Moreover, attempts to falsify a claim, by replicating an experiment, are hard and problematic for it involves tacit knowledge (i.e. unarticulated knowledge), matters of time and money and replication of exact similar conditions, which is hard. Tacit knowledge can never be fully articulated or translated into a set of rules. Some commentators have argued that Collins's "experimenter's regress" is foreshadowed by Sextus Empiricus' argument that "if we shall judge the intellects by the senses, and the senses by the intellect, this involves circular reasoning inasmuch as it is required that the intellects should be judged first in order that the intellects may be tested  we possess no means by which to judge objects" (quoted after Godin & Gingras 2002: 140). Others have extended Collins's argument to the cases of theoretical practice ("theoretician's regress"; Kennefick 2000) and computer simulation studies ("simulationist's regress"; Gelfert 2011; Tolk 2017).


In mathematics and computer science, symbolic-numeric computation is the use of software that combines symbolic and numeric methods to solve problems.


If T denotes the entry in the table for nonterminal A and terminal a, then T contains the rule A → w if and only if a is in Fi(w) or ε is in Fi(w) and a is in Fo(A).Equivalently: T contains the rule A → w for each a ∈ Fi(w)·Fo(A). If the table contains at most one rule in every one of its cells, then the parser will always know which rule it has to use and can therefore parse strings without backtracking. It is in precisely this case that the grammar is called an LL(1) grammar.


Depending on the context, the conditional expectation can be either a random variable or a function. The random variable is denoted E ( X ∣ Y ) {\displaystyle E(X\mid Y)} analogously to conditional probability. The function form is either denoted E ( X ∣ Y = y ) {\displaystyle E(X\mid Y=y)} or a separate function symbol such as f ( y ) {\displaystyle f(y)} is introduced with the meaning E ( X ∣ Y ) = f ( Y ) {\displaystyle E(X\mid Y)=f(Y)} .


In probability theory and statistics, the normal-Wishart distribution (or Gaussian-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and precision matrix (the inverse of the covariance matrix).


In the United States and Canada, caller ID information is sent to the called party by the telephone switch as an analog data stream (similar to data passed between two modems), using Bell 202 modulation between the first and second rings, while the telephone unit is still on hook. If the telephone call is answered too quickly after the first ring, caller ID information may not be transmitted to the recipient. Also, in the United States and Canada a caller may block the display of the number they are calling from by dialling *67 before dialling the phone number. This will not work when dialling an "800" number, where the receiver of the call pays for the call or when 911 emergency calls are made.


Non-transparent communication systems have one or both of the following problems: user data may be incorrectly interpreted as internal commands. For example, modems with a Time Independent Escape Sequence or 20th century Signaling System No. 5 and R2 signalling telephone systems, which occasionally incorrectly interpreted user data (from a "blue box") as commands. output "user data" may not always be the same as input user data.


In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" (in units such as shannons (bits), nats or hartleys) obtained about one random variable by observing the other random variable. The concept of mutual information is intimately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable. Not limited to real-valued random variables and linear dependence like the correlation coefficient, MI is more general and determines how different the joint distribution of the pair ( X , Y ) {\displaystyle (X,Y)} is from the product of the marginal distributions of X {\displaystyle X} and Y {\displaystyle Y} .


In parallel computer architectures, a systolic array is a homogeneous network of tightly coupled data processing units (DPUs) called cells or nodes. Each node or DPU independently computes a partial result as a function of the data received from its upstream neighbours, stores the result within itself and passes it downstream. Systolic arrays were first used in Colossus, which was an early computer used to break German Lorenz ciphers during World War II. Due to the classified nature of Colossus, they were independently invented or rediscovered by H. T. Kung and Charles Leiserson who described arrays for many dense linear algebra computations (matrix product, solving systems of linear equations, LU decomposition, etc.) for banded matrices.


For most common relations in mathematics, special symbols are introduced, like "<" for "is less than", and "|" for "is a nontrivial divisor of", and, most popular "=" for "is equal to". For example, "1<3", "1 is less than 3", and "(1,3) ∈ Rless" mean all the same; some authors also write "(1,3) ∈ (<)".


In mathematics, a Suslin tree is a tree of height ω1 such that every branch and every antichain is at most countable. They are named after Mikhail Yakovlevich Suslin. Every Suslin tree is an Aronszajn tree. The existence of a Suslin tree is independent of ZFC, and is equivalent to the existence of a Suslin line (shown by Kurepa (1935)) or a Suslin algebra.


In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent. The equation is named after Srinivasa Ramanujan, who conjectured that it has only five integer solutions, and after Trygve Nagell, who proved the conjecture. It implies non-existence of perfect binary codes with the minimum Hamming distance 5 or 6.


Cray decided that the 8600 would include four complete CPUs sharing the main memory. To improve overall throughput, the machine could operate in a special mode that sent a single instruction to all four processors with different data. This technique, today known as SIMD, reduced the total number of memory accesses because the instruction was only read once, instead of four times.


In mathematics, a pseudo-finite field F is an infinite model of the first-order theory of finite fields. This is equivalent to the condition that F is quasi-finite (perfect with a unique extension of every positive degree) and pseudo algebraically closed (every absolutely irreducible variety over F has a point defined over F). Every hyperfinite field is pseudo-finite and every pseudo-finite field is quasifinite. Every non-principal ultraproduct of finite fields is pseudo-finite. Pseudo-finite fields were introduced by Ax (1968).


In probabilistic combinatorics, the questions are of the following type: what is the probability of a certain property for a random discrete object, such as a random graph? For instance, what is the average number of triangles in a random graph? Probabilistic methods are also used to determine the existence of combinatorial objects with certain prescribed properties (for which explicit examples might be difficult to find) by observing that the probability of randomly selecting an object with those properties is greater than 0.


In project management, there are generally considered to be three constraints: time (sometimes schedule), cost (sometimes budget), and scope. (Quality is often added as a fourth constraint---represented as the middle of a triangle.) The assumption is that a change in one constraint will affect the others.Without timeboxing, projects usually work to a fixed scope, in which case when it becomes clear that some deliverables cannot be completed within the planned timescales, either the deadline has to be extended (to allow more time to complete the fixed scope) or more people are involved (to complete the fixed scope in the same time). Often both happen, resulting in delayed delivery, increased costs, and often reduced quality (as per The Mythical Man-Month principle). With timeboxing, the deadline is fixed, meaning that the scope would have to be reduced. As this means organizations have to focus on completing the most important deliverables first, timeboxing often goes hand-in-hand with a scheme for prioritizing of deliverables (such as with the MoSCoW method).


In mathematics, the quadratic bottleneck assignment problem (QBAP) is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research, from the category of the facilities location problems.It is related to the quadratic assignment problem in the same way as the linear bottleneck assignment problem is related to the linear assignment problem, the "sum" is replaced with "max" in the objective function. The problem models the following real-life problem: There are a set of n facilities and a set of n locations. For each pair of locations, a distance is specified and for each pair of facilities a weight or flow is specified (e.g., the amount of supplies transported between the two facilities). The problem is to assign all facilities to different locations with the goal of minimizing the maximum of the distances multiplied by the corresponding flows.


The Ping-Pong scheme described in Transaction Processing eliminates this problem by alternately writing the contents of said (logical) last page to two different physical pages inside the log file (the actual last page i and its empty successor i+1). Once said logical log page is no longer the last page (i.e. it is completely filled with log data), it is written one last time to the regular physical position (i) inside the log file. This scheme requires the usage of time stamps for each page in order to distinguish the most recent version of the logical last page one from its predecessor.


In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form x 2 ≡ a mod p {\displaystyle x^{2}\equiv a{\bmod {p}}} for an odd prime p {\displaystyle p} ; that is, to determine the "perfect squares" modulo p {\displaystyle p} . However, this is a non-constructive result: it gives no help at all for finding a specific solution; for this, other methods are required. For example, in the case p ≡ 3 mod 4 {\displaystyle p\equiv 3{\bmod {4}}} using Euler's criterion one can give an explicit formula for the "square roots" modulo p {\displaystyle p} of a quadratic residue a {\displaystyle a} , namely, ± a p + 1 4 {\displaystyle \pm a^{\frac {p+1}{4}}} indeed, ( ± a p + 1 4 ) 2 = a p + 1 2 = a ⋅ a p − 1 2 ≡ a ( a p ) = a mod p .


However, the program will neither spin nor poll; thus terms like wait, hang or yield may also convey the behaviour; also in the context that it will not block other independent processes from running.) Examples (c is a variable): keyboard ?


. , x k − x k − 1 ) {\displaystyle U=(x_{2}-x_{1},x_{3}-x_{2},...,x_{k}-x_{k-1})} whose columns are the k − 1 {\displaystyle k-1} differences. Then, one computes the vector c = − U + ( x k + 1 − x k ) {\displaystyle c=-U^{+}(x_{k+1}-x_{k})} where U + {\displaystyle U^{+}} denotes the Moore–Penrose pseudoinverse of U {\displaystyle U} .


In monotone drawing of graphs, in 2-visibility representation of graphs.


In social network analysis, the co-stardom network represents the collaboration graph of film actors i.e. movie stars. The co-stardom network can be represented by an undirected graph of nodes and links. Nodes correspond to the movie star actors and two nodes are linked if they co-starred (performed) in the same movie. The links are un-directed, and can be weighted or not depending on the goals of study.


Using the theory of elliptic functions, it can be shown that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane. The torus is also an abelian group, and this correspondence is also a group isomorphism. Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in Andrew Wiles's proof of Fermat's Last Theorem.


A proactive fault management method to deal with the software aging incident is software rejuvenation. This method can be classified as an environment diversity technique that usually is implemented through software rejuvenation agents (SRA). The phenomenon was first identified by David Parnas, in an essay that explored what to do about it:"Programs, like people, get old.


Therefore, whenever an ambiguity is possible, the synonym used for "recursive language" is Turing-decidable language, rather than simply decidable. The class of all recursive languages is often called R, although this name is also used for the class RP. This type of language was not defined in the Chomsky hierarchy of (Chomsky 1959). All recursive languages are also recursively enumerable. All regular, context-free and context-sensitive languages are recursive.


In mathematical analysis and computer science, functions which are Z-order, Lebesgue curve, Morton space-filling curve, Morton order or Morton code map multidimensional data to one dimension while preserving locality of the data points. It is named in France after Henri Lebesgue, who studied it in 1904, and named in the United States after Guy Macdonald Morton, who first applied the order to file sequencing in 1966. The z-value of a point in multidimensions is simply calculated by interleaving the binary representations of its coordinate values. Once the data are sorted into this ordering, any one-dimensional data structure can be used, such as simple one dimensional arrays, binary search trees, B-trees, skip lists or (with low significant bits truncated) hash tables. The resulting ordering can equivalently be described as the order one would get from a depth-first traversal of a quadtree or octree.


This value is the average of the values of k nearest neighbors. If k = 1, then the output is simply assigned to the value of that single nearest neighbor.k-NN is a type of classification where the function is only approximated locally and all computation is deferred until function evaluation. Since this algorithm relies on distance for classification, if the features represent different physical units or come in vastly different scales then normalizing the training data can improve its accuracy dramatically.Both for classification and regression, a useful technique can be to assign weights to the contributions of the neighbors, so that the nearer neighbors contribute more to the average than the more distant ones.


Structure descriptions are presented in terms of better known types of semigroups. The best known type of semigroup is the group. A (necessarily incomplete) list of various special classes of semigroups is presented below. To the extent possible the defining properties are formulated in terms of the binary operations in the semigroups. The references point to the locations from where the defining properties are sourced.


However, fundamental process ontologies are becoming more important in recent times, because the progress in the discovery of the foundations of physics spurred the development of a basic concept able to integrate such boundary notions as "energy," "object", and those of the physical dimensions of space and time. In computer science, a process ontology is a description of the components and their relationships that make up a process. A formal process ontology is an ontology in the knowledge domain of operations.


Various object-oriented programming languages offer similar facilities for abstraction, all to support a general strategy of polymorphism in object-oriented programming, which includes the substitution of one type for another in the same or similar role. Although not as generally supported, a configuration or image or package may predetermine a great many of these bindings at compile-time, link-time, or loadtime. This would leave only a minimum of such bindings to change at run-time.


In mathematics the one-sided limit x → a+ means x approaches a from the right (i.e., right-sided limit), and x → a− means x approaches a from the left (i.e., left-sided limit). For example, 1/x → + ∞ {\displaystyle \infty } as x → 0+ but 1/x → − ∞ {\displaystyle \infty } as x → 0−.


Define: S = { i: 1 ⩽ i ⩽ N , c i 1 ≠ c i 2 } . {\displaystyle S=\left\{i\ :\ 1\leqslant i\leqslant N,c_{i}^{1}\neq c_{i}^{2}\right\}.}


In telecommunications, an area code split is the practice of introducing a new telephone area code by geographically dividing an existing numbering plan area (NPA), and assigning area codes to the resulting divisions, but retaining the existing area code only for one of the divisions. The purpose of this practice is to provide more central office prefixes, and therefore more telephone numbers, in an area with high demand for telephone services, and prevent a shortage of telephone numbers. An increasing demand for telephone numbers has existed since the development of automatic telephony in the early 20th century, but was spurred especially since the 1990s, with the proliferation of fax machines, pager systems, mobile telephones, computer modems, and finally smart phones. When an area code split is implemented, the telephone numbers in the affected area are typically changed to a new area code only, but this still requires the printing of new stationery, advertisements, and signage for many customers, and the dissemination of the new area code to family, friends, and customers.


Return x n {\displaystyle \mathbf {x} _{n}} as the minimizing position and f ( x n ) {\displaystyle f(\mathbf {x} _{n})} as the function minimum.To assure good behavior, it is necessary that some conditions must be satisfied by p n {\displaystyle \mathbf {p} _{n}} . Roughly speaking p n {\displaystyle \mathbf {p} _{n}} should not be too far away from ∇ f ( x n ) {\displaystyle \nabla f(\mathbf {x} _{n})} . A precise version is as follows (see e.g. Bertsekas (2016)).


An arrow → in column next signifies that the rebalancing is complete with this step. If the column after determines exactly one case, this case is given as the subsequent one, otherwise there are question marks. The loop is contained in the sections "Insert case I1" and "Insert case I2", where in case I2 the problem of rebalancing is escalated Δ h = 2 {\displaystyle \Delta h=2} tree levels or 1 black level higher in the tree, in that the grandfather G becomes the new current node N. So it takes maximally h 2 {\displaystyle {\tfrac {h}{2}}} steps of iteration to repair the tree (where h {\displaystyle h} is the height of the tree).


The α- series process. Given a sequence of non-negative random variables: { X k , k = 1 , 2 , … } {\displaystyle \{X_{k},k=1,2,\dots \}} , if they are independent and the cdf of X k k a {\displaystyle {\frac {X_{k}}{k^{a}}}} is given by F ( x ) {\displaystyle F(x)} for k = 1 , 2 , … {\displaystyle k=1,2,\dots } , where a {\displaystyle a} is a positive constant, then { X k , k = 1 , 2 , … } {\displaystyle \{X_{k},k=1,2,\ldots \}} is called an α- series process. The threshold geometric process.


In queueing theory, a discipline within the mathematical theory of probability, a layered queueing network (or rendezvous network) is a queueing network model where the service time for each job at each service node is given by the response time of a queueing network (and those service times in turn may also be determined by further nested networks). Resources can be nested and queues form along the nodes of the nesting structure. The nesting structure thus defines "layers" within the queueing model.Layered queueing has applications in a wide range of distributed systems which involve different master/slave, replicated services and client-server components, allowing each local node to be represented by a specific queue, then orchestrating the evaluation of these queues.For large population of jobs, a fluid limit has been shown in PEPA to be a give good approximation of performance measures.


In some ways it is related to the concepts of place attachment and sense of place. Place identity is largely related to the concepts of community formation because it recognizes that geographical spaces do not solely bond a community together but rather there are social bonds that account for community formation. Those social forces often are feelings of belonging and security, which involve theoretical formations of community. Theoretical formations of community, which were identified in Community: Seeking Safety in an Insecure World (Bauman, 2001) act as bonds formed by similar locality, culture, language, kinship and/or experiences.


In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n. Each prime factor of n occurs exactly once as a factor of this product: The radical plays a central role in the statement of the abc conjecture.


A rival programming-language usage was pioneered by the original version of ALGOL, which was designed in 1958 and implemented in 1960. ALGOL included a relational operator that tested for equality, allowing constructions like if x = 2 with essentially the same meaning of = as the conditional usage in mathematics. The equal sign was reserved for this usage.


In the AGM framework, a belief set is represented by a deductively closed set of propositional formulae. While such sets are infinite, they can always be finitely representable. However, working with deductively closed sets of formulae leads to the implicit assumption that equivalent belief sets should be considered equal when revising. This is called the principle of irrelevance of syntax.


{\displaystyle \|\nabla v\|_{L^{2}(\Omega )}^{2}=\|v\|_{L^{p+1}(\Omega )}^{p+1}>0.} Solutions to the original variational problem that lie in the Nehari manifold are (constrained) minimizers of the energy, and so direct methods in the calculus of variations can be brought to bear. More generally, given a suitable functional J, the associated Nehari manifold is defined as the set of functions u in an appropriate function space for which ⟨ J ′ ( u ) , u ⟩ = 0. {\displaystyle \langle J'(u),u\rangle =0.} Here J′ is the functional derivative of J.


In software development, distributed version control (also known as distributed revision control) is a form of version control in which the complete codebase, including its full history, is mirrored on every developer's computer. Compared to centralized version control, this enables automatic management branching and merging, speeds up most operations (except pushing and pulling), improves the ability to work offline, and does not rely on a single location for backups. Git, the world's most popular version control system, is a distributed version control system. In 2010, software development author Joel Spolsky described distributed version control systems as "possibly the biggest advance in software development technology in the  ten years".


In some simple cases, an analysis of the recipe can reveal the maximum production rate and the rate limiting unit. In the process example above if a number of batches or lots of Product C are to be produced, it is useful to calculate the minimum time between consecutive batch starts (cycle-time). If a batch is allowed to start before the end of the prior batch the minimum cycle-time is given by the following relationship: Where CTmin is the shortest possible cycle time for a process with M unit-procedures and τj is the total duration for the jth unit-procedure. The unit-procedure with the maximum duration is sometimes referred to as the bottleneck.


In most cases, place the items in order of usage, with the most-used meanings appearing at the top and less common meanings below. A recommended order is: Articles with a clarifier in parentheses (Anticipation (music)) Articles with the item as part of the name (Computer keyboard as part of a Keyboard dab page) Synonyms Larger subject articles which treat this item in a section (Medieval art from a Fresco dab page)Unless the list is quite short, separate the articles in categories (1) and (2) from those in (3) and (4), with the "may also be" line shown below:


One rule to rewrite a term could be applied in many different ways to that term, or more than one rule could be applicable. Rewriting systems then do not provide an algorithm for changing one term to another, but a set of possible rule applications. When combined with an appropriate algorithm, however, rewrite systems can be viewed as computer programs, and several theorem provers and declarative programming languages are based on term rewriting.


In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of the center. Median income, for example, may be a better way to describe center of the income distribution because increases in the largest incomes alone have no effect on median. For this reason, the median is of central importance in robust statistics.


In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence X1, X2, X3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. Thus, for example the sequences X 1 , X 2 , X 3 , X 4 , X 5 , X 6 and X 3 , X 6 , X 1 , X 5 , X 2 , X 4 {\displaystyle X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}\quad {\text{ and }}\quad X_{3},X_{6},X_{1},X_{5},X_{2},X_{4}} both have the same joint probability distribution. It is closely related to the use of independent and identically distributed random variables in statistical models. Exchangeable sequences of random variables arise in cases of simple random sampling.


In mathematics, a permutation polynomial (for a given ring) is a polynomial that acts as a permutation of the elements of the ring, i.e. the map x ↦ g ( x ) {\displaystyle x\mapsto g(x)} is a bijection. In case the ring is a finite field, the Dickson polynomials, which are closely related to the Chebyshev polynomials, provide examples. Over a finite field, every function, so in particular every permutation of the elements of that field, can be written as a polynomial function. In the case of finite rings Z/nZ, such polynomials have also been studied and applied in the interleaver component of error detection and correction algorithms.


Details of the history of this search, as well as the sequences leading to home primes for all other numbers through 100, are maintained at Patrick De Geest's worldofnumbers website. A wiki primarily associated with the Great Internet Mersenne Prime Search maintains the complete known data through 1000 in base 10 and also has lists for the bases 2 through 9. The primes in HP(n) are 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, ... (sequence A037274 in the OEIS)Aside from the computational problems that have had so much time devoted to them, it appears absolute proof of existence of a home prime for any specific number might entail its effective computation. In purely heuristic terms, the existence has probability 1 for all numbers, but such heuristics make assumptions about numbers drawn from a wide variety of processes that, though they are likely correct, fall short of the standard of proof usually required of mathematical claims.


In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts. Therefore, the distribution of one complex random variable may be interpreted as the joint distribution of two real random variables.


For example, per capita income is the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic: it is greatly influenced by outliers (values much larger or smaller than most others). For skewed distributions, such as the distribution of income for which a few people's incomes are substantially higher than most people's, the arithmetic mean may not coincide with one's notion of "middle". In that case, robust statistics, such as the median, may provide a better description of central tendency.


In mathematics, a matrix of ones or all-ones matrix is a matrix where every entry is equal to one. Examples of standard notation are given below: J 2 = ( 1 1 1 1 ) ; J 3 = ( 1 1 1 1 1 1 1 1 1 ) ; J 2 , 5 = ( 1 1 1 1 1 1 1 1 1 1 ) ; J 1 , 2 = ( 1 1 ) . {\displaystyle J_{2}={\begin{pmatrix}1&1\\1&1\end{pmatrix}};\quad J_{3}={\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}};\quad J_{2,5}={\begin{pmatrix}1&1&1&1&1\\1&1&1&1&1\end{pmatrix}};\quad J_{1,2}={\begin{pmatrix}1&1\end{pmatrix}}.\quad } Some sources call the all-ones matrix the unit matrix, but that term may also refer to the identity matrix, a different type of matrix. A vector of ones or all-ones vector is matrix of ones having row or column form; it should not be confused with unit vectors.


In mathematics and computer science, an algorithmic technique is a general approach for implementing a process or computation.


In the Microsoft DOS environment, daemon-like programs were implemented as terminate-and-stay-resident programs (TSR).


In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to convex functions which are not necessarily differentiable. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization. Let f: I → R {\displaystyle f:I\to \mathbb {R} } be a real-valued convex function defined on an open interval of the real line. Such a function need not be differentiable at all points: For example, the absolute value function f ( x ) = | x | {\displaystyle f(x)=|x|} is non-differentiable when x = 0 {\displaystyle x=0} . However, as seen in the graph on the right (where f ( x ) {\displaystyle f(x)} in blue has non-differentiable kinks similar to the absolute value function), for any x 0 {\displaystyle x_{0}} in the domain of the function one can draw a line which goes through the point ( x 0 , f ( x 0 ) ) {\displaystyle (x_{0},f(x_{0}))} and which is everywhere either touching or below the graph of f. The slope of such a line is called a subderivative.


In mathematics, a unique sink orientation is an orientation of the edges of a polytope such that, in every face of the polytope (including the whole polytope as one of the faces), there is exactly one vertex for which all adjoining edges are oriented inward (i.e. towards that vertex). If a polytope is given together with a linear objective function, and edges are oriented from vertices with smaller objective function values to vertices with larger objective values, the result is a unique sink orientation. Thus, unique sink orientations can be used to model linear programs as well as certain nonlinear programs such as the smallest circle problem.


There exist n-ary groups with more than one neutral element. If the set of all neutral elements of an n-ary group is non-empty it forms an n-ary subgroup.Some authors include an identity in the definition of an n-ary group but as mentioned above such n-ary operations are just repeated binary operations. Groups with intrinsically n-ary operations do not have an identity element.


Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates 10−1 (0.1), the second position 10−2 (0.01), and so on for each successive position. As an example, the number 2674 in a base-10 numeral system is: (2 × 103) + (6 × 102) + (7 × 101) + (4 × 100)or (2 × 1000) + (6 × 100) + (7 × 10) + (4 × 1).


In the CP/M 8-bit operating system used on early personal computers, the standard DUMP program would list a file 16 bytes per line with the hex offset at the start of the line and the ASCII equivalent of each byte at the end. Bytes outside the standard range of printable ASCII characters (20 to 7E) would be displayed as a single period for visual alignment. This same format was used to display memory when invoking the D command in the standard CP/M debugger DDT. Later incarnations of the format (e.g. in the DOS debugger DEBUG) changed the space between the 8th and 9th byte to a dash, without changing the overall width.


As such, this generalizes the same process when done with the Lyndon words. Hall trees can also be used to give a total order to the elements of a group, via the commutator collecting process, which is a special case of the general construction given below.


In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K2, the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions (x, y) for: y 2 = x 3 + a x + b {\displaystyle y^{2}=x^{3}+ax+b} for some coefficients a and b in K. The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition 4a3 + 27b2 ≠ 0, that is, being square-free in x.) It is always understood that the curve is really sitting in the projective plane, with the point O being the unique point at infinity.


Since this relaxation in US export restrictions, and because most personal computers connected to the Internet include US-sourced web browsers such as Firefox or Internet Explorer, almost every Internet user worldwide has potential access to quality cryptography via their browsers (e.g., via Transport Layer Security). The Mozilla Thunderbird and Microsoft Outlook E-mail client programs similarly can transmit and receive emails via TLS, and can send and receive email encrypted with S/MIME. Many Internet users don't realize that their basic application software contains such extensive cryptosystems. These browsers and email programs are so ubiquitous that even governments whose intent is to regulate civilian use of cryptography generally don't find it practical to do much to control distribution or use of cryptography of this quality, so even when such laws are in force, actual enforcement is often effectively impossible.


In telephony, call progress tones are audible tones that provide an indication of the status of a telephone call to the user. The tones are generated by a central office or a private branch exchange (PBX) to the calling party. Telecommunication equipment such as fax machines and modems are designed to recognize certain tones, such as dial tone and busy tone.


In the cost-to-cost method, a project's cost to date is compared to the total expected cost of the project. The costs of products already bought for a contract, but not installed, should not be added in calculating the percentage of completion (unless they were specifically obtained for that contract). Furthermore, the cost of equipment is assigned over the course of the contract, rather than directly, unless title to the supplies is being transported to the customer.


In particular, unfair coins would have p ≠ 1 / 2. {\displaystyle p\neq 1/2.} The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted (so n would be 1 for such a binomial distribution). It is also a special case of the two-point distribution, for which the possible outcomes need not be 0 and 1.


An alternative hypothesis can be defined under which each value π i {\displaystyle ~\pi _{i}~} is replaced by its maximum likelihood estimate p i = x i N . {\displaystyle ~p_{i}={\frac {\;x_{i}\,}{N}}~.}


In number theory, a practical number or panarithmic number is a positive integer n {\displaystyle n} such that all smaller positive integers can be represented as sums of distinct divisors of n {\displaystyle n} . For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2. The sequence of practical numbers (sequence A005153 in the OEIS) begins Practical numbers were used by Fibonacci in his Liber Abaci (1202) in connection with the problem of representing rational numbers as Egyptian fractions. Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators.The name "practical number" is due to Srinivasan (1948).


The elements of G conventionally have circular symmetry such that E  = 0 {\displaystyle \mathbb {E} =0} . Inverse Complex Wishart The distribution of the inverse complex Wishart distribution of Y = S − 1 {\displaystyle \mathbf {Y} =\mathbf {S^{-1}} } according to Goodman, Shaman is f Y ( Y ) = | Y | − ( n + p ) e − tr  ( M Y − 1 ) | M | − n ⋅ C Γ ~ p ( n ) , n ≥ p , det ( Y ) > 0 {\displaystyle f_{Y}(\mathbf {Y} )={\frac {\left|\mathbf {Y} \right|^{-(n+p)}e^{-\operatorname {tr} (\mathbf {M} \mathbf {Y^{-1}} )}}{\left|\mathbf {M} \right|^{-n}\cdot {\mathcal {C}}{\widetilde {\Gamma }}_{p}(n)}},\;\;\;n\geq p,\;\;\;\det \left(\mathbf {Y} \right)>0} where M = Γ − 1 {\displaystyle \mathbf {M} =\mathbf {\Gamma ^{-1}} } . If derived via a matrix inversion mapping, the result depends on the complex Jacobian determinant C J Y ( Y − 1 ) = | Y | − 2 p − 2 {\displaystyle {\mathcal {C}}J_{Y}(Y^{-1})=\left|Y\right|^{-2p-2}} Goodman and others discuss such complex Jacobians.


In queueing theory, a discipline within the mathematical theory of probability, an M/G/k queue is a queue model where arrivals are Markovian (modulated by a Poisson process), service times have a General distribution and there are k servers. The model name is written in Kendall's notation, and is an extension of the M/M/c queue, where service times must be exponentially distributed and of the M/G/1 queue with a single server. Most performance metrics for this queueing system are not known and remain an open problem.


In the birthday problem, neither of the two people is chosen in advance. By contrast, the probability q(n) that someone in a room of n other people has the same birthday as a particular person (for example, you) is given by q ( n ) = 1 − ( 365 − 1 365 ) n {\displaystyle q(n)=1-\left({\frac {365-1}{365}}\right)^{n}} and for general d by q ( n ; d ) = 1 − ( d − 1 d ) n . {\displaystyle q(n;d)=1-\left({\frac {d-1}{d}}\right)^{n}.} In the standard case of d = 365, substituting n = 23 gives about 6.1%, which is less than 1 chance in 16. For a greater than 50% chance that one person in a roomful of n people has the same birthday as you, n would need to be at least 253. This number is significantly higher than 365/2 = 182.5: the reason is that it is likely that there are some birthday matches among the other people in the room.


In the Web Annotation standard, n annotation is considered to be a set of connected resources, typically including a body and target, and conveys that the body is related to the target. The exact nature of this relationship changes according to the intention of the annotation, but the body is most frequently somehow "about" the target. (...) The (...) model supports additional functionality, enabling content to be embedded within the annotation, selecting arbitrary segments of resources, choosing the appropriate representation of a resource and providing styling hints to help clients render the annotation appropriately. The basic data structures of Web Annotation (Fig. 1) are target (the element being annotated, e.g., a web document or a part of it), body (the content of the annotation, e.g., a string value), and annotation (the element that serves to relate body and target of an annotation)The body can be a literal value or structured content (a URI).


There are a number of ways to deal with this. One is to use a pseudo inverse instead of the usual matrix inverse in the above formulae. However, better numeric stability may be achieved by first projecting the problem onto the subspace spanned by Σ b {\displaystyle \Sigma _{b}} .


In order to expose bugs, a fuzzer must be able to distinguish expected (normal) from unexpected (buggy) program behavior. However, a machine cannot always distinguish a bug from a feature. In automated software testing, this is also called the test oracle problem.Typically, a fuzzer distinguishes between crashing and non-crashing inputs in the absence of specifications and to use a simple and objective measure.


The number of elements in a Tamari lattice for a sequence of n + 1 objects is the nth Catalan number Cn. The Tamari lattice can also be described in several other equivalent ways: It is the poset of sequences of n integers a1, ..., an, ordered coordinatewise, such that i ≤ ai ≤ n and if i ≤ j ≤ ai then aj ≤ ai (Huang & Tamari 1972). It is the poset of binary trees with n leaves, ordered by tree rotation operations. It is the poset of ordered forests, in which one forest is earlier than another in the partial order if, for every j, the jth node in a preorder traversal of the first forest has at least as many descendants as the jth node in a preorder traversal of the second forest (Knuth 2005). It is the poset of triangulations of a convex n-gon, ordered by flip operations that substitute one diagonal of the polygon for another.


The set of tempered distributions forms a vector subspace of the space of distributions D ′ ( U ) {\displaystyle {\mathcal {D}}^{\prime }(U)} and is thus one example of a space of distributions; there are many other spaces of distributions. There also exist other major classes of test functions that are not subsets of C c ∞ ( U ) , {\displaystyle C_{c}^{\infty }(U),} such as spaces of analytic test functions, which produce very different classes of distributions. The theory of such distributions has a different character from the previous one because there are no analytic functions with non-empty compact support. Use of analytic test functions leads to Sato's theory of hyperfunctions.


. ", which may be denoted "S", between the various "numerals". "We must now consider the serial character of the natural numbers in the order 0, 1, 2, 3, .


Otherwise these operations can alter the result of summation. Further, the terms of Grandi's series can be rearranged to have its accumulation points at any interval of two or more consecutive integer numbers, not only 0 or 1. For instance, the series 1 + 1 + 1 + 1 + 1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 + 1 − ⋯ {\displaystyle 1+1+1+1+1-1-1+1+1-1-1+1+1-1-1+1+1-\cdots } (in which, after five initial +1 terms, the terms alternate in pairs of +1 and −1 terms–the infinitude of both +1’s and -1’s allows any finite number of 1’s or -1’s to be prepended, by Hilbert's paradox of the Grand Hotel) is a permutation of Grandi's series in which each value in the rearranged series corresponds to a value that is at most four positions away from it in the original series; its accumulation points are 3, 4, and 5.


The overall accuracy would be 95%, but in more detail the classifier would have a 100% recognition rate (sensitivity) for the cancer class but a 0% recognition rate for the non-cancer class. F1 score is even more unreliable in such cases, and here would yield over 97.4%, whereas informedness removes such bias and yields 0 as the probability of an informed decision for any form of guessing (here always guessing cancer). According to Davide Chicco and Giuseppe Jurman, the most informative metric to evaluate a confusion matrix is the Matthews correlation coefficient (MCC).Other metrics can be included in a confusion matrix, each of them having their significance and use.


In the early 20th century, several electrical engineers intuitively recognized that Boolean algebra was analogous to the behavior of certain types of electrical circuits. Claude Shannon formally proved such behavior was logically equivalent to Boolean algebra in his 1937 master's thesis, A Symbolic Analysis of Relay and Switching Circuits. Today, all modern general purpose computers perform their functions using two-value Boolean logic; that is, their electrical circuits are a physical manifestation of two-value Boolean logic. They achieve this in various ways: as voltages on wires in high-speed circuits and capacitive storage devices, as orientations of a magnetic domain in ferromagnetic storage devices, as holes in punched cards or paper tape, and so on.


Using a pairing function on ω (such as (n,k) goes to (n2+2·n·k+k2+n+3·k)/2), we can map the powerset of ω×ω into the powerset of ω. And we can map the powerset of ω into the Cantor set, a subset of the real numbers. So statements about H ℵ 1 {\displaystyle H_{\aleph _{1}}} can be converted into statements about the reals.


In physics, a zero mode is an eigenvector with a vanishing eigenvalue. In various subfields of physics zero modes appear whenever a physical system possesses a certain symmetry. For example, normal modes of multidimensional harmonic oscillator (e.g. a system of beads arranged around the circle, connected with springs) corresponds to elementary vibrational modes of the system. In such a system zero modes typically occur and are related with a rigid rotation around the circle. The kernel of an operator consists of left zero modes, and the cokernel consists of the right zero modes.


In natural language processing, a corpus is a set of sentences or texts, and a language model is a probability distribution over entire sentences or texts. Consequently, in NLP, the more commonly used measure is perplexity per word, defined as: where s 1 , . . .


Numbers of the form Mn = 2n − 1 without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. The smallest composite Mersenne number with prime exponent n is 211 − 1 = 2047 = 23 × 89.


In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity a 2 − b 2 = ( a + b ) ( a − b ) {\displaystyle a^{2}-b^{2}=(a+b)(a-b)} in elementary algebra.


In some languages, class variables and class methods are either statically resolved, not via dynamic dispatch, or their memory statically allocated at compile time (once for the entire class, as static variables), not dynamically allocated at run time (at every instantiation of an object). In other cases, however, either or both of these are dynamic. For example, if classes can be dynamically defined (at run time), class variables of these classes are allocated dynamically when the class is defined, and in some languages class methods are also dispatched dynamically. Thus in some languages, static member variable or static member function are used synonymously with or in place of "class variable" or "class function", but these are not synonymous across languages. These terms are commonly used in Java, C# , and C++, where class variables and class methods are declared with the static keyword, and referred to as static member variables or static member functions.


The term subscriber trunk dialling is used in the United Kingdom, the Republic of Ireland, Australia, India and South East Asia. In the UK, the term is obsolescent, better known as the UK area codes. The introduction in the UK of subscriber dialling of long-distance calls removed the distinction that had existed between trunk and toll calls.


If N = 1, then there is roughly one recursive subroutine call for every input, but more generally there is one recursive call for (roughly) every N/2 inputs if the recursion stops at exactly n = N. By making N sufficiently large, the overhead of recursion can be made negligible (precisely this technique of a large base case for recursive summation is employed by high-performance FFT implementations). Regardless of N, exactly n−1 additions are performed in total, the same as for naive summation, so if the recursion overhead is made negligible then pairwise summation has essentially the same computational cost as for naive summation. A variation on this idea is to break the sum into b blocks at each recursive stage, summing each block recursively, and then summing the results, which was dubbed a "superblock" algorithm by its proposers. The above pairwise algorithm corresponds to b = 2 for every stage except for the last stage which is b = N.


Analytic solutions for this mechanism (also similar to the solution of Price) were presented in 2000 by Dorogovtsev, Mendes and Samukhin and independently by Krapivsky, Redner, and Leyvraz, and later rigorously proved by mathematician Béla Bollobás. Notably, however, this mechanism only produces a specific subset of networks in the scale-free class, and many alternative mechanisms have been discovered since.The history of scale-free networks also includes some disagreement. On an empirical level, the scale-free nature of several networks has been called into question.


The trivial case is checking to see whether the return value of a function indicated success or failure, and to therefore cease further processing upon failure. This return value is actually often itself the result of a sanity check. For example, if the function attempted to open, write to, and close a file, a sanity check may be used to ensure that it did not fail on any of these actions—which is a sanity check often ignored by programmers.These kinds of sanity checks may be used during development for debugging purposes and also to aid in troubleshooting software runtime errors.


For an n {\displaystyle n} -card deck, the number of riffle shuffles needed grows as 1.5 log 2  n {\displaystyle 1.5\log _{2}n} . The most developed theory concerns randomized algorithms for #P-Complete algorithmic counting problems such as the number of graph colorings of a given n {\displaystyle n} vertex graph. Such problems can, for sufficiently large number of colors, be answered using the Markov chain Monte Carlo method and showing that the mixing time grows only as n log  ( n ) {\displaystyle n\log(n)} (Jerrum 1995).


In the Standard Generalized Markup Language (SGML), an entity is a primitive data type, which associates a string with either a unique alias (such as a user-specified name) or an SGML reserved word (such as #DEFAULT). Entities are foundational to the organizational structure and definition of SGML documents. The SGML specification defines numerous entity types, which are distinguished by keyword qualifiers and context. An entity string value may variously consist of plain text, SGML tags, and/or references to previously defined entities. Certain entity types may also invoke external documents. Entities are called by reference.


In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. The Dedekind number M(n) is the number of monotone boolean functions of n variables. Equivalently, it is the number of antichains of subsets of an n-element set, the number of elements in a free distributive lattice with n generators, and one more than the number of abstract simplicial complexes on a set with n elements. Accurate asymptotic estimates of M(n) and an exact expression as a summation are known. However Dedekind's problem of computing the values of M(n) remains difficult: no closed-form expression for M(n) is known, and exact values of M(n) have been found only for n ≤ 9 (sequence A000372 in the OEIS).


In order to be able to meticulously study the English language, an annotated text corpus was much needed. The Penn Treebank was one of the most used corpora. It consisted of IBM computer manuals, transcribed telephone conversations, and other texts, together containing over 4.5 million words of American English, annotated using both part-of-speech tagging and syntactic bracketing.Japanese sentence corpora were analyzed and a pattern of log-normality was found in relation to sentence length.


In the C programming language, operations can be performed on a bit level using bitwise operators. Bitwise operations are contrasted by byte-level operations which characterize the bitwise operators' logical counterparts, the AND, OR, NOT operators. Instead of performing on individual bits, byte-level operators perform on strings of eight bits (known as bytes) at a time.


In syllogistic logic, there are 256 possible ways to construct categorical syllogisms using the A, E, I, and O statement forms in the square of opposition. Of the 256, only 24 are valid forms. Of the 24 valid forms, 15 are unconditionally valid, and 9 are conditionally valid.


In system software, a monolithic kernel is an operating system (OS) architecture where the entire OS is working in kernel space. == References ==


In the derivation of the van Cittert–Zernike theorem we write the direction cosines l {\displaystyle l} and m {\displaystyle m} as 1 2 ( x 1 + x 2 ) / R {\displaystyle {\frac {1}{2}}(x_{1}+x_{2})/R} and 1 2 ( y 1 + y 2 ) / R {\displaystyle {\frac {1}{2}}(y_{1}+y_{2})/R} . There is, however, a third direction cosine which is neglected since R ≫ 1 2 ( x 1 + x 2 ) {\displaystyle R\gg {\frac {1}{2}}(x_{1}+x_{2})} and R ≫ 1 2 ( y 1 + y 2 ) {\displaystyle R\gg {\frac {1}{2}}(y_{1}+y_{2})} ; under these assumptions it is very close to unity. But if the source has a large angular extent, we cannot neglect this third direction cosine and the van Cittert–Zernike theorem no longer holds. Because most astronomical sources subtend very small angles on the sky (typically much less than a degree), this assumption of the theorem is easily fulfilled in the domain of radio astronomy.


In statistics, a maximum-entropy Markov model (MEMM), or conditional Markov model (CMM), is a graphical model for sequence labeling that combines features of hidden Markov models (HMMs) and maximum entropy (MaxEnt) models. An MEMM is a discriminative model that extends a standard maximum entropy classifier by assuming that the unknown values to be learnt are connected in a Markov chain rather than being conditionally independent of each other. MEMMs find applications in natural language processing, specifically in part-of-speech tagging and information extraction.


In measurement of the I/O performance of five filesystems with five storage configurations—single SSD, RAID 0, RAID 1, RAID 10, and RAID 5 it was shown that F2FS on RAID 0 and RAID 5 with eight SSDs outperforms EXT4 by 5 times and 50 times, respectively. The measurements also suggest that the RAID controller can be a significant bottleneck in building a RAID system with high speed SSDs.


Francis Galton established the first tests in London for measuring IQ. He tested thousands of people, examining their physical characteristics as a basis for his results and many of the records remain today. James Cattell studied with him, and eventually worked on his own with brass instruments for evaluation.


In the belief revision framework, counterfactuals are treated using a formal implementation of the Ramsey test. In these systems, a counterfactual A > B holds if and only if the addition of A to the current body of knowledge has B as a consequence. This condition relates counterfactual conditionals to belief revision, as the evaluation of A > B can be done by first revising the current knowledge with A and then checking whether B is true in what results. Revising is easy when A is consistent with the current beliefs, but can be hard otherwise. Every semantics for belief revision can be used for evaluating conditional statements. Conversely, every method for evaluating conditionals can be seen as a way for performing revision.


The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals. It was later generalized to Darboux's formula.


In practice, the output from many common PRNGs exhibit artifacts that cause them to fail statistical pattern-detection tests. These include: Shorter-than-expected periods for some seed states (such seed states may be called "weak" in this context); Lack of uniformity of distribution for large quantities of generated numbers; Correlation of successive values; Poor dimensional distribution of the output sequence; Distances between where certain values occur are distributed differently from those in a random sequence distribution.Defects exhibited by flawed PRNGs range from unnoticeable (and unknown) to very obvious. An example was the RANDU random number algorithm used for decades on mainframe computers. It was seriously flawed, but its inadequacy went undetected for a very long time.


In probability theory, the output of the softargmax function can be used to represent a categorical distribution – that is, a probability distribution over K different possible outcomes.


In software development, a mistake or error may be introduced at any stage. Bugs arise from oversight or misunderstanding by a software team during specification, design, coding, configuration, data entry or documentation. For example, a relatively simple program to alphabetize a list of words, the design might fail to consider what should happen when a word contains a hyphen. Or when converting an abstract design into code, the coder might inadvertently create an off-by-one error which can be a "<" where "<=" was intended, and fail to sort the last word in a list.


Algorithms which use context-free grammars often rely on some variant of the CYK algorithm, usually with some heuristic to prune away unlikely analyses to save time. (See chart parsing.) However some systems trade speed for accuracy using, e.g., linear-time versions of the shift-reduce algorithm. A somewhat recent development has been parse reranking in which the parser proposes some large number of analyses, and a more complex system selects the best option. In natural language understanding applications, semantic parsers convert the text into a representation of its meaning.


"The sky is blue", "grass is green", etc. when there are actually myriad variations in hue, chroma, within these areas. In order to represent objects realistically, painters must look beyond the simplifications of local color. Demonstrations of color constancy show how flawed local color assumptions can be when the light source has a color shift. In contemporary sculpture local color is the original color of the raw material that remains unpainted in the completed work.


In the context of harming privacy, information collection means gathering whatever information can be obtained by doing something to obtain it. Examples include surveillance and interrogation. Another example is how consumers and marketers also collect information in the business context through facial recognition which has recently caused a concern for things such as privacy. There is currently research being done related to this topic.


Some binary data corresponds to computer instructions, such as the data within processor registers decoded by the control unit along the fetch-decode-execute cycle. Computers rarely modify individual bits for performance reasons. Instead, data is aligned in groups of a fixed number of bits, usually 1 byte (8 bits).


While there are exponentially-many such cycles, a separation oracle that works in polynomial time can be implemented by just finding an odd cycle of minimum length, which can be done in polynomial time. The dual of the configuration linear program for the bin packing problem. It can be approximated by an LP with a constraint for each feasible configuration. While there are exponentially-many such cycles, a separation oracle that works in pseudopolynomial time can be implemented by solving a knapsack problem. This is used by the Karmarkar-Karp bin packing algorithms.


Includes user views and descriptions of how the system is expected to behave.Information viewpointMetamodel view – An abstract view that defines information model elements and their structures and relationships. Defines the classes of data that are created and managed by the system and the data architecture. Information view – Describes the actual data and information as it is realized and manipulated within the system.


In telecommunication, an n-ary code is a code that has n significant conditions, where n is a positive integer greater than 1. The integer substituted for n indicates the specific number of significant conditions, i.e., quantization states, in the code. For example, an 8-ary code has eight significant conditions and can convey three bits per code symbol. A prefix that indicates an integer, e.g., "bin", "tern," or "quatern", may be used in lieu of a numeral, to produce "binary", "ternary", or "quaternary" (2, 3, and 4 states respectively).


This paper also introduced concepts from convex geometry, including "Elfving sets" and Elfving's theorem. Being symmetric, Elfving sets are formed by the union of a set and its reflection through the origin, −S ∪ S. According to Chernoff (1999, p. 204), Elfving was generous in crediting others' results: His paper in the Cramér-festschrift acknowledged unpublished notes of L. J. Savage; Elfving was a referee for the fundamental paper on optimal designs by Kiefer and Wolfowitz.


{\displaystyle M=B^{*}B.} A matrix is positive semi-definite if it satisfies similar equivalent conditions where "positive" is replaced by "nonnegative", "invertible matrix" is replaced by "matrix", and the word "leading" is removed. Positive-definite and positive-semidefinite real matrices are at the basis of convex optimization, since, given a function of several real variables that is twice differentiable, then if its Hessian matrix (matrix of its second partial derivatives) is positive-definite at a point p, then the function is convex near p, and, conversely, if the function is convex near p, then the Hessian matrix is positive-semidefinite at p. Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ones.


{\displaystyle \sum _{k=0}^{\infty }\left|\Pr(S_{n}=k)-{\lambda _{n}^{k}e^{-\lambda _{n}} \over k! }\right|<2\left(\sum _{i=1}^{n}p_{i}^{2}\right).} In other words, the sum has approximately a Poisson distribution and the above inequality bounds the approximation error in terms of the total variation distance.


In testing the null hypothesis that the population mean is equal to a specified value μ0, one uses the statistic t = x ¯ − μ 0 s / n {\displaystyle t={\frac {{\bar {x}}-\mu _{0}}{s/{\sqrt {n}}}}} where x ¯ {\displaystyle {\bar {x}}} is the sample mean, s is the sample standard deviation and n is the sample size. The degrees of freedom used in this test are n − 1. Although the parent population does not need to be normally distributed, the distribution of the population of sample means x ¯ {\displaystyle {\bar {x}}} is assumed to be normal. By the central limit theorem, if the observations are independent and the second moment exists, then t {\displaystyle t} will be approximately normal N(0;1).


Let a = ( a 1 , … , a k − 1 ) {\displaystyle \mathbf {a} =(a_{1},\ldots ,a_{k-1})} be a vector of positive integers and V {\displaystyle V} be an n {\displaystyle n} -element vertex set. We say that a family of partitions P = P ( k − 1 , a ) = { P ( 1 ) … , P ( k − 1 ) } {\displaystyle {\mathcal {P}}={\mathcal {P}}(k-1,\mathbf {a} )=\{{\mathcal {P}}^{(1)}\,\ldots ,{\mathcal {P}}^{(k-1)}\}} on V {\displaystyle V} is ( μ , δ , d , r ) {\displaystyle (\mu ,\delta ,\mathbf {d} ,r)} -equitable if it satisfies the following: P ( 1 ) = { V i: i ∈  } {\displaystyle {\mathcal {P}}^{(1)}=\{V_{i}\colon i\in \}} is equitable vertex partition of V {\displaystyle V} . That is | V 1 | ≤ … ≤ | V a 1 | ≤ | V 1 | + 1 {\displaystyle |V_{1}|\leq \ldots \leq |V_{a_{1}}|\leq |V_{1}|+1} .


In mathematics, in the area of abstract algebra known as group theory, an A-group is a type of group that is similar to abelian groups. The groups were first studied in the 1940s by Philip Hall, and are still studied today. A great deal is known about their structure.


In the Linux kernel, configfs and sysfs provide files that can be used to query the kernel for information and configure entities in the kernel. procfs maps processes and, on Linux, other operating system structures into a filespace.


This is exactly the situation in computers with cache memory, one of the earliest commercial examples of which was the IBM System/360 Model 85. In the Model 85 all addresses were real addresses referring to the main core store. A semiconductor cache store, invisible to the user, held the contents of parts of the main store in use by the currently executing program.


The validation sample is used to construct a classification matrix which contains the number of correctly classified and incorrectly classified cases. The percentage of correctly classified cases is called the hit ratio. Plot the results on a two dimensional map, define the dimensions, and interpret the results.


In music, pieces or sections which confound expectations and may be or are interpreted simultaneously in different ways are ambiguous, such as some polytonality, polymeter, other ambiguous meters or rhythms, and ambiguous phrasing, or (Stein 2005, p. 79) any aspect of music. The music of Africa is often purposely ambiguous. To quote Sir Donald Francis Tovey (1935, p. 195), "Theorists are apt to vex themselves with vain efforts to remove uncertainty just where it has a high aesthetic value."


In the 1950s, attempts started to apply computers to the recognition, interpretation and translation of natural languages, such as English and Russian. This requires a machine-readable description of the phrase structure of sentences, that can be used to parse and interpret them, and to generate them. Context-free grammars, a concept from structural linguistics, were adopted for this purpose; their rules can express how sentences are recursively built out of parts of speech, such as noun phrases and verb phrases, and ultimately, words, such as nouns, verbs, and pronouns. This work influenced the design and implementation of programming languages, most notably, of ALGOL 60, which introduced a syntax description in Backus–Naur form.


Such an element is called a modular element. Even more generally, the modular law may hold for any a and a fixed pair (x, b). Such a pair is called a modular pair, and there are various generalizations of modularity related to this notion and to semimodularity. Modular lattices are sometimes called Dedekind lattices after Richard Dedekind, who discovered the modular identity in several motivating examples.


"Memoryless" means that if X {\displaystyle X} is a random variable with such a distribution, then for any numbers 0 < y < x {\displaystyle 0 x ∣ X > y ) = Pr ( X > x − y ) {\displaystyle \Pr(X>x\mid X>y)=\Pr(X>x-y)} . Verification of conditions of characterization theorems in practice is possible only with some error ϵ {\displaystyle \epsilon } , i.e., only to a certain degree of accuracy. Such a situation is observed, for instance, in the cases where a sample of finite size is considered.


The method of casting out nines offers a quick check of decimal arithmetic computations performed by hand. It is based on modular arithmetic modulo 9, and specifically on the crucial property that 10 ≡ 1 (mod 9). Arithmetic modulo 7 is used in algorithms that determine the day of the week for a given date. In particular, Zeller's congruence and the Doomsday algorithm make heavy use of modulo-7 arithmetic. More generally, modular arithmetic also has application in disciplines such as law (e.g., apportionment), economics (e.g., game theory) and other areas of the social sciences, where proportional division and allocation of resources plays a central part of the analysis.


Mobile UI context includes signal cues from user activity, such as the location where or the time when the device is in use, that can be observed from user interactions within a mobile app. Such context clues can be used to provide automatic suggestions when scheduling an appointment or activity or to filter a list of various services for the user. The user is often the focus of interaction with their device, and the interface entails components of both hardware and software.


The owner has a distributive right to exclude others (i.e. the right to command a "fair share" of personal property). Private property is a social relationship between the owner and persons deprived, i.e. not a relationship between person and thing. Private property may include artifacts, factories, mines, dams, infrastructure, natural vegetation, mountains, deserts and seas—these generate capital for the owner without the owner necessarily having to perform any physical labor. Conversely, those who perform labor using somebody else's private property are considered deprived of the value of their work in Marxist doctrine, and are instead given a salary that is disjointed from the value generated by the worker. In Marxist theory, private property typically refers to capital or the means of production, while personal property refers to consumer and non-capital goods and services.


However, this heuristic can also produce errors. When people are asked whether there are more English words with K in the first position or with K in the third position, they use the same process. It is easy to think of words that begin with K, such as kangaroo, kitchen, or kept.


In philosophy, the term formal ontology is used to refer to an ontology defined by axioms in a formal language with the goal to provide an unbiased (domain- and application-independent) view on reality, which can help the modeler of domain- or application-specific ontologies (information science) to avoid possibly erroneous ontological assumptions encountered in modeling large-scale ontologies. By maintaining an independent view on reality a formal (upper level) ontology gains the following properties: indefinite expandability: the ontology remains consistent with increasing content. content and context independence: any kind of 'concept' can find its place. accommodate different levels of granularity.


In statistics, the order of a kernel is the degree of the first non-zero moment of a kernel.


In some languages, allophonic palatalization developed into phonemic palatalization by phonemic split. In other languages, phonemes that were originally phonetically palatalized changed further: palatal secondary place of articulation developed into changes in manner of articulation or primary place of articulation. Phonetic palatalization of a consonant sometimes causes surrounding vowels to change by coarticulation or assimilation. In Russian, "soft" (palatalized) consonants are usually followed by vowels that are relatively more front (that is, closer to  or ), and vowels following "hard" (unpalatalized) consonants are further back. See Russian phonology § Allophony for more information.


Measurement of productivity is at its most accurate in business because of the availability of all elementary data of the quantities and prices of the inputs and the output in production. The more comprehensive the entity we want to analyse by measurements, the more data need to be aggregated. In productivity measurement, combining and aggregating the data always involves reduced measurement accuracy.


In the beginning, the tree is first created in time Θ(k). In each step of merging, only the games on the path from the new element to the root need to be replayed. In each layer, only one comparison is needed.


In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an information-bearing signal (called the message signal or modulation signal) for the purpose of conveying information.This carrier wave usually has a much higher frequency than the message signal does. This is because it is impractical to transmit signals with low frequencies. The purpose of the carrier is usually either to transmit the information through space as an electromagnetic wave (as in radio communication), or to allow several carriers at different frequencies to share a common physical transmission medium by frequency division multiplexing (as in a cable television system). The term originated in radio communication, where the carrier wave creates the waves which carry the information (modulation) through the air from the transmitter to the receiver. The term is also used for an unmodulated emission in the absence of any modulating signal.In music production, carrier signals can be controlled by a modulating signal to change the sound property of an audio recording and add a sense of depth and movement.


In mathematics, the rearrangement inequality states that for every choice of real numbers and every permutation y σ ( 1 ) , … , y σ ( n ) {\displaystyle y_{\sigma (1)},\ldots ,y_{\sigma (n)}} of y 1 , … , y n . {\displaystyle y_{1},\ldots ,y_{n}.} If the numbers x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are different, meaning that x 1 < ⋯ < x n , {\displaystyle x_{1}<\cdots


In mathematics, a moment matrix is a special symmetric square matrix whose rows and columns are indexed by monomials. The entries of the matrix depend on the product of the indexing monomials only (cf. Hankel matrices.) Moment matrices play an important role in polynomial fitting, polynomial optimization (since positive semidefinite moment matrices correspond to polynomials which are sums of squares) and econometrics.


Haxe uses structural typing, but classes are not structurally subtyped. In languages which support subtype polymorphism, a similar dichotomy can be formed based on how the subtype relationship is defined.


The notion was introduced by George E. Collins in 1975, together with an algorithm for computing it. Collins' algorithm has a computational complexity that is double exponential in n. This is an upper bound, which is reached on most entries.


It is possible to construct a unit distance graph efficiently, given its points. Finding all unit distances has applications in pattern matching, where it can be a first step in finding congruent copies of larger patterns. However, determining whether a given graph can be represented as a unit distance graph is NP-hard, and more specifically complete for the existential theory of the reals.


Constructed languages have been included in standardized tests such as the SAT, where they were used to test the applicant's ability to infer and apply grammatical rules. By the same token, a constructed language might also be used to restrict thought, as in George Orwell's Newspeak, or to simplify thought, as in Toki Pona. However, linguists such as Steven Pinker argue that ideas exist independently of language.


In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one or more independent variables (often called 'predictors', 'covariates', 'explanatory variables' or 'features'). The most common form of regression analysis is linear regression, in which one finds the line (or a more complex linear combination) that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line (or hyperplane) that minimizes the sum of squared differences between the true data and that line (or hyperplane). For specific mathematical reasons (see linear regression), this allows the researcher to estimate the conditional expectation (or population average value) of the dependent variable when the independent variables take on a given set of values.


In the early 1990s, MIPS began to license their designs to third-party vendors. This proved fairly successful due to the simplicity of the core, which allowed it to have many uses that would have formerly used much less able complex instruction set computer (CISC) designs of similar gate count and price; the two are strongly related: the price of a CPU is generally related to the number of gates and the number of external pins. Sun Microsystems attempted to enjoy similar success by licensing their SPARC core but was not nearly as successful. By the late 1990s, MIPS was a powerhouse in the embedded processor field.


In some programming languages, an assignment statement returns a value, while in others it does not. In most expression-oriented programming languages (for example, C), the assignment statement returns the assigned value, allowing such idioms as x = y = a, in which the assignment statement y = a returns the value of a, which is then assigned to x. In a statement such as while ((ch = getchar()) != EOF) {…}, the return value of a function is used to control a loop while assigning that same value to a variable. In other programming languages, Scheme for example, the return value of an assignment is undefined and such idioms are invalid.


In mechanical engineering, a key is a machine element used to connect a rotating machine element to a shaft. The key prevents relative rotation between the two parts and may enable torque transmission. For a key to function, the shaft and rotating machine element must have a keyway and a keyseat, which is a slot and pocket in which the key fits.


In numerical analysis, order of accuracy quantifies the rate of convergence of a numerical approximation of a differential equation to the exact solution. Consider u {\displaystyle u} , the exact solution to a differential equation in an appropriate normed space ( V , | | | | ) {\displaystyle (V,||\ ||)} . Consider a numerical approximation u h {\displaystyle u_{h}} , where h {\displaystyle h} is a parameter characterizing the approximation, such as the step size in a finite difference scheme or the diameter of the cells in a finite element method. The numerical solution u h {\displaystyle u_{h}} is said to be n {\displaystyle n} th-order accurate if the error E ( h ) := | | u − u h | | {\displaystyle E(h):=||u-u_{h}||} is proportional to the step-size h {\displaystyle h} to the n {\displaystyle n} th power: E ( h ) = | | u − u h | | ≤ C h n {\displaystyle E(h)=||u-u_{h}||\leq Ch^{n}} where the constant C {\displaystyle C} is independent of h {\displaystyle h} and usually depends on the solution u {\displaystyle u} .


This algorithm for APR is required for some but not all forms of consumer debt in the EU. For example, this EU directive is limited to agreements of €50,000 and below and excludes all mortgages.In the Netherlands the formula above is also used for mortgages. In many cases the mortgage is not always paid back completely at the end of period N, but for instance when the borrower sells his house or dies.


The English verb is to krige, and the most common noun is kriging. The word is sometimes capitalized as Kriging in the literature. Though computationally intensive in its basic formulation, kriging can be scaled to larger problems using various approximation methods.


In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to the elements of a list, a vector, or a matrix, depending on whether one is writing a formal mathematical paper for publication, or when one is writing a computer program.


Section 230 of the Communications Decency Act gives protection from any liability as a result from the publication provided by another party. Common issues include defamation, but many courts have expanded it to include other claims as well.Online communities of various kinds (social networking sites, blogs, media sharing sites, etc.) are posing new challenges for all levels of law enforcement in combating many kinds of crimes including harassment, identity theft, copyright infringement, etc. Copyright law is being challenged and debated with the shift in how individuals now disseminate their intellectual property. Individuals come together via online communities in collaborative efforts to create.


In mathematics, a ridge function is any function f: R d → R {\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} } that can be written as the composition of a univariate function with an affine transformation, that is: f ( x ) = g ( x ⋅ a ) {\displaystyle f({\boldsymbol {x}})=g({\boldsymbol {x}}\cdot {\boldsymbol {a}})} for some g: R → R {\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } and a ∈ R d {\displaystyle {\boldsymbol {a}}\in \mathbb {R} ^{d}} . Coinage of the term 'ridge function' is often attributed to B.F. Logan and L.A. Shepp.


In mathematics, specifically abstract algebra, a square class of a field F {\displaystyle F} is an element of the square class group, the quotient group F × / F × 2 {\displaystyle F^{\times }/F^{\times 2}} of the multiplicative group of nonzero elements in the field modulo the square elements of the field. Each square class is a subset of the nonzero elements (a coset of the multiplicative group) consisting of the elements of the form xy2 where x is some particular fixed element and y ranges over all nonzero field elements.For instance, if F = R {\displaystyle F=\mathbb {R} } , the field of real numbers, then F × {\displaystyle F^{\times }} is just the group of all nonzero real numbers (with the multiplication operation) and F × 2 {\displaystyle F^{\times 2}} is the subgroup of positive numbers (as every positive number has a real square root). The quotient of these two groups is a group with two elements, corresponding to two cosets: the set of positive numbers and the set of negative numbers. Thus, the real numbers have two square classes, the positive numbers and the negative numbers.Square classes are frequently studied in relation to the theory of quadratic forms.


In professional typography, the term typeface is not interchangeable with the word font (originally "fount" in British English, and pronounced "font"), because the term font has historically been defined as a given alphabet and its associated characters in a single size. For example, 8-point Caslon Italic was one font, and 10-point Caslon Italic was another. Historically, a font came from a type foundry as a set of "sorts", with number of copies of each character included. As the range of typeface designs increased and requirements of publishers broadened over the centuries, fonts of specific weight (blackness or lightness) and stylistic variants (most commonly regular or roman as distinct to italic, as well as condensed) have led to font families, collections of closely related typeface designs that can include hundreds of styles.


Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups and categories) explicitly require their binary operations to be associative. However, many important and interesting operations are non-associative; some examples include subtraction, exponentiation, and the vector cross product. In contrast to the theoretical properties of real numbers, the addition of floating point numbers in computer science is not associative, and the choice of how to associate an expression can have a significant effect on rounding error.


The choice of syntax is affected by both linguistic and computational concerns; for instance some parsing systems use lexical functional grammar, but in general, parsing for grammars of this type is known to be NP-complete. Head-driven phrase structure grammar is another linguistic formalism which has been popular in the parsing community, but other research efforts have focused on less complex formalisms such as the one used in the Penn Treebank. Shallow parsing aims to find only the boundaries of major constituents such as noun phrases.


Characterization is not unique to mathematics, but since the science is abstract, much of the activity can be described as "characterization". For instance, in Mathematical Reviews, as of 2018, more than 24,000 articles contain the word in the article title, and 93,600 somewhere in the review. In an arbitrary context of objects and features, characterizations have been expressed via the heterogeneous relation aRb, meaning that object a has feature b. For example, b may mean abstract or concrete. The objects can be considered the extensions of the world, while the features are expression of the intensions. A continuing program of characterization of various objects leads to their categorization.


In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other. A function maps elements from its domain to elements in its codomain. Given a function f: X → Y {\displaystyle f\colon X\to Y}: The function is injective, or one-to-one, if each element of the codomain is mapped to by at most one element of the domain, or equivalently, if distinct elements of the domain map to distinct elements in the codomain. An injective function is also called an injection.


In telecommunications, point-to-multipoint communication (P2MP, PTMP or PMP) is communication which is accomplished via a distinct type of one-to-many connection, providing multiple paths from a single location to multiple locations.Point-to-multipoint telecommunications is typically used in wireless Internet and IP telephony via gigahertz radio frequencies. P2MP systems have been designed with and without a return channel from the multiple receivers. A central antenna or antenna array broadcasts to several receiving antennas and the system uses a form of time-division multiplexing to allow for the return channel traffic.


In statistics the Cramér–von Mises criterion is a criterion used for judging the goodness of fit of a cumulative distribution function F ∗ {\displaystyle F^{*}} compared to a given empirical distribution function F n {\displaystyle F_{n}} , or for comparing two empirical distributions. It is also used as a part of other algorithms, such as minimum distance estimation. It is defined as ω 2 = ∫ − ∞ ∞  2 d F ∗ ( x ) {\displaystyle \omega ^{2}=\int _{-\infty }^{\infty }^{2}\,\mathrm {d} F^{*}(x)} In one-sample applications F ∗ {\displaystyle F^{*}} is the theoretical distribution and F n {\displaystyle F_{n}} is the empirically observed distribution.


In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are sometimes excluded from the data set. An outlier can be an indication of exciting possibility, but can also cause serious problems in statistical analyses. Outliers can occur by chance in any distribution, but they can indicate novel behaviour or structures in the data-set, measurement error, or that the population has a heavy-tailed distribution.


It is still possible to check GCC-style assembly for clobber mistakes with knowledge of the instruction set.GNAT (Ada language frontend of the GCC suite), and LLVM uses the GCC syntax. The D programming language uses a DSL similar to the MSVC extension officially for x86_64, but the LLVM-based LDC also provides the GCC-style syntax on every architecture. MSVC only supports inline assembler on 32-bit x86.The Rust language has since migrated to a syntax abstracting away inline assembly options further than the LLVM (GCC-style) version. It provides enough information to allow transforming the block into an externally-assembled function if the backend could not handle embedded assembly.


In statistics and in particular statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated value of the standard deviation (a measure of statistical dispersion) of a population of values, in such a way that the expected value of the calculation equals the true value. Except in some important situations, outlined later, the task has little relevance to applications of statistics since its need is avoided by standard procedures, such as the use of significance tests and confidence intervals, or by using Bayesian analysis. However, for statistical theory, it provides an exemplar problem in the context of estimation theory which is both simple to state and for which results cannot be obtained in closed form. It also provides an example where imposing the requirement for unbiased estimation might be seen as just adding inconvenience, with no real benefit.


Instead of using all the elements contained in the selected clusters, the researcher randomly selects elements from each cluster. Constructing the clusters is the first stage. Deciding what elements within the cluster to use is the second stage.


In general, however, Catalan is an SVO language. Sentences can be simple or compound, depending on whether they contain just one verb or more than one. In the sentences with more than one verb, they can be on an equal footing (juxtaposition or coordination), or there may be one main verb and other subordinate ones.


"The concept of alt-metrics was introduced in 2009 by Cameron Neylon and Shirly Wu as article-level metrics. In contrast with the focus of leading metrics on journals (impact factor) or, more recently, on individual researchers (h-index), the article-level metrics makes it possible to track the circulation of individual publications: "(an) article that used to live on a shelf now lives in Mendeley, CiteULike, or Zotero – where we can see and count it" As such they are more compatible with the diversity of publication strategies that has characterized open science: preprints, reports or even non-textual outputs like dataset or software may also have associated metrics. In their original research proposition, Neylon and Wu favored the use of data from reference management software like Zotero or Mendeley.


Square-free factorization is the first step of the polynomial factorization algorithms that are implemented in computer algebra systems. Therefore, the algorithm of square-free factorization is basic in computer algebra. Over a field of characteristic 0, the quotient of f {\displaystyle f} by its GCD with its derivative is the product of the a i {\displaystyle a_{i}} in the above square-free decomposition.


In the fair item allocation problem, there are n items and k people, each of which assigns a possibly different value to each item. The goal is to partition the items among the people in as fair way as possible. The natural generalization of the greedy number partitioning algorithm is the envy-graph algorithm.


One important principle is that if two sets X and Y have the same finite number of elements, and a function f: X → Y is known to be injective, then it is also surjective, and vice versa. A related fact is known as the pigeonhole principle, which states that if two sets X and Y have finite numbers of elements n and m with n > m, then any map f: X → Y is not injective (so there exist two distinct elements of X that f sends to the same element of Y); this follows from the former principle, since if f were injective, then so would its restriction to a strict subset S of X with m elements, which restriction would then be surjective, contradicting the fact that for x in X outside S, f(x) cannot be in the image of the restriction. Similar counting arguments can prove the existence of certain objects without explicitly providing an example. In the case of infinite sets this can even apply in situations where it is impossible to give an example.The domain of enumerative combinatorics deals with computing the number of elements of finite sets, without actually counting them; the latter usually being impossible because infinite families of finite sets are considered at once, such as the set of permutations of {1, 2, ..., n} for any natural number n.


In response to enterprise environment needs, OpenSocial added support for advanced mashup scenarios. It enabled gadgets to "securely message each other in a loosely coupled manner." This new feature was called Inter-Gadget Communication.


Step 2: Collect "similar" classes into 'bundles': These above collections can be put into a "binary relation" (comparing for) similarity by "equinumerosity", symbolized here by ≈, i.e. one-one correspondence of the elements, and thereby create Russellian classes of classes or what Russell called "bundles". "We can suppose all couples in one bundle, all trios in another, and so on. In this way we obtain various bundles of collections, each bundle consisting of all the collections that have a certain number of terms.


In operating-system-level virtualization, a physical server is virtualized at the operating system level, enabling multiple isolated and secure virtualized servers to run on a single physical server. The "guest" operating system environments share the same running instance of the operating system as the host system. Thus, the same operating system kernel is also used to implement the "guest" environments, and applications running in a given "guest" environment view it as a stand-alone system.


The depth may also be given as the number of memory elements v in the polynomial or the maximum possible number of states of the encoder (typically: 2 v {\displaystyle 2^{v}} ). Convolutional codes are often described as continuous. However, it may also be said that convolutional codes have arbitrary block length, rather than being continuous, since most real-world convolutional encoding is performed on blocks of data.


In the beginning of the COVID-19 pandemic, former US President Donald Trump delivered a very dangerous message to the public on the use of disinfectants, which was immediately rejected and refuted by health professionals. In essence, and as mentioned above, virucides are usually toxic depending on concentrations, mixture, etc., and can be deadly not just to viruses, but also if inside a human or animal body or on surface of body.With regards to the COVID-19 pandemic, some of the mentioned agents are still under research about their microbicidal activity and effectivity against SARS-CoV-2 e. g. on surfaces, as mouth-washes, hand-washing, etc. A mixture of 62–71% ethanol, 0.5% hydrogen peroxide or 0.1% sodium hypochlorite is found to be able to deactivate the novel Coronavirus on surfaces within 1 minute.A 2020 systematic review on hydrogen peroxide (H2O2) mouth-washes concludes, that they don't have an effect on virucidal activity, recommending that "dental care protocols during the COVID-19 pandemic should be revised." Additional research with relation to the Coronavirus virucidal efficacy is on-going.Various information and overview of light-based strategies (UV-C and other types of light sources; see also Ultraviolet germicidal irradiation) to combat the COVID-19 pandemic are available.As systematic review of 16 studies by Cochrane on Antimicrobial mouthwashes (gargling) and nasal sprays concludes that "there is currently no evidence relating to the benefits and risks of patients with COVID‐19 using antimicrobial mouthwashes or nasal sprays."


In the Tikhonov regularization setting, the filter function for RLS is described below. As shown in, in this setting, c = ( K + n λ I ) − 1 Y {\displaystyle c=(K+n\lambda I)^{-1}Y} . Thus, c = ( K + n λ I ) − 1 Y = Q ( Σ + n λ I ) − 1 Q T Y = ∑ i = 1 n 1 σ i + n λ < q i , Y > q i . {\displaystyle c=(K+n\lambda I)^{-1}Y=Q(\Sigma +n\lambda I)^{-1}Q^{T}Y=\sum _{i=1}^{n}{\frac {1}{\sigma _{i}+n\lambda }} q_{i}.}


When Mozilla redirected funding away from the project in 2020, it was forked by its original developers as Coqui STT using the same open-source license.Google Gboard supports speech recognition on all Android applications. It can be activated through the microphone icon.The commercial cloud based speech recognition APIs are broadly available. For more software resources, see List of speech recognition software.


In superintuitionistic and modal logics, a logic is structurally complete if every admissible rule is derivable. == References ==


In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.


In statistical classification, Bayes error rate is the lowest possible error rate for any classifier of a random outcome (into, for example, one of two categories) and is analogous to the irreducible error.A number of approaches to the estimation of the Bayes error rate exist. One method seeks to obtain analytical bounds which are inherently dependent on distribution parameters, and hence difficult to estimate. Another approach focuses on class densities, while yet another method combines and compares various classifiers.The Bayes error rate finds important use in the study of patterns and machine learning techniques.