CbD essentially coincides with the probabilistic part of Abramsky's sheaf-theoretic approach if the system is strongly consistently connected, which means that the joint distributions of { R q 1 c , … , R q k c } {\displaystyle \left\{R_{q_{1}}^{c},\ldots ,R_{q_{k}}^{c}\right\}} and { R q 1 c ′ , … , R q k c ′ } {\displaystyle \left\{R_{q_{1}}^{c'},\ldots ,R_{q_{k}}^{c'}\right\}} coincide whenever q 1 , … , q k {\displaystyle q_{1},\ldots ,q_{k}} are measured in contexts c , c ′ {\displaystyle c,c'} . However, unlike most approaches to contextuality, CbD allows for inconsistent connectedness, with R q c {\displaystyle R_{q}^{c}} and R q c ′ {\displaystyle R_{q}^{c'}} differently distributed. This makes CbD applicable to physics experiments in which no-disturbance condition is violated, as well as to human behavior where this condition is violated as a rule. In particular, Vctor Cervantes, Ehtibar Dzhafarov, and colleagues have demonstrated that random variables describing certain paradigms of simple decision making form contextual systems, whereas many other decision-making systems are noncontextual once their inconsistent connectedness is properly taken into account.


In the English language, a compound sentence is composed of at least two independent clauses. It does not require a dependent clause. The clauses are joined by a coordinating conjunction, a semicolon that functions as a conjunction, a colon instead of a semicolon between two sentences when the second sentence explains or illustrates the first sentence and no coordinating conjunction is being used to connect the sentences, or a conjunctive adverb preceded by a semicolon. A conjunction can be used to make a compound sentence.


In quantum computing, Grover's algorithm, also known as the quantum search algorithm, refers to a quantum algorithm for unstructured search that finds with high probability the unique input to a black box function that produces a particular output value, using just O ( N ) {\displaystyle O({\sqrt {N}})} evaluations of the function, where N {\displaystyle N} is the size of the function's domain. It was devised by Lov Grover in 1996.The analogous problem in classical computation cannot be solved in fewer than O ( N ) {\displaystyle O(N)} evaluations (because, on average, one has to check half of the domain to get a 50% chance of finding the right input). Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani proved that any quantum solution to the problem needs to evaluate the function Ω ( N ) {\displaystyle \Omega ({\sqrt {N}})} times, so Grover's algorithm is asymptotically optimal.


Modern x86 is relatively uncommon in embedded systems, however, and small low power applications (using tiny batteries), and low-cost microprocessor markets, such as home appliances and toys, lack significant x86 presence. Simple 8- and 16-bit based architectures are common here, as well as simpler RISC architectures like RISC-V, although the x86-compatible VIA C7, VIA Nano, AMD's Geode, Athlon Neo and Intel Atom are examples of 32- and 64-bit designs used in some relatively low-power and low-cost segments. There have been several attempts, including by Intel, to end the market dominance of the "inelegant" x86 architecture designed directly from the first simple 8-bit microprocessors.


Since entrenched CEO pay their workers high salaries, the CEO-worker relationship improves, making workers less likely to unionize. Often workers perceive managers' benefits to be more beneficial for them than unions. This leads us to the conclusion that entrenched CEOs have the characteristic of being very competitive when it comes to work loyalty. == References ==


In predicate logic, existential generalization (also known as existential introduction, ∃I) is a valid rule of inference that allows one to move from a specific statement, or one instance, to a quantified generalized statement, or existential proposition. In first-order logic, it is often used as a rule for the existential quantifier ( ∃ {\displaystyle \exists } ) in formal proofs. Example: "Rover loves to wag his tail. Therefore, something loves to wag its tail."


Differential equations or difference equations on such graphs can be employed to leverage the graph's structure for tasks such as image segmentation (where the vertices represent pixels and the weighted edges encode pixel similarity based on comparisons of Moore neighborhoods or larger windows), data clustering, data classification, or community detection in a social network (where the vertices represent users of the network, the edges represent links between users, and the weight function indicates the strength of interactions between users). The main advantage of finite weighted graphs is that by not being restricted to highly regular structures such as discrete regular grids, lattice graphs, or meshes, they can be applied to represent abstract data with irregular interrelationships. If a finite weighted graph is geometrically embedded in a Euclidean space, i.e., the graph vertices represent points of this space, then it can be interpreted as a discrete approximation of a related nonlocal operator in the continuum setting.


(In some accounts that take this approach, the constituent lacking the determiner – that called N-bar above – may be referred to as a noun phrase.) This analysis of noun phrases is widely referred to as the DP hypothesis. It has been the preferred analysis of noun phrases in the minimalist program from its start (since the early 1990s), though the arguments in its favor tend to be theory-internal.


Likewise, lattices are directed sets both upward and downward. In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory.


In several languages, the grave accent distinguishes both homophones and words that otherwise would be homographs: In Bulgarian and Macedonian, it distinguishes the conjunction и ("and") from the short-form feminine possessive pronoun ѝ. In Catalan, it distinguishes homophone words such as ma ("my (f)") and mà ("hand"). In French the grave accent on the letters a and u has no effect on pronunciation and just distinguishes homonyms otherwise spelled the same, for example the preposition à ("to/belonging to/towards") from the verb a (" has") as well as the adverb là ("there") and the feminine definite article la; it is also used in the words déjà ("already"), deçà (preceded by en or au, and meaning "closer than" or "inferior to (a given value)"), the phrase çà et là ("hither and thither"; without the accents, it would literally mean "it and the") and its functional synonym deçà, delà. It is used on the letter u only to distinguish où ("where") and ou ("or"). È is rarely used to distinguish homonyms except in dès/des ("since/some"), ès/es ("in/(thou) art"), and lès/les ("near/the").


In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory).


Recursive languages are also called decidable. The concept of decidability may be extended to other models of computation. For example, one may speak of languages decidable on a non-deterministic Turing machine.


In consequence, mutually exclusive events have the property: P(A ∩ B) = 0.For example, in a standard 52-card deck with two colors it is impossible to draw a card that is both red and a club because clubs are always black. If just one card is drawn from the deck, either a red card (heart or diamond) or a black card (club or spade) will be drawn. When A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B).


This process is known as regularization. Tikhonov regularization is one of the most commonly used for regularization of linear ill-posed problems. The definition of a well-posed problem comes from the work of Jacques Hadamard on mathematical modeling of physical phenomena.


In some languages, stative and dynamic verbs will use entirely different morphological markers on the verbs themselves. For example, in the Mantauran dialect of Rukai, an indigenous language of Taiwan, the two types of verbs take different prefixes in their finite forms, with dynamic verbs taking o- and stative verbs taking ma-. Thus, the dynamic verb "jump" is o-coroko in the active voice, while the stative verb "love" is ma-ðalamə. This sort of marking is characteristic of other Formosan languages as well.


short cable lengths and limited physical extent, avoiding signal runtime performance degradation. majority / quorum mechanisms to guarantee data consistency whenever parts of the cluster become inaccessible.Indicators for eventually consistent designs (not suitable for transactional applications!)


The residual, or error, sum-of-squares is SSE = ∑ i = 1 n ( X i − X ¯ ) 2 + ∑ i = 1 n ( Y i − Y ¯ ) 2 + ∑ i = 1 n ( Z i − Z ¯ ) 2 {\displaystyle {\text{SSE}}=\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}+\sum _{i=1}^{n}(Y_{i}-{\bar {Y}})^{2}+\sum _{i=1}^{n}(Z_{i}-{\bar {Z}})^{2}} with 3(n−1) degrees of freedom. Of course, introductory books on ANOVA usually state formulae without showing the vectors, but it is this underlying geometry that gives rise to SS formulae, and shows how to unambiguously determine the degrees of freedom in any given situation.


In practice, stacks of indices can count and remember what rules were applied and in which order. For example, indexed grammars can describe the context-sensitive language of word triples { www: w ∈ {a,b}* }: A derivation of abbabbabb is then S ⇒ S ⇒ S ⇒ S ⇒ T T T ⇒ a T T T ⇒ ab T T T ⇒ abb T T T ⇒ abb T T ⇒ ... ⇒ abb abb T ⇒ ... ⇒ abb abb abb.As another example, the grammar G = ⟨ {S,T,A,B,C}, {a,b,c}, {f,g}, P, S ⟩ produces the language { anbncn: n ≥ 1 }, where the production set P consists of An example derivation is S ⇒ T ⇒ T ⇒ A B C ⇒ aA B C ⇒ aA bB C ⇒ aA bB cC ⇒ aa bB cC ⇒ aa bb cC ⇒ aa bb cc.Both example languages are not context-free by the pumping lemma.


In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that were part of the RSA Factoring Challenge. The challenge was to find the prime factors of each number. It was created by RSA Laboratories in March 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers. The challenge was ended in 2007.RSA Laboratories (which is an acronym of the creators of the technique; Rivest, Shamir and Adleman) published a number of semiprimes with 100 to 617 decimal digits.


In metalogic, formal languages are sometimes called object languages. The language used to make statements about an object language is called a metalanguage. This distinction is a key difference between logic and metalogic. While logic deals with proofs in a formal system, expressed in some formal language, metalogic deals with proofs about a formal system which are expressed in a metalanguage about some object language.


In fact, the consecutive weightings of any digital number system can be employed since they all constitute geometric progressions. For example, the binary, ternary, octal and decimal number systems use a radix ‘R’ of 2, 3, 8 and 10 respectively. The value ‘R’ is also the common ratio of the geometric progression going up in rank order while ‘r’ is the complementary common ratio descending in rank.


x z )  = ( λ x . y z ) {\displaystyle (\lambda x.x\ z)=(\lambda x.y\ z)} The new rules block this substitution so that it remains as, ( λ x . x z )  = ( λ x . x z ) {\displaystyle (\lambda x.x\ z)=(\lambda x.x\ z)}


In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: 0 , 1 , {\displaystyle 0,1,} or e {\displaystyle e} depending on the context. If the group operation is denoted ⋅ {\displaystyle \,\cdot \,} then it is defined by e ⋅ e = e .


It limits the ways that entities can use patients' personal information and protects the privacy of all medical information no matter what form it is in. Quality teleradiology must abide by important HIPAA rules to ensure patients' privacy is protected. Also State laws governing the licensing requirements and medical malpractice insurance coverage required for physicians vary from state to state.


{\displaystyle pq-\varphi (pq)=pq-(p-1)(q-1)=p+q-1=n-1.\,} It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 5 is a noncototient. The remaining odd numbers are covered by the observations 1 = 2 − ϕ ( 2 ) , 3 = 9 − ϕ ( 9 ) {\displaystyle 1=2-\phi (2),3=9-\phi (9)} and 5 = 25 − ϕ ( 25 ) {\displaystyle 5=25-\phi (25)} . For even numbers, it can be shown 2 p q − φ ( 2 p q ) = 2 p q − ( p − 1 ) ( q − 1 ) = p q + p + q − 1 = ( p + 1 ) ( q + 1 ) − 2 {\displaystyle 2pq-\varphi (2pq)=2pq-(p-1)(q-1)=pq+p+q-1=(p+1)(q+1)-2} Thus, all even numbers n such that n+2 can be written as (p+1)*(q+1) with p, q primes are cototients.


In the ISO 5725-1 standard accuracy for measuring instruments is defined as “the closeness of agreement between a test result and the accepted reference value”. This term “accuracy” includes both the systematic error and the bias component. Each device has its manufacturer stated accuracy specification and its tested accuracy. Uncertainty takes all the metering system factors that impact measurement accuracy into account.


as a result. The theory is the answer the scientist creates using logic and reason to explain the phenomenon. The scientist then focuses on how to conduct experiments to test the theory incrementally and the theory is either proven to be true or false through repeatable and legitimate experimentation.


Thus, the maximum entropy principle is not merely an alternative way to view the usual methods of inference of classical statistics, but represents a significant conceptual generalization of those methods. However these statements do not imply that thermodynamical systems need not be shown to be ergodic to justify treatment as a statistical ensemble. In ordinary language, the principle of maximum entropy can be said to express a claim of epistemic modesty, or of maximum ignorance. The selected distribution is the one that makes the least claim to being informed beyond the stated prior data, that is to say the one that admits the most ignorance beyond the stated prior data.


Failing usually results in the student leaving the program or re-taking the test after some time has passed. A second failure normally guarantees dismissal from the graduate program, though progress on previous attempts may convince the student's program to grant a third, final attempt. Some schools have an intermediate category, passing at the master's level, which does not permit the student to continue doctoral study, but does allow the student to leave with a master's degree despite not having completed a thesis.In some U.S.


In software engineering, a pipeline consists of a chain of processing elements (processes, threads, coroutines, functions, etc.), arranged so that the output of each element is the input of the next; the name is by analogy to a physical pipeline. Usually some amount of buffering is provided between consecutive elements. The information that flows in these pipelines is often a stream of records, bytes, or bits, and the elements of a pipeline may be called filters; this is also called the pipe(s) and filters design pattern.


Native data types can be mapped to/from Java data types. For compound types such as objects, arrays and strings the native code must explicitly convert the data by calling methods in the JNIEnv. A JNI environment pointer (JNIEnv*) is passed as an argument for each native function mapped to a Java method, allowing for interaction with the JNI environment within the native method.


In morphology and syntax, words are often organized into lexical categories or word classes, such as "noun", "verb", "adjective", and so on. These word classes have grammatical features (also called categories or inflectional categories), which can have one of a set of potential values (also called the property, meaning, or feature of the category).For example, consider the pronoun in English. Pronouns are a lexical category. Pronouns have the person feature, which can have a value of "first", "second", or "third".


Despite that, the purely theoretical existence results are nevertheless ubiquitous in contemporary mathematics. For example, John Nash's original proof of the existence of a Nash equilibrium in 1951 was such an existence theorem. An approach which is constructive was also later found in 1962.


In software development, a given software system's ability to tolerate failures while still ensuring adequate quality of service—often generalized as resilience—is typically specified as a requirement. However, development teams often fail to meet this requirement due to factors such as short deadlines or lack of knowledge of the field. Chaos engineering is a technique to meet the resilience requirement. Chaos engineering can be used to achieve resilience against infrastructure failures, network failures, and application failures.


The same convention is adopted in functional programming, particularly in Haskell. In geometry, geography and astronomy, prime and double prime are used as abbreviations for minute and second of arc (and thus latitude, longitude, elevation and right ascension). In physics, the prime is used to denote variables after an event.


In probability theory, Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest case, let X1, ..., Xn be independent Bernoulli random variables taking values +1 and −1 with probability 1/2 (this distribution is also known as the Rademacher distribution), then for every positive ε {\displaystyle \varepsilon } , P ( | 1 n ∑ i = 1 n X i | > ε ) ≤ 2 exp  ( − n ε 2 2 ( 1 + ε 3 ) ) . {\displaystyle \mathbb {P} \left(\left|{\frac {1}{n}}\sum _{i=1}^{n}X_{i}\right|>\varepsilon \right)\leq 2\exp \left(-{\frac {n\varepsilon ^{2}}{2(1+{\frac {\varepsilon }{3}})}}\right).}


In spherical coordinates, with θ the angle with the z axis and φ the rotation around the z axis, and F again written in local unit coordinates, the divergence is div  F = ∇ ⋅ F = 1 r 2 ∂ ∂ r ( r 2 F r ) + 1 r sin  θ ∂ ∂ θ ( sin  θ F θ ) + 1 r sin  θ ∂ F φ ∂ φ . {\displaystyle \operatorname {div} \mathbf {F} =\nabla \cdot \mathbf {F} ={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}F_{r}\right)+{\frac {1}{r\sin \theta }}{\frac {\partial }{\partial \theta }}(\sin \theta \,F_{\theta })+{\frac {1}{r\sin \theta }}{\frac {\partial F_{\varphi }}{\partial \varphi }}.}


While much of the notation may be applied with any tensors, operations relating to a differential structure are only applicable to tensor fields. Where needed, the notation extends to components of non-tensors, particularly multidimensional arrays.


In more general mathematical setting, the Boltzmann distribution is also known as the Gibbs measure. In statistics and machine learning it is called a log-linear model. In deep learning the Boltzmann distribution is used in the sampling distribution of stochastic neural networks such as the Boltzmann machine.


All bounds are asymptotically tight when the number of groups is constant. The proofs use discrepancy theory.Unanimous envy-freeness with high probability: When all k groups contain the same number of agents, and their valuations are drawn at random, an envy-free allocation exists with high probability if the number of goods is in Ω ( n log  n ) {\displaystyle \Omega (n\log n)} , and can be attained by a greedy algorithm that maximizes the sum of utilities. The results can be extended to two groups with different sizes.


The SUS has been widely used in the evaluation of a range of systems. Bangor, Kortum and Miller have used the scale extensively over a ten-year period and have produced normative data that allow SUS ratings to be positioned relative to other systems. They propose an extension to SUS to provide an adjective rating that correlates with a given score. Based on a review of hundreds of usability studies, Sauro and Lewis proposed a curved grading scale for mean SUS scores.


In the field of telecommunications, "data retention" generally refers to the storage of call detail records (CDRs) of telephony and internet traffic and transaction data (IPDRs) by governments and commercial organisations. In the case of government data retention, the data that is stored is usually of telephone calls made and received, emails sent and received, and websites visited. Location data is also collected. The primary objective in government data retention is traffic analysis and mass surveillance.


One can logically define a semigroup in which the underlying set S is empty. The binary operation in the semigroup is the empty function from S × S to S. This operation vacuously satisfies the closure and associativity axioms of a semigroup. Not excluding the empty semigroup simplifies certain results on semigroups.


In number theory, the totient summatory function Φ ( n ) {\displaystyle \Phi (n)} is a summatory function of Euler's totient function defined by: Φ ( n ) := ∑ k = 1 n φ ( k ) , n ∈ N {\displaystyle \Phi (n):=\sum _{k=1}^{n}\varphi (k),\quad n\in \mathbf {N} } It is the number of coprime integer pairs {p, q}, 1 ≤ p ≤ q ≤ n.


The formulation of the theorem uses the following definition. A word in two variables, say x and y, is an expression of the form W ( x , y ) = x m 1 y n 1 x m 2 y n 2 ⋯ x m p , {\displaystyle W(x,y)=x^{m_{1}}y^{n_{1}}x^{m_{2}}y^{n_{2}}\cdots x^{m_{p}},} where m1, n1, m2, n2, …, mp are non-negative integers. The degree of this word is m 1 + n 1 + m 2 + n 2 + ⋯ + m p .


Further inputs to the algorithm are the marginal sample distribution p ( x ) {\displaystyle p(x)\,} which has already been determined by the dominant eigenvector of P {\displaystyle P\,} and the matrix valued Kullback–Leibler divergence function D i , j K L = D K L  ) {\displaystyle D_{i,j}^{KL}=D^{KL}{\Big }{\Big )}} derived from the sample spacings and transition probabilities. The matrix p ( y i | c j ) {\displaystyle p(y_{i}|c_{j})\,} can be initialized randomly or with a reasonable guess, while matrix p ( c i | x j ) {\displaystyle p(c_{i}|x_{j})\,} needs no prior values. Although the algorithm converges, multiple minima may exist that would need to be resolved.


Under such assumptions, the case n < m {\displaystyle n m {\displaystyle n>m} and { E i } {\displaystyle \{E_{i}\}} does not consist of mutually orthogonal projections.For the second possibility, the problem of finding a suitable projection-valued measure now becomes the following problem. By assumption, the non-square matrix M =  {\displaystyle M={\begin{bmatrix}x_{1}\cdots x_{n}\end{bmatrix}}} is a co-isometry, that is M M ∗ = I {\displaystyle MM^{*}=I} . If we can find a ( n − m ) × n {\displaystyle (n-m)\times n} matrix N where U =  {\displaystyle U={\begin{bmatrix}M\\N\end{bmatrix}}} is a n × n unitary matrix, the projection-valued measure whose elements are projections onto the column vectors of U will then have the desired properties. In principle, such a N can always be found.


In the beginning of a prefix sum operation, each processing element i {\displaystyle i} owns a message m i {\displaystyle m_{i}} . The goal is to compute ⨁ 0 ≤ j ≤ i m j {\displaystyle \bigoplus _{0\leq j\leq i}m_{j}} , where ⊕ {\displaystyle \oplus } is an associative operation. The following pseudo code describes the algorithm. Input: message m i {\displaystyle m_{i}} of processor i {\displaystyle i} .


In n dimensions, when all i1, ...,in, j1, ..., jn take values 1, 2, ..., n: where the exclamation mark (!) denotes the factorial, and δα...β... is the generalized Kronecker delta. For any n, the property ∑ i , j , k , ⋯ = 1 n ε i j k … ε i j k … = n ! {\displaystyle \sum _{i,j,k,\dots =1}^{n}\varepsilon _{ijk\dots }\varepsilon _{ijk\dots }=n!} follows from the facts that every permutation is either even or odd, (+1)2 = (−1)2 = 1, and the number of permutations of any n-element set number is exactly n!.The particular case of (8) with k = n − 2 {\textstyle k=n-2} is


As a consequence, code quality without the context of the whole system, as W. Edwards Deming described it, has limited value.To view, explore, analyze, and communicate software quality measurements, concepts and techniques of information visualization provide visual, interactive means useful, in particular, if several software quality measures have to be related to each other or to components of a software or system. For example, software maps represent a specialized approach that "can express and combine information about software development, software quality, and system dynamics".Software quality also plays a role in the release phase of a software project. Specifically, the quality and establishment of the release processes (also patch processes), configuration management are important parts of an overall software engineering process.


The latter usually applies to a form of argument that does not comply with the valid inference rules of logic, whereas the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption. Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (e.g., the introduction of Pasch's axiom of Euclidean geometry, the five colour theorem of graph theory).


In the early 1990s NeXT Computer recognized that connecting to databases was essential to most businesses and yet also potentially complex. Every data source has a different data-access language (or API), driving up the costs to learn and use each vendor's product. The NeXT engineers wanted to apply the advantages of object-oriented programming, by getting objects to "talk" to relational databases.


Items should be empirically related to one another, which leads to the second step of examining their multivariate relationships. Third, indexes scores are designed, which involves determining their score ranges and weights for the items.


Qualcomm announced a partnership that will allow supported Android phones, starting with Snapdragon 8 Gen 2 chipset, to send and receive text messages via Iridium satellites. In 2022, T-Mobile formed a partnership to use Starlink services via existing LTE/5G NR spectrum. AST SpaceMobile aims to build a cellular space network from scratch. It will allow existing, unmodified smartphones to connect to satellites in areas with coverage gaps.


In number theory, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by Ewald Warning (1935) and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by Chevalley (1935). Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are quasi-algebraically closed fields (Artin 1982, page x).


One way to state the conjecture is that if R has no nil ideal, other than {0}, then it has no nil one-sided ideal, other than {0}. This question was posed in 1930 by Gottfried Köthe (1905–1989). The Köthe conjecture has been shown to be true for various classes of rings, such as polynomial identity rings and right Noetherian rings, but a general solution remains elusive.


The Hexadecimal notation for null is 00. Decoding the Base64 string AA== also yields the null character. In documentation, the null character is sometimes represented as a single-em-width symbol containing the letters "NUL". In Unicode, there is a character for this: U+2400 ␀ .


In some games, it is possible to partition the elements of X (or a subset of them) into a set of pairwise-disjoint pairs. Under certain conditions, a player can win using the following greedy strategy: "whenever your opponent picks an element of pair i, pick the other element of pair i". The "certain conditions" are different for Maker and for Breaker; see pairing strategy.


In the OMG vision of modeling, data is split into abstract models and concrete models. The abstract models represent the semantic information, whereas the concrete models represent visual diagrams. Abstract models are instances of arbitrary MOF-based modeling languages such as UML or SysML. For diagrams, the Diagram Interchange (DI, XMI) standard is used.


In multilevel processor sharing a finite set of thresholds are defined and jobs partitioned according to how much service they have received. The lowest level (containing jobs which have received the least service) has the highest priority and higher levels monotonically decreasing priorities. Within each level an internal discipline is used. == References ==


In the United States, the National Weather Service issues flood watches and warnings for large-scale, gradual river flooding. Watches are issued when flooding is possible or expected within 12–48 hours, and warnings are issued when flooding over a large area or river flooding is imminent or occurring. Both can be issued on a county-by-county basis or for specific rivers or points along a river. When rapid flooding from heavy rain or a dam failure is expected, flash flood watches and warnings are issued.


Developers of mobile applications must also consider a large array of devices with different screen sizes, hardware specifications, and configurations because of intense competition in mobile hardware and changes within each of the platforms. Today, mobile apps are usually distributed via an official online outlet or marketplace (e.g. Apple - The App Store, Google - Google Play) and there is a formalized process by which developers submit their apps for approval and inclusion in those marketplaces.


In practice, it is difficult to use Kuratowski's criterion to quickly decide whether a given graph is planar. However, there exist fast algorithms for this problem: for a graph with n vertices, it is possible to determine in time O(n) (linear time) whether the graph may be planar or not (see planarity testing). For a simple, connected, planar graph with v vertices and e edges and f faces, the following simple conditions hold for v ≥ 3: Theorem 1. e ≤ 3v – 6; Theorem 2. If there are no cycles of length 3, then e ≤ 2v – 4.


In this context "information" is used to refer to all the details of the state and the statement that information must be preserved means that details corresponding to an earlier time can always be reconstructed at a later time. Mathematically, the Schrödinger equation implies that the wavefunction at a time t1 can be related to the wavefunction at a time t2 by means of a unitary operator. Since the unitary operator is bijective, the wavefunction at t2 can be obtained from the wavefunction at t1 and vice versa.


Cantor's theorem had immediate and important consequences for the philosophy of mathematics. For instance, by iteratively taking the power set of an infinite set and applying Cantor's theorem, we obtain an endless hierarchy of infinite cardinals, each strictly larger than the one before it. Consequently, the theorem implies that there is no largest cardinal number (colloquially, "there's no largest infinity").


In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician Cahit Arf (1941) when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf invariant is the substitute, in characteristic 2, for the discriminant for quadratic forms in characteristic not 2. Arf used his invariant, among others, in his endeavor to classify quadratic forms in characteristic 2. In the special case of the 2-element field F2 the Arf invariant can be described as the element of F2 that occurs most often among the values of the form.


In the Feasibility stage, costs of the requirements are determined. For user requirements, the current cost of work is compared to the future projected costs once the new system is in place. Questions such as these are asked: “What are data entry errors costing us now?” Or “What is the cost of scrap due to operator error with the current interface?” Actually, the need for the new tool is often recognized as these questions come to the attention of financial people in the organization. Business costs would include, “What department has the budget for this?” “What is the expected rate of return on the new product in the marketplace?” “What’s the internal rate of return in reducing costs of training and support if we make a new, easier-to-use system?” Technical costs are related to software development costs and hardware costs.


In quantum information theory, the Lieb conjecture is a theorem concerning the Wehrl entropy of quantum systems for which the classical phase space is a sphere. It states that no state of such a system has a lower Wehrl entropy than the SU(2) coherent states. The analogous property for quantum systems for which the classical phase space is a plane was conjectured by Alfred Wehrl in 1978 and proven soon afterwards by Elliott H. Lieb, who at the same time extended it to the SU(2) case. The conjecture was only proven in 2012, by Lieb and Jan Philip Solovej.


In programming languages, monomorphization is a compile-time process where polymorphic functions are replaced by many monomorphic functions for each unique instantiation. It is considered beneficial to undergo the mentioned transformation because it results in the output intermediate representation (IR) having specific types, which allows for more effective optimization. Additionally, many IRs are intended to be low-level and do not accommodate polymorphism. The resulting code is generally faster than dynamic dispatch, but may require more compilation time and storage space due to duplicating the function body.


In the US, the PPI was known as the Wholesale Price Index, or WPI, up to 1978. The PPI is one of the oldest continuous systems of statistical data published by the Bureau of Labor Statistics, as well as one of the oldest economic time series compiled by the Federal Government. The origins of the index can be found in an 1891 U.S. Senate resolution authorizing the Senate Committee on Finance to investigate the effects of the tariff laws "upon the imports and exports, the growth, development, production, and prices of agricultural and manufactured articles at home and abroad".


Computations using this algorithm form part of the cryptographic protocols that are used to secure internet communications, and in methods for breaking these cryptosystems by factoring large composite numbers. The Euclidean algorithm may be used to solve Diophantine equations, such as finding numbers that satisfy multiple congruences according to the Chinese remainder theorem, to construct continued fractions, and to find accurate rational approximations to real numbers. Finally, it can be used as a basic tool for proving theorems in number theory such as Lagrange's four-square theorem and the uniqueness of prime factorizations. The original algorithm was described only for natural numbers and geometric lengths (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. This led to modern abstract algebraic notions such as Euclidean domains.


In statistics, the Ramsey Regression Equation Specification Error Test (RESET) test is a general specification test for the linear regression model. More specifically, it tests whether non-linear combinations of the explanatory variables help to explain the response variable. The intuition behind the test is that if non-linear combinations of the explanatory variables have any power in explaining the response variable, the model is misspecified in the sense that the data generating process might be better approximated by a polynomial or another non-linear functional form. The test was developed by James B. Ramsey as part of his Ph.D. thesis at the University of Wisconsin–Madison in 1968, and later published in the Journal of the Royal Statistical Society in 1969.


In mathematics, an orthodox semigroup is a regular semigroup whose set of idempotents forms a subsemigroup. In more recent terminology, an orthodox semigroup is a regular E-semigroup. The term orthodox semigroup was coined by T. E. Hall and presented in a paper published in 1969. Certain special classes of orthodox semigroups had been studied earlier. For example, semigroups that are also unions of groups, in which the sets of idempotents form subsemigroups were studied by P. H. H. Fantham in 1960.


In the above definition, we are concerned with the number of bits that must be deterministically transmitted between two parties. If both the parties are given access to a random number generator, can they determine the value of f {\displaystyle f} with much less information exchanged? Yao, in his seminal paper answers this question by defining randomized communication complexity. A randomized protocol R {\displaystyle R} for a function f {\displaystyle f} has two-sided error.


In the following example the result of the subtraction has fewer digits than x {\displaystyle x}: 123410  - 123401  Using the first method the sum of the nines' complement of x {\displaystyle x} and y {\displaystyle y} is 876589  + 123401  ———————— 999990 The nines' complement of 999990 is 000009. Removing the leading zeros gives 9 the desired result. If the subtrahend, y {\displaystyle y} , has fewer digits than the minuend, x {\displaystyle x} , leading zeros must be added in the second method. These zeros become leading nines when the complement is taken. For example: 48032  - 391  can be rewritten 48032  - 00391  Replacing 00391 with its nines' complement and adding 1 produces the sum: 48032  + 99608  + 1 ——————— 147641 Dropping the leading 1 gives the correct answer: 47641.


In order to make an accurate comparison and determine if there is a match, the system requires a shape or points measurement to be compared against the information in the database. This process must be discriminating, quick to compute, concise to store, pose-independent and efficient to match.


In number theory, the Padovan sequence is the sequence of integers P(n) defined by the initial values P ( 0 ) = P ( 1 ) = P ( 2 ) = 1 , {\displaystyle P(0)=P(1)=P(2)=1,} and the recurrence relation P ( n ) = P ( n − 2 ) + P ( n − 3 ) . {\displaystyle P(n)=P(n-2)+P(n-3).} The first few values of P(n) are 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ... (sequence A000931 in the OEIS)A Padovan prime is a Padovan number that is prime. The first Padovan primes are: 2, 3, 5, 7, 37, 151, 3329, 23833, 13091204281, 3093215881333057, 1363005552434666078217421284621279933627102780881053358473, 1558877695141608507751098941899265975115403618621811951868598809164180630185566719, ... (sequence A100891 in the OEIS).The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay Dom.


In mathematics, particularly in computer algebra, Abramov's algorithm computes all rational solutions of a linear recurrence equation with polynomial coefficients. The algorithm was published by Sergei A. Abramov in 1989.


1 _ 01 ϕ 0 1 _ 0 → 1 _ 01 = 210 .011 ϕ 1 1 _ 0 → 001 = 1011 .011 ϕ 200 → 1001 = 1100 .100 ϕ 011 → 100 = 10000 .1 ϕ 011 → 100 {\displaystyle {\begin{aligned}211.0{\underline {1}}0_{\phi }=211&. {\underline {1}}01_{\phi }&0{\underline {1}}0\rightarrow {\underline {1}}01\\=210&.011_{\phi }&1{\underline {1}}0\rightarrow 001\\=1011&.011_{\phi }&200\rightarrow 1001\\=1100&.100_{\phi }&011\rightarrow 100\\=10000&.1_{\phi }&011\rightarrow 100\\\end{aligned}}} Any positive number with a non-standard terminating base-φ representation can be uniquely standardized in this manner. If we get to a point where all digits are "0" or "1", except for the first digit being negative, then the number is negative.


In mathematics, Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/π. They devised several other algorithms. They published the book Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity.


Pseudaria, an ancient lost book of false proofs, is attributed to Euclid.Mathematical fallacies exist in many branches of mathematics. In elementary algebra, typical examples may involve a step where division by zero is performed, where a root is incorrectly extracted or, more generally, where different values of a multiple valued function are equated. Well-known fallacies also exist in elementary Euclidean geometry and calculus.


There are two common types of mode scramblers: the "Step-Graded-Step" (S-G-S) and the "step index with bends". The S-G-S mode scrambler is actually an assembly, a fusion-spliced concatenation of a step-index profile, a graded-index profile and another step-index profile fiber. Typically, each segment is approximately 1 meter long, and may use segments of unconventional size to produce the distribution required according to core size of fiber to be tested.


This can be proved by reduction from Planar SAT. For the case in which all holes are single points, several constant-factor approximations have been developed: A (3+sqrt(3)) approximation in time O ( n 2 ) {\displaystyle O(n^{2})} ; A (3+sqrt(3)) approximation in time O ( n log  n ) {\displaystyle O(n\log {n})} ; A 4 approximation in time O ( n log  n ) {\displaystyle O(n\log {n})} (more generally, in d dimensions, it is a 2 d {\displaystyle 2d} approximation in time O ( d n log  n ) {\displaystyle O(dn\log {n})} ), A 3 approximation in time O ( n 4 ) {\displaystyle O(n^{4})} ; A 1.75 approximation in time O ( n 5 ) {\displaystyle O(n^{5})} (more generally, in d dimensions, it is a 2 d − 4 + 4 / d {\displaystyle 2d-4+4/d} approximation in time O ( d n 2 d + 1 ) {\displaystyle O(dn^{2d+1})} ); the latter approximation uses a restricted variant of the problem called guillotine partitioning, in which the cuts must be guillotine cuts (edge-to-edge cuts). Several polynomial-time approximation schemes using sophisticated guillotine cuts.


In this case, the factorization can be done with root-finding algorithms. The case of polynomials with integer coefficients is fundamental for computer algebra. There are efficient computer algorithms for computing (complete) factorizations within the ring of polynomials with rational number coefficients (see factorization of polynomials).


In some linear optimization problems, even though the number of constraints is exponential, one can still write a custom separation oracle that works in polynomial time. Some examples are: The minimum-cost arborescence problem: given a weighted directed graph and a vertex r in it, find a subgraph of minimum cost that contains a directed path from r to any other vertex. The problem can be presented as an LP with a constraint for each subset of vertices, which is an exponential number of constraints.


{\displaystyle \pi \sigma \neq \sigma \pi .} As a bijection from a set to itself, a permutation is a function that performs a rearrangement of a set, and is not an arrangement itself. An older and more elementary viewpoint is that permutations are the arrangements themselves.


In the Japanese kantenji braille, the standard 8-dot Braille patterns 2378, 12378, 23478, and 123478 are the patterns related to Braille pattern dots-1236, since the two additional dots of kantenji patterns 01236, 12367, and 012367 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


In mathematics, the probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects from a specified class, the probability that the result is of the prescribed kind is strictly greater than zero. Although the proof uses probability, the final conclusion is determined for certain, without any possible error. This method has now been applied to other areas of mathematics such as number theory, linear algebra, and real analysis, as well as in computer science (e.g. randomized rounding), and information theory.


In the context of program correctness, static analysis can discover vulnerabilities during the development phase of the program. These vulnerabilities are easier to correct than the ones found during the testing phase since static analysis leads to the root of the vulnerability. Due to many forms of static analysis being computationally undecidable, the mechanisms for doing it will not always terminate with the right answer – either because they sometimes return a false negative ("no problems found" when the code does in fact have problems) or a false positive, or because they never return the wrong answer but sometimes never terminate. Despite their limitations, the first type of mechanism might reduce the number of vulnerabilities, while the second can sometimes give strong assurance of the lack of a certain class of vulnerabilities.


In the 1960s, swapping was an early virtual memory technique. An entire program or entire segment would be "swapped out" (or "rolled out") from RAM to disk or drum, and another one would be swapped in (or rolled in). A swapped-out program would be current but its execution would be suspended while its RAM was in use by another program; a program with a swapped-out segment could continue running until it needed that segment, at which point it would be suspended until the segment was swapped in. A program might include multiple overlays that occupy the same memory at different times.


The purpose of sampling is to reduce the cost and/or the amount of work that it would take to survey the entire target population. A survey that measures the entire target population is called a census. A sample refers to a group or section of a population from which information is to be obtained Survey samples can be broadly divided into two types: probability samples and super samples.


In the following description, the terminology used will be particular to the Windows NT operating system, but the concepts are applicable to other virtual memory operating systems.When a new application on a 32-bit OS is executed, the process has a 4 GiB VAS: each one of the memory addresses (from 0 to 232 − 1) in that space can have a single byte as a value. Initially, none of them have values ('-' represents no value). Using or setting values in such a VAS would cause a memory exception. 0 4 GiB VAS |----------------------------------------------| Then the application's executable file is mapped into the VAS.


Ego integrity is the psychological concept of the ego's accumulated assurance of its capacity for order and meaning. Ego identity is the accrued confidence that the inner sameness and continuity prepared in the past are matched by the sameness and continuity of one's meaning for others, as evidenced in the promise of a career. Body and ego control organ expressions and of the other attributes of the dynamics of a physical system to face the emotions of ego death in circumstances which can summon, sometimes, anti-theonymistic self-abandonment.


The database, where all managed objects are stored, is called Management Information Base. In contrast with a CI, a managed object is "dynamic" and communicates with other network resources that are managed. Note: A managed object may represent a physical entity, a network service, or an abstraction of a resource that exists independently of its use in management.


In the United States, the Hearing Aid Compatibility Act of 1988 requires that the Federal Communications Commission (FCC) ensure that all telephones manufactured or imported for use in the United States after August 1989, and all "essential" telephones, be hearing aid-compatible (through the use of a telecoil). "Essential" phones are defined as "coin-operated telephones, telephones provided for emergency use, and other telephones frequently needed for use by persons using such hearing aids." These might include workplace telephones, telephones in confined settings (like hospitals and nursing homes), and telephones in hotel and motel rooms. Secure telephones, as well as telephones used with public mobile and private radio services, are exempt from the HAC Act.


In order to use InfoPath to fill in a form, a designer must develop an InfoPath template first. According to Jean Paoli, one of its developers, a key architectural design decision was "to adhere to the XML paradigm of separating the data in a document from the formatting." A patent filed in 2000 by Adriana Neagu and Jean Paoli describes the technology as "authoring XML using DHTML views and XSLT. "All the data stored in InfoPath forms are stored in an XML format, which is referred to as the "data source".


This is intended to improve readability of the code, by ensuring that statements that are at the same level of nesting within a block are aligned to the same column. Other types of secondary notation, however, are not part of the formal notation. For example, when wrapping long lines, every line that is a continuation of a previous line can be indented arbitrarily.


In phrase-based translation, the aim was to reduce the restrictions of word-based translation by translating whole sequences of words, where the lengths may differ. The sequences of words were called blocks or phrases, however, typically they were not linguistic phrases, but phrasemes that were found using statistical methods from corpora. It has been shown that restricting the phrases to linguistic phrases (syntactically motivated groups of words, see syntactic categories) decreased the quality of translation.The chosen phrases were further mapped one-to-one based on a phrase translation table, and could be reordered. This table could be learnt based on word-alignment, or directly from a parallel corpus. The second model was trained using the expectation maximization algorithm, similarly to the word-based IBM model.


This replaces the instruction with a jump to a VM-safe equivalent compiled code fragment in hypervisor memory. The guest user-mode code, running in ring 3, generally runs directly on the host hardware in ring 3.In both cases, VirtualBox uses CSAM and PATM to inspect and patch the offending instructions whenever a fault occurs. VirtualBox also contains a dynamic recompiler, based on QEMU to recompile any real mode or protected mode code entirely (e.g. BIOS code, a DOS guest, or any operating system startup).Using these techniques, VirtualBox could achieve performance comparable to that of VMware in its later versions.The feature was dropped starting with VirtualBox 6.1.


In mathematics, Euclidean relations are a class of binary relations that formalize "Axiom 1" in Euclid's Elements: "Magnitudes which are equal to the same are equal to each other."


In order for equation A T A v = A T b {\displaystyle A^{T}Av=A^{T}b} to be solvable, A T A {\displaystyle A^{T}A} should be invertible, or A T A {\displaystyle A^{T}A} 's eigenvalues satisfy λ 1 ≥ λ 2 > 0 {\displaystyle \lambda _{1}\geq \lambda _{2}>0} . To avoid noise issue, usually λ 2 {\displaystyle \lambda _{2}} is required to not be too small. Also, if λ 1 / λ 2 {\displaystyle \lambda _{1}/\lambda _{2}} is too large, this means that the point p {\displaystyle p} is on an edge, and this method suffers from the aperture problem.


In the following code: a = b * c + g; d = b * c * e; it may be worth transforming the code to: tmp = b * c; a = tmp + g; d = tmp * e; if the cost of storing and retrieving tmp is less than the cost of calculating b * c an extra time.


In participant governance a network is governed by its members themselves. They call such networks that involve most or all network members interacting on a relatively equal basis in the process of governance "shared participant governance".


Periphrastic Hindi verb forms consist of two elements. The first of these two elements is the aspect marker and the second element (the copula) is the common tense/mood marker. In literary Arabic (الْفُصْحَى al-fuṣḥā) the verb has two aspect-tenses: perfective (past), and imperfective (non-past).


In the case of k-nearest neighbors regression, when the expectation is taken over the possible labeling of a fixed training set, a closed-form expression exists that relates the bias–variance decomposition to the parameter k:: 37, 223 E   = ( f ( x ) − 1 k ∑ i = 1 k f ( N i ( x ) ) ) 2 + σ 2 k + σ 2 {\displaystyle \operatorname {E} =\left(f(x)-{\frac {1}{k}}\sum _{i=1}^{k}f(N_{i}(x))\right)^{2}+{\frac {\sigma ^{2}}{k}}+\sigma ^{2}} where N 1 ( x ) , … , N k ( x ) {\displaystyle N_{1}(x),\dots ,N_{k}(x)} are the k nearest neighbors of x in the training set. The bias (first term) is a monotone rising function of k, while the variance (second term) drops off as k is increased. In fact, under "reasonable assumptions" the bias of the first-nearest neighbor (1-NN) estimator vanishes entirely as the size of the training set approaches infinity.


In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, namely a tail event of independent σ-algebras, will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one. Tail events are defined in terms of countably infinite families of σ-algebras. For illustrative purposes, we present here the special case in which each sigma algebra is generated by a random variable X k {\displaystyle X_{k}} for k ∈ N {\displaystyle k\in \mathbb {N} } . Let F {\displaystyle {\mathcal {F}}} be the sigma-algebra generated jointly by all of the X k {\displaystyle X_{k}} .


In process improvement efforts, the process performance index is an estimate of the process capability of a process during its initial set-up, before it has been brought into a state of statistical control.Formally, if the upper and lower specifications of the process are USL and LSL, the estimated mean of the process is μ ^ {\displaystyle {\hat {\mu }}} , and the estimated variability of the process (expressed as a standard deviation) is σ ^ {\displaystyle {\hat {\sigma }}} , then the process performance index is defined as: P ^ p k = min  {\displaystyle {\hat {P}}_{pk}=\min {\Bigg }} σ ^ {\displaystyle {\hat {\sigma }}} is estimated using the sample standard deviation. Ppk may be negative if the process mean falls outside the specification limits (because the process is producing a large proportion of defective output). Some specifications may only be one sided (for example, strength). For specifications that only have a lower limit, P ^ p , l o w e r = μ ^ − L S L 3 σ ^ {\displaystyle {\hat {P}}_{p,lower}={{\hat {\mu }}-LSL \over 3{\hat {\sigma }}}} ; for those that only have an upper limit, P ^ p , u p p e r = U S L − μ ^ 3 σ ^ {\displaystyle {\hat {P}}_{p,upper}={USL-{\hat {\mu }} \over 3{\hat {\sigma }}}} . Practitioners may also encounter P ^ p = U S L − L S L 6 σ ^ {\displaystyle {\hat {P}}_{p}={\frac {USL-LSL}{6{\hat {\sigma }}}}} , a metric that does not account for process performance not exactly centered between the specification limits, and therefore is interpreted as what the process would be capable of achieving if it could be centered and stabilized.


The reinforced concrete will continue to carry the load provided that sufficient reinforcement is present. A typical design problem is to find the smallest amount of reinforcement that can carry the stresses on a small cube (Fig. 1). This can be formulated as an optimization problem.


In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: E →B, an orientation of E means: for each fiber Ex, there is an orientation of the vector space Ex and one demands that each trivialization map (which is a bundle map) ϕ U: π − 1 ( U ) → U × R n {\displaystyle \phi _{U}:\pi ^{-1}(U)\to U\times \mathbf {R} ^{n}} is fiberwise orientation-preserving, where Rn is given the standard orientation. In more concise terms, this says that the structure group of the frame bundle of E, which is the real general linear group GLn(R), can be reduced to the subgroup consisting of those with positive determinant. If E is a real vector bundle of rank n, then a choice of metric on E amounts to a reduction of the structure group to the orthogonal group O(n). In that situation, an orientation of E amounts to a reduction from O(n) to the special orthogonal group SO(n).


It turned out that descendant trees of a particular kind, the so-called pruned coclass trees whose infinitely many vertices share a common coclass r {\displaystyle r} , reveal a repeating finite pattern. These two crucial properties of finiteness and periodicity admit a characterization of all members of the tree by finitely many parametrized presentations.


Replacing each vertex of the graph by a point and each edge of the graph by a point produces a non-Hausdorff space in which the open sets are the sets S {\displaystyle S} with the property that, if a vertex v {\displaystyle v} of G {\displaystyle G} belongs to S {\displaystyle S} , then so does every edge having v {\displaystyle v} as one of its endpoints.In either case, every finite subgraph of G {\displaystyle G} corresponds to a compact subspace of the topological space, and every compact subspace corresponds to a finite subgraph together with, in the Hausdorff case, finitely many compact proper subsets of edges. Thus, a graph may be covered by a nested sequence of compact sets if and only if it is locally finite, having a finite number of edges at every vertex. If a graph G {\displaystyle G} is connected and locally finite, then it has a compact cover in which the set κ i {\displaystyle \kappa _{i}} is the set of vertices at distance at most i {\displaystyle i} from some arbitrarily chosen starting vertex.


In mathematics, and more precisely in semigroup theory, a variety of finite semigroups is a class of semigroups having some nice algebraic properties. Those classes can be defined in two distinct ways, using either algebraic notions or topological notions. Varieties of finite monoids, varieties of finite ordered semigroups and varieties of finite ordered monoids are defined similarly. This notion is very similar to the general notion of variety in universal algebra.


In symmetric cryptography, the padding oracle attack can be applied to the CBC mode of operation, where the "oracle" (usually a server) leaks data about whether the padding of an encrypted message is correct or not. Such data can allow attackers to decrypt (and sometimes encrypt) messages through the oracle using the oracle's key, without knowing the encryption key.


In statistical hypothesis testing, a type I error is the mistaken rejection of an actually true null hypothesis (also known as a "false positive" finding or conclusion; example: "an innocent person is convicted"), while a type II error is the failure to reject a null hypothesis that is actually false (also known as a "false negative" finding or conclusion; example: "a guilty person is not convicted"). Much of statistical theory revolves around the minimization of one or both of these errors, though the complete elimination of either is a statistical impossibility if the outcome is not determined by a known, observable causal process. By selecting a low threshold (cut-off) value and modifying the alpha (α) level, the quality of the hypothesis test can be increased. The knowledge of type I errors and type II errors is widely used in medical science, biometrics and computer science.Intuitively, type I errors can be thought of as errors of commission (i.e., the researcher unluckily concludes that something is the fact).


In some jurisdictions the law also requires customers to collect the receipt and keep it at least for a short while after leaving the shop, again to check that the shop records sales, so that it cannot evade sales taxes. Often cash registers are attached to scales, barcode scanners, checkstands, and debit card or credit card terminals. Increasingly, dedicated cash registers are being replaced with general purpose computers with POS software. Today, point of sale systems scan the barcode (usually EAN or UPC) for each item, retrieve the price from a database, calculate deductions for items on sale (or, in British retail terminology, "special offer", "multibuy" or "buy one, get one free"), calculate the sales tax or VAT, calculate differential rates for preferred customers, actualize inventory, time and date stamp the transaction, record the transaction in detail including each item purchased, record the method of payment, keep totals for each product or type of product sold as well as total sales for specified periods, and do other tasks as well.


It may also result in smoother convergence, as the gradient computed at each step is averaged over more training samples. The convergence of stochastic gradient descent has been analyzed using the theories of convex minimization and of stochastic approximation. Briefly, when the learning rates η {\displaystyle \eta } decrease with an appropriate rate, and subject to relatively mild assumptions, stochastic gradient descent converges almost surely to a global minimum when the objective function is convex or pseudoconvex, and otherwise converges almost surely to a local minimum. This is in fact a consequence of the Robbins–Siegmund theorem.


In practice, one can perform multiple stochastic gradient passes (also called cycles or epochs) over the data. The algorithm thus obtained is called incremental gradient method and corresponds to an iteration w i = w i − 1 − γ i ∇ V ( ⟨ w i − 1 , x t i ⟩ , y t i ) {\displaystyle \textstyle w_{i}=w_{i-1}-\gamma _{i}\nabla V(\langle w_{i-1},x_{t_{i}}\rangle ,y_{t_{i}})} The main difference with the stochastic gradient method is that here a sequence t i {\displaystyle t_{i}} is chosen to decide which training point is visited in the i {\displaystyle i} -th step. Such a sequence can be stochastic or deterministic.


Different from abbreviations, those are universally recognizable across language barriers as distinct and well-defined symbols. At the coding level, prosigns admit any form the Morse code can take, unlike abbreviations which have to be sent as a sequence of individual letters, like ordinary text. On the other hand, most prosigns codes are much longer than typical codes for letters and numbers.


Table to translate raw ASCII values (A,D,M,S) to new subroutine index (1,4,3,2) in constant time using one-dimensional array (gaps in the range are shown as '..' for this example, meaning 'all hex values up to next row'. The first two columns are not part of the array) In automata-based programming and pseudoconversational transaction processing, if the number of distinct program states is small, a "dense sequence" control variable can be used to efficiently dictate the entire flow of the main program loop. A two byte raw data value would require a minimum table size of 65,536 bytes – to handle all input possibilities – whilst allowing just 256 different output values. However, this direct translation technique provides an extremely fast validation & conversion to a (relative) subroutine pointer if the heuristics, together with sufficient fast access memory, permits its use.


In probability theory, to postselect is to condition a probability space upon the occurrence of a given event. In symbols, once we postselect for an event E {\displaystyle E} , the probability of some other event F {\displaystyle F} changes from Pr   {\textstyle \operatorname {Pr} } to the conditional probability Pr   {\displaystyle \operatorname {Pr} } . For a discrete probability space, Pr   = Pr   Pr   {\textstyle \operatorname {Pr} ={\frac {\operatorname {Pr} }{\operatorname {Pr} }}} , and thus we require that Pr   {\textstyle \operatorname {Pr} } be strictly positive in order for the postselection to be well-defined. See also PostBQP, a complexity class defined with postselection.


Regulators and carriers have also been considering blocks of the 300 MHz spectrum which is normally used for television broadcasters. If a company agrees to volunteer their spectrum, the FCC will ask for 120 MHz of it. Also, the FCC has been thinking about spectrum sharing which would allow wireless ISPs to purchase DTV licenses In January 2011, Clearwire agreed to sell off its unused spectrum in order to raise money for company spectrum and to seemingly allow other companies to pick up on some unused space.


In the course of its use, the Latin alphabet was adapted for use in new languages, sometimes representing phonemes not found in languages that were already written with the Roman characters. To represent these new sounds, extensions were therefore created, be it by adding diacritics to existing letters, by joining multiple letters together to make ligatures, by creating completely new forms, or by assigning a special function to pairs or triplets of letters. These new forms are given a place in the alphabet by defining an alphabetical order or collation sequence, which can vary with the particular language.


Chip-level multiprocessing (CMP or multicore): integrates two or more processors into one chip, each executing threads independently. Any combination of multithreaded/SMT/CMP.The key factor to distinguish them is to look at how many instructions the processor can issue in one cycle and how many threads from which the instructions come. For example, Sun Microsystems' UltraSPARC T1 is a multicore processor combined with fine-grain multithreading technique instead of simultaneous multithreading because each core can only issue one instruction at a time.


In mathematics, the concept of groupoid algebra generalizes the notion of group algebra.


Denote by C the set of different configurations (and their number). For each size s in S and configuration c in C, denote: ns - the number of items of size s. as,c - the number of occurrences of size s in configuration c. xc - a variable denoting the number of bins with configuration c. Then, the configuration LP of bin-packing is: minimize ∑ c ∈ C x c subject to {\displaystyle {\text{minimize}}~~~\sum _{c\in C}x_{c}~~~{\text{subject to}}} ∑ c ∈ C a s , c x c ≥ n s {\displaystyle \sum _{c\in C}a_{s,c}x_{c}\geq n_{s}} for all s in S (- all ns items of size s are packed).


1-D structures such as this are easier to fabricate compared with constructing a rigid 2-D structure.The Symmetrical-Ring Structure is another classic example. Described by the nomenclature these are two rectangular square D type configurations, exactly the same size, lying flat, side by side, in the unit cell. Also these are not concentric.


In statistics, principal component regression (PCR) is a regression analysis technique that is based on principal component analysis (PCA). More specifically, PCR is used for estimating the unknown regression coefficients in a standard linear regression model. In PCR, instead of regressing the dependent variable on the explanatory variables directly, the principal components of the explanatory variables are used as regressors. One typically uses only a subset of all the principal components for regression, making PCR a kind of regularized procedure and also a type of shrinkage estimator.


Code-based modularity allows developers to reuse and repair parts of the application, but development tools are required to perform these maintenance functions (e.g. the application may need to be recompiled). Object-based modularity provides the application as a collection of separate executable files that may be independently maintained and replaced without redeploying the entire application (e.g. Microsoft's Dynamic-link library (DLL); Sun/UNIX shared object files). Some object messaging capabilities allow object-based applications to be distributed across multiple computers (e.g. Microsoft's Component Object Model (COM)).


For all possible assignments of false (✗) or true (✓) to P, Q, and R (columns 1, 3, 5), each subformula is evaluated according to the rules for material conditional, the result being shown below its main operator. Column 6 shows that the whole formula evaluates to true in every case, i.e. that it is a tautology. In fact, its antecedent (column 2) and its consequent (column 10) are even equivalent.


In recent versions of Unix-like operating systems such as Linux, OpenBSD, NetBSD, Solaris and FreeBSD, up to 255-character passphrases can be used.


{\displaystyle K(x,y;0)=\delta (x-y)~.} As a fundamental solution, the kernel is additive, ∫ d y K ( x , y ; t ) K ( y , z ; t ′ ) = K ( x , z ; t + t ′ ) . {\displaystyle \int dyK(x,y;t)K(y,z;t')=K(x,z;t+t')~.}


The order of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p and every positive integer k there are fields of order pk, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory.


A simple solution is to simply avoid taking such unreliable words into account as well. Applying again Bayes' theorem, and assuming the classification between spam and ham of the emails containing a given word ("replica") is a random variable with beta distribution, some programs decide to use a corrected probability: Pr ′ ( S | W ) = s ⋅ Pr ( S ) + n ⋅ Pr ( S | W ) s + n {\displaystyle \Pr '(S|W)={\frac {s\cdot \Pr(S)+n\cdot \Pr(S|W)}{s+n}}} where: Pr ′ ( S | W ) {\displaystyle \Pr '(S|W)} is the corrected probability for the message to be spam, knowing that it contains a given word ; s {\displaystyle s} is the strength we give to background information about incoming spam ; Pr ( S ) {\displaystyle \Pr(S)} is the probability of any incoming message to be spam ; n {\displaystyle n} is the number of occurrences of this word during the learning phase ; Pr ( S | W ) {\displaystyle \Pr(S|W)} is the spamicity of this word. (Demonstration:) This corrected probability is used instead of the spamicity in the combining formula.


The advantages of salt nitriding are: Quick processing time - usually in the order of 4 hours or so to achieve Simple operation - heat the salt and workpieces to temperature and submerge until the duration has transpired.The disadvantages are: The salts used are highly toxic - Disposal of salts are controlled by stringent environmental laws in western countries and has increased the costs involved in using salt baths. This is one of the most significant reasons the process has fallen out of favor in recent decades. Only one process possible with a particular salt type - since the nitrogen potential is set by the salt, only one type of process is possible


Linguistic concepts take time to settle in the memory. The learner must use the new concepts frequently after presentation, either by thinking or by speaking, in order to master them. His last observation was that language was learned in sentences with the verb as the most crucial component.


In sentences with Subject, Object, and Verb, any order is possible. However, some orders are more common than others. In a sample of 568 sentences of Caesar containing all three elements examined by Pinkster, the proportions were: SOV: 63% OSV: 21% OVS: 6% VOS: 5% SVO: 4% VSO: 1%An example of the typical Subject-Object-Verb (SOV) word order in Caesar is: Caesar suās cōpiās in proximum collem subdūcit.


In Knaster's original publication on dendroids, in 1961, he posed the problem of characterizing the dendroids which can be embedded into the Euclidean plane. This problem remains open.


In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable y is modelled as an nth degree polynomial in x. Polynomial regression fits a nonlinear relationship between the value of x and the corresponding conditional mean of y, denoted E(y |x). Although polynomial regression fits a nonlinear model to the data, as a statistical estimation problem it is linear, in the sense that the regression function E(y | x) is linear in the unknown parameters that are estimated from the data. For this reason, polynomial regression is considered to be a special case of multiple linear regression. The explanatory (independent) variables resulting from the polynomial expansion of the "baseline" variables are known as higher-degree terms. Such variables are also used in classification settings.


Sensitivity auditing is especially suitable in an adversarial context, where not only the nature of the evidence, but also the degree of certainty and uncertainty associated to the evidence, is the subject of partisan interests. These are the settings considered in Post-normal science or in Mode 2 science. Post-normal science (PNS) is a concept developed by Silvio Funtowicz and Jerome Ravetz, which proposes a methodology of inquiry that is appropriate when “facts are uncertain, values in dispute, stakes high and decisions urgent” (Funtowicz and Ravetz, 1992: 251–273).


In telecommunication, common-channel signaling (CCS), or common-channel interoffice signaling (CCIS), is the transmission of control information (signaling) via a separate channel than that used for the messages, The signaling channel usually controls multiple message channels.In the public switched telephone network (PSTN) one channel of a communications link is typically used for the sole purpose of carrying signaling for establishment and tear down of telephone calls. The remaining channels are used entirely for the transmission of voice messages. In most cases, a single 64 kbit/s channel is sufficient to handle the call setup and call clear-down traffic for numerous bearer (voice and data) channels.The technical alternative to CCS is channel-associated signaling (CAS), in which each bearer channel has a dedicated signaling channel.


The form that a proposition takes depends on the type of logic. The type of logic called propositional, sentential, or statement logic includes only operators and propositional constants as symbols in its language. The propositions in this language are propositional constants, which are considered atomic propositions, and composite (or compound) propositions, which are composed by recursively applying operators to propositions.


Consequently, descendant trees play a fundamental role in the classification of finite p-groups. By means of kernels and targets of Artin transfer homomorphisms, descendant trees can be endowed with additional structure. An important question is how the descendant tree T ( R ) {\displaystyle {\mathcal {T}}(R)} can actually be constructed for an assigned starting group which is taken as the root R {\displaystyle R} of the tree. The p-group generation algorithm is a recursive process for constructing the descendant tree of a given finite p-group playing the role of the tree root. This algorithm is implemented in the computational algebra systems GAP and Magma.


Then, someone hands you a huge number of KCBS pentagrams, but at first, all of the colorings are hidden. You're told you can only uncover 2 vertices at most, and only if they share a common edge. So, for each pentagram, you randomly pick an edge and uncover the colors on its vertices.


Two years later, Sergei Adian showed that certain Burnside groups are also counterexamples. None of these counterexamples are finitely presented, and for some years it was considered possible that the conjecture held for finitely presented groups.


S+consonant clusters may vary between aspirated and nonaspirated depending upon if the cluster crosses a morpheme boundary or not. For instance, distend has unaspirated  since it is not analyzed as two morphemes, but distaste has an aspirated middle  because it is analyzed as dis- + taste and the word taste has an aspirated initial t. Word-final voiceless stops are sometimes aspirated. Voiceless stops in Pashto are slightly aspirated prevocalically in a stressed syllable.


In probability theory, a sub-Gaussian distribution is a probability distribution with strong tail decay. Informally, the tails of a sub-Gaussian distribution are dominated by (i.e. decay at least as fast as) the tails of a Gaussian. This property gives sub-Gaussian distributions their name. Formally, the probability distribution of a random variable X {\displaystyle X} is called sub-Gaussian if there is a positive constant C such that for every t ≥ 0 {\displaystyle t\geq 0} , P  ( | X | ≥ t ) ≤ 2 exp  ( − t 2 / C 2 ) {\textstyle \operatorname {P} (|X|\geq t)\leq 2\exp {(-t^{2}/C^{2})}} .


Regarding diagnosis, a proposed taxonomy of service tools is as follows: Level 1: Service tool that indicates if a product is functional or not functional. Describing computer servers, the states are often referred to as ‘up’ or ‘down’. This is a binary value.


In number theory, the Pólya conjecture (or Pólya's conjecture) stated that "most" (i.e., 50% or more) of the natural numbers less than any given number have an odd number of prime factors. The conjecture was set forth by the Hungarian mathematician George Pólya in 1919, and proved false in 1958 by C. Brian Haselgrove. Though mathematicians typically refer to this statement as the Pólya conjecture, Pólya never actually conjectured that the statement was true; rather, he showed that the truth of the statement would imply the Riemann hypothesis. For this reason, it is more accurately called "Pólya's problem". The size of the smallest counterexample is often used to demonstrate the fact that a conjecture can be true for many cases and still fail to hold in general, providing an illustration of the strong law of small numbers.


In the design of programming languages, an erroneous program is one whose semantics are not well-defined, but where the language implementation is not obligated to signal an error either at compile or at execution time. For example, in Ada: In addition to bounded errors, the language rules define certain kinds of errors as leading to erroneous execution. Like bounded errors, the implementation need not detect such errors either prior to or during run time. Unlike bounded errors, there is no language-specified bound on the possible effect of erroneous execution; the effect is in general not predictable.Defining a condition as "erroneous" means that the language implementation need not perform a potentially expensive check (e.g. that a global variable refers to the same object as a subroutine parameter) but may nonetheless depend on a condition being true in defining the semantics of the program. == Notes ==


In the decimal (base-10) Hindu–Arabic numeral system, each position starting from the right is a higher power of 10. The first position represents 100 (1), the second position 101 (10), the third position 102 (10 × 10 or 100), the fourth position 103 (10 × 10 × 10 or 1000), and so on. Fractional values are indicated by a separator, which can vary in different locations. Usually this separator is a period or full stop, or a comma.


In sequence design, a Feedback with Carry Shift Register (or FCSR) is the arithmetic or with carry analog of a linear-feedback shift register (LFSR). If N > 1 {\displaystyle N>1} is an integer, then an N-ary FCSR of length r {\displaystyle r} is a finite state device with a state ( a ; z ) = ( a 0 , a 1 , … , a r − 1 ; z ) {\displaystyle (a;z)=(a_{0},a_{1},\dots ,a_{r-1};z)} consisting of a vector of elements a i {\displaystyle a_{i}} in { 0 , 1 , … , N − 1 } = S {\displaystyle \{0,1,\dots ,N-1\}=S} and an integer z {\displaystyle z} . The state change operation is determined by a set of coefficients q 1 , … , q n {\displaystyle q_{1},\dots ,q_{n}} and is defined as follows: compute s = q r a 0 + q r − 1 a 1 + ⋯ + q 1 a r − 1 + z {\displaystyle s=q_{r}a_{0}+q_{r-1}a_{1}+\dots +q_{1}a_{r-1}+z} . Express s as s = a r + N z ′ {\displaystyle s=a_{r}+Nz'} with a r {\displaystyle a_{r}} in S {\displaystyle S} .


(binary, 6 multiplications) a 15 = (  2 × a ) 3 {\displaystyle a^{15}=(^{2}\times a)^{3}\!} (shortest addition chain, 5 multiplications). a 15 = a 3 × (  2 ) 2 {\displaystyle a^{15}=a^{3}\times (^{2})^{2}\!}


For this purpose several test procedures have been devised. The test procedure due to M.S.E (Mean Square Error/Estimator) Bartlett test is represented here. This test procedure is based on the statistic whose sampling distribution is approximately a Chi-Square distribution with (k − 1) degrees of freedom, where k is the number of random samples, which may vary in size and are each drawn from independent normal distributions.


In the 1970s the National Bureau of Standards (NBS) defined a set of convenient numbers to ease metrication in the United States. This system of metric values was described as 1–2–5 series in reverse, with assigned preferences for those numbers which are multiples of 5, 2, and 1 (plus their powers of 10), excluding linear dimensions above 100 mm.


For many modern packet-switched communication systems, even a single uncorrected bit error is enough to cause the loss of a data packet by causing its CRC check to fail; whether that packet loss was caused by a single bit error or a hundred-bit-long error burst is irrelevant. For systems using large amounts of forward error correction, the reverse applies; a single low-level bit error will almost never occur, since any small errors will almost always be corrected, but any error sufficiently large to cause the forward error correction to fail will almost always result in a large burst error. More specialist and precise definitions of errored seconds exist in standards such as the T1 and DS1 transport systems.


In mathematics, a binary relation on a set may, or may not, hold between two given set members. For example, "is less than" is a relation on the set of natural numbers; it holds e.g. between 1 and 3 (denoted as 1<3) , and likewise between 3 and 4 (denoted as 3<4), but neither between 3 and 1 nor between 4 and 4. As another example, "is sister of" is a relation on the set of all people, it holds e.g. between Marie Curie and Bronisława Dłuska, and likewise vice versa. Set members may not be in relation "to a certain degree" - either they are in relation or they are not.


If frame pointers are being used, the prologue will typically set the new value of the frame pointer register from the stack pointer. Space on the stack for local variables can then be allocated by incrementally changing the stack pointer. The Forth programming language allows explicit winding of the call stack (called there the "return stack").


That may take many clock cycles, since the operand of X may in turn depend on previous instructions that fetch data from main memory.Rather than halt while waiting for X to be finished, stage A may guess whether the branch will be taken or not, and fetch the next instruction Y based on that guess. If the guess later turns out to be incorrect (hopefully rarely), the system would have to backtrack and resume with the correct choice. Namely, all the changes that were made to the machine's state by stage A and subsequent stages based on that guess would have to be undone, the instructions following X already in the pipeline would have to be flushed, and stage A would have to restart with the correct instruction pointer. This branch prediction strategy is a special case of speculative execution.


5. Clustering We can then cluster different documents based on the features we have generated. See the algorithm section in cluster analysis for different types of clustering methods.


The objective is to achieve the minimum perturbation to the item size vector so that all the items can be packed into the prescribed number of bins. In the maximum resource bin packing problem, the goal is to maximize the number of bins used, such that, for some ordering of the bins, no item in a later bin fits in an earlier bin. In a dual problem, the number of bins is fixed, and the goal is to minimize the total number or the total size of items placed into the bins, such that no remaining item fits into an unfilled bin.


In quantum mechanics, information theory, and Fourier analysis, the entropic uncertainty or Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies. It turns out that Heisenberg's uncertainty principle can be expressed as a lower bound on the sum of these entropies. This is stronger than the usual statement of the uncertainty principle in terms of the product of standard deviations. In 1957, Hirschman considered a function f and its Fourier transform g such that g ( y ) ≈ ∫ − ∞ ∞ exp  ( − 2 π i x y ) f ( x ) d x , f ( x ) ≈ ∫ − ∞ ∞ exp  ( 2 π i x y ) g ( y ) d y , {\displaystyle g(y)\approx \int _{-\infty }^{\infty }\exp(-2\pi ixy)f(x)\,dx,\qquad f(x)\approx \int _{-\infty }^{\infty }\exp(2\pi ixy)g(y)\,dy~,} where the "≈" indicates convergence in L2, and normalized so that (by Plancherel's theorem), ∫ − ∞ ∞ | f ( x ) | 2 d x = ∫ − ∞ ∞ | g ( y ) | 2 d y = 1 .


Pell's equation: x 2 − P y 2 = 1 , {\displaystyle \ x^{2}-Py^{2}=1,} where P {\displaystyle P} is a given integer that is not a square number, and in which the variables x {\displaystyle x} and y {\displaystyle y} are required to be integers. The equation of Pythagorean triples: x 2 + y 2 = z 2 , {\displaystyle x^{2}+y^{2}=z^{2},} in which the variables x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} are required to be positive integers. The equation of the Fermat–Catalan conjecture: a m + b n = c k , {\displaystyle a^{m}+b^{n}=c^{k},} in which the variables a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} are required to be coprime positive integers, and the variables m {\displaystyle m} , n {\displaystyle n} , and k {\displaystyle k} are required to be positive integers satisfying the following equation: 1 m + 1 n + 1 k < 1. {\displaystyle {\frac {1}{m}}+{\frac {1}{n}}+{\frac {1}{k}}<1.}


In the case of complete ranks, a commonly used significance test for W against a null hypothesis of no agreement (i.e. random rankings) is given by Kendall and Gibbons (1990) χ 2 = m ( n − 1 ) W {\displaystyle \chi ^{2}=m(n-1)W} Where the test statistic takes a chi-squared distribution with d f = n − 1 {\displaystyle df=n-1} degrees of freedom. In the case of incomplete rankings (see above), this becomes χ 2 = λ ( n 2 − 1 ) k + 1 W {\displaystyle \chi ^{2}={\frac {\lambda (n^{2}-1)}{k+1}}W} Where again, there are d f = n − 1 {\displaystyle df=n-1} degrees of freedom. Legendre compared via simulation the power of the chi-square and permutation testing approaches to determining significance for Kendall's W. Results indicated the chi-square method was overly conservative compared to a permutation test when m < 20 {\displaystyle m<20} . Marozzi extended this by also considering the F test, as proposed in the original publication introducing the W statistic by Kendall & Babington Smith (1939): F = W ( m − 1 ) 1 − W {\displaystyle F={\frac {W(m-1)}{1-W}}} Where the test statistic follows an F distribution with v 1 = n − 1 − ( 2 / m ) {\displaystyle v_{1}=n-1-(2/m)} and v 2 = ( m − 1 ) v 1 {\displaystyle v_{2}=(m-1)v_{1}} degrees of freedom. Marozzi found the F test performs approximately as well as the permutation test method, and may be preferred to when m {\displaystyle m} is small, as it is computationally simpler.


In telecommunications, a protocol data unit (PDU) is a single unit of information transmitted among peer entities of a computer network. It is composed of protocol-specific control information and user data. In the layered architectures of communication protocol stacks, each layer implements protocols tailored to the specific type or mode of data exchange. For example, the Transmission Control Protocol (TCP) implements a connection-oriented transfer mode, and the PDU of this protocol is called a segment, while the User Datagram Protocol (UDP) uses datagrams as protocol data units for connectionless communication. A layer lower in the Internet protocol suite, at the Internet layer, the PDU is called a packet, irrespective of its payload type.


In the context of computer programming, instrumentation refers to the measure of a product's performance, in order to diagnose errors and to write trace information. Instrumentation can be of two types: source instrumentation and binary instrumentation.


Since, for each method of type Queue, type Stack provides a method with a matching name and signature, this check would succeed. However, clients accessing a Stack object through a reference of type Queue would, based on Queue's documentation, expect FIFO behavior but observe LIFO behavior, invalidating these clients' correctness proofs and potentially leading to incorrect behavior of the program as a whole. This example violates behavioral subtyping because type Stack is not a behavioral subtype of type Queue: it is not the case that the behavior described by the documentation of type Stack (i.e. LIFO behavior) complies with the documentation of type Queue (which requires FIFO behavior).


The fact that a map f: X → Y {\displaystyle f:X\rightarrow Y} is an embedding is often indicated by the use of a "hooked arrow" (U+21AA ↪ RIGHTWARDS ARROW WITH HOOK); thus: f: X ↪ Y . {\displaystyle f:X\hookrightarrow Y.} (On the other hand, this notation is sometimes reserved for inclusion maps.)


Consequently the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles and formally published in 1995.


In mathematics education, a number sentence is an equation or inequality expressed using numbers and mathematical symbols. The term is used in primary level mathematics teaching in the US, Canada, UK, Australia, New Zealand and South Africa.


In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent. Any collection of mutually independent random variables is pairwise independent, but some pairwise independent collections are not mutually independent. Pairwise independent random variables with finite variance are uncorrelated. A pair of random variables X and Y are independent if and only if the random vector (X, Y) with joint cumulative distribution function (CDF) F X , Y ( x , y ) {\displaystyle F_{X,Y}(x,y)} satisfies F X , Y ( x , y ) = F X ( x ) F Y ( y ) , {\displaystyle F_{X,Y}(x,y)=F_{X}(x)F_{Y}(y),} or equivalently, their joint density f X , Y ( x , y ) {\displaystyle f_{X,Y}(x,y)} satisfies f X , Y ( x , y ) = f X ( x ) f Y ( y ) .


In mathematics, especially in category theory, the codensity monad is a fundamental construction associating a monad to a wide class of functors.


Consider the overcategory Top / X {\displaystyle {\text{Top}}/X} of topological spaces over X {\displaystyle X} , that is, the category of topological spaces together with fixed continuous maps to X {\displaystyle X} . Every object of this category is a continuous map f: Y → X {\displaystyle f:Y\to X} , and a morphism from Y → X {\displaystyle Y\to X} to Z → X {\displaystyle Z\to X} is a continuous map Y → Z {\displaystyle Y\to Z} that commutes with the two maps to X {\displaystyle X} . There is a functor Γ: Top / X → Sets {\displaystyle \Gamma :{\text{Top}}/X\to {\text{Sets}}} sending an object f: Y → X {\displaystyle f:Y\to X} to f − 1 F ( Y ) {\displaystyle f^{-1}F(Y)} .


In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the) finite-dimensional linear representations of the group. Elements of the representation ring are sometimes called virtual representations. For a given group, the ring will depend on the base field of the representations. The case of complex coefficients is the most developed, but the case of algebraically closed fields of characteristic p where the Sylow p-subgroups are cyclic is also theoretically approachable.


The notion of residuated map can be generalized to a binary operator (or any higher arity) via component-wise residuation. This approach gives rise to notions of left and right division in a partially ordered magma, additionally endowing it with a quasigroup structure. (One speaks only of residuated algebra for higher arities). A binary (or higher arity) residuated map is usually not residuated as a unary map.


Many other disciplines use similar criteria or have specific measures geared toward the objectives of the field. Optimality criteria include maximum likelihood, Bayesian, maximum parsimony, sum of squared residuals, least absolute deviations, and many others. == References ==


Calls from mobile phones are usually routed based on the location of the base station rather than the calling party, so these (along with landline calls to non-emergency telephone numbers) must be handled manually by the telephone operator or dispatcher, determining whether the caller or incident is within a particular city limit or not so that the proper authorities may be sent. A city's limit may extend into more than one county, which can complicate certain matters of policing and taxation. (For example, sales tax revenue collected in a city by one county may not be spent in another part of the city outside that county.)


. . , a n ⟩ {\displaystyle q_{1}\cong \langle a_{1},...,a_{n}\rangle } q 2 ≅ ⟨ b 1 , .


In mathematics, binary splitting is a technique for speeding up numerical evaluation of many types of series with rational terms. In particular, it can be used to evaluate hypergeometric series at rational points.


Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links (degrees) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size N of the network and average degree constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree k is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree k while the other nodes have a small number of links.


In network science, a hub is a node with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks. : 27 While hubs cannot be observed in a random network, they are expected to emerge in scale-free networks. The uprise of hubs in scale-free networks is associated with power-law distribution.


In modern businesses, the design of file servers is complicated by competing demands for storage space, access speed, recoverability, ease of administration, security, and budget. This is further complicated by a constantly changing environment, where new hardware and technology rapidly obsolesces old equipment, and yet must seamlessly come online in a fashion compatible with the older machinery. To manage throughput, peak loads, and response time, vendors may utilize queuing theory to model how the combination of hardware and software will respond over various levels of demand.


In network science, the configuration model is a method for generating random networks from a given degree sequence. It is widely used as a reference model for real-life social networks, because it allows the modeler to incorporate arbitrary degree distributions.


Our example is based on a modified Post–Turing machine model of a Turing Machine. This model prints only the symbols 0 and 1. The blank tape is considered to be all b's.


In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form M ( λ ) x = 0 , {\displaystyle M(\lambda )x=0,} where x ≠ 0 {\displaystyle x\neq 0} is a vector, and M {\displaystyle M} is a matrix-valued function of the number λ {\displaystyle \lambda } . The number λ {\displaystyle \lambda } is known as the (nonlinear) eigenvalue, the vector x {\displaystyle x} as the (nonlinear) eigenvector, and ( λ , x ) {\displaystyle (\lambda ,x)} as the eigenpair. The matrix M ( λ ) {\displaystyle M(\lambda )} is singular at an eigenvalue λ {\displaystyle \lambda } .


The first few measured and corrected moments about the mean are then related as follows: μ ^ 2 = m 2 − 1 12 c 2 μ ^ 3 = m 3 μ ^ 4 = m 4 − 1 2 m 2 c 2 + 7 240 c 4 . {\displaystyle {\begin{aligned}{\hat {\mu }}_{2}&=m_{2}-{\frac {1}{12}}c^{2}\\{\hat {\mu }}_{3}&=m_{3}\\{\hat {\mu }}_{4}&=m_{4}-{\frac {1}{2}}m_{2}c^{2}+{\frac {7}{240}}c^{4}.\end{aligned}}} When the data come from a normally distributed population, then binning and using the midpoint of the bin as the observed value results in an overestimate of the variance. That is why the correction to the variance is negative.


In post-order we always recursively traverse the current node's left subtree, next we recursively traverse the current node's right subtree and then visit the current node. Post-order traversal can be useful to get postfix expression of a binary expression tree.


For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, they are the GCD of the original two numbers.


In statistics and computational geometry, the notion of centerpoint is a generalization of the median to data in higher-dimensional Euclidean space. Given a set of points in d-dimensional space, a centerpoint of the set is a point such that any hyperplane that goes through that point divides the set of points in two roughly equal subsets: the smaller part should have at least a 1/(d + 1) fraction of the points. Like the median, a centerpoint need not be one of the data points. Every non-empty set of points (with no duplicates) has at least one centerpoint.


In operating systems, processes are loaded into memory, and wait for their turn to be executed by the central processing unit (CPU). CPU scheduling manages process states and decides when a process will be executed next by using the input queue.