this upper bound was rather loose; a widely-held conjecture about the distribution of the Sophie Germain primes would, if true, immediately cut the worst case down to O ~ ( log  ( n ) 6 ) {\displaystyle {\tilde {O}}(\log(n)^{6})} . In the months following the discovery, new variants appeared (Lenstra 2002, Pomerance 2002, Berrizbeitia 2002, Cheng 2003, Bernstein 2003a/b, Lenstra and Pomerance 2003), which improved the speed of computation greatly. Owing to the existence of the many variants, Crandall and Papadopoulos refer to the "AKS-class" of algorithms in their scientific paper "On the implementation of AKS-class primality tests", published in March 2003.


In the early years of knowledge-based systems the knowledge-bases were fairly small. The knowledge-bases that were meant to actually solve real problems rather than do proof of concept demonstrations needed to focus on well defined problems. So for example, not just medical diagnosis as a whole topic, but medical diagnosis of certain kinds of diseases. As knowledge-based technology scaled up, the need for larger knowledge bases and for modular knowledge bases that could communicate and integrate with each other became apparent.