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Incompilerdesign,static single assignment form(often abbreviated asSSA formor simplySSA) is a type ofintermediate representation(IR) where eachvariableisassignedexactly once. SSA is used in most high-quality optimizing compilers for imperative languages, includingLLVM, theGNU Compiler Collection, and many commercial compilers.
There are efficient algorithms for converting programs into SSA form. To convert to SSA, existing variables in the original IR are split into versions, new variables typically indicated by the original name with a subscript, so that every definition gets its own version. Additional statements that assign to new versions of variables may also need to be introduced at the join point of two control flow paths. Converting from SSA form to machine code is also efficient.
SSA makes numerous analyses needed for optimizations easier to perform, such as determininguse-define chains, because when looking at a use of a variable there is only one place where that variable may have received a value. Most optimizations can be adapted to preserve SSA form, so that one optimization can be performed after another with no additional analysis. The SSA based optimizations are usually more efficient and more powerful than their non-SSA form prior equivalents.
Infunctional languagecompilers, such as those forSchemeandML,continuation-passing style(CPS) is generally used. SSA is formally equivalent to a well-behaved subset of CPS excluding non-local control flow, so optimizations and transformations formulated in terms of one generally apply to the other. Using CPS as the intermediate representation is more natural for higher-order functions and interprocedural analysis. CPS also easily encodescall/cc, whereas SSA does not.[1]
SSA was developed in the 1980s by several researchers atIBM. Kenneth Zadeck, a key member of the team, moved to Brown University as development continued.[2][3]A 1986 paper introduced birthpoints, identity assignments, and variable renaming such that variables had a single static assignment.[4]A subsequent 1987 paper byJeanne Ferranteand Ronald Cytron[5]proved that the renaming done in the previous paper removes all false dependencies for scalars.[3]In 1988, Barry Rosen,Mark N. Wegman, and Kenneth Zadeck replaced the identity assignments with Φ-functions, introduced the name "static single-assignment form", and demonstrated a now-common SSA optimization.[6]The name Φ-function was chosen by Rosen to be a more publishable version of "phony function".[3]Alpern, Wegman, and Zadeck presented another optimization, but using the name "static single assignment".[7]Finally, in 1989, Rosen, Wegman, Zadeck, Cytron, and Ferrante found an efficient means of converting programs to SSA form.[8]
The primary usefulness of SSA comes from how it simultaneously simplifies and improves the results of a variety ofcompiler optimizations, by simplifying the properties of variables. For example, consider this piece of code:
Humans can see that the first assignment is not necessary, and that the value ofybeing used in the third line comes from the second assignment ofy. A program would have to performreaching definition analysisto determine this. But if the program is in SSA form, both of these are immediate:
Compiler optimizationalgorithms that are either enabled or strongly enhanced by the use of SSA include:
Converting ordinary code into SSA form is primarily a matter of replacing the target of each assignment with a new variable, and replacing each use of a variable with the "version" of the variablereachingthat point. For example, consider the followingcontrol-flow graph:
Changing the name on the left hand side of "x←{\displaystyle \leftarrow }x - 3" and changing the following uses ofxto that new name would leave the program unaltered. This can be exploited in SSA by creating two new variables:x1andx2, each of which is assigned only once. Likewise, giving distinguishing subscripts to all the other variables yields:
It is clear which definition each use is referring to, except for one case: both uses ofyin the bottom block could be referring to eithery1ory2, depending on which path the control flow took.
To resolve this, a special statement is inserted in the last block, called aΦ (Phi) function. This statement will generate a new definition ofycalledy3by "choosing" eithery1ory2, depending on the control flow in the past.
Now, the last block can simply usey3, and the correct value will be obtained either way. A Φ function forxis not needed: only one version ofx, namelyx2is reaching this place, so there is no problem (in other words, Φ(x2,x2)=x2).
Given an arbitrary control-flow graph, it can be difficult to tell where to insert Φ functions, and for which variables. This general question has an efficient solution that can be computed using a concept calleddominance frontiers(see below).
Φ functions are not implemented as machine operations on most machines. A compiler can implement a Φ function by inserting "move" operations at the end of every predecessor block. In the example above, the compiler might insert a move fromy1toy3at the end of the middle-left block and a move fromy2toy3at the end of the middle-right block. These move operations might not end up in the final code based on the compiler'sregister allocationprocedure. However, this approach may not work when simultaneous operations are speculatively producing inputs to a Φ function, as can happen onwide-issuemachines. Typically, a wide-issue machine has a selection instruction used in such situations by the compiler to implement the Φ function.
In a control-flow graph, a node A is said tostrictlydominatea different node B if it is impossible to reach B without passing through A first. In other words, if node B is reached, then it can be assumed that A has run. A is said todominateB (or B tobe dominated byA) if either A strictly dominates B or A = B.
A node which transfers control to a node A is called animmediate predecessorof A.
Thedominance frontierof node A is the set of nodes B where Adoes notstrictly dominate B, but does dominate some immediate predecessor of B. These are the points at which multiple control paths merge back together into a single path.
For example, in the following code:
Node 1 strictly dominates 2, 3, and 4 and the immediate predecessors of node 4 are nodes 2 and 3.
Dominance frontiers define the points at which Φ functions are needed. In the above example, when control is passed to node 4, the definition ofresultused depends on whether control was passed from node 2 or 3. Φ functions are not needed for variables defined in a dominator, as there is only one possible definition that can apply.
There is an efficient algorithm for finding dominance frontiers of each node. This algorithm was originally described in "Efficiently Computing Static Single Assignment Form and the Control Graph" by Ron Cytron, Jeanne Ferrante, et al. in 1991.[10]
Keith D. Cooper, Timothy J. Harvey, and Ken Kennedy ofRice Universitydescribe an algorithm in their paper titledA Simple, Fast Dominance Algorithm:[11]
In the code above,idom(b)is theimmediate dominatorof b, the unique node that strictly dominates b but does not strictly dominate any other node that strictly dominates b.
"Minimal" SSA inserts the minimal number of Φ functions required to ensure that each name is assigned a value exactly once and that each reference (use) of a name in the original program can still refer to a unique name. (The latter requirement is needed to ensure that the compiler can write down a name for each operand in each operation.)
However, some of these Φ functions could bedead. For this reason, minimal SSA does not necessarily produce the fewest Φ functions that are needed by a specific procedure. For some types of analysis, these Φ functions are superfluous and can cause the analysis to run less efficiently.
Pruned SSA form is based on a simple observation: Φ functions are only needed for variables that are "live" after the Φ function. (Here, "live" means that the value is used along some path that begins at the Φ function in question.) If a variable is not live, the result of the Φ function cannot be used and the assignment by the Φ function is dead.
Construction of pruned SSA form useslive-variable informationin the Φ function insertion phase to decide whether a given Φ function is needed. If the original variable name isn't live at the Φ function insertion point, the Φ function isn't inserted.
Another possibility is to treat pruning as adead-code eliminationproblem. Then, a Φ function is live only if any use in the input program will be rewritten to it, or if it will be used as an argument in another Φ function. When entering SSA form, each use is rewritten to the nearest definition that dominates it. A Φ function will then be considered live as long as it is the nearest definition that dominates at least one use, or at least one argument of a live Φ.
Semi-pruned SSA form[12]is an attempt to reduce the number of Φ functions without incurring the relatively high cost of computing live-variable information. It is based on the following observation: if a variable is never live upon entry into a basic block, it never needs a Φ function. During SSA construction, Φ functions for any "block-local" variables are omitted.
Computing the set of block-local variables is a simpler and faster procedure than full live-variable analysis, making semi-pruned SSA form more efficient to compute than pruned SSA form. On the other hand, semi-pruned SSA form will contain more Φ functions.
Block arguments are an alternative to Φ functions that is representationally identical but in practice can be more convenient during optimization. Blocks are named and take a list of block arguments, notated as function parameters. When calling a block the block arguments are bound to specified values.MLton,SwiftSIL, and LLVMMLIRuse block arguments.[13]
SSA form is not normally used for direct execution (although it is possible to interpret SSA[14]), and it is frequently used "on top of" another IR with which it remains in direct correspondence. This can be accomplished by "constructing" SSA as a set of functions that map between parts of the existing IR (basic blocks, instructions, operands,etc.) and its SSA counterpart. When the SSA form is no longer needed, these mapping functions may be discarded, leaving only the now-optimized IR.
Performing optimizations on SSA form usually leads to entangled SSA-Webs, meaning there are Φ instructions whose operands do not all have the same root operand. In such casescolor-outalgorithms are used to come out of SSA. Naive algorithms introduce a copy along each predecessor path that caused a source of different root symbol to be put in Φ than the destination of Φ. There are multiple algorithms for coming out of SSA with fewer copies, most use interference graphs or some approximation of it to do copy coalescing.[15]
Extensions to SSA form can be divided into two categories.
Renaming schemeextensions alter the renaming criterion. Recall that SSA form renames each variable when it is assigned a value. Alternative schemes include static single use form (which renames each variable at each statement when it is used) and static single information form (which renames each variable when it is assigned a value, and at the post-dominance frontier).
Feature-specificextensions retain the single assignment property for variables, but incorporate new semantics to model additional features. Some feature-specific extensions model high-level programming language features like arrays, objects and aliased pointers. Other feature-specific extensions model low-level architectural features like speculation and predication.
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The replication crisis, also known as the reproducibility or replicability crisis, refers to the growing number of published scientific results that other researchers have been unable to reproduce or verify. Because the reproducibility of empirical results is an essential part of thescientific method,[2]such failures undermine the credibility of theories that build on them and can call into question substantial parts of scientific knowledge.
The replication crisis is frequently discussed in relation topsychologyandmedicine, where considerable efforts have been undertaken to reinvestigate classic results, to determine whether they are reliable, and if they turn out not to be, the reasons for the failure.[3][4]Data strongly indicates that othernaturalandsocial sciencesare affected as well.[5]
The phrasereplication crisiswas coined in the early 2010s[6]as part of a growing awareness of the problem. Considerations of causes and remedies have given rise to a new scientific discipline,metascience,[7]which uses methods of empirical research to examine empirical research practice.[8]
Considerations about reproducibility can be placed into two categories.Reproducibilityin the narrow sense refers to re-examining and validating the analysis of a given set of data.Replicationrefers to repeating an existing experiment or study using new, independent data with the goal of verifying the original conclusions.
Replicationhas been called "the cornerstone of science".[9][10]Environmental health scientist Stefan Schmidt began a 2009 review with this description of replication:
Replication is one of the central issues in any empirical science. To confirm results or hypotheses by a repetition procedure is at the basis of any scientific conception. A replication experiment to demonstrate that the same findings can be obtained in any other place by any other researcher is conceived as an operationalization of objectivity. It is the proof that the experiment reflects knowledge that can be separated from the specific circumstances (such as time, place, or persons) under which it was gained.[11]
But there is limited consensus on how to definereplicationand potentially related concepts.[12][13][11]A number of types of replication have been identified:
Reproducibilitycan also be distinguished fromreplication, as referring to reproducing the same results using the same data set. Reproducibility of this type is why many researchers make their data available to others for testing.[15]
The replication crisis does not necessarily mean these fields are unscientific.[16][17][18]Rather, this process is part of the scientific process in which old ideas or those that cannot withstand careful scrutiny are pruned,[19][20]although this pruning process is not always effective.[21][22]
A hypothesis is generally considered to be supported when the results match the predicted pattern and that pattern of results is found to bestatistically significant. Results are considered significant whenever the relative frequency of the observed pattern falls below an arbitrarily chosen value (i.e. thesignificance level) when assuming thenull hypothesisis true. This generally answers the question of how unlikely results would be if no difference existed at the level of thestatistical population. If the probability associated with thetest statisticexceeds the chosencritical value, the results are considered statistically significant.[23]The corresponding probability of exceeding the critical value is depicted asp< 0.05, wherep(typically referred to as the "p-value") is the probability level. This should result in 5% of hypotheses that are supported being false positives (an incorrect hypothesis being erroneously found correct), assuming the studies meet all of the statistical assumptions. Some fields use smaller p-values, such asp< 0.01 (1% chance of a false positive) orp< 0.001 (0.1% chance of a false positive). But a smaller chance of a false positive often requires greater sample sizes or a greater chance of afalse negative (a correct hypothesis being erroneously found incorrect). Althoughp-value testing is the most commonly used method, it is not the only method.
Certain terms commonly used in discussions of the replication crisis have technically precise meanings, which are presented here.[1]
In the most common case,null hypothesis testing, there are two hypotheses, anull hypothesisH0{\displaystyle H_{0}}and analternative hypothesisH1{\displaystyle H_{1}}. The null hypothesis is typically of the form "X and Y arestatistically independent". For example, the null hypothesis might be "taking drug X doesnotchange 1-year recovery rate from disease Y", and the alternative hypothesis is that it does change.
As testing for full statistical independence is difficult, the full null hypothesis is often reduced to asimplifiednull hypothesis "the effect size is 0", where "effect size" is a real number that is 0 if thefullnull hypothesis is true, and the larger the effect size is, the more the null hypothesis is false.[24]For example, if X is binary, then the effect size might be defined as the change in the expectation of Y upon a change of X:(effect size)=E[Y|X=1]−E[Y|X=0]{\displaystyle ({\text{effect size}})=\mathbb {E} [Y|X=1]-\mathbb {E} [Y|X=0]}Note that the effect size as defined above might be zero even if X and Y are not independent, such as whenY∼N(0,1+X){\displaystyle Y\sim {\mathcal {N}}(0,1+X)}. Since different definitions of "effect size" capture different ways for X and Y to be dependent, there are many different definitions of effect size.
In practice, effect sizes cannot be directly observed, but must be measured bystatistical estimators. For example, the above definition of effect size is often measured byCohen's destimator. The same effect size might have multiple estimators, as they have tradeoffs betweenefficiency,bias,variance, etc. This further increases the number of possible statistical quantities that can be computed on a single dataset. When an estimator for an effect size is used for statistical testing, it is called atest statistic.
A null hypothesistestis a decision procedure which takes in some data, and outputs eitherH0{\displaystyle H_{0}}orH1{\displaystyle H_{1}}. If it outputsH1{\displaystyle H_{1}}, it is usually stated as "there is a statistically significant effect" or "the null hypothesis is rejected".
Often, the statistical test is a (one-sided)threshold test, which is structured as follows:
A two-sided threshold test is similar, but with two thresholds, such that it outputsH1{\displaystyle H_{1}}if eithert[D]<tthreshold−{\displaystyle t[D]<t_{\text{threshold}}^{-}}ort[D]>tthreshold+{\displaystyle t[D]>t_{\text{threshold}}^{+}}
There are 4 possible outcomes of a null hypothesis test: false negative, true negative, false positive, true positive. A false negative means thatH0{\displaystyle H_{0}}is true, but the test outcome isH1{\displaystyle H_{1}}; a true negative means thatH0{\displaystyle H_{0}}is true, and the test outcome isH0{\displaystyle H_{0}}, etc.
Significance level, false positive rate, or the alpha level, is the probability of finding the alternative to be true when the null hypothesis is true:(significance):=α:=Pr(findH1|H0){\displaystyle ({\text{significance}}):=\alpha :=Pr({\text{find }}H_{1}|H_{0})}For example, when the test is a one-sided threshold test, thenα=PrD∼H0(t[D]>tthreshold){\displaystyle \alpha =Pr_{D\sim H_{0}}(t[D]>t_{\text{threshold}})}whereD∼H0{\displaystyle D\sim H_{0}}means "the data is sampled fromH0{\displaystyle H_{0}}".
Statistical power, true positive rate, is the probability of finding the alternative to be true when the alternative hypothesis is true:(power):=1−β:=Pr(findH1|H1){\displaystyle ({\text{power}}):=1-\beta :=Pr({\text{find }}H_{1}|H_{1})}whereβ{\displaystyle \beta }is also called the false negative rate. For example, when the test is a one-sided threshold test, then1−β=PrD∼H1(t[D]>tthreshold){\displaystyle 1-\beta =Pr_{D\sim H_{1}}(t[D]>t_{\text{threshold}})}.
Given a statistical test and a data setD{\displaystyle D}, the correspondingp-valueis the probability that the test statistic is at least as extreme, conditional onH0{\displaystyle H_{0}}. For example, for a one-sided threshold test,p[D]=PrD′∼H0(t[D′]>t[D]){\displaystyle p[D]=Pr_{D'\sim H_{0}}(t[D']>t[D])}If the null hypothesis is true, then the p-value is distributed uniformly on[0,1]{\displaystyle [0,1]}. Otherwise, it is typically peaked atp=0.0{\displaystyle p=0.0}and roughly exponential, though the precise shape of the p-value distribution depends on what the alternative hypothesis is.[25][26]
Since the p-value is distributed uniformly on[0,1]{\displaystyle [0,1]}conditional on the null hypothesis, one may construct a statistical test with any significance levelα{\displaystyle \alpha }by simply computing the p-value, then outputH1{\displaystyle H_{1}}ifp[D]<α{\displaystyle p[D]<\alpha }. This is usually stated as "the null hypothesis is rejected at significance levelα{\displaystyle \alpha }", or "H1(p<α){\displaystyle H_{1}\;(p<\alpha )}", such as "smoking is correlated with cancer (p < 0.001)".
The beginning of the replication crisis can be traced to a number of events in the early 2010s. Philosopher of science and social epistemologist Felipe Romero identified four events that can be considered precursors to the ongoing crisis:[27]
This series of events generated a great deal of skepticism about the validity of existing research in light of widespread methodological flaws and failures to replicate findings. This led prominent scholars to declare a "crisis of confidence" in psychology and other fields,[42]and the ensuing situation came to be known as the "replication crisis".
Although the beginning of the replication crisis can be traced to the early 2010s, some authors point out that concerns about replicability and research practices in the social sciences had been expressed much earlier. Romero notes that authors voiced concerns about the lack of direct replications in psychological research in the late 1960s and early 1970s.[43][44]He also writes that certain studies in the 1990s were already reporting that journal editors and reviewers are generally biased against publishing replication studies.[45][46]
In the social sciences, the blogData Colada(whose three authors coined the term "p-hacking" in a 2014 paper) has been credited with contributing to the start of the replication crisis.[47][48][49]
University of Virginia professor and cognitive psychologistBarbara A. Spellmanhas written that many criticisms of research practices and concerns about replicability of research are not new.[50]She reports that between the late 1950s and the 1990s, scholars were already expressing concerns about a possible crisis of replication,[51]a suspiciously high rate of positive findings,[52]questionable research practices (QRPs),[53]the effects of publication bias,[54]issues with statistical power,[55][56]and bad standards of reporting.[51]
Spellman also identifies reasons that the reiteration of these criticisms and concerns in recent years led to a full-blown crisis and challenges to the status quo. First, technological improvements facilitated conducting and disseminating replication studies, and analyzing large swaths of literature for systemic problems. Second, the research community's increasing size and diversity made the work of established members more easily scrutinized by other community members unfamiliar with them. According to Spellman, these factors, coupled with increasingly limited resources and misaligned incentives for doing scientific work, led to a crisis in psychology and other fields.[50]
According toAndrew Gelman,[57]the works ofPaul Meehl,Jacob Cohen, andTverskyandKahnemanin the 1960s-70s were early warnings of replication crisis. In discussing the origins of the problem, Kahneman himself noted historical precedents insubliminal perceptionanddissonance reductionreplication failures.[58]
It had been repeatedly pointed out since 1962[55]that most psychological studies have low power (true positive rate), but low power persisted for 50 years, indicating a structural and persistent problem in psychological research.[59][60]
Several factors have combined to put psychology at the center of the conversation.[61][62]Some areas of psychology once considered solid, such associal primingandego depletion,[63]have come under increased scrutiny due to failed replications.[64]Much of the focus has been onsocial psychology,[65]although other areas of psychology such asclinical psychology,[66][67][68]developmental psychology,[69][70][71]andeducational researchhave also been implicated.[72][73][74][75][76]
In August 2015, the first openempirical studyof reproducibility in psychology was published, calledThe Reproducibility Project: Psychology. Coordinated by psychologistBrian Nosek, researchers redid 100 studies in psychological science from three high-ranking psychology journals (Journal of Personality and Social Psychology,Journal of Experimental Psychology: Learning, Memory, and Cognition, andPsychological Science). 97 of the original studies had significant effects, but of those 97, only 36% of the replications yielded significant findings (pvalue below 0.05).[12]The meaneffect sizein the replications was approximately half the magnitude of the effects reported in the original studies. The same paper examined the reproducibility rates and effect sizes by journal and discipline. Study replication rates were 23% for theJournal of Personality and Social Psychology, 48% forJournal of Experimental Psychology: Learning, Memory, and Cognition, and 38% forPsychological Science. Studies in the field of cognitive psychology had a higher replication rate (50%) than studies in the field of social psychology (25%).[77]
Of the 64% of non-replications, only 25% disproved the original result (at statistical significance). The other 49% were inconclusive, neither supporting nor contradicting the original result. This is because many replications were underpowered, with a sample 2.5 times smaller than the original.[78]
A study published in 2018 inNature Human Behaviourreplicated 21 social and behavioral science papers fromNatureandScience,finding that only about 62% could successfully reproduce original results.[79][80]
Similarly, in a study conducted under the auspices of theCenter for Open Science, a team of 186 researchers from 60 different laboratories (representing 36 different nationalities from six different continents) conducted replications of 28 classic and contemporary findings in psychology.[81][82]The study's focus was not only whether the original papers' findings replicated but also the extent to which findings varied as a function of variations in samples and contexts. Overall, 50% of the 28 findings failed to replicate despite massive sample sizes. But if a finding replicated, then it replicated in most samples. If a finding was not replicated, then it failed to replicate with little variation across samples and contexts. This evidence is inconsistent with a proposed explanation that failures to replicate in psychology are likely due to changes in the sample between the original and replication study.[82]
Results of a 2022 study suggest that many earlierbrain–phenotypestudies ("brain-wide association studies" (BWAS)) produced invalid conclusions as the replication of such studies requires samples from thousands of individuals due to smalleffect sizes.[83][84]
Of 49 medical studies from 1990 to 2003 with more than 1000 citations, 92% found that the studied therapies were effective. Of these studies, 16% were contradicted by subsequent studies, 16% had found stronger effects than did subsequent studies, 44% were replicated, and 24% remained largely unchallenged.[85]A 2011 analysis by researchers with pharmaceutical companyBayerfound that, at most, a quarter of Bayer's in-house findings replicated the original results.[86]But the analysis of Bayer's results found that the results that did replicate could often be successfully used for clinical applications.[87]
In a 2012 paper,C. Glenn Begley, a biotech consultant working atAmgen, and Lee Ellis, a medical researcher at the University of Texas, found that only 11% of 53 pre-clinical cancer studies had replications that could confirm conclusions from the original studies.[38]In late 2021, The Reproducibility Project: Cancer Biology examined 53 top papers about cancer published between 2010 and 2012 and showed that among studies that provided sufficient information to be redone, the effect sizes were 85% smaller on average than the original findings.[88][89]A survey of cancer researchers found that half of them had been unable to reproduce a published result.[90]Another report estimated that almost half of randomized controlled trials contained flawed data (based on the analysis of anonymized individual participant data (IPD) from more than 150 trials).[91]
In nutrition science, for most food ingredients, there were studies that found that the ingredient has an effect on cancer risk. Specifically, out of a random sample of 50 ingredients from a cookbook, 80% had articles reporting on their cancer risk. Statistical significance decreased for meta-analyses.[92]
Economicshas lagged behind other social sciences and psychology in its attempts to assess replication rates and increase the number of studies that attempt replication.[13]A 2016 study in the journalSciencereplicated 18experimental studiespublished in two leading economics journals,The American Economic Reviewand theQuarterly Journal of Economics, between 2011 and 2014. It found that about 39% failed to reproduce the original results.[93][94][95]About 20% of studies published inThe American Economic Revieware contradicted by other studies despite relying on the same or similar data sets.[96]A study of empirical findings in theStrategic Management Journalfound that about 30% of 27 retested articles showed statistically insignificant results for previously significant findings, whereas about 4% showed statistically significant results for previously insignificant findings.[97]
A 2019 study inScientific Dataestimated with 95% confidence that of 1,989 articles on water resources and management published in 2017, study results might be reproduced for only 0.6% to 6.8%, largely because the articles did not provide sufficient information to allow for replication.[98]
A 2016 survey byNatureon 1,576 researchers who took a brief online questionnaire on reproducibility found that more than 70% of researchers have tried and failed to reproduce another scientist's experiment results (including 87% ofchemists, 77% ofbiologists, 69% ofphysicistsandengineers, 67% ofmedical researchers, 64% ofearthandenvironmental scientists, and 62% of all others), and more than half have failed to reproduce their own experiments. But fewer than 20% had been contacted by another researcher unable to reproduce their work. The survey found that fewer than 31% of researchers believe that failure to reproduce results means that the original result is probably wrong, although 52% agree that a significant replication crisis exists. Most researchers said they still trust the published literature.[5][99]In 2010, Fanelli (2010)[100]found that 91.5% of psychiatry/psychology studies confirmed the effects they were looking for, and concluded that the odds of this happening (a positive result) was around five times higher than in fields such asastronomyorgeosciences. Fanelli argued that this is because researchers in "softer" sciences have fewer constraints to their conscious and unconscious biases.
Early analysis ofresult-blind peer review, which is less affected by publication bias, has estimated that 61% of result-blind studies in biomedicine and psychology have led tonull results, in contrast to an estimated 5% to 20% in earlier research.[101]
In 2021, a study conducted byUniversity of California, San Diegofound that papers that cannot be replicated are more likely to be cited.[102]Nonreplicable publications are often cited more even after a replication study is published.[103]
There are many proposed causes for the replication crisis.
The replication crisis may be triggered by the "generation of new data and scientific publications at an unprecedented rate" that leads to "desperation to publish or perish" and failure to adhere to good scientific practice.[104]
Predictions of an impending crisis in the quality-control mechanism of science can be traced back several decades.Derek de Solla Price—considered the father ofscientometrics, thequantitative studyof science—predicted in 1963 that science could reach "senility" as a result of its own exponential growth.[105]Some present-day literature seems to vindicate this "overflow" prophecy, lamenting the decay in both attention and quality.[106][107]
HistorianPhilip Mirowskiargues that the decline of scientific quality can be connected to its commodification, especially spurred by major corporations' profit-driven decision to outsource their research to universities andcontract research organizations.[108]
Socialsystems theory, as expounded in the work of German sociologistNiklas Luhmann, inspires a similar diagnosis. This theory holds that each system, such as economy, science, religion, and media, communicates using its own code:trueandfalsefor science,profitandlossfor the economy,newsandno-newsfor the media, and so on.[109][110]According to some sociologists, science'smediatization,[111]commodification,[108]and politicization,[111][112]as a result of the structural coupling among systems, have led to a confusion of the original system codes.
A major cause of low reproducibility is thepublication biasstemming from the fact that statistically non-significant results and seemingly unoriginal replications are rarely published. Only a very small proportion of academic journals in psychology and neurosciences explicitly welcomed submissions of replication studies in their aim and scope or instructions to authors.[113][114]This does not encourage reporting on, or even attempts to perform, replication studies. Among 1,576 researchersNaturesurveyed in 2016, only a minority had ever attempted to publish a replication, and several respondents who had published failed replications noted that editors and reviewers demanded that they play down comparisons with the original studies.[5][99]An analysis of 4,270 empirical studies in 18 business journals from 1970 to 1991 reported that less than 10% of accounting, economics, and finance articles and 5% of management and marketing articles were replication studies.[93][115]Publication bias is augmented by thepressure to publishand the author's ownconfirmation bias,[a]and is an inherent hazard in the field, requiring a certain degree of skepticism on the part of readers.[41]
Publication bias leads to what psychologistRobert Rosenthalcalls the "file drawer effect". The file drawer effect is the idea that as a consequence of the publication bias, a significant number of negative results[b]are not published. According to philosopher of science Felipe Romero, this tends to produce "misleading literature and biased meta-analytic studies",[27]and when publication bias is considered along with the fact that a majority of tested hypotheses might be falsea priori, it is plausible that a considerable proportion of research findings might be false positives, as shown by metascientist John Ioannidis.[1]In turn, a high proportion of false positives in the published literature can explain why many findings are nonreproducible.[27]
Another publication bias is that studies that do not reject the null hypothesis are scrutinized asymmetrically. For example, they are likely to be rejected as being difficult to interpret or having a Type II error. Studies that do reject the null hypothesis are not likely to be rejected for those reasons.[117]
In popular media, there is another element of publication bias: the desire to make research accessible to the public led to oversimplification and exaggeration of findings, creating unrealistic expectations and amplifying the impact of non-replications. In contrast, null results and failures to replicate tend to go unreported. This explanation may apply topower posing's replication crisis.[118]
Even high-impact journals have a significant fraction of mathematical errors in their use of statistics. For example, 11% of statistical results published inNatureandBMJin 2001 are "incongruent", meaning that the reported p-value is mathematically different from what it should be if it were correctly calculated from the reported test statistic. These errors were likely from typesetting, rounding, and transcription errors.[119]
Among 157 neuroscience papers published in five top-ranking journals that attempt to show that two experimental effects are different, 78 erroneously tested instead for whether one effect is significant while the other is not, and 79 correctly tested for whether their difference is significantly different from 0.[120]
The consequences for replicability of the publication bias are exacerbated by academia's "publish or perish" culture. As explained by metascientist Daniele Fanelli, "publish or perish" culture is a sociological aspect of academia whereby scientists work in an environment with very high pressure to have their work published in recognized journals. This is the consequence of the academic work environment being hypercompetitive and of bibliometric parameters (e.g., number of publications) being increasingly used to evaluate scientific careers.[121]According to Fanelli, this pushes scientists to employ a number of strategies aimed at making results "publishable". In the context of publication bias, this can mean adopting behaviors aimed at making results positive or statistically significant, often at the expense of their validity (see QRPs, section 4.3).[121]
According to Center for Open Science founder Brian Nosek and his colleagues, "publish or perish" culture created a situation whereby the goals and values of single scientists (e.g., publishability) are not aligned with the general goals of science (e.g., pursuing scientific truth). This is detrimental to the validity of published findings.[122]
Philosopher Brian D. Earp and psychologist Jim A. C. Everett argue that, although replication is in the best interests of academics and researchers as a group, features of academic psychological culture discourage replication by individual researchers. They argue that performing replications can be time-consuming, and take away resources from projects that reflect the researcher's original thinking. They are harder to publish, largely because they are unoriginal, and even when they can be published they are unlikely to be viewed as major contributions to the field. Replications "bring less recognition and reward, including grant money, to their authors".[123]
In his 1971 bookScientific Knowledge and Its Social Problems, philosopher and historian of scienceJerome R. Ravetzpredicted that science—in its progression from "little" science composed of isolated communities of researchers to "big" science or "techno-science"—would suffer major problems in its internal system of quality control. He recognized that the incentive structure for modern scientists could become dysfunctional, creatingperverse incentivesto publish any findings, however dubious. According to Ravetz, quality in science is maintained only when there is a community of scholars, linked by a set of shared norms and standards, who are willing and able to hold each other accountable.
Certain publishing practices also make it difficult to conduct replications and to monitor the severity of the reproducibility crisis, for articles often come with insufficient descriptions for other scholars to reproduce the study. The Reproducibility Project: Cancer Biology showed that of 193 experiments from 53 top papers about cancer published between 2010 and 2012, only 50 experiments from 23 papers have authors who provided enough information for researchers to redo the studies, sometimes with modifications. None of the 193 papers examined had its experimental protocols fully described and replicating 70% of experiments required asking for key reagents.[88][89]The aforementioned study of empirical findings in theStrategic Management Journalfound that 70% of 88 articles could not be replicated due to a lack of sufficient information for data or procedures.[93][97]Inwater resourcesandmanagement, most of 1,987 articles published in 2017 were not replicable because of a lack of available information shared online.[98]In studies ofevent-related potentials, only two-thirds the information needed to replicate a study were reported in a sample of 150 studies, highlighting that there are substantial gaps in reporting.[124]
By theDuhem-Quine thesis, scientific results are interpreted by both a substantive theory and a theory of instruments. For example, astronomical observations depend both on the theory of astronomical objects and the theory of telescopes. A large amount of non-replicable research might accumulate if there is a bias of the following kind: faced with a null result, a scientist prefers to treat the data as saying the instrument is insufficient; faced with a non-null result, a scientist prefers to accept the instrument as good, and treat the data as saying something about the substantive theory.[125]
Smaldino and McElreath[60]proposed a simple model for thecultural evolutionof scientific practice. Each lab randomly decides to produce novel research or replication research, at different fixed levels of false positive rate, true positive rate, replication rate, and productivity (its "traits"). A lab might use more "effort", making theROC curvemore convex but decreasing productivity. A lab accumulates a score over its lifetime that increases with publications and decreases when another lab fails to replicate its results. At regular intervals, a random lab "dies" and another "reproduces" a child lab with a similar trait as its parent. Labs with higher scores are more likely to reproduce. Under certain parameter settings, the population of labs converge to maximum productivity even at the price of very high false positive rates.
Questionable research practices (QRPs) are intentional behaviors that capitalize on the gray area of acceptable scientific behavior or exploit theresearcher degrees of freedom(researcher DF), which can contribute to the irreproducibility of results by increasing the probability of false positive results.[126][127][41]Researcher DF are seen inhypothesisformulation,design of experiments,data collectionandanalysis, andreporting of research.[127]But in many analyst studies involving several researchers or research teams analyzing the same data, analysts obtain different and sometimes conflicting results, even without incentives to report statistically significant findings across psychology, linguistics, and ecology.[128][129][130]This is because research design and data analysis entail numerous decisions that are not sufficiently constrained by a field’s best practices and statistical methodologies. As a result, researcher DF can lead to situations where some failed replication attempts use a different, yet plausible, research design or statistical analysis; such studies do not necessarily undermine previous findings.[131]Multiverse analysis, a method that makes inferences based on all plausible data-processing pipelines, provides a solution to the problem of analytical flexibility.[132]
Instead, estimating many statistical models (known asdata dredging[127][133][40][c]),selective reportingonly statistically significant findings,[126][127][133][40][d]andHARKing(hypothesizing after results are known) are examples of questionable research practices.[127][133][40][e]In medicine, irreproducible studies have six features in common: investigators not being blinded to the experimental versus the control arms; failure to repeat experiments; lack ofpositiveandnegative controls; failing to report all the data; inappropriate use of statistical tests; and use of reagents that were not appropriatelyvalidated.[135]
QRPs do not include more explicit violations of scientific integrity, such as data falsification.[126][127]Fraudulent research does occur, as in the case of scientific fraud by social psychologistDiederik Stapel,[136][14]cognitive psychologistMarc Hauserand social psychologist Lawrence Sanna,[14]but it appears to be uncommon.[14]
According toIUprofessor Ernest O’Boyle and psychologist Martin Götz, around 50% of researchers surveyed across various studies admitted engaging in HARKing.[137]In a survey of 2,000 psychologists by behavioral scientist Leslie K. John and colleagues, around 94% of psychologists admitted having employed at least one QRP. More specifically, 63% admitted failing to report all of a study's dependent measures, 28% to report all of a study's conditions, and 46% to selectively reporting studies that produced the desired pattern of results. In addition, 56% admitted having collected more data after having inspected already collected data, and 16% to having stopped data collection because the desired result was already visible.[40]According to biotechnology researcher J. Leslie Glick's estimate in 1992, 10% to 20% of research and development studies involved either QRPs or outright fraud.[138]The methodology used to estimate QRPs has been contested, and more recent studies suggested lower prevalence rates on average.[139]
A 2009 meta-analysis found that 2% of scientists across fields admitted falsifying studies at least once and 14% admitted knowing someone who did. Such misconduct was, according to one study, reported more frequently by medical researchers than by others.[140]
According toDeakin Universityprofessor Tom Stanley and colleagues, one plausible reason studies fail to replicate is lowstatistical power. This happens for three reasons. First, a replication study with low power is unlikely to succeed since, by definition, it has a low probability to detect a true effect. Second, if the original study has low power, it will yield biasedeffect sizeestimates. When conductinga priori power analysisfor the replication study, this will result in underestimation of the required sample size. Third, if the original study has low power, the post-study odds of a statistically significant finding reflecting a true effect are quite low. It is therefore likely that a replication attempt of the original study would fail.[15]
Mathematically, the probability of replicating a previous publication that rejected a null hypothesisH0{\displaystyle H_{0}}in favor of an alternativeH1{\displaystyle H_{1}}is(significance)Pr(H0|publication)+(power)Pr(H1|publication)≤(power){\displaystyle ({\text{significance}})Pr(H_{0}|{\text{publication}})+({\text{power}})Pr(H_{1}|{\text{publication}})\leq ({\text{power}})}assuming significance is less than power. Thus, low power implies low probability of replication, regardless of how the previous publication was designed, and regardless of which hypothesis is really true.[78]
Stanley and colleagues estimated the average statistical power of psychological literature by analyzing data from 200meta-analyses. They found that on average, psychology studies have between 33.1% and 36.4% statistical power. These values are quite low compared to the 80% considered adequate statistical power for an experiment. Across the 200 meta-analyses, the median of studies with adequate statistical power was between 7.7% and 9.1%, implying that a positive result would replicate with probability less than 10%, regardless of whether the positive result was a true positive or a false positive.[15]
The statistical power ofneurosciencestudies is quite low. The estimated statistical power offMRIresearch is between .08 and .31,[141]and that of studies ofevent-related potentialswas estimated as .72‒.98 for large effect sizes, .35‒.73 for medium effects, and .10‒.18 for small effects.[124]
In a study published inNature, psychologist Katherine Button and colleagues conducted a similar study with 49 meta-analyses in neuroscience, estimating a median statistical power of 21%.[142]Meta-scientistJohn Ioannidisand colleagues computed an estimate of average power for empirical economic research, finding a median power of 18% based on literature drawing upon 6.700 studies.[143]In light of these results, it is plausible that a major reason for widespread failures to replicate in several scientific fields might be very low statistical power on average.
The same statistical test with the same significance level will have lower statistical power if the effect size is small under the alternative hypothesis. Complex inheritable traits are typically correlated with a large number of genes, each of small effect size, so high power requires a large sample size. In particular, many results from thecandidate geneliterature suffered from small effect sizes and small sample sizes and would not replicate. More data fromgenome-wide association studies(GWAS) come close to solving this problem.[144][145]As a numeric example, most genes associated with schizophrenia risk have low effect size (genotypic relative risk, GRR). A statistical study with 1000 cases and 1000 controls has 0.03% power for a gene with GRR = 1.15, which is already large for schizophrenia. In contrast, the largest GWAS to date has ~100% power for it.[146]
Even when the study replicates, the replication typically have smaller effect size. Underpowered studies have a large effect size bias.[147]
In studies that statistically estimate a regression factor, such as thek{\displaystyle k}inY=kX+b{\displaystyle Y=kX+b}, when the dataset is large, noise tends to cause the regression factor to be underestimated, but when the dataset is small, noise tends to cause the regression factor to be overestimated.[148]
Meta-analyses have their own methodological problems and disputes, which leads to rejection of the meta-analytic method by researchers whose theory is challenged by meta-analysis.[117]
Rosenthal proposed the "fail-safe number" (FSN)[54]to avoid the publication bias against null results. It is defined as follows: Suppose the null hypothesis is true; how many publications would be required to make the current result indistinguishable from the null hypothesis?
Rosenthal's point is that certain effect sizes are large enough, such that even if there is a total publication bias against null results (the "file drawer problem"), the number of unpublished null results would be impossibly large to swamp out the effect size. Thus, the effect size must be statistically significant even after accounting for unpublished null results.
One objection to the FSN is that it is calculated as if unpublished results are unbiased samples from the null hypothesis. But if the file drawer problem is true, then unpublished results would have effect sizes concentrated around 0. Thus fewer unpublished null results would be necessary to swap out the effect size, and so the FSN is an overestimate.[117]
Another problem with meta-analysis is that bad studies are "infectious" in the sense that one bad study might cause the entire meta-analysis to overestimate statistical significance.[78]
Various statistical methods can be applied to make the p-value appear smaller than it really is. This need not be malicious, as moderately flexible data analysis, routine in research, can increase the false-positive rate to above 60%.[41]
For example, if one collects some data, applies several different significance tests to it, and publishes only the one that happens to have a p-value less than 0.05, then the total p-value for "at least one significance test reaches p < 0.05" can be much larger than 0.05, because even if the null hypothesis were true, the probability that one out of many significance tests is extreme is not itself extreme.
Typically, a statistical study has multiple steps, with several choices at each step, such as during data collection, outlier rejection, choice of test statistic, choice of one-tailed or two-tailed test, etc. These choices in the "garden of forking paths" multiply, creating many "researcher degrees of freedom". The effect is similar to the file-drawer problem, as the paths not taken are not published.[149]
Consider a simple illustration. Suppose the null hypothesis is true, and we have 20 possible significance tests to apply to the dataset. Also suppose the outcomes to the significance tests are independent. By definition of "significance", each test has probability 0.05 to pass with significance level 0.05. The probability that at least 1 out of 20 is significant is, by assumption of independence,1−(1−0.05)20=0.64{\displaystyle 1-(1-0.05)^{20}=0.64}.[150]
Another possibility is themultiple comparisons problem. In 2009, it was twice noted that fMRI studies had a suspicious number of positive results with large effect sizes, more than would be expected since the studies have low power (one example[151]had only 13 subjects). It pointed out that over half of the studies would test for correlation between a phenomenon and individual fMRI voxels, and only report on voxels exceeding chosen thresholds.[152]
Optional stopping is a practice where one collects data until some stopping criterion is reached. Though a valid procedure, it is easily misused. The problem is that p-value of an optionally stopped statistical test is larger than it seems. Intuitively, this is because the p-value is supposed to be the sum of all events at least as rare as what is observed. With optional stopping, there are even rarer events that are difficult to account for, i.e. not triggering the optional stopping rule, and collecting even more data before stopping. Neglecting these events leads to a p-value that is too low. In fact, if the null hypothesis is true,anysignificance level can be reached if one is allowed to keep collecting data and stop when the desired p-value (calculated as if one has always been planning to collect exactly this much data) is obtained.[153]For a concrete example of testing for a fair coin, seep-value#optional stopping.
More succinctly, the proper calculation of p-value requires accounting for counterfactuals, that is, what the experimentercouldhave done in reaction to data thatmighthave been. Accounting for what might have been is hard even for honest researchers.[153]One benefit of preregistration is to account for all counterfactuals, allowing the p-value to be calculated correctly.[154]
The problem of early stopping is not just limited to researcher misconduct. There is often pressure to stop early if the cost of collecting data is high. Some animal ethics boards even mandate early stopping if the study obtains a significant result midway.[150]
Such practices are widespread in psychology. In a 2012 survey, 56% of psychologists admitted to early stopping, 46% to only reporting analyses that "worked", and 38% topost hocexclusion, that is, removing some dataafteranalysis was already performed on the data before reanalyzing the remaining data (often on the premise of "outlier removal").[40]
As also reported by Stanley and colleagues, a further reason studies might fail to replicate is highheterogeneityof the to-be-replicated effects. In meta-analysis, "heterogeneity" refers to the variance in research findings that results from there being no single true effect size. Instead, findings in such cases are better seen as a distribution of true effects.[15]Statistical heterogeneity is calculated using the I-squared statistic,[155]defined as "the proportion (or percentage) of observed variation among reported effect sizes that cannot be explained by the calculated standard errors associated with these reported effect sizes".[15]This variation can be due to differences in experimental methods, populations, cohorts, and statistical methods between replication studies. Heterogeneity poses a challenge to studies attempting to replicate previously foundeffect sizes. When heterogeneity is high, subsequent replications have a high probability of finding an effect size radically different than that of the original study.[f]
Importantly, significant levels of heterogeneity are also found in direct/exact replications of a study. Stanley and colleagues discuss this while reporting a study by quantitative behavioral scientist Richard Klein and colleagues, where the authors attempted to replicate 15 psychological effects across 36 different sites in Europe and the U.S. In the study, Klein and colleagues found significant amounts of heterogeneity in 8 out of 16 effects (I-squared = 23% to 91%). Importantly, while the replication sites intentionally differed on a variety of characteristics, such differences could account for very little heterogeneity . According to Stanley and colleagues, this suggested that heterogeneity could have been a genuine characteristic of the phenomena being investigated. For instance, phenomena might be influenced by so-called "hidden moderators" – relevant factors that were previously not understood to be important in the production of a certain effect.
In their analysis of 200 meta-analyses of psychological effects, Stanley and colleagues found a median percent of heterogeneity of I-squared = 74%. According to the authors, this level of heterogeneity can be considered "huge". It is three times larger than the random sampling variance of effect sizes measured in their study. If considered alongsampling error, heterogeneity yields astandard deviationfrom one study to the next even larger than the median effect size of the 200 meta-analyses they investigated.[g]The authors conclude that if replication is defined by a subsequent study finding a sufficiently similar effect size to the original, replication success is not likely even if replications have very large sample sizes. Importantly, this occurs even if replications are direct or exact since heterogeneity nonetheless remains relatively high in these cases.
Within economics, the replication crisis may be also exacerbated because econometric results are fragile:[156]using different but plausibleestimation proceduresordata preprocessingtechniques can lead to conflicting results.[157][158][159]
New York Universityprofessor Jay Van Bavel and colleagues argue that a further reason findings are difficult to replicate is the sensitivity to context of certain psychological effects. On this view, failures to replicate might be explained by contextual differences between the original experiment and the replication, often called "hiddenmoderators".[160]Van Bavel and colleagues tested the influence of context sensitivity by reanalyzing the data of the widely cited Reproducibility Project carried out by the Open Science Collaboration.[12]They re-coded effects according to their sensitivity to contextual factors and then tested the relationship between context sensitivity and replication success in variousregression models.
Context sensitivity was found to negatively correlate with replication success, such that higher ratings of context sensitivity were associated with lower probabilities of replicating an effect.[h]Importantly, context sensitivity significantly correlated with replication success even when adjusting for other factors considered important for reproducing results (e.g., effect size and sample size of original, statistical power of the replication, methodological similarity between original and replication).[i]In light of the results, the authors concluded that attempting a replication in a different time, place or with a different sample can significantly alter an experiment's results. Context sensitivity thus may be a reason certain effects fail to replicate in psychology.[160]
In the framework of Bayesian probability, byBayes' theorem, rejecting the null hypothesis at significance level 5% does not mean that the posterior probability for the alternative hypothesis is 95%, and the posterior probability is also different from the probability of replication.[161][153]Consider a simplified case where there are only two hypotheses. Let the prior probability of the null hypothesis bePr(H0){\displaystyle Pr(H_{0})}, and the alternativePr(H1)=1−Pr(H0){\displaystyle Pr(H_{1})=1-Pr(H_{0})}. For a given statistical study, let its false positive rate (significance level) bePr(findH1|H0){\displaystyle Pr({\text{find }}H_{1}|H_{0})}, and true positive rate (power) bePr(findH1|H1){\displaystyle Pr({\text{find }}H_{1}|H_{1})}. For illustrative purposes, let significance level be 0.05 and power be 0.45 (underpowered).
Now, by Bayes' theorem, conditional on the statistical studying findingH1{\displaystyle H_{1}}to be true, the posterior probability ofH1{\displaystyle H_{1}}actually being true is not1−Pr(findH1|H0)=0.95{\displaystyle 1-Pr({\text{find }}H_{1}|H_{0})=0.95}, but
Pr(H1|findH1)=Pr(findH1|H1)Pr(H1)Pr(findH1|H0)Pr(H0)+Pr(findH1|H1)Pr(H1){\displaystyle Pr(H_{1}|{\text{ find }}H_{1})={\frac {Pr({\text{ find }}H_{1}|H_{1})Pr(H_{1})}{Pr({\text{ find }}H_{1}|H_{0})Pr(H_{0})+Pr({\text{ find }}H_{1}|H_{1})Pr(H_{1})}}}
and the probability of replicating the statistical study isPr(replication|findH1)=Pr(findH1|H1)Pr(H1|findH1)+Pr(findH1|H0)Pr(H0|findH1){\displaystyle Pr({\text{replication}}|{\text{ find }}H_{1})=Pr({\text{find }}H_{1}|H_{1})Pr(H_{1}|{\text{ find }}H_{1})+Pr({\text{find }}H_{1}|H_{0})Pr(H_{0}|{\text{ find }}H_{1})}which is also different fromPr(H1|findH1){\displaystyle Pr(H_{1}|{\text{ find }}H_{1})}. In particular, for a fixed level of significance, the probability of replication increases with power, and prior probability forH1{\displaystyle H_{1}}. If the prior probability forH1{\displaystyle H_{1}}is small, then one would require a high power for replication.
For example, if the prior probability of the null hypothesis isPr(H0)=0.9{\displaystyle Pr(H_{0})=0.9}, and the study found a positive result, then the posterior probability forH1{\displaystyle H_{1}}isPr(H1|findH1)=0.50{\displaystyle Pr(H_{1}|{\text{ find }}H_{1})=0.50}, and the replication probability isPr(replication|findH1)=0.25{\displaystyle Pr({\text{replication}}|{\text{ find }}H_{1})=0.25}.
Some argue that null hypothesis testing is itself inappropriate, especially in "soft sciences" like social psychology.[162][163]
As repeatedly observed by statisticians,[164]in complex systems, such as social psychology, "the null hypothesis is always false", or "everything is correlated". If so, then if the null hypothesis is not rejected, that does not show that the null hypothesis is true, but merely that it was a false negative, typically due to low power.[165]Low power is especially prevalent in subject areas where effect sizes are small and data is expensive to acquire, such as social psychology.[162][166]
Furthermore, when the null hypothesis is rejected, it might not be evidence for the substantial alternative hypothesis. In soft sciences, many hypotheses can predict a correlation between two variables. Thus, evidenceagainstthe null hypothesis "there is no correlation" is no evidenceforone of the many alternative hypotheses that equally well predict "there is a correlation". Fisher developed the NHST for agronomy, where rejecting the null hypothesis is usually good proof of the alternative hypothesis, since there are not many of them. Rejecting the hypothesis "fertilizer does not help" is evidence for "fertilizer helps". But in psychology, there are many alternative hypotheses for every null hypothesis.[166][167]
In particular, when statistical studies on extrasensory perception reject the null hypothesis at extremely low p-value (as in the case ofDaryl Bem), it does not imply the alternative hypothesis "ESP exists". Far more likely is that there was a small (non-ESP) signal in the experiment setup that has been measured precisely.[168]
Paul Meehlnoted that statistical hypothesis testing is used differently in "soft" psychology (personality, social, etc.) from physics. In physics, a theory makes a quantitative prediction and is tested by checking whether the prediction falls within the statistically measured interval. In soft psychology, a theory makes a directional prediction and is tested by checking whether the null hypothesis is rejected in the right direction. Consequently, improved experimental technique makes theories more likely to be falsified in physics but less likely to be falsified in soft psychology, as the null hypothesis is always false since any two variables are correlated by a "crud factor" of about 0.30. The net effect is an accumulation of theories that remainunfalsified, but with no empirical evidence for preferring one over the others.[23][167]
According to philosopherAlexander Bird, a possible reason for the low rates of replicability in certain scientific fields is that a majority of tested hypotheses are falsea priori.[169]On this view, low rates of replicability could be consistent with quality science. Relatedly, the expectation that most findings should replicate would be misguided and, according to Bird, a form of base rate fallacy. Bird's argument works as follows. Assuming an ideal situation of a test of significance, whereby the probability of incorrectly rejecting the null hypothesis is 5% (i.e.Type I error) and the probability of correctly rejecting the null hypothesis is 80% (i.e.Power), in a context where a high proportion of tested hypotheses are false, it is conceivable that the number of false positives would be high compared to those of true positives.[169]For example, in a situation where only 10% of tested hypotheses are actually true, one can calculate that as many as 36% of results will be false positives.[j]
The claim that the falsity of most tested hypotheses can explain low rates of replicability is even more relevant when considering that the average power for statistical tests in certain fields might be much lower than 80%. For example, the proportion of false positives increases to a value between 55.2% and 57.6% when calculated with the estimates of an average power between 34.1% and 36.4% for psychology studies, as provided by Stanley and colleagues in their analysis of 200 meta-analyses in the field.[15]A high proportion of false positives would then result in many research findings being non-replicable.
Bird notes that the claim that a majority of tested hypotheses are falsea prioriin certain scientific fields might be plausible given factors such as the complexity of the phenomena under investigation, the fact that theories are seldom undisputed, the "inferential distance" between theories and hypotheses, and the ease with which hypotheses can be generated. In this respect, the fields Bird takes as examples are clinical medicine, genetic and molecular epidemiology, and social psychology. This situation is radically different in fields where theories have outstanding empirical basis and hypotheses can be easily derived from theories (e.g., experimental physics).[169]
When effects are wrongly stated as relevant in the literature, failure to detect this by replication will lead to the canonization of such false facts.[170]
A 2021 study found that papers in leading general interest, psychology and economicsjournalswith findings that could not be replicated tend to be cited more over time than reproducible research papers, likely because these results are surprising or interesting. The trend is not affected by publication of failed reproductions, after which only 12% of papers that cite the original research will mention the failed replication.[171][172]Further, experts are able to predict which studies will be replicable, leading the authors of the 2021 study, Marta Serra-Garcia andUri Gneezy, to conclude that experts apply lower standards to interesting results when deciding whether to publish them.[172]
Concerns have been expressed within the scientific community that the general public may consider science less credible due to failed replications.[173]Research supporting this concern is sparse, but a nationally representative survey in Germany showed that more than 75% of Germans have not heard of replication failures in science.[174]The study also found that most Germans have positive perceptions of replication efforts: only 18% think that non-replicability shows that science cannot be trusted, while 65% think that replication research shows that science applies quality control, and 80% agree that errors and corrections are part of science.[174]
With the replication crisis of psychology earning attention, Princeton University psychologistSusan Fiskedrew controversy for speaking against critics of psychology for what she called bullying and undermining the science.[175][176][177][178]She called these unidentified "adversaries" names such as "methodological terrorist" and "self-appointed data police", saying that criticism of psychology should be expressed only in private or by contacting the journals.[175]Columbia University statistician and political scientistAndrew Gelmanresponded to Fiske, saying that she had found herself willing to tolerate the "dead paradigm" of faulty statistics and had refused to retract publications even when errors were pointed out.[175]He added that her tenure as editor had been abysmal and that a number of published papers she edited were found to be based on extremely weak statistics; one of Fiske's own published papers had a major statistical error and "impossible" conclusions.[175]
Some researchers inpsychologyindicate that the replication crisis is a foundation for a "credibility revolution", where changes in standards by which psychological science are evaluated may include emphasizing transparency and openness, preregistering research projects, and replicating research with higher standards for evidence to improve the strength of scientific claims.[179]Such changes may diminish the productivity of individual researchers, but this effect could be avoided by data sharing and greater collaboration.[179]A credibility revolution could be good for the research environment.[180]
Focus on the replication crisis has led to renewed efforts in psychology to retest important findings.[41][181]A 2013 special edition of the journalSocial Psychologyfocused on replication studies.[13]
Standardizationas well as (requiring) transparency of the used statistical and experimental methods have been proposed.[182]Carefuldocumentationof the experimental set-up is considered crucial for replicability of experiments and various variables may not be documented and standardized such as animals' diets in animal studies.[183]
A 2016 article byJohn Ioannidiselaborated on "Why Most Clinical Research Is Not Useful".[184]Ioannidis describes what he views as some of the problems and calls for reform, characterizing certain points for medical research to be useful again; one example he makes is the need for medicine to be patient-centered (e.g. in the form of thePatient-Centered Outcomes Research Institute) instead of the current practice to mainly take care of "the needs of physicians, investigators, or sponsors".
Metascience is the use ofscientific methodologyto study science itself. It seeks to increase the quality of scientific research while reducing waste. It is also known as "research on research" and "the science of science", as it usesresearch methodsto study howresearchis done and where improvements can be made. Metascience is concerned with all fields of research and has been called "a bird's eye view of science."[185]In Ioannidis's words, "Science is the best thing that has happened to human beings ... but we can do it better."[186]
Meta-research continues to be conducted to identify the roots of the crisis and to address them. Methods of addressing the crisis includepre-registrationof scientific studies andclinical trialsas well as the founding of organizations such asCONSORTand theEQUATOR Networkthat issue guidelines for methodology and reporting. Efforts continue to reform the system of academic incentives, improve thepeer reviewprocess, reduce themisuse of statistics, combat bias in scientific literature, and increase the overall quality and efficiency of the scientific process.
Some authors have argued that the insufficient communication of experimental methods is a major contributor to the reproducibility crisis and that better reporting of experimental design and statistical analyses would improve the situation. These authors tend to plead for both a broad cultural change in the scientific community of how statistics are considered and a more coercive push from scientific journals and funding bodies.[187]But concerns have been raised about the potential for standards for transparency and replication to be misapplied to qualitative as well as quantitative studies.[188]
Business and management journals that have introduced editorial policies on data accessibility, replication, and transparency include theStrategic Management Journal, theJournal of International Business Studies, and theManagement and Organization Review.[93]
In response to concerns in psychology about publication bias anddata dredging, more than 140 psychology journals have adopted result-blind peer review. In this approach, studies are accepted not on the basis of their findings and after the studies are completed, but before they are conducted and on the basis of themethodological rigorof their experimental designs, and the theoretical justifications for their statistical analysis techniques before data collection or analysis is done.[189]Early analysis of this procedure has estimated that 61% of result-blind studies have led tonull results, in contrast to an estimated 5% to 20% in earlier research.[101]In addition, large-scale collaborations between researchers working in multiple labs in different countries that regularly make their data openly available for different researchers to assess have become much more common in psychology.[190]
Scientific publishing has begun usingpre-registration reportsto address the replication crisis.[191][192]The registered report format requires authors to submit a description of the study methods and analyses prior to data collection. Once the method and analysis plan is vetted through peer-review, publication of the findings is provisionally guaranteed, based on whether the authors follow the proposed protocol. One goal of registered reports is to circumvent thepublication biastoward significant findings that can lead to implementation of questionable research practices. Another is to encourage publication of studies with rigorous methods.
The journalPsychological Sciencehas encouraged thepreregistrationof studies and the reporting of effect sizes and confidence intervals.[193]The editor in chief also noted that the editorial staff will be asking for replication of studies with surprising findings from examinations using small sample sizes before allowing the manuscripts to be published.
It has been suggested that "a simple way to check how often studies have been repeated, and whether or not the original findings are confirmed" is needed.[171]Categorizations and ratings of reproducibility at the study or results level, as well as addition of links to and rating of third-party confirmations, could be conducted by the peer-reviewers, the scientific journal, or by readers in combination with novel digital platforms or tools.
Many publications require ap-valueofp< 0.05 to claimstatistical significance. The paper "Redefine statistical significance",[194]signed by a large number of scientists and mathematicians, proposes that in "fields where the threshold for defining statistical significance for new discoveries isp< 0.05, we propose a change top< 0.005. This simple step would immediately improve the reproducibility of scientific research in many fields." Their rationale is that "a leading cause of non-reproducibility (is that the) statistical standards of evidence for claiming new discoveries in many fields of science are simply too low. Associating 'statistically significant' findings withp< 0.05 results in a high rate of false positives even in the absence of other experimental, procedural and reporting problems."[194]
This call was subsequently criticised by another large group, who argued that "redefining" the threshold would not fix current problems, would lead to some new ones, and that in the end, all thresholds needed to be justified case-by-case instead of following general conventions.[195]A 2022 followup study examined these competing recommendations' practical impact. Despite high citation rates of both proposals, researchers found limited implementation of either the p < 0.005 threshold or the case-by-case justification approach in practice. This revealed what the authors called a "vicious cycle", in which scientists reject recommendations because they are not standard practice, while the recommendations fail to become standard practice because few scientists adopt them.[196]
Although statisticians are unanimous that the use of "p< 0.05" as a standard for significance provides weaker evidence than is generally appreciated, there is a lack of unanimity about what should be done about it. Some have advocated thatBayesian methodsshould replacep-values. This has not happened on a wide scale, partly because it is complicated and partly because many users distrust the specification of prior distributions in the absence of hard data. A simplified version of the Bayesian argument, based on testing a point null hypothesis was suggested by pharmacologistDavid Colquhoun.[197][198]The logical problems of inductive inference were discussed in "The Problem with p-values" (2016).[199]
The hazards of reliance onp-values arises partly because even an observation ofp= 0.001 is not necessarily strong evidence against the null hypothesis.[198]Despite the fact that the likelihood ratio in favor of the alternative hypothesis over the null is close to 100, if the hypothesis was implausible, with a prior probability of a real effect being 0.1, even the observation ofp= 0.001 would have a false positive risk of 8 percent. It would still fail to reach the 5 percent level.
It was recommended that the terms "significant" and "non-significant" should not be used.[198]p-values and confidence intervals should still be specified, but they should be accompanied by an indication of the false-positive risk. It was suggested that the best way to do this is to calculate the prior probability that would be necessary to believe in order to achieve a false positive risk of a certain level, such as 5%. The calculations can be done with various computer software.[198][200]This reverse Bayesian approach, which physicistRobert Matthewssuggested in 2001,[201]is one way to avoid the problem that the prior probability is rarely known.
To improve the quality of replications, largersample sizesthan those used in the original study are often needed.[202]Larger sample sizes are needed because estimates ofeffect sizesin published work are often exaggerated due to publication bias and large sampling variability associated with small sample sizes in an original study.[203][204][205]Further, usingsignificance thresholdsusually leads to inflated effects, because particularly with small sample sizes, only the largest effects will become significant.[163]
One common statistical problem isoverfitting, that is, when researchers fit a regression model over a large number of variables but a small number of data points. For example, a typical fMRI study of emotion, personality, and social cognition has fewer than 100 subjects, but each subject has 10,000 voxels. The study would fit a sparse linear regression model that uses the voxels to predict a variable of interest, such as self-reported stress. But the study would then report on the p-value of the modelon the same datait was fitted to. The standard approach in statistics, where data is split into atraining and a validation set, is resisted because test subjects are expensive to acquire.[152][206]
One possible solution iscross-validation, which allows model validation while also allowing the whole dataset to be used for model-fitting.[207]
In July 2016, theNetherlands Organisation for Scientific Researchmade €3 million available for replication studies. The funding is for replication based on reanalysis of existing data and replication by collecting and analysing new data. Funding is available in the areas of social sciences, health research and healthcare innovation.[208]
In 2013, theLaura and John Arnold Foundationfunded the launch ofThe Center for Open Sciencewith a $5.25 million grant. By 2017, it provided an additional $10 million in funding.[209]It also funded the launch of theMeta-Research Innovation Center at Stanfordat Stanford University run by Ioannidis and medical scientistSteven Goodmanto study ways to improve scientific research.[209]It also provided funding for theAllTrialsinitiative led in part by medical scientistBen Goldacre.[209]
Based on coursework in experimental methods at MIT, Stanford, and theUniversity of Washington, it has been suggested that methods courses in psychology and other fields should emphasize replication attempts rather than original studies.[210][211][212]Such an approach would help students learn scientific methodology and provide numerous independent replications of meaningful scientific findings that would test the replicability of scientific findings. Some have recommended that graduate students should be required to publish a high-quality replication attempt on a topic related to their doctoral research prior to graduation.[213]
There has been a concern that replication attempts have been growing.[214][215][216]As a result, this may lead to lead to research waste.[217]In turn, this has led to a need to systematically track replication attempts. As a result, several databases have been created (e.g.[218][219]). The databases have created a Replication Database that includes psychology and speech-language therapy, among other disciplines, to promote theory-driven research and optimize the use of academic and institutional resource, while promoting trust in science.[220]
Some institutions requireundergraduatestudents to submit a final year thesis that consists of an original piece of research. Daniel Quintana, a psychologist at the University of Oslo in Norway, has recommended that students should be encouraged to perform replication studies in thesis projects, as well as being taught aboutopen science.[221]
Researchers demonstrated a way of semi-automated testing for reproducibility: statements about experimental results were extracted from, as of 2022non-semantic, gene expression cancer research papers and subsequently reproduced viarobot scientist"Eve".[222][223]Problems of this approach include that it may not be feasible for many areas of research and that sufficient experimental data may not get extracted from some or many papers even if available.
PsychologistDaniel Kahnemanargued that, in psychology, the original authors should be involved in the replication effort because the published methods are often too vague.[224][225]Others, such as psychologist Andrew Wilson, disagree, arguing that the original authors should write down the methods in detail.[224]An investigation of replication rates in psychology in 2012 indicated higher success rates of replication in replication studies when there was author overlap with the original authors of a study[226](91.7% successful replication rates in studies with author overlap compared to 64.6% successful replication rates without author overlap).
The replication crisis has led to the formation and development of various large-scale and collaborative communities to pool their resources to address a single question across cultures, countries and disciplines.[227]The focus is on replication, to ensure that the effect generalizes beyond a specific culture and investigate whether the effect is replicable and genuine.[228]This allows interdisciplinary internal reviews, multiple perspectives, uniform protocols across labs, and recruiting larger and more diverse samples.[228]Researchers can collaborate by coordinating data collection or fund data collection by researchers who may not have access to the funds, allowing larger sample sizes and increasing the robustness of the conclusions.
PsychologistMarcus R. Munafòand EpidemiologistGeorge Davey Smithargue, in a piece published byNature, that research should emphasizetriangulation, not just replication, to protect against flawed ideas. They claim that,
replication alone will get us only so far (and) might actually make matters worse ... [Triangulation] is the strategic use of multiple approaches to address one question. Each approach has its own unrelated assumptions, strengths and weaknesses. Results that agree across different methodologies are less likely to beartefacts. ... Maybe one reason replication has captured so much interest is the often-repeated idea thatfalsificationis at the heart of the scientific enterprise. This idea was popularized byKarl Popper's 1950s maxim that theories can never be proved, only falsified. Yet an overemphasis on repeating experiments could provide an unfounded sense of certainty about findings that rely on a single approach. ... philosophers of science have moved on since Popper. Better descriptions of how scientists actually work include what epistemologistPeter Liptoncalled in 1991 "inference to the best explanation".[229]
The dominant scientific and statistical model of causation is the linear model.[230]The linear model assumes that mental variables are stable properties which are independent of each other. In other words, these variables are not expected to influence each other. Instead, the model assumes that the variables will have an independent, linear effect on observable outcomes.[230]
Social scientists Sebastian Wallot and Damian Kelty-Stephen argue that the linear model is not always appropriate.[230]An alternative is the complex system model which assumes that mental variables are interdependent. These variables are not assumed to be stable, rather they will interact and adapt to each specific context.[230]They argue that the complex system model is often more appropriate in psychology, and that the use of the linear model when the complex system model is more appropriate will result in failed replications.[230]
...psychology may be hoping for replications in the very measurements and under the very conditions where a growing body of psychological evidence explicitly discourages predicting replication. Failures to replicate may be plainly baked into the potentially incomplete, but broadly sweeping failure of human behavior to conform to the standard of independen[ce] ...[230]
Replication is fundamental for scientific progress to confirm original findings. However, replication alone is not sufficient to resolve the replication crisis. Replication efforts should seek not just to support or question the original findings, but also to replace them with revised, stronger theories with greater explanatory power. This approach therefore involves pruning existing theories, comparing all the alternative theories, and making replication efforts more generative and engaged in theory-building.[231][232]However, replication alone is not enough, it is important to assess the extent that results generalise across geographical, historical and social contexts is important for several scientific fields, especially practitioners and policy makers to make analyses in order to guide important strategic decisions. Reproducible and replicable findings was the best predictor of generalisability beyond historical and geographical contexts, indicating that for social sciences, results from a certain time period and place can meaningfully drive as to what is universally present in individuals.[233]
Open data, open source software and open source hardware all are critical to enabling reproducibility in the sense of validation of the original data analysis. The use of proprietary software, the lack of the publication of analysis software and the lack of open data prevents the replication of studies. Unless software used in research is open source, reproducing results with different software and hardware configurations is impossible.[234]CERNhas both Open Data and CERN Analysis Preservation projects for storing data, all relevant information, and all software and tools needed to preserve an analysis at the large experiments of theLHC. Aside from all software and data, preserved analysis assets include metadata that enable understanding of the analysis workflow, related software, systematic uncertainties, statistics procedures and meaningful ways to search for the analysis, as well as references to publications and to backup material.[235]CERN software is open source and available for use outside ofparticle physicsand there is some guidance provided to other fields on the broad approaches and strategies used for open science in contemporary particle physics.[236]
Online repositories where data, protocols, and findings can be stored and evaluated by the public seek to improve the integrity and reproducibility of research. Examples of such repositories include theOpen Science Framework,Registry of Research Data Repositories, and Psychfiledrawer.org. Sites like Open Science Framework offer badges for using open science practices in an effort to incentivize scientists. However, there have been concerns that those who are most likely to provide their data and code for analyses are the researchers that are likely the most sophisticated.[237]Ioannidis suggested that "the paradox may arise that the most meticulous and sophisticated and method-savvy and careful researchers may become more susceptible to criticism and reputation attacks by reanalyzers who hunt for errors, no matter how negligible these errors are".[237]
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https://en.wikipedia.org/wiki/Replication_crisis
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Inlogic, a true/falsedecision problemisdecidableif there exists aneffective methodfor deriving the correct answer.Zeroth-order logic(propositional logic) is decidable, whereasfirst-orderandhigher-orderlogic are not.Logical systemsare decidable if membership in their set oflogically validformulas (or theorems) can be effectively determined. Atheory(set of sentencesclosedunderlogical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems areundecidable, that is, it has been proven that no effective method for determining membership (returning a correct answer after finite, though possibly very long, time in all cases) can exist for them.
Eachlogical systemcomes with both asyntactic component, which among other things determines the notion ofprovability, and asemantic component, which determines the notion oflogical validity. The logically valid formulas of a system are sometimes called thetheoremsof the system, especially in the context of first-order logic whereGödel's completeness theoremestablishes the equivalence of semantic and syntactic consequence. In other settings, such aslinear logic, the syntactic consequence (provability) relation may be used to define the theorems of a system.
A logical system is decidable if there is an effective method for determining whether arbitrary formulas are theorems of the logical system. For example,propositional logicis decidable, because thetruth-tablemethod can be used to determine whether an arbitrarypropositional formulais logically valid.
First-order logicis not decidable in general; in particular, the set of logical validities in anysignaturethat includes equality and at least one other predicate with two or more arguments is not decidable.[1]Logical systems extending first-order logic, such assecond-order logicandtype theory, are also undecidable.
The validities ofmonadic predicate calculuswith identity are decidable, however. This system is first-order logic restricted to those signatures that have no function symbols and whose relation symbols other than equality never take more than one argument.
Some logical systems are not adequately represented by the set of theorems alone. (For example,Kleene's logichas no theorems at all.) In such cases, alternative definitions of decidability of a logical system are often used, which ask for an effective method for determining something more general than just validity of formulas; for instance, validity ofsequents, or theconsequence relation{(Г,A) | Г ⊧A} of the logic.
Atheoryis a set of formulas, often assumed to beclosedunderlogical consequence. Decidability for a theory concerns whether there is an effective procedure that decides whether the formula is a member of the theory or not, given an arbitrary formula in the signature of the theory. The problem of decidability arises naturally when a theory is defined as the set of logical consequences of a fixed set of axioms.
There are several basic results about decidability of theories. Every (non-paraconsistent) inconsistent theory is decidable, as every formula in the signature of the theory will be a logical consequence of, and thus a member of, the theory. Everycompleterecursively enumerablefirst-order theory is decidable. An extension of a decidable theory may not be decidable. For example, there are undecidable theories in propositional logic, although the set of validities (the smallest theory) is decidable.
A consistent theory that has the property that every consistent extension is undecidable is said to beessentially undecidable. In fact, every consistent extension will be essentially undecidable. The theory of fields is undecidable but not essentially undecidable.Robinson arithmeticis known to be essentially undecidable, and thus every consistent theory that includes or interprets Robinson arithmetic is also (essentially) undecidable.
Examples of decidable first-order theories include the theory ofreal closed fields, andPresburger arithmetic, while the theory ofgroupsandRobinson arithmeticare examples of undecidable theories.
Some decidable theories include (Monk 1976, p. 234):[2]
Methods used to establish decidability includequantifier elimination,model completeness, and theŁoś-Vaught test.
Some undecidable theories include:[2]
Theinterpretabilitymethod is often used to establish undecidability of theories. If an essentially undecidable theoryTis interpretable in a consistent theoryS, thenSis also essentially undecidable. This is closely related to the concept of amany-one reductionincomputability theory.
A property of a theory or logical system weaker than decidability issemidecidability. A theory is semidecidable if there is a well-defined method whose result, given an arbitrary formula, arrives as positive, if the formula is in the theory; otherwise, may never arrive at all; otherwise, arrives as negative. A logical system is semidecidable if there is a well-defined method for generating a sequence of theorems such that each theorem will eventually be generated. This is different from decidability because in a semidecidable system there may be no effective procedure for checking that a formula isnota theorem.
Every decidable theory or logical system is semidecidable, but in general the converse is not true; a theory is decidable if and only if both it and its complement are semi-decidable. For example, the set of logical validitiesVof first-order logic is semi-decidable, but not decidable. In this case, it is because there is no effective method for determining for an arbitrary formulaAwhetherAis not inV. Similarly, the set of logical consequences of anyrecursively enumerable setof first-order axioms is semidecidable. Many of the examples of undecidable first-order theories given above are of this form.
Decidability should not be confused withcompleteness. For example, the theory ofalgebraically closed fieldsis decidable but incomplete, whereas the set of all true first-order statements about nonnegative integers in the language with + and × is complete but undecidable.
Unfortunately, as a terminological ambiguity, the term "undecidable statement" is sometimes used as a synonym forindependent statement.
As with the concept of adecidable set, the definition of a decidable theory or logical system can be given either in terms ofeffective methodsor in terms ofcomputable functions. These are generally considered equivalent perChurch's thesis. Indeed, the proof that a logical system or theory is undecidable will use the formal definition of computability to show that an appropriate set is not a decidable set, and then invoke Church's thesis to show that the theory or logical system is not decidable by any effective method (Enderton 2001, pp. 206ff.).
Somegameshave been classified as to their decidability:
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https://en.wikipedia.org/wiki/Decidability_(logic)
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Information Processing Language(IPL) is aprogramming languagecreated byAllen Newell,Cliff Shaw, andHerbert A. SimonatRAND Corporationand theCarnegie Institute of Technologyabout 1956. Newell had the job of language specifier-application programmer, Shaw was the system programmer, and Simon had the job of application programmer-user.
The code includes features intended to help with programs that perform simple problem solving actions such as lists,dynamic memory allocation,data types,recursion,functionsas arguments, generators, andcooperative multitasking. IPL invented the concept of list processing, albeit in anassembly-languagestyle.
An IPL computer has:
The data structure of IPL is the list, but lists are more intricate structures than in many languages. A list consists of a singly linked sequence of symbols, as might be expected—plus somedescription lists, which are subsidiary singly linked lists interpreted as alternating attribute names and values. IPL provides primitives to access and mutate attribute value by name. The description lists are given local names (of the form 9–1). So, a list named L1 containing the symbols S4 and S5, and described by associating value V1 to attribute A1 and V2 to A2, would be stored as follows. 0 indicates the end of a list; the cell names 100, 101, etc. are automatically generated internal symbols whose values are irrelevant. These cells can be scattered throughout memory; only L1, which uses a regional name that must be globally known, needs to reside in a specific place.
IPL is anassembly languagefor manipulating lists. It has a few cells which are used as special-purpose registers. H1, for example, is the program counter. The SYMB field of H1 is the name of the current instruction. However, H1 is interpreted as a list; the LINK of H1 is, in modern terms, a pointer to the beginning of the call stack. For example, subroutine calls push the SYMB of H1 onto this stack.
H2 is the free-list. Procedures which need to allocate memory grab cells off of H2; procedures which are finished with memory put it on H2. On entry to a function, the list of parameters is given in H0; on exit, the results should be returned in H0. Many procedures return a boolean result indicating success or failure, which is put in H5. Ten cells, W0-W9, are reserved for public working storage. Procedures are "morally bound" (to quote the CACM article) to save and restore the values of these cells.
There are eight instructions, based on the values of P: subroutine call, push/pop S to H0; push/pop the symbol in S to the list attached to S; copy value to S; conditional branch. In these instructions, S is the target. S is either the value of the SYMB field if Q=0, the symbol in the cell named by SYMB if Q=1, or the symbol in the cell named by the symbol in the cell named by SYMB if Q=2. In all cases but conditional branch, the LINK field of the cell tells which instruction to execute next.
IPL has a library of some 150 basic operations. These include such operations as:
IPL was first utilized to demonstrate that the theorems inPrincipia Mathematicawhich were proven laboriously by hand, byBertrand RussellandAlfred North Whitehead, could in fact beproven by computation. According to Simon's autobiographyModels of My Life, this application was originally developed first by hand simulation, using his children as the computing elements, while writing on and holding up note cards as the registers which contained the state variables of the program.
IPL was used to implement several earlyartificial intelligenceprograms, also by the same authors: theLogic Theorist(1956), theGeneral Problem Solver(1957), and theircomputer chessprogramNSS(1958).
Several versions of IPL were created: IPL-I (never implemented), IPL-II (1957 forJOHNNIAC), IPL-III (existed briefly), IPL-IV, IPL-V (1958, forIBM 650,IBM 704,IBM 7090,Philco model 212, many others. Widely used). IPL-VI was a proposal for an IPL hardware.[1][2][3]
A co-processor “IPL-VC” for the CDC 3600 at Argonne National Libraries was developed which could run IPL-V commands.[4][5]It was used to implement another checker-playing program.[6]This hardware implementation did not improve running times sufficiently to “compete favorably with a language more directly oriented to the structure of present-day machines”.[7]
IPL was soon displaced byLisp, which had much more powerful features, a simpler syntax, and the benefit of automaticgarbage collection.
IPL arguably introduced several programming language features:
Many of these features were generalized, rationalized, and incorporated into Lisp[8]and from there into many other programming languages during the next several decades.
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https://en.wikipedia.org/wiki/Information_Processing_Language
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Agrowing self-organizing map (GSOM)is a growing variant of aself-organizing map(SOM). The GSOM was developed to address the issue of identifying a suitable map size in theSOM. It starts with a minimal number of nodes (usually 4) and grows new nodes on the boundary based on a heuristic. By using the value called Spread Factor (SF), the data analyst has the ability to control the growth of the GSOM.
All the starting nodes of the GSOM are boundary nodes, i.e. each node has the freedom to grow in its own direction at the beginning. (Fig. 1) New Nodes are grown from the boundary nodes. Once a node is selected for growing all its free neighboring positions will be grown new nodes. The figure shows the three possible node growth options for a rectangular GSOM.
The GSOM process is as follows:
The GSOM can be used for many preprocessing tasks inData mining, forNonlinear dimensionality reduction, for approximation of principal curves and manifolds, forclusteringandclassification. It gives often the better representation of the data geometry than the SOM (see the classical benchmark for principal curves on the left).
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https://en.wikipedia.org/wiki/Growing_self-organizing_map
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Inlogic, aconditional quantifieris a kind ofLindström quantifier(orgeneralized quantifier)QAthat, relative to a classical modelA, satisfies some or all of the following conditions ("X" and "Y" range over arbitrary formulas in onefree variable):
(The implication arrow denotes material implication in the metalanguage.) Theminimal conditional logicMis characterized by the first six properties, and stronger conditional logics include some of the other ones. For example, the quantifier ∀A, which can be viewed as set-theoretic inclusion, satisfies all of the above except [symmetry]. Clearly [symmetry] holds for ∃Awhile e.g. [contraposition] fails.
A semantic interpretation of conditional quantifiers involves a relation between sets of subsets of a given structure—i.e. a relation between properties defined on the structure. Some of the details can be found in the articleLindström quantifier.
Conditional quantifiers are meant to capture certain properties concerning conditional reasoning at an abstract level. Generally, it is intended to clarify the role of conditionals in a first-order language as they relate to otherconnectives, such as conjunction or disjunction. While they can cover nested conditionals, the greater complexity of the formula, specifically the greater the number of conditional nesting, the less helpful they are as a methodological tool for understanding conditionals, at least in some sense. Compare this methodological strategy for conditionals with that offirst-degree entailmentlogics.
Serge Lapierre.Conditionals and Quantifiers, inQuantifiers, Logic, and Language, Stanford University, pp. 237–253, 1995.
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https://en.wikipedia.org/wiki/Conditional_quantifier
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Inastrophysics, agravastar(ablend wordof "gravitationalvacuumstar") is an object hypothesized in a 2001 paper byPawel O. MazurandEmil Mottolaas an alternative to theblack holetheory.[1]It has the usual black holemetricoutside of thehorizon, butde Sitter metricinside. On the horizon there is a thin shell ofexotic matter. Thissolutionto theEinstein equationsis stable and has nosingularities.[2]Further theoretical considerations of gravastars include the notion of anestar(a second gravastarnestedwithin the first one).[3][4]
In the original formulation by Mazur and Mottola,[5]a gravastar is composed of three regions, differentiated by the relationship between pressurepand energy densityρ[jargon]. The central region consists offalse vacuumor "dark energy", and in this regionp= −ρ. Surrounding it is a thin shell ofperfect fluidwherep=ρ. On the exterior is true vacuum, wherep=ρ= 0.
The dark-energy-like behavior of the inner region prevents collapse to a singularity, and the presence of the thin shell prevents the formation of anevent horizon, avoiding the infiniteblue shift[jargon]. The inner region has thermodynamically noentropyand may be thought of as a gravitationalBose–Einstein condensate. Severe red-shifting of photons as they climb out of the gravity well would make the fluid shell also seem very cold, almostabsolute zero.
In addition to the original thin-shell formulation, gravastars with continuous pressure have been proposed. These objects must containanisotropicstress.[6]
Externally, a gravastar appears similar to a black hole: it is visible by the high-energy radiation it emits while consuming matter, and by theHawking radiationit creates.[citation needed]Astronomers search the sky forX-raysemitted by infalling matter to detect black holes. A gravastar would produce an identical signature. It is also possible, if the thin shell is transparent to radiation, that gravastars may be distinguished from ordinary black holes by differentgravitational lensingproperties, as photon like particles' paths[jargon]may pass through.[7]
Mazur and Mottola suggest that the violent creation of a gravastar might be an explanation for the origin of ouruniverseand many other universes because all the matter from a collapsing star would implode "through" the central hole and explode into a new dimension and expand forever, which would be consistent with the current theories regarding theBig Bang.[8]This "new dimension" exerts an outward pressure on the Bose-Einstein condensate layer and prevents it from collapsing further.
Gravastars also could provide a mechanism for describing howdark energyaccelerates theexpansion of the universe. One possible hypothesis uses Hawking radiation as a means to exchange energy between the "parent" universe and the "child" universe, and so cause the rate of expansion to accelerate, but this area is under much speculation.[citation needed]
Gravastar formation may provide an alternative explanation for sudden and intensegamma-ray burststhroughout space.[citation needed]
LIGO's observations of gravitational waves from colliding objects have been found either to not be consistent with the gravastar concept,[9][10][11]or to be indistinguishable from ordinary black holes.[12][13]
By taking quantum physics into account, the gravastar hypothesis attempts to resolve contradictions caused by conventionalblack holetheories.[14]
In a gravastar, the event horizon is not present. The layer of positive-pressure fluid would lie just outside the "event horizon", being prevented from complete collapse by the inner false vacuum.[2]Due to the absence of an event horizon, the time coordinate of the exterior vacuum geometry is everywhere valid.
In 2007, theoretical work indicated that under certain conditions, gravastars as well as other alternative black hole models are not stable when they rotate.[15]Theoretical work has also shown that certain rotating gravastars are stable assuming certain angular velocities, shell thicknesses, and compactnesses. It is also possible that some gravastars which are mathematically unstable may be physically stable over cosmological timescales.[16]Theoretical support for the feasibility of gravastars does not exclude the existence of black holes as shown in other theoretical studies.[17]
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https://en.wikipedia.org/wiki/Gravastar
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Inphysics,chemistry, andmaterials science,percolation(fromLatinpercolare'to filter, trickle through') refers to the movement andfilteringof fluids through porous materials. It is described byDarcy's law. Broader applications have since been developed that cover connectivity of many systems modeled as lattices or graphs, analogous to connectivity of lattice components in the filtration problem that modulates capacity for percolation.
During the last decades,percolation theory, the mathematical study ofpercolation, has brought new understanding and techniques to a broad range of topics in physics, materials science,complex networks,epidemiology, and other fields. For example, ingeology, percolation refers to filtration of water through soil and permeable rocks. The water flows torechargethegroundwaterin thewater tableandaquifers. In places whereinfiltration basinsorseptic drain fieldsare planned to dispose of substantial amounts of water, apercolation testis needed beforehand to determine whether the intended structure is likely to succeed or fail.
In two dimensional square lattice percolation is defined as follows. A site is "occupied" with
probability p or "empty" (in which case its edges are removed) with probability 1 – p; the
corresponding problem is called site percolation, see Fig. 2.
Percolation typically exhibitsuniversality.Statistical physicsconcepts such as scaling theory,renormalization,phase transition,critical phenomenaandfractalsare used to characterize percolation properties.Combinatoricsis commonly employed to studypercolation thresholds.
Due to the complexity involved in obtaining exact results from analytical models of percolation, computer simulations are typically used. The current fastest algorithm for percolation was published in 2000 byMark Newmanand Robert Ziff.[1]
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https://en.wikipedia.org/wiki/Percolation
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Stigler's lawconcerns the supposed tendency ofeponymousexpressions for scientific discoveries to honor people other than their respective originators.
Examples include:
Since that definition predated Boyce and Codd's own definition by some three years, it seems to me that BCNF ought by rights to be calledHeathnormal form. But it isn't.[12]
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https://en.wikipedia.org/wiki/List_of_examples_of_Stigler%27s_law
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Sibilants(fromLatin:sibilans'hissing') arefricativeconsonants of higheramplitudeandpitch, made bydirectinga stream of air with the tongue towards theteeth.[1]Examples of sibilants are the consonants at the beginning of theEnglishwordssip,zip,ship, andgenre. The symbols in theInternational Phonetic Alphabetused to denote the sibilant sounds in these words are, respectively,[s][z][ʃ][ʒ]. Sibilants have a characteristically intense sound, which accounts for theirparalinguisticuse in getting one's attention (e.g. calling someone using "psst!" or quieting someone using "shhhh!").
In thealveolarhissingsibilants[s]and[z], the back of the tongue forms a narrow channel (isgrooved) to focus the stream of air more intensely, resulting in a high pitch. With thehushingsibilants (occasionally termedshibilants), such as English[ʃ],[tʃ],[ʒ], and[dʒ], the tongue is flatter, and the resulting pitch lower.[2][3]
A broader category isstridents, which include more fricatives than sibilants such asuvulars. Sibilants are a higher pitched subset of the stridents. The English sibilants are:
while the English stridents are:
as/f/and/v/are stridents but not sibilants because they are lower in pitch.[4][5]
Some linguistics use the terms "stridents" and "sibilants" interchangeably to refer to the greateramplitudeandpitchcompared to other fricatives.[6]
"Stridency" refers to theperceptualintensityof the sound of a sibilant consonant, orobstacle fricativesoraffricates, which refers to the critical role of the teeth in producing the sound as an obstacle to the airstream. Non-sibilant fricatives and affricates produce their characteristic sound directly with the tongue or lips etc. and the place of contact in the mouth, without secondary involvement of the teeth.[citation needed]
The characteristic intensity of sibilants means that small variations in tongue shape and position are perceivable, with the result that there are many sibilant types that contrast in various languages.
Sibilants are louder than their non-sibilant counterparts, and most of their acoustic energy occurs at higher frequencies than non-sibilant fricatives—usually around 8,000 Hz.[7]
All sibilants arecoronal consonants(made with the tip or front part of the tongue). However, there is a great deal of variety among sibilants as to tongue shape, point of contact on the tongue, and point of contact on the upper side of the mouth.
The following variables affect sibilant sound quality, and, along with their possible values, are ordered from sharpest (highest-pitched) to dullest (lowest-pitched):
Generally, the values of the different variables co-occur so as to produce an overall sharper or duller sound. For example, a laminal denti-alveolar grooved sibilant occurs inPolish, and a subapical palatal retroflex sibilant occurs inToda.
The main distinction is the shape of the tongue. Most sibilants have agrooverunning down the centerline of the tongue that helps focus the airstream, but it is not known how widespread this is. In addition, the following tongue shapes are described, from sharpest and highest-pitched to dullest and lowest-pitched:
The latter three post-alveolar types of sounds are often known as "hushing" sounds because of their quality, as opposed to the "hissing" alveolar sounds. The alveolar sounds in fact occur in several varieties, in addition to the normal sound of Englishs:
Speaking non-technically, the retroflex consonant[ʂ]sounds somewhat like a mixture between the regular English[ʃ]of "ship" and a strong American "r"; while the alveolo-palatal consonant[ɕ]sounds somewhat like a mixture of English[ʃ]of "ship" and the[sj]in the middle of "miss you".
Sibilants can be made at anycoronalarticulation[citation needed], i.e. the tongue can contact the upper side of the mouth anywhere from the upper teeth (dental) to thehard palate(palatal), with the in-between articulations beingdenti-alveolar,alveolarandpostalveolar.
The tongue can contact the upper side of the mouth with the very tip of the tongue (anapicalarticulation, e.g.[ʃ̺]); with the surface just behind the tip, called thebladeof the tongue (alaminalarticulation, e.g.[ʃ̻]); or with the underside of the tip (asubapicalarticulation). Apical and subapical articulations are alwaystongue-up, with the tip of the tongue above the teeth, while laminal articulations can be either tongue-up ortongue-down, with the tip of the tongue behind the lower teeth. This distinction is particularly important forretroflexsibilants, because all three varieties can occur, with noticeably different sound qualities.
For tongue-down laminal articulations, an additional distinction can be made depending on where exactly behind the lower teeth the tongue tip is placed. A little ways back from the lower teeth is a hollow area (or pit) in the lower surface of the mouth. When the tongue tip rests in this hollow area, there is an empty space below the tongue (asublingual cavity), which results in a relatively duller sound. When the tip of the tongue rests against the lower teeth, there is no sublingual cavity, resulting in a sharper sound. Usually, the position of the tip of the tongue correlates with the grooved vs. hushing tongue shape so as to maximize the differences. However, the palato-alveolar sibilants in theNorthwest Caucasian languagessuch asUbykhare an exception. These sounds have the tongue tip resting directly against the lower teeth, which gives the sounds a quality that Catford describes as "hissing-hushing". Ladefoged and Maddieson[1]term this a "closedlaminal postalveolar" articulation, and transcribe them (following Catford) as[ŝ,ẑ], although this is not an IPA notation.
The following table shows the types of sibilant fricatives defined in theInternational Phonetic Alphabet:
Diacritics can be used for finer detail. For example, apical and laminal alveolars can be specified as[s̺]vs[s̻]; adental(or more likelydenti-alveolar) sibilant as[s̪]; a palatalized alveolar as[sʲ]; and a generic "retracted sibilant" as[s̠], a transcription frequently used for the sharper-quality types of retroflex consonants (e.g. the laminal "flat" type and the "apico-alveolar" type). There is no diacritic to denote the laminal "closed" articulation of palato-alveolars in theNorthwest Caucasian languages, but they are sometimes provisionally transcribed as[ŝẑ].
The attested possibilities, with exemplar languages, are as follows. Note that the IPA diacritics are simplified; some articulations would require two diacritics to be fully specified, but only one is used in order to keep the results legible without the need forOpenTypeIPA fonts. Also,Ladefogedhas resurrected an obsolete IPA symbol, the under dot, to indicateapical postalveolar(normally included in the category ofretroflex consonants), and that notation is used here. (Note that the notations̠,ṣis sometimes reversed; either may also be called 'retroflex' and writtenʂ.)
^1⟨ŝẑ⟩is an ad-hoc transcription. The old IPA letters⟨ʆʓ⟩are also available.
^2These sounds are usually just transcribed⟨ʂʐ⟩. Apical postalveolar and subapical palatal sibilants do not contrast in any language, but if necessary, apical postalveolars can be transcribed with an apical diacritic, as⟨s̠̺z̠̺⟩or⟨ʂ̺ʐ̺⟩. Ladefoged resurrects the old retroflex sub-dot for apical retroflexes,⟨ṣẓ⟩Also seen in the literature on e.g. Hindi and Norwegian is⟨ᶘᶚ⟩– the domed articulation of[ʃʒ]precludes a subapical realization.
Whistled sibilants occur phonemically in several southern Bantu languages, the best known beingShona. However, they also occur in speech pathology and may be caused by dental prostheses or orthodontics.
The whistled sibilants of Shona have been variously described—aslabializedbut not velarized, as retroflex, etc., but none of these features are required for the sounds.[10]Using theExtended IPA, Shonasvandzvmay be transcribed⟨s͎⟩and⟨z͎⟩. Other transcriptions seen include purely labialized⟨s̫⟩and⟨z̫⟩(Ladefoged and Maddieson 1996) and labially co-articulated⟨sᶲ⟩and⟨zᵝ⟩(or⟨s͡ɸ⟩and⟨z͜β⟩). In the otherwise IPA transcription of Shona in Doke (1967), the whistled sibilants are transcribed with the non-IPA letters⟨ȿɀ⟩and⟨tȿdɀ⟩.
Besides Shona, whistled sibilants have been reported as phonemes inKalanga,Tsonga,Changana,Tswa—all of which are Southern African languages—andTabasaran. The articulation of whistled sibilants may differ between languages. In Shona, the lips arecompressedthroughout, and the sibilant may be followed by normal labialization upon release. (That is, there is a contrast amongs, sw, ȿ, ȿw.) In Tsonga, the whistling effect is weak; the lips are narrowed but also the tongue isretroflex. Tswa may be similar. In Changana, the lips are rounded (protruded), but so is /s/ in the sequence /usu/, so there is evidently some distinct phonetic phenomenon occurring here that has yet to be formally identified and described.[11]
Not including differences inmanner of articulationorsecondary articulation, some languages have as many as four different types of sibilants. For example,Northern QiangandSouthern Qianghave a four-way distinction among sibilant affricates/ts//tʂ//tʃ//tɕ/, with one for each of the four tongue shapes.[citation needed]Todaalso has a four-way sibilant distinction, with one alveolar, one palato-alveolar, and two retroflex (apical postalveolar and subapical palatal).[citation needed]
The now-extinctUbykh languagewas particularly complex, with a total of 27 sibilant consonants. Not only all four tongue shapes were represented (with the palato-alveolar appearing in the laminal "closed" variation) but also both the palato-alveolars and alveolo-palatals could additionally appearlabialized. Besides, there was a five-way manner distinction among voiceless and voiced fricatives, voiceless and voiced affricates, andejectiveaffricates. (The three labialized palato-alveolar affricates were missing, which is why the total was 27, not 30.)[citation needed]The Bzyp dialect of the relatedAbkhaz languagealso has a similar inventory.[citation needed]
Some languages have four types whenpalatalizationis considered.Polishis one example, with both palatalized and non-palatalized laminal denti-alveolars, laminal postalveolar (or "flat retroflex"), and alveolo-palatal ([s̪z̪][s̪ʲz̪ʲ][s̠z̠][ɕʑ]).[citation needed]Russianhas the same surface contrasts, but the alveolo-palatals are arguably not phonemic. They occur only geminate, and the retroflex consonants never occur geminate, which suggests that both are allophones of the same phoneme.[citation needed]
Somewhat more common are languages with three sibilant types, including one hissing and two hushing. As with Polish and Russian, the two hushing types are usually postalveolar and alveolo-palatal since these are the two most distinct from each other.Mandarin Chineseis an example of such a language.[citation needed]However, other possibilities exist.Serbo-Croatianhas alveolar, flat postalveolar and alveolo-palatal affricates whereasBasquehas palato-alveolar and laminal and apical alveolar (apico-alveolar) fricatives and affricates (late Medieval peninsularSpanishandPortuguesehad the same distinctions among fricatives).
Many languages, such asEnglishorArabic, have two sibilant types, one hissing and one hushing. A wide variety of languages across the world have this pattern. Perhaps most common is the pattern, as in English and Arabic, with alveolar and palato-alveolar sibilants. Modern northern peninsularSpanishhas a singleapico-alveolarsibilant fricative[s̠], as well as a single palato-alveolar sibilant affricate[tʃ]. However, there are also languages with alveolar and apical retroflex sibilants (such as StandardVietnamese) and with alveolar and alveolo-palatal postalveolars (e.g. alveolar and laminal palatalized[ʃʒtʃdʒ]i.e.[ʃʲʒʲtʃʲdʒʲ]inCatalanandBrazilian Portuguese, the latter probably through Amerindian influence,[12]and alveolar and dorsal i.e.[ɕʑtɕdʑ]proper inJapanese).[13]
Only a few languages with sibilants lack the hissing type.Middle Vietnameseis normally reconstructed with two sibilant fricatives, both hushing (one retroflex, one alveolo-palatal). Some languages have only a single hushing sibilant and no hissing sibilant. That occurs in southern Peninsular Spanish dialects of the "ceceo" type, which have replaced the former hissing fricative with[θ], leaving only[tʃ].
Languages with no sibilants are fairly rare. Most have no fricatives at all or only the fricative/h/. Examples include mostAustralian languages, andRotokas, and what is generally reconstructed forProto-Bantu. Languages with fricatives but no sibilants, however, do occur, such asUkueinNigeria, which has only the fricatives/f,v,h/. Also, almost all EasternPolynesian languageshave no sibilants but do have the fricatives/v/and/or/f/:Māori,Hawaiian,Tahitian,Rapa Nui, mostCook Islands Māoridialects,Marquesan, andTuamotuan.
Tamilonly has the sibilant/ʂ/and fricative/f/in loanwords, and they are frequently replaced by native sounds. The sibilants[s,ɕ]exist as allophones of/t͡ɕ/and the fricative[h]as an allophone of/k/.
Authors includingChomskyandHallegroup[f]and[v]as sibilants. However, they do not have the grooved articulation and high frequencies of other sibilants, and most phoneticians[1]continue to group them together withbilabial[ɸ],[β]and (inter)dental[θ],[ð]as non-sibilantanteriorfricatives. For a grouping of sibilants and[f,v], the termstridentis more common. Some researchers judge[f]to be non-strident in English, based on measurements of its comparative amplitude, but to be strident in other languages (for example, in the African languageEwe, where it contrasts with non-strident[ɸ]).
The nature ofsibilantsas so-called 'obstacle fricatives' is complicated – there is a continuum of possibilities relating to the angle at which the jet of air may strike an obstacle. The grooving often considered necessary for classification as asibilanthas been observed in ultrasound studies of the tongue for the supposedlynon-sibilantvoiceless alveolar fricative[θ̠]of English.[14]
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Inmathematics, alinear approximationis an approximation of a generalfunctionusing alinear function(more precisely, anaffine function). They are widely used in the method offinite differencesto produce first order methods for solving or approximating solutions to equations.
Given a twice continuously differentiable functionf{\displaystyle f}of onerealvariable,Taylor's theoremfor the casen=1{\displaystyle n=1}states thatf(x)=f(a)+f′(a)(x−a)+R2{\displaystyle f(x)=f(a)+f'(a)(x-a)+R_{2}}whereR2{\displaystyle R_{2}}is the remainder term. The linear approximation is obtained by dropping the remainder:f(x)≈f(a)+f′(a)(x−a).{\displaystyle f(x)\approx f(a)+f'(a)(x-a).}
This is a good approximation whenx{\displaystyle x}is close enough toa{\displaystyle a};since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for thetangent lineto the graph off{\displaystyle f}at(a,f(a)){\displaystyle (a,f(a))}. For this reason, this process is also called thetangent line approximation. Linear approximations in this case are further improved when thesecond derivativeof a,f″(a){\displaystyle f''(a)}, is sufficiently small (close to zero) (i.e., at or near aninflection point).
Iff{\displaystyle f}isconcave downin the interval betweenx{\displaystyle x}anda{\displaystyle a}, the approximation will be an overestimate (since the derivative is decreasing in that interval). Iff{\displaystyle f}isconcave up, the approximation will be an underestimate.[1]
Linear approximations forvectorfunctions of a vector variable are obtained in the same way, with the derivative at a point replaced by theJacobianmatrix. For example, given a differentiable functionf(x,y){\displaystyle f(x,y)}with real values, one can approximatef(x,y){\displaystyle f(x,y)}for(x,y){\displaystyle (x,y)}close to(a,b){\displaystyle (a,b)}by the formulaf(x,y)≈f(a,b)+∂f∂x(a,b)(x−a)+∂f∂y(a,b)(y−b).{\displaystyle f\left(x,y\right)\approx f\left(a,b\right)+{\frac {\partial f}{\partial x}}\left(a,b\right)\left(x-a\right)+{\frac {\partial f}{\partial y}}\left(a,b\right)\left(y-b\right).}
The right-hand side is the equation of the plane tangent to the graph ofz=f(x,y){\displaystyle z=f(x,y)}at(a,b).{\displaystyle (a,b).}
In the more general case ofBanach spaces, one hasf(x)≈f(a)+Df(a)(x−a){\displaystyle f(x)\approx f(a)+Df(a)(x-a)}whereDf(a){\displaystyle Df(a)}is theFréchet derivativeoff{\displaystyle f}ata{\displaystyle a}.
Gaussian opticsis a technique ingeometrical opticsthat describes the behaviour of light rays in optical systems by using theparaxial approximation, in which only rays which make small angles with theoptical axisof the system are considered.[2]In this approximation, trigonometric functions can be expressed as linear functions of the angles. Gaussian optics applies to systems in which all the optical surfaces are either flat or are portions of asphere. In this case, simple explicit formulae can be given for parameters of an imaging system such as focal distance, magnification and brightness, in terms of the geometrical shapes and material properties of the constituent elements.
The period of swing of asimple gravity pendulumdepends on itslength, the localstrength of gravity, and to a small extent on the maximumanglethat the pendulum swings away from vertical,θ0, called theamplitude.[3]It is independent of themassof the bob. The true periodTof a simple pendulum, the time taken for a complete cycle of an ideal simple gravity pendulum, can be written in several different forms (seependulum), one example being theinfinite series:[4][5]T=2πLg(1+116θ02+113072θ04+⋯){\displaystyle T=2\pi {\sqrt {L \over g}}\left(1+{\frac {1}{16}}\theta _{0}^{2}+{\frac {11}{3072}}\theta _{0}^{4}+\cdots \right)}
whereLis the length of the pendulum andgis the localacceleration of gravity.
However, if one takes the linear approximation (i.e. if the amplitude is limited to small swings,[Note 1]) theperiodis:[6]
In the linear approximation, the period of swing is approximately the same for different size swings: that is,the period is independent of amplitude. This property, calledisochronism, is the reason pendulums are so useful for timekeeping.[7]Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.
The electrical resistivity of most materials changes with temperature. If the temperatureTdoes not vary too much, a linear approximation is typically used:ρ(T)=ρ0[1+α(T−T0)]{\displaystyle \rho (T)=\rho _{0}[1+\alpha (T-T_{0})]}whereα{\displaystyle \alpha }is called thetemperature coefficient of resistivity,T0{\displaystyle T_{0}}is a fixed reference temperature (usually room temperature), andρ0{\displaystyle \rho _{0}}is the resistivity at temperatureT0{\displaystyle T_{0}}. The parameterα{\displaystyle \alpha }is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation,α{\displaystyle \alpha }is different for different reference temperatures. For this reason it is usual to specify the temperature thatα{\displaystyle \alpha }was measured at with a suffix, such asα15{\displaystyle \alpha _{15}}, and the relationship only holds in a range of temperatures around the reference.[8]When the temperature varies over a large temperature range, the linear approximation is inadequate and a more detailed analysis and understanding should be used.
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TheWi-Fi Allianceis a non-profit[1]organization that owns theWi-Fitrademark. Manufacturers may use the trademark to brand products certified for Wi-Fi interoperability. It is based inAustin, Texas.
Early802.11products suffered frominteroperabilityproblems because theInstitute of Electrical and Electronics Engineers(IEEE) had no provision for testing equipment for compliance with its standards.
In 1999, pioneers of a new, higher-speed variant endorsed theIEEE 802.11bspecification to form the Wireless Ethernet Compatibility Alliance (WECA) and branded the new technology Wi-Fi.[2][3]
The group of companies included3Com, Aironet (acquired byCisco),Harris Semiconductor(nowIntersil),Lucent Technologies(the WLAN part was renamed as Orinoco, become part ofAvaya, then acquired byExtreme Networks),NokiaandSymbol Technologies(acquired byMotorola,Zebra Technologies, and nowExtreme Networks).[4]
The alliance listsApple,Comcast,Samsung,Sony,LG,Intel,Dell,Broadcom,Cisco,Qualcomm,Motorola,Microsoft,Texas Instruments, andT-Mobileas key sponsors. The charter for this independent organization was to perform testing, certify interoperability of products, and to promote the technology.[5]
WECA renamed itself theWi-Fi Alliancein 2002.[6]
Most producers of 802.11 equipment became members, and as of 2012,[update]the Wi-Fi Alliance included over 550 member companies. The Wi-Fi Alliance extended Wi-Fi beyondwireless local area networkapplications into point-to-point and personal area networking and enabled specific applications such asMiracast.
The Wi-Fi Alliance owns and controls the "Wi-Fi Certified"logo, a registeredtrademark, which is permitted only on equipment which has passed testing. Purchasers relying on that trademark may have greater chances of interoperation than otherwise. Testing involves not only radio and data format interoperability, butsecurity protocols, as well as optional testing for quality of service and power management protocols.[7]Wi-Fi Certified products have to demonstrate that they can perform well in networks with other Wi-Fi Certified products, running common applications, in situations similar to those encountered in everyday use.
Certification employs 3 principles:
The Wi-Fi Alliance definition of interoperability demands that products have to show satisfactory performance levels in typicalnetworkconfigurations and have to support both established and emerging applications.
The Wi-Fi Alliance certification process includes three types of tests to ensure interoperability. Wi-Fi Certified products are tested for:
The Wi-Fi Alliance provides certification testing in two levels:[8]
Mandatory:
Optional:
There are a number of certification programs by Wi-Fi alliance:[14]
The 802.11 protocols are IEEE standards, identified as 802.11b, 11g, 11n, 11ac, etc. In 2018 The Wi-Fi Alliance created the simpler generation labels Wi-Fi 4 - 6 beginning with Wi-Fi 5, retroactively added Wi-Fi 4 and later added Wi-Fi 6 and Wi-Fi 6E.[18][19][20]Wi-Fi 5 had Wave 1 and Wave 2 phases. Wi-Fi 6E extends the 2.4/5 GHz range to 6 GHz, where licensed. Listed in historical and capacity order. See the individual 802.11 articles for version details or802.11for a composite summary.
WiGigrefers to 60 GHz wireless local area network connection. It was initially announced in 2013 byWireless Gigabit Alliance, and was adopted by the Wi-Fi Alliance in 2013. They started certifying in 2016. The first version of WiGig isIEEE 802.11ad, and a newer versionIEEE 802.11aywas released in 2021.[21][22][23]
In October 2010, the Alliance began to certifyWi-Fi Direct, that allows Wi-Fi-enabled devices to communicate directly with each other by setting up ad-hoc networks, without going through awireless access pointor hotspot.[24][25]Since 2009 when it was first announced, some suggested Wi-Fi Direct might replace the need forBluetoothon applications that do not rely on Bluetooth low energy.[26][27]
Wi-Fi Protected Accessis a security mechanism based on IEEE 802.11i amendment to the standard that the Wi-Fi Alliance started to certify from the year of 2003.[28]
IBSS with Wi-Fi Protected Setup would enable the creation ofad hoc networkbetween devices directly without central access point.[29]
Wi-Fi Passpoint, alternatively known asHotspot 2.0, is a solution for enabling inter-carrier roaming.[30]It utilizesIEEE 802.11u.
Wi-Fi Easy Connect is a protocol that would enable easily establishing connections viaQR code.[31]
Wi-Fi Protected Setup(WPS) is a network security standard to simply create a securewireless home network, created and introduced by Wi-Fi Alliance in 2006.
Miracast, introduced in 2012, is a standard for wireless display connections from devices such as laptops, tablets, or smartphones. Its goal is to replace cables connecting from the device to the display.[32]
Wi-Fi Aware is an interoperability certification program announced in January 2015 that enables device users, when in the range of a particular access point or another compatible device, to receive notifications of applications or services available in the proximity.[33][34]Later versions of this standard included new features such as the capability to establish a peer-to-peer data connection for file transfer.[35]
Fears were voiced immediately in media that it would be predominantly used forproximity marketing.[36]
Wi-Fi Location is a type ofWi-Fi positioning system, and the certification could help providing accuracy to in-door positioning.[37]
TDLS, or Tunneled Direct Link Setup, is "a seamless way to stream media and other data faster between devices already on the same Wi-Fi network" based onIEEE 802.11zand added to Wi-Fi Alliance certification program in 2012. Devices using it communicate directly with one another, without involving the wireless network's router.[38]
The certification of Wi-Fi Agile Multiband indicate devices can automatically connect and maintain connection in the most suitable way. It covers theIEEE 802.11kstandard about access point information report, theIEEE 802.11vstandard that enable exchanging information about state of network,IEEE 802.11ustandard about additional information of a Wi-Fi network,IEEE 802.11rabout fast transition roaming between different access points, as well as other technologies specified by Wi-Fi alliance.
Wi-Fi EasyMesh is a certification program based on its Multi-Access Point specification for creating Wi-Fi meshes from products by different vendors,[39]based onIEEE 1905.1. It is intended to address the problem of Wi-Fi systems that need to cover large areas where several routers serve as multiple access points, working together to form a larger/extended and unified network.[40][41][42]
Formerly known as Carrier Wi-Fi, Wi-Fi Vantage is a certification program for operators to maintain and manage quality Wi-Fi connections in high usage environment.[43]It includes a number of certification, such as Wi-Fi certified ac (as in 802.11ac), Passpoint, Agile Multiband, and Optimized Connectivity.[44]
Wi-Fi Multimedia (WMM) or known asWireless Multimedia Extensionsis a Wi-Fi Alliance interoperability certification based on theIEEE 802.11estandard. It provides basicquality of service(QoS) features toIEEE 802.11networks.
Wi-Fi Home Design is a set of guidelines released by Wi-Fi alliance for inclusion of wireless network in home design.[45]
Wi-Fi HaLowis a standard for low-power wide-area (LPWA) connection standard using sub-1 GHz spectrum forIoTdevices. It is based onIEEE 802.11ah.[46]
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Viable system theory(VST) concernscyberneticprocesses in relation to the development/evolution ofdynamical systems: it can be used to explainliving systems, which are considered to be complex andadaptive, can learn, and are capable of maintaining an autonomous existence, at least within the confines of their constraints. These attributes involve the maintenance ofinternal stabilitythroughadaptationto changingenvironments. One can distinguish between two strands such theory:formal systemsand principally non-formal system. Formal viable system theory is normally referred to asviability theory, and provides a mathematical approach to explore the dynamics ofcomplex systemsset within the context ofcontrol theory. In contrast, principally non-formal viable system theory is concerned with descriptive approaches to the study of viability through the processes ofcontrol and communication, though these theories may have mathematical descriptions associated with them.
The concept of viability arose withStafford Beerin the 1950s through hisparadigmof management systems.[1][2][3]Its formal relative,viability theorybegan its life in 1976 with the mathematical interpretation of a book byJacques Monodpublished in 1971 and entitledChance and Necessity, and which concerned processes ofevolution.[4]Viability theory is concerned with dynamic adaptation of uncertain evolutionary systems to environments defined by constraints, the values of which determine the viability of the system. Both formal and non-formal approaches ultimately concern the structure and evolutionary dynamics of viability incomplex systems.
An alternative non-formal paradigm arose in the late 1980s through the work of Eric Schwarz,[5]which increases the dimensionality of Beer's paradigm[6][7]
The viable system theory of Beer is most well known through hisviable system model[8]and is concerned with viable organisations capable of evolving.[9]Through both internal and external analysis it is possible to identify the relationships and modes of behaviour that constitute viability. The model is underpinned by the realisation that organisations are complex, and recognising the existence of complexity is inherent to processes of analysis. Beer's management systems paradigm is underpinned by a set of propositions, sometimes referred to as cybernetic laws. Sitting within this is his viable systems model (VSM) and one of its laws is a principle ofrecursion, so that just as the model can be applied to divisions in a department, it can also be applied to the departments themselves. This is permitted through Beer's viability law which states thatevery viable system contains and is contained in a viable system.[10]The cybernetic laws are applied to all types of human activity systems[11]like organisations and institutions.
Now, paradigms are concerned with not only theory but also modes of behaviour within inquiry. One significant part of Beer's paradigm is the development of his Viable Systems Model (VSM) that addresses problem situations in terms of control and communication processes, seeking to ensure system viability within the object of attention. Another is Beer'sSyntegrityprotocol which centres on the means by which effectivecommunicationsin complex situations can occur. VSM has been used successfully to diagnose organisational pathologies (conditions of social ill-health). The model involves not only an operative system that has both structure (e.g., divisions in an organisation or departments in a division) from whichbehaviouremanates that is directed towards an environment, but also a meta-system, which some have called the observer of the system.[12]The system and meta-system areontologicallydifferent, so that for instance where in a production company the system is concerned with production processes and their immediate management, the meta-system is more concerned with the management of the production system as a whole. The connection between the system and meta-system is explained through Beer's Cybernetic map.[13]Beer considered that viable social systems should be seen as living systems.[14]Humberto Maturanaused the term orautopoiesis(self-production) to explain biological living systems, but was reluctant to accept that social systems were living.
The viable system theory of Schwarz is more directed towards the explicit examination of issues of complexity than is that of Beer. The theory begins with the idea ofdissipative systems. While all isolatedsystemsconserveenergy, in non-isolated systems, one can distinguish between conservative systems (in which thekinetic energyis conserved) and dissipative systems (where the total kinetic andpotential energyis conserved, but where part of the energy is changed in form and lost). If dissipated systems are far from equilibrium they "try" to recoverequilibriumso quickly that they form dissipative structures to accelerate the process. Dissipative systems can create structured spots whereentropylocally decreases and sonegentropylocally increases to generate order and organisation. Dissipative systems involve far-from-equilibrium process that are inherently dynamically unstable, though they survive through the creation of order that is beyond the thresholds of instability.
Schwarz explicitly defined the living system in terms of its metastructure[15]involving a system, a metasystem and a meta-meta-system, this latter being an essential attribute. As with Beer, the system is concerned with operative attributes. Schwarz's meta-system is essentially concerned with relationships, and the meta-meta system is concerned with all forms ofknowledgeand its acquisition. Thus, where in Beer's theorylearningprocesses can only be discussed in terms of implicit processes, in Schwarz's theory they can be discussed in explicit terms.
Schwarz's living system model is a summary of much of the knowledge ofcomplex adaptive systems, but succinctly compressed as a graphical genericmetamodel. It is this capacity of compression that establishes it as a new theoretical structure that is beyond the concept of autopoiesis/self-production proposed byHumberto Maturana, through the concept of autogenesis. While the concept of autogenesis has not had the collective coherence that autopoiesis has,[16][17]Schwarz clearly defined it as a network of self-creation processes and firmly integrated it with relevant theory in complexity in a way not previously done. The outcome illustrates how a complex and adaptive viable system is able to survive - maintaining an autonomous durable existence within the confines of its own constraints. The nature of viable systems is that they should have at least potential independence in their processes of regulation, organisation, production, and cognition. The generic model provides a holistic relationship between the attributes that explains the nature of viable systems and how they survive. It addresses the emergence and possible evolution of organisations towards complexity and autonomy intended to refer to any domain of system (e.g., biological, social, or cognitive).
Systems in general, but also human activity systems, are able to survive (in other words they become viable) when they develop:
(a) patterns of self-organisation that lead to self-organisation through morphogenesis and complexity;
(b) patterns for long term evolution towards autonomy;
(c) patterns that lead to the functioning of viable systems.
This theory was intended to embrace the dynamics of dissipative systems using three planes.
Each of the three planes (illustrated in Figure 1 below) is an independent ontological domain, interactively connected through networks of processes, and it shows the basic ontological structure of the viable system.
Connected with this is an evolutionary spiral of self-organisation (adapted from Schwarz's 1997 paper), shown in Figure 2 below.
Here, there are 4 phases or modes that a viable system can pass through. Mode 3 occurs with one of three possible outcomes (trifurcation): system death when viability is lost; more of the same; and metamorphosis when the viable system survives because it changes form.
The dynamic process that viable living systems have, as they move from stability to instability and back again, is explained in Table 1, referring to aspects of both Figures 1 and 2.
Schwarz's VST has been further developed, set within a social knowledge context, and formulated asautonomous agency theory.[18][19]
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Inmathematicsandmathematical logic,Boolean algebrais a branch ofalgebra. It differs fromelementary algebrain two ways. First, the values of thevariablesare thetruth valuestrueandfalse, usually denoted by 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra useslogical operatorssuch asconjunction(and) denoted as∧,disjunction(or) denoted as∨, andnegation(not) denoted as¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal way of describinglogical operationsin the same way that elementary algebra describes numerical operations.
Boolean algebra was introduced byGeorge Boolein his first bookThe Mathematical Analysis of Logic(1847),[1]and set forth more fully in hisAn Investigation of the Laws of Thought(1854).[2]According toHuntington, the termBoolean algebrawas first suggested byHenry M. Shefferin 1913,[3]althoughCharles Sanders Peircegave the title "A Boolian [sic] Algebra with One Constant" to the first chapter of his "The Simplest Mathematics" in 1880.[4]Boolean algebra has been fundamental in the development ofdigital electronics, and is provided for in all modernprogramming languages. It is also used inset theoryandstatistics.[5]
A precursor of Boolean algebra wasGottfried Wilhelm Leibniz'salgebra of concepts. The usage of binary in relation to theI Chingwas central to Leibniz'scharacteristica universalis. It eventually created the foundations of algebra of concepts.[6]Leibniz's algebra of concepts is deductively equivalent to the Boolean algebra of sets.[7]
Boole's algebra predated the modern developments inabstract algebraandmathematical logic; it is however seen as connected to the origins of both fields.[8]In an abstract setting, Boolean algebra was perfected in the late 19th century byJevons,Schröder,Huntingtonand others, until it reached the modern conception of an (abstract)mathematical structure.[8]For example, the empirical observation that one can manipulate expressions in thealgebra of sets, by translating them into expressions in Boole's algebra, is explained in modern terms by saying that the algebra of sets isaBoolean algebra(note theindefinite article). In fact,M. H. Stoneproved in 1936that every Boolean algebra isisomorphicto afield of sets.[9][10]
In the 1930s, while studyingswitching circuits,Claude Shannonobserved that one could also apply the rules of Boole's algebra in this setting,[11]and he introducedswitching algebraas a way to analyze and design circuits by algebraic means in terms oflogic gates. Shannon already had at his disposal the abstract mathematical apparatus, thus he cast his switching algebra as thetwo-element Boolean algebra. In modern circuit engineering settings, there is little need to consider other Boolean algebras, thus "switching algebra" and "Boolean algebra" are often used interchangeably.[12][13][14]
Efficient implementationofBoolean functionsis a fundamental problem in thedesignofcombinational logiccircuits. Modernelectronic design automationtools forvery-large-scale integration(VLSI) circuits often rely on an efficient representation of Boolean functions known as (reduced ordered)binary decision diagrams(BDD) forlogic synthesisandformal verification.[15]
Logic sentences that can be expressed in classicalpropositional calculushave anequivalent expressionin Boolean algebra. Thus,Boolean logicis sometimes used to denote propositional calculus performed in this way.[16][17][18]Boolean algebra is not sufficient to capture logic formulas usingquantifiers, like those fromfirst-order logic.
Although the development ofmathematical logicdid not follow Boole's program, the connection between his algebra and logic was later put on firm ground in the setting ofalgebraic logic, which also studies the algebraic systems of many other logics.[8]Theproblem of determining whetherthe variables of a given Boolean (propositional) formula can be assigned in such a way as to make the formula evaluate to true is called theBoolean satisfiability problem(SAT), and is of importance totheoretical computer science, being the first problem shown to beNP-complete. The closely relatedmodel of computationknown as aBoolean circuitrelatestime complexity(of analgorithm) tocircuit complexity.
Whereas expressions denote mainlynumbersin elementary algebra, in Boolean algebra, they denote thetruth valuesfalseandtrue. These values are represented with thebits, 0 and 1. They do not behave like theintegers0 and 1, for which1 + 1 = 2, but may be identified with the elements of thetwo-element fieldGF(2), that is,integer arithmetic modulo 2, for which1 + 1 = 0. Addition and multiplication then play the Boolean roles of XOR (exclusive-or) and AND (conjunction), respectively, with disjunctionx∨y(inclusive-or) definable asx+y−xyand negation¬xas1 −x. InGF(2),−may be replaced by+, since they denote the same operation; however, this way of writing Boolean operations allows applying the usual arithmetic operations of integers (this may be useful when using a programming language in whichGF(2)is not implemented).
Boolean algebra also deals withfunctionswhich have their values in the set{0,1}. Asequence of bitsis a commonly used example of such a function. Another common example is the totality of subsets of a setE: to a subsetFofE, one can define theindicator functionthat takes the value1onF, and0outsideF. The most general example is the set elements of aBoolean algebra, with all of the foregoing being instances thereof.
As with elementary algebra, the purely equational part of the theory may be developed, without considering explicit values for the variables.[19]
While Elementary algebra has four operations (addition, subtraction, multiplication, and division), the Boolean algebra has only three basic operations:conjunction,disjunction, andnegation, expressed with the correspondingbinary operatorsAND(∧{\displaystyle \land }) and OR (∨{\displaystyle \lor }) and theunary operatorNOT(¬{\displaystyle \neg }), collectively referred to asBoolean operators.[20]Variables in Boolean algebra that store the logical value of 0 and 1 are called theBoolean variables. They are used to store either true or false values.[21]The basic operations on Boolean variablesxandyare defined as follows:
Alternatively, the values ofx∧y,x∨y, and ¬xcan be expressed by tabulating their values withtruth tablesas follows:[22]
When used in expressions, the operators are applied according to the precedence rules. As with elementary algebra, expressions in parentheses are evaluated first, following the precedence rules.[23]
If the truth values 0 and 1 are interpreted as integers, these operations may be expressed with the ordinary operations of arithmetic (wherex+yuses addition andxyuses multiplication), or by the minimum/maximum functions:
One might consider that only negation and one of the two other operations are basic because of the following identities that allow one to define conjunction in terms of negation and the disjunction, and vice versa (De Morgan's laws):[24]
Operations composed from the basic operations include, among others, the following:
These definitions give rise to the following truth tables giving the values of these operations for all four possible inputs.
Alawof Boolean algebra is anidentitysuch asx∨ (y∨z) = (x∨y) ∨zbetween two Boolean terms, where aBoolean termis defined as an expression built up from variables and the constants 0 and 1 using the operations ∧, ∨, and ¬. The concept can be extended to terms involving other Boolean operations such as ⊕, →, and ≡, but such extensions are unnecessary for the purposes to which the laws are put. Such purposes include the definition of aBoolean algebraas anymodelof the Boolean laws, and as a means for deriving new laws from old as in the derivation ofx∨ (y∧z) =x∨ (z∧y)fromy∧z=z∧y(as treated in§ Axiomatizing Boolean algebra).
Boolean algebra satisfies many of the same laws as ordinary algebra when one matches up ∨ with addition and ∧ with multiplication. In particular the following laws are common to both kinds of algebra:[25][26]
The following laws hold in Boolean algebra, but not in ordinary algebra:
Takingx= 2in the third law above shows that it is not an ordinary algebra law, since2 × 2 = 4. The remaining five laws can be falsified in ordinary algebra by taking all variables to be 1. For example, in absorption law 1, the left hand side would be1(1 + 1) = 2, while the right hand side would be 1 (and so on).
All of the laws treated thus far have been for conjunction and disjunction. These operations have the property that changing either argument either leaves the output unchanged, or the output changes in the same way as the input. Equivalently, changing any variable from 0 to 1 never results in the output changing from 1 to 0. Operations with this property are said to bemonotone. Thus the axioms thus far have all been for monotonic Boolean logic. Nonmonotonicity enters via complement ¬ as follows.[5]
The complement operation is defined by the following two laws.
All properties of negation including the laws below follow from the above two laws alone.[5]
In both ordinary and Boolean algebra, negation works by exchanging pairs of elements, hence in both algebras it satisfies the double negation law (also called involution law)
But whereasordinary algebrasatisfies the two laws
Boolean algebra satisfiesDe Morgan's laws:
The laws listed above define Boolean algebra, in the sense that they entail the rest of the subject. The lawscomplementation1 and 2, together with the monotone laws, suffice for this purpose and can therefore be taken as one possiblecompleteset of laws oraxiomatizationof Boolean algebra. Every law of Boolean algebra follows logically from these axioms. Furthermore, Boolean algebras can then be defined as themodelsof these axioms as treated in§ Boolean algebras.
Writing down further laws of Boolean algebra cannot give rise to any new consequences of these axioms, nor can it rule out any model of them. In contrast, in a list of some but not all of the same laws, there could have been Boolean laws that did not follow from those on the list, and moreover there would have been models of the listed laws that were not Boolean algebras.
This axiomatization is by no means the only one, or even necessarily the most natural given that attention was not paid as to whether some of the axioms followed from others, but there was simply a choice to stop when enough laws had been noticed, treated further in§ Axiomatizing Boolean algebra. Or the intermediate notion of axiom can be sidestepped altogether by defining a Boolean law directly as anytautology, understood as an equation that holds for all values of its variables over 0 and 1.[27][28]All these definitions of Boolean algebra can be shown to be equivalent.
Principle: If {X, R} is apartially ordered set, then {X, R(inverse)} is also a partially ordered set.
There is nothing special about the choice of symbols for the values of Boolean algebra. 0 and 1 could be renamed toαandβ, and as long as it was done consistently throughout, it would still be Boolean algebra, albeit with some obvious cosmetic differences.
But suppose 0 and 1 were renamed 1 and 0 respectively. Then it would still be Boolean algebra, and moreover operating on the same values. However, it would not be identical to our original Boolean algebra because now ∨ behaves the way ∧ used to do and vice versa. So there are still some cosmetic differences to show that the notation has been changed, despite the fact that 0s and 1s are still being used.
But if in addition to interchanging the names of the values, the names of the two binary operations are also interchanged,nowthere is no trace of what was done. The end product is completely indistinguishable from what was started with. The columns forx∧yandx∨yin the truth tables have changed places, but that switch is immaterial.
When values and operations can be paired up in a way that leaves everything important unchanged when all pairs are switched simultaneously, the members of each pair are calleddualto each other. Thus 0 and 1 are dual, and ∧ and ∨ are dual. Theduality principle, also calledDe Morgan duality, asserts that Boolean algebra is unchanged when all dual pairs are interchanged.
One change not needed to make as part of this interchange was to complement. Complement is aself-dualoperation. The identity or do-nothing operationx(copy the input to the output) is also self-dual. A more complicated example of a self-dual operation is(x∧y) ∨ (y∧z) ∨ (z∧x). There is no self-dual binary operation that depends on both its arguments. A composition of self-dual operations is a self-dual operation. For example, iff(x,y,z) = (x∧y) ∨ (y∧z) ∨ (z∧x), thenf(f(x,y,z),x,t)is a self-dual operation of four argumentsx,y,z,t.
The principle of duality can be explained from agroup theoryperspective by the fact that there are exactly four functions that are one-to-one mappings (automorphisms) of the set ofBoolean polynomialsback to itself: the identity function, the complement function, the dual function and the contradual function (complemented dual). These four functions form agroupunderfunction composition, isomorphic to theKlein four-group,actingon the set of Boolean polynomials.Walter Gottschalkremarked that consequently a more appropriate name for the phenomenon would be theprinciple(orsquare)of quaternality.[5]: 21–22
AVenn diagram[29]can be used as a representation of a Boolean operation using shaded overlapping regions. There is one region for each variable, all circular in the examples here. The interior and exterior of regionxcorresponds respectively to the values 1 (true) and 0 (false) for variablex. The shading indicates the value of the operation for each combination of regions, with dark denoting 1 and light 0 (some authors use the opposite convention).
The three Venn diagrams in the figure below represent respectively conjunctionx∧y, disjunctionx∨y, and complement ¬x.
For conjunction, the region inside both circles is shaded to indicate thatx∧yis 1 when both variables are 1. The other regions are left unshaded to indicate thatx∧yis 0 for the other three combinations.
The second diagram represents disjunctionx∨yby shading those regions that lie inside either or both circles. The third diagram represents complement ¬xby shading the regionnotinside the circle.
While we have not shown the Venn diagrams for the constants 0 and 1, they are trivial, being respectively a white box and a dark box, neither one containing a circle. However, we could put a circle forxin those boxes, in which case each would denote a function of one argument,x, which returns the same value independently ofx, called a constant function. As far as their outputs are concerned, constants and constant functions are indistinguishable; the difference is that a constant takes no arguments, called azeroaryornullaryoperation, while a constant function takes one argument, which it ignores, and is aunaryoperation.
Venn diagrams are helpful in visualizing laws. The commutativity laws for ∧ and ∨ can be seen from the symmetry of the diagrams: a binary operation that was not commutative would not have a symmetric diagram because interchangingxandywould have the effect of reflecting the diagram horizontally and any failure of commutativity would then appear as a failure of symmetry.
Idempotenceof ∧ and ∨ can be visualized by sliding the two circles together and noting that the shaded area then becomes the whole circle, for both ∧ and ∨.
To see the first absorption law,x∧ (x∨y) =x, start with the diagram in the middle forx∨yand note that the portion of the shaded area in common with thexcircle is the whole of thexcircle. For the second absorption law,x∨ (x∧y) =x, start with the left diagram forx∧yand note that shading the whole of thexcircle results in just thexcircle being shaded, since the previous shading was inside thexcircle.
The double negation law can be seen by complementing the shading in the third diagram for ¬x, which shades thexcircle.
To visualize the first De Morgan's law,(¬x) ∧ (¬y) = ¬(x∨y), start with the middle diagram forx∨yand complement its shading so that only the region outside both circles is shaded, which is what the right hand side of the law describes. The result is the same as if we shaded that region which is both outside thexcircleandoutside theycircle, i.e. the conjunction of their exteriors, which is what the left hand side of the law describes.
The second De Morgan's law,(¬x) ∨ (¬y) = ¬(x∧y), works the same way with the two diagrams interchanged.
The first complement law,x∧ ¬x= 0, says that the interior and exterior of thexcircle have no overlap. The second complement law,x∨ ¬x= 1, says that everything is either inside or outside thexcircle.
Digital logic is the application of the Boolean algebra of 0 and 1 to electronic hardware consisting oflogic gatesconnected to form acircuit diagram. Each gate implements a Boolean operation, and is depicted schematically by a shape indicating the operation. The shapes associated with the gates for conjunction (AND-gates), disjunction (OR-gates), and complement (inverters) are as follows:[30]
The lines on the left of each gate represent input wires orports. The value of the input is represented by a voltage on the lead. For so-called "active-high" logic, 0 is represented by a voltage close to zero or "ground," while 1 is represented by a voltage close to the supply voltage; active-low reverses this. The line on the right of each gate represents the output port, which normally follows the same voltage conventions as the input ports.
Complement is implemented with an inverter gate. The triangle denotes the operation that simply copies the input to the output; the small circle on the output denotes the actual inversion complementing the input. The convention of putting such a circle on any port means that the signal passing through this port is complemented on the way through, whether it is an input or output port.
Theduality principle, orDe Morgan's laws, can be understood as asserting that complementing all three ports of an AND gate converts it to an OR gate and vice versa, as shown in Figure 4 below. Complementing both ports of an inverter however leaves the operation unchanged.
More generally, one may complement any of the eight subsets of the three ports of either an AND or OR gate. The resulting sixteen possibilities give rise to only eight Boolean operations, namely those with an odd number of 1s in their truth table. There are eight such because the "odd-bit-out" can be either 0 or 1 and can go in any of four positions in the truth table. There being sixteen binary Boolean operations, this must leave eight operations with an even number of 1s in their truth tables. Two of these are the constants 0 and 1 (as binary operations that ignore both their inputs); four are the operations that depend nontrivially on exactly one of their two inputs, namelyx,y, ¬x, and ¬y; and the remaining two arex⊕y(XOR) and its complementx≡y.
The term "algebra" denotes both a subject, namely the subject ofalgebra, and an object, namely analgebraic structure. Whereas the foregoing has addressed the subject of Boolean algebra, this section deals with mathematical objects called Boolean algebras, defined in full generality as any model of the Boolean laws. We begin with a special case of the notion definable without reference to the laws, namely concrete Boolean algebras, and then givethe formal definitionof the general notion.
Aconcrete Boolean algebraorfield of setsis any nonempty set of subsets of a given setXclosed under the set operations ofunion,intersection, andcomplementrelative toX.[5]
(HistoricallyXitself was required to be nonempty as well to exclude the degenerate or one-element Boolean algebra, which is the one exception to the rule that all Boolean algebras satisfy the same equations since the degenerate algebra satisfies every equation. However, this exclusion conflicts with the preferred purely equational definition of "Boolean algebra", there being no way to rule out the one-element algebra using only equations— 0 ≠ 1 does not count, being a negated equation. Hence modern authors allow the degenerate Boolean algebra and letXbe empty.)
Example 1.Thepower set2XofX, consisting of allsubsetsofX. HereXmay be any set: empty, finite, infinite, or evenuncountable.
Example 2.The empty set andX. This two-element algebra shows that a concrete Boolean algebra can be finite even when it consists of subsets of an infinite set. It can be seen that every field of subsets ofXmust contain the empty set andX. Hence no smaller example is possible, other than the degenerate algebra obtained by takingXto be empty so as to make the empty set andXcoincide.
Example 3.The set of finite andcofinitesets of integers, where a cofinite set is one omitting only finitely many integers. This is clearly closed under complement, and is closed under union because the union of a cofinite set with any set is cofinite, while the union of two finite sets is finite. Intersection behaves like union with "finite" and "cofinite" interchanged. This example is countably infinite because there are only countably many finite sets of integers.
Example 4.For a less trivial example of the point made by example 2, consider aVenn diagramformed bynclosed curvespartitioningthe diagram into 2nregions, and letXbe the (infinite) set of all points in the plane not on any curve but somewhere within the diagram. The interior of each region is thus an infinite subset ofX, and every point inXis in exactly one region. Then the set of all 22npossible unions of regions (including the empty set obtained as the union of the empty set of regions andXobtained as the union of all 2nregions) is closed under union, intersection, and complement relative toXand therefore forms a concrete Boolean algebra. Again, there are finitely many subsets of an infinite set forming a concrete Boolean algebra, with example 2 arising as the casen= 0 of no curves.
A subsetYofXcan be identified with anindexed familyof bits withindex setX, with the bit indexed byx∈Xbeing 1 or 0 according to whether or notx∈Y. (This is the so-calledcharacteristic functionnotion of a subset.) For example, a 32-bit computer word consists of 32 bits indexed by the set {0,1,2,...,31}, with 0 and 31 indexing the low and high order bits respectively. For a smaller example, ifX={a,b,c}{\displaystyle X=\{a,b,c\}}wherea, b, care viewed as bit positions in that order from left to right, the eight subsets {}, {c}, {b}, {b,c}, {a}, {a,c}, {a,b}, and {a,b,c} ofXcan be identified with the respective bit vectors 000, 001, 010, 011, 100, 101, 110, and 111. Bit vectors indexed by the set of natural numbers are infinitesequencesof bits, while those indexed by therealsin theunit interval[0,1] are packed too densely to be able to write conventionally but nonetheless form well-defined indexed families (imagine coloring every point of the interval [0,1] either black or white independently; the black points then form an arbitrary subset of [0,1]).
From this bit vector viewpoint, a concrete Boolean algebra can be defined equivalently as a nonempty set of bit vectors all of the same length (more generally, indexed by the same set) and closed under the bit vector operations ofbitwise∧, ∨, and ¬, as in1010∧0110 = 0010,1010∨0110 = 1110, and¬1010 = 0101, the bit vector realizations of intersection, union, and complement respectively.
The set {0,1} and its Boolean operations as treated above can be understood as the special case of bit vectors of length one, which by the identification of bit vectors with subsets can also be understood as the two subsets of a one-element set. This is called theprototypicalBoolean algebra, justified by the following observation.
This observation is proved as follows. Certainly any law satisfied by all concrete Boolean algebras is satisfied by the prototypical one since it is concrete. Conversely any law that fails for some concrete Boolean algebra must have failed at a particular bit position, in which case that position by itself furnishes a one-bit counterexample to that law. Nondegeneracy ensures the existence of at least one bit position because there is only one empty bit vector.
The final goal of the next section can be understood as eliminating "concrete" from the above observation. That goal is reached via the stronger observation that, up to isomorphism, all Boolean algebras are concrete.
The Boolean algebras so far have all been concrete, consisting of bit vectors or equivalently of subsets of some set. Such a Boolean algebra consists of a set and operations on that set which can beshownto satisfy the laws of Boolean algebra.
Instead of showing that the Boolean laws are satisfied, we can instead postulate a setX, two binary operations onX, and one unary operation, andrequirethat those operations satisfy the laws of Boolean algebra. The elements ofXneed not be bit vectors or subsets but can be anything at all. This leads to the more generalabstractdefinition.
For the purposes of this definition it is irrelevant how the operations came to satisfy the laws, whether by fiat or proof. All concrete Boolean algebras satisfy the laws (by proof rather than fiat), whence every concrete Boolean algebra is a Boolean algebra according to our definitions. This axiomatic definition of a Boolean algebra as a set and certain operations satisfying certain laws or axiomsby fiatis entirely analogous to the abstract definitions ofgroup,ring,fieldetc. characteristic of modern orabstract algebra.
Given any complete axiomatization of Boolean algebra, such as the axioms for acomplementeddistributive lattice, a sufficient condition for analgebraic structureof this kind to satisfy all the Boolean laws is that it satisfy just those axioms. The following is therefore an equivalent definition.
The section onaxiomatizationlists other axiomatizations, any of which can be made the basis of an equivalent definition.
Although every concrete Boolean algebra is a Boolean algebra, not every Boolean algebra need be concrete. Letnbe asquare-freepositive integer, one not divisible by the square of an integer, for example 30 but not 12. The operations ofgreatest common divisor,least common multiple, and division inton(that is, ¬x=n/x), can be shown to satisfy all the Boolean laws when their arguments range over the positive divisors ofn. Hence those divisors form a Boolean algebra. These divisors are not subsets of a set, making the divisors ofna Boolean algebra that is not concrete according to our definitions.
However, if each divisor ofnisrepresentedby the set of its prime factors, this nonconcrete Boolean algebra isisomorphicto the concrete Boolean algebra consisting of all sets of prime factors ofn, with union corresponding to least common multiple, intersection to greatest common divisor, and complement to division inton. So this example, while not technically concrete, is at least "morally" concrete via this representation, called anisomorphism. This example is an instance of the following notion.
The next question is answered positively as follows.
That is, up to isomorphism, abstract and concrete Boolean algebras are the same thing. This result depends on theBoolean prime ideal theorem, a choice principle slightly weaker than theaxiom of choice. This strong relationship implies a weaker result strengthening the observation in the previous subsection to the following easy consequence of representability.
It is weaker in the sense that it does not of itself imply representability. Boolean algebras are special here, for example arelation algebrais a Boolean algebra with additional structure but it is not the case that every relation algebra is representable in the sense appropriate to relation algebras.
The above definition of an abstract Boolean algebra as a set together with operations satisfying "the" Boolean laws raises the question of what those laws are. A simplistic answer is "all Boolean laws", which can be defined as all equations that hold for the Boolean algebra of 0 and 1. However, since there are infinitely many such laws, this is not a satisfactory answer in practice, leading to the question of it suffices to require only finitely many laws to hold.
In the case of Boolean algebras, the answer is "yes": the finitely many equations listed above are sufficient. Thus, Boolean algebra is said to befinitely axiomatizableorfinitely based.
Moreover, the number of equations needed can be further reduced. To begin with, some of the above laws are implied by some of the others. A sufficient subset of the above laws consists of the pairs of associativity, commutativity, and absorption laws, distributivity of ∧ over ∨ (or the other distributivity law—one suffices), and the two complement laws. In fact, this is the traditional axiomatization of Boolean algebra as acomplementeddistributive lattice.
By introducing additional laws not listed above, it becomes possible to shorten the list of needed equations yet further; for instance, with the vertical bar representing theSheffer strokeoperation, the single axiom((a∣b)∣c)∣(a∣((a∣c)∣a))=c{\displaystyle ((a\mid b)\mid c)\mid (a\mid ((a\mid c)\mid a))=c}is sufficient to completely axiomatize Boolean algebra. It is also possible to find longer single axioms using more conventional operations; seeMinimal axioms for Boolean algebra.[32]
Propositional logicis alogical systemthat is intimately connected to Boolean algebra.[5]Many syntactic concepts of Boolean algebra carry over to propositional logic with only minor changes in notation and terminology, while the semantics of propositional logic are defined via Boolean algebras in a way that the tautologies (theorems) of propositional logic correspond to equational theorems of Boolean algebra.
Syntactically, every Boolean term corresponds to apropositional formulaof propositional logic. In this translation between Boolean algebra and propositional logic, Boolean variablesx, y,... becomepropositional variables(oratoms)P, Q, ... Boolean terms such asx∨ybecome propositional formulasP∨Q; 0 becomesfalseor⊥, and 1 becomestrueorT. It is convenient when referring to generic propositions to use Greek letters Φ, Ψ, ... as metavariables (variables outside the language of propositional calculus, used when talkingaboutpropositional calculus) to denote propositions.
The semantics of propositional logic rely ontruth assignments. The essential idea of a truth assignment is that the propositional variables are mapped to elements of a fixed Boolean algebra, and then thetruth valueof a propositional formula using these letters is the element of the Boolean algebra that is obtained by computing the value of the Boolean term corresponding to the formula. In classical semantics, only the two-element Boolean algebra is used, while inBoolean-valued semanticsarbitrary Boolean algebras are considered. Atautologyis a propositional formula that is assigned truth value1by every truth assignment of its propositional variables to an arbitrary Boolean algebra (or, equivalently, every truth assignment to the two element Boolean algebra).
These semantics permit a translation between tautologies of propositional logic and equational theorems of Boolean algebra. Every tautology Φ of propositional logic can be expressed as the Boolean equation Φ = 1, which will be a theorem of Boolean algebra. Conversely, every theorem Φ = Ψ of Boolean algebra corresponds to the tautologies (Φ ∨ ¬Ψ) ∧ (¬Φ ∨ Ψ) and (Φ ∧ Ψ) ∨ (¬Φ ∧ ¬Ψ). If → is in the language, these last tautologies can also be written as (Φ → Ψ) ∧ (Ψ → Φ), or as two separate theorems Φ → Ψ and Ψ → Φ; if ≡ is available, then the single tautology Φ ≡ Ψ can be used.
One motivating application of propositional calculus is the analysis of propositions and deductive arguments in natural language.[33]Whereas the proposition "ifx= 3, thenx+ 1 = 4" depends on the meanings of such symbols as + and 1, the proposition "ifx= 3, thenx= 3" does not; it is true merely by virtue of its structure, and remains true whether "x= 3" is replaced by "x= 4" or "the moon is made of green cheese." The generic or abstract form of this tautology is "ifP, thenP," or in the language of Boolean algebra,P→P.[citation needed]
ReplacingPbyx= 3 or any other proposition is calledinstantiationofPby that proposition. The result of instantiatingPin an abstract proposition is called aninstanceof the proposition. Thus,x= 3 →x= 3 is a tautology by virtue of being an instance of the abstract tautologyP→P. All occurrences of the instantiated variable must be instantiated with the same proposition, to avoid such nonsense asP→x= 3 orx= 3 →x= 4.
Propositional calculus restricts attention to abstract propositions, those built up from propositional variables using Boolean operations. Instantiation is still possible within propositional calculus, but only by instantiating propositional variables by abstract propositions, such as instantiatingQbyQ→PinP→ (Q→P) to yield the instanceP→ ((Q→P) →P).
(The availability of instantiation as part of the machinery of propositional calculus avoids the need for metavariables within the language of propositional calculus, since ordinary propositional variables can be considered within the language to denote arbitrary propositions. The metavariables themselves are outside the reach of instantiation, not being part of the language of propositional calculus but rather part of the same language for talking about it that this sentence is written in, where there is a need to be able to distinguish propositional variables and their instantiations as being distinct syntactic entities.)
An axiomatization of propositional calculus is a set of tautologies calledaxiomsand one or more inference rules for producing new tautologies from old. Aproofin an axiom systemAis a finite nonempty sequence of propositions each of which is either an instance of an axiom ofAor follows by some rule ofAfrom propositions appearing earlier in the proof (thereby disallowing circular reasoning). The last proposition is thetheoremproved by the proof. Every nonempty initial segment of a proof is itself a proof, whence every proposition in a proof is itself a theorem. An axiomatization issoundwhen every theorem is a tautology, andcompletewhen every tautology is a theorem.[34]
Propositional calculus is commonly organized as aHilbert system, whose operations are just those of Boolean algebra and whose theorems are Boolean tautologies, those Boolean terms equal to the Boolean constant 1. Another form issequent calculus, which has two sorts, propositions as in ordinary propositional calculus, and pairs of lists of propositions calledsequents, such asA∨B,A∧C, ... ⊢A,B→C, ....The two halves of a sequent are called the antecedent and the succedent respectively. The customary metavariable denoting an antecedent or part thereof is Γ, and for a succedent Δ; thus Γ,A⊢ Δ would denote a sequent whose succedent is a list Δ and whose antecedent is a list Γ with an additional propositionAappended after it. The antecedent is interpreted as the conjunction of its propositions, the succedent as the disjunction of its propositions, and the sequent itself as theentailmentof the succedent by the antecedent.
Entailment differs from implication in that whereas the latter is a binaryoperationthat returns a value in a Boolean algebra, the former is a binaryrelationwhich either holds or does not hold. In this sense, entailment is anexternalform of implication, meaning external to the Boolean algebra, thinking of the reader of the sequent as also being external and interpreting and comparing antecedents and succedents in some Boolean algebra. The natural interpretation of ⊢ is as ≤ in the partial order of the Boolean algebra defined byx≤yjust whenx∨y=y. This ability to mix external implication ⊢ and internal implication → in the one logic is among the essential differences between sequent calculus and propositional calculus.[35]
Boolean algebra as the calculus of two values is fundamental to computer circuits, computer programming, and mathematical logic, and is also used in other areas of mathematics such as set theory and statistics.[5]
In the early 20th century, several electrical engineers[who?]intuitively recognized that Boolean algebra was analogous to the behavior of certain types of electrical circuits.Claude Shannonformally proved such behavior was logically equivalent to Boolean algebra in his 1937 master's thesis,A Symbolic Analysis of Relay and Switching Circuits.
Today, all modern general-purposecomputersperform their functions using two-value Boolean logic; that is, their electrical circuits are a physical manifestation of two-value Boolean logic. They achieve this in various ways: asvoltages on wiresin high-speed circuits and capacitive storage devices, as orientations of amagnetic domainin ferromagnetic storage devices, as holes inpunched cardsorpaper tape, and so on. (Some early computers used decimal circuits or mechanisms instead of two-valued logic circuits.)
Of course, it is possible to code more than two symbols in any given medium. For example, one might use respectively 0, 1, 2, and 3 volts to code a four-symbol alphabet on a wire, or holes of different sizes in a punched card. In practice, the tight constraints of high speed, small size, and low power combine to make noise a major factor. This makes it hard to distinguish between symbols when there are several possible symbols that could occur at a single site. Rather than attempting to distinguish between four voltages on one wire, digital designers have settled on two voltages per wire, high and low.
Computers use two-value Boolean circuits for the above reasons. The most common computer architectures use ordered sequences of Boolean values, called bits, of 32 or 64 values, e.g. 01101000110101100101010101001011. When programming inmachine code,assembly language, and certain otherprogramming languages, programmers work with the low-level digital structure of thedata registers. These registers operate on voltages, where zero volts represents Boolean 0, and a reference voltage (often +5 V, +3.3 V, or +1.8 V) represents Boolean 1. Such languages support both numeric operations and logical operations. In this context, "numeric" means that the computer treats sequences of bits asbinary numbers(base two numbers) and executes arithmetic operations like add, subtract, multiply, or divide. "Logical" refers to the Boolean logical operations of disjunction, conjunction, and negation between two sequences of bits, in which each bit in one sequence is simply compared to its counterpart in the other sequence. Programmers therefore have the option of working in and applying the rules of either numeric algebra or Boolean algebra as needed. A core differentiating feature between these families of operations is the existence of thecarryoperation in the first but not the second.
Other areas where two values is a good choice are the law and mathematics. In everyday relaxed conversation, nuanced or complex answers such as "maybe" or "only on the weekend" are acceptable. In more focused situations such as a court of law or theorem-based mathematics, however, it is deemed advantageous to frame questions so as to admit a simple yes-or-no answer—is the defendant guilty or not guilty, is the proposition true or false—and to disallow any other answer. However, limiting this might prove in practice for the respondent, the principle of the simple yes–no question has become a central feature of both judicial and mathematical logic, makingtwo-valued logicdeserving of organization and study in its own right.
A central concept of set theory is membership. An organization may permit multiple degrees of membership, such as novice, associate, and full. With sets, however, an element is either in or out. The candidates for membership in a set work just like the wires in a digital computer: each candidate is either a member or a nonmember, just as each wire is either high or low.
Algebra being a fundamental tool in any area amenable to mathematical treatment, these considerations combine to make the algebra of two values of fundamental importance to computer hardware, mathematical logic, and set theory.
Two-valued logic can be extended tomulti-valued logic, notably by replacing the Boolean domain {0, 1} with the unit interval [0,1], in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1 −x, conjunction (AND) is replaced with multiplication (xy), and disjunction (OR) is defined viaDe Morgan's law. Interpreting these values as logicaltruth valuesyields a multi-valued logic, which forms the basis forfuzzy logicandprobabilistic logic. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.
The original application for Boolean operations wasmathematical logic, where it combines the truth values, true or false, of individual formulas.
Natural languages such as English have words for several Boolean operations, in particular conjunction (and), disjunction (or), negation (not), and implication (implies).But notis synonymous withand not. When used to combine situational assertions such as "the block is on the table" and "cats drink milk", which naïvely are either true or false, the meanings of theselogical connectivesoften have the meaning of their logical counterparts. However, with descriptions of behavior such as "Jim walked through the door", one starts to notice differences such as failure of commutativity, for example, the conjunction of "Jim opened the door" with "Jim walked through the door" in that order is not equivalent to their conjunction in the other order, sinceandusually meansand thenin such cases. Questions can be similar: the order "Is the sky blue, and why is the sky blue?" makes more sense than the reverse order. Conjunctive commands about behavior are like behavioral assertions, as inget dressed and go to school. Disjunctive commands suchlove me or leave meorfish or cut baittend to be asymmetric via the implication that one alternative is less preferable. Conjoined nouns such astea and milkgenerally describe aggregation as with set union whiletea or milkis a choice. However, context can reverse these senses, as inyour choices are coffee and teawhich usually means the same asyour choices are coffee or tea(alternatives). Double negation, as in "I don't not like milk", rarely means literally "I do like milk" but rather conveys some sort of hedging, as though to imply that there is a third possibility. "Not not P" can be loosely interpreted as "surely P", and althoughPnecessarily implies "not notP," the converse is suspect in English, much as withintuitionistic logic. In view of the highly idiosyncratic usage of conjunctions in natural languages, Boolean algebra cannot be considered a reliable framework for interpreting them.
Boolean operations are used indigital logicto combine the bits carried on individual wires, thereby interpreting them over {0,1}. When a vector ofnidentical binary gates are used to combine two bit vectors each ofnbits, the individual bit operations can be understood collectively as a single operation on values from aBoolean algebrawith 2nelements.
Naive set theoryinterprets Boolean operations as acting on subsets of a given setX. As we saw earlier this behavior exactly parallels the coordinate-wise combinations of bit vectors, with the union of two sets corresponding to the disjunction of two bit vectors and so on.
The 256-element free Boolean algebra on three generators is deployed incomputer displaysbased onraster graphics, which usebit blitto manipulate whole regions consisting ofpixels, relying on Boolean operations to specify how the source region should be combined with the destination, typically with the help of a third region called themask. Modernvideo cardsoffer all223= 256ternary operations for this purpose, with the choice of operation being a one-byte (8-bit) parameter. The constantsSRC = 0xaaor0b10101010,DST = 0xccor0b11001100, andMSK = 0xf0or0b11110000allow Boolean operations such as(SRC^DST)&MSK(meaning XOR the source and destination and then AND the result with the mask) to be written directly as a constant denoting a byte calculated at compile time,0x80in the(SRC^DST)&MSKexample,0x88if justSRC^DST, etc. At run time the video card interprets the byte as the raster operation indicated by the original expression in a uniform way that requires remarkably little hardware and which takes time completely independent of the complexity of the expression.
Solid modelingsystems forcomputer aided designoffer a variety of methods for building objects from other objects, combination by Boolean operations being one of them. In this method the space in which objects exist is understood as a setSofvoxels(the three-dimensional analogue of pixels in two-dimensional graphics) and shapes are defined as subsets ofS, allowing objects to be combined as sets via union, intersection, etc. One obvious use is in building a complex shape from simple shapes simply as the union of the latter. Another use is in sculpting understood as removal of material: any grinding, milling, routing, or drilling operation that can be performed with physical machinery on physical materials can be simulated on the computer with the Boolean operationx∧ ¬yorx−y, which in set theory is set difference, remove the elements ofyfrom those ofx. Thus given two shapes one to be machined and the other the material to be removed, the result of machining the former to remove the latter is described simply as their set difference.
Search engine queries also employ Boolean logic. For this application, each web page on the Internet may be considered to be an "element" of a "set." The following examples use a syntax supported byGoogle.[NB 1]
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https://en.wikipedia.org/wiki/Boolean_algebra_(logic)
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Inmilitaryterminology, acountersignis a sign, word, or any other signal previously agreed upon and required to be exchanged between apicketor guard and anybody approaching his or her post. The term usually encompasses both the sign given by the approaching party as well as the sentry's reply. However, in some militaries, the countersign is strictly the reply of the sentry to the password given by the person approaching.[1]
A well-known sign/countersign used by theAllied forcesonD-DayduringWorld War II: the challenge/sign was "flash", thepassword"thunder" and the countersign (to challenge the person giving the first codeword) "Welcome".[2]
Some countersigns include words that are difficult for an enemy to pronounce. For instance, in the above example, the word "thunder" contains avoiceless dental fricative(/θ/),[3]which does not exist inGerman.[4]
The opening lines ofWilliam Shakespeare's playHamletare between soldiers on duty are viewed as representing a crude sign in which the line "Long live the King!" was a sign between soldiers:
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https://en.wikipedia.org/wiki/Countersign_(military)
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Thebasic exchange telephone radio serviceorBETRSis a fixed radio service where amultiplexed,digital radiolink is used as the last segment of thelocal loopto providewirelesstelephone service to subscribers in remote areas. BETRS technology was developed in the mid-1980s and allows up to four subscribers to use a single radiochannelpair,simultaneously, withoutinterferingwith one another.
In the US, this service may operate in the paired 152/158 and 454/459MHzbands and on 10channel blocksin the 816-820/861-865 MHz bands. BETRS may be licensed only to state-certified carriers in the area where the service is provided and is considered a part of thepublic switched telephone network(PSTN) by state regulators.
Regulation of this service currently resides in parts 1 and 22 of theCode of Federal Regulations(CFR), Subtitle 47 onTelecommunications, and may be researched or ordered through theGovernment Printing Office(GPO).
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https://en.wikipedia.org/wiki/Basic_exchange_telephone_radio_service
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Anidentity scoreis a system for detectingidentity theft. Identity scores are increasingly being adopted as a means to preventfraudin business[1]and as a tool to verify and correctpublic records.
Identity scores incorporate, a broad set ofconsumerdata that gauges a person's legitimacy. Identity score components can include (but are not limited to)personal identifiers,public records,Internetdata,governmentrecords,corporatedata, predicted behavior patterns based on empirical data, self-assessed behavior patterns, andcredit records.
Identity scoring was originally developed for use by financial services firms to measure the fraud risk for new customers opening accounts. Typical external credit and fraud checks often fail to detect erroneous background information.
Identity scoring is also being tested as a means for financial institutions to comply with criminal investigations and antiterrorism measures, such as theBank Secrecy Act (BSA)and theUSA PATRIOT Act. Usage of fraudverificationtools and third-partyauthenticationsystems to verify identities and “red flag” suspicious activity is greatly enhanced by identity scoring.
Identity scores are built from collecting information from a variety of sources and analyzing discernible patterns from the total information. These records can generally be broken down into three categories:Public records, private records, andcredit records.
Public records can include (but are not limited to) any of the following sources:
Private (non-credit) records can include (but are not limited to) any of the following sources:
Private (credit) records can include (but are not limited to) any of the following sources:
Each identity scoring system uses individual data components to generate their score, meaning that results can vary wildly even for the same individual.
Typical identity score components can include (but are not limited to):
Identity scores are sometimes calculated usingpredictive analytics, the science of taking behavioral data and comparing it against historical patterns to identify potentially risky or fraudulent activity.
By compiling publicly available information and using predictive analytics to gauge the patterns of how the information is used, identity scoring systems can measure theauthenticityof a particular identity.
Identity scoring can be used in a variety of ways, from identity verification and measuring fraud risk on theenterpriselevel, to preventing fraudulent use of identities and syntheticidentity thefton the consumer level. Identity scoring can theoretically provide much more definitive proof of an identity's legitimacy, because of the amount of identifying data it utilizes. Virtually all public information about an individual can be used as data in their identity score.
Credit scoresare compiled from information sources relating to credit, such as number of credit accounts held, balances on each account, dates of collection activity, and so on. Credit scores do not measure any financial or personal activity that is not related to credit, and identity fraud that does not involve credit will not appear on yourcredit reportor affect your credit score. Credit scores and the credit scoring system are also very predictable—there are specific steps you follow to improve your credit score, dispute errors in credit reports, etc.
Identity scores are compiled from much larger sources of information, including criminal records, property records, and so on. Identity scoring enables “grading” of patterns of behavior via predictive analytics, from which an identity monitoring service can track an individual's or criminal group's activity across several enterprises, instead of being confined to monitoring just one area.
Identity scores are also much more mutable and “fuzzy” than credit scores, because the source information—public records and personally identifying information—is constantly changing. Every time an individual changes a job, buys or sells property, or has an encounter with law enforcement, this person's public records are altered. Coordinating the information across so many different sources makes it very difficult to fix errors in one's information once they occur.
Where credit scores have a generally accepted model of a three-digit-number (used for theFICO score, the newVantageScore, and credit bureaus' proprietary scores), identity scoring models vary wildly from product to product.
Identity scoring works by matching the information the user provides against billions of records in publicdatabases, ranging from property and tax records to Internetsearch engines, and calculating it against patterns designed to recognize fraud or identity theft.
Example: John's name andSocial Security numberwere stolen by identity thieves who hacked a stolenlaptop. They take her Social Security number and combine it with another stolen name, and use it to open a series of new accounts, including credit cards and retailgift cards. An identity protection system that used identity scoring would alert Wendy that her Social Security number had been compromised.
Because identity scores include much more accurate information and can predict behavior patterns more definitively than credit scores, theGartnerresearch firm predicted that identity scoring will surpass credit monitoring as the leading identity theft prevention measure by 2009. However, Gartner research analyst Avivah Litan warned that identity scoring was not a foolproof system, as it still relied on the underlying accuracy of the information used.
There are three types of breeder documents, which are documents designed to verify other identification documents.[2]
Reliance on these documents to verify identities is flawed, as there is no standardized means to verify that information contained in breeder documents is legitimate. Identity scoring can be used as a tool to authenticate identities on an independent level in cases of employment hiring and information verification.
Currently there is no standard means to verify that information provided on anI-9work document is legitimate, for example. The desire for industries to quickly hire cheap labor trumps any incentive a business has to check the credentials of their new hires, leading to a “gray market” for stolen identities and contributing to continuing surges inillegal immigration. Tools that employ identity scoring to verify that a person's name and Social Security number match, or that their I-9 data is correct, could cut down on the sale and misuse of personal information while enabling better enforcement of immigration law.
The following companies make use of identity scoring products or systems in their businesses:
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https://en.wikipedia.org/wiki/Identity_score
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Wiles's proof of Fermat's Last Theoremis aproofby British mathematicianSir Andrew Wilesof a special case of themodularity theoremforelliptic curves. Together withRibet's theorem, it provides a proof forFermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be impossible to prove using previous knowledge by almost all living mathematicians at the time.[1]: 203–205, 223, 226
Wiles first announced his proof on 23 June 1993 at a lecture inCambridgeentitled "Modular Forms, Elliptic Curves and Galois Representations".[2]However, in September 1993 the proof was found to contain an error. One year later on 19 September 1994, in what he would call "the most important moment of [his] working life", Wiles stumbled upon a revelation that allowed him to correct the proof to the satisfaction of the mathematical community. The corrected proof was published in 1995.[3]
Wiles's proof uses many techniques fromalgebraic geometryandnumber theoryand has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry such as thecategoryofschemes, significant number theoretic ideas fromIwasawa theory, and other 20th-century techniques which were not available to Fermat. The proof's method of identification of adeformation ringwith aHecke algebra(now referred to as anR=T theorem) to provemodularity liftingtheorems has been an influential development inalgebraic number theory.
Together, the two papers which contain the proof are 129 pages long[4][5]and consumed more than seven years of Wiles's research time.John Coatesdescribed the proof as one of the highest achievements of number theory, andJohn Conwaycalled it "the proof of the [20th] century."[6]Wiles's path to proving Fermat's Last Theorem, by way of proving the modularity theorem for the special case ofsemistable elliptic curves, established powerful modularity lifting techniques and opened up entire new approaches to numerous other problems. For proving Fermat's Last Theorem, he wasknighted, and received other honours such as the 2016Abel Prize. When announcing that Wiles had won the Abel Prize, theNorwegian Academy of Science and Lettersdescribed his achievement as a "stunning proof".[3]
Fermat's Last Theorem, formulated in 1637, states that no three positive integersa,b, andccan satisfy the equation
ifnis an integer greater than two (n> 2).
Over time, this simple assertion became one of the most famousunproved claimsin mathematics. Between its publication and Andrew Wiles's eventual solution more than 350 years later, many mathematicians and amateurs attempted to prove this statement, either for all values ofn> 2, or for specific cases. It spurred the development of entire new areas withinnumber theory. Proofs were eventually found for all values ofnup to around 4 million, first by hand, and later by computer. However, no general proof was found that would be valid for all possible values ofn, nor even a hint how such a proof could be undertaken.
Separately from anything related to Fermat's Last Theorem, in the 1950s and 1960s Japanese mathematicianGoro Shimura, drawing on ideas posed byYutaka Taniyama, conjectured that a connection might exist betweenelliptic curvesandmodular forms. These were mathematical objects with no known connection between them. Taniyama and Shimura posed the question whether, unknown to mathematicians, the two kinds of object were actually identical mathematical objects, just seen in different ways.
They conjectured that everyrationalelliptic curve is alsomodular. This became known as the Taniyama–Shimura conjecture. In the West, this conjecture became well known through a 1967 paper byAndré Weil, who gave conceptual evidence for it; thus, it is sometimes called the Taniyama–Shimura–Weil conjecture.
By around 1980, much evidence had been accumulated to form conjectures about elliptic curves, and many papers had been written which examined the consequences if the conjecture were true, but the actual conjecture itself was unproven and generally considered inaccessible—meaning that mathematicians believed a proof of the conjecture was probably impossible using current knowledge.
For decades, the conjecture remained an important but unsolved problem in mathematics. Around 50 years after first being proposed, the conjecture was finally proven and renamed themodularity theorem, largely as a result of Andrew Wiles's work described below.
On yet another separate branch of development, in the late 1960s, Yves Hellegouarch came up with the idea of associating hypothetical solutions (a,b,c) of Fermat's equation with a completely different mathematical object: an elliptic curve.[7]The curve consists of all points in the plane whose coordinates (x,y) satisfy the relation
Such an elliptic curve would enjoy very special properties due to the appearance of high powers of integers in its equation and the fact thatan+bn=cnwould be annth power as well.
In 1982–1985,Gerhard Freycalled attention to the unusual properties of this same curve, now called aFrey curve. He showed that it was likely that the curve could link Fermat and Taniyama, since anycounterexampleto Fermat's Last Theorem would probably also imply that an elliptic curve existed that was notmodular. Frey showed that there were good reasons to believe that any set of numbers (a,b,c,n) capable of disproving Fermat's Last Theorem could also probably be used to disprove the Taniyama–Shimura–Weil conjecture. Therefore, if the Taniyama–Shimura–Weil conjecture were true, no set of numbers capable of disproving Fermat could exist, so Fermat's Last Theorem would have to be true as well.
The conjecture says that each elliptic curve withrationalcoefficients can be constructed in an entirely different way, not by giving its equation but by usingmodular functionstoparametrisecoordinatesxandyof the points on it. Thus, according to the conjecture, any elliptic curve overQwould have to be amodular elliptic curve, yet if a solution to Fermat's equation with non-zeroa,b,candngreater than 2 existed, the corresponding curve would not be modular, resulting in a contradiction. If the link identified by Frey could be proven, then in turn, it would mean that a disproof of Fermat's Last Theorem would disprove the Taniyama–Shimura–Weil conjecture, or by contraposition, a proof of the latter would prove the former as well.[8]
To complete this link, it was necessary to show that Frey's intuition was correct: that a Frey curve, if it existed, could not be modular. In 1985,Jean-Pierre Serreprovided a partial proof that a Frey curve could not be modular. Serre did not provide a complete proof of his proposal; the missing part (which Serre had noticed early on[9]: 1) became known as the epsilon conjecture (sometimes written ε-conjecture; now known asRibet's theorem). Serre's main interest was in an even more ambitious conjecture,Serre's conjectureon modularGalois representations, which would imply the Taniyama–Shimura–Weil conjecture. However his partial proof came close to confirming the link between Fermat and Taniyama.
In the summer of 1986,Ken Ribetsucceeded in proving the epsilon conjecture, now known asRibet's theorem. His article was published in 1990. In doing so, Ribet finally proved the link between the two theorems by confirming, as Frey had suggested, that a proof of the Taniyama–Shimura–Weil conjecture for the kinds of elliptic curves Frey had identified, together with Ribet's theorem, would also prove Fermat's Last Theorem.
In mathematical terms, Ribet's theorem showed that if the Galois representation associated with an elliptic curve has certain properties (which Frey's curve has), then that curve cannot be modular, in the sense that there cannot exist a modular form which gives rise to the same Galois representation.[10]
Following the developments related to the Frey curve, and its link to both Fermat and Taniyama, a proof of Fermat's Last Theorem would follow from a proof of the Taniyama–Shimura–Weil conjecture—or at least a proof of the conjecture for the kinds of elliptic curves that included Frey's equation (known assemistable elliptic curves).
However, despite the progress made by Serre and Ribet, this approach to Fermat was widely considered unusable as well, since almost all mathematicians saw the Taniyama–Shimura–Weil conjecture itself as completely inaccessible to proof with current knowledge.[1]: 203–205, 223, 226For example, Wiles's ex-supervisorJohn Coatesstated that it seemed "impossible to actually prove",[1]: 226and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".[1]: 223
Hearing of Ribet's 1986 proof of the epsilon conjecture, English mathematician Andrew Wiles, who had studied elliptic curves and had a childhood fascination with Fermat, decided to begin working in secret towards a proof of the Taniyama–Shimura–Weil conjecture, since it was now professionally justifiable,[11]as well as because of the enticing goal of proving such a long-standing problem.
Ribet later commented that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]."[1]: 223
Wiles initially presented his proof in 1993. It was finally accepted as correct, and published, in 1995, following the correction of a subtle error in one part of his original paper. His work was extended to a full proof of the modularity theorem over the following six years by others, who built on Wiles's work.
During 21–23 June 1993, Wiles announced and presented his proof of the Taniyama–Shimura conjecture for semistable elliptic curves, and hence of Fermat's Last Theorem, over the course of three lectures delivered at theIsaac Newton Institute for Mathematical SciencesinCambridge, England.[2]There was a relatively large amount of press coverage afterwards.[12]
After the announcement,Nick Katzwas appointed as one of the referees toreviewWiles's manuscript. In the course of his review, he asked Wiles a series of clarifying questions that led Wiles to recognise that the proof contained a gap. There was an error in one critical portion of the proof which gave a bound for the order of a particular group: theEuler systemused to extendKolyvaginandFlach's method was incomplete. The error would not have rendered his work worthless—each part of Wiles's work was highly significant and innovative by itself, as were the many developments and techniques he had created in the course of his work, and only one part was affected.[1]: 289, 296–297Without this part proved, however, there was no actual proof of Fermat's Last Theorem.
Wiles spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former studentRichard Taylor, without success.[13][14][15]By the end of 1993, rumours had spread that under scrutiny, Wiles's proof had failed, but how seriously was not known. Mathematicians were beginning to pressure Wiles to disclose his work whether or not complete, so that the wider community could explore and use whatever he had managed to accomplish. Instead of being fixed, the problem, which had originally seemed minor, now seemed very significant, far more serious, and less easy to resolve.[16]
Wiles states that on the morning of 19 September 1994, he was on the verge of giving up and was almost resigned to accepting that he had failed, and to publishing his work so that others could build on it and find the error. He states that he was having a final look to try to understand the fundamental reasons why his approach could not be made to work, when he had a sudden insight that the specific reason why the Kolyvagin–Flach approach would not work directly also meant that his original attempt usingIwasawa theorycould be made to work if he strengthened it using experience gained from the Kolyvagin–Flach approach since then. Each was inadequate by itself, but fixing one approach with tools from the other would resolve the issue and produce aclass number formula(CNF) valid for all cases that were not already proven by his refereed paper:[13][17]
I was sitting at my desk examining the Kolyvagin–Flach method. It wasn't that I believed I could make it work, but I thought that at least I could explain why it didn't work. Suddenly I had this incredible revelation. I realised that, the Kolyvagin–Flach method wasn't working, but it was all I needed to make my original Iwasawa theory work from three years earlier. So out of the ashes of Kolyvagin–Flach seemed to rise the true answer to the problem. It was so indescribably beautiful; it was so simple and so elegant. I couldn't understand how I'd missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I'd keep coming back to my desk looking to see if it was still there. It was still there. I couldn't contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much.
On 6 October Wiles asked three colleagues (includingGerd Faltings) to review his new proof,[19]and on 24 October 1994 Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem"[4]and "Ring theoretic properties of certain Hecke algebras",[5]the second of which Wiles had written with Taylor and proved that certain conditions were met which were needed to justify the corrected step in the main paper.
The two papers were vetted and finally published as the entirety of the May 1995 issue of theAnnals of Mathematics. The new proof was widely analysed and became accepted as likely correct in its major components.[6][10][11]These papers established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.
Fermat claimed to "... have discovered a truly marvelous proof of this, which this margin is too narrow to contain".[20][21]Wiles's proof is very complex, and incorporates the work of so many other specialists that it was suggested in 1994 that only a small number of people were capable of fully understanding at that time all the details of what he had done.[2][22]The complexity of Wiles's proof motivated a 10-day conference atBoston University; the resulting book of conference proceedings aimed to make the full range of required topics accessible to graduate students in number theory.[9]
As noted above, Wiles proved the Taniyama–Shimura–Weil conjecture for the special case of semistable elliptic curves, rather than for all elliptic curves. Over the following years,Christophe Breuil,Brian Conrad,Fred Diamond, andRichard Taylor(sometimes abbreviated as "BCDT") carried the work further, ultimately proving the Taniyama–Shimura–Weil conjecture for all elliptic curves in a 2001 paper.[23]Now proven, the conjecture became known as themodularity theorem.
In 2005, Dutchcomputer scientistJan Bergstraposed the problem of formalizing Wiles's proof in such a way that it could beverified by computer.[24]
Wiles proved the modularity theorem for semistable elliptic curves, from which Fermat’s last theorem follows usingproof by contradiction. In this proof method, one assumes the opposite of what is to be proved, and shows if that were true, it would create a contradiction. The contradiction shows that the assumption (that the conclusion is wrong) must have been incorrect, requiring the conclusion to hold.
The proof falls roughly in two parts: In the first part, Wiles proves a general result about "lifts", known as the "modularity lifting theorem". This first part allows him to prove results about elliptic curves by converting them to problems aboutGalois representationsof elliptic curves. He then uses this result to prove that all semistable curves are modular, by proving that theGalois representationsof these curves are modular.
Wiles aims first of all to prove a result about these representations, that he will use later: that if a semistable elliptic curveEhas a Galois representationρ(E,p)that is modular, the elliptic curve itself must be modular.
Proving this is helpful in two ways: it makes counting and matching easier, and, significantly, to prove the representation is modular, we would only have to prove it for one single prime numberp, and we can do this usingany primethat makes our work easy – it does not matter which prime we use.
This is the most difficult part of the problem – technically it means proving that if the Galois representationρ(E,p)is a modular form, so are all the other related Galois representationsρ(E,p∞)for all powers ofp.[3]This is the so-called "modular liftingproblem", and Wiles approached it usingdeformations.
Together, these allow us to work with representations of curves rather than directly with elliptic curves themselves. Our original goal will have been transformed into proving the modularity of geometric Galois representations of semistable elliptic curves, instead. Wiles described this realization as a "key breakthrough".
A Galois representation of an elliptic curve isG→ GL(Zp). To show that a geometric Galois representation of an elliptic curve is a modular form, we need to find anormalized eigenformwhoseeigenvalues(which are also itsFourier seriescoefficients) satisfy acongruence relationshipfor all but a finite number of primes.
This is Wiles'slifting theorem(ormodularity lifting theorem), a major and revolutionary accomplishment at the time.
So we can try to prove all of our elliptic curves are modular by using one prime number asp- but if we do not succeed in proving this for all elliptic curves, perhaps we can prove the rest by choosing different prime numbers as 'p' for the difficult cases.
The proof must cover the Galois representations of all semistable elliptic curvesE, but for each individual curve, we only need to prove it is modular using one prime numberp.)
From above, it does not matter which prime is chosen for the representations. We can use any one prime number that is easiest. 3 is the smallest prime number more than 2, and some work has already been done on representations of elliptic curves usingρ(E, 3), so choosing 3 as our prime number is a helpful starting point.
Wiles found that it was easier to prove the representation was modular by choosing a primep= 3in the cases where the representationρ(E, 3)is irreducible, but the proof whenρ(E, 3)is reducible was easier to prove by choosingp= 5. So, the proof splits in two at this point.
The switch betweenp= 3andp= 5has since opened a significant area of study in its own right(seeSerre's modularity conjecture).
Wiles uses his modularity lifting theorem to make short work of this case:
This existing result forp= 3is crucial to Wiles's approach and is one reason for initially usingp= 3.
Wiles found that when the representation of an elliptic curve usingp= 3is reducible, it was easier to work withp= 5and use his new lifting theorem to prove thatρ(E, 5)will always be modular, than to try and prove directly thatρ(E, 3)itself is modular (remembering that we only need to prove it for one prime).
Wiles showed that in this case, one could always find another semistable elliptic curveFsuch that the representationρ(F, 3)is irreducible and also the representationsρ(E, 5)andρ(F, 5)areisomorphic(they have identical structures).
This proves:
Wiles opted to attempt to match elliptic curves to acountableset of modular forms. He found that this direct approach was not working, so he transformed the problem by instead matching theGalois representationsof the elliptic curves to modular forms. Wiles denotes this matching (or mapping) that, more specifically, is aring homomorphism:
R{\displaystyle R}is a deformation ring andT{\displaystyle \mathbf {T} }is aHecke ring.
Wiles had the insight that in many cases this ringhomomorphismcould be a ringisomorphism(Conjecture 2.16 in Chapter 2, §3 of the 1995 paper[4]). He realised that the map betweenR{\displaystyle R}andT{\displaystyle \mathbf {T} }is an isomorphism if and only if twoabelian groupsoccurring in the theory are finite and have the samecardinality. This is sometimes referred to as the "numerical criterion". Given this result, Fermat's Last Theorem is reduced to the statement that two groups have the same order. Much of the text of the proof leads into topics and theorems related toring theoryandcommutation theory. Wiles's goal was to verify that the mapR→T{\displaystyle R\rightarrow \mathbf {T} }is an isomorphism and ultimately thatR=T{\displaystyle R=\mathbf {T} }. In treating deformations, Wiles defined four cases, with theflatdeformation case requiring more effort to prove and treated in a separate article in the same volume entitled "Ring-theoretic properties of certain Hecke algebras".
Gerd Faltings, in his bulletin, gives the followingcommutative diagram(p. 745):
or ultimately thatR=T{\displaystyle R=\mathbf {T} }, indicating acomplete intersection. Since Wiles could not show thatR=T{\displaystyle R=\mathbf {T} }directly, he did so throughZ3,F3{\displaystyle \mathbf {Z} _{3},\mathbf {F} _{3}}andT/m{\displaystyle \mathbf {T} /{\mathfrak {m}}}vialifts.
In order to perform this matching, Wiles had to create aclass number formula(CNF). He first attempted to use horizontalIwasawa theorybut that part of his work had an unresolved issue such that he could not create a CNF. At the end of the summer of 1991, he learned about anEuler systemrecently developed byVictor KolyvaginandMatthias Flachthat seemed "tailor made" for the inductive part of his proof, which could be used to create a CNF, and so Wiles set his Iwasawa work aside and began working to extend Kolyvagin and Flach's work instead, in order to create the CNF his proof would require.[25]By the spring of 1993, his work had covered all but a few families of elliptic curves, and in early 1993, Wiles was confident enough of his nearing success to let one trusted colleague into his secret. Since his work relied extensively on using the Kolyvagin–Flach approach, which was new to mathematics and to Wiles, and which he had also extended, in January 1993 he asked his Princeton colleague,Nick Katz, to help him review his work for subtle errors. Their conclusion at the time was that the techniques Wiles used seemed to work correctly.[1]: 261–265[26]
Wiles's use of Kolyvagin–Flach would later be found to be the point of failure in the original proof submission, and he eventually had to revert to Iwasawa theory and a collaboration with Richard Taylor to fix it. In May 1993, while reading a paper by Mazur, Wiles had the insight that the 3/5 switch would resolve the final issues and would then cover all elliptic curves.
Given an elliptic curveE{\displaystyle E}over the fieldQ{\displaystyle \mathbb {Q} }of rational numbers, for every prime powerℓn{\displaystyle \ell ^{n}}, there exists ahomomorphismfrom theabsolute Galois group
to
the group ofinvertible2 by 2 matrices whose entries are integers moduloℓn{\displaystyle \ell ^{n}}. This is becauseE(Q¯){\displaystyle E({\bar {\mathbb {Q} }})}, the points ofE{\displaystyle E}overQ¯{\displaystyle {\bar {\mathbb {Q} }}}, form anabelian groupon whichGal(Q¯/Q){\displaystyle \operatorname {Gal} ({\bar {\mathbb {Q} }}/\mathbb {Q} )}acts; the subgroup of elementsx{\displaystyle x}such thatℓnx=0{\displaystyle \ell ^{n}x=0}is just(Z/ℓnZ)2{\displaystyle (\mathbb {Z} /\ell ^{n}\mathbb {Z} )^{2}}, and anautomorphismof this group is a matrix of the type described.
Less obvious is that given a modular form of a certain special type, aHecke eigenformwith eigenvalues inQ{\displaystyle \mathbb {Q} }, one also gets a homomorphism
This goes back to Eichler and Shimura. The idea is that the Galois group acts first on the modular curve on which the modular form is defined, thence on theJacobian varietyof the curve, and finally on the points ofℓn{\displaystyle \ell ^{n}}power order on that Jacobian. The resulting representation is not usually 2-dimensional, but theHecke operatorscut out a 2-dimensional piece. It is easy to demonstrate that these representations come from some elliptic curve but the converse is the difficult part to prove.
Instead of trying to go directly from the elliptic curve to the modular form, one can first pass to themodℓn{\displaystyle {\bmod {\ell ^{n}}}}representation for someℓ{\displaystyle \ell }andn{\displaystyle n}, and from that to the modular form. In the case whereℓ=3{\displaystyle \ell =3}andn=1{\displaystyle n=1}, results of theLanglands–Tunnell theoremshow that themod3{\displaystyle {\bmod {3}}}representation of any elliptic curve overQ{\displaystyle \mathbb {Q} }comes from a modular form. The basic strategy is to use induction onn{\displaystyle n}to show that this is true forℓ=3{\displaystyle \ell =3}and anyn{\displaystyle n}, that ultimately there is a single modular form that works for alln. To do this, one uses a counting argument, comparing the number of ways in which one canlifta modℓn{\displaystyle \ell ^{n}}Galois representation to one modℓn+1{\displaystyle \ell ^{n+1}}and the number of ways in which one can lift a modℓn{\displaystyle \ell ^{n}}modular form. An essential point is to impose a sufficient set of conditions on the Galois representation; otherwise, there will be too many lifts and most will not be modular. These conditions should be satisfied for the representations coming from modular forms and those coming from elliptic curves.
If the original(mod3){\displaystyle (\mathrm {mod} \,3)}representation has an image which is too small, one runs into trouble with the lifting argument, and in this case, there is a final trick which has since been studied in greater generality in the subsequent work on theSerre modularity conjecture. The idea involves the interplay between the(mod3){\displaystyle (\mathrm {mod} \,3)}and(mod5){\displaystyle (\mathrm {mod} \,5)}representations. In particular, if the mod-5 Galois representationρ¯E,5{\displaystyle {\overline {\rho }}_{E,5}}associated to an semistable elliptic curveEoverQis irreducible, then there is another semistable elliptic curveE'overQsuch that its associated mod-5 Galois representationρ¯E′,5{\displaystyle {\overline {\rho }}_{E',5}}is isomorphic toρ¯E,5{\displaystyle {\overline {\rho }}_{E,5}}andsuch that its associated mod-3 Galois representationρ¯E,3{\displaystyle {\overline {\rho }}_{E,3}}is irreducible (and therefore modular by Langlands–Tunnell).[27]
In his 108-page article published in 1995, Wiles divides the subject matter up into the following chapters (preceded here by page numbers):
Gerd Faltingssubsequently provided some simplifications to the 1995 proof, primarily in switching from geometric constructions to rather simpler algebraic ones.[19][28]The book of the Cornell conference also contained simplifications to the original proof.[9]
Wiles's paper is more than 100 pages long and often uses the specialised symbols and notations ofgroup theory,algebraic geometry,commutative algebra, andGalois theory. The mathematicians who helped to lay the groundwork for Wiles often created new specialised concepts and technicaljargon.
Among the introductory presentations are an email which Ribet sent in 1993;[29][30]Hesselink's quick review of top-level issues, which gives just the elementary algebra and avoids abstract algebra;[24]or Daney's web page, which provides a set of his own notes and lists the current books available on the subject. Weston attempts to provide a handy map of some of the relationships between the subjects.[31]F. Q. Gouvêa's 1994 article "A Marvelous Proof", which reviews some of the required topics, won a Lester R. Ford award from theMathematical Association of America.[32][33]Faltings' 5-page technical bulletin on the matter is a quick and technical review of the proof for the non-specialist.[34]For those in search of a commercially available book to guide them, he recommended that those familiar with abstract algebra read Hellegouarch, then read the Cornell book,[9]which is claimed to be accessible to "a graduate student in number theory". The Cornell book does not cover the entirety of the Wiles proof.[12]
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https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem
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Worldwide Interoperability for Microwave Access(WiMAX) is a family ofwireless broadbandcommunication standards based on theIEEE 802.16set of standards, which provide physical layer (PHY) andmedia access control(MAC) options.
TheWiMAX Forumwas formed in June 2001 to promote conformity and interoperability, including the definition of system profiles for commercial vendors.[1]The forum describes WiMAX as "a standards-based technology enabling the delivery oflast milewireless broadband accessas an alternative tocableandDSL".[2]
WiMAX was initially designed to provide 30 to 40 megabit-per-second data rates,[3]with the 2011 update providing up to 1 Gbit/s[3]for fixed stations.IEEE 802.16mor Wireless MAN-Advanced was a candidate for4G, in competition with theLTE Advancedstandard. WiMAX release 2.1, popularly branded asWiMAX 2+, is a backwards-compatible transition from previous WiMAX generations. It is compatible and interoperable withTD-LTE. Newer versions, still backward compatible, include WiMAX release 2.2 (2014) and WiMAX release 3 (2021, adds interoperation with5G NR).
WiMAX refers to interoperable implementations of theIEEE 802.16family of wireless-networks standards ratified by the WiMAX Forum. (Similarly,Wi-Firefers to interoperable implementations of theIEEE 802.11Wireless LAN standards certified by theWi-Fi Alliance.) WiMAX Forum certification allows vendors to sell fixed or mobile products as WiMAX certified, thus ensuring a level of interoperability with other certified products, as long as they fit the same profile.
The original IEEE 802.16 standard (now called "Fixed WiMAX") was published in 2001.
WiMAX adopted some of its technology fromWiBro, a service marketed in Korea.[4]
Mobile WiMAX (originally based on 802.16e-2005) is the revision that was deployed in many countries and is the basis for future revisions such as 802.16m-2011.
WiMAX was sometimes referred to as "Wi-Fi on steroids"[5]and can be used for a number of applications including broadband connections, cellularbackhaul,hotspots, etc. It is similar toLong-range Wi-Fi, but it can enable usage at much greater distances.[6]
The scalable physical layer architecture that allows for data rate to scale easily with available channel bandwidth and range of WiMAX make it suitable for the following potential applications:
WiMAX can provide at-home or mobileInternet accessacross whole cities or countries. In many cases, this has resulted in competition in markets which typically only had access through an existing incumbent DSL (or similar) operator.
Additionally, given the relatively low costs associated with the deployment of a WiMAX network (in comparison with3G,HSDPA,xDSL,HFCorFTTx), it is now economically viable to provide last-mile broadband Internet access in remote locations.
Mobile WiMAX was a replacement candidate forcellular phonetechnologies such asGSMandCDMA, or can be used as an overlay to increase capacity. Fixed WiMAX is also considered as a wirelessbackhaultechnology for2G,3G, and4Gnetworks in both developed and developing nations.[7][8]
In North America, backhaul for urban operations is typically provided via one or morecopper wireline connections, whereas remote cellular operations are sometimes backhauled via satellite. In other regions, urban and rural backhaul is usually provided bymicrowave links. (The exception to this is where the network is operated by an incumbent with ready access to the copper network.) WiMAX has more substantial backhaul bandwidth requirements than legacy cellular applications. Consequently, the use of wireless microwave backhaul is on the rise in North America and existing microwave backhaul links in all regions are being upgraded.[9]Capacities of between 34 Mbit/s and 1 Gbit/s[10]are routinely being deployed with latencies in the order of 1 ms.
In many cases, operators are aggregating sites using wireless technology and then presenting traffic on to fiber networks where convenient. WiMAX in this application competes withmicrowave radio,E-lineand simple extension of the fiber network itself.
WiMAX directly supports the technologies that maketriple-playservice offerings possible (such asquality of serviceandmulticast). These are inherent to the WiMAX standard rather than being added on ascarrier Ethernetis toEthernet.
On May 7, 2008, in the United States,Sprint Nextel,Google,Intel,Comcast,Bright House, andTime Warnerannounced a pooling of an average of 120 MHz of spectrum and merged withClearwireto market the service. The new company hoped to benefit from combined services offerings and network resources as a springboard past its competitors. The cable companies were expected to provide media services to other partners while gaining access to the wireless network as aMobile virtual network operatorto provide triple-play services.
Some wireless industry analysts, such as Ken Dulaney and Todd Kort at Gartner, were skeptical how the deal would work out: Although fixed-mobile convergence had been a recognized factor in the industry, prior attempts to form partnerships among wireless and cable companies had generally failed to lead to significant benefits for the participants. Other analysts at IDC favored the deal, pointing out that as wireless progresses to higher bandwidth, it inevitably competes more directly with cable, DSL and fiber, inspiring competitors into collaboration. Also, as wireless broadband networks grow denser and usage habits shift, the need for increased backhaul and media services accelerate, therefore the opportunity to leverage high bandwidth assets was expected to increase.
The Aeronautical Mobile Airport Communication System (AeroMACS) is a wireless broadband network for the airport surface intended to link the control tower, aircraft, and fixed assets. In 2007, AeroMACS obtained a worldwide frequency allocation in the 5 GHz aviation band. As of 2018, there were 25 AeroMACS deployments in 8 countries, with at least another 25 deployments planned.[11]
IEEE 802.16REVd and IEEE 802.16e standards support bothtime-division duplexingandfrequency-division duplexingas well as a half duplex FDD, that allows for a low cost implementation.
Devices that provide connectivity to a WiMAX network are known assubscriber stations(SS).
Portable units include handsets (similar to cellularsmartphones); PC peripherals (PC Cards or USB dongles); and embedded devices in laptops, which are now available for Wi-Fi services. In addition, there is much emphasis by operators on consumer electronics devices such as Gaming consoles, MP3 players and similar devices. WiMAX is more similar to Wi-Fi than to other3Gcellular technologies.
The WiMAX Forum website provides a list of certified devices. However, this is not a complete list of devices available as certified modules are embedded into laptops, MIDs (Mobile Internet devices), and other private labeled devices.
WiMAX gateway devices are available as both indoor and outdoor versions from manufacturers includingVecima Networks,Alvarion,Airspan,ZyXEL,Huawei, andMotorola. Thelist of WiMAX networksand WiMAX Forum[12]provide more links to specific vendors, products and installations.
Many of the WiMAX gateways that are offered by manufactures such as these are stand-alone self-install indoor units. Such devices typically sit near the customer's window with the best signal, and provide:
Indoor gateways are convenient, but radio losses mean that the subscriber may need to be significantly closer to the WiMAX base station than with professionally installed external units.
Outdoor units are roughly the size of a laptop PC, and their installation is comparable to the installation of a residentialsatellite dish. A higher-gaindirectional outdoor unit will generally result in greatly increased range and throughput but with the obvious loss of practical mobility of the unit.
USBcan provide connectivity to a WiMAX network through adongle. Generally, these devices are connected to a notebook or net book computer. Dongles typically have omnidirectional antennas which are of lower gain compared to other devices. As such, these devices are best used in areas of good coverage.
HTC announced the first WiMAX enabledmobile phone, theMax 4G, on November 12, 2008.[13]The device was only available to certain markets in Russia on theYotanetwork until 2010.[14]
HTC andSprint Nextelreleased the second WiMAX enabled mobile phone, theHTC Evo 4G, March 23, 2010 at the CTIA conference in Las Vegas. The device, made available on June 4, 2010,[15]is capable of both EV-DO(3G) and WiMAX(pre-4G) as well as simultaneous data & voice sessions. Sprint Nextel announced at CES 2012 that it will no longer be offering devices using the WiMAX technology due to financial circumstances, instead, along with its network partnerClearwire, Sprint Nextel rolled out a 4G network having decided to shift and utilizeLTE4G technology instead.
WiMAX is based uponIEEE802.16e-2005,[16]approved in December 2005. It is a supplement to the IEEE Std 802.16-2004,[17]and so the actual standard is 802.16-2004 as amended by 802.16e-2005. Thus, these specifications need to be considered together.
IEEE 802.16e-2005 improves upon IEEE 802.16-2004 by:
SOFDMA (used in 802.16e-2005) and OFDM256 (802.16d) are not compatible thus equipment will have to be replaced if an operator is to move to the later standard (e.g., Fixed WiMAX to Mobile WiMAX).
The original version of the standard on which WiMAX is based (IEEE 802.16) specified a physical layer operating in the 10 to 66 GHz range. 802.16a, updated in 2004 to 802.16-2004, added specifications for the 2 to 11 GHz range. 802.16-2004 was updated by 802.16e-2005 in 2005 and uses scalableorthogonal frequency-division multiple access[18](SOFDMA), as opposed to the fixedorthogonal frequency-division multiplexing(OFDM) version with 256 sub-carriers (of which 200 are used) in 802.16d. More advanced versions, including 802.16e, also bring multiple antenna support throughMIMO. (SeeWiMAX MIMO) This brings potential benefits in terms of coverage, self installation, power consumption, frequency re-use and bandwidth efficiency. WiMax is the most energy-efficient pre-4G technique amongLTEandHSPA+.[19]
The WiMAX MAC uses ascheduling algorithmfor which the subscriber station needs to compete only once for initial entry into the network. After network entry is allowed, the subscriber station is allocated an access slot by the base station. The time slot can enlarge and contract, but remains assigned to the subscriber station, which means that other subscribers cannot use it. In addition to being stable under overload and over-subscription, the scheduling algorithm can also be morebandwidthefficient. The scheduling algorithm also allows the base station to control QoS parameters by balancing the time-slot assignments among the application needs of the subscriber station.
As a standard intended to satisfy needs of next-generation data networks (4G), WiMAX is distinguished by its dynamic burst algorithm modulation adaptive to the physical environment the RF signal travels through. Modulation is chosen to be more spectrally efficient (more bits perOFDM/SOFDMAsymbol). That is, when the bursts have a highsignal strengthand a highcarrier to noiseplus interference ratio (CINR), they can be more easily decoded usingdigital signal processing(DSP). In contrast, operating in less favorable environments for RF communication, the system automatically steps down to a more robust mode (burst profile) which means fewer bits per OFDM/SOFDMA symbol; with the advantage that power per bit is higher and therefore simpler accurate signal processing can be performed.
Burst profiles are used inverse (algorithmically dynamic) to low signal attenuation; meaning throughput between clients and the base station is determined largely by distance. Maximum distance is achieved by the use of the most robust burst setting; that is, the profile with the largest MAC frame allocation trade-off requiring more symbols (a larger portion of the MAC frame) to be allocated in transmitting a given amount of data than if the client were closer to the base station.
The client's MAC frame and their individual burst profiles are defined as well as the specific time allocation. However, even if this is done automatically then the practical deployment should avoid high interference and multipath environments. The reason for which is obviously that too much interference causes the network to function poorly and can also misrepresent the capability of the network.
The system is complex to deploy as it is necessary to track not only the signal strength and CINR (as in systems likeGSM) but also how the available frequencies will be dynamically assigned (resulting in dynamic changes to the available bandwidth.) This could lead to cluttered frequencies with slow response times or lost frames.
As a result, the system has to be initially designed in consensus with the base station product team to accurately project frequency use, interference, and general product functionality.
The Asia-Pacific region has surpassed the North American region in terms of 4G broadband wireless subscribers. There were around 1.7 million pre-WiMAX and WiMAX customers in Asia – 29% of the overall market – compared to 1.4 million in the US and Canada.[20]
The WiMAX Forum has proposed an architecture that defines how a WiMAX network can be connected with an IP based core network, which is typically chosen by operators that serve as Internet Service Providers (ISP); Nevertheless, the WiMAX BS provide seamless integration capabilities with other types of architectures as with packet switched Mobile Networks.
The WiMAX forum proposal defines a number of components, plus some of the interconnections (or reference points) between these, labeled R1 to R5 and R8:
The functional architecture can be designed into various hardware configurations rather than fixed configurations. For example, the architecture is flexible enough to allow remote/mobile stations of varying scale and functionality and Base Stations of varying size – e.g. femto, pico, and mini BS as well as macros.
WiMAX 2.1 and above can be integrated with a LTE TDD network and perform handovers from/to LTE TDD.[22]WiMAX 3 expands the integration to5G NR.[23]
There is no uniform global licensed spectrum for WiMAX, however the WiMAX Forum published three licensed spectrum profiles: 2.3 GHz, 2.5 GHz and 3.5 GHz, in an effort to drive standardisation and decrease cost.
In the US, the biggest segment available was around 2.5 GHz,[24]and is already assigned, primarily toSprint NextelandClearwire. Elsewhere in the world, the most-likely bands used will be the Forum approved ones, with 2.3 GHz probably being most important in Asia. Some countries in Asia likeIndiaandIndonesiawill use a mix of 2.5 GHz, 3.3 GHz and other frequencies.Pakistan'sWateen Telecomuses 3.5 GHz.
Analog TV bands (700 MHz) may become available, but await the completedigital television transition, and other uses have been suggested for that spectrum. In the USA the FCCauction for this spectrumbegan in January 2008 and, as a result, the biggest share of the spectrum went to Verizon Wireless and the next biggest to AT&T.[25]Both of these companies stated their intention of supportingLTE, a technology which competes directly with WiMAX. EU commissionerViviane Redinghas suggested re-allocation of 500–800 MHz spectrum for wireless communication, including WiMAX.[26]
WiMAX profiles define channel size,TDD/FDDand other necessary attributes in order to have interoperating products. The current fixed profiles are defined for both TDD and FDD profiles. At this point, all of the mobile profiles are TDD only. The fixed profiles have channel sizes of 3.5 MHz, 5 MHz, 7 MHz and 10 MHz. The mobile profiles are 5 MHz, 8.75 MHz and 10 MHz. (Note: the 802.16 standard allows a far wider variety of channels, but only the above subsets are supported as WiMAX profiles.)
Since October 2007, the Radio communication Sector of the International Telecommunication Union (ITU-R) has decided to include WiMAX technology in the IMT-2000 set of standards.[27]This enables spectrum owners (specifically in the 2.5–2.69 GHz band at this stage) to use WiMAX equipment in any country that recognizes the IMT-2000.
WiMAX cannot deliver 70Mbit/sover 50 km (31 mi). Like all wireless technologies, WiMAX can operate at higher bitrates or over longer distances but not both. Operating at the maximum range of 50 km (31 mi) increasesbit error rateand thus results in a much lower bitrate. Conversely, reducing the range (to under 1 km) allows a device to operate at higher bitrates.
A citywide deployment of WiMAX inPerth,Australiademonstrated that customers at the cell-edge with an indoorCustomer-premises equipment(CPE) typically obtain speeds of around 1–4 Mbit/s, with users closer to the cell site obtaining speeds of up to 30 Mbit/s.[citation needed]
Like all wireless systems, available bandwidth is shared between users in a given radio sector, so performance could deteriorate in the case of many active users in a single sector. However, with adequate capacity planning and the use of WiMAX's QoS, a minimum guaranteed throughput for each subscriber can be put in place. In practice, most users will have a range of 4–8 Mbit/s services and additional radio cards will be added to the base station to increase the number of users that may be served as required.
A number of specialized companies produced baseband ICs and integrated RFICs for WiMAX Subscriber Stations in the 2.3, 2.5 and 3.5 GHz bands (refer to 'Spectrum allocation' above). These companies include, but are not limited to, Beceem,Sequans, andPicoChip.
Comparisons and confusion between WiMAX andWi-Fiare frequent, because both are related to wireless connectivity and Internet access.[28]
Although Wi-Fi and WiMAX are designed for different situations, they are complementary. WiMAX network operators typically provide a WiMAX Subscriber Unit that connects to the metropolitan WiMAX network and provides Wi-Fi connectivity within the home or business for computers and smartphones. This enables the user to place the WiMAX Subscriber Unit in the best reception area, such as a window, and have date access throughout their property.
TTCN-3test specification language is used for the purposes of specifying conformance tests for WiMAX implementations. The WiMAX test suite is being developed by a Specialist Task Force atETSI(STF 252).[29]
The WiMAX Forum is a non profit organization formed to promote the adoption of WiMAX compatible products and services.[30]
A major role for the organization is to certify the interoperability of WiMAX products.[31]Those that pass conformance and interoperability testing achieve the "WiMAX Forum Certified" designation, and can display this mark on their products and marketing materials. Some vendors claim that their equipment is "WiMAX-ready", "WiMAX-compliant", or "pre-WiMAX", if they are not officially WiMAX Forum Certified.
Another role of the WiMAX Forum is to promote the spread of knowledge about WiMAX. In order to do so, it has a certified training program that is currently offered in English and French. It also offers a series of member events and endorses some industry events.
WiSOA was the first global organization composed exclusively of owners of WiMAX spectrum with plans to deploy WiMAX technology in those bands. WiSOA focused on the regulation, commercialisation, and deployment of WiMAX spectrum in the 2.3–2.5 GHz and the 3.4–3.5 GHz ranges. WiSOA merged with theWireless Broadband Alliancein April 2008.[32]
In 2011, theTelecommunications Industry Associationreleased three technical standards (TIA-1164, TIA-1143, and TIA-1140) that cover the air interface and core networking aspects of Wi-MaxHigh-Rate Packet Data(HRPD) systems using a Mobile Station/Access Terminal (MS/AT) with a single transmitter.[33]
Within the marketplace, WiMAX's main competition came from existing, widely deployed wireless systems such asUniversal Mobile Telecommunications System(UMTS),CDMA2000, existing Wi-Fi, mesh networking and eventually 4G (LTE).
In the future, competition will be from the evolution of the major cellular standards to4G,[needs update]high-bandwidth, low-latency, all-IP networks with voice services built on top. The worldwide move to 4G for GSM/UMTS andAMPS/TIA(including CDMA2000) is the3GPP Long Term Evolution(LTE) effort.
The LTE Standard was finalized in December 2008, with the first commercial deployment of LTE carried out by TeliaSonera in Oslo and Stockholm in December, 2009. Henceforth, LTE saw rapidly increasing adoption by mobile carriers around the world.
Although WiMax was much earlier to market than LTE, LTE was an upgrade and extension of previous 3G (GSM and CDMA) standards, whereas WiMax was a relatively new and different technology without a large user base. Ultimately, LTE won the war to become the 4G standard because mobile operators such as Verizon, AT&T, Vodafone, NTT, and Deutsche Telekom chose to extend their investments in know-how, equipment and spectrum from 3G to LTE, rather than adopt a new technology standard. It would never have been cost-effective for WiMax network operators to compete against fixed-line broadband networks based on 4G technologies. By 2009, most mobile operators began to realize that mobile connectivity (not fixed 802.16e) was the future, and that LTE was going to become the new worldwide mobile connectivity standard, so they chose to wait for LTE to develop rather than switch from 3G to WiMax.
WiMax was a superior technology in terms of speed (roughly 25 Mbit/s) for a few years (2005-2009), and it pioneered some new technologies such as MIMO. But the mobile version of WiMax (802.16m), intended to compete with GSM and CDMA technologies, was too little/too late in getting established, and by the time the LTE standard was finalized in December 2008, the fate of WiMax as a mobile solution was doomed and it was clear that LTE (not WiMax) would become the world's new 4G standard. The largest wireless broadband partner using WiMax, Clearwire, announced in 2008 that they would begin overlaying their existing WiMax network with LTE technology, which was necessary for Clearwire to obtain investments they needed to stay in business.
In some areas of the world, the wide availability of UMTS and a general desire for standardization meant spectrum was not allocated for WiMAX: in July 2005, theEU-wide frequency allocation for WiMAX was blocked.[citation needed]
Early WirelessMAN standards, The European standardHiperMANand Korean standardWiBrowere harmonized as part of WiMAX and are no longer seen as competition but as complementary.[citation needed]All networks now being deployed in South Korea, the home of the WiBro standard, are now WiMAX.[citation needed]
The IEEE 802.16m-2011 standard[34]was the core technology for WiMAX 2. The IEEE 802.16m standard was submitted to the ITU forIMT-Advancedstandardization.[35]IEEE 802.16m is one of the major candidates for IMT-Advanced technologies by ITU. Among many enhancements, IEEE 802.16m systems can provide four times faster[clarification needed]data speed than the WiMAX Release 1.
WiMAX Release 2 provided backward compatibility with Release 1. WiMAX operators could migrate from release 1 to release 2 by upgrading channel cards or software. The WiMAX 2 Collaboration Initiative was formed to help this transition.[36]
It was anticipated that using 4X2MIMOin the urban microcell scenario with only a single 20 MHzTDDchannel available system wide, the 802.16m system can support both 120 Mbit/s downlink and 60 Mbit/s uplink per site simultaneously.
It was expected that the WiMAX Release 2 would be available commercially in the 2011–2012 timeframe.[37]
WiMAX Release 2.1 was released in early-2010s which broke compatibility with earlier WiMAX networks.[citation needed]Significant number of operators have migrated to the new standard that is compatible with TD-LTE by the end of 2010s.
A field test conducted in 2007 by SUIRG (Satellite Users Interference Reduction Group) with support from the U.S. Navy, the Global VSAT Forum, and several member organizations yielded results showing interference at 12 km when using the same channels for both the WiMAX systems and satellites inC-band.[38]
As of October 2010, the WiMAX Forum claimed over 592 WiMAX (fixed and mobile) networks deployed in over 148 countries, covering over 621 million people.[39]By February 2011, the WiMAX Forum cited coverage of over 823 million people, and estimated coverage to over 1 billion people by the end of the year. Note that coverage means the offer of availability of WiMAX service to populations within various geographies, not the number of WiMAX subscribers.[40]
South Korea launched a WiMAX network in the second quarter of 2006. Spain delivered full coverage in two cities Seville and Málaga in 2008 reaching 20,000 portable units. By the end of 2008 there were 350,000 WiMAX subscribers in Korea.[41]
Worldwide, by early 2010 WiMAX seemed to be ramping quickly relative to other available technologies, though access in North America lagged.[42]Yota, the largest WiMAX network operator in the world in 4Q 2009,[43][44]announced in May 2010 that it would move new network deployments to LTE and, subsequently, change its existing networks as well.[citation needed]
A study published in September 2010 by Blycroft Publishing estimated 800 management contracts from 364 WiMAX operations worldwide offering active services (launched or still trading as opposed to just licensed and still to launch).[45]The WiMAX Forum announced on Aug 16, 2011 that there were more than 20 million WiMAX subscribers worldwide, the high-water mark for this technology.http://wimaxforum.org/Page/News/PR/20110816_WiMAX_Subscriptions_Surpass_20_Million_Globally
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CDMA2000++(also known asC2KorIMT Multi‑Carrier(IMT‑MC)) is a family of3G[1]mobile technology standards for sending voice, data, andsignalingdata betweenmobile phonesandcell sites. It is developed by3GPP2as a backwards-compatible successor tosecond-generationcdmaOne(IS-95) set of standards and used especially in North America and South Korea.
CDMA2000 compares toUMTS, a competing set of3Gstandards, which is developed by3GPPand used in Europe, Japan, China, and Singapore.
The name CDMA2000 denotes a family of standards that represent the successive, evolutionary stages of the underlying technology. These are:
All are approved radio interfaces for theITU'sIMT-2000. In the United States,CDMA2000is a registered trademark of theTelecommunications Industry Association(TIA-USA).[2]
CDMA2000 1X (IS-2000), also known as1xand1xRTT, is the core CDMA2000 wireless air interface standard. The designation "1x", meaning1 times radio transmission technology, indicates the sameradio frequency(RF) bandwidth asIS-95: aduplexpair of 1.25 MHz radio channels. 1xRTT almost doubles the capacity of IS-95 by adding 64 more traffic channels to theforward link,orthogonalto (inquadraturewith) the original set of 64. The 1X standard supports packet data speeds of up to 153kbit/swith real world data transmission averaging 80–100 kbit/s in most commercial applications.[3]IMT-2000 also made changes to thedata link layerfor greater use of data services, including medium and link access control protocols andquality of service(QoS). The IS-95 data link layer only providedbest-effort deliveryfor data and circuit switched channel for voice (i.e., a voice frame once every 20 ms).
CDMA2000 1xEV-DO (Evolution-Data Optimized), often abbreviated asEV-DOorEV, is atelecommunicationsstandard for thewirelesstransmission of data throughradiosignals, typically forbroadband Internet access. It usesmultiplexingtechniques includingcode-division multiple access(CDMA) as well astime-division multiple accessto maximize both individual user's throughput and the overall system throughput. It is standardized (IS-856) by3rd Generation Partnership Project 2(3GPP2) as part of the CDMA2000 family of standards and has been adopted by manymobile phoneservice providers around the world – particularly those previously employing CDMA networks.
1X Advanced (Rev.E)[4][5]is the evolution of CDMA2000 1X. It provides up to four times the capacity and 70% more coverage compared to 1X.[6]
The CDMA Development Group states that, as of April 2014, there are 314operatorsin 118 countries offering CDMA2000 1X and/or 1xEV-DO service.[7][needs update]
CDMA2000 technology was developed byQualcommin the late 1990s as an enhancement to the CDMA standard.
The intended4Gsuccessor to CDMA2000 wasUMB (Ultra Mobile Broadband); however, in November 2008,Qualcommannounced it was ending development of the technology, favoringLTEinstead.[8]
In 2007, Qualcomm provided a global patent license for CDMA2000 to the Chinese company Teleepoch.[9]
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https://en.wikipedia.org/wiki/CDMA2000
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Apivot tableis atableof values which are aggregations of groups of individual values from a more extensive table (such as from adatabase,spreadsheet, orbusiness intelligence program) within one or more discrete categories. The aggregations or summaries of the groups of the individual terms might include sums, averages, counts, or other statistics. A pivot table is the outcome of the statistical processing of tabularized raw data and can be used for decision-making.
Althoughpivot tableis a generic term,Microsoftheld a trademark on the term in the United States from 1994 to 2020.[1]
In their bookPivot Table Data Crunching,[2]Bill Jelen and Mike Alexander refer toPito Salasas the "father of pivot tables". While working on a concept for a new program that would eventually becomeLotus Improv, Salas noted that spreadsheets have patterns of data. A tool that could help the user recognize these patterns would help to build advanced data models quickly. With Improv, users could define and store sets of categories, then change views by dragging category names with the mouse. This core functionality would provide the model for pivot tables.
Lotus Developmentreleased Improv in 1991 on theNeXTplatform. A few months after the release of Improv,Brio Technologypublished a standaloneMacintoshimplementation, called DataPivot (with technology eventually patented in 1999).[3]Borlandpurchased the DataPivot technology in 1992 and implemented it in their own spreadsheet application,Quattro Pro.
In 1993 the Microsoft Windows version of Improv appeared. Early in 1994Microsoft Excel5[4]brought a new functionality called a "PivotTable" to market. Microsoft further improved this feature in later versions of Excel:
In 2007 Oracle Corporation madePIVOTandUNPIVOToperators available inOracle Database11g.[6]
For typical data entry and storage, data usually appear inflattables, meaning that they consist of only columns and rows, as in the following portion of a sample spreadsheet showing data on shirt types:
While tables such as these can contain many data items, it can be difficult to get summarized information from them. A pivot table can help quickly summarize the data and highlight the desired information. The usage of a pivot table is extremely broad and depends on the situation. The first question to ask is, "What am I seeking?" In the example here, let us ask, "How manyUnitsdid we sell in eachRegionfor everyShip Date?":
A pivot table usually consists ofrow,columnanddata(orfact) fields. In this case, the column isship date, the row isregionand the data we would like to see is (sum of)units. These fields allow several kinds ofaggregations, including: sum, average,standard deviation, count, etc. In this case, the total number of units shipped is displayed here using asumaggregation.
Using the example above, the software will find all distinct values forRegion. In this case, they are:North,South,East,West. Furthermore, it will find all distinct values forShip date. Based on the aggregation type,sum, it will summarize the fact, the quantities ofUnit, and display them in a multidimensional chart. In the example above, the first datum is 66. This number was obtained by finding all records where bothRegionwasEastandShip Datewas2005-01-31, and adding theUnitsof that collection of records (i.e., cells E2 to E7) together to get a final result.
Pivot tables are not created automatically. For example, in Microsoft Excel one must first select the entire data in the original table and then go to the Insert tab and select "Pivot Table" (or "Pivot Chart"). The user then has the option of either inserting the pivot table into an existing sheet or creating a new sheet to house the pivot table. A pivot table field list is provided to the user which lists all the column headers present in the data. For instance, if a table represents sales data of a company, it might include Date of sale, Sales person, Item sold, Color of item, Units sold, Per unit price, and Total price. This makes the data more readily accessible.
The fields that would be created will be visible on the right hand side of the worksheet. By default, the pivot table layout design will appear below this list.
Pivot Table fields are the building blocks of pivot tables. Each of the fields from the list can be dragged on to this layout, which has four options:
Some uses of pivot tables are related to the analysis of questionnaires with optional responses but some implementations of pivot tables do not allow these use cases. For example the implementation inLibreOffice Calcsince 2012 is not able to process empty cells.[7][8]
Report filter is used to apply a filter to an entire table. For example, if the "Color of Item" field is dragged to this area, then the table constructed will have a report filter inserted above the table. This report filter will have drop-down options (Black, Red, and White in the example above). When an option is chosen from thisdrop-down list("Black" in this example), then the table that would be visible will contain only the data from those rows that have the "Color of Item= Black".
Column labels are used to apply a filter to one or more columns that have to be shown in the pivot table. For instance if the "Salesperson" field is dragged to this area, then the table constructed will have values from the column "Sales Person",i.e., one will have a number of columns equal to the number of "Salesperson". There will also be one added column of Total. In the example above, this instruction will create five columns in the table — one for each salesperson, and Grand Total. There will be a filter above the data — column labels — from which one can select or deselect a particular salesperson for the pivot table.
This table will not have any numerical values as no numerical field is selected but when it is selected, the values will automatically get updated in the column of "Grand total".
Row labels are used to apply a filter to one or more rows that have to be shown in the pivot table. For instance, if the "Salesperson" field is dragged on this area then the other output table constructed will have values from the column "Salesperson",i.e., one will have a number of rows equal to the number of "Sales Person". There will also be one added row of "Grand Total". In the example above, this instruction will create five rows in the table — one for each salesperson, and Grand Total. There will be a filter above the data — row labels — from which one can select or deselect a particular salesperson for the Pivot table.
This table will not have any numerical values, as no numerical field is selected, but when it is selected, the values will automatically get updated in the Row of "Grand Total".
This usually takes a field that has numerical values that can be used for different types of calculations. However, using text values would also not be wrong; instead of Sum, it will give a count. So, in the example above, if the "Units sold" field is dragged to this area along with the row label of "Salesperson", then the instruction will add a new column, "Sum of units sold", which will have values against each salesperson.
Pivot tables or pivot functionality are an integral part of manyspreadsheet applicationsand somedatabase software, as well as being found in other data visualization tools andbusiness intelligencepackages.
Programming languages and libraries suited to work with tabular data contain functions that allow the creation and manipulation of pivot tables.
Excel pivot tables include the feature to directly query anonline analytical processing(OLAP) server for retrieving data instead of getting the data from an Excel spreadsheet. On this configuration, a pivot table is a simple client of an OLAP server. Excel's PivotTable not only allows for connecting to Microsoft's Analysis Service, but to anyXML for Analysis(XMLA) OLAP standard-compliant server.
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https://en.wikipedia.org/wiki/Pivot_table
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Blum Blum Shub(B.B.S.) is apseudorandom number generatorproposed in 1986 byLenore Blum,Manuel BlumandMichael Shub[1]that is derived fromMichael O. Rabin's one-way function.
Blum Blum Shub takes the form
whereM=pqis the product of two largeprimespandq. At each step of the algorithm, some output is derived fromxn+1; the output is commonly either thebit parityofxn+1or one or more of the least significant bits ofxn+1.
Theseedx0should be an integer that is co-prime toM(i.e.pandqare not factors ofx0) and not 1 or 0.
The two primes,pandq, should both becongruentto 3 (mod 4) (this guarantees that eachquadratic residuehas onesquare rootwhich is also a quadratic residue), and should besafe primeswith a smallgcd((p-3)/2, (q-3)/2) (this makes the cycle length large).
An interesting characteristic of the Blum Blum Shub generator is the possibility to calculate anyxivalue directly (viaEuler's theorem):
whereλ{\displaystyle \lambda }is theCarmichael function. (Here we haveλ(M)=λ(p⋅q)=lcm(p−1,q−1){\displaystyle \lambda (M)=\lambda (p\cdot q)=\operatorname {lcm} (p-1,q-1)}).
There is a proof reducing its security to thecomputational difficultyof factoring.[1]When the primes are chosen appropriately, andO(loglogM) lower-order bits of eachxnare output, then in the limit asMgrows large, distinguishing the output bits from random should be at least as difficult as solving thequadratic residuosity problemmoduloM.
The performance of the BBS random-number generator depends on the size of the modulusMand the number of bits per iterationj. While loweringMor increasingjmakes the algorithm faster, doing so also reduces the security. A 2005 paper gives concrete, as opposed to asymptotic, security proof of BBS, for a givenMandj. The result can also be used to guide choices of the two numbers by balancing expected security against computational cost.[2]
Letp=11{\displaystyle p=11},q=23{\displaystyle q=23}ands=3{\displaystyle s=3}(wheres{\displaystyle s}is the seed). We can expect to get a large cycle length for those small numbers, becausegcd((p−3)/2,(q−3)/2)=2{\displaystyle {\rm {gcd}}((p-3)/2,(q-3)/2)=2}.
The generator starts to evaluatex0{\displaystyle x_{0}}by usingx−1=s{\displaystyle x_{-1}=s}and creates the sequencex0{\displaystyle x_{0}},x1{\displaystyle x_{1}},x2{\displaystyle x_{2}},…{\displaystyle \ldots }x5{\displaystyle x_{5}}= 9, 81, 236, 36, 31, 202. The following table shows the output (in bits) for the different bit selection methods used to determine the output.
The following is aPythonimplementation that does check for primality.
The followingCommon Lispimplementation provides a simple demonstration of the generator, in particular regarding the three bit selection methods. It is important to note that the requirements imposed upon the parametersp,qands(seed) are not checked.
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https://en.wikipedia.org/wiki/Blum_Blum_Shub
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Participatory GIS(PGIS) orpublic participation geographic information system(PPGIS) is aparticipatoryapproach tospatial planningand spatial information andcommunications management.[1][2]
PGIS combinesParticipatory Learning and Action(PLA) methods withgeographic information systems(GIS).[3]PGIS combines a range of geo-spatial information management tools and methods such as sketch maps,participatory 3D modelling(P3DM),aerial photography,satellite imagery, andglobal positioning system(GPS) data to represent peoples' spatial knowledge in the forms of (virtual or physical) two- or three-dimensional maps used as interactive vehicles for spatial learning, discussion, information exchange, analysis, decision making and advocacy.[4]Participatory GIS implies making geographic technologies available to disadvantaged groups in society in order to enhance their capacity in generating, managing, analysing and communicating spatial information.
PGIS practice is geared towards community empowerment through measured, demand-driven, user-friendly and integrated applications of geo-spatial technologies.[citation needed]GIS-based maps and spatial analysis become major conduits in the process. A good PGIS practice is embedded into long-lasting spatial decision-making processes, is flexible, adapts to different socio-cultural and bio-physical environments, depends on multidisciplinary facilitation and skills and builds essentially on visual language. The practice integrates several tools and methods whilst often relying on the combination of 'expert' skills with socially differentiated local knowledge. It promotes interactive participation of stakeholders in generating and managing spatial information and it uses information about specific landscapes to facilitate broadly-based decision making processes that support effective communication and community advocacy.
If appropriately utilized,[5]the practice could exert profound impacts on community empowerment, innovation and social change.[6]More importantly, by placing control of access and use ofculturally sensitivespatial information in the hands of those who generated them, PGIS practice could protect traditional knowledge and wisdom from external exploitation.
PPGIS is meant to bring the academic practices ofGISand mapping to the local level in order to promote knowledge production by local and non-governmental groups.[7]The idea behind PPGIS is empowerment and inclusion of marginalized populations, who have little voice in the public arena, through geographic technology education and participation. PPGIS uses and produces digital maps, satellite imagery, sketch maps, and multiple other spatial and visual tools, to change geographic involvement and awareness on a local level. The term was coined in 1996 at the meetings of theNational Center for Geographic Information and Analysis(NCGIA).[8][9][10]
Attendees to theMapping for Change International Conference on Participatory Spatial Information Management and Communicationconferred to at least three potential implications of PPGIS; it can: (1) enhance capacity in generating, managing, and communicating spatial information; (2) stimulate innovation; and ultimately; (3) encourage positive social change.[11][12]This reflects on the rather nebulous definition of PPGIS as referenced in theEncyclopedia of GIS[13]which describes PPGIS as having a definition problem.
There are a range of applications for PPGIS. The potential outcomes can be applied from community andneighborhood planningand development to environmental andnatural resource management. Marginalized groups, be theygrassrootsorganizations toindigenous populationscould benefit from GIS technology.
Governments,non-government organizationsandnon-profit groupsare a big force behind multiple programs. The current extent of PPGIS programs in the US has been evaluated by Sawicki and Peterman.[14]They catalog over 60 PPGIS programs who aid in "public participation in community decision making by providing local-area data to community groups," in the United States.[15]: 24The organizations providing these programs are mostly universities, localchambers of commerce, non-profitfoundations.
In general, neighborhood empowerment groups can form and gain access to information that is normally easy for the official government and planning offices to obtain. It is easier for this to happen than for individuals of lower-income neighborhoods just working by themselves. There have been several projects where university students help implement GIS in neighborhoods and communities. It is believed[by whom?]that access to information is the doorway to more effective government for everybody and community empowerment. In a case study of a group in Milwaukee, residents of aninner cityneighborhood became active participants in building a community information system, learning to access public information and create and analyze new databases derived from their own surveys, all with the purpose of making these residents useful actors in city management and in the formation of public policy.[16]In a number of cases, there are providers of data for community groups, but the groups may not know that such entities exist. Getting the word out would be beneficial.[citation needed]
Some of the spatial data that the neighborhood wanted was information on abandoned or boarded-up buildings and homes, vacant lots, and properties that contained garbage, rubbish and debris that contributed to health and safety issues in the area. They also appreciated being able to find landlords that were not keeping up the properties. The university team and the community were able to build databases and make maps that would help them find these areas and perform the spatial analysis that they needed. Community members learned how to use the computer resources, ArcView 1.0, and build a theme or land use map of the surrounding area. They were able to perform spatial queries and analyze neighborhood problems. Some of these problems included finding absentee landlords and finding code violations for the buildings on the maps.[16]
There are two approaches to PPGIS use and application. These two perspectives, top–down and bottom–up, are the currently debated schism in PPGIS.
According to Sieber (2006), PPGIS was first envisioned as a means of mapping individuals by multiple social and economic demographic factors in order to analyze the spatial differences in access to social services. She refers to this kind of PPGIS astop-down, being that it is less hands on for the public, but theoretically serves the public by making adjustments for the deficiencies, and improvements in public management.[17]
A current trend with academic involvement in PPGIS, is researching existing programs, and or starting programs in order to collect data on the effectiveness of PPGIS. Elwood (2006) inThe Professional Geographer, talks in depth about the "everyday inclusions, exclusions, and contradictions of Participatory GIS research."[18]The research is being conducted in order to evaluate if PPGIS is involving the public equally. In reference to Sieber's top-down PPGIS, this is a counter method of PPGIS, rightly referred to asbottom-upPPGIS. Its purpose is to work with the public to let them learn the technologies, then producing their own GIS.
Public participation GIS is defined by Sieber as the use of geographic information systems to broaden public involvement in policymaking as well as to the value of GIS to promote the goals of nongovernmental organizations, grassroots groups and community-based organizations.[17]It would seem on the surface that PPGIS, as it is commonly referred to, in this sense would be of a beneficial nature to those in the community or area that is being represented. But in truth only certain groups or individuals will be able to obtain the technology and use it. Is PPGIS becoming more available to the underprivileged sector of the community? The question of "who benefits?" should always be asked, and does this harm a community or group of individuals.
The local, participatory management of urban neighborhoods usually follows on from 'claiming the territory', and has to be made compatible with national or local authority regulations on administering, managing and planning urban territory.[19]PPGIS applied to participatory community/neighborhood planning has been examined by many others.[20][21][22]Specific attention has been given to applications such as housing issues[23]or neighborhood revitalization.[24]Spatial databases along with the P-mapping are used to maintain a public records GIS or community land information systems.[25]These are just a few of the uses of GIS in the community.
Public Participation in decision making processes works not only to identify areas of common values or variability, but also as an illustrative and instructional tool. One example of effective dialogue and building trust between the community and decision makers comes from pre-planning for development in the United Kingdom. It involves using GIS andmulti-criteria decision analysis(MCDA) to make a decision about wind farm siting. This method hinges upon taking all stakeholder perspectives into account to improve chances of reaching consensus . This also creates a more transparent process and adds weight to the final decision by building upon traditional methods such as public meetings and hearings, surveys, focus groups, and deliberative processes enabling participants more insights and more informed opinions on environmental issues.[26]
Collaborative processes that consider objective and subjective inputs have the potential to efficiently address some of the conflict between development and nature as they involve a fuller justification by wind farm developers for location, scale, and design. Spatial tools such as creation of 3D view sheds offer participants new ways of assessing visual intrusion to make a more informed decision. Higgs et al.[26]make a telling statement when analyzing the success of this project – "the only way of accommodating people's landscape concerns is to site wind farms in places that people find more acceptable". This implies that developers recognize the validity of citizens' concerns and are willing to compromise in identifying sites where wind farms will not only be successful financially, but also successful politically and socially. This creates greater accountability and facilitates the incorporation of stakeholder values to resolve differences and gain public acceptance for vital development projects.
In another planning example, Simao et al.[27]analyzed how to create sustainable development options with widespread community support. They determined that stakeholders need to learn likely outcomes that result from stated preferences, which can be supported through enhanced access to information and incentives to increase public participation. Through a multi-criteria spatialdecision support systemstakeholders were able to voice concerns and work on a compromise solution to have final outcome accepted by majority when siting wind farms. This differs from the work of Higgs et al. in that the focus was on allowing users to learn from the collaborative process, both interactively and iteratively about the nature of the problem and their own preferences for desirable characteristics of solution.
This stimulated sharing of opinions and discussion of interests behind preferences. After understanding the problem more fully, participants could discuss alternative solutions and interact with other participants to come to a compromise solution.[27]Similar work has been done to incorporate public participation in spatial planning for transportation system development,[28]and this method of two-way benefits is even beginning to move towards web-based mapping services to further simplify and extend the process into the community.[29]
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https://en.wikipedia.org/wiki/Public_participation_GIS
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This is a list ofpublicationsinstatistics, organized by field.
Some reasons why a particular publication might be regarded as important:
Mathematical Methods of Statistics
Statistical Decision Functions
Testing Statistical Hypotheses
An Essay Towards Solving a Problem in the Doctrine of Chances
On Small Differences in Sensation
Truth and Probability
Bayesian Inference in Statistical Analysis
Theory of Probability
Introduction to statistical decision theory
An Introduction to Multivariate Analysis
Time Series Analysis Forecasting and Control
Statistical Methods for Research Workers
Statistical Methods
Principles and Procedures of Statistics with Special Reference to the Biological Sciences.
Biometry: The Principles and Practices of Statistics in Biological Research
On the uniform convergence of relative frequencies of events to their probabilities
On the mathematical foundations of theoretical statistics
Estimation of variance and covariance components
Maximum-likelihood estimation for the mixed analysis of variance model
Recovery of inter-block information when block sizes are unequal
Estimation of Variance and Covariance Components in Linear Models
Nonparametric estimation from incomplete observations
A generalized Wilcoxon test for comparing arbitrarily singly-censored samples
Evaluation of survival data and two new rank order statistics arising in its consideration
Regression Models and Life Tables
The Statistical Analysis of Failure Time Data
Report on Certain Enteric Fever Inoculation Statistics
The Probability Integral Transformation for Testing Goodness of Fit and Combining Independent Tests of Significance
Combining Independent Tests of Significance
The combination of estimates from different experiments
On Small Differences in Sensation
The Design of Experiments
The Design and Analysis of Experiments
On the Experimental Attainment of Optimum Conditions (with discussion)
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https://en.wikipedia.org/wiki/List_of_important_publications_in_statistics#Multivariate_analysis
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Aglossary(fromAncient Greek:γλῶσσα,glossa; language, speech, wording), also known as avocabularyorclavis, is an alphabetical list oftermsin a particulardomain of knowledgewith thedefinitionsfor those terms.[citation needed]Traditionally, a glossary appears at the end of abookand includes terms within that book that are either newly introduced, uncommon, or specialized. While glossaries are most commonly associated withnon-fictionbooks, in some cases,fictionnovels sometimes include a glossary for unfamiliar terms.
A bilingual glossary is a list of terms in one language defined in a second language orglossedbysynonyms(or at least near-synonyms) in another language.
In a general sense, a glossary contains explanations ofconceptsrelevant to a certain field of study or action. In this sense, the term is related to the notion ofontology. Automatic methods have been also provided that transform a glossary into an ontology[1]or a computational lexicon.[2]
Acore glossaryis a simple glossary orexplanatory dictionarythat enables definition of other concepts, especially for newcomers to a language or field of study. It contains a small working vocabulary and definitions for important or frequently encountered concepts, usually including idioms or metaphors useful in a culture.
Computational approachesto the automated extraction of glossaries from corpora[3]or the Web[4][5]have been developed in the recent years[timeframe?]. These methods typically start from domainterminologyand extract one or more glosses for each term of interest. Glosses can then be analyzed to extracthypernymsof the defined term and other lexical and semantic relations.
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https://en.wikipedia.org/wiki/Glossary
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Alanguage modelis amodelof natural language.[1]Language models are useful for a variety of tasks, includingspeech recognition,[2]machine translation,[3]natural language generation(generating more human-like text),optical character recognition,route optimization,[4]handwriting recognition,[5]grammar induction,[6]andinformation retrieval.[7][8]
Large language models(LLMs), currently their most advanced form, are predominantly based ontransformerstrained on larger datasets (frequently using wordsscrapedfrom the publicinternet). They have supersededrecurrent neural network-based models, which had previously superseded the purely statistical models, such aswordn-gram language model.
Noam Chomskydid pioneering work on language models in the 1950s by developing a theory offormal grammars.[9]
In 1980, statistical approaches were explored and found to be more useful for many purposes than rule-based formal grammars. Discrete representations likewordn-gram language models, with probabilities for discrete combinations of words, made significant advances.
In the 2000s, continuous representations for words, such asword embeddings, began to replace discrete representations.[10]Typically, the representation is areal-valuedvector that encodes the meaning of the word in such a way that the words that are closer in the vector space are expected to be similar in meaning, and common relationships between pairs of words like plurality or gender.
In 1980, the first significant statistical language model was proposed, and during the decade IBM performed ‘Shannon-style’ experiments, in which potential sources for language modeling improvement were identified by observing and analyzing the performance of human subjects in predicting or correcting text.[11]
Awordn-gram language modelis a purely statistical model of language. It has been superseded byrecurrent neural network–based models, which have been superseded bylarge language models.[12]It is based on an assumption that the probability of the next word in a sequence depends only on a fixed size window of previous words. If only one previous word is considered, it is called a bigram model; if two words, a trigram model; ifn− 1 words, ann-gram model.[13]Special tokens are introduced to denote the start and end of a sentence⟨s⟩{\displaystyle \langle s\rangle }and⟨/s⟩{\displaystyle \langle /s\rangle }.
Maximum entropylanguage models encode the relationship between a word and then-gram history using feature functions. The equation is
P(wm∣w1,…,wm−1)=1Z(w1,…,wm−1)exp(aTf(w1,…,wm)){\displaystyle P(w_{m}\mid w_{1},\ldots ,w_{m-1})={\frac {1}{Z(w_{1},\ldots ,w_{m-1})}}\exp(a^{T}f(w_{1},\ldots ,w_{m}))}
whereZ(w1,…,wm−1){\displaystyle Z(w_{1},\ldots ,w_{m-1})}is thepartition function,a{\displaystyle a}is the parameter vector, andf(w1,…,wm){\displaystyle f(w_{1},\ldots ,w_{m})}is the feature function. In the simplest case, the feature function is just an indicator of the presence of a certainn-gram. It is helpful to use a prior ona{\displaystyle a}or some form ofregularization.
The log-bilinear model is another example of an exponential language model.
Skip-gram language model is an attempt at overcoming the data sparsity problem that the preceding model (i.e. wordn-gram language model) faced. Words represented in an embedding vector were not necessarily consecutive anymore, but could leave gaps that areskippedover (thus the name "skip-gram").[14]
Formally, ak-skip-n-gram is a length-nsubsequence where the components occur at distance at mostkfrom each other.
For example, in the input text:
the set of 1-skip-2-grams includes all the bigrams (2-grams), and in addition the subsequences
In skip-gram model, semantic relations between words are represented bylinear combinations, capturing a form ofcompositionality. For example, in some such models, ifvis the function that maps a wordwto itsn-d vector representation, then
v(king)−v(male)+v(female)≈v(queen){\displaystyle v(\mathrm {king} )-v(\mathrm {male} )+v(\mathrm {female} )\approx v(\mathrm {queen} )}
Continuous representations orembeddings of wordsare produced inrecurrent neural network-based language models (known also ascontinuous space language models).[17]Such continuous space embeddings help to alleviate thecurse of dimensionality, which is the consequence of the number of possible sequences of words increasingexponentiallywith the size of the vocabulary, further causing a data sparsity problem. Neural networks avoid this problem by representing words as non-linear combinations of weights in a neural net.[18]
Alarge language model(LLM) is a type ofmachine learningmodeldesigned fornatural language processingtasks such as languagegeneration. LLMs are language models with many parameters, and are trained withself-supervised learningon a vast amount of text.
Although sometimes matching human performance, it is not clear whether they are plausiblecognitive models. At least for recurrent neural networks, it has been shown that they sometimes learn patterns that humans do not, but fail to learn patterns that humans typically do.[22]
Evaluation of the quality of language models is mostly done by comparison to human created sample benchmarks created from typical language-oriented tasks. Other, less established, quality tests examine the intrinsic character of a language model or compare two such models. Since language models are typically intended to be dynamic and to learn from data they see, some proposed models investigate the rate of learning, e.g., through inspection of learning curves.[23]
Various data sets have been developed for use in evaluating language processing systems.[24]These include:
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Aquineis acomputer programthat takes no input and produces a copy of its ownsource codeas its only output. The standard terms for these programs in thecomputability theoryandcomputer scienceliterature are "self-replicating programs", "self-reproducing programs", and "self-copying programs".
A quine is afixed pointof an execution environment, when that environment is viewed as afunctiontransforming programs into their outputs. Quines are possible in anyTuring-completeprogramming language, as a direct consequence ofKleene's recursion theorem. For amusement, programmers sometimes attempt to develop the shortest possible quine in any givenprogramming language.
The name "quine" was coined byDouglas Hofstadter, in his popular 1979 science bookGödel, Escher, Bach, in honor of philosopherWillard Van Orman Quine(1908–2000), who made an extensive study ofindirect self-reference, and in particular for the following paradox-producing expression, known asQuine's paradox:
"Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.
John von Neumanntheorized aboutself-reproducing automatain the 1940s. Later, Paul Bratley and Jean Millo's article "Computer Recreations: Self-Reproducing Automata" discussed them in 1972.[1]Bratley first became interested in self-reproducing programs after seeing the first known such program written inAtlas Autocodeat Edinburgh in the 1960s by theUniversity of Edinburghlecturer and researcherHamish Dewar.
The "download source" requirement of theGNU Affero General Public Licenseis based on the idea of a quine.[2]
In general, the method used to create a quine in any programming language is to have, within the program, two pieces: (a)codeused to do the actual printing and (b)datathat represents the textual form of the code. The code functions by using the data to print the code (which makes sense since the data represents the textual form of the code), but it also uses the data, processed in a simple way, to print the textual representation of the data itself.
Here are three small examples in Python3:
The followingJavacode demonstrates the basic structure of a quine.
The source code contains a string array of itself, which is output twice, once inside quotation marks.
This code was adapted from an original post from c2.com, where the author, Jason Wilson, posted it as a minimalistic version of a Quine, without Java comments.[3]
Thanks to newtext blocksfeature in Java 15 (or newer), a more readable and simpler version is possible:[4]
The same idea is used in the followingSQLquine:
Some programming languages have the ability to evaluate a string as a program. Quines can take advantage of this feature. For example, thisRubyquine:
Luacan do:
In Python 3.8:
In many functional languages, includingSchemeand otherLisps, and interactive languages such asAPL, numbers are self-evaluating. InTI-BASIC, if the last line of a program returns a value, the returned value is displayed on the screen. Therefore, in such languages a program consisting of only a single digit results in a 1-byte quine. Since such code does notconstructitself, this is often considered cheating.
In some languages, particularlyscripting languagesbut alsoC, an empty source file is a fixed point of the language, being a valid program that produces no output.[a]Such an empty program, submitted as "the world's smallest self reproducing program", once won the "worst abuse of the rules" prize in theInternational Obfuscated C Code Contest.[5]The program was not actually compiled, but usedcpto copy the file into another file, which could be executed to print nothing.[6]
Quines, per definition, cannot receiveanyform of input, including reading a file, which means a quine is considered to be "cheating" if it looks at its own source code. The followingshellscript is not a quine:
A shorter variant, exploiting the behaviour ofshebangdirectives:
Other questionable techniques include making use of compiler messages; for example, in theGW-BASICenvironment, entering "Syntax Error" will cause the interpreter to respond with "Syntax Error".
Quine code can also be outputted visually, for example it's used to visualize the neutral zone inYars' Revenge, along withsyntactic saccharin, to obfuscate the source code.
The quine concept can be extended to multiple levels of recursion, giving rise to "ouroborosprograms", or quine-relays. This should not be confused withmultiquines.
This Java program outputs the source for a C++ program that outputs the original Java code.
Such programs have been produced with various cycle lengths:
David Madore, creator ofUnlambda, describes multiquines as follows:[16]
"A multiquine is a set of r different programs (in r different languages – without this condition we could take them all equal to a single quine), each of which is able to print any of the r programs (including itself) according to the command line argument it is passed. (Cheating is not allowed: the command line arguments must not be too long – passing the full text of a program is considered cheating)."
A multiquine consisting of 2 languages (or biquine) would be a program which:
A biquine could then be seen as a set of two programs, both of which are able to print either of the two, depending on the command line argument supplied.
Theoretically, there is no limit on the number of languages in a multiquine.
A 5-part multiquine (or pentaquine) has been produced withPython,Perl,C,NewLISP, andF#[17]and there is also a 25-language multiquine.[18]
Similar to, but unlike a multiquine, apolyglotprogram is a computer program or script written in a valid form of multiple programming languages or file formats by combining their syntax. A polyglot program is not required to have a self-reproducing quality, although a polyglot program can also be a quine in one or more of its possible ways to execute.
Unlike quines and multiquines, polyglot programs are not guaranteed to exist between arbitrary sets of languages as a result of Kleene's recursion theorem, because they rely on the interplay between the syntaxes, and not a provable property that one can always be embedded within another.
A radiation-hardened quine is a quine that can have any single character removed and still produces the original program with no missing character. Of necessity, such quines are much more convoluted than ordinary quines, as is seen by the following example inRuby:[19]
Usingrelational programmingtechniques, it is possible to generate quines automatically by transforming the interpreter (or equivalently, the compiler and runtime) of a language into a relational program, and then solving for afixed point.[20]
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Inbroadcasting, the termblackoutrefers to the non-airing oftelevisionorradioprogramming in a certainmedia market.[1]
It is particularly prevalent in thebroadcasting of sports events, although other television or radio programs may be blacked out as well. Most blackout policies serve to protect local broadcasters (primarilyregional sports networks) from competition by "out-of-market" networks that carry different teams, by only allowing viewers to watch non-national telecasts of teams within their designated markets (with television providers blacking out regional telecasts of teams that are outside their market; in turn, encouraging viewers to purchase subscription-basedout-of-market sports packages), and by allowing teams to black out national telecasts of games that are also being shown by a local broadcaster. In these situations, the national stations would close in those areas for the duration of the game, and in some cases be replaced with other stations until the game ends.
By contrast, some blackout policies, such as those of the U.S.National Football Leagueand Englishassociation football(soccer), serve to encourage attendance to games by respectively requiring that a specific percentage of tickets be sold in order for a game to be televised in the home team's market, or by enforcing a blanket prohibition on any domestic telecasts of the sport during specific windows.
The term is also used in relation to situations where programming is removed or replaced on international feeds of a television service, because the broadcaster does not hold the territorial rights to air the programs outside of their home country. In some cases, replacement programming airs, but when there's no replacement programming required, the feed would temporarily close, and would not resume broadcasting until the next programme was due to begin.
Perhaps the most notable non-sports-related blackout in television was the blackout ofCanadian federal electioncoverage. Because there are sixtime zonesacross Canada, polls close in different parts of the country at different times.Section 329 of the Canada Elections Actoutlawed disseminating election results from other ridings in constituencies where polls were still open, ostensibly to prevent the results fromthe Eastfrom influencing voters inwestern ridings.[2]
However, in thefederal election in 2000, Paul Charles Bryan published results fromAtlantic Canadaonline despite being told not to by the authorities. Bryan was charged before theProvincial Court of British Columbia, but fought the charges as unconstitutional undersection 2of theCanadian Charter of Rights and Freedoms, which protects freedom of expression andfreedom of association. Bryan's victory before theBritish Columbia Supreme Courtmeant that voters inBritish Columbiaand the rest of Canada legally learned of election results in other ridings during thefederal election in 2004. However, Elections Canada appealed, and Bryan lost his case before theBritish Columbia Court of Appeal. Bryan further appealed to theSupreme Court of Canada, but in a ruling made on March 15, 2007 (R. v. Bryan), in a 5–4 ruling, the Court ruled that Section 329 of the Canada Elections Act isconstitutionaland justified undersection 1of theCanadian Charter of Rights and Freedoms.Stephen Harper, who later became Prime Minister, labelledElections Canada"jackasses" and tried to raise money for Bryan. TheCanadian Broadcasting Corporationalso supported Bryan, hoping to "make election night a bigger event than it already is".[3]
Before the 2000 election, Elections Canada moved to reduce the effects of the blackout and the influence of unauthorized knowledge of election results in Western ridings by altering the times that polls close, so that polls no longer close at the same local time throughout the country. Polls inAtlantic Canadaclose at 9 p.m.Atlantic(9:30in Newfoundland), polls fromAlbertatoQuebecclose an hour later (9 p.m.Eastern, 8 p.m.Centraland 7 p.m.Mountain) and finally, polls in British Columbia close an hour after that (7 p.m.Pacific). Historically, the results of the election are often not decisively known until more than an hour after polls close in the Eastern Time Zone, but are usually known within two hours of these polls closing.
Provincial elections are not subject to blackout restrictions – in provinces that have two time zones, the vast majority of the population lives in one time zone or the other. Election laws in these provinces stipulate that all polls are to close at the same time – this time invariably being 8:00 p.m. (or 9:00 p.m. in Ontario beginning with the2007 provincial election) in the time zone of the majority.
On August 17, 2011, Elections Canada Chief Electoral OfficerMarc Mayrandsuggested improvements of the voting system to Parliament; among them were a proposal to remove the blackout rule. Mayrand argued that "the growing use ofsocial mediaputs in question not only the practical enforceability of the rule, but also its very intelligibility and usefulness in a world where the distinction between private communication and public transmission is quickly eroding. The time has come for Parliament to consider revoking the current rule."[4][5]On January 13, 2012, it was announced that the federal government would introduce legislation that would repeal the blackout rule, citing the increased use of social media. The blackout rule was officially repealed in October 2015, prior to the2015 Canadian federal election.[2]
TheCanadian Football League's constitution does provide the option for teams to black out games in their home markets in order to encourage attendance; at one point, the CFL required games to be blacked out within a radius of 120 kilometres (75 miles) around the closest over-the-air signal carrying the game, or 56 kilometres (35 miles) of the stadium for cable broadcasts (and, for theSaskatchewan Roughriders, the entirety of theprovince).[6][7]
The policy received significant criticism in 2002 when theHamilton Tiger-Catsenforced a blackout on a game against theToronto Argonautsthat had playoff implications; the range of the blackout was considered too wide for the market.[7]
Under the league's 2008–2013 contract withTSN, teams were given a cap on the number of blackouts they could impose per-season (with the number varying by media and CFL reports, ranging from 2 for Hamilton and Toronto, and 5 for teams in Western Canada), and final decisions were assigned to the league if at least 90% of tickets were sold out within 48 hours of the game. Although the CFL stated that the league's current contract with TSN (which began in 2014) does allow for blackouts, they have been seldom-used, if not at all.[6][8]
As in the U.S., National Hockey League games that are not scheduled as national telecasts bySportsnetorTVA Sportsare broadcast byregional feedsof either Sportsnet,TSN, orRDS(French), and are blacked out for viewers outside the team's home market. Sportsnet's four regional feeds correspond with each of its NHL teams' designated markets; the Ontario and Pacific feeds are designated to theToronto Maple Leafs, andVancouver Canucksrespectively, while Sportsnet West and its corresponding market (which includes all ofAlbertaandSaskatchewan) is shared by theEdmonton OilersandCalgary Flames. Although West is also the main feed forManitoba, Flames and Oilers games are blacked out there to protect theWinnipeg Jets. As of August 2014, TSN is similarly structured, with the Ottawa Senators on TSN5 (East), Maple Leafs on TSN4 (Ontario), and Jets on TSN3 (Manitoba and Saskatchewan). TheMontreal Canadienswere added in 2017 onTSN2(which was originally promoted as being a secondary national channel).[9]The Canadiens and Senators share the same market, which includes parts of Eastern Ontario (primarily theOttawa Valley), and the entirety of Quebec and Atlantic Canada, while Saskatchewan is shared by the Jets, Flames, and Oilers.[10][11]
Until the 2014–15 season, allFrench-languagebroadcasts of theMontreal Canadienswere available nationally on RDS, which was previously the national French-language rightsholder of the NHL in Canada. As RDS was, until 2011, the only French-language cable sports channel in Canada,[12]the team forwent a separate regional rights deal and allowed all of its games to be broadcast as part of the national package. As of the 2014–15 season,Quebecor MediaandTVA Sportsis the national French rightsholder as part of a sub-licensing agreement withRogers Communications.[13][14][15]RDS negotiated a 12-year deal with the team for regional rights to the Canadiens: games are now blacked out for viewers outside Quebec, Atlantic Canada, and parts of Eastern Ontario.[11][16]
Out-of-market games can be viewed using the subscription-basedNHL Centre Iceand Sportsnet+; in-market games are blacked out from Centre Ice to protect local broadcasters,[17][18][19]but Sportsnet+ does not black out in-market broadcasts of games televised by Sportsnet since it is a direct-to-consumer version of the Sportsnet channels themselves.[20][21]
Many programs carried onInternet televisionin other parts of the world are not available in Canada because the major broadcast networks in Canada secure exclusive rights to them and prevent Internet television aggregators, one notable example beingHulu, from distributing them in Canada. TheNational Football League, for example, sold worldwide Internet broadcast rights to a package of itsThursday Night Footballgames during the2016 seasontoTwitter; however,Rogers Mediaforced Twitter to block the streams in Canada by virtue of its holding of terrestrial television rights in the country.[22]Numerous organizations have attempted to establish workarounds that route Canadians' Internet traffic through the United States, workarounds that local broadcasters have opposed, with one,Bell Media, calling such practices "stealing",[23]and that aggregators such asNetflixhave actively fought against.[24]
Indian lawrequiresall sporting events of "national importance", whose broadcast rights are owned by a pay television service, to be simulcast by the state broadcasterDoordarshan(DD) on itsDD NationalTV channel.Tata Sky(which is partially owned by the parent company ofStar India, owner of theStar Sportsnetworks) filed a lawsuit over the rule, arguing that these simulcasts devalued the exclusive broadcast rights because DD National is a must-carry channel. In 2017, theSupreme Court of Indiaruled that pay television services must black out DD National when it is airing such events in order to protect the pay TV broadcaster, restricting availability of DD's simulcasts of such events to terrestrial television andDD Free Dish.[25][26]
UEFAArticle 48.2 and the majorassociation footballleagues of the United Kingdom enforce a blackout on all television broadcasts of football between 2:45 p.m. and 5:15 p.m. on Saturday matchdays. This applies to all matches, regardless of whether they are a domestic or international competition. A match which kicks off within the window may be joined in progress once the blackout window ends.[27][28]
This policy is ostensibly intended to encourage fans to attend football matches in-person, especially in lower divisions that compete with top-flight matches on television. The practice originated in the 1960s;BurnleychairmanBob Lordwas opposed to television broadcasts of football matches — going as far as banning theBBCfrom televisingMatch of the DayfromTurf Moorfor a time. He pushed theFootball Leagueto adopt this stance as an organization-wide policy; it has since been adopted byThe Football Associationand the currentPremier League, which broke away from the Football League in 1992 to become the highest level of club football in England.[29][30][31]
Affected matches can still be broadcast internationally, hence more Premier League matches can be shown outside the United Kingdom by other rightsholders than within. This intricacy created a "grey market" for obtaining the broadcasts from alternative sources, such as foreign satellite providers or unofficial online streaming services. The Premier League and other stakeholders have historically considered this practice to be a violation of thecopyrightof the broadcasts. In 2014, for taking inadequate steps to prevent unauthorized retransmissions from its streaming broadcasts online, the Premier League briefly restrictedMENAregion rightsholderbeIN Sportsto one 3 p.m. match per week on television only.[29][30][31]
Critics, includingAdvocate Generalat theCourt of Justice of the European UnionJuliane Kokott, have argued that 3 p.m. blackouts are outdated, as its purpose is hindered — especially within the Premier League — by the high demand for the few tickets available to the public, and that there was little evidence that television broadcasts actually affected attendance.[32][33][31][34][35]To preserve the value of its domestic broadcast rights and allow more games to be televised, the Premier League has added more matches in windows outside of Saturday afternoons, such as weekdays and Sundays — including thefinal matchday of the season.[34][35]
In 2018, after complying by blacking out the first 15 minutes of aSerie Amatch that sawCristiano Ronaldo's on-field debut forJuventus, streaming serviceEleven Sports UK & Irelandbegan to defy the ban and show selected Serie A andLa Ligamatches during this period. On 17 October 2018, Eleven announced that it would cease its telecasts of 3 p.m. kickoffs, but argued that the rule was outdated because only the UK andMontenegrohave such blackout rules, and that the blackout period encourages illegal streaming. A representative of La Liga has backed Eleven Sports' position.[36][37][38]
In April 2020, due to theCOVID-19 pandemic, UEFA authorised the suspension of the blackout rule for the remainder of the season.[39][40]Upon the resumption of the2019–20 Premier League, all matches were shown on domestic television due to them being playedbehind closed doors, while a number of free-to-air broadcasts (viaSky Sports' sister channelsPickandSky One,Amazon Prime Videoand its sister serviceTwitch, and the BBC — which usually holds rights to free-to-air highlights programmes) were also aired.[41][42][43]This arrangement continued into the first month of the2020–21 Premier League.[44][45]After an attemptedpay-per-viewscheme folded in November 2020, the Premier League returned to allocating the matches to the four broadcasters through at least the end of 2020.[46][47][48][49]
In 2023, the Premier League sought a rare private prosecution against members of a fraud "gang" who sold £10-a-month subscriptions to retransmitted games. The illegal streams brought in more than £7m in revenue from more than 50,000 subscribers, with five members receiving jail sentences between three and eleven years.[50]
Major League Baseballand theNational Hockey Leaguehave very similar blackout rules. Unlike theNational Football League, the blackout of games has nothing to do with attendance, but instead is implemented to protect broadcasters with contracts to air games. Unless one of MLB's national partners hold exclusive rights to a certain regular season game (such as ESPN'sSunday Night Baseballor Apple TV+’sFriday Night Baseball), the local broadcaster of a game has priority over a national broadcaster, and the national broadcast would be blacked out in markets where a local broadcaster is also showing coverage.[51][52]The blackout rules do not apply during the postseason, as there are no regional television broadcasts.
The NHL utilizes a similar policy of exclusive and non-exclusive national games; with the new broadcast deals enacted with2021–22 season, all regular season games carried byABC, ESPN, and ESPN+are exclusive national broadcasts. AllTNTgames were exclusive national broadcasts during the 2021-22 season, but became subject to blackouts the following season.[53]In some cases, national games are scheduled in windows where no other games involving U.S. teams are being played. NHL Network still carries non-exclusive national games, most of which are simulcast from one of the regional broadcasts or aCanadian national broadcast.[54]All games in the first round of theStanley Cup playoffsare non-exclusive national games (though with no blackouts of the national broadcaster), after which they are exclusive to ESPN, TNT, or TBS.[55][56]
Out-of-market games can be viewed using the subscription-basedMLB Extra Innings,MLB.tv, andNHL Center Iceservices, as well as ESPN+ for the NHL. In-market games are blacked out from all four services to protect local broadcasters, and they do not offer nationally televised games (except for NHL games exclusively carried or simulcast by ESPN+).
In Major League Baseball, there are no radio blackouts. However, for many years, the local radio networks of the two participating ballclubs in theWorld Serieswere not allowed to air games, forcingflagshipstations, if they wanted to carry the Series, to simulcast the network broadcast. As an example, whileBoston Red Soxradio flagshipWHDHandSt. Louis Cardinalsflagship stationKMOXboth broadcast the1967 World Series, both stations had to simulcast theNBC Radiobroadcast along with Boston'sWCOPand St. Louis'sKSD, the nominal NBC Radio affiliates in those cities.
This changed after1980, as fans of thePhiladelphia Phillieswere angry that they could not hear their popular broadcasting team ofHarry KalasandRichie Ashburncall the team's appearance in that year's World Series. Their complaints led to a provision in Major League Baseball's next broadcasting contract permitting the radio flagships of the participating ballclubs to produce and air their own Series broadcasts locally.[57]Since then, only the flagship stations of the two participating ballclubs can originate coverage (though their broadcasts, as well as the national English and Spanish broadcasts, are also available out of market via subscription-based packages on such platforms asMLB.com,Sirius XM, andTuneIn). Flagship stations are required to make mention of the presenting sponsor of the national ESPN Radio broadcasts as also sponsoring the team's own broadcasts during the World Series (as of 2016 this isAutoZone). All other network affiliates of the two clubs must carry the feed from MLB's national partner (currently ESPN Radio). Should another ESPN Radio affiliate exist in the same market, that station can claim exclusivity, forcing a blackout of the team network affiliate from carrying the game, although this is rarely done as listener pushback against the ESPN Radio affiliate blocking the local play-by-play would likely be untenable (for instance in2016, ESPN Radio O&OWMVPinChicagobroadcast the national ESPN feed as expected, but made no move to block the officialCubsbroadcasterWSCRfrom carrying local play-by-play, to the point of only mentioning the national coverage existed on their station through promos in national ESPN Radio programming).
Additionally, radio stations (including flagships) may not include MLB games in the liveInternetstreamsof their station programming. MLB itself offers radio feeds as a pay service via the league and team websites, along with being a part of the monthly premium fee service from streaming providerTuneIn. Some stations will simply stream the station's regularly scheduled programming that is being pre-empted by the game.
The NHL has no radio blackouts for local broadcasts, althoughNBC Sports Radiobroadcasts are, similarly to some cable broadcasts, not carried within the local markets of participating teams. Internet streaming of radio calls from the NHL's team radio networks, unlike MLB, are allowed to be broadcast for free nationwide with no geoblocking. Also, unlike other leagues, the Stanley Cup Finals (should a team make it to that point in the playoffs) can also be carried on all affiliates of that team's radio network with no restrictions.
Prior to the1998-99 NBA lockout, the NBA and the WNBA used to black out nationally televised games oncable televisionwithin 35 miles (56 km) of the home team's market; however, these are now restricted to games onNBA TV, WatchESPN and other streaming providers.
The NFL has engaged in various blackout policies to protect both local ticket sales, and local rightsholders of specific games.
In the NFL, any broadcaster that has a signal that hits any area within a 75 miles (121 km) radius of an NFLstadiummay only broadcast a game if that game is a road game (also known as an away game), or if the game sells out 72 hours or more before the start time for the game.[58][59]If sold out in less than 72 hours, or is close to being sold out by the deadline, the team can sometimes request a time extension. Furthermore, broadcasters with NFL contracts are required to show their markets' road games, even if the secondary markets have substantial fanbases for other teams (like inHarrisburg, Pennsylvania, officially aBaltimore Ravenssecondary market, but home to manyPittsburgh Steelersfans[citation needed]). Sometimes[when?]if a game is within a few hundred tickets of selling out, a broadcaster[example needed]with rights to show the nearly sold-out game will buy the remaining tickets (and give them to local charities) so it can broadcast the game. Other teams elect to close off sections of their stadium, but cannot sell these tickets for any game that season if they choose to do so.[60]As a result, if the home team's game is a Sunday day game, both networks can air only one game each in that market (until 2000, this rule applied whether or not the game was blacked out; however, this was changed because some markets virtually never aired doubleheaders as a result). Usually, but not always, when each network can show only one game each in a market, the two stations work out between themselves which will show an early game and which will show a late game. This only affects the primary market, and not markets in a 75-mile (121 km) radius, which always get a doubleheader each Sunday. For theNFL International Series, the network broadcasting an International Series game will not have the game blacked out for the team's markets as the game is played outside of the United States; however, some blackout regulations do apply.
There have been two exceptions to the rule, of which one has never been implemented and the other no longer applies. The first is for theGreen Bay Packers, which have two overlapping 75-mile blackout zones – one surroundingthe team's stadiuminGreen Bayand another surroundingMilwaukee. The team'sradio flagship stationis in Milwaukee, and the Packers played part of their home schedule in Milwaukee from 1953 through 1994. However, this policy has never been implemented in the Packers' case, as they have sold out every home game in Green Bay since 1960 and have a decades-long season-ticket waiting list (games in Milwaukee also sold out during this period). The second exception was for theBills Toronto Series; by a technicality,Rogers Communications(the team's lessee) owned all tickets to those games and resold them to potential fans. Even when Rogers failed to sell all of the tickets, they were still technically defined to be sellouts by the league since Rogers was said to have "bought" the tickets. The technicality came into play for both Toronto Series preseason games, and again for the last two regular season games of the series.[61][62]The Bills Toronto Series was cancelled after the 2013 season, largely due to the aforementioned lackluster attendance.
In June 2012, NFL blackout regulations were revised in which, for the first time in NFL history, home games would no longer require a total sellout to be televised locally; instead, teams would be allowed to set a benchmark anywhere from 85 to 100 percent of the stadium's non-premium seats. Any seats sold beyond that benchmark are subject to heavierrevenue sharingwith the league.[63]Four teams, the Buffalo Bills, theCleveland Browns, theIndianapolis Coltsand theSan Diego Chargers, opted out of the new rules, as it would require the teams to pay a higher percentage of gate fees to the NFL's revenue fund.[64]In the2013 NFL season, theOakland Raidersbegan to artificially limit the capacity ofOakland Coliseumby 11,000 in order to improve their chances of meeting the 85% threshold; the seats comprised sections of "Mount Davis", an extended upper deck that had originally been built as part of the Raiders' 1995 return to Oakland. Under NFL rules, the stadium had to remain in this configuration for the entirety of the season.[65]
In the2015 NFL season, the league, after no games were blacked out at all in the2014 season, voted to "suspend" the blackout policy as an experiment.[66]The suspension continued into the2016 season(a season that included the return of theRamsto theLos Angeles Memorial Coliseumas an interim home until the completion ofSoFi Stadium; the Coliseum has had long-standing issues with NFL sell-outs); commissioner Roger Goodell stated that the league needed to further investigate the impact of removing the blackout rules before such a change is made permanent.[67]The suspension quietly continued into the2017 NFL seasonas well, which saw theSan Diego Chargersalso relocate to Los Angeles, temporarily using the 27,000-seat,soccer-specificDignity Health Sports Park(known as StubHub Center before 2019) as an interim venue until the completion of SoFi Stadium for the2020 season, which is shared with the Rams.[68]
The suspension came a year after theFederal Communications Commission(FCC) ended a policy that formally forbade multichannel television providers from distributing telecasts of sporting events that had been blacked out by local broadcast television stations. Then-FCC chairmanTom Wheelerconsidered such policies to be "obsolete".[69]The policies are still enforced via contractual agreements between the NFL and its media partners.[70][71][72]
Per NFL policies, all games that are exclusively televised on pay television or streaming, includingESPN'sMonday Night FootballandAmazon Prime Video’sThursday Night Footballaresyndicatedto over-the-air broadcasters in the markets of the teams involved, and blacked out on the cable channel in defense of the local simulcast. The local market for these rights is defined as any station within the 75-mile (121 km) radius of a team's respective stadium. When this happens, the cable network affected closes in the region, with cable operators choosing to either leave the space blank for the duration of the game, or replacing it with a relay of another station.
This policy attracted controversy in December 2007, whenHartford, ConnecticutCBSaffiliateWFSBwas refused permission to air the local simulcast of aNew England Patriots-New York Giantsgame on December 29, 2007. The game, which was part of theThursday Night Footballpackage on NFL Network, would see the Patriots attempt to become the first NFL team since 1972 and the expansion of the regular season to 16 games, to finish the regular season undefeated. At the time, NFL Network was available only on a sports tier of cable providerComcastin the immediate viewing areas of the Patriots and Giants.[73]SenatorJohn Kerryand Rep.Ed Markey, both of the state ofMassachusettsand fans of the Patriots team, wrote to the NFL as well as Comcast andTime Warner Cable, to request that the Patriots-Giants game be aired at least onbasic cablein order to reach the highest possible number of television-viewing fans, citing the "potentially historic" nature of the game.[74]Kerry clarified the next week that he did not intend to interrupt current negotiations between the cable operators and NFL.[75]
On December 19, 2007,Joe Courtneyand other members of the Connecticut Congressional Delegation wrote to NFL commissionerRoger Goodellto try to have the NFL allow wider broadcast access to the game.[73]Consequently, on December 26, the NFL announced that the game would be simulcast nationally onCBSandNBC, in addition toWCVB-TV(ABC) in Boston andWWOR-TV(MyNetworkTV) in Secaucus, New Jersey (which is part of the New York City media market)—which had both acquired the local rights to the game.[76]
Although NFL Network would later become more established, in 2014 the NFL began to sub-license the right to produce theThursday Night Footballtelecasts, and air selected games from the package in simulcast with NFL Network, to a broadcast television rightsholder (initiallyCBS). This was part of a move to help heighten the profile of the fledgling Thursday night games.[77][78]
For radio broadcasts, the NFL follows a nearly identical policy to MLB. There are no radio blackouts, but only each team's flagship station can carry local broadcasts during the conference championships orSuper Bowl. All other markets must carry theNFL on Westwood Onefeed for those games. For all other weeks, within 75 miles of a team's stadium, only stations the team or its flagship station contracts with can carry those games, regardless if the team is home or away. Thus, any competing station that carries Westwood One broadcasts cannot air those games. Like MLB, the NFL makes local broadcasts (except for those of theTennessee Titans) available on NFL's Game Pass service andSirius Satellite Radio; as a result, radio stations that carry NFL games, from any source, and stream on the Internet are prohibited from streaming games online outside of their DMA, although it seems this provision is loosely enforced in some cases;WBBMin Chicago andWWL (AM)in New Orleans regularly air live broadcasts of their teams' games over their Internet stream, as doesWTMJin Milwaukee with the Packers, though both stations went to a desktop-only streaming policy in 2015 due to the introduction of GamePass and the absorption of theNFL Audio Passstreaming system into Game Pass. Since the 2022-23 season, WXTB (the Bucs' flagship station) blacks out coverage on all devices unless in the station's coverage area, likely due to the launch of NFL+.
In order to protecthigh schoolandcollege football, the federalSports Broadcasting Act of 1961cancels antitrust protection for television broadcasts of any professional football game on Friday evenings or Saturdays by television stations within 75 miles (121 km) of the venue of a college or high school game, that had been announced in a general circulation newspaper prior to August 1 of the calendar year. This lasts from the second Friday in September through the second Friday in December.[79][80]
To comply with this law, the NFL largely avoids scheduling games on Saturdays altogether until the final weeks of the regular season (which begin in mid-December), which usually feature several Saturday double- or triple-headers.[81][79][82][83]A notable effect of this law occurred in the2004 NFL season, where aTennessee Titans/Miami Dolphinsgame in week 1—which had been moved up to Saturday, September 11, due toHurricane Ivan; presumably to comply with the Act, the game was only broadcast locally, and blacked out onNFL Sunday Ticket.[79]
To encourage local attendance, the live television broadcast of theIndianapolis 500is blacked out on theIndianapolisaffiliate of its broadcaster if the race is not a sellout. Since1992, the station that airs the race in the Indianapolis market (ABCaffiliateWRTVfrom 1986 to 2018,NBCaffiliateWTHRfrom 2019 to 2024, andFoxaffiliateWXINfrom 2025 onwards) airs the race on tape delay in prime time, and carries the network's prime time programming in the race's timeslot under special dispensation from the network.[84]Prior to this, from 1986 to 1991, local and syndicated Sunday programming would continue to air in their regular timeslots.
The blackout has only been lifted five times since live flag-to-flag coverage of the 500 officially began in1986:
Prior to 1986,ABChad aired an edited broadcast of the race in prime time.[88][89]
Until 2001, the same blackout policy applied to theBrickyard 400, aNASCAR Cup Seriesevent also held atIndianapolis Motor Speedway; at the time, television rights to NASCAR events were sold by the owners of their respective tracks,[90]and IMS had packaged the 400 withABC's rights to the Indianapolis 500.[91]This policy ended in 2001 when NASCAR centralized the television rights to all events, and sold them in two packages toFox SportsandNBC/TNTrespectively.[90]
A 1963 episode of theCBStelevision drama seriesEast Side/West Side, focusing on an African-American couple inHarlem, was blacked out by network affiliates inShreveport, Louisiana(KSLA) andAtlanta, Georgia(WAGA-TV).[92]
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Inlinguistics,declension(verb:todecline) is the changing of the form of aword, generally to express itssyntactic functionin the sentence by way of aninflection. Declension may apply tonouns,pronouns,adjectives,adverbs, anddeterminers. It serves to indicatenumber(e.g. singular, dual, plural),case(e.g.nominative,accusative,genitive, ordative),gender(e.g. masculine, feminine, or neuter), and a number of othergrammatical categories. Inflectional change ofverbsis calledconjugation.
Declension occurs in many languages. It is an important aspect of language families likeQuechuan(i.e., languages native to theAndes),Indo-European(e.g.German,Icelandic,Irish,Lithuanian and Latvian,Slavic,Sanskrit,Latin,AncientandModern Greek,Albanian,Romanian,Kurdish,ClassicalandModern Armenian),[excessive detail?]Bantu(e.g.Swahili,Zulu,Kikuyu),Semitic(e.g.Modern Standard Arabic),Finno-Ugric(e.g.Hungarian,Finnish,Estonian), andTurkic(e.g.Turkish).
Old Englishwas aninflectional language, but largely abandoned inflectional changes as it evolved intoModern English. Though traditionally classified assynthetic, Modern English has become a mostlyanalytic language.
Unlike English, many languages usesuffixesto specify subjects and objects and word cases in general.Inflected languageshave a freer word order than modern English, ananalytic languagein whichword orderidentifies the subject and object.[1][2]As an example, even though both of the following sentences consist of the same words, the meaning is different:[1]
Hypothetically speaking, suppose English were a language with a more complex declension system in which cases were formed by adding the suffixes:
The first sentence above could be formed with any of the followingword ordersand would have the same meaning:[1]
As a more complex example, the sentence:
becomes nonsensical in English if the words are rearranged (because there are no cases):
But if English were a highly inflected language, likeLatinor someSlavic languagessuch asCroatian, both sentences could mean the same thing.[1]They would both contain five nouns in five different cases:mum– vocative (hey!),dog– nominative (who?),boy– genitive (of whom?),cat– accusative (whom?),street– locative (where?);[3]the adjectivelittlewould be in the same case as the noun it modifies (boy), and the case of the determinerourwouldagreewith the case of the noun it determines (street).[4]
Using the case suffixes invented for this example, the original sentence would read:
And like other inflected languages, the sentence rearranged in the following ways would mean virtually the same thing, but with different expressiveness:[5]
Instead of thelocative, theinstrumental formof "down our street" could also be used:[6]
Different word orders preserving the original meaning are possible in an inflected language,[5]while modern English relies on word order for meaning, with a little flexibility.[1]This is one of the advantages of an inflected language. The English sentences above, when read without the made-up case suffixes, are confusing.
These contrived examples are relatively simple, whereas actual inflected languages have a far more complicated set of declensions, where the suffixes (or prefixes orinfixes) change depending on thegender of the noun, thequantity of the noun, and other possible factors. This complexity and the possible lengthening of words is one of the disadvantages of inflected languages. Notably, many of these languages lackarticles. There may also beirregular nounswhere the declensions are unique for each word (likeirregular verbswithconjugation). In inflected languages, otherparts of speechsuch asnumerals,demonstratives,adjectives,[7]andarticles[8]are also declined.
It is agreed thatAncient Greekshad a "vague" idea of the forms of a noun in their language. A fragment ofAnacreonseems to confirm this idea. Nevertheless, it cannot be concluded that the Ancient Greeks actually knew what the cases were. TheStoicsdeveloped many basic notions that today are the rudiments oflinguistics. The idea of grammatical cases is also traced back to the Stoics, but it is still not completely clear what the Stoics exactly meant with their notion of cases.[9][10]
InModern English, the system of declensions is so simple compared to some other languages that the termdeclensionis rarely used.
Most nouns in English have distinctsingularandpluralforms. Nouns and most noun phrases can form apossessiveconstruction. Plurality is most commonly shown by theending-s(or-es), whereas possession is always shown by the enclitic-'sor, for plural forms ending ins, by just an apostrophe.
Consider, for example, the forms of the noungirl. Most speakers pronounce all forms other than the singular plain form (girl) exactly the same.[note 1]
By contrast, a few irregular nouns (likeman/men) are slightly more complex in their forms. In this example, all four forms are pronounced distinctly.
For nouns, in general, gender is not declined in Modern English. There are isolated situations where certain nouns may be modified to reflect gender, though not in a systematic fashion. Loan words from other languages, particularly Latin and the Romance languages, often preserve their gender-specific forms in English, e.g.alumnus(masculine singular) andalumna(feminine singular). Similarly, names borrowed from other languages show comparable distinctions:AndrewandAndrea,PaulandPaula, etc. Additionally, suffixes such as-ess,-ette, and-erare sometimes applied to create overtly gendered versions of nouns, with marking for feminine being much more common than marking for masculine. Many nouns can actually function as members of two genders or even all three, and the gender classes of English nouns are usually determined by their agreement with pronouns, rather than marking on the nouns themselves.
There can be other derivations from nouns that are not considered declensions. For example, the proper nounBritainhas the associated descriptive adjectiveBritishand thedemonymBriton. Though these words are clearly related, and are generally consideredcognates, they are not specifically treated as forms of thesame word, and thus are not declensions.
Pronounsin English have more complex declensions. For example, thefirst person"I":
Whereas nouns do not distinguish between thesubjective (nominative)andobjective (oblique)cases, some pronouns do; that is, they decline to reflect their relationship to averborpreposition, orcase. Consider the difference betweenhe(subjective) andhim(objective), as in "He saw it" and "It saw him"; similarly, considerwho, which is subjective, and the objectivewhom(although it is increasingly common to usewhofor both).
The one situation wheregender[note 2]is still clearly part of the English language is in the pronouns for the third person singular. Consider the following:
The distinguishing of neuter for persons and non-persons is peculiar to English. This has existed since the 14th century.[11][12]However, the use ofsingular theyis often restricted to specific contexts, depending on the dialect or the speaker. It is most typically used to refer to a single person of unknown gender (e.g. "someone left their jacket behind") or a hypothetical person where gender is insignificant (e.g. "If someone wants to, then they should"). Its use has expanded in recent years due to increasing social recognition of persons who do not identify themselves as male or female[13](seegender-nonbinary). Thesingular theystill uses plural verb forms, reflecting its origins.
Some English adjectives and adverbs are declined fordegree of comparison. The unmarked form is thepositiveform, such asquick. Comparative forms are formed with the ending-er(quicker), while superlative forms are formed with-est(quickest). Some are uncomparable; the remainder are usually periphrastic constructions withmore(more beautiful) andmost(most modestly). Seedegree of comparisonfor more.
Adjectives are not declined for case in Modern English (though they were in Old English), nor number nor gender.[note 3]
The demonstrative determinersthisandthatare declined for number, astheseandthose.
Thearticleis never regarded as declined in Modern English, although formally, the wordsthatand possiblyshecorrespond to forms of the predecessor ofthe(sēm.,þætn.,sēof.) as it was declined in Old English.
Just as verbs in Latin are conjugated to indicate grammatical information, Latin nouns and adjectives that modify them are declined to signal their roles in sentences. There are five important cases for Latin nouns:nominative,genitive,dative,accusative, andablative. Since thevocative caseusuallytakes the same form as the nominative, it is seldom spelt out in grammar books.[dubious–discuss]Yet another case, thelocative, is limited to a small number of words.
The usual basic functions of these cases are as follows:
The genitive, dative, accusative, and ablative also have important functions to indicate the object of a preposition.
Given below is the declension paradigm of Latinpuer'boy' andpuella'girl':
From the provided examples we can see how cases work:
liber
book
puerī
boy.GEN
liber puerī
book boy.GEN
the book of the boy
puer
boy.NOM
puellae
girl.DAT
rosam
rose.ACC
dat
give.3SG.PRES
puer puellae rosam dat
boy.NOM girl.DAT rose.ACC give.3SG.PRES
the boy gives the girl a rose
Sanskrit, another Indo-European language, has eight cases:nominative,vocative,accusative,genitive,dative,ablative,locativeandinstrumental.[14]Some do not count vocative as a separate case, despite it having a distinctive ending in the singular, but consider it as a different use of the nominative.[15]
Sanskrit grammatical cases have been analyzed extensively. The grammarianPāṇiniidentified six semanticrolesorkaraka, which correspond closely to the eight cases:[16]
For example, consider the following sentence:
vṛkṣ-āt
from the tree
parṇ-aṁ
a leaf
bhūm-āu
to the ground
patati
falls
vṛkṣ-ātparṇ-aṁbhūm-āupatati
{from the tree} {a leaf} {to the ground} falls
"a leaf falls from the tree to the ground"
Hereleafis the agent,treeis the source, andgroundis the locus. The endings-aṁ,-at,-āumark the cases associated with these meanings.
Verse 37 of the Rāmarakṣāstotram gives an example of all 8 types of declensions in Sanskrit for the singular proper noun Rāma.[17]
The case declension here isRāmaḥbut thevisargahas undergonesandhi.
Both words 'Rāma Rameśa'are individually declined as 'rāmaṃ rameśaṃ
Rāmeṇais the declension that underwent sandhi with the wordabhihatā
Dative case is used here to show thatRāmais the receiver of the reverence.
The declension here isRāmātthat has undergone sandhi withnāsti.
Ablative case is also used for comparisons in Sanskrit
Normal declension without sandhi.
Locative case to indicate the 'focus of thoughts'
Vocative case uses the plain stem, unlike Nominative which adds a visarga.
Sometimes vocative is considered to be a different use of nominative.[15]
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Inalgebra(in particular inalgebraic geometryoralgebraic number theory), avaluationis afunctionon afieldthat provides a measure of the size or multiplicity of elements of the field. It generalizes tocommutative algebrathe notion of size inherent in consideration of the degree of apoleormultiplicityof azeroincomplex analysis, the degree of divisibility of a number by a prime number innumber theory, and the geometrical concept ofcontactbetween twoalgebraicoranalytic varietiesin algebraic geometry. A field with a valuation on it is called avalued field.
One starts with the following objects:
The ordering andgroup lawonΓare extended to the setΓ ∪ {∞}[a]by the rules
Then avaluation ofKis anymap
that satisfies the following properties for alla,binK:
A valuationvistrivialifv(a) = 0 for allainK×, otherwise it isnon-trivial.
The second property asserts that any valuation is agroup homomorphismonK×. The third property is a version of thetriangle inequalityonmetric spacesadapted to an arbitrary Γ (seeMultiplicative notationbelow). For valuations used ingeometricapplications, the first property implies that any non-emptygermof an analytic variety near a point contains that point.
The valuation can be interpreted as the order of theleading-order term.[b]The third property then corresponds to the order of a sum being the order of the larger term,[c]unless the two terms have the same order, in which case they may cancel and the sum may have larger order.
For many applications,Γis an additive subgroup of thereal numbersR{\displaystyle \mathbb {R} }[d]in which case ∞ can be interpreted as +∞ in theextended real numbers; note thatmin(a,+∞)=min(+∞,a)=a{\displaystyle \min(a,+\infty )=\min(+\infty ,a)=a}for any real numbera, and thus +∞ is the unit under the binary operation of minimum. The real numbers (extended by +∞) with the operations of minimum and addition form asemiring, called the mintropical semiring,[e]and a valuationvis almost a semiring homomorphism fromKto the tropical semiring, except that the homomorphism property can fail when two elements with the same valuation are added together.
The concept was developed byEmil Artinin his bookGeometric Algebrawriting the group inmultiplicative notationas(Γ, ·, ≥):[1]
Instead of ∞, we adjoin a formal symbolOto Γ, with the ordering and group law extended by the rules
Then avaluationofKis any map
satisfying the following properties for alla,b∈K:
(Note that the directions of the inequalities are reversed from those in the additive notation.)
IfΓis a subgroup of thepositive real numbersunder multiplication, the last condition is theultrametricinequality, a stronger form of thetriangle inequality|a+b|v≤|a|v+|b|v, and| ⋅ |vis anabsolute value. In this case, we may pass to the additive notation with value groupΓ+⊆(R,+){\displaystyle \Gamma _{+}\subseteq (\mathbb {R} ,+)}by takingv+(a) = −log|a|v.
Each valuation onKdefines a corresponding linearpreorder:a≼b⇔|a|v≤|b|v. Conversely, given a "≼" satisfying the required properties, we can define valuation|a|v= {b:b≼a∧a≼b}, with multiplication and ordering based onKand≼.
In this article, we use the terms defined above, in the additive notation. However, some authors use alternative terms:
There are several objects defined from a given valuationv:K→ Γ ∪ {∞};
Two valuationsv1andv2ofKwith valuation group Γ1and Γ2, respectively, are said to beequivalentif there is an order-preservinggroup isomorphismφ: Γ1→ Γ2such thatv2(a) = φ(v1(a)) for allainK×. This is anequivalence relation.
Two valuations ofKare equivalent if and only if they have the same valuation ring.
Anequivalence classof valuations of a field is called aplace.Ostrowski's theoremgives a complete classification of places of the field ofrational numbersQ:{\displaystyle \mathbb {Q} :}these are precisely the equivalence classes of valuations for thep-adiccompletionsofQ.{\displaystyle \mathbb {Q} .}
Letvbe a valuation ofKand letLbe afield extensionofK. Anextension ofv(toL) is a valuationwofLsuch that therestrictionofwtoKisv. The set of all such extensions is studied in theramification theory of valuations.
LetL/Kbe afinite extensionand letwbe an extension ofvtoL. Theindexof Γvin Γw, e(w/v) = [Γw: Γv], is called thereduced ramification indexofwoverv. It satisfies e(w/v) ≤ [L:K] (thedegreeof the extensionL/K). Therelative degreeofwovervis defined to bef(w/v) = [Rw/mw:Rv/mv] (the degree of the extension of residue fields). It is also less than or equal to the degree ofL/K. WhenL/Kisseparable, theramification indexofwovervis defined to be e(w/v)pi, wherepiis theinseparable degreeof the extensionRw/mwoverRv/mv.
When the ordered abelian groupΓis the additive group of theintegers, the associated valuation is equivalent to an absolute value, and hence induces ametricon the fieldK. IfKiscompletewith respect to this metric, then it is called acomplete valued field. IfKis not complete, one can use the valuation to construct itscompletion, as in the examples below, and different valuations can define different completion fields.
In general, a valuation induces auniform structureonK, andKis called a complete valued field if it iscompleteas a uniform space. There is a related property known asspherical completeness: it is equivalent to completeness ifΓ=Z,{\displaystyle \Gamma =\mathbb {Z} ,}but stronger in general.
The most basic example is thep-adic valuationνpassociated to a prime integerp, on the rational numbersK=Q,{\displaystyle K=\mathbb {Q} ,}with valuation ringR=Z(p),{\displaystyle R=\mathbb {Z} _{(p)},}whereZ(p){\displaystyle \mathbb {Z} _{(p)}}is the localization ofZ{\displaystyle \mathbb {Z} }at the prime ideal(p){\displaystyle (p)}. The valuation group is the additive integersΓ=Z.{\displaystyle \Gamma =\mathbb {Z} .}For an integera∈R=Z,{\displaystyle a\in R=\mathbb {Z} ,}the valuation νp(a) measures the divisibility ofaby powers ofp:
and for a fraction, νp(a/b) = νp(a) − νp(b).
Writing this multiplicatively yields thep-adic absolute value, which conventionally has as base1/p=p−1{\displaystyle 1/p=p^{-1}}, so|a|p:=p−νp(a){\displaystyle |a|_{p}:=p^{-\nu _{p}(a)}}.
ThecompletionofQ{\displaystyle \mathbb {Q} }with respect to νpis the fieldQp{\displaystyle \mathbb {Q} _{p}}ofp-adic numbers.
Let K =F(x), the rational functions on the affine lineX=F1, and take a pointa∈ X. For a polynomialf(x)=ak(x−a)k+ak+1(x−a)k+1+⋯+an(x−a)n{\displaystyle f(x)=a_{k}(x{-}a)^{k}+a_{k+1}(x{-}a)^{k+1}+\cdots +a_{n}(x{-}a)^{n}}withak≠0{\displaystyle a_{k}\neq 0}, defineva(f) = k, the order of vanishing atx=a; andva(f/g) =va(f) −va(g). Then the valuation ringRconsists of rational functions with no pole atx=a, and the completion is theformal Laurent seriesringF((x−a)). This can be generalized to the field ofPuiseux seriesK{{t}} (fractional powers), theLevi-Civita field(its Cauchy completion), and the field ofHahn series, with valuation in all cases returning the smallest exponent oftappearing in the series.
Generalizing the previous examples, letRbe aprincipal ideal domain,Kbe itsfield of fractions, andπbe anirreducible elementofR. Since every principal ideal domain is aunique factorization domain, every non-zero elementaofRcan be written (essentially) uniquely as
where thee's are non-negative integers and thepiare irreducible elements ofRthat are notassociatesofπ. In particular, the integereais uniquely determined bya.
Theπ-adic valuation ofKis then given by
If π' is another irreducible element ofRsuch that (π') = (π) (that is, they generate the same ideal inR), then the π-adic valuation and the π'-adic valuation are equal. Thus, the π-adic valuation can be called theP-adic valuation, whereP= (π).
The previous example can be generalized toDedekind domains. LetRbe a Dedekind domain,Kits field of fractions, and letPbe a non-zero prime ideal ofR. Then, thelocalizationofRatP, denotedRP, is a principal ideal domain whose field of fractions isK. The construction of the previous section applied to the prime idealPRPofRPyields theP-adic valuation ofK.
Suppose thatΓ∪ {0} is the set of non-negative real numbers under multiplication. Then we say that the valuation isnon-discreteif its range (the valuation group) is infinite (and hence has an accumulation point at 0).
Suppose thatXis a vector space overKand thatAandBare subsets ofX. Then we say thatAabsorbsBif there exists aα∈Ksuch thatλ∈Kand|λ| ≥ |α|implies thatB ⊆ λ A.Ais calledradialorabsorbingifAabsorbs every finite subset ofX. Radial subsets ofXare invariant under finite intersection. Also,Ais calledcircledifλinKand|λ| ≥ |α|impliesλ A ⊆ A. The set of circled subsets ofLis invariant under arbitrary intersections. Thecircled hullofAis the intersection of all circled subsets ofXcontainingA.
Suppose thatXandYare vector spaces over a non-discrete valuation fieldK, letA ⊆ X,B ⊆ Y, and letf : X → Ybe a linear map. IfBis circled or radial then so isf−1(B){\displaystyle f^{-1}(B)}. IfAis circled then so isf(A)but ifAis radial thenf(A)will be radial under the additional condition thatfis surjective.
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Anintegrated development environment(IDE) is asoftware applicationthat provides comprehensive facilities forsoftware development. An IDE normally consists of at least asource-code editor,build automationtools, and adebugger. Some IDEs, such asIntelliJ IDEA,EclipseandLazaruscontain the necessarycompiler,interpreteror both; others, such asSharpDevelopandNetBeans, do not.
The boundary between an IDE and other parts of the broader software development environment is not well-defined; sometimes aversion control systemor various tools to simplify the construction of agraphical user interface(GUI) are integrated. Many modern IDEs also have aclass browser, anobject browser, and aclass hierarchy diagramfor use inobject-oriented software development.
Integrated development environments are designed to maximize programmer productivity by providing tight-knit components with similaruser interfaces. IDEs present a single program in which all development is done. This program typically provides many features for authoring, modifying, compiling, deploying and debugging software. This contrasts with software development using unrelated tools, such asvi,GDB,GNU Compiler Collection, ormake.
One aim of the IDE is to reduce the configuration necessary to piece together multiple development utilities. Instead, it provides the same set of capabilities as one cohesive unit. Reducing setup time can increase developer productivity, especially in cases where learning to use the IDE is faster than manually integrating and learning all of the individual tools. Tighter integration of all development tasks has the potential to improve overall productivity beyond just helping with setup tasks. For example, code can be continuously parsed while it is being edited, providing instant feedback when syntax errors are introduced, thus allowing developers to debug code much faster and more easily with an IDE.
Some IDEs are dedicated to a specificprogramming language, allowing a feature set that most closely matches theprogramming paradigmsof the language. However, there are many multiple-language IDEs.
While most modern IDEs are graphical, text-based IDEs such asTurbo Pascalwere in popular use before the availability of windowing systems likeMicrosoft Windowsand theX Window System(X11). They commonly use function keys orhotkeysto execute frequently used commands or macros.
IDEs initially became possible when developing via aconsoleorterminal. Early systems could not support one, since programs were submitted to acompilerorassemblerviapunched cards,paper tape, etc.Dartmouth BASICwas the first language to be created with an IDE (and was also the first to be designed for use while sitting in front of a console or terminal).[citation needed]Its IDE (part of theDartmouth Time-Sharing System) was command-based, and therefore did not look much like the menu-driven, graphical IDEs popular after the advent of thegraphical user interface. However it integrated editing, file management, compilation, debugging and execution in a manner consistent with a modern IDE.
Maestro Iis a product from Softlab Munich and was the world's first integrated development environment[1]for software.Maestro Iwas installed for 22,000 programmers worldwide. Until 1989, 6,000 installations existed in theFederal Republic of Germany. Maestro was arguably the world leader in this field during the 1970s and 1980s. Today one of the last Maestro I can be found in the Museum of Information Technology at Arlington in Texas.
One of the first IDEs with a plug-in concept wasSoftbench. In 1995Computerwochecommented that the use of an IDE was not well received by developers since it would fence in their creativity.
As of August 2023[update], the most commonly searched for IDEs onGoogle SearchwereVisual Studio,Visual Studio Code, andEclipse.[2]
The IDE editor usually providessyntax highlighting, it can show both the structures, the language keywords and the syntax errors with visually distinct colors and font effects.[3]
Code completion is an important IDE feature, intended to speed up programming. Modern IDEs even haveintelligent code completion.
Code completionis anautocompletionfeature in many integrated development environments (IDEs) that speeds up the process of coding applications by fixing common mistakes and suggesting lines of code. This usually happens through popups while typing, querying parameters of functions, and query hints related to syntax errors. Modern code completion software typically usesgenerative artificial intelligencesystems to predict lines of code[citation needed]. Code completion and related tools serve as documentation and disambiguation forvariablenames,functions, andmethods, usingstatic analysis.[4][5]
Advanced IDEs provide support forautomated refactoring.[3]
An IDE is expected to provide integratedversion control, in order to interact with source repositories.[3]
IDEs are also used for debugging, using an integrateddebugger, with support for setting breakpoints in the editor, visual rendering of steps, etc.[9]
IDEs may provide support for code search. Code search has two different meanings. First, it means searching for class and function declarations, usages, variable and field read/write, etc. IDEs can use different kinds of user interface for code search, for example form-based widgets[10]and natural-language based interfaces.
Second, it means searching for a concrete implementation of some specified functionality.[11]
Visual programmingis a usage scenario in which an IDE is generally required. Visual Basic allows users to create new applications by moving programming, building blocks, or code nodes to create flowcharts or structure diagrams that are then compiled or interpreted. These flowcharts often are based on theUnified Modeling Language.
This interface has been popularized with theLego Mindstormssystem and is being actively perused by a number of companies wishing to capitalize on the power of custom browsers like those found atMozilla.KTechlabsupports flowcode and is a popular open-source IDE and Simulator for developing software for microcontrollers. Visual programming is also responsible for the power ofdistributed programming(cf.LabVIEWand EICASLAB software). An early visual programming system,Max, was modeled after an analogsynthesizerdesign and has been used to develop real-time music performance software since the 1980s. Another early example wasPrograph, adataflow-based system originally developed for theMacintosh. The graphical programming environment "Grape" is used to programqfix robot kits.
This approach is also used in specialist software such as Openlab, where the end-users want the flexibility of a full programming language, without the traditional learning curve associated with one.
Some IDEs support multiple languages, such asGNU Emacs,IntelliJ IDEA,Eclipse,MyEclipse,NetBeans,MonoDevelop, JDoodle or PlayCode.
Support for alternative languages is often provided byplugins, allowing them to be installed on the same IDE at the same time. For example,Flycheckis a modern on-the-fly syntax checking extension forGNU Emacs24 with support for 39 languages.[12]Another example is JDoodle, an online cloud-based IDE that supports 88 languages.[1]Eclipse, andNetbeanshave plugins forC/C++,Ada,GNAT(for exampleAdaGIDE),Perl,Python,Ruby, andPHP, which are selected between automatically based on file extension, environment or project settings.
IDEs can be implemented in various languages, for example:
Unixprogrammers can combinecommand-linePOSIXtools into a complete development environment, capable of developing large programs such as theLinux kerneland its environment.[13]In this sense, the entire Unix system functions as an IDE.[14]The free softwareGNU toolchain(includingGNU Compiler Collection(GCC),GNU Debugger(GDB), andGNU make) is available on many platforms, including Windows.[15]The pervasive Unix philosophy of "everything is a text stream" enables developers who favorcommand-lineoriented tools to use editors with support for many of the standard Unix and GNU build tools, building an IDE with programs likeEmacs[16][17][18]orVim.Data Display Debuggeris intended to be an advanced graphical front-end for many text-baseddebuggerstandard tools. Some programmers prefer managingmakefilesand their derivatives to the similar code building tools included in a full IDE. For example, most contributors to thePostgreSQLdatabase usemakeandGDBdirectly to develop new features.[19]Even when building PostgreSQL forMicrosoft WindowsusingVisual C++,Perlscripts are used as a replacement formakerather than relying on any IDE features.[20]Some Linux IDEs such asGeanyattempt to provide a graphical front end to traditional build operations.
On the variousMicrosoft Windowsplatforms, command-line tools for development are seldom used. Accordingly, there are many commercial and non-commercial products. However, each has a different design commonly creating incompatibilities. Most major compiler vendors for Windows still provide free copies of their command-line tools, includingMicrosoft(Visual C++,Platform SDK,.NET FrameworkSDK,nmakeutility).
IDEs have always been popular on the Apple Macintosh'sclassic Mac OSandmacOS, dating back toMacintosh Programmer's Workshop,Turbo Pascal, THINK Pascal andTHINK Cenvironments of the mid-1980s. Currently macOS programmers can choose between native IDEs likeXcodeand open-source tools such asEclipseandNetbeans.ActiveState Komodois a proprietary multilanguage IDE supported on macOS.
Anonline integrated development environment, also known as a web IDE or cloud IDE, is abrowserbased IDE that allows for software development or web development.[21]An online IDE can be accessed from a web browser, allowing for a portable work environment. An online IDE does not usually contain all of the same features as a traditional or desktop IDE although all of the basic IDE features, such as syntax highlighting, are typically present.
A Mobile-Based Integrated Development Environment (IDE) is a software application that provides a comprehensive suite of tools for software development on mobile platforms. Unlike traditional desktop IDEs, mobile-based IDEs are designed to run on smartphones and tablets, allowing developers to write, debug, and deploy code directly from their mobile devices.
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Asoftware license manageris asoftware management toolused byindependent software vendorsor by end-user organizations to control where and how software products are able to run. License managers protect software vendors from losses due tosoftware piracyand enable end-user organizations to comply withsoftware licenseagreements. License managers enable software vendors to offer a wide range of usage-centric software licensing models, such asproduct activation,trial licenses,subscription licenses, feature-based licenses, andfloating licensingfrom the same software package they provide to all users.
A license manager is different from asoftware asset managementtool, which end-user organizations employ to manage the software they have licensed from many software vendors. However, some software asset management tools include license manager functions. These are used to reconcile software licenses and installed software, and generally include device discovery, software inventory, license compliance, andreportingfunctions.
An additional benefit of these software management tools are that they reduce the difficulty, cost, and time required for reporting and can increase operational transparency in order to prevent litigation costs associated with software misuse, as set forth by theSarbanes-Oxley Act.[1][2]
License managementsolutions provided by non-vendor companies are more valuable to the end-users, since most vendors do not provide enough license usage information. A vendor license manager provides limited information, while non-vendor license management solutions are developed for end-users in order to maximally optimize the licenses they have.[3]
Most license managers can cover differentsoftware licensing modelsaslicense donglesor license USB keys,floating licenses,network licenses,concurrent licenseetc.
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Instatistics,scaled correlationis a form of a coefficient ofcorrelationapplicable to data that have a temporal component such astime series. It is the average short-term correlation. If the signals have multiple components (slow and fast), scaled coefficient of correlation can be computed only for the fast components of the signals, ignoring the contributions of the slow components.[1]Thisfiltering-likeoperation has the advantages of not having to make assumptions about the sinusoidal nature of the signals.
For example, in the studies of brain signals researchers are often interested in the high-frequency components (beta and gamma range; 25–80 Hz), and may not be interested in lower frequency ranges (alpha, theta, etc.). In that case scaled correlation can be computed only for frequencies higher than 25 Hz by choosing the scale of the analysis,s, to correspond to the period of that frequency (e.g.,s= 40 ms for 25 Hz oscillation).
Scaled correlation between two signals is defined as the average correlation computed across short segments of those signals. First, it is necessary to determine the number of segmentsK{\displaystyle K}that can fit into the total lengthT{\displaystyle T}of the signals for a given scales{\displaystyle s}:
Next, ifrk{\displaystyle r_{k}}isPearson's coefficient of correlationfor segmentk{\displaystyle k}, the scaled correlation across the entire signalsr¯s{\displaystyle {\bar {r}}_{s}}is computed as
In a detailed analysis, Nikolić et al.[1]showed that the degree to which the contributions of the slow components will be attenuated depends on three factors, the choice of the scale, the amplitude ratios between the slow and the fast component, and the differences in their oscillation frequencies. The larger the differences in oscillation frequencies, the more efficiently will the contributions of the slow components be removed from the computed correlation coefficient. Similarly, the smaller the power of slow components relative to the fast components, the better will scaled correlation perform.
Scaled correlation can be applied toauto-andcross-correlationin order to investigate how correlations of high-frequency components change at different temporal delays. To compute cross-scaled-correlation for every time shift properly, it is necessary to segment the signals anew after each time shift. In other words, signals are always shiftedbeforethe segmentation is applied. Scaled correlation has been subsequently used to investigate synchronization hubs in the visual cortex.[2]Scaled correlation can be also used to extract functional networks.[3]
Scaled correlation should be in many cases preferred over signal filtering based on spectral methods. The advantage of scaled correlation is that it does not make assumptions about the spectral properties of the signal (e.g., sinusoidal shapes of signals). Nikolić et al.[1]have shown that the use ofWiener–Khinchin theoremto remove slow components is inferior to results obtained by scaled correlation. These advantages become obvious especially when the signals are non-periodic or when they consist of discrete events such as the time stamps at which neuronal action potentials have been detected.
A detailed insight into a correlation structure across different scales can be provided by visualization using multiresolution correlation analysis.[4]
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Boolean grammars, introduced byOkhotin[Wikidata], are a class offormal grammarsstudied informal languagetheory. They extend the basic type of grammars, thecontext-free grammars, withconjunctionandnegationoperations. Besides these explicit operations, Boolean grammars allow implicitdisjunctionrepresented by multiple rules for a single nonterminal symbol, which is the only logical connective expressible in context-free grammars. Conjunction and negation can be used, in particular, to specify intersection and complement of languages. An intermediate class of grammars known asconjunctive grammarsallows conjunction and disjunction, but not negation.
The rules of a Boolean grammar are of the form
A→α1&…&αm&¬β1&…&¬βn{\displaystyle A\to \alpha _{1}\And \ldots \And \alpha _{m}\And \lnot \beta _{1}\And \ldots \And \lnot \beta _{n}}
whereA{\displaystyle A}is a nonterminal,m+n≥1{\displaystyle m+n\geq 1}andα1{\displaystyle \alpha _{1}}, ...,αm{\displaystyle \alpha _{m}},β1{\displaystyle \beta _{1}}, ...,βn{\displaystyle \beta _{n}}are strings formed of symbols inΣ{\displaystyle \Sigma }andN{\displaystyle N}. Informally, such a rule asserts that every stringw{\displaystyle w}overΣ{\displaystyle \Sigma }that satisfies each of the syntactical conditions represented byα1{\displaystyle \alpha _{1}}, ...,αm{\displaystyle \alpha _{m}}and none of the syntactical conditions represented byβ1{\displaystyle \beta _{1}}, ...,βn{\displaystyle \beta _{n}}therefore satisfies the condition defined byA{\displaystyle A}.
There exist several formal definitions of the language generated by a Boolean grammar. They have one thing in common: if the grammar is represented as a system oflanguage equationswith union, intersection, complementation and concatenation, the languages generated by the grammar must be the solution of this system. The semantics differ in details, some define the languages using language equations, some draw upon ideas from the field oflogic programming. However, these nontrivial issues of formal definition are mostly irrelevant for practical considerations, and one can construct grammars according to the given informal semantics. The practical properties of the model are similar to those ofconjunctive grammars, while the descriptional capabilities are further improved. In particular, some practically useful properties inherited fromcontext-free grammars, such as efficient parsing algorithms, are retained, seeOkhotin (2010).
Thisformal methods-related article is astub. You can help Wikipedia byexpanding it.
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https://en.wikipedia.org/wiki/Boolean_grammar
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Inmathematics, well-behavedtopological spacescan belocalizedat primes, in a similar way to thelocalization of a ringat a prime. This construction was described byDennis Sullivanin 1970 lecture notes that were finally published in (Sullivan 2005).
The reason to do this was in line with an idea of makingtopology, more preciselyalgebraic topology, more geometric. Localization of a spaceXis a geometric form of the algebraic device of choosing 'coefficients' in order to simplify the algebra, in a given problem. Instead of that, the localization can be applied to the spaceX, directly, giving a second spaceY.
We letAbe asubringof therational numbers, and letXbe asimply connectedCW complex. Then there is a simply connected CW complexYtogether with a map fromXtoYsuch that
This spaceYis unique up tohomotopy equivalence, and is called thelocalizationofXatA.
IfAis the localization ofZat a primep, then the spaceYis called thelocalizationofXatp.
The map fromXtoYinducesisomorphismsfrom theA-localizations of the homology and homotopy groups ofXto the homology and homotopy groups ofY.
Category:Localization (mathematics)
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Goldbach's conjectureis one of the oldest and best-knownunsolved problemsinnumber theoryand all ofmathematics. It states that everyevennatural numbergreater than 2 is the sum of twoprime numbers.
The conjecture has been shown to hold for all integers less than4×1018but remains unproven despite considerable effort.
On 7 June 1742, thePrussianmathematicianChristian Goldbachwrote a letter toLeonhard Euler(letter XLIII),[2]in which he proposed the following conjecture:
Goldbach was following the now-abandoned convention ofconsidering 1to be aprime number,[3]so that a sum of units would be a sum of primes.
He then proposed a second conjecture in the margin of his letter, which implies the first:[4]
Es scheinet wenigstens, dass eine jede Zahl, die grösser ist als 2, ein aggregatum trium numerorum primorum sey.It seems at least, that every integer greater than 2 can be written as the sum of three primes.
Euler replied in a letter dated 30 June 1742[5]and reminded Goldbach of an earlier conversation they had had ("... so Ew vormals mit mir communicirt haben ..."), in which Goldbach had remarked that the first of those two conjectures would follow from the statement
This is in fact equivalent to his second, marginal conjecture.
In the letter dated 30 June 1742, Euler stated:[6][7]
Dass ... ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann.That ... every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it.
René Descarteswrote that "Every even number can be expressed as the sum of at most three primes."[8]The proposition is similar to, but weaker than, Goldbach's conjecture.Paul Erdőssaid that "Descartes actually discovered this before Goldbach... but it is better that the conjecture was named for Goldbach because, mathematically speaking, Descartes was infinitely rich and Goldbach was very poor."[9]
The strong Goldbach conjecture is much more difficult than theweak Goldbach conjecture, which says that every odd integer greater than 5 is the sum of three primes. UsingVinogradov's method,Nikolai Chudakov,[10]Johannes van der Corput,[11]andTheodor Estermann[12]showed (1937–1938) thatalmost alleven numbers can be written as the sum of two primes (in the sense that the fraction of even numbers up to someNwhich can be so written tends towards 1 asNincreases). In 1930,Lev Schnirelmannproved that anynatural numbergreater than 1 can be written as the sum of not more thanCprime numbers, whereCis an effectively computable constant; seeSchnirelmann density.[13][14]Schnirelmann's constant is the lowest numberCwith this property. Schnirelmann himself obtainedC<800000. This result was subsequently enhanced by many authors, such asOlivier Ramaré, who in 1995 showed that every even numbern≥ 4is in fact the sum of at most 6 primes. The best known result currently stems from the proof of the weak Goldbach conjecture byHarald Helfgott,[15]which directly implies that every even numbern≥ 4is the sum of at most 4 primes.[16][17]
In 1924, Hardy and Littlewood showed under the assumption of thegeneralized Riemann hypothesisthat the number of even numbers up toXviolating the Goldbach conjecture ismuch less thanX1⁄2+cfor smallc.[18]
In 1948, usingsieve theorymethods,Alfréd Rényishowed that every sufficiently large even number can be written as the sum of a prime and analmost primewith at mostKfactors.[19]Chen Jingrunshowed in 1973 using sieve theory that everysufficiently largeeven number can be written as the sum of either two primes, or a prime and asemiprime(the product of two primes).[20]SeeChen's theoremfor further information.
In 1975,Hugh Lowell MontgomeryandBob Vaughanshowed that "most" even numbers are expressible as the sum of two primes. More precisely, they showed that there exist positive constantscandCsuch that for all sufficiently large numbersN, every even number less thanNis the sum of two primes, with at mostCN1 −cexceptions. In particular, the set of even integers that are not the sum of two primes hasdensityzero.
In 1951,Yuri Linnikproved the existence of a constantKsuch that every sufficiently large even number is the sum of two primes and at mostKpowers of 2.János PintzandImre Ruzsafound in 2020 thatK= 8works.[21]Assuming thegeneralized Riemann hypothesis,K= 7also works, as shown byRoger Heath-BrownandJan-Christoph Schlage-Puchtain 2002.[22]
A proof for the weak conjecture was submitted in 2013 byHarald HelfgotttoAnnals of Mathematics Studiesseries. Although the article was accepted, Helfgott decided to undertake the major modifications suggested by the referee. Despite several revisions, Helfgott's proof has not yet appeared in a peer-reviewed publication.[23][24][25]The weak conjecture is implied by the strong conjecture, as ifn− 3is a sum of two primes, thennis a sum of three primes. However, the converse implication and thus the strong Goldbach conjecture would remain unproven if Helfgott's proof is correct.
For small values ofn, the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. For instance, in 1938, Nils Pipping laboriously verified the conjecture up ton=100000.[26]With the advent of computers, many more values ofnhave been checked; T. Oliveira e Silva ran a distributed computer search that has verified the conjecture forn≤4×1018(and double-checked up to4×1017) as of 2013. One record from this search is that3325581707333960528is the smallest number that cannot be written as a sum of two primes where one is smaller than 9781.[27]
Goldbach's Conjecture(Chinese:哥德巴赫猜想) is the title of the biography of Chinese mathematician and number theoristChen Jingrun, written byXu Chi.
The conjecture is a central point in the plot of the 1992 novelUncle Petros and Goldbach's Conjectureby Greek authorApostolos Doxiadis, in the short story "Sixty Million Trillion Combinations" byIsaac Asimovand also in the 2008 mystery novelNo One You KnowbyMichelle Richmond.[28]
Goldbach's conjecture is part of the plot of the 2007 Spanish filmFermat's Room.
Goldbach's conjecture is featured as the main topic of research of the titular character Marguerite in the 2023 French-Swiss filmMarguerite's Theorem.[29]
Each of the three conjectures has a natural analog in terms of the modern definition of a prime, under which 1 is excluded. A modern version of the first conjecture is:
A modern version of the marginal conjecture is:
And a modern version of Goldbach's older conjecture of which Euler reminded him is:
These modern versions might not be entirely equivalent to the corresponding original statements. For example, if there were an even integerN=p+ 1larger than 4, forpa prime, that could not be expressed as the sum of two primes in the modern sense, then it would be a counterexample to the modern version of the third conjecture (without being a counterexample to the original version). The modern version is thus probably stronger (but in order to confirm that, one would have to prove that the first version, freely applied to any positive even integern, could not possibly rule out the existence of such a specific counterexampleN). In any case, the modern statements have the same relationships with each other as the older statements did. That is, the second and third modern statements are equivalent, and either implies the first modern statement.
The third modern statement (equivalent to the second) is the form in which the conjecture is usually expressed today. It is also known as the "strong", "even", or "binary" Goldbach conjecture. A weaker form of the second modern statement, known as "Goldbach's weak conjecture", the "odd Goldbach conjecture", or the "ternary Goldbach conjecture", asserts that
Statistical considerations that focus on theprobabilistic distribution of prime numberspresent informal evidence in favour of the conjecture (in both the weak and strong forms) forsufficiently largeintegers: the greater the integer, the more ways there are available for that number to be represented as the sum of two or three other numbers, and the more "likely" it becomes that at least one of these representations consists entirely of primes.
A very crude version of theheuristicprobabilistic argument (for the strong form of the Goldbach conjecture) is as follows. Theprime number theoremasserts that an integermselected at random has roughly a1/lnmchance of being prime. Thus ifnis a large even integer andmis a number between 3 andn/2, then one might expect the probability ofmandn−msimultaneously being prime to be1/lnmln(n−m). If one pursues this heuristic, one might expect the total number of ways to write a large even integernas the sum of two odd primes to be roughly
Sincelnn≪√n, this quantity goes to infinity asnincreases, and one would expect that every large even integer has not just one representation as the sum of two primes, but in fact very many such representations.
This heuristic argument is actually somewhat inaccurate because it assumes that the events ofmandn−mbeing prime arestatistically independentof each other. For instance, ifmis odd, thenn−mis also odd, and ifmis even, thenn−mis even, a non-trivial relation because, besides the number 2, only odd numbers can be prime. Similarly, ifnis divisible by 3, andmwas already a prime other than 3, thenn−mwould also becoprimeto 3 and thus be slightly more likely to be prime than a general number. Pursuing this type of analysis more carefully,G. H. HardyandJohn Edensor Littlewoodin 1923 conjectured (as part of theirHardy–Littlewood prime tuple conjecture) that for any fixedc≥ 2, the number of representations of a large integernas the sum ofcprimesn=p1+ ⋯ +pcwithp1≤ ⋯ ≤pcshould beasymptoticallyequal to
where the product is over all primesp, andγc,p(n)is the number of solutions to the equationn=q1+ ⋯ +qcmodpinmodular arithmetic, subject to theconstraintsq1, …,qc≠ 0 modp. This formula has been rigorously proven to be asymptotically valid forc≥ 3from the work ofIvan Matveevich Vinogradov, but is still only a conjecture whenc= 2.[citation needed]In the latter case, the above formula simplifies to 0 whennis odd, and to
whennis even, whereΠ2isHardy–Littlewood's twin prime constant
This is sometimes known as theextended Goldbach conjecture. The strong Goldbach conjecture is in fact very similar to thetwin primeconjecture, and the two conjectures are believed to be of roughly comparable difficulty.
TheGoldbach partition functionis the function that associates to each even integer the number of ways it can be decomposed into a sum of two primes. Its graph looks like acometand is therefore calledGoldbach's comet.[30]
Goldbach's comet suggests tight upper and lower bounds on the number of representations of an even number as the sum of two primes, and also that the number of these representations depend strongly on the value modulo 3 of the number.
Although Goldbach's conjecture implies that every positive integer greater than one can be written as a sum of at most three primes, it is not always possible to find such a sum using agreedy algorithmthat uses the largest possible prime at each step. ThePillai sequencetracks the numbers requiring the largest number of primes in their greedy representations.[31]
Similar problems to Goldbach's conjecture exist in which primes are replaced by other particular sets of numbers, such as the squares:
Goldbach's conjecture is used when studying computation complexity.[37]The connection is made through theBusy Beaverfunction, where BB(n) is the maximum number of steps taken by anynstateTuring machinethat halts. There is a 27-state Turing machine that halts if and only if Goldbach's conjecture is false.[37]Hence if BB(27) was known, and the Turing machine did not stop in that number of steps, it would be known to run forever and hence no counterexamples exist (which proves the conjecture true). This is a completely impractical way to settle the conjecture; instead it is used to suggest that BB(27) will be very hard to compute, at least as difficult as settling the Goldbach conjecture.
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In the mathematical theory ofnon-standard positional numeral systems, theKomornik–Loreti constantis amathematical constantthat represents the smallest baseqfor which the number 1 has a unique representation, called itsq-development. The constant is named afterVilmos KomornikandPaola Loreti, who defined it in 1998.[1]
Given a real numberq> 1, the series
is called theq-expansion, orβ{\displaystyle \beta }-expansion, of the positive real numberxif, for alln≥0{\displaystyle n\geq 0},0≤an≤⌊q⌋{\displaystyle 0\leq a_{n}\leq \lfloor q\rfloor }, where⌊q⌋{\displaystyle \lfloor q\rfloor }is thefloor functionandan{\displaystyle a_{n}}need not be aninteger. Any real numberx{\displaystyle x}such that0≤x≤q⌊q⌋/(q−1){\displaystyle 0\leq x\leq q\lfloor q\rfloor /(q-1)}has such an expansion, as can be found using thegreedy algorithm.
The special case ofx=1{\displaystyle x=1},a0=0{\displaystyle a_{0}=0}, andan=0{\displaystyle a_{n}=0}or1{\displaystyle 1}is sometimes called aq{\displaystyle q}-development.an=1{\displaystyle a_{n}=1}gives the only 2-development. However, for almost all1<q<2{\displaystyle 1<q<2}, there are an infinite number of differentq{\displaystyle q}-developments. Even more surprisingly though, there exist exceptionalq∈(1,2){\displaystyle q\in (1,2)}for which there exists only a singleq{\displaystyle q}-development. Furthermore, there is a smallest number1<q<2{\displaystyle 1<q<2}known as the Komornik–Loreti constant for which there exists a uniqueq{\displaystyle q}-development.[2]
The Komornik–Loreti constant is the valueq{\displaystyle q}such that
wheretk{\displaystyle t_{k}}is theThue–Morse sequence, i.e.,tk{\displaystyle t_{k}}is the parity of the number of 1's in thebinary representationofk{\displaystyle k}. It has approximate value
The constantq{\displaystyle q}is also the unique positive real solution to the equation
This constant istranscendental.[4]
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Inmathematics, aprojective planeis a geometric structure that extends the concept of aplane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thusanytwo distinct lines in a projective plane intersect at exactly one point.
Renaissance artists, in developing the techniques of drawing inperspective, laid the groundwork for this mathematical topic. The archetypical example is thereal projective plane, also known as theextended Euclidean plane.[1]This example, in slightly different guises, is important inalgebraic geometry,topologyandprojective geometrywhere it may be denoted variously byPG(2,R),RP2, orP2(R), among other notations. There are many other projective planes, both infinite, such as thecomplex projective plane, and finite, such as theFano plane.
A projective plane is a 2-dimensionalprojective space. Not all projective planes can beembeddedin 3-dimensional projective spaces; such embeddability is a consequence of a property known asDesargues' theorem, not shared by all projective planes.
Aprojective planeis a rank 2incidence structure(P,L,I){\displaystyle ({\mathcal {P}},{\mathcal {L}},I)}consisting of a set ofpointsP{\displaystyle {\mathcal {P}}}, a set oflinesL{\displaystyle {\mathcal {L}}}, and a symmetric relationI{\displaystyle I}on the setP∪L{\displaystyle {\mathcal {P}}\cup {\mathcal {L}}}calledincidence, having the following properties:[2]
The second condition means that there are noparallel lines. The last condition excludes the so-calleddegeneratecases (seebelow). The term "incidence" is used to emphasize the symmetric nature of the relationship between points and lines. Thus the expression "pointPis incident with lineℓ" is used instead of either "Pis onℓ" or "ℓpasses throughP".
It follows from the definition that the number of pointss+1{\displaystyle s+1}incident with any given line in a projective plane is the same as the number of lines incident with any given point. The (possibly infinite) cardinal numbers{\displaystyle s}is calledorderof the plane.
To turn the ordinary Euclidean plane into a projective plane, proceed as follows:
The extended structure is a projective plane and is called theextended Euclidean planeor thereal projective plane. The process outlined above, used to obtain it, is called "projective completion" orprojectivization. This plane can also be constructed by starting fromR3viewed as a vector space, see§ Vector space constructionbelow.
The points of theMoulton planeare the points of the Euclidean plane, with coordinates in the usual way. To create the Moulton plane from the Euclidean plane some of the lines are redefined. That is, some of their point sets will be changed, but other lines will remain unchanged. Redefine all the lines with negative slopes so that they look like "bent" lines, meaning that these lines keep their points with negativex-coordinates, but the rest of their points are replaced with the points of the line with the samey-intercept but twice the slope wherever theirx-coordinate is positive.
The Moulton plane has parallel classes of lines and is anaffine plane. It can be projectivized, as in the previous example, to obtain theprojective Moulton plane.Desargues' theoremis not a valid theorem in either the Moulton plane or the projective Moulton plane.
This example has just thirteen points and thirteen lines. We label the points P1, ..., P13and the lines m1, ..., m13. Theincidence relation(which points are on which lines) can be given by the followingincidence matrix. The rows are labelled by the points and the columns are labelled by the lines. A 1 in rowiand columnjmeans that the point Piis on the line mj, while a 0 (which we represent here by a blank cell for ease of reading) means that they are not incident. The matrix is in Paige–Wexler normal form.
To verify the conditions that make this a projective plane, observe that every two rows have exactly one common column in which 1s appear (every pair of distinct points are on exactly one common line) and that every two columns have exactly one common row in which 1s appear (every pair of distinct lines meet at exactly one point). Among many possibilities, the points P1, P4, P5, and P8, for example, will satisfy the third condition. This example is known as theprojective plane of order three.
Though the line at infinity of the extended real plane may appear to have a different nature than the other lines of that projective plane, this is not the case. Another construction of the same projective plane shows that no line can be distinguished (on geometrical grounds) from any other. In this construction, each "point" of the real projective plane is the one-dimensional subspace (ageometricline) through the origin in a 3-dimensional vector space, and a "line" in the projective plane arises from a (geometric) plane through the origin in the 3-space. This idea can be generalized and made more precise as follows.[3]
LetKbe anydivision ring(skewfield). LetK3denote the set of all triplesx=(x0,x1,x2)of elements ofK(aCartesian productviewed as avector space). For any nonzeroxinK3, the minimal subspace ofK3containingx(which may be visualized as all the vectors in a line through the origin) is the subset
ofK3. Similarly, letxandybe linearly independent elements ofK3, meaning thatkx+my= 0implies thatk=m= 0. The minimal subspace ofK3containingxandy(which may be visualized as all the vectors in a plane through the origin) is the subset
ofK3. This 2-dimensional subspace contains various 1-dimensional subspaces through the origin that may be obtained by fixingkandmand taking the multiples of the resulting vector. Different choices ofkandmthat are in the same ratio will give the same line.
Theprojective planeoverK, denoted PG(2,K) orKP2, has a set ofpointsconsisting of all the 1-dimensional subspaces inK3. A subsetLof the points of PG(2,K) is alinein PG(2,K) if there exists a 2-dimensional subspace ofK3whose set of 1-dimensional subspaces is exactlyL.
Verifying that this construction produces a projective plane is usually left as a linear algebra exercise.
An alternate (algebraic) view of this construction is as follows. The points of this projective plane are the equivalence classes of the setK3\ {(0, 0, 0)}modulo theequivalence relation
Lines in the projective plane are defined exactly as above.
The coordinates(x0,x1,x2)of a point in PG(2,K) are calledhomogeneous coordinates. Each triple(x0,x1,x2)represents a well-defined point in PG(2,K), except for the triple(0, 0, 0), which represents no point. Each point in PG(2,K), however, is represented by many triples.
IfKis atopological space, thenKP2inherits a topology via theproduct,subspace, andquotienttopologies.
Thereal projective planeRP2arises whenKis taken to be thereal numbers,R. As a closed, non-orientable real 2-manifold, it serves as a fundamental example in topology.[4]
In this construction, consider the unit sphere centered at the origin inR3. Each of theR3lines in this construction intersects the sphere at two antipodal points. Since theR3line represents a point ofRP2, we will obtain the same model ofRP2by identifying the antipodal points of the sphere. The lines ofRP2will be the great circles of the sphere after this identification of antipodal points. This description gives the standard model ofelliptic geometry.
Thecomplex projective planeCP2arises whenKis taken to be thecomplex numbers,C. It is a closed complex 2-manifold, and hence a closed, orientable real 4-manifold. It and projective planes over otherfields(known aspappian planes) serve as fundamental examples inalgebraic geometry.[5]
Thequaternionic projective planeHP2is also of independent interest.[6]
ByWedderburn's Theorem, a finite division ring must be commutative and so be a field. Thus, the finite examples of this construction are known as "field planes". TakingKto be thefinite fieldofq=pnelements with primepproduces a projective plane ofq2+q+ 1points. The field planes are usually denoted by PG(2,q) where PG stands for projective geometry, the "2" is the dimension andqis called theorderof the plane (it is one less than the number of points on any line). The Fano plane, discussed below, is denoted by PG(2, 2). Thethird example aboveis the projective plane PG(2, 3).
TheFano planeis the projective plane arising from the field of two elements. It is the smallest projective plane, with only seven points and seven lines. In the figure at right, the seven points are shown as small balls, and the seven lines are shown as six line segments and a circle. However, one could equivalently consider the balls to be the "lines" and the line segments and circle to be the "points" – this is an example ofdualityin the projective plane: if the lines and points are interchanged, the result is still a projective plane (seebelow). A permutation of the seven points that carriescollinearpoints (points on the same line) to collinear points is called acollineationorsymmetryof the plane. The collineations of a geometry form agroupunder composition, and for the Fano plane this group (PΓL(3, 2) = PGL(3, 2)) has 168 elements.
Thetheorem of Desarguesis universally valid in a projective plane if and only if the plane can be constructed from a three-dimensional vector space over a skewfield asabove.[7]These planes are calledDesarguesian planes, named afterGirard Desargues. The real (or complex) projective plane and the projective plane of order 3 givenaboveare examples of Desarguesian projective planes. The projective planes that can not be constructed in this manner are callednon-Desarguesian planes, and theMoulton planegivenaboveis an example of one. The PG(2,K) notation is reserved for the Desarguesian planes. WhenKis afield, a very common case, they are also known asfield planesand if the field is afinite fieldthey can be calledGalois planes.
Asubplaneof a projective plane(P,L,I){\displaystyle ({\mathcal {P}},{\mathcal {L}},I)}is a pair of subsets(P′,L′){\displaystyle ({{\mathcal {P}}'},{{\mathcal {L}}'})}whereP′⊆P{\displaystyle {{\mathcal {P}}'}\subseteq {\mathcal {P}}},L′⊆L{\displaystyle {{\mathcal {L}}'}\subseteq {\mathcal {L}}}and(P′,L′,I′){\displaystyle ({{\mathcal {P}}'},{{\mathcal {L}}'},I')}is itself a projective plane with respect to the restrictionI′{\displaystyle I'}of the incidence relationI{\displaystyle I}to(P′∪L′)×(P′∪L′){\displaystyle ({{\mathcal {P}}'}\cup {{\mathcal {L}}'})\times ({{\mathcal {P}}'}\cup {{\mathcal {L}}'})}.
(Bruck 1955) proves the following theorem. Let Π be a finite projective plane of orderNwith a proper subplane Π0of orderM. Then eitherN=M2orN≥M2+M.
A subplane(P′,L′){\displaystyle ({{\mathcal {P}}'},{{\mathcal {L}}'})}of(P,L,I){\displaystyle ({\mathcal {P}},{\mathcal {L}},I)}is aBaer subplaneif every line inL∖L′{\displaystyle {\mathcal {L}}\setminus {{\mathcal {L}}'}}is incident with exactly one point inP′{\displaystyle {\mathcal {P}}'}and every point inP∖P′{\displaystyle {\mathcal {P}}\setminus {{\mathcal {P}}'}}is incident with exactly one line ofL′{\displaystyle {\mathcal {L}}'}.
A finite Desarguesian projective plane of orderq{\displaystyle q}admits Baer subplanes (all necessarily Desarguesian) if and
only ifq{\displaystyle q}is square; in this
case the order of the Baer subplanes isq{\displaystyle {\sqrt {q}}}.
In the finite Desarguesian planes PG(2,pn), the subplanes have orders which are the orders of the subfields of the finite field GF(pn), that is,piwhereiis a divisor ofn. In non-Desarguesian planes however, Bruck's theorem gives the only information about subplane orders. The case of equality in the inequality of this theorem is not known to occur. Whether or not there exists a subplane of orderMin a plane of orderNwithM2+M=Nis an open question. If such subplanes existed there would be projective planes of composite (non-prime power) order.
AFano subplaneis a subplane isomorphic to PG(2, 2), the unique projective plane of order 2.
If you consider aquadrangle(a set of 4 points no three collinear) in this plane, the points determine six of the lines of the plane. The remaining three points (called thediagonal pointsof the quadrangle) are the points where the lines that do not intersect at a point of the quadrangle meet. The seventh line consists of all the diagonal points (usually drawn as a circle or semicircle).
In finite desarguesian planes, PG(2,q), Fano subplanes exist if and only ifqis even (that is, a power of 2). The situation in non-desarguesian planes is unsettled. They could exist in any non-desarguesian plane of order greater than 6, and indeed, they have been found in all non-desarguesian planes in which they have been looked for (in both odd and even orders).
An open question, apparently due toHanna Neumannthough not published by her, is: Does every non-desarguesian plane contain a Fano subplane?
A theorem concerning Fano subplanes due to (Gleason 1956) is:
Projectivization of the Euclidean plane produced the real projective plane. The inverse operation—starting with a projective plane, remove one line and all the points incident with that line—produces anaffine plane.
More formally anaffine planeconsists of a set oflinesand a set ofpoints, and a relation between points and lines calledincidence, having the following properties:
The second condition means that there areparallel linesand is known asPlayfair'saxiom. The expression "does not meet" in this condition is shorthand for "there does not exist a point incident with both lines".
The Euclidean plane and the Moulton plane are examples of infinite affine planes. A finite projective plane will produce a finite affine plane when one of its lines and the points on it are removed. Theorderof a finite affine plane is the number of points on any of its lines (this will be the same number as the order of the projective plane from which it comes). The affine planes which arise from the projective planes PG(2,q) are denoted by AG(2,q).
There is a projective plane of orderNif and only if there is anaffine planeof orderN. When there is only one affine plane of orderNthere is only one projective plane of orderN, but the converse is not true. The affine planes formed by the removal of different lines of the projective plane will be isomorphic if and only if the removed lines are in the same orbit of the collineation group of the projective plane. These statements hold for infinite projective planes as well.
The affine planeK2overKembeds intoKP2via the map which sends affine (non-homogeneous) coordinates tohomogeneous coordinates,
The complement of the image is the set of points of the form(0,x1,x2). From the point of view of the embedding just given, these points are thepoints at infinity. They constitute a line inKP2—namely, the line arising from the plane
inK3—called theline at infinity. The points at infinity are the "extra" points where parallel lines intersect in the construction of the extended real plane; the point (0,x1,x2) is where all lines of slopex2/x1intersect. Consider for example the two lines
in the affine planeK2. These lines have slope 0 and do not intersect. They can be regarded as subsets ofKP2via the embedding above, but these subsets are not lines inKP2. Add the point(0, 1, 0)to each subset; that is, let
These are lines inKP2; ū arises from the plane
inK3, while ȳ arises from the plane
The projective lines ū and ȳ intersect at(0, 1, 0). In fact, all lines inK2of slope 0, when projectivized in this manner, intersect at(0, 1, 0)inKP2.
The embedding ofK2intoKP2given above is not unique. Each embedding produces its own notion of points at infinity. For example, the embedding
has as its complement those points of the form(x0, 0,x2), which are then regarded as points at infinity.
When an affine plane does not have the form ofK2withKa division ring, it can still be embedded in a projective plane, but the construction used above does not work. A commonly used method for carrying out the embedding in this case involves expanding the set of affine coordinates and working in a more general "algebra".
One can construct a coordinate "ring"—a so-calledplanar ternary ring(not a genuine ring)—corresponding to any projective plane. A planar ternary ring need not be a field or division ring, and there are many projective planes that are not constructed from a division ring. They are callednon-Desarguesian projective planesand are an active area of research. TheCayley plane(OP2), a projective plane over theoctonions, is one of these because the octonions do not form a division ring.[8]
Conversely, given a planar ternary ring (R,T), a projective plane can be constructed (see below). The relationship is not one to one. A projective plane may be associated with several non-isomorphic planar ternary rings. The ternary operatorTcan be used to produce two binary operators on the setR, by:
The ternary operator islinearifT(x,m,k) =x⋅m+k. When the set of coordinates of a projective plane actually form a ring, a linear ternary operator may be defined in this way, using the ring operations on the right, to produce a planar ternary ring.
Algebraic properties of this planar ternary coordinate ring turn out to correspond to geometric incidence properties of the plane. For example,Desargues' theoremcorresponds to the coordinate ring being obtained from adivision ring, whilePappus's theoremcorresponds to this ring being obtained from acommutativefield. A projective plane satisfying Pappus's theorem universally is called aPappian plane.Alternative, not necessarilyassociative, division algebras like the octonions correspond toMoufang planes.
There is no known purely geometric proof of the purely geometric statement that Desargues' theorem implies Pappus' theorem in a finite projective plane (finite Desarguesian planes are Pappian). (The converse is true in any projective plane and is provable geometrically, but finiteness is essential in this statement as there are infinite Desarguesian planes which are not Pappian.) The most common proof uses coordinates in a division ring andWedderburn's theoremthat finite division rings must be commutative;Bamberg & Penttila (2015)give a proof that uses only more "elementary" algebraic facts about division rings.
To describe a finite projective plane of orderN(≥ 2) using non-homogeneous coordinates and a planar ternary ring:
On these points, construct the following lines:
For example, forN= 2we can use the symbols {0, 1} associated with the finite field of order 2. The ternary operation defined byT(x,m,k) =xm+kwith the operations on the right being the multiplication and addition in the field yields the following:
Degenerate planes do not fulfill thethird conditionin the definition of a projective plane. They are not structurally complex enough to be interesting in their own right, but from time to time they arise as special cases in general arguments. There are seven kinds of degenerate plane according to (Albert & Sandler 1968). They are:
These seven cases are not independent, the fourth and fifth can be considered as special cases of the sixth, while the second and third are special cases of the fourth and fifth respectively. The special case of the seventh plane with no additional lines can be seen as an eighth plane. All the cases can therefore be organized into two families of degenerate planes as follows (this representation is for finite degenerate planes, but may be extended to infinite ones in a natural way):
1) For any number of pointsP1, ...,Pn, and linesL1, ...,Lm,
2) For any number of pointsP1, ...,Pn, and linesL1, ...,Ln, (same number of points as lines)
Acollineationof a projective plane is abijective mapof the plane to itself which maps points to points and lines to lines that preserves incidence, meaning that ifσis a bijection and pointPis on linem, thenPσis onmσ.[9]
Ifσis a collineation of a projective plane, a pointPwithP=Pσis called afixed pointofσ, and a linemwithm=mσis called afixed lineofσ. The points on a fixed line need not be fixed points, their images underσare just constrained to lie on this line. The collection of fixed points and fixed lines of a collineation form aclosed configuration, which is a system of points and lines that satisfy the first two but not necessarily the third condition in thedefinitionof a projective plane. Thus, the fixed point and fixed line structure for any collineation either form a projective plane by themselves, or adegenerate plane. Collineations whose fixed structure forms a plane are calledplanar collineations.
Ahomography(orprojective transformation) of PG(2,K) is a collineation of this type of projective plane which is a linear transformation of the underlying vector space. Using homogeneous coordinates they can be represented by invertible3 × 3matrices overKwhich act on the points of PG(2,K) byy=MxT, wherexandyare points inK3(vectors) andMis an invertible3 × 3matrix overK.[10]Two matrices represent the same projective transformation if one is a constant multiple of the other. Thus the group of projective transformations is the quotient of thegeneral linear groupby the scalar matrices called theprojective linear group.
Another type of collineation of PG(2,K) is induced by anyautomorphismofK, these are calledautomorphic collineations. Ifαis an automorphism ofK, then the collineation given by(x0,x1, x2) → (x0α,x1α,x2α)is an automorphic collineation. Thefundamental theorem of projective geometrysays that all the collineations of PG(2,K) are compositions of homographies and automorphic collineations. Automorphic collineations are planar collineations.
A projective plane is defined axiomatically as anincidence structure, in terms of a setPof points, a setLof lines, and anincidence relationIthat determines which points lie on which lines. AsPandLare only sets one can interchange their roles and define aplane dual structure.
By interchanging the role of "points" and "lines" in
we obtain the dual structure
whereI* is theconverse relationofI.
In a projective plane a statement involving points, lines and incidence between them that is obtained from another such statement by interchanging the words "point" and "line" and making whatever grammatical adjustments that are necessary, is called theplane dual statementof the first. The plane dual statement of "Two points are on a unique line." is "Two lines meet at a unique point." Forming the plane dual of a statement is known asdualizingthe statement.
If a statement is true in a projective planeC, then the plane dual of that statement must be true in the dual planeC*. This follows since dualizing each statement in the proof "inC" gives a statement of the proof "inC*."
In the projective planeC, it can be shown that there exist four lines, no three of which are concurrent. Dualizing this theorem and the first two axioms in the definition of a projective plane shows that the plane dual structureC* is also a projective plane, called thedual planeofC.
IfCandC* are isomorphic, thenCis calledself-dual. The projective planes PG(2,K) for any division ringKare self-dual. However, there arenon-Desarguesian planeswhich are not self-dual, such as the Hall planes and some that are, such as theHughes planes.
ThePrinciple of plane dualitysays that dualizing any theorem in a self-dual projective planeCproduces another theorem valid inC.
Adualityis a map from a projective planeC= (P,L,I)to its dual planeC* = (L,P,I*)(seeabove) which preserves incidence. That is, a dualityσwill map points to lines and lines to points (Pσ=LandLσ=P) in such a way that if a pointQis on a linem(denoted byQIm) thenQσI*mσ⇔mσIQσ. A duality which is an isomorphism is called acorrelation.[11]If a correlation exists then the projective planeCis self-dual.
In the special case that the projective plane is of thePG(2,K)type, withKa division ring, a duality is called areciprocity.[12]These planes are always self-dual. By thefundamental theorem of projective geometrya reciprocity is the composition of anautomorphic functionofKand ahomography. If the automorphism involved is the identity, then the reciprocity is called aprojective correlation.
A correlation of order two (aninvolution) is called apolarity. If a correlationφis not a polarity thenφ2is a nontrivial collineation.
It can be shown that a projective plane has the same number of lines as it has points (infinite or finite). Thus, for every finite projective plane there is anintegerN≥ 2 such that the plane has
The numberNis called theorderof the projective plane.
The projective plane of order 2 is called theFano plane. See also the article onfinite geometry.
Using the vector space construction with finite fields there exists a projective plane of orderN=pn, for each prime powerpn. In fact, for all known finite projective planes, the orderNis a prime power.[citation needed]
The existence of finite projective planes of other orders is an open question. The only general restriction known on the order is theBruck–Ryser–Chowla theoremthat if the orderNiscongruentto 1 or 2 mod 4, it must be the sum of two squares. This rules outN= 6. The next caseN= 10has been ruled out by massive computer calculations.[13]Nothing more is known; in particular, the question of whether there exists a finite projective plane of orderN= 12is still open.[citation needed]
Another longstanding open problem is whether there exist finite projective planes ofprimeorder which are not finite field planes (equivalently, whether there exists a non-Desarguesian projective plane of prime order).[citation needed]
A projective plane of orderNis a SteinerS(2,N+ 1,N2+N+ 1)system
(seeSteiner system). Conversely, one can prove that all Steiner systems of this form (λ= 2) are projective planes.
Automorphismsfor PG(n,k), withk=pm,p=prime is (m!)(kn+1− 1)(kn+1−k)(kn+1−k2)...(kn+1−kn)/(k− 1).
The number of mutuallyorthogonal Latin squaresof orderNis at mostN− 1.N− 1exist if and only if there is a projective plane of orderN.
While the classification of all projective planes is far from complete, results are known for small orders:
Projective planes may be thought of asprojective geometriesof dimension two.[15]Higher-dimensional projective geometries can be defined in terms of incidence relations in a manner analogous to the definition of a projective plane.
The smallest projective space of dimension 3 isPG(3,2).
These turn out to be "tamer" than the projective planes since the extra degrees of freedom permitDesargues' theoremto be proved geometrically in the higher-dimensional geometry. This means that the coordinate "ring" associated to the geometry must be adivision ring(skewfield)K, and the projective geometry is isomorphic to the one constructed from the vector spaceKd+1, i.e. PG(d,K). As in the construction given earlier, the points of thed-dimensionalprojective spacePG(d,K) are the lines through the origin inKd+1and a line in PG(d,K) corresponds to a plane through the origin inKd+1. In fact, eachi-dimensional object in PG(d,K), withi<d, is an(i+ 1)-dimensional (algebraic) vector subspace ofKd+1("goes through the origin"). The projective spaces in turn generalize to theGrassmannian spaces.
It can be shown that if Desargues' theorem holds in a projective space of dimension greater than two, then it must also hold in all planes that are contained in that space. Since there are projective planes in which Desargues' theorem fails (non-Desarguesian planes), these planes can not be embedded in a higher-dimensional projective space. Only the planes from the vector space construction PG(2,K) can appear in projective spaces of higher dimension. Some disciplines in mathematics restrict the meaning of projective plane to only this type of projective plane since otherwise general statements about projective spaces would always have to mention the exceptions when the geometric dimension is two.[16]
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Inmathematicsandprobability theory,continuum percolation theoryis a branch of mathematics that extends discretepercolation theorytocontinuous space(oftenEuclidean spaceℝn). More specifically, the underlying points of discrete percolation form types of lattices whereas the underlying points of continuum percolation are often randomly positioned in some continuous space and form a type ofpoint process. For each point, a random shape is frequently placed on it and the shapes overlap each with other to form clumps or components. As in discrete percolation, a common research focus of continuum percolation is studying the conditions of occurrence for infinite or giant components.[1][2]Other shared concepts and analysis techniques exist in these two types of percolation theory as well as the study ofrandom graphsandrandom geometric graphs.
Continuum percolation arose from an early mathematical model forwireless networks,[2][3]which, with the rise of several wireless network technologies in recent years, has been generalized and studied in order to determine the theoretical bounds ofinformation capacityand performance in wireless networks.[4][5]In addition to this setting, continuum percolation has gained application in other disciplines including biology, geology, and physics, such as the study ofporous materialandsemiconductors, while becoming a subject of mathematical interest in its own right.[6]
In the early 1960sEdgar Gilbert[3]proposed a mathematical model in wireless networks that gave rise to the field of continuum percolation theory, thus generalizing discrete percolation.[2]The underlying points of this model, sometimes known as the Gilbert disk model, were scattered uniformly in the infinite planeℝ2according to a homogeneousPoisson process. Gilbert, who had noticed similarities between discrete and continuum percolation,[7]then used concepts and techniques from the probability subject ofbranching processesto show that athreshold valueexisted for the infinite or "giant" component.
The exact names, terminology, and definitions of these models may vary slightly depending on the source, which is also reflected in the use ofpoint process notation.
A number of well-studied models exist in continuum percolation, which are often based on homogeneousPoisson point processes.
Consider a collection of points{xi}in the planeℝ2that form a homogeneous Poisson processΦwith constant (point) densityλ. For each point of the Poisson process (i.e.xi∈Φ), place a diskDiwith its center located at the pointxi. If each diskDihas a random radiusRi(from a commondistribution) that isindependentof all the other radii and all the underlying points{xi}, then the resulting mathematical structure is known as a random disk model.
Given a random disk model, if the set union of all the disks{Di}is taken, then the resulting structure⋃iDiis known as a Boolean–Poisson model (also known as simply theBoolean model),[8]which is a commonly studied model in continuum percolation[1]as well asstochastic geometry.[8]If all the radii are set to some common constant, say,r> 0, then the resulting model is sometimes known as the Gilbert disk (Boolean) model.[9]
The disk model can be generalized to more arbitrary shapes where, instead of a disk, a randomcompact(hence bounded and closed inℝ2) shapeSiis placed on each pointxi. Again, each shapeSihas a commondistributionandindependentto all other shapes and the underlying (Poisson) point process. This model is known as the germ–grain model where the underlying points{xi}are thegermsand the random compact shapesSiare thegrains. Theset unionof all the shapes forms a Boolean germ-grain model. Typical choices for the grains include disks, randompolygonand segments of random length.[8]
Boolean models are also examples ofstochastic processesknown as coverage processes.[10]The above models can be extended from the planeℝ2to general Euclidean spaceℝn.
In the Boolean–Poisson model, disks there can be isolated groups or clumps of disks that do not contact any other clumps of disks. These clumps are known as components. If the area (or volume in higher dimensions) of a component is infinite, one says it is an infinite or "giant" component. A major focus of percolation theory is establishing the conditions when giant components exist in models, which has parallels with the study of random networks. If no big component exists, the model is said to be subcritical. The conditions of giant component criticality naturally depend on parameters of the model such as the density of the underlying point process.
The excluded area of a placed object is defined as the minimal area around the object into which an additional object cannot be placed without overlapping with the first object. For example, in a system of randomly oriented homogeneous rectangles of lengthl, widthwand aspect ratior=l/w, the average excluded area is given by:[11]
In a system of identical ellipses with semi-axesaandband ratior=a/b, and perimeterC, the average excluded areas is given by:[12]
The excluded area theory states that the critical number density (percolation threshold)Ncof a system is inversely proportional to the average excluded areaAr:
It has been shown via Monte-Carlo simulations that percolation threshold in both homogeneous and heterogeneous systems of rectangles or ellipses is dominated by the average excluded areas and can be approximated fairly well by the linear relation
with a proportionality constant in the range 3.1–3.5.[11][12]
The applications of percolation theory are various and range from material sciences towireless communicationsystems. Often the work involves showing that a type ofphase transitionoccurs in the system.
Wireless networks are sometimes best represented with stochastic models owing to their complexity and unpredictability, hence continuum percolation have been used to developstochastic geometry models of wireless networks. For example, the tools of continuous percolation theory and coverage processes have been used to study the coverage and connectivity ofsensor networks.[13][14]One of the main limitations of these networks is energy consumption where usually each node has a battery and an embedded form of energy harvesting. To reduce energy consumption in sensor networks, various sleep schemes have been suggested that entail having a subcollection of nodes go into a low energy-consuming sleep mode. These sleep schemes obviously affect the coverage and connectivity of sensor networks. Simple power-saving models have been proposed such as the simple uncoordinated 'blinking' model where (at each time interval) each node independently powers down (or up) with some fixed probability. Using the tools of percolation theory, a blinking Boolean Poisson model has been analyzed to study thelatencyand connectivity effects of such a simple power scheme.[13]
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Incomputer programming,bounds checkingis any method of detecting whether avariableis within someboundsbefore it is used. It is usually used to ensure that a number fits into a given type (range checking), or that a variable being used as anarrayindex is within the bounds of the array (index checking). A failed bounds check usually results in the generation of some sort ofexceptionsignal.
As performing bounds checking during each use can be time-consuming, it is not always done.Bounds-checking eliminationis acompiler optimizationtechnique that eliminates unneeded bounds checking.
A range check is a check to make sure a number is within a certain range; for example, to ensure that a value about to be assigned to a 16-bit integer is within the capacity of a 16-bit integer (i.e. checking againstwrap-around). This is not quite the same astype checking.[how?]Other range checks may be more restrictive; for example, a variable to hold the number of a calendar month may be declared to accept only the range 1 to 12.
Example inPython:
Index checking means that, in allexpressionsindexing an array, the index value is checked against the bounds of the array (which were established when the array was defined), and if the index is out-of-bounds, further execution is suspended via some sort of error. Because reading or especially writing a value outside the bounds of an array may cause the program to malfunction or crash or enable security vulnerabilities (seebuffer overflow), index checking is a part of manyhigh-level languages.
Early compiledprogramming languageswith index checking ability includedALGOL 60,ALGOL 68andPascal, as well as interpreted programming languages such asBASIC.
Many programming languages, such asC, never perform automatic bounds checking to raise speed. However, this leaves manyoff-by-one errorsandbuffer overflowsuncaught. Many programmers believe these languages sacrifice too much for rapid execution.[1]In his 1980Turing Awardlecture,C. A. R. Hoaredescribed his experience in the design ofALGOL 60, a language that included bounds checking, saying:
A consequence of this principle is that every occurrence of every subscript of every subscripted variable was on every occasion checked at run time against both the upper and the lower declared bounds of the array. Many years later we asked our customers whether they wished us to provide an option to switch off these checks in the interest of efficiency on production runs. Unanimously, they urged us not to—they already knew how frequently subscript errors occur on production runs where failure to detect them could be disastrous. I note with fear and horror that even in 1980, language designers and users have not learned this lesson. In any respectable branch of engineering, failure to observe such elementary precautions would have long been against the law.
Mainstream languages that enforce run time checking includeAda,C#,Haskell,Java,JavaScript,Lisp,PHP,Python,Ruby,Rust, andVisual Basic. TheDandOCamllanguages have run time bounds checking that is enabled or disabled with a compiler switch. InC++run time checking is not part of the language, but part of theSTLand is enabled with a compiler switch (_GLIBCXX_DEBUG=1 or _LIBCPP_DEBUG=1). C# also supportsunsafe regions: sections of code that (among other things) temporarily suspend bounds checking to raise efficiency. These are useful for speeding up small time-critical bottlenecks without sacrificing the safety of a whole program.
TheJS++programming language is able to analyze if an array index or map key is out-of-bounds at compile time usingexistent types, which is anominal typedescribing whether the index or key is within-bounds or out-of-bounds and guides code generation. Existent types have been shown to add only1ms overhead[clarify]to compile times.[2]
The safety added by bounds checking necessarily costs CPU time if the checking is performed in software; however, if the checks could be performed by hardware, then the safety can be provided "for free" with no runtime cost. An early system with hardware bounds checking was theICL 2900 Seriesmainframe announced in 1974.[3]TheVAXcomputer has an INDEX assembly instruction for array index checking which takes six operands, all of which can use any VAX addressing mode. The B6500 and similarBurroughscomputers performed bound checking via hardware, irrespective of which computer language had been compiled to produce the machine code. A limited number of laterCPUshave specialised instructions for checking bounds, e.g., the CHK2 instruction on theMotorola 68000series.
Research has been underway since at least 2005 regarding methods to use x86's built-in virtual memory management unit to ensure safety of array and buffer accesses.[4]In 2015 Intel provided theirIntel MPXextensions in theirSkylakeprocessor architecture which stores bounds in a CPU register and table in memory. As of early 2017 at leastGCCsupports MPX extensions.
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Counterexample-guided abstraction refinement(CEGAR) is a technique forsymbolic model checking.[1][2]It is also applied inmodal logictableau calculialgorithms to optimise their efficiency.[3]
In computer-aided verification and analysis of programs, models of computation often consist ofstates. Models for even small programs, however, may have an enormous number of states. This is identified as the state explosion problem.[4]CEGAR addresses this problem with two stages —abstraction, which simplifies a model by grouping states, andrefinement, which increases the precision of the abstraction to better approximate the original model.
If a desired property for a program is not satisfied in the abstract model, a counterexample is generated. The CEGAR process then checks whether the counterexample is spurious, i.e., if the counterexample also applies to the under-abstraction but not the actual program. If this is the case, it concludes that the counterexample is attributed to inadequate precision of the abstraction. Otherwise, the process finds a bug in the program. Refinement is performed when a counterexample is found to be spurious.[5]The iterative procedure terminates either if a bug is found or when the abstraction has been refined to the extent that it is equivalent to the original model.
To reason about the correctness of a program, particularly those involving the concept of time forconcurrency, state transition models are used. In particular, finite-state models can be used along withtemporal logicin automatic verification.[6]The concept of abstraction is thus founded upon a mapping between twoKripke structures. Specifically, programs can be described withcontrol-flow automata(CFA).[7]
Define a Kripke structureM{\displaystyle M}as⟨S,s0,R,L⟩{\displaystyle \langle S,s_{0},R,L\rangle }, where
An abstraction ofM{\displaystyle M}is defined by⟨Sα,s0α,Rα,Lα⟩{\displaystyle \langle S_{\alpha },s_{0}^{\alpha },R_{\alpha },L_{\alpha }\rangle }whereα{\displaystyle \alpha }is an abstraction mapping that maps every state inS{\displaystyle S}to a state inSα{\displaystyle S_{\alpha }}.[5]
To preserve the critical properties of the model, the abstraction mapping maps the initial state in the original models0{\displaystyle s_{0}}to its counterparts0α{\displaystyle s_{0}^{\alpha }}in the abstract model. The abstraction mapping also guarantees that the transition relations between two states are preserved.
In each iteration,model checkingis performed for the abstract model. Bounded model checking, for instance, generates a propositional formula that is then checked forBoolean satisfiabilityby aSAT solver.[5]
When counterexamples are found, they are examined to determine if they are spurious examples, i.e., they are unauthentic ones that emerge from the under-abstraction of the model. A non-spurious counterexample reflects the incorrectness of the program, which may be sufficient to terminate the program verification process and conclude that the program is incorrect. The main objective of the refinement process handle spurious counterexamples. It eliminates them by increasing the granularity of the abstraction.
The refinement process ensures that the dead-end states and the bad states do not belong to the same abstract state. A dead-end state is a reachable one with no outgoing transition whereas a bad-state is one with transitions causing the counterexample.[2]
Sincemodal logicis often interpreted withKripke semantics, where a Kripke frame resembles the structure of state transition systems concerned in program verification, the CEGAR technique is also implemented forautomated theorem proving.[3]
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Asprachbund(/ˈsprɑːkbʊnd/, fromGerman:Sprachbund[ˈʃpʁaːxbʊnt]ⓘ,lit.'language federation'), also known as alinguistic area,area of linguistic convergence, ordiffusion area, is a group oflanguagesthat shareareal featuresresulting from geographical proximity andlanguage contact. The languages may begenetically unrelated, or only distantly related, but the sprachbund characteristics might give a false appearance of relatedness.
A grouping of languages that share features can only be defined as a sprachbund if the features are shared for some reason other than the genetic history of the languages. Without knowledge of the history of a regional group of similar languages, it may be difficult to determine whether sharing indicates a language family or a sprachbund.[1]
In a 1904 paper,Jan Baudouin de Courtenayemphasised the need to distinguish between language similarities arising from a genetic relationship (rodstvo) and those arising fromconvergencedue to language contact (srodstvo).[2][3]
Nikolai Trubetzkoyintroduced the Russian termязыковой союз(yazykovoy soyuz'language union') in a 1923 article.[4]In a paper presented to the firstInternational Congress of Linguistsin 1928, he used a Germancalqueof this term,Sprachbund, defining it as a group of languages with similarities insyntax, morphological structure, cultural vocabulary and sound systems, but without systematic sound correspondences, shared basic morphology or shared basic vocabulary.[5][3]
Later workers, starting with Trubetzkoy's colleagueRoman Jakobson,[6][7]have relaxed the requirement of similarities in all four of the areas stipulated by Trubetzkoy.[8][9][10]
A rigorous set of principles for what evidence is valid for establishing a linguistic area has been presented by Campbell, Kaufman, and Smith-Stark.[11]
The idea of areal convergence is commonly attributed toJernej Kopitar's description in 1830 ofAlbanian,BulgarianandRomanianas giving the impression of"nur eine Sprachform ... mit dreierlei Sprachmaterie"(lit.'only one language form with three kinds of language material'),[12]which has been rendered byVictor Friedmanas "one grammar with the [sic] three lexicons".[13][14]
TheBalkan Sprachbundcomprises Albanian, Romanian, theSouth Slavic languagesof the southern Balkans (Bulgarian,Macedonianand to a lesser degreeSerbo-Croatian),Greek, BalkanTurkish, andRomani.
All but one of these areIndo-European languagesbut from very divergent branches, and Turkish is aTurkic language. Yet they have exhibited several signs of grammatical convergence, such as avoidance of theinfinitive,future tenseformation, and others.
The same features are not found in other languages that are otherwise closely related, such as the other Romance languages in relation to Romanian, and the other Slavic languages such as Polish in relation to Bulgaro-Macedonian.[9][14]
Languages of theMainland Southeast Asia linguistic areahave such great surface similarity that early linguists tended to group them all into a single family, although the modern consensus places them into numerous unrelated families. The area stretches from Thailand to China and is home to speakers of languages of theSino-Tibetan,Hmong–Mien(or Miao–Yao),Tai–Kadai,Austronesian(represented byChamic) andMon–Khmerfamilies.[15]
Neighbouring languages across these families, though presumed unrelated, often have similar features, which are believed to have spread by diffusion.
A well-known example is the similartonesystems inSinitic languages(Sino-Tibetan), Hmong–Mien,Tai languages(Kadai) andVietnamese(Austroasiatic). Most of these languages passed through an earlier stage with three tones on most syllables (but no tonal distinctions onchecked syllablesending in astop consonant), which was followed by atone splitwhere the distinction between voiced and voiceless consonants disappeared but in compensation the number of tones doubled. These parallels led to confusion over the classification of these languages, untilAndré-Georges Haudricourtshowed in 1954 that tone was not an invariant feature, by demonstrating that Vietnamese tones corresponded to certain final consonants in other languages of the Mon–Khmer family, and proposed that tone in the other languages had a similar origin.[15]
Similarly, the unrelatedKhmer(Mon–Khmer),Cham(Austronesian) andLao(Kadai) languages have almost identical vowel systems.
Many languages in the region are of theisolating(or analytic) type, with mostly monosyllabic morphemes and little use ofinflectionoraffixes, though a number of Mon–Khmer languages havederivational morphology.
Shared syntactic features includeclassifiers,object–verb orderandtopic–commentstructure, though in each case there are exceptions in branches of one or more families.[15]
In a classic 1956 paper titled "India as a Linguistic Area",Murray Emeneaulaid the groundwork for the general acceptance of the concept of a sprachbund. In the paper, Emeneau observed that the subcontinent'sDravidianandIndo-Aryan languagesshared a number of features that were not inherited from a common source, but wereareal features, the result of diffusion during sustained contact. These includeretroflex consonants,echo words,subject–object–verbword order,discourse markers, and thequotative.[16]
Emeneau specified the tools to establish that language and culture had fused for centuries on the Indian soil to produce an integrated mosaic of structural convergence of four distinct language families:Indo-Aryan,Dravidian,MundaandTibeto-Burman. This concept provided scholarly substance for explaining the underlying Indian-ness of apparently divergent cultural and linguistic patterns. With his further contributions, this area has now become a major field of research in language contact and convergence.[9][17][18]
Some linguists, such asMatthias Castrén,G. J. Ramstedt,Nicholas PoppeandPentti Aalto, supported the idea that theMongolic,Turkic, andTungusicfamilies of Asia (and some small parts of Europe) have a common ancestry, in a controversial group they callAltaic.[citation needed]KoreanicandJaponiclanguages, which are also hypothetically related according to some scholars likeWilliam George Aston, Shōsaburō Kanazawa,Samuel MartinandSergei Starostin, are sometimes included as part of the purported Altaic family. This latter hypothesis was supported by people includingRoy Andrew Miller, John C. Street andKarl Heinrich Menges.Gerard Clauson,Gerhard Doerfer,Juha Janhunen,Stefan Georgand others dispute or reject this.[citation needed]A common alternative explanation for similarities among the "Altaic" languages, such asvowel harmonyandagglutination, is that they are due to areal diffusion.[19]
TheQinghai–Gansu sprachbund, in the northeastern part of theTibetan plateauspanning the Chinese provinces ofQinghaiandGansu, is an area of interaction between varieties of northwestMandarin Chinese,Amdo TibetanandMongolicandTurkic languages.[citation needed]
Standard Average European(SAE) is a concept introduced in 1939 byBenjamin Whorfto group the modernIndo-Europeanlanguages of Europewhich shared common features.[20]Whorf argued that theselanguageswere characterized by a number of similarities includingsyntaxandgrammar,vocabularyand its use as well as the relationship between contrasting words and their origins, idioms and word order which all made them stand out from many other language groups around the world which do not share these similarities; in essence creating a continentalsprachbund. His point was to argue that the disproportionate degree of knowledge of SAE languages biasedlinguiststowards considering grammatical forms to be highly natural or even universal, when in fact they were only peculiar to the SAElanguage group.
Whorf likely consideredRomanceandWest Germanicto form the core of the SAE, i.e. theliterary languagesofEuropewhich have seen substantial cultural influence fromLatinduring themedieval period. TheNorth GermanicandBalto-Slavic languagestend to be more peripheral members.
Alexander Gode, who was instrumental in the development ofInterlingua, characterized it as "Standard Average European".[21]The Romance,Germanic, andSlaviccontrol languages of Interlingua are reflective of the language groups most often included in the SAESprachbund.
The Standard Average EuropeanSprachbundis most likely the result of ongoinglanguage contactin the time of theMigration Period[22][full citation needed]and later, continuing during theMiddle Agesand theRenaissance.[citation needed]Inheritance of the SAE features fromProto-Indo-Europeancan be ruled out because Proto-Indo-European, as currently reconstructed, lacked most of the SAE features.[23][full citation needed]
Language families that have been proposed to actually be sprachbunds
The work began to assume the character of a comparison betweenHopiand western European languages. It also became evident that even the grammar of Hopi bore a relation to Hopi culture, and the grammar of European tongues to our own "Western" or "European" culture. And it appeared that the interrelation brought in those large subsummations of experience by language, such as our own terms "time", "space", "substance", and "matter". Since, with respect to the traits compared, there is little difference betweenEnglish,French,German, or otherEuropean languageswith the 'possible' (but doubtful) exception ofBalto-Slavicandnon-Indo-European, I have lumped these languages into one group called SAE, or "Standard Average European."
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Software cracking(known as "breaking" mostly in the 1980s[1]) is an act of removingcopy protectionfrom a software.[2]Copy protection can be removed by applying a specificcrack. Acrackcan mean any tool that enables breaking software protection, a stolen product key, or guessed password. Cracking software generally involves circumventing licensing and usage restrictions on commercial software by illegal methods. These methods can include modifying code directly through disassembling and bit editing, sharing stolen product keys, or developing software to generate activation keys.[3]Examples ofcracks are: applying apatchor by creating reverse-engineered serial number generators known askeygens, thus bypassing software registration and payments or converting a trial/demo version of the software into fully-functioning software without paying for it.[4]Software cracking contributes to the rise ofonline piracywhere pirated software is distributed to end-users[2]through filesharing sites likeBitTorrent,One click hosting(OCH), or viaUsenetdownloads, or by downloading bundles of the original software with cracks or keygens.[4]
Some of these tools are calledkeygen,patch,loader, orno-disc crack. A keygen is a handmade product serial number generator that often offers the ability to generate working serial numbers in your own name. A patch is a small computer program that modifies the machine code of another program. This has the advantage for a cracker to not include a large executable in a release when only a few bytes are changed.[5]A loader modifies the startup flow of a program and does not remove the protection but circumvents it.[6][7]A well-known example of a loader is atrainerused to cheat in games.[8]Fairlightpointed out in one of their.nfofiles that these type of cracks are not allowed forwarez scenegame releases.[9][6][10]Anukewarhas shown that the protection may not kick in at any point for it to be a valid crack.[11]
Software cracking is closely related toreverse engineeringbecause the process of attacking a copy protection technology, is similar to the process of reverse engineering.[12]The distribution of cracked copies is illegal in most countries. There have been lawsuits over cracking software.[13]It might be legal to use cracked software in certain circumstances.[14]Educational resources for reverse engineering and software cracking are, however, legal and available in the form ofCrackmeprograms.
Software are inherently expensive to produce but cheap to duplicate and distribute. Therefore, software producers generally tried to implement some form ofcopy protectionbefore releasing it to the market. In 1984, Laind Huntsman, the head of software development for Formaster, a software protection company, commented that "no protection system has remained uncracked by enterprising programmers for more than a few months".[2]In 2001, Dan S. Wallach, a professor fromRice University, argued that "those determined to bypass copy-protection have always found ways to do so – and always will".[15]
Most of the early software crackers were computer hobbyists who often formed groups that competed against each other in the cracking and spreading of software. Breaking a new copy protection scheme as quickly as possible was often regarded as an opportunity to demonstrate one's technical superiority rather than a possibility of money-making. Software crackers usually did not benefit materially from their actions and their motivation was the challenge itself of removing the protection.[2]Some low skilled hobbyists would take already cracked software and edit various unencrypted strings of text in it to change messages a game would tell a game player, often something considered vulgar. Uploading the altered copies on file sharing networks provided a source of laughs for adult users. The cracker groups of the 1980s started to advertise themselves and their skills by attaching animated screens known ascrack introsin the software programs they cracked and released.[16]Once the technical competition had expanded from the challenges of cracking to the challenges of creating visually stunning intros, the foundations for a new subculture known asdemoscenewere established. Demoscene started to separate itself from the illegal "warez scene" during the 1990s and is now regarded as a completely different subculture. Many software crackers have later grown into extremely capable software reverse engineers; the deep knowledge of assembly required in order to crack protections enables them toreverse engineerdriversin order to port them from binary-only drivers forWindowsto drivers with source code forLinuxand otherfreeoperating systems. Also because music and game intro was such an integral part of gaming the music format and graphics became very popular when hardware became affordable for the home user.
With the rise of theInternet, software crackers developed secretive online organizations. In the latter half of the nineties, one of the most respected sources of information about "software protection reversing" wasFravia's website.
In 2017, a group of software crackers started a project to preserveApple IIsoftware by removing thecopy protection.[17]
TheHigh Cracking University(+HCU) was founded byOld Red Cracker(+ORC), considered a genius of reverse engineering and a legendary figure inReverse Code Engineering(RCE), to advance research into RCE. He had also taught and authored many papers on the subject, and his texts are considered classics in the field and are mandatory reading for students of RCE.[18]
The addition of the "+" sign in front of the nickname of a reverser signified membership in the +HCU. Amongst the students of +HCU were the top of the elite Windows reversers worldwide.[18]+HCU published a new reverse engineering problem annually and a small number of respondents with the best replies qualified for an undergraduate position at the university.[18]
+Fravia was a professor at +HCU. Fravia's website was known as "+Fravia's Pages of Reverse Engineering" and he used it to challenge programmers as well as the wider society to "reverse engineer" the "brainwashing of a corrupt and rampant materialism". In its heyday, his website received millions of visitors per year and its influence was "widespread".[18]On his site, +Fravia also maintained a database of the tutorials generated by +HCU students for posterity.[19]
Nowadays most of the graduates of +HCU have migrated to Linux and few have remained as Windows reversers. The information at the university has been rediscovered by a new generation of researchers and practitioners of RCE who have started new research projects in the field.[18]
The most common software crack is the modification of an application's binary to cause or prevent a specific key branch in the program's execution. This is accomplished byreverse engineeringthe compiled program code using adebuggersuch asx64dbg,SoftICE,[20]OllyDbg,GDB, orMacsBuguntil the software cracker reaches thesubroutinethat contains the primary method of protecting the software (or bydisassemblingan executable file with a program such asIDA).[21]The binary is then modified using thedebuggeror ahex editorsuch asHIEW[22]ormonitorin a manner that replaces a prior branchingopcodewith its complement or aNOPopcodeso the key branch will either always execute a specificsubroutineor skip over it. Almost all common software cracks are a variation of this type. A region of code that must not be entered is often called a "bad boy" while one that should be followed is a "good boy".[23]
Proprietary softwaredevelopers are constantly developing techniques such ascode obfuscation,encryption, andself-modifying codeto make binary modification increasingly difficult.[24]Even with these measures being taken, developers struggle to combat software cracking. This is because it is very common for a professional to publicly release a simple cracked EXE or Retrium Installer for public download, eliminating the need for inexperienced users to crack the software themselves.
A specific example of this technique is a crack that removes the expiration period from a time-limited trial of an application. These cracks are usually programs that alter the program executable and sometimes the.dll or .solinked to the application and the process of altering the original binary files is called patching.[12]Similar cracks are available for software that requires a hardwaredongle. A company can also break the copy protection of programs that they have legally purchased but that arelicensedto particular hardware, so that there is no risk of downtime due to hardware failure (and, of course, no need to restrict oneself to running the software on bought hardware only).
Another method is the use of special software such asCloneCDto scan for the use of a commercial copy protection application. After discovering the software used to protect the application, another tool may be used to remove the copy protection from the software on theCDorDVD. This may enable another program such asAlcohol 120%, CloneDVD,Game Jackal, orDaemon Toolsto copy the protected software to a user's hard disk. Popular commercial copy protection applications which may be scanned for includeSafeDiscandStarForce.[25]
In other cases, it might be possible todecompilea program in order to get access to the originalsource codeor code on alevel higherthanmachine code. This is often possible withscripting languagesand languages utilizingJITcompilation. An example is cracking (or debugging) on the .NET platform where one might consider manipulatingCILto achieve one's needs.Java'sbytecodealso works in a similar fashion in which there is an intermediate language before the program is compiled to run on the platform dependentmachine code.[26]
Advanced reverse engineering for protections such asSecuROM,SafeDisc,StarForce, orDenuvorequires a cracker, or many crackers to spend much more time studying the protection, eventually finding every flaw within the protection code, and then coding their own tools to "unwrap" the protection automatically from executable (.EXE) and library (.DLL) files.
There are a number of sites on the Internet that let users download cracks produced bywarez groupsfor popular games and applications (although at the danger of acquiring malicious software that is sometimes distributed via such sites).[27]Although these cracks are used by legal buyers of software, they can also be used by people who have downloaded or otherwise obtained unauthorized copies (often throughP2Pnetworks).
Software cracking led to the distribution of pirated software around the world (software piracy). It was estimated that the United States lost US$2.3 billion in business application software in 1996. Software piracy rates were especially prevalent in African, Asian, Eastern European, and Latin American countries. In certain countries such as Indonesia, Pakistan, Kuwait, China, and El Salvador,[28]90% of the software used was pirated.[29]
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Inprobability theory, atelescoping Markov chain (TMC)is a vector-valuedstochastic processthat satisfies aMarkov propertyand admits a hierarchical format through a network of transition matrices with cascading dependence.[1]
For anyN>1{\displaystyle N>1}consider the set of spaces{Sℓ}ℓ=1N{\displaystyle \{{\mathcal {S}}^{\ell }\}_{\ell =1}^{N}}. The hierarchical processθk{\displaystyle \theta _{k}}defined in the product-space
is said to be a TMC if there is a set of transition probability kernels{Λn}n=1N{\displaystyle \{\Lambda ^{n}\}_{n=1}^{N}}such that
Thisprobability-related article is astub. You can help Wikipedia byexpanding it.
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Business System 12, or simplyBS12, was one of the first fullyrelational database management systems, designed and implemented byIBM'sBureau Servicesubsidiary at the company's international development centre inUithoorn,Netherlands. Programming started in 1978 and the first version was delivered in 1982. It was never widely used and essentially disappeared soon after the division was shut down in 1985, possibly because IBM and other companies settled onSQLas the standard.
BS12's lasting contribution to history was the use of a new query language based onISBL, created at IBM's UKScientific Centre. Developers of the famousSystem Runderway in the US at the same time were also consulted on certain matters concerning the engine, but the BS12 team rejectedSQLunequivocally, being convinced that this apparently unsound and difficult-to-use language (which at that time was also relationally incomplete) would never catch on.
BS12 included a number of interesting features that have yet to appear on most SQL-based systems, some a consequence of following the ISBL precedent, others due to deliberate design. For instance, a view could be parameterised andparameterscould be of type TABLE. Thus, a view could in effect be a newrelational operatordefined in terms of the existing operators.Codd'sDIVIDE operatorwas in fact implemented that way.
Another feature that could have easily been included in SQL systems was the support for update operations on the catalog tables (system tables describing the structure of the database, as in SQL). A new table could be created by inserting a row into theTABLEScatalog, and then columns added to it by inserting intoCOLUMNS.
In addition, BS12 was ahead of SQL in supporting user-defined functions and procedures, using aTuring completesublanguage,triggers, and a simple "call" interface for use by application programs, all in its very first release in 1982.
Sample query for determining which departments are over their salary budgets:[1]
Note the "natural join" on the common column,DEPTNUM. Although some SQL dialects support natural joins, for familiarity, the example will show only a "traditional" join. Here is the equivalent SQL for comparison:
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Speech analyticsis the process of analyzing recorded calls to gather customer information to improve communication and future interaction. The process is primarily used by customer contact centers to extract information buried in client interactions with an enterprise.[1]Although speech analytics includes elements ofautomatic speech recognition, it is known for analyzing the topic being discussed, which is weighed against the emotional character of the speech and the amount and locations of speech versus non-speech during the interaction. Speech analytics in contact centers can be used to mine recorded customer interactions to surface the intelligence essential for building effective cost containment and customer service strategies. The technology can pinpoint cost drivers, trend analysis, identify strengths and weaknesses with processes and products, and help understand how the marketplace perceives offerings.[2]
Speech analytics provides a Complete analysis of recorded phone conversations between a company and its customers.[3]It provides advanced functionality and valuable intelligence from customer calls. This information can be used to discover information relating to strategy, product, process, operational issues and contact center agent performance.[4]In addition, speech analytics can automatically identify areas in which contact center agents may need additional training or coaching,[5]and can automatically monitor the customer service provided on calls.[6]
The process can isolate the words and phrases used most frequently within a given time period, as well as indicate whether usage is trending up or down. This information is useful for supervisors, analysts, and others in an organization to spot changes in consumer behavior and take action to reduce call volumes—and increase customer satisfaction. It allows insight into a customer's thought process, which in turn creates an opportunity for companies to make adjustments.[7]
Speech analytics applications can spot spoken keywords or phrases, either as real-time alerts on live audio or as a post-processing step on recorded speech. This technique is also known asaudio mining. Other uses include categorization of speech in the contact center environment to identify calls from unsatisfied customers.[8]
Measures such asPrecision and recall, commonly used in the field ofInformation retrieval, are typical ways of quantifying the response of a speech analytics search system.[9]Precision measures the proportion of search results that are relevant to the query. Recall measures the proportion of the total number of relevant items that were returned by the search results. Where a standardised test set has been used, measures such as precision and recall can be used to directly compare the search performance of different speech analytics systems.
Making a meaningful comparison of the accuracy of different speech analytics systems can be difficult. The output of LVCSR systems can be scored against reference word-level transcriptions to produce a value for the word error rate (WER), but because phonetic systems use phones as the basic recognition unit, rather than words, comparisons using this measure cannot be made. When speech analytics systems are used to search for spoken words or phrases, what matters to the user is the accuracy of the search results that are returned. Because the impact of individual recognition errors on these search results can vary greatly, measures such as word error rate are not always helpful in determining overall search accuracy from the user perspective.
According to the US Government Accountability Office,[10]“data reliability refers to the accuracy and completeness of computer-processed data, given the uses they are intended for.” In the realm of Speech Recognition and Analytics, “completeness” is measured by the “detection rate”, and usually as accuracy goes up, the detection rate goes down.[11]
Speech analytics vendors use the "engine" of a 3rd party and others develop proprietary engines. The technology mainly uses three approaches. The phonetic approach is the fastest for processing, mostly because the size of the grammar is very small, with a phoneme as the basic recognition unit. There are only few tens of unique phonemes in most languages, and the output of this recognition is a stream (text) of phonemes, which can then be searched. Large-vocabulary continuous speech recognition (LVCSR, more commonly known as speech-to-text, full transcription or ASR - automatic speech recognition) uses a set of words (bi-grams, tri-grams etc.) as the basic unit. This approach requires hundreds of thousands of words to match the audio against. It can surface new business issues, the queries are much faster, and the accuracy is higher than the phonetic approach.[12]
Extendedspeech emotion recognitionand prediction is based on three main classifiers: kNN, C4.5 and SVM RBF Kernel. This set achieves better performance than each basic classifier taken separately. It is compared with two other sets of classifiers: one-against-all (OAA) multiclass SVM with Hybrid kernels and the set of classifiers which consists of the following two basic classifiers: C5.0 and Neural Network. The proposed variant achieves better performance than the other two sets of classifiers.[13]
Market research indicates that speech analytics is projected to become a billion dollar industry by 2020 withNorth Americahaving the largest market share.[14]The growth rate is attributed to rising requirements for compliance and risk management as well as an increase in industry competition through market intelligence.[15]Thetelecommunications,ITandoutsourcingsegments of the industry are considered to hold the largest market share with expected growth from the travel and hospitality segments.[14]
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https://en.wikipedia.org/wiki/Speech_analytics
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Apage tableis adata structureused by avirtual memorysystem in acomputerto store mappings betweenvirtual addressesandphysical addresses. Virtual addresses are used by the program executed by the accessingprocess, while physical addresses are used by the hardware, or more specifically, by therandom-access memory(RAM) subsystem. The page table is a key component ofvirtual address translationthat is necessary to accessdatain memory. The page table is set up by the computer'soperating system, and may be read and written during the virtual address translation process by thememory management unitor by low-level system software or firmware.
In operating systems that use virtual memory, every process is given the impression that it is working with large, contiguous sections of memory. Physically, the memory of each process may be dispersed across different areas of physical memory, or may have been moved (paged out) to secondary storage, typically to ahard disk drive(HDD) orsolid-state drive(SSD).
When a process requests access to data in its memory, it is the responsibility of the operating system to map the virtual address provided by the process to the physical address of the actual memory where that data is stored. The page table is where mappings of virtual addresses to physical addresses are stored, with each mapping also known as apage table entry(PTE).[1][2]
Thememory management unit(MMU) inside the CPU stores a cache of recently used mappings from the operating system's page table. This is called thetranslation lookaside buffer(TLB), which is an associative cache.
When a virtual address needs to be translated into a physical address, the TLB is searched first. If a match is found, which is known as aTLB hit, the physical address is returned and memory access can continue. However, if there is no match, which is called aTLB miss, the MMU, the system firmware, or the operating system's TLB miss handler will typically look up the address mapping in the page table to see whether a mapping exists, which is called apage walk. If one exists, it is written back to the TLB, which must be done because the hardware accesses memory through the TLB in a virtual memory system, and the faulting instruction is restarted, which may happen in parallel as well. The subsequent translation will result in a TLB hit, and the memory access will continue.
The page table lookup may fail, triggering apage fault, for two reasons:
When physical memory is not full this is a simple operation; the page is written back into physical memory, the page table and TLB are updated, and the instruction is restarted. However, when physical memory is full, one or more pages in physical memory will need to be paged out to make room for the requested page. The page table needs to be updated to mark that the pages that were previously in physical memory are no longer there, and to mark that the page that was on disk is now in physical memory. The TLB also needs to be updated, including removal of the paged-out page from it, and the instruction restarted. Which page to page out is the subject ofpage replacement algorithms.
Some MMUs trigger a page fault for other reasons, whether or not the page is currently resident in physical memory and mapped into the virtual address space of a process:
The simplest page table systems often maintain aframetable and a page table. The frame table holds information about which frames are mapped. In more advanced systems, the frame table can also hold information about which address space a page belongs to, statistics information, or other background information.
The page table is an array of page table entries.
Each page table entry (PTE) holds the mapping between a virtual address of a page and the address of a physical frame. There is also auxiliary information about the page such as a present bit, adirtyor modified bit, address space or process ID information, amongst others.
Secondary storage, such as a hard disk drive, can be used to augment physical memory. Pages can be paged in and out of physical memory and the disk. The present bit can indicate what pages are currently present in physical memory or are on disk, and can indicate how to treat these different pages, i.e. whether to load a page from disk and page another page in physical memory out.
The dirty bit allows for a performance optimization. A page on disk that is paged in to physical memory, then read from, and subsequently paged out again does not need to be written back to disk, since the page has not changed. However, if the page was written to after it is paged in, its dirty bit will be set, indicating that the page must be written back to the backing store. This strategy requires that the backing store retain a copy of the page after it is paged in to memory. When a dirty bit is not used, the backing store need only be as large as the instantaneous total size of all paged-out pages at any moment. When a dirty bit is used, at all times some pages will exist in both physical memory and the backing store.
In operating systems that are notsingle address space operating systems, address space or process ID information is necessary so the virtual memory management system knows what pages to associate to what process. Two processes may use two identical virtual addresses for different purposes. The page table must supply different virtual memory mappings for the two processes. This can be done by assigning the two processes distinct address map identifiers, or by using process IDs. Associating process IDs with virtual memory pages can also aid in selection of pages to page out, as pages associated with inactive processes, particularly processes whose code pages have been paged out, are less likely to be needed immediately than pages belonging to active processes.
As an alternative to tagging page table entries with process-unique identifiers, the page table itself may occupy a different virtual-memory page for each process so that the page table becomes a part of the process context. In such an implementation, the process's page table can be paged out whenever the process is no longer resident in memory.
There are several types of page tables, which are optimized for different requirements. Essentially, a bare-bones page table must store the virtual address, the physical address that is "under" this virtual address, and possibly some address space information.
Aninverted page table(IPT) is best thought of as an off-chip extension of theTLBwhich uses normal system RAM. Unlike a true page table, it is not necessarily able to hold all current mappings. The operating system must be prepared to handle misses, just as it would with a MIPS-style software-filled TLB.
The IPT combines a page table and aframe tableinto one data structure. At its core is a fixed-size table with the number of rows equal to the number of frames in memory. If there are 4,000 frames, the inverted page table has 4,000 rows. For each row there is an entry for the virtual page number (VPN), the physical page number (not the physical address), some other data and a means for creating acollisionchain, as we will see later.
Searching through all entries of the core IPT structure is inefficient, and ahash tablemay be used to map virtual addresses (and address space/PID information if need be) to an index in the IPT - this is where the collision chain is used. This hash table is known as ahash anchor table. The hashing function is not generally optimized for coverage - raw speed is more desirable. Of course, hash tables experience collisions. Due to this chosen hashing function, we may experience a lot of collisions in usage, so for each entry in the table the VPN is provided to check if it is the searched entry or a collision.
In searching for a mapping, the hash anchor table is used. If no entry exists, a page fault occurs. Otherwise, the entry is found. Depending on the architecture, the entry may be placed in the TLB again and the memory reference is restarted, or the collision chain may be followed until it has been exhausted and a page fault occurs.
A virtual address in this schema could be split into two, the first half being a virtual page number and the second half being the offset in that page.
A major problem with this design is poorcache localitycaused by thehash function. Tree-based designs avoid this by placing the page table entries for adjacent pages in adjacent locations, but an inverted page table destroys spatiallocality of referenceby scattering entries all over. An operating system may minimize the size of the hash table to reduce this problem, with the trade-off being an increased miss rate.
There is normally one hash table, contiguous in physical memory, shared by all processes. A per-process identifier is used to disambiguate the pages of different processes from each other. It is somewhat slow to remove the page table entries of a given process; the OS may avoid reusing per-process identifier values to delay facing this. Alternatively, per-process hash tables may be used, but they are impractical because ofmemory fragmentation, which requires the tables to be pre-allocated.
Inverted page tables are used for example on thePowerPC, theUltraSPARCand theIA-64architecture.[4]
The inverted page table keeps a listing of mappings installed for all frames in physical memory. However, this could be quite wasteful. Instead of doing so, we could create a page table structure that contains mappings for virtual pages. It is done by keeping several page tables that cover a certain block of virtual memory. For example, we can create smaller 1024-entry 4 KB pages that cover 4 MB of virtual memory.
This is useful since often the top-most parts and bottom-most parts of virtual memory are used in running a process - the top is often used for text and data segments while the bottom for stack, with free memory in between. The multilevel page table may keep a few of the smaller page tables to cover just the top and bottom parts of memory and create new ones only when strictly necessary.
Now, each of these smaller page tables are linked together by a master page table, effectively creating atreedata structure. There need not be only two levels, but possibly multiple ones. For example, a virtual address in this schema could be split into three parts: the index in the root page table, the index in the sub-page table, and the offset in that page.
Multilevel page tables are also referred to as "hierarchical page tables".
It was mentioned that creating a page table structure that contained mappings for every virtual page in the virtual address space could end up being wasteful. But, we can get around the excessive space concerns by putting the page table in virtual memory, and letting the virtual memory system manage the memory for the page table.
However, part of this linear page table structure must always stay resident in physical memory in order to prevent circularpage faultsand look for a key part of the page table that is not present in the page table.
Nested page tables can be implemented to increase the performance ofhardware virtualization. By providing hardware support forpage-table virtualization, the need to emulate is greatly reduced. Forx86 virtualizationthe current choices areIntel'sExtended Page Tablefeature andAMD'sRapid Virtualization Indexingfeature.
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Incategory theory, a branch ofmathematics, asectionis aright inverseof somemorphism.Dually, aretractionis aleft inverseof somemorphism.
In other words, iff:X→Y{\displaystyle f:X\to Y}andg:Y→X{\displaystyle g:Y\to X}are morphisms whose compositionf∘g:Y→Y{\displaystyle f\circ g:Y\to Y}is theidentity morphismonY{\displaystyle Y}, theng{\displaystyle g}is a section off{\displaystyle f}, andf{\displaystyle f}is a retraction ofg{\displaystyle g}.[1]
Every section is amonomorphism(every morphism with a left inverse isleft-cancellative), and every retraction is anepimorphism(every morphism with a right inverse isright-cancellative).
Inalgebra, sections are also calledsplit monomorphismsand retractions are also calledsplit epimorphisms. In anabelian category, iff:X→Y{\displaystyle f:X\to Y}is a split epimorphism with split monomorphismg:Y→X{\displaystyle g:Y\to X}, thenX{\displaystyle X}isisomorphicto thedirect sumofY{\displaystyle Y}and thekerneloff{\displaystyle f}. The synonymcoretractionfor section is sometimes seen in the literature, although rarely in recent work.
The concept of a retraction in category theory comes from the essentially similar notion of aretractionintopology:f:X→Y{\displaystyle f:X\to Y}whereY{\displaystyle Y}is a subspace ofX{\displaystyle X}is a retraction in the topological sense, if it's a retraction of the inclusion mapi:Y↪X{\displaystyle i:Y\hookrightarrow X}in the category theory sense. The concept in topology was defined byKarol Borsukin 1931.[2]
Borsuk's student,Samuel Eilenberg, was withSaunders Mac Lanethe founder of category theory, and (as the earliest publications on category theory concerned various topological spaces) one might have expected this term to have initially be used. In fact, their earlier publications, up to, e.g., Mac Lane (1963)'sHomology, used the term right inverse. It was not until 1965 when Eilenberg andJohn Coleman Moorecoined the dual term 'coretraction' that Borsuk's term was lifted to category theory in general.[3]The term coretraction gave way to the term section by the end of the 1960s.
Both use of left/right inverse and section/retraction are commonly seen in the literature: the former use has the advantage that it is familiar from the theory ofsemigroupsandmonoids; the latter is considered less confusing by some because one does not have to think about 'which way around' composition goes, an issue that has become greater with the increasing popularity of the synonymf∘g{\displaystyle f\circ g}forg∘f{\displaystyle g\circ f}.[4]
In thecategory of sets, every monomorphism (injectivefunction) with anon-emptydomainis a section, and every epimorphism (surjective function) is a retraction; the latter statement is equivalent to theaxiom of choice.
In thecategory of vector spacesover afieldK, every monomorphism and every epimorphism splits; this follows from the fact thatlinear mapscan be uniquely defined by specifying their values on abasis.
In thecategory of abelian groups, the epimorphismZ→Z/2Zwhich sends everyintegerto its remaindermodulo 2does not split; in fact the only morphismZ/2Z→Zis thezero map. Similarly, the natural monomorphismZ/2Z→Z/4Zdoesn't split even though there is a non-trivial morphismZ/4Z→Z/2Z.
The categorical concept of a section is important inhomological algebra, and is also closely related to the notion of asectionof afiber bundleintopology: in the latter case, a section of a fiber bundle is a section of the bundle projection map of the fiber bundle.
Given aquotient spaceX¯{\displaystyle {\bar {X}}}with quotient mapπ:X→X¯{\displaystyle \pi \colon X\to {\bar {X}}}, a section ofπ{\displaystyle \pi }is called atransversal.
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https://en.wikipedia.org/wiki/Section_(category_theory)
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Algebraic geometryis a branch ofmathematicswhich usesabstract algebraictechniques, mainly fromcommutative algebra, to solvegeometrical problems. Classically, it studieszerosofmultivariate polynomials; the modern approach generalizes this in a few different aspects.
The fundamental objects of study in algebraic geometry arealgebraic varieties, which are geometric manifestations ofsolutionsofsystems of polynomial equations. Examples of the most studied classes of algebraic varieties arelines,circles,parabolas,ellipses,hyperbolas,cubic curveslikeelliptic curves, and quartic curves likelemniscatesandCassini ovals. These areplane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a givenpolynomial equation. Basic questions involve the study of points of special interest likesingular points,inflection pointsandpoints at infinity. More advanced questions involve thetopologyof the curve and the relationship between curves defined by different equations.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields ascomplex analysis, topology andnumber theory. As a study of systems of polynomial equations in several variables, the subject of algebraic geometry begins with finding specific solutions viaequation solving, and then proceeds to understand the intrinsic properties of the totality of solutions of a system of equations. This understanding requires both conceptual theory and computational technique.
In the 20th century, algebraic geometry split into several subareas.
Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology,differentialandcomplex geometry. One key achievement of this abstract algebraic geometry isGrothendieck'sscheme theorywhich allows one to usesheaf theoryto study algebraic varieties in a way which is very similar to its use in the study ofdifferentialandanalytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, throughHilbert's Nullstellensatz, with amaximal idealof thecoordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory.Wiles' proofof the longstanding conjecture calledFermat's Last Theoremis an example of the power of this approach.
In classical algebraic geometry, the main objects of interest are the vanishing sets of collections ofpolynomials, meaning the set of all points that simultaneously satisfy one or morepolynomial equations. For instance, thetwo-dimensionalsphereof radius 1 in three-dimensionalEuclidean spaceR3could be defined as the set of all points(x,y,z){\displaystyle (x,y,z)}with
A "slanted" circle inR3can be defined as the set of all points(x,y,z){\displaystyle (x,y,z)}which satisfy the two polynomial equations
First we start with afieldk. In classical algebraic geometry, this field was always the complex numbersC, but many of the same results are true if we assume only thatkisalgebraically closed. We consider theaffine spaceof dimensionnoverk, denotedAn(k) (or more simplyAn, whenkis clear from the context). When one fixes a coordinate system, one may identifyAn(k) withkn. The purpose of not working withknis to emphasize that one "forgets" the vector space structure thatkncarries.
A functionf:An→A1is said to bepolynomial(orregular) if it can be written as a polynomial, that is, if there is a polynomialpink[x1,...,xn] such thatf(M) =p(t1,...,tn) for every pointMwith coordinates (t1,...,tn) inAn. The property of a function to be polynomial (or regular) does not depend on the choice of a coordinate system inAn.
When a coordinate system is chosen, the regular functions on the affinen-space may be identified with the ring ofpolynomial functionsinnvariables overk. Therefore, the set of the regular functions onAnis a ring, which is denotedk[An].
We say that a polynomialvanishesat a point if evaluating it at that point gives zero. LetSbe a set of polynomials ink[An]. Thevanishing set of S(orvanishing locusorzero set) is the setV(S) of all points inAnwhere every polynomial inSvanishes. Symbolically,
A subset ofAnwhich isV(S), for someS, is called analgebraic set. TheVstands forvariety(a specific type of algebraic set to be defined below).
Given a subsetUofAn, can one recover the set of polynomials which generate it? IfUisanysubset ofAn, defineI(U) to be the set of all polynomials whose vanishing set containsU. TheIstands forideal: if two polynomialsfandgboth vanish onU, thenf+gvanishes onU, and ifhis any polynomial, thenhfvanishes onU, soI(U) is always an ideal of the polynomial ringk[An].
Two natural questions to ask are:
The answer to the first question is provided by introducing theZariski topology, a topology onAnwhose closed sets are the algebraic sets, and which directly reflects the algebraic structure ofk[An]. ThenU=V(I(U)) if and only ifUis an algebraic set or equivalently a Zariski-closed set. The answer to the second question is given byHilbert's Nullstellensatz. In one of its forms, it says thatI(V(S)) is theradicalof the ideal generated byS. In more abstract language, there is aGalois connection, giving rise to twoclosure operators; they can be identified, and naturally play a basic role in the theory; theexampleis elaborated at Galois connection.
For various reasons we may not always want to work with the entire ideal corresponding to an algebraic setU.Hilbert's basis theoremimplies that ideals ink[An] are always finitely generated.
An algebraic set is calledirreducibleif it cannot be written as the union of two smaller algebraic sets. Any algebraic set is a finite union of irreducible algebraic sets and this decomposition is unique. Thus its elements are called theirreducible componentsof the algebraic set. An irreducible algebraic set is also called avariety. It turns out that an algebraic set is a variety if and only if it may be defined as the vanishing set of aprime idealof thepolynomial ring.
Some authors do not make a clear distinction between algebraic sets and varieties and useirreducible varietyto make the distinction when needed.
Just ascontinuous functionsare the natural maps ontopological spacesandsmooth functionsare the natural maps ondifferentiable manifolds, there is a natural class of functions on an algebraic set, calledregular functionsorpolynomial functions. A regular function on an algebraic setVcontained inAnis the restriction toVof a regular function onAn. For an algebraic set defined on the field of the complex numbers, the regular functions are smooth and evenanalytic.
It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in anormaltopological space, where theTietze extension theoremguarantees that a continuous function on a closed subset always extends to the ambient topological space.
Just as with the regular functions on affine space, the regular functions onVform a ring, which we denote byk[V]. This ring is called thecoordinate ringof V.
Since regular functions on V come from regular functions onAn, there is a relationship between the coordinate rings. Specifically, if a regular function onVis the restriction of two functionsfandgink[An], thenf−gis a polynomial function which is null onVand thus belongs toI(V). Thusk[V] may be identified withk[An]/I(V).
Using regular functions from an affine variety toA1, we can defineregular mapsfrom one affine variety to another. First we will define a regular map from a variety into affine space: LetVbe a variety contained inAn. Choosemregular functions onV, and call themf1, ...,fm. We define aregular mapffromVtoAmby lettingf= (f1, ...,fm). In other words, eachfidetermines one coordinate of therangeoff.
IfV′ is a variety contained inAm, we say thatfis aregular mapfromVtoV′ if the range offis contained inV′.
The definition of the regular maps apply also to algebraic sets.
The regular maps are also calledmorphisms, as they make the collection of all affine algebraic sets into acategory, where the objects are the affine algebraic sets and themorphismsare the regular maps. The affine varieties is a subcategory of the category of the algebraic sets.
Given a regular mapgfromVtoV′ and a regular functionfofk[V′], thenf∘g∈k[V]. The mapf→f∘gis aring homomorphismfromk[V′] tok[V]. Conversely, every ring homomorphism fromk[V′] tok[V] defines a regular map fromVtoV′. This defines anequivalence of categoriesbetween the category of algebraic sets and theopposite categoryof the finitely generatedreducedk-algebras. This equivalence is one of the starting points ofscheme theory.
In contrast to the preceding sections, this section concerns only varieties and not algebraic sets. On the other hand, the definitions extend naturally to projective varieties (next section), as an affine variety and its projective completion have the same field of functions.
IfVis an affine variety, its coordinate ring is anintegral domainand has thus afield of fractionswhich is denotedk(V) and called thefield of the rational functionsonVor, shortly, thefunction fieldofV. Its elements are the restrictions toVof therational functionsover the affine space containingV. Thedomainof a rational functionfis notVbut thecomplementof the subvariety (a hypersurface) where the denominator offvanishes.
As with regular maps, one may define arational mapfrom a varietyVto a varietyV'. As with the regular maps, the rational maps fromVtoV' may be identified to thefield homomorphismsfromk(V') tok(V).
Two affine varieties arebirationally equivalentif there are two rational functions between them which areinverseone to the other in the regions where both are defined. Equivalently, they are birationally equivalent if their function fields are isomorphic.
An affine variety is arational varietyif it is birationally equivalent to an affine space. This means that the variety admits arational parameterization, that is aparametrizationwithrational functions. For example, the circle of equationx2+y2−1=0{\displaystyle x^{2}+y^{2}-1=0}is a rational curve, as it has theparametric equation
which may also be viewed as a rational map from the line to the circle.
The problem ofresolution of singularitiesis to know if every algebraic variety is birationally equivalent to a variety whose projective completion is nonsingular (see alsosmooth completion). It was solved in the affirmative incharacteristic0 byHeisuke Hironakain 1964 and is yet unsolved in finite characteristic.
Just as the formulas for the roots of second, third, and fourth degree polynomials suggest extending real numbers to the more algebraically complete setting of the complex numbers, many properties of algebraic varieties suggest extending affine space to a more geometrically complete projective space. Whereas the complex numbers are obtained by adding the numberi, a root of the polynomialx2+ 1, projective space is obtained by adding in appropriate points "at infinity", points where parallel lines may meet.
To see how this might come about, consider the varietyV(y−x2). If we draw it, we get aparabola. Asxgoes to positive infinity, the slope of the line from the origin to the point (x,x2) also goes to positive infinity. Asxgoes to negative infinity, the slope of the same line goes to negative infinity.
Compare this to the varietyV(y−x3). This is acubic curve. Asxgoes to positive infinity, the slope of the line from the origin to the point (x,x3) goes to positive infinity just as before. But unlike before, asxgoes to negative infinity, the slope of the same line goes to positive infinity as well; the exact opposite of the parabola. So the behavior "at infinity" ofV(y−x3) is different from the behavior "at infinity" ofV(y−x2).
The consideration of theprojective completionof the two curves, which is their prolongation "at infinity" in theprojective plane, allows us to quantify this difference: the point at infinity of the parabola is aregular point, whose tangent is theline at infinity, while the point at infinity of the cubic curve is acusp. Also, both curves are rational, as they are parameterized byx, and theRiemann-Roch theoremimplies that the cubic curve must have a singularity, which must be at infinity, as all its points in the affine space are regular.
Thus many of the properties of algebraic varieties, including birational equivalence and all the topological properties, depend on the behavior "at infinity" and so it is natural to study the varieties in projective space. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example,Bézout's theoremon the number of intersection points between two varieties can be stated in its sharpest form only in projective space. For these reasons, projective space plays a fundamental role in algebraic geometry.
Nowadays, theprojective spacePnof dimensionnis usually defined as the set of the lines passing through a point, considered as the origin, in the affine space of dimensionn+ 1, or equivalently to the set of the vector lines in a vector space of dimensionn+ 1. When a coordinate system has been chosen in the space of dimensionn+ 1, all the points of a line have the same set of coordinates, up to the multiplication by an element ofk. This defines thehomogeneous coordinatesof a point ofPnas a sequence ofn+ 1elements of the base fieldk, defined up to the multiplication by a nonzero element ofk(the same for the whole sequence).
A polynomial inn+ 1variables vanishes at all points of a line passing through the origin if and only if it ishomogeneous. In this case, one says that the polynomialvanishesat the corresponding point ofPn. This allows us to define aprojective algebraic setinPnas the setV(f1, ...,fk), where a finite set of homogeneous polynomials{f1, ...,fk}vanishes. Like for affine algebraic sets, there is abijectionbetween the projective algebraic sets and the reducedhomogeneous idealswhich define them. Theprojective varietiesare the projective algebraic sets whose defining ideal is prime. In other words, a projective variety is a projective algebraic set, whosehomogeneous coordinate ringis anintegral domain, theprojective coordinates ringbeing defined as the quotient of the graded ring or the polynomials inn+ 1variables by the homogeneous (reduced) ideal defining the variety. Every projective algebraic set may be uniquely decomposed into a finite union of projective varieties.
The only regular functions which may be defined properly on a projective variety are the constant functions. Thus this notion is not used in projective situations. On the other hand, thefield of the rational functionsorfunction fieldis a useful notion, which, similarly to the affine case, is defined as the set of the quotients of two homogeneous elements of the same degree in the homogeneous coordinate ring.
Real algebraic geometry is the study of real algebraic varieties.
The fact that the field of the real numbers is anordered fieldcannot be ignored in such a study. For example, the curve of equationx2+y2−a=0{\displaystyle x^{2}+y^{2}-a=0}is a circle ifa>0{\displaystyle a>0}, but has no real points ifa<0{\displaystyle a<0}. Real algebraic geometry also investigates, more broadly,semi-algebraic sets, which are the solutions of systems of polynomial inequalities. For example, neither branch of thehyperbolaof equationxy−1=0{\displaystyle xy-1=0}is a real algebraic variety. However, the branch in the first quadrant is a semi-algebraic set defined byxy−1=0{\displaystyle xy-1=0}andx>0{\displaystyle x>0}.
One open problem in real algebraic geometry is the following part ofHilbert's sixteenth problem: Decide which respective positions are possible for the ovals of a nonsingularplane curveof degree 8.
One may date the origin of computational algebraic geometry to meeting EUROSAM'79 (International Symposium on Symbolic and Algebraic Manipulation) held atMarseille, France, in June 1979. At this meeting,
Since then, most results in this area are related to one or several of these items either by using or improving one of these algorithms, or by finding algorithms whose complexity is simply exponential in the number of the variables.
A body of mathematical theory complementary to symbolic methods callednumerical algebraic geometryhas been developed over the last several decades. The main computational method ishomotopy continuation. This supports, for example, a model offloating pointcomputation for solving problems of algebraic geometry.
AGröbner basisis a system ofgeneratorsof a polynomialidealwhose computation allows the deduction of many properties of the affine algebraic variety defined by the ideal.
Given an idealIdefining an algebraic setV:
Gröbner basis computations do not allow one to compute directly theprimary decompositionofInor the prime ideals defining the irreducible components ofV, but most algorithms for this involve Gröbner basis computation. The algorithms which are not based on Gröbner bases useregular chainsbut may need Gröbner bases in some exceptional situations.
Gröbner bases are deemed to be difficult to compute. In fact they may contain, in the worst case, polynomials whose degree is doubly exponential in the number of variables and a number of polynomials which is also doubly exponential. However, this is only a worst case complexity, and the complexity bound of Lazard's algorithm of 1979 may frequently apply.Faugère F5 algorithmrealizes this complexity, as it may be viewed as an improvement of Lazard's 1979 algorithm. It follows that the best implementations allow one to compute almost routinely with algebraic sets of degree more than 100. This means that, presently, the difficulty of computing a Gröbner basis is strongly related to the intrinsic difficulty of the problem.
CAD is an algorithm which was introduced in 1973 by G. Collins to implement with an acceptable complexity theTarski–Seidenberg theoremonquantifier eliminationover the real numbers.
This theorem concerns the formulas of thefirst-order logicwhoseatomic formulasare polynomial equalities or inequalities between polynomials with real coefficients. These formulas are thus the formulas which may be constructed from the atomic formulas by the logical operatorsand(∧),or(∨),not(¬),for all(∀) andexists(∃). Tarski's theorem asserts that, from such a formula, one may compute an equivalent formula without quantifier (∀, ∃).
The complexity of CAD is doubly exponential in the number of variables. This means that CAD allows, in theory, to solve every problem of real algebraic geometry which may be expressed by such a formula, that is almost every problem concerning explicitly given varieties and semi-algebraic sets.
While Gröbner basis computation has doubly exponential complexity only in rare cases, CAD has almost always this high complexity. This implies that, unless if most polynomials appearing in the input are linear, it may not solve problems with more than four variables.
Since 1973, most of the research on this subject is devoted either to improving CAD or finding alternative algorithms in special cases of general interest.
As an example of the state of art, there are efficient algorithms to find at least a point in every connected component of a semi-algebraic set, and thus to test if a semi-algebraic set is empty. On the other hand, CAD is yet, in practice, the best algorithm to count the number of connected components.
The basic general algorithms of computational geometry have a double exponential worst casecomplexity. More precisely, ifdis the maximal degree of the input polynomials andnthe number of variables, their complexity is at mostd2cn{\displaystyle d^{2^{cn}}}for some constantc, and, for some inputs, the complexity is at leastd2c′n{\displaystyle d^{2^{c'n}}}for another constantc′.
During the last 20 years of the 20th century, various algorithms have been introduced to solve specific subproblems with a better complexity. Most of these algorithms have a complexitydO(n2){\displaystyle d^{O(n^{2})}}.[1]
Among these algorithms which solve a sub problem of the problems solved by Gröbner bases, one may citetesting if an affine variety is emptyandsolving nonhomogeneous polynomial systems which have a finite number of solutions.Such algorithms are rarely implemented because, on most entriesFaugère's F4 and F5 algorithmshave a better practical efficiency and probably a similar or better complexity (probablybecause the evaluation of the complexity of Gröbner basis algorithms on a particular class of entries is a difficult task which has been done only in a few special cases).
The main algorithms of real algebraic geometry which solve a problem solved by CAD are related to the topology of semi-algebraic sets. One may citecounting the number of connected components,testing if two points are in the same componentsorcomputing aWhitney stratificationof a real algebraic set. They have a complexity ofdO(n2){\displaystyle d^{O(n^{2})}}, but the constant involved byOnotation is so high that using them to solve any nontrivial problem effectively solved by CAD, is impossible even if one could use all the existing computing power in the world. Therefore, these algorithms have never been implemented and this is an active research area to search for algorithms with have together a good asymptotic complexity and a good practical efficiency.
The modern approaches to algebraic geometry redefine and effectively extend the range of basic objects in various levels of generality to schemes,formal schemes,ind-schemes,algebraic spaces,algebraic stacksand so on. The need for this arises already from the useful ideas within theory of varieties, e.g. the formal functions of Zariski can be accommodated by introducing nilpotent elements in structure rings; considering spaces of loops and arcs, constructing quotients by group actions and developing formal grounds for naturalintersection theoryanddeformation theorylead to some of the further extensions.
Most remarkably, in the early 1960s,algebraic varietieswere subsumed intoAlexander Grothendieck's concept of ascheme. Their local objects are affine schemes or prime spectra which are locally ringed spaces which form a category which is antiequivalent to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a fieldk, and the category of finitely generated reducedk-algebras. The gluing is along Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set theoretic sense is then replaced by aGrothendieck topology. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely theétale topology, and the two flat Grothendieck topologies: fppf and fpqc; nowadays some other examples became prominent includingNisnevich topology. Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions leading toArtin stacksand, even finer,Deligne–Mumford stacks, both often called algebraic stacks.
Sometimes other algebraic sites replace the category of affine schemes. For example,Nikolai Durovhas introduced commutative algebraic monads as a generalization of local objects in a generalized algebraic geometry. Versions of atropical geometry, of anabsolute geometryover a field of one element and an algebraic analogue ofArakelov's geometrywere realized in this setup.
Another formal generalization is possible touniversal algebraic geometryin which everyvariety of algebrashas its own algebraic geometry. The termvariety of algebrasshould not be confused withalgebraic variety.
The language of schemes, stacks and generalizations has proved to be a valuable way of dealing with geometric concepts and became cornerstones of modern algebraic geometry.
Algebraic stacks can be further generalized and for many practical questions like deformation theory and intersection theory, this is often the most natural approach. One can extend theGrothendieck siteof affine schemes to ahigher categoricalsite ofderived affine schemes, by replacing the commutative rings with an infinity category ofdifferential graded commutative algebras, or of simplicial commutative rings or a similar category with an appropriate variant of a Grothendieck topology. One can also replace presheaves of sets by presheaves of simplicial sets (or of infinity groupoids). Then, in presence of an appropriate homotopic machinery one can develop a notion of derived stack as such a presheaf on the infinity category of derived affine schemes, which is satisfying certain infinite categorical version of a sheaf axiom (and to be algebraic, inductively a sequence of representability conditions).Quillen model categories, Segal categories andquasicategoriesare some of the most often used tools to formalize this yielding thederived algebraic geometry, introduced by the school ofCarlos Simpson, including Andre Hirschowitz,Bertrand Toën, Gabrielle Vezzosi, Michel Vaquié and others; and developed further byJacob Lurie,Bertrand Toën, andGabriele Vezzosi. Another (noncommutative) version of derived algebraic geometry, using A-infinity categories has been developed from the early 1990s byMaxim Kontsevichand followers.
Some of the roots of algebraic geometry date back to the work of theHellenistic Greeksfrom the 5th century BC. TheDelian problem, for instance, was to construct a lengthxso that the cube of sidexcontained the same volume as the rectangular boxa2bfor given sidesaandb.Menaechmus(c.350 BC) considered the problem geometrically by intersecting the pair of plane conicsay=x2andxy=ab.[2]In the 3rd century BC,ArchimedesandApolloniussystematically studied additional problems onconic sectionsusing coordinates.[2][3]Apolloniusin the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work ofDescartesby some 1800 years.[4]His application of reference lines, adiameterand atangentis essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding coordinates using geometric methods like using parabolas and curves.[5][6][7]Medieval mathematicians, includingOmar Khayyam,Leonardo of Pisa,GersonidesandNicole Oresmein theMedieval Period,[8]solved certain cubic and quadratic equations by purely algebraic means and then interpreted the results geometrically. ThePersianmathematicianOmar Khayyám(born 1048 AD) believed that there was a relationship betweenarithmetic,algebraandgeometry.[9][10][11]This was criticized by Jeffrey Oaks, who claims that the study of curves by means of equations originated with Descartes in the seventeenth century.[12]
Such techniques of applying geometrical constructions to algebraic problems were also adopted by a number ofRenaissancemathematicians such asGerolamo CardanoandNiccolò Fontana "Tartaglia"on their studies of the cubic equation. The geometrical approach to construction problems, rather than the algebraic one, was favored by most 16th and 17th century mathematicians, notablyBlaise Pascalwho argued against the use of algebraic and analytical methods in geometry.[13]The French mathematiciansFranciscus Vietaand laterRené DescartesandPierre de Fermatrevolutionized the conventional way of thinking about construction problems through the introduction ofcoordinate geometry. They were interested primarily in the properties ofalgebraic curves, such as those defined byDiophantine equations(in the case of Fermat), and the algebraic reformulation of the classical Greek works on conics and cubics (in the case of Descartes).
During the same period, Blaise Pascal andGérard Desarguesapproached geometry from a different perspective, developing thesyntheticnotions ofprojective geometry. Pascal and Desargues also studied curves, but from the purely geometrical point of view: the analog of the Greekruler and compass construction. Ultimately, theanalytic geometryof Descartes and Fermat won out, for it supplied the 18th century mathematicians with concrete quantitative tools needed to study physical problems using the new calculus ofNewtonandLeibniz. However, by the end of the 18th century, most of the algebraic character of coordinate geometry was subsumed by thecalculus of infinitesimalsofLagrangeandEuler.
It took the simultaneous 19th century developments ofnon-Euclidean geometryandAbelian integralsin order to bring the old algebraic ideas back into the geometrical fold. The first of these new developments was seized up byEdmond LaguerreandArthur Cayley, who attempted to ascertain the generalized metric properties of projective space. Cayley introduced the idea ofhomogeneous polynomial forms, and more specificallyquadratic forms, on projective space. Subsequently,Felix Kleinstudied projective geometry (along with other types of geometry) from the viewpoint that the geometry on a space is encoded in a certain class oftransformationson the space. By the end of the 19th century, projective geometers were studying more general kinds of transformations on figures in projective space. Rather than the projective linear transformations which were normally regarded as giving the fundamentalKleinian geometryon projective space, they concerned themselves also with the higher degreebirational transformations. This weaker notion of congruence would later lead members of the 20th centuryItalian school of algebraic geometryto classifyalgebraic surfacesup tobirational isomorphism.
The second early 19th century development, that of Abelian integrals, would leadBernhard Riemannto the development ofRiemann surfaces.
In the same period began the algebraization of the algebraic geometry throughcommutative algebra. The prominent results in this direction areHilbert's basis theoremandHilbert's Nullstellensatz, which are the basis of the connection between algebraic geometry and commutative algebra, andMacaulay'smultivariate resultant, which is the basis ofelimination theory. Probably because of the size of the computation which is implied by multivariate resultants, elimination theory was forgotten during the middle of the 20th century until it was renewed bysingularity theoryand computational algebraic geometry.[a]
B. L. van der Waerden,Oscar ZariskiandAndré Weildeveloped a foundation for algebraic geometry based on contemporarycommutative algebra, includingvaluation theoryand the theory ofideals. One of the goals was to give a rigorous framework for proving the results of theItalian school of algebraic geometry. In particular, this school used systematically the notion ofgeneric pointwithout any precise definition, which was first given by these authors during the 1930s.
In the 1950s and 1960s,Jean-Pierre SerreandAlexander Grothendieckrecast the foundations making use ofsheaf theory. Later, from about 1960, and largely led by Grothendieck, the idea ofschemeswas worked out, in conjunction with a very refined apparatus ofhomological techniques. After a decade of rapid development the field stabilized in the 1970s, and new applications were made, both tonumber theoryand to more classical geometric questions on algebraic varieties,singularities,moduli, andformal moduli.
An important class of varieties, not easily understood directly from their defining equations, are theabelian varieties, which are the projective varieties whose points form an abeliangroup. The prototypical examples are theelliptic curves, which have a rich theory. They were instrumental in the proof ofFermat's Last Theoremand are also used inelliptic-curve cryptography.
In parallel with the abstract trend of the algebraic geometry, which is concerned with general statements about varieties, methods for effective computation with concretely-given varieties have also been developed, which lead to the new area of computational algebraic geometry. One of the founding methods of this area is the theory ofGröbner bases, introduced byBruno Buchbergerin 1965. Another founding method, more specially devoted to real algebraic geometry, is thecylindrical algebraic decomposition, introduced byGeorge E. Collinsin 1973.
See also:derived algebraic geometry.
Ananalytic varietyover the field of real or complex numbers is defined locally as the set of common solutions of several equations involvinganalytic functions. It is analogous to the concept ofalgebraic varietyin that it carries a structure sheaf of analytic functions instead of regular functions. Anycomplex manifoldis a complex analytic variety. Since analytic varieties may havesingular points, not all complex analytic varieties are manifolds. Over a non-archimedean field analytic geometry is studied viarigid analytic spaces.
Modern analytic geometry over the field of complex numbers is closely related to complex algebraic geometry, as has been shown byJean-Pierre Serrein his paperGAGA,[14]the name of which is French forAlgebraic geometry and analytic geometry. The GAGA results over the field of complex numbers may be extended to rigid analytic spaces over non-archimedean fields.[15]
Algebraic geometry now finds applications instatistics,[16]control theory,[17][18]robotics,[19]error-correcting codes,[20]phylogenetics[21]andgeometric modelling.[22]There are also connections tostring theory,[23]game theory,[24]graph matchings,[25]solitons[26]andinteger programming.[27]
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Aserviceis an act or use for which aconsumer,company, orgovernmentis willing topay.[1]Examples include work done by barbers, doctors, lawyers, mechanics, banks, insurance companies, and so on. Public services are those that society (nation state, fiscal union or region) as a whole pays for. Usingresources, skill, ingenuity, and experience, service providers benefit service consumers. Services may be defined as intangible acts or performances whereby the service provider provides value to the customer.
Services have three key characteristics:[2]
Services are by definition intangible. They are not manufactured, transported or stocked.
One cannot store services for future use. They are produced and consumed simultaneously.
Services are perishable in two regards:
The service provider must deliver the service at the exact time of service consumption. The service is not manifested in a physical object that is independent of the provider. The service consumer is also inseparable from service delivery. Examples: The service consumer must sit in the hairdresser's chair, or in the airplane seat. Correspondingly, the hairdresser or the pilot must be in the shop or plane, respectively, to deliver the service.
Each service is unique. It can never be exactly repeated as the time, location, circumstances, conditions, current configurations or assigned resources are different for the next delivery, even if the same service is requested by the consumer. Many services are regarded as heterogeneous and are typically modified for each service-consumer or for each service-context.[2]Example: The taxi service which transports the service consumer from home to work is different from the taxi service which transports the same service consumer from work to home – another point in time, the other direction, possibly another route, probably another taxi-driver and cab. Another and more commontermfor this isheterogeneity.[citation needed]
Mass generation and delivery of services must be mastered for a service provider to expand. This can be seen as a problem ofservice quality. Both inputs and outputs to the processes involved providing services are highly variable, as are the relationships between these processes, making it difficult to maintain consistent service quality. Many services involve variable human activity, rather than a precisely determined process; exceptions includeutilities. The human factor is often the key success factor in service provision. Demand can vary byseason,timeof day,business cycle, etc. Consistency is necessary to create enduring business relationships.
Any service can be clearly and completely, consistently and concisely specified by means of standard attributes that conform to theMECE principle(Mutually Exclusive, Collectively Exhaustive).
The delivery of a service typically involves six factors:
The service encounter is defined as all activities involved in the service delivery process. Some service managers use the term "moment of truth" to indicate that point in a service encounter where interactions are most intense.[citation needed]
Manybusiness theoristsview service provision as a performance or act (sometimes humorously referred to asdramalurgy, perhaps in reference todramaturgy). The location of the service delivery is referred to as thestageand the objects that facilitate the service process are calledprops. A script is a sequence ofbehaviorsfollowed by those involved, including the client(s). Some servicedramasare tightly scripted, others are moread lib. Role congruence occurs when eachactorfollows a script that harmonizes with therolesplayed by the other actors.[citation needed]
In some service industries, especially health care, dispute resolution and social services, a popular concept is the idea of the caseload, which refers to the total number of patients, clients, litigants, or claimants for which a given employee is responsible. Employees must balance the needs of each individual case against the needs of all other current cases as well as their own needs.[citation needed]
UnderEnglish law, if a service provider is induced to deliver services to adishonestclient by a deception, this is an offence under theTheft Act 1978.[citation needed]
Lovelock used the number of delivery sites (whether single or multiple) and the method of delivery to classify services in a 2 x 3 matrix. Then implications are that the convenience of receiving the service is the lowest when the customer has to come to the service and must use a single or specific outlet. Convenience increases (to a point) as the number of service points increase.[citation needed]
The distinction between a good and a service remains disputed. The perspective in the late-eighteenth and early-nineteenth centuries focused on creation and possession of wealth. Classical economists contended that goods were objects of value over which ownership rights could be established and exchanged. Ownership implied tangible possession of an object that had been acquired through purchase, barter or gift from the producer or previous owner and was legally identifiable as the property of the current owner.
Adam Smith's famous book,The Wealth of Nations, published in1776, distinguished between the outputs of what he termed "productive" and "unproductive" labor. The former, he stated, produced goods that could be stored after production and subsequently exchanged for money or other items of value. The latter, however useful or necessary, created services that perished at the time of production and therefore did not contribute to wealth. Building on this theme, French economist Jean-Baptiste Say argued that production and consumption were inseparable in services, coining the term "immaterial products" to describe them.
In the modern day, Gustofsson & Johnson describe a continuum with pure service on one terminal point and purecommodity goodon the other.[3]Mostproductsfall between these two extremes. For example, arestaurantprovides a physical good (thefood), but also provides services in the form of ambience, the setting and clearing of the table, etc. And although some utilities actually deliver physical goods — like water utilities that deliver water — utilities are usually treated as services.[citation needed]
The following is a list of service industries, grouped into sectors. Parenthetical notations indicate how specificoccupationsandorganizationscan be regarded as service industries to the extent they provide an intangible service, as opposed to a tangible good.
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The following tables list thecomputational complexityof variousalgorithmsfor commonmathematical operations.
Here, complexity refers to thetime complexityof performing computations on amultitape Turing machine.[1]Seebig O notationfor an explanation of the notation used.
Note: Due to the variety of multiplication algorithms,M(n){\displaystyle M(n)}below stands in for the complexity of the chosen multiplication algorithm.
This table lists the complexity of mathematical operations on integers.
On stronger computational models, specifically apointer machineand consequently also aunit-cost random-access machineit is possible to multiply twon-bit numbers in timeO(n).[6]
Here we consider operations over polynomials andndenotes their degree; for the coefficients we use aunit-costmodel, ignoring the number of bits in a number. In practice this means that we assume them to be machine integers.
Many of the methods in this section are given in Borwein & Borwein.[7]
Theelementary functionsare constructed by composing arithmetic operations, theexponential function(exp{\displaystyle \exp }), thenatural logarithm(log{\displaystyle \log }),trigonometric functions(sin,cos{\displaystyle \sin ,\cos }), and their inverses. The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions areanalyticand hence invertible by means of Newton's method. In particular, if eitherexp{\displaystyle \exp }orlog{\displaystyle \log }in the complex domain can be computed with some complexity, then that complexity is attainable for all other elementary functions.
Below, the sizen{\displaystyle n}refers to the number of digits of precision at which the function is to be evaluated.
It is not known whetherO(M(n)logn){\displaystyle O(M(n)\log n)}is the optimal complexity for elementary functions. The best known lower bound is the trivial boundΩ{\displaystyle \Omega }(M(n)){\displaystyle (M(n))}.
This table gives the complexity of computing approximations to the given constants ton{\displaystyle n}correct digits.
Algorithms fornumber theoreticalcalculations are studied incomputational number theory.
The following complexity figures assume that arithmetic with individual elements has complexityO(1), as is the case with fixed-precisionfloating-point arithmeticor operations on afinite field.
In 2005,Henry Cohn,Robert Kleinberg,Balázs Szegedy, andChris Umansshowed that either of two different conjectures would imply that the exponent of matrix multiplication is 2.[34]
Algorithms for computingtransformsof functions (particularlyintegral transforms) are widely used in all areas of mathematics, particularlyanalysisandsignal processing.
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Inmathematics,convergence testsare methods of testing for theconvergence,conditional convergence,absolute convergence,interval of convergenceor divergence of aninfinite series∑n=1∞an{\displaystyle \sum _{n=1}^{\infty }a_{n}}.
If the limit of the summand is undefined or nonzero, that islimn→∞an≠0{\displaystyle \lim _{n\to \infty }a_{n}\neq 0}, then the series must diverge. In this sense, the partial sums areCauchyonly ifthis limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as thenth-term test,test for divergence, orthe divergence test.
This is also known asd'Alembert's criterion.
This is also known as thenth root testorCauchy's criterion.
The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely.[1]
The series can be compared to an integral to establish convergence or divergence. Letf:[1,∞)→R+{\displaystyle f:[1,\infty )\to \mathbb {R} _{+}}be a non-negative andmonotonically decreasing functionsuch thatf(n)=an{\displaystyle f(n)=a_{n}}. If∫1∞f(x)dx=limt→∞∫1tf(x)dx<∞,{\displaystyle \int _{1}^{\infty }f(x)\,dx=\lim _{t\to \infty }\int _{1}^{t}f(x)\,dx<\infty ,}then the series converges. But if the integral diverges, then the series does so as well.
In other words, the seriesan{\displaystyle {a_{n}}}convergesif and only ifthe integral converges.
A commonly-used corollary of the integral test is the p-series test. Letk>0{\displaystyle k>0}. Then∑n=k∞(1np){\displaystyle \sum _{n=k}^{\infty }{\bigg (}{\frac {1}{n^{p}}}{\bigg )}}converges ifp>1{\displaystyle p>1}.
The case ofp=1,k=1{\displaystyle p=1,k=1}yields the harmonic series, which diverges. The case ofp=2,k=1{\displaystyle p=2,k=1}is theBasel problemand the series converges toπ26{\displaystyle {\frac {\pi ^{2}}{6}}}. In general, forp>1,k=1{\displaystyle p>1,k=1}, the series is equal to theRiemann zeta functionapplied top{\displaystyle p}, that isζ(p){\displaystyle \zeta (p)}.
If the series∑n=1∞bn{\displaystyle \sum _{n=1}^{\infty }b_{n}}is anabsolutely convergentseries and|an|≤|bn|{\displaystyle |a_{n}|\leq |b_{n}|}for sufficiently largen, then the series∑n=1∞an{\displaystyle \sum _{n=1}^{\infty }a_{n}}converges absolutely.
If{an},{bn}>0{\displaystyle \{a_{n}\},\{b_{n}\}>0}, (that is, each element of the two sequences is positive) and the limitlimn→∞anbn{\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}}exists, is finite and non-zero, then either both series converge or both series diverge.
Let{an}{\displaystyle \left\{a_{n}\right\}}be a non-negative non-increasing sequence. Then the sumA=∑n=1∞an{\displaystyle A=\sum _{n=1}^{\infty }a_{n}}convergesif and only ifthe sumA∗=∑n=0∞2na2n{\displaystyle A^{*}=\sum _{n=0}^{\infty }2^{n}a_{2^{n}}}converges. Moreover, if they converge, thenA≤A∗≤2A{\displaystyle A\leq A^{*}\leq 2A}holds.
Suppose the following statements are true:
Then∑anbn{\displaystyle \sum a_{n}b_{n}}is also convergent.
Everyabsolutely convergentseries converges.
Suppose the following statements are true:
Then∑n=1∞(−1)nan{\displaystyle \sum _{n=1}^{\infty }(-1)^{n}a_{n}}and∑n=1∞(−1)n+1an{\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}a_{n}}are convergent series.
This test is also known as theLeibniz criterion.
If{an}{\displaystyle \{a_{n}\}}is asequenceofreal numbersand{bn}{\displaystyle \{b_{n}\}}a sequence ofcomplex numberssatisfying
whereMis some constant, then the series
converges.
A series∑i=0∞ai{\displaystyle \sum _{i=0}^{\infty }a_{i}}is convergent if and only if for everyε>0{\displaystyle \varepsilon >0}there is a natural numberNsuch that
holds for alln>Nand allp≥ 1.
Let(an)n≥1{\displaystyle (a_{n})_{n\geq 1}}and(bn)n≥1{\displaystyle (b_{n})_{n\geq 1}}be two sequences of real numbers. Assume that(bn)n≥1{\displaystyle (b_{n})_{n\geq 1}}is astrictly monotoneand divergent sequence and the following limit exists:
Then, the limit
Suppose that (fn) is a sequence of real- or complex-valued functions defined on a setA, and that there is a sequence of non-negative numbers (Mn) satisfying the conditions
Then the series
converges absolutely anduniformlyonA.
The ratio test may be inconclusive when the limit of the ratio is 1. Extensions to the ratio test, however, sometimes allows one to deal with this case.
Let {an} be a sequence of positive numbers.
Define
If
exists there are three possibilities:
An alternative formulation of this test is as follows. Let{an} be a series of real numbers. Then ifb> 1 andK(a natural number) exist such that
for alln>Kthen the series {an} is convergent.
Let {an} be a sequence of positive numbers.
Define
If
exists, there are three possibilities:[2][3]
Let {an} be a sequence of positive numbers. Ifanan+1=1+αn+O(1/nβ){\displaystyle {\frac {a_{n}}{a_{n+1}}}=1+{\frac {\alpha }{n}}+O(1/n^{\beta })}for some β > 1, then∑an{\displaystyle \sum a_{n}}converges ifα > 1and diverges ifα ≤ 1.[4]
Let {an} be a sequence of positive numbers. Then:[5][6][7]
(1)∑an{\displaystyle \sum a_{n}}converges if and only if there is a sequencebn{\displaystyle b_{n}}of positive numbers and a real numberc> 0 such thatbk(ak/ak+1)−bk+1≥c{\displaystyle b_{k}(a_{k}/a_{k+1})-b_{k+1}\geq c}.
(2)∑an{\displaystyle \sum a_{n}}diverges if and only if there is a sequencebn{\displaystyle b_{n}}of positive numbers such thatbk(ak/ak+1)−bk+1≤0{\displaystyle b_{k}(a_{k}/a_{k+1})-b_{k+1}\leq 0}
and∑1/bn{\displaystyle \sum 1/b_{n}}diverges.
Let∑n=1∞an{\displaystyle \sum _{n=1}^{\infty }a_{n}}be an infinite series with real terms and letf:R→R{\displaystyle f:\mathbb {R} \to \mathbb {R} }be any real function such thatf(1/n)=an{\displaystyle f(1/n)=a_{n}}for all positive integersnand the second derivativef″{\displaystyle f''}exists atx=0{\displaystyle x=0}. Then∑n=1∞an{\displaystyle \sum _{n=1}^{\infty }a_{n}}converges absolutely iff(0)=f′(0)=0{\displaystyle f(0)=f'(0)=0}and diverges otherwise.[8]
Consider the series
Cauchy condensation testimplies that (i) is finitely convergent if
is finitely convergent. Since
(ii) is a geometric series with ratio2(1−α){\displaystyle 2^{(1-\alpha )}}. (ii) is finitely convergent if its ratio is less than one (namelyα>1{\displaystyle \alpha >1}).Thus, (i) is finitely convergentif and only ifα>1{\displaystyle \alpha >1}.
While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence ofinfinite products. This can be achieved using following theorem: Let{an}n=1∞{\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }}be a sequence of positive numbers. Then the infinite product∏n=1∞(1+an){\displaystyle \prod _{n=1}^{\infty }(1+a_{n})}convergesif and only ifthe series∑n=1∞an{\displaystyle \sum _{n=1}^{\infty }a_{n}}converges. Also similarly, if0≤an<1{\displaystyle 0\leq a_{n}<1}holds, then∏n=1∞(1−an){\displaystyle \prod _{n=1}^{\infty }(1-a_{n})}approaches a non-zero limit if and only if the series∑n=1∞an{\displaystyle \sum _{n=1}^{\infty }a_{n}}converges .
This can be proved by taking the logarithm of the product and using limit comparison test.[9]
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Īhām(ایهام) inPersian,Urdu,KurdishandArabic poetryis a literary device in which an author uses a word, or an arrangement of words, that can be read in several ways. Each of the meanings may be logically sound, equally true and intended.[1]
In the 12th century,Rashid al-Din Vatvatdefinedīhāmas follows: "Īhāmin Persian means to create doubt. This is a literary device, also calledtakhyīl[to make one suppose and fancy], whereby a writer (dabīr), in prose, or a poet, in verse, employs a word with two different meanings, one direct and immediate (qarīb) and the other remote and strange (gharīb), in such a manner that the listener, as soon as he hears that word, thinks of its direct meaning while in actuality the remote meaning is intended."[1]
Amir Khusrow(1253–1325 CE) introduced the notion that any of the several meanings of a word, or phrase, might be equally true and intended, creating a multilayered text.[2]Discerning the various layers of meanings would be a challenge to the reader, who has to focus on and keep turning over the passage in his mind, applying his erudition and imagination to perceive alternative meanings.[1]
Another idea associated withīhāmis that a verse may function as a mirror of the reader's condition, as expressed by the 14th-century authorShaykh Maneri: "A verse by itself has no fixed meaning. It is the reader/listener who picks up an idea consistent with the subjective condition of his mind."[1]The 15th-century poetFawhr-e Din Nizamiconsideredīhāman essential element of any good work of poetry: "A poem that doesn't have dual-meaning words, such a poem does not attract anyone at all—a poem without words of two senses."[3]
Īhāmis an important stylistic device inSufiliterature, perfected by writers such asHafez(1325/1326–1389/1390 CE).[1][4]Nalîis an example of another poet who has usedīhāmwidely in his poetry. Applications of this "art of ambiguity" or "amphibology" include texts that can be read as descriptions of earthly or divine love.[4][5][6]
Haleh Pourafzal and Roger Montgomery, writing inHaféz: Teachings of the Philosopher of Love(1998), discussīhāmin terms of "biluminosity", simultaneous illumination from two directions, describing it as "a technique of comparison involving wordplay, sound association, and double entendre, keeping the reader in doubt as to the 'right' meaning of the word. Biluminosity removes the burden of choice and invites the reader to enter a more empowering dimension ofīhāmthat embraces the quality of amphibians [...]—beings capable of living equally well in two radically different environments. As a result, the reader is freed from the obsession to find the 'right answer' through speculation and instead can concentrate on enjoying nuances and being awed by how the slightest shift in perception creates a new meaning. [...] From the perspective of Haféz as the composer of poetry, biluminosity allows two different points of view to shed light upon each other."[7]
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Verification-based message-passingalgorithms(VB-MPAs)incompressed sensing(CS), a branch ofdigital signal processingthat deals with measuringsparse signals, are some methods to efficiently solve the recovery problem in compressed sensing. One of the main goal in compressed sensing is the recovery process. Generally speaking, recovery process in compressed sensing is a method by which the original signal is estimated using the knowledge of the compressed signal and the measurement matrix.[1]Mathematically, the recovery process in Compressed Sensing is finding the sparsest possible solution of an under-determinedsystem of linear equations. Based on the nature of the measurement matrix one can employ different reconstruction methods. If the measurement matrix is also sparse, one efficient way is to use Message Passing Algorithms for signal recovery. Although there are message passing approaches that deals with dense matrices, the nature of those algorithms are to some extent different from the algorithms working on sparse matrices.[1][2]
The main problem in recovery process in CS is to find the sparsest possible solution to the following under-determined system of linear equationsAx=y{\displaystyle Ax=y}whereA{\displaystyle A}is the measurement matrix,x{\displaystyle x}is the original signal to be recovered andy{\displaystyle y}is the compresses known signal. When the matrixA{\displaystyle A}is sparse, one can represent this matrix by abipartite graphG=(Vl∪Vr,E){\displaystyle G=(V_{l}\cup V_{r},E)}for better understanding.[2][3][4][5]Vl{\displaystyle V_{l}}is the set of variable nodes inG{\displaystyle G}which represents the set of elements ofx{\displaystyle x}and alsoVr{\displaystyle V_{r}}is the set of check nodes corresponding to the set of elements ofy{\displaystyle y}. Besides, there is an edgee=(u,v){\displaystyle e=(u,v)}betweenu∈Vl{\displaystyle u\in V_{l}}andv∈Vr{\displaystyle v\in V_{r}}if the corresponding elements inA{\displaystyle A}is non-zero, i.e.Av,u≠0{\displaystyle A_{v,u}\neq 0}. Moreover, the weight of the edgew(e)=Av,u{\displaystyle w(e)=A_{v,u}}.[6]Here is an example of a binary sparse measurement matrix where the weights of the edges are either zero or one.
A=[001000001010000101010000100001000010111000000000000100100001000010100001000000011100010010000100]{\displaystyle A=\left[{\begin{array}{c c c c c c c c c c c c}0&0&1&0&0&0&0&0&1&0&1&0\\0&0&0&1&0&1&0&1&0&0&0&0\\1&0&0&0&0&1&0&0&0&0&1&0\\1&1&1&0&0&0&0&0&0&0&0&0\\0&0&0&1&0&0&1&0&0&0&0&1\\0&0&0&0&1&0&1&0&0&0&0&1\\0&0&0&0&0&0&0&1&1&1&0&0\\0&1&0&0&1&0&0&0&0&1&0&0\end{array}}\right]}
The basic idea behind message passing algorithms in CS is to transmit appropriate messages between variable nodes and check nodes in aniterative mannerin order to efficiently find signalx{\displaystyle x}. These messages are different for variable nodes and check nodes. However, the basic nature of the messages for all variable node and check nodes are the same in all of the verification based message passing algorithms.[6]The messagesμv(vi):Vl↦R×{0,1}{\displaystyle \mu ^{v}(v_{i}):~V_{l}\mapsto \mathbb {R} \times \{0,1\}}emanating from variable nodevi{\displaystyle v_{i}}contains the value of the check node and an indicator which shows if the variable node is verified or not. Moreover, the messagesμc(ci):Vr↦R×Z+{\displaystyle \mu ^{c}(c_{i}):~V_{r}\mapsto \mathbb {R} \times \mathbb {Z} ^{+}}emanating from check nodeci{\displaystyle c_{i}}contains the value of the check node and the remaining degree of the check node in the graph.[6][7]
In each iteration, every variable node and check node produce a new message to be transmitted to all of its neighbors based on the messages that they have received from their own neighbors. This local property of the message passing algorithms enables them to be implemented as parallel processing algorithms and makes the time complexity of these algorithm so efficient.[8]
The common rule between all verification based message passing algorithms is the fact that once a variable node become verified then this variable node can be removed from the graph and the algorithm can be executed to solve the rest of the graph. Different verification bases message passing algorithms use different combinations of verification rules.[6]
The verification rules are as follows:
The message passing rules given above are the basic and only rules that should be used in any verification based message passing algorithm. It is shown that these simple rules can efficiently recover the original signal provided that certain conditions are satisfied.[8][6]
There are four algorithms known as VB-MPA's, namely Genie, LM, XH, and SBB.[6]All of these algorithms use the same strategy for recovery of the original signal; however, they use different combination of the message passing rules to verify variable nodes.
Genie algorithm is thebenchmarkin this topic. Firstly, Genie algorithm is assumed to have the knowledge of thesupportset of the signal, i.e. the set of non-zero elements of the original signal. Using this knowledge, Genie should not care about the zero variable nodes in the graph, and the only task of the Genie algorithm is to recover the values of the non-zero elements of the original signal. Although, Genie does not have any practical aspect, it can be regarded as the benchmark of the problem especially in the sense that this algorithm outperforms other algorithms in this category and one can measure how successful one algorithms is by comparing that to the Genie algorithm.
Since Genie only wants to find the value of the non-zero elements of the signal it is not necessary to employ rules that are responsible for zero valued variable node in this algorithm. Therefore, Genie only uses D1CN as the verification rule.
This algorithm unlike the Genie algorithm does not have any knowledge about the support set of signal, and it uses D1CN and ZCN together to solve the recovery process in CS. In fact, ZCN is the rule that attempts to verify the zero valued variable nodes and D1CN is responsible for non-zero valued variable nodes. This usage of this algorithm is when one does not have non-binary matrix. In such cases, employing the third rule violated the locality nature of the algorithms. This issue will be considered in SBB algorithm.[6]
This algorithm is the same as LM, but it only uses ECN instead of D1CN for the verification of the non-zero variable nodes. If the non-zero elements of the measurement matrix arebinary, then this algorithm cannot be implemented efficiently and the locality of the algorithm will be violated.
The most powerful practical algorithm among all of the verification message passing algorithms is the SBB algorithm that employs all of the verification rules for the recovery of the original signal. In this algorithm, D1CN and ECN are responsible for the verification of the non-zero elements of the signal and ZCN and ECN will verify zero variable nodes.
Thepseudo codeof the VB-MPAs is as follows. In the following algorithmμi{\displaystyle \mu _{i}}represents theith{\displaystyle i^{th}}component of the messages emanating from variable and check nodes.VN{\displaystyle VN}is in fact a variable that keeps the labels of the verified variable nodes.VN′{\displaystyle VN'}is also used to keep the set of verified variable nodes in the previous iteration. By using these two variables one can see if there is any progress in the number of verified variable nodes in the algorithm, and if there is no progress then the algorithm will terminate.[6][9]
In all of the algorithms the messages emanating from check nodes are the same; however, since the verification rules are different for different algorithms the messages produced by variable nodes will be different in each algorithm.[6]The algorithm given above works for all of the VB-MPA's, and different algorithms use different rules in half round 2 of round 1 and 2. For instance, Genie algorithm uses D1CN rule in Half round 2 of round 1, and in fact the half round 2 of round 2 which uses ZCN rule is useless in Genie algorithm. LM algorithm uses D1CN in Half round 2 of round 1 and XH algorithm uses ECN rule in this stage instead of D1CN. SBB algorithm also uses both D1CN and ECN rule in the second half round of round 1. All of these rules can be efficiently implemented inupdate_rulefunction in the second half round of round 1.
Although there is no guarantee that these algorithms succeed in all of the cases but we can guarantee that if some of the variable nodes become verified during these algorithms then the values of those variable nodes are correctalmost surely. In order to show that it is enough to show that all of the verification rules work perfectly and withoutfalse verification.[6][8]
The algebraic point of view of ZCN rule is that if in asystem of linear equationsthe right hand side of the equation is zero thenalmost surelyall of the unknowns in that equations are zero. This is due to the fact that the original signal is assumed to be sparse, besides, we also should have the assumption that the non-zero elements of the signals are chosen form acontinuous distribution. Suppose that there ared{\displaystyle d}variables in that equation, if some of them ind−1{\displaystyle d-1}elements are non-zero then the otherdth{\displaystyle d^{th}}variable node value should have exactly the negative value of the summation of thosed−1{\displaystyle d-1}variable nodes. If the non-zero elements of the original signal are chosen from acontinuous distributionthen the probability of this to occur is zero. Therefore, ZCN rule works perfectly.[6][8]
D1CN says that if a variable node is the only unknown variable in an equation then the value of that variable equals theright hand side of that equation. In fact, an equation with just one unknown variable is a check node with degree one, i.e. a check node with just one unverified variable node in its neighborhood.[6][8]
This rule has two parts, the first part deals with non-zero elements of the signal while the second one is responsible for the zero elements of the original signal. For the first part, it says that if we have two or more equations with the sameright hand side, and if we only have one singleunknown variablev{\displaystyle v}common in all of those equations then the value of this common variable should be the value of theright hand sideof those equations. Besides, it says that all other variables in those equations should be zero. Suppose that one of those variablesv′{\displaystyle v'}is not zero, then theright hand sideof the equation which contains bothv,v′{\displaystyle v,v'}should bex(v′)+x(v){\displaystyle x(v')+x(v)}(For simplicity assume that theedge weightsare all 1 or zero). Besides, since we know thatv{\displaystyle v}is the only unique variable in all of these equations then there should be one equationc{\displaystyle c}in whichv{\displaystyle v}exists andv′{\displaystyle v'}does not exist. On the other hand, we know that theright hand sideof these equations are the same; therefore, theright hand sideof equationc{\displaystyle c}should also bex(v)+x(v′){\displaystyle x(v)+x(v')}. If we removev′{\displaystyle v'}from this equation we should have the summation of some unknown variables to be a non-zero valuex(v′){\displaystyle x(v')}. Since the non-zero elements ofx{\displaystyle x}are chosen randomly from acontinuous distributionthe probability that this summation equals exactlyx(v′){\displaystyle x(v')}is zero. Therefore,almost surelythe value ofv{\displaystyle v}is zero and all other variables in these equations have value zero.[6][8][7]
There is just one scenario remained for the second part of the ECN rule as most of it has been covered in the first part. This scenario is the one that we have some equations with the sameright hand sidebut there is two or more variable node common in all of those equations. In this case, we can say nothing about those common variable nodes; however, we can say that all the other variable nodes in those equations are zero. The proof of this claim can be achieved by achange of variablein those equations. Suppose thatv1,v2,...,vq{\displaystyle v_{1},v_{2},...,v_{q}}are the common variable nodes in those equations. If we setv′=v1+v2+...+vq{\displaystyle v'=v_{1}+v_{2}+...+v_{q}}then the problem will be changed to the first part where we only have one common variable node in all of those equations. Therefore, with the same reasoning as in the first part we can see that all other variable nodes that are not common in all of those equations can be verified with value zeroalmost surely.[6][8][7]
When the non-zero elements of the measurement matrix are chosen randomly from acontinuous distribution, then it can be shown that if one variable node receives equal messages divided by theedge weightsfrom its neighbors then this variable node is the only unique variable connected to all of those check nodes, therefore, the rule can be applied using a local decision approach, and the variable node can verify itself without further knowledge about the other connections of those check nodes. Moreover, the second part of the ECN rule is not necessary to be implemented as the non-zero verified variable node in the ECN rule will be removed from thebipartite graphin the nextiterationand ZCN rule will be enough to verify all the zero valued variable nodes remained from those equations with the sameright hand side. All in all, when the non-zero elements of the measurement matrix are chosen form acontinuous distributionthen the SBB and XH algorithm that use ECN rule can be implemented efficiently.[6]
Every minor loop in the main loop of the algorithm can be executed inparallel processors, if we consider each variable and check node as a separate processor. Therefore, every minor loop in the algorithm can be executed inconstant timeO(1){\displaystyle O(1)}. Moreover, since the algorithm will terminate when there is no progress in verification of the variable nodes then the if in the worst case in each iteration of the main loop there is only one variable node to be verified, then the maximum number of times that the main loop will be executed is|Vl|{\displaystyle |V_{l}|}. Therefore, the whole algorithm will be executed inO(|Vl|){\displaystyle O(|V_{l}|)}time.[7]
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Lexicologyis the branch oflinguisticsthat analyzes thelexiconof a specificlanguage. A word is the smallest meaningful unit of alanguagethat can stand on its own, and is made up of small components calledmorphemesand even smaller elements known asphonemes, or distinguishing sounds. Lexicology examines every feature of a word – includingformation,spelling,origin,usage, anddefinition.[1]
Lexicology also considers the relationships that exist between words. In linguistics, thelexiconof a language is composed oflexemes, which are abstract units of meaning that correspond to a set of related forms of a word. Lexicology looks at how words can be broken down as well as identifies common patterns they follow.[2]
Lexicology is associated withlexicography, which is the practice of compilingdictionaries.[3]
The termlexicologyderives from theGreekword λεξικόνlexicon(neuter of λεξικόςlexikos, "of or for words",[4]from λέξιςlexis, "speech" or "word"[5]) and -λογία-logia, "the study of" (asuffixderived from λόγοςlogos, amongst others meaning "learning, reasoning, explanation, subject-matter").[6]Etymology as a science is actually a focus of lexicology. Since lexicology studies the meaning of words and their semantic relations, it often explores the history and development of a word. Etymologists analyze related languages using thecomparative method, which is a set of techniques that allow linguists to recover the ancestral phonological, morphological, syntactic, etc., components of modern languages by comparing theircognatematerial.[7]This means manyword rootsfrom different branches of the Indo-Europeanlanguage familycan be traced back to single words from theProto-Indo-European language. TheEnglish language, for instance, contains moreborrowed words(or loan words) in itsvocabularythan native words.[8]Examples includeparkourfromFrench,karaokefromJapanese,coconutfromPortuguese,mangofromHindi, etc. A lot ofmusic terminology, likepiano,solo, andopera, is borrowed fromItalian. These words can be further classified according to the linguistic element that is borrowed: phonemes, morphemes, and semantics.[7]
General lexicologyis the broad study of words regardless of a language's specific properties. It is concerned with linguistic features that are common among all languages, such as phonemes and morphemes.Special lexicology, on the other hand, looks at what a particular language contributes to its vocabulary, such asgrammars.[2]Altogether lexicological studies can be approached two ways:
These complementary perspectives were proposed bySwisslinguistFerdinand de Saussure.[10]Lexicology can have both comparative and contrastive methodologies.Comparative lexicologysearches for similar features that are shared among two or more languages.Contrastive lexicologyidentifies the linguistic characteristics which distinguish between related and unrelated languages.[9]
Thesubfieldof semantics that pertains especially to lexicological work is calledlexical semantics. In brief, lexical semantics contemplates the significance of words and their meanings through several lenses, includingsynonymy,antonymy,hyponymy, andpolysemy, among others. Semantic analysis of lexical material may involve both thecontextualizationof the word(s) andsyntactic ambiguity.Semasiologyandonomasiologyare relevant linguistic disciplines associated with lexical semantics.[9]
A word can have two kinds of meaning: grammatical and lexical.Grammatical meaningrefers to a word's function in a language, such astenseorplurality, which can be deduced fromaffixes.Lexical meaningis not limited to a single form of a word, but rather what the word denotes as a base word. For example, theverbto walkcan becomewalks,walked, andwalking –each word has a different grammatical meaning, but the same lexical meaning ("to move one's feet at a regular pace").[11]
Another focus of lexicology isphraseology, which studies multi-word expressions, oridioms, like 'raining cats and dogs.' The meaning of the phrase as a whole has a different meaning than each word does on its own and is often unpredictable when considering its components individually. Phraseology examines how and why such meanings exist, and analyzes the laws that govern these word combinations.[12]
Idioms and other phraseological units can be classified according to content and/ or meaning. They are difficult to translate word-for-word from one language to another.[13]
Lexicographyis the study oflexiconsand the art of compiling dictionaries.[14]It is divided into two separateacademic disciplines:
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Adivision algorithmis analgorithmwhich, given twointegersNandD(respectively the numerator and the denominator), computes theirquotientand/orremainder, the result ofEuclidean division. Some are applied by hand, while others are employed by digital circuit designs and software.
Division algorithms fall into two main categories: slow division and fast division. Slow division algorithms produce one digit of the final quotient per iteration. Examples of slow division includerestoring, non-performing restoring,non-restoring, andSRTdivision. Fast division methods start with a close approximation to the final quotient and produce twice as many digits of the final quotient on each iteration.[1]Newton–RaphsonandGoldschmidtalgorithms fall into this category.
Variants of these algorithms allow using fastmultiplication algorithms. It results that, for large integers, thecomputer timeneeded for a division is the same, up to a constant factor, as the time needed for a multiplication, whichever multiplication algorithm is used.
Discussion will refer to the formN/D=(Q,R){\displaystyle N/D=(Q,R)}, where
is the input, and
is the output.
The simplest division algorithm, historically incorporated into agreatest common divisoralgorithm presented inEuclid'sElements, Book VII, Proposition 1, finds the remainder given two positive integers using only subtractions and comparisons:
The proof that the quotient and remainder exist and are unique (described atEuclidean division) gives rise to a complete division algorithm, applicable to both negative and positive numbers, using additions, subtractions, and comparisons:
This procedure always produces R ≥ 0. Although very simple, it takes Ω(Q) steps, and so is exponentially slower than even slow division algorithms like long division. It is useful if Q is known to be small (being anoutput-sensitive algorithm), and can serve as an executable specification.
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
When used with a binary radix, this method forms the basis for the (unsigned) integer division with remainder algorithm below.Short divisionis an abbreviated form of long division suitable for one-digit divisors.Chunking– also known as the partial quotients method or the hangman method – is a less-efficient form of long division which may be easier to understand. By allowing one to subtract more multiples than what one currently has at each stage, a more freeform variant of long division can be developed as well.
The following algorithm, the binary version of the famouslong division, will divideNbyD, placing the quotient inQand the remainder inR. In the following pseudo-code, all values are treated as unsigned integers.
If we take N=11002(1210) and D=1002(410)
Step 1: Set R=0 and Q=0Step 2: Take i=3 (one less than the number of bits in N)Step 3: R=00 (left shifted by 1)Step 4: R=01 (setting R(0) to N(i))Step 5: R < D, so skip statement
Step 2: Set i=2Step 3: R=010Step 4: R=011Step 5: R < D, statement skipped
Step 2: Set i=1Step 3: R=0110Step 4: R=0110Step 5: R>=D, statement enteredStep 5b: R=10 (R−D)Step 5c: Q=10 (setting Q(i) to 1)
Step 2: Set i=0Step 3: R=100Step 4: R=100Step 5: R>=D, statement enteredStep 5b: R=0 (R−D)Step 5c: Q=11 (setting Q(i) to 1)
endQ=112(310) and R=0.
Slow division methods are all based on a standard recurrence equation[2]
where:
Restoring division operates onfixed-pointfractional numbers and depends on the assumption 0 <D<N.[citation needed]
The quotient digitsqare formed from the digit set {0,1}.
The basic algorithm for binary (radix 2) restoring division is:
Non-performing restoring division is similar to restoring division except that the value of 2R is saved, soDdoes not need to be added back in for the case of R < 0.
Non-restoring division uses the digit set {−1, 1} for the quotient digits instead of {0, 1}. The algorithm is more complex, but has the advantage when implemented in hardware that there is only one decision and addition/subtraction per quotient bit; there is no restoring step after the subtraction,[3]which potentially cuts down the numbers of operations by up to half and lets it be executed faster.[4]The basic algorithm for binary (radix 2) non-restoring division of non-negative numbers is:[verification needed]
Following this algorithm, the quotient is in a non-standard form consisting of digits of −1 and +1. This form needs to be converted to binary to form the final quotient. Example:
If the −1 digits ofQ{\displaystyle Q}are stored as zeros (0) as is common, thenP{\displaystyle P}isQ{\displaystyle Q}and computingM{\displaystyle M}is trivial: perform a ones' complement (bit by bit complement) on the originalQ{\displaystyle Q}.
Finally, quotients computed by this algorithm are always odd, and the remainder in R is in the range −D ≤ R < D. For example, 5 / 2 = 3 R −1. To convert to a positive remainder, do a single restoring stepafterQ is converted from non-standard form to standard form:
The actual remainder is R >> n. (As with restoring division, the low-order bits of R are used up at the same rate as bits of the quotient Q are produced, and it is common to use a single shift register for both.)
SRT division is a popular method for division in manymicroprocessorimplementations.[5][6]The algorithm is named after D. W. Sweeney ofIBM, James E. Robertson ofUniversity of Illinois, andK. D. TocherofImperial College London. They all developed the algorithm independently at approximately the same time (published in February 1957, September 1958, and January 1958 respectively).[7][8][9]
SRT division is similar to non-restoring division, but it uses alookup tablebased on the dividend and the divisor to determine each quotient digit.
The most significant difference is that aredundant representationis used for the quotient. For example, when implementing radix-4 SRT division, each quotient digit is chosen fromfivepossibilities: { −2, −1, 0, +1, +2 }. Because of this, the choice of a quotient digit need not be perfect; later quotient digits can correct for slight errors. (For example, the quotient digit pairs (0, +2) and (1, −2) are equivalent, since 0×4+2 = 1×4−2.) This tolerance allows quotient digits to be selected using only a few most-significant bits of the dividend and divisor, rather than requiring a full-width subtraction. This simplification in turn allows a radix higher than 2 to be used.
Like non-restoring division, the final steps are a final full-width subtraction to resolve the last quotient bit, and conversion of the quotient to standard binary form.
TheIntel Pentiumprocessor'sinfamous floating-point division bugwas caused by an incorrectly coded lookup table. Five of the 1066 entries had been mistakenly omitted.[10][11][12]
Newton–Raphson usesNewton's methodto find thereciprocalofD{\displaystyle D}and multiply that reciprocal byN{\displaystyle N}to find thefinal quotientQ{\displaystyle Q}.
The steps of Newton–Raphson division are:
In order to apply Newton's method to find the reciprocal ofD{\displaystyle D}, it is necessary to find a functionf(X){\displaystyle f(X)}that has a zero atX=1/D{\displaystyle X=1/D}. The obvious such function isf(X)=DX−1{\displaystyle f(X)=DX-1}, but the Newton–Raphson iteration for this is unhelpful, since it cannot be computed without already knowing the reciprocal ofD{\displaystyle D}(moreover it attempts to compute the exact reciprocal in one step, rather than allow for iterative improvements). A function that does work isf(X)=(1/X)−D{\displaystyle f(X)=(1/X)-D}, for which the Newton–Raphson iteration gives
which can be calculated fromXi{\displaystyle X_{i}}using only multiplication and subtraction, or using twofused multiply–adds.
From a computation point of view, the expressionsXi+1=Xi+Xi(1−DXi){\displaystyle X_{i+1}=X_{i}+X_{i}(1-DX_{i})}andXi+1=Xi(2−DXi){\displaystyle X_{i+1}=X_{i}(2-DX_{i})}are not equivalent. To obtain a result with a precision of 2nbits while making use of the second expression, one must compute the product betweenXi{\displaystyle X_{i}}and(2−DXi){\displaystyle (2-DX_{i})}with double the given precision ofXi{\displaystyle X_{i}}(nbits).[citation needed]In contrast, the product betweenXi{\displaystyle X_{i}}and(1−DXi){\displaystyle (1-DX_{i})}need only be computed with a precision ofnbits, because the leadingnbits (after the binary point) of(1−DXi){\displaystyle (1-DX_{i})}are zeros.
If the error is defined asεi=1−DXi{\displaystyle \varepsilon _{i}=1-DX_{i}}, then:
This squaring of the error at each iteration step – the so-calledquadratic convergenceof Newton–Raphson's method – has the effect that the number of correct digits in the result roughlydoubles for every iteration, a property that becomes extremely valuable when the numbers involved have many digits (e.g. in the large integer domain). But it also means that the initial convergence of the method can be comparatively slow, especially if the initial estimateX0{\displaystyle X_{0}}is poorly chosen.
For the subproblem of choosing an initial estimateX0{\displaystyle X_{0}}, it is convenient to apply a bit-shift to the divisorDto scale it so that 0.5 ≤D≤ 1. Applying the same bit-shift to the numeratorNensures the quotient does not change. Once within a bounded range, a simple polynomialapproximationcan be used to find an initial estimate.
The linearapproximationwith minimum worst-case absolute error on the interval[0.5,1]{\displaystyle [0.5,1]}is:
The coefficients of the linear approximationT0+T1D{\displaystyle T_{0}+T_{1}D}are determined as follows. The absolute value of the error is|ε0|=|1−D(T0+T1D)|{\displaystyle |\varepsilon _{0}|=|1-D(T_{0}+T_{1}D)|}. The minimum of the maximum absolute value of the error is determined by theChebyshev equioscillation theoremapplied toF(D)=1−D(T0+T1D){\displaystyle F(D)=1-D(T_{0}+T_{1}D)}. The local minimum ofF(D){\displaystyle F(D)}occurs whenF′(D)=0{\displaystyle F'(D)=0}, which has solutionD=−T0/(2T1){\displaystyle D=-T_{0}/(2T_{1})}. The function at that minimum must be of opposite sign as the function at the endpoints, namely,F(1/2)=F(1)=−F(−T0/(2T1)){\displaystyle F(1/2)=F(1)=-F(-T_{0}/(2T_{1}))}. The two equations in the two unknowns have a unique solutionT0=48/17{\displaystyle T_{0}=48/17}andT1=−32/17{\displaystyle T_{1}=-32/17}, and the maximum error isF(1)=1/17{\displaystyle F(1)=1/17}. Using this approximation, the absolute value of the error of the initial value is less than
The best quadratic fit to1/D{\displaystyle 1/D}in the interval is
It is chosen to make the error equal to a re-scaled third orderChebyshev polynomialof the first kind, and gives an absolute value of the error less than or equal to 1/99. This improvement is equivalent tolog2(log99/log17)≈0.7{\displaystyle \log _{2}(\log 99/\log 17)\approx 0.7}Newton–Raphson iterations, at a computational cost of less than one iteration.
It is possible to generate a polynomial fit of degree larger than 2, computing the coefficients using theRemez algorithm. The trade-off is that the initial guess requires more computational cycles but hopefully in exchange for fewer iterations of Newton–Raphson.
Since for this method theconvergenceis exactly quadratic, it follows that, from an initial errorε0{\displaystyle \varepsilon _{0}},S{\displaystyle S}iterations will give an answer accurate to
binary places. Typical values are:
A quadratic initial estimate plus two iterations is accurate enough for IEEEsingle precision, but three iterations are marginal fordouble precision. A linear initial estimate plus four iterations is sufficient for both double anddouble extendedformats.
The following computes the quotient ofNandDwith a precision ofPbinary places:
For example, for a double-precision floating-point division, this method uses 10 multiplies, 9 adds, and 2 shifts.
There is an iteration which uses three multiplications to cube the error:
TheYiεiterm is new.
Expanding out the above,Xi+1{\displaystyle X_{i+1}}can be written as
with the result that the error term
This is 3/2 the computation of the quadratic iteration, but achieveslog3/log2≈1.585{\displaystyle \log 3/\log 2\approx 1.585}as much convergence, so is slightly more efficient. Put another way, two iterations of this method raise the error to the ninth power at the same computational cost as three quadratic iterations, which only raise the error to the eighth power.
The number of correct bits afterS{\displaystyle S}iterations is
binary places. Typical values are:
A quadratic initial estimate plus two cubic iterations provides ample precision for an IEEE double-precision result. It is also possible to use a mixture of quadratic and cubic iterations.
Using at least one quadratic iteration ensures that the error is positive, i.e. the reciprocal is underestimated.[13]: 370This can simplify a following rounding step if an exactly-rounded quotient is required.
Using higher degree polynomials in either the initialization or the iteration results in a degradation of performance because the extra multiplications required would be better spent on doing more iterations.[citation needed]
Goldschmidt division[14](after Robert Elliott Goldschmidt)[15]uses an iterative process of repeatedly multiplying both the dividend and divisor by a common factorFi, chosen such that the divisor converges to 1. This causes the dividend to converge to the sought quotientQ:
The steps for Goldschmidt division are:
AssumingN/Dhas been scaled so that 0 <D< 1, eachFiis based onD:
Multiplying the dividend and divisor by the factor yields:
After a sufficient numberkof iterationsQ=Nk{\displaystyle Q=N_{k}}.
The Goldschmidt method is used inAMDAthlon CPUs and later models.[16][17]It is also known as Anderson Earle Goldschmidt Powers (AEGP) algorithm and is implemented by variousIBMprocessors.[18][19]Although it converges at the same rate as a Newton–Raphson implementation, one advantage of the Goldschmidt method is that the multiplications in the numerator and in the denominator can be done in parallel.[19]
The Goldschmidt method can be used with factors that allow simplifications by thebinomial theorem.
AssumeN/D{\displaystyle N/D}has been scaled by apower of twosuch thatD∈(12,1]{\displaystyle D\in \left({\tfrac {1}{2}},1\right]}.
We chooseD=1−x{\displaystyle D=1-x}andFi=1+x2i{\displaystyle F_{i}=1+x^{2^{i}}}.
This yields
Afternsteps(x∈[0,12)){\displaystyle \left(x\in \left[0,{\tfrac {1}{2}}\right)\right)}, the denominator1−x2n{\displaystyle 1-x^{2^{n}}}can be rounded to1with arelative error
which is maximum at2−2n{\displaystyle 2^{-2^{n}}}whenx=12{\displaystyle x={\tfrac {1}{2}}}, thus providing a minimum precision of2n{\displaystyle 2^{n}}binary digits.
Methods designed for hardware implementation generally do not scale to integers with thousands or millions of decimal digits; these frequently occur, for example, inmodularreductions incryptography. For these large integers, more efficient division algorithms transform the problem to use a small number of multiplications, which can then be done using an asymptotically efficientmultiplication algorithmsuch as theKaratsuba algorithm,Toom–Cook multiplicationor theSchönhage–Strassen algorithm. The result is that thecomputational complexityof the division is of the same order (up to a multiplicative constant) as that of the multiplication. Examples include reduction to multiplication byNewton's methodasdescribed above,[20]as well as the slightly fasterBurnikel-Ziegler division,[21]Barrett reductionandMontgomery reductionalgorithms.[22][verification needed]Newton's method is particularly efficient in scenarios where one must divide by the same divisor many times, since after the initial Newton inversion only one (truncated) multiplication is needed for each division.
The division by a constantDis equivalent to the multiplication by itsreciprocal.
Since the denominator is constant, so is its reciprocal (1/D). Thus it is possible to compute the value of (1/D) once at compile time, and at run time perform the multiplicationN·(1/D) rather than the divisionN/D. Infloating-pointarithmetic the use of (1/D) presents little problem,[a]but inintegerarithmetic the reciprocal will always evaluate to zero (assuming |D| > 1).
It is not necessary to use specifically (1/D); any value (X/Y) that reduces to (1/D) may be used. For example, for division by 3, the factors 1/3, 2/6, 3/9, or 194/582 could be used. Consequently, ifYwere a power of two the division step would reduce to a fast right bit shift. The effect of calculatingN/Das (N·X)/Yreplaces a division with a multiply and a shift. Note that the parentheses are important, asN·(X/Y) will evaluate to zero.
However, unlessDitself is a power of two, there is noXandYthat satisfies the conditions above. Fortunately, (N·X)/Ygives exactly the same result asN/Din integer arithmetic even when (X/Y) is not exactly equal to 1/D, but "close enough" that the error introduced by the approximation is in the bits that are discarded by the shift operation.[23][24][25]Barrett reductionuses powers of 2 for the value ofYto make division byYa simple right shift.[b]
As a concretefixed-point arithmeticexample, for 32-bit unsigned integers, division by 3 can be replaced with a multiply by2863311531/233, a multiplication by 2863311531 (hexadecimal0xAAAAAAAB) followed by a 33 right bit shift. The value of 2863311531 is calculated as233/3, then rounded up. Likewise, division by 10 can be expressed as a multiplication by 3435973837 (0xCCCCCCCD) followed by division by 235(or 35 right bit shift).[27]: p230-234OEISprovides sequences of the constants for multiplication asA346495and for the right shift asA346496.
For generalx-bit unsigned integer division where the divisorDis not a power of 2, the following identity converts the division into twox-bit addition/subtraction, onex-bit byx-bit multiplication (where only the upper half of the result is used) and several shifts, after precomputingk=x+⌈log2D⌉{\displaystyle k=x+\lceil \log _{2}{D}\rceil }anda=⌈2kD⌉−2x{\displaystyle a=\left\lceil {\frac {2^{k}}{D}}\right\rceil -2^{x}}:
In some cases, division by a constant can be accomplished in even less time by converting the "multiply by a constant" into aseries of shifts and adds or subtracts.[28]Of particular interest is division by 10, for which the exact quotient is obtained, with remainder if required.[29]
When a division operation is performed, the exactquotientq{\displaystyle q}andremainderr{\displaystyle r}are approximated to fit within the computer’s precision limits. The Division Algorithm states:
[a=bq+r]{\displaystyle [a=bq+r]}
where0≤r<|b|{\displaystyle 0\leq r<|b|}.
Infloating-point arithmetic, the quotientq{\displaystyle q}is represented asq~{\displaystyle {\tilde {q}}}and the remainderr{\displaystyle r}asr~{\displaystyle {\tilde {r}}}, introducingrounding errorsϵq{\displaystyle \epsilon _{q}}ϵq{\displaystyle \epsilon _{q}}andϵr{\displaystyle \epsilon _{r}}:
[q~=q+ϵq][r~=r+ϵr]{\displaystyle [{\tilde {q}}=q+\epsilon _{q}][{\tilde {r}}=r+\epsilon _{r}]}
This rounding causes a small error, which can propagate and accumulate through subsequent calculations. Such errors are particularly pronounced in iterative processes and when subtracting nearly equal values - is toldloss of significance. To mitigate these errors, techniques such as the use ofguard digitsorhigher precision arithmeticare employed.[30][31]
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Instatistics,sampling biasis abiasin which a sample is collected in such a way that some members of the intendedpopulationhave a lower or highersampling probabilitythan others. It results in abiased sample[1]of a population (or non-human factors) in which all individuals, or instances, were not equally likely to have been selected.[2]If this is not accounted for, results can be erroneously attributed to the phenomenon under study rather than to the method ofsampling.
Medical sources sometimes refer to sampling bias asascertainment bias.[3][4]Ascertainment bias has basically the same definition,[5][6]but is still sometimes classified as a separate type of bias.[5]
Sampling bias is usually classified as a subtype ofselection bias,[7]sometimes specifically termedsample selection bias,[8][9][10]but some classify it as a separate type of bias.[11]A distinction, albeit not universally accepted, of sampling bias is that it undermines theexternal validityof a test (the ability of its results to be generalized to the entire population), whileselection biasmainly addressesinternal validityfor differences or similarities found in the sample at hand. In this sense, errors occurring in the process of gathering the sample or cohort cause sampling bias, while errors in any process thereafter cause selection bias.
However, selection bias and sampling bias are often used synonymously.[12]
The study of medical conditions begins with anecdotal reports. By their nature, such reports only include those referred for diagnosis and treatment. A child who can't function in school is more likely to be diagnosed withdyslexiathan a child who struggles but passes. A child examined for one condition is more likely to be tested for and diagnosed with other conditions, skewingcomorbiditystatistics. As certain diagnoses become associated with behavior problems orintellectual disability, parents try to prevent their children from being stigmatized with those diagnoses, introducing further bias. Studies carefully selected from whole populations are showing that many conditions are much more common and usually much milder than formerly believed.
Geneticists are limited in how they can obtain data from human populations. As an example, consider a human characteristic. We are interested in deciding if the characteristic is inherited as asimple Mendeliantrait. Following the laws ofMendelian inheritance, if the parents in a family do not have the characteristic, but carry the allele for it, they are carriers (e.g. a non-expressiveheterozygote). In this case their children will each have a 25% chance of showing the characteristic. The problem arises because we can't tell which families have both parents as carriers (heterozygous) unless they have a child who exhibits the characteristic. The description follows the textbook by Sutton.[13]
The figure shows the pedigrees of all the possible families with two children when the parents are carriers (Aa).
The probabilities of each of the families being selected is given in the figure, with the sample frequency of affected children also given. In this simple case, the researcher will look for a frequency of4⁄7or5⁄8for the characteristic, depending on the type of truncate selection used.
An example of selection bias is called the "caveman effect". Much of our understanding ofprehistoricpeoples comes from caves, such ascave paintingsmade nearly 40,000 years ago. If there had been contemporary paintings on trees, animal skins or hillsides, they would have been washed away long ago. Similarly, evidence of fire pits,middens,burial sites, etc. are most likely to remain intact to the modern era in caves. Prehistoric people are associated with caves because that is where the data still exists, not necessarily because most of them lived in caves for most of their lives.[14]
Sampling bias is problematic because it is possible that astatisticcomputed of the sample is systematically erroneous. Sampling bias can lead to a systematic over- or under-estimation of the correspondingparameterin the population. Sampling bias occurs in practice as it is practically impossible to ensure perfect randomness in sampling. If the degree of misrepresentation is small, then the sample can be treated as a reasonable approximation to a random sample. Also, if the sample does not differ markedly in the quantity being measured, then a biased sample can still be a reasonable estimate.
The wordbiashas a strong negative connotation. Indeed, biases sometimes come from deliberate intent to mislead or otherscientific fraud. In statistical usage, bias merely represents a mathematical property, no matter if it is deliberate or unconscious or due to imperfections in the instruments used for observation. While some individuals might deliberately use a biased sample to produce misleading results, more often, a biased sample is just a reflection of the difficulty in obtaining a truly representative sample, or ignorance of the bias in their process of measurement or analysis. An example of how ignorance of a bias can exist is in the widespread use of a ratio (a.k.a.fold change) as a measure of difference in biology. Because it is easier to achieve a large ratio with two small numbers with a given difference, and relatively more difficult to achieve a large ratio with two large numbers with a larger difference, large significant differences may be missed when comparing relatively large numeric measurements. Some have called this a 'demarcation bias' because the use of a ratio (division) instead of a difference (subtraction) removes the results of the analysis from science into pseudoscience (SeeDemarcation Problem).
Some samples use a biased statistical design which nevertheless allows the estimation of parameters. The U.S.National Center for Health Statistics, for example, deliberately oversamples from minority populations in many of its nationwide surveys in order to gain sufficient precision for estimates within these groups.[15]These surveys require the use of sample weights (see later on) to produce proper estimates across all ethnic groups. Provided that certain conditions are met (chiefly that the weights are calculated and used correctly) these samples permit accurate estimation of population parameters.
A classic example of a biased sample and the misleading results it produced occurred in 1936. In the early days of opinion polling, the AmericanLiterary Digestmagazine collected over two million postal surveys and predicted that the Republican candidate in theU.S. presidential election,Alf Landon, would beat the incumbent president,Franklin Roosevelt, by a large margin. The result was the exact opposite. The Literary Digest survey represented a sample collected from readers of the magazine, supplemented by records of registered automobile owners and telephone users. This sample included an over-representation of wealthy individuals, who, as a group, were more likely to vote for the Republican candidate. In contrast, a poll of only 50 thousand citizens selected byGeorge Gallup's organization successfully predicted the result, leading to the popularity of theGallup poll.
Another classic example occurred in the1948 presidential election. On election night, theChicago Tribuneprinted the headlineDEWEY DEFEATS TRUMAN, which turned out to be mistaken. In the morning the grinningpresident-elect,Harry S. Truman, was photographed holding a newspaper bearing this headline. The reason the Tribune was mistaken is that their editor trusted the results of aphone survey. Survey research was then in its infancy, and few academics realized that a sample of telephone users was not representative of the general population. Telephones were not yet widespread, and those who had them tended to be prosperous and have stable addresses. (In many cities, theBell Systemtelephone directorycontained the same names as theSocial Register). In addition, the Gallup poll that the Tribune based its headline on was over two weeks old at the time of the printing.[17]
Inair qualitydata, pollutants (such ascarbon monoxide,nitrogen monoxide,nitrogen dioxide, orozone) frequently show highcorrelations, as they stem from the same chemical process(es). These correlations depend on space (i.e., location) and time (i.e., period). Therefore, a pollutant distribution is not necessarily representative for every location and every period. If a low-cost measurement instrument is calibrated with field data in a multivariate manner, more precisely by collocation next to a reference instrument, the relationships between the different compounds are incorporated into the calibration model. By relocation of the measurement instrument, erroneous results can be produced.[18]
A twenty-first century example is theCOVID-19 pandemic, where variations in sampling bias inCOVID-19 testinghave been shown to account for wide variations in bothcase fatality ratesand theage distributionof cases across countries.[19][20]
If entire segments of the population are excluded from a sample, then there are no adjustments that can produce estimates that are representative of the entire population. But if some groups are underrepresented and the degree of underrepresentation can be quantified, then sample weights can correct the bias. However, the success of the correction is limited to the selection model chosen. If certain variables are missing the methods used to correct the bias could be inaccurate.[21]
For example, a hypothetical population might include 10 million men and 10 million women. Suppose that a biased sample of 100 patients included 20 men and 80 women. A researcher could correct for this imbalance byattaching a weightof 2.5 for each male and 0.625 for each female. This would adjust any estimates to achieve the same expected value as a sample that included exactly 50 men and 50 women, unless men and women differed in their likelihood of taking part in the survey.[citation needed]
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Psycholinguisticsorpsychology of languageis the study of the interrelation between linguistic factors and psychological aspects.[1]The discipline is mainly concerned with the mechanisms by which language is processed and represented in the mind and brain; that is, thepsychologicalandneurobiologicalfactors that enablehumansto acquire, use, comprehend, and producelanguage.[2]
Psycholinguistics is concerned with the cognitive faculties and processes that are necessary to produce the grammatical constructions of language. It is also concerned with the perception of these constructions by a listener.
Initial forays into psycholinguistics were in the philosophical and educational fields, mainly due to their location in departments other thanapplied sciences(e.g., cohesive data on how thehuman brainfunctioned). Modern research makes use ofbiology,neuroscience,cognitive science,linguistics, andinformation scienceto study how the mind-brain processes language, and less so the known processes ofsocial sciences,human development, communication theories, andinfant development, among others.
There are several subdisciplines with non-invasive techniques for studying the neurological workings of the brain. For example,neurolinguisticshas become a field in its own right, anddevelopmental psycholinguistics, as a branch of psycholinguistics, concerns itself with a child's ability to learn language.
Psycholinguistics is an interdisciplinary field that consists of researchers from a variety of different backgrounds, includingpsychology,cognitive science,linguistics,speech and language pathology, anddiscourse analysis. Psycholinguists study how people acquire and use language, according to the following main ways:
A researcher interested in language comprehension may studywordrecognition duringreading, to examine the processes involved in the extraction oforthographic,morphological,phonological, andsemanticinformation from patterns in printed text. A researcher interested in language production might study how words are prepared to be spoken starting from the conceptual or semantic level (this concerns connotation, and possibly can be examined through the conceptual framework concerned with thesemantic differential).Developmental psycholinguistsstudy infants' and children's ability to learn and process language.[3]
Psycholinguistics further divide their studies according to the different components that make up humanlanguage.
Linguistics-related areas include:
In seeking to understand the properties of language acquisition, psycholinguistics has roots in debates regarding innate versus acquired behaviors (both in biology and psychology). For some time, the concept of an innate trait was something that was not recognized in studying the psychology of the individual.[4]However, with the redefinition of innateness as time progressed, behaviors considered innate could once again be analyzed as behaviors that interacted with the psychological aspect of an individual. After the diminished popularity of thebehavioristmodel,ethologyreemerged as a leading train of thought within psychology, allowing the subject of language, aninnate human behavior, to be examined once more within the scope of psychology.[4]
The theoretical framework for psycholinguistics ostensibly began to be developed near the end of the 19th century as the "psychology of language". The work ofEdward ThorndikeandFrederic Bartlettlaid the foundations[citation needed]of what would come to be known[according to whom?]as "psycholinguistics."
The use of the term "psycholinguistic" is first encountered inadjectiveform in psychologistJacob Kantor1936 bookAn Objective Psychology of Grammar.[5]: 260
The term "psycholinguistics" came into wider usage in 1946 when Kantor's student Nicholas Pronko published an article entitled "Psycholinguistics: A Review".[6]Pronko's intention was to unify related theoretical approaches under a single name.[5][6]The term was used for the first time to talk about an interdisciplinary field "that could be coherent",[5]inCharles E. OsgoodandThomas A. Sebeok'sPsycholinguistics: A Survey of Theory and Research Problems(1954).[7]: 1679–1692
Though there is still much debate, there are two primary theories on childhood language acquisition:
The innatist perspective began in 1959 withNoam Chomsky's highly critical review ofB.F. Skinner'sVerbal Behavior(1957).[8]This review helped start what has been called thecognitive revolutionin psychology. Chomsky posited that humans possess a special, innate ability for language, and thatcomplex syntactic features, such asrecursion, are "hard-wired" in the brain. These abilities are thought to be beyond the grasp of even the most intelligent and social non-humans. When Chomsky asserted that children acquiring a language have a vast search space to explore among all possible human grammars, there was no evidence that children receivedsufficient input to learnall the rules of their language. Hence, there must be some other innate mechanism that endows humans with the ability to learn language. According to the "innateness hypothesis", such a language faculty is what defines human language and makes that faculty different from even the most sophisticated forms of animal communication.
The field of linguistics and psycholinguistics has since been defined by pro-and-con reactions to Chomsky. The view in favor of Chomsky still holds that the human ability to use language (specifically the ability to use recursion) is qualitatively different from any sort of animal ability.[9]
The view that language must be learned was especially popular before 1960 and is well represented by thementalistictheories ofJean Piagetand the empiricistRudolf Carnap. Likewise, the behaviorist school of psychology puts forth the point of view that language is a behavior shaped by conditioned response; hence it is learned. The view that language can be learned has had a recent resurgence inspired byemergentism. This view challenges the "innate" view as scientificallyunfalsifiable; that is to say, it cannot be tested. With the increase in computer technology since the 1980s, researchers have been able to simulate language acquisition using neural network models.[10]
The structures and uses of language are related to the formation of ontological insights.[11]Some see this system as "structured cooperation between language-users" who use conceptual andsemantic differencein order to exchange meaning and knowledge, as well as give meaning to language, thereby examining and describing "semantic processes bound by a 'stopping' constraint which are not cases of ordinary deferring." Deferring is normally done for a reason, and a rational person is always disposed to defer if there is good reason.[12]
The theory of the "semantic differential" supposes universal distinctions, such as:[13]
One question in the realm of language comprehension is how people understand sentences as they read (i.e.,sentence processing). Experimental research has spawned several theories about the architecture and mechanisms of sentence comprehension. These theories are typically concerned with the types of information, contained in the sentence, that the reader can use to build meaning and the point at which that information becomes available to the reader. Issues such as "modular" versus "interactive" processing have been theoretical divides in the field.
A modular view of sentence processing assumes that the stages involved in reading a sentence function independently as separate modules. These modules have limited interaction with one another. For example, one influential theory of sentence processing, the "garden-path theory", states that syntactic analysis takes place first. Under this theory, as the reader is reading a sentence, he or she creates the simplest structure possible, to minimize effort and cognitive load.[14]This is done without any input fromsemantic analysisor context-dependent information. Hence, in the sentence "The evidence examined by the lawyer turned out to be unreliable", by the time the reader gets to the word "examined" he or she has committed to a reading of the sentence in which the evidence is examining something because it is the simplest parsing. This commitment is made even though it results in an implausible situation: evidence cannot examine something. Under this "syntax first" theory, semantic information is processed at a later stage. It is only later that the reader will recognize that he or she needs to revise the initial parsing into one in which "the evidence" is being examined. In this example, readers typically recognize their mistake by the time they reach "by the lawyer" and must go back and reevaluate the sentence.[15]This reanalysis is costly and contributes to slower reading times. A 2024 study found that during self-paced reading tasks, participants progressively read faster and recalled information more accurately, suggesting that task adaptation is driven by learning processes rather than by declining motivation.[16]
In contrast to the modular view, an interactive theory of sentence processing, such as aconstraint-basedlexical approach assumes that all available information contained within a sentence can be processed at any time.[17]Under an interactive view, the semantics of a sentence (such as plausibility) can come into play early on to help determine the structure of a sentence. Hence, in the sentence above, the reader would be able to make use of plausibility information in order to assume that "the evidence" is being examined instead of doing the examining. There are data to support both modular and interactive views; which view is correct is debatable.
When reading,saccadescan cause the mind to skip over words because it does not see them as important to the sentence, and the mind completely omits it from the sentence or supplies the wrong word in its stead. This can be seen in "Paris in thethe Spring". This is a common psychological test, where the mind will often skip the second "the", especially when there is a line break in between the two.[18]
Language production refers to how people produce language, either in written or spoken form, in a way that conveys meanings comprehensible to others. One of the most effective ways to explain the way people represent meanings using rule-governed languages is by observing and analyzing instances ofspeech errors, which include speech disfluencies like false starts, repetition, reformulation and constant pauses in between words or sentences, as well as slips of the tongue, like-blendings, substitutions, exchanges (e.g.Spoonerism), and various pronunciation errors.
These speech errors have significant implications for understanding how language is produced, in that they reflect that:[19]
It is useful to differentiate between three separate phases of language production:[20]
Psycholinguistic research has largely concerned itself with the study of formulation because the conceptualization phase remains largely elusive and mysterious.[20]
Linguistic relativity, often associated with the Sapir-Whorf hypothesis, posits that the structure of a language influences cognitive processes and world perception. While early formulations of this idea were largely speculative, modern psycholinguistic research has reframed it as a testable hypothesis within the broader study of language and thought.
Contemporary approaches to linguistic relativity are often discussed into following perspectives:
A key refinement of linguistic relativity is Slobin’s (1996) "Thinking for Speaking" hypothesis, which argues that language influences cognition most strongly when individuals prepare to communicate. Unlike traditional views of linguistic relativity, which suggest that language passively shapes thought, "Thinking for Speaking" proposes that speakers actively engage with linguistic categories and structures while constructing utterances.[23]
From a psycholinguistic standpoint, research on linguistic relativity intersects with conceptual representations, perceptual learning, and cognitive flexibility. Experimental studies have tested these ideas by examining how speakers of different languages categorize the world differently. For instance, cross-linguistic comparisons in spatial cognition reveal that languages with absolute spatial frames (e.g., Guugu Yimithirr) encourage speakers to encode space differently than languages with relative spatial frames (e.g., English).[21]
In the domain of bilingual cognition, psycholinguistic research suggests that bilinguals may experience cognitive restructuring, where language context modulates perception and categorization. Recent studies indicate that bilinguals can flexibly switch between different conceptual systems, depending on the language they are using, particularly in domains such as motion perception, event construal, and time perception.[24]
Overall, linguistic relativity in psycholinguistics is no longer seen as a rigid determinism of thought by language, but rather as a gradual, experience-based modulation of cognition by linguistic structures. This perspective has led to a shift from a purely linguistic hypothesis to an integrative cognitive science framework incorporating evidence from experimental psychology, neuroscience, and computational modeling.[25]
Many of the experiments conducted in psycholinguistics, especially early on, are behavioral in nature. In these types of studies, subjects are presented with linguistic stimuli and asked to respond. For example, they may be asked to make a judgment about a word (lexical decision), reproduce the stimulus, or say a visually presented word aloud. Reaction times to respond to the stimuli (usually on the order of milliseconds) and proportion of correct responses are the most often employed measures of performance in behavioral tasks. Such experiments often take advantage ofpriming effects, whereby a "priming" word or phrase appearing in the experiment can speed up the lexical decision for a related "target" word later.[26]
As an example of how behavioral methods can be used in psycholinguistics research, Fischler (1977) investigated word encoding, using a lexical-decision task.[27]He asked participants to make decisions about whether two strings of letters were English words. Sometimes the strings would be actual English words requiring a "yes" response, and other times they would be non-words requiring a "no" response. A subset of the licit words were related semantically (e.g., cat–dog) while others were unrelated (e.g., bread–stem). Fischler found that related word pairs were responded to faster, compared to unrelated word pairs, which suggests that semantic relatedness can facilitate word encoding.[27]
Recently,eye trackinghas been used to study onlinelanguage processing. Beginning with Rayner (1978), the importance of understanding eye-movements during reading was established.[28]Later, Tanenhaus et al. (1995) used a visual-world paradigm to study the cognitive processes related to spoken language.[29]Assuming that eye movements are closely linked to the current focus of attention, language processing can be studied by monitoring eye movements while a subject is listening to spoken language.
Theanalysisof systematicerrors in speech, as well as the writing andtypingof language, can provide evidence of the process that has generated it. Errors of speech, in particular, grant insight into how the mind produces language while a speaker is mid-utterance. Speech errors tend to occur in thelexical,morpheme, andphonemeencoding steps of language production, as seen by the ways errors can manifest themselves.[30]
The types of speech errors, with some examples, include:[30][31][32]
Speech errors will usually occur in the stages that involve lexical, morpheme, or phoneme encoding, and usually not in the first step ofsemantic encoding.[33]This can be attributed to a speaker still conjuring the idea of what to say; and unless he changes his mind, can not be mistaken for what he wanted to say.
Until the recent advent ofnon-invasivemedical techniques, brain surgery was the preferred way for language researchers to discover how language affects the brain. For example, severing thecorpus callosum(the bundle of nerves that connects the two hemispheres of the brain) was at one time a treatment for some forms ofepilepsy. Researchers could then study the ways in which the comprehension and production of language were affected by such drastic surgery. When an illness made brain surgery necessary, language researchers had an opportunity to pursue their research.
Newer, non-invasive techniques now include brain imaging bypositron emission tomography(PET);functional magnetic resonance imaging(fMRI);event-related potentials(ERPs) inelectroencephalography(EEG) andmagnetoencephalography(MEG); andtranscranial magnetic stimulation(TMS). Brain imaging techniques vary in their spatial and temporal resolutions (fMRI has a resolution of a few thousand neurons per pixel, and ERP has millisecond accuracy). Each methodology has advantages and disadvantages for the study of psycholinguistics.[34]
Computational modelling, such as theDRC modelof reading and word recognition proposed byMax Coltheartand colleagues,[35]is another methodology, which refers to the practice of setting up cognitive models in the form of executable computer programs. Such programs are useful because they require theorists to be explicit in their hypotheses and because they can be used to generate accurate predictions for theoretical models that are so complex thatdiscursive analysisis unreliable. Other examples of computational modelling areMcClellandandElman'sTRACEmodel ofspeech perception[36]and Franklin Chang's Dual-Path model of sentence production.[37]
The psychophysical approach in psycholinguistics applies quantitative measurement techniques to investigate how linguistic structures influence perception and cognitive processes. Unlike traditional behavioral experiments that rely on categorical judgments or reaction times, psychophysical methods allow for precise, continuous measurement of perceptual and cognitive changes induced by language.
A key advantage of psychophysical methods is their ability to capture fine-grained perceptual effects of language. For instance, studies on color perception have used just-noticeable difference (JND) thresholds to show that speakers of languages with finer color distinctions (e.g., Russian for light vs. dark blue) exhibit heightened perceptual sensitivity at linguistic category boundaries.[22]
Recent psychophysical research has also been applied to time perception, investigating how bilinguals process temporal information differently based on their linguistic background. Using psychophysical duration estimation tasks, researchers have demonstrated that bilinguals may exhibit different time perception patterns depending on which language they are using at the moment.[24]
These methods provide insights into how linguistic categories shape cognitive processing at a perceptual level, distinguishing between effects that arise from language structure itself and those that emerge from general cognitive mechanisms. As psycholinguistics continues to integrate computational and neuroscientific approaches, psychophysical techniques offer a bridge between language processing and sensory cognition, refining our understanding of how language interacts with perception.
Psycholinguistics is concerned with the nature of the processes that the brain undergoes in order to comprehend and produce language. For example, thecohort modelseeks to describe how words are retrieved from themental lexiconwhen an individual hears or sees linguistic input.[26][38]Using newnon-invasiveimaging techniques, recent research seeks to shed light on the areas of the brain involved in language processing.
Another unanswered question in psycholinguistics is whether the human ability to use syntax originates from innate mental structures or social interaction, and whether or not some animals can be taught the syntax of human language.
Two other major subfields of psycholinguistics investigatefirst language acquisition, the process by which infants acquire language, andsecond language acquisition. It is much more difficult for adults to acquiresecond languagesthan it is for infants to learn their first language (infants are able to learn more than one native language easily). Thus,sensitive periodsmay exist during which language can be learned readily.[39]A great deal of research in psycholinguistics focuses on how this ability develops and diminishes over time. It also seems to be the case that the more languages one knows, the easier it is to learn more.[40]
The field ofaphasiologydeals with language deficits that arise because of brain damage. Studies in aphasiology can offer both advances in therapy for individuals suffering from aphasia and further insight into how the brain processes language.
A short list of books that deal with psycholinguistics, written in language accessible to the non-expert, includes:
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Atreatiseis aformaland systematic writtendiscourseon some subject concerned with investigating or exposing the main principles of the subject and its conclusions.[1]Amonographis a treatise on a specialized topic.[2]
The word "treatise" has its origins in the early 14th century, derived from the Anglo-French termtretiz, which itself comes from the Old Frenchtraitis, meaning "treatise" or "account." This Old French term is rooted in the verbtraitier, which means "to deal with" or "to set forth in speech or writing".[3]
The etymological lineage can be traced further back to the Latin wordtractatus, which is a form of the verbtractare, meaning "to handle," "to manage," or "to deal with".[4][5]The Latin roots suggest a connotation of engaging with or discussing a subject in depth, which aligns with the modern understanding of a treatise as a formal and systematic written discourse on a specific topic.[6]
The works presented here have been identified as influential by scholars on the development of human civilization.
Euclid'sElementshas appeared in more editions than any other books except theBibleand is one of the most important mathematical treatises ever. It has been translated to numerous languages and remains continuously in print since the beginning of printing. Before the invention of the printing press, it was manually copied and widely circulated. When scholars recognized its excellence, they removed inferior works from circulation in its favor. Many subsequent authors, such asTheon of Alexandria, made their own editions, with alterations, comments, and new theorems or lemmas. Many mathematicians were influenced and inspired by Euclid's masterpiece. For example,Archimedes of SyracuseandApollonius of Perga, the greatest mathematicians of their time, received their training from Euclid's students and hisElementsand were able to solve many open problems at the time of Euclid. It is a prime example of how to write a text in pure mathematics, featuring simple and logical axioms, precise definitions, clearly stated theorems, and logical deductive proofs. TheElementsconsists of thirteen books dealing with geometry (including the geometry of three-dimensional objects such as polyhedra), number theory, and the theory of proportions. It was essentially a compilation of all mathematics known to the Greeks up until Euclid's time.[10]
Drawing on the work of his predecessors, especially the experimental research ofMichael Faraday, the analogy with heat flow byWilliam Thomson(later Lord Kelvin) and the mathematical analysis ofGeorge Green, James Clerk Maxwell synthesized all that was known about electricity and magnetism into a single mathematical framework,Maxwell's equations. Originally, there were 20 equations in total. In hisTreatise on Electricity and Magnetism(1873), Maxwell reduced them to eight.[11]Maxwell used his equations to predict the existence of electromagnetic waves, which travel at the speed of light. In other words, light is but one kind of electromagnetic wave. Maxwell's theory predicted there ought to be other types, with different frequencies. After some ingenious experiments, Maxwell's prediction was confirmed byHeinrich Hertz. In the process, Hertz generated and detected what are now called radio waves and built crude radio antennas and the predecessors of satellite dishes.[12]Hendrik Lorentzderived, using suitable boundary conditions,Fresnel's equationsfor the reflection and transmission of light in different media from Maxwell's equations. He also showed that Maxwell's theory succeeded in illuminating the phenomenon of light dispersion where other models failed.John William Strutt(Lord Rayleigh) andJosiah Willard Gibbsthen proved that the optical equations derived from Maxwell's theory are the only self-consistent description of the reflection, refraction, and dispersion of light consistent with experimental results.Opticsthus found a new foundation inelectromagnetism.[11]
Hertz's experimental work in electromagnetism stimulated interest in the possibility of wireless communication, which did not require long and expensive cables and was faster than even the telegraph.Guglielmo Marconiadapted Hertz's equipment for this purpose in the 1890s. He achieved the first international wireless transmission between England and France in 1900 and by the following year, he succeeded in sending messages inMorse codeacross the Atlantic. Seeing its value, the shipping industry adopted this technology at once.Radio broadcastingbecame extremely popular in the twentieth century and remains in common use in the early twenty-first.[12]But it wasOliver Heaviside, an enthusiastic supporter of Maxwell's electromagnetic theory, who deserves most of the credit for shaping how people understood and applied Maxwell's work for decades to come; he was responsible for considerable progress in electrical telegraphy, telephony, and the study of the propagation of electromagnetic waves. Independent of Gibbs, Heaviside assembled a set of mathematical tools known asvector calculusto replace thequaternions, which were in vogue at the time but which Heaviside dismissed as "antiphysical and unnatural."[13]
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Inlinear algebra, theHermite normal formis an analogue ofreduced echelon formformatricesover theintegersZ{\displaystyle \mathbb {Z} }. Just asreduced echelon formcan be used to solve problems about the solution to the linear systemAx=b{\displaystyle Ax=b}wherex∈Rn{\displaystyle x\in \mathbb {R} ^{n}}, the Hermite normal form can solve problems about the solution to the linear systemAx=b{\displaystyle Ax=b}where this timex{\displaystyle x}is restricted to have integer coordinates only. Other applications of the Hermite normal form includeinteger programming,[1]cryptography,[2]andabstract algebra.[3]
Various authors may prefer to talk about Hermite normal form in either row-style or column-style. They are essentially the same up to transposition.
A matrixA∈Zm×n{\displaystyle A\in \mathbb {Z} ^{m\times n}}has a (row) Hermite normal formH{\displaystyle H}if there is a squareunimodular matrixU{\displaystyle U}whereH=UA{\displaystyle H=UA}.
H{\displaystyle H}has the following restrictions:[4][5][6]
The third condition is not standard among authors, for example some sources force non-pivots to be nonpositive[7][8]or place no sign restriction on them.[9]However, these definitions are equivalent by using a different unimodular matrixU{\displaystyle U}. A unimodular matrix is a square integer matrix whosedeterminantis 1 or −1 (and henceinvertible). In fact, a unimodular matrix is invertible over the integers, as can be seen, for example, fromCramer's Rule.
A matrixA∈Zm×n{\displaystyle A\in \mathbb {Z} ^{m\times n}}has a (column) Hermite normal formH{\displaystyle H}if there is a squareunimodular matrixU{\displaystyle U}whereH=AU{\displaystyle H=AU}andH{\displaystyle H}has the following restrictions:[8][10]
Note that the row-style definition has a unimodular matrixU{\displaystyle U}multiplyingA{\displaystyle A}on the left (meaningU{\displaystyle U}is acting on the rows ofA{\displaystyle A}), while the column-style definition has the unimodular matrix action on the columns ofA{\displaystyle A}. The two definitions of Hermite normal forms are simply transposes of each other.
Every full row rankm-by-nmatrixAwith integer entries has a uniquem-by-nmatrixHin Hermite normal form, such thatH=UAfor some square unimodular matrixU.[5][11][12]
In the examples below,His the Hermite normal form of the matrixA, andUis a unimodular matrix such thatUA=H.A=(331401000019160003)H=(30110100001910003)U=(1−30−10100001−50001){\displaystyle A={\begin{pmatrix}3&3&1&4\\0&1&0&0\\0&0&19&16\\0&0&0&3\end{pmatrix}}\qquad H={\begin{pmatrix}3&0&1&1\\0&1&0&0\\0&0&19&1\\0&0&0&3\end{pmatrix}}\qquad U=\left({\begin{array}{rrrr}1&-3&0&-1\\0&1&0&0\\0&0&1&-5\\0&0&0&1\end{array}}\right)}
A=(236256168311)H=(1050−110328−20061−13)U=(9−515−2011−61){\displaystyle A={\begin{pmatrix}2&3&6&2\\5&6&1&6\\8&3&1&1\end{pmatrix}}\qquad H=\left({\begin{array}{rrrr}1&0&50&-11\\0&3&28&-2\\0&0&61&-13\end{array}}\right)\qquad U=\left({\begin{array}{rrr}9&-5&1\\5&-2&0\\11&-6&1\end{array}}\right)}
IfAhas only one row then eitherH=AorH= −A, depending on whether the single row ofAhas a positive or negative leading coefficient.
There are many algorithms for computing the Hermite normal form, dating back to 1851. One such algorithm is described in.[13]: 43--45But only in 1979 an algorithm for computing the Hermite normal form that ran instrongly polynomial timewas first developed;[14]that is, the number of steps to compute the Hermite normal form is bounded above by a polynomial in the dimensions of the input matrix, and the space used by the algorithm (intermediate numbers) is bounded by a polynomial in the binary encoding size of the numbers in the input matrix.
One class of algorithms is based onGaussian eliminationin that special elementary matrices are repeatedly used.[11][15][16]TheLLLalgorithm can also be used to efficiently compute the Hermite normal form.[17][18]
A typicallatticeinRnhas the formL={∑i=1nαiai|αi∈Z}{\textstyle L=\left\{\left.\sum _{i=1}^{n}\alpha _{i}\mathbf {a} _{i}\;\right\vert \;\alpha _{i}\in {\textbf {Z}}\right\}}where theaiare inRn. If thecolumnsof a matrixAare theai, the lattice can be associated with the columns of a matrix, andAis said to be a basis ofL. Because the Hermite normal form is unique, it can be used to answer many questions about two lattice descriptions. For what follows,LA{\displaystyle L_{A}}denotes the lattice generated by the columns of A. Because the basis is in the columns of the matrixA, the column-style Hermite normal form must be used. Given two bases for a lattice,AandA', the equivalence problem is to decide ifLA=LA′.{\displaystyle L_{A}=L_{A'}.}This can be done by checking if the column-style Hermite normal form ofAandA'are the same up to the addition of zero columns. This strategy is also useful for deciding if a lattice is a subset (LA⊆LA′{\displaystyle L_{A}\subseteq L_{A'}}if and only ifL[A∣A′]=LA′{\displaystyle L_{[A\mid A']}=L_{A'}}), deciding if a vector v is in a lattice (v∈LA{\displaystyle v\in L_{A}}if and only ifL[v∣A]=LA{\displaystyle L_{[v\mid A]}=L_{A}}), and for other calculations.[19]
The linear systemAx=bhas an integer solutionxif and only if the systemHy=bhas an integer solutionywherey=U−1xandHis the column-style Hermite normal form ofA. Checking thatHy=bhas an integer solution is easier thanAx=bbecause the matrixHis triangular.[11]: 55
Many mathematical software packages can compute the Hermite normal form:
Hermite normal form can be defined when we replaceZby an arbitraryDedekind domain.[21](for instance, anyprincipal-ideal domain). For instance, incontrol theoryit can be useful to consider Hermite normal form for the polynomialsF[x]over a given fieldF.
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In mathematics, aQ-categoryoralmost quotient category[1]is acategorythat is a "milder version of a Grothendieck site."[2]A Q-category is acoreflective subcategory.[1][clarification needed]The Q stands for a quotient.
The concept of Q-categories was introduced by Alexander Rosenberg in 1988.[2]The motivation for the notion was its use innoncommutative algebraic geometry; in this formalism,noncommutative spacesare defined assheaveson Q-categories.
A Q-category is defined by the formula[1][further explanation needed]A:(u∗⊣u∗):A¯→u∗←u∗A{\displaystyle \mathbb {A} :(u^{*}\dashv u_{*}):{\bar {A}}{\stackrel {\overset {u^{*}}{\leftarrow }}{\underset {u_{*}}{\to }}}A}whereu∗{\displaystyle u^{*}}is the left adjoint in a pair ofadjoint functorsand is afull and faithful functor.
Thiscategory theory-related article is astub. You can help Wikipedia byexpanding it.
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Inmathematics(in particular,functional analysis),convolutionis amathematical operationon twofunctionsf{\displaystyle f}andg{\displaystyle g}that produces a third functionf∗g{\displaystyle f*g}, as theintegralof the product of the two functions after one is reflected about the y-axis and shifted. The termconvolutionrefers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (seecommutativity). Graphically, it expresses how the 'shape' of one function is modified by the other.
Some features of convolution are similar tocross-correlation: for real-valued functions, of a continuous or discrete variable, convolutionf∗g{\displaystyle f*g}differs from cross-correlationf⋆g{\displaystyle f\star g}only in that eitherf(x){\displaystyle f(x)}org(x){\displaystyle g(x)}is reflected about the y-axis in convolution; thus it is a cross-correlation ofg(−x){\displaystyle g(-x)}andf(x){\displaystyle f(x)}, orf(−x){\displaystyle f(-x)}andg(x){\displaystyle g(x)}.[A]For complex-valued functions, the cross-correlation operator is theadjointof the convolution operator.
Convolution has applications that includeprobability,statistics,acoustics,spectroscopy,signal processingandimage processing,geophysics,engineering,physics,computer visionanddifferential equations.[1]
The convolution can be defined for functions onEuclidean spaceand othergroups(asalgebraic structures).[citation needed]For example,periodic functions, such as thediscrete-time Fourier transform, can be defined on acircleand convolved byperiodic convolution. (See row 18 atDTFT § Properties.) Adiscrete convolutioncan be defined for functions on the set ofintegers.
Generalizations of convolution have applications in the field ofnumerical analysisandnumerical linear algebra, and in the design and implementation offinite impulse responsefilters in signal processing.[citation needed]
Computing theinverseof the convolution operation is known asdeconvolution.
The convolution off{\displaystyle f}andg{\displaystyle g}is writtenf∗g{\displaystyle f*g}, denoting the operator with the symbol∗{\displaystyle *}.[B]It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. As such, it is a particular kind ofintegral transform:
An equivalent definition is (seecommutativity):
While the symbolt{\displaystyle t}is used above, it need not represent the time domain. At eacht{\displaystyle t}, the convolution formula can be described as the area under the functionf(τ){\displaystyle f(\tau )}weighted by the functiong(−τ){\displaystyle g(-\tau )}shifted by the amountt{\displaystyle t}. Ast{\displaystyle t}changes, the weighting functiong(t−τ){\displaystyle g(t-\tau )}emphasizes different parts of the input functionf(τ){\displaystyle f(\tau )}; Ift{\displaystyle t}is a positive value, theng(t−τ){\displaystyle g(t-\tau )}is equal tog(−τ){\displaystyle g(-\tau )}that slides or is shifted along theτ{\displaystyle \tau }-axis toward the right (toward+∞{\displaystyle +\infty }) by the amount oft{\displaystyle t}, while ift{\displaystyle t}is a negative value, theng(t−τ){\displaystyle g(t-\tau )}is equal tog(−τ){\displaystyle g(-\tau )}that slides or is shifted toward the left (toward−∞{\displaystyle -\infty }) by the amount of|t|{\displaystyle |t|}.
For functionsf{\displaystyle f},g{\displaystyle g}supportedon only[0,∞){\displaystyle [0,\infty )}(i.e., zero for negative arguments), the integration limits can be truncated, resulting in:
For the multi-dimensional formulation of convolution, seedomain of definition(below).
A common engineering notational convention is:[2]
which has to be interpreted carefully to avoid confusion. For instance,f(t)∗g(t−t0){\displaystyle f(t)*g(t-t_{0})}is equivalent to(f∗g)(t−t0){\displaystyle (f*g)(t-t_{0})}, butf(t−t0)∗g(t−t0){\displaystyle f(t-t_{0})*g(t-t_{0})}is in fact equivalent to(f∗g)(t−2t0){\displaystyle (f*g)(t-2t_{0})}.[3]
Given two functionsf(t){\displaystyle f(t)}andg(t){\displaystyle g(t)}withbilateral Laplace transforms(two-sided Laplace transform)
and
respectively, the convolution operation(f∗g)(t){\displaystyle (f*g)(t)}can be defined as theinverse Laplace transformof the product ofF(s){\displaystyle F(s)}andG(s){\displaystyle G(s)}.[4][5]More precisely,
Lett=u+v{\displaystyle t=u+v}, then
Note thatF(s)⋅G(s){\displaystyle F(s)\cdot G(s)}is the bilateral Laplace transform of(f∗g)(t){\displaystyle (f*g)(t)}. A similar derivation can be done using theunilateral Laplace transform(one-sided Laplace transform).
The convolution operation also describes the output (in terms of the input) of an important class of operations known aslinear time-invariant(LTI). SeeLTI system theoryfor a derivation of convolution as the result of LTI constraints. In terms of theFourier transformsof the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as atransfer function). SeeConvolution theoremfor a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms.
The resultingwaveform(not shown here) is the convolution of functionsf{\displaystyle f}andg{\displaystyle g}.
Iff(t){\displaystyle f(t)}is aunit impulse, the result of this process is simplyg(t){\displaystyle g(t)}. Formally:
One of the earliest uses of the convolution integral appeared inD'Alembert's derivation ofTaylor's theoreminRecherches sur différents points importants du système du monde,published in 1754.[6]
Also, an expression of the type:
is used bySylvestre François Lacroixon page 505 of his book entitledTreatise on differences and series, which is the last of 3 volumes of the encyclopedic series:Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800.[7]Soon thereafter, convolution operations appear in the works ofPierre Simon Laplace,Jean-Baptiste Joseph Fourier,Siméon Denis Poisson, and others. The term itself did not come into wide use until the 1950s or 1960s. Prior to that it was sometimes known asFaltung(which meansfoldinginGerman),composition product,superposition integral, andCarson's integral.[8]Yet it appears as early as 1903, though the definition is rather unfamiliar in older uses.[9][10]
The operation:
is a particular case of composition products considered by the Italian mathematicianVito Volterrain 1913.[11]
When a functiongT{\displaystyle g_{T}}is periodic, with periodT{\displaystyle T}, then for functions,f{\displaystyle f}, such thatf∗gT{\displaystyle f*g_{T}}exists, the convolution is also periodic and identical to:
wheret0{\displaystyle t_{0}}is an arbitrary choice. The summation is called aperiodic summationof the functionf{\displaystyle f}.
WhengT{\displaystyle g_{T}}is a periodic summation of another function,g{\displaystyle g}, thenf∗gT{\displaystyle f*g_{T}}is known as acircularorcyclicconvolution off{\displaystyle f}andg{\displaystyle g}.
And if the periodic summation above is replaced byfT{\displaystyle f_{T}}, the operation is called aperiodicconvolution offT{\displaystyle f_{T}}andgT{\displaystyle g_{T}}.
For complex-valued functionsf{\displaystyle f}andg{\displaystyle g}defined on the setZ{\displaystyle \mathbb {Z} }of integers, thediscrete convolutionoff{\displaystyle f}andg{\displaystyle g}is given by:[12]
or equivalently (seecommutativity) by:
The convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers. When the sequences are the coefficients of twopolynomials, then the coefficients of theordinary product of the two polynomialsare the convolution of the original two sequences. This is known as theCauchy productof the coefficients of the sequences.
Thus whenghas finite support in the set{−M,−M+1,…,M−1,M}{\displaystyle \{-M,-M+1,\dots ,M-1,M\}}(representing, for instance, afinite impulse response), a finite summation may be used:[13]
When a functiongN{\displaystyle g_{_{N}}}is periodic, with periodN,{\displaystyle N,}then for functions,f,{\displaystyle f,}such thatf∗gN{\displaystyle f*g_{_{N}}}exists, the convolution is also periodic and identical to:
The summation onk{\displaystyle k}is called aperiodic summationof the functionf.{\displaystyle f.}
IfgN{\displaystyle g_{_{N}}}is a periodic summation of another function,g,{\displaystyle g,}thenf∗gN{\displaystyle f*g_{_{N}}}is known as acircular convolutionoff{\displaystyle f}andg.{\displaystyle g.}
When the non-zero durations of bothf{\displaystyle f}andg{\displaystyle g}are limited to the interval[0,N−1],{\displaystyle [0,N-1],}f∗gN{\displaystyle f*g_{_{N}}}reduces to these common forms:
The notationf∗Ng{\displaystyle f*_{N}g}forcyclic convolutiondenotes convolution over thecyclic groupofintegers moduloN.
Circular convolution arises most often in the context of fast convolution with afast Fourier transform(FFT) algorithm.
In many situations, discrete convolutions can be converted to circular convolutions so that fast transforms with a convolution property can be used to implement the computation. For example, convolution of digit sequences is the kernel operation inmultiplicationof multi-digit numbers, which can therefore be efficiently implemented with transform techniques (Knuth 1997, §4.3.3.C;von zur Gathen & Gerhard 2003, §8.2).
Eq.1requiresNarithmetic operations per output value andN2operations forNoutputs. That can be significantly reduced with any of several fast algorithms.Digital signal processingand other applications typically use fast convolution algorithms to reduce the cost of the convolution to O(NlogN) complexity.
The most common fast convolution algorithms usefast Fourier transform(FFT) algorithms via thecircular convolution theorem. Specifically, thecircular convolutionof two finite-length sequences is found by taking an FFT of each sequence, multiplying pointwise, and then performing an inverse FFT. Convolutions of the type defined above are then efficiently implemented using that technique in conjunction with zero-extension and/or discarding portions of the output. Other fast convolution algorithms, such as theSchönhage–Strassen algorithmor the Mersenne transform,[14]use fast Fourier transforms in otherrings. The Winograd method is used as an alternative to the FFT.[15]It significantly speeds up 1D,[16]2D,[17]and 3D[18]convolution.
If one sequence is much longer than the other, zero-extension of the shorter sequence and fast circular convolution is not the most computationally efficient method available.[19]Instead, decomposing the longer sequence into blocks and convolving each block allows for faster algorithms such as theoverlap–save methodandoverlap–add method.[20]A hybrid convolution method that combines block andFIRalgorithms allows for a zero input-output latency that is useful for real-time convolution computations.[21]
The convolution of two complex-valued functions onRdis itself a complex-valued function onRd, defined by:
and is well-defined only iffandgdecay sufficiently rapidly at infinity in order for the integral to exist. Conditions for the existence of the convolution may be tricky, since a blow-up ingat infinity can be easily offset by sufficiently rapid decay inf. The question of existence thus may involve different conditions onfandg:
Iffandgarecompactly supportedcontinuous functions, then their convolution exists, and is also compactly supported and continuous (Hörmander 1983, Chapter 1). More generally, if either function (sayf) is compactly supported and the other islocally integrable, then the convolutionf∗gis well-defined and continuous.
Convolution offandgis also well defined when both functions are locally square integrable onRand supported on an interval of the form[a, +∞)(or both supported on[−∞,a]).
The convolution offandgexists iffandgare bothLebesgue integrable functionsinL1(Rd), and in this casef∗gis also integrable (Stein & Weiss 1971, Theorem 1.3). This is a consequence ofTonelli's theorem. This is also true for functions inL1, under the discrete convolution, or more generally for theconvolution on any group.
Likewise, iff∈L1(Rd) andg∈Lp(Rd) where1 ≤p≤ ∞, thenf*g∈Lp(Rd), and
In the particular casep= 1, this shows thatL1is aBanach algebraunder the convolution (and equality of the two sides holds iffandgare non-negative almost everywhere).
More generally,Young's inequalityimplies that the convolution is a continuous bilinear map between suitableLpspaces. Specifically, if1 ≤p,q,r≤ ∞satisfy:
then
so that the convolution is a continuous bilinear mapping fromLp×LqtoLr.
The Young inequality for convolution is also true in other contexts (circle group, convolution onZ). The preceding inequality is not sharp on the real line: when1 <p,q,r< ∞, there exists a constantBp,q< 1such that:
The optimal value ofBp,qwas discovered in 1975[22]and independently in 1976,[23]seeBrascamp–Lieb inequality.
A stronger estimate is true provided1 <p,q,r< ∞:
where‖g‖q,w{\displaystyle \|g\|_{q,w}}is theweakLqnorm. Convolution also defines a bilinear continuous mapLp,w×Lq,w→Lr,w{\displaystyle L^{p,w}\times L^{q,w}\to L^{r,w}}for1<p,q,r<∞{\displaystyle 1<p,q,r<\infty }, owing to the weak Young inequality:[24]
In addition to compactly supported functions and integrable functions, functions that have sufficiently rapid decay at infinity can also be convolved. An important feature of the convolution is that iffandgboth decay rapidly, thenf∗galso decays rapidly. In particular, iffandgarerapidly decreasing functions, then so is the convolutionf∗g. Combined with the fact that convolution commutes with differentiation (see#Properties), it follows that the class ofSchwartz functionsis closed under convolution (Stein & Weiss 1971, Theorem 3.3).
Iffis a smooth function that iscompactly supportedandgis a distribution, thenf∗gis a smooth function defined by
More generally, it is possible to extend the definition of the convolution in a unique way withφ{\displaystyle \varphi }the same asfabove, so that the associative law
remains valid in the case wherefis a distribution, andga compactly supported distribution (Hörmander 1983, §4.2).
The convolution of any twoBorel measuresμandνofbounded variationis the measureμ∗ν{\displaystyle \mu *\nu }defined by (Rudin 1962)
In particular,
whereA⊂Rd{\displaystyle A\subset \mathbf {R} ^{d}}is a measurable set and1A{\displaystyle 1_{A}}is theindicator functionofA{\displaystyle A}.
This agrees with the convolution defined above when μ and ν are regarded as distributions, as well as the convolution of L1functions when μ and ν are absolutely continuous with respect to the Lebesgue measure.
The convolution of measures also satisfies the following version of Young's inequality
where the norm is thetotal variationof a measure. Because the space of measures of bounded variation is aBanach space, convolution of measures can be treated with standard methods offunctional analysisthat may not apply for the convolution of distributions.
The convolution defines a product on thelinear spaceof integrable functions. This product satisfies the following algebraic properties, which formally mean that the space of integrable functions with the product given by convolution is a commutativeassociative algebrawithoutidentity(Strichartz 1994, §3.3). Other linear spaces of functions, such as the space of continuous functions of compact support, areclosedunder the convolution, and so also form commutative associative algebras.
Proof (usingconvolution theorem):
q(t)⟺FQ(f)=R(f)S(f){\displaystyle q(t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ Q(f)=R(f)S(f)}
q(−t)⟺FQ(−f)=R(−f)S(−f){\displaystyle q(-t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ Q(-f)=R(-f)S(-f)}
q(−t)=F−1{R(−f)S(−f)}=F−1{R(−f)}∗F−1{S(−f)}=r(−t)∗s(−t){\displaystyle {\begin{aligned}q(-t)&={\mathcal {F}}^{-1}{\bigg \{}R(-f)S(-f){\bigg \}}\\&={\mathcal {F}}^{-1}{\bigg \{}R(-f){\bigg \}}*{\mathcal {F}}^{-1}{\bigg \{}S(-f){\bigg \}}\\&=r(-t)*s(-t)\end{aligned}}}
Iffandgare integrable functions, then the integral of their convolution on the whole space is simply obtained as the product of their integrals:[25]
This follows fromFubini's theorem. The same result holds iffandgare only assumed to be nonnegative measurable functions, byTonelli's theorem.
In the one-variable case,
whereddx{\displaystyle {\frac {d}{dx}}}is thederivative. More generally, in the case of functions of several variables, an analogous formula holds with thepartial derivative:
A particular consequence of this is that the convolution can be viewed as a "smoothing" operation: the convolution offandgis differentiable as many times asfandgare in total.
These identities hold for example under the condition thatfandgare absolutely integrable and at least one of them has an absolutely integrable (L1) weak derivative, as a consequence ofYoung's convolution inequality. For instance, whenfis continuously differentiable with compact support, andgis an arbitrary locally integrable function,
These identities also hold much more broadly in the sense of tempered distributions if one offorgis arapidly decreasing tempered distribution, a
compactly supported tempered distribution or a Schwartz function and the other is a tempered distribution. On the other hand, two positive integrable and infinitely differentiable functions may have a nowhere continuous convolution.
In the discrete case, thedifference operatorDf(n) =f(n+ 1) −f(n) satisfies an analogous relationship:
Theconvolution theoremstates that[26]
whereF{f}{\displaystyle {\mathcal {F}}\{f\}}denotes theFourier transformoff{\displaystyle f}.
Versions of this theorem also hold for theLaplace transform,two-sided Laplace transform,Z-transformandMellin transform.
IfW{\displaystyle {\mathcal {W}}}is theFourier transform matrix, then
where∙{\displaystyle \bullet }isface-splitting product,[27][28][29][30][31]⊗{\displaystyle \otimes }denotesKronecker product,∘{\displaystyle \circ }denotesHadamard product(this result is an evolving ofcount sketchproperties[32]).
This can be generalized for appropriate matricesA,B{\displaystyle \mathbf {A} ,\mathbf {B} }:
from the properties of theface-splitting product.
The convolution commutes with translations, meaning that
where τxf is the translation of the functionfbyxdefined by
Iffis aSchwartz function, thenτxfis the convolution with a translated Dirac delta functionτxf=f∗τxδ. So translation invariance of the convolution of Schwartz functions is a consequence of the associativity of convolution.
Furthermore, under certain conditions, convolution is the most general translation invariant operation. Informally speaking, the following holds
Thus some translation invariant operations can be represented as convolution. Convolutions play an important role in the study oftime-invariant systems, and especiallyLTI system theory. The representing functiongSis theimpulse responseof the transformationS.
A more precise version of the theorem quoted above requires specifying the class of functions on which the convolution is defined, and also requires assuming in addition thatSmust be acontinuous linear operatorwith respect to the appropriatetopology. It is known, for instance, that every continuous translation invariant continuous linear operator onL1is the convolution with a finiteBorel measure. More generally, every continuous translation invariant continuous linear operator onLpfor 1 ≤p< ∞ is the convolution with atempered distributionwhoseFourier transformis bounded. To wit, they are all given by boundedFourier multipliers.
IfGis a suitablegroupendowed with ameasureλ, and iffandgare real or complex valuedintegrablefunctions onG, then we can define their convolution by
It is not commutative in general. In typical cases of interestGis alocally compactHausdorfftopological groupand λ is a (left-)Haar measure. In that case, unlessGisunimodular, the convolution defined in this way is not the same as∫f(xy−1)g(y)dλ(y){\textstyle \int f\left(xy^{-1}\right)g(y)\,d\lambda (y)}. The preference of one over the other is made so that convolution with a fixed functiongcommutes with left translation in the group:
Furthermore, the convention is also required for consistency with the definition of the convolution of measures given below. However, with a right instead of a left Haar measure, the latter integral is preferred over the former.
Onlocally compact abelian groups, a version of theconvolution theoremholds: the Fourier transform of a convolution is the pointwise product of the Fourier transforms. Thecircle groupTwith the Lebesgue measure is an immediate example. For a fixedginL1(T), we have the following familiar operator acting on theHilbert spaceL2(T):
The operatorTiscompact. A direct calculation shows that its adjointT*is convolution with
By the commutativity property cited above,Tisnormal:T*T=TT* . Also,Tcommutes with the translation operators. Consider the familySof operators consisting of all such convolutions and the translation operators. ThenSis a commuting family of normal operators. According tospectral theory, there exists an orthonormal basis {hk} that simultaneously diagonalizesS. This characterizes convolutions on the circle. Specifically, we have
which are precisely thecharactersofT. Each convolution is a compactmultiplication operatorin this basis. This can be viewed as a version of the convolution theorem discussed above.
A discrete example is a finitecyclic groupof ordern. Convolution operators are here represented bycirculant matrices, and can be diagonalized by thediscrete Fourier transform.
A similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finite-dimensionalunitary representationsform an orthonormal basis inL2by thePeter–Weyl theorem, and an analog of the convolution theorem continues to hold, along with many other aspects ofharmonic analysisthat depend on the Fourier transform.
LetGbe a (multiplicatively written) topological group.
If μ and ν areRadon measuresonG, then their convolutionμ∗νis defined as thepushforward measureof thegroup actionand can be written as[33]
for each measurable subsetEofG. The convolution is also a Radon measure, whosetotal variationsatisfies
In the case whenGislocally compactwith (left-)Haar measureλ, and μ and ν areabsolutely continuouswith respect to a λ,so that each has a density function, then the convolution μ∗ν is also absolutely continuous, and its density function is just the convolution of the two separate density functions. In fact, ifeithermeasure is absolutely continuous with respect to the Haar measure, then so is their convolution.[34]
If μ and ν areprobability measureson the topological group(R,+),then the convolutionμ∗νis theprobability distributionof the sumX+Yof twoindependentrandom variablesXandYwhose respective distributions are μ and ν.
Inconvex analysis, theinfimal convolutionof proper (not identically+∞{\displaystyle +\infty })convex functionsf1,…,fm{\displaystyle f_{1},\dots ,f_{m}}onRn{\displaystyle \mathbb {R} ^{n}}is defined by:[35](f1∗⋯∗fm)(x)=infx{f1(x1)+⋯+fm(xm)|x1+⋯+xm=x}.{\displaystyle (f_{1}*\cdots *f_{m})(x)=\inf _{x}\{f_{1}(x_{1})+\cdots +f_{m}(x_{m})|x_{1}+\cdots +x_{m}=x\}.}It can be shown that the infimal convolution of convex functions is convex. Furthermore, it satisfies an identity analogous to that of the Fourier transform of a traditional convolution, with the role of the Fourier transform is played instead by theLegendre transform:φ∗(x)=supy(x⋅y−φ(y)).{\displaystyle \varphi ^{*}(x)=\sup _{y}(x\cdot y-\varphi (y)).}We have:(f1∗⋯∗fm)∗(x)=f1∗(x)+⋯+fm∗(x).{\displaystyle (f_{1}*\cdots *f_{m})^{*}(x)=f_{1}^{*}(x)+\cdots +f_{m}^{*}(x).}
Let (X, Δ, ∇,ε,η) be abialgebrawith comultiplication Δ, multiplication ∇, unit η, and counitε. The convolution is a product defined on theendomorphism algebraEnd(X) as follows. Letφ,ψ∈ End(X), that is,φ,ψ:X→Xare functions that respect all algebraic structure ofX, then the convolutionφ∗ψis defined as the composition
The convolution appears notably in the definition ofHopf algebras(Kassel 1995, §III.3). A bialgebra is a Hopf algebra if and only if it has an antipode: an endomorphismSsuch that
Convolution and related operations are found in many applications in science, engineering and mathematics.
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Incomputer science,arrayis adata typethat represents a collection ofelements(valuesorvariables), each selected by one or more indices (identifying keys) that can be computed atrun timeduring program execution. Such a collection is usually called anarray variableorarray value.[1]By analogy with the mathematical conceptsvectorandmatrix, array types with one and two indices are often calledvector typeandmatrix type, respectively. More generally, a multidimensional array type can be called atensor type, by analogy with the mathematical concept,tensor.[2]
Language support for array types may include certainbuilt-inarray data types, some syntactic constructions (array type constructors) that theprogrammermay use to define such types and declare array variables, and special notation for indexing array elements.[1]For example, in thePascal programming language, the declarationtypeMyTable=array[1..4,1..2]ofinteger, defines a new array data type calledMyTable. The declarationvar A: MyTablethen defines a variableAof that type, which is an aggregate of eight elements, each being an integer variable identified by two indices. In the Pascal program, those elements are denotedA[1,1],A[1,2],A[2,1], …,A[4,2].[3]Special array types are often defined by the language's standardlibraries.
Dynamic listsare also more common and easier to implement[dubious–discuss]thandynamic arrays. Array types are distinguished fromrecordtypes mainly because they allow the element indices to be computed atrun time, as in the PascalassignmentA[I,J] := A[N-I,2*J]. Among other things, this feature allows a single iterativestatementto process arbitrarily many elements of an array variable.
In more theoretical contexts, especially intype theoryand in the description of abstractalgorithms, the terms "array" and "array type" sometimes refer to anabstract data type(ADT) also calledabstract arrayor may refer to anassociative array, amathematicalmodel with the basic operations and behavior of a typical array type in most languages – basically, a collection of elements that are selected by indices computed at run-time.
Depending on the language, array types may overlap (or be identified with) other data types that describe aggregates of values, such aslistsandstrings. Array types are often implemented byarray data structures, but sometimes by other means, such ashash tables,linked lists, orsearch trees.
Heinz Rutishauser's programming language Superplan (1949–1951) included multi-dimensional arrays. However, although Rutishauser described how a compiler for his language should be built, did not implement one.
Assembly languages and low-level languages like BCPL[4]generally have no syntactic support for arrays.
Because of the importance of array structures for efficient computation, the earliest high-level programming languages, includingFORTRAN(1957),COBOL(1960), andAlgol 60(1960), provided support for multi-dimensional arrays.
An array data structure can be mathematically modeled as anabstract data structure(anabstract array) with two operations
These operations are required to satisfy theaxioms[5]
for any array stateA, any valueV, and any tuplesI,Jfor which the operations are defined.
The first axiom means that each element behaves like a variable. The second axiom means that elements with distinct indices behave asdisjointvariables, so that storing a value in one element does not affect the value of any other element.
These axioms do not place any constraints on the set of valid index tuplesI, therefore this abstract model can be used fortriangular matricesand other oddly-shaped arrays.
In order to effectively implement variables of such types asarray structures(with indexing done bypointer arithmetic), many languages restrict the indices tointegerdata types[6][7](or other types that can be interpreted as integers, such asbytesandenumerated types), and require that all elements have the same data type and storage size. Most of those languages also restrict each index to a finiteintervalof integers, that remains fixed throughout the lifetime of the array variable. In somecompiledlanguages, in fact, the index ranges may have to be known atcompile time.
On the other hand, some programming languages provide more liberal array types, that allow indexing by arbitrary values, such asfloating-point numbers,strings,objects,references, etc.. Such index values cannot be restricted to an interval, much less a fixed interval. So, these languages usually allow arbitrary new elements to be created at any time. This choice precludes the implementation of array types as array data structures. That is, those languages use array-like syntax to implement a more generalassociative arraysemantics, and must therefore be implemented by ahash tableor some othersearch data structure.
The number of indices needed to specify an element is called thedimension,dimensionality, orrankof the array type. (This nomenclature conflicts with the concept of dimension in linear algebra, which expresses theshape of a matrix. Thus, an array of numbers with 5 rows and 4 columns, hence 20 elements, is said to have dimension 2 in computing contexts, but represents a matrix that is said to be 4×5-dimensional. Also, the computer science meaning of "rank" conflicts with the notion oftensor rank, which is a generalization of the linear algebra concept ofrank of a matrix.)
Many languages support only one-dimensional arrays. In those languages, a multi-dimensional array is typically represented by anIliffe vector, a one-dimensional array ofreferencesto arrays of one dimension less. A two-dimensional array, in particular, would be implemented as a vector of pointers to its rows. Thus an element in rowiand columnjof an arrayAwould be accessed by double indexing (A[i][j]in typical notation). This way of emulating multi-dimensional arrays allows the creation ofjagged arrays, where each row may have a different size – or, in general, where the valid range of each index depends on the values of all preceding indices.
This representation for multi-dimensional arrays is quite prevalent in C and C++ software. However, C and C++ will use a linear indexing formula for multi-dimensional arrays that are declared with compile time constant size, e.g. byintA[10][20]orintA[m][n], instead of the traditionalint**A.[8]
The C99 standard introduced Variable Length Array types that let define array types with dimensions computed in run time. The dynamic 4D array can be constructed using a pointer to 4d array, e.g.int(*arr)[t][u][v][w]=malloc(sizeof*arr);. The individual elements are accessed by first de-referencing an array pointer followed by indexing, e.g.(*arr)[i][j][k][l]. Alternatively, n-d arrays can be declared as pointers to its first element which is a (n-1) dimensional array, e.g.int(*arr)[u][v][w]=malloc(t*sizeof*arr);and accessed using more idiomatic syntax, e.g.arr[i][j][k][l].
Most programming languages that support arrays support thestoreandselectoperations, and have special syntax for indexing. Early languages used parentheses, e.g.A(i,j), as in FORTRAN; others choose square brackets, e.g.A[i,j]orA[i][j], as in Algol 60 and Pascal (to distinguish from the use of parentheses forfunction calls).
Array data types are most often implemented as array structures: with the indices restricted to integer (or totally ordered) values, index ranges fixed at array creation time, and multilinear element addressing. This was the case in most"third generation"languages, and is still the case of mostsystems programming languagessuch asAda,C, andC++. In some languages, however, array data types have the semantics of associative arrays, with indices of arbitrary type and dynamic element creation. This is the case in somescripting languagessuch asAwkandLua, and of some array types provided by standardC++libraries.
Some languages (like Pascal and Modula) performbounds checkingon every access, raising anexceptionor aborting the program when any index is out of its valid range. Compilers may allow these checks to be turned off to trade safety for speed. Other languages (like FORTRAN and C) trust the programmer and perform no checks. Good compilers may also analyze the program to determine the range of possible values that the index may have, and this analysis may lead tobounds-checking elimination.
Some languages, such as C, provide onlyzero-basedarray types, for which the minimum valid value for any index is 0.[9]This choice is convenient for array implementation and address computations. With a language such as C, a pointer to the interior of any array can be defined that will symbolically act as a pseudo-array that accommodates negative indices. This works only because C does not check an index against bounds when used.
Other languages provide onlyone-basedarray types, where each index starts at 1; this is the traditional convention in mathematics for matrices and mathematicalsequences. A few languages, such as Pascal and Lua, supportn-basedarray types, whose minimum legal indices are chosen by the programmer. The relative merits of each choice have been the subject of heated debate. Zero-based indexing can avoidoff-by-oneorfencepost errors.[10]
The relation between numbers appearing in an array declaration and the index of that array's last element also varies by language. In many languages (such as C), one should specify the number of elements contained in the array; whereas in others (such as Pascal andVisual Basic .NET) one should specify the numeric value of the index of the last element. Needless to say, this distinction is immaterial in languages where the indices start at 1, such asLua.
Some programming languages supportarray programming, where operations and functions defined for certain data types are implicitly extended to arrays of elements of those types. Thus one can writeA+Bto add corresponding elements of two arraysAandB. Usually these languages provide both theelement-by-element multiplicationand the standardmatrix productoflinear algebra, and which of these is represented by the*operator varies by language.
Languages providing array programming capabilities have proliferated since the innovations in this area ofAPL. These are core capabilities ofdomain-specific languagessuch asGAUSS,IDL,Matlab, andMathematica. They are a core facility in newer languages, such asJuliaand recent versions ofFortran. These capabilities are also provided via standard extension libraries for other general purpose programming languages (such as the widely usedNumPylibrary forPython).
Many languages provide a built-instringdata type, with specialized notation ("string literals") to build values of that type. In some languages (such as C), a string is just an array of characters, or is handled in much the same way. Other languages, likePascal, may provide vastly different operations for strings and arrays.
Some programming languages provide operations that return the size (number of elements) of a vector, or, more generally, range of each index of an array. InCandC++arrays do not support thesizefunction, so programmers often have to declare separate variable to hold the size, and pass it to procedures as a separate parameter.
Elements of a newly created array may have undefined values (as in C), or may be defined to have a specific "default" value such as 0 or anull pointer(as in Java).
InC++astd::vectorobject supports thestore,select, andappendoperations with the performance characteristics discussed above. Vectors can be queried for their size and can be resized. Slower operations like inserting an element in the middle are also supported.
Anarray slicingoperation takes a subset of the elements of an array-typed entity (value or variable) and then assembles them as another array-typed entity, possibly with other indices. If array types are implemented as array structures, many useful slicing operations (such as selecting a sub-array, swapping indices, or reversing the direction of the indices) can be performed very efficiently by manipulating thedope vectorof the structure. The possible slicings depend on the implementation details: for example,Fortranallows slicing off one column of a matrix variable, but not a row, and treat it as a vector.
On the other hand, other slicing operations are possible when array types are implemented in other ways.
Some languages allowdynamic arrays(also called resizable, growable, or extensible): array variables whose index ranges may be expanded at any time after creation, without changing the values of its current elements.
For one-dimensional arrays, this facility may be provided as an operationappend(A,x)that increases the size of the arrayAby one and then sets the value of the last element tox. Other array types (such as Pascal strings) provide a concatenation operator, which can be used together with slicing to achieve that effect and more. In some languages, assigning a value to an element of an array automatically extends the array, if necessary, to include that element. In other array types, a slice can be replaced by an array of different size, with subsequent elements being renumbered accordingly – as in Python's list assignmentA[5:5] = [10,20,30], that inserts three new elements (10, 20, and 30) before element "A[5]". Resizable arrays are conceptually similar tolists, and the two concepts are synonymous in some languages.
An extensible array can be implemented as a fixed-size array, with a counter that records how many elements are actually in use. Theappendoperation merely increments the counter; until the whole array is used, when theappendoperation may be defined to fail. This is an implementation of adynamic arraywith a fixed capacity, as in thestringtype of Pascal. Alternatively, theappendoperation may re-allocate the underlying array with a larger size, and copy the old elements to the new area.
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Abiometric passport(also known as anelectronic passport,e-passportor adigital passport) is apassportthat has an embedded electronicmicroprocessorchip, which containsbiometricinformation that can be used to authenticate the identity of the passport holder. It usescontactless smart cardtechnology, including a microprocessor chip (computer chip) and antenna (for both power to the chip and communication) embedded in the front or back cover, or centre page, of the passport. The passport's critical information is printed on the data page of the passport, repeated on themachine readable linesand stored in the chip.Public key infrastructure(PKI) is used to authenticate the data stored electronically in the passport chip, making it expensive and difficult to forge when all security mechanisms are fully and correctly implemented.
Most countries are issuing biometric passports to their citizens.Malaysiawas the first country to issuebiometric passportsin 1998.[1]By the end of 2008, 60 countries were issuing such passports,[2]which increased to over 150 by mid-2019.[3]
The currently standardised biometrics used for this type of identification system arefacial recognition,fingerprint recognition, andiris recognition. These were adopted after assessment of several different kinds of biometrics includingretinal scan. Document and chip characteristics are documented in theInternational Civil Aviation Organization's (ICAO) Doc 9303 (ICAO 9303).[4]The ICAO defines the biometric file formats and communication protocols to be used in passports. Only the digital image (usually inJPEGorJPEG 2000format) of each biometric feature is actually stored in the chip. The comparison of biometric features is performed outside the passport chip by electronic border control systems (e-borders). To store biometric data on the contactless chip, it includes a minimum of 32 kilobytes ofEEPROMstorage memory, and runs on an interface in accordance with theISO/IEC 14443international standard, amongst others. These standards intend interoperability between different countries and different manufacturers of passport books.
Somenational identity cards, such as those fromAlbania,Brazil, theNetherlands, andSaudi Arabiaare fully ICAO 9303 compliant biometrictravel documents. However others, such as theUnited States passport card, are not.[5]
Biometric passports have protection mechanisms to avoid and/or detect attacks:
To assure interoperability and functionality of the security mechanisms listed above, ICAO andGermanFederal Office for Information Security(BSI) have specified several test cases. These test specifications are updated with every new protocol and are covering details starting from the paper used and ending in the chip that is included.[9]
Since the introduction of biometric passports, several attacks have been presented and demonstrated.
Privacyproponents in many countries question and protest the lack of information about exactly what the passports' chip will contain, and whether they affectcivil liberties. The main problem they point out is that data on the passports can be transferred with wirelessRFIDtechnology, which can become a major vulnerability. Although this could allowID-check computers to obtain a person's information without a physical connection, it may also allow anyone with the necessary equipment to perform the same task. If the personal information and passport numbers on the chip are notencrypted, the information might wind up in the wrong hands.
On 15 December 2006, theBBCpublished an article[26]on the British ePassport, citing the above stories and adding that:
and adding that the Future of Identity in the Information Society (FIDIS) network's research team (a body of IT security experts funded by the European Union) has "also come out against the ePassport scheme... [stating that] European governments have forced a document on its people that dramatically decreases security and increases the risk of identity theft."[27]
Most security measures are designed against untrusted citizens (the "provers"), but the scientific security community recently also addressed the threats from untrustworthy verifiers, such as corrupt governmental organizations, or nations using poorly implemented, unsecure electronic systems.[28]New cryptographic solutions such asprivate biometricsare being proposed to mitigate threats of mass theft of identity. These are under scientific study, but not yet implemented in biometric passports.
It was planned that, except for Denmark andIreland,EU passportswould have digital imaging andfingerprintscan biometrics placed on their RFID chips.[116]This combination ofbiometricsaims to create an unrivaled level of security and protection against fraudulent identification papers[vague]. Technical specifications for the new passports have been established by the European Commission.[117]The specifications are binding for theSchengen agreementparties, i.e. the EU countries, except Ireland, and the fourEuropean Free Trade Associationcountries—Iceland, Liechtenstein,[118][119]Norway and Switzerland.[120]These countries are obliged to implement machine readable facial images in the passports by 28 August 2006, and fingerprints by 26 June 2009.[121]TheEuropean Data Protection Supervisorhas stated that the current legal framework fails to "address all the possible and relevant issues triggered by the inherent imperfections of biometric systems".[122]
Irish biometric passports only used a digital image and not fingerprinting. German passports printed after 1 November 2007 contain two fingerprints, one from each hand, in addition to a digital photograph. Romanian passports will also contain two fingerprints, one from each hand. The Netherlands also takes fingerprints and was[123]the only EU member that had plans to store these fingerprints centrally.[124]According to EU requirements, only nations that are signatories to theSchengen acquisare required to add fingerprint biometrics.[125]
In the EU nations, passport prices will be:
In the EFTA, passport prices will be:
The ICAO standard sets a 35x45 mm image with adequate resolution with the following requirements:
Though some countries like USA use a 2x2 inch photo format (51x51 mm), they usually crop it to be closer to 35:45 in ratio when issuing a passport.
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In the mathematicaltheory of probability, theIonescu-Tulcea theorem, sometimes called theIonesco Tulcea extension theorem, deals with the existence ofprobability measuresfor probabilistic events consisting of a countably infinite number of individual probabilistic events. In particular, the individual events may beindependentor dependent with respect to each other. Thus, the statement goes beyond the mere existence of countableproduct measures. The theorem was proved byCassius Ionescu-Tulceain 1949.[1][2]
Suppose that(Ω0,A0,P0){\displaystyle (\Omega _{0},{\mathcal {A}}_{0},P_{0})}is aprobability spaceand(Ωi,Ai){\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}fori∈N{\displaystyle i\in \mathbb {N} }is a sequence ofmeasurable spaces. For eachi∈N{\displaystyle i\in \mathbb {N} }let
be theMarkov kernelderived from(Ωi−1,Ai−1){\displaystyle (\Omega ^{i-1},{\mathcal {A}}^{i-1})}and(Ωi,Ai),{\displaystyle (\Omega _{i},{\mathcal {A}}_{i}),}, where
Then there exists a sequence of probability measures
and there exists a uniquely defined probability measureP{\displaystyle P}on(∏k=0∞Ωk,⨂k=0∞Ak){\displaystyle \left(\prod _{k=0}^{\infty }\Omega _{k},\bigotimes _{k=0}^{\infty }{\mathcal {A}}_{k}\right)}, so that
is satisfied for eachA∈Ai{\displaystyle A\in {\mathcal {A}}^{i}}andi∈N{\displaystyle i\in \mathbb {N} }. (The measureP{\displaystyle P}hasconditional probabilitiesequal to the stochastic kernels.)[3]
The construction used in the proof of the Ionescu-Tulcea theorem is often used in the theory ofMarkov decision processes, and, in particular, the theory ofMarkov chains.[3]
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Inalgebra, afieldkisperfectif any one of the following equivalent conditions holds:
Otherwise,kis calledimperfect.
In particular, all fields of characteristic zero and allfinite fieldsare perfect.
Perfect fields are significant becauseGalois theoryover these fields becomes simpler, since the general Galois assumption of field extensions being separable is automatically satisfied over these fields (see third condition above).
Another important property of perfect fields is that they admitWitt vectors.
More generally, aringof characteristicp(paprime) is calledperfectif theFrobenius endomorphismis anautomorphism.[1](When restricted tointegral domains, this is equivalent to the above condition "every element ofkis apth power".)
Examples of perfect fields are:
Most fields that are encountered in practice are perfect. The imperfect case arises mainly inalgebraic geometryin characteristicp> 0. Every imperfect field is necessarilytranscendentalover itsprime subfield(the minimal subfield), because the latter is perfect.
An example of an imperfect field is the fieldFp(x){\displaystyle \mathbf {F} _{p}(x)}of rational polynomials in an unknown elementx{\displaystyle x}. This can be seen from the fact that the Frobenius endomorphism sendsx↦xp{\displaystyle x\mapsto x^{p}}and therefore is not surjective. Equivalently, one can show that the polynomialf(X)=Xp−x{\displaystyle f(X)=X^{p}-x}, which is an element of(Fp(x))[X]{\displaystyle (\mathbf {F} _{p}(x))[X]}, is irreducible but inseparable.
This field embeds into the perfect field
called itsperfection. Imperfect fields cause technical difficulties because irreducible polynomials can become reducible in the algebraic closure of the base field. For example,[4]considerf(x,y)=xp+ayp∈k[x,y]{\displaystyle f(x,y)=x^{p}+ay^{p}\in k[x,y]}fork{\displaystyle k}an imperfect field of characteristicp{\displaystyle p}andanot ap-th power ink. Then in its algebraic closurekalg[x,y]{\displaystyle k^{\operatorname {alg} }[x,y]}, the following equality holds:
wherebp=aand suchbexists in this algebraic closure. Geometrically, this means thatf{\displaystyle f}does not define anaffineplane curveink[x,y]{\displaystyle k[x,y]}.
Anyfinitely generated field extensionKover a perfect fieldkis separably generated, i.e. admits a separatingtranscendence base, that is, a transcendence base Γ such thatKis separably algebraic overk(Γ).[5]
One of the equivalent conditions says that, in characteristicp, a field adjoined with allpr-th roots (r≥ 1) is perfect; it is called theperfect closureofkand usually denoted bykp−∞{\displaystyle k^{p^{-\infty }}}.
The perfect closure can be used in a test for separability. More precisely, a commutativek-algebraAis separable if and only ifA⊗kkp−∞{\displaystyle A\otimes _{k}k^{p^{-\infty }}}is reduced.[6]
In terms ofuniversal properties, theperfect closureof a ringAof characteristicpis a perfect ringApof characteristicptogether with aring homomorphismu:A→Apsuch that for any other perfect ringBof characteristicpwith a homomorphismv:A→Bthere is a unique homomorphismf:Ap→Bsuch thatvfactors throughu(i.e.v=fu). The perfect closure always exists; the proof involves "adjoiningp-th roots of elements ofA", similar to the case of fields.[7]
Theperfectionof a ringAof characteristicpis the dual notion (though this term is sometimes used for the perfect closure). In other words, the perfectionR(A) ofAis a perfect ring of characteristicptogether with a mapθ:R(A) →Asuch that for any perfect ringBof characteristicpequipped with a mapφ:B→A, there is a unique mapf:B→R(A)such thatφfactors throughθ(i.e.φ=θf). The perfection ofAmay be constructed as follows. Consider theprojective system
where the transition maps are the Frobenius endomorphism. Theinverse limitof this system isR(A) and consists of sequences (x0,x1, ... ) of elements ofAsuch thatxi+1p=xi{\displaystyle x_{i+1}^{p}=x_{i}}for alli. The mapθ:R(A) →Asends (xi) tox0.[8]
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Inmathematical statistics, theKullback–Leibler(KL)divergence(also calledrelative entropyandI-divergence[1]), denotedDKL(P∥Q){\displaystyle D_{\text{KL}}(P\parallel Q)}, is a type ofstatistical distance: a measure of how much a modelprobability distributionQis different from a true probability distributionP.[2][3]Mathematically, it is defined as
DKL(P∥Q)=∑x∈XP(x)logP(x)Q(x).{\displaystyle D_{\text{KL}}(P\parallel Q)=\sum _{x\in {\mathcal {X}}}P(x)\,\log {\frac {P(x)}{Q(x)}}.}
A simpleinterpretationof the KL divergence ofPfromQis theexpectedexcesssurprisefrom usingQas a model instead ofPwhen the actual distribution isP. While it is a measure of how different two distributions are and is thus a distance in some sense, it is not actually ametric, which is the most familiar and formal type of distance. In particular, it is not symmetric in the two distributions (in contrast tovariation of information), and does not satisfy thetriangle inequality. Instead, in terms ofinformation geometry, it is a type ofdivergence,[4]a generalization ofsquared distance, and for certain classes of distributions (notably anexponential family), it satisfies a generalizedPythagorean theorem(which applies to squared distances).[5]
Relative entropy is always a non-negativereal number, with value 0 if and only if the two distributions in question are identical. It has diverse applications, both theoretical, such as characterizing the relative(Shannon) entropyin information systems, randomness in continuoustime-series, and information gain when comparing statistical models ofinference; and practical, such as applied statistics,fluid mechanics,neuroscience,bioinformatics, andmachine learning.
Consider two probability distributionsPandQ. Usually,Prepresents the data, the observations, or a measured probability distribution. DistributionQrepresents instead a theory, a model, a description or an approximation ofP. The Kullback–Leibler divergenceDKL(P∥Q){\displaystyle D_{\text{KL}}(P\parallel Q)}is then interpreted as the average difference of the number of bits required for encoding samples ofPusing a code optimized forQrather than one optimized forP. Note that the roles ofPandQcan be reversed in some situations where that is easier to compute, such as with theexpectation–maximization algorithm (EM)andevidence lower bound (ELBO)computations.
The relative entropy was introduced bySolomon KullbackandRichard LeiblerinKullback & Leibler (1951)as "the mean information for discrimination betweenH1{\displaystyle H_{1}}andH2{\displaystyle H_{2}}per observation fromμ1{\displaystyle \mu _{1}}",[6]where one is comparing two probability measuresμ1,μ2{\displaystyle \mu _{1},\mu _{2}}, andH1,H2{\displaystyle H_{1},H_{2}}are the hypotheses that one is selecting from measureμ1,μ2{\displaystyle \mu _{1},\mu _{2}}(respectively). They denoted this byI(1:2){\displaystyle I(1:2)}, and defined the "'divergence' betweenμ1{\displaystyle \mu _{1}}andμ2{\displaystyle \mu _{2}}" as the symmetrized quantityJ(1,2)=I(1:2)+I(2:1){\displaystyle J(1,2)=I(1:2)+I(2:1)}, which had already been defined and used byHarold Jeffreysin 1948.[7]InKullback (1959), the symmetrized form is again referred to as the "divergence", and the relative entropies in each direction are referred to as a "directed divergences" between two distributions;[8]Kullback preferred the termdiscrimination information.[9]The term "divergence" is in contrast to a distance (metric), since the symmetrized divergence does not satisfy the triangle inequality.[10]Numerous references to earlier uses of the symmetrized divergence and to otherstatistical distancesare given inKullback (1959, pp. 6–7, §1.3 Divergence). The asymmetric "directed divergence" has come to be known as the Kullback–Leibler divergence, while the symmetrized "divergence" is now referred to as theJeffreys divergence.
Fordiscrete probability distributionsPandQdefined on the samesample space,X{\displaystyle {\mathcal {X}}},the relative entropy fromQtoPis defined[11]to be
DKL(P∥Q)=∑x∈XP(x)logP(x)Q(x),{\displaystyle D_{\text{KL}}(P\parallel Q)=\sum _{x\in {\mathcal {X}}}P(x)\,\log {\frac {P(x)}{Q(x)}}\,,}
which is equivalent to
DKL(P∥Q)=−∑x∈XP(x)logQ(x)P(x).{\displaystyle D_{\text{KL}}(P\parallel Q)=-\sum _{x\in {\mathcal {X}}}P(x)\,\log {\frac {Q(x)}{P(x)}}\,.}
In other words, it is theexpectationof the logarithmic difference between the probabilitiesPandQ, where the expectation is taken using the probabilitiesP.
Relative entropy is only defined in this way if, for allx,Q(x)=0{\displaystyle Q(x)=0}impliesP(x)=0{\displaystyle P(x)=0}(absolute continuity). Otherwise, it is often defined as+∞{\displaystyle +\infty },[1]but the value+∞{\displaystyle \ +\infty \ }is possible even ifQ(x)≠0{\displaystyle Q(x)\neq 0}everywhere,[12][13]provided thatX{\displaystyle {\mathcal {X}}}is infinite in extent. Analogous comments apply to the continuous and general measure cases defined below.
WheneverP(x){\displaystyle P(x)}is zero the contribution of the corresponding term is interpreted as zero because
limx→0+xlog(x)=0.{\displaystyle \lim _{x\to 0^{+}}x\,\log(x)=0\,.}
For distributionsPandQof acontinuous random variable, relative entropy is defined to be the integral[14]
DKL(P∥Q)=∫−∞∞p(x)logp(x)q(x)dx,{\displaystyle D_{\text{KL}}(P\parallel Q)=\int _{-\infty }^{\infty }p(x)\,\log {\frac {p(x)}{q(x)}}\,dx\,,}
wherepandqdenote theprobability densitiesofPandQ.
More generally, ifPandQare probabilitymeasureson ameasurable spaceX,{\displaystyle {\mathcal {X}}\,,}andPisabsolutely continuouswith respect toQ, then the relative entropy fromQtoPis defined as
DKL(P∥Q)=∫x∈XlogP(dx)Q(dx)P(dx),{\displaystyle D_{\text{KL}}(P\parallel Q)=\int _{x\in {\mathcal {X}}}\log {\frac {P(dx)}{Q(dx)}}\,P(dx)\,,}
whereP(dx)Q(dx){\displaystyle {\frac {P(dx)}{Q(dx)}}}is theRadon–Nikodym derivativeofPwith respect toQ, i.e. the uniqueQalmost everywhere defined functionronX{\displaystyle {\mathcal {X}}}such thatP(dx)=r(x)Q(dx){\displaystyle P(dx)=r(x)Q(dx)}which exists becausePis absolutely continuous with respect toQ. Also we assume the expression on the right-hand side exists. Equivalently (by thechain rule), this can be written as
DKL(P∥Q)=∫x∈XP(dx)Q(dx)logP(dx)Q(dx)Q(dx),{\displaystyle D_{\text{KL}}(P\parallel Q)=\int _{x\in {\mathcal {X}}}{\frac {P(dx)}{Q(dx)}}\ \log {\frac {P(dx)}{Q(dx)}}\ Q(dx)\,,}
which is theentropyofPrelative toQ. Continuing in this case, ifμ{\displaystyle \mu }is any measure onX{\displaystyle {\mathcal {X}}}for which densitiespandqwithP(dx)=p(x)μ(dx){\displaystyle P(dx)=p(x)\mu (dx)}andQ(dx)=q(x)μ(dx){\displaystyle Q(dx)=q(x)\mu (dx)}exist (meaning thatPandQare both absolutely continuous with respect toμ{\displaystyle \mu }),then the relative entropy fromQtoPis given as
DKL(P∥Q)=∫x∈Xp(x)logp(x)q(x)μ(dx).{\displaystyle D_{\text{KL}}(P\parallel Q)=\int _{x\in {\mathcal {X}}}p(x)\,\log {\frac {p(x)}{q(x)}}\ \mu (dx)\,.}
Note that such a measureμ{\displaystyle \mu }for which densities can be defined always exists, since one can takeμ=12(P+Q){\textstyle \mu ={\frac {1}{2}}\left(P+Q\right)}although in practice it will usually be one that applies in the context likecounting measurefor discrete distributions, orLebesgue measureor a convenient variant thereof likeGaussian measureor the uniform measure on thesphere,Haar measureon aLie groupetc. for continuous distributions.
The logarithms in these formulae are usually taken tobase2 if information is measured in units ofbits, or to baseeif information is measured innats. Most formulas involving relative entropy hold regardless of the base of the logarithm.
Various conventions exist for referring toDKL(P∥Q){\displaystyle D_{\text{KL}}(P\parallel Q)}in words. Often it is referred to as the divergencebetweenPandQ, but this fails to convey the fundamental asymmetry in the relation. Sometimes, as in this article, it may be described as the divergence ofPfromQor as the divergencefromQtoP. This reflects theasymmetryinBayesian inference, which startsfromapriorQand updatestotheposteriorP. Another common way to refer toDKL(P∥Q){\displaystyle D_{\text{KL}}(P\parallel Q)}is as the relative entropy ofPwith respect toQor theinformation gainfromPoverQ.
Kullback[3]gives the following example (Table 2.1, Example 2.1). LetPandQbe the distributions shown in the table and figure.Pis the distribution on the left side of the figure, abinomial distributionwithN=2{\displaystyle N=2}andp=0.4{\displaystyle p=0.4}.Qis the distribution on the right side of the figure, adiscrete uniform distributionwith the three possible outcomesx=0,1,2(i.e.X={0,1,2}{\displaystyle {\mathcal {X}}=\{0,1,2\}}), each with probabilityp=1/3{\displaystyle p=1/3}.
Relative entropiesDKL(P∥Q){\displaystyle D_{\text{KL}}(P\parallel Q)}andDKL(Q∥P){\displaystyle D_{\text{KL}}(Q\parallel P)}are calculated as follows. This example uses thenatural logwith basee, designatedlnto get results innats(seeunits of information):
DKL(P∥Q)=∑x∈XP(x)lnP(x)Q(x)=925ln9/251/3+1225ln12/251/3+425ln4/251/3=125(32ln2+55ln3−50ln5)≈0.0852996,{\displaystyle {\begin{aligned}D_{\text{KL}}(P\parallel Q)&=\sum _{x\in {\mathcal {X}}}P(x)\,\ln {\frac {P(x)}{Q(x)}}\\&={\frac {9}{25}}\ln {\frac {9/25}{1/3}}+{\frac {12}{25}}\ln {\frac {12/25}{1/3}}+{\frac {4}{25}}\ln {\frac {4/25}{1/3}}\\&={\frac {1}{25}}\left(32\ln 2+55\ln 3-50\ln 5\right)\\&\approx 0.0852996,\end{aligned}}}
DKL(Q∥P)=∑x∈XQ(x)lnQ(x)P(x)=13ln1/39/25+13ln1/312/25+13ln1/34/25=13(−4ln2−6ln3+6ln5)≈0.097455.{\displaystyle {\begin{aligned}D_{\text{KL}}(Q\parallel P)&=\sum _{x\in {\mathcal {X}}}Q(x)\,\ln {\frac {Q(x)}{P(x)}}\\&={\frac {1}{3}}\,\ln {\frac {1/3}{9/25}}+{\frac {1}{3}}\,\ln {\frac {1/3}{12/25}}+{\frac {1}{3}}\,\ln {\frac {1/3}{4/25}}\\&={\frac {1}{3}}\left(-4\ln 2-6\ln 3+6\ln 5\right)\\&\approx 0.097455.\end{aligned}}}
In the field of statistics, theNeyman–Pearson lemmastates that the most powerful way to distinguish between the two distributionsPandQbased on an observationY(drawn from one of them) is through the log of the ratio of their likelihoods:logP(Y)−logQ(Y){\displaystyle \log P(Y)-\log Q(Y)}. The KL divergence is the expected value of this statistic ifYis actually drawn fromP. Kullback motivated the statistic as an expected log likelihood ratio.[15]
In the context ofcoding theory,DKL(P∥Q){\displaystyle D_{\text{KL}}(P\parallel Q)}can be constructed by measuring the expected number of extrabitsrequired tocodesamples fromPusing a code optimized forQrather than the code optimized forP.
In the context ofmachine learning,DKL(P∥Q){\displaystyle D_{\text{KL}}(P\parallel Q)}is often called theinformation gainachieved ifPwould be used instead ofQwhich is currently used. By analogy with information theory, it is called therelative entropyofPwith respect toQ.
Expressed in the language ofBayesian inference,DKL(P∥Q){\displaystyle D_{\text{KL}}(P\parallel Q)}is a measure of the information gained by revising one's beliefs from theprior probability distributionQto theposterior probability distributionP. In other words, it is the amount of information lost whenQis used to approximateP.[16]
In applications,Ptypically represents the "true" distribution of data, observations, or a precisely calculated theoretical distribution, whileQtypically represents a theory, model, description, orapproximationofP. In order to find a distributionQthat is closest toP, we can minimize the KL divergence and compute aninformation projection.
While it is astatistical distance, it is not ametric, the most familiar type of distance, but instead it is adivergence.[4]While metrics are symmetric and generalizelineardistance, satisfying thetriangle inequality, divergences are asymmetric and generalizesquareddistance, in some cases satisfying a generalizedPythagorean theorem. In generalDKL(P∥Q){\displaystyle D_{\text{KL}}(P\parallel Q)}does not equalDKL(Q∥P){\displaystyle D_{\text{KL}}(Q\parallel P)}, and the asymmetry is an important part of the geometry.[4]Theinfinitesimalform of relative entropy, specifically itsHessian, gives ametric tensorthat equals theFisher information metric; see§ Fisher information metric. Fisher information metric on the certain probability distribution let determine the natural gradient for information-geometric optimization algorithms.[17]Its quantum version is Fubini-study metric.[18]Relative entropy satisfies a generalized Pythagorean theorem forexponential families(geometrically interpreted asdually flat manifolds), and this allows one to minimize relative entropy by geometric means, for example byinformation projectionand inmaximum likelihood estimation.[5]
The relative entropy is theBregman divergencegenerated by the negative entropy, but it is also of the form of anf-divergence. For probabilities over a finitealphabet, it is unique in being a member of both of these classes ofstatistical divergences. The application of Bregman divergence can be found in mirror descent.[19]
Consider a growth-optimizing investor in a fair game with mutually exclusive outcomes
(e.g. a “horse race” in which the official odds add up to one).
The rate of return expected by such an investor is equal to the relative entropy
between the investor's believed probabilities and the official odds.[20]This is a special case of a much more general connection between financial returns and divergence measures.[21]
Financial risks are connected toDKL{\displaystyle D_{\text{KL}}}via information geometry.[22]Investors' views, the prevailing market view, and risky scenarios form triangles on the relevant manifold of probability distributions. The shape of the triangles determines key financial risks (both qualitatively and quantitatively). For instance, obtuse triangles in which investors' views and risk scenarios appear on “opposite sides” relative to the market describe negative risks, acute triangles describe positive exposure, and the right-angled situation in the middle corresponds to zero risk. Extending this concept, relative entropy can be hypothetically utilised to identify the behaviour of informed investors, if one takes this to be represented by the magnitude and deviations away from the prior expectations of fund flows, for example.[23]
In information theory, theKraft–McMillan theoremestablishes that any directly decodable coding scheme for coding a message to identify one valuexi{\displaystyle x_{i}}out of a set of possibilitiesXcan be seen as representing an implicit probability distributionq(xi)=2−ℓi{\displaystyle q(x_{i})=2^{-\ell _{i}}}overX, whereℓi{\displaystyle \ell _{i}}is the length of the code forxi{\displaystyle x_{i}}in bits. Therefore, relative entropy can be interpreted as the expected extra message-length per datum that must be communicated if a code that is optimal for a given (wrong) distributionQis used, compared to using a code based on the true distributionP: it is theexcessentropy.
DKL(P∥Q)=∑x∈Xp(x)log1q(x)−∑x∈Xp(x)log1p(x)=H(P,Q)−H(P){\displaystyle {\begin{aligned}D_{\text{KL}}(P\parallel Q)&=\sum _{x\in {\mathcal {X}}}p(x)\log {\frac {1}{q(x)}}-\sum _{x\in {\mathcal {X}}}p(x)\log {\frac {1}{p(x)}}\\[5pt]&=\mathrm {H} (P,Q)-\mathrm {H} (P)\end{aligned}}}
whereH(P,Q){\displaystyle \mathrm {H} (P,Q)}is thecross entropyofQrelative toPandH(P){\displaystyle \mathrm {H} (P)}is theentropyofP(which is the same as the cross-entropy of P with itself).
The relative entropyDKL(P∥Q){\displaystyle D_{\text{KL}}(P\parallel Q)}can be thought of geometrically as astatistical distance, a measure of how far the distributionQis from the distributionP. Geometrically it is adivergence: an asymmetric, generalized form of squared distance. The cross-entropyH(P,Q){\displaystyle H(P,Q)}is itself such a measurement (formally aloss function), but it cannot be thought of as a distance, sinceH(P,P)=:H(P){\displaystyle H(P,P)=:H(P)}is not zero. This can be fixed by subtractingH(P){\displaystyle H(P)}to makeDKL(P∥Q){\displaystyle D_{\text{KL}}(P\parallel Q)}agree more closely with our notion of distance, as theexcessloss. The resulting function is asymmetric, and while this can be symmetrized (see§ Symmetrised divergence), the asymmetric form is more useful. See§ Interpretationsfor more on the geometric interpretation.
Relative entropy relates to "rate function" in the theory oflarge deviations.[24][25]
Arthur Hobson proved that relative entropy is the only measure of difference between probability distributions that satisfies some desired properties, which are the canonical extension to those appearing in a commonly usedcharacterization of entropy.[26]Consequently,mutual informationis the only measure of mutual dependence that obeys certain related conditions, since it can be definedin terms of Kullback–Leibler divergence.
In particular, ifP(dx)=p(x)μ(dx){\displaystyle P(dx)=p(x)\mu (dx)}andQ(dx)=q(x)μ(dx){\displaystyle Q(dx)=q(x)\mu (dx)}, thenp(x)=q(x){\displaystyle p(x)=q(x)}μ{\displaystyle \mu }-almost everywhere. The entropyH(P){\displaystyle \mathrm {H} (P)}thus sets a minimum value for the cross-entropyH(P,Q){\displaystyle \mathrm {H} (P,Q)}, theexpectednumber ofbitsrequired when using a code based onQrather thanP; and the Kullback–Leibler divergence therefore represents the expected number of extra bits that must be transmitted to identify a valuexdrawn fromX, if a code is used corresponding to the probability distributionQ, rather than the "true" distributionP.
Denotef(α):=DKL((1−α)Q+αP∥Q){\displaystyle f(\alpha ):=D_{\text{KL}}((1-\alpha )Q+\alpha P\parallel Q)}and note thatDKL(P∥Q)=f(1){\displaystyle D_{\text{KL}}(P\parallel Q)=f(1)}. The first derivative off{\displaystyle f}may be derived and evaluated as followsf′(α)=∑x∈X(P(x)−Q(x))(log((1−α)Q(x)+αP(x)Q(x))+1)=∑x∈X(P(x)−Q(x))log((1−α)Q(x)+αP(x)Q(x))f′(0)=0{\displaystyle {\begin{aligned}f'(\alpha )&=\sum _{x\in {\mathcal {X}}}(P(x)-Q(x))\left(\log \left({\frac {(1-\alpha )Q(x)+\alpha P(x)}{Q(x)}}\right)+1\right)\\&=\sum _{x\in {\mathcal {X}}}(P(x)-Q(x))\log \left({\frac {(1-\alpha )Q(x)+\alpha P(x)}{Q(x)}}\right)\\f'(0)&=0\end{aligned}}}Further derivatives may be derived and evaluated as followsf″(α)=∑x∈X(P(x)−Q(x))2(1−α)Q(x)+αP(x)f″(0)=∑x∈X(P(x)−Q(x))2Q(x)f(n)(α)=(−1)n(n−2)!∑x∈X(P(x)−Q(x))n((1−α)Q(x)+αP(x))n−1f(n)(0)=(−1)n(n−2)!∑x∈X(P(x)−Q(x))nQ(x)n−1{\displaystyle {\begin{aligned}f''(\alpha )&=\sum _{x\in {\mathcal {X}}}{\frac {(P(x)-Q(x))^{2}}{(1-\alpha )Q(x)+\alpha P(x)}}\\f''(0)&=\sum _{x\in {\mathcal {X}}}{\frac {(P(x)-Q(x))^{2}}{Q(x)}}\\f^{(n)}(\alpha )&=(-1)^{n}(n-2)!\sum _{x\in {\mathcal {X}}}{\frac {(P(x)-Q(x))^{n}}{\left((1-\alpha )Q(x)+\alpha P(x)\right)^{n-1}}}\\f^{(n)}(0)&=(-1)^{n}(n-2)!\sum _{x\in {\mathcal {X}}}{\frac {(P(x)-Q(x))^{n}}{Q(x)^{n-1}}}\end{aligned}}}Hence solving forDKL(P∥Q){\displaystyle D_{\text{KL}}(P\parallel Q)}via the Taylor expansion off{\displaystyle f}about0{\displaystyle 0}evaluated atα=1{\displaystyle \alpha =1}yieldsDKL(P∥Q)=∑n=0∞f(n)(0)n!=∑n=2∞1n(n−1)∑x∈X(Q(x)−P(x))nQ(x)n−1{\displaystyle {\begin{aligned}D_{\text{KL}}(P\parallel Q)&=\sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}\\&=\sum _{n=2}^{\infty }{\frac {1}{n(n-1)}}\sum _{x\in {\mathcal {X}}}{\frac {(Q(x)-P(x))^{n}}{Q(x)^{n-1}}}\end{aligned}}}P≤2Q{\displaystyle P\leq 2Q}a.s. is a sufficient condition for convergence of the series by the following absolute convergence argument∑n=2∞|1n(n−1)∑x∈X(Q(x)−P(x))nQ(x)n−1|=∑n=2∞1n(n−1)∑x∈X|Q(x)−P(x)||1−P(x)Q(x)|n−1≤∑n=2∞1n(n−1)∑x∈X|Q(x)−P(x)|≤∑n=2∞1n(n−1)=1{\displaystyle {\begin{aligned}\sum _{n=2}^{\infty }\left\vert {\frac {1}{n(n-1)}}\sum _{x\in {\mathcal {X}}}{\frac {(Q(x)-P(x))^{n}}{Q(x)^{n-1}}}\right\vert &=\sum _{n=2}^{\infty }{\frac {1}{n(n-1)}}\sum _{x\in {\mathcal {X}}}\left\vert Q(x)-P(x)\right\vert \left\vert 1-{\frac {P(x)}{Q(x)}}\right\vert ^{n-1}\\&\leq \sum _{n=2}^{\infty }{\frac {1}{n(n-1)}}\sum _{x\in {\mathcal {X}}}\left\vert Q(x)-P(x)\right\vert \\&\leq \sum _{n=2}^{\infty }{\frac {1}{n(n-1)}}\\&=1\end{aligned}}}P≤2Q{\displaystyle P\leq 2Q}a.s. is also a necessary condition for convergence of the series by the following proof by contradiction. Assume thatP>2Q{\displaystyle P>2Q}with measure strictly greater than0{\displaystyle 0}. It then follows that there must exist some valuesε>0{\displaystyle \varepsilon >0},ρ>0{\displaystyle \rho >0}, andU<∞{\displaystyle U<\infty }such thatP≥2Q+ε{\displaystyle P\geq 2Q+\varepsilon }andQ≤U{\displaystyle Q\leq U}with measureρ{\displaystyle \rho }. The previous proof of sufficiency demonstrated that the measure1−ρ{\displaystyle 1-\rho }component of the series whereP≤2Q{\displaystyle P\leq 2Q}is bounded, so we need only concern ourselves with the behavior of the measureρ{\displaystyle \rho }component of the series whereP≥2Q+ε{\displaystyle P\geq 2Q+\varepsilon }. The absolute value of then{\displaystyle n}th term of this component of the series is then lower bounded by1n(n−1)ρ(1+εU)n{\displaystyle {\frac {1}{n(n-1)}}\rho \left(1+{\frac {\varepsilon }{U}}\right)^{n}}, which is unbounded asn→∞{\displaystyle n\to \infty }, so the series diverges.
The following result, due to Donsker and Varadhan,[29]is known asDonsker and Varadhan's variational formula.
Theorem [Duality Formula for Variational Inference]—LetΘ{\displaystyle \Theta }be a set endowed with an appropriateσ{\displaystyle \sigma }-fieldF{\displaystyle {\mathcal {F}}}, and two probability measuresPandQ, which formulate twoprobability spaces(Θ,F,P){\displaystyle (\Theta ,{\mathcal {F}},P)}and(Θ,F,Q){\displaystyle (\Theta ,{\mathcal {F}},Q)}, withQ≪P{\displaystyle Q\ll P}. (Q≪P{\displaystyle Q\ll P}indicates thatQis absolutely continuous with respect toP.) Lethbe a real-valued integrablerandom variableon(Θ,F,P){\displaystyle (\Theta ,{\mathcal {F}},P)}. Then the following equality holds
logEP[exph]=supQ≪P{EQ[h]−DKL(Q∥P)}.{\displaystyle \log E_{P}[\exp h]=\operatorname {sup} _{Q\ll P}\{E_{Q}[h]-D_{\text{KL}}(Q\parallel P)\}.}
Further, the supremum on the right-hand side is attained if and only if it holds
Q(dθ)P(dθ)=exph(θ)EP[exph],{\displaystyle {\frac {Q(d\theta )}{P(d\theta )}}={\frac {\exp h(\theta )}{E_{P}[\exp h]}},}
almost surely with respect to probability measureP, whereQ(dθ)P(dθ){\displaystyle {\frac {Q(d\theta )}{P(d\theta )}}}denotes the Radon-Nikodym derivative ofQwith respect toP.
For a short proof assuming integrability ofexp(h){\displaystyle \exp(h)}with respect toP, letQ∗{\displaystyle Q^{*}}haveP-densityexph(θ)EP[exph]{\displaystyle {\frac {\exp h(\theta )}{E_{P}[\exp h]}}}, i.e.Q∗(dθ)=exph(θ)EP[exph]P(dθ){\displaystyle Q^{*}(d\theta )={\frac {\exp h(\theta )}{E_{P}[\exp h]}}P(d\theta )}Then
DKL(Q∥Q∗)−DKL(Q∥P)=−EQ[h]+logEP[exph].{\displaystyle D_{\text{KL}}(Q\parallel Q^{*})-D_{\text{KL}}(Q\parallel P)=-E_{Q}[h]+\log E_{P}[\exp h].}
Therefore,
EQ[h]−DKL(Q∥P)=logEP[exph]−DKL(Q∥Q∗)≤logEP[exph],{\displaystyle E_{Q}[h]-D_{\text{KL}}(Q\parallel P)=\log E_{P}[\exp h]-D_{\text{KL}}(Q\parallel Q^{*})\leq \log E_{P}[\exp h],}
where the last inequality follows fromDKL(Q∥Q∗)≥0{\displaystyle D_{\text{KL}}(Q\parallel Q^{*})\geq 0}, for which equality occurs if and only ifQ=Q∗{\displaystyle Q=Q^{*}}. The conclusion follows.
Suppose that we have twomultivariate normal distributions, with meansμ0,μ1{\displaystyle \mu _{0},\mu _{1}}and with (non-singular)covariance matricesΣ0,Σ1.{\displaystyle \Sigma _{0},\Sigma _{1}.}If the two distributions have the same dimension,k, then the relative entropy between the distributions is as follows:[30]
DKL(N0∥N1)=12[tr(Σ1−1Σ0)−k+(μ1−μ0)TΣ1−1(μ1−μ0)+lndetΣ1detΣ0].{\displaystyle D_{\text{KL}}\left({\mathcal {N}}_{0}\parallel {\mathcal {N}}_{1}\right)={\frac {1}{2}}\left[\operatorname {tr} \left(\Sigma _{1}^{-1}\Sigma _{0}\right)-k+\left(\mu _{1}-\mu _{0}\right)^{\mathsf {T}}\Sigma _{1}^{-1}\left(\mu _{1}-\mu _{0}\right)+\ln {\frac {\det \Sigma _{1}}{\det \Sigma _{0}}}\right].}
Thelogarithmin the last term must be taken to baseesince all terms apart from the last are base-elogarithms of expressions that are either factors of the density function or otherwise arise naturally. The equation therefore gives a result measured innats. Dividing the entire expression above byln(2){\displaystyle \ln(2)}yields the divergence inbits.
In a numerical implementation, it is helpful to express the result in terms of the Cholesky decompositionsL0,L1{\displaystyle L_{0},L_{1}}such thatΣ0=L0L0T{\displaystyle \Sigma _{0}=L_{0}L_{0}^{T}}andΣ1=L1L1T{\displaystyle \Sigma _{1}=L_{1}L_{1}^{T}}. Then withMandysolutions to the triangular linear systemsL1M=L0{\displaystyle L_{1}M=L_{0}}, andL1y=μ1−μ0{\displaystyle L_{1}y=\mu _{1}-\mu _{0}},
DKL(N0∥N1)=12(∑i,j=1k(Mij)2−k+|y|2+2∑i=1kln(L1)ii(L0)ii).{\displaystyle D_{\text{KL}}\left({\mathcal {N}}_{0}\parallel {\mathcal {N}}_{1}\right)={\frac {1}{2}}\left(\sum _{i,j=1}^{k}{\left(M_{ij}\right)}^{2}-k+|y|^{2}+2\sum _{i=1}^{k}\ln {\frac {(L_{1})_{ii}}{(L_{0})_{ii}}}\right).}
A special case, and a common quantity invariational inference, is the relative entropy between a diagonal multivariate normal, and a standard normal distribution (with zero mean and unit variance):
DKL(N((μ1,…,μk)T,diag(σ12,…,σk2))∥N(0,I))=12∑i=1k[σi2+μi2−1−ln(σi2)].{\displaystyle D_{\text{KL}}\left({\mathcal {N}}\left(\left(\mu _{1},\ldots ,\mu _{k}\right)^{\mathsf {T}},\operatorname {diag} \left(\sigma _{1}^{2},\ldots ,\sigma _{k}^{2}\right)\right)\parallel {\mathcal {N}}\left(\mathbf {0} ,\mathbf {I} \right)\right)={\frac {1}{2}}\sum _{i=1}^{k}\left[\sigma _{i}^{2}+\mu _{i}^{2}-1-\ln \left(\sigma _{i}^{2}\right)\right].}
For two univariate normal distributionspandqthe above simplifies to[31]DKL(p∥q)=logσ1σ0+σ02+(μ0−μ1)22σ12−12{\displaystyle D_{\text{KL}}\left({\mathcal {p}}\parallel {\mathcal {q}}\right)=\log {\frac {\sigma _{1}}{\sigma _{0}}}+{\frac {\sigma _{0}^{2}+{\left(\mu _{0}-\mu _{1}\right)}^{2}}{2\sigma _{1}^{2}}}-{\frac {1}{2}}}
In the case of co-centered normal distributions withk=σ1/σ0{\displaystyle k=\sigma _{1}/\sigma _{0}}, this simplifies[32]to:
DKL(p∥q)=log2k+(k−2−1)/2/ln(2)bits{\displaystyle D_{\text{KL}}\left({\mathcal {p}}\parallel {\mathcal {q}}\right)=\log _{2}k+(k^{-2}-1)/2/\ln(2)\mathrm {bits} }
Consider two uniform distributions, with the support ofp=[A,B]{\displaystyle p=[A,B]}enclosed withinq=[C,D]{\displaystyle q=[C,D]}(C≤A<B≤D{\displaystyle C\leq A<B\leq D}). Then the information gain is:
DKL(p∥q)=logD−CB−A{\displaystyle D_{\text{KL}}\left({\mathcal {p}}\parallel {\mathcal {q}}\right)=\log {\frac {D-C}{B-A}}}
Intuitively,[32]the information gain to aktimes narrower uniform distribution containslog2k{\displaystyle \log _{2}k}bits. This connects with the use of bits in computing, wherelog2k{\displaystyle \log _{2}k}bits would be needed to identify one element of aklong stream.
Theexponential familyof distribution is given by
pX(x|θ)=h(x)exp(θTT(x)−A(θ)){\displaystyle p_{X}(x|\theta )=h(x)\exp \left(\theta ^{\mathsf {T}}T(x)-A(\theta )\right)}
whereh(x){\displaystyle h(x)}is reference measure,T(x){\displaystyle T(x)}is sufficient statistics,θ{\displaystyle \theta }is canonical natural parameters, andA(θ){\displaystyle A(\theta )}is the log-partition function.
The KL divergence between two distributionsp(x|θ1){\displaystyle p(x|\theta _{1})}andp(x|θ2){\displaystyle p(x|\theta _{2})}is given by[33]
DKL(θ1∥θ2)=(θ1−θ2)Tμ1−A(θ1)+A(θ2){\displaystyle D_{\text{KL}}(\theta _{1}\parallel \theta _{2})={\left(\theta _{1}-\theta _{2}\right)}^{\mathsf {T}}\mu _{1}-A(\theta _{1})+A(\theta _{2})}
whereμ1=Eθ1[T(X)]=∇A(θ1){\displaystyle \mu _{1}=E_{\theta _{1}}[T(X)]=\nabla A(\theta _{1})}is the mean parameter ofp(x|θ1){\displaystyle p(x|\theta _{1})}.
For example, for the Poisson distribution with meanλ{\displaystyle \lambda }, the sufficient statisticsT(x)=x{\displaystyle T(x)=x}, the natural parameterθ=logλ{\displaystyle \theta =\log \lambda }, and log partition functionA(θ)=eθ{\displaystyle A(\theta )=e^{\theta }}. As such, the divergence between two Poisson distributions with meansλ1{\displaystyle \lambda _{1}}andλ2{\displaystyle \lambda _{2}}is
DKL(λ1∥λ2)=λ1logλ1λ2−λ1+λ2.{\displaystyle D_{\text{KL}}(\lambda _{1}\parallel \lambda _{2})=\lambda _{1}\log {\frac {\lambda _{1}}{\lambda _{2}}}-\lambda _{1}+\lambda _{2}.}
As another example, for a normal distribution with unit varianceN(μ,1){\displaystyle N(\mu ,1)}, the sufficient statisticsT(x)=x{\displaystyle T(x)=x}, the natural parameterθ=μ{\displaystyle \theta =\mu }, and log partition functionA(θ)=μ2/2{\displaystyle A(\theta )=\mu ^{2}/2}. Thus, the divergence between two normal distributionsN(μ1,1){\displaystyle N(\mu _{1},1)}andN(μ2,1){\displaystyle N(\mu _{2},1)}is
DKL(μ1∥μ2)=(μ1−μ2)μ1−μ122+μ222=(μ2−μ1)22.{\displaystyle D_{\text{KL}}(\mu _{1}\parallel \mu _{2})=\left(\mu _{1}-\mu _{2}\right)\mu _{1}-{\frac {\mu _{1}^{2}}{2}}+{\frac {\mu _{2}^{2}}{2}}={\frac {{\left(\mu _{2}-\mu _{1}\right)}^{2}}{2}}.}
As final example, the divergence between a normal distribution with unit varianceN(μ,1){\displaystyle N(\mu ,1)}and a Poisson distribution with meanλ{\displaystyle \lambda }is
DKL(μ∥λ)=(μ−logλ)μ−μ22+λ.{\displaystyle D_{\text{KL}}(\mu \parallel \lambda )=(\mu -\log \lambda )\mu -{\frac {\mu ^{2}}{2}}+\lambda .}
While relative entropy is astatistical distance, it is not ametricon the space of probability distributions, but instead it is adivergence.[4]While metrics are symmetric and generalizelineardistance, satisfying thetriangle inequality, divergences are asymmetric in general and generalizesquareddistance, in some cases satisfying a generalizedPythagorean theorem. In generalDKL(P∥Q){\displaystyle D_{\text{KL}}(P\parallel Q)}does not equalDKL(Q∥P){\displaystyle D_{\text{KL}}(Q\parallel P)}, and while this can be symmetrized (see§ Symmetrised divergence), the asymmetry is an important part of the geometry.[4]
It generates atopologyon the space ofprobability distributions. More concretely, if{P1,P2,…}{\displaystyle \{P_{1},P_{2},\ldots \}}is a sequence of distributions such that
limn→∞DKL(Pn∥Q)=0,{\displaystyle \lim _{n\to \infty }D_{\text{KL}}(P_{n}\parallel Q)=0,}
then it is said that
Pn→DQ.{\displaystyle P_{n}\xrightarrow {D} \,Q.}
Pinsker's inequalityentails that
Pn→DP⇒Pn→TVP,{\displaystyle P_{n}\xrightarrow {D} P\Rightarrow P_{n}\xrightarrow {TV} P,}
where the latter stands for the usual convergence intotal variation.
Relative entropy is directly related to theFisher information metric. This can be made explicit as follows. Assume that the probability distributionsPandQare both parameterized by some (possibly multi-dimensional) parameterθ{\displaystyle \theta }. Consider then two close by values ofP=P(θ){\displaystyle P=P(\theta )}andQ=P(θ0){\displaystyle Q=P(\theta _{0})}so that the parameterθ{\displaystyle \theta }differs by only a small amount from the parameter valueθ0{\displaystyle \theta _{0}}. Specifically, up to first order one has (using theEinstein summation convention)P(θ)=P(θ0)+ΔθjPj(θ0)+⋯{\displaystyle P(\theta )=P(\theta _{0})+\Delta \theta _{j}\,P_{j}(\theta _{0})+\cdots }
withΔθj=(θ−θ0)j{\displaystyle \Delta \theta _{j}=(\theta -\theta _{0})_{j}}a small change ofθ{\displaystyle \theta }in thejdirection, andPj(θ0)=∂P∂θj(θ0){\displaystyle P_{j}\left(\theta _{0}\right)={\frac {\partial P}{\partial \theta _{j}}}(\theta _{0})}the corresponding rate of change in the probability distribution. Since relative entropy has an absolute minimum 0 forP=Q{\displaystyle P=Q}, i.e.θ=θ0{\displaystyle \theta =\theta _{0}}, it changes only tosecondorder in the small parametersΔθj{\displaystyle \Delta \theta _{j}}. More formally, as for any minimum, the first derivatives of the divergence vanish
∂∂θj|θ=θ0DKL(P(θ)∥P(θ0))=0,{\displaystyle \left.{\frac {\partial }{\partial \theta _{j}}}\right|_{\theta =\theta _{0}}D_{\text{KL}}(P(\theta )\parallel P(\theta _{0}))=0,}
and by theTaylor expansionone has up to second order
DKL(P(θ)∥P(θ0))=12ΔθjΔθkgjk(θ0)+⋯{\displaystyle D_{\text{KL}}(P(\theta )\parallel P(\theta _{0}))={\frac {1}{2}}\,\Delta \theta _{j}\,\Delta \theta _{k}\,g_{jk}(\theta _{0})+\cdots }
where theHessian matrixof the divergence
gjk(θ0)=∂2∂θj∂θk|θ=θ0DKL(P(θ)∥P(θ0)){\displaystyle g_{jk}(\theta _{0})=\left.{\frac {\partial ^{2}}{\partial \theta _{j}\,\partial \theta _{k}}}\right|_{\theta =\theta _{0}}D_{\text{KL}}(P(\theta )\parallel P(\theta _{0}))}
must bepositive semidefinite. Lettingθ0{\displaystyle \theta _{0}}vary (and dropping the subindex 0) the Hessiangjk(θ){\displaystyle g_{jk}(\theta )}defines a (possibly degenerate)Riemannian metricon theθparameter space, called the Fisher information metric.
Whenp(x,ρ){\displaystyle p_{(x,\rho )}}satisfies the following regularity conditions:
∂log(p)∂ρ,∂2log(p)∂ρ2,∂3log(p)∂ρ3{\displaystyle {\frac {\partial \log(p)}{\partial \rho }},{\frac {\partial ^{2}\log(p)}{\partial \rho ^{2}}},{\frac {\partial ^{3}\log(p)}{\partial \rho ^{3}}}}exist,|∂p∂ρ|<F(x):∫x=0∞F(x)dx<∞,|∂2p∂ρ2|<G(x):∫x=0∞G(x)dx<∞|∂3log(p)∂ρ3|<H(x):∫x=0∞p(x,0)H(x)dx<ξ<∞{\displaystyle {\begin{aligned}\left|{\frac {\partial p}{\partial \rho }}\right|&<F(x):\int _{x=0}^{\infty }F(x)\,dx<\infty ,\\\left|{\frac {\partial ^{2}p}{\partial \rho ^{2}}}\right|&<G(x):\int _{x=0}^{\infty }G(x)\,dx<\infty \\\left|{\frac {\partial ^{3}\log(p)}{\partial \rho ^{3}}}\right|&<H(x):\int _{x=0}^{\infty }p(x,0)H(x)\,dx<\xi <\infty \end{aligned}}}
whereξis independent ofρ∫x=0∞∂p(x,ρ)∂ρ|ρ=0dx=∫x=0∞∂2p(x,ρ)∂ρ2|ρ=0dx=0{\displaystyle \left.\int _{x=0}^{\infty }{\frac {\partial p(x,\rho )}{\partial \rho }}\right|_{\rho =0}\,dx=\left.\int _{x=0}^{\infty }{\frac {\partial ^{2}p(x,\rho )}{\partial \rho ^{2}}}\right|_{\rho =0}\,dx=0}
then:D(p(x,0)∥p(x,ρ))=cρ22+O(ρ3)asρ→0.{\displaystyle {\mathcal {D}}(p(x,0)\parallel p(x,\rho ))={\frac {c\rho ^{2}}{2}}+{\mathcal {O}}\left(\rho ^{3}\right){\text{ as }}\rho \to 0.}
Another information-theoretic metric isvariation of information, which is roughly a symmetrization ofconditional entropy. It is a metric on the set ofpartitionsof a discreteprobability space.
MAUVE is a measure of the statistical gap between two text distributions, such as the difference between text generated by a model and human-written text. This measure is computed using Kullback–Leibler divergences between the two distributions in a quantized embedding space of a foundation model.
Many of the other quantities of information theory can be interpreted as applications of relative entropy to specific cases.
Theself-information, also known as theinformation contentof a signal, random variable, oreventis defined as the negative logarithm of theprobabilityof the given outcome occurring.
When applied to adiscrete random variable, the self-information can be represented as[citation needed]
I(m)=DKL(δim∥{pi}),{\displaystyle \operatorname {\operatorname {I} } (m)=D_{\text{KL}}\left(\delta _{\text{im}}\parallel \{p_{i}\}\right),}
is the relative entropy of the probability distributionP(i){\displaystyle P(i)}from aKronecker deltarepresenting certainty thati=m{\displaystyle i=m}— i.e. the number of extra bits that must be transmitted to identifyiif only the probability distributionP(i){\displaystyle P(i)}is available to the receiver, not the fact thati=m{\displaystyle i=m}.
Themutual information,
I(X;Y)=DKL(P(X,Y)∥P(X)P(Y))=EX{DKL(P(Y∣X)∥P(Y))}=EY{DKL(P(X∣Y)∥P(X))}{\displaystyle {\begin{aligned}\operatorname {I} (X;Y)&=D_{\text{KL}}(P(X,Y)\parallel P(X)P(Y))\\[5pt]&=\operatorname {E} _{X}\{D_{\text{KL}}(P(Y\mid X)\parallel P(Y))\}\\[5pt]&=\operatorname {E} _{Y}\{D_{\text{KL}}(P(X\mid Y)\parallel P(X))\}\end{aligned}}}
is the relative entropy of thejoint probability distributionP(X,Y){\displaystyle P(X,Y)}from the productP(X)P(Y){\displaystyle P(X)P(Y)}of the twomarginal probability distributions— i.e. the expected number of extra bits that must be transmitted to identifyXandYif they are coded using only their marginal distributions instead of the joint distribution. Equivalently, if the joint probabilityP(X,Y){\displaystyle P(X,Y)}isknown, it is the expected number of extra bits that must on average be sent to identifyYif the value ofXis not already known to the receiver.
TheShannon entropy,
H(X)=E[IX(x)]=logN−DKL(pX(x)∥PU(X)){\displaystyle {\begin{aligned}\mathrm {H} (X)&=\operatorname {E} \left[\operatorname {I} _{X}(x)\right]\\&=\log N-D_{\text{KL}}{\left(p_{X}(x)\parallel P_{U}(X)\right)}\end{aligned}}}
is the number of bits which would have to be transmitted to identifyXfromNequally likely possibilities,lessthe relative entropy of the uniform distribution on therandom variatesofX,PU(X){\displaystyle P_{U}(X)}, from the true distributionP(X){\displaystyle P(X)}— i.e.lessthe expected number of bits saved, which would have had to be sent if the value ofXwere coded according to the uniform distributionPU(X){\displaystyle P_{U}(X)}rather than the true distributionP(X){\displaystyle P(X)}. This definition of Shannon entropy forms the basis ofE.T. Jaynes's alternative generalization to continuous distributions, thelimiting density of discrete points(as opposed to the usualdifferential entropy), which defines the continuous entropy aslimN→∞HN(X)=logN−∫p(x)logp(x)m(x)dx,{\displaystyle \lim _{N\to \infty }H_{N}(X)=\log N-\int p(x)\log {\frac {p(x)}{m(x)}}\,dx,}which is equivalent to:log(N)−DKL(p(x)||m(x)){\displaystyle \log(N)-D_{\text{KL}}(p(x)||m(x))}
Theconditional entropy[34],
H(X∣Y)=logN−DKL(P(X,Y)∥PU(X)P(Y))=logN−DKL(P(X,Y)∥P(X)P(Y))−DKL(P(X)∥PU(X))=H(X)−I(X;Y)=logN−EY[DKL(P(X∣Y)∥PU(X))]{\displaystyle {\begin{aligned}\mathrm {H} (X\mid Y)&=\log N-D_{\text{KL}}(P(X,Y)\parallel P_{U}(X)P(Y))\\[5pt]&=\log N-D_{\text{KL}}(P(X,Y)\parallel P(X)P(Y))-D_{\text{KL}}(P(X)\parallel P_{U}(X))\\[5pt]&=\mathrm {H} (X)-\operatorname {I} (X;Y)\\[5pt]&=\log N-\operatorname {E} _{Y}\left[D_{\text{KL}}\left(P\left(X\mid Y\right)\parallel P_{U}(X)\right)\right]\end{aligned}}}
is the number of bits which would have to be transmitted to identifyXfromNequally likely possibilities,lessthe relative entropy of the product distributionPU(X)P(Y){\displaystyle P_{U}(X)P(Y)}from the true joint distributionP(X,Y){\displaystyle P(X,Y)}— i.e.lessthe expected number of bits saved which would have had to be sent if the value ofXwere coded according to the uniform distributionPU(X){\displaystyle P_{U}(X)}rather than the conditional distributionP(X|Y){\displaystyle P(X|Y)}ofXgivenY.
When we have a set of possible events, coming from the distributionp, we can encode them (with alossless data compression) usingentropy encoding. This compresses the data by replacing each fixed-length input symbol with a corresponding unique, variable-length,prefix-free code(e.g.: the events (A, B, C) with probabilities p = (1/2, 1/4, 1/4) can be encoded as the bits (0, 10, 11)). If we know the distributionpin advance, we can devise an encoding that would be optimal (e.g.: usingHuffman coding). Meaning the messages we encode will have the shortest length on average (assuming the encoded events are sampled fromp), which will be equal toShannon's Entropyofp(denoted asH(p){\displaystyle \mathrm {H} (p)}). However, if we use a different probability distribution (q) when creating the entropy encoding scheme, then a larger number ofbitswill be used (on average) to identify an event from a set of possibilities. This new (larger) number is measured by thecross entropybetweenpandq.
Thecross entropybetween twoprobability distributions(pandq) measures the average number ofbitsneeded to identify an event from a set of possibilities, if a coding scheme is used based on a given probability distributionq, rather than the "true" distributionp. The cross entropy for two distributionspandqover the sameprobability spaceis thus defined as follows.
H(p,q)=Ep[−logq]=H(p)+DKL(p∥q).{\displaystyle \mathrm {H} (p,q)=\operatorname {E} _{p}[-\log q]=\mathrm {H} (p)+D_{\text{KL}}(p\parallel q).}
For explicit derivation of this, see theMotivationsection above.
Under this scenario, relative entropies (kl-divergence) can be interpreted as the extra number of bits, on average, that are needed (beyondH(p){\displaystyle \mathrm {H} (p)}) for encoding the events because of usingqfor constructing the encoding scheme instead ofp.
InBayesian statistics, relative entropy can be used as a measure of the information gain in moving from aprior distributionto aposterior distribution:p(x)→p(x∣I){\displaystyle p(x)\to p(x\mid I)}. If some new factY=y{\displaystyle Y=y}is discovered, it can be used to update the posterior distribution forXfromp(x∣I){\displaystyle p(x\mid I)}to a new posterior distributionp(x∣y,I){\displaystyle p(x\mid y,I)}usingBayes' theorem:
p(x∣y,I)=p(y∣x,I)p(x∣I)p(y∣I){\displaystyle p(x\mid y,I)={\frac {p(y\mid x,I)p(x\mid I)}{p(y\mid I)}}}
This distribution has a newentropy:
H(p(x∣y,I))=−∑xp(x∣y,I)logp(x∣y,I),{\displaystyle \mathrm {H} {\big (}p(x\mid y,I){\big )}=-\sum _{x}p(x\mid y,I)\log p(x\mid y,I),}
which may be less than or greater than the original entropyH(p(x∣I)){\displaystyle \mathrm {H} (p(x\mid I))}. However, from the standpoint of the new probability distribution one can estimate that to have used the original code based onp(x∣I){\displaystyle p(x\mid I)}instead of a new code based onp(x∣y,I){\displaystyle p(x\mid y,I)}would have added an expected number of bits:
DKL(p(x∣y,I)∥p(x∣I))=∑xp(x∣y,I)logp(x∣y,I)p(x∣I){\displaystyle D_{\text{KL}}{\big (}p(x\mid y,I)\parallel p(x\mid I){\big )}=\sum _{x}p(x\mid y,I)\log {\frac {p(x\mid y,I)}{p(x\mid I)}}}
to the message length. This therefore represents the amount of useful information, or information gain, aboutX, that has been learned by discoveringY=y{\displaystyle Y=y}.
If a further piece of data,Y2=y2{\displaystyle Y_{2}=y_{2}}, subsequently comes in, the probability distribution forxcan be updated further, to give a new best guessp(x∣y1,y2,I){\displaystyle p(x\mid y_{1},y_{2},I)}. If one reinvestigates the information gain for usingp(x∣y1,I){\displaystyle p(x\mid y_{1},I)}rather thanp(x∣I){\displaystyle p(x\mid I)}, it turns out that it may be either greater or less than previously estimated:
∑xp(x∣y1,y2,I)logp(x∣y1,y2,I)p(x∣I){\displaystyle \sum _{x}p(x\mid y_{1},y_{2},I)\log {\frac {p(x\mid y_{1},y_{2},I)}{p(x\mid I)}}}may be ≤ or > than∑xp(x∣y1,I)logp(x∣y1,I)p(x∣I){\textstyle \sum _{x}p(x\mid y_{1},I)\log {\frac {p(x\mid y_{1},I)}{p(x\mid I)}}}
and so the combined information gain doesnotobey the triangle inequality:
DKL(p(x∣y1,y2,I)∥p(x∣I)){\displaystyle D_{\text{KL}}{\big (}p(x\mid y_{1},y_{2},I)\parallel p(x\mid I){\big )}}may be <, = or > thanDKL(p(x∣y1,y2,I)∥p(x∣y1,I))+DKL(p(x∣y1,I)∥p(x∣I)){\displaystyle D_{\text{KL}}{\big (}p(x\mid y_{1},y_{2},I)\parallel p(x\mid y_{1},I){\big )}+D_{\text{KL}}{\big (}p(x\mid y_{1},I)\parallel p(x\mid I){\big )}}
All one can say is that onaverage, averaging usingp(y2∣y1,x,I){\displaystyle p(y_{2}\mid y_{1},x,I)}, the two sides will average out.
A common goal inBayesian experimental designis to maximise the expected relative entropy between the prior and the posterior.[35]When posteriors are approximated to be Gaussian distributions, a design maximising the expected relative entropy is calledBayes d-optimal.
Relative entropyDKL(p(x∣H1)∥p(x∣H0)){\textstyle D_{\text{KL}}{\bigl (}p(x\mid H_{1})\parallel p(x\mid H_{0}){\bigr )}}can also be interpreted as the expecteddiscrimination informationforH1{\displaystyle H_{1}}overH0{\displaystyle H_{0}}: the mean information per sample for discriminating in favor of a hypothesisH1{\displaystyle H_{1}}against a hypothesisH0{\displaystyle H_{0}}, when hypothesisH1{\displaystyle H_{1}}is true.[36]Another name for this quantity, given to it byI. J. Good, is the expected weight of evidence forH1{\displaystyle H_{1}}overH0{\displaystyle H_{0}}to be expected from each sample.
The expected weight of evidence forH1{\displaystyle H_{1}}overH0{\displaystyle H_{0}}isnotthe same as the information gain expected per sample about the probability distributionp(H){\displaystyle p(H)}of the hypotheses,
DKL(p(x∣H1)∥p(x∣H0))≠IG=DKL(p(H∣x)∥p(H∣I)).{\displaystyle D_{\text{KL}}(p(x\mid H_{1})\parallel p(x\mid H_{0}))\neq IG=D_{\text{KL}}(p(H\mid x)\parallel p(H\mid I)).}
Either of the two quantities can be used as autility functionin Bayesian experimental design, to choose an optimal next question to investigate: but they will in general lead to rather different experimental strategies.
On the entropy scale ofinformation gainthere is very little difference between near certainty and absolute certainty—coding according to a near certainty requires hardly any more bits than coding according to an absolute certainty. On the other hand, on thelogitscale implied by weight of evidence, the difference between the two is enormous – infinite perhaps; this might reflect the difference between being almost sure (on a probabilistic level) that, say, theRiemann hypothesisis correct, compared to being certain that it is correct because one has a mathematical proof. These two different scales ofloss functionfor uncertainty arebothuseful, according to how well each reflects the particular circumstances of the problem in question.
The idea of relative entropy as discrimination information led Kullback to propose the Principle ofMinimum Discrimination Information(MDI): given new facts, a new distributionfshould be chosen which is as hard to discriminate from the original distributionf0{\displaystyle f_{0}}as possible; so that the new data produces as small an information gainDKL(f∥f0){\displaystyle D_{\text{KL}}(f\parallel f_{0})}as possible.
For example, if one had a prior distributionp(x,a){\displaystyle p(x,a)}overxanda, and subsequently learnt the true distribution ofawasu(a){\displaystyle u(a)}, then the relative entropy between the new joint distribution forxanda,q(x∣a)u(a){\displaystyle q(x\mid a)u(a)}, and the earlier prior distribution would be:
DKL(q(x∣a)u(a)∥p(x,a))=Eu(a){DKL(q(x∣a)∥p(x∣a))}+DKL(u(a)∥p(a)),{\displaystyle D_{\text{KL}}(q(x\mid a)u(a)\parallel p(x,a))=\operatorname {E} _{u(a)}\left\{D_{\text{KL}}(q(x\mid a)\parallel p(x\mid a))\right\}+D_{\text{KL}}(u(a)\parallel p(a)),}
i.e. the sum of the relative entropy ofp(a){\displaystyle p(a)}the prior distribution forafrom the updated distributionu(a){\displaystyle u(a)}, plus the expected value (using the probability distributionu(a){\displaystyle u(a)}) of the relative entropy of the prior conditional distributionp(x∣a){\displaystyle p(x\mid a)}from the new conditional distributionq(x∣a){\displaystyle q(x\mid a)}. (Note that often the later expected value is called theconditional relative entropy(orconditional Kullback–Leibler divergence) and denoted byDKL(q(x∣a)∥p(x∣a)){\displaystyle D_{\text{KL}}(q(x\mid a)\parallel p(x\mid a))}[3][34]) This is minimized ifq(x∣a)=p(x∣a){\displaystyle q(x\mid a)=p(x\mid a)}over the whole support ofu(a){\displaystyle u(a)}; and we note that this result incorporates Bayes' theorem, if the new distributionu(a){\displaystyle u(a)}is in fact a δ function representing certainty thatahas one particular value.
MDI can be seen as an extension ofLaplace'sPrinciple of Insufficient Reason, and thePrinciple of Maximum EntropyofE.T. Jaynes. In particular, it is the natural extension of the principle of maximum entropy from discrete to continuous distributions, for which Shannon entropy ceases to be so useful (seedifferential entropy), but the relative entropy continues to be just as relevant.
In the engineering literature, MDI is sometimes called thePrinciple of Minimum Cross-Entropy(MCE) orMinxentfor short. Minimising relative entropy frommtopwith respect tomis equivalent to minimizing the cross-entropy ofpandm, since
H(p,m)=H(p)+DKL(p∥m),{\displaystyle \mathrm {H} (p,m)=\mathrm {H} (p)+D_{\text{KL}}(p\parallel m),}
which is appropriate if one is trying to choose an adequate approximation top. However, this is just as oftennotthe task one is trying to achieve. Instead, just as often it ismthat is some fixed prior reference measure, andpthat one is attempting to optimise by minimisingDKL(p∥m){\displaystyle D_{\text{KL}}(p\parallel m)}subject to some constraint. This has led to some ambiguity in the literature, with some authors attempting to resolve the inconsistency by redefining cross-entropy to beDKL(p∥m){\displaystyle D_{\text{KL}}(p\parallel m)}, rather thanH(p,m){\displaystyle \mathrm {H} (p,m)}[citation needed].
Surprisals[37]add where probabilities multiply. The surprisal for an event of probabilitypis defined ass=−klnp{\displaystyle s=-k\ln p}. Ifkis{1,1/ln2,1.38×10−23}{\displaystyle \left\{1,1/\ln 2,1.38\times 10^{-23}\right\}}then surprisal is in{{\displaystyle \{}nats, bits, orJ/K}{\displaystyle J/K\}}so that, for instance, there areNbits of surprisal for landing all "heads" on a toss ofNcoins.
Best-guess states (e.g. for atoms in a gas) are inferred by maximizing theaverage surprisalS(entropy) for a given set of control parameters (like pressurePor volumeV). This constrainedentropy maximization, both classically[38]and quantum mechanically,[39]minimizesGibbsavailability in entropy units[40]A≡−klnZ{\displaystyle A\equiv -k\ln Z}whereZis a constrained multiplicity orpartition function.
When temperatureTis fixed, free energy (T×A{\displaystyle T\times A}) is also minimized. Thus ifT,V{\displaystyle T,V}and number of moleculesNare constant, theHelmholtz free energyF≡U−TS{\displaystyle F\equiv U-TS}(whereUis energy andSis entropy) is minimized as a system "equilibrates." IfTandPare held constant (say during processes in your body), theGibbs free energyG=U+PV−TS{\displaystyle G=U+PV-TS}is minimized instead. The change in free energy under these conditions is a measure of availableworkthat might be done in the process. Thus available work for an ideal gas at constant temperatureTo{\displaystyle T_{o}}and pressurePo{\displaystyle P_{o}}isW=ΔG=NkToΘ(V/Vo){\displaystyle W=\Delta G=NkT_{o}\Theta (V/V_{o})}whereVo=NkTo/Po{\displaystyle V_{o}=NkT_{o}/P_{o}}andΘ(x)=x−1−lnx≥0{\displaystyle \Theta (x)=x-1-\ln x\geq 0}(see alsoGibbs inequality).
More generally[41]thework availablerelative to some ambient is obtained by multiplying ambient temperatureTo{\displaystyle T_{o}}by relative entropy ornet surprisalΔI≥0,{\displaystyle \Delta I\geq 0,}defined as the average value ofkln(p/po){\displaystyle k\ln(p/p_{o})}wherepo{\displaystyle p_{o}}is the probability of a given state under ambient conditions. For instance, the work available in equilibrating a monatomic ideal gas to ambient values ofVo{\displaystyle V_{o}}andTo{\displaystyle T_{o}}is thusW=ToΔI{\displaystyle W=T_{o}\Delta I}, where relative entropy
ΔI=Nk[Θ(VVo)+32Θ(TTo)].{\displaystyle \Delta I=Nk\left[\Theta {\left({\frac {V}{V_{o}}}\right)}+{\frac {3}{2}}\Theta {\left({\frac {T}{T_{o}}}\right)}\right].}
The resulting contours of constant relative entropy, shown at right for a mole of Argon at standard temperature and pressure, for example put limits on the conversion of hot to cold as in flame-powered air-conditioning or in the unpowered device to convert boiling-water to ice-water discussed here.[42]Thus relative entropy measures thermodynamic availability in bits.
Fordensity matricesPandQon aHilbert space, thequantum relative entropyfromQtoPis defined to be
DKL(P∥Q)=Tr(P(logP−logQ)).{\displaystyle D_{\text{KL}}(P\parallel Q)=\operatorname {Tr} (P(\log P-\log Q)).}
Inquantum information sciencethe minimum ofDKL(P∥Q){\displaystyle D_{\text{KL}}(P\parallel Q)}over all separable statesQcan also be used as a measure ofentanglementin the stateP.
Just as relative entropy of "actual from ambient" measures thermodynamic availability, relative entropy of "reality from a model" is also useful even if the only clues we have about reality are some experimental measurements. In the former case relative entropy describesdistance to equilibriumor (when multiplied by ambient temperature) the amount ofavailable work, while in the latter case it tells you about surprises that reality has up its sleeve or, in other words,how much the model has yet to learn.
Although this tool for evaluating models against systems that are accessible experimentally may be applied in any field, its application to selecting astatistical modelviaAkaike information criterionare particularly well described in papers[43]and a book[44]by Burnham and Anderson. In a nutshell the relative entropy of reality from a model may be estimated, to within a constant additive term, by a function of the deviations observed between data and the model's predictions (like themean squared deviation) . Estimates of such divergence for models that share the same additive term can in turn be used to select among models.
When trying to fit parametrized models to data there are various estimators which attempt to minimize relative entropy, such asmaximum likelihoodandmaximum spacingestimators.[citation needed]
Kullback & Leibler (1951)also considered the symmetrized function:[6]
DKL(P∥Q)+DKL(Q∥P){\displaystyle D_{\text{KL}}(P\parallel Q)+D_{\text{KL}}(Q\parallel P)}
which they referred to as the "divergence", though today the "KL divergence" refers to the asymmetric function (see§ Etymologyfor the evolution of the term). This function is symmetric and nonnegative, and had already been defined and used byHarold Jeffreysin 1948;[7]it is accordingly called theJeffreys divergence.
This quantity has sometimes been used forfeature selectioninclassificationproblems, wherePandQare the conditionalpdfsof a feature under two different classes. In the Banking and Finance industries, this quantity is referred to asPopulation Stability Index(PSI), and is used to assess distributional shifts in model features through time.
An alternative is given via theλ{\displaystyle \lambda }-divergence,
Dλ(P∥Q)=λDKL(P∥λP+(1−λ)Q)+(1−λ)DKL(Q∥λP+(1−λ)Q),{\displaystyle D_{\lambda }(P\parallel Q)=\lambda D_{\text{KL}}(P\parallel \lambda P+(1-\lambda )Q)+(1-\lambda )D_{\text{KL}}(Q\parallel \lambda P+(1-\lambda )Q),}
which can be interpreted as the expected information gain aboutXfrom discovering which probability distributionXis drawn from,PorQ, if they currently have probabilitiesλ{\displaystyle \lambda }and1−λ{\displaystyle 1-\lambda }respectively.[clarification needed][citation needed]
The valueλ=0.5{\displaystyle \lambda =0.5}gives theJensen–Shannon divergence, defined by
DJS=12DKL(P∥M)+12DKL(Q∥M){\displaystyle D_{\text{JS}}={\tfrac {1}{2}}D_{\text{KL}}(P\parallel M)+{\tfrac {1}{2}}D_{\text{KL}}(Q\parallel M)}
whereMis the average of the two distributions,
M=12(P+Q).{\displaystyle M={\tfrac {1}{2}}\left(P+Q\right).}
We can also interpretDJS{\displaystyle D_{\text{JS}}}as the capacity of a noisy information channel with two inputs giving the output distributionsPandQ. The Jensen–Shannon divergence, like allf-divergences, islocallyproportional to theFisher information metric. It is similar to theHellinger metric(in the sense that it induces the same affine connection on astatistical manifold).
Furthermore, the Jensen–Shannon divergence can be generalized using abstract statistical M-mixtures relying on an abstract mean M.[45][46]
There are many other important measures ofprobability distance. Some of these are particularly connected with relative entropy. For example:
Other notable measures of distance include theHellinger distance,histogram intersection,Chi-squared statistic,quadratic form distance,match distance,Kolmogorov–Smirnov distance, andearth mover's distance.[49]
Just asabsoluteentropy serves as theoretical background fordatacompression,relativeentropy serves as theoretical background fordatadifferencing– the absolute entropy of a set of data in this sense being the data required to reconstruct it (minimum compressed size), while the relative entropy of a target set of data, given a source set of data, is the data required to reconstruct the targetgiventhe source (minimum size of apatch).
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TheAMDAthlon X2processor family consists of processors based on both theAthlon 64 X2and thePhenomprocessor families. The original Athlon X2 processors were low-power Athlon 64 X2Brisbaneprocessors, while newer processors released in Q2 2008 are based on theK10Kumaprocessor.
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Predictive policingis the usage of mathematics,predictive analytics, and other analytical techniques inlaw enforcementto identify potential criminal activity.[1][2][3]A report published by theRAND Corporationidentified four general categories predictive policing methods fall into: methods for predicting crimes, methods for predicting offenders, methods for predicting perpetrators' identities, and methods for predicting victims of crime.[4]
Predictive policing uses data on the times, locations and nature of past crimes to provide insight to police strategists concerning where, and at what times,police patrolsshould patrol, or maintain a presence, in order to make the best use of resources or to have the greatest chance of deterring or preventing future crimes. This type of policing detects signals and patterns in crime reports to anticipate if crime will spike, when a shooting may occur, where the next car will be broken into, and who the next crimevictimwill be.Algorithmsare produced by taking into account these factors, which consist of large amounts of data that can be analyzed.[5][6]The use of algorithms creates a more effective approach that speeds up the process of predictive policing since it can quickly factor in different variables to produce an automated outcome. From the predictions the algorithm generates, they should be coupled with a prevention strategy, which typically sends an officer to the predicted time and place of the crime.[7]The use of automated predictive policing supplies a more accurate and efficient process when looking at future crimes because there is data to back up decisions, rather than just the instincts of police officers. By having police use information from predictive policing, they are able to anticipate the concerns of communities, wisely allocate resources to times and places, and prevent victimization.[8]
Police may also use data accumulated on shootings and thesounds of gunfireto identify locations of shootings. The city ofChicagouses data blended from population mappingcrime statisticsto improve monitoring and identify patterns.[9]
Rather than predicting crime, predictive policing can be used to prevent it. The "AIEthics of Care" approach recognizes that some locations have greater crime rates as a result of negative environmental conditions.Artificial intelligencecan be used to minimize crime by addressing the identified demands.[10]
At the conclusion of intense combat operations in April 2003, Improvised Explosive Devices (IEDs) were dispersed throughout Iraq’s streets. These devices were deployed to monitor and counteract U.S. military activities using predictive policing tactics. However, the extensive areas covered by these IEDs made it impractical for Iraqi forces to respond to every American presence within the region. This challenge led to the concept of Actionable Hot Spots—zones experiencing high levels of activity yet too vast for effective control. This situation presented difficulties for the Iraqi military in selecting optimal locations for surveillance, sniper placements, and route patrols along areas monitored by IEDs.[citation needed]
The roots of predictive policing can be traced to the policy approach of social governance, in whichleader of the Chinese Communist PartyXi Jinpingannounced at a security conference in 2016 is the Chinese regime’s agenda to promote a harmonious and prosperous country through an extensive use of information systems.[11]A common instance of social governance is the development of thesocial credit system, where big data is used to digitize identities and quantify trustworthiness. There is no other comparably comprehensive and institutionalized system of citizen assessment in the West.[12]
The increase in collecting and assessing aggregate public and private information by China’spolice forceto analyze past crime and forecast future criminal activity is part of the government’s mission to promote social stability by converting intelligence-led policing (i.e. effectively using information) into informatization (i.e. using information technologies) of policing.[11]The increase in employment of big data through the policegeographical information system(PGIS) is withinChina’spromise to better coordinate information resources across departments and regions to transform analysis of past crime patterns and trends into automated prevention and suppression of crime.[13][14]PGIS was first introduced in 1970s and was originally used for internal government management and research institutions for city surveying and planning. Since the mid-1990s PGIS has been introduced into the Chinese public security industry to empower law enforcement by promoting police collaboration and resource sharing.[13][15]The current applications of PGIS are still contained within the stages of public map services,spatial queries, andhot spotmapping. Its application in crime trajectory analysis and prediction is still in the exploratory stage; however, the promotion of informatization of policing has encouraged cloud-based upgrades to PGIS design, fusion of multi-sourcespatiotemporaldata, and developments to police spatiotemporalbig dataanalysis and visualization.[16]
Although there is no nationwide police prediction program in China, local projects between 2015 and 2018 have also been undertaken in regions such asZhejiang,Guangdong,Suzhou, andXinjiang, that are either advertised as or are building blocks towards a predictive policing system.[11][17]
Zhejiang and Guangdong had established prediction and prevention oftelecommunicationfraud through the real-time collection and surveillance of suspicious online or telecommunication activities and the collaboration with private companies such as theAlibaba Groupfor the identification of potential suspects.[18]The predictive policing and crime prevention operation involves forewarning to specific victims, with 9,120 warning calls being made in 2018 by theZhongshanpolice force along with direct interception of over 13,000 telephone calls and over 30,000 text messages in 2017.[11]
Substance-related crime is also investigated in Guangdong, specifically theZhongshanpolice force who were the first city in 2017 to utilize wastewater analysis and data models that included water and electricity usage to locate hotspots for drug crime. This method led to the arrest of 341 suspects in 45 different criminal investigations by 2019.[19]
InChina, Suzhou Police Bureau has adopted predictive policing since 2013. During 2015–2018, several cities in China have adopted predictive policing.[20]China has used predictive policing to identify and target people for sent toXinjiang internment camps.[21][22]
The integrated joint operations platform (IJOP) predictive policing system is operated by theCentral Political and Legal Affairs Commission.[23]
In Europe there has been significant pushback against predictive policing and the broader use of artificial intelligence in policing on both a national and European Union level.[24]
The DanishPOL-INTELproject has been operational since 2017 and is based on theGothamsystem fromPalantir Technologies. The Gotham system has also been used by German state police andEuropol.[24]
Predictive policing has been used inthe Netherlands.[24]
In theUnited States, the practice of predictive policing has been implemented by police departments in several states such as California, Washington, South Carolina, Alabama, Arizona, Tennessee, New York, and Illinois.[25][26]
In New York, the NYPD has begun implementing a new crime tracking program calledPatternizr. The goal of the Patternizr was to help aid police officers in identifying commonalities in crimes committed by the same offenders or same group of offenders. With the help of the Patternizr, officers are able to save time and be more efficient as the program generates the possible "pattern" of different crimes. The officer then has to manually search through the possible patterns to see if the generated crimes are related to the current suspect. If the crimes do match, the officer will launch a deeper investigation into the pattern crimes.[27]
In India, various state police forces have adopted AI technologies to enhance their law enforcement capabilities. For instance, the Maharashtra Police have launchedMaharashtra Advanced Research and Vigilance for Enhanced Law Enforcement (MARVEL), the country's first state-level police AI system, to improve crime prediction and detection.[28]Additionally, the Uttar Pradesh Police utilize the AI-powered mobile application 'Trinetra' for facial recognition and criminal tracking.[29]
Predictive policing faces issues that affect its effectiveness. Obioha mentions several concerns raised about predictive policing. High costs and limited use prevent more widespread use, especially among poorer countries. Another issue that affects predictive policing is that it relies on human input to determine patterns. Flawed data can lead to biased and possibly racist results.[30]Technology cannot predict crime, it can only weaponize proximity to policing. Though it is claimed to be unbiased data, communities of color and low income are the most targeted.[31]It should also be noted that not all crime is reported, making the data faulty[further explanation needed]and inaccurate.[citation needed]
In 2020, followingprotests against police brutality, a group of mathematicians published a letter inNotices of the American Mathematical Societyurging colleagues to stop work on predictive policing. Over 1,500 other mathematicians joined the proposed boycott.[32]
Some applications of predictive policing have targeted minority neighborhoods and lack feedback loops.[33]
Cities throughout the United States are enacting legislation to restrict the use of predictive policing technologies and other “invasive” intelligence-gathering techniques within their jurisdictions.
Following the introduction of predictive policing as a crime reduction strategy, via the results of an algorithm created through the use of the software PredPol, the city ofSanta Cruz, Californiaexperienced a decline in the number of burglaries reaching almost 20% in the first six months the program was in place. Despite this, in late June 2020 in the aftermath of the murder ofGeorge FloydinMinneapolis, Minnesotaalong with a growing call for increased accountability amongst police departments, the Santa Cruz City Council voted in favor of a complete ban on the use of predictive policing technology.[34]
Accompanying the ban on predictive policing, was a similar prohibition offacial recognition technology. Facial recognition technology has been criticized for its reduced accuracy on darker skin tones - which can contribute to cases of mistaken identity and potentially,wrongful convictions.[35]
In 2019, Michael Oliver, ofDetroit, Michigan, was wrongfully accused oflarcenywhen his face registered as a “match” in theDataWorks Plussoftware to the suspect identified in a video taken by the victim of the alleged crime. Oliver spent months going to court arguing for his innocence - and once the judge supervising the case viewed the video footage of the crime, it was clear that Oliver was not the perpetrator. In fact, the perpetrator and Oliver did not resemble each other at all - except for the fact that they are both African-American which makes it more likely that the facial recognition technology will make an identification error.[35]
With regards to predictive policing technology, the mayor of Santa Cruz, Justin Cummings, is quoted as saying, “this is something that targets people who are like me,” referencing the patterns ofracial biasand discrimination that predictive policing can continue rather than stop.[36]
For example, asDorothy Robertsexplains in her academic journal article, Digitizing the Carceral State, the data entered into predictive policing algorithms to predict where crimes will occur or who is likely to commit criminal activity, tends to contain information that has been impacted by racism. For example, the inclusion of arrest or incarceration history, neighborhood of residence, level of education, membership ingangsor organized crime groups,911call records, among other features, can produce algorithms that suggest the over-policing ofminorityorlow-incomecommunities.[35]
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Incryptography, theTiny Encryption Algorithm(TEA) is ablock ciphernotable for its simplicity of description andimplementation, typically a few lines of code. It was designed byDavid WheelerandRoger Needhamof theCambridge Computer Laboratory; it was first presented at theFast Software Encryptionworkshop inLeuvenin 1994, and first published in the proceedings of that workshop.[4]
The cipher is not subject to anypatents.
TEA operates on two 32-bitunsigned integers(could be derived from a 64-bit datablock) and uses a 128-bitkey. It has aFeistel structurewith a suggested 64 rounds, typically implemented in pairs termedcycles. It has an extremely simplekey schedule, mixing all of the key material in exactly the same way for each cycle. Different multiples of amagic constantare used to prevent simple attacks based on thesymmetryof the rounds. The magic constant, 2654435769 or 0x9E3779B9 is chosen to be⌊232⁄𝜙⌋, where𝜙is thegolden ratio(as anothing-up-my-sleeve number).[4]
TEA has a few weaknesses. Most notably, it suffers from equivalent keys—each key is equivalent to three others, which means that the effective key size is only 126bits.[5]As a result, TEA is especially bad as acryptographic hash function. This weakness led to a method forhackingMicrosoft'sXboxgame console, where the cipher was used as a hash function.[6]TEA is also susceptible to arelated-key attackwhich requires 223chosen plaintextsunder a related-key pair, with 232time complexity.[2]Because of these weaknesses, theXTEAcipher was designed.
The first published version of TEA was supplemented by a second version that incorporated extensions to make it more secure.Block TEA(which was specified along withXTEA) operates on arbitrary-size blocks in place of the 64-bit blocks of the original.
A third version (XXTEA), published in 1998, described further improvements for enhancing the security of the Block TEA algorithm.
Following is an adaptation of the reference encryption and decryption routines inC, released into the public domain by David Wheeler and Roger Needham:[4]
Note that the reference implementation acts on multi-byte numeric values. The original paper does not specify how to derive the numbers it acts on from binary or other content.
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Capgras delusionorCapgras syndromeis apsychiatric disorderin which a person holds adelusionthat a friend, spouse, parent, other close family member, or pet has been replaced by an identicalimpostor. It is named afterJoseph Capgras(1873–1950), the French psychiatrist who first described the disorder.
The Capgras delusion is classified as adelusional misidentification syndrome, a class of beliefs that involves the misidentification of people, places, or objects.[2]It can occur inacute,transient, orchronicforms. Cases in which patients hold the belief that time has been "warped" or "substituted" have also been reported.[3]
The delusion most commonly occurs in individuals diagnosed with apsychotic disorder, usuallyschizophrenia;[4]it has also been seen inbrain injury,[5]dementia with Lewy bodies,[6][7]and other forms ofdementia.[8]It presents often in individuals with aneurodegenerative disease, particularly at an older age;[9]it has also been reported as occurring in association withdiabetes,hypothyroidism, andmigraine attacks.[10]In one isolated case, the Capgras delusion was temporarily induced in a healthy subject by administration ofketamine.[11]It occurs more frequently in females, with a female to male ratio of approximately 3∶2.[12]
Capgras syndrome is named afterJoseph Capgras, a Frenchpsychiatristwho first described the disorder in 1923 in his paper co-authored by Jean Reboul-Lachaux.[13]They described the case of a French woman, "Madame Macabre," who complained that corresponding "doubles" had taken the places of her husband and other people she knew.[5]Capgras and Reboul-Lachaux first called the syndrome "l'illusion des sosies", which can be translated literally as "the illusion of Doppelgänger."[14]
The syndrome was initially considered a purely psychiatric disorder, the delusion of a double seen as symptomatic ofschizophrenia, and purely a female disorder (though this is now known not to be the case[15]) often noted as a symptom ofhysteria. Most of the proposed explanations initially following that of Capgras and Reboul-Lachaux werepsychoanalyticalin nature. It was not until the 1980s that attention turned to the usually co-existing organic brain lesions originally thought to be essentially unrelated or coincidental. Today, the Capgras syndrome is understood as a neurological disorder, in which the delusion primarily results from organic brain lesions or degeneration.[16]
Compared to otherdelusional misidentification syndromes, like theFregoli delusion, the Capgras delusion is more widely documented.[17]
It is generally agreed[18]that the Capgras delusion has a complex and organic basis caused by structural damage to organs[19]and can be better understood by examining neuroanatomical damage associated with the syndrome.[20]
In one of the first papers to consider the cerebral basis of the Capgras delusion, Alexander,Stussand Benson pointed out in 1979 that the disorder might be related to a combination of frontal lobe damage causing problems with familiarity and right hemisphere damage causing problems with visual recognition.[21]
Further clues to the possible causes of the Capgras delusion were suggested by the study of brain-injured patients who had developedprosopagnosia. In this condition, patients are unable torecognize faces consciously, despite being able to recognize other types of visual objects. However, a 1984 study by Bauer showed that even though conscious face recognition was impaired, patients with the condition showedautonomicarousal (measured by agalvanic skin responsemeasure) to familiar faces,[22]suggesting there are two pathways to face recognition—one conscious and one unconscious.
In a 1990 paper published in theBritish Journal of Psychiatry, psychologistsHadyn Ellisand Andy Young hypothesized that patients with Capgras delusion may have a "mirror image" ordouble dissociationofprosopagnosia, in that their conscious ability to recognize faces was intact, but they might have damage to the system which produces the automatic emotional arousal to familiar faces.[23]This might lead to the experience of recognizing someone while feeling something was not "quite right" about them. In 1997, Ellis and his colleagues published a study of five patients with Capgras delusion (all diagnosed with schizophrenia) and confirmed that although they could consciously recognize the faces, they did not show the normal automatic emotional arousal response.[24]The same low level of autonomic response was shown in the presence of strangers. Young (2008) has theorized that this means that patients with the disease experience a "loss" of familiarity, not a "lack" of it.[25]Further evidence for this explanation comes from other studies measuring galvanic skin responses (GSR) to faces. A patient with Capgras delusion showed reduced GSRs to faces in spite of normal face recognition.[26]This theory for the causes of Capgras delusion was summarised inTrends in Cognitive Sciencesin 2001.[2]
William HirsteinandVilayanur S. Ramachandranreported similar findings in a paper published on a single case of a patient with Capgras delusion after brain injury.[27]Ramachandran portrayed this case in his bookPhantoms in the Brain[28]and gave a talk about it atTED2007.[29]Since the patient was capable of feeling emotions and recognizing faces but could not feel emotions when recognizing familiar faces, Ramachandran hypothesizes the origin of Capgras syndrome is a disconnection between the temporalcortex, where faces are usually recognized (seetemporal lobe), and thelimbic system, involved inemotions. More specifically, he emphasizes the disconnection between theamygdalaand theinferotemporal cortex.[5]
In 2010, Hirstein revised this theory to explain why a person with Capgras syndrome would have the particular reaction of not recognizing a familiar person.[30]Hirstein explained the theory as being "a more specific version of the earlier position I took in the 1997 article with V. S. Ramachandran," and elaborated:
According to my current approach, we represent the people we know well with hybrid representations containing two parts. One part represents them externally: how they look, sound, etc. The other part represents them internally: their personalities, beliefs, characteristic emotions, preferences, etc. Capgras syndrome occurs when the internal portion of the representation is damaged or inaccessible. This produces the impression of someone who looks right on the outside, but seems different on the inside, i.e., an impostor. This gives a much more specific explanation that fits well with what the patients actually say. It corrects a problem with the earlier hypothesis in that there are many possible responses to the lack of an emotion upon seeing someone.[31]
Furthermore, Ramachandran suggests a relationship between the Capgras syndrome and a more general difficulty in linking successiveepisodic memoriesbecause of the crucial role emotion plays in creating memories. Since the patient could not put together memories and feelings, he believed objects in a photograph were new on every viewing, even though they normally should have evoked feelings (e.g., a person close to him, a familiar object, or even himself).[32]Others like Merrin and Silberfarb (1976)[15]have also proposed links between the Capgras syndrome and deficits in aspects of memory. They suggest that an important and familiar person (the usual subject of the delusion) has many layers of visual, auditory, tactile, and experiential memories associated with them, so the Capgras delusion can be understood as a failure ofobject constancyat a high perceptual level.[15]
Most likely, more than a mere impairment of the automatic emotional arousal response is necessary to form the Capgras delusion, as the same pattern has been reported in patients showing no signs of delusions.[33]Ellis suggested that a second factor explains why this unusual experience is transformed into a delusional belief; this second factor is thought to be an impairment in reasoning, although no specific impairment has been found to explain all cases.[34]Many have argued for the inclusion of the role of patientphenomenologyin explanatory models of the Capgras syndrome in order to better understand the mechanisms that enable the creation and maintenance of delusional beliefs.[35][36]
Capgras syndrome has also been linked toreduplicative paramnesia, another delusional misidentification syndrome in which a person believes a location has been duplicated or relocated. Since these two syndromes are highly associated, it has been proposed that they affect similar areas of the brain and therefore have similar neurological implications.[37]Reduplicative paramnesia is understood to affect the frontal lobe, and thus it is believed that Capgras syndrome is also associated with the frontal lobe.[38]Even if the damage is not directly to the frontal lobe, an interruption of signals between other lobes and the frontal lobe could result in Capgras syndrome.[9]Some authors have highlighted cannabis consumption as a trigger for Capgras syndrome.[39]
Because it is a rare and poorly understood condition, there is no established way to diagnose the Capgras delusion. Diagnosis is primarily made on apsychiatric evaluationof the patient, who is most likely brought to a psychiatrist's attention by a family member or friend believed to be an imposter by the person under the delusion. The patient may undergo mental skills tests to check fordementiaor other conditions, and brain imaging tests likeMRIorEEGthat look for lesions or other brain changes.[40]
Treatment of Capgras delusion has not been well studied, so there is no evidence-based approach.[41]Typically, treatment of delusional disorders is challenging due to poor patient insight and lack of empirical data.[37]Treatment is generally therapy, often with support ofantipsychoticmedication.[41][42][43]As manifestation of Capgras delusion is often a symptom rather than a syndrome itself, treatment may focus on the accompanying condition.[4]A study has shown that using medications appropriately to target the underlying disorder's core symptoms can be an effective management strategy. Hospitalization may be necessary, if the patient is engaging in self-harm or violence.[37]
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Vocal pedagogyis the study of the art and science ofvoiceinstruction. It is used in the teaching ofsingingand assists in defining what singing is, how singing works, and howsinging techniqueis accomplished.
Vocal pedagogy covers a broad range of aspects of singing, ranging from the physiological process of vocal production to the artistic aspects of interpretation of songs from different genres or historical eras. Typical areas of study include:[1]
All of these different concepts are a part of developingvocal technique. Not allvoice teachershave the same opinions within every topic of study which causes variations in pedagogical approaches and vocal technique.
Within Western culture, the study of vocal pedagogy began inAncient Greece. Scholars such asAlypiusandPythagorasstudied and made observations on the art of singing. It is unclear, however, whether the Greeks ever developed a systematic approach to teaching singing as little writing on the subject survives today.[2]
The first surviving record of a systematized approach to teaching singing was developed in themedievalmonasteriesof theRoman Catholic Churchsometime near the beginning of the 13th century. As with other fields of study, the monasteries were the center of musical intellectual life during the medieval period and many men within the monasteries devoted their time to the study of music and the art of singing. Highly influential in the development of a vocal pedagogical system were monksJohannes de GarlandiaandJerome of Moraviawho were the first to develop a concept ofvocal registers. These men identified three registers:chest voice,throat voice, andhead voice(pectoris, guttoris, and capitis). Their concept of head voice, however, is much more similar to the modern pedagogists understanding of thefalsetto register. Other concepts discussed in the monastic system includedvocal resonance,voice classification, breath support, diction, and tone quality to name a few. The ideas developed within the monastic system highly influenced the development of vocal pedagogy over the next several centuries including theBel Cantostyle of singing.[2]
With the onset of theRenaissancein the 15th century, the study of singing began to move outside of the church. The courts of rich patrons, such as the Dukes ofBurgundywho supported theBurgundian Schooland theFranco-Flemish School, became secular centers of study for singing and all other areas of musical study. The vocal pedagogical methods taught in these schools, however, were based on the concepts developed within the monastic system. Many of the teachers within these schools had their initial musical training from singing in church choirs as children. The church also remained at the forefront of musical composition at this time and remained highly influential in shaping musical tastes and practices both in and outside the church. It was the Catholic Church that first popularized the use ofcastratosingers in the 16th century, which ultimately led to the popularity of castrato voices inBaroqueand Classicaloperas.[3]
While the church maintained its dominance on intellectual and cultural life, there are individual examples of writers on voice pedagogy from this period who were from outside the church who put forward new ways of thinking and talking about the art of singing; although they lacked the wider influence of the monastic writers. The physician and court singerGiovanni Camillo Maffeiwas the first writer on vocal pedagogy to incorporate knowledge of the physiology of the voice into a theory of singing in his treatiseDiscorso delta voce e del modo d'apparare di cantar di garganta, and Scala naturale, overo Fantasia dolcissima, intorno alle cose occulte e desiderate nella filosofia(Venice, 1564).[4]
It was not until the development of opera in the 17th century that vocal pedagogy began to break away from some of the established thinking of the monastic writers and develop deeper understandings of the physical process of singing and its relation to key concepts likevocal registrationandvocal resonation. It was also during this time that notedvoice teachersbegan to emerge.Giulio Cacciniis an example of an important early Italian voice teacher.[2]In the late 17th century, thebel cantomethod of singing began to develop in Italy. This style of singing had a huge impact on the development of opera and the development of vocal pedagogy during theClassicalandRomanticperiods. It was during this time that teachers and composers first began to identify singers by and write roles for more specificvoice types. However, it was not until the 19th century that more clearly defined voice classification systems like the GermanFachsystem emerged. Within these systems, more descriptive terms were used in classifying voices such ascoloratura sopranoandlyric soprano.[3]
Voice teachers in the 19th century continued to train singers for careers in opera.Manuel Patricio Rodríguez Garcíais often considered one of the most important voice teachers of the 19th century, and is credited with the development of thelaryngoscopeand the beginning of modern voice pedagogy.
The field of voice pedagogy became more fully developed in the middle of the 20th century. A few American voice teachers began to study the science, anatomy, and physiology of singing, especiallyRalph AppelmanatIndiana University,Oren Brownat theWashington University School of Medicineand later theJuilliard School, andWilliam Vennardat theUniversity of Southern California. This shift in approach to the study of singing led to the rejection of many of the assertions of thebel cantosinging method, most particularly in the areas ofvocal registrationandvocal resonation.[5]As a result, there are currently two predominating schools of thought among voice teachers today, those who maintain the historical positions of the bel canto method and those who choose to embrace more contemporary understandings based in current knowledge of human anatomy and physiology. There are also those teachers who borrow ideas from both perspectives, creating a hybrid of the two.[6][7]
Appelman and Vennard were also part of a group of voice instructors who developed courses of study for beginning voice teachers, adding these scientific ideas to the standard exercises and empirical ways to improve vocal technique, and by 1980 the subject of voice pedagogy was beginning to be included in many college music degree programs for singers and vocal music educators.[5]
More recent works by authors such asRichard Millerand Johan Sundberg have increased the general knowledge of voice teachers, and scientific and practical aspects of voice pedagogy continue to be studied and discussed by professionals. In addition, the creation of organisations such as theNational Association of Teachers of Singing(now an international organization of Vocal Instructors) has enabled voice teachers to establish more of a consensus about their work, and has expanded the understanding of what singing teachers do.[1][8]
There are basically three major approaches to vocal pedagogy. They're all related to how the mechanistic and psychological controls are employed while singing. Some voice instructors advocate an extreme mechanistic approach that believes that singing is largely a matter of getting the right physical parts in the right places at the right time, and that correcting vocal faults is accomplished by calling direct attention to the parts which are not working well. On the other extreme, is the school of thought that believes that attention should never be directed to any part of the vocal mechanism—that singing is a matter of producing the right mental images of the desired tone, and that correcting vocal faults is achieved by learning to think the right thoughts and by releasing the emotions through interpretation of the music. Most voice teachers, however, believe that the truth lies somewhere in between the two extremes and adopt a composite of those two approaches.[9]
There are four physical processes involved in producing vocal sound:respiration,phonation,resonation, andarticulation. These processes occur in the following sequence:
Although these four processes are to be considered separately, in actual practice they merge into one coordinated function. With an effective singer or speaker, one should rarely be reminded of the process involved as their mind and body are so coordinated that one only perceives the resulting unified function. Many vocal problems result from a lack of coordination within this process.[7]
In its most basic sense, respiration is the process of moving air in and out of the body—inhalation and exhalation. Sound is produced in the larynx. But producing the sound would not be possible without a power source: the flow of air from the lungs. This flow sets the vocal folds into motion to produce sound.[10]Breathing for singing and speaking is a more controlled process than is the ordinary breathing used for sustaining life. The controls applied to exhalation are particularly important in good vocal technique.[7]
Phonationis the process of producing vocal sound by the vibration of thevocal foldsthat is in turn modified by the resonance of thevocal tract.[11][12]It takes place in thelarynxwhen thevocal foldsare brought together and breath pressure is applied to them in such a way that vibration ensues causing an audible source of acoustic energy, i.e., sound, which can then be modified by the articulatory actions of the rest of thevocal apparatus. The vocal folds are brought together primarily by the action of the interarytenoid muscles, which pull thearytenoid cartilagestogether.[1]
Vocal resonationis the process by which the basic product of phonation is enhanced in timbre and/or intensity by the air-filled cavities through which it passes on its way to the outside air. Various terms related to the resonation process include amplification, enrichment, enlargement, improvement, intensification, and prolongation, although in strictly scientific usage acoustic authorities would question most of them. The main point to be drawn from these terms by a singer or speaker is that the result of resonation is, or should be, to make a better sound.[1]
There are seven areas that may be listed as possible vocal resonators. In sequence from the lowest within the body to the highest, these areas are thechest, thetracheal tree, thelarynxitself, thepharynx, theoral cavity, thenasal cavity, and thesinuses.[9]
Research has shown that the larynx, the pharynx and the oral cavity are the main resonators of vocal sound, with the nasal cavity only coming into play in nasal consonants, or nasal vowels, such as those found in French. This main resonating space, from above the vocal folds to the lips is known as thevocal tract. Many voice users experience sensations in the sinuses that may be misconstrued as resonance. However, these sensations are caused by sympathetic vibrations, and are a result, rather than a cause, of efficient vocal resonance.[8]
Articulation is the process by which the joint product of the vibrator and the resonators is shaped into recognizable speech sounds through the muscular adjustments and movements of the speech organs. These adjustments and movements of the articulators result in verbal communication and thus form the essential difference between the human voice and other musical instruments. Singing without understandable words limits the voice to nonverbal communication.[9]In relation to the physical process of singing, vocal instructors tend to focus more on active articulation as opposed to passive articulation. There are five basic active articulators: the lip ("labial consonants"), the flexible front of the tongue ("coronal consonants"), the middle/back of the tongue ("dorsal consonants"), the root of the tongue together with theepiglottis("pharyngeal consonants"), and theglottis("glottal consonants"). These articulators can act independently of each other, and two or more may work together in what is calledcoarticulation.
Unlike active articulation, passive articulation is a continuum without many clear-cut boundaries. The places linguolabial and interdental, interdental and dental, dental and alveolar, alveolar and palatal, palatal and velar, velar and uvular merge into one another, and a consonant may be pronounced somewhere between the named places.
In addition, when the front of the tongue is used, it may be the upper surface orbladeof the tongue that makes contact ("laminal consonants"), the tip of the tongue ("apical consonants"), or the under surface ("sub-apical consonants"). These articulations also merge into one another without clear boundaries.
Interpretation is sometimes listed by voice teachers as a fifth physical process even though strictly speaking it is not a physical process. The reason for this is that interpretation does influence the kind of sound a singer makes which is ultimately achieved through a physical action the singer is doing. Although teachers may acquaint their students with musical styles and performance practices and suggest certain interpretive effects, most voice teachers agree that interpretation can not be taught. Students who lack a natural creative imagination and aesthetic sensibility can not learn it from someone else. Failure to interpret well is not a vocal fault, even though it may affect vocal sound significantly.[1]
Vocal sounds are divided into two basic categories—vowelsandconsonants—with a wide variety of sub-classifications. Voice teachers and serious voice students spend a great deal of time studying how the voice forms vowels and consonants, and studying the problems that certain consonants or vowels may cause while singing. TheInternational Phonetic Alphabetis used frequently by voice teachers and their students.[9]
Describing vocal sound is an inexact science largely because thehuman voiceis a self-contained instrument. Since the vocal instrument is internal, the singer's ability to monitor the sound produced is complicated by the vibrations carried to the ear through the Eustachean (auditory) tube and the bony structures of the head and neck. In other words, most singers hear something different in their ears/head than what a person listening to them hears. As a result, voice teachers often focus less on how it "sounds" and more on how it "feels". Vibratory sensations resulting from the closely related processes of phonation and resonation, and kinesthetic ones arising from muscle tension, movement, body position, and weight serve as a guide to the singer on correct vocal production.
Another problem in describing vocal sound lies in the vocal vocabulary itself. There are many schools of thought within vocal pedagogy and different schools have adopted different terms, sometimes from other artistic disciplines. This has led to the use of a plethora of descriptive terms applied to the voice which are not always understood to mean the same thing. Some terms sometimes used to describe a quality of a voice's sound are: warm, white, dark, light, round, reedy, spread, focused, covered, swallowed, forward, ringing, hooty, bleaty, plummy, mellow, pear-shaped, and so forth.[7]
The singing process functions best when certain physical conditions of the body exist. The ability to move air in and out of the body freely and to obtain the needed quantity of air can be seriously affected by the body alignment of the various parts of the breathing mechanism. A sunken chest position will limit the capacity of the lungs, and a tense abdominal wall will inhibit the downward travel of thediaphragm. Good body alignment allows the breathing mechanism to fulfill its basic function efficiently without any undue expenditure of energy. Good body alignment also makes it easier to initiate phonation and to tune the resonators as proper alignment prevents unnecessary tension in the body. Voice Instructors have also noted that when singers assume good body alignment it often provides them with a greater sense of self-assurance and poise while performing. Audiences also tend to respond better to singers with good body alignment. Habitual good body alignment also ultimately improves the overall health of the body by enabling better blood circulation and preventing fatigue and stress on the body.[6]
All singing begins with breath. All vocal sounds are created byvibrationsin thelarynxcaused by air from thelungs.Breathingin everyday life is asubconsciousbodily function which occurs naturally; however, thesingermust have control of the intake and exhalation of breath to achieve maximum results from their voice.
Natural breathing has three stages: a breathing-in period, a breathing-out period, and a resting or recovery period; these stages are not usually consciously controlled. Within singing there are four stages of breathing:
These stages must be under conscious control by the singer until they become conditioned reflexes. Many singers abandon conscious controls before their reflexes are fully conditioned which ultimately leads to chronic vocal problems.[13]
In Europeanclassical musicandopera, voices are treated likemusical instruments.Composerswho write vocal music must have an understanding of the skills, talents, and vocal properties of singers.Voice classificationis the process by which human singing voices are evaluated and are thereby designated intovoice types. These qualities include but are not limited to:vocal range,vocal weight,vocal tessitura, vocaltimbre, andvocal transition pointssuch as breaks and lifts within the voice. Other considerations are physical characteristics, speech level, scientific testing, andvocal registration.[14]The science behind voice classification developed within European classical music and has been slow in adapting to more modern forms of singing. Voice classification is often used withinoperato associate possible roles with potential voices. There are currently several different systems in use within classical music including: the GermanFachsystem and the choral music system among many others. No system is universally applied or accepted.[3]
However, most classical music systems acknowledge seven different major voice categories. Women are typically divided into three groups:soprano,mezzo-soprano, andcontralto. Men are usually divided into four groups:countertenor,tenor,baritone, andbass. When considering children's voices, an eighth term,treble, can be applied. Within each of these major categories there are several sub-categories that identify specific vocal qualities likecoloraturafacility and vocal weight to differentiate between voices.[1]
Withinchoral music, singers voices are divided solely on the basis of vocal range. Choral music most commonly divides vocal parts into high and low voices within each sex (SATB). As a result, the typical choral situation affords many opportunities for misclassification to occur.[1]Since most people have medium voices, they must be assigned to a part that is either too high or too low for them; the mezzo-soprano must sing soprano or alto and the baritone must sing tenor or bass. Either option can present problems for the singer, but for most singers there are fewer dangers in singing too low than in singing too high.[15]
Within contemporary forms of music (sometimes referred to asContemporary Commercial Music), singers are classified by thestyle of musicthey sing, such as jazz, pop, blues, soul, country, folk, and rock styles. There is currently no authoritative voice classification system within non-classical music.[16]Attempts have been made to adopt classicalvoice typeterms to other forms of singing but such attempts have been met with controversy. The development of voice categorizations were made with the understanding that the singer would be using classical vocal technique within a specified range using unamplified (no microphones) vocal production. Since contemporary musicians use different vocal techniques, microphones, and are not forced to fit into a specific vocal role, applying such terms as soprano, tenor, baritone, etc. can be misleading or even inaccurate.[7]
Many voice teachers warn of the dangers of quick identification. Premature concern with classification can result in misclassification, with all its attendant dangers. Vennard says:
"I never feel any urgency about classifying a beginning student. So many premature diagnoses have been proved wrong, and it can be harmful to the student and embarrassing to the teacher to keep striving for an ill-chosen goal. It is best to begin in the middle part of the voice and work upward and downward until the voice classifies itself."[6]
Most voice teachers believe that it is essential to establish good vocal habits within a limited and comfortable range before attempting to classify the voice. When techniques of posture, breathing,phonation, resonation, and articulation have become established in this comfortable area, the true quality of the voice will emerge and the upper and lower limits of the range can be explored safely. Only then can a tentative classification be arrived at, and it may be adjusted as the voice continues to develop.[9]Many acclaimed voice instructors suggest that teachers begin by assuming that a voice is of a medium classification until it proves otherwise. The reason for this is that the majority of individuals possess medium voices and therefore this approach is less likely to misclassify or damage the voice.[1]
Vocal registrationrefers to the system of vocal registers within the human voice. A register in the human voice is a particular series of tones, produced in the same vibratory pattern of thevocal folds, and possessing the same quality. Registers originate inlaryngealfunction. They occur because the vocal folds are capable of producing several different vibratory patterns. Each of these vibratory patterns appears within a particular range ofpitchesand produces certain characteristic sounds.[17]The term register can be somewhat confusing as it encompasses several aspects of the human voice. The term register can be used to refer to any of the following:[1]
Inlinguistics, aregister languageis a language which combinestoneand vowelphonationinto a singlephonologicalsystem.
Withinspeech pathologythe term vocal register has three constituent elements: a certain vibratory pattern of the vocal folds, a certain series of pitches, and a certain type of sound. Speech pathologists identify four vocal registers based on the physiology of laryngeal function: thevocal fry register, themodal register, thefalsetto register, and thewhistle register. This view is also adopted by many teachers of singing.[1]
Some voice teachers, however, organize registers differently. There are over a dozen different constructs of vocal registers in use within the field. The confusion which exists concerning what a register is, and how many registers there are, is due in part to what takes place in the modal register when a person sings from the lowestpitchesof that register to the highest pitches. The frequency of vibration of the vocal folds is determined by their length, tension, and mass. As pitch rises, the vocal folds are lengthened, tension increases, and their thickness decreases. In other words, all three of these factors are in a state of flux in the transition from the lowest to the highest tones.[17]
If a singer holds any of these factors constant and interferes with their progressive state of change, his laryngeal function tends to become static and eventually breaks occur with obvious changes of tone quality. These breaks are often identified as register boundaries or as transition areas between registers. The distinct change or break between registers is called apassaggioor aponticello.[18]Vocal instructors teach that with study a singer can move effortlessly from one register to the other with ease and consistent tone. Registers can even overlap while singing. Teachers who like to use this theory of "blending registers" usually help students through the "passage" from one register to another by hiding their "lift" (where the voice changes).
However, many voice instructors disagree with this distinction of boundaries blaming such breaks on vocal problems which have been created by a static laryngeal adjustment that does not permit the necessary changes to take place. This difference of opinion has effected the different views on vocal registration.[1]
Singing is an integrated and coordinated act and it is difficult to discuss any of the individual technical areas and processes without relating them to the others. For example, phonation only comes into perspective when it is connected with respiration; the articulators affect resonance; the resonators affect the vocal folds; the vocal folds affect breath control; and so forth. Vocal problems are often a result of a breakdown in one part of this coordinated process which causes voice teachers to frequently focus in, intensively, on one area of the process with their student until that issue is resolved. However, some areas of the art of singing are so much the result of coordinated functions that it is hard to discuss them under a traditional heading like phonation, resonation, articulation, or respiration.
Once the voice student has become aware of the physical processes that make up the act of singing and of how those processes function, the student begins the task of trying to coordinate them. Inevitably, students and teachers will become more concerned with one area of the technique than another. The various processes may progress at different rates, with a resulting imbalance or lack of coordination. The areas of vocal technique which seem to depend most strongly on the student's ability to coordinate various functions are:[1]
Some consider that singing is not a natural process but is a skill that requires highly developed muscle reflexes, but others consider that some ways of singing can be considered as natural.[19]Singing does not require much muscle strength but it does require a high degree of muscle coordination. Individuals can develop their voices further through the careful and systematic practice of both songs and vocal exercises. Voice teachers instruct their students to exercise their voices in an intelligent manner. Singers should be thinking constantly about the kind of sound they are making and the kind of sensations they are feeling while they are singing.[7]
There are several purposes for vocal exercises, including:[1]
An important goal of vocal development is to learn to sing to the natural limits of one's vocal range without any undesired changes of quality or technique. Voice instructors teach that a singer can only achieve this goal when all of the physical processes involved in singing (such as laryngeal action, breath support, resonance adjustment, and articulatory movement) are effectively working together. Most voice teachers believe that the first step in coordinating these processes is by establishing good vocal habits in the most comfortable tessitura of the voice first before slowly expanding the range beyond that.[6]
There are three factors which significantly affect the ability to sing higher or lower:
McKinney says, "These three factors can be expressed in three basic rules: (1) As you sing higher, you must use more energy; as you sing lower, you must use less. (2) As you sing higher, you must use more space; as you sing lower, you must use less. (3) As you sing higher, you must use more depth; as you sing lower, you must use less."[1]
Some voice teachers will spend time working with their students on general music knowledge and skills, particularlymusic theory,music history, and musical styles and practices as it relates to the vocal literature being studied. If required they may also spend time helping their students become better sight readers, often adoptingsolfège, which assigns certain syllables to the notes of the scale.
Since singing is a performing art, voice teachers spend some of their time preparing their students for performance. This includes teaching their students etiquette of behavior on stage such as bowing, learning to manage stage fright, addressing problems like nervous tics, and the use of equipment such as microphones. Some students may also be preparing for careers in the fields ofoperaormusical theaterwhere acting skills are required. Many voice instructors will spend time on acting techniques and audience communication with students in these fields of interest. Students of opera also spend a great deal of time with their voice teachers learning foreign language pronunciations.
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Inmathematics, asetiscountableif either it isfiniteor it can be made inone to one correspondencewith the set ofnatural numbers.[a]Equivalently, a set iscountableif there exists aninjective functionfrom it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.
In more technical terms, assuming theaxiom of countable choice, a set iscountableif itscardinality(the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to becountably infinite.
The concept is attributed toGeorg Cantor, who proved the existence ofuncountable sets, that is, sets that are not countable; for example the set of thereal numbers.
Although the terms "countable" and "countably infinite" as defined here are quite common, the terminology is not universal.[1]An alternative style usescountableto mean what is here called countably infinite, andat most countableto mean what is here called countable.[2][3]
The termsenumerable[4]anddenumerable[5][6]may also be used, e.g. referring to countable and countably infinite respectively,[7]definitions vary and care is needed respecting the difference withrecursively enumerable.[8]
A setS{\displaystyle S}iscountableif:
All of these definitions are equivalent.
A setS{\displaystyle S}iscountablyinfiniteif:
A set isuncountableif it is not countable, i.e. its cardinality is greater thanℵ0{\displaystyle \aleph _{0}}.[9]
In 1874, inhis first set theory article, Cantor proved that the set ofreal numbersis uncountable, thus showing that not all infinite sets are countable.[16]In 1878, he used one-to-one correspondences to define and compare cardinalities.[17]In 1883, he extended the natural numbers with his infiniteordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities.[18]
Asetis a collection ofelements, and may be described in many ways. One way is simply to list all of its elements; for example, the set consisting of the integers 3, 4, and 5 may be denoted{3,4,5}{\displaystyle \{3,4,5\}}, called roster form.[19]This is only effective for small sets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element, sometimes an ellipsis ("...") is used to represent many elements between the starting element and the end element in a set, if the writer believes that the reader can easily guess what ... represents; for example,{1,2,3,…,100}{\displaystyle \{1,2,3,\dots ,100\}}presumably denotes the set ofintegersfrom 1 to 100. Even in this case, however, it is stillpossibleto list all the elements, because the number of elements in the set is finite. If we number the elements of the set 1, 2, and so on, up ton{\displaystyle n}, this gives us the usual definition of "sets of sizen{\displaystyle n}".
Some sets areinfinite; these sets have more thann{\displaystyle n}elements wheren{\displaystyle n}is any integer that can be specified. (No matter how large the specified integern{\displaystyle n}is, such asn=101000{\displaystyle n=10^{1000}}, infinite sets have more thann{\displaystyle n}elements.) For example, the set of natural numbers, denotable by{0,1,2,3,4,5,…}{\displaystyle \{0,1,2,3,4,5,\dots \}},[a]has infinitely many elements, and we cannot use any natural number to give its size. It might seem natural to divide the sets into different classes: put all the sets containing one element together; all the sets containing two elements together; ...; finally, put together all infinite sets and consider them as having the same size. This view works well for countably infinite sets and was the prevailing assumption before Georg Cantor's work. For example, there are infinitely many odd integers, infinitely many even integers, and also infinitely many integers overall. We can consider all these sets to have the same "size" because we can arrange things such that, for every integer, there is a distinct even integer:…−2→−4,−1→−2,0→0,1→2,2→4⋯{\displaystyle \ldots \,-\!2\!\rightarrow \!-\!4,\,-\!1\!\rightarrow \!-\!2,\,0\!\rightarrow \!0,\,1\!\rightarrow \!2,\,2\!\rightarrow \!4\,\cdots }or, more generally,n→2n{\displaystyle n\rightarrow 2n}(see picture). What we have done here is arrange the integers and the even integers into aone-to-one correspondence(orbijection), which is afunctionthat maps between two sets such that each element of each set corresponds to a single element in the other set. This mathematical notion of "size", cardinality, is that two sets are of the same size if and only if there is a bijection between them. We call all sets that are in one-to-one correspondence with the integerscountably infiniteand say they have cardinalityℵ0{\displaystyle \aleph _{0}}.
Georg Cantorshowed that not all infinite sets are countably infinite. For example, the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers). The set of real numbers has a greater cardinality than the set of natural numbers and is said to be uncountable.
By definition, a setS{\displaystyle S}iscountableif there exists abijectionbetweenS{\displaystyle S}and a subset of thenatural numbersN={0,1,2,…}{\displaystyle \mathbb {N} =\{0,1,2,\dots \}}. For example, define the correspondencea↔1,b↔2,c↔3{\displaystyle a\leftrightarrow 1,\ b\leftrightarrow 2,\ c\leftrightarrow 3}Since every element ofS={a,b,c}{\displaystyle S=\{a,b,c\}}is paired withprecisely oneelement of{1,2,3}{\displaystyle \{1,2,3\}},andvice versa, this defines a bijection, and shows thatS{\displaystyle S}is countable. Similarly we can show all finite sets are countable.
As for the case of infinite sets, a setS{\displaystyle S}is countably infinite if there is abijectionbetweenS{\displaystyle S}and all ofN{\displaystyle \mathbb {N} }. As examples, consider the setsA={1,2,3,…}{\displaystyle A=\{1,2,3,\dots \}}, the set of positiveintegers, andB={0,2,4,6,…}{\displaystyle B=\{0,2,4,6,\dots \}}, the set of even integers. We can show these sets are countably infinite by exhibiting a bijection to the natural numbers. This can be achieved using the assignmentsn↔n+1{\displaystyle n\leftrightarrow n+1}andn↔2n{\displaystyle n\leftrightarrow 2n}, so that0↔1,1↔2,2↔3,3↔4,4↔5,…0↔0,1↔2,2↔4,3↔6,4↔8,…{\displaystyle {\begin{matrix}0\leftrightarrow 1,&1\leftrightarrow 2,&2\leftrightarrow 3,&3\leftrightarrow 4,&4\leftrightarrow 5,&\ldots \\[6pt]0\leftrightarrow 0,&1\leftrightarrow 2,&2\leftrightarrow 4,&3\leftrightarrow 6,&4\leftrightarrow 8,&\ldots \end{matrix}}}Every countably infinite set is countable, and every infinite countable set is countably infinite. Furthermore, any subset of the natural numbers is countable, and more generally:
Theorem—A subset of a countable set is countable.[20]
The set of allordered pairsof natural numbers (theCartesian productof two sets of natural numbers,N×N{\displaystyle \mathbb {N} \times \mathbb {N} }is countably infinite, as can be seen by following a path like the one in the picture:
The resultingmappingproceeds as follows:
0↔(0,0),1↔(1,0),2↔(0,1),3↔(2,0),4↔(1,1),5↔(0,2),6↔(3,0),…{\displaystyle 0\leftrightarrow (0,0),1\leftrightarrow (1,0),2\leftrightarrow (0,1),3\leftrightarrow (2,0),4\leftrightarrow (1,1),5\leftrightarrow (0,2),6\leftrightarrow (3,0),\ldots }This mapping covers all such ordered pairs.
This form of triangular mappingrecursivelygeneralizes ton{\displaystyle n}-tuplesof natural numbers, i.e.,(a1,a2,a3,…,an){\displaystyle (a_{1},a_{2},a_{3},\dots ,a_{n})}whereai{\displaystyle a_{i}}andn{\displaystyle n}are natural numbers, by repeatedly mapping the first two elements of ann{\displaystyle n}-tuple to a natural number. For example,(0,2,3){\displaystyle (0,2,3)}can be written as((0,2),3){\displaystyle ((0,2),3)}. Then(0,2){\displaystyle (0,2)}maps to 5 so((0,2),3){\displaystyle ((0,2),3)}maps to(5,3){\displaystyle (5,3)}, then(5,3){\displaystyle (5,3)}maps to 39. Since a different 2-tuple, that is a pair such as(a,b){\displaystyle (a,b)}, maps to a different natural number, a difference between two n-tuples by a single element is enough to ensure the n-tuples being mapped to different natural numbers. So, an injection from the set ofn{\displaystyle n}-tuples to the set of natural numbersN{\displaystyle \mathbb {N} }is proved. For the set ofn{\displaystyle n}-tuples made by the Cartesian product of finitely many different sets, each element in each tuple has the correspondence to a natural number, so every tuple can be written in natural numbers then the same logic is applied to prove the theorem.
Theorem—TheCartesian productof finitely many countable sets is countable.[21][b]
The set of allintegersZ{\displaystyle \mathbb {Z} }and the set of allrational numbersQ{\displaystyle \mathbb {Q} }may intuitively seem much bigger thanN{\displaystyle \mathbb {N} }. But looks can be deceiving. If a pair is treated as thenumeratoranddenominatorof avulgar fraction(a fraction in the form ofa/b{\displaystyle a/b}wherea{\displaystyle a}andb≠0{\displaystyle b\neq 0}are integers), then for every positive fraction, we can come up with a distinct natural number corresponding to it. This representation also includes the natural numbers, since every natural numbern{\displaystyle n}is also a fractionn/1{\displaystyle n/1}. So we can conclude that there are exactly as many positive rational numbers as there are positive integers. This is also true for all rational numbers, as can be seen below.
Theorem—Z{\displaystyle \mathbb {Z} }(the set of all integers) andQ{\displaystyle \mathbb {Q} }(the set of all rational numbers) are countable.[c]
In a similar manner, the set ofalgebraic numbersis countable.[23][d]
Sometimes more than one mapping is useful: a setA{\displaystyle A}to be shown as countable is one-to-one mapped (injection) to another setB{\displaystyle B}, thenA{\displaystyle A}is proved as countable ifB{\displaystyle B}is one-to-one mapped to the set of natural numbers. For example, the set of positiverational numberscan easily be one-to-one mapped to the set of natural number pairs (2-tuples) becausep/q{\displaystyle p/q}maps to(p,q){\displaystyle (p,q)}. Since the set of natural number pairs is one-to-one mapped (actually one-to-one correspondence or bijection) to the set of natural numbers as shown above, the positive rational number set is proved as countable.
Theorem—Any finiteunionof countable sets is countable.[24][25][e]
With the foresight of knowing that there are uncountable sets, we can wonder whether or not this last result can be pushed any further. The answer is "yes" and "no", we can extend it, but we need to assume a new axiom to do so.
Theorem—(Assuming theaxiom of countable choice) The union of countably many countable sets is countable.[f]
For example, given countable setsa,b,c,…{\displaystyle {\textbf {a}},{\textbf {b}},{\textbf {c}},\dots }, we first assign each element of each set a tuple, then we assign each tuple an index using a variant of the triangular enumeration we saw above:IndexTupleElement0(0,0)a01(0,1)a12(1,0)b03(0,2)a24(1,1)b15(2,0)c06(0,3)a37(1,2)b28(2,1)c19(3,0)d010(0,4)a4⋮{\displaystyle {\begin{array}{c|c|c }{\text{Index}}&{\text{Tuple}}&{\text{Element}}\\\hline 0&(0,0)&{\textbf {a}}_{0}\\1&(0,1)&{\textbf {a}}_{1}\\2&(1,0)&{\textbf {b}}_{0}\\3&(0,2)&{\textbf {a}}_{2}\\4&(1,1)&{\textbf {b}}_{1}\\5&(2,0)&{\textbf {c}}_{0}\\6&(0,3)&{\textbf {a}}_{3}\\7&(1,2)&{\textbf {b}}_{2}\\8&(2,1)&{\textbf {c}}_{1}\\9&(3,0)&{\textbf {d}}_{0}\\10&(0,4)&{\textbf {a}}_{4}\\\vdots &&\end{array}}}
We need theaxiom of countable choiceto indexallthe setsa,b,c,…{\displaystyle {\textbf {a}},{\textbf {b}},{\textbf {c}},\dots }simultaneously.
Theorem—The set of all finite-lengthsequencesof natural numbers is countable.
This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, and so on, each of which is a countable set (finite Cartesian product). Thus the set is a countable union of countable sets, which is countable by the previous theorem.
Theorem—The set of all finitesubsetsof the natural numbers is countable.
The elements of any finite subset can be ordered into a finite sequence. There are only countably many finite sequences, so also there are only countably many finite subsets.
Theorem—LetS{\displaystyle S}andT{\displaystyle T}be sets.
These follow from the definitions of countable set as injective / surjective functions.[g]
Cantor's theoremasserts that ifA{\displaystyle A}is a set andP(A){\displaystyle {\mathcal {P}}(A)}is itspower set, i.e. the set of all subsets ofA{\displaystyle A}, then there is no surjective function fromA{\displaystyle A}toP(A){\displaystyle {\mathcal {P}}(A)}. A proof is given in the articleCantor's theorem. As an immediate consequence of this and the Basic Theorem above we have:
Proposition—The setP(N){\displaystyle {\mathcal {P}}(\mathbb {N} )}is not countable; i.e. it isuncountable.
For an elaboration of this result seeCantor's diagonal argument.
The set ofreal numbersis uncountable,[h]and so is the set of all infinitesequencesof natural numbers.
If there is a set that is a standard model (seeinner model) of ZFC set theory, then there is a minimal standard model (seeConstructible universe). TheLöwenheim–Skolem theoremcan be used to show that this minimal model is countable. The fact that the notion of "uncountability" makes sense even in this model, and in particular that this modelMcontains elements that are:
was seen as paradoxical in the early days of set theory; seeSkolem's paradoxfor more.
The minimal standard model includes all thealgebraic numbersand all effectively computabletranscendental numbers, as well as many other kinds of numbers.
Countable sets can betotally orderedin various ways, for example:
In both examples of well orders here, any subset has aleast element; and in both examples of non-well orders,somesubsets do not have aleast element.
This is the key definition that determines whether a total order is also a well order.
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Automatic bug-fixingis the automaticrepairofsoftware bugswithout the intervention of a human programmer.[1][2][3]It is also commonly referred to asautomatic patch generation,automatic bug repair, orautomatic program repair.[3]The typical goal of such techniques is to automatically generate correctpatchesto eliminate bugs insoftware programswithout causingsoftware regression.[4]
Automatic bug fixing is made according to a specification of the expected behavior which can be for instance aformal specificationor atest suite.[5]
A test-suite – the input/output pairs specify the functionality of the program, possibly captured inassertionscan be used as atest oracleto drive the search. This oracle can in fact be divided between thebug oraclethat exposes the faulty behavior, and theregression oracle, which encapsulates the functionality any program repair method must preserve. Note that a test suite is typically incomplete and does not cover all possible cases. Therefore, it is often possible for a validated patch to produce expected outputs for all inputs in the test suite but incorrect outputs for other inputs.[6]The existence of such validated but incorrect patches is a major challenge for generate-and-validate techniques.[6]Recent successful automatic bug-fixing techniques often rely on additional information other than the test suite, such as information learned from previous human patches, to further identify correct patches among validated patches.[7]
Another way to specify the expected behavior is to useformal specifications[8][9]Verification against full specifications that specify the whole program behavior including functionalities is less common because such specifications are typically not available in practice and the computation cost of suchverificationis prohibitive. For specific classes of errors, however, implicit partial specifications are often available. For example, there are targeted bug-fixing techniques validating that the patched program can no longer trigger overflow errors in the same execution path.[10]
Generate-and-validate approaches compile and test each candidate patch to collect all validated patches that produce expected outputs for all inputs in the test suite.[5][6]Such a technique typically starts with a test suite of the program, i.e., a set oftest cases, at least one of which exposes the bug.[5][7][11][12]An early generate-and-validate bug-fixing systems is GenProg.[5]The effectiveness of generate-and-validate techniques remains controversial, because they typically do not providepatch correctness guarantees.[6]Nevertheless, the reported results of recent state-of-the-art techniques are generally promising. For example, on systematically collected 69 real world bugs in eight largeC software programs, the state-of-the-art bug-fixing system Prophet generates correct patches for 18 out of the 69 bugs.[7]
One way to generate candidate patches is to applymutation operatorson the original program. Mutation operators manipulate the original program, potentially via itsabstract syntax treerepresentation, or a more coarse-grained representation such as operating at thestatement-level orblock-level. Earliergenetic improvementapproaches operate at the statement level and carry out simple delete/replace operations such as deleting an existing statement or replacing an existing statement with another statement in the same source file.[5][13]Recent approaches use more fine-grained operators at theabstract syntax treelevel to generate more diverse set of candidate patches.[12]Notably, the statement deletion mutation operator, and more generally removing code, is a reasonable repair strategy, or at least a good fault localization strategy.[14]
Another way to generate candidate patches consists of using fix templates. Fix templates are typically predefined changes for fixing specific classes of bugs.[15]Examples of fix templates include inserting aconditional statementto check whether the value of a variable is null to fix null pointer exception, or changing an integer constant by one to fix off-by-one errors.[15]
Repair techniques exist that are based on symbolic execution. For example, Semfix[16]uses symbolic execution to extract a repair constraint. Angelix[17]introduced the concept of angelic forest in order to deal with multiline patches.
Under certain assumptions, it is possible to state the repair problem as a synthesis problem.
SemFix[16]uses component-based synthesis.[18]Dynamoth uses dynamic synthesis.[19]S3[20]is based onsyntax-guided synthesis.[21]SearchRepair[22]converts potential patches into an SMT formula and queries candidate patches that allow the patched program to pass all supplied test cases.
Machine learningtechniques can improve the effectiveness of automatic bug-fixing systems.[7]One example of such techniques learns from past successful patches from human developers collected fromopen sourcerepositoriesinGitHubandSourceForge.[7]It then use the learned information to recognize and prioritize potentially correct patches among all generated candidate patches.[7]Alternatively, patches can be directly mined from existing sources. Example approaches include mining patches from donor applications[10]or from QA web sites.[23]
Getafix[24]is a language-agnostic approach developed and used in production atFacebook. Given a sample ofcode commitswhere engineers fixed a certain kind of bug, it learns human-like fix patterns that apply to future bugs of the same kind. Besides using Facebook's owncode repositoriesas training data, Getafix learnt some fixes fromopen sourceJava repositories. When new bugs get detected, Getafix applies its previously learnt patterns to produce candidate fixes and ranks them within seconds. It presents only the top-ranked fix for final validation by tools or an engineer, in order to save resources and ideally be so fast that no human time was spent on fixing the same bug, yet.
For specific classes of errors, targeted automatic bug-fixing techniques use specialized templates:
Comparing to generate-and-validate techniques, template-based techniques tend to have better bug-fixing accuracy but a much narrowed scope.[6][27]
There are multiple uses of automatic bug fixing:
In essence, automatic bug fixing is a search activity, whether deductive-based or heuristic-based. The search space of automatic bug fixing is composed of all edits that can be possibly made to a program. There have been studies to understand the structure of this search space. Qi et al.[30]showed that the original fitness function of Genprog is not better than random search to drive the search. Long et al.'s[31]study indicated that correct patches can be considered as sparse in the search space and that incorrect overfitting patches are vastly more abundant (see also discussion about overfitting below).
Sometimes, in test-suite based program repair, tools generate patches that pass the test suite, yet are actually incorrect, this is known as the "overfitting" problem.[32]"Overfitting" in this context refers to the fact that the patch overfits to the test inputs. There are different kinds of overfitting: incomplete fixing means that only some buggy inputs are fixed, regression introduction means some previously working features are broken after the patch (because they were poorly tested). Early prototypes for automatic repair suffered a lot from overfitting: on the Manybugs C benchmark, Qi et al.[6]reported that 104/110 of plausible GenProg patches were overfitting. In the context of synthesis-based repair, Le et al.[33]obtained more than 80% of overfitting patches.
One way to avoid overfitting is to filter out the generated patches. This can be done based on dynamic analysis.[34]Alternatively, Tian et al. propose heuristic approaches to assess patch correctness.[35][36]
Automatic bug-fixing techniques that rely on a test suite do not provide patch correctness guarantees, because the test suite is incomplete and does not cover all cases.[6]A weak test suite may cause generate-and-validate techniques to produce validated but incorrect patches that have negative effects such as eliminating desirable functionalities, causing memory leaks, and introducing security vulnerabilities.[6]One possible approach is to amplify the failing test suite by automatically generating further test cases that are then labelled as passing or failing. To minimize the human labelling effort, an automatictest oraclecan be trained that gradually learns to automatically classify test cases as passing or failing and only engages the bug-reporting user for uncertain cases.[37]
A limitation of generate-and-validate repair systems is the search space explosion.[31]For a program, there are a large number of statements to change and for each statement there are a large number of possible modifications. State-of-the-art systems address this problem by assuming that a small modification is enough for fixing a bug, resulting in a search space reduction.
The limitation of approaches based on symbolic analysis[16][17]is that real world programs are often converted to intractably large formulas especially for modifying statements withside effects.
Benchmarks of bugs typically focus on one specific programming language.
In C, the Manybugs benchmark collected by GenProg authors contains 69 real world defects and it is widely used to evaluate many other bug-fixing tools for C.[13][7][12][17]
InJava, the main benchmark is Defects4J now extensively used in most research papers on program repair for Java.[38][39]Alternative benchmarks exist, such as the Quixbugs benchmark,[40]which contains original bugs for program repair. Other benchmarks of Java bugs include Bugs.jar,[41]based on past commits.
Automatic bug-fixing is an active research topic in computer science. There are many implementations of various bug-fixing techniques especially for C and Java programs. Note that most of these implementations are research prototypes for demonstrating their techniques, i.e., it is unclear whether their current implementations are ready for industrial usage or not.
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BlueKeep(CVE-2019-0708) is asecurity vulnerabilitythat was discovered inMicrosoft'sRemote Desktop Protocol(RDP) implementation, which allows for the possibility ofremote code execution.
First reported in May 2019, it is present in all unpatched Windows NT-based versions of Microsoft Windows fromWindows 2000throughWindows Server 2008 R2andWindows 7. Microsoft issued a security patch (including an out-of-band update for several versions of Windows that have reached their end-of-life, such asWindows XP) on 14 May 2019. On 13 August 2019, related BlueKeep security vulnerabilities, collectively namedDejaBlue, were reported to affectnewerWindows versions, includingWindows 7and all recent versions up toWindows 10of the operating system, as well as the older Windows versions.[3]On 6 September 2019, aMetasploitexploit of thewormableBlueKeep security vulnerability was announced to have been released into the public realm.[4]
The BlueKeep security vulnerability was first noted by theUK National Cyber Security Centre[2]and, on 14 May 2019, reported byMicrosoft. The vulnerability was named BlueKeep by computer security expert Kevin Beaumont onTwitter. BlueKeep is officially tracked as: CVE-2019-0708and is a "wormable"remote code executionvulnerability.[5][6]
Both the U.S.National Security Agency(which issued its own advisory on the vulnerability on 4 June 2019)[7]and Microsoft stated that this vulnerability could potentially be used byself-propagating worms, with Microsoft (based on a security researcher's estimation that nearly 1 million devices were vulnerable) saying that such a theoretical attack could be of a similar scale toEternalBlue-based attacks such asNotPetyaandWannaCry.[8][9][7]
On the same day as the NSA advisory, researchers of theCERT Coordination Centerdisclosed a separateRDP-related security issue inthe Windows 10 May 2019 UpdateandWindows Server 2019, citing a new behaviour where RDPNetwork Level Authentication(NLA) login credentials are cached on the client system, and the user can re-gain access to their RDP connection automatically if their network connection is interrupted. Microsoft dismissed this vulnerability as being intended behaviour, and it can be disabled viaGroup Policy.[10]
As of 1 June 2019, no activemalwareof the vulnerability seemed to be publicly known; however, undisclosedproof of concept(PoC) codes exploiting the vulnerability may have been available.[8][11][12][13]On 1 July 2019,Sophos, a British security company, reported on a working example of such a PoC, in order to emphasize the urgent need to patch the vulnerability.[14][15][16]On 22 July 2019, more details of an exploit were purportedly revealed by a conference speaker from a Chinese security firm.[17]On 25 July 2019, computer experts reported that a commercial version of the exploit may have been available.[18][19]On 31 July 2019, computer experts reported a significant increase in malicious RDP activity and warned, based on histories of exploits from similar vulnerabilities, that an active exploit of the BlueKeep vulnerability in the wild might be imminent.[20]
On 13 August 2019, related BlueKeep security vulnerabilities, collectively namedDejaBlue, were reported to affect newer Windows versions, includingWindows 7and all recent versions of the operating system up toWindows 10, as well as the older Windows versions.[3]
On 6 September 2019, an exploit of the wormable BlueKeep security vulnerability was announced to have been released into the public realm.[4]The initial version of this exploit was, however, unreliable, being known to cause "blue screen of death" (BSOD) errors. A fix was later announced, removing the cause of the BSOD error.[21]
On 2 November 2019, the first BlueKeep hacking campaign on a mass scale was reported, and included an unsuccessfulcryptojackingmission.[22]
On 8 November 2019, Microsoft confirmed a BlueKeep attack, and urged users to immediately patch their Windows systems.[23]
The RDP protocol uses "virtual channels", configured before authentication, as a data path between the client and server for providing extensions. RDP 5.1 defines 32 "static" virtual channels, and "dynamic" virtual channels are contained within one of these static channels. If a server binds the virtual channel "MS_T120" (a channel for which there is no legitimate reason for a client to connect to) with a static channel other than 31,heap corruptionoccurs that allows forarbitrary code executionat the system level.[24]
Windows XP,Windows Vista,Windows 7,Windows Server 2003,Windows Server 2008, andWindows Server 2008 R2were named by Microsoft as being vulnerable to this attack. Versions newer than 7, such asWindows 8,Windows 10andWindows 11, were not affected. TheCybersecurity and Infrastructure Security Agencystated that it had also successfully achieved code execution via the vulnerability onWindows 2000.[25]
Microsoft released patches for the vulnerability on 14 May 2019, forWindows XP,Windows Vista,Windows 7,Windows Server 2003,Windows Server 2008, andWindows Server 2008 R2. This included versions of Windows that have reached theirend-of-life(such as Vista, XP, and Server 2003) and thus are no longer eligible for security updates.[8]The patch forces the aforementioned "MS_T120" channel to always be bound to 31 even if requested otherwise by an RDP server.[24]
The NSA recommended additional measures, such as disablingRemote Desktop Servicesand its associatedport(TCP3389) if it is not being used, and requiringNetwork Level Authentication(NLA) for RDP.[26]According to computer security companySophos, two-factor authentication may make the RDP issue less of a vulnerability. However, the best protection is to take RDP off the Internet: switch RDP off if not needed and, if needed, make RDP accessible only via aVPN.[27]
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In linguistics,selectiondenotes the ability ofpredicatesto determine the semantic content of theirarguments.[1]Predicates select their arguments, which means they limit the semantic content of their arguments. A distinction may sometimes be drawn between types of selection; viz.,s(emantic)-selectionversusc(ategory)-selection. Selection in general stands in contrast tosubcategorization:[2]selection is a semantic concept, whereas subcategorization is a syntactic one;[3]predicates bothselectandsubcategorizefor theircomplementarguments, but onlyselecttheir subject arguments.
Selection is closely related tovalency, a term used in grammars other than the Chomskian generative grammar for a similar phenomenon.
The following pairs of sentences illustrate the concept of selection; the # indicates semantic deviance:
The predicateis wiltingselects a subject argument that is a plant or is plant-like. Similarly, the predicatedrankselects an object argument that is a liquid or is liquid-like. A building cannot normally be understood as wilting, just as a car cannot normally be interpreted as a liquid. The b-sentences are possible only given an unusual context that establishes appropriate metaphorical meaning. The deviance of the b-sentences is thus attributed to violation of those selectional restrictions determined by the predicatesis wiltinganddrank.
When a mismatch between a selector and a selected element triggers reinterpretation of the meaning of those elements, that process is referred to ascoercion.[4]
One sometimes encounters the termss(emantic)-selectionandc(ategory)-selection.[5]The concept of c-selection overlaps to an extent with subcategorization. Predicates c-select thesyntactic categoryof their complement arguments—e.g., noun (phrase), verb (phrase), adjective (phrase), etc.; that is, they determine thesyntactic categoryof their complements. In contrast, predicates s-select thesemantic contentof their arguments; thus, s-selection is a semantic concept, whereas c-selection is a syntactic one. (Note that when the termsselectionandselectional restrictionsappear without thec-ors-prefixes, they are usually understood to refer to s-selection.)[6][7]
The b-sentences above do not contain violations of the c-selectional restrictions of the predicatesis wiltinganddrank; they are, rather, well-formed from a syntactic point of view (hence #, not *), for the argumentsthe buildinganda carsatisfy the c-selectional restrictions of their respective predicates (i.e., in this case, the arguments are required to be nouns or noun phrases). Only the s-selectional restrictions of the predicatesis wiltinganddrankare violated in the b-sentences.
Selectional constraintsorselectional preferencesdescribe the degree of s-selection, in contrast toselectional restrictions, which treat s-selection as a binary yes-or-no.[8]Selectional preferences have often been used as a source of linguistic information innatural language processingapplications.[9]Thematic fitis a measure of how much a particular word in a particular role (like subject or direct object) matches the selectional preference of a particular predicate. For example, the wordcakehas a high thematic fit as a direct object forcut.[10]
The concepts of c-selection and subcategorization overlap in meaning and use to a significant degree.[11]If there is a difference between these concepts, it resides with the status of the subject argument. Traditionally, predicates are interpreted as NOT subcategorizing for their subject argument, because the subject argument appears outside of the minimal VP containing the predicate.[12]Predicates do, however, c-select their subject arguments; e.g.:
The predicateeatsc-selects both its subject argumentFredand its object argumentbeans, but as far as subcategorization is concerned,eatssubcategorizes for only its object argument,beans. This difference between c-selection and subcategorization depends, crucially, upon the understanding of subcategorization: an approach to subcategorization that sees predicates as subcategorizing for their subject argumentsas well asfor their object arguments will draw no distinction between c-selection and subcategorization; the two concepts are then synonymous.
Selection can be closely associated withthematic relations(e.g. agent, patient, theme, goal, etc.).[13]By limiting the semantic content of their arguments, predicates are determining the thematic relations/roles that their arguments bear.
Several linguistic theories make explicit use of selection. These include:
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In computer data, asubstitute character(␚) is acontrol characterthat is used to pad transmitted data in order to send it in blocks of fixed size, or to stand in place of a character that is recognized to be invalid, erroneous or unrepresentable on a given device. It is also used as an escape sequence in someprogramming languages.
In theASCII character set, this character is encoded by the number 26 (1Ahex). Standardkeyboardstransmit this code when theCtrlandZkeys are pressed simultaneously (Ctrl+Z, often documented by convention as^Z).[1]Unicodeinherits this character from ASCII, but recommends that thereplacement character(�, U+FFFD) be used instead to represent un-decodable inputs, when the output encoding is compatible with it.
Historically, underPDP-6monitor,[2]RT-11,VMS, andTOPS-10,[3]and in early PCCP/M1 and 2operating systems(and derivatives likeMP/M) it was necessary to explicitly mark theend of a file(EOF) because the nativefilesystemcould not record the exact file size by itself; files were allocated in extents (records) of a fixed size, typically leaving some allocated but unused space at the end of each file.[4][5][6][7]This extra space was filled with1A16(hex) characters under CP/M. The extended CP/M filesystems used by CP/M 3 and higher (and derivatives likeConcurrent CP/M,Concurrent DOS, andDOS Plus) did support byte-granular files,[8][9]so this was no longer a requirement, but it remained as a convention (especially fortext files) in order to ensure backward compatibility.
InCP/M,86-DOS,MS-DOS,PC DOS,DR-DOS, and their various derivatives, the SUB character was also used to indicate the end of a character stream,[citation needed]and thereby used to terminate user input in an interactivecommand linewindow (and as such, often used to finish console input redirection, e.g. as instigated by the commandCOPYCON: TYPEDTXT.TXT).
While no longer technically required to indicate the end of a file, as of 2017, many text editors[which?]and program languages still support this convention, or can be configured to insert this character at the end of a file when editing, or at least properly cope with them in text files.[citation needed]In such cases, it is often termed a "soft" EOF, as it does not necessarily represent the physical end of the file, but is more a marker indicating that "there is no useful data beyond this point". In reality, more data may exist beyond this character up to the actual end of the data in the file system, thus it can be used to hide file content when the file is entered at the console or opened in editors. Many file format standards (e.g.PNGorGIF) include the SUB character in their headers to perform precisely this function. Some modern text file formats (e.g.CSV-1203[10]) still recommend a trailing EOF character to be appended as the last character in the file. However, typingControl+Zdoes not embed an EOF character into a file in eitherDOSorWindows, nor do theAPIsof those systems use the character to denote the actual end of a file.
Some programming languages (e.g.Visual Basic) will not read past a "soft" EOF when using the built-in text file reading primitives (INPUT, LINE INPUT etc.),[citation needed]and alternate methods must be adopted, e.g. opening the file in binary mode or using the File System Object to progress beyond it.
Character 26 was used to mark "End of file" even though ASCII calls this character Substitute, and has other characters to indicate "End of file". Number 28 which is called "File Separator" has also been used for similar purposes.
InUnix-like operating systems, this character is typically used inshellsas a way for the user tosuspendthe currently executing interactive process.[11]The suspended process can then be resumed inforeground(interactive) mode, or be made to resume execution inbackgroundmode, or beterminated. When entered by a user at theircomputer terminal, the currently running foreground process is sent a "terminal stop" (SIGTSTP) signal, which generally causes the process to suspend its execution. The user can later continue the process execution by using the "foreground" command (fg) or the "background" command (bg).
The Unicode Security Considerations report[12]recommends this character as a safe replacement for unmappable characters during character set conversion.
In many GUIs and applications,Control+Z(⌘ Command+ZonmacOS) can be used toundothe last action. In many applications, earlier actions than the last one can also be undone by pressingControl+Zmultiple times.Control+Zwas one of a handful ofkeyboardsequences chosen by the program designers atXerox PARCto controltext editing.
ASCIIandUnicoderepresentation of "substitute":
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TheIndian numbering systemis used inIndia,Pakistan,Nepal,Sri Lanka, andBangladeshto express large numbers, which differs from theInternational System of Units. Commonly used quantities includelakh(one hundred thousand) andcrore(ten million) – written as 100,000 and 10,000,000 respectively in somelocales.[1]For example: 150,000rupeesis "1.5lakhrupees" which can be written as "1,50,000 rupees", and 30,000,000 (thirty million) rupees is referred to as "3crorerupees" which can be written as "3,00,00,000 rupees".
There are names for numbers larger thancrore, but they are less commonly used. These includearab(100crore, 109),kharab(100arab, 1011),nilor sometimestransliteratedasneel(100 kharab, 1013),padma(100 nil, 1015),shankh(100 padma, 1017), andmahashankh(100 shankh, 1019). In common parlance (though inconsistent), thelakhandcroreterminology repeats for larger numbers. Thuslakh croreis 1012.
In the ancient Indian system, still in use in regional languages of India, there are words for (1062). These names respectively starting at 1000 aresahasra,ayuta,laksha,niyuta,koti,arbhudha,abhja,karva,nikarva,mahapadma,shanmkhu,jaladhi,amtya,madhya,paraardha. In the Indian system, now prevalent in the northern parts,[clarification needed]the next powers of ten are onelakh, tenlakh, onecrore, tencrore, onearab(or one hundredcrore), and so on.
The Indian system isdecimal(base-10), same as in theInternational System of Units, and the first fiveorders of magnitudeare named in a similar way: one (100), ten (101), one hundred (102), one thousand (103), and ten thousand (104). For higher powers of ten, naming diverges. The Indian system uses names for everysecondpower of ten:lakh(105),crore(107),arab(109),kharab(1011), etc. In the rest of the world,long and short scales, there are names for everythirdpower of ten. The short scale uses million (106), billion (109), trillion (1012), etc.
The Indian system groups digits of a large decimal representation differently than theInternational System of Units. The Indian system does group the first three digits to the left of the decimal point. But thereafter, groups by two digits to align with the naming of quantities at multiples of 100.[2]
Like English and other locales, the Indian system uses aperiodas thedecimal separatorand thecommafor grouping, while others use a comma for decimal separator and athin spaceor point to group digits.[3]
When speakers of indigenous Indian languages are speaking English, the pronunciations may be closer to their mother tongue; e.g. "lakh" and "crore" might be pronounced /lɑkʰ/, /kɑrɔːr/, respectively.
The table below includes the spelling and pronunciation of numbers in various Indian languages along with corresponding short scale names.
(bongo)দশ হাজার লাখ কোটি(dôś hāzār lākh kōṭi)
(mohabongo)শত হাজার লাখ কোটি(śoto hāzār lākh kōṭi)
There are various systems of numeration found in various ancient epic literature of India (itihasas). The following table gives one such system used in the ValmikiRamayana.[4]
The denominations by which land was measured in theKumaon Kingdomwere based on arable lands and thus followed an approximate system with local variations. The most common of these was avigesimal(base-20) numbering system with the main denomination called abisi(seeHindustani numberbīs), which corresponded to the land required to sow 20nalisof seed. Consequently, its actual land measure varied based on the quality of the soil.[5]This system became the established norm in Kumaon by 1891.[6]
Below is a list of translations for the words lakh and crore in other languages spoken in the Indian subcontinent:
Formal written publications in English in India tend to use lakh/crore for Indian currency and International numbering for foreign currencies.[7]
The official usage of this system is limited to the nations ofIndia,PakistanandBangladesh. It is universally employed within these countries, and is preferred to the International numbering system.[8]
Sri LankaandNepalused this system in the past but has switched to the International numbering system in recent years. In theMaldives, the term lakh is widely used in official documents and local speech. However, theInternational System of Unitsis preferred for higher denominations (such as millions).
Most institutions and citizens in India use the Indian number system. TheReserve Bank of Indiawas noted as a rare exception in 2015,[9]whereas by 2024 the Indian system was used for amounts in rupees and the International system for foreign currencies throughout the Reserve Bank's website.[10]
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Insyntacticanalysis, aconstituentis a word or a group of words that function as a single unit within a hierarchical structure. The constituent structure of sentences is identified usingtests for constituents.[1]These tests apply to a portion of a sentence, and the results provide evidence about the constituent structure of the sentence. Many constituents arephrases. A phrase is a sequence of one or more words (in some theories two or more) built around aheadlexical itemand working as a unit within a sentence. A word sequence is shown to be a phrase/constituent if it exhibits one or more of the behaviors discussed below. The analysis of constituent structure is associated mainly withphrase structure grammars, althoughdependency grammarsalso allow sentence structure to be broken down into constituent parts.
Tests for constituents are diagnostics used to identify sentence structure. There are numerous tests for constituents that are commonly used to identify the constituents of English sentences. 15 of the most commonly used tests are listed next: 1)coordination(conjunction), 2) pro-form substitution (replacement), 3)topicalization(fronting), 4)do-so-substitution, 5)one-substitution, 6)answer ellipsis(question test), 7)clefting, 8)VP-ellipsis, 9) pseudoclefting, 10) passivization, 11) omission (deletion), 12) intrusion, 13) wh-fronting, 14) general substitution, 15)right node raising(RNR).
The order in which these 15 tests are listed here corresponds to the frequency of use, coordination being the most frequently used of the 15 tests and RNR being the least frequently used. A general word of caution is warranted when employing these tests, since they often deliver contradictory results. The tests are merely rough-and-ready tools that grammarians employ to reveal clues about syntactic structure. Some syntacticians even arrange the tests on a scale of reliability, with less-reliable tests treated as useful to confirm constituency though not sufficient on their own. Failing to pass a single test does not mean that the test string is not a constituent, and conversely, passing a single test does not necessarily mean the test string is a constituent. It is best to apply as many tests as possible to a given string in order to prove or to rule out its status as a constituent.
The 15 tests are introduced, discussed, and illustrated below mainly relying on the same one sentence:[2]
By restricting the introduction and discussion of the tests for constituents below mainly to this one sentence, it becomes possible to compare the results of the tests. To aid the discussion and illustrations of the constituent structure of this sentence, the following two sentence diagrams are employed (D = determiner, N = noun, NP = noun phrase, Pa = particle, S = sentence, V = Verb, VP =verb phrase):
These diagrams show two potential analyses of the constituent structure of the sentence. A given node in a tree diagram is understood as marking a constituent, that is, a constituent is understood as corresponding to a given node and everything that that node exhaustively dominates. Hence the first tree, which shows the constituent structure according todependency grammar, marks the following words and word combinations as constituents:Drunks,off,the,the customers, andput off the customers.[3]The second tree, which shows the constituent structure according tophrase structure grammar, marks the following words and word combinations as constituents:Drunks,could,put,off,the,customers,the customers,put off the customers, andcould put off the customers. The analyses in these two tree diagrams provide orientation for the discussion of tests for constituents that now follows.
Thecoordinationtest assumes that only constituents can be coordinated, i.e., joined by means of a coordinator such asand,or, orbut:[4]The next examples demonstrate that coordination identifies individual words as constituents:
The square brackets mark the conjuncts of the coordinate structures. Based on these data, one might assume thatdrunks,could,put off, andcustomersare constituents in the test sentence because these strings can be coordinated withbums,would,drive away, andneighbors, respectively. Coordination also identifies multi-word strings as constituents:
These data suggest thatthe customers,put off the customers, andcould put off the customersare constituents in the test sentence.
Examples such as (a-g) are not controversial insofar as many theories of sentence structure readily view the strings tested in sentences (a-g) as constituents. However, additional data are problematic, since they suggest that certain strings are also constituents even though most theories of syntax do not acknowledge them as such, e.g.
These data suggest thatcould put off,put off these, andDrunks couldare constituents in the test sentence. Most theories of syntax reject the notion that these strings are constituents, though. Data such as (h-j) are sometimes addressed in terms of theright node raising(RNR) mechanism.
The problem for the coordination test represented by examples (h-j) is compounded when one looks beyond the test sentence, for one quickly finds that coordination suggests that a wide range of strings are constituents that most theories of syntax do not acknowledge as such, e.g.
The stringsfrom home on Tuesdayandfrom home on Tuesday on his bicycleare not viewed as constituents in most theories of syntax, and concerning sentence (m), it is very difficult there to even discern how one should delimit the conjuncts of the coordinate structure. The coordinate structures in (k-l) are sometimes characterized in terms of non-constituent conjuncts (NCC), and the instance of coordination in sentence (m) is sometimes discussed in terms of stripping and/orgapping.
Due to the difficulties suggested with examples (h-m), many grammarians view coordination skeptically regarding its value as a test for constituents. The discussion of the other tests for constituents below reveals that this skepticism is warranted, since coordination identifies many more strings as constituents than the other tests for constituents.[5]
Proformsubstitution, or replacement, involves replacing the test string with the appropriate proform (e.g. pronoun, pro-verb, pro-adjective, etc.). Substitution normally involves using a definite proform likeit,he,there,here, etc. in place of a phrase or a clause. If such a change yields a grammatical sentence where the general structure has not been altered, then the test string is likely a constituent:[6]
These examples suggest thatDrunks,the customers, andput off the customersin the test sentence are constituents. An important aspect of the proform test is the fact that it fails to identify most subphrasal strings as constituents, e.g.
These examples suggest that the individual wordscould,put,off, andcustomersshould not be viewed as constituents. This suggestion is of course controversial, since most theories of syntax assume that individual words are constituents by default. The conclusion one can reach based on such examples, however, is that proform substitution using a definite proform identifies phrasal constituents only; it fails to identify sub-phrasal strings as constituents.
Topicalizationinvolves moving the test string to the front of the sentence. It is a simple movement operation.[7]Many instances of topicalization seem only marginally acceptable when taken out of context. Hence to suggest a context, an instance of topicalization can be preceded by...andand a modal adverb can be added as well (e.g.certainly):
These examples suggest thatthe customersandput off the customersare constituents in the test sentence. Topicalization is like many of the other tests in that it identifies phrasal constituents only. When the test sequence is a sub-phrasal string, topicalization fails:
These examples demonstrate thatcustomers,could,put,off, andthefail the topicalization test. Since these strings are all sub-phrasal, one can conclude that topicalization is unable to identify sub-phrasal strings as constituents.
Do-so-substitution is a test that substitutes a form ofdo so(does so,did so,done so,doing so) into the test sentence for the target string. This test is widely used to probe the structure of strings containing verbs (becausedois a verb).[8]The test is limited in its applicability, though, precisely because it is only applicable to strings containing verbs:
The 'a' example suggests thatput off the customersis a constituent in the test sentence, whereas the b example fails to suggest thatcould put off the customersis a constituent, fordo socannot include the meaning of themodal verbcould. To illustrate more completely how thedo sotest is employed, another test sentence is now used, one that contains two post-verbal adjunct phrases:
These data suggest thatmet them,met them in the pub, andmet them in the pub because we had timeare constituents in the test sentence. Taken together, such examples seem to motivate a structure for the test sentence that has a left-branching verb phrase, because only a left-branching verb phrase can view each of the indicated strings as a constituent. There is a problem with this sort of reasoning, however, as the next example illustrates:
In this case,did soappears to stand in for the discontinuous word combination consisting ofmet themandbecause we had time. Such a discontinuous combination of words cannot be construed as a constituent. That such an interpretation ofdid sois indeed possible is seen in a fuller sentence such asYou met them in the cafe because you had time, and we did so in the pub. In this case, the preferred reading ofdid sois that it indeed simultaneously stands in for bothmet themandbecause we had time.
Theone-substitution test replaces the test string with the indefinite pronounoneorones.[9]If the result is acceptable, then the test string is deemed a constituent. Sinceoneis a type of pronoun,one-substitution is only of value when probing the structure of noun phrases. In this regard, the test sentence from above is expanded in order to better illustrate the manner in which one-substitution is generally employed:
These examples suggest thatcustomers,loyal customers,customers around here,loyal customers around here, andcustomers around here who we rely onare constituents in the test sentence. Some have pointed to a problem associated with theone-substitution in this area, however. This problem is that it is impossible to produce a single constituent structure of the noun phrasethe loyal customers around here who we rely onthat could simultaneous view all of the indicated strings as constituents.[10]Another problem that has been pointed out concerning theone-substitution as a test for constituents is the fact that it at times suggests that non-string word combinations are constituents,[11]e.g.
The word combination consisting of bothloyal customersandwho we rely onis discontinuous in the test sentence, a fact that should motivate one to generally question the value ofone-substitution as a test for constituents.
The answer fragment test involves forming a question that contains a single wh-word (e.g.who,what,where, etc.). If the test string can then appear alone as the answer to such a question, then it is likely a constituent in the test sentence:[12]
These examples suggest thatDrunks,the customers, andput off the customersare constituents in the test sentence. The answer fragment test is like most of the other tests for constituents in that it does not identify sub-phrasal strings as constituents:
These answer fragments are all grammatically unacceptable, suggesting thatcould,put,off, andcustomersare not constituents. Note as well that the latter two questions themselves are ungrammatical. It is apparently often impossible to form the question in a way that could successfully elicit the indicated strings as answer fragments. The conclusion, then, is that the answer fragment test is like most of the other tests in that it fails to identify sub-phrasal strings as constituents.
Cleftinginvolves placing the test string X within the structure beginning withIt is/was:It was X that....[13]The test string appears as the pivot of the cleft sentence:
These examples suggest thatDrunksandthe customersare constituents in the test sentence. Example c is of dubious acceptability, suggesting thatput off the customersmay not be constituent in the test string. Clefting is like most of the other tests for constituents in that it fails to identify most individual words as constituents:
The examples suggest that each of the individual wordscould,put,off,the, andcustomersare not constituents, contrary to what most theories of syntax assume. In this respect, clefting is like many of the other tests for constituents in that it only succeeds at identifying certain phrasal strings as constituents.
The VP-ellipsis test checks to see which strings containing one or more predicative elements (usually verbs) can be elided from a sentence. Strings that can be elided are deemed constituents:[14]The symbol ∅ is used in the following examples to mark the position of ellipsis:
These examples suggest thatput offis not a constituent in the test sentence, but thatimmediately put off the customers,put off the customers when they arrive, andimmediately put off the customers when they arriveare constituents. Concerning the stringput off the customersin (b), marginal acceptability makes it difficult to draw a conclusion aboutput off the customers.
There are various difficulties associated with this test. The first of these is that it can identify too many constituents, such as in this case here where it is impossible to produce a single constituent structure that could simultaneously view each of the three acceptable examples (c-e) as having elided a constituent. Another problem is that the test can at times suggest that a discontinuous word combination is a constituent, e.g.:
In this case, it appears as though the elided material corresponds to the discontinuous word combination includinghelpandin the office.
Pseudoclefting is similar to clefting in that it puts emphasis on a certain phrase in a sentence. There are two variants of the pseudocleft test. One variant inserts the test string X in a sentence starting with a free relative clause:What.....is/are X; the other variant inserts X at the start of the sentence followed by theit/areand then the free relative clause:X is/are what/who...Only the latter of these two variants is illustrated here.[15]
These examples suggest thatDrunks,the customers, andput off the customersare constituents in the test sentence. Pseudoclefting fails to identify most individual words as constituents:
The pseudoclefting test is hence like most of the other tests insofar as it identifies phrasal strings as constituents, but does not suggest that sub-phrasal strings are constituents.
Passivization involves changing an active sentence to a passive sentence, or vice versa. Theobjectof the active sentence is changed to thesubjectof the corresponding passive sentence:[16]
The fact that sentence (b), the passive sentence, is acceptable, suggests thatDrunksandthe customersare constituents in sentence (a). The passivization test used in this manner is only capable of identifying subject and object words, phrases, and clauses as constituents. It does not help identify other phrasal or sub-phrasal strings as constituents. In this respect, the value of passivization as test for constituents is very limited.
Omission checks whether the target string can be omitted without influencing the grammaticality of the sentence. In most cases, local and temporal adverbials, attributive modifiers, and optional complements can be safely omitted and thus qualify as constituents.[17]
This sentence suggests that the definite articletheis a constituent in the test sentence. Regarding the test sentence, however, the omission test is very limited in its ability to identify constituents, since the strings that one wants to check do not appear optionally. Therefore, the test sentence is adapted to better illustrate the omission test:
The ability to omitobnoxious,immediately, andwhen they arrivesuggests that these strings are constituents in the test sentence. Omission used in this manner is of limited applicability, since it is incapable of identifying any constituent that appears obligatorily. Hence there are many target strings that most accounts of sentence structure take to be constituents but that fail the omission test because these constituents appear obligatorily, such as subject phrases.
Intrusion probes sentence structure by having an adverb "intrude" into parts of the sentence. The idea is that the strings on either side of the adverb are constituents.[18]
Example (a) suggests thatDrunksandcould put off the customersare constituents. Example (b) suggests thatDrunks couldandput off the customersare constituents. The combination of (a) and (b) suggest in addition thatcouldis a constituent. Sentence (c) suggests thatDrunks could putandoff the customersare not constituents. Example (d) suggests thatDrunks could put offandthe customersare not constituents. And example (e) suggests thatDrunks could put off theandcustomersare not constituents.
Those that employ the intrusion test usually use a modal adverb likedefinitely. This aspect of the test is problematic, though, since the results of the test can vary based upon the choice of adverb. For instance, manner adverbs distribute differently than modal adverbs and will hence suggest a distinct constituent structure from that suggested by modal adverbs.
Wh-fronting checks to see if the test string can be fronted as a wh-word.[19]This test is similar to the answer fragment test insofar it employs just the first half of that test, disregarding the potential answer to the question.
These examples suggest thatDrunks,the customers, andput off the customersare constituents in the test sentence. Wh-fronting is like a number of the other tests in that it fails to identify many subphrasal strings as constituents:
These examples demonstrate a lack of evidence for viewing the individual wordswould,put,off,the, andcustomersas constituents.
The general substitution test replaces the test string with some other word or phrase.[20]It is similar to proform substitution, the only difference being that the replacement word or phrase is not a proform, e.g.
These examples suggest that the stringsDrunks,the customers, andcouldare constituents in the test sentence. There is a major problem with this test, for it is easily possible to find a replacement word for strings that the other tests suggest are clearly not constituents, e.g.
These examples suggest thatcould put,Drunks could, andcould put off theare constituents in the test sentence. This is contrary to what the other tests reveal and to what most theories of sentence structure assume. The value of general substitution as test for constituents is therefore suspect. It is like the coordination test in that it suggests that too many strings are constituents.
Right node raising, abbreviated as RNR, is a test that isolates the test string on the right side of a coordinate structure.[21]The assumption is that only constituents can be shared by the conjuncts of a coordinate structure, e.g.
These examples suggest thatcould put off the customers,put off the customers, andthe customersare constituents in the test sentence. There are two problems with the RNR diagnostic as a test for constituents. The first is that it is limited in its applicability, since it is only capable of identifying strings as constituents if they appear on the right side of the test sentence. The second is that it can suggest strings to be constituents that most of the other tests suggest are not constituents. To illustrate this point, a different example must be used:
These examples suggest thattheir bicycles (his bicycle) to us to use if need be,to us to use if need be, andto use if need beare constituents in the test sentence. Most theories of syntax do not view these strings as constituents, and more importantly, most of the other tests suggest that they are not constituents.
In short, these tests are not taken for granted because a constituent may pass one test and fail to pass many others. We need to consult our intuitive thinking when judging the constituency of any set of words.
A word of caution is warranted concerning the tests for constituents as just discussed above. These tests are found in textbooks on linguistics and syntax that are written mainly with the syntax of English in mind, and the examples that are discussed are mainly from English. The tests may or may not be valid and useful when probing the constituent structure of other languages. Ideally, a battery of tests for constituents can and should be developed for each language, catered to the idiosyncrasies of the language at hand.
Constituent structure analyses of sentences are a central concern for theories of syntax. A given theory can produce an analysis of constituent structure that is quite unlike the next. This point is evident with the two tree diagrams above of the sentenceDrunks could put off the customers, where the dependency grammar analysis of constituent structure looks very much unlike the phrase structure analysis. The crucial difference across the two analyses is that the phrase structure analysis views every individual word as a constituent by default, whereas the dependency grammar analysis sees only those individual words as constituents that do not dominate other words. Phrase structure grammars therefore acknowledge many more constituents than dependency grammars.
A second example further illustrates this point (D = determiner, N = noun, NP = noun phrase, Pa = particle, S = sentence, V = Verb, V' = verb-bar, VP = verb phrase):
The dependency grammar tree shows five words and word combinations as constituents:who,these,us,these diagrams, andshow us. The phrase structure tree, in contrast, shows nine words and word combinations as constituents:what,do,these,diagrams,show,us,these diagrams,show us, anddo these diagrams show us. The two diagrams thus disagree concerning the status ofdo,diagrams,show, anddo these diagrams show us, the phrase structure diagram showing them as constituents and the dependency grammar diagram showing them as non-constituents. To determine which analysis is more plausible, one turns to the tests for constituents discussed above.[22]
Within phrase structure grammars, views about of constituent structure can also vary significantly. Many modern phrase structure grammars assume that syntactic branching is always binary, that is, each greater constituent is necessarily broken down into two lesser constituents. More dated phrase structures analyses are, however, more likely to allow n-ary branching, that is, each greater constituent can be broken down into one, two, or more lesser constituents. The next two trees illustrate the distinction (Aux =auxiliary verb, AuxP = auxiliary verb phrase, Aux' = Aux-bar, D = determiner, N = noun, NP = noun phrase, P = preposition, PP = prepositional phrase, Pa = particle, S = sentence, t = trace, V = Verb, V' = verb-bar, VP = verb phrase):
The details in the second diagram here not crucial to the point at hand. This point is that the all branching there is strictly binary, whereas in the first tree diagram ternary branching is present twice, for the AuxP and for the VP. Observe in this regard that strictly binary branching analyses increase the number of (overt) constituents to what is possible. The word combinationshave sent many things to usandmany things to usare shown as constituents in the second tree diagram but not in the first. Which of these two analyses is better is again at least in part a matter of what the tests for constituents can reveal.
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In mathematics,arithmetic geometryis roughly the application of techniques fromalgebraic geometryto problems innumber theory.[1]Arithmetic geometry is centered aroundDiophantine geometry, the study ofrational pointsofalgebraic varieties.[2][3]
In more abstract terms, arithmetic geometry can be defined as the study ofschemesoffinite typeover thespectrumof thering of integers.[4]
The classical objects of interest in arithmetic geometry are rational points:sets of solutionsof asystem of polynomial equationsovernumber fields,finite fields,p-adic fields, orfunction fields, i.e.fieldsthat are notalgebraically closedexcluding thereal numbers. Rational points can be directly characterized byheight functionswhich measure their arithmetic complexity.[5]
The structure of algebraic varieties defined over non-algebraically closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry. Over finite fields,étale cohomologyprovidestopological invariantsassociated to algebraic varieties.[6]p-adic Hodge theorygives tools to examine when cohomological properties of varieties over thecomplex numbersextend to those overp-adic fields.[7]
In the early 19th century,Carl Friedrich Gaussobserved that non-zerointegersolutions tohomogeneous polynomialequations withrationalcoefficients exist if non-zero rational solutions exist.[8]
In the 1850s,Leopold Kroneckerformulated theKronecker–Weber theorem, introduced the theory ofdivisors, and made numerous other connections between number theory andalgebra. He then conjectured his "liebster Jugendtraum" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as histwelfth problem, which outlines a goal to have number theory operate only with rings that are quotients ofpolynomial ringsover the integers.[9]
In the late 1920s,André Weildemonstrated profound connections between algebraic geometry and number theory with his doctoral work leading to theMordell–Weil theoremwhich demonstrates that the set of rational points of anabelian varietyis afinitely generated abelian group.[10]
Modern foundations of algebraic geometry were developed based on contemporarycommutative algebra, includingvaluation theoryand the theory ofidealsbyOscar Zariskiand others in the 1930s and 1940s.[11]
In 1949,André Weilposed the landmarkWeil conjecturesabout thelocal zeta-functionsof algebraic varieties over finite fields.[12]These conjectures offered a framework between algebraic geometry and number theory that propelledAlexander Grothendieckto recast the foundations making use ofsheaf theory(together withJean-Pierre Serre), and later scheme theory, in the 1950s and 1960s.[13]Bernard Dworkproved one of the four Weil conjectures (rationality of the local zeta function) in 1960.[14]Grothendieck developed étale cohomology theory to prove two of the Weil conjectures (together withMichael ArtinandJean-Louis Verdier) by 1965.[6][15]The last of the Weil conjectures (an analogue of theRiemann hypothesis) would be finally proven in 1974 byPierre Deligne.[16]
Between 1956 and 1957,Yutaka TaniyamaandGoro Shimuraposed theTaniyama–Shimura conjecture(now known as the modularity theorem) relatingelliptic curvestomodular forms.[17][18]This connection would ultimately lead tothe first proofofFermat's Last Theoremin number theory through algebraic geometry techniques ofmodularity liftingdeveloped byAndrew Wilesin 1995.[19]
In the 1960s, Goro Shimura introducedShimura varietiesas generalizations ofmodular curves.[20]Since the 1979, Shimura varieties have played a crucial role in theLanglands programas a natural realm of examples for testing conjectures.[21]
In papers in 1977 and 1978,Barry Mazurproved thetorsion conjecturegiving a complete list of the possible torsion subgroups of elliptic curves over the rational numbers. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certainmodular curves.[22][23]In 1996, the proof of the torsion conjecture was extended to all number fields byLoïc Merel.[24]
In 1983,Gerd Faltingsproved theMordell conjecture, demonstrating that a curve of genus greater than 1 has only finitely many rational points (where the Mordell–Weil theorem only demonstratesfinite generationof the set of rational points as opposed to finiteness).[25][26]
In 2001, the proof of thelocal Langlands conjectures for GLnwas based on the geometry of certain Shimura varieties.[27]
In the 2010s,Peter Scholzedevelopedperfectoid spacesand new cohomology theories in arithmetic geometry over p-adic fields with application toGalois representationsand certain cases of theweight-monodromy conjecture.[28][29]
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Informal semantics,Strawson entailmentis a variant of the concept ofentailmentwhich is insensitive topresuppositionfailures. Formally, a sentencePStrawson-entails a sentenceQiffQis always true whenPis true andQs presuppositions are satisfied. For example, "Maria loves every cat" Strawson-entails "Maria loves her cat" because Maria could not love every cat without loving her own, assuming that she has one. This would not be an ordinary entailment, since the first sentence could be true while the second is undefined on account of a presupposition failure; loving every cat would not guarantee that she owns a cat.[1][2][3]
Strawson entailment has played an important role in semantic theory since somenatural languageexpressions have been argued to be sensitive to Strawson-entailment rather than pure entailment. For instance, the textbook theory of weaknegative polarity itemsholds that they are licensed only in Strawson-downward entailingenvironments. Other phenomena that have been analyzed using Strawson entailment include temporal adverbials, covert reciprocals, andscalar implicature.[1][2][3][4]Although the concept is widely used within formal semantics, it is not universally adopted and alternative proposals have argued both for returning to pure entailment and for generalizing the notion further to consider not-at-issue content beyond presupposition.[3][1][5]
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Thelaw of trivialityisC. Northcote Parkinson's 1957 argument that people within an organization commonly give disproportionate weight to trivial issues.[1]Parkinson provides the example of a fictional committee whose job was to approve the plans for anuclear power plantspending the majority of its time on discussions about relatively minor but easy-to-grasp issues, such as what materials to use for the staff bicycle shed, while neglecting the proposed design of the plant itself, which is far more important and a far more difficult and complex task.
The law has been applied tosoftware developmentand other activities.[2]The termsbicycle-shed effect,bike-shed effect, andbike-sheddingwere coined based on Parkinson's example; it was popularized in theBerkeley Software Distributioncommunity by the Danish software developerPoul-Henning Kampin 1999[3]and, due to that, has since become popular within the field of software development generally.
The concept was first presented as a corollary of his broader "Parkinson's law" spoof of management. He dramatizes this "law of triviality" with the example of a committee's deliberations on an atomic reactor, contrasting it to deliberations on a bicycle shed. As he put it: "The time spent on any item of the agenda will be in inverse proportion to the sum [of money] involved." A reactor is so vastly expensive and complicated that an average person cannot understand it (seeambiguity aversion), so one assumes that those who work on it understand it. However, everyone can visualize a cheap, simple bicycle shed, so planning one can result in endless discussions because everyone involved wants to implement their own proposal and demonstrate personal contribution.[4]
After a suggestion of building something new for the community, like a bike shed, problems arise when everyone involved argues about the details. This is a metaphor indicating that it is not necessary to argue about every little feature based simply on having the knowledge to do so. Some people have commented that the amount of noise generated by a change is inversely proportional to the complexity of the change.[3]
Behavioral research has produced evidence which confirms theories proposed by the law of triviality. People tend to spend more time on small decisions than they should, and less time on big decisions than they should. A simple explanation is that during the process of making a decision, one has to assess whether enough information has been collected to make the decision. If people make mistakes about whether they have enough information, then they will tend to feel overwhelmed by large and complex matters and stop collecting information too early to adequately inform their big decisions. The reason is that big decisions require collecting information for a long time and working hard to understand its complex ramifications. This leaves more of an opportunity to make a mistake (and stop) before getting enough information. Conversely, for small decisions, where people should devote little attention and act without hesitation, they may inefficiently continue to ponder for too long, partly because they are better able to understand the subject.[5]
There are several other principles, well known in specific problem domains, which express a similar sentiment.
Sayre's lawis a more general principle, which holds (among other formulations) that "In any dispute, the intensity of feeling is inversely proportional to the value of the issues at stake"; many formulations of the principle focus onacademia.
Wadler's law, named for computer scientistPhilip Wadler,[6]is a principle which asserts that the bulk of discussion onprogramming-language designcenters onsyntax(which, for purposes of the argument, is considered a solved problem), as opposed tosemantics.[7]
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In computing, ashell builtinis acommandor afunction, exposed by ashell, that is implemented in the shell itself, instead of an externalprogramwhich the shell would load and execute.[1][2][3][4]
A shell builtin starts faster than an external program because there is no program loading overhead. However, its implementation code is in the shell program, and thus modifying it requires modifying the shell. Therefore, a shell builtin is usually only used for simple, almost trivial, commands, such as text output.
Some commands must be implemented as builtins due to the nature of theoperating system.
Notably, thecdcommand, which changes theworking directoryof the shell is often a builtin since a program runs in a separateprocessand working directory is specific to each process. Runningcdas an external program would not affect the working directory of the shell that loaded it.[5]
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Inchemistry,molecular symmetrydescribes thesymmetrypresent inmoleculesand the classification of these molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can be used to predict or explain many of a molecule'schemical properties, such as whether or not it has adipole moment, as well as its allowedspectroscopic transitions. To do this it is necessary to usegroup theory. This involves classifying the states of the molecule using theirreducible representationsfrom thecharacter tableof the symmetry group of the molecule. Symmetry is useful in the study ofmolecular orbitals, with applications to theHückel method, toligand field theory, and to theWoodward–Hoffmann rules.[1][2]Many university level textbooks onphysical chemistry,quantum chemistry,spectroscopy[3]andinorganic chemistrydiscuss symmetry.[4][5][6][7][8]Another framework on a larger scale is the use ofcrystal systemsto describecrystallographicsymmetry in bulk materials.
There are many techniques for determining the symmetry of a given molecule, includingX-ray crystallographyand various forms ofspectroscopy.Spectroscopic notationis based on symmetry considerations.
The point group symmetry of a molecule is defined by the presence or absence of 5 types ofsymmetry element.
The five symmetry elements have associated with them five types ofsymmetry operation, which leave the geometry of the molecule indistinguishable from the starting geometry. They are sometimes distinguished from symmetry elements by acaretorcircumflex. Thus,Ĉnis the rotation of a molecule around an axis andÊis the identity operation. A symmetry element can have more than one symmetry operation associated with it. For example, theC4axis of thesquarexenon tetrafluoride(XeF4) molecule is associated with twoĈ4rotations in opposite directions (90° and 270°), aĈ2rotation (180°) andĈ1(0° or 360°). BecauseĈ1is equivalent toÊ,Ŝ1to σ andŜ2toî, all symmetry operations can be classified as either proper or improper rotations.
For linear molecules, either clockwise or counterclockwise rotation about the molecular axis by any angle Φ is a symmetry operation.
The symmetry operations of a molecule (or other object) form agroup. In mathematics, a group is a set with abinary operationthat satisfies the four properties listed below.
In asymmetry group, the group elements are the symmetry operations (not the symmetry elements), and the binary combination consists of applying first one symmetry operation and then the other. An example is the sequence of aC4rotation about the z-axis and a reflection in the xy-plane, denoted σ(xy)C4. By convention the order of operations is from right to left.
A symmetry group obeys the defining properties of any group.
Theorderof a group is the number of elements in the group. For groups of small orders, the group properties can be easily verified by considering its composition table, a table whose rows and columns correspond to elements of the group and whose entries correspond to their products.
The successive application (orcomposition) of one or more symmetry operations of a molecule has an effect equivalent to that of some single symmetry operation of the molecule. For example, aC2rotation followed by a σvreflection is seen to be a σv' symmetry operation: σv*C2= σv'. ("OperationAfollowed byBto formC" is writtenBA=C).[11]Moreover, the set of all symmetry operations (including this composition operation) obeys all the properties of a group, given above. So (S,*) is a group, whereSis the set of all symmetry operations of some molecule, and * denotes the composition (repeated application) of symmetry operations.
This group is called thepoint groupof that molecule, because the set of symmetry operations leave at least one point fixed (though for some symmetries an entire axis or an entire plane remains fixed). In other words, a point group is a group that summarises all symmetry operations that all molecules in that category have.[11]The symmetry of a crystal, by contrast, is described by aspace groupof symmetry operations, which includestranslationsin space.
Assigning each molecule a point group classifies molecules into categories with similar symmetry properties. For example, PCl3, POF3, XeO3, and NH3all share identical symmetry operations.[12]They all can undergo the identity operationE, two differentC3rotation operations, and three different σvplane reflections without altering their identities, so they are placed in one point group,C3v, with order 6.[11]Similarly, water (H2O) and hydrogen sulfide (H2S) also share identical symmetry operations. They both undergo the identity operationE, oneC2rotation, and two σvreflections without altering their identities, so they are both placed in one point group,C2v, with order 4.[13]This classification system helps scientists to study molecules more efficiently, since chemically related molecules in the same point group tend to exhibit similar bonding schemes, molecular bonding diagrams, and spectroscopic properties.[11]Point group symmetry describes the symmetry of a molecule when fixed at its equilibrium configuration in a particular electronic state. It does not allow for tunneling between minima nor for the change in shape that can come about from the centrifugal distortion effects of molecular rotation.
The following table lists many of thepoint groupsapplicable to molecules, labelled using theSchoenflies notation, which is common in chemistry and molecular spectroscopy. The descriptions include common shapes of molecules, which can be explained by theVSEPR model. In each row, the descriptions and examples have no higher symmetries, meaning that the named point group capturesallof the point symmetries.
All of the group operations described above and the symbols for crystallographic point groups themselves were first published byArthur Schoenfliesin 1891 but the groups had been applied by other researchers to the external morphology of crystals much earlier in the 19th century.
In 1914Max von Lauepublished the results of experiments using x-ray diffraction to elucidate the internal structures of crystals producing a limited version of the table of "Laue classes" shown. When adapted for molecular work this table first divides point groups into three kinds: asymmetric, symmetric and spherical tops. These are categories based on the angular momentum of molecules, having respectively 3, 2 and 1 distinct values of angular momentum, becoming more symmetrical down the table. A further sub-division into systems is defined by the rotational groupGin the leftmost column then into rows of Laue classes that take the form of cyclic and dihedral groups in the first two categories and tetrahedral and octahedral classes in the third. Rotational groups occur in the first column and define the non-rotational groups in their class. The second and third columns contain non-rotational groups belonging to the same abstract group as that in the first column A fourth column contains groups that are a direct product of their defining rotational group with space inversion (parity inversion) and so are of twice the order of other members of the class. Groups in this column contain the inversion operation itself as a member. For example, seven groups in the hexagonal system all contain the C6cyclic system, mostly as physical rotational group but in the third column of the table as an abstract group. So, C6and C3hare distinct manifestations of the same group while C6his simply C6xi. Groups D6, C6vand D3hare also example of the same abstract group and D6his the direct product D6xi.
It is difficult to overstate the importance of the Laue class in the applications of point groups to the description of physical properties at the molecular level. Since all the point groups of a Laue class have the same abstract structure, they also have exactly the same irreducible representations and character tables. Any representation of one is automatically a representation of the class and any group in it. One important point is that higher symmetry molecules do not cease to have the lower symmetry of their subgroups. Using the hexagonal example again, picture the series of groups C6of order 6, D6of order 12 and D6hof order 24, each group being of twice the order of the previous one. Once the irreducible representations of a physical example of a cyclic group have been found it is usually a simple process to extend this to the higher order groups. Laue found that x-ray diffraction was unable to distinguish between such groups Tetrahedral and octahedral point groups have a relationship similar to that between cyclic and dihedral groups and the tetrahedral g occurs in all cubic groups.
A set ofmatricesthat multiply together in a way that mimics the multiplication table of the elements of a group is called arepresentationof the group. The simplest method of obtaining a representation of molecular group transformations is to trace the movements of atoms in a molecule when symmetry operations are applied. For example, a water molecule belonging to theC2vpoint group might have an oxygen atom labelled 1 and two hydrogen atoms labelled 2 and 3 as shown in the right hand column vector below. If the hydrogen atoms are imagined to rotate by 180 degrees about an axis passing through the oxygen atom we have the familiarC2operation of this point group. The oxygen atom in position number 1 stays in position but the atoms in positions 2 and 3 are moved to positions 3 and 2 in the resulting column vector. The matrix connecting the two provides a 3 x 3 representation for this operation.
This point group only contains four operations and matrices for the other three operations are obtained similarly, including the identity matrix which just contains 1's on the leading diagonal (top left to bottom right) and 0's elsewhere. Having obtained the representation matrices in this way it is not difficult to show that they multiply out in exactly the same way as the operations themselves.
Although an infinite number of such representations exist, theirreducible representations(or "irreps") of the group are all that are needed as all other representations of the group can be described as adirect sumof the irreducible representations. The first step in finding the irreps making up a given representation is to sum up the values of the leading diagonals for each matrix so, taking the identity matrix first then the matrices in the order above, one obtains (3, 1, 3, 1). These values are the traces or characters of the four matrices. Asymmetric point groups such asC2vonly have 1-dimensional irreps so the character of an irrep is exactly the same is the irrep itself and the following table can be interpreted as irreps or characters.
Looking again at the characters obtained for the 3D representation above (3, 1, 3, 1), we only need simple arithmetic to break this down into irreps. Clearly, E = 3 means there are three irreps and a C2representation sum of 1 means there must be two A and one B irreps so the only combination that adds up to the characters derived is 2A1+ B1
Robert Mullikenwas the first to publish character tables in English and so the notation used to label irreps in the above table is called Mulliken notation. For asymmetric groups it consists of letters A and B with subscripts 1 and 2 as above and subscripts g and u as in the C2hexample below. (Subscript 3 also appears in D2) The irreducible representations are those matrix representations in which the matrices are in their most diagonal form possible and for asymmetric groups this means totally diagonal. One further thing to note about the irrep/character table above is the appearance of polar and axial base vector symbols on the right hand side. This tells us that, for example, cartesian base vector x transforms as irrep B1under the operations of this group. The same collection of product base vectors is used for all asymmetric groups but symmetric and spherical groups use different sets of product base vectors.
Point group C2hhas the operations {E, C2, i, σh} and the 1,5-dibromonapthalene (C10H6Br2) shown in the figure belongs to this symmetry group. It is possible to construct four 18 x 18 matrices representing the transformations of atoms during its symmetry operations in the style of the water molecule example above and reduce it to 18 1D irreps. Notice however that carbon atom number 1 either stays in place or it is exchanged with carbon atom number 5 and these two atoms can be analysed separately from all the other atoms in the molecule. The transformation matrix for these two toms alone during the molecular C2rotation is
with character 0. When this computation is carried out for each of the operations above the characters obtained are (2,0,0.2) because two operations leave the atoms in place and two move them. The irrep table for this group is below. The first column tells us there are two1D irreps, the second column (C2) that there is one A and one B while columns 3 and 4 reveal that one irrep has subscript g the other has to have subscript u. This means that the irreps resulting from the two atoms are Ag+ Bu. In fact, the 18 atoms in this molecule are paired off in exactly the same way as carbon atoms 1 and 5 so that, from a symmetry perspective, the atom consists of 9 pairs of equivalent atoms related through symmetry. It follows that each pair contributes the same irreps as the pair examined above to give a total 18 dimensional irrep result of 9(Ag+ Bu).
Symmetric point groups are divided into systems based on the increasing order of the main rotational axis from three to infinity. Systems are in turn divided into cyclic and dihedral groups and within a system the order of the dihedral group is twice that of the cyclic group. Cyclic groups only have one dimensional representations as shown in the table of irreps and the number of irreps is equal to the order of the group. The irreps shown use standard notation for the rotational group of a class but Mulliken sometimes gave different symbols to other members of the same class even though they belong to the same abstract group and therefore have the same irreps.
Dihedral point groups contain a cyclic group of the same rotational order: so group Dnalways contains group Cnas an index-2 subgroup. It follows that dihedral irreps are superimposed on cyclic irreps because the cyclic group within a dihedral one does not cease to be a cyclic group. A dihedral group also contains a 2-fold rotational axis at right angles to the main cyclic axis and this has two consequences. Firstly, the A and B cyclic irreps are split into pairs of one dimensional irreps identified by subscripts 1 and 2. Secondly, pairs of 1D E−xand E+xcyclic irreps combine to form single Ex2D irreps in the dihedral group because the 2-fold horizontal rotation makes pairs of rotations equivalent. For example, a 60 degree rotation about the main axis becomes equivalent to a (360 − 60) degree rotation because the 2-fold horizontal rotation makes them equivalent. Combinations of this kind are said to form a class. Infinite order dihedral group irreps sometimes use Greek symbol descriptions,Σ,Π,Δthat follow from early linear molecule calculations.
Spherical point group representations
Spherical classes are defined by the tetrahedral, octahedral and icosahedral rotational groups T, O and I. The first two of these, T and O, are related in much the same way as cyclic and dihedral groups are related in symmetric groups. Both tetrahedral and octahedral molecules are often shown with their atoms inscribed in the apices or faces of cubes and might be considered as a single "cubic" system. Every point group in this system contains the simple tetrahedral rotational group as a subgroup. Methane (CH4) is often used as an example and, although often described as a tetrahedral molecule because of the very visible rotational symmetry, it really belongs to the octahedral symmetry class. Considering methane first as a tetrahedral molecule the 12 operations of group T are {E, 3 x c, 4 x b, 4 x b3} where c is a 180 degree rotation along x,y and z axes and b is a 120 degree rotation about the apices of a cube. Character tables under these four headings exhibit the corresponding four irreps A, E+1, E-1and T and it would it is not difficult to convert the transformations of atoms during the symmetry operations to reducible matrices and thence to molecular irreps but this not necessary. Methane has two sets of equivalent atoms that are transformed into each other during operations: a single carbon atom and 4 hydrogen atoms. A single atom can only ever be transformed into itself and therefore always contributes the most symmetrical irrep to the end total irrep count. Additionally, there is a rule of group theory that the most symmetrical irrep must occur once and only once in the irreps of any equivalent atom set so the five dimensions of irreps being sought contain 2A and three others. The only way of filling the remaining three dimensions is to adopt 3D irrep T so the irreps are 2A + T. (E irreps have to be taken in pairs in physical molecular applications).
methane sulfur hexafluoride
Extending this treatment to the octahedral group Tdrequires six 4-fold roto-inversion operations (f) about the main axes and six 2-fold roto-inversions (a), appearing as mirror reflections through opposite edges of the imaginary cube in which methane is placed. So half the operations of this group are rotational and half non-rotational. Rotational group T exists within non-rotational group Td= {E, 3 x c, 8 x b/b3, 6 x f, 6 x a} T irreps A, E+1, E-1and T are promoted to Tdirreps A1, A2, E1and T1and T2. (E irreps collapse into one because the 3-fold irreps become equivalent while A and T expand into two). Reasoning as above we know that the irreps in Tdmust be 2A1+ Txso the last step is to find the 3D subscript. A brief look at the 4 x 4 transformation matrix for the 4-fold rotation operation f shows character Ch(f) = 0 and the subscript x has to be 2 to balance the 1 on the A irrep.
Sulfur hexafluoride can be treated first as a tetrahedral molecule T, then as octahedral O and finally as centred molecule Oi.There are two sets of equivalent atoms consisting of a single sulfur atom and six fluorine atoms. Transformations of the fluorine atoms generate a six dimensional representation that can only reduce into the direct sum of tetrahedral irreps A, E+1, E-1and T because the direct sum must include the most symmetrical irrep once and only once. A direct sum of 5 can only be made up from a 2 and a 3 - no other combination is possible. These irreps are "promoted" to 2A1+ E1+ Txin group O. To get the x observe that the 4-fold rotation in SF6 has character Ch(f) = 2 because two atoms stay in position and a glance at this column of the table suggests A1+ E1+ T1. Finally the inversion operation (i) applied go the fluorine atoms has character Ch(i) = 0 indicating equal numbers ofgandusubscripts (because none of the atoms remains in position). Since the most symmetrical irrep must occur once the only possible result is A1g+ E1g+ T1u. The single sulfur atom always has the most symmetric irrep to the final reduction of the seven dimensional matrices to a direct sum is 2A1g+ E1g+ T1u.
The following reference of character tables uses symbols Zxfor abstract cyclic groups Cxwith A4and S4(alternating and symmetric permutations of 4 objects for T and O. Many authors just use C, T and O in two senses, making it clear which is intended.
Much of our understanding of quantum theory was developed during the early years of the 20th century, leading up to Schrodinger's three dimensional wave equation.
E. Bright Wilsonused character tables in 1934 to predict the symmetry of vibrationalnormal modes.[16]
Hans Betheused characters of point group operations in his study ofligand field theoryin 1929, andEugene Wignerused group theory to explain the selection rules ofatomic spectroscopy.[17]The first character tables were compiled byLászló Tisza(1933), in connection to vibrational spectra.
The complete set of 32 crystallographic point groups was published in 1936 by Rosenthal and Murphy[18]
When Schrodinger's 3D wave equation is applied to a one-electron atom it provides a number of solutions called wave functions that are then used to label the allowed energy levels in that atom. Exact solutions of this kind are usually described by three quantum numbers, n, l and m from which the probable radial and angular distribution of the electron around the atom can be computed. This type of deduction leads to the familiar s, p, d, f, ... description of atomic orbitals based on the l and m quantum numbers. Each solution is a base vector from which more complex structures may be constructed. Descriptions of many-electron atoms use the one-electron model to build models that are sometimes pictured as multiple electrons in the simple structure. Molecular orbitals then take linear combinations of atomic orbitals (LCAOs) to explain the distribution of electrons over multiple atoms within a molecule. Atomic orbital symmetry follows from the angular part of the wave function which increases in complexity in the series s,p,d,f,... so that s orbitals only have radial symmetry while p orbital base vectors have a symmetry identical to that of the Cartesian polar base vectors.
Consider the example of water (H2O), which has theC2vsymmetry described above. The 2pxorbitalof oxygen has, like the x base vector, B1symmetry. It is oriented perpendicular to the plane of the molecule and switches sign with aC2and a σv'(yz) operation, but remains unchanged with the other two operations (obviously, the character for the identity operation is always +1). This orbital's character set is thus {1, −1, 1, −1}, corresponding to the B1irreducible representation. Likewise, the 2pzorbital is seen to have the symmetry of the A1irreducible representation (i.e.: none of the symmetry operations change it), 2pyB2, and the 3dxyorbital A2. These assignments are noted in the rightmost columns of the table.
Eachmolecular orbitalalso has the symmetry of one irreducible representation. For example,ethylene(C2H4) has symmetry group D2h, and its highest occupied molecular orbital (HOMO) is the bonding pi orbital which forms a basis for its irreducible representation B1u.[19]
One can determine the symmetry operations of the point group for a particular molecule by considering the geometrical symmetry of its molecular model. However, when one uses a point group to classify molecular states, the operations in it are not to be interpreted in the same way. Instead the operations are interpreted as rotating and/or reflecting the vibronic (vibration-electronic) coordinates and these operations commute with the vibronic Hamiltonian.[20]They are "symmetry operations" for that vibronic Hamiltonian. The point group is used to classify by symmetry the vibronic eigenstates of a rigid molecule. The symmetry classification of the rotational levels, the eigenstates of the full (rotation-vibration-electronic) Hamiltonian, can be achieved through the use of the appropriate permutation-inversion group (called themolecular symmetry group), as introduced byLonguet-Higgins.[21]
Each normal mode ofmolecular vibrationhas a symmetry which forms a basis for one irreducible representation of the molecular symmetry group.[22]For example, the water molecule has three normal modes of vibration: symmetric stretch in which the two O-H bond lengths vary in phase with each other, asymmetric stretch in which they vary out of phase, and bending in which the bond angle varies. The molecular symmetry of water is C2vwith four irreducible representations A1, A2, B1and B2. The symmetric stretching and the bending modes have symmetry A1, while the asymmetric mode has symmetry B2. The overall symmetry of the three vibrational modes is therefore Γvib= 2A1+ B2.[22][23]
The molecular symmetry ofammonia(NH3) is C3v, withsymmetry operationsE, C3and σv.[9]For N = 4 atoms, the number of vibrational modes for a non-linear molecule is 3N-6 = 6, due to the relative motion of thenitrogenatom and the three hydrogen atoms. All threehydrogenatoms travel symmetrically along the N-H bonds, either in the direction of the nitrogen atom or away from it. This mode is known assymmetric stretch(v₁) and reflects the symmetry in the N-H bond stretching. Of the three vibrational modes, this one has the highestfrequency.[24]
In theBending(ν₂) vibration, the nitrogen atom stays on the axis of symmetry, while the three hydrogen atoms move in different directions from one another, leading to changes in the bond angles. The hydrogen atoms move like an umbrella, so this mode is often referred to as the "umbrella mode".[26]
There is also anAsymmetric Stretchmode (ν₃) in which one hydrogen atom approaches the nitrogen atom while the other two hydrogens move away.
The total number of degrees of freedom for each symmetry species (orirreducible representation) can be determined. Ammonia has four atoms, and each atom is associated with threevector components. The symmetry group C3vfor NH3has the three symmetry species A1, A2and E. The modes of vibration include the vibrational, rotational and translational modes.
Total modes = 3A1+ A2+ 4E. This is a total of 12 modes because each E corresponds to 2 degenerate modes (at the same energy).
Rotational modes= A2+ E (3 modes)
Translational modes = A1+ E
Vibrational modes= Total modes - Rotational modes - Translational modes = 3A1+ A2+ 4E - A2- E - A1- E = 2A1+ 2E (6 modes).
As discussed above in§ The molecular symmetry group, point groups are useful for classifying the vibrational and electronic states ofrigidmolecules (sometimes calledsemi-rigidmolecules) which undergo only small oscillations about a single equilibrium geometry.Longuet-Higginsintroduced the molecular symmetry group (a more general type of symmetry group)[21]suitable not only for classifying the vibrational and electronic states of rigid molecules but also for classifying their rotational and nuclear spin states. Further, such groups can be used to classify the states ofnon-rigid(orfluxional) molecules that tunnel between equivalent geometries[27]and to allow for the distorting effects of molecular rotation. The symmetry operations in the molecular symmetry group are so-called 'feasible' permutations of identical nuclei, or inversion with respect to the center of mass (theparityoperation), or a combination of the two, so that the group is sometimes called a "permutation-inversion group".[21][28]
Examples of molecular nonrigidity abound. For example,ethane(C2H6) has three equivalentstaggered conformations. Tunneling between the conformations occurs at ordinary temperatures byinternal rotation of one methyl grouprelative to the other. This is not a rotation of the entire molecule about theC3axis, although each conformation hasD3dsymmetry, as in the table above. The molecule2-butyne(dimethylacetylene) has the same molecular symmetry group (G36} as ethane but a very much
lower torsional barrier. Similarly,ammonia(NH3) has two equivalent pyramidal (C3v) conformations which are interconverted by the process known asnitrogen inversion.
Additionally, themethanemolecule (CH4) andTrihydrogen cation{H3+) have highly symmetric equilibrium structures withTdandD3hpoint group symmetries respectively; they lack permanent electric dipole moments but they do have very weak pure rotation spectra because of rotational centrifugal distortion.[29][30]
Sometimes it is necessary to consider together electronic states having different point group symmetries at equilibrium. For example, in its ground (N) electronic state the ethylene molecule C2H4hasD2hpoint group symmetry whereas in the excited (V) state it hasD2dsymmetry. To treat these two states together it is necessary to
allow torsion and to use thedouble groupof the molecular symmetry groupG16.[31]
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Acourtesy name(Chinese:字;pinyin:zì;lit.'character'), also known as astyle name, is an additional name bestowed upon individuals at adulthood, complementing their given name.[1]This tradition is prevalent in theEast Asian cultural sphere, particularly inChina,Japan, andVietnam.[2]Courtesy names are a marker of adulthood and were historically given to men at the age of 20, and sometimes to women upon marriage.
Unlikeart names, which are more akin topseudonymsorpen names, courtesy names served a formal and respectful purpose.[1]In traditional Chinese society, using someone's given name in adulthood was considered disrespectful among peers, making courtesy names essential for formal communication and writing.
Courtesy names often reflect the meaning of the given name or use homophonic characters, and were typically disyllabic after theQin dynasty. The practice also extended to other East Asian cultures, and was sometimes adopted byMongolsandManchusduring theQing dynasty. The choice of a courtesy name was significant, intended to express moral integrity and respect within the cultural context.
A courtesy name is a name traditionally given to Chinese men at the age of 20sui, marking theircoming of age. It was sometimes given to women, usually upon marriage.[1]The practice is no longer common in modern Chinese society. According to theBook of Rites, after a man reached adulthood, it was disrespectful for others of the same generation to address him by hisgiven name.[3]Thus, the given name was reserved for oneself and one's elders, whereas the courtesy name would be used by adults of the same generation to refer to one another on formal occasions or in writing. Another translation ofziis "style name", but this translation has been criticised as misleading, because it could imply an official or legal title.[1]
Generally speaking, courtesy names before theQin dynastywere one syllable, and from the Qin to the 20th century they were mostlydisyllabic, consisting of twoChinese characters.[1]Courtesy names were often relative to the meaning of the person's given name, the relationship could be synonyms, relative affairs, or rarely but sometimes antonym. For example,Chiang Kai-shek's given name (中正,romanizedas Chung-cheng) and courtesy name (介石, romanized as Kai-shek) are both from theyù(豫) hexagram 16 ofI Ching.[4]
Another way to form a courtesy name is to use the homophonic characterzi(子) – a respectful title for a man – as the first character of the disyllabic courtesy name. Thus, for example,Gongsun Qiao's courtesy name was Zichan (子產), andDu Fu's was Zimei (子美). It was also common to construct a courtesy name by using as the first character one which expresses the bearer's birth order among male siblings in his family. ThusConfucius, whose name was Kong Qiu (孔丘), was given the courtesy name Zhongni (仲尼), where the first characterzhongindicates that he was the second son born into his family. The characters commonly used arebo(伯) for the first,zhong(仲) for the second,shu(叔) for the third, andji(季) typically for the youngest, if the family consists of more than three sons. GeneralSun Jian's four sons, for instance, wereSun Ce(伯符, Bófú),Sun Quan(仲謀, Zhòngmóu),Sun Yi(叔弼, Shūbì) andSun Kuang(季佐, Jìzuǒ).[5]
Reflecting a general cultural tendency toregard names as significant, the choice of what name to bestow upon one's children was considered very important in traditional China.[6]Yan Zhituiof theNorthern Qidynasty asserted that whereas the purpose of a given name was to distinguish one person from another, a courtesy name should express the bearer's moral integrity.[citation needed]
Prior to the twentieth century,sinicizedKoreans,Vietnamese, andJapanesewere also referred to by their courtesy name. The practice was also adopted by someMongolsandManchusafter the Qing conquest of China.[citation needed]
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Simicsis afull-system simulatoror virtual platform used to run unchanged production binaries of the target hardware. Simics was originally developed by theSwedish Institute of Computer Science(SICS), and then spun off toVirtutechfor commercial development in 1998. Virtutech was acquired byIntelin 2010. Currently, Simics is provided byIntelin a public release[1]and sold commercially byWind River Systems, which was in the past a subsidiary of Intel.
Simics contains bothinstruction set simulatorsand hardware models, and is or has been used to simulate systems such asAlpha,ARM(32- and 64-bit),IA-64,MIPS(32- and 64-bit),MSP430,PowerPC(32-and64-bit),RISC-V(32-and64-bit),SPARC-V8 and V9, andx86andx86-64CPUs.
Many different operating systems have been run on various simulated virtual platforms, includingLinux,MS-DOS,Windows,VxWorks,OSE,Solaris,FreeBSD,QNX,RTEMS,UEFI, andZephyr.
TheNetBSDAMD64 port was initially developed using Simics before the public release of the chip.[2]The purpose of simulation in Simics is often to develop software for a particular type of hardware without requiring access to that precise hardware, using Simics as avirtual platform. This can applied both to pre-release and pre-silicon software development for future hardware, as well as for existing hardware. Intel uses Simics to provide its ecosystem with access to future platform months or years ahead of the hardware launch.[3]
The current version of Simics is 6 which was released publicly in 2019.[4][5]Simics runs on 64-bit x86-64 machines runningMicrosoft WindowsandLinux(32-bit support was dropped with the release of Simics 5, since 64-bit provides significant performance advantages and is universally available on current hardware). The previous version, Simics 5, was released in 2015.[6]
Simics has the ability to execute a system in forward and reverse direction.[7]Reverse debuggingcan illuminate how an exceptional condition orbugoccurred. When executing an OS such asLinuxin reverse using Simics, previously deleted files reappear when the deletion point is passed in reverse and scrolling and other graphical display and console updates occur backwards as well.
Simics is built for high performance execution of full-system models, and uses bothbinary translationandhardware-assisted virtualizationto increase simulation speed. It is natively multithreaded and can simulate multiple target (or guest) processors and boards using multiple host threads. It has been used to run simulations containing hundreds of target processors.
Thisemulation-related article is astub. You can help Wikipedia byexpanding it.
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In thehistory of cryptography, theECM Mark IIwas acipher machineused by the United States for messageencryptionfromWorld War IIuntil the 1950s. The machine was also known as theSIGABAorConverter M-134by the Army, orCSP-888/889by the Navy, and a modified Navy version was termed theCSP-2900.
Like many machines of the era it used an electromechanical system ofrotorsto encipher messages, but with a number of security improvements over previous designs. No successfulcryptanalysisof the machine during its service lifetime is publicly known.
It was clear to US cryptographers well before World War II that the single-stepping mechanical motion of rotor machines (e.g. theHebern machine) could be exploited by attackers. In the case of the famousEnigma machine, these attacks were supposed to be upset by moving the rotors to random locations at the start of each new message. This, however, proved not to be secure enough, and German Enigma messages were frequently broken bycryptanalysisduring World War II.
William Friedman, director of theUS Army'sSignals Intelligence Service, devised a system to correct for this attack by truly randomizing the motion of the rotors. His modification consisted of apaper tapereader from ateletypemachine attached to a small device with metal "feelers" positioned to pass electricity through the holes. When a letter was pressed on the keyboard the signal would be sent through the rotors as it was in the Enigma, producing an encrypted version. In addition, the current would also flow through the paper tape attachment, and any holes in the tape at its current location would cause the corresponding rotor to turn, and then advance the paper tape one position. In comparison, the Enigma rotated its rotors one position with each key press, a much less random movement. The resulting design went into limited production as theM-134 Converter, and its message settings included the position of the tape and the settings of a plugboard that indicated which line of holes on the tape controlled which rotors. However, there were problems using fragile paper tapes under field conditions.
Friedman's associate,Frank Rowlett, then came up with a different way to advance the rotors, using another set of rotors. In Rowlett's design, each rotor must be constructed such that between one and four output signals were generated, advancing one or more of the rotors (rotors normally have one output for every input). There was little money for encryption development in the US before the war, so Friedman and Rowlett built a series of "add on" devices called theSIGGOO(or M-229) that were used with the existing M-134s in place of the paper tape reader. These were external boxes containing a three rotor setup in which five of the inputs were live, as if someone had pressed five keys at the same time on an Enigma, and the outputs were "gathered up" into five groups as well — that is all the letters from A to E would be wired together for instance. That way the five signals on the input side would be randomized through the rotors, and come out the far side with power in one of five lines. Now the movement of the rotors could be controlled with a day code, and the paper tape was eliminated. They referred to the combination of machines as the M-134-C.
In 1935 they showed their work toJoseph Wenger, a cryptographer in theOP-20-Gsection of theU.S. Navy. He found little interest for it in the Navy until early 1937, when he showed it to CommanderLaurance Safford, Friedman's counterpart in theOffice of Naval Intelligence. He immediately saw the potential of the machine, and he and Commander Seiler then added a number of features to make the machine easier to build, resulting in theElectric Code Machine Mark II(orECM Mark II), which the navy then produced as the CSP-889 (or 888).
Oddly, the Army was unaware of either the changes or the mass production of the system, but were "let in" on the secret in early 1940. In 1941 the Army and Navy joined in a joint cryptographic system, based on the machine. The Army then started using it as theSIGABA. Just over 10,000 machines were built.[1]: p. 152
On 26 June 1942, the Army and Navy agreed not to allow SIGABA machines to be placed in foreign territory except where armed American personnel were able to protect the machine.[2]The SIGABA would be made available to another Allied country only if personnel of that country were denied direct access to the machine or its operation by an American liaison officer who would operate it.[2]
SIGABA was similar to the Enigma in basic theory, in that it used a series of rotors to encipher every character of the plaintext into a different character of ciphertext. Unlike Enigma's three rotors however, the SIGABA included fifteen, and did not use a reflecting rotor.
The SIGABA had three banks of five rotors each; the action of two of the banks controlled the stepping of the third.
The SIGABA advanced one or more of its main rotors in a complex, pseudorandom fashion. This meant that attacks which could break other rotor machines with simpler stepping (for example, Enigma) were made much more complex. Even with the plaintext in hand, there were so many potential inputs to the encryption that it was difficult to work out the settings.
On the downside, the SIGABA was also large, heavy, expensive, difficult to operate, mechanically complex, and fragile. It was nowhere near as practical a device as the Enigma, which was smaller and lighter than the radios with which it was used. It found widespread use in the radio rooms of US Navy ships, but as a result of these practical problems the SIGABA simply couldn't be used in the field. In most theatres other systems were used instead, especially for tactical communications. One of the most famous was the use ofNavajo code talkersfor tactical field communications in the Pacific Theater. In other theatres, less secure, but smaller, lighter, and sturdier machines were used, such as theM-209. SIGABA, impressive as it was, was overkill for tactical communications. This said, new speculative evidence emerged more recently that the M-209 code was broken by German cryptanalysts during World War II.[3]
Because SIGABA did not have a reflector, a 26+ pole switch was needed to change the signal paths through the alphabet maze between the encryption and decryption modes. The long “controller” switch was mounted vertically, with its knob on the top of the housing. See image. It had five positions, O, P, R, E and D. Besides encrypt (E) and decrypt (D), it had a plain text position (P) that printed whatever was typed on the output tape, and a reset position (R) that was used to set the rotors and to zeroize the machine. The O position turned the machine off. The P setting was used to print the indicators and date/time groups on the output tape. It was the only mode that printed numbers. No printing took place in the R setting, but digit keys were active to increment rotors.
During encryption, the Z key was connected to the X key and the space bar produced a Z input to the alphabet maze. A Z was printed as a space on decryption. The reader was expected to understand that a word like “xebra” in a decrypted message was actually “zebra.” The printer automatically added a space between each group of five characters during encryption.
The SIGABA was zeroized when all the index rotors read zero in their low order digit and all the alphabet and code rotors were set to the letter O. Each rotor had a cam that caused the rotor to stop in the proper position during the zeroize process.
SIGABA's rotors were all housed in a removable frame held in place by four thumb screws. This allowed the most sensitive elements of the machine to be stored in more secure safes and to be quickly thrown overboard or otherwise destroyed if capture was threatened. It also allowed a machine to quickly switch between networks that used different rotor orders. Messages had two 5- character indicators, an exterior indicator that specified the system being used and the security classification and an interior indicator that determined the initial settings of the code and alphabet rotors. The key list included separate index rotor settings for each security classification. This prevented lower classification messages from being used as cribs to attack higher classification messages.
The Navy and Army had different procedures for the interior indicator. Both started by zeroizing the machine and having the operator select a random 5-character string for each new message. This was then encrypted to produce the interior indicator. Army key lists included an initial setting for the rotors that was used to encrypt the random string. The Navy operators used the keyboard to increment the code rotors until they matched the random character string. The alphabet rotor would move during this process and their final position was the internal indicator. In case of joint operations, the Army procedures were followed.
The key lists included a “26-30” check string. After the rotors were reordered according to the current key, the operator would zeroize the machine, encrypt 25 characters and then encrypt “AAAAA”. The ciphertext resulting from the five A's had to match the check string. The manual warned that typographical errors were possible in key lists and that a four character match should be accepted.
The manual also gave suggestions on how to generate random strings for creating indicators. These included using playing cards and poker chips, to selecting characters from cipher texts and using the SIGABA itself as a random character generator.[4]
Although the SIGABA was extremely secure, the US continued to upgrade its capability throughout the war, for fear of the Axis cryptanalytic ability to break SIGABA's code. When the German'sENIGMAmessages and Japan'sType B Cipher Machinewere broken, the messages were closely scrutinized for signs that Axis forces were able to read the US cryptography codes. Axisprisoners of war(POWs) were also interrogated with the goal of finding evidence that US cryptography had been broken. However, neither the Germans nor the Japanese were making any progress in breaking the SIGABA code. A decrypted JN-A-20 message, dated 24 January 1942, sent from the navalattachéin Berlin to vice chief of Japanese Naval General Staff in Tokyo stated that "joint Jap[anese]-German cryptanalytical efforts" to be "highly satisfactory", since the "German[s] have exhibited commendable ingenuity and recently experienced some success on English Navy systems", but are "encountering difficulty in establishing successful techniques of attack on 'enemy' code setup". In another decrypted JN-A-20 message, the Germans admitted that their progress in breaking US communications was unsatisfactory. The Japanese also admitted in their own communications that they had made no real progress against the American cipher system. In September 1944, when the Allies were advancing steadily on the Western front, the war diary of the German Signal Intelligence Group recorded: "U.S. 5-letter traffic: Work discontinued as unprofitable at this time".[5]
SIGABA systems were closely guarded at all times, with separate safes for the system base and the code-wheel assembly, but there was one incident where a unit was lost for a time. On February 3, 1945, a truck carrying a SIGABA system in three safes was stolen while its guards were visiting a brothel in recently liberatedColmar, France.General Eisenhowerordered an extensive search, which finally discovered the safes six weeks later in a nearby river.[6]: pp.510–512
The need for cooperation among US, British, and Canadian forces in carrying out joint military operations against Axis forces gave rise to the need for a cipher system that could be used by all Allied forces. This functionality was achieved in three different ways. Firstly, the ECM Adapter (CSP 1000), which could beretrofittedon Allied cipher machines, was produced at the Washington Naval Yard ECM Repair Shop. A total of 3,500 adapters were produced.[5]The second method was to adapt the SIGABA for interoperation with a modified British machine, theTypex. The common machine was known as theCombined Cipher Machine(CCM), and was used from November 1943.[2]Because of the high cost of production, only 631 CCMs were made. The third way was the most common and most cost-effective. It was the "X" Adapter manufactured by theTeletype Corporationin Chicago. A total of 4,500 of these adapters were installed at depot-level maintenance facilities.[5]
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Inmathematics, asurjective function(also known assurjection, oronto function/ˈɒn.tuː/) is afunctionfsuch that, for every elementyof the function'scodomain, there existsat leastone elementxin the function'sdomainsuch thatf(x) =y. In other words, for a functionf:X→Y, the codomainYis theimageof the function's domainX.[1][2]It is not required thatxbeunique; the functionfmay map one or more elements ofXto the same element ofY.
The termsurjectiveand the related termsinjectiveandbijectivewere introduced byNicolas Bourbaki,[3][4]a group of mainlyFrench20th-centurymathematicianswho, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French wordsurmeansoverorabove, and relates to the fact that theimageof the domain of a surjective function completely covers the function's codomain.
Any function induces a surjection byrestrictingits codomain to the image of its domain. Every surjective function has aright inverseassuming theaxiom of choice, and every function with a right inverse is necessarily a surjection. Thecompositionof surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.
Asurjective functionis afunctionwhoseimageis equal to itscodomain. Equivalently, a functionf{\displaystyle f}withdomainX{\displaystyle X}and codomainY{\displaystyle Y}is surjective if for everyy{\displaystyle y}inY{\displaystyle Y}there exists at least onex{\displaystyle x}inX{\displaystyle X}withf(x)=y{\displaystyle f(x)=y}.[1]Surjections are sometimes denoted by a two-headed rightwards arrow (U+21A0↠RIGHTWARDS TWO HEADED ARROW),[5]as inf:X↠Y{\displaystyle f\colon X\twoheadrightarrow Y}.
Symbolically,
A function isbijectiveif and only if it is both surjective andinjective.
If (as is often done) a function is identified with itsgraph, then surjectivity is not a property of the function itself, but rather a property of themapping.[7]This is, the function together with its codomain. Unlike injectivity, surjectivity cannot be read off of the graph of the function alone.
The functiong:Y→Xis said to be aright inverseof the functionf:X→Yiff(g(y)) =yfor everyyinY(gcan be undone byf). In other words,gis a right inverse offif thecompositionfogofgandfin that order is theidentity functionon the domainYofg. The functiongneed not be a completeinverseoffbecause the composition in the other order,gof, may not be the identity function on the domainXoff. In other words,fcan undo or "reverse"g, but cannot necessarily be reversed by it.
Every function with a right inverse is necessarily a surjection. The proposition that every surjective function has a right inverse is equivalent to theaxiom of choice.
Iff:X→Yis surjective andBis asubsetofY, thenf(f−1(B)) =B. Thus,Bcan be recovered from itspreimagef−1(B).
For example, in the first illustration in thegallery, there is some functiongsuch thatg(C) = 4. There is also some functionfsuch thatf(4) =C. It doesn't matter thatgis not unique (it would also work ifg(C) equals 3); it only matters thatf"reverses"g.
A functionf:X→Yis surjective if and only if it isright-cancellative:[8]given any functionsg,h:Y→Z, whenevergof=hof, theng=h. This property is formulated in terms of functions and theircompositionand can be generalized to the more general notion of themorphismsof acategoryand their composition. Right-cancellative morphisms are calledepimorphisms. Specifically, surjective functions are precisely the epimorphisms in thecategory of sets. The prefixepiis derived from the Greek prepositionἐπίmeaningover,above,on.
Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right inversegof a morphismfis called asectionoff. A morphism with a right inverse is called asplit epimorphism.
Any function with domainXand codomainYcan be seen as aleft-totalandright-uniquebinary relation betweenXandYby identifying it with itsfunction graph. A surjective function with domainXand codomainYis then a binary relation betweenXandYthat is right-unique and both left-total andright-total.
Thecardinalityof the domain of a surjective function is greater than or equal to the cardinality of its codomain: Iff:X→Yis a surjective function, thenXhas at least as many elements asY, in the sense ofcardinal numbers. (The proof appeals to theaxiom of choiceto show that a functiong:Y→Xsatisfyingf(g(y)) =yfor allyinYexists.gis easily seen to be injective, thus theformal definitionof |Y| ≤ |X| is satisfied.)
Specifically, if bothXandYarefinitewith the same number of elements, thenf:X→Yis surjective if and only iffisinjective.
Given two setsXandY, the notationX≤*Yis used to say that eitherXis empty or that there is a surjection fromYontoX. Using the axiom of choice one can show thatX≤*YandY≤*Xtogether imply that|Y| = |X|,a variant of theSchröder–Bernstein theorem.
Thecompositionof surjective functions is always surjective: Iffandgare both surjective, and the codomain ofgis equal to the domain off, thenfogis surjective. Conversely, iffogis surjective, thenfis surjective (butg, the function applied first, need not be). These properties generalize from surjections in thecategory of setsto anyepimorphismsin anycategory.
Any function can be decomposed into a surjection and aninjection: For any functionh:X→Zthere exist a surjectionf:X→Yand an injectiong:Y→Zsuch thath=gof. To see this, defineYto be the set ofpreimagesh−1(z)wherezis inh(X). These preimages aredisjointandpartitionX. Thenfcarries eachxto the element ofYwhich contains it, andgcarries each element ofYto the point inZto whichhsends its points. Thenfis surjective since it is a projection map, andgis injective by definition.
Any function induces a surjection by restricting its codomain to its range. Any surjective function induces a bijection defined on aquotientof its domain by collapsing all arguments mapping to a given fixed image. More precisely, every surjectionf:A→Bcan be factored as a projection followed by a bijection as follows. LetA/~ be theequivalence classesofAunder the followingequivalence relation:x~yif and only iff(x) =f(y). Equivalently,A/~ is the set of all preimages underf. LetP(~) :A→A/~ be theprojection mapwhich sends eachxinAto its equivalence class [x]~, and letfP:A/~ →Bbe the well-defined function given byfP([x]~) =f(x). Thenf=fPoP(~).
Given fixed finite setsAandB, one can form the set of surjectionsA↠B. Thecardinalityof this set is one of the twelve aspects of Rota'sTwelvefold way, and is given by|B|!{|A||B|}{\textstyle |B|!{\begin{Bmatrix}|A|\\|B|\end{Bmatrix}}}, where{|A||B|}{\textstyle {\begin{Bmatrix}|A|\\|B|\end{Bmatrix}}}denotes aStirling number of the second kind.
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Afriend-to-friend(orF2F) computer network is a type ofpeer-to-peer networkin which users only make direct connections with people they know.Passwordsordigital signaturescan be used forauthentication.
Unlike other kinds ofprivate P2P, users in a friend-to-friend network cannot find out who else is participating beyond their own circle of friends, so F2F networks can grow in size without compromising their users' anonymity.Retroshare,WASTE,GNUnet,FreenetandOneSwarmare examples of software that can be used to build F2F networks, though RetroShare is the only one of these configured for friend-to-friend operation by default.
Many F2F networks support indirectanonymousorpseudonymouscommunication between users who do not know or trust one another. For example, anodein a friend-to-friendoverlaycan automatically forward a file (or a request for a file) anonymously between two friends, without telling either of them the other's name orIP address. These friends can in turn automatically forward the same file (or request) to their own friends, and so on.
Dan Bricklincoined the term "friend-to-friend network" in 2000.[1]
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TheAssociation of College and Research Librariesdefinesinformation literacyas a "set of integrated abilities encompassing the reflective discovery of information, the understanding of how information is produced and valued and the use of information in creating new knowledge and participating ethically in communities of learning".[1][2][3][4]In the United Kingdom, theChartered Institute of Library and Information Professionals' definition also makes reference to knowing both "when" and "why" information is needed.[5]
The 1989American Library Association(ALA) Presidential Committee on Information Literacy formally defined information literacy (IL) as attributes of an individual, stating that "to be information literate, a person must be able to recognize when information is needed and have the ability to locate, evaluate and use effectively the needed information".[6][7]In 1990, academic Lori Arp published a paper asking, "Are information literacy instruction and bibliographic instruction the same?"[8]Arp argued that neither term was particularly well defined by theoreticians or practitioners in the field. Further studies were needed to lessen the confusion and continue to articulate the parameters of the question.[8]
The Alexandria Proclamation of 2005 defined the term as a human rights issue: "Information literacy empowers people in all walks of life to seek, evaluate, use and create information effectively to achieve their personal, social, occupational and educational goals. It is a basic human right in a digital world and promotes social inclusion in all nations."[9]The United States National Forum on Information Literacy defined information literacy as "the ability to know when there is a need for information, to be able to identify, locate, evaluate, and effectively use that information for the issue or problem at hand."[10][11]Meanwhile, in the UK, the library professional bodyCILIP, define information literacy as "the ability to think critically and make balanced judgements about any information we find and use. It empowers us as citizens to develop informed views and to engage fully with society.”[12]
A number of other efforts have been made to better define the concept and its relationship to other skills and forms ofliteracy. Other pedagogical outcomes related to information literacy include traditional literacy,computer literacy, research skills andcritical thinkingskills. Information literacy as a sub-discipline is an emerging topic of interest and counter measure among educators and librarians with the prevalence ofmisinformation,fake news, anddisinformation.
Scholars have argued that in order to maximize people's contributions to a democratic andpluralisticsociety, educators should be challenging governments and the business sector to support and fund educational initiatives in information literacy.[13]
The phrase "information literacy" first appeared in print in a 1974 report written on behalf of theNational Commission on Libraries and Information Scienceby Paul G. Zurkowski, who was at the time president of the Information Industry Association (now theSoftware and Information Industry Association). Zurkowski used the phrase to describe the "techniques and skills" learned by the information literate "for utilizing the wide range of information tools as well as primary sources in molding information solutions to their problems" and drew a relatively firm line between the "literates" and "information illiterates."[14]
The concept of information literacy appeared again in a 1976 paper by Lee Burchina presented at theTexas A&M Universitylibrary's symposium. Burchina identified a set of skills needed to locate and use information for problem solving and decision making.[15]In another 1976 article inLibrary Journal, M.R. Owens applied the concept to political information literacy and civic responsibility, stating, "All [people] are created equal but voters with information resources are in a position to make more intelligent decisions than citizens who are information illiterates. The application of information resources to the process of decision-making to fulfill civic responsibilities is a vital necessity."[16]
In a literature review published in an academic journal in 2020,Oral Roberts Universityprofessor Angela Sample cites several conceptual waves of information literacy definitions as defining information as a way of thinking, a set of skills, and asocial practice.[17][18][19]The introduction of these concepts led to the adoption of a mechanism calledmetaliteracyand the creation of threshold concepts and knowledge dispositions, which led to the creation of the ALA's Information Literacy Framework.[18][17]
TheAmerican Library Association's Presidential Committee on Information Literacy released a report on January 10, 1989. Titled as the Presidential Committee on Information Literacy: Final Report,[20]the article outlines the importance of information literacy, opportunities to develop it, and the idea of an Information Age School. The recommendations of the Committee led to establishment of the National Forum on Information Literacy, a coalition of more than 90 national and international organizations.[10]
In 1998, theAmerican Association of School Librariansand theAssociation for Educational Communications and TechnologypublishedInformation Power: Building Partnerships for Learning, which further established specific goals for information literacy education, defining some nine standards in the categories of "information literacy," "independent learning," and "social responsibility."[21]
Also in 1998, the Presidential Committee on Information Literacy updated its final report.[22]The report outlined six recommendations from the original report, and examined areas of challenge and progress.
In 1999, the Society of College, National and University Libraries (SCONUL) in the UK publishedThe Seven Pillars of Information Literacyto model the relationship between information skills and IT skills, and the idea of the progression of information literacy into the curriculum of higher education.
In 2003, the National Forum on Information Literacy, along withUNESCOand theNational Commission on Libraries and Information Science, sponsored an international conference in Prague.[23]Representatives from twenty-three countries gathered to discuss the importance of information literacy in a global context. The resulting Prague Declaration[24]described information literacy as a "key to social, cultural, and economic development of nations and communities, institutions and individuals in the 21st century" and declared its acquisition as "part of the basichuman rightof lifelong learning".[24]
In the United States specifically, information literacy was prioritized in 2009 during PresidentBarack Obama's first term. In effort to stress the value information literacy has on everyday communication, he designated Octoberas National Information Literacy Awareness Monthin his released proclamation.[25]
TheAmerican Library Association's Presidential Committee on Information Literacy defined information literacy as the ability "to recognize when information is needed and have the ability to locate, evaluate, and use effectively the needed information" and highlighted information literacy as a skill essential forlifelong learningand the production of an informed and prosperous citizenry.[20]
The committee outlined six principal recommendations. Included were recommendations like "Reconsider the ways we have organized information institutionally, structured information access, and defined information's role in our lives at home in the community, and in the work place"; to promote "public awareness of the problems created by information illiteracy"; to develop a national research agenda related to information and its use; to ensure the existence of "a climate conducive to students' becoming information literate"; to include information literacy concerns inteacher educationdemocracy.[26]
In the updated report, the committee ended with an invitation, asking the National Forum and regular citizens to recognize that "the result of these combined efforts will be a citizenry which is made up of effective lifelong learners who can always find the information needed for the issue or decision at hand. This new generation of information literate citizens will truly be America's most valuable resource," and to continue working toward an information literate world.[27]
The Presidential Committee on Information Literacy resulted in the creation of the National Forum on Information Literacy.
In 1983, United States published "A Nation at Risk: The Imperative for Educational Reform", a report declaring that a "rising tide of mediocrity" was eroding the foundation of the American educational system.[28]The report has been regarded as the genesis of the current educational reform movement within the United States.[citation needed]
This report, in conjunction with the rapid emergence of the information society, led theAmerican Library Association (ALA)to convene a panel of educators and librarians in 1987. The Forum,UNESCOandInternational Federation of Library Associations and Institutions(IFLA) collaborated to organize several "experts meetings" that resulted in the Prague Declaration (2003) and the Alexandria Proclamation (2005). Both statements underscore the importance of information literacy as a basic, fundamental human right, and consider IL as a lifelong learning skill.
IFLAhas established an Information Literacy Section. The Section has, in turn, developed and mounted an Information Literacy Resources Directory, called InfoLit Global. Librarians, educators and information professionals may self-register and upload information-literacy-related materials. (IFLA, Information Literacy Section, n.d.) According to the IFLA website, "The primary purpose of the Information Literacy Section is to foster international cooperation in the development of information literacy education in all types of libraries and information institutions."[29]
This alliance was created from the recommendation of the Prague Conference of Information Literacy Experts in 2003. One of its goals is to allow for the sharing of information literacy research and knowledge between nations. The IAIL also sees "lifelong learning" as a basic human right, and their ultimate goal is to use information literacy as a way to allow everyone to participate in the "Information Society" as a way of fulfilling this right.[30]The following organizations are founding members of IAIL:
UNESCO’s MIL aims to promote critical thinking and enhance an individual’s ability to access, evaluate, use and create media in various forms.
According to the UNESCO website, their "action to provide people with the skills and abilities for critical reception, assessment and use of information and media in their professional and personal lives."[36]Their goal is to create information literate societies by creating and maintaining educational policies for information literacy. They work with teachers around the world, training them in the importance of information literacy and providing resources for them to use in their classrooms.
UNESCO publishes studies in multiple countries, looking at how information literacy is currently taught, how it differs in different demographics, and how to raise awareness. They also publish tools and curricula for school boards and teachers to implement.[37]
In "Information Literacy as a Liberal Art," Jeremy J. Shapiro and Shelley K. Hughes (1996) advocated a more holistic approach to information literacy education, one that encouraged not merely the addition of information technology courses as an adjunct to existing curricula, but rather a radically new conceptualization of "our entire educational curriculum in terms of information."
Drawing upon Enlightenment ideals like those articulated byEnlightenmentphilosopherCondorcet, Shapiro and Hughes argued that information literacy education is "essential to the future ofdemocracy, if citizens are to be intelligent shapers of theinformation societyrather than its pawns, and to humanistic culture, if information is to be part of a meaningful existence rather than a routine of production and consumption."
To this end, Shapiro and Hughes outlined a "prototype curriculum" that encompassed the concepts ofcomputer literacy, library skills, and "a broader, critical conception of a more humanistic sort," suggesting seven important components of a holistic approach to information literacy:
Ira Shor further defines critical literacy as "[habits] of thought, reading, writing, and speaking which go beneath surface meaning, first impressions, dominant myths, official pronouncements, traditional clichés,received wisdom, and mere opinions, to understand the deep meaning, root causes, social context, ideology, and personal consequences of any action, event, object, process, organization, experience, text, subject matter, policy, mass media, or discourse."[40]
Big6 (Eisenberg and Berkowitz 1990) is a six-step process that provides support in the activities required to solve information-based problems: task definition, information seeking strategies, location and access, use of information, synthesis, and evaluation.[41][42]The Big6 skills have been used in a variety of settings to help those with a variety of needs. For example, the library of Dubai Women's College, in Dubai, United Arab Emirates which is an English as a second language institution, uses the Big6 model for its information literacy workshops. According to Story-Huffman (2009), using Big6 at the college "has transcended cultural and physical boundaries to provide a knowledge base to help students become information literate" (para. 8). In primary grades, Big6 has been found to work well with variety of cognitive and language levels found in the classroom.
Differentiated instruction and the Big6 appear to be made for each other. While it seems as though all children will be on the same Big6 step at the same time during a unit of instruction, there is no reason students cannot work through steps at an individual pace. In addition, the Big 6 process allows for seamless differentiation by interest.[43]
Issues to consider in the Big6 approach have been highlighted by Philip Doty:
This approach is problem-based, is designed to fit into the context of Benjamin Bloom's taxonomy of cognitive objectives, and aims toward the development of critical thinking. While the Big6 approach has a great deal of power, it also has serious weaknesses. Chief among these are the fact that users often lack well-formed statements of information needs, as well as the model's reliance on problem-solving rhetoric. Often, the need for information and its use are situated in circumstances that are not as well-defined, discrete, and monolithic as problems.[44]
Eisenberg (2004) has recognized that there are a number of challenges to effectively applying the Big6 skills, not the least of which isinformation overloadwhich can overwhelm students. Part of Eisenberg's solution is for schools to help students become discriminating users of information.
This conception, used primarily in thelibrary and information studiesfield, and rooted in the concepts oflibrary instructionand bibliographic instruction, is the ability "to recognize when information is needed and have the ability to locate, evaluate and use effectively the needed information."[45]In this view, information literacy is the basis for lifelong learning. It is also the basis for evaluating contemporary sources of information.
In the publicationInformation Power: Building Partnerships for Learning(AASL and AECT, 1998), three categories, nine standards, and twenty-nine indicators are used to describe the information literate student.
The categories and their standards are as follows:
Standards: The student who is information literate
Standards: The student who is an independent learner is information literate and
Standards: The student who contributes positively to the learning community and to society is information literate and
Since information may be presented in a number of formats, the term "information" applies to more than just the printed word. Otherliteraciessuch as visual, media, computer, network, and basic literacies are implicit in information literacy.
Many of those who are in most need of information literacy are often amongst those least able to access the information they require:
Minority and at-risk students, illiterate adults, people with English as a second language, and economically disadvantaged people are among those most likely to lack access to the information that can improve their situations. Most are not even aware of the potential help that is available to them.[47]
As the Presidential Committee report points out, members of these disadvantaged groups are often unaware that libraries can provide them with the access, training and information they need. In Osborne (2004), many libraries around the country are finding numerous ways to reach many of these disadvantaged groups by discovering their needs in their own environments (including prisons) and offering them specific services in the libraries themselves.
The rapidly evolving information landscape has demonstrated a need for education methods and practices to evolve and adapt accordingly. Information literacy is a key focus of educational institutions at all levels and in order to uphold this standard, institutions are promoting a commitment to lifelong learning and an ability to seek out and identify innovations that will be needed to keep pace with or outpace changes.[48]
Educational methods and practices, within our increasingly information-centric society, must facilitate and enhance a student's ability to harness the power of information. Key to harnessing the power of information is the ability to evaluate information, to ascertain among other things its relevance, authenticity and modernity. The information evaluation process is crucial life skill and a basis for lifelong learning.[49]According to Lankshear and Knobel, what is needed in our education system is a new understanding of literacy, information literacy and on literacy teaching. Educators need to learn to account for the context of our culturally and linguistically diverse and increasingly globalized societies. We also need to take account of the burgeoning variety of text forms associated with information and multimedia technologies.[50]
Evaluation consists of several component processes including metacognition, goals, personal disposition, cognitive development, deliberation, and decision-making. This is both a difficult and complex challenge and underscores the importance of being able to think critically.
Critical thinking is an important educational outcome for students.[49]Education institutions have experimented with several strategies to help foster critical thinking, as a means to enhance information evaluation and information literacy among students. When evaluating evidence, students should be encouraged to practice formal argumentation.[51]Debates and formal presentations must also be encouraged to analyze and critically evaluate information.
Education professionals must underscore the importance of high information quality. Students must be trained to distinguish between fact and opinion. They must be encouraged to use cue words such as "I think" and "I feel" to help distinguish between factual information and opinions. Information related skills that are complex or difficult to comprehend must be broken down into smaller parts. Another approach would be to train students in familiar contexts. Education professionals should encourage students to examine "causes" of behaviors, actions and events. Research shows that people evaluate more effectively if causes are revealed, where available.[48]
Information in any format is produced to convey a message and is shared via a selected delivery method. The iterative processes of researching, creating, revising, and disseminating information vary, and the resulting product reflects these differences (Association of College, p. 5).
Some call for increased critical analysis in Information Literacy instruction. Smith (2013) identifies this as beneficial "to individuals, particularly young
people during their period of formal education. It could equip them with the skills they need to understand the political system and their place within it, and, where necessary, to challenge this" (p. 16).[52]
National content standards, state standards and information literacy skills terminology may vary, but all have common components relating to information literacy.
Information literacy skills are critical to several of the National Education Goals outlined in theGoals 2000: Educate America Act, particularly in the act's aims to increase "school readiness," "student achievement andcitizenship," and "adult literacyandlifelong learning."[53]Of specific relevance are the "focus onlifelong learning, the ability tothink critically, and on the use of new and existing information forproblem solving," all of which are important components of information literacy.[54]
In 1998, theAmerican Association of School Librariansand theAssociation for Educational Communications and Technologypublished "Information Literacy Standards for Student Learning," which identified nine standards that librarians and teachers in K–12 schools could use to describe information literate students and define the relationship of information literacy to independent learning and social responsibility:
In 2007 AASL expanded and restructured the standards that school librarians should strive for in their teaching. These were published as "Standards for the 21st Century Learner" and address several literacies: information, technology, visual, textual, and digital. These aspects of literacy were organized within four key goals: that "learners use of skills, resources, & tools" to "inquire, think critically, and gain knowledge"; to "draw conclusions, make informed decisions, apply knowledge to new situations, and create new knowledge"; to "share knowledge and participate ethically and productively as members of our democratic society"; and to "pursue personal and aesthetic growth."[55]
In 2000, the Association of College and Research Libraries (ACRL), a division of theAmerican Library Association(ALA), released "Information Literacy Competency Standards for Higher Education," describing five standards and numerous performance indicators considered best practices for the implementation and assessment of postsecondary information literacy programs.[56]The five standards are:
These standards were meant to span from the simple to more complicated, or in terms of Bloom'sTaxonomy of Educational Objectives, from the "lower order" to the "higher order." Lower order skills would involve for instance being able to use an online catalog to find a book relevant to an information need in an academic library. Higher order skills would involve critically evaluating and synthesizing information from multiple sources into a coherent interpretation or argument.[57]
In 2016, the Association of College and Research Librarians (ACRL) rescinded the Standards and replaced them with the Framework for Information Literacy for Higher Education, which offers the following set of core ideas:
The Framework is based on a cluster of interconnected core concepts, with flexible options for implementation, rather than on a set of standards or learning outcomes, or any prescriptive enumeration of skills. At the[58]heart of this Framework are conceptual understandings that organize many other concepts and ideas about information, research, and scholarship into a coherent whole.[59]
Today instruction methods have changed drastically from the mostly one-directional teacher-student model, to a more collaborative approach where the students themselves feel empowered. Much of this challenge is now being informed by theAmerican Association of School Librariansthat published new standards for student learning in 2007.
Within the K–12 environment, effective curriculum development is vital to imparting Information Literacy skills to students. Given the already heavy load on students, efforts must be made to avoid curriculum overload.[60]Eisenberg strongly recommends adopting a collaborative approach to curriculum development among classroom teachers, librarians, technology teachers, and other educators. Staff must be encouraged to work together to analyze student curriculum needs, develop a broad instruction plan, set information literacy goals, and design specific unit and lesson plans that integrate the information skills and classroom content. These educators can also collaborate on teaching and assessment duties
Educators are selecting various forms ofresource-based learning(authentic learning, problem-based learning and work-based learning) to help students focus on the process and to help students learn from the content. Information literacy skills are necessary components of each. Within a school setting, it is very important that a students' specific needs as well as the situational context be kept in mind when selecting topics for integrated information literacy skills instruction. The primary goal should be to provide frequent opportunities for students to learn and practice information problem solving.[60]To this extent, it is also vital to facilitate repetition of information seeking actions and behavior. The importance of repetition in information literacy lesson plans cannot be underscored, since we tend to learn through repetition. A students' proficiency will improve over time if they are afforded regular opportunities to learn and to apply the skills they have learnt.
The process approach to education is requiring new forms of student assessment. Students demonstrate their skills, assess their own learning, and evaluate the processes by which this learning has been achieved by preparing portfolios, learning and research logs, and using rubrics.
Information literacy efforts are underway on individual, local, and regional bases.
Many states have either fully adopted AASL information literacy standards or have adapted them to suit their needs.[48]States such as Oregon (OSLIS, 2009)[61]increasing rely on these guidelines for curriculum development and setting information literacy goals. Virginia,[62]on the other hand, chose to undertake a comprehensive review, involving all relevant stakeholders and formulate its own guidelines and standards for information literacy. At an international level, two framework documents jointly produced by UNESCO and the IFLA (International Federation of Library Associations and Institutions) developed two framework documents that laid the foundations in helping define the educational role to be played by school libraries: the School library manifesto (1999).[63]
Another immensely popular approach to imparting information literacy is the Big6 set of skills.[60]Eisenberg claims that the Big6 is the most widely used model in K–12 education. This set of skills seeks to articulate the entire information seeking life cycle. The Big6 is made up of six major stages and two sub-stages under each major stages. It defines the six steps as being: task definition, information seeking strategies, location and access, use of information, synthesis, and evaluation. Such approaches seek to cover the full range of information problem-solving actions that a person would normally undertake, when faced with an information problem or with making a decision based on available resources.
Information literacy instruction in higher education can take a variety of forms: stand-alone courses or classes, online tutorials, workbooks, course-related instruction, or course-integrated instruction.
The six regional accreditation boards have added information literacy to their standards.[64]Librarians often are required to teach the concepts of information literacy during "one shot" classroom lectures. There are also credit courses offered by academic librarians to instruct college students in becoming more information literate. Additionally, information literacy instruction is usually tailored to a specific disciplines. One such attempt in the area of physics was published in 2009 but there are many many more published.[65]
In 2016, theAssociation of College and Research Libraries(ACRL, part of theAmerican Library Association) adopted a new "Framework for Information Literacy for Higher Education,"[66]replacing the ACRL's "Information Literacy Standards for Higher Education" that had been approved in 2000. The standards were largely criticized by proponents ofcritical information literacy, a concept deriving fromcritical pedagogy, for being too prescriptive.[67]It's termed a "framework" because it consists of interconnected core concepts designed to be interpreted and implemented locally depending on the context and needs of the audience. The framework draws on recent research around threshold concepts, or the ideas that are gateways to broader understanding or skills in a given discipline.[68]It also draws on newer research around metaliteracy, and assumes a more holistic view of information literacy that includes creation and collaboration in addition to consumption, so is appropriate for current practices around social media and Web 2.0.[69]The six concepts, or frames, are:
This draws from the concept ofmetaliteracy,[69]which offers a renewed vision of information literacy as an overarching set of abilities in which students are consumers and creators of information who can participate successfully in collaborative spaces (Association of College, p. 2)
There is a growing body of scholarly research describing faculty-librarian collaboration to bring information literacy skills practice into higher education curriculum, moving beyond "one shot" lectures to an integrated model in which librarians help design assignments, create guides to useful course resources, and provide direct support to students throughout courses.[70][71][72][73][74][75]A recent literature review indicates that there is still a lack of evidence concerning the unique information literacy practices of doctoral students, especially within disciplines such as the health sciences.[76]
There have also been efforts in higher education to highlight issues of data privacy, as they relate to information literacy. For example, at theUniversity of North Florida, in 2021, data privacy was added to their Library and Information Studies curriculum. The history of data privacy was included in this change, as well as topics such as, "data collection, data brokers, browser fingerprinting, cookies, data security, IP ranges, SSO, http vs https, anonymization, encryption, opt out vs opt in. These are all areas in which information professionals can improve information literacy, through understanding data privacy, practicing good techniques for data privacy, and teaching patrons about the importance/techniques for data privacy.[77]Additionally, information literacy instruction has been offered focusing onfake news.[78]
Now that information literacy has become a part of the core curriculum at many post-secondary institutions, the library community is charged to provide information literacy instruction in a variety of formats, includingonline learningand distance education. TheAssociation of College and Research Libraries(ACRL) addresses this need in its Guidelines for Distance Education Services (2000):
Library resources and services in institutions of higher education must meet the needs of all their faculty, students, and academic support staff, wherever these individuals are located, whether on a main campus, off campus, in distance education or extended campus programs—or in the absence of a campus at all, in courses taken for credit or non-credit; in continuing education programs; in courses attended in person or by means of electronic transmission; or any other means of distance education.
Within the e-learning anddistance educationworlds, providing effective information literacy programs brings together the challenges of both distance librarianship and instruction. With the prevalence of course management systems such asWebCTandBlackboard, library staff are embedding information literacy training within academic programs and within individual classes themselves.[79]
In October 2013, theNational Library of Singapore(NLB) created the S.U.R.E, (Source, Understand, Research, Evaluate) campaign.[80]The objectives and strategies of the S.U.R.E. campaign were first presented at the 2014 IFLA WLIC.[81]It is summarised by NLB as simplifying information literacy into four basic building blocks, to "promote and educate the importance of Information Literacy and discernment in information searching."[82]
Public events furthering the S.U.R.E. campaign were organised 2015. This was called the "Super S.U.R.E. Show" involving speakers to engage the public with their anecdotes and other learning points, for example the ability to separate fact from opinion.[83]
Information literacy is taught by librarians at institutes of higher education. Some components of information literacy are embedded in the undergraduate curriculum at theNational University of Singapore.[84]
Many academic libraries are participating in a culture of assessment, and attempt to show the value of their information literacy interventions to their students. Librarians use a variety of techniques for this assessment, some of which aim to empower students and librarians and resist adherence to unquestioned norms.[85]Information literacy instruction has been shown to improve student outcomes in higher education.[86]Oakleaf describes the benefits and dangers of various assessment approaches:fixed-choice tests, performance assessments, andrubrics.[87]
In public libraries, information literacy is connected tolifelong learning, development of employability skills, personal health management, andinformal learning.[88]
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Ingraph theory, anisomorphism ofgraphsGandHis abijectionbetween the vertex sets ofGandH
such that any two verticesuandvofGareadjacentinGif and only iff(u){\displaystyle f(u)}andf(v){\displaystyle f(v)}are adjacent inH. This kind of bijection is commonly described as "edge-preserving bijection", in accordance with the general notion ofisomorphismbeing a structure-preserving bijection.
If anisomorphismexists between two graphs, then the graphs are calledisomorphicand denoted asG≃H{\displaystyle G\simeq H}. In the case when the isomorphism is a mapping of a graph onto itself, i.e., whenGandHare one and the same graph, the isomorphism is called anautomorphismofG.
Graph isomorphism is anequivalence relationon graphs and as such it partitions theclassof all graphs intoequivalence classes. A set of graphs isomorphic to each other is called anisomorphism classof graphs. The question of whether graph isomorphism can be determined in polynomial time is a major unsolved problem in computer science, known as thegraph isomorphism problem.[1][2]
The two graphs shown below are isomorphic, despite their different lookingdrawings.
f(b) = 6
f(c) = 8
f(d) = 3
f(g) = 5
f(h) = 2
f(i) = 4
f(j) = 7
In the above definition, graphs are understood to beundirectednon-labelednon-weightedgraphs. However, the notion of isomorphism may be applied to all other variants of the notion of graph, by adding the requirements to preserve the corresponding additional elements of structure: arc directions, edge weights, etc., with the following exception.
Forlabeled graphs, two definitions of isomorphism are in use.
Under one definition, an isomorphism is a vertex bijection which is both edge-preserving and label-preserving.[3][4]
Under another definition, an isomorphism is an edge-preserving vertex bijection which preserves equivalence classes of labels, i.e., vertices with equivalent (e.g., the same) labels are mapped onto the vertices with equivalent labels and vice versa; same with edge labels.[5]
For example, theK2{\displaystyle K_{2}}graph with the two vertices labelled with 1 and 2 has a single automorphism under the first definition, but under the second definition there are two auto-morphisms.
The second definition is assumed in certain situations when graphs are endowed withunique labelscommonly taken from the integer range 1,...,n, wherenis the number of the vertices of the graph, used only to uniquely identify the vertices. In such cases two labeled graphs are sometimes said to be isomorphic if the corresponding underlying unlabeled graphs are isomorphic (otherwise the definition of isomorphism would be trivial).
The formal notion of "isomorphism", e.g., of "graph isomorphism", captures the informal notion that some objects have "the same structure" if one ignores individual distinctions of "atomic" components of objects in question. Whenever individuality of "atomic" components (vertices and edges, for graphs) is important for correct representation of whatever is modeled by graphs, the model is refined by imposing additional restrictions on the structure, and other mathematical objects are used:digraphs,labeled graphs,colored graphs,rooted treesand so on. The isomorphism relation may also be defined for all these generalizations of graphs: the isomorphism bijection must preserve the elements of structure which define the object type in question:arcs, labels, vertex/edge colors, the root of the rooted tree, etc.
The notion of "graph isomorphism" allows us to distinguishgraph propertiesinherent to the structures of graphs themselves from properties associated with graph representations:graph drawings,data structures for graphs,graph labelings, etc. For example, if a graph has exactly onecycle, then all graphs in its isomorphism class also have exactly one cycle. On the other hand, in the common case when the vertices of a graph are (representedby) theintegers1, 2,...N, then the expression
may be different for two isomorphic graphs.
TheWhitney graph isomorphism theorem,[6]shown byHassler Whitney, states that two connected graphs are isomorphic if and only if theirline graphsare isomorphic, with a single exception:K3, thecomplete graphon three vertices, and thecomplete bipartite graphK1,3, which are not isomorphic but both haveK3as their line graph. The Whitney graph theorem can be extended tohypergraphs.[7]
While graph isomorphism may be studied in a classical mathematical way, as exemplified by the Whitney theorem, it is recognized that it is a problem to be tackled with an algorithmic approach. The computational problem of determining whether two finite graphs are isomorphic is called the graph isomorphism problem.
Its practical applications include primarilycheminformatics,mathematical chemistry(identification of chemical compounds), andelectronic design automation(verification of equivalence of various representations of the design of anelectronic circuit).
The graph isomorphism problem is one of few standard problems incomputational complexity theorybelonging toNP, but not known to belong to either of its well-known (and, ifP ≠ NP, disjoint) subsets:PandNP-complete. It is one of only two, out of 12 total, problems listed inGarey & Johnson (1979)whose complexity remains unresolved, the other beinginteger factorization. It is however known that if the problem is NP-complete then thepolynomial hierarchycollapses to a finite level.[8]
In November 2015,László Babai, a mathematician and computer scientist at the University of Chicago, claimed to have proven that the graph isomorphism problem is solvable inquasi-polynomial time.[9][10]He published preliminary versions of these results in the proceedings of the 2016Symposium on Theory of Computing,[11]and of the 2018International Congress of Mathematicians.[12]In January 2017, Babai briefly retracted the quasi-polynomiality claim and stated asub-exponential timecomplexity bound instead. He restored the original claim five days later.[13]As of 2024[update], the full journal version of Babai's paper has not yet been published.
Its generalization, thesubgraph isomorphism problem, is known to be NP-complete.
The main areas of research for the problem are design of fast algorithms and theoretical investigations of itscomputational complexity, both for the general problem and for special classes of graphs.
TheWeisfeiler Leman graph isomorphism testcan be used to heuristically test for graph isomorphism.[14]If the test fails the two input graphs are guaranteed to be non-isomorphic. If the test succeeds the graphs may or may not be isomorphic. There are generalizations of the test algorithm that are guaranteed to detect isomorphisms, however their run time is exponential.
Another well-known algorithm for graph isomorphism is the vf2 algorithm, developed by Cordella et al. in 2001.[15]The vf2 algorithm is a depth-first search algorithm that tries to build an isomorphism between two graphs incrementally. It uses a set of feasibility rules to prune the search space, allowing it to efficiently handle graphs with thousands of nodes. The vf2 algorithm has been widely used in various applications, such as pattern recognition, computer vision, and bioinformatics. While it has a worst-case exponential time complexity, it performs well in practice for many types of graphs.
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Inmathematics, anorder topologyis a specifictopologythat can be defined on anytotally ordered set. It is a natural generalization of the topology of thereal numbersto arbitrary totally ordered sets.
IfXis a totally ordered set, theorder topologyonXis generated by thesubbaseof "open rays"
for alla, binX. ProvidedXhas at least two elements, this is equivalent to saying that the openintervals
together with the above rays form abasefor the order topology. Theopen setsinXare the sets that are aunionof (possibly infinitely many) such open intervals and rays.
Atopological spaceXis calledorderableorlinearly orderable[1]if there exists a total order on its elements such that the order topology induced by that order and the given topology onXcoincide. The order topology makesXinto acompletely normalHausdorff space.
The standard topologies onR,Q,Z, andNare the order topologies.
IfYis a subset ofX,Xa totally ordered set, thenYinherits a total order fromX. The setYtherefore has an order topology, theinduced order topology. As a subset ofX,Yalso has asubspace topology. The subspace topology is always at least asfineas the induced order topology, but they are not in general the same.
For example, consider the subsetY= {−1} ∪ {1/n}n∈Nof therationals. Under the subspace topology, thesingleton set{−1} is open inY, but under the induced order topology, any open set containing −1 must contain all but finitely many members of the space.
Though the subspace topology ofY= {−1} ∪ {1/n}n∈Nin the section above is shown not to be generated by the induced order onY, it is nonetheless an order topology onY; indeed, in the subspace topology every point is isolated (i.e., singleton {y} is open inYfor everyyinY), so the subspace topology is thediscrete topologyonY(the topology in which every subset ofYis open), and the discrete topology on any set is an order topology. To define a total order onYthat generates the discrete topology onY, simply modify the induced order onYby defining −1 to be the greatest element ofYand otherwise keeping the same order for the other points, so that in this new order (call it say <1) we have 1/n<1−1 for alln∈N. Then, in the order topology onYgenerated by <1, every point ofYis isolated inY.
We wish to define here a subsetZof a linearly ordered topological spaceXsuch that no total order onZgenerates the subspace topology onZ, so that the subspace topology will not be an order topology even though it is the subspace topology of a space whose topology is an order topology.
LetZ={−1}∪(0,1){\displaystyle Z=\{-1\}\cup (0,1)}in thereal line. The same argument as before shows that the subspace topology onZis not equal to the induced order topology onZ, but one can show that the subspace topology onZcannot be equal to any order topology onZ.
An argument follows. Suppose by way of contradiction that there is somestrict total order< onZsuch that the order topology generated by < is equal to the subspace topology onZ(note that we are not assuming that < is the induced order onZ, but rather an arbitrarily given total order onZthat generates the subspace topology).
LetM=Z\ {−1} = (0,1), thenMisconnected, soMis dense on itself and has no gaps, in regards to <. If −1 is not the smallest or the largest element ofZ, then(−∞,−1){\displaystyle (-\infty ,-1)}and(−1,∞){\displaystyle (-1,\infty )}separateM, a contradiction. Assume without loss of generality that −1 is the smallest element ofZ. Since {−1} is open inZ, there is some pointpinMsuch that the interval (−1,p) isempty, sopis the minimum ofM. ThenM\ {p} = (0,p) ∪ (p,1) is not connected with respect to the subspace topology inherited fromR. On the other hand, the subspace topology ofM\ {p} inherited from the order topology ofZcoincides with the order topology ofM\ {p} induced by <, which is connected since there are no gaps inM\ {p} and it is dense. This is a contradiction.
Several variants of the order topology can be given:
These topologies naturally arise when working withsemicontinuous functions, in that a real-valued function on a topological space is lower semicontinuous if and only if it iscontinuouswhen the reals are equipped with the right order.[3]The (natural)compact open topologyon the resulting set of continuous functions is sometimes referred to as thesemicontinuous topology[4].
Additionally, these topologies can be used to givecounterexamplesin general topology. For example, the left or right order topology on a bounded set provides an example of acompact spacethat is not Hausdorff.
The left order topology is the standard topology used for manyset-theoreticpurposes on aBoolean algebra.[clarification needed]
For anyordinal numberλone can consider the spaces of ordinal numbers
together with the natural order topology. These spaces are calledordinal spaces. (Note that in the usual set-theoretic construction of ordinal numbers we haveλ= [0,λ) andλ+ 1 = [0,λ]). Obviously, these spaces are mostly of interest whenλis an infinite ordinal; for finite ordinals, the order topology is simply thediscrete topology.
Whenλ= ω (the first infinite ordinal), the space [0,ω) is justNwith the usual (still discrete) topology, while [0,ω] is theone-point compactificationofN.
Of particular interest is the case whenλ= ω1, the set of all countable ordinals, and thefirst uncountable ordinal. The element ω1is alimit pointof the subset [0,ω1) even though nosequenceof elements in [0,ω1) has the element ω1as its limit. In particular, [0,ω1] is notfirst-countable. The subspace [0,ω1) is first-countable however, since the only point in [0,ω1] without a countablelocal baseis ω1. Some further properties include
Anyordinal numbercan be viewed as a topological space by endowing it with the order topology (indeed, ordinals arewell-ordered, so in particulartotally ordered). Unless otherwise specified, this is the usual topology given to ordinals. Moreover, if we are willing to accept aproper classas a topological space, then we may similarly view the class of all ordinals as a topological space with the order topology.
The set oflimit pointsof an ordinalαis precisely the set oflimit ordinalsless thanα.Successor ordinals(and zero) less thanαareisolated pointsinα. In particular, the finite ordinals and ω arediscretetopological spaces, and no ordinal beyond that is discrete. The ordinalαiscompactas a topological space if and only ifαis either asuccessor ordinalor zero.
Theclosed setsof a limit ordinalαare just the closed sets in the sense that we have already defined, namely, those that contain a limit ordinal whenever they contain all sufficiently large ordinals below it.
Any ordinal is, of course, an open subset of any larger ordinal. We can also define the topology on the ordinals in the followinginductiveway: 0 is the empty topological space,α+1 is obtained by taking theone-point compactificationofα, and forδa limit ordinal,δis equipped with theinductive limittopology. Note that ifαis a successor ordinal, thenαis compact, in which case its one-point compactificationα+1 is thedisjoint unionofαand a point.
As topological spaces, all the ordinals areHausdorffand evennormal. They are alsototally disconnected(connected components are points),scattered(every non-empty subspace has an isolated point; in this case, just take the smallest element),zero-dimensional(the topology has aclopenbasis: here, write an open interval (β,γ) as the union of the clopen intervals (β,γ'+1) = [β+1,γ'] forγ'<γ). However, they are notextremally disconnectedin general (there are open sets, for example the even numbers from ω, whoseclosureis not open).
The topological spaces ω1and its successor ω1+1 are frequently used as textbook examples of uncountable topological spaces. For example, in the topological space ω1+1, the element ω1is in the closure of the subset ω1even though no sequence of elements in ω1has the element ω1as its limit: an element in ω1is a countable set; for any sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable; this union is an upper bound of the elements of the sequence, and therefore of the limit of the sequence, if it has one.
The space ω1isfirst-countablebut notsecond-countable, and ω1+1 has neither of these two properties, despite beingcompact. It is also worthy of note that anycontinuous functionfrom ω1toR(thereal line) is eventually constant: so theStone–Čech compactificationof ω1is ω1+1, just as its one-point compactification (in sharp contrast to ω, whose Stone–Čech compactification is muchlargerthan ω).
Ifαis a limit ordinal andXis a set, anα-indexed sequence of elements ofXmerely means a function fromαtoX. This concept, atransfinite sequenceorordinal-indexed sequence, is a generalization of the concept of asequence. An ordinary sequence corresponds to the caseα= ω.
IfXis a topological space, we say that anα-indexed sequence of elements ofXconvergesto a limitxwhen it converges as anet, in other words, when given anyneighborhoodUofxthere is an ordinalβ<αsuch thatxιis inUfor allι≥β.
Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω1is a limit point of ω1+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω1-indexed sequence which maps any ordinal less than ω1to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω1, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable.
However, ordinal-indexed sequences are not powerful enough to replace nets (orfilters) in general: for example, on theTychonoff plank(the product space(ω1+1)×(ω+1){\displaystyle (\omega _{1}+1)\times (\omega +1)}), the corner point(ω1,ω){\displaystyle (\omega _{1},\omega )}is a limit point (it is in the closure) of the open subsetω1×ω{\displaystyle \omega _{1}\times \omega }, but it is not the limit of an ordinal-indexed sequence.
This article incorporates material from Order topology onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.
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ThePusey–Barrett–Rudolph(PBR)theorem[1]is ano-go theoreminquantum foundationsdue to Matthew Pusey, Jonathan Barrett, andTerry Rudolph(for whom the theorem is named) in 2012. It has particular significance for how one may interpret the nature of thequantum state.
With respect to certain realisthidden variable theoriesthat attempt to explain the predictions ofquantum mechanics, the theorem rules that pure quantum states must be "ontic" in the sense that they correspond directly to states of reality, rather than "epistemic" in the sense that they represent probabilistic or incomplete states of knowledge about reality.
The PBR theorem may also be compared with other no-go theorems likeBell's theoremand theBell–Kochen–Specker theorem, which, respectively, rule out the possibility of explaining the predictions of quantum mechanics withlocalhidden variable theories and noncontextual hidden variable theories. Similarly, the PBR theorem could be said to rule outpreparation independenthidden variable theories, in which quantum states that are prepared independently have independent hidden variable descriptions.
This result was cited by theoretical physicistAntony Valentinias "the most important general theorem relating to the foundations of quantum mechanics sinceBell's theorem".[2]
This theorem, which first appeared as anarXivpreprint[3]and was subsequently published inNature Physics,[1]concerns the interpretational status of pure quantum states. Under the classification of hidden variable models of Harrigan and Spekkens,[4]the interpretation of the quantum wavefunction|ψ⟩{\displaystyle |\psi \rangle }can be categorized as eitherψ-ontic if "every complete physical state or ontic state in the theory is consistent with only one pure quantum state" andψ-epistemic "if there exist ontic states that are consistent with more than one pure quantum state." The PBR theorem proves that either the quantum state|ψ⟩{\displaystyle |\psi \rangle }isψ-ontic, or else non-entangledquantum states violate the assumption of preparation independence, which would entailaction at a distance.
In conclusion, we have presented ano-go theorem, which—modulo assumptions—shows that models in which the quantum state is interpreted as mereinformationabout an objective physical state of a system cannot reproduce the predictions of quantum theory. The result is in the same spirit as Bell’s theorem, which states that no local theory can reproduce the predictions of quantum theory.
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Atrade secretis a form ofintellectual property(IP) comprising confidential information that is not generally known or readily ascertainable, derives economic value from its secrecy, and is protected by reasonable efforts to maintain its confidentiality.[1][2][3]Well-known examples include theCoca-Cola formulaand the recipe forKentucky Fried Chicken.
Unlike other forms of IP, trade secrets do not require formal registration and can be protected indefinitely, as long as they remain undisclosed.[4]Instead,non-disclosure agreements(NDAs), among other measures, are commonly used to keep the information secret.[5][6]
Like other IP assets, trade secrets may be sold or licensed.[7]Unauthorized acquisition, use, or disclosure of a trade secret by others in a manner contrary to honest commercial practices is considered misappropriation of the trade secret. If trade secretmisappropriationhappens, the trade secret holder can seek variouslegal remedies.[7]
The precise definition of a trade secret varies byjurisdiction, as do the types of information eligible trade secret protection. However, in general, trade secrets are confidential information that is:
All three elements are required. If any element ceases to exist, then the trade secret will also cease to exist.[4]
Trade secret protection covers confidential information, which can include technical and scientific data, business and commercial information, and financial records.[3]Even “negative” information, like failed experiments, can be valuable by helping companies avoid repeating costly mistakes.[3]
In international law, while "trade secrets" and "confidential information" are often used interchangeably, trade secrets are technically a subset of confidential information.[8]To qualify as a trade secret, confidential information must meet the specific requirements set by a country's national laws, which are often influenced by Article 39 of theTRIPS Agreement.[8][9]
Commentators likeA. Arthur Schillerhave argued that trade secrets were protected underRoman lawby a claim known asactio servi corrupti, meaning an "action for making a slave worse" or "an action for corrupting a servant." The Roman law is described as follows:
[T]he Roman owner of a mark or firm name was legally protected against unfair usage by a competitor through theactio servi corrupti... which the Roman jurists used to grant commercial relief under the guise of private law actions. "If, as the writer believes [writes Schiller], various private cases of action were available in satisfying commercial needs, the state was acting in exactly the same fashion as it does at the present day."[10]
The suggestion that trade secret law has its roots in Roman law was introduced in 1929 in aColumbia Law Reviewarticle called "Trade Secrets and the Roman Law: TheActio Servi Corrupti", which has been reproduced in Schiller's,An American Experience in Roman Law1 (1971).
However, theUniversity of Georgia Law SchoolprofessorAlan Watsonargued inTrade Secrets and Roman Law: The Myth Explodedthat theactio servi corruptiwas not used to protect trade secrets. Rather, he explained:
Schiller is sadly mistaken as to what was going on. ... Theactio servi corruptipresumably or possibly could be used to protect trade secrets and other similar commercial interests. That was not its purpose and was, at most, an incidental spin-off. But there is not the slightest evidence that the action was ever so used. In this regard theactio servi corruptiis not unique. Exactly the same can be said of many private law actions including those for theft, damage to property, deposit, and production of property. All of these could, I suppose, be used to protect trade secrets, etc., but there is no evidence they were. It is bizarre to see any degree the Romanactio servi corruptias the counterpart of modern law for the protection of trade secrets and other such commercial interests.[10]
Modern trade secret law is primarily rooted in Anglo-Americancommon law.[11](p6)The earliest recorded court case was the 1817 English caseNewbery v. James,which involved a secret formula for gout treatment.[12][11](p5)[13]In the United States, this concept was first recognized in the 1837 caseVickery v. Welch, involving the sale of a chocolate factory and the seller’s agreement to keep the secret recipe confidential.[14][15]
NewberyandVickeryonly awarded compensation for losses (damages) and did not issue orders to prevent the misuse of secrets (injunctive relief).[11](p5)The first English case involving injunctive relief wasYovatt v. Winyardin 1820, where the court issued an injunction to prevent a former employee from using or disclosing recipes he had secretly copied from his employer's veterinary medicine practice.[16][17]
In the United States, the 1868 Massachusetts Supreme Court decision inPeabody v. Norfolkis one of the most well-known and well-reasoned early trade secret case, establishing foundational legal principles that continue to be central to common law.[18][19]In this case, the court ruled that Peabody’s confidential manufacturing process was a protectable trade secret and issued an injunction preventing former employees from using or disclosing it after they shared it with a competitor.[18]
In 1939, theRestatement of Torts,published by theAmerican Law Institute, offered, among other things, one of the earliest formal definitions of a trade secret. According to Section 757, Comment b, a trade secret may consist of "any formula, pattern, device, or compilation of information which is used in one's business, and which gives the business an opportunity to obtain an advantage over competitors who do not know or use it."[20](p278)This definition became widely used by courts across the United States.[20](p278)As the first attempt to outline the accepted principles of trade secret law, theRestatementserved as the primary authority adopted in virtually every reported case.[20](p282)
Trade secret law saw further development in 1979 when theUniform Law Commission(ULC) introduced a model law known as theUniform Trade Secrets Act(UTSA), which was later amended in 1985. The UTSA defines the types of information eligible for trade secret protection, establishes a private cause of action for misappropriation, and outlines remedies such as injunctions, damages, and, in certain cases, attorneys' fees.[21]It has since been adopted by 48 states, along with the District of Columbia, Puerto Rico, and the U.S. Virgin Islands, with New York and North Carolina as the exceptions.[22][23]
The UTSA influenced theDefend Trade Secrets Act(DTSA) of 2016, which created a federal civilcause of actionfor trade secret misappropriation, allowing plaintiffs to file cases directly in federal courts if "the trade secret is related to a product or service used in ... interstate or foreign commerce."[23]
Trade secret law is governed by national legal systems.[24]However, international standards for protecting secrets (called “undisclosed information”) were established as part of theTRIPS Agreementin 1995.[24]Article 39 of TRIPS obligates member countries to protect “undisclosed information” from unauthorized use conducted “in a manner contrary to honest commercial practices,” including actions such as breach of contract, breach of confidence, and unfair competition. For the information to qualify, it must not be generally known or easily accessible, must hold value due to its secrecy, and must be safeguarded through “reasonable steps” to keep it secret.[24][25]
Trade secrets are an important, but invisible component of a company'sintellectual property(IP). Their contribution to a company's value can be major.[26]Being invisible, that contribution is hard to measure.[27]Still, research shows that changes in trade secrets laws affect business spending onR&Dandpatents.[28][29]This research provides indirect evidence of the value of trade secrecy.
Unlike other forms ofintellectual property, trade secrets do not require formal registration and can be protected indefinitely, as long as they remain secret.[4]Maintaining secrecy is both a practical necessity and a legal obligation, as trade secret owners must take "reasonable" measures to protect the confidentiality of their trade secrets to qualify for legal protection.[30](p4)"Reasonable" efforts are decided case by case, considering factors like the type and value of the secret, its importance to the business, the company’s size, and its organizational complexity.[30](p4)
The most common reason for trade secret disputes to arise is when former employees of trade secret-bearing companies leave to work for a competitor and are suspected of taking or using valuable confidential information belonging to their former employer.[31]Legal protections includenon-disclosure agreements(NDAs), andwork-for-hireandnon-compete clauses. In other words, in exchange for an opportunity to be employed by the holder of secrets, an employee may agree to not reveal their prospective employer's proprietary information, to surrender or assign to their employer ownership rights to intellectual work and work-products produced during the course (or as a condition) of employment, and to not work for a competitor for a given period of time (sometimes within a given geographic region).
Violating the agreement generally carries the possibility of heavy financial penalties, thus disincentivizing the revealing of trade secrets. Trade secret information can be protected through legal action including an injunction preventingbreaches of confidentiality, monetary damages, and, in some instances, punitive damages and attorneys’ fees too. In extraordinary circumstances, anex parte seizureunder theDefend Trade Secrets Act(DTSA) also allows for the court to seize property to prevent the propagation or dissemination of the trade secret.[31]
However, proving a breach of an NDA by a former stakeholder who is legally working for a competitor or prevailing in a lawsuit for breaching a non-compete clause can be very difficult.[32]A holder of a trade secret may also require similar agreements from other parties, such as vendors, licensees, and board members.
As a company can protect its confidential information through NDA, work-for-hire, and non-compete contracts with its stakeholders (within the constraints of employment law, including only restraint that is reasonable in geographic- and time-scope), these protective contractual measures effectively create a monopoly on secret information that does not expire as would apatentorcopyright. The lack of formal protection associated with registered intellectual property rights, however, means that a third party not bound by a signed agreement is not prevented from independently duplicating and using the secret information once it is discovered, such as throughreverse engineering.
Therefore, trade secrets such as secret formulae are often protected by restricting the key information to a few trusted individuals. Famous examples of products protected by trade secrets areChartreuse liqueurandCoca-Cola.[33]
Because protection of trade secrets can, in principle, extend indefinitely, it may provide an advantage over patent protection and other registered intellectual property rights, which last for a limited duration. For example, the Coca-Cola company has no patent for theformula of Coca-Colaand has been effective in protecting it for many more years than the 20 years of protection that a patent would have provided. In fact, Coca-Cola refused to reveal its trade secret under at least two judges' orders.[34]
Trade secret legal protection can reduce the knowledge spillover, which enhances the knowledge spread and technology improvement.[35]Therefore, while trade secret laws strengthen R&D exclusivity and encourage firms to engage in innovative activities, broadly reducingknowledge spilloverscan harm economic growth.
In general, trade secret misappropriation occurs when someone improperly acquires, discloses, or uses a trade secret without the trade secret holder's consent.[36][37][38]Common scenarios include former employees taking proprietary data to a new employer in violation ofnon-disclosure agreements(NDAs), espionage, or unauthorized disclosure.[39][36][40]
To prove misappropriation, the trade secret holder must generally show—subject to the specific requirements of the applicable jurisdiction—that:
While the improper, dishonest, or unlawful acquisition, use, or disclosure of trade secret information by unauthorized third parties is generally prohibited, there are exceptions to this rule. The scope of these exceptions and limitations varies across jurisdictions:
InCommonwealthcommon lawjurisdictions, confidentiality and trade secrets are regarded as anequitableright rather than apropertyright.[44]
TheCourt of Appeal of England and Walesin the case ofSaltman Engineering Co Ltd v. Campbell Engineering Ltd[45]held that the action for breach of confidence is based on a principle of preserving "good faith".
The test for a cause of action for breach of confidence in thecommon lawworld is set out in the case ofCoco v. A.N. Clark (Engineers) Ltd:[46]
The "quality of confidence" highlights that trade secrets are a legal concept. With sufficient effort or through illegal acts (such as breaking and entering), competitors can usually obtain trade secrets. However, so long as the owner of the trade secret can prove that reasonable efforts have been made to keep the information confidential, the information remains a trade secret and generally remains legally protected. Conversely, trade secret owners who cannot evidence reasonable efforts at protecting confidential information risk losing the trade secret, even if the information is obtained by competitors illegally. It is for this reason that trade secret owners shred documents and do not simply recycle them.[citation needed]
A successful plaintiff is entitled to various forms ofjudicial relief, including:
Hong Kongdoes not follow the traditional commonwealth approach, instead recognizing trade secrets where a judgment of the High Court indicates that confidential information may be a property right.[47]
The EU adopted aDirective on the Protection of Trade Secretson 27 May 2016.[48]The goal of the directive is to harmonize the definition of trade secrets in accordance with existing international standards, and the means of obtaining protection of trade secrets within the EU.[48]
Unlike other protections, like in the US, the trade secrets in the EU are not absolutely seen as an IP right, as it gives the holder no exclusive rights. It is more a protection against the unfair use or publication of the secret information.[48]
Within the U.S., trade secrets generally encompass a company's proprietary information that is not generally known to its competitors, and which provides the company with a competitive advantage.[49]
Although trade secrets law evolved under state common law, prior to 1974, the question of whether patent law preempted state trade secrets law had been unanswered. In 1974, theUnited States Supreme Courtissued the landmark decision,Kewanee Oil Co. v. Bicron Corp.,which resolved the question in favor of allowing the states to freely develop their own trade secret laws.[50]
In 1979, several U.S. states adopted theUniform Trade Secrets Act(UTSA), which was further amended in 1985, with approximately 47 states having adopted some variation of it as the basis for trade secret law. Another significant development is theEconomic Espionage Act (EEA) of 1996(18 U.S.C.§§ 1831–1839), which makes the theft or misappropriation of a trade secret a federal crime.
This law contains two provisions criminalizing two sorts of activity:
The statutory penalties are different for the two offenses. The EEA was extended in 2016 to allow companies to file civil suits in federal court.[51]
On May 11, 2016, President Obama signed theDefend Trade Secrets Act(DTSA), 18 U.S.C. §§ 1839 et seq., which for the first time created a federal cause of action for misappropriating trade secrets.[52]The DTSA provides for both a private right of action for damages and injunction and a civil action for injunction brought by the Attorney General.[53]
The statute followed state laws on liability in significant part, defining trade secrets in the same way as the Uniform Trade Secrets Act as,
"all forms and types of financial, business, scientific, technical, economic, or engineering information, including patterns, plans, compilations, program devices, formulas, designs, prototypes, methods, techniques, processes, procedures, programs, or codes, whether tangible or intangible, and whether or how stored, compiled, or memorialized physically, electronically, graphically, photographically, or in writing if (A) the owner thereof has taken reasonable measures to keep such information secret; and (B) the information derives independent economic value, actual or potential, from not being generally known to, and not being readily ascertainable through proper means by, another person who can obtain economic value from the disclosure or use of the information."
However, the law contains several important differences from prior law:
The DTSA also clarifies that a United States resident (including a company) can be liable for misappropriation that takes place outside the United States, and any person can be liable as long as an act in furtherance of the misappropriation takes place in the United States, 18 U.S.C. §1837. The DTSA provides the courts with broad injunctive powers. 18 U.S.C. §1836(b)(3).
The DTSA does not preempt or supplant state laws, but provides an additional cause of action. Because states vary significantly in their approach to the "inevitable disclosure" doctrine,[54]its use has limited, if any, application under the DTSA, 18 U.S.C.§1836(b)(3)(A).[55]
In the United States, trade secrets are not protected by law in the same way aspatentsortrademarks. While the US Constitution explicitlyauthorizesthe existence of and the federal jurisdiction overpatentsandcopyrights, it is silent on trade secrets,trademarks, etc. For this reason, Federal Law for the latter types of intellectual property is based on theCommerce Clause(rather than theCopyright Clause) under a theory, that these IP types are used forinterstate commerce. On other hand, the application of the Interstate Commerce Theory did not find much judicial support in regulating trade secrets: since a trade secret process is used in a State, where it is protected by state law, federal protection may be needed only whenindustrial espionageby a foreign entity is involved (given that the States themselves cannot regulate commerce with foreign powers).
Due to these Constitutional requirements, patents and trademarks enjoy a strong federal protection in the USA (theLanham ActandPatent Act, respectively), while trade secrets usually have to rely on more limitedstate laws. Most states have adopted theUniform Trade Secrets Act(UTSA), except forMassachusetts,New York, andNorth Carolina. However, since 2016 with the enactment of theDefend Trade Secrets Act(DTSA), some additional trade secrets protection has become also available under federal law. One of the differences between patents and trademarks, on the one hand, and trade secrets, on the other, is that a trade secret is protected only when the owner has taken reasonable measures to protect the information as a secret (see18 U.S.C.§ 1839(3)(A)).
Nations have different trademark policies. Assuming the mark in question meets certain other standards of protectibility, trademarks are generally protected from infringement on the grounds that other uses might confuse consumers as to the origin or nature of the goods once the mark has been associated with a particular supplier. Similar considerations apply toservice marksandtrade dress. By definition, a trademark enjoys no protection (quatrademark) until and unless it is "disclosed" to consumers, for only then are consumers able to associate it with a supplier or source in the requisite manner. (That a company plans tousea certain trademark might itself be protectable as a trade secret, however, until the mark is actually made public.)[56]To acquire atrademark rights under U.S. law, one must simply use the mark "in commerce".[57]It is possible to register a trademark in the United States, both at the federal and state levels. Registration of trademarks confers some advantages, including stronger protection in certain respects, but registration is not required in order to get protection.[57]Registration may be required in order to file a lawsuit for trademark infringement.
To acquire a patent,enablinginformation about the method or product has to be supplied to a patent office and upon publication (usually, years before issuance of a patent), it becomes available to all. After expiration of the patent, competitors can copy the method or product legally. The most important advantage of patents (compared to trade secrets) is that patents assure the monopoly of their owners, even when the patented subject matter is independently invented by others later (there aresome exceptions), as well as when the patented subject matter was invented by others prior to the patent'spriority date, kept as a trade secret, and used by the other in its business. Although it is legally possible to "convert" a trade secret into a patent, the claims in such patent would be limited to things, that are easily discernable from examining such things. This means, thatcompositions of matterandarticles of manufacturecan not be patented after they become available to public, whileprocessescan.
The temporarymonopolyon the patented invention is regarded as apay-offfor disclosing the information to the public.[citation needed]In order to obtain a patent, the inventor mustdisclose the invention, so that others will be able to both make and use the invention. Often, an invention will be improved after filing of the patent application, and additional information will be learned. None of that additional information must be disclosed through the patent application process, and it may thus be kept as a trade secret.[58]That nondisclosed information will often increase the commercial viability of the patent. Most patent licenses include clauses that require the inventor to disclose any trade secrets they have, and patent licensors must be careful to maintain their trade secrets while licensing a patent through such means as the use of anon-disclosure agreement. Compared to patents, the advantages of trade secrets are that a trade secret is not time limited (it "continues indefinitely as long as the secret is not revealed to the public", whereas a patent is only in force for a specified time, after which others may freely copy the invention), a trade secret does not imply any registration costs,[59]has an immediate effect, does not require compliance with any formalities, and does not imply any disclosure of the invention to the public.[59]The disadvantages of trade secrets include that "others may be able to legally discover the secret and be thereafter entitled to use it", "others may obtain patent protection for legally discovered secrets", and a trade secret is more difficult to enforce than a patent.[60]
TheFreedom of Information Actof 1966 (FOIA), which requires federal agencies to provide documents to the public on request, includes a discretionary exemption for trade secrets.[61]Thus, trade secret regulations can mask the composition of chemical agents inconsumer products, which has long been criticized for allowing the trade secret holders to hide the presence of potentially harmful andtoxic substances. It has been argued that the public is being denied a clear picture of such products' safety, whereas competitors are well positioned to analyze its chemical composition.[62]In 2004, the National Environmental Trust tested 40 common consumer products; in more than half of them they found toxic substances not listed on theproduct label.[62]
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Anapplication programming interface(API) is a connection betweencomputersor betweencomputer programs. It is a type of softwareinterface, offering a service to other pieces ofsoftware.[1]A document or standard that describes how to build such a connection or interface is called anAPI specification. A computer system that meets this standard is said toimplementorexposean API. The term API may refer either to the specification or to the implementation.
In contrast to auser interface, which connects a computer to a person, an application programming interface connects computers or pieces of software to each other. It is not intended to be used directly by a person (theend user) other than acomputer programmer[1]who is incorporating it into software. An API is often made up of different parts which act as tools or services that are available to the programmer. A program or a programmer that uses one of these parts is said tocallthat portion of the API. The calls that make up the API are also known assubroutines, methods, requests, orendpoints. An API specificationdefinesthese calls, meaning that it explains how to use or implement them.
One purpose of APIs is tohide the internal detailsof how a system works, exposing only those parts a programmer will find useful and keeping them consistent even if the internal details later change. An API may be custom-built for a particular pair of systems, or it may be a shared standard allowinginteroperabilityamong many systems.
The term API is often used to refer toweb APIs,[2]which allow communication between computers that are joined by theinternet. There are also APIs forprogramming languages,software libraries, computeroperating systems, andcomputer hardware. APIs originated in the 1940s, though the term did not emerge until the 1960s and 70s.
An API opens a software system to interactions from the outside. It allows two software systems to communicate across a boundary — an interface — using mutually agreed-upon signals.[3]In other words, an API connects software entities together. Unlike auser interface, an API is typically not visible to users. It is an "under the hood" portion of a software system, used for machine-to-machine communication.[4]
A well-designed API exposes only objects or actions needed by software or software developers. It hides details that have no use. Thisabstractionsimplifies programming.[5]
Building software using APIs has been compared to using building-block toys, such asLegobricks. Software services or software libraries are analogous to the bricks; they may be joined together via their APIs, composing a new software product.[6]The process of joining is calledintegration.[3]
As an example, consider a weather sensor that offers an API. When a certain message is transmitted to the sensor, it will detect the current weather conditions and reply with a weather report. The message that activates the sensor is an APIcall, and the weather report is an APIresponse.[7]A weather forecasting app might integrate with a number of weather sensor APIs, gathering weather data from throughout a geographical area.
An API is often compared to acontract. It represents an agreement between parties: a service provider who offers the API and the software developers who rely upon it. If the API remains stable, or if it changes only in predictable ways, developers' confidence in the API will increase. This may increase their use of the API.[8]
The termAPIinitially described an interface only for end-user-facing programs, known asapplication programs. This origin is still reflected in the name "application programming interface." Today, the term is broader, including alsoutility softwareand evenhardware interfaces.[10]
The idea of the API is much older than the term itself. British computer scientistsMaurice WilkesandDavid Wheelerworked on a modularsoftware libraryin the 1940s forEDSAC, an early computer. Thesubroutinesin this library were stored onpunched paper tapeorganized in afiling cabinet. This cabinet also contained what Wilkes and Wheeler called a "library catalog" of notes about each subroutine and how to incorporate it into a program. Today, such a catalog would be called an API (or an API specification or API documentation) because it instructs a programmer on how to use (or "call") each subroutine that the programmer needs.[10]
Wilkes and Wheeler's bookThe Preparation of Programs for an Electronic Digital Computercontains the first published API specification.Joshua Blochconsiders that Wilkes and Wheeler "latently invented" the API, because it is more of a concept that is discovered than invented.[10]
The term "application program interface" (without an-ingsuffix) is first recorded in a paper calledData structures and techniques for remotecomputer graphicspresented at anAFIPSconference in 1968.[12][10]The authors of this paper use the term to describe the interaction of anapplication—a graphics program in this case—with the rest of the computer system. A consistent application interface (consisting ofFortransubroutine calls) was intended to free the programmer from dealing with idiosyncrasies of the graphics display device, and to providehardware independenceif the computer or the display were replaced.[11]
The term was introduced to the field ofdatabasesbyC. J. Date[13]in a 1974 paper calledTheRelationalandNetworkApproaches: Comparison of the Application Programming Interface.[14]An API became a part of theANSI/SPARC frameworkfordatabase management systems. This framework treated the application programming interface separately from other interfaces, such as the query interface. Database professionals in the 1970s observed these different interfaces could be combined; a sufficiently rich application interface could support the other interfaces as well.[9]
This observation led to APIs that supported all types of programming, not just application programming. By 1990, the API was defined simply as "a set of services available to a programmer for performing certain tasks" by technologistCarl Malamud.[15]
The idea of the API was expanded again with the dawn ofremote procedure callsandweb APIs. Ascomputer networksbecame common in the 1970s and 80s, programmers wanted to call libraries located not only on their local computers, but on computers located elsewhere. These remote procedure calls were well supported by theJavalanguage in particular. In the 1990s, with the spread of theinternet, standards likeCORBA,COM, andDCOMcompeted to become the most common way to expose API services.[16]
Roy Fielding's dissertationArchitectural Styles and the Design of Network-based Software ArchitecturesatUC Irvinein 2000 outlinedRepresentational state transfer(REST) and described the idea of a "network-based Application Programming Interface" that Fielding contrasted with traditional "library-based" APIs.[17]XMLandJSONweb APIs saw widespread commercial adoption beginning in 2000 and continuing as of 2021. The web API is now the most common meaning of the term API.[2]
TheSemantic Webproposed byTim Berners-Leein 2001 included "semantic APIs" that recast the API as anopen, distributed data interface rather than a software behavior interface.[18]Proprietaryinterfaces and agents became more widespread than open ones, but the idea of the API as a data interface took hold. Because web APIs are widely used to exchange data of all kinds online, API has become a broad term describing much of the communication on the internet.[16]When used in this way, the term API has overlap in meaning with the termcommunication protocol.
The interface to asoftware libraryis one type of API. The API describes and prescribes the "expected behavior" (a specification) while the library is an "actual implementation" of this set of rules.
A single API can have multiple implementations (or none, being abstract) in the form of different libraries that share the same programming interface.
The separation of the API from its implementation can allow programs written in one language to use a library written in another. For example, becauseScalaandJavacompile to compatiblebytecode, Scala developers can take advantage of any Java API.[19]
API use can vary depending on the type of programming language involved.
An API for aprocedural languagesuch asLuacould consist primarily of basic routines to execute code, manipulate data or handle errors while an API for anobject-oriented language, such as Java, would provide a specification of classes and itsclass methods.[20][21]Hyrum's law states that "With a sufficient number of users of an API, it does not matter what you promise in the contract: all observable behaviors of your system will be depended on by somebody."[22]Meanwhile, several studies show that most applications that use an API tend to use a small part of the API.[23]
Language bindingsare also APIs. By mapping the features and capabilities of one language to an interface implemented in another language, a language binding allows a library or service written in one language to be used when developing in another language.[24]Tools such asSWIGand F2PY, aFortran-to-Pythoninterface generator, facilitate the creation of such interfaces.[25]
An API can also be related to asoftware framework: a framework can be based on several libraries implementing several APIs, but unlike the normal use of an API, the access to the behavior built into the framework is mediated by extending its content with new classes plugged into the framework itself.
Moreover, the overall program flow of control can be out of the control of the caller and in the framework's hands byinversion of controlor a similar mechanism.[26][27]
An API can specify the interface between an application and theoperating system.[28]POSIX, for example, specifies a set of common APIs that aim to enable an application written for a POSIX conformant operating system to becompiledfor another POSIX conformant operating system.
LinuxandBerkeley Software Distributionare examples of operating systems that implement the POSIX APIs.[29]
Microsofthas shown a strong commitment to a backward-compatible API, particularly within itsWindows API(Win32) library, so older applications may run on newer versions of Windows using an executable-specific setting called "Compatibility Mode".[30]
An API differs from anapplication binary interface(ABI) in that an API is source code based while an ABI isbinarybased. For instance,POSIXprovides APIs while theLinux Standard Baseprovides an ABI.[31][32]
Remote APIs allow developers to manipulate remote resources throughprotocols, specific standards for communication that allow different technologies to work together, regardless of language or platform.
For example, the Java Database Connectivity API allows developers to query many different types ofdatabaseswith the same set of functions, while theJava remote method invocationAPI uses the Java Remote Method Protocol to allowinvocationof functions that operate remotely, but appear local to the developer.[33][34]
Therefore, remote APIs are useful in maintaining the object abstraction inobject-oriented programming; amethod call, executed locally on aproxyobject, invokes the corresponding method on the remote object, using the remoting protocol, and acquires the result to be used locally as a return value.
A modification of the proxy object will also result in a corresponding modification of the remote object.[35]
Web APIs are the defined interfaces through which interactions happen between an enterprise and applications that use its assets, which also is aService Level Agreement(SLA) to specify the functional provider and expose the service path or URL for its API users. An API approach is an architectural approach that revolves around providing a program interface to a set of services to different applications serving different types of consumers.[36]
When used in the context ofweb development, an API is typically defined as a set of specifications, such asHypertext Transfer Protocol(HTTP) request messages, along with a definition of the structure of response messages, usually in an Extensible Markup Language (XML) or JavaScript Object Notation (JSON) format. An example might be a shipping company API that can be added to an eCommerce-focused website to facilitate ordering shipping services and automatically include current shipping rates, without the site developer having to enter the shipper's rate table into a web database. While "web API" historically has been virtually synonymous withweb service, the recent trend (so-calledWeb 2.0) has been moving away from Simple Object Access Protocol (SOAP) based web services andservice-oriented architecture(SOA) towards more directrepresentational state transfer(REST) styleweb resourcesandresource-oriented architecture(ROA).[37]Part of this trend is related to theSemantic Webmovement towardResource Description Framework(RDF), a concept to promote web-basedontology engineeringtechnologies. Web APIs allow the combination of multiple APIs into new applications known asmashups.[38]In the social media space, web APIs have allowed web communities to facilitate sharing content and data between communities and applications. In this way, content that is created in one place dynamically can be posted and updated to multiple locations on the web.[39]For example, Twitter's REST API allows developers to access core Twitter data and the Search API provides methods for developers to interact with Twitter Search and trends data.[40]
The design of an API has significant impact on its usage.[5]The principle ofinformation hidingdescribes the role of programming interfaces as enablingmodular programmingby hiding the implementation details of the modules so that users of modules need not understand the complexities inside the modules.[41]Thus, the design of an API attempts to provide only the tools a user would expect.[5]The design of programming interfaces represents an important part ofsoftware architecture, the organization of a complex piece of software.[42]
APIs are one of the more common ways technology companies integrate. Those that provide and use APIs are considered as being members of a business ecosystem.[43]
The main policies for releasing an API are:[44]
An important factor when an API becomes public is its "interface stability". Changes to the API—for example adding new parameters to a function call—could break compatibility with the clients that depend on that API.[48]
When parts of a publicly presented API are subject to change and thus not stable, such parts of a particular API should be documented explicitly as "unstable". For example, in theGoogle Guavalibrary, the parts that are considered unstable, and that might change soon, are marked with theJava annotation@Beta.[49]
A public API can sometimes declare parts of itself asdeprecatedor rescinded. This usually means that part of the API should be considered a candidate for being removed, or modified in a backward incompatible way. Therefore, these changes allow developers to transition away from parts of the API that will be removed or not supported in the future.[50]
Client code may contain innovative or opportunistic usages that were not intended by the API designers. In other words, for a library with a significant user base, when an element becomes part of the public API, it may be used in diverse ways.[51]On February 19, 2020,Akamaipublished their annual “State of the Internet” report, showcasing the growing trend of cybercriminals targeting public API platforms at financial services worldwide. From December 2017 through November 2019, Akamai witnessed 85.42 billion credential violation attacks. About 20%, or 16.55 billion, were against hostnames defined as API endpoints. Of these, 473.5 million have targeted financial services sector organizations.[52]
API documentation describes what services an API offers and how to use those services, aiming to cover everything a client would need to know for practical purposes.
Documentation is crucial for the development and maintenance of applications using the API.[53]API documentation is traditionally found in documentation files but can also be found in social media such as blogs, forums, and Q&A websites.[54]
Traditional documentation files are often presented via a documentation system, such as Javadoc or Pydoc, that has a consistent appearance and structure.
However, the types of content included in the documentation differs from API to API.[55]
In the interest of clarity, API documentation may include a description of classes and methods in the API as well as "typical usage scenarios, code snippets, design rationales, performance discussions, and contracts", but implementation details of the API services themselves are usually omitted. It can take a number of forms, including instructional documents, tutorials, and reference works. It'll also include a variety of information types, including guides and functionalities.
Restrictions and limitations on how the API can be used are also covered by the documentation. For instance, documentation for an API function could note that its parameters cannot be null, that the function itself is notthread safe.[56]Because API documentation tends to be comprehensive, it is a challenge for writers to keep the documentation updated and for users to read it carefully, potentially yielding bugs.[48]
API documentation can be enriched with metadata information likeJava annotations. This metadata can be used by the compiler, tools, and by therun-timeenvironment to implement custom behaviors or custom handling.[57]
It is possible to generate API documentation in a data-driven manner. By observing many programs that use a given API, it is possible to infer the typical usages, as well the required contracts and directives.[58]Then, templates can be used to generate natural language from the mined data.
In 2010, Oracle Corporation sued Google for having distributed a new implementation of Java embedded in the Android operating system.[59]Google had not acquired any permission to reproduce the Java API, although permission had been given to the similar OpenJDK project. JudgeWilliam Alsupruled in theOracle v. Googlecase that APIs cannot becopyrightedin the U.S. and that a victory for Oracle would have widely expanded copyright protection to a "functional set of symbols" and allowed the copyrighting of simple software commands:
To accept Oracle's claim would be to allow anyone to copyright one version of code to carry out a system of commands and thereby bar all others from writing its different versions to carry out all or part of the same commands.[60][61]
Alsup's ruling was overturned in 2014 on appeal to theCourt of Appeals for the Federal Circuit, though the question of whether such use of APIs constitutesfair usewas left unresolved.[62][63]
In 2016, following a two-week trial, a jury determined that Google's reimplementation of the Java API constitutedfair use, but Oracle vowed to appeal the decision.[64]Oracle won on its appeal, with the Court of Appeals for the Federal Circuit ruling that Google's use of the APIs did not qualify for fair use.[65]In 2019, Google appealed to theSupreme Court of the United Statesover both the copyrightability and fair use rulings, and the Supreme Court granted review.[66]Due to theCOVID-19 pandemic, the oral hearings in the case were delayed until October 2020.[67]
The case was decided by the Supreme Court in Google's favor.[68]
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Incomputer science,canonicalization(sometimesstandardizationornormalization) is a process for convertingdatathat has more than one possible representation into a "standard", "normal", orcanonical form. This can be done to compare different representations for equivalence, to count the number of distinct data structures, to improve the efficiency of variousalgorithmsby eliminating repeated calculations, or to make it possible to impose a meaningfulsortingorder.
Files infile systemsmay in most cases be accessed through multiplefilenames. For instance inUnix-like systems, the string "/./" can be replaced by "/". In theC standard library, the functionrealpath()performs this task. Other operations performed by this function to canonicalize filenames are the handling of/..components referring to parent directories, simplification of sequences of multiple slashes, removal of trailing slashes, and the resolution ofsymbolic links.
Canonicalization of filenames is important for computer security. For example, a web server may have a restriction that only files under the cgi directoryC:\inetpub\wwwroot\cgi-binmay be executed. This rule is enforced by checking that the path starts withC:\inetpub\wwwroot\cgi-bin\and only then executing it. While the fileC:\inetpub\wwwroot\cgi-bin\..\..\..\Windows\System32\cmd.exeinitially appears to be in the cgi directory, it exploits the..path specifier to traverse back up the directory hierarchy in an attempt to execute a file outside ofcgi-bin. Permittingcmd.exeto execute would be an error caused by a failure to canonicalize the filename to the simplest representation,C:\Windows\System32\cmd.exe, and is called adirectory traversalvulnerability. With the path canonicalized, it is clear the file should not be executed.
InUnicode, many accented letters can be represented in more than one way. For example,écan be represented in Unicode as the Unicode character U+0065 (LATIN SMALL LETTER E) followed by the character U+0301 (COMBINING ACUTE ACCENT), but it can also be represented as the precomposed character U+00E9 (LATIN SMALL LETTER E WITH ACUTE). This makes string comparison more complicated, since every possible representation of a string containing such glyphs must be considered. To deal with this, Unicode provides the mechanism ofcanonical equivalence. In this context, canonicalization isUnicode normalization.
Variable-width encodingsin the Unicode standard, in particularUTF-8, may cause an additional need for canonicalization in some situations. Namely, by the standard, in UTF-8 there is only one valid byte sequence for any Unicode character,[1]but some byte sequences are invalid, i.e., they cannot be obtained by encoding any string of Unicode characters into UTF-8. Some sloppy decoder implementations may accept invalid byte sequences as input and produce a valid Unicode character as output for such a sequence. If one uses such a decoder, some Unicode characters effectively have more than one corresponding byte sequence: the valid one and some invalid ones. This could lead to security issues similar to the one described in the previous section. Therefore, if one wants to apply some filter (e.g., a regular expression written in UTF-8) to UTF-8 strings that will later be passed to a decoder that allows invalid byte sequences, one should canonicalize the strings before passing them to the filter. In this context, canonicalization is the process of translating every string character to its single valid byte sequence. An alternative to canonicalization is to reject any strings containing invalid byte sequences.
Acanonical URLis aURLfor defining thesingle source of truthforduplicate content.
A canonical URL is the URL of the page that Google thinks is most representative from a set of duplicate pages on your site. For example, if you have URLs for the same page, such ashttps://example.com/?dress=1234andhttps://example.com/dresses/1234, Google chooses one as canonical. Note that the pages do not need to be absolutely identical; minor changes in sorting or filtering of list pages do not make the page unique (for example, sorting by price or filtering by item color).
The canonical can be in a different domain than a duplicate.[2]
With the help of canonical URLs, a search engine knows which link should be provided in a query result.
Acanonical link elementcan get used to define a canonical URL.
Inintranets, manual searching for information is predominant. In this case, canonical URLs can be defined in a non-machine-readable form, too. For example in aguideline.
Canonical URLs are usually the URLs that get used for theshare action.
Since the Canonical URL gets used in the search result of search engines, they are in most cases alanding page.
In web search andsearch engine optimization(SEO),URL canonicalizationdeals with web content that has more than one possible URL. Having multiple URLs for the same web content can cause problems for search engines - specifically in determining which URL should be shown in search results.[3]Most search engines support theCanonical link elementas a hint to which URL should be treated as the true version. As indicated by John Mueller of Google, having other directives in a page, like therobots noindexelement can give search engines conflicting signals about how to handle canonicalization[4]
Example:
All of these URLs point to the homepage of Wikipedia, but a search engine will only consider one of them to be the canonical form of the URL.
ACanonical XMLdocument is by definition an XML document that is in XML Canonical form, defined byThe Canonical XML specification. Briefly, canonicalization removes whitespace within tags, uses particular character encodings, sorts namespace references and eliminates redundant ones, removes XML and DOCTYPE declarations, and transforms relative URIs into absolute URIs.
A simple example would be the following two snippets of XML:
The first example contains extra spaces in the closing tag of the first node. The second example, which has been canonicalized, has had these spaces removed. Note that only the spaces within the tags are removed under W3C canonicalization, not those between tags.
A full summary of canonicalization changes is listed below:
Inmorphologyandlexicography, alemmais thecanonical formof a set ofwords. InEnglish, for example,run,runs,ran, andrunningare forms of the samelexeme, so we can select one of them; ex.run, to represent all the forms.Lexical databasessuch asUnitexuse this kind of representation.
Lemmatisationis the process of converting a word to itscanonical form.
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https://en.wikipedia.org/wiki/Canonicalization
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Inmathematics, aPadé approximantis the "best" approximation of a function near a specific point by arational functionof given order. Under this technique, the approximant'spower seriesagrees with the power series of the function it is approximating. The technique was developed around 1890 byHenri Padé, but goes back toGeorg Frobenius, who introduced the idea and investigated the features of rational approximations of power series.
The Padé approximant often gives better approximation of the function than truncating itsTaylor series, and it may still work where the Taylor series does notconverge. For these reasons Padé approximants are used extensively in computercalculations. They have also been used asauxiliary functionsinDiophantine approximationandtranscendental number theory, though for sharp results ad hoc methods—in some sense inspired by the Padé theory—typically replace them. Since a Padé approximant is a rational function, an artificial singular point may occur as an approximation, but this can be avoided byBorel–Padé analysis.
The reason the Padé approximant tends to be a better approximation than a truncating Taylor series is clear from the viewpoint of the multi-point summation method. Since there are many cases in which the asymptotic expansion at infinity becomes 0 or a constant, it can be interpreted as the "incomplete two-point Padé approximation", in which the ordinary Padé approximation improves on the method of truncating a Taylor series.
Given a functionfand twointegersm≥ 0andn≥ 1, thePadé approximantof order[m/n]is the rational function
R(x)=∑j=0majxj1+∑k=1nbkxk=a0+a1x+a2x2+⋯+amxm1+b1x+b2x2+⋯+bnxn,{\displaystyle R(x)={\frac {\sum _{j=0}^{m}a_{j}x^{j}}{1+\sum _{k=1}^{n}b_{k}x^{k}}}={\frac {a_{0}+a_{1}x+a_{2}x^{2}+\dots +a_{m}x^{m}}{1+b_{1}x+b_{2}x^{2}+\dots +b_{n}x^{n}}},}which agrees withf(x)to the highest possible order, which amounts tof(0)=R(0),f′(0)=R′(0),f″(0)=R″(0),⋮f(m+n)(0)=R(m+n)(0).{\displaystyle {\begin{aligned}f(0)&=R(0),\\f'(0)&=R'(0),\\f''(0)&=R''(0),\\&\mathrel {\;\vdots } \\f^{(m+n)}(0)&=R^{(m+n)}(0).\end{aligned}}}
Equivalently, ifR(x){\displaystyle R(x)}is expanded in a Maclaurin series (Taylor seriesat 0), its firstm+n{\displaystyle m+n}terms would equal the firstm+n{\displaystyle m+n}terms off(x){\displaystyle f(x)}, and thusf(x)−R(x)=cm+n+1xm+n+1+cm+n+2xm+n+2+⋯{\displaystyle f(x)-R(x)=c_{m+n+1}x^{m+n+1}+c_{m+n+2}x^{m+n+2}+\cdots }
When it exists, the Padé approximant is unique as a formal power series for the givenmandn.[1]
The Padé approximant defined above is also denoted as[m/n]f(x).{\displaystyle [m/n]_{f}(x).}
For givenx, Padé approximants can be computed byWynn's epsilon algorithm[2]and also othersequence transformations[3]from the partial sumsTN(x)=c0+c1x+c2x2+⋯+cNxN{\displaystyle T_{N}(x)=c_{0}+c_{1}x+c_{2}x^{2}+\cdots +c_{N}x^{N}}of theTaylor seriesoff, i.e., we haveck=f(k)(0)k!.{\displaystyle c_{k}={\frac {f^{(k)}(0)}{k!}}.}fcan also be aformal power series, and, hence, Padé approximants can also be applied to the summation ofdivergent series.
One way to compute a Padé approximant is via theextended Euclidean algorithmfor thepolynomial greatest common divisor.[4]The relationR(x)=P(x)/Q(x)=Tm+n(x)modxm+n+1{\displaystyle R(x)=P(x)/Q(x)=T_{m+n}(x){\bmod {x}}^{m+n+1}}is equivalent to the existence of some factorK(x){\displaystyle K(x)}such thatP(x)=Q(x)Tm+n(x)+K(x)xm+n+1,{\displaystyle P(x)=Q(x)T_{m+n}(x)+K(x)x^{m+n+1},}which can be interpreted as theBézout identityof one step in the computation of the extended greatest common divisor of the polynomialsTm+n(x){\displaystyle T_{m+n}(x)}andxm+n+1{\displaystyle x^{m+n+1}}.
Recall that, to compute the greatest common divisor of two polynomialspandq, one computes via long division the remainder sequencer0=p,r1=q,rk−1=qkrk+rk+1,{\displaystyle r_{0}=p,\;r_{1}=q,\quad r_{k-1}=q_{k}r_{k}+r_{k+1},}k= 1, 2, 3, ...withdegrk+1<degrk{\displaystyle \deg r_{k+1}<\deg r_{k}\,}, untilrk+1=0{\displaystyle r_{k+1}=0}. For the Bézout identities of the extended greatest common divisor one computes simultaneously the two polynomial sequencesu0=1,v0=0,u1=0,v1=1,uk+1=uk−1−qkuk,vk+1=vk−1−qkvk{\displaystyle u_{0}=1,\;v_{0}=0,\quad u_{1}=0,\;v_{1}=1,\quad u_{k+1}=u_{k-1}-q_{k}u_{k},\;v_{k+1}=v_{k-1}-q_{k}v_{k}}to obtain in each step the Bézout identityrk(x)=uk(x)p(x)+vk(x)q(x).{\displaystyle r_{k}(x)=u_{k}(x)p(x)+v_{k}(x)q(x).}
For the[m/n]approximant, one thus carries out the extended Euclidean algorithm forr0=xm+n+1,r1=Tm+n(x){\displaystyle r_{0}=x^{m+n+1},\;r_{1}=T_{m+n}(x)}and stops it at the last instant thatvk{\displaystyle v_{k}}has degreenor smaller.
Then the polynomialsP=rk,Q=vk{\displaystyle P=r_{k},\;Q=v_{k}}give the[m/n]Padé approximant. If one were to compute all steps of the extended greatest common divisor computation, one would obtain an anti-diagonal of thePadé table.
To study the resummation of adivergent series, say∑z=1∞f(z),{\displaystyle \sum _{z=1}^{\infty }f(z),}it can be useful to introduce the Padé or simply rational zeta function asζR(s)=∑z=1∞R(z)zs,{\displaystyle \zeta _{R}(s)=\sum _{z=1}^{\infty }{\frac {R(z)}{z^{s}}},}whereR(x)=[m/n]f(x){\displaystyle R(x)=[m/n]_{f}(x)}is the Padé approximation of order(m,n)of the functionf(x). Thezeta regularizationvalue ats= 0is taken to be the sum of the divergent series.
The functional equation for this Padé zeta function is∑j=0najζR(s−j)=∑j=0mbjζ0(s−j),{\displaystyle \sum _{j=0}^{n}a_{j}\zeta _{R}(s-j)=\sum _{j=0}^{m}b_{j}\zeta _{0}(s-j),}whereajandbjare the coefficients in the Padé approximation. The subscript '0' means that the Padé is of order [0/0] and hence, we have theRiemann zeta function.
Padé approximants can be used to extract critical points and exponents of functions.[5][6]In thermodynamics, if a functionf(x)behaves in a non-analytic way near a pointx=rlikef(x)∼|x−r|p{\displaystyle f(x)\sim |x-r|^{p}}, one callsx=ra critical point andpthe associated critical exponent off. If sufficient terms of the series expansion offare known, one can approximately extract the critical points and the critical exponents from respectively the poles and residues of the Padé approximants[n/n+1]g(x){\displaystyle [n/n+1]_{g}(x)}, whereg=f′/f{\displaystyle g=f'/f}.
A Padé approximant approximates a function in one variable. An approximant in two variables is called a Chisholm approximant (afterJ. S. R. Chisholm),[7]in multiple variables a Canterbury approximant (after Graves-Morris at the University of Kent).[8]
The conventional Padé approximation is determined to reproduce the Maclaurin expansion up to a given order. Therefore, the approximation at the value apart from the expansion point may be poor. This is avoided by the 2-point Padé approximation, which is a type of multipoint summation method.[9]Atx=0{\displaystyle x=0}, consider a case that a functionf(x){\displaystyle f(x)}which is expressed by asymptotic behaviorf0(x){\displaystyle f_{0}(x)}:f∼f0(x)+o(f0(x)),x→0,{\displaystyle f\sim f_{0}(x)+o{\big (}f_{0}(x){\big )},\quad x\to 0,}and atx→∞{\displaystyle x\to \infty }, additional asymptotic behaviorf∞(x){\displaystyle f_{\infty }(x)}:f(x)∼f∞(x)+o(f∞(x)),x→∞.{\displaystyle f(x)\sim f_{\infty }(x)+o{\big (}f_{\infty }(x){\big )},\quad x\to \infty .}
By selecting the major behavior off0(x),f∞(x){\displaystyle f_{0}(x),f_{\infty }(x)}, approximate functionsF(x){\displaystyle F(x)}such that simultaneously reproduce asymptotic behavior by developing the Padé approximation can be found in various cases. As a result, at the pointx→∞{\displaystyle x\to \infty }, where the accuracy of the approximation may be the worst in the ordinary Padé approximation, good accuracy of the 2-point Padé approximant is guaranteed. Therefore, the 2-point Padé approximant can be a method that gives a good approximation globally forx=0∼∞{\displaystyle x=0\sim \infty }.
In cases wheref0(x),f∞(x){\displaystyle f_{0}(x),f_{\infty }(x)}are expressed by polynomials or series of negative powers, exponential function, logarithmic function orxlnx{\displaystyle x\ln x}, we can apply 2-point Padé approximant tof(x){\displaystyle f(x)}. There is a method of using this to give an approximate solution of a differential equation with high accuracy.[9]Also, for the nontrivial zeros of the Riemann zeta function, the first nontrivial zero can be estimated with some accuracy from the asymptotic behavior on the real axis.[9]
A further extension of the 2-point Padé approximant is the multi-point Padé approximant.[9]This method treats singularity pointsx=xj(j=1,2,3,…,N){\displaystyle x=x_{j}(j=1,2,3,\dots ,N)}of a functionf(x){\displaystyle f(x)}which is to be approximated. Consider the cases when singularities of a function are expressed with indexnj{\displaystyle n_{j}}byf(x)∼Aj(x−xj)nj,x→xj.{\displaystyle f(x)\sim {\frac {A_{j}}{(x-x_{j})^{n_{j}}}},\quad x\to x_{j}.}
Besides the 2-point Padé approximant, which includes information atx=0,x→∞{\displaystyle x=0,x\to \infty }, this method approximates to reduce the property of diverging atx∼xj{\displaystyle x\sim x_{j}}. As a result, since the information of the peculiarity of the function is captured, the approximation of a functionf(x){\displaystyle f(x)}can be performed with higher accuracy.
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https://en.wikipedia.org/wiki/Pad%C3%A9_approximant
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Psychobabble(aportmanteauof "psychology" or "psychoanalysis" and "babble") is a derogatory name for therapy speech or writing that usespsychologicaljargon,buzzwords, andesotericlanguage to create an impression of truth orplausibility. The term implies that the speaker or writer lacks the experience and understanding necessary for the proper use of psychological terms. Additionally, it may imply that the content of speech deviates markedly from common sense and good judgement.
Some buzzwords that are commonly heard in psychobabble have come into widespread use inbusiness management,motivational seminars,self-help,folk psychology, andpopular psychology.
Frequent use of psychobabble can associate a clinical, psychological word with meaningless, or less meaningful, buzzword definitions. Laypersons often use such words when they describelife problemsas clinical maladies even though the clinical terms are not meaningful or appropriate.
Mostprofessionsdevelop a uniquevocabularyorjargonwhich, with frequent use, may become commonplace buzzwords. Professional psychologists may reject the "psychobabble" label when it is applied to their own special terminology.
The allusions to psychobabble imply that some psychological concepts lack precision and have become meaningless orpseudoscientific.
Psychobabble was defined by the writer who coined the word, R.D. Rosen,[1][2]as
a set of repetitive verbal formalities that kills off the very spontaneity, candour, and understanding it pretends to promote. It’s an idiom that reduces psychological insight to a collection of standardized observations that provides a frozen lexicon to deal with an infinite variety of problems.[3]
The word itself came into popular use after his 1977 publication ofPsychobabble: Fast Talk and Quick Cure in the Era of Feeling.[4]
Rosen coined the word in 1975 in a book review forThe Boston Phoenix, then featured it in a cover story for the magazineNew Timestitled "Psychobabble: The New Language of Candor."[5]His bookPsychobabbleexplores the dramatic expansion of psychological treatments and terminology in both professional and non-professional settings.
Certain terms considered to bepsychological jargonmay be dismissed as psychobabble when they are used by laypersons or in discussions ofpopular psychologythemes.New Age philosophies,self-helpgroups,personal developmentcoaching, andlarge-group awareness trainingare often said to employ psychobabble.
The word "psychobabble" may refer contemptuously to pretentious psychologicalgibberish. Automated talk-therapy offered by variousELIZAcomputer programs produce notable examples of conversational patterns that are psychobabble, even though they may not be loaded with jargon. ELIZA programs parody clinical conversations in which a therapist replies to a statement with a question that requires little or no specific knowledge.
"Neurobabble" is a related term. Beyerstein (1990)[6]wrote that neurobabble can appear in "ads [that] suggest that brain 'repatterning' will foster effortless learning, creativity, and prosperity." He associated neuromythologies ofleft/right brain pseudosciencewith specific New Age products and techniques. He stated that "the purveyors of neurobabble urge us to equate truth with what feels right and to abandon the commonsense insistence that those who would enlighten us provide at least as much evidence as we demand of politicians or used-car salesmen."
Psychobabble terms are typically words or phrases which have their roots inpsychotherapeutic practice. Psychobabblers commonly overuse such terms as if they possessed some special value or meaning.
Rosen has suggested that the following terms often appear in psychobabble:co-dependent,delusion,denial,dysfunctional,empowerment,holistic,meaningful relationship,multiple personality disorder,narcissism,psychosis,self-actualization,synergy, andmindfulness.
Extensive examples of psychobabble appear inCyra McFadden's satirical novelThe Serial: A Year in the Life of Marin County(1977).[7]In his collection of critical essays,Working with Structuralism(1981), the British scholar and novelistDavid Lodgegives a structural analysis of the language used in the novel and notes that McFadden endorsed the use of the term.[8]
In 2010,Theodore Dalrympledefined psychobabble as "the means by which people talk about themselves without revealing anything."[9]
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https://en.wikipedia.org/wiki/Psychobabble
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Price's laworPrice's square root lawis abibliometrichypothesis proposed byDerek J. de Solla Pricesuggesting that in any scientific field, half of the published research comes from the square root of the total number of authors in that field.
The law specifically states that if n represents the total number of authors in a scientific domain, then √n authors will be responsible for producing approximately 50% of the total publications in that field. For example, if 100 papers are written by 25 authors, then25=5{\displaystyle {\sqrt {25}}=5}out of the 25 authors will have contributed 50 papers.
Derek J. de Solla Price introduced this concept in his 1963 book "Little Science, Big Science" as part of his broader research on scientific productivity and information dynamics.[1]The law was intended to describe the uneven distribution of scientific output across researchers.
Subsequent research has largely contradicted Price's original hypothesis. Multiple studies across various scientific disciplines have found that the actual distribution of publications is more skewed than Price's law predicted. Most empirical analyses suggest that a much smaller proportion of researchers produce a significantly larger percentage of publications. The relatedLotka's law,[2]for example, is a better fit.[3][4]
Despite its empirical limitations, Price's law remains important in various fields,[5][6]for example to understand scientific productivity patterns, analyze or research output distributions, or highlight the concentration of scientific work among a small number of researchers
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https://en.wikipedia.org/wiki/Price%27s_law
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Apersonal air vehicle(PAV) is a proposed class of passengeraircraftproviding on-demand air transport.
The emergence of this alternative to traditional ground transport methods has been enabled byunmanned aerial vehicletechnologies andelectric propulsion.
Barriers includeaviation safety,airworthiness,operating costs,usability,airspaceintegration,aircraft noiseandemissions, tackled first by smallUAScertification then experience.[1]
There is no fully accepted definition as yet of apersonal air vehicle(PAV). Typically it is understood to be an autonomous electric aircraft with point-to-point VTOL capability. It may or may not be treated as a single-seat autonomous electric vehicle, as distinguished from the multi-seateVTOL.[2]It is intended to provide flight convenience similar to the private car in terms of accessibility and ease of operation, while also offering the speed and routing efficiencies made possible by direct point-to-point flight. The PAV differs from conventionalgeneral aviationtypes in being usable by people with no pilot qualifications.[3]
Besides the fabrication of personal air vehicles, the creation of autonomous systems for PAVs is also being researched. First off,synthetic vision electronic flight instrument systems(EFIS) asHighway in the sky(HITS) makes it much easier to control aircraft.[4]Also,Phantom Worksis working on designing a system that allows to automate PAVs. The PAVs are designated their own "lanes" in the sky, thereby ensuring the avoidance of possible collisions. In addition, the different PAVs are also capable of detecting each other and communicating with each other, further decreasing the risk of collisions.[5]
TheFederal Aviation Administration(FAA) infrastructure is not currently capable of handling the increase in aircraft traffic that would be generated by PAVs. The FAA plan to upgrade forms theNext Generation Air Transportation System, planned for 2025.[6]An interim plan is to use smaller airports. Modeling by NASA and others have shown that PAVs using smaller community airports would not interfere with commercial traffic at larger airports. Currently there are over 10,000 public and private small airports in the United States that could be used for this type of transportation. This infrastructure is currently underutilized, used primarily by recreational aircraft.
Noise from PAVs could also upset communities if they operate near homes and businesses. Without lower noise levels that enable residential landings, any PAV must take off and land at an FAA-controlled airfield, where higher sound levels have been approved.
Studies have explored ways to make helicopters and aircraft less noisy, but noise levels remain high. In 2005 a simple method of reducing noise was identified: Keep aircraft at a higher altitude during landing. This is called aContinuous Descent Approach(CDA).[7]
Many proposed PAV aircraft are based onelectric batteries, however they have low range due to the lowspecific energyof current batteries.[8]This range may be insufficient to provide adequate safety margin to find a landing site in an emergency.
Fuel cellaircraft have been proposed as a solution to this issue, owing to the much higher specific energy ofhydrogen.[8][9]
Urban flight safety is a well-known problem for regulators and industry. On May 16, 1977, theNew York Airways accidentof aSikorsky S-61helicopter shuttle fromJohn F. Kennedy International Airport, which landed on the roof of the Pan Am Building (nowMetLife Building) when a landing gear collapsed and a detached rotor blade killed several people on the helipad and one woman onMadison Avenue, ending that business for decades almost around the world. Currenthelicopteraccident rates would be insufficient for urban mobility. TheSikorsky S-92's safety-focused design still allows one fatal accident per million flight hours. This rate would lead to 150 accidents per year for 50,000 eVTOLs flying 3,000 hours a year.[10]
For Sikorsky Innovations, the emerging $30 billion urban air mobility market needs safety at least as good asFAR Part 29governing over 7,000 lb (3.2 t) helicopters.
By May 2018, Sikorsky flew anS-76120 hours with full point-to-point, real timeautonomous flightandterrainavoidance the hard way, withLevel A softwareandredundancy, with a safety pilot.[11]Sikorsky Aircraftwant to reach a verticalflight safetyof one failure per 10 million hours on high-utilization platforms by combining currentrotorcraftexperience with advances in autonomous flight,airspaceintegration andelectric propulsion.[10]
NASAestablished the Personal Air Vehicle Sector Project in 2002, as part of their Vehicle Systems Program (VSP). This project was part of the NASA Vehicle Integration, Strategy, and Technology Assessment (VISTA) office, which also included sectors for Subsonic Transports, VTOL Aircraft, Supersonic Aircraft, and High Altitude Long Endurance Aircraft. The objective of each sector was to establish vehicle capability goals and the required technology investment strategies to achieve those breakthroughs.[12]
The difference in vehicle characteristics between PAVs and existing General Aviation single engine piston aircraft was set out in 2003 at an American Institute of Aeronautics and Astronautics (AIAA) conference.[13]Advanced concepts would be needed to dramatically enhance ease of use, safety, efficiency, field length performance, and affordability.
In 2006 the VSP was replaced by new NASA Aeronautics initiatives. PAV technology development efforts at NASA shifted to a prize-based investment, with NASA Centennial Challenge Prize funds of $250,000 being provided for a Personal Air Vehicle Challenge in 2007.[citation needed]
TheEuropean Unionis funding a 3-leg€4.2m study (under theSeventh Framework Programme) of technologies and impacts for PAVs; Human-aircraft interaction, Automation of aerial systems in cluttered environments, and Exploring the socio-technological environment.[14][15]
NASA Langley has researched and prototyped the necessary PAV technologies and has dedicated the largest cash prize in the history of GA to the PAV that can demonstrate the best overall combination of performance. The PAV flight competition for this prize, known as the first annualPAV Challenge, was held Aug 4-12, 2007 and hosted theCAFE Foundationin Santa Rosa, California.[16]
In 2008 the challenge was renamed as the General Aviation Technology Challenge.
The new prizes were:
The winners were:
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https://en.wikipedia.org/wiki/Personal_air_vehicle
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