In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function f: R d → R {\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} } that for all x , y ∈ R d {\displaystyle x,y\in \mathbb {R} ^{d}} such that x {\displaystyle x} is majorized by y {\displaystyle y} , one has that f ( x ) ≤ f ( y ) {\displaystyle f(x)\leq f(y)} . Named after Issai Schur, Schur-convex functions are used in the study of majorization. Every function that is convex and symmetric is also Schur-convex. The opposite implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments).


In UPSR, the data is transmitted in both directions, clock and counter clock wise, at the source ADM. At the destination then, both signals are compared and the best one of the two is selected. If a failure occurs then the destination just needs to switch to the unaffected path.


In telecommunication, a basic exchange telecommunications radio service (BETRS) is a commercial service that can extend telephone service to rural areas by replacing the local loop with radio communications. In the BETRS, non-government ultra high frequency (UHF) and very high frequency (VHF) common carrier and the private radio service frequencies are shared.


Convergence is usually non-monotone, that is, neither the objective function nor the residual or gradient magnitude necessarily decrease with each iteration along a successful convergence toward the solution. If f {\displaystyle f} is a quadratic function with Hessian A {\displaystyle A} , 1 / α L O N G {\displaystyle 1/\alpha ^{LONG}} is the Rayleigh quotient of A {\displaystyle A} by vector Δ x {\displaystyle \Delta x} , and 1 / α S H O R T {\displaystyle 1/\alpha ^{SHORT}} is the Rayleigh quotient of A {\displaystyle A} by vector A Δ x {\displaystyle {\sqrt {A}}\Delta x} (here taking A {\displaystyle {\sqrt {A}}} as a solution to ( A ) T A = A {\displaystyle ({\sqrt {A}})^{T}{\sqrt {A}}=A} , more at Definite matrix). Fletcher compared its computational performance to conjugate gradient (CG) methods, finding CG tending faster for linear problems, but BB often faster for non-linear problems versus applicable CG-based methods. BB has low storage requirements, suitable for large systems with millions of elements in x {\displaystyle x} . α S H O R T α L O N G = c o s 2 ( {\displaystyle {\frac {\alpha ^{SHORT}}{\alpha ^{LONG}}}=cos^{2}(} angle between Δ x {\displaystyle \Delta x} and Δ g ) {\displaystyle \Delta g)} .


The CP decomposition has found some applications in linguistics and chemometrics. The CP rank was introduced by Frank Lauren Hitchcock in 1927 and later rediscovered several times, notably in psychometrics. The CP decomposition is referred to as CANDECOMP, PARAFAC, or CANDECOMP/PARAFAC (CP). PARAFAC2 rank decomposition is yet to explore. Another popular generalization of the matrix SVD known as the higher-order singular value decomposition computes orthonormal mode matrices and has found applications in econometrics, signal processing, computer vision, computer graphics, psychometrics.


{\displaystyle \left(\pm a^{\frac {p+1}{4}}\right)^{2}=a^{\frac {p+1}{2}}=a\cdot a^{\frac {p-1}{2}}\equiv a\left({\frac {a}{p}}\right)=a{\bmod {p}}.} This formula only works if it is known in advance that a {\displaystyle a} is a quadratic residue, which can be checked using the law of quadratic reciprocity. The quadratic reciprocity theorem was conjectured by Euler and Legendre and first proved by Gauss, who referred to it as the "fundamental theorem" in his Disquisitiones Arithmeticae and his papers, writing The fundamental theorem must certainly be regarded as one of the most elegant of its type.


In numerical mathematics, artificial precision is a source of error that occurs when a numerical value or semantic is expressed with more precision than was initially provided from measurement or user input. For example, a person enters their birthday as the date 1984-01-01 but it is stored in a database as 1984-01-01T00:00:00Z which introduces the artificial precision of the hour, minute, and second they were born, and may even affect the date, depending on the user's actual place of birth. This is also an example of false precision, which is artificial precision specifically of numerical quantities or measures.


Bob proceeds to generate a string of random bits b ′ {\displaystyle b'} of the same length as b {\displaystyle b} and then measures the qubits he has received from Alice, obtaining a bit string a ′ {\displaystyle a'} . At this point, Bob announces publicly that he has received Alice's transmission. Alice then knows she can now safely announce b {\displaystyle b} , i.e., the bases in which the qubits were prepared.


In practice, we can construct one specific rank factorization as follows: we can compute B {\textstyle B} , the reduced row echelon form of A {\textstyle A} . Then C {\textstyle C} is obtained by removing from A {\textstyle A} all non-pivot columns (which can be determined by looking for columns in B {\textstyle B} which do not contain a pivot), and F {\textstyle F} is obtained by eliminating any all-zero rows of B {\textstyle B} . Note: For a full-rank square matrix (i.e. when n = m = r {\textstyle n=m=r} ), this procedure will yield the trivial result C = A {\textstyle C=A} and F = B = I n {\textstyle F=B=I_{n}} (the n × n {\textstyle n\times n} identity matrix).


In the Unicode standard, a plane is a continuous group of 65,536 (216) code points. There are 17 planes, identified by the numbers 0 to 16, which corresponds with the possible values 00–1016 of the first two positions in six position hexadecimal format (U+hhhhhh). Plane 0 is the Basic Multilingual Plane (BMP), which contains most commonly used characters.


The network does this by utilising the Authentication Center and is accomplished without transmitting the key directly. Every GSM phone contains a unique identifier (different from the phone number), called the International Mobile Equipment Identity (IMEI). This can be found by dialing *#06#. When a phone contacts the network, its IMEI may be checked against the Equipment Identity Register to locate stolen phones and facilitate monitoring.


Concretely, in a linear regression where the errors are identically distributed, the variability of residuals of inputs in the middle of the domain will be higher than the variability of residuals at the ends of the domain: linear regressions fit endpoints better than the middle. This is also reflected in the influence functions of various data points on the regression coefficients: endpoints have more influence. Thus to compare residuals at different inputs, one needs to adjust the residuals by the expected variability of residuals, which is called studentizing. This is particularly important in the case of detecting outliers, where the case in question is somehow different from the others in a dataset. For example, a large residual may be expected in the middle of the domain, but considered an outlier at the end of the domain.


These are set on objects by the higher-level storage systems that use the OSD for persistent storage. For example, attributes might be used to classify objects, or to capture relationships among different objects stored on different OSDs. A list command returns a list of identifiers for objects within a partition, optionally filtered by matches against their attribute values.


In the following, we have n kinds of items, a 1 {\displaystyle a_{1}} through a n {\displaystyle a_{n}} and m kinds of bins b 1 {\displaystyle b_{1}} through b m {\displaystyle b_{m}} . Each bin b i {\displaystyle b_{i}} is associated with a budget t i {\displaystyle t_{i}} . For a bin b i {\displaystyle b_{i}} , each item a j {\displaystyle a_{j}} has a profit p i j {\displaystyle p_{ij}} and a weight w i j {\displaystyle w_{ij}} . A solution is an assignment from items to bins.


OpenSocial Markup Language (OSML Markup) is a new set of standardized tags to accomplish common tasks or safely perform normally unsafe operations within templates. OSML is extensible. Developers can create a library of their own custom tags.


There are two ways to give consent: explicit consent or implied consent. Explicit consent is when a patient clearly communicates to a healthcare worker, verbally or in writing or in some other way, that relevant confidential information can be shared. Implied consent, means that a patient's consent to share personal confidential information is assumed.


Subsequences can contain consecutive elements which were not consecutive in the original sequence. A subsequence which consists of a consecutive run of elements from the original sequence, such as ⟨ B , C , D ⟩ , {\displaystyle \langle B,C,D\rangle ,} from ⟨ A , B , C , D , E , F ⟩ , {\displaystyle \langle A,B,C,D,E,F\rangle ,} is a substring. The substring is a refinement of the subsequence. The list of all subsequences for the word "apple" would be "a", "ap", "al", "ae", "app", "apl", "ape", "ale", "appl", "appe", "aple", "apple", "p", "pp", "pl", "pe", "ppl", "ppe", "ple", "pple", "l", "le", "e", "" (empty string).


In the case of integrating supplemental data source, KG(Knowledge graph) formally represents the meaning involved in information by describing concepts, relationships between things, and categories of things. These embedded semantics with the data offer significant advantages such as reasoning over data and dealing with heterogeneous data sources. The rules can be applied on KG more efficiently using graph query.


In number theory, the Chinese hypothesis is a disproven conjecture stating that an integer n is prime if and only if it satisfies the condition that 2 n − 2 {\displaystyle 2^{n}-2} is divisible by n—in other words, that an integer n is prime if and only if 2 n ≡ 2 mod n {\displaystyle 2^{n}\equiv 2{\bmod {n}}} . It is true that if n is prime, then 2 n ≡ 2 mod n {\displaystyle 2^{n}\equiv 2{\bmod {n}}} (this is a special case of Fermat's little theorem), however the converse (if 2 n ≡ 2 mod n {\displaystyle 2^{n}\equiv 2{\bmod {n}}} then n is prime) is false, and therefore the hypothesis as a whole is false. The smallest counterexample is n = 341 = 11×31. Composite numbers n for which 2 n − 2 {\displaystyle 2^{n}-2} is divisible by n are called Poulet numbers. They are a special class of Fermat pseudoprimes.


Now if the domain is a set, the function comprehension principle, also called axiom of unique choice or non-choice, says that a function as a set, with some codomain, exists well. (And this principle is valid in a theory like C Z F {\displaystyle {\mathsf {CZF}}} . Also compare with the Replacement axiom.)


In 1912, British engineer Arthur Pollen developed the first electrically powered mechanical analogue computer (called at the time the Argo Clock). It was used by the Imperial Russian Navy in World War I. The alternative Dreyer Table fire control system was fitted to British capital ships by mid-1916. Mechanical devices were also used to aid the accuracy of aerial bombing.


In robot motion planning, a Pfaffian constraint is a set of k linearly independent constraints linear in velocity, i.e., of the form One source of Pfaffian constraints is rolling without slipping in wheeled robots. == References ==


In mathematics, a relative scalar (of weight w) is a scalar-valued function whose transform under a coordinate transform, on an n-dimensional manifold obeys the following equation where that is, the determinant of the Jacobian of the transformation. A scalar density refers to the w = 1 {\displaystyle w=1} case. Relative scalars are an important special case of the more general concept of a relative tensor.


In 1969 Frank DeRemer invented the LALR and Simple LR parsers, both based on the LR parser and having greatly reduced memory requirements at the cost of less language recognition power. The LALR parser was the stronger alternative. These two parsers have since been widely used in compilers of many computer languages. Recent research has identified methods by which canonical LR parsers may be implemented with dramatically reduced table requirements over Knuth's table-building algorithm.


Branching processes can also be used to model other systems with similar dynamics, e.g., the spread of surnames in genealogy or the propagation of neutrons in a nuclear reactor. A central question in the theory of branching processes is the probability of ultimate extinction, where no individuals exist after some finite number of generations. Using Wald's equation, it can be shown that starting with one individual in generation zero, the expected size of generation n equals μn where μ is the expected number of children of each individual.


In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desarguesian planes, and the vast majority of known non-Desarguesian planes are either translation planes, or can be obtained from a translation plane via successive iterations of dualization and/or derivation.In a projective plane, let P represent a point, and l represent a line. A central collineation with center P and axis l is a collineation fixing every point on l and every line through P. It is called an elation if P is on l, otherwise it is called a homology. The central collineations with center P and axis l form a group.


In mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, currying a function f {\displaystyle f} that takes three arguments creates a nested unary function g {\displaystyle g} , so that the code let x = f ( a , b , c ) {\displaystyle {\text{let }}x=f(a,b,c)} gives x {\displaystyle x} the same value as the code let h = g ( a ) let i = h ( b ) let x = i ( c ) , {\displaystyle {\begin{aligned}{\text{let }}h=g(a)\\{\text{let }}i=h(b)\\{\text{let }}x=i(c),\end{aligned}}} or called in sequence, let x = g ( a ) ( b ) ( c ) . {\displaystyle {\text{let }}x=g(a)(b)(c).} In a more mathematical language, a function that takes two arguments, one from X {\displaystyle X} and one from Y {\displaystyle Y} , and produces outputs in Z , {\displaystyle Z,} by currying is translated into a function that takes a single argument from X {\displaystyle X} and produces as outputs functions from Y {\displaystyle Y} to Z .


In mathematical analysis, the maximum and minimum of a function are, respectively, the largest and smallest value taken by the function. Known generically as extremum, they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.


In socio-hydrology, it is often assumed that societies build flood memory after extreme events. Flood memory is considered as a primary mechanism explaining the emergence of levee effects. It is hyphosised to be built after flooding and proportional to associated losses. Flood memory does decay over time. It is very difficult to observe, so proxy variable such as flood insurance coverage are used.


. , k − 1 {\displaystyle 1,2,...,k-1} , find the best cover that does not violate the budget. Call this cover H 1 {\displaystyle H_{1}} .


In quantum mechanics, different possibilities can cancel. In probability theory with a finite number of states, the probabilities can always be multiplied by a positive number to make their sum equal to one.


In the Germanic languages, a strong verb is a verb that marks its past tense by means of changes to the stem vowel. The majority of the remaining verbs form the past tense by means of a dental suffix, and are known as weak verbs. In modern English, strong verbs include sing (present I sing, past I sang, past participle I have sung) and drive (present I drive, past I drove, past participle I have driven), as opposed to weak verbs such as open (present I open, past I opened, past participle I have opened). Not all verbs with a change in the stem vowel are strong verbs, however; they may also be irregular weak verbs such as bring, brought, brought or keep, kept, kept.


Next is that the errors in behavior suggest internal plans for what will be done later. Also, the time to initiate a movement sequence can increase with the length or complexity of the sequence. The next line is the properties of movements occurring early in a sequence can anticipate later features.


In mathematics, especially in order theory, a maximal element of a subset S of some preordered set is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some preordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum. The maximum of a subset S {\displaystyle S} of a preordered set is an element of S {\displaystyle S} which is greater than or equal to any other element of S , {\displaystyle S,} and the minimum of S {\displaystyle S} is again defined dually. In the particular case of a partially ordered set, while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements. Specializing further to totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide.


In music theory, the scale degree is the position of a particular note on a scale relative to the tonic—the first and main note of the scale from which each octave is assumed to begin. Degrees are useful for indicating the size of intervals and chords and whether an interval is major or minor. In the most general sense, the scale degree is the number given to each step of the scale, usually starting with 1 for tonic. Defining it like this implies that a tonic is specified.


In mathematics, informally speaking, Euclid's orchard is an array of one-dimensional "trees" of unit height planted at the lattice points in one quadrant of a square lattice. More formally, Euclid's orchard is the set of line segments from (x, y, 0) to (x, y, 1), where x and y are positive integers. The trees visible from the origin are those at lattice points (x, y, 0), where x and y are coprime, i.e., where the fraction x/y is in reduced form. The name Euclid's orchard is derived from the Euclidean algorithm.


For this problem: Auletta, De Prisco, Penna and Persiano presented a 4-approximation monotone algorithm, which runs in polytime when the number of machines is fixed. Ambrosio and Auletta proved that the Longest Processing Time algorithm is monotone whenever the machine speeds are powers of some c ≥ 2, but not when c ≤ 1.78. In contrast, List scheduling is not monotone for c > 2. Andelman, Azar and Sorani presented a 5-approximation monotone algorithm, which runs in polytime even when the number of machines is variable. Kovacz presented a 3-approximation monotone algorithm.


In the first and second editions of A Dictionary of Modern English Usage Fowler uses the heading false scent to explain writing that causes the reader to second-guess: because the writer knows what is coming ahead, he may forget that his reader does not, and unwittingly "lay false scent" by writing something ambiguous that can only be disambiguated later in the text (for example "I looked at the man with the telescope, and watched him put the telescope away"). The reader, once he realises he has been distracted, must go back and rescan the sentence or paragraph to understand the writer's intended meaning. == References ==


This can be compared with the classical discrete Fourier transform, which takes O ( n 2 n ) {\displaystyle O(n2^{n})} gates (where n {\displaystyle n} is the number of bits), which is exponentially more than O ( n 2 ) {\displaystyle O(n^{2})} . The quantum Fourier transform acts on a quantum state vector (a quantum register), and the classical Fourier transform acts on a vector. Both types of vectors can be written as lists of complex numbers.


In order for the conditions for atomic broadcast to be satisfied, the participants must effectively "agree" on the order of receipt of the messages. Participants recovering from failure, after the other participants have "agreed" an order and started to receive the messages, must be able to learn and comply with the agreed order. Such considerations indicate that in systems with crash failures, atomic broadcast and consensus are equivalent problems.A value can be proposed by a process for consensus by atomically broadcasting it, and a process can decide a value by selecting the value of the first message which it atomically receives. Thus, consensus can be reduced to atomic broadcast.


In mathematical finite group theory, a group of GF(2)-type is a group with an involution centralizer whose generalized Fitting subgroup is a group of symplectic type (Gorenstein 1982, definition 1.45). As the name suggests, many of the groups of Lie type over the field with 2 elements are groups of GF(2)-type. Also 16 of the 26 sporadic groups are of GF(2)-type, suggesting that in some sense sporadic groups are somehow related to special properties of the field with 2 elements. Timmesfeld (1978) showed roughly that groups of GF(2)-type can be subdivided into 8 types.


In mathematical optimization, the cutting-plane method is any of a variety of optimization methods that iteratively refine a feasible set or objective function by means of linear inequalities, termed cuts. Such procedures are commonly used to find integer solutions to mixed integer linear programming (MILP) problems, as well as to solve general, not necessarily differentiable convex optimization problems. The use of cutting planes to solve MILP was introduced by Ralph E. Gomory. Cutting plane methods for MILP work by solving a non-integer linear program, the linear relaxation of the given integer program.


In probability theory, conditional independence describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. Conditional independence is usually formulated in terms of conditional probability, as a special case where the probability of the hypothesis given the uninformative observation is equal to the probability without. If A {\displaystyle A} is the hypothesis, and B {\displaystyle B} and C {\displaystyle C} are observations, conditional independence can be stated as an equality: P ( A ∣ B , C ) = P ( A ∣ C ) {\displaystyle P(A\mid B,C)=P(A\mid C)} where P ( A ∣ B , C ) {\displaystyle P(A\mid B,C)} is the probability of A {\displaystyle A} given both B {\displaystyle B} and C {\displaystyle C} . Since the probability of A {\displaystyle A} given C {\displaystyle C} is the same as the probability of A {\displaystyle A} given both B {\displaystyle B} and C {\displaystyle C} , this equality expresses that B {\displaystyle B} contributes nothing to the certainty of A {\displaystyle A} .


In queueing theory, a discipline within the mathematical theory of probability, a heavy traffic approximation (sometimes heavy traffic limit theorem or diffusion approximation) is the matching of a queueing model with a diffusion process under some limiting conditions on the model's parameters. The first such result was published by John Kingman who showed that when the utilisation parameter of an M/M/1 queue is near 1 a scaled version of the queue length process can be accurately approximated by a reflected Brownian motion.


In the basic SIP specification, only requests and final responses (i.e. 2XX response codes) are transmitted reliably, this is, they are retransmitted by the sender until the acknowledge message arrives (i.e. the corresponding response code to a request, or the ACK request corresponding to a 2XX response code). This mechanism is necessary since SIP can run not only over reliable transport protocols (TCP) that assure that the message is delivered, but also over unreliable ones (UDP) that offer no delivery guarantees, and it is even possible that both kinds of protocols are present in different parts of the transport network. However, in such an scenario as the IMS framework, it is necessary to extend this reliability to provisional responses to INVITE requests (for session establishment, this is, to start a call). The reliability of provisional responses extension provides a mechanism to confirm that provisional responses such as the 180 Ringing response code, that lets the caller know that the callee is being alerted, are successfully received.


Two regularization parameters are used in this framework: λ {\displaystyle \lambda } for the estimation of C ^ Y X π , C ^ X X π = Υ D Υ T {\displaystyle {\widehat {\mathcal {C}}}_{YX}^{\pi },{\widehat {\mathcal {C}}}_{XX}^{\pi }={\boldsymbol {\Upsilon }}\mathbf {D} {\boldsymbol {\Upsilon }}^{T}} and λ ~ {\displaystyle {\widetilde {\lambda }}} for the estimation of the final conditional embedding operator C ^ Y ∣ X π = C ^ Y X π ( ( C ^ X X π ) 2 + λ ~ I ) − 1 C ^ X X π . {\displaystyle {\widehat {\mathcal {C}}}_{Y\mid X}^{\pi }={\widehat {\mathcal {C}}}_{YX}^{\pi }\left(\left({\widehat {\mathcal {C}}}_{XX}^{\pi }\right)^{2}+{\widetilde {\lambda }}\mathbf {I} \right)^{-1}{\widehat {\mathcal {C}}}_{XX}^{\pi }.} The latter regularization is done on square of C ^ X X π {\displaystyle {\widehat {\mathcal {C}}}_{XX}^{\pi }} because D {\displaystyle D} may not be positive definite.


On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory. Due to their general nature and versatility, sheaves have several applications in topology and especially in algebraic and differential geometry.


As an example, the greatest common divisor of 15 and 69 is 3, and 3 can be written as a combination of 15 and 69 as 3 = 15 × (−9) + 69 × 2, with Bézout coefficients −9 and 2. Many other theorems in elementary number theory, such as Euclid's lemma or the Chinese remainder theorem, result from Bézout's identity. A Bézout domain is an integral domain in which Bézout's identity holds. In particular, Bézout's identity holds in principal ideal domains. Every theorem that results from Bézout's identity is thus true in all principal ideal domains.


In other words, each quantifier is a family of properties on dom(A), so each is called a monadic quantifier. Any quantifier defined as an n > 0-ary relation between properties on dom(A) is called monadic. Lindström introduced polyadic ones that are n > 0-ary relations between relations on domains of structures.


In mathematics in general, a characterization theorem says that a particular object – a function, a space, etc. – is the only one that possesses properties specified in the theorem. A characterization of a probability distribution accordingly states that it is the only probability distribution that satisfies specified conditions. More precisely, the model of characterization of probability distribution was described by V.M. Zolotarev in such manner.


While this allows non-standard interpretations of symbols such as + {\displaystyle +} , one can restrict their meaning by providing additional axioms. The satisfiability modulo theories problem considers satisfiability of a formula with respect to a formal theory, which is a (finite or infinite) set of axioms.


In non-relational systems, hierarchical databases, the distant counterpart of a table is a structured file, representing the rows of a table in each row of the file and each column in a row. This structure implies that a row can have repeating information, generally in the child data segments. Data are stored in sequence of physical records.


In the field of retrieval systems, established approaches include: Data retrieval systems, such as database management systems, are well suitable for the storage and retrieval of structured data. Information retrieval systems, such as web search engines, are very effective in finding the relevant documents or web pages.Both approaches require a user to read and analyze often long lists of data sets or documents in order to extract meaning. The goal of knowledge retrieval systems is to reduce the burden of those processes by improved search and representation. This improvement is needed to leverage the increasing data volumes available on the Internet.


Most such devices include a tiny postage-stamp-sized LCD screen for viewing simplified ladder logic (only a very small portion of the program being visible at a given time) and status of I/O points, and typically these screens are accompanied by a 4-way rocker push-button plus four more separate push-buttons, similar to the key buttons on a VCR remote control, and used to navigate and edit the logic. Most have a small plug for connecting via RS-232 or RS-485 to a personal computer so that programmers can use simple applications in general-purpose OS like MS Windows, macOS or Linux, that have user-friendly (G)UIs, for programming instead of being forced to use the tiny LCD and push-button set for this purpose. Unlike regular PLCs that are usually modular and greatly expandable, the PLRs are usually not modular or expandable, but their price can be two orders of magnitude less than a PLC, and they still offer robust design and deterministic execution of the logic.


In mathematics and geographic information science, a shortest-path graph is an undirected graph defined from a set of points in the Euclidean plane. The shortest-path graph is proposed with the idea of inferring edges between a point set such that the shortest path taken over the inferred edges will roughly align with the shortest path taken over the imprecise region represented by the point set. The edge set of the shortest-path graph varies based on a single parameter t ≥ 1. When the weight of an edge is defined as its Euclidean length raised to the power of the parameter t ≥ 1, the edge is present in the shortest-path graph if and only if it is the least weight path between its endpoints.


There are also examples for which the minimal number of cells is doubly exponential, showing that every general algorithm for cylindrical algebraic decomposition has a double exponential complexity. CAD provides an effective version of quantifier elimination over the reals that has a much better computational complexity than that resulting from the original proof of Tarski–Seidenberg theorem. It is efficient enough to be implemented on a computer. It is one of the most important algorithms of computational real algebraic geometry. Searching to improve Collins' algorithm, or to provide algorithms that have a better complexity for subproblems of general interest, is an active field of research.


{\displaystyle \textstyle {\frac {n!}{k!(n-k)!}}} whenever k ≤ n {\displaystyle k\leq n} , and which is zero when k > n {\displaystyle k>n} . This formula can be derived from the fact that each k-combination of a set S of n members has k !


In practice, Wilson's theorem is useless as a primality test because computing (n − 1)! modulo n for large n is computationally complex, and much faster primality tests are known (indeed, even trial division is considerably more efficient).Used in the other direction, to determine the primality of the successors of large factorials, it is indeed a very fast and effective method. This is of limited utility, however.


In mathematics, and in particular game theory, Sion's minimax theorem is a generalization of John von Neumann's minimax theorem, named after Maurice Sion. It states: Let X {\displaystyle X} be a compact convex subset of a linear topological space and Y {\displaystyle Y} a convex subset of a linear topological space. If f {\displaystyle f} is a real-valued function on X × Y {\displaystyle X\times Y} with f ( x , ⋅ ) {\displaystyle f(x,\cdot )} upper semicontinuous and quasi-concave on Y {\displaystyle Y} , ∀ x ∈ X {\displaystyle \forall x\in X} , and f ( ⋅ , y ) {\displaystyle f(\cdot ,y)} lower semicontinuous and quasi-convex on X {\displaystyle X} , ∀ y ∈ Y {\displaystyle \forall y\in Y} then, min x ∈ X sup y ∈ Y f ( x , y ) = sup y ∈ Y min x ∈ X f ( x , y ) . {\displaystyle \min _{x\in X}\sup _{y\in Y}f(x,y)=\sup _{y\in Y}\min _{x\in X}f(x,y).}


In set theory, #S is one possible notation for the cardinality or size of the set S, instead of | S | {\displaystyle |S|} . That is, for a set S = { s 1 , s 2 , s 3 , … , s n } {\displaystyle S=\{s_{1},s_{2},s_{3},\dots ,s_{n}\}} , in which all s i {\displaystyle s_{i}} are mutually distinct, # S = n = | S | . {\displaystyle \#S=n=|S|.}


The same argument repeated (by symmetry of the problem) is valid when ω {\displaystyle \omega } starts with a negative rotation about the z axis, or a rotation about the x axis. This shows that if ω {\displaystyle \omega } is given by a non-trivial word in A and B, then ω ≠ e {\displaystyle \omega \neq e} . Therefore, the group H is a free group, isomorphic to F2.


In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } , the matrix A can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix L and an upper triangular matrix U. The systems L ( U x ) = b {\displaystyle L(U\mathbf {x} )=\mathbf {b} } and U x = L − 1 b {\displaystyle U\mathbf {x} =L^{-1}\mathbf {b} } require fewer additions and multiplications to solve, compared with the original system A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } , though one might require significantly more digits in inexact arithmetic such as floating point. Similarly, the QR decomposition expresses A as QR with Q an orthogonal matrix and R an upper triangular matrix. The system Q(Rx) = b is solved by Rx = QTb = c, and the system Rx = c is solved by 'back substitution'. The number of additions and multiplications required is about twice that of using the LU solver, but no more digits are required in inexact arithmetic because the QR decomposition is numerically stable.


In the context of network theory a scale-free ideal network is a random network with a degree distribution following the scale-free ideal gas density distribution. These networks are able to reproduce city-size distributions and electoral results by unraveling the size distribution of social groups with information theory on complex networks when a competitive cluster growth process is applied to the network. In models of scale-free ideal networks it is possible to demonstrate that Dunbar's number is the cause of the phenomenon known as the 'six degrees of separation'.


In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed.


Due to this superposition, measurement of the qubit will "collapse" it into one of its basis states with a given probability. Because of the entanglement, measurement of one qubit will "collapse" the other qubit to a state whose measurement will yield one of two possible values, where the value depends on which Bell's state the two qubits are in initially. Bell's states can be generalized to certain quantum states of multi-qubit systems, such as the GHZ state for 3 or more subsystems. Understanding of Bell's states is useful in analysis of quantum communication, such as superdense coding and quantum teleportation. The no-communication theorem prevents this behavior from transmitting information faster than the speed of light.


In contrast to the steady incremental improvements of the past few decades, the application of deep learning decreased word error rate by 30%. This innovation was quickly adopted across the field.


In metric graph theory, a convex subgraph of an undirected graph G is a subgraph that includes every shortest path in G between two of its vertices. Thus, it is analogous to the definition of a convex set in geometry, a set that contains the line segment between every pair of its points. Convex subgraphs play an important role in the theory of partial cubes and median graphs. In particular, in median graphs, the convex subgraphs have the Helly property: if a family of convex subgraphs has the property that all pairwise intersections are nonempty, then the whole family has a nonempty intersection.


In principle, a 64-bit microprocessor can address 16 EiB (16 × 10246 = 264 = 18,446,744,073,709,551,616 bytes, or about 18.4 exabytes) of memory. However, not all instruction sets, and not all processors implementing those instruction sets, support a full 64-bit virtual or physical address space. The x86-64 architecture (as of 2016) allows 48 bits for virtual memory and, for any given processor, up to 52 bits for physical memory. These limits allow memory sizes of 256 TiB (256 × 10244 bytes) and 4 PiB (4 × 10245 bytes), respectively.


In mathematics, pentation (or hyper-5) is the next hyperoperation after tetration and before hexation. It is defined as iterated (repeated) tetration (assuming right-associativity), just as tetration is iterated right-associative exponentiation. It is a binary operation defined with two numbers a and b, where a is tetrated to itself b-1 times. For instance, using hyperoperation notation for pentation and tetration, 2  3 {\displaystyle 23} means tetrating 2 to itself 2 times, or 2  ( 2  2 ) {\displaystyle 2(22)} . This can then be reduced to 2  ( 2 2 ) = 2  4 = 2 2 2 2 = 2 2 4 = 2 16 = 65 , 536. {\displaystyle 2(2^{2})=24=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65,536.}


In sociology, command hierarchy is seen as the most visible element of a "power network." In this model, social capital is viewed as being mobilized in response to orders that move through the hierarchy leading to the phrase "command and control".


Later it was demonstrated that all basic mathematical structures either are some kinds of named sets or are built of named sets. According to Anellis, Burgin & Kaloujnine introduced set-theoretical named sets in 1983 and Burgin introduced named sets in the most general form in 1990. Since then Burgin continued to develop this theory in a series of papers and a book. In 2011, Zellweger applied the theory of named sets to model data relations in the relational database for an end-user interface.


Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of number of elements. In the case of infinite sets, the behavior is more complex.


In the area of modern algebra known as group theory, the Suzuki groups, denoted by Sz(22n+1), 2B2(22n+1), Suz(22n+1), or G(22n+1), form an infinite family of groups of Lie type found by Suzuki (1960), that are simple for n ≥ 1. These simple groups are the only finite non-abelian ones with orders not divisible by 3.


This eliminates noncredible threats, which are threats that a player would not carry out if they were ever called upon to do so. For example, consider a dynamic game with an incumbent firm and a potential entrant to the industry. The incumbent has a monopoly and wants to maintain its market share.


In nonstandard analysis, a branch of mathematics, overspill (referred to as overflow by Goldblatt (1998, p. 129)) is a widely used proof technique. It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set *N of hypernatural numbers. By applying the induction principle for the standard integers N and the transfer principle we get the principle of internal induction: For any internal subset A of *N, if 1 is an element of A, and for every element n of A, n + 1 also belongs to A,then A = *NIf N were an internal set, then instantiating the internal induction principle with N, it would follow N = *N which is known not to be the case.


In this case, the voting circuit can output the correct result, and discard the erroneous version. After this, the internal state of the erroneous replication is assumed to be different from that of the other two, and the voting circuit can switch to a DMR mode. This model can be applied to any larger number of replications.


In statistics, binary data is a statistical data type consisting of categorical data that can take exactly two possible values, such as "A" and "B", or "heads" and "tails". It is also called dichotomous data, and an older term is quantal data. The two values are often referred to generically as "success" and "failure". As a form of categorical data, binary data is nominal data, meaning the values are qualitatively different and cannot be compared numerically.


In network theory, the Wiener connector is a means of maximizing efficiency in connecting specified "query vertices" in a network. Given a connected, undirected graph and a set of query vertices in a graph, the minimum Wiener connector is an induced subgraph that connects the query vertices and minimizes the sum of shortest path distances among all pairs of vertices in the subgraph. In combinatorial optimization, the minimum Wiener connector problem is the problem of finding the minimum Wiener connector. It can be thought of as a version of the classic Steiner tree problem (one of Karp's 21 NP-complete problems), where instead of minimizing the size of the tree, the objective is to minimize the distances in the subgraph.The minimum Wiener connector was first presented by Ruchansky et al. in 2015.The minimum Wiener connector has applications in many domains where there is a graph structure and an interest in learning about connections between sets of individuals.


Early scrambling or encryption methods required a hard line for authorization of receive sites. Today, a digital cellular telephone is sufficient for most situations. C-Band transportable service remains a prevalent source of long-haul transmission because of its immunity to the "rain fade" that Ku band experiences in significant rainstorms.


In principle, there can be more than one such code for a given word length, but the term Gray code was first applied to a particular binary code for non-negative integers, the binary-reflected Gray code, or BRGC. Bell Labs researcher George R. Stibitz described such a code in a 1941 patent application, granted in 1943. Frank Gray introduced the term reflected binary code in his 1947 patent application, remarking that the code had "as yet no recognized name". He derived the name from the fact that it "may be built up from the conventional binary code by a sort of reflection process".


In mathematics, a surjective function (also known as surjection, or onto function ) is a function f such that every element y can be mapped from some element x such that f(x) = y. In other words, every element of the function's codomain is the image of at least one element of its domain. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y. The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word sur means over or above, and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by restricting its codomain to the image of its domain.


. . , n } → { 0 , 1 } {\displaystyle f:\{0,1,...,n\}\rightarrow \{0,1\}} . Symmetric Boolean functions are used to classify Boolean satisfiability problems.


In mathematics, in the areas of group theory and combinatorics, Hall words provide a unique monoid factorisation of the free monoid. They are also totally ordered, and thus provide a total order on the monoid. This is analogous to the better-known case of Lyndon words; in fact, the Lyndon words are a special case, and almost all properties possessed by Lyndon words carry over to Hall words. Hall words are in one-to-one correspondence with Hall trees.


As told by Hardy: I remember once going to see him  when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two  cubes in two different ways."


In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insufficient to alter the behavior of the rest of the system. The concept of a test particle often simplifies problems, and can provide a good approximation for physical phenomena. In addition to its uses in the simplification of the dynamics of a system in particular limits, it is also used as a diagnostic in computer simulations of physical processes.


Kernel adaptive filters implement a nonlinear transfer function using kernel methods. In these methods, the signal is mapped to a high-dimensional linear feature space and a nonlinear function is approximated as a sum over kernels, whose domain is the feature space. If this is done in a reproducing kernel Hilbert space, a kernel method can be a universal approximator for a nonlinear function.


A semigroup without an identity element can be easily turned into a monoid by just adding an identity element. Consequently, monoids are studied in the theory of semigroups rather than in group theory. Semigroups should not be confused with quasigroups, which are a generalization of groups in a different direction; the operation in a quasigroup need not be associative but quasigroups preserve from groups a notion of division.


In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers R ≥ 0, but in distribution functions.Let D+ be the set of all probability distribution functions F such that F(0) = 0 (F is a nondecreasing, left continuous mapping from R into  such that max(F) = 1). Then given a non-empty set S and a function F: S × S → D+ where we denote F(p, q) by Fp,q for every (p, q) ∈ S × S, the ordered pair (S, F) is said to be a probabilistic metric space if: For all u and v in S, u = v if and only if Fu,v(x) = 1 for all x > 0. For all u and v in S, Fu,v = Fv,u. For all u, v and w in S, Fu,v(x) = 1 and Fv,w(y) = 1 ⇒ Fu,w(x + y) = 1 for x, y > 0.


In modern news, social media has become a major source of amateur reporting. The Arab Spring is believed to have been aided by citizen journalism that was reported using and disseminated by Facebook and Twitter. The Top 15 Most Popular Social Media Sites range from 15 million users to 900 million users, and as their user base grows stronger, amateur reporting's base expands as well.


In particular, this problem arises when we attempt to build a so-called RASP, a "universal machine" (see more at Universal Turing machine) that uses its finite-state machine to interpret a "program of instructions" located in its registers – i.e. we are building what is nowadays called a computer with the von Neumann architecture.Observe that the counter machine's finite state machine must call out a register explicitly (directly) by its name/number: INC (65,356) calls out register number "65,365" explicitly. If the number of registers exceeds the capability of the finite state machine to address them, then registers outside the bounds will be unreachable. For example, if the finite state machine can only reach 65,536 = 216 registers then how can it reach the 65,537th?So how do we address a register beyond the bounds of the finite state machine?


In order to achieve a precise and effective typestate analysis, it is necessary to address the problem of aliasing. Aliasing occurs when an object has more than one reference or pointer that points to it. For the analysis to be correct, state changes to a given object must be reflected in all references that point to that object, but in general it is a difficult problem to track all such references. This becomes especially hard if the analysis needs to be modular, that is, applicable to each part of a large program separately without taking the rest of the program into account.


The Coordinating Committee admitted that they "wrestled with articulating a business case for implementing RDA", nevertheless the report recommended that RDA be adopted by the three national libraries, contingent on several improvements being made. The earliest possible date for implementation was given as January 2013, as the consensus emerging from the analysis of the test data showed that while there were discernible benefits to implementing RDA, these benefits would not be realized without further changes to current cataloging practices, including developing a successor to the MARC format.Several other institutions were involved in the RDA test. Many of these institutions documented their findings in a special issue of Cataloging & Classification Quarterly.


In supervised learning, a random sub-sample of all records is taken and manually classified as either 'fraudulent' or 'non-fraudulent' (task can be decomposed on more classes to meet algorithm requirements). Relatively rare events such as fraud may need to be over sampled to get a big enough sample size. These manually classified records are then used to train a supervised machine learning algorithm. After building a model using this training data, the algorithm should be able to classify new records as either fraudulent or non-fraudulent.


In the cellular phone industry, mobile phones and their networks sometimes support concatenated short message service (or concatenated SMS) to overcome the limitation on the number of characters that can be sent in a single SMS text message transmission (which is usually 160). Using this method, long messages are split into smaller messages by the sending device and recombined at the receiving end. Each message is then billed separately.


In set theory, functions are identified with their function graphs. Using set builder notation, a collection of pairs may be characterized, f := { ⟨ x , y ⟩ ∣ x ∈ A ∧ ψ ( x , y ) } . {\displaystyle f:={\big \{}\langle x,y\rangle \mid x\in A\land \psi (x,y){\big \}}.} The axiom of replacement in Zermelo–Fraenkel set theory implies that this is actually a set and a function in the above sense.


This notation has been retained in operating systems that were directly or indirectly derived from CP/M, including DR-DOS, MS-DOS, OS/2 and Windows. On Linux systems, the command hexcat produces this classic output format too.


In the fields of databases and transaction processing (transaction management), a schedule (or history) of a system is an abstract model to describe execution of transactions running in the system. Often it is a list of operations (actions) ordered by time, performed by a set of transactions that are executed together in the system. If the order in time between certain operations is not determined by the system, then a partial order is used. Examples of such operations are requesting a read operation, reading, writing, aborting, committing, requesting a lock, locking, etc. Not all transaction operation types should be included in a schedule, and typically only selected operation types (e.g., data access operations) are included, as needed to reason about and describe certain phenomena. Schedules and schedule properties are fundamental concepts in database concurrency control theory.


In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesian product of two partially ordered sets are (see Fig.4): the lexicographical order: (a, b) ≤ (c, d) if a < c or (a = c and b ≤ d); the product order: (a, b) ≤ (c, d) if a ≤ c and b ≤ d; the reflexive closure of the direct product of the corresponding strict orders: (a, b) ≤ (c, d) if (a < c and b < d) or (a = c and b = d).All three can similarly be defined for the Cartesian product of more than two sets. Applied to ordered vector spaces over the same field, the result is in each case also an ordered vector space. See also orders on the Cartesian product of totally ordered sets.


In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function f: X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph.This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily continuous. One particularly well-known class of closed graph theorems are the closed graph theorems in functional analysis.


Attack - An attempt by a player to win a point by hitting the ball over the net. Attack line - In indoor volleyball, a line three metres from the net which marks the limit for where a back-row player may advance to hit a ball from above the net. Back-row player - In indoor volleyball, any of three players positioned at the back of the court.


Army used in for tactical group messaging in its Force Battle Command Brigade and Below (FBCB2) system.Several other approaches to reliable multicast were being developed at approximately the same time, and in April 1999, the IETF chartered the Reliable Multicast Transport Working Group (RMTWG) to standardize reliable multicast transport.The RMTWG pursued the strategy of developing building blocks and protocol instantiations. This strategy avoided a "one size fits all" protocol, which in turn could accommodate the large number of applications and types of applications that reliable multicast could support. Building blocks were defined as “a set of easily-separable coarse-grained modular components that are common to multiple protocols along with abstract APIs that define a building block's access methods and their arguments.” Initial building blocks included negative acknowledgments, forward error correction, a generic signaling mechanism for router assist, and transport protection Protocol instantiations were defined as “specifications that define the necessary gluing logic and minimal additional functionality required to realize a working protocol from one or more building blocks.” Those specifications would also include an abstract API that defined the interface between the protocol implementation and an application. Two protocol instantiations were chosen: A NACK-based protocol An Asynchronous Layered Coding protocolIn July 2005 the NACK-based protocol building blocks and protocol instantiation were submitted as “Experimental” in RFC 3940, and in November 2009 “NACK-Oriented Reliable Multicast (NORM) Transport Protocol” was approved in RFC 5740.The RMTWG was disestablished in September 2013.


In probability and statistics, Student's t-distribution (or simply the t-distribution) t ν {\displaystyle t_{\nu }} is a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped. However, t ν {\displaystyle t_{\nu }} has heavier tails and the amount of probability mass in the tails is controlled by the parameter ν {\displaystyle \nu } .


In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as q p ( a ) = a p − 1 − 1 p , {\displaystyle q_{p}(a)={\frac {a^{p-1}-1}{p}},} or δ p ( a ) = a − a p p {\displaystyle \delta _{p}(a)={\frac {a-a^{p}}{p}}} .This article is about the former; for the latter see p-derivation. The quotient is named after Pierre de Fermat. If the base a is coprime to the exponent p then Fermat's little theorem says that qp(a) will be an integer. If the base a is also a generator of the multiplicative group of integers modulo p, then qp(a) will be a cyclic number, and p will be a full reptend prime.


The challenge is to reduce the number of colors from n to, e.g., Δ + 1. The more colors are employed, e.g. O(Δ) instead of Δ + 1, the fewer communication rounds are required.A straightforward distributed version of the greedy algorithm for (Δ + 1)-coloring requires Θ(n) communication rounds in the worst case − information may need to be propagated from one side of the network to another side. The simplest interesting case is an n-cycle.


In the 1993 film Jurassic Park, Connection Machines (non-functioning dummies) are visible in the park's control room, programmer Dennis Nedry mentions "eight Connection Machines" and a video about dinosaur cloning mentions "Thinking Machines supercomputers". In the 1996 film Mission Impossible, Luther Stickell asks Franz Krieger for "Thinking Machine laptops" to help hack into the CIA's Langley supercomputer.Tom Clancy's novel Rainbow Six speaks of the NSA's "star machine from a company gone bankrupt, the Super-Connector from Thinking Machines, Inc., of Cambridge, Massachusetts" in the NSA's basement. In addition, in The Bear and the Dragon says the National Security Agency could crack nearly any book or cipher with one of three custom operating systems designed for a Thinking Machines supercomputer. In the 2008 video game Fallout 3, it is mentioned that the pre-war firm that made the computer systems for Vaults is called Think Machine.


In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers R {\displaystyle \mathbb {R} } , sometimes called the continuum. It is an infinite cardinal number and is denoted by c {\displaystyle {\mathfrak {c}}} (lowercase Fraktur "c") or | R | {\displaystyle |\mathbb {R} |} .The real numbers R {\displaystyle \mathbb {R} } are more numerous than the natural numbers N {\displaystyle \mathbb {N} } . Moreover, R {\displaystyle \mathbb {R} } has the same number of elements as the power set of N . {\displaystyle \mathbb {N} .}


In mathematics, a Gödel numbering for sequences provides an effective way to represent each finite sequence of natural numbers as a single natural number. While a set theoretical embedding is surely possible, the emphasis is on the effectiveness of the functions manipulating such representations of sequences: the operations on sequences (accessing individual members, concatenation) can be "implemented" using total recursive functions, and in fact by primitive recursive functions. It is usually used to build sequential “data types” in arithmetic-based formalizations of some fundamental notions of mathematics. It is a specific case of the more general idea of Gödel numbering. For example, recursive function theory can be regarded as a formalization of the notion of an algorithm, and can be regarded as a programming language to mimic lists by encoding a sequence of natural numbers in a single natural number.


In mathematical finite group theory, the uniqueness case is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem. The uniqueness case covers groups G of characteristic 2 type with e(G) ≥ 3 that have an almost strongly p-embedded maximal 2-local subgroup for all primes p whose 2-local p-rank is sufficiently large (usually at least 3). Aschbacher (1983a, 1983b) proved that there are no finite simple groups in the uniqueness case.


However, since only Alice knows b {\displaystyle b} , it makes it virtually impossible for either Bob or Eve to distinguish the states of the qubits. Also, after Bob has received the qubits, we know that Eve cannot be in possession of a copy of the qubits sent to Bob, by the no-cloning theorem, unless she has made measurements. Her measurements, however, risk disturbing a particular qubit with probability ½ if she guesses the wrong basis.


In regards to issues with connectivity, it is important to consider the musical score of composite films. Much like visual repetition or thematic similarities, the score offers another medium through which stories can be linked. Rather than leaving viewers with content to process, decode, intellectualize and then react to, the film score can produce emotional reactions immediately, before viewers have a chance to analyze their own responses. There are three main categories of composite film scores:


In other words, a "sum" s 1 {\displaystyle s_{1}} with only one term evaluates to that one term, while a "sum" s 0 {\displaystyle s_{0}} with no terms evaluates to 0. Allowing a "sum" with only 1 or 0 terms reduces the number of cases to be considered in many mathematical formulas.


In mathematics, a weighing matrix of order n {\displaystyle n} and weight w {\displaystyle w} is a matrix W {\displaystyle W} with entries from the set { 0 , 1 , − 1 } {\displaystyle \{0,1,-1\}} such that: W W T = w I n {\displaystyle WW^{\mathsf {T}}=wI_{n}} Where W T {\displaystyle W^{\mathsf {T}}} is the transpose of W {\displaystyle W} and I n {\displaystyle I_{n}} is the identity matrix of order n {\displaystyle n} . The weight w {\displaystyle w} is also called the degree of the matrix. For convenience, a weighing matrix of order n {\displaystyle n} and weight w {\displaystyle w} is often denoted by W ( n , w ) {\displaystyle W(n,w)} .Weighing matrices are so called because of their use in optimally measuring the individual weights of multiple objects. When the weighing device is a balance scale, the statistical variance of the measurement can be minimized by weighing multiple objects at once, including some objects in the opposite pan of the scale where they subtract from the measurement.


For example, a model might be selected by maximizing its performance on some set of training data, and yet its suitability might be determined by its ability to perform well on unseen data; then over-fitting occurs when a model begins to "memorize" training data rather than "learning" to generalize from a trend. As an extreme example, if the number of parameters is the same as or greater than the number of observations, then a model can perfectly predict the training data simply by memorizing the data in its entirety. (For an illustration, see Figure 2.)


In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind Un(P, Q) with relatively prime parameters P, Q and positive discriminant, an element Un with n ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12th Fibonacci number F(12) = U12(1, −1) = 144 and its equivalent U12(−1, −1) = −144. In particular, for n greater than 12, the nth Fibonacci number F(n) has at least one prime divisor that does not divide any earlier Fibonacci number. Carmichael (1913, Theorem 21) proved this theorem. Recently, Yabuta (2001) gave a simple proof.


In the context of primitive recursive functions, it is convenient to have a means to represent finite sequences of natural numbers as single natural numbers. One such method, Gödel's encoding, represents a sequence of positive integers ⟨ n 0 , n 1 , n 2 , … , n k ⟩ {\displaystyle \langle n_{0},n_{1},n_{2},\ldots ,n_{k}\rangle } as ∏ i = 0 k p i n i {\displaystyle \prod _{i=0}^{k}p_{i}^{n_{i}}} ,where pi represent the ith prime. It can be shown that, with this representation, the ordinary operations on sequences are all primitive recursive. These operations include Determining the length of a sequence, Extracting an element from a sequence given its index, Concatenating two sequences.Using this representation of sequences, it can be seen that if h(m) is primitive recursive then the function f ( n ) = h ( ⟨ f ( 0 ) , f ( 1 ) , f ( 2 ) , … , f ( n − 1 ) ⟩ ) {\displaystyle f(n)=h(\langle f(0),f(1),f(2),\ldots ,f(n-1)\rangle )} .is also primitive recursive. When the sequence ⟨ n 0 , n 1 , n 2 , … , n k ⟩ {\displaystyle \langle n_{0},n_{1},n_{2},\ldots ,n_{k}\rangle } is allowed to include zeros, it is instead represented as ∏ i = 0 k p i ( n i + 1 ) {\displaystyle \prod _{i=0}^{k}p_{i}^{(n_{i}+1)}} ,which makes it possible to distinguish the codes for the sequences ⟨ 0 ⟩ {\displaystyle \langle 0\rangle } and ⟨ 0 , 0 ⟩ {\displaystyle \langle 0,0\rangle } .


In the following example, statement b explicitly negates statement a: Statements can also be mutually exclusive, without explicitly negating each other as in the following example:


In the classic problem the fuel in the jeep and at fuel dumps is treated as a continuous quantity. More complex variations on the problem have been proposed in which the fuel can only be left or collected in discrete amounts.In the camel and bananas problem, the merchant has n units of bananas. The camel can carry at most 1 unit of bananas at any time, and can travel 1 unit of distance on 1 unit of bananas. The market is at m units of distance away.


Another square-difference-free set is obtained by doubling the Moser–de Bruijn sequence. The best known upper bound on the size of a square-difference-free set of numbers up to n {\displaystyle n} is only slightly sublinear, but the largest known sets of this form are significantly smaller, of size ≈ n 0.733412 {\displaystyle \approx n^{0.733412}} . Closing the gap between these upper and lower bounds remains an open problem. The sublinear size bounds on square-difference-free sets can be generalized to sets where certain other polynomials are forbidden as differences between pairs of elements.


In signal processing, pre-emphasis is a technique to protect against anticipated noise. The idea is to boost (and hence distort) the frequency range that is most susceptible to noise beforehand, so that after a noisy process (transmission over cable, tape recording...) more information can be recovered from that frequency range. Removal of the distortion caused by pre-emphasis is called de-emphasis, making the output accurately reproduce the original input. Emphasis is commonly used in FM broadcasting (preemphasis improvement) and vinyl (e.g. LP) records. For example, high-frequency signal components may be emphasized to produce a more equal modulation index for a transmitted frequency spectrum, and therefore a better signal-to-noise ratio for the entire frequency range.


In protected mode, the segment_part is replaced by a 16-bit selector, in which the 13 upper bits (bit 3 to bit 15) contain the index of an entry inside a descriptor table. The next bit (bit 2) specifies whether the operation is used with the GDT or the LDT. The lowest two bits (bit 1 and bit 0) of the selector are combined to define the privilege of the request, where the values of 0 and 3 represent the highest and the lowest privilege, respectively. This means that the byte offset of descriptors in the descriptor table is the same as the 16-bit selector, provided the lower three bits are zeroed. The descriptor table entry defines the real linear address of the segment, a limit value for the segment size, and some attribute bits (flags).


{\displaystyle x|(y\mid (x\mid z))=((z\mid y)\mid y)\mid x.} In 1973, Padmanabhan and Quackenbush demonstrated a method that, in principle, would yield a 1-basis for Boolean algebra.


In mathematics, the rational sieve is a general algorithm for factoring integers into prime factors. It is a special case of the general number field sieve. While it is less efficient than the general algorithm, it is conceptually simpler. It serves as a helpful first step in understanding how the general number field sieve works.


In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. A coarse problem with one or few unknowns per subdomain is used to further coordinate the solution between the subdomains globally. The problems on the subdomains are independent, which makes domain decomposition methods suitable for parallel computing. Domain decomposition methods are typically used as preconditioners for Krylov space iterative methods, such as the conjugate gradient method, GMRES, and LOBPCG.


In neural networks, each neuron receives input from some number of locations in the previous layer. In a convolutional layer, each neuron receives input from only a restricted area of the previous layer called the neuron's receptive field. Typically the area is a square (e.g. 5 by 5 neurons). Whereas, in a fully connected layer, the receptive field is the entire previous layer.


According to Kunegis, Blattner, and Moser several online networks follow a non-linear preferential attachment model. Communication networks and online contact networks are sub-linear while interaction networks are super-linear. The co-author network among scientists also shows the signs of sub-linear preferential attachment.


In social systems, deterministic chaos is infrequent, because the elements of the system include individuals whose values, awareness, will, foresight, and fallibility, affect the dynamic behavior of the system. However, this does not completely exclude any notional possibility of deterministic chaos in social systems. In fact some authorities argue an increase in the development of nonlinear dynamics and instabilities of social systems.


The classification efficiency is usually indicated by Receiver operating characteristics. In the original SIMCA method, the ends of the hyper-plane of each class are closed off by setting statistical control limits along the retained principal components axes (i.e., score value between plus and minus 0.5 times score standard deviation). More recent adaptations of the SIMCA method close off the hyper-plane by construction of ellipsoids (e.g. Hotelling's T2 or Mahalanobis distance). With such modified SIMCA methods, classification of an object requires both that its orthogonal distance from the model and its projection within the model (i.e. score value within the region defined by the ellipsoid) are not significant.


This is not to say that all software data ought to be manipulated as graphs, but rather that they can be exchanged as graphs. It can be used to represent instance data as well as schemas for describing the structure of the data. Moreover, the schema can be explicitly stated along with instance data.


In the Hindley–Milner type system, expressions can be given multiple types through parametric polymorphism. But naively giving multiple types to references breaks type safety. The following are typing rules for references and related operators in ML-like languages. r e f: ∀ α .


In markup languages and the digital humanities, overlap occurs when a document has two or more structures that interact in a non-hierarchical manner. A document with overlapping markup cannot be represented as a tree. This is also known as concurrent markup. Overlap happens, for instance, in poetry, where there may be a metrical structure of feet and lines; a linguistic structure of sentences and quotations; and a physical structure of volumes and pages and editorial annotations.


In number theory, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution x to the equation (or congruence) x k ≡ 1 ( mod n ) {\displaystyle x^{k}\equiv 1{\pmod {n}}} . If k is the smallest such exponent for x, then x is called a primitive kth root of unity modulo n. See modular arithmetic for notation and terminology. The roots of unity modulo n are exactly the integers that are coprime with n. In fact, these integers are roots of unity modulo n by Euler's theorem, and the other integers cannot be roots of unity modulo n, because they are zero divisors modulo n. A primitive root modulo n, is a generator of the group of units of the ring of integers modulo n. There exist primitive roots modulo n if and only if λ ( n ) = φ ( n ) , {\displaystyle \lambda (n)=\varphi (n),} where λ {\displaystyle \lambda } and φ {\displaystyle \varphi } are respectively the Carmichael function and Euler's totient function. A root of unity modulo n is a primitive kth root of unity modulo n for some divisor k of λ ( n ) , {\displaystyle \lambda (n),} and, conversely, there are primitive kth roots of unity modulo n if and only if k is a divisor of λ ( n ) . {\displaystyle \lambda (n).}


In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture. As of April 2023, the only known Wieferich primes are 1093 and 3511 (sequence A001220 in the OEIS).


Very often, and in this article, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring. Integer linear programming is a collection of methods for finding the "best" integer solution (when there are many). Gröbner basis theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure.


In simple terms, software verification is: "Assuming we should build X, does our software achieve its goals without any bugs or gaps?" On the other hand, software validation is: "Was X what we should have built? Does X meet the high-level requirements?"


A bijective proof. Two sets are shown to have the same number of members by exhibiting a bijection, i.e. a one-to-one correspondence, between them.The term "combinatorial proof" may also be used more broadly to refer to any kind of elementary proof in combinatorics. However, as Glass (2003) writes in his review of Benjamin & Quinn (2003) (a book about combinatorial proofs), these two simple techniques are enough to prove many theorems in combinatorics and number theory.


Supercritical p > ( 1 + ε ) / n {\displaystyle p>(1+\varepsilon )/n} There is a single giant component containing a linear number of vertices. For large values of p {\displaystyle p} its size approaches the whole graph: | C 1 | ≈ y n {\displaystyle |C_{1}|\approx yn} where y {\displaystyle y} is the positive solution to the equation e − p n y = 1 − y {\displaystyle e^{-pny}=1-y} . The remaining components are small, with logarithmic size.In the same model of random graphs, there will exist multiple connected components with high probability for values of p {\displaystyle p} below a significantly higher threshold, p < ( 1 − ε ) ( log  n ) / n {\displaystyle p<(1-\varepsilon )(\log n)/n} , and a single connected component for values above the threshold, p > ( 1 + ε ) ( log  n ) / n {\displaystyle p>(1+\varepsilon )(\log n)/n} .


For example, if x 5 {\displaystyle x_{5}} is non-basic and its coefficient in r {\displaystyle r} is positive, then increasing it above 0 may make z {\displaystyle z} larger. If it is possible to do so without violating other constraints, then the increased variable becomes basic (it "enters the basis"), while some basic variable is decreased to 0 to keep the equality constraints and thus becomes non-basic (it "exits the basis"). If this process is done carefully, then it is possible to guarantee that z {\displaystyle z} increases until it reaches an optimal BFS.


The concept is not unlike the limited licensing approach for computer software, which places rigid restrictions on resale and reproduction. The intent is to make users understand that the content of any textbook is the intellectual property of the author and/or the publisher, and that as such, subject to copyright. Obviously, this idea is completely opposed to the millennia-old tradition of the sale of used books, and would make that entire industry illegal.


{\displaystyle M^{\text{T}}\Omega M=\Omega .} Under a change of basis, represented by a matrix A, we have Ω ↦ A T Ω A {\displaystyle \Omega \mapsto A^{\text{T}}\Omega A} M ↦ A − 1 M A . {\displaystyle M\mapsto A^{-1}MA.} One can always bring Ω {\displaystyle \Omega } to either the standard form given in the introduction or the block diagonal form described below by a suitable choice of A.


In personality pathology, dimensional models of personality disorders (also known as the dimensional approach to personality disorders, dimensional classification, and dimensional assessments) conceptualize personality disorders as quantitatively rather than qualitatively different from normal personality. They consist of extreme, maladaptive levels of certain personality characteristics (these characteristics are commonly described as facets within broader personality factors or traits). Within the context of personality psychology, a "dimension" refers to a continuum on which an individual can have various levels of a characteristic, in contrast to the dichotomous categorical approach in which an individual does or does not possess a characteristic.


In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper.The term map may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial.


In mathematics, Euclid numbers are integers of the form En = pn # + 1, where pn # is the nth primorial, i.e. the product of the first n prime numbers. They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theorem that there are infinitely many prime numbers.


In the Japanese encodings ISO 646-JP (a 7-bit code based on ASCII), JIS X 0201 (an 8-bit code), and Shift JIS (a multi-byte encoding which is 8-bit for ASCII), the code point 0x5C that would be used for backslash in ASCII is instead rendered as a yen sign ¥. Due to extensive use of the 005C code point to represent the yen sign, even today some fonts such as MS Mincho render the backslash character as a ¥, so the characters at Unicode code points 00A5 (¥) and 005C (\) both render as ¥ when these fonts are selected. Computer programs still treat 005C as a backslash in these environments but display it as a yen sign, causing confusion, especially in MS-DOS filenames.Several other ISO 646 versions also replace backslash with other characters, including ₩ (Korean), Ö (German, Swedish), Ø (Danish, Norwegian), ç (French) and Ñ (Spanish), leading to similar problems, though with less lasting impact compared to the yen sign. In 1991, RFC 1345 suggested // as a unique two-character mnemonic that might be used in internet standards as "a practical way of identifying  character, without reference to a coded character set and its code in  coded character set". Consequently, this style may be seen in early Internet Engineering Task Force documents.


In most computer programming languages a do while loop is a control flow statement that executes a block of code and then either repeats the block or exits the loop depending on a given boolean condition. The do while construct consists of a process symbol and a condition. First the code within the block is executed. Then the condition is evaluated.


In the 1980s, the Graphical Kernel System (GKS) library, based on a 1970s specification with a similar basic geometry and command structure to NAPLPS, was widely implemented on microcomputers, and became the basis of Digital Research's GSX graphics system used in their GEM GUI. GKS was later extended into a 3D version, and additions to this resulted in PHIGS (Programmer's Hierarchical Interactive Graphics System), a competitor to OpenGL.


As a result, it is often considered to be a more intuitive, but a less systematic approach to divisions – where the efficiency is highly dependent upon one's numeracy skills. To calculate the whole number quotient of dividing a large number by a small number, the student repeatedly takes away "chunks" of the large number, where each "chunk" is an easy multiple (for example 100×, 10×, 5× 2×, etc.) of the small number, until the large number has been reduced to zero – or the remainder is less than the small number itself.


And symmetrically (when Xk is a sub-martingale): P ( X N − X 0 ≤ − ϵ ) ≤ exp  ( − ϵ 2 2 ∑ k = 1 N c k 2 ) . {\displaystyle {\text{P}}(X_{N}-X_{0}\leq -\epsilon )\leq \exp \left({-\epsilon ^{2} \over 2\sum _{k=1}^{N}c_{k}^{2}}\right).} If X is a martingale, using both inequalities above and applying the union bound allows one to obtain a two-sided bound: P ( | X N − X 0 | ≥ ϵ ) ≤ 2 exp  ( − ϵ 2 2 ∑ k = 1 N c k 2 ) . {\displaystyle {\text{P}}(|X_{N}-X_{0}|\geq \epsilon )\leq 2\exp \left({-\epsilon ^{2} \over 2\sum _{k=1}^{N}c_{k}^{2}}\right).}


As in discrete percolation, a common research focus of continuum percolation is studying the conditions of occurrence for infinite or giant components. Other shared concepts and analysis techniques exist in these two types of percolation theory as well as the study of random graphs and random geometric graphs. Continuum percolation arose from an early mathematical model for wireless networks, which, with the rise of several wireless network technologies in recent years, has been generalized and studied in order to determine the theoretical bounds of information capacity and performance in wireless networks. In addition to this setting, continuum percolation has gained application in other disciplines including biology, geology, and physics, such as the study of porous material and semiconductors, while becoming a subject of mathematical interest in its own right.


In mathematics, Light's associativity test is a procedure invented by F. W. Light for testing whether a binary operation defined in a finite set by a Cayley multiplication table is associative. The naive procedure for verification of the associativity of a binary operation specified by a Cayley table, which compares the two products that can be formed from each triple of elements, is cumbersome. Light's associativity test simplifies the task in some instances (although it does not improve the worst-case runtime of the naive algorithm, namely O ( n 3 ) {\displaystyle {\mathcal {O}}\left(n^{3}\right)} for sets of size n {\displaystyle n} ).


For example, the proton spin puzzle, the EMC effect, the distributions of electric charges inside the nucleons, as found by Hofstadter in 1956, and the ad hoc CKM matrix elements. When the term "preon" was coined, it was primarily to explain the two families of spin-1/2 fermions: quarks and leptons. More recent preon models also account for spin-1 bosons, and are still called "preons". Each of the preon models postulates a set of fewer fundamental particles than those of the Standard Model, together with the rules governing how those fundamental particles combine and interact. Based on these rules, the preon models try to explain the Standard Model, often predicting small discrepancies with this model and generating new particles and certain phenomena which do not belong to the Standard Model.


In the 20th century, following the development of formal logic, the ampersand became a commonly used logical notation for the binary operator or sentential connective AND. This usage was adopted in computing. Many languages with syntax derived from C, including C++, Perl, and more differentiate between: & for bitwise AND. (4 & 2) is zero, (4 & 5) is 4.


In the US, the CDC recommends essential components of AMS programs (ASP) for acute care hospitals, small and critical access hospitals, resource-limited facilities, long-term care facilities, and outpatient facilities.As of 2014, thirteen internet-based institutional ASP resources in US academic medical centers had been published. An ASP has the following tasks, in line with quality improvement theory:


An arrow from the node representing a candidate X to the one representing a candidate Y is labelled with d. To avoid cluttering the diagram, an arrow has only been drawn from X to Y when d > d (i.e. the table cells with light green background), omitting the one in the opposite direction (the table cells with light red background). One example of computing the strongest path strength is p = 33: the strongest path from B to D is the direct path (B, D) which has strength 33. But when computing p, the strongest path from A to C is not the direct path (A, C) of strength 26, rather the strongest path is the indirect path (A, D, C) which has strength min(30, 28) = 28.


In mathematical logic, a formal theory is a set of sentences expressed in a formal language. A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules, which may be interpreted as valid rules of inference, or a set of axioms, or have both.


These directories contain files with names such as "ABCD1234.JPG" that consist of four alphanumeric characters (often "100_", "DSC0", "DSCF", "IMG_", "MOV_", or "P000"), followed by a number. Handling of directories with possibly user-created duplicate numbers may vary among camera firmwares. DCF 2.0 adds support for DCF optional files recorded in an optional color space (that is, Adobe RGB rather than sRGB). Such files must be indicated by a leading "_" (as in "_DSC" instead of "100_" or "DSC0").


In structure mining, a graph kernel is a kernel function that computes an inner product on graphs. Graph kernels can be intuitively understood as functions measuring the similarity of pairs of graphs. They allow kernelized learning algorithms such as support vector machines to work directly on graphs, without having to do feature extraction to transform them to fixed-length, real-valued feature vectors. They find applications in bioinformatics, in chemoinformatics (as a type of molecule kernels), and in social network analysis.Concepts of graph kernels have been around since the 1999, when D. Haussler introduced convolutional kernels on discrete structures.


In statistical analysis of binary classification, the F-score or F-measure is a measure of a test's accuracy. It is calculated from the precision and recall of the test, where the precision is the number of true positive results divided by the number of all positive results, including those not identified correctly, and the recall is the number of true positive results divided by the number of all samples that should have been identified as positive. Precision is also known as positive predictive value, and recall is also known as sensitivity in diagnostic binary classification.


In order to measure the information of a string relative to another there is the need to rely on relative semi-distances (NRC). These are measures that do not need to respect symmetry and triangle inequality distance properties. Although the NCD and the NRC seem very similar, they address different questions. The NCD measures how similar both strings are, mostly using the information content, while the NRC indicates the fraction of a target string that cannot be constructed using information from another string. For a comparison, with application to the evolution of primate genomes, see.


According to the Curry–Howard isomorphism, lambda calculus on its own can express theorems in intuitionistic logic only, and several classical logical theorems can't be written at all. However with these new operators one is able to write terms that have the type of, for example, Peirce's law. Semantically these operators correspond to continuations, found in some functional programming languages.


In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets X and Y is a new set of ordered pairs (x, y) consisting of elements x in X and y in Y. It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element x is related to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. A binary relation is the most studied special case n = 2 of an n-ary relation over sets X1, ..., Xn, which is a subset of the Cartesian product X 1 × ⋯ × X n .


Unlike Schoof's algorithm, the SEA algorithm is typically implemented as a probabilistic algorithm (of the Las Vegas type), so that root-finding and other operations can be performed more efficiently. Its computational complexity is dominated by the cost of computing the modular polynomials Ψ ℓ ( X , Y ) {\displaystyle \Psi _{\ell }(X,Y)} , but as these do not depend on E {\displaystyle E} , they may be computed once and reused. Under the heuristic assumption that there are sufficiently many small Elkies primes, and excluding the cost of computing modular polynomials, the asymptotic running time of the SEA algorithm is O ( n 2 M ( n 2 ) / log  n ) = O ( n 4 + o ( 1 ) ) {\displaystyle O(n^{2}M(n^{2})/\log {n})=O(n^{4+o(1)})} , where n = log  q {\displaystyle n=\log {q}} . Its space complexity is O ( n 3 log  n ) {\displaystyle O(n^{3}\log {n})} , but when precomputed modular polynomials are used this increases to O ( n 4 ) {\displaystyle O(n^{4})} .


In order to test plastic welds, there are several requirements for both the inspector as well as the test method. Furthermore, there are two different types of testing weld quality. These two types are destructive and non-destructive testing. Destructive testing serves to qualify and quantify the weld joint whereas nondestructive testing serves to identify anomalies, discontinuities, cracks, and/or crevices.


In most of these cases the OEMs are open about their use of such software and fulfil the requirements of their Free software licenses, such as the GNU General Public License (GPL), but in a small number of cases this use is masked, either deliberately or through professed ignorance or misunderstanding. Violators are usually found through public records, where they may be forced to declare their implementations, or through their own advertising, for example "Embedded Software Engineers with Mandatory Linux Experience Required" on their careers pages, and yet their site or product documentation offers no source download or offer to supply the software source as required by the license GPL. Organizations such as gpl-violations.org, the Free Software Foundation (FSF) and the Software Freedom Law Center (SFLC) are now more organized at pursuing such violators and obtaining compliance. Usually, they seek voluntary compliance as a first step and only enter legal proceedings when blocked. When notified of violations they confirm them by asking the supplier, examining available product samples, or even going so far as to make blind purchases of the product through front companies.


The interpretation commonly taken, however, is that an underlying morpheme |-i| palatalizes the consonant and is subsequently deleted. Palatalization may also occur as a morphological feature. For example, although Russian makes phonemic contrasts between palatalized and unpalatalized consonants, alternations across morpheme boundaries are normal: ответ ('answer') vs. ответить ('to answer') несу (' carry') vs. несёт ('carries') голод ('hunger') vs. голоден  ('hungry' masc.)


When p = 1 / 2 {\displaystyle p=1/2} , the uncertainty is at a maximum; if one were to place a fair bet on the outcome in this case, there is no advantage to be gained with prior knowledge of the probabilities. In this case, the entropy is maximum at a value of 1 bit. Intermediate values fall between these cases; for instance, if p = 1 / 4 {\displaystyle p=1/4} , there is still a measure of uncertainty on the outcome, but one can still predict the outcome correctly more often than not, so the uncertainty measure, or entropy, is less than 1 full bit.


In mathematics, Hall's marriage theorem, proved by Philip Hall (1935), is a theorem with two equivalent formulations. In each case, the theorem gives a necessary and sufficient condition for an object to exist: The combinatorial formulation answers whether a finite collection of sets has a transversal—that is, whether an element can be chosen from each set without repetition. Hall's condition is that for any group of sets from the collection, the total unique elements they contain is at least as large as the number of sets in the group. The graph theoretic formulation answers whether a finite bipartite graph has a perfect matching—that is, a way to match each vertex from one group uniquely to an adjacent vertex from the other group. Hall's condition is that any subset of vertices from one group has a neighbourhood of equal or greater size.


In natural language processing, language identification or language guessing is the problem of determining which natural language given content is in. Computational approaches to this problem view it as a special case of text categorization, solved with various statistical methods.


In numerical analysis, the Peano kernel theorem is a general result on error bounds for a wide class of numerical approximations (such as numerical quadratures), defined in terms of linear functionals. It is attributed to Giuseppe Peano.


In mathematics, the interval chromatic number X<(H) of an ordered graph H is the minimum number of intervals the (linearly ordered) vertex set of H can be partitioned into so that no two vertices belonging to the same interval are adjacent in H.


In mathematical combinatorics, the Transylvania lottery is a lottery where players selected three numbers from 1-14 for each ticket, and then three numbers are chosen randomly. A ticket wins if two of the numbers match the random ones. The problem asks how many tickets the player must buy in order to be certain of winning. (Javier Martínez, Gloria Gutiérrez & Pablo Cordero et al. 2008, p.85)(Mazur 2010, p.280 problem 15) An upper bound can be given using the Fano plane with a collection of 14 tickets in two sets of seven.


The question is: how should we choose the prior parameters ( s , G 0 ) {\displaystyle \left(s,G_{0}\right)} of the DP, in particular the infinite dimensional one G 0 {\displaystyle G_{0}} , in case of lack of prior information? To address this issue, the only prior that has been proposed so far is the limiting DP obtained for s → 0 {\displaystyle s\rightarrow 0} , which has been introduced under the name of Bayesian bootstrap by Rubin; in fact it can be proven that the Bayesian bootstrap is asymptotically equivalent to the frequentist bootstrap introduced by Bradley Efron. The limiting Dirichlet process s → 0 {\displaystyle s\rightarrow 0} has been criticized on diverse grounds.


In software engineering, syntactic methods are techniques for developing correct software programs. The techniques attempt to detect, and thus prevent, certain kinds of defects (bugs) by examining the structure of the code being produced at its syntactic rather than semantic level.


In statistics, the power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis ( H 0 {\displaystyle H_{0}} ) when a specific alternative hypothesis ( H 1 {\displaystyle H_{1}} ) is true. It is commonly denoted by 1 − β {\displaystyle 1-\beta } , and represents the chances of a true positive detection conditional on the actual existence of an effect to detect. Statistical power ranges from 0 to 1, and as the power of a test increases, the probability β {\displaystyle \beta } of making a type II error by wrongly failing to reject the null hypothesis decreases.


In order to speed up web server responses by lowering average HTTP response times and hardware resources used, many popular web servers implement one or more content caches, each one specialized in a content category.Content is usually cached by its origin, e.g.: static content: file cache; dynamic content: dynamic cache (module / program output).


, FN), and let Q {\displaystyle \mathbb {Q} } denote an equivalent martingale measure. Let U = (U0, U1, . .


Defining Child-Related OperationsThere are two design variants for defining and implementing child-related operations like adding/removing a child component to/from the container (add(child)/remove(child)) and accessing a child component (getChild()): Design for uniformity: Child-related operations are defined in the Component interface. This enables clients to treat Leaf and Composite objects uniformly. But type safety is lost because clients can perform child-related operations on Leaf objects.


In mathematical cryptography, a Kleinian integer is a complex number of the form m + n 1 + − 7 2 {\displaystyle m+n{\frac {1+{\sqrt {-7}}}{2}}} , with m and n rational integers. They are named after Felix Klein. The Kleinian integers form a ring called the Kleinian ring, which is the ring of integers in the imaginary quadratic field Q ( − 7 ) {\displaystyle \mathbb {Q} ({\sqrt {-7}})} . This ring is a unique factorization domain.


In some jurisdictions, false statement is a crime similar to perjury.