in mathematics, the support function ha of a non - empty closed convex set a in r n { \ displaystyle \ mathbb { r } ^ { n } } describes the ( signed ) distances of supporting hyperplanes of a from the origin. the support function is a convex function on r n { \ displaystyle \ mathbb { r } ^ { n } }. any non - empty closed convex set a is uniquely determined by ha. furthermore, the support function, as a function of the set a, is compatible with many natural geometric operations, like scaling, translation, rotation and minkowski addition. due to these properties, the support function is one of the most central basic concepts in convex geometry.


in protected mode a segment cannot be both writable and executable. therefore, when implementing the tiny memory model the code segment register must point to the same physical address and have the same limit as the data segment register. this defeated one of the features of the 80286, which makes sure data segments are never executable and code segments are never writable ( which means that self - modifying code is never allowed ). however, on the 80386, with its paged memory management unit it is possible to protect individual memory pages against writing. memory models are not limited to 16 - bit programs.


an ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes. networks rich in structural holes are a form of social capital in that they offer information benefits. the main player in a network that bridges structural holes is able to access information from diverse sources and clusters. for example, in business networks, this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries / sectors. this concept is similar to mark granovetter's theory of weak ties, which rests on the basis that having a broad range of contacts is most effective for job attainment.


less common are unilateral contracts, in which one party makes a promise, but the other side does not promise anything. in these cases, those accepting the offer are not required to communicate their acceptance to the offeror. in a reward contract, for example, a person who has lost a dog could promise a reward if the dog is found, through publication or orally.


more generally the transition rates are given in form of a finite measure c λ ( η, d ξ ) { \ displaystyle c _ { \ lambda } ( \ eta, d \ xi ) } on s λ { \ displaystyle s ^ { \ lambda } }. the generator l { \ displaystyle l } of an ips has the following form. first, the domain of l { \ displaystyle l } is a subset of the space of " observables ", that is, the set of real valued continuous functions on the configuration space ω { \ displaystyle \ omega }.


there are many key distribution schemes in the literature that are designed to maintain an easy and at the same time secure communication among sensor nodes. the most accepted method of key distribution in wsns is key predistribution, where secret keys are placed in sensor nodes before deployment. when the nodes are deployed over the target area, the secret keys are used to create the network. for more info see : key distribution in wireless sensor networks.


in the following example expressed in c, a program has two variables which are adjacent in memory : an 8 - byte - long string buffer, a, and a two - byte big - endian integer, b. initially, a contains nothing but zero bytes, and b contains the number 1979. now, the program attempts to store the null - terminated string " excessive " with ascii encoding in the a buffer. " excessive " is 9 characters long and encodes to 10 bytes including the null terminator, but a can take only 8 bytes.


woodall's conjecture, an unsolved problem in this area, states that in any directed graph the minimum number of edges in a dicut ( the unweighted minimum closure ) equals the maximum number of disjoint dijoins that can be found in the graph ( a packing of dijoins ). a fractional weighted version of the conjecture, posed by jack edmonds and rick giles, was refuted by alexander schrijver. in the other direction, the lucchesi – younger theorem states that the minimum size of a dijoin equals the maximum number of disjoint dicuts that can be found in a given graph. = = references = =


in mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by stanisław zaremba and david hilbert around 1900. the method relies on methods of functional analysis and topology. as well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy.


in the fall of 2004, the exam became computerized ; the mat is now solely a computer - based test ( cbt ). out of the 120 questions, only 100 count in the test - taker's score. the remaining 20 questions are experimental. there is no way for test - takers to identify any of the 20 experimental questions on a given test form, as the two types of questions are intermingled. tests taken before october 2004 were scored simply by the number of questions the test - taker answered correctly, with a range from 0 - 100.


in practice, most failures can be traced back to some type of human error, for example in : management decisions ( e. g. in budgeting, timing, and required tasks ) systems engineering : use studies ( load cases ) systems engineering : requirement analysis / setting systems engineering : configuration control assumptions calculations / simulations / fem analysis design design drawings testing ( e. g. incorrect load settings or failure measurement ) statistical analysis manufacturing quality control maintenance maintenance manuals training classifying and ordering of information feedback of field information ( e. g. incorrect or too vague ) etc. however, humans are also very good at detecting such failures, correcting for them, and improvising when abnormal situations occur. therefore, policies that completely rule out human actions in design and production processes to improve reliability may not be effective. some tasks are better performed by humans and some are better performed by machines. furthermore, human errors in management ; the organization of data and information ; or the misuse or abuse of items, may also contribute to unreliability. this is the core reason why high levels of reliability for complex systems can only be achieved by following a robust systems engineering process with proper planning and execution of the validation and verification tasks. this also includes careful organization of data and information sharing and creating a " reliability culture ", in the same way that having a " safety culture " is paramount in the development of safety critical systems.


this can be used to handle queries in the following fashion : when we look up cell t for some k, we can check if k is in the range { 1,..., m } : if it is not, then t is uninitialized. otherwise, we check v ], and verify that the first component of this pair is equal to k. if it is not, then t is uninitialized ( and just happened by accident to fall in the range { 1,..., m } ). otherwise, we know that t is indeed one of the initialized cells, and the corresponding value is the second component of the pair.


in multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form lim m → ∞ lim n → ∞ a n, m = lim m → ∞ ( lim n → ∞ a n, m ) { \ displaystyle \ lim _ { m \ to \ infty } \ lim _ { n \ to \ infty } a _ { n, m } = \ lim _ { m \ to \ infty } \ left ( \ lim _ { n \ to \ infty } a _ { n, m } \ right ) }, lim y → b lim x → a f ( x, y ) = lim y → b ( lim x → a f ( x, y ) ) { \ displaystyle \ lim _ { y \ to b } \ lim _ { x \ to a } f ( x, y ) = \ lim _ { y \ to b } \ left ( \ lim _ { x \ to a } f ( x, y ) \ right ) }, or other similar forms. an iterated limit is only defined for an expression whose value depends on at least two variables. to evaluate such a limit, one takes the limiting process as one of the two variables approaches some number, getting an expression whose value depends only on the other variable, and then one takes the limit as the other variable approaches some number.


the definition of the term named entity is therefore not strict and often has to be explained in the context in which it is used. certain hierarchies of named entity types have been proposed in the literature. bbn categories, proposed in 2002, is used for question answering and consists of 29 types and 64 subtypes. sekine's extended hierarchy, proposed in 2002, is made of 200 subtypes. more recently, in 2011 ritter used a hierarchy based on common freebase entity types in ground - breaking experiments on ner over social media text.


in more recent years, since the development of the top - down methodologies, ia have been used as a basis for the development of a knowledge audit, which itself in - turn contributes to an organisation's knowledge management strategy. once complete, the ia allows examination into where knowledge is produced, where there may be need for further input and where knowledge transfer is required. furthermore, this analysis develops strategy for knowledge capture, access, storage, dissemination and validation. dissimilarly to the ia, the objectives of the knowledge audit are to identify any people - related issues which impact the ways in which knowledge is created, transferred and shared and to identify where knowledge could be captured, where it is required and then determine how best to undertake a knowledge transfer as " unlike information, knowledge is bound to a person, organisation or community. " similarities between the knowledge and information audit methodologies can be noted however, as questionnaires, the development of an inventory, analysis of flow and a data map are here again used. the importance of this audit therefore is to understand the strategic significance of an organisation's knowledge assets to ensure management is focused to those areas it is specifically required.


in psychology and sociology, a trust metric is a measurement or metric of the degree to which one social actor ( an individual or a group ) trusts another social actor. trust metrics may be abstracted in a manner that can be implemented on computers, making them of interest for the study and engineering of virtual communities, such as friendster and livejournal. trust escapes a simple measurement because its meaning is too subjective for universally reliable metrics, and the fact that it is a mental process, unavailable to instruments. there is a strong argument against the use of simplistic metrics to measure trust due to the complexity of the process and the'embeddedness'of trust that makes it impossible to isolate trust from related factors.


now, θ j = h − 1 ( s, z 1 j, …, z m j ) { \ displaystyle { \ breve { \ theta } } _ { j } = h ^ { - 1 } ( s, { \ breve { z } } _ { 1 } ^ { j }, \ ldots, { \ breve { z } } _ { m } ^ { j } ) } is a solution of ( 1 ) with the seed { z 1 j, …, z m j } { \ displaystyle \ { { \ breve { z } } _ { 1 } ^ { j }, \ ldots, { \ breve { z } } _ { m } ^ { j } \ } }. since the seeds are equally distributed, the sole caveat comes from their independence or, conversely from their dependence on?


in mathematics, in number theory, gauss composition law is a rule, invented by carl friedrich gauss, for performing a binary operation on integral binary quadratic forms ( ibqfs ). gauss presented this rule in his disquisitiones arithmeticae, a textbook on number theory published in 1801, in articles 234 - 244. gauss composition law is one of the deepest results in the theory of ibqfs and gauss's formulation of the law and the proofs its properties as given by gauss are generally considered highly complicated and very difficult. several later mathematicians have simplified the formulation of the composition law and have presented it in a format suitable for numerical computations. the concept has also found generalisations in several directions.


in other words, if a model defines type colour : : = red | blue | green and some operation uses c of type colour, then by applying this tactic each test class will by divided into three new test classes : one in which c equals red, the other in which c equals blue, and the third where c equals green. proper subset of set extension ( psse ). this tactic uses the same concept of ise but applied to set inclusions.


mood can be bound up with tense, aspect, or both, in particular verb forms. hence, certain languages are sometimes analysed as having a single tense – aspect – mood ( tam ) system, without separate manifestation of the three categories. the term tense, then, particularly in less formal contexts, is sometimes used to denote any combination of tense proper, aspect, and mood.


storable resources remain available unless depleted by usage, and may be replenished by project tasks that produce them. nonstorable resources must be renewed for each time period, even if not used in previous periods. resource scheduling, availability, and optimisation are considered key to successful project management. allocation of limited resources is based on the priority given to each of the project activities. their priorities are calculated using the critical path method and heuristic analysis. for a case with a constraint on the available resources, the objective is to create the most efficient schedule possible - minimising project duration and maximising the use of the resources available.


in the case r = 2, the hypergraph becomes a bipartite graph, and the conjecture becomes τ ( h ) ≤ ν ( h ) { \ displaystyle \ tau ( h ) \ leq \ nu ( h ) }. this is known to be true by konig's theorem. in the case r = 3, the conjecture has been proved by ron aharoni. the proof uses the aharoni - haxell theorem for matching in hypergraphs. in the cases r = 4 and r = 5, the following weaker version has been proved by penny haxell and scott : there exists some ε > 0 such that τ ( h ) ≤ ( r − ε ) ⋅ ν ( h ) { \ displaystyle \ tau ( h ) \ leq ( r - \ varepsilon ) \ cdot \ nu ( h ) }. moreover, in the cases r = 4 and r = 5, ryser's conjecture has been proved by tuza ( 1978 ) in the special case ν ( h ) = 1 { \ displaystyle \ nu ( h ) = 1 }, i. e. : ν ( h ) = 1 τ ( h ) ≤ r − 1 { \ displaystyle \ nu ( h ) = 1 \ implies \ tau ( h ) \ leq r - 1 }.


in mathematics, the entropy influence conjecture is a statement about boolean functions originally conjectured by ehud friedgut and gil kalai in 1996.


in mathematics, and in particular model theory, a prime model is a model that is as simple as possible. specifically, a model p { \ displaystyle p } is prime if it admits an elementary embedding into any model m { \ displaystyle m } to which it is elementarily equivalent ( that is, into any model m { \ displaystyle m } satisfying the same complete theory as p { \ displaystyle p } ).


euler's sum of powers conjecture ( disproved ) concerns situations in which the sum of n integers, each a kth power of an integer, equals another kth power. the fermat - catalan conjecture asks whether there are an infinitude of examples in which the sum of two coprime integers, each a power of an integer, with the powers not necessarily equal, can equal another integer that is a power, with the reciprocals of the three powers summing to less than 1. beal's conjecture concerns the question of whether the sum of two coprime integers, each a power greater than 2 of an integer, with the powers not necessarily equal, can equal another integer that is a power greater than 2.


instead, the receiving'client'( object or function ) is provided with its dependencies by external code ( an'injector'), which it is not aware of. dependency injection makes implicit dependencies explicit and helps solve the following problems : how can a class be independent from the creation of the objects it depends on? how can an application, and the objects it uses support different configurations?


the record size is stored on a file - by - file basis in special entries in the directory table. sequential access methods for ibm's z / os and z / vse mainframe operating systems : basic sequential access method ( bsam ), basic partitioned access method ( bpam ) and queued sequential access method ( qsam ) ; see access methods and data set ( ibm mainframe ) for more examples pick operating system – a record - oriented filesystem and database that uses hash - coding to store data. shared file system ( sfs ) for ibm's vm virtual storage access method ( vsam ) – for ibm's z / os and z / vse mainframe operating systems


in mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof, a kind of proof whose semantic properties are exposed. when all the theorems of a logic formalised in a structural proof theory have analytic proofs, then the proof theory can be used to demonstrate such things as consistency, provide decision procedures, and allow mathematical or computational witnesses to be extracted as counterparts to theorems, the kind of task that is more often given to model theory.


the dual graph of a directed graph, embedded in the plane, is a graph with a vertex for each face of the given graph, and a dual edge between two dual vertices when the corresponding two faces are separated by an edge. each dual edge crosses one of the original graph edges, turned by 90° clockwise. for a dicut in the given graph, the duals of the edges that cross the dicut form a directed cycle in the dual graph, and vice versa. a dijoin can be defined as a set of edges that crosses all dicuts ; when the edges of a dijoin are contracted, the result is a strongly connected graph.


in number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n ; i. e., if there exists an integer x such that : x 2 ≡ q ( mod n ). { \ displaystyle x ^ { 2 } \ equiv q { \ pmod { n } }. } otherwise, q is called a quadratic nonresidue modulo n. originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.


in statistical classification, the bayes classifier minimizes the probability of misclassification.


there is some centralization ( a general assembly ) and some central decision ( over local decision ) : there is no choice of " each room decision " or " each regulars'community decision ". so it is a central decision. the subsidiarity principle can be applied : there is an " embryonic local governance " connecting the cyclists, and the other people ( voters ) of the condominium recognise the group, transferring some ( little ) responsibility to them ( the keys of the gym room and right to advocate their cycling activities to other residents ). there is no " enforced minoritarianism " ; it seems a legitimate characterization of a relevant ( and not dominant ) minority. this is a tyranny of the majority situation because : there is a little " global gain " in a global decision ( where x wins ), and a good " local gain " in local decision ( local y preference ) ; there is relevant voting for a local decision : 6 voters ( 46 % ) are gym room regulars, 5 that voted y. the majority of them ( 83 % ) voted y. in this situation, even with no formal federation structure, the minority and a potential local governance emerged : the tyranny perception arrives with it.


in statistics, the neyman – pearson lemma was introduced by jerzy neyman and egon pearson in a paper in 1933. the neyman - pearson lemma is part of the neyman - pearson theory of statistical testing, which introduced concepts like errors of the second kind, power function, and inductive behavior. the previous fisherian theory of significance testing postulated only one hypothesis. by introducing a competing hypothesis, the neyman - pearsonian flavor of statistical testing allows investigating the two types of errors.


in stack - oriented programming languages ( and concatenative ones, most of which are stack based ), point - free methods are commonly used. for example, a procedure to compute the fibonacci numbers might look like the following in postscript :


in order to effectively implement variables of such types as array structures ( with indexing done by pointer arithmetic ), many languages restrict the indices to integer data types ( or other types that can be interpreted as integers, such as bytes and enumerated types ), and require that all elements have the same data type and storage size. most of those languages also restrict each index to a finite interval of integers, that remains fixed throughout the lifetime of the array variable. in some compiled languages, in fact, the index ranges may have to be known at compile time. on the other hand, some programming languages provide more liberal array types, that allow indexing by arbitrary values, such as floating - point numbers, strings, objects, references, etc.. such index values cannot be restricted to an interval, much less a fixed interval.


in number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. a more general definition includes all positive rational numbers with this property. the sequence of ( integer ) congruent numbers starts with 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120,... ( sequence a003273 in the oeis ) for example, 5 is a congruent number because it is the area of a ( 20 / 3, 3 / 2, 41 / 6 ) triangle. similarly, 6 is a congruent number because it is the area of a ( 3, 4, 5 ) triangle. 3 and 4 are not congruent numbers.


in philosophy, ontological maximalism ( or metaontological maximalism ) is a ontological realist position that asserts, " whatever can exist does in some sense exist ".


in mathematics, a natural number n is a blum integer if n = p × q is a semiprime for which p and q are distinct prime numbers congruent to 3 mod 4. that is, p and q must be of the form 4t + 3, for some integer t. integers of this form are referred to as blum primes. this means that the factors of a blum integer are gaussian primes with no imaginary part. the first few blum integers are 21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497,... ( sequence a016105 in the oeis ) the integers were named for computer scientist manuel blum.


georg cantor, the inventor of set theory, showed in 1874 that there is more than one kind of infinity, specifically that the collection of all natural numbers and the collection of all real numbers, while both infinite, are not equinumerous ( see cantor's first uncountability proof ). in his controversial 1878 paper, cantor explicitly defined the notion of " power " of sets and used it to prove that the set of all natural numbers and the set of all rational numbers are equinumerous ( an example where a proper subset of an infinite set is equinumerous to the original set ), and that the cartesian product of even a countably infinite number of copies of the real numbers is equinumerous to a single copy of the real numbers. cantor's theorem from 1891 implies that no set is equinumerous to its own power set ( the set of all its subsets ).


this recovers both the inclusion of finite sets in sets ( where a cogenerator is the set of two elements ), and also the inclusion of finite - dimensional vector spaces in vector spaces ( where the cogenerator is the ground field ). sipos showed that the algebras over the codensity monad of the inclusion of finite sets ( regarded as discrete topological spaces ) into topological spaces are equivalent to stone spaces. avery shows that the giry monad arises as the codensity monad of natural forgetful functors between certain categories of convex vector spaces to measurable spaces.


a theory consists of some basis statements called axioms, and some deducing rules ( sometimes included in the axioms ). the theorems of the theory are the statements that can be derived from the axioms by using the deducing rules. this formalization led to proof theory, which allows proving general theorems about theorems and proofs. in particular, godel's incompleteness theorems show that every consistent theory containing the natural numbers has true statements on natural numbers that are not theorems of the theory ( that is they cannot be proved inside the theory ). as the axioms are often abstractions of properties of the physical world, theorems may be considered as expressing some truth, but in contrast to the notion of a scientific law, which is experimental, the justification of the truth of a theorem is purely deductive.


here the matrix completion problem does not obey the restricted isometry property ( rip ). for matrices, the rip would assume that the sampling operator obeys ( 1 − δ ) ‖ x ‖ f 2 ≤ 1 p ‖ p ω ( x ) ‖ f 2 ≤ ( 1 + δ ) ‖ x ‖ f 2 { \ displaystyle ( 1 - \ delta ) \ | x \ | _ { f } ^ { 2 } \ leq { \ frac { 1 } { p } } \ | p _ { \ omega } ( x ) \ | _ { f } ^ { 2 } \ leq ( 1 + \ delta ) \ | x \ | _ { f } ^ { 2 } } for all matrices x { \ displaystyle x } with sufficiently small rank and δ < 1 { \ displaystyle \ delta < 1 } sufficiently small. the methods are also applicable to sparse signal recovery problems in which the rip does not hold.


in the definitions above, the functions t ( x ), η ( θ ), and a ( η ) were arbitrary. however, these functions have important interpretations in the resulting probability distribution. t ( x ) is a sufficient statistic of the distribution. for exponential families, the sufficient statistic is a function of the data that holds all information the data x provides with regard to the unknown parameter values.


as a byproduct of this latter work, she proved sophie germain's theorem, which verified the first case of fermat's last theorem ( namely, the case in which p { \ displaystyle p } does not divide x y z { \ displaystyle xyz } ) for every odd prime exponent less than 270 { \ displaystyle 270 }, and for all primes p { \ displaystyle p } such that at least one of 2 p + 1 { \ displaystyle 2p + 1 }, 4 p + 1 { \ displaystyle 4p + 1 }, 8 p + 1 { \ displaystyle 8p + 1 }, 10 p + 1 { \ displaystyle 10p + 1 }, 14 p + 1 { \ displaystyle 14p + 1 } and 16 p + 1 { \ displaystyle 16p + 1 } is prime ( specially, the primes p { \ displaystyle p } such that 2 p + 1 { \ displaystyle 2p + 1 } is prime are called sophie germain primes ). germain tried unsuccessfully to prove the first case of fermat's last theorem for all even exponents, specifically for n = 2 p { \ displaystyle n = 2p }, which was proved by guy terjanian in 1977. in 1985, leonard adleman, roger heath - brown and etienne fouvry proved that the first case of fermat's last theorem holds for infinitely many odd primes p { \ displaystyle p }.


it implies a coarse - grained authentication, given that domains appear on the right part of email addresses, after the at sign. fine - grain authentication, at user level, can be achieved by other means, such as pretty good privacy and s / mime. at present, digital identity needs to be managed by each individual.


in probability theory, a member of the ( a, b, 0 ) class of distributions is any distribution of a discrete random variable n whose values are nonnegative integers whose probability mass function satisfies the recurrence formula p k p k − 1 = a + b k, k = 1, 2, 3, … { \ displaystyle { \ frac { p _ { k } } { p _ { k - 1 } } } = a + { \ frac { b } { k } }, \ qquad k = 1, 2, 3, \ dots } for some real numbers a and b, where p k = p ( n = k ) { \ displaystyle p _ { k } = p ( n = k ) }. only the poisson, binomial and negative binomial distributions satisfy the full form of this relationship. these are also the three discrete distributions among the six members of the natural exponential family with quadratic variance functions ( nef – qvf ). more general distributions can be defined by fixing some initial values of pj and applying the recursion to define subsequent values.


the n - dimensional parallelotope spanned by the rows of an n×n hadamard matrix has the maximum possible n - dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1. equivalently, a hadamard matrix has maximal determinant among matrices with entries of absolute value less than or equal to 1 and so is an extremal solution of hadamard's maximal determinant problem. certain hadamard matrices can almost directly be used as an error - correcting code using a hadamard code ( generalized in reed – muller codes ), and are also used in balanced repeated replication ( brr ), used by statisticians to estimate the variance of a parameter estimator.


an alternative way to compute c x { \ displaystyle { \ textbf { c } } _ { x } }, is by using the n ′ × m { \ displaystyle n'\ times m } " trajectory matrix " d { \ displaystyle { \ textbf { d } } } that is formed by m { \ displaystyle m } lag - shifted copies of x ( t ) { \ displaystyle { \ it { x ( t ) } } }, which are n ′ = n − m + 1 { \ displaystyle n'= n - m + 1 } long ; then c x = 1 n ′ d t d. { \ displaystyle { \ textbf { c } } _ { x } = { \ frac { 1 } { n'} } { \ textbf { d } } ^ { \ rm { t } } { \ textbf { d } }. } the m { \ displaystyle m } eigenvectors e k { \ displaystyle { \ textbf { e } } _ { k } } of the lag - covariance matrix c x { \ displaystyle { \ textbf { c } } _ { x } } are called temporal empirical orthogonal functions ( eofs ).


in technical applications of 3d computer graphics ( cax ) such as computer - aided design and computer - aided manufacturing, surfaces are one way of representing objects. the other ways are wireframe ( lines and curves ) and solids. point clouds are also sometimes used as temporary ways to represent an object, with the goal of using the points to create one or more of the three permanent representations.


in non - germanic indo - european languages, past marking is typically combined with a distinction between perfective and imperfective aspect, with the former reserved for single completed actions in the past. french for instance, has an imperfect tense form similar to that of german but used only for past habitual or past progressive contexts like " i used to... " or " i was doing... ". similar patterns extend across most languages of the indo - european family right through to the indic languages.


the discovery of deuterium stood, however. urey and washburn attempted to use electrolysis to create pure heavy water. their technique was sound, but they were beaten to it in 1933 by lewis, who had the resources of the university of california at his disposal.


in software architecture, a service mesh is a dedicated infrastructure layer for facilitating service - to - service communications between services or microservices using a proxy. a dedicated communication layer can provide numerous benefits, such as providing observability into communications, providing secure connections or automating retries and backoff for failed requests. a service mesh consists of network proxies paired with each service in an application and a set of task - management processes. the proxies are called the data plane and the management processes are called the control plane. the data plane intercepts calls between different services and processes them ; the control plane is the brain of the mesh that coordinates the behavior of proxies and provides apis for operations and maintenance personnel to manipulate and observe the entire network. the service mesh architecture is implemented by software products such as istio, cilium, linkerd, consul, aws app mesh, kuma, traefik mesh, greymatter. io and open service mesh. many service meshes use the envoy proxy on the data plane.


in number theory, the law of quadratic reciprocity, like the pythagorean theorem, has lent itself to an unusually large number of proofs. several hundred proofs of the law of quadratic reciprocity have been published.


in mathematics, the bauer – fike theorem is a standard result in the perturbation theory of the eigenvalue of a complex - valued diagonalizable matrix. in its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a properly chosen eigenvalue of the exact matrix. informally speaking, what it says is that the sensitivity of the eigenvalues is estimated by the condition number of the matrix of eigenvectors. the theorem was proved by friedrich l. bauer and c. t. fike in 1960.


( gauss'conjecture was proven more than one hundred years later by kurt heegner, alan baker and harold stark. ) however, this was understood ( only ) in the language of equivalence classes of quadratic forms, so that in particular the analogy between quadratic forms and the fermat equation seems not to have been perceived.


in mathematics, linearization is finding the linear approximation to a function at a given point. the linear approximation of a function is the first order taylor expansion around the point of interest. in the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. this method is used in fields such as engineering, physics, economics, and ecology.


in probability theory, a markov model is a stochastic model used to model pseudo - randomly changing systems. it is assumed that future states depend only on the current state, not on the events that occurred before it ( that is, it assumes the markov property ). generally, this assumption enables reasoning and computation with the model that would otherwise be intractable. for this reason, in the fields of predictive modelling and probabilistic forecasting, it is desirable for a given model to exhibit the markov property.


they may also be used for ordering ( as in " this is the third largest city in the country " ), in which case they serve as ordinal numbers. natural numbers are sometimes used as labels, known as nominal numbers, having none of the properties of numbers in a mathematical sense ( e. g. sports jersey numbers ). the natural numbers form a set, often symbolized as n { \ textstyle \ mathbb { n } }. many other number sets are built by successively extending the set of natural numbers : the integers, by including an additive identity 0 ( if not yet in ) and an additive inverse −n for each nonzero natural number n ; the rational numbers, by including a multiplicative inverse 1 / n { \ displaystyle 1 / n } for each nonzero integer n ( and also the product of these inverses by integers ) ; the real numbers by including the limits of cauchy sequences of rationals ; the complex numbers, by adjoining to the real numbers a square root of −1 ( and also the sums and products thereof ) ; and so on.


{ \ displaystyle 0, 1, 2, 3, \ ldots, n, \ ldots ; \ aleph _ { 0 }, \ aleph _ { 1 }, \ aleph _ { 2 }, \ ldots, \ aleph _ { \ alpha }, \ ldots. \ } this sequence starts with the natural numbers including zero ( finite cardinals ), which are followed by the aleph numbers. the aleph numbers are indexed by ordinal numbers. if the axiom of choice is true, this transfinite sequence includes every cardinal number.


however, although it is easy to score a phylogenetic tree ( by counting the number of character - state changes ), there is no algorithm to quickly generate the most - parsimonious tree. instead, the most - parsimonious tree must be sought in " tree space " ( i. e., amongst all possible trees ). for a small number of taxa ( i. e., fewer than nine ) it is possible to do an exhaustive search, in which every possible tree is scored, and the best one is selected.


in mathematics, a cardinal function ( or cardinal invariant ) is a function that returns cardinal numbers.


in order to demonstrate that ip ⊆ pspace, we present a simulation of an interactive proof system by a polynomial space machine. now, we can define : pr = max p pr { \ displaystyle \ pr = \ max \ nolimits _ { p } \ pr \ left } and for every 0 ≤ j ≤ p and every message history mj, we inductively define the function nmj : n m j = { 0 j = p and m p = reject 1 j = p and m p = accept max m j + 1 n m j + 1 j < p and j is odd wt - avg m j + 1 n m j + 1 j < p and j is even { \ displaystyle n _ { m _ { j } } = { \ begin { cases } 0 & j = p { \ text { and } } m _ { p } = { \ text { reject } } \ \ 1 & j = p { \ text { and } } m _ { p } = { \ text { accept } } \ \ \ max _ { m _ { j + 1 } } n _ { m _ { j + 1 } } & j


since alice chooses a { \ displaystyle a } randomly, and since b is independent of a { \ displaystyle a }, alice is guaranteed that the bit d = a ⊕ b { \ displaystyle d = a \ oplus b } is random. with similar arguments, bob is also guaranteed that the bit d is random. variations of coin tossing have been investigated in relativistic cryptography by colbeck and kent.


although many functions do not have an inverse, every relation does have a unique converse. the unary operation that maps a relation to the converse relation is an involution, so it induces the structure of a semigroup with involution on the binary relations on a set, or, more generally, induces a dagger category on the category of relations as detailed below. as a unary operation, taking the converse ( sometimes called conversion or transposition ) commutes with the order - related operations of the calculus of relations, that is it commutes with union, intersection, and complement.


let t 1 { \ displaystyle t _ { 1 } } be a theory obtained from t { \ displaystyle t } by extending its language with new constants a 1, …, a m { \ displaystyle a _ { 1 }, \ ldots, a _ { m } } and adding a new axiom φ ( a 1, …, a m ) { \ displaystyle \ varphi ( a _ { 1 }, \ ldots, a _ { m } ) }. then t 1 { \ displaystyle t _ { 1 } } is a conservative extension of t { \ displaystyle t }, which means that the theory t 1 { \ displaystyle t _ { 1 } } has the same set of theorems in the original language ( i. e., without constants a i { \ displaystyle a _ { i } } ) as the theory t { \ displaystyle t }. such a theory can also be conservatively extended by introducing a new functional symbol : suppose that a closed formula x → y φ ( y, x → ) { \ displaystyle \ forall { \ vec { x } } \, \ exists y \, \! \, \ varphi ( y, { \ vec { x } } ) } is a theorem of a first - order theory t { \ displaystyle t }, where we denote x → : = ( x 1, …, x n ) { \ displaystyle { \ vec { x } } : = ( x _ { 1 }, \ ldots, x _ { n } ) }. let t 1 { \ displaystyle t _ { 1 } } be a theory obtained from t { \ displaystyle t } by extending its language with a new functional symbol f { \ displaystyle f } ( of arity n { \ displaystyle n } ) and adding a new axiom x → φ ( f ( x → ), x → ) { \ displaystyle \ forall { \ vec { x } } \, \ varphi ( f ( { \ vec { x } } ), { \ vec { x } } ) }.


in the digital imaging community the term annotation is commonly used for visible metadata superimposed on an image without changing the underlying master image, such as sticky notes, virtual laser pointers, circles, arrows, and black - outs ( cf. redaction ). in the medical imaging community, an annotation is often referred to as a region of interest and is encoded in dicom format.


in quantum information theory, quantum channels, or quantum operations, are defined to be completely positive maps between c * - algebras. being a classification for all such maps, stinespring's theorem is important in that context. for example, the uniqueness part of the theorem has been used to classify certain classes of quantum channels. for the comparison of different channels and computation of their mutual fidelities and information another representation of the channels by their " radon – nikodym " derivatives introduced by belavkin is useful.


the ability to perform economical maximum likelihood soft decision decoding is one of the major benefits of convolutional codes. this is in contrast to classic block codes, which are generally represented by a time - variant trellis and therefore are typically hard - decision decoded. convolutional codes are often characterized by the base code rate and the depth ( or memory ) of the encoder { \ displaystyle }.


page faults are not only used for memory protection. the operating system may manage the page table in such a way that a reference to a page that has been previously paged out to secondary storage causes a page fault. the operating system intercepts the page fault, loads the required memory page, and the application continues as if no fault had occurred.


in the field of mathematics called combinatorial optimization, the method of symmetry - breaking constraints can be used to take advantage of symmetries in many constraint satisfaction and optimization problems, by adding constraints that eliminate symmetries and reduce the search space size. symmetries in a combinatorial problem increase the size of the search space and therefore, time is wasted in visiting new solutions which are symmetric to the already visited solutions. the solution time of a combinatorial problem can be reduced by adding new constraints, referred as symmetry breaking constraints, such that some of the symmetric solutions are eliminated from the search space while preserving the existence of at least one solution. symmetry is common in many real - life combinatorial problems.


in recursive function theory, double recursion is an extension of primitive recursion which allows the definition of non - primitive recursive functions like the ackermann function. raphael m. robinson called functions of two natural number variables g ( n, x ) double recursive with respect to given functions, if g ( 0, x ) is a given function of x. g ( n + 1, 0 ) is obtained by substitution from the function g ( n, · ) and given functions. g ( n + 1, x + 1 ) is obtained by substitution from g ( n + 1, x ), the function g ( n, · ) and given functions. robinson goes on to provide a specific double recursive function ( originally defined by rozsa peter ) g ( 0, x ) = x + 1 g ( n + 1, 0 ) = g ( n, 1 ) g ( n + 1, x + 1 ) = g ( n, g ( n + 1, x ) ) where the given functions are primitive recursive, but g is not primitive recursive. in fact, this is precisely the function now known as the ackermann function.


in the average case, if the input numbers are distributed uniformly in, then the largest sum in an lpt schedule satisfies the following properties : the expected largest sum for m = 2 machines is at least n 4 + 1 4 n + 4 { \ displaystyle { \ frac { n } { 4 } } + { \ frac { 1 } { 4n + 4 } } } and at most n 4 + e 2 n + 2 { \ displaystyle { \ frac { n } { 4 } } + { \ frac { e } { 2n + 2 } } }, where n is the number of inputs. the largest sum is at most 1 + o ( log log n / n ) { \ displaystyle 1 + o ( \ log { \ log { n } } / n ) } times the optimum almost surely, and 1 + o ( 1 / n ) { \ displaystyle 1 + o ( 1 / n ) } in expectation, where n { \ displaystyle n } is the number of inputs. the difference between the lpt largest sum and the optimal largest sum is at most o ( log n / n ) { \ displaystyle o ( \ log { n } / n ) } almost surely ( for uniform or negative exponential distributions ), and at most o ( m 2 / n ) { \ displaystyle o ( m ^ { 2 } / n ) } in expectation ( for uniform distribution ). these results hold also for machines with different speeds.


very long numbers can be further grouped by doubling up separators. typically decimal numbers ( base - 10 ) are grouped in three digit groups ( representing one of 1000 possible values ), binary numbers ( base - 2 ) in four digit groups ( one nibble, representing one of 16 possible values ), and hexadecimal numbers ( base - 16 ) in two digit groups ( each digit is one nibble, so two digits are one byte, representing one of 256 possible values ). numbers from other systems ( such as id numbers ) are grouped following whatever convention is in use.


for instance, the following properties hold : p i 2 = 1, i = 1, 2,..


in systems theory, the black box is an abstraction representing a class of concrete open system which can be viewed solely in terms of its stimuli inputs and output reactions : the constitution and structure of the box are altogether irrelevant to the approach under consideration, which is purely external or phenomenological. in other words, only the behavior of the system will be accounted for. the understanding of a black box is based on the " explanatory principle ", the hypothesis of a causal relation between the input and the output. this principle states that input and output are distinct, that the system has observable ( and relatable ) inputs and outputs and that the system is black to the observer ( non - openable ).


now operator decides the number of vehicles ( n ), which use alternative route. the optimal number of vehicles ( n ) can be obtained by calculus of variation, to make marginal cost of each route equal. thus, optimal condition is t0 = t1 + ∆1.


in mathematics, computer science and especially graph theory, a distance matrix is a square matrix ( two - dimensional array ) containing the distances, taken pairwise, between the elements of a set. depending upon the application involved, the distance being used to define this matrix may or may not be a metric. if there are n elements, this matrix will have size n×n. in graph - theoretic applications, the elements are more often referred to as points, nodes or vertices.


the process was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics. the poisson point process is often defined on the real line, where it can be considered as a stochastic process. in this setting, it is used, for example, in queueing theory to model random events, such as the arrival of customers at a store, phone calls at an exchange or occurrence of earthquakes, distributed in time. in the plane, the point process, also known as a spatial poisson process, can represent the locations of scattered objects such as transmitters in a wireless network, particles colliding into a detector, or trees in a forest.


in printing and publishing, proofs are the preliminary versions of publications meant for review by authors, editors, and proofreaders, often with extra - wide margins. galley proofs may be uncut and unbound, or in some cases electronically transmitted. they are created for proofreading and copyediting purposes, but may also be used for promotional and review purposes.


for example, on a social platform a supplier of sports gear promises the first 20 callers a free bicycle helmet. the lapse of this information is significant. then, after the 20th reader has called, the information value drops down to zero.


in the eidas regulation trust services are defined as electronic services, normally provided by tsps, which consist of electronic signatures, electronic seals, electronic time stamps, electronic registered delivery services and website authentication. in essence, the eidas regulation provides a framework to promote : transparency and accountability : well - defined minimal obligations for tsps and liability. guarantee of trustworthiness of the services together with security requirements for tsps. technological neutrality : avoiding requirements which could only be met by a specific technology. market rules and standardization certainty.


in mathematical logic and computer science, some type theories and type systems include a top type that is commonly denoted with top or the symbol. the top type is sometimes called also universal type, or universal supertype as all other types in the type system of interest are subtypes of it, and in most cases, it contains every possible object of the type system. it is in contrast with the bottom type, or the universal subtype, which every other type is supertype of and it is often that the type contains no members at all.


in order theory, an embedding of partially ordered sets is a function f { \ displaystyle f } between partially ordered sets x { \ displaystyle x } and y { \ displaystyle y } such that x 1, x 2 ∈ x : x 1 ≤ x 2 f ( x 1 ) ≤ f ( x 2 ). { \ displaystyle \ forall x _ { 1 }, x _ { 2 } \ in x : x _ { 1 } \ leq x _ { 2 } \ iff f ( x _ { 1 } ) \ leq f ( x _ { 2 } ). } injectivity of f { \ displaystyle f } follows quickly from this definition. in domain theory, an additional requirement is that y ∈ y : { x f ( x ) ≤ y } { \ displaystyle \ forall y \ in y : \ { x \ mid f ( x ) \ leq y \ } } is directed.


in programming language theory, the poplmark challenge ( from " principles of programming languages benchmark ", formerly mechanized metatheory for the masses! ) ( aydemir, 2005 ) is a set of benchmarks designed to evaluate the state of automated reasoning ( or mechanization ) in the metatheory of programming languages, and to stimulate discussion and collaboration among a diverse cross section of the formal methods community. very loosely speaking, the challenge is about measurement of how well programs may be proven to match a specification of how they are intended to behave ( and the many complex issues that this involves ). the challenge was initially proposed by the members of the pl club at the university of pennsylvania, in association with collaborators around the world.


first, one typically starts by defining a term as follows : a variable, or a function symbol applied to the number of terms required by the function symbol's arity. for example, if + is a binary function symbol and x, y, and z are variables, then x + ( y + z ) is a term, which might be written with the symbols in various orders. once a term is defined, a proposition can then be defined as follows : a predicate symbol applied to the number of terms required by its arity, or an operator applied to the number of propositions required by its arity, or a quantifier applied to a proposition. for example, if = is a binary predicate symbol and is a quantifier, then, y, z is a proposition. this more complex structure of propositions allows these logics to make finer distinctions between inferences, i. e., to have greater expressive power.


in mathematics, the stone functor is a functor s : topop → bool, where top is the category of topological spaces and bool is the category of boolean algebras and boolean homomorphisms. it assigns to each topological space x the boolean algebra s ( x ) of its clopen subsets, and to each morphism fop : x → y in topop ( i. e., a continuous map f : y → x ) the homomorphism s ( f ) : s ( x ) → s ( y ) given by s ( f ) ( z ) = f−1.


the human body, being a physiological universal, provides an ideal domain for research into semantic and lexical universals. in a seminal study, cecil h. brown ( 1976 ) proposed a number of universals in the semantics of body part terminology, including the following : in any language, there will be distinct terms for body, head, arm, eyes, nose, and mouth ; if there is a distinct term for foot, there will be a distinct term for hand ; similarly, if there are terms for individual toes, then there are terms for individual fingers. subsequent research has shown that most of these features have to be considered cross - linguistic tendencies rather than true universals.


in particular, using the properties of the cumulant generating function, e ( t j ) = ∂ a ( η ) ∂ η j { \ displaystyle \ operatorname { e } ( t _ { j } ) = { \ frac { \ partial a ( \ eta ) } { \ partial \ eta _ { j } } } } and cov ( t i, t j ) = ∂ 2 a ( η ) ∂ η i ∂ η j. { \ displaystyle \ operatorname { cov } \ left ( t _ { i }, \ t _ { j } \ right ) = { \ frac { \ partial ^ { 2 } a ( \ eta ) } { \ partial \ eta _ { i } \, \ partial \ eta _ { j } } }. } the first two raw moments and all mixed second moments can be recovered from these two identities.


in mathematics, an injective function ( also known as injection, or one - to - one function ) is a function f that maps distinct elements of its domain to distinct elements ; that is, x1 = x2 implies f ( x1 ) = f ( x2 ). ( equivalently, f ( x1 ) = f ( x2 ) implies x1 = x2 in the equivalent contrapositive statement. ) in other words, every element of the function's codomain is the image of at most one element of its domain.


in software engineering, the mediator pattern defines an object that encapsulates how a set of objects interact. this pattern is considered to be a behavioral pattern due to the way it can alter the program's running behavior. in object - oriented programming, programs often consist of many classes. business logic and computation are distributed among these classes.


different theoretical explanations of these inconsistencies are discussed later in the article. there are various types of phrase in which the ordering of head and complement ( s ) may be considered when attempting to determine the head directionality of a language, including : verb phrase : the head of verb phrase ( vp ) is a verb, and the complement ( s ) are most commonly objects of various types. the ordering here is related to one of the chief questions in the word order typology of languages, namely the normal order of subject, verb and object within a clause ( languages are classed on this basis as svo, sov, vso, etc. ). noun phrase : the head of a noun phrase ( np ) is a noun ; various kinds of complementizer phrases ( cps ) and adpositional phrases ( pps ) can be complements. adjective phrase : the head of an adjective phrase ( ap ) is an adjective, which can take as a complement, for example, an adverbial phrase or adpositional phrase ( pp ). adpositional phrase : the head of an adpositional phrase ( pp ) is an apposition.


in the cc4 network, which is a three - stage network, the number of input nodes is one more than the size of the training vector, with the extra node serving as the biasing node whose input is always 1. for binary input vectors, the weights from the input nodes to the hidden neuron ( say of index j ) corresponding to the trained vector is given by the following formula : w i j = { − 1, for x i = 0 + 1, for x i = 1 r − s + 1, for i = n + 1 { \ displaystyle w _ { ij } = { \ begin { cases } - 1, & { \ mbox { for } } x _ { i } = 0 \ \ + 1, & { \ mbox { for } } x _ { i } = 1 \ \ r - s + 1, & { \ mbox { for } } i = n + 1 \ end { cases } } } where r { \ displaystyle r } is the radius of generalization and s { \ displaystyle s } is the hamming weight ( the number of 1s ) of the binary sequence. from the hidden layer to the output layer the weights are 1 or - 1 depending on whether the vector belongs to a given output class or not. the neurons in the hidden and output layers output 1 if the weighted sum to the input is 0 or positive and 0, if the weighted sum to the input is negative : y = { 1 if x i ≥ 0 0 if x i < 0 { \ displaystyle y = \ left \ { { \ begin { matrix } 1 & { \ mbox { if } } \ sum x _ { i } \ geq 0 \ \ 0 & { \ mbox { if } } \ sum x _ { i } < 0 \ end { matrix } } \ right. }


camel case is sometimes used for abbreviated names of certain neighborhoods, e. g. new york city neighborhoods soho ( south of houston street ) and tribeca ( triangle below canal street ) and san francisco's soma ( south of market ). such usages erode quickly, so the neighborhoods are now typically rendered as soho, tribeca, and soma. internal capitalization has also been used for other technical codes like hela ( 1983 ).


in the context of education, one - to - one computing ( sometimes abbreviated as " 1 : 1 " ) refers to academic institutions, such as schools or colleges, that allow each enrolled student to use an electronic device in order to access the internet, digital course materials, and digital textbooks. the concept has been actively explored and sporadically implemented since the late 1990s. one - to - one computing used to be contrasted with a policy of " bring your own device " ( byod ), which encourages or requires students to use their own laptops, smartphones or other electronic devices in class.


rather than a complicated script running a set of unit tests, if this simple programme fails to compile or execute, it proves that the supporting environment likely has a configuration problem that will prevent any code from compiling or executing. but if " hello world " executes, then any problems experienced with other programmes likely can be attributed to errors in that application's code rather than the environment. the association for computing machinery, and software projects such as android, mediawiki and twitter, discourage use of the phrase sanity check in favour of other terms such as confidence test, coherence check, or simply test, as part of a wider attempt to avoid ableist language and increase inclusivity.


in the following, matrices will be indicated by indexed variables. " subject " indices will be indicated using letters a { \ displaystyle a }, b { \ displaystyle b } and c { \ displaystyle c }, with values running from 1 { \ displaystyle 1 } to p { \ displaystyle p } which is equal to 10 { \ displaystyle 10 } in the above example. " factor " indices will be indicated using letters p { \ displaystyle p }, q { \ displaystyle q } and r { \ displaystyle r }, with values running from 1 { \ displaystyle 1 } to k { \ displaystyle k } which is equal to 2 { \ displaystyle 2 } in the above example. " instance " or " sample " indices will be indicated using letters i { \ displaystyle i }, j { \ displaystyle j } and k { \ displaystyle k }, with values running from 1 { \ displaystyle 1 } to n { \ displaystyle n }.


in the example s = { e, f } with the equalities given, s is a semigroup. it demonstrates the possibility for ( s, ∗ ) to have several left identities. in fact, every element can be a left identity. in a similar manner, there can be several right identities.


in 1881, the company changed its name to the champion bridge company in 1881 and began to also manufacture farm implements, iron fences, and some machinery. in 1893, the company moved to its present location on east sugartree street in wilmington. the company was among the first to use and promote steel for the construction of smaller highway bridges.


in video gaming, players may be given a ranking. to " rank up " is to achieve a higher ranking relative to other players, especially with strategies that do not depend on the player's skill.


it is also difficult to tell when exactly a grammatical structure has been learned, as learners may use structures correctly in some situations but not in others. thus it is more accurate to speak of sequences of acquisition, in which specific grammatical features in a language are acquired before or after certain others but the overall order of acquisition is less rigid. for example, if neither feature b nor feature d can be acquired until feature a has been acquired ( feature b and d depend on a ) and feature c depends on b, but d does not depend on b ( or, therefore, on c ), then acquisition orders ( a, b, c, d ) and ( a, d, b, c ) are possible, as they are both valid topological orderings.


in the case n = 6, the unique optimal polygon is not regular. the solution to this case was published in 1975 by ronald graham, answering a question posed in 1956 by hanfried lenz ; it takes the form of an irregular equidiagonal pentagon with an obtuse isosceles triangle attached to one of its sides, with the distance from the apex of the triangle to the opposite pentagon vertex equal to the diagonals of the pentagon. its area is 0. 674981.... ( sequence a111969 in the oeis ), a number that satisfies the equation 4096 x10 + 8192x9 − 3008x8 − 30848x7 + 21056x6 + 146496x5 − 221360x4 + 1232x3 + 144464x2 − 78488x + 11993 = 0. graham conjectured that the optimal solution for the general case of even values of n consists in the same way of an equidiagonal ( n − 1 ) - gon with an isosceles triangle attached to one of its sides, its apex at unit distance from the opposite ( n − 1 ) - gon vertex. in the case n = 8 this was verified by a computer calculation by audet et al. graham's proof that his hexagon is optimal, and the computer proof of the n = 8 case, both involved a case analysis of all possible n - vertex thrackles with straight edges. the full conjecture of graham, characterizing the solution to the biggest little polygon problem for all even values of n, was proven in 2007 by foster and szabo.


in programming language theory and proof theory, the curry – howard correspondence ( also known as the curry – howard isomorphism or equivalence, or the proofs - as - programs and propositions - or formulae - as - types interpretation ) is the direct relationship between computer programs and mathematical proofs. it is a generalization of a syntactic analogy between systems of formal logic and computational calculi that was first discovered by the american mathematician haskell curry and the logician william alvin howard. it is the link between logic and computation that is usually attributed to curry and howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by l. e. j. brouwer, arend heyting and andrey kolmogorov ( see brouwer – heyting – kolmogorov interpretation ) and stephen kleene ( see realizability ). the relationship has been extended to include category theory as the three - way curry – howard – lambek correspondence.


the distinction came into prominence on modern character displays. the digit 0 with a dot in the centre seems to have originated as an option on ibm 3270 displays. its appearance has continued with taligent's command line typeface andale mono. one variation used a short vertical bar instead of the dot.


in numerical analysis and linear algebra, lower – upper ( lu ) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix ( see matrix decomposition ). the product sometimes includes a permutation matrix as well. lu decomposition can be viewed as the matrix form of gaussian elimination.


in mathematics, a filter on a set x { \ displaystyle x } informally gives a notion of which subsets a ⊆ x { \ displaystyle a \ subseteq x } are " large ". filter quantifiers are a type of logical quantifier which, informally, say whether or not a statement is true for " most " elements of x. { \ displaystyle x. } such quantifiers are often used in combinatorics, model theory ( such as when dealing with ultraproducts ), and in other fields of mathematical logic where ( ultra ) filters are used.


some drawbacks of writing use cases include the fact that each action, by the actor or the world, consist of little detail, and is simply a small action. this may possibly lead to further imagination and different interpretation of action from different designers. also, during the process, it is really easy to oversimplify a task, since a small task derived from a larger task may still consist of even smaller tasks which were missed.


fe'i bananas, grown and eaten in the islands of the pacific, are derived from entirely different wild species than traditional bananas and plantains. most fe'i bananas are cooked, but karat bananas, which are short and squat with bright red skins, very different from the usual yellow dessert bananas, are eaten raw. in the spanish market, the distinction is among platano, applied to the cavendish cultivars produced in the spanish canary islands under the protected geographical indication platano de canarias, banana, applied to dessert imports from africa and the americas, and platano macho ( literally, " male banana " ), applied to imports that are to be cooked. in summary, in commerce in europe and the americas ( although not in small - scale cultivation ), it is possible to distinguish between " bananas ", which are eaten raw, and " plantains ", which are cooked. in other regions of the world, particularly india, southeast asia and the islands of the pacific, there are many more kinds of banana and the two - fold distinction is not useful and not made in local languages. plantains are one of many kinds of cooking bananas, which are not always distinct from dessert bananas.


the thirteenth loaf is called the vantage loaf because it is considered advantageous overall to get 13 loaves for the price of 12. in arthurian legend, which was recorded in medieval texts, king arthur is resting in avalon with the twelve greatest knights of the round table, totalling 13, and will return when his country is in peril. the thirteen treasures of britain are a series of magical items listed in late medieval texts. the thirteen postures of tai chi are thirteen postures ( consisting of eight gates and five steps ) which are considered to be of fundamental importance in the practice of tai chi. in astronomy there are 13 star constellations in the zodiac ( including ophiuchus ) ; this can be compared with astrology where there are 12 signs of the zodiac. in judaism, 13 signifies the age at which a boy matures and becomes a bar mitzvah, i. e., a full member of the jewish faith ( counts as a member of minyan ).


in mathematics, fermat's theorem ( also known as interior extremum theorem ) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point ( the function's derivative is zero at that point ). fermat's theorem is a theorem in real analysis, named after pierre de fermat. by using fermat's theorem, the potential extrema of a function f { \ displaystyle \ displaystyle f }, with derivative f ′ { \ displaystyle \ displaystyle f'}, are found by solving an equation in f ′ { \ displaystyle \ displaystyle f'}. fermat's theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points ( not a maximum or minimum ). the function's second derivative, if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum.


: xiv the maximum possible clock rate is capped by the logic path with the longest propagation delay, called the critical path. because of that, the paths that may operate quickly are idle most of the time. a widely distributed clock network dissipates a lot of useful power and must run whether the circuit is receiving inputs or not. because of this level of complexity, testing and debugging takes over half of development time in all dimensions for synchronous circuits.


in the above uml class diagram, the caretaker class refers to the originator class for saving ( creatememento ( ) ) and restoring ( restore ( memento ) ) originator's internal state. the originator class implements ( 1 ) creatememento ( ) by creating and returning a memento object that stores originator's current internal state and ( 2 ) restore ( memento ) by restoring state from the passed in memento object. the uml sequence diagram shows the run - time interactions : ( 1 ) saving originator's internal state : the caretaker object calls creatememento ( ) on the originator object, which creates a memento object, saves its current internal state ( setstate ( ) ), and returns the memento to the caretaker. ( 2 ) restoring originator's internal state : the caretaker calls restore ( memento ) on the originator object and specifies the memento object that stores the state that should be restored. the originator gets the state ( getstate ( ) ) from the memento to set its own state.


in mathematics, thabit derived an equation for determining amicable numbers. his proof of this rule is presented in the treatise on the derivation of the amicable numbers in an easy way. this was done while writing on the theory of numbers, extending their use to describe the ratios between geometrical quantities, a step which the greeks did not take. thabit's work on amicable numbers and number theory helped him to invest more heavily into the geometrical relations of numbers establishing his transversal ( geometry ) theorem. thabit described a generalized proof of the pythagorean theorem.


in telecommunications, guard intervals are used to ensure that distinct transmissions do not interfere with one another, or otherwise cause overlapping transmissions. these transmissions may belong to different users ( as in tdma ) or to the same user ( as in ofdm ). the purpose of the guard interval is to introduce immunity to propagation delays, echoes and reflections, to which digital data is normally very sensitive.


for example, when j = 3 { \ displaystyle j = 3 }, the quantity d ( g ( 3 ) | q ( 2 ) ) { \ displaystyle d ( { \ mathcal { g } } ^ { ( 3 ) } \ vert \ mathbf { q } ^ { ( 2 ) } ) } is equal to the fraction of triangles formed by 2 - edges in the subhypergraph that are 3 - edges. definition. for j ≥ 3 { \ displaystyle j \ geq 3 }, fix some classes v i 1, …, v i j { \ displaystyle v _ { i _ { 1 } }, \ ldots, v _ { i _ { j } } } of g ( 1 ) { \ displaystyle { \ mathcal { g } } ^ { ( 1 ) } } with 1 ≤ i 1 < … < i j ≤ l { \ displaystyle 1 \ leq i _ { 1 } < \ ldots 0 { \ displaystyle \ mu > 0 } be a real number, r ≥ 1 { \ displaystyle r \ geq 1 } be an integer, and δ = ( δ 2, …, δ k − 1 ) { \ displaystyle \ delta = ( \ delta _ { 2 }, \ ldots, \ delta _ { k - 1 } ) }, d = ( d 2, …, d k − 1 ) { \ displaystyle \ mathbf { d } = ( d _ { 2 }, \ ldots, d _ { k - 1 } ) } be vectors of positive reals.


in statistics and data mining, x - means clustering is a variation of k - means clustering that refines cluster assignments by repeatedly attempting subdivision, and keeping the best resulting splits, until a criterion such as the akaike information criterion ( aic ) or bayesian information criterion ( bic ) is reached.


in the context of combinatorial game theory, which typically studies sequential games with perfect information, a game tree is a graph representing all possible game states within such a game. such games include well - known ones such as chess, checkers, go, and tic - tac - toe. this can be used to measure the complexity of a game, as it represents all the possible ways a game can pan out.


in the first version in 2003, north eastern scotland ( which then included part of moray ) was coded ukm1, and highlands and islands was coded ukm4. the current nuts level 1 codes start with " c " ( following " uk " ) rather than " 1 " because the new list reflected the revised regions of england and local government changes throughout the uk ; " 1 " to " b " had been used for the 11 regions in the previous coding system.


also, the fact that q ( t ) { \ displaystyle q ( t ) } is random with probability density given by the square modulus of ψ ( t, ⋅ ) { \ displaystyle \ psi ( t, \ cdot ) } implies that the conditional probability density of q i ( t ) { \ displaystyle q ^ { \ text { i } } ( t ) } given q ii ( t ) { \ displaystyle q ^ { \ text { ii } } ( t ) } is given by the square modulus of the ( normalized ) conditional wavefunction ψ i ( t, ⋅ ) { \ displaystyle \ psi ^ { \ text { i } } ( t, \ cdot ) } ( in the terminology of durr et al. this fact is called the fundamental conditional probability formula ). unlike the universal wavefunction, the conditional wavefunction of a subsystem does not always evolve by the schrodinger equation, but in many situations it does. for instance, if the universal wavefunction factors as ψ ( t, q i, q ii ) = ψ i ( t, q i ) ψ ii ( t, q ii ), { \ displaystyle \ psi ( t, q ^ { \ text { i } }, q ^ { \ text { ii } } ) = \ psi ^ { \ text { i } } ( t, q ^ { \ text { i } } ) \ psi ^ { \ text { ii } } ( t, q ^ { \ text { ii } } ), } then the conditional wavefunction of subsystem ( i ) is ( up to an irrelevant scalar factor ) equal to ψ i { \ displaystyle \ psi ^ { \ text { i } } } ( this is what standard quantum theory would regard as the wavefunction of subsystem ( i ) ).


in mathematics, specifically number theory, granville numbers, also known as s { \ displaystyle { \ mathcal { s } } } - perfect numbers, are an extension of the perfect numbers.


in the following examples, the natural numbers refer to the set of positive integers. the equation x = ( y 1 + 1 ) ( y 2 + 1 ) { \ displaystyle x = ( y _ { 1 } + 1 ) ( y _ { 2 } + 1 ) } is an example of a diophantine equation with a parameter x and unknowns y1 and y2. the equation has a solution in y1 and y2 precisely when x can be expressed as a product of two integers greater than 1, in other words x is a composite number. namely, this equation provides a diophantine definition of the set { 4, 6, 8, 9, 10, 12, 14, 15, 16, 18,... } consisting of the composite numbers.


in multitype branching processes, individuals are not identical, but can be classified into n types. after each time step, an individual of type i will produce individuals of different types, and x i { \ displaystyle \ mathbf { x } _ { i } }, a random vector representing the numbers of children in different types, satisfies a probability distribution on n n { \ displaystyle \ mathbb { n } ^ { n } }. for example, consider the population of cancer stem cells ( cscs ) and non - stem cancer cells ( nsccs ). after each time interval, each csc has probability p 1 { \ displaystyle p _ { 1 } } to produce two cscs ( symmetric division ), probability p 2 { \ displaystyle p _ { 2 } } to produce one csc and one nscc ( asymmetric division ), probability p 3 { \ displaystyle p _ { 3 } } to produce one csc ( stagnation ), and probability 1 − p 1 − p 2 − p 3 { \ displaystyle 1 - p _ { 1 } - p _ { 2 } - p _ { 3 } } to produce nothing ( death ) ; each nscc has probability p 4 { \ displaystyle p _ { 4 } } to produce two nsccs ( symmetric division ), probability p 5 { \ displaystyle p _ { 5 } } to produce one nscc ( stagnation ), and probability 1 − p 4 − p 5 { \ displaystyle 1 - p _ { 4 } - p _ { 5 } } to produce nothing ( death ).


the systems are similar in that they were all produced in the same way. this infinite sequence is an ensemble.


in mathematics, bertrand's postulate ( actually now a theorem ) states that for each n ≥ 2 { \ displaystyle n \ geq 2 } there is a prime p { \ displaystyle p } such that n < p < 2 n { \ displaystyle n


a daemon is usually created either by a process forking a child process and then immediately exiting, thus causing init to adopt the child process, or by the init process directly launching the daemon. in addition, a daemon launched by forking and exiting typically must perform other operations, such as dissociating the process from any controlling terminal ( tty ).


in the context of dikw, data is conceived of as symbols or signs, representing stimuli or signals, that are " of no use until... in a usable ( that is, relevant ) form ". zeleny characterized this non - usable characteristic of data as " know - nothing ". in some cases, data is understood to refer not only to symbols, but also to signals or stimuli referred to by said symbols — what zins terms subjective data. where universal data, for zins, are " the product of observation " ( italics in original ), subjective data are the observations. this distinction is often obscured in definitions of data in terms of " facts ".


for example, moreover, since s × s will never exceed m2 < 22p, this simple technique converges in at most 1 p - bit addition ( and possibly a carry from the pth bit to the 1st bit ), which can be done in linear time. this algorithm has a small exceptional case. it will produce 2n−1 for a multiple of the modulus rather than the correct value of 0.


in mathematics, differential galois theory studies the galois groups of differential equations.


as a result, different parameterized types are implemented by the same class or interface at run time. all invocations of a given generic type declaration share a single run - time implementation. this results in the possibility of heap pollution. under certain conditions, a variable of a parameterized type may refer to an object that is not of that parameterized type. the variable will always refer to an object that is an instance of a class that implements the parameterized type. heap pollution in a non - varargs context


in probability theory, the law of large numbers ( lln ) is a theorem that describes the result of performing the same experiment a large number of times. according to the law, the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed. the lln is important because it guarantees stable long - term results for the averages of some random events. for example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. any winning streak by a player will eventually be overcome by the parameters of the game.


alcohol - based hand sanitizers are almost entirely ineffective against norovirus ( or norwalk ) type viruses, the most common cause of contagious gastroenteritis. us centers for disease control and prevention recommend hand washing with soap over hand sanitizer rubs, particularly when hands are visibly dirty. the increasing use of these agents is based on their ease of use and rapid killing activity against micro - organisms ; however, they should not serve as a replacement for proper hand washing unless soap and water are unavailable. despite their effectiveness, non - water agents do not cleanse the hands of organic material, but simply disinfect them. it is for this reason that hand sanitizers are not as effective as soap and water at preventing the spread of many pathogens, since the pathogens remain on the hands.


in the drupal community, " core " refers to the collaboratively built codebase that can be extended through contributory modules and — for versions prior to drupal 8 — is kept outside of the " sites " folder of a drupal installation. ( starting with version 8, the core is kept in its own'core'sub - directory. ) drupal core is the stock element of drupal. common drupal - specific libraries, as well as the bootstrap process, are defined as drupal core ; all other functionality is defined as drupal modules including the system module itself.


in number theory, the multiplicative digital root of a natural number n { \ displaystyle n } in a given number base b { \ displaystyle b } is found by multiplying the digits of n { \ displaystyle n } together, then repeating this operation until only a single - digit remains, which is called the multiplicative digital root of n { \ displaystyle n }. the multiplicative digital root for the first few positive integers are : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, 8, 4, 0. ( sequence a031347 in the oeis ) multiplicative digital roots are the multiplicative equivalent of digital roots.


division in semigroups ( or in monoids ) is not possible in general. the formal study of semigroups began in the early 20th century. early results include a cayley theorem for semigroups realizing any semigroup as transformation semigroup, in which arbitrary functions replace the role of bijections from group theory.


the graphic presented in taxon sampling, bioinformatics, and phylogenomics, compares the correctness of phylogenetic trees generated using fewer taxa and more sites per taxon on the x - axis to more taxa and fewer sites per taxon on the y - axis. with fewer taxa, more genes are sampled amongst the taxonomic group ; in comparison, with more taxa added to the taxonomic sampling group, fewer genes are sampled. each method has the same total number of nucleotide sites sampled.


in the following case, one of the ( redundant ) learners fails, but the basic paxos protocol still succeeds. client proposer acceptor learner | | | | | | | x - - - - - - - - > | | | | | | request | x - - - - - - - - - > | - > | - > | | | prepare ( 1 ) | | < - - - - - - - - - x - - x - - x | | promise ( 1, { va, vb, vc } ) | x - - - - - - - - - > | - > | - > | | | accept! ( 1, v ) | | < - - - - - - - - - x - - x - - x - - - - - - > | - > | accepted ( 1, v ) | | | | | |!!! fail!! | < - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - x response | | | | | |


for example, a player whose score is three strokes over par after a given hole would appear as " + 3 " on the scoreboard. if two or more players have the same number of strokes, it may be desired to determine an outright winner. two of the more common methods are a playoff and scorecard count back.


in statistics, the mean squared error ( mse ) or mean squared deviation ( msd ) of an estimator ( of a procedure for estimating an unobserved quantity ) measures the average of the squares of the errors — that is, the average squared difference between the estimated values and the actual value. mse is a risk function, corresponding to the expected value of the squared error loss. the fact that mse is almost always strictly positive ( and not zero ) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate.


{ \ displaystyle h _ { ji } = { \ begin { cases } { \ boldsymbol { v } } _ { j } ^ { \ mathrm { t } } { \ boldsymbol { av } } _ { i } & { \ text { if } } j \ leq i { \ text {, } } \ \ \ lvert { \ boldsymbol { w } } _ { i + 1 } \ rvert _ { 2 } & { \ text { if } } j = i + 1 { \ text {, } } \ \ 0 & { \ text { if } } j > i + 1 { \ text {. } } \ end { cases } } } when applying the arnoldi iteration to solving linear systems, one starts with r 0 = b − a x 0 { \ displaystyle { \ boldsymbol { r } } _ { 0 } = { \ boldsymbol { b } } - { \ boldsymbol { ax } } _ { 0 } }, the residual corresponding to an initial guess x 0 { \ displaystyle { \ boldsymbol { x } } _ { 0 } }. after each step of iteration, one computes y i = h i − 1 ( ‖ r 0 ‖ 2 e 1 ) { \ displaystyle { \ boldsymbol { y } } _ { i } = { \ boldsymbol { h } } _ { i } ^ { - 1 } ( \ lvert { \ boldsymbol { r } } _ { 0 } \ rvert _ { 2 } { \ boldsymbol { e } } _ { 1 } ) } and the new iterate x i = x 0 + v i y i { \ displaystyle { \ boldsymbol { x } } _ { i } = { \ boldsymbol { x } } _ { 0 } + { \ boldsymbol { v } } _ { i } { \ boldsymbol { y } } _ { i } }.


in the 1970s, software engineers needed language support to break large projects down into modules. one obvious feature was to decompose large projects physically into separate files. a less obvious feature was to decompose large projects logically into abstract datatypes. at the time, languages supported concrete ( scalar ) datatypes like integer numbers, floating - point numbers, and strings of characters.


constructivism is a mathematical philosophy that rejects all proof methods that involve the existence of objects that are not explicitly built. this excludes, in particular, the use of the law of the excluded middle, the axiom of infinity, and the axiom of choice, and induces a different meaning for some terminology ( for example, the term " or " has a stronger meaning in constructive mathematics than in classical ). some non - constructive proofs show that if a certain proposition is false, a contradiction ensues ; consequently the proposition must be true ( proof by contradiction ). however, the principle of explosion ( ex falso quodlibet ) has been accepted in some varieties of constructive mathematics, including intuitionism. constructive proofs can be seen as defining certified mathematical algorithms : this idea is explored in the brouwer – heyting – kolmogorov interpretation of constructive logic, the curry – howard correspondence between proofs and programs, and such logical systems as per martin - lof's intuitionistic type theory, and thierry coquand and gerard huet's calculus of constructions.


in set theory, an arbitrary permutation of the elements of a set x is an automorphism. the automorphism group of x is also called the symmetric group on x. in elementary arithmetic, the set of integers, z, considered as a group under addition, has a unique nontrivial automorphism : negation. considered as a ring, however, it has only the trivial automorphism. generally speaking, negation is an automorphism of any abelian group, but not of a ring or field.


in mathematics, a euclidean field is an ordered field k for which every non - negative element is a square : that is, x ≥ 0 in k implies that x = y2 for some y in k. the constructible numbers form a euclidean field. it is the smallest euclidean field, as every euclidean field contains it as an ordered subfield. in other words, the constructible numbers form the euclidean closure of the rational numbers.


once we have determined which case we are in, instead of using division polynomials, we are able to work with a polynomial that has lower degree than the corresponding division polynomial : o ( l ) { \ displaystyle o ( l ) } rather than o ( l 2 ) { \ displaystyle o ( l ^ { 2 } ) }. for efficient implementation, probabilistic root - finding algorithms are used, which makes this a las vegas algorithm rather than a deterministic algorithm. under the heuristic assumption that approximately half of the primes up to an o ( log q ) { \ displaystyle o ( \ log q ) } bound are elkies primes, this yields an algorithm that is more efficient than schoof's, with an expected running time of o ( log 6 q ) { \ displaystyle o ( \ log ^ { 6 } q ) } using naive arithmetic, and o ~ ( log 4 q ) { \ displaystyle { \ tilde { o } } ( \ log ^ { 4 } q ) } using fast arithmetic. although this heuristic assumption is known to hold for most elliptic curves, it is not known to hold in every case, even under the grh.


in the linux kernel, the fat, hfs, hpfs, ntfs, and udf file system drivers support a umask mount option, which controls how the disk information is mapped to permissions. this is not the same as the per - process mask described above, although the permissions are calculated in a similar way. some of these file system drivers also support separate masks for files and directories, using mount options such as fmask.


every tree of fixed size occurs linearly many times. critical p ≈ 1 / n { \ displaystyle p \ approx 1 / n } the largest connected component has a number of vertices proportional to n 2 / 3 { \ displaystyle n ^ { 2 / 3 } }. there may exist several other large components ; however, the total number of vertices in non - tree components is again proportional to n 2 / 3 { \ displaystyle n ^ { 2 / 3 } }.


there are numerous special classes of semigroups, semigroups with additional properties, which appear in particular applications. some of these classes are even closer to groups by exhibiting some additional but not all properties of a group. of these we mention : regular semigroups, orthodox semigroups, semigroups with involution, inverse semigroups and cancellative semigroups. there are also interesting classes of semigroups that do not contain any groups except the trivial group ; examples of the latter kind are bands and their commutative subclass — semilattices, which are also ordered algebraic structures.


there are two ways this is represented in the ipa : ( a ) the same way as ejectives, with an apostrophe ; or, ( b ) more properly with a superscript glottal stop or with an under - tilde for creaky voice. for example, the yapese word for sick with a glottalized m could be transcribed, or. ( in some conventions, the apostrophe can occur above the em. )


such sets can be either disjoint or non - disjoint sets. spatially near sets are also descriptively near sets.


an important property of the pearson correlation is that it is invariant to application of separate linear transformations to the two variables being compared. thus, if we are correlating x and y, where, say, y = 2x + 1, the pearson correlation between x and y is 1 — a perfect correlation. this property does not make sense for the icc, since there is no basis for deciding which transformation is applied to each value in a group. however, if all the data in all groups are subjected to the same linear transformation, the icc does not change.


ordinary consumer machines and game consoles began to have parallel processors like the intel core, amd k10, and cell. graphics card companies like nvidia and amd began introducing large parallel systems like cuda and opencl. it appears, however, that these new technologies do not cite fgcs research.


in programming languages ( especially functional programming languages ) and type theory, an option type or maybe type is a polymorphic type that represents encapsulation of an optional value ; e. g., it is used as the return type of functions which may or may not return a meaningful value when they are applied. it consists of a constructor which either is empty ( often named none or nothing ), or which encapsulates the original data type a ( often written just a or some a ). a distinct, but related concept outside of functional programming, which is popular in object - oriented programming, is called nullable types ( often expressed as a? ). the core difference between option types and nullable types is that option types support nesting ( e. g. maybe ( maybe string ) = maybe string ), while nullable types do not ( e. g.


in an undirected simple graph of order n, the maximum degree of each vertex is n − 1 and the maximum size of the graph is n ( n − 1 ) / 2. the edges of an undirected simple graph permitting loops g { \ displaystyle g } induce a symmetric homogeneous relation { \ displaystyle \ sim } on the vertices of g { \ displaystyle g } that is called the adjacency relation of g { \ displaystyle g }. specifically, for each edge ( x, y ) { \ displaystyle ( x, y ) }, its endpoints x { \ displaystyle x } and y { \ displaystyle y } are said to be adjacent to one another, which is denoted x y { \ displaystyle x \ sim y }.


other operators run their own speaking clocks, with broadly similar formats, or redirect to bt's service. virgin media have their own service available by dialling 123 from a virgin media line. sky also have their own service accessible by dialling 123 from a sky telephone line.


in object - oriented programming, a class is an extensible program - code - template for creating objects, providing initial values for state ( member variables ) and implementations of behavior ( member functions or methods ). in many languages, the class name is used as the name for the class ( the template itself ), the name for the default constructor of the class ( a subroutine that creates objects ), and as the type of objects generated by instantiating the class ; these distinct concepts are easily conflated. although, to the point of conflation, one could argue that is a feature inherent in a language because of its polymorphic nature and why these languages are so powerful, dynamic and adaptable for use compared to languages without polymorphism present. thus they can model dynamic systems ( i. e. the real world, machine learning, ai ) more easily.


in signal processing theory, gaussian noise, named after carl friedrich gauss, is a kind of signal noise that has a probability density function ( pdf ) equal to that of the normal distribution ( which is also known as the gaussian distribution ). in other words, the values that the noise can take are gaussian - distributed. the probability density function p { \ displaystyle p } of a gaussian random variable z { \ displaystyle z } is given by : φ ( z ) = 1 σ 2 π e − ( z − μ ) 2 / ( 2 σ 2 ) { \ displaystyle \ varphi ( z ) = { \ frac { 1 } { \ sigma { \ sqrt { 2 \ pi } } } } e ^ { - ( z - \ mu ) ^ { 2 } / ( 2 \ sigma ^ { 2 } ) } } where z { \ displaystyle z } represents the grey level, μ { \ displaystyle \ mu } the mean grey value and σ { \ displaystyle \ sigma } its standard deviation. a special case is white gaussian noise, in which the values at any pair of times are identically distributed and statistically independent ( and hence uncorrelated ). in communication channel testing and modelling, gaussian noise is used as additive white noise to generate additive white gaussian noise. in telecommunications and computer networking, communication channels can be affected by wideband gaussian noise coming from many natural sources, such as the thermal vibrations of atoms in conductors ( referred to as thermal noise or johnson – nyquist noise ), shot noise, black - body radiation from the earth and other warm objects, and from celestial sources such as the sun.


if a m > t { \ displaystyle a _ { m } > t }, set r { \ displaystyle r } to m − 1 { \ displaystyle m - 1 }. else, a m ≤ t { \ displaystyle a _ { m } \ leq t } ; set l { \ displaystyle l } to m { \ displaystyle m }. now l = r { \ displaystyle l = r }, the search is done. if a l = t { \ displaystyle a _ { l } = t }, return l { \ displaystyle l }. otherwise, the search terminates as unsuccessful. where ceil is the ceiling function, the pseudocode for this version is : function binary _ search _ alternative ( a, n, t ) is l : = 0 r : = n − 1 while l! = r do m : = ceil ( ( l + r ) / 2 ) if a > t then r : = m − 1 else : l : = m if a = t then return l return unsuccessful


in most forms of latin scribal abbreviation, an overline or macron indicates omitted letters similar to use of apostrophes in english contractions. letters with macrons or overlines continue to be used in medical abbreviations in various european languages, particularly for prescriptions. common examples include a, a, or a for ante ( " before " ) c, c, or c for cum ( " with " ) p, p, or p for post ( " after " ) q, q, or q for quisque and its inflections ( " every ", " each " ) s, s, or s for sine ( " without " ) x, x, or x for exceptus and its inflections ( " except " ) note, however, that abbreviations involving the letter h take their macron halfway up the ascending line rather than at the normal height for unicode overlines and macrons : ħ. this is separately encoded in unicode with the symbols using bar diacritics and appears shorter than other overlines in many fonts.


the contents of the posterior analytics may be summarised as follows : all demonstration must be founded on principles already known. the principles on which it is founded must either themselves be demonstrable, or be so - called first principles, which cannot be demonstrated, nor need to be, being evident in themselves ( " nota per se " ). we cannot demonstrate things in a circular way, supporting the conclusion by the premises, and the premises by the conclusion.


in the case when the factor graph is a tree, the belief propagation algorithm will compute the exact marginals. furthermore, with proper scheduling of the message updates, it will terminate after two full passes through the tree. this optimal scheduling can be described as follows : before starting, the graph is oriented by designating one node as the root ; any non - root node which is connected to only one other node is called a leaf.


appropriate for their use hardware resources, such as computers, mobile phones and pagers. as its core, the provisioning process monitors access rights and privileges to ensure the security of an enterprise's resources and user privacy. as a secondary responsibility, it ensures compliance and minimizes the vulnerability of systems to penetration and abuse. as a tertiary responsibility, it tries to reduce the amount of custom configuration using boot image control and other methods that radically reduce the number of different configurations involved.


in statistics, censoring is a condition in which the value of a measurement or observation is only partially known. for example, suppose a study is conducted to measure the impact of a drug on mortality rate. in such a study, it may be known that an individual's age at death is at least 75 years ( but may be more ). such a situation could occur if the individual withdrew from the study at age 75, or if the individual is currently alive at the age of 75.


in microscopy, negative staining is an established method, often used in diagnostic microscopy, for contrasting a thin specimen with an optically opaque fluid. in this technique, the background is stained, leaving the actual specimen untouched, and thus visible. this contrasts with positive staining, in which the actual specimen is stained.


in mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, : pp. 160 is a value of an evolving variable at some point in time designated as the initial time ( typically denoted t = 0 ). for a system of order k ( the number of time lags in discrete time, or the order of the largest derivative in continuous time ) and dimension n ( that is, with n different evolving variables, which together can be denoted by an n - dimensional coordinate vector ), generally nk initial conditions are needed in order to trace the system's variables forward through time. in both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables ( state variables ) at any future time.


in probability theory, the inverse gaussian distribution ( also known as the wald distribution ) is a two - parameter family of continuous probability distributions with support on ( 0, ∞ ). its probability density function is given by f ( x ; μ, λ ) = λ 2 π x 3 exp ( − λ ( x − μ ) 2 2 μ 2 x ) { \ displaystyle f ( x ; \ mu, \ lambda ) = { \ sqrt { \ frac { \ lambda } { 2 \ pi x ^ { 3 } } } } \ exp { \ biggl ( } - { \ frac { \ lambda ( x - \ mu ) ^ { 2 } } { 2 \ mu ^ { 2 } x } } { \ biggr ) } } for x > 0, where μ > 0 { \ displaystyle \ mu > 0 } is the mean and λ > 0 { \ displaystyle \ lambda > 0 } is the shape parameter. the inverse gaussian distribution has several properties analogous to a gaussian distribution. the name can be misleading : it is an " inverse " only in that, while the gaussian describes a brownian motion's level at a fixed time, the inverse gaussian describes the distribution of the time a brownian motion with positive drift takes to reach a fixed positive level.


in the event a radio is stolen, or a user's permission to access the radio system is revoked, a data packet can be sent to the radio's id to disable the radio. this prevents the radio from transmitting or receiving until either an un - inhibit packet is sent to the radio or in some cases re - programmed using the appropriate service software. some literature refers to this feature as'stunning'and'un - stunning'the radio, or a'radiokill '.


in mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms ( 1 ) decrease in absolute value, and ( 2 ) approach zero in the limit. the test was used by gottfried leibniz and is sometimes known as leibniz's test, leibniz's rule, or the leibniz criterion. the test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test.