however, with introduction of font formats such as opentype, those supplemental glyphs were merged into the main fonts, relying on specific software capabilities to access the alternate glyphs. since apple's and microsoft's operating systems supported different character sets in the platform related fonts, some foundries used expert fonts in a different way. these fonts included the characters which were missing on either macintosh or windows computers, e. g. fractions, ligatures or some accented glyphs. the goal was to deliver the whole character set to the customer regardless of which operating system was used.


the grounds advanced for the'hypothesis'are worthless. the authors proposing such opinions might be competent, decent, honest individuals, but the views they present are demonstrably wrong.


in mathematics, computational group theory is the study of groups by means of computers. it is concerned with designing and analysing algorithms and data structures to compute information about groups. the subject has attracted interest because for many interesting groups ( including most of the sporadic groups ) it is impractical to perform calculations by hand.


in the calculus of variations the legendre – clebsch condition is a second - order condition which a solution of the euler – lagrange equation must satisfy in order to be a minimum. for the problem of minimizing a b l ( t, x, x ′ ) d t. { \ displaystyle \ int _ { a } ^ { b } l ( t, x, x') \, dt. \, } the condition is l x ′ x ′ ( t, x ( t ), x ′ ( t ) ) ≥ 0, t ∈ { \ displaystyle l _ { x'x'} ( t, x ( t ), x'( t ) ) \ geq 0, \, \ forall t \ in }


performing this operation gives us 2 1 × 0. 0001 2 { \ displaystyle 2 ^ { 1 } \ times 0. 0001 _ { 2 } } or 2 − 2 × 0. 100 2 { \ displaystyle 2 ^ { - 2 } \ times 0. 100 _ { 2 } }. without using a guard digit we have 2 1 × 0. 100 2 − 2 1 × 0. 011 2 { \ displaystyle 2 ^ { 1 } \ times 0. 100 _ { 2 } - 2 ^ { 1 } \ times 0. 011 _ { 2 } }, yielding 2 1 × 0. 001 2 = { \ displaystyle 2 ^ { 1 } \ times 0. 001 _ { 2 } = } or 2 − 1 × 0. 100 2 { \ displaystyle 2 ^ { - 1 } \ times 0. 100 _ { 2 } }. this gives us a relative error of 1.


in networked systems where competitive decision making takes place, game theory is often used to model system dynamics, and convergence towards equilibria can be considered as a driver of topological evolution. for example, kasthurirathna and piraveenan have shown that when individuals in a system display varying levels of rationality, improving the overall system rationality might be an evolutionary reason for the emergence of scale - free networks. they demonstrated this by applying evolutionary pressure on an initially random network which simulates a range of classic games, so that the network converges towards nash equilibria while being allowed to re - wire. the networks become increasingly scale - free during this process.


in most cases, the annual maintenance costs dwarf the purchase price. for example, when policy servers are used ( in both the plugin and proxy - based architectures ), high - end hardware is needed in order to handle the workload required to run the web access management infrastructure. centralized administration is an additional hidden cost, because customers will need to hire and train staff to exclusively manage policy entitlements for the underlying web applications. a final hidden cost relates to regulatory compliance. since web access management is similar in concept to a firewall ( more closely aligned to an application - layer firewall ), it must be able to handle major audit requirements, especially for public companies subject to the sarbanes - oxley act ( not to mention those that are bound by the health insurance portability and accountability act, pci, or cpni ). larger companies spend tremendous amounts of time and money auditing these web access management infrastructures since they are the enforcement points for many internal and external applications.


in mathematics, a bijection, also known as a bijective function, one - to - one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set ; there are no unpaired elements between the two sets. in mathematical terms, a bijective function f : x → y is a one - to - one ( injective ) and onto ( surjective ) mapping of a set x to a set y. the term one - to - one correspondence must not be confused with one - to - one function ( an injective function ; see figures ). a bijection from the set x to the set y has an inverse function from y to x. if x and y are finite sets, then the existence of a bijection means they have the same number of elements.


in statistics and markov modeling, an ancestral graph is a type of mixed graph to provide a graphical representation for the result of marginalizing one or more vertices in a graphical model that takes the form of a directed acyclic graph.


in programming languages, a delimited continuation, composable continuation or partial continuation, is a " slice " of a continuation frame that has been reified into a function. unlike regular continuations, delimited continuations return a value, and thus may be reused and composed. control delimiters, the basis of delimited continuations, were introduced by matthias felleisen in 1988 though early allusions to composable and delimited continuations can be found in carolyn talcott's stanford 1984 dissertation, felleisen et al., felleisen's 1987 dissertation, and algorithms for functional backtracking, e. g., for pattern matching, for parsing, in the algebraic logic functional programming language, and in the functional implementations of prolog where the failure continuation is often kept implicit and the reason of being for the success continuation is that it is composable.


due to the subtyping relation, a term may belong to more than one type. subtyping is therefore a form of type polymorphism. in object - oriented programming the term'polymorphism'is commonly used to refer solely to this subtype polymorphism, while the techniques of parametric polymorphism would be considered generic programming.


in probability theory and statistics, a mixture is a probabilistic combination of two or more probability distributions. the concept arises mostly in two contexts : a mixture defining a new probability distribution from some existing ones, as in a mixture distribution or a compound distribution. here a major problem often is to derive the properties of the resulting distribution. a mixture used as a statistical model such as is often used for statistical classification. the model may represent the population from which observations arise as a mixture of several components, and the problem is that of a mixture model, in which the task is to infer from which of a discrete set of sub - populations each observation originated.


in number theory, mills'constant is defined as the smallest positive real number a such that the floor function of the double exponential function a 3 n { \ displaystyle \ lfloor a ^ { 3 ^ { n } } \ rfloor } is a prime number for all positive natural numbers n. this constant is named after william harold mills who proved in 1947 the existence of a based on results of guido hoheisel and albert ingham on the prime gaps. its value is unproven, but if the riemann hypothesis is true, it is approximately 1. 3063778838630806904686144926... ( sequence a051021 in the oeis ).


in mathematics, an evasive boolean function ƒ ( of n variables ) is a boolean function for which every decision tree algorithm has running time of exactly n. consequently, every decision tree algorithm that represents the function has, at worst case, a running time of n.


in military cryptography, a codress message is an encrypted message whose address is also encrypted. this is usually done to prevent traffic analysis. = = references = =


forward chaining starts with the known facts and asserts new facts. backward chaining starts with goals, and works backward to determine what facts must be asserted so that the goals can be achieved. additionally, the concept of'inference'has expanded to include the process through which trained neural networks generate predictions or decisions.


with the development of formal logic, hilbert asked whether it would be possible to prove that an axiom system is consistent by analyzing the structure of possible proofs in the system, and showing through this analysis that it is impossible to prove a contradiction. this idea led to the study of proof theory. moreover, hilbert proposed that the analysis should be entirely concrete, using the term finitary to refer to the methods he would allow but not precisely defining them.


engaging in a structured dialogue or repeated discussion, to exchange ideas about how to get specific about what it means and how to clear it up ( scrum method ). 13. allocating different applications of the concept to different but related sets ( boolean logic ).


in the data warehousing strategy, the data from different sources are extracted and integrated in a single database. for example, various'omics'datasets may be integrated to provide biological insights into biological systems. examples include data from genomics, transcriptomics, proteomics, interactomics, metabolomics. ideally, changes in these sources are regularly synchronized to the integrated database.


in mathematics and computing, universal hashing ( in a randomized algorithm or data structure ) refers to selecting a hash function at random from a family of hash functions with a certain mathematical property ( see definition below ). this guarantees a low number of collisions in expectation, even if the data is chosen by an adversary. many universal families are known ( for hashing integers, vectors, strings ), and their evaluation is often very efficient. universal hashing has numerous uses in computer science, for example in implementations of hash tables, randomized algorithms, and cryptography.


in mathematics, a random minimum spanning tree may be formed by assigning random weights from some distribution to the edges of an undirected graph, and then constructing the minimum spanning tree of the graph. when the given graph is a complete graph on n vertices, and the edge weights have a continuous distribution function whose derivative at zero is d > 0, then the expected weight of its random minimum spanning trees is bounded by a constant, rather than growing as a function of n. more precisely, this constant tends in the limit ( as n goes to infinity ) to ζ ( 3 ) / d, where ζ is the riemann zeta function and ζ ( 3 ) is apery's constant. for instance, for edge weights that are uniformly distributed on the unit interval, the derivative is d = 1, and the limit is just ζ ( 3 ). in contrast to uniformly random spanning trees of complete graphs, for which the typical diameter is proportional to the square root of the number of vertices, random minimum spanning trees of complete graphs have typical diameter proportional to the cube root. random minimum spanning trees of grid graphs may be used for invasion percolation models of liquid flow through a porous medium, and for maze generation. = = references = =


while avoiding the problems of depending on delayed recordings, this method was not without its own inherent pitfalls. as the originating station was usually somewhat — as far as 50 – 100 miles ( 80 – 160 km ) or more — the signal could frequently be degraded or interfered with by other stations on the same channel during periods of ionospheric or tropospheric disturbance. for that matter, if the originating station experienced technical difficulties and had to leave the air, the receiving station would be left without network programming during the outage as well.


the statement,'if x exists, then x has most of the φ's'expresses a necessary truth ( in the idiolect of the speaker ). ( c ) for any successful theory, the account must not be circular. the properties which are used in the vote must not themselves involve the notion of reference in such a way that it is ultimately impossible to eliminate.


when the active array contains no more processes, the scheduler swaps the active and expired arrays, hence the name o ( 1 ) scheduler. in unix or linux, the sar command is used to check the run queue. the vmstat unix or linux command can also be used to determine the number of processes that are queued to run or waiting to run. these appear in the'r'column. example : $ vmstat procs - - - - - - - - - - - memory - - - - - - - - - - - - - swap - - - - - - - io - - - - - system - - - - - - - - cpu - - - - - r b swpd free buff cache si so bi bo in cs us sy id wa st 2 0 0 4579152 324416 4619528 0 0 402 236 3357 15 20 2 78 0 0 there are two models for run queues : one that assigns a run queue to each physical processor, and the other has only one run queue in the system


by analysing the retained data, governments can identify the locations of individuals, an individual's associates and the members of a group such as political opponents. these activities may or may not be lawful, depending on the constitutions and laws of each country. in many jurisdictions, access to these databases may be made by a government with little or no judicial oversight. in the case of commercial data retention, the data retained will usually be on transactions and web sites visited. data retention also covers data collected by other means ( e. g., by automatic number - plate recognition systems ) and held by government and commercial organisations.


in number theory, a balanced prime is a prime number with equal - sized prime gaps above and below it, so that it is equal to the arithmetic mean of the nearest primes above and below. or to put it algebraically, given a prime number p n { \ displaystyle p _ { n } }, where n is its index in the ordered set of prime numbers, p n = p n − 1 + p n + 1 2. { \ displaystyle p _ { n } = { { p _ { n - 1 } + p _ { n + 1 } } \ over 2 }. } for example, 53 is the sixteenth prime ; the fifteenth and seventeenth primes, 47 and 59, add up to 106, and half of that is 53 ; thus 53 is a balanced prime.


the substitutions may be applied in any order we like, as the result will be the same. below, the substitutions applied to the number on the previous line are on the right, the resulting number on the left. 211. 0 1 _ 0 = 211.


in one instance, trojan horses were used as a targeted threat so that israeli companies could conduct corporate espionage on each other. the hotword trojan, the ginwui and the ppdropper trojans are additional examples of trojans used for corporate espionage. targeted destination attacks use harvested ip addresses to send messages directly to recipients without an mx record lookup. it aims for specific sites and users by defeating hosted protection services and internal gateways to deliver e - mail with malicious payloads.


in multilevel security mode of operation ( also called controlled security mode ), all users must have : signed nda for all information on the system. proper clearance for some information on the system. formal access approval for some information on the system. a valid need to know for some information on the system. all users can access some data, based on their need to know, clearance and formal access approval


at injection, the compressed air would then press the fuel through the disc - type atomisers into the combustion chamber. manufacturing engines featuring the open nozzle design was considerably cheaper and easier than making them with a closed nozzle design. it also allows for using tar as fuel.


in social network analysis, betweenness centrality can have different implications. from a macroscopic perspective, bridging positions or " structural holes " ( indicated by high betweenness centrality ) reflect power, because they allow the person on the bridging position to exercise control ( e. g., decide whether to share information or not ) over the persons it connects between. from the microscopic perspective of ego networks ( i. e., only considering first - degree connections ), in online social networks a high betweenness centrality coincides with nominations of closest friends ( i. e., strong interpersonal ties ), because it reflects social capital investments into the relationship when distant social circles ( e. g., family and university ) are bridged ( often resulting from an introduction by ego ).


a proper treatment of this complication calls for the conception of something slightly more general than a sign relation proper, namely, a sign relational complex. in effect, expressed in the roughest practical terms, this allows for missing data in the columns of the relational database table for the sign relation in question. typically one operates on the default assumption that all of the roles of elementary sign relations are filled, but remains wary enough of the possible exceptions to deal with them on an ad hoc basis.


analyses differ in how much they rely on morphology to explain them. some have questioned their linguistic status, as well as the very use of the term " classifier ". not much is known yet about their syntax or phonology.


in the construction of database applications, it can be useful to introduce an additional layer of data dictionary software, i. e. middleware, which communicates with the underlying dbms data dictionary. such a " high - level " data dictionary may offer additional features and a degree of flexibility that goes beyond the limitations of the native " low - level " data dictionary, whose primary purpose is to support the basic functions of the dbms, not the requirements of a typical application. for example, a high - level data dictionary can provide alternative entity - relationship models tailored to suit different applications that share a common database. extensions to the data dictionary also can assist in query optimization against distributed databases.


one exception to the single - entry - point paradigm is android. android applications do not have a single entry point – there is no special main function. instead, they have essential components ( activities and services ) which the system can load and run as needed. an occasionally used technique is the fat binary, which consists of several executables for different targets packaged in a single file. most commonly, this is implemented by a single overall entry point, which is compatible with all targets and branches to the target - specific entry point. alternative techniques include storing separate executables in separate forks, each with its own entry point, which is then selected by the operating system.


the nine lemma can be proved by direct diagram chasing, or by applying the snake lemma ( to the two bottom rows in the first case, and to the two top rows in the second case ). linderholm ( p.


in the second sentence, " is " can however be interpreted as an ordinary copula and the past participle as an adjective. sentences of the second type are called false passives by some linguists, who feel that such sentences are simply confused with the passive voice due to their outward similarity. other linguists consider the second type to be a different kind of passive – a stative passive ( rarely called statal, static, or resultative passive ), in contrast to the dynamic or eventive passive illustrated by the first sentence.


in terms of the quartic residue symbol, the law of quartic reciprocity for gaussian integers states that if π and θ are primary ( congruent to 1 mod ( 1 + i ) 3 ) gaussian primes then − 1 = ( − 1 ) n π − 1 4 n θ − 1 4. { \ displaystyle { \ bigg } \ left ^ { - 1 } = ( - 1 ) ^ { { \ frac { n \ pi - 1 } { 4 } } { \ frac { n \ theta - 1 } { 4 } } }. }


another way of thinking of cluster states is as a particular instance of graph states, where the underlying graph is a connected subset of a d - dimensional lattice. cluster states are especially useful in the context of the one - way quantum computer. for a comprehensible introduction to the topic see. formally, cluster states | { κ } ⟩ c { \ displaystyle | \ phi _ { \ { \ kappa \ } } \ rangle _ { c } } are states which obey the set eigenvalue equations : k ( a ) | { κ } ⟩ c = ( − 1 ) κ a | { κ } ⟩ c { \ displaystyle k ^ { ( a ) } { \ left | \ phi _ { \ { \ kappa \ } } \ right \ rangle _ { c } } = ( - 1 ) ^ { \ kappa _ { a } } { \ left | \ phi _ { \ { \ kappa \ } } \ right \ rangle _ { c } } } where k ( a ) { \ displaystyle k ^ { ( a ) } } are the correlation operators k ( a ) = σ x ( a ) b ∈ n ( a ) σ z ( b ) { \ displaystyle k ^ { ( a ) } = \ sigma _ { x } ^ { ( a ) } \ bigotimes _ { b \ in \ mathrm { n } ( a ) } \ sigma _ { z } ^ { ( b ) } } with σ x { \ displaystyle \ sigma _ { x } } and σ z { \ displaystyle \ sigma _ { z } } being pauli matrices, n ( a ) { \ displaystyle n ( a ) } denoting the neighbourhood of a { \ displaystyle a } and { κ a ∈ { 0, 1 } | a ∈ c } { \ displaystyle \ { \ kappa _ { a } \ in \ { 0, 1 \ } | a \ in c \ } } being a set of binary parameters specifying the particular instance of a cluster state.


in the context of searching for information, researchers have identified two forms of information overload : outcome overload where there are too many sources of information and textual overload where the individual sources are too long. this form of information overload may cause searchers to be less systematic. disillusionment when a search is more challenging than expected may result in an individual being less able to search effectively. information overload when searching can result in a satisficing strategy. : 7


in the russian peasant method, the powers of two in the decomposition of the multiplicand are found by writing it on the left and progressively halving the left column, discarding any remainder, until the value is 1 ( or −1, in which case the eventual sum is negated ), while doubling the right column as before. lines with even numbers on the left column are struck out, and the remaining numbers on the right are added together.


suppose we have a preparation procedure for a system in a physics lab : for example, the procedure might involve a physical apparatus and some protocols for manipulating the apparatus. as a result of this preparation procedure, some system is produced and maintained in isolation for some small period of time. by repeating this laboratory preparation procedure we obtain a sequence of systems x1, x2,...., xk, which in our mathematical idealization, we assume is an infinite sequence of systems.


in mathematical physics, especially quantum mechanics, it is common to write the inner product between elements as ⟨ a | b ⟩, as a short version of ⟨ a | · | b ⟩, or ⟨ a | o | b ⟩, where o is an operator. this is known as dirac notation or bra – ket notation, to note vectors from the dual spaces of the bra ⟨ a | and the ket | b ⟩. but there are other notations used. in continuum mechanics, chevrons may be used as macaulay brackets.


the process of search can be divided in three parts : generate descriptors for the media which we are going to use as query and the descriptors for the media in our database. compare descriptors of the query and our database ’ s media. list the media sorted by maximum coincidence.


in probability theory, there exist several different notions of convergence of random variables. the convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. the same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. the different possible notions of convergence relate to how such a behavior can be characterized : two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution.


in the japanese kantenji braille, the standard 8 - dot braille patterns 38, 138, 348, and 1348 are the patterns related to braille pattern dots - 26, since the two additional dots of kantenji patterns 026, 267, and 0267 are placed above the base 6 - dot cell, instead of below, as in standard 8 - dot braille.


in the japanese kantenji braille, the standard 8 - dot braille patterns 23, 123, 234, and 1234 are the 8 - dot braille patterns related to braille pattern dots - 12, since the two additional dots of kantenji patterns 012, 127, and 0127 are placed above the base 6 - dot cell, instead of below, as in standard 8 - dot braille.


further, giles and pulleyblank showed that if p { \ displaystyle p } is a polytope whose vertices are all integer valued, then p { \ displaystyle p } is the solution set of some tdi system a x ≤ b { \ displaystyle ax \ leq b }, where b { \ displaystyle b } is integer valued. note that tdi is a weaker sufficient condition for integrality than total unimodularity. = = references = =


in philosophy of science, confirmation holism, also called epistemological holism, is the view that no individual statement can be confirmed or disconfirmed by an empirical test, but rather that only a set of statements ( a whole theory ) can be so. it is attributed to willard van orman quine who motivated his holism through extending pierre duhem's problem of underdetermination in physical theory to all knowledge claims. duhem's idea was, roughly, that no theory of any type can be tested in isolation but only when embedded in a background of other hypotheses, e. g. hypotheses about initial conditions. quine thought that this background involved not only such hypotheses but also our whole web of belief, which, among other things, includes our mathematical and logical theories and our scientific theories. this last claim is sometimes known as the duhem – quine thesis. a related claim made by quine, though contested by some ( see adolf grunbaum 1962 ), is that one can always protect one's theory against refutation by attributing failure to some other part of our web of belief. in his own words, " any statement can be held true come what may, if we make drastic enough adjustments elsewhere in the system. "


in natural language processing, multinomial lr classifiers are commonly used as an alternative to naive bayes classifiers because they do not assume statistical independence of the random variables ( commonly known as features ) that serve as predictors. however, learning in such a model is slower than for a naive bayes classifier, and thus may not be appropriate given a very large number of classes to learn. in particular, learning in a naive bayes classifier is a simple matter of counting up the number of co - occurrences of features and classes, while in a maximum entropy classifier the weights, which are typically maximized using maximum a posteriori ( map ) estimation, must be learned using an iterative procedure ; see # estimating the coefficients.


in natural language processing, dependency - based parsing can be formulated as an asp problem. the following code parses the latin sentence " puella pulchra in villa linguam latinam discit ", " the pretty girl is learning latin in the villa ". the syntax tree is expressed by the arc predicates which represent the dependencies between the words of the sentence. the computed structure is a linearly ordered rooted tree.


one of the motivations of establishing a distinct, higher level of linguistic analysis is, then, to explain the structural ambiguity due to the constructional homonymities at a lower level. on the other hand, each linguistic level also captures some structural similarities within the level that are not explained in lower levels. chomsky uses this argument as well to motivate the establishment of distinct levels of linguistic analysis. chomsky then shows that a grammar which analyzes sentences up to the phrase structure level contains many constructional homonymities at the phrase structure level where the resulting ambiguities need to be explained at a higher level. then he shows how his newly invented “ transformational level ” can naturally and successfully function as that higher level. he further claims that any phrase structure grammar which cannot explain these ambiguities as successfully as transformational grammar does must be considered " inadequate ".


in mathematics, given a non - empty set of objects of finite extension in d { \ displaystyle d } - dimensional space, for example a set of points, a bounding sphere, enclosing sphere or enclosing ball for that set is an d { \ displaystyle d } - dimensional solid sphere containing all of these objects. used in computer graphics and computational geometry, a bounding sphere is a special type of bounding volume. there are several fast and simple bounding sphere construction algorithms with a high practical value in real - time computer graphics applications. in statistics and operations research, the objects are typically points, and generally the sphere of interest is the minimal bounding sphere, that is, the sphere with minimal radius among all bounding spheres.


in the japanese kantenji braille, the standard 8 - dot braille patterns 368, 1368, 3468, and 13468 are the patterns related to braille pattern dots - 256, since the two additional dots of kantenji patterns 0256, 2567, and 02567 are placed above the base 6 - dot cell, instead of below, as in standard 8 - dot braille.


in early versions of the problem, the gambler begins with no initial knowledge about the machines. herbert robbins in 1952, realizing the importance of the problem, constructed convergent population selection strategies in " some aspects of the sequential design of experiments ". a theorem, the gittins index, first published by john c. gittins, gives an optimal policy for maximizing the expected discounted reward.


it arose in connection with james watt's pioneering work on the steam engine. the equation of the curve can be given in polar coordinates as r 2 = b 2 − 2. { \ displaystyle r ^ { 2 } = b ^ { 2 } - \ left ^ { 2 }. }


which is why it is important to consider encryption, hashing, and other security mechanisms in your design to ensure that information collected from a potential attacker won't allow access. another key feature to client - server security design is good coding practices.


in the shrikhande graph, any two vertices i and j have two distinct neighbors in common ( excluding the two vertices i and j themselves ), which holds true whether or not i is adjacent to j. in other words, it is strongly regular and its parameters are : { 16, 6, 2, 2 }, i. e., λ = μ = 2 { \ displaystyle \ lambda = \ mu = 2 }. this equality implies that the graph is associated with a symmetric bibd. the shrikhande graph shares these parameters with exactly one other graph, the 4×4 rook's graph, i. e., the line graph l ( k4, 4 ) of the complete bipartite graph k4, 4. the latter graph is the only line graph l ( kn, n ) for which the strong regularity parameters do not determine that graph uniquely but are shared with a different graph, namely the shrikhande graph ( which is not a rook's graph ). the shrikhande graph is locally hexagonal ; that is, the neighbors of each vertex form a cycle of six vertices.


in mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. a graph in this context is made up of vertices ( also called nodes or points ) which are connected by edges ( also called links or lines ). a distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. graphs are one of the principal objects of study in discrete mathematics.


this was partly because differences in their architectures required these changes to optimize unix to each architecture. thus as general purpose operating systems became stable, supercomputers began to borrow and adapt the critical system code from them and relied on the rich set of secondary functions that came with them, not having to reinvent the wheel. however, at the same time the size of the code for general purpose operating systems was growing rapidly. by the time unix - based code had reached 500, 000 lines long, its maintenance and use was a challenge.


in practice, the functions f { \ displaystyle f } and f { \ displaystyle f } are often not known or assumed. however, they can be estimated from an observed frequency distribution. in this section, we give an example. consider the following table, representing a sample of 3, 800 ( discrete - valued ) observations : because the observations are discrete - valued, constructing the exact distribution of the median is not an immediate translation of the above expression for pr ( median = v ) { \ displaystyle \ pr ( \ operatorname { median } = v ) } ; one may ( and typically does ) have multiple instances of the median in one's sample.


between the two extremes the realizations are discrete distributions with less and less concentration as α { \ displaystyle \ alpha } increases. the dirichlet process can also be seen as the infinite - dimensional generalization of the dirichlet distribution. in the same way as the dirichlet distribution is the conjugate prior for the categorical distribution, the dirichlet process is the conjugate prior for infinite, nonparametric discrete distributions.


they are distinct because they possess a variety of desirable properties, most importantly the existence of a sufficient statistic. the concept of exponential families is credited to e. j. g. pitman, g. darmois, and b. o. koopman in 1935 – 1936. exponential families of distributions provides a general framework for selecting a possible alternative parameterisation of a parametric family of distributions, in terms of natural parameters, and for defining useful sample statistics, called the natural sufficient statistics of the family.


however, only the receiving party has access to the decryption key that enables messages to be read. public - key encryption was first described in a secret document in 1973 ; beforehand, all encryption schemes were symmetric - key ( also called private - key ). : 478 although published subsequently, the work of diffie and hellman was published in a journal with a large readership, and the value of the methodology was explicitly described.


" depending upon how appropriate the upgrades are considered by other owners of the same model, this may reduce or enhance the value of the car. if the car is in regular use, non - original upgrades are likely to be more acceptable ; if the car is a stored collector's piece, originality would be more important. it is important as a restorer or owner to know what is acceptable to the potential market for the finished car, in order not to de - value it.


in mathematics, veblen's theorem, introduced by oswald veblen ( 1912 ), states that the set of edges of a finite graph can be written as a union of disjoint simple cycles if and only if every vertex has even degree. thus, it is closely related to the theorem of euler ( 1736 ) that a finite graph has an euler tour ( a single non - simple cycle that covers the edges of the graph ) if and only if it is connected and every vertex has even degree. indeed, a representation of a graph as a union of simple cycles may be obtained from an euler tour by repeatedly splitting the tour into smaller cycles whenever there is a repeated vertex.


therefore, φ ( 9 ) = 6. as another example, φ ( 1 ) = 1 since for n = 1 the only integer in the range from 1 to n is 1 itself, and gcd ( 1, 1 ) = 1.


the list of revisions from the start to head ( in graph theory terms, the unique path in the tree, which forms a linear graph as before ) is the trunk or mainline. conversely, when a revision can be based on more than one previous revision ( when a node can have more than one parent ), the resulting process is called a merge, and is one of the most complex aspects of revision control. this most often occurs when changes occur in multiple branches ( most often two, but more are possible ), which are then merged into a single branch incorporating both changes.


( z ∈ w ↔ ( z ∈ x ∨ z = y ) ). { \ displaystyle \ forall x. \ forall y. \ exists w. \ forall z. { \ big ( } z \ in w \ leftrightarrow ( z \ in x \ lor z = y ) { \ big ) }. }


in statistics, some monte carlo methods require independent observations in a sample to be drawn from a one - dimensional distribution in sorted order. in other words, all n order statistics are needed from the n observations in a sample. the naive method performs a sort and takes o ( n log n ) time. there are also o ( n ) algorithms which are better suited for large n. the special case of drawing n sorted observations from the uniform distribution on is equivalent to drawing from the uniform distribution on an n - dimensional simplex ; this task is a part of sequential importance resampling.


in order to represent any network, it is necessary to characterize the properties of the corresponding graph of nodes and links. studies on the collaboration network of movie actors have been described in literature such as the work done by ( watts and strogatz, 1998 ), and barabasi and albert in ( 1999 ) and ( 2000 ). the general characteristics are described below. according to watts and strogatz ( 1998 ), the movie / actor network indicated the following characteristics showing a small - world property of the underlying network : size : 225 226 average degree : 61 average path length : 3. 65 clustering coefficient : 0. 79compared to a random graph of the same size and average degree, the average path length is close in value.


in statistics and information theory, a maximum entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions. according to the principle of maximum entropy, if nothing is known about a distribution except that it belongs to a certain class ( usually defined in terms of specified properties or measures ), then the distribution with the largest entropy should be chosen as the least - informative default. the motivation is twofold : first, maximizing entropy minimizes the amount of prior information built into the distribution ; second, many physical systems tend to move towards maximal entropy configurations over time.


in program analysis, shape analysis is a static code analysis technique that discovers and verifies properties of linked, dynamically allocated data structures in ( usually imperative ) computer programs. it is typically used at compile time to find software bugs or to verify high - level correctness properties of programs. in java programs, it can be used to ensure that a sort method correctly sorts a list. for c programs, it might look for places where a block of memory is not properly freed.


from an a - priori point of view, the main criticism is that taking s → 0 { \ displaystyle s \ rightarrow 0 } is far from leading to a noninformative prior. moreover, a - posteriori, it assigns zero probability to any set that does not include the observations. the imprecise dirichlet process has been proposed to overcome these issues. the basic idea is to fix s > 0 { \ displaystyle s > 0 } but do not choose any precise base measure g 0 { \ displaystyle g _ { 0 } }. more precisely, the imprecise dirichlet process ( idp ) is defined as follows : i d p : { d p ( s, g 0 ) : g 0 ∈ p } { \ displaystyle ~ ~ \ mathrm { idp } : ~ \ left \ { \ mathrm { dp } \ left ( s, g _ { 0 } \ right ) : ~ ~ g _ { 0 } \ in \ mathbb { p } \ right \ } } where p { \ displaystyle \ mathbb { p } } is the set of all probability measures. in other words, the idp is the set of all dirichlet processes ( with a fixed s > 0 { \ displaystyle s > 0 } ) obtained by letting the base measure g 0 { \ displaystyle g _ { 0 } } to span the set of all probability measures.


consider a computer program for recognizing dogs ( the relevant element ) in a digital photograph. upon processing a picture which contains ten cats and twelve dogs, the program identifies eight dogs. of the eight elements identified as dogs, only five actually are dogs ( true positives ), while the other three are cats ( false positives ).


each bit completes a small task, which then builds up to the final bigger task. like writing code on a computer, it is easier to write the basic smaller parts and make them work first, and then put them together to finish the larger more complicated code, instead of tackling the entire code from the very beginning. the first solution is less risky because if something goes wrong with the code, it is easier to look for the problem in the smaller bits, since the segment with the problem will be the one that does not work, while in the latter solution, the programmer may have to look through the entire code to search for a single error, which proves time - consuming.


in mathematics, an almost perfect number ( sometimes also called slightly defective or least deficient number ) is a natural number n such that the sum of all divisors of n ( the sum - of - divisors function σ ( n ) ) is equal to 2n − 1, the sum of all proper divisors of n, s ( n ) = σ ( n ) − n, then being equal to n − 1. the only known almost perfect numbers are powers of 2 with non - negative exponents ( sequence a000079 in the oeis ). therefore the only known odd almost perfect number is 20 = 1, and the only known even almost perfect numbers are those of the form 2k for some positive integer k ; however, it has not been shown that all almost perfect numbers are of this form. it is known that an odd almost perfect number greater than 1 would have at least six prime factors. if m is an odd almost perfect number then m ( 2m − 1 ) is a descartes number. moreover if a and b are positive odd integers such that b + 3 < a < m / 2 { \ displaystyle b + 3


in mathematics, hensel's lemma, also known as hensel's lifting lemma, named after kurt hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number p, then this root can be lifted to a unique root modulo any higher power of p. more generally, if a polynomial factors modulo p into two coprime polynomials, this factorization can be lifted to a factorization modulo any higher power of p ( the case of roots corresponds to the case of degree 1 for one of the factors ). by passing to the " limit " ( in fact this is an inverse limit ) when the power of p tends to infinity, it follows that a root or a factorization modulo p can be lifted to a root or a factorization over the p - adic integers. these results have been widely generalized, under the same name, to the case of polynomials over an arbitrary commutative ring, where p is replaced by an ideal, and " coprime polynomials " means " polynomials that generate an ideal containing 1 ". hensel's lemma is fundamental in p - adic analysis, a branch of analytic number theory. the proof of hensel's lemma is constructive, and leads to an efficient algorithm for hensel lifting, which is fundamental for factoring polynomials, and gives the most efficient known algorithm for exact linear algebra over the rational numbers.


in statistics, the bootstrap is a method for inferring the variability of data that has an unknown distribution using pseudoreplications of the original data. for example, given a set of 100 data points, a pseudoreplicate is a data set of the same size ( 100 points ) randomly sampled from the original data, with replacement. that is, each original data point may be represented more than once in the pseudoreplicate, or not at all. statistical support involves evaluation of whether the original data has similar properties to a large set of pseudoreplicates.


participants were asked to watch a 20 - minute news video ( half of the participants saw the negative images and the other half did not ) and an additional ten - minute video. they were instructed to pay attention because they would be tested afterwards. a follow up survey was sent 6 to 7 weeks later to measure memory and recall from the news video.


in queueing theory, a discipline within the mathematical theory of probability, the decomposition method is an approximate method for the analysis of queueing networks where the network is broken into subsystems which are independently analyzed. the individual queueing nodes are considered to be independent g / g / 1 queues where arrivals are governed by a renewal process and both service time and arrival distributions are parametrised to match the first two moments of data. = = references = =


the sample code for the banked xios implementation published in the mp / m ii system implementors guide was written by altos ( and carries a disclaimer that it only works as - is with their sun series 8000 ). the " 8000 " contained in the name of altos'first series of computer did cause some confusion in the marketplace because its name may have suggested the inclusion of the 16 - bit zilog z8000 processor, which had just been released in 1979, although altos'acs - 8000 did not use this processor, but the older 8 - bit z80. a 1981 review in computerworld, comparing the acs 8000 with other multi - user systems, found that altos'z80 processor was underpowered, especially for cpu - intensive tasks ( most other multi - user systems used 16 - bit processors by then ), but the acs - 8000 was found adequate for multi - user order entry systems. a configuration with a 10 - mb hard - drive plus a 1 - mb 8 " floppy drive, bundled with a printer and one terminal was priced at $ 12, 340 ( the same machine but with four terminals was $ 15, 625 ), which was considerably less than most other multi - user systems, which were typically priced in the $ 25, 000 – $ 50, 000 range.


.., n } { \ displaystyle q _ { i } \ in \ { 0, 1,..., k \ }, \ forall i \ in \ { 1, 2,..., n \ } } and that at least one component q i = k { \ displaystyle q _ { i } = k } let s n d { \ displaystyle s _ { n } ^ { d } } represent a n - dimensional hypersphere with radius of d = ‖ q → ‖ { \ displaystyle d = \ left \ vert { \ vec { q } } \ right \ vert }.


in some cases a kbs could not be built because the organization did not have all the knowledge needed to support all their activities. in these cases logico - linguistic modeling showed shortcomings in the supply of information and where more was needed. for example, a planning department in a telecoms company


in mathematics, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors. particularly, it's defined by a ratio between the number of divisors an integer has and that integer raised to some positive power. for any possible exponent, whichever integer has the highest ratio is a superior highly composite number. it is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer.


in telecommunication, a tactical communications system is a communications system that ( a ) is used within, or in direct support of, tactical forces, ( b ) is designed to meet the requirements of changing tactical situations and varying environmental conditions, ( c ) provides securable communications, such as voice, data, and video, among mobile users to facilitate command and control within, and in support of, tactical forces, and ( d ) usually requires extremely short installation times, usually on the order of hours, in order to meet the requirements of frequent relocation.


in portfolio theory in finance, an objective often is to choose a portfolio of risky assets such that the distribution of the random portfolio return has desirable properties. for example, one might want to choose the portfolio return having the lowest variance for a given expected value. here the random vector is the vector r { \ displaystyle \ mathbf { r } } of random returns on the individual assets, and the portfolio return p ( a random scalar ) is the inner product of the vector of random returns with a vector w of portfolio weights — the fractions of the portfolio placed in the respective assets. since p = wt r { \ displaystyle \ mathbf { r } }, the expected value of the portfolio return is wte ( r { \ displaystyle \ mathbf { r } } ) and the variance of the portfolio return can be shown to be wtcw, where c is the covariance matrix of r { \ displaystyle \ mathbf { r } }.


loosely speaking, rejection of the null hypothesis implies that there is sufficient evidence against it. as a particular example, if a null hypothesis states that a certain summary statistic t { \ displaystyle t } follows the standard normal distribution n ( 0, 1 ), then the rejection of this null hypothesis could mean that ( i ) the mean of t { \ displaystyle t } is not 0, or ( ii ) the variance of t { \ displaystyle t } is not 1, or ( iii ) t { \ displaystyle t } is not normally distributed. different tests of the same null hypothesis would be more or less sensitive to different alternatives. however, even if we do manage to reject the null hypothesis for all 3 alternatives, and even if we know the distribution is normal and variance is 1, the null hypothesis test does not tell us which non - zero values of the mean are now most plausible. the more independent observations from the same probability distribution one has, the more accurate the test will be, and the higher the precision with which one will be able to determine the mean value and show that it is not equal to zero ; but this will also increase the importance of evaluating the real - world or scientific relevance of this deviation.


+ t 2 ⟨ ⟨ i, j ⟩ ⟩ ( c i σ † c j σ + h. c. ) + j ⟨ i, j ⟩ ( s i ⋅ s j − n i n j 4 ) − μ i n i, { \ displaystyle { \ mathcal { \ hat { h } } } = t _ { 1 } \ sum \ limits _ { \ langle i, j \ rangle } \ left ( c _ { i \ sigma } ^ { \ dagger } c _ { j \ sigma } + \ mathrm { h. c. } \ right ) \ + \ t _ { 2 } \ sum \ limits _ { \ langle \ langle i, j \ rangle \ rangle } \ left ( c _ { i \ sigma } ^ { \ dagger } c _ { j \ sigma } + \ mathrm { h. c. } \ right ) \ + \ j \ sum \ limits _ { \ langle i, j \ rangle } \ left ( \ mathbf { s } _ { i } \ cdot \ mathbf { s } _ { j } - { \ frac { n _ { i } n _ { j } } { 4 } } \ right ) - \ \ mu \ sum \ limits _ { i } n _ { i }, } where ⟨... ⟩ and ⟨ ⟨... ⟩ ⟩ denote the nearest and next - nearest neighbors, respectively, with two different values for the hopping integral ( t1 and t2 ) and μ is the chemical potential.


ground truth also helps with atmospheric correction. since images from satellites have to pass through the atmosphere, they can get distorted because of absorption in the atmosphere. so ground truth can help fully identify objects in satellite photos.


permissible station content is defined by the fcc as : "... only noncommercial voice information pertaining to traffic and road conditions, traffic hazard and travel advisories, directions, availability of lodging, rest stops and service stations, and descriptions of local points of interest. it is not permissible to identify the commercial name of any business whose service may be available within or outside the coverage area of a travelers'information station. however, to facilitate announcements concerning departures / arrivals and parking areas at air, train, and bus terminals, the trade name identification of carriers is permitted.


in the fields of predictive modelling and probabilistic forecasting, the markov property is considered desirable since it may enable the reasoning and resolution of the problem that otherwise would not be possible to be resolved because of its intractability. such a model is known as a markov model.


in polyhedral combinatorics, the hypersimplex δ d, k { \ displaystyle \ delta _ { d, k } } is a convex polytope that generalizes the simplex. it is determined by two integers d { \ displaystyle d } and k { \ displaystyle k }, and is defined as the convex hull of the d { \ displaystyle d } - dimensional vectors whose coefficients consist of k { \ displaystyle k } ones and d − k { \ displaystyle d - k } zeros. equivalently, δ d, k { \ displaystyle \ delta _ { d, k } } can be obtained by slicing the d { \ displaystyle d } - dimensional unit hypercube d { \ displaystyle ^ { d } } with the hyperplane of equation x 1 + + x d = k { \ displaystyle x _ { 1 } + \ cdots + x _ { d } = k } and, for this reason, it is a ( d − 1 ) { \ displaystyle ( d - 1 ) } - dimensional polytope when 0 < k < d { \ displaystyle 0


must be nowhere near as talented as the devoted and serious method actors that aren't so popular like. " in general, the reversal usually goes : most people believe a and b are both true. b is false.


evaluating the loop condition can have side effects, so an additional evaluation by the if construct should be compensated by replacing the while loop with a do { } while. if the code used do { } while in the first place, the whole guarding process is not needed, as the loop body is guaranteed to execute at least once. this code can be optimized further. for example, strength reduction could remove the two multiplications inside the loop ( 6 * i and a ), and induction variable elimination could then elide i completely. since 6 * i must be in lock step with i itself, there is no need to have both.


for example, the cake may be the 1 - dimensional interval and each piece is an interval ; or, the cake may be a rectangle cut along its longer side so that each piece is a rectangle. every cut - set can be represented by n numbers xi, i = 1,..., n, where xi is the length of the ith piece. we assume that the total length of the cake is 1, so x1 +... + xn = 1.


exact tests that are based on discrete test statistics may be conservative, indicating that the actual rejection rate lies below the nominal significance level α { \ displaystyle \ alpha }. as an example, this is the case for fisher's exact test and its more powerful alternative, boschloo's test. if the test statistic is continuous, it will reach the significance level exactly. parametric tests, such as those used in exact statistics, are exact tests when the parametric assumptions are fully met, but in practice, the use of the term exact ( significance ) test is reserved for non - parametric tests, i. e., tests that do not rest on parametric assumptions.


in number theory, an additive function is an arithmetic function f ( n ) of the positive integer variable n such that whenever a and b are coprime, the function applied to the product ab is the sum of the values of the function applied to a and b :


in the sprague – grundy theory the minimum excluded ordinal is used to determine the nimber of a normal - play impartial game. in such a game, either player has the same moves in each position and the last player to move wins. the nimber is equal to 0 for a game that is lost immediately by the first player, and is equal to the mex of the nimbers of all possible next positions for any other game. for example, in a one - pile version of nim, the game starts with a pile of n stones, and the player to move may take any positive number of stones.


memory controllers such as the intel 945 chipset list the configurations they support : " 256 - mib, 512 - mib, and 1 - gib ddr2 technologies for ×8 and ×16 devices ", " four ranks for all ddr2 devices up to 512 - mibit density ", " eight ranks for 1 - gibit ddr2 devices ". as an example, take an i945 memory controller with four kingston khx6400d2 / 1g memory modules, where each module has a capacity of 1 gib. kingston describes each module as composed of 16 " 64m×8 - bit " chips with each chip having an 8 - bit - wide data bus.


equivalent to bitwise addition without use of a carry bit. | | : concatenation operator. combine the strings on either side of the operator. 0a : a string of a 0 bits.


from the 1920s through the 1970s, typing speed ( along with shorthand speed ) was an important secretarial qualification and typing contests were popular and often publicized by typewriter companies as promotional tools. a less common measure of the speed of a typist, cpm is used to identify the number of characters typed per minute. this is a common measurement for typing programs, or typing tutors, as it can give a more accurate measure of a person's typing speed without having to type for a prolonged period of time.


the numbers less than ( n k ) { \ displaystyle { \ tbinom { n } { k } } } correspond to all k - combinations of { 0, 1,..., n − 1 }. the correspondence does not depend on the size n of the set that the k - combinations are taken from, so it can be interpreted as a map from n to the k - combinations taken from n ; in this view the correspondence is a bijection. the number n corresponding to ( ck,..., c2, c1 ) is given by n = ( c k k ) + + ( c 2 2 ) + ( c 1 1 ) { \ displaystyle n = { \ binom { c _ { k } } { k } } + \ cdots + { \ binom { c _ { 2 } } { 2 } } + { \ binom { c _ { 1 } } { 1 } } }. the fact that a unique sequence corresponds to any non - negative number n was first observed by d. h. lehmer.


in number theory, bertrand's postulate is a theorem stating that for any integer n > 1 { \ displaystyle n > 1 }, there always exists at least one prime number such that n < p < 2 n. { \ displaystyle n


atari's 8 - bit line used a system reset button for this same purpose. debugging nmis have appeared in a number of forms, including the apple macintosh's " programmers'button ", and certain key combinations on sun workstations. with the introduction of windows 2000, microsoft allowed the use of an nmi to cause a system to either break into a debugger, or dump the contents of memory to disk and reboot. debugging nmis have also been used by devices that allow leisure users and gamers to manipulate running programs.


in mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space ( for example, a bilinear form or a multilinear form ) that is zero whenever any pair of arguments is equal. more generally, the vector space may be a module over a commutative ring. the notion of alternatization ( or alternatisation ) is used to derive an alternating multilinear map from any multilinear map with all arguments belonging to the same space.


in military operations, high - value targets such as command centers are frequently protected by layers of defense systems, which may in turn be protected by other systems. in order to reach a target, all of its defenses must be taken down, making it into a secondary target. each target needs a certain amount of resources to be allocated to it in order to perform a successful attack. the optimal set of targets to attack, to obtain the most value for the resources expended, can be modeled as a closure problem.


in syntactic ambiguity, the same sequence of words is interpreted as having different syntactic structures. in contrast, in semantic ambiguity the structure remains the same, but the individual words are interpreted differently. controlled natural languages are often designed to be unambiguous so that they can be parsed into a logical form.


in terms of composition of the differential operator di which takes the partial derivative with respect to xi : d i ∘ d j = d j ∘ d i { \ displaystyle d _ { i } \ circ d _ { j } = d _ { j } \ circ d _ { i } }. from this relation it follows that the ring of differential operators with constant coefficients, generated by the di, is commutative ; but this is only true as operators over a domain of sufficiently differentiable functions. it is easy to check the symmetry as applied to monomials, so that one can take polynomials in the xi as a domain. in fact smooth functions are another valid domain.


category theory deals with abstract objects and morphisms between those objects. in category theory, an automorphism is an endomorphism ( i. e., a morphism from an object to itself ) which is also an isomorphism ( in the categorical sense of the word, meaning there exists a right and left inverse endomorphism ). this is a very abstract definition since, in category theory, morphisms are not necessarily functions and objects are not necessarily sets. in most concrete settings, however, the objects will be sets with some additional structure and the morphisms will be functions preserving that structure.


in mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. specifically, the commutation matrix k ( m, n ) is the nm × mn matrix which, for any m × n matrix a, transforms vec ( a ) into vec ( at ) : k ( m, n ) vec ( a ) = vec ( at ). here vec ( a ) is the mn × 1 column vector obtain by stacking the columns of a on top of one another : vec ( a ) = t { \ displaystyle \ operatorname { vec } ( \ mathbf { a } ) = ^ { \ mathrm { t } } } where a =. in other words, vec ( a ) is the vector obtained by vectorizing a in column - major order. similarly, vec ( at ) is the vector obtaining by vectorizing a in row - major order. in the context of quantum information theory, the commutation matrix is sometimes referred to as the swap matrix or swap operator


the maximum entropy principle is also needed to guarantee the uniqueness and consistency of probability assignments obtained by different methods, statistical mechanics and logical inference in particular. the maximum entropy principle makes explicit our freedom in using different forms of prior data. as a special case, a uniform prior probability density ( laplace's principle of indifference, sometimes called the principle of insufficient reason ), may be adopted.


active format description is occasionally incorrectly referred to as " active format descriptor ". there is no " descriptor " ( descriptor has a specific meaning in iso / iec 13818 - 1, mpeg syntax ). the afd data is carried in the video layer of mpeg, iso / iec 13818 - 2.


in the book the humane interface, jef raskin championed what he termed quasimodes, which are modes that are kept in place only through some constant action on the part of the user ; such modes are also called spring - loaded modes. the term quasimode is a composite of the latin prefix quasi - ( which means almost, to some degree ) and the english word " mode ". modifier keys on the keyboard, such as the shift key, the alt key and the control key, are all examples of a quasimodal interface. the application enters into that mode as long as the user is performing a conscious action, like pressing a key and keeping it pressed while invoking a command.


for example, if one wishes to prove that every girl in the united states ( a ) has brown hair ( b ), one can either try to directly prove a → b { \ displaystyle a \ to b } by checking that all girls in the united states do indeed have brown hair, or try to prove ¬ b → ¬ a { \ displaystyle \ neg b \ to \ neg a } by checking that all girls without brown hair are indeed all outside the us. in particular, if one were to find at least one girl without brown hair within the us, then one would have disproved ¬ b → ¬ a { \ displaystyle \ neg b \ to \ neg a }, and equivalently a → b { \ displaystyle a \ to b }. in general, for any statement where a implies b, not b always implies not a. as a result, proving or disproving either one of these statements automatically proves or disproves the other, as they are logically equivalent to each other.


in the era of the electrical telegraph, its principal users were post offices, railway stations, the more important governmental centers ( ministries ), stock exchanges, very few nationally distributed newspapers, the largest internationally important corporations, and wealthy individuals. despite the fact that telephone devices existed before the invention of the telephone exchange, their success and economical operation would have been impossible on the same schema and structure of the contemporary telegraph, as prior to the invention of the telephone exchange switchboard, early telephones were hardwired to and communicated with only a single other telephone ( such as from an individual's home to the person's business ). a telephone exchange is a telephone system for a small geographic area that provides the switching ( interconnection ) of subscriber lines for calls made between them. telephone exchanges replaced small telephone systems that connected its users with direct lines between each and every subscriber station.


while such sources are still at a developmental stage qkd has been carried out successfully with them. however, as current sources operate at a low efficiency and frequency key rates and transmission distances are limited. another solution is to modify the bb84 protocol, as is done for example in the sarg04 protocol, in which the secure key rate scales as t 3 / 2 { \ displaystyle t ^ { 3 / 2 } }.


it also entails two - way explicit key confirmation, making it an authenticated key agreement with key confirmation ( akc ) protocol. sts was originally presented in 1987 in the context of isdn security ( o'higgins et al. 1987 ), finalized in 1989 and generally presented by whitfield diffie, paul c. van oorschot and michael j. wiener in 1992. the historical context for the protocol is also discussed in diffie ( 1988 ).


the apex api provides services to manage partitions, processes and timing, as well as partition / process communication and error handling. the partitioning environment can be implemented by using a hypervisor to map partitions to virtual machines, but this is not required. the standard is overseen by the aeec apex subcommittee.


in software development, agile practices ( sometimes written " agile " ) include requirements discovery and solutions improvement through the collaborative effort of self - organizing and cross - functional teams with their customer ( s ) / end user ( s ), popularized in the 2001 manifesto for agile software development, these values and principles were derived from and underpin a broad range of software development frameworks, including scrum and kanban. while there is much anecdotal evidence that adopting agile practices and values improves the effectiveness of software professionals, teams and organizations, the empirical evidence is mixed and hard to find.


in set theory, an element of a subset inherits all the attributes contained in the superset. for example, a student is a person. therefore, the set of students is a subset of the set of persons.


it is assumed that g is simple, that is, it does not contain loops or parallel edges. let a be the adjacency matrix of g and let λ i { \ displaystyle \ lambda _ { i } }, i = 1, …, n { \ displaystyle i = 1, \ ldots, n }, be the eigenvalues of a. then the energy of the graph is defined as : e ( g ) = i = 1 n | λ i |. { \ displaystyle e ( g ) = \ sum _ { i = 1 } ^ { n } | \ lambda _ { i } |. }


this necessary distributivity of • over ∨ does not in general entail distributivity of ∧ over ∨, that is, a residuated lattice need not be a distributive lattice. however distributivity of ∧ over ∨ is entailed when • and ∧ are the same operation, a special case of residuated lattices called a heyting algebra. alternative notations for x • y include, x ; y ( relation algebra ), and x⊗y ( linear logic ).


in mathematics, addition and multiplication of real numbers is associative. by contrast, in computer science, the addition and multiplication of floating point numbers is not associative, as rounding errors are introduced when dissimilar - sized values are joined together. to illustrate this, consider a floating point representation with a 4 - bit mantissa : even though most computers compute with 24 or 53 bits of mantissa, this is an important source of rounding error, and approaches such as the kahan summation algorithm are ways to minimise the errors. it can be especially problematic in parallel computing.


in the case of bipartite graphs or multigraphs with maximum degree δ, the optimal number of colors is exactly δ. cole, ost & schirra ( 2001 ) showed that an optimal edge coloring of these graphs can be found in the near - linear time bound o ( m log δ ), where m is the number of edges in the graph ; simpler, but somewhat slower, algorithms are described by cole & hopcroft ( 1982 ) and alon ( 2003 ). the algorithm of alon ( 2003 ) begins by making the input graph regular, without increasing its degree or significantly increasing its size, by merging pairs of vertices that belong to the same side of the bipartition and then adding a small number of additional vertices and edges. then, if the degree is odd, alon finds a single perfect matching in near - linear time, assigns it a color, and removes it from the graph, causing the degree to become even. finally, alon applies an observation of gabow ( 1976 ), that selecting alternating subsets of edges in an euler tour of the graph partitions it into two regular subgraphs, to split the edge coloring problem into two smaller subproblems, and his algorithm solves the two subproblems recursively.


in mathematics, the layer cake representation of a non - negative, real - valued measurable function f { \ displaystyle f } defined on a measure space ( ω, a, μ ) { \ displaystyle ( \ omega, { \ mathcal { a } }, \ mu ) } is the formula f ( x ) = 0 ∞ 1 l ( f, t ) ( x ) d t, { \ displaystyle f ( x ) = \ int _ { 0 } ^ { \ infty } 1 _ { l ( f, t ) } ( x ) \, \ mathrm { d } t, } for all x ∈ ω { \ displaystyle x \ in \ omega }, where 1 e { \ displaystyle 1 _ { e } } denotes the indicator function of a subset e ⊆ ω { \ displaystyle e \ subseteq \ omega } and l ( f, t ) { \ displaystyle l ( f, t ) } denotes the super - level set l ( f, t ) = { y ∈ ω f ( y ) ≥ t }. { \ displaystyle l ( f, t ) = \ { y \ in \ omega \ mid f ( y ) \ geq t \ }. } the layer cake representation follows easily from observing that 1 l ( f, t ) ( x ) = 1 ( t ) { \ displaystyle 1 _ { l ( f, t ) } ( x ) = 1 _ { } ( t ) } and then using the formula f ( x ) = 0 f ( x ) d t. { \ displaystyle f ( x ) = \ int _ { 0 } ^ { f ( x ) } \, \ mathrm { d } t. }


in marketing, customers can be grouped into fuzzy clusters based on their needs, brand choices, psycho - graphic profiles, or other marketing related partitions.


other examples include the functions sin ( x ) x { \ displaystyle { \ frac { \ sin ( x ) } { x } } } and x x. { \ displaystyle x ^ { x }. } liouville's theorem states that elementary antiderivatives, if they exist, are in the same differential field as the function, plus possibly a finite number of applications of the logarithm function.


initially, every element in the matrix is zeros. then for each “ edge ” point in the original space, we can formulate a circle in the parameter space and increase the voting number of the grid cell which the circle passes through. this process is called “ voting ”. after voting, we can find local maxima in the accumulator matrix. the positions of the local maxima are corresponding to the circle centers in the original space.


sometimes the term noncommutative ring is used instead of ring to refer to an unspecified ring which is not necessarily commutative, and hence may be commutative. generally, this is for emphasizing that the studied properties are not restricted to commutative rings, as, in many contexts, ring is used as a shorthand for commutative ring. although some authors do not assume that rings have a multiplicative identity, in this article we make that assumption unless stated otherwise.


such non - subclassable classes restrict reusability, particularly when developers only have access to precompiled binaries and not source code. a non - subclassable class has no subclasses, so it can be easily deduced at compile time that references or pointers to objects of that class are actually referencing instances of that class and not instances of subclasses ( they don't exist ) or instances of superclasses ( upcasting a reference type violates the type system ). because the exact type of the object being referenced is known before execution, early binding ( also called static dispatch ) can be used instead of late binding ( also called dynamic dispatch ), which requires one or more virtual method table lookups depending on whether multiple inheritance or only single inheritance are supported in the programming language that is being used.


in statistics the trimean ( tm ), or tukey's trimean, is a measure of a probability distribution's location defined as a weighted average of the distribution's median and its two quartiles : t m = q 1 + 2 q 2 + q 3 4 { \ displaystyle tm = { \ frac { q _ { 1 } + 2q _ { 2 } + q _ { 3 } } { 4 } } } this is equivalent to the average of the median and the midhinge : t m = 1 2 ( q 2 + q 1 + q 3 2 ) { \ displaystyle tm = { \ frac { 1 } { 2 } } \ left ( q _ { 2 } + { \ frac { q _ { 1 } + q _ { 3 } } { 2 } } \ right ) } the foundations of the trimean were part of arthur bowley's teachings, and later popularized by statistician john tukey in his 1977 book which has given its name to a set of techniques called exploratory data analysis. like the median and the midhinge, but unlike the sample mean, it is a statistically resistant l - estimator with a breakdown point of 25 %. this beneficial property has been described as follows : an advantage of the trimean as a measure of the center ( of a distribution ) is that it combines the median's emphasis on center values with the midhinge's attention to the extremes.


in multivariate analysis, canonical correspondence analysis ( cca ) is an ordination technique that determines axes from the response data as a linear combination of measured predictors. cca is commonly used in ecology in order to extract gradients that drive the composition of ecological communities. cca extends correspondence analysis ( ca ) with regression, in order to incorporate predictor variables.


the term is also used for the goal that is achieved, when such a design has reached the end of the flow and its timing requirements are satisfied. the main steps of the design flow, which may be involved in this process, are logic synthesis, placement, clock - tree synthesis and routing. a single reference clock is often cascaded and synthesized into many different output blocks of clocks resulting into a tree structure. with present technologies all of them need to be timing - aware for a design to properly meet its timing requirements, but with technologies in the range of the micrometre only logic synthesis eda tools had such a prerequisite.


in order to minimize the risk of dangerous failures, safety - related electronic systems have to be developed following the applicable product liability requirements. disregard for, or inadequate application of these standards can lead to not only personal injuries, but also severe legal and economic consequences such as product cancellations or recalls. the iec 61508 standard, generally applicable to electrical / electronic / programmable safety - related products, is only partially adequate for automotive - development requirements.


in some implementations the performance of the physical storage can actually be improved, mainly due to caching. caching however requires the visibility of the data contained within the i / o request and so is limited to in - band and symmetric virtualization software and devices. however these implementations also directly influence the latency of an i / o request ( cache miss ), due to the i / o having to flow through the software or device. assuming the software or device is efficiently designed this impact should be minimal when compared with the latency associated with physical disk accesses.


the necessary conditions are sufficient for optimality if the objective function f { \ displaystyle f } of a maximization problem is a differentiable concave function, the inequality constraints g j { \ displaystyle g _ { j } } are differentiable convex functions, the equality constraints h i { \ displaystyle h _ { i } } are affine functions, and slater's condition holds. similarly, if the objective function f { \ displaystyle f } of a minimization problem is a differentiable convex function, the necessary conditions are also sufficient for optimality. it was shown by martin in 1985 that the broader class of functions in which kkt conditions guarantees global optimality are the so - called type 1 invex functions.


they are not allowed access to any identifying information, however. risk of death or harm information within the record can be shared with authorities without permission when failure to do so would result in death or harm, either to the patient or to others. information cannot be used, however, to initiate or substantiate a charge unless the previous criteria are met ( i. e., information from illicit drug testing cannot be used to bring charges of possession against a patient ). this rule was established in the united states supreme court case jaffe v. redmond.


the first term on the right is the " reduced correlation matrix " and will be equal to the correlation matrix except for its diagonal values which will be less than unity. these diagonal elements of the reduced correlation matrix are called " communalities " ( which represent the fraction of the variance in the observed variable that is accounted for by the factors ) : h a 2 = 1 − ψ a = j ℓ a j ℓ a j { \ displaystyle h _ { a } ^ { 2 } = 1 - \ psi _ { a } = \ sum _ { j } \ ell _ { aj } \ ell _ { aj } } the sample data z a i { \ displaystyle z _ { ai } } will not exactly obey the fundamental equation given above due to sampling errors, inadequacy of the model, etc. the goal of any analysis of the above model is to find the factors f p i { \ displaystyle f _ { pi } } and loadings ℓ a p { \ displaystyle \ ell _ { ap } } which give a " best fit " to the data. in factor analysis, the best fit is defined as the minimum of the mean square error in the off - diagonal residuals of the correlation matrix : ε 2 = a = b 2 { \ displaystyle \ varepsilon ^ { 2 } = \ sum _ { a \ neq b } \ left ^ { 2 } } this is equivalent to minimizing the off - diagonal components of the error covariance which, in the model equations have expected values of zero.


{ \ displaystyle ( s \ cdot f ) ( x ) = f ( xs ). } it is well - defined ( i. e., s ⋅ f { \ displaystyle s \ cdot f } is r - linear ) since ( s ⋅ f ) ( r x ) = f ( r x s ) = r f ( x s ) = r ( s ⋅ f ) ( x ), { \ displaystyle ( s \ cdot f ) ( rx ) = f ( rxs ) = rf ( xs ) = r ( s \ cdot f ) ( x ), } and s ⋅ f { \ displaystyle s \ cdot f } is a ring action since ( s t ⋅ f ) ( x ) = f ( x s t ) = ( t ⋅ f ) ( x s ) = s ⋅ ( t ⋅ f ) ( x ) { \ displaystyle ( st \ cdot f ) ( x ) = f ( xst ) = ( t \ cdot f ) ( xs ) = s \ cdot ( t \ cdot f ) ( x ) }. note : the above verification would " fail " if one used the left r - action in place of the right s - action. in this sense, hom is often said to " use up " the r - action. similarly, if m is a left r - module and n is an ( r, s ) - module, then hom r ( m, n ) { \ displaystyle \ operatorname { hom } _ { r } ( m, n ) } is a right s - module by ( f ⋅ s ) ( x ) = f ( x ) s { \ displaystyle ( f \ cdot s ) ( x ) = f ( x ) s }.


the proof was accepted for publication in the annals of mathematics studies series in 2015, and has been undergoing further review and revision since ; fully - refereed chapters in close to final form are being made public in the process. some state the conjecture as every odd number greater than 7 can be expressed as the sum of three odd primes. this version excludes 7 = 2 + 2 + 3 because this requires the even prime 2. on odd numbers larger than 7 it is slightly stronger as it also excludes sums like 17 = 2 + 2 + 13, which are allowed in the other formulation. helfgott's proof covers both versions of the conjecture. like the other formulation, this one also immediately follows from goldbach's strong conjecture.


in mathematics and computing, the levenberg – marquardt algorithm ( lma or just lm ), also known as the damped least - squares ( dls ) method, is used to solve non - linear least squares problems. these minimization problems arise especially in least squares curve fitting. the lma interpolates between the gauss – newton algorithm ( gna ) and the method of gradient descent.


to support his position that methodological rules generally do not contribute to scientific success, feyerabend analyzed counterexamples to the claim that ( good ) science operates according to the methodological standards invoked by philosophers during feyerabend's time ( namely, inductivism and falsificationism ). starting from episodes in science that are generally regarded as indisputable instances of progress ( e. g. the copernican revolution ), he argued that these episodes violated all common prescriptive rules of science. moreover, he claimed that applying such rules in these historical situations would actually have prevented scientific revolution. his primary case study is galileo's hypothesis that the earth rotates on its axis.


in mathematics education at the primary school level, chunking ( sometimes also called the partial quotients method ) is an elementary approach for solving simple division questions by repeated subtraction. it is also known as the hangman method with the addition of a line separating the divisor, dividend, and partial quotients. it has a counterpart in the grid method for multiplication as well. in general, chunking is more flexible than the traditional method in that the calculation of quotient is less dependent on the place values.


in the case of symmetric - key algorithm cryptosystems, an adversary must not be able to compute any information about a plaintext from its ciphertext. this may be posited as an adversary, given two plaintexts of equal length and their two respective ciphertexts, cannot determine which ciphertext belongs to which plaintext.


in music theory, the term mode or modus is used in a number of distinct senses, depending on context. its most common use may be described as a type of musical scale coupled with a set of characteristic melodic and harmonic behaviors. it is applied to major and minor keys as well as the seven diatonic modes ( including the former as ionian and aeolian ) which are defined by their starting note or tonic. ( olivier messiaen's modes of limited transposition are strictly a scale type. )


however, for certain systems and algorithms, complete pivoting ( or maximal pivoting ) may be required for acceptable accuracy. complete pivoting interchanges both rows and columns in order to use the largest ( by absolute value ) element in the matrix as the pivot. complete pivoting is usually not necessary to ensure numerical stability and, due to the additional cost of searching for the maximal element, the improvement in numerical stability that it provides is typically outweighed by its reduced efficiency for all but the smallest matrices.


in the context of ibm pc compatible and wintel platforms, a 16 - bit application is any software written for ms - dos, os / 2 1. x or early versions of microsoft windows which originally ran on the 16 - bit intel 8088 and intel 80286 microprocessors. such applications used a 20 - bit or 24 - bit segment or selector - offset address representation to extend the range of addressable memory locations beyond what was possible using only 16 - bit addresses. programs containing more than 216 bytes ( 65, 536 bytes ) of instructions and data therefore required special instructions to switch between their 64 - kilobyte segments, increasing the complexity of programming 16 - bit applications.


in practice, cuckoo hashing is about 20 – 30 % slower than linear probing, which is the fastest of the common approaches. the reason is that cuckoo hashing often causes two cache misses per search, to check the two locations where a key might be stored, while linear probing usually causes only one cache miss per search. however, because of its worst case guarantees on search time, cuckoo hashing can still be valuable when real - time response rates are required. one advantage of cuckoo hashing is its link - list free property, which fits gpu processing well.


in mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme. for this reason, formal schemes frequently appear in topics such as deformation theory. but the concept is also used to prove a theorem such as the theorem on formal functions, which is used to deduce theorems of interest for usual schemes.


in the following examples : let f, g, h be predicate letters ; let a, b, c be individual constants ; let x, y, z be variables.


the technique is used frequently when a complete list of all members of the population does not exist and is inappropriate. in some cases, several levels of cluster selection may be applied before the final sample elements are reached. for example, household surveys conducted by the australian bureau of statistics begin by dividing metropolitan regions into'collection districts'and selecting some of these collection districts ( first stage ).


in the 1980s the french thomson company produced a range of 8 - bit computers based on the 6809e cpu. they were released in several variations ( mostly concerning the keyboard or color of the casing ) covering the mo and to series from late 1982 to 1989. while mo and to models are incompatible in software, most of the peripherals and hardware were compatible. these machines were common in france due to the 1980s governmental educational program computing for all ( informatique pour tous ). around 100, 000 mo5 and to7 / 70 computers were ordered and installed in schools.


a document that is stored in such a database, typically would contain more than one normalized data unit and often the relationships between the units as well. if all the data units and the relationships in question are often retrieved together, then this approach optimizes the number of retrieves. it also simplifies how data gets replicated, because now there is a clearly identifiable unit of data whose consistency is self - contained.


in mid - 2002 the cfsci was established as a non - membership 501 ( c ) ( 3 ) corporation under non - profit law. the cfsci board of directors and its staff comprise the cfsci. ( subsequently, the cfsci board amended the bylaws to create a class of " at - large members, " however the at - large members have no voting rights. ) the cfsci was set up as a non - membership organization because the aag felt it would be too cumbersome to include representatives from all local fscs in future voting, due to how many local councils there were and how difficult it was to determine how the various fscs would be represented, given how many different types of fscs there were.


this architecture may replace the traditional pigment based colour filters, leading to enhanced efficiencies, viewing angles and contrast, as well as the highly saturated colour delivered by the use of qds. the display landscape is constantly changing, but with the shift to μ - and mini - leds, quantum dots are looking likely to provide the only solution. μleds are leds with a very small chip size ( down to single digit microns ) which can be directly used as pixels on a display device.


the pseudoinverse facilitates the statement and proof of results in linear algebra. the pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. it can be computed using the singular value decomposition. in the special case where a { \ displaystyle a } is a normal matrix ( for example, a hermitian matrix ), the pseudoinverse a + { \ displaystyle a ^ { + } } annihilates the kernel of a { \ displaystyle a } and acts as a traditional inverse of a { \ displaystyle a } on the subspace orthogonal to the kernel.


in mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. for example, the converse of the relation'child of'is the relation'parent of '. in formal terms, if x { \ displaystyle x } and y { \ displaystyle y } are sets and l ⊆ x × y { \ displaystyle l \ subseteq x \ times y } is a relation from x { \ displaystyle x } to y, { \ displaystyle y, } then l t { \ displaystyle l ^ { \ operatorname { t } } } is the relation defined so that y l t x { \ displaystyle yl ^ { \ operatorname { t } } x } if and only if x l y. { \ displaystyle xly. }


in mathematics, given a category c, a quotient of an object x by an equivalence relation f : r → x × x { \ displaystyle f : r \ to x \ times x } is a coequalizer for the pair of maps r → f x × x → pr i x, i = 1, 2, { \ displaystyle r \ { \ overset { f } { \ to } } \ x \ times x \ { \ overset { \ operatorname { pr } _ { i } } { \ to } } \ x, \ \ i = 1, 2, } where r is an object in c and " f is an equivalence relation " means that, for any object t in c, the image ( which is a set ) of f : r ( t ) = mor ( t, r ) → x ( t ) × x ( t ) { \ displaystyle f : r ( t ) = \ operatorname { mor } ( t, r ) \ to x ( t ) \ times x ( t ) } is an equivalence relation ; that is, a reflexive, symmetric and transitive relation. the basic case in practice is when c is the category of all schemes over some scheme s. but the notion is flexible and one can also take c to be the category of sheaves.


in this case, the expectation value is the probability that the experiment results in " 1 ", and it can be computed as in quantum theory, it is also possible for an operator to have a non - discrete spectrum, such as the position operator x { \ displaystyle x } in quantum mechanics. this operator has a completely continuous spectrum, with eigenvalues and eigenvectors depending on a continuous parameter, x { \ displaystyle x }. specifically, the operator x { \ displaystyle x } acts on a spatial vector | x ⟩ { \ displaystyle | x \ rangle } as x | x ⟩ = x | x ⟩ { \ displaystyle x | x \ rangle = x | x \ rangle }.


there is no generally agreed set of properties that make a particular trust metric better than others, as each metric is designed to serve different purposes, e. g. provides certain classification scheme for trust metrics. two groups of trust metrics can be identified : empirical metrics focusing on supporting the capture of values of trust in a reliable and standardized way ; formal metrics that focus on formalization leading to the ease of manipulation, processing and reasoning about trust. formal metrics can be further classified depending on their properties. trust metrics enable trust modelling and reasoning about trust.


in statistics, the matrix t - distribution ( or matrix variate t - distribution ) is the generalization of the multivariate t - distribution from vectors to matrices. the matrix t - distribution shares the same relationship with the multivariate t - distribution that the matrix normal distribution shares with the multivariate normal distribution. for example, the matrix t - distribution is the compound distribution that results from sampling from a matrix normal distribution having sampled the covariance matrix of the matrix normal from an inverse wishart distribution. in a bayesian analysis of a multivariate linear regression model based on the matrix normal distribution, the matrix t - distribution is the posterior predictive distribution.


this tactic associates a set of constant values to a vis'variable and generates as many test classes as elements are in the set. each test class is characterized by a predicate of the form v a r = v a l { \ displaystyle var = val } where var is the name of the variable and val is one of the values of the set. numeric ranges ( nr ).


( see example below. ) programming languages have the analogous notions of typing and scoping. a compiler or interpreter for the language must recognize which uses of a variable belong together ( refer to the same variable ).


a practical implementation of these strategies with refinements is available at the ace website. the knuth – bendix algorithm also can perform coset enumeration, and unlike the todd – coxeter algorithm, it can sometimes solve the word problem for infinite groups. the main practical difficulties in producing a coset enumerator are that it is difficult or impossible to predict how much memory or time will be needed to complete the process.


in practice, " gray code " almost always refers to a binary - reflected gray code ( brgc ). however, mathematicians have discovered other kinds of gray codes. like brgcs, each consists of a list of words, where each word differs from the next in only one digit ( each word has a hamming distance of 1 from the next word ).


in the following, r { \ displaystyle \ mathbb { r } } represents the real numbers with their usual topology. the subspace topology of the natural numbers, as a subspace of r { \ displaystyle \ mathbb { r } }, is the discrete topology. the rational numbers q { \ displaystyle \ mathbb { q } } considered as a subspace of r { \ displaystyle \ mathbb { r } } do not have the discrete topology ( { 0 } for example is not an open set in q { \ displaystyle \ mathbb { q } } ). if a and b are rational, then the intervals ( a, b ) and are respectively open and closed, but if a and b are irrational, then the set of all rational x with a < x < b is both open and closed.


for example, twitter can use machine learning applications to flag content that does not comply with its terms of service and identify extremist posts encouraging terrorism. facebook and google have developed a content hierarchy system where fact - checkers can identify and de - rank possible disinformation and adjust algorithms accordingly. companies are considering using procedural legal systems to regulate content on their platforms as well.


, a n { \ displaystyle a _ { 1 },..., a _ { n } } are attribute names. its result is defined as the set obtained when the components of the tuples in r { \ displaystyle r } are restricted to the set { a 1,..


in mathematics, the symmetric closure of a binary relation r { \ displaystyle r } on a set x { \ displaystyle x } is the smallest symmetric relation on x { \ displaystyle x } that contains r. { \ displaystyle r. } for example, if x { \ displaystyle x } is a set of airports and x r y { \ displaystyle xry } means " there is a direct flight from airport x { \ displaystyle x } to airport y { \ displaystyle y } ", then the symmetric closure of r { \ displaystyle r } is the relation " there is a direct flight either from x { \ displaystyle x } to y { \ displaystyle y } or from y { \ displaystyle y } to x { \ displaystyle x } ". or, if x { \ displaystyle x } is the set of humans and r { \ displaystyle r } is the relation'parent of ', then the symmetric closure of r { \ displaystyle r } is the relation " x { \ displaystyle x } is a parent or a child of y { \ displaystyle y } ".


in the diagram, mgf is the mask generating function, usually mgf1, hash is the chosen hash function, hlen is the length of the output of the hash function in bytes, k is the length of the rsa modulus n in bytes, m is the message to be padded ( at most k − 2 ⋅ h l e n − 2 { \ displaystyle k - 2 \ cdot \ mathrm { hlen } - 2 } bytes ), l is an optional label to be associated with the message ( the label is the empty string by default and can be used to authenticate data without requiring encryption ), ps is a byte string of k − m l e n − 2 ⋅ h l e n − 2 { \ displaystyle k - \ mathrm { mlen } - 2 \ cdot \ mathrm { hlen } - 2 } null - bytes. ⊕ is an xor - operation.


sometimes, a set comes equipped with a natural hierarchical structure. for example, the set of natural numbers n is equipped with a natural pre - order structure, where n ≤ n ′ { \ displaystyle n \ leq n'} whenever we can find some other number m { \ displaystyle m } so that n + m = n ′ { \ displaystyle n + m = n'}. that is, n ′ { \ displaystyle n'} is bigger than n { \ displaystyle n } only because we can get to n ′ { \ displaystyle n'} from n { \ displaystyle n } using m { \ displaystyle m }.


consistent use of templates. producing a consistent set of models and templates to document the requirements. documenting dependencies.


a block is usually subjected to some type of block processing, such as multidimensional parity checking, associated with it. a block transfer attempt is a coordinated sequence of user and telecommunication system activities undertaken to effect transfer of an individual block from a source user to a destination user. a block transfer attempt begins when the first bit of the block crosses the functional interface between the source user and the telecommunication system. a block transfer attempt ends either in successful block transfer or in block transfer failure.


in numerically challenging applications, sophisticated preconditioners are used, which may lead to variable preconditioning, changing between iterations. even if the preconditioner is symmetric positive - definite on every iteration, the fact that it may change makes the arguments above invalid, and in practical tests leads to a significant slow down of the convergence of the algorithm presented above. using the polak – ribiere formula β k : = r k + 1 t ( z k + 1 − z k ) r k t z k { \ displaystyle \ beta _ { k } : = { \ frac { \ mathbf { r } _ { k + 1 } ^ { \ mathsf { t } } \ left ( \ mathbf { z } _ { k + 1 } - \ mathbf { z } _ { k } \ right ) } { \ mathbf { r } _ { k } ^ { \ mathsf { t } } \ mathbf { z } _ { k } } } } instead of the fletcher – reeves formula β k : = r k + 1 t z k + 1 r k t z k { \ displaystyle \ beta _ { k } : = { \ frac { \ mathbf { r } _ { k + 1 } ^ { \ mathsf { t } } \ mathbf { z } _ { k + 1 } } { \ mathbf { r } _ { k } ^ { \ mathsf { t } } \ mathbf { z } _ { k } } } } may dramatically improve the convergence in this case. this version of the preconditioned conjugate gradient method can be called flexible, as it allows for variable preconditioning.


one of the functions of a static map is to allow a traveler to decide upon a course of action suitable for getting from one point to another. in times of war, the terrain and the troops and weapons deployed upon it can be changed much more rapidly than cartographers can change their maps. a commander with fingerspitzengefuhl would hold such a map in their mind, and adjust it by incorporating any significant information that was received. colonel mehta basti ram was said to have fingerspitzengefuhl.