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Instatistics,economics, andeconophysics, theking effectis the phenomenon in which the top one or two members of a ranked set show up as clearoutliers. These top one or two members are unexpectedly large because they do not conform to the statisticaldistributionorrank-distributionwhich the remainder of the set obeys.
Distributions typically followed include thepower-law distribution,[2]that is a basis for thestretched exponential function,[1][3]andparabolic fractal distribution.
The King effect has been observed in the distribution of:
Note, however, that the king effect is not limited to outliers with a positive evaluation attached to their rank: for rankings on an undesirable attribute, there may exist apauper effect, with a similar detachment of extremely ranked data points from the reasonably distributed portion of the data set.[citation needed]
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https://en.wikipedia.org/wiki/King_effect
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Ineconomics, theLorenz curveis a graphical representation of thedistribution of incomeor ofwealth. It was developed byMax O. Lorenzin 1905 for representinginequalityof thewealth distribution.
The curve is agraphshowing the proportion of overall income or wealth assumed by the bottomx%of the people, although this is not rigorously true for a finite population (see below). It is often used to representincome distribution, where it shows for the bottomx%of households, what percentage(y%)of the total income they have. Thepercentageof households is plotted on thex-axis, the percentage of income on they-axis. It can also be used to show distribution ofassets. In such use, many economists consider it to be a measure ofsocial inequality.
The concept is useful in describing inequality among the size of individuals inecology[1]and in studies ofbiodiversity, where the cumulative proportion of species is plotted against the cumulative proportion of individuals.[2]It is also useful inbusiness modeling: e.g., inconsumer finance, to measure the actual percentagey%ofdelinquenciesattributable to thex%of people with worstrisk scores. Lorenz curves were also applied toepidemiologyandpublic health, e.g., to measure pandemic inequality as the distribution of nationalcumulative incidence(y%) generated by the population residing in areas (x%) ranked with respect to their local epidemicattack rate.[3]
Data from 2005.
Points on the Lorenz curve represent statements such as, "the bottom 20% of all households have 10% of the total income."
A perfectly equal income distribution would be one in which every person has the same income. In this case, the bottomN%of society would always haveN%of the income. This can be depicted by the straight liney=x; called the "line of perfect equality."
By contrast, a perfectly unequal distribution would be one in which one person has all the income and everyone else has none. In that case, the curve would be aty= 0%for allx< 100%, andy= 100%whenx= 100%. This curve is called the "line of perfect inequality."
TheGini coefficientis the ratio of the area between the line of perfect equality and the observed Lorenz curve to the area between the line of perfect equality and the line of perfect inequality. The higher the coefficient, the more unequal the distribution is. In the diagram on the right, this is given by the ratioA/ (A+B), whereAandBare the areas of regions as marked in the diagram.
The Lorenz curve is a probability plot (aP–P plot) comparing the distribution of avariableagainst a hypothetical uniform distribution of that variable. It can usually be represented by a functionL(F), whereF, the cumulative portion of the population, is represented by the horizontal axis, andL, the cumulative portion of the total wealth or income, is represented by the vertical axis.
The curveLneed not be a smoothly increasing function ofF, For wealth distributions there may be oligarchies or people with negative wealth for instance.[4]
For a discrete distribution of Y given by valuesy1, ...,ynin non-decreasing order(yi≤yi+1)and their probabilitiesf(yj):=Pr(Y=yj){\displaystyle f(y_{j}):=\Pr(Y=y_{j})}the Lorenz curve is thecontinuouspiecewise linear functionconnecting the points(Fi,Li),i= 0 ton, whereF0= 0,L0= 0, and fori= 1 ton:Fi:=∑j=1if(yj)Si:=∑j=1if(yj)yjLi:=SiSn{\displaystyle {\begin{aligned}F_{i}&:=\sum _{j=1}^{i}f(y_{j})\\S_{i}&:=\sum _{j=1}^{i}f(y_{j})\,y_{j}\\L_{i}&:={\frac {S_{i}}{S_{n}}}\end{aligned}}}
When allyiare equally probable with probabilities1 /nthis simplifies toFi=inSi=1n∑j=1iyjLi=SiSn{\displaystyle {\begin{aligned}F_{i}&={\frac {i}{n}}\\S_{i}&={\frac {1}{n}}\sum _{j=1}^{i}\;y_{j}\\L_{i}&={\frac {S_{i}}{S_{n}}}\end{aligned}}}
For acontinuous distributionwith theprobability density functionfand thecumulative distribution functionF, the Lorenz curveLis given by:L(F(x))=∫−∞xtf(t)dt∫−∞∞tf(t)dt=∫−∞xtf(t)dtμ{\displaystyle L(F(x))={\frac {\int _{-\infty }^{x}t\,f(t)\,dt}{\int _{-\infty }^{\infty }t\,f(t)\,dt}}={\frac {\int _{-\infty }^{x}t\,f(t)\,dt}{\mu }}}whereμ{\displaystyle \mu }denotes the average. The Lorenz curveL(F)may then be plotted as a function parametric inx:L(x)vs.F(x). In other contexts, the quantity computed here is known as the length biased (or size biased) distribution; it also has an important role in renewal theory.
Alternatively, for acumulative distribution functionF(x)with inversex(F), the Lorenz curveL(F)is directly given by:L(F)=∫0Fx(F1)dF1∫01x(F1)dF1{\displaystyle L(F)={\frac {\int _{0}^{F}x(F_{1})\,dF_{1}}{\int _{0}^{1}x(F_{1})\,dF_{1}}}}
The inversex(F)may not exist because the cumulative distribution function has intervals of constant values. However, the previous formula can still apply by generalizing the definition ofx(F):x(F1)=inf{y:F(y)≥F1}{\displaystyle x(F_{1})=\inf\{y:F(y)\geq F_{1}\}}whereinfis theinfimum.
For an example of a Lorenz curve, seePareto distribution.
A Lorenz curve always starts at (0,0) and ends at (1,1).
The Lorenz curve is not defined if the mean of the probability distribution is zero or infinite.
The Lorenz curve for a probability distribution is acontinuous function. However, Lorenz curves representing discontinuous functions can be constructed as the limit of Lorenz curves of probability distributions, the line of perfect inequality being an example.
The information in a Lorenz curve may be summarized by theGini coefficientand theLorenz asymmetry coefficient.[1]
The Lorenz curve cannot rise above the line of perfect equality.
A Lorenz curve that never falls beneath a second Lorenz curve and at least once runs above it, has Lorenz dominance over the second one.[5]
If the variable being measured cannot take negative values, the Lorenz curve:
Note however that a Lorenz curve fornet worthwould start out by going negative due to the fact that some people have a negative net worth because of debt.
The Lorenz curve is invariant under positive scaling. IfXis a random variable, for any positive numbercthe random variablecXhas the same Lorenz curve asX.
The Lorenz curve is flipped twice, once aboutF= 0.5and once aboutL= 0.5, by negation. IfXis a random variable with Lorenz curveLX(F), then−Xhas the Lorenz curve:
The Lorenz curve is changed by translations so that the equality gapF−L(F)changes in proportion to the ratio of the original and translated means. IfXis a random variable with a Lorenz curveLX(F)and meanμX, then for any constantc≠ −μX,X+chas a Lorenz curve defined by:F−LX+c(F)=μXμX+c(F−LX(F)){\displaystyle F-L_{X+c}(F)={\frac {\mu _{X}}{\mu _{X}+c}}(F-L_{X}(F))}
For a cumulative distribution functionF(x)with meanμand (generalized) inversex(F), then for anyFwith 0 <F< 1:
BothL(F) =FPandL(F) = 1 − (1 −F)1/P, forP≥ 1, are well-known functional forms for the Lorenz curve.[6]
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https://en.wikipedia.org/wiki/Lorenz_curve
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Lotka's law,[1]named afterAlfred J. Lotka, is one of a variety of special applications ofZipf's law. It describes the frequency of publication by authors in any given field.
LetX{\displaystyle X}be the number of publications,Y{\displaystyle Y}be the number of authors withX{\displaystyle X}publications, andk{\displaystyle k}be a constant depending on the specific field. Lotka's law states thatY∝X−k{\displaystyle Y\propto X^{-k}}.
In Lotka's original publication, he claimedk=2{\displaystyle k=2}. Subsequent research showed thatk{\displaystyle k}varies depending on the discipline.
Equivalently, Lotka's law can be stated asY′∝X−(k−1){\displaystyle Y'\propto X^{-(k-1)}}, whereY′{\displaystyle Y'}is the number of authors withat leastX{\displaystyle X}publications. Their equivalence can be proved by taking thederivative.
Assume that n=2 in a discipline, then as the number of articles published increases, authors producing that many publications become less frequent. There are 1/4 as many authors publishing two articles within a specified time period as there are single-publication authors, 1/9 as many publishing three articles, 1/16 as many publishing four articles, etc.
And if 100 authors wroteexactlyone article each over a specific period in the discipline, then:
That would be a total of 294 articles and 155 writers, with an average of 1.9 articles for each writer.
Lotka's law may be described using theZeta distribution:
forx=1,2,3,4,…{\displaystyle x=1,2,3,4,\dots }and where
is theRiemann zeta function. It is the limiting case ofZipf's lawwhere an individual's maximum number of publications is infinite.
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Menzerath's law, also known as theMenzerath–Altmann law(named afterPaul MenzerathandGabriel Altmann), is alinguisticlaw according to which the increase of the size of a linguistic construct results in a decrease of the size of its constituents, and vice versa.[1][2]
For example, the longer a sentence (measured in terms of the number of clauses), the shorter the clauses (measured in terms of the number of words), or: the longer a word (in syllables or morphs), the shorter the syllables or morphs in sounds.
In the 19th century,Eduard Sieversobserved that vowels in short words are pronounced longer than the same vowels in long words.[3][4]: 122Menzerath & de Oleza (1928)[5]expanded this observation to state that, as the number of syllables in words increases, the syllables themselves become shorter on average.
From this, the following hypothesis developed:
The larger the whole, the smaller its parts.
In particular, for linguistics:
The larger a linguistic construct, the smaller its constituents.
In the early 1980s, Altmann, Heups,[6]and Köhler[7]demonstrated using quantitative methods that this postulate can also be applied to larger constructs of natural language: the larger the sentence, the smaller the individual clauses, etc. A prerequisite for such relationships is that a relationship between units (here: sentence) and their direct constituents (here: clause) is examined.[8][9][1]: Übersichten
According to Altmann (1980),[8]it can be mathematically stated as:
y=a⋅xb⋅e−cx{\displaystyle y=a\cdot x^{b}\cdot e^{-cx}}
where:
The law can be explained by assuming that linguistic segments contain information about their structure (besides the information that needs to be communicated).[7]The assumption that the length of the structure information is independent of the length of the other content of the segment yields the alternative formula that was also successfully empirically tested.[10]
Gerlach (1982)[11]checked a German dictionary[12]with about 15,000 entries:
Wherex{\displaystyle x}is the number of morphs per word,n{\displaystyle n}is the number of words in the dictionary with lengthx{\displaystyle x};y{\displaystyle y}is the observed average length of morphs (number of phonemes per morph);y∗{\displaystyle y^{*}}is the prediction according toy=axb{\displaystyle y=ax^{b}}wherea,b{\displaystyle a,b}are fited to data. TheF-testhasp<0.001{\displaystyle p<0.001}.
As another example, the simplest form of Menzerath's law,y=axb{\displaystyle y=ax^{b}}, holds for the duration of vowels in Hungarian words:[13]
More examples are on the German Wikipedia pages onphoneme duration,syllable duration,word length,clause length, andsentence length.
This law also seems to hold true for at least a subclass of JapaneseKanjicharacters.[14]
Beyondquantitative linguistics, Menzerath's law can be discussed in any multi-level complex systems. Given three levels,x{\displaystyle x}is the number of middle-level units contained in a high-level unit,y{\displaystyle y}is the averaged number of low-level units contained in middle-level units, Menzerath's law claims a negativecorrelationbetweeny{\displaystyle y}andx{\displaystyle x}.
Menzerath's law is shown to be true for both thebase-exon-genelevels in thehuman genome,[15]andbase-chromosome-genomelevels in genomes from a collection of species.[16]In addition, Menzerath's law was shown to accurately predict the distribution of protein lengths in terms of amino acid number in theproteomeof ten organisms.[17]
Furthermore, studies have shown that the social behavior of baboon groups also corresponds to Menzerath's Law: the larger the entire group, the smaller the subordinate social groups.[1]: 99 ff
In 2016, a research group at theUniversity of Michiganfound that the calls ofgeladasobey Menzerath's law, observing that calls are abbreviated when used in longer sequences.[18][19]
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ThePareto principle(also known as the80/20 rule, thelaw of the vital fewand theprinciple of factor sparsity[1][2]) states that for many outcomes, roughly 80% of consequences come from 20% of causes (the "vital few").[1]
In 1941,management consultantJoseph M. Jurandeveloped the concept in the context of quality control and improvement after reading the works of ItaliansociologistandeconomistVilfredo Pareto, who wrote in 1906 about the 80/20 connection while teaching at theUniversity of Lausanne.[3]In his first work,Cours d'économie politique, Pareto showed that approximately 80% of the land in theKingdom of Italywas owned by 20% of the population. The Pareto principle is only tangentially related to thePareto efficiency.
Mathematically, the 80/20 rule is roughly described by apower lawdistribution (also known as aPareto distribution) for a particular set of parameters. Many natural phenomena are distributed according to power law statistics.[4]It is anadageofbusiness managementthat "80% of sales come from 20% ofclients."[5]
In 1941, Joseph M. Juran, a Romanian-born American engineer, came across the work of Italian polymathVilfredo Pareto. Pareto noted that approximately 80% of Italy's land was owned by 20% of the population.[6][4]Juran applied the observation that 80% of an issue is caused by 20% of the causes to quality issues. Later during his career, Juran preferred to describe this as "the vital few and the useful many" to highlight that the contribution of the remaining 80% should not be discarded entirely.[7]
The demonstration of the Pareto principle is explained by a large proportion of process variation being associated with a small proportion of process variables.[2]This is a special case of the wider phenomenon ofPareto distributions. If thePareto indexα, which is one of the parameters characterizing a Pareto distribution, is chosen asα= log45 ≈ 1.16, then one has 80% of effects coming from 20% of causes.[8]
The term 80/20 is only a shorthand for the general principle at work. In individual cases, the distribution could be nearer to 90/5 or 70/40. Note that there is no need for the two numbers to add up to the number 100, as they are measures of different things. The Pareto principle is an illustration of a "power law" relationship, which also occurs in phenomena such asbush firesand earthquakes.[9]Because it is self-similar over a wide range of magnitudes, it produces outcomes completely different fromNormal or Gaussian distributionphenomena. This fact explains the frequent breakdowns of sophisticated financial instruments, which are modeled on the assumption that a Gaussian relationship is appropriate to something like stock price movements.[10]
Using the "A:B" notation (for example, 0.8:0.2) and withA+B= 1,inequality measureslike theGini index(G)andtheHoover index(H) can be computed. In this case both are the same:
Pareto analysis is a formal technique useful where many possible courses of action are competing for attention. In essence, the problem-solver estimates the benefit delivered by each action, then selects a number of the most effective actions that deliver a total benefit reasonably close to the maximal possible one.
Pareto analysis is a creative way of looking at causes of problems because it helps stimulate thinking and organize thoughts. However, it can be limited by its exclusion of possibly important problems which may be small initially, but will grow with time. It should be combined with other analytical tools such asfailure mode and effects analysisandfault tree analysisfor example.[citation needed]
This technique helps to identify the top portion of causes that need to be addressed to resolve the majority of problems. Once the predominant causes are identified, then tools like theIshikawa diagramor Fish-bone Analysis can be used to identify the root causes of the problems. While it is common to refer to pareto as "80/20" rule, under the assumption that, in all situations, 20% of causes determine 80% of problems, this ratio is merely a convenient rule of thumb and is not, nor should it be considered, an immutable law of nature.
The application of the Pareto analysis in risk management allows management to focus on those risks that have the most impact on the project.[11]
Steps to identify the important causes using 80/20 rule:[12]
Pareto's observation was in connection withpopulation and wealth. Pareto noticed that approximately 80% of Italy's land was owned by 20% of the population.[6]He then carried out surveys on a variety of other countries and found to his surprise that a similar distribution applied.[citation needed]
A chart that demonstrated the effect appeared in the 1992United Nations Development ProgramReport, which showed that the richest 20% of the world's population receives 82.7% of the world's income.[13]However, among nations, theGini indexshows that wealth distributions vary substantially around this norm.[14]
The principle also holds within the tails of the distribution. The physicist Victor Yakovenko of theUniversity of Maryland, College Parkand AC Silva analyzed income data from the US Internal Revenue Service from 1983 to 2001 and found that theincome distributionof the richest 1–3% of the population also follows Pareto's principle.[16]
InTalent: How to Identify Entrepreneurs, economistTyler Cowenand entrepreneurDaniel Grosssuggest that the Pareto Principle can be applied to the role of the 20% most talented individuals in generating the majority ofeconomic growth.[17]According to theNew York Timesin 1988, manyvideo rental shopsreported that 80% of revenue came from 20% of videotapes (although rarely rented classics such asGone with the Windmust be stocked to appear to have a good selection).[18]
Incomputer sciencethe Pareto principle can be applied tooptimizationefforts.[19]For example,Microsoftnoted that by fixing the top 20% of the most-reported bugs, 80% of the related errors and crashes in a given system would be eliminated.[20]Lowell Arthur expressed that "20% of the code has 80% of the errors. Find them, fix them!"[21]
Occupational health and safetyprofessionals use the Pareto principle to underline the importance of hazard prioritization. Assuming 20% of the hazards account for 80% of the injuries, and by categorizing hazards, safety professionals can target those 20% of the hazards that cause 80% of the injuries or accidents. Alternatively, if hazards are addressed in random order, a safety professional is more likely to fix one of the 80% of hazards that account only for some fraction of the remaining 20% of injuries.[22]
Aside from ensuring efficient accident prevention practices, the Pareto principle also ensures hazards are addressed in an economical order, because the technique ensures the utilized resources are best used to prevent the most accidents.[23]
The Pareto principle is the basis for thePareto chart, one of the key tools used intotal quality controlandSix Sigmatechniques. The Pareto principle serves as a baseline forABC-analysisand XYZ-analysis, widely used inlogisticsand procurement for the purpose of optimizing stock of goods, as well as costs of keeping and replenishing that stock.[24]In engineering control theory, such as for electromechanical energy converters, the 80/20 principle applies to optimization efforts.[19]
The remarkable success of statistically based searches for root causes is based upon a combination of an empirical principle and mathematical logic. The empirical principle is usually known as the Pareto principle.[25]With regard to variation causality, this principle states that there is a non-random distribution of the slopes of the numerous (theoretically infinite) terms in the general equation.
All of the terms are independent of each other by definition. Interdependent factors appear as multiplication terms. The Pareto principle states that the effect of the dominant term is very much greater than the second-largest effect term, which in turn is very much greater than the third, and so on.[26]There is no explanation for this phenomenon; that is why we refer to it as an empirical principle.
The mathematical logic is known as the square-root-of-the-sum-of-the-squares axiom. This states that the variation caused by the steepest slope must be squared, and then the result added to the square of the variation caused by the second-steepest slope, and so on. The total observed variation is then the square root of the total sum of the variation caused by individual slopes squared. This derives from the probability density function for multiple variables or the multivariate distribution (we are treating each term as an independent variable).
The combination of the Pareto principle and the square-root-of-the-sum-of-the-squares axiom means that the strongest term in the general equation totally dominates the observed variation of effect. Thus, the strongest term will dominate the data collected for hypothesis testing.
In the systems science discipline,Joshua M. EpsteinandRobert Axtellcreated anagent-based simulationmodel calledSugarscape, from adecentralized modelingapproach, based on individual behavior rules defined for each agent in the economy. Wealth distribution and Pareto's 80/20 principle emerged in their results, which suggests the principle is a collective consequence of these individual rules.[27]
In 2009, theAgency for Healthcare Research and Qualitysaid 20% of patients incurred 80% of healthcare expenses due to chronic conditions.[28]A 2021 analysis showed unequal distribution of healthcare costs, with older patients and those with poorer health incurring more costs.[29]The 80/20 rule has been proposed as a rule of thumb for the infection distribution insuperspreading events.[30][31]However, the degree of infectiousness has been found to be distributed continuously in the population.[31]Inepidemicswith super-spreading, the majority of individuals infect relatively fewsecondary contacts.
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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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Theprinciple of least effortis a broadtheorythat covers diverse fields fromevolutionary biologytowebpage design. It postulates that animals, people, and even well-designed machines will naturally choose thepath of least resistanceor "effort". It is closely related to many other similar principles (seeprinciple of least actionor other articles listedbelow).
This is perhaps best known, or at least documented, among researchers in the field oflibrary and information science. Their principle states that aninformation-seekingclient will tend to use the most convenient search method in the least exacting mode available. Information-seeking behavior stops as soon as minimally acceptable results are found. This theory holds true regardless of the user's proficiency as a searcher, or their level of subject expertise. Also, this theory takes into account the user's previous information-seeking experience. The user will use the tools that are most familiar and easy to use that find results. The principle of least effort is known as a "deterministic description of human behavior".[1]
The principle of least effort applies not only in the library context, but also to any information-seeking activity. For example, one might consult a generalist co-worker down the hall rather than a specialist in another building, so long as the generalist's answers were within the threshold of acceptability.
The principle of least effort is analogous to thepath of least resistance.
The principle was first articulated by the Italian philosopherGuillaume Ferreroin an article in theRevue philosophique de la France et de l'étranger, 1 January 1894.[2]About fifty years later, this principle was studied bylinguistGeorge Kingsley Zipfwho wroteHuman Behaviour and the Principle of Least Effort: An Introduction to Human Ecology, first published in 1949. He theorised that the distribution of word use was due to tendency to communicate efficiently with least effort and this theory is known asZipf's Law.[3]
Within the context of information seeking, the principle of least effort was studied by Herbert Poole who wroteTheories of the Middle Rangein 1985.[4]LibrarianThomas Mann lists the principle of least effort as one of several principles guidinginformation seekingbehavior in his 1987 book,A Guide to Library Research Methods.[5]
Likewise, one of the most common measures of information-seeking behavior, library circulation statistics, also follows the80-20 rule. This suggests that information-seeking behavior is a manifestation not of anormal distributioncurve, but of apower lawcurve.
(Regarding the Law of Least Effort, seeWilliam James,The Principles of Psychology, 1890, page 944.)
The principle of least effort is especially important when considering design for libraries and research in the context of the modern library. Libraries must take into consideration the user's desire to find information quickly and easily. The principle must be considered when designing individualOnline Public Access Catalogs (OPACs)as well as other library tools.
The principle is a guiding force for the push to provide access toelectronic mediain libraries. The principle of least effort was further explored in a study of library behavior of graduate students by Zao Liu and Zheng Ye (Lan) Lang published in 2004. The study sampledTexas A&Mdistance learning graduate students to test what library resources they used, and why they used those particular resources. In this study the Internet was used the most, while libraries were the next most used resource for conducting class research. The study found that most students used these resources due to their quickness and ability to access from home. The study found that the principle of least effort was the primary behavior model of mostdistance learningstudents.[6]This means that modern libraries, especially academic libraries, need to analyze their electronic databases in order to successfully cater to the needs of the changing realities of information science.
Professional writers employ the principle of least effort duringaudience analysis. The writer analyzes the reader's environment, previous knowledge, and other similar information which the reader may already know. Intechnical writing, recursive organization, where parts resemble the organization of the whole, helps readers find their way. Consistency of navigational features is a common concern in software design.
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Rank–size distributionis the distribution of size by rank, in decreasing order of size. For example, if a data set consists of items of sizes 5, 100, 5, and 8, the rank-size distribution is 100, 8, 5, 5 (ranks 1 through 4). This is also known as therank–frequency distribution, when the source data are from afrequency distribution. These are particularly of interest when the data vary significantly in scales, such as city size or word frequency. These distributions frequently follow apower lawdistribution, or less well-known ones such as astretched exponential functionorparabolic fractal distribution, at least approximately for certain ranges of ranks; see below.
A rank-size distribution is not aprobability distributionorcumulative distribution function. Rather, it is a discrete form of aquantile function(inverse cumulative distribution) in reverse order, giving the size of the element at a given rank.
In the case of city populations, the resulting distribution in a country, a region, or the world will be characterized by its largest city, with other cities decreasing in size respective to it, initially at a rapid rate and then more slowly. This results in a few large cities and a much larger number of cities orders of magnitude smaller. For example, a rank 3 city would have one-third the population of a country's largest city, a rank 4 city would have one-fourth the population of the largest city, and so on.[2]
A rank-size (or rank–frequency) distribution is often segmented into ranges. This is frequently done somewhat arbitrarily or due to external factors, particularly formarket segmentation, but can also be due to distinct behavior as rank varies.
Most simply and commonly, a distribution may be split in two pieces, termed theheadandtail. If a distribution is broken into three pieces, the third (middle) piece has several terms, genericallymiddle,[3]alsobelly,[4]torso,[5]andbody.[6]These frequently have some adjectives added, most significantlylong tail, alsofat belly,[4]chunky middle, etc. In more traditional terms, these may be calledtop-tier,mid-tier, andbottom-tier.
The relative sizes and weights of these segments (how many ranks in each segment, and what proportion of the total population is in a given segment) qualitatively characterize a distribution, analogously to theskewnessorkurtosisof a probability distribution. Namely: is it dominated by a few top members (head-heavy, like profits in the recorded music industry), or is it dominated by many small members (tail-heavy, like internet search queries), or distributed in some other way? Practically, this determines strategy: where should attention be focused?
These distinctions may be made for various reasons. For example, they may arise from differing properties of the population, as in the90–9–1 principle, which posits that in an internet community, 90% of the participants of a community only view content, 9% of the participants edit content, and 1% of the participants actively create new content. As another example, in marketing, one may pragmatically consider the head as all members that receive personalized attention, such as personal phone calls; while the tail is everything else, which does not receive personalized attention, for example receivingform letters; and the line is simply set at a point that resources allow, or where it makes business sense to stop.
Purely quantitatively, a conventional way of splitting a distribution into head and tail is to consider the head to be the firstpportion of ranks, which account for1−p{\displaystyle 1-p}of the overall population, as in the 80:20Pareto principle, where the top 20% (head) comprises 80% of the overall population. The exact cutoff depends on the distribution – each distribution has a single such cutoff point—and for power, laws can be computed from thePareto index.
Segments may arise naturally due to actual changes in the behavior of the distribution as rank varies. Most common is theking effect, where the behavior of the top handful of items does not fit the pattern of the rest, as illustrated at the top for country populations, and above for most common words in English Wikipedia. For higher ranks, behavior may change at some point, and be well-modeled by different relations in different regions; on the whole by apiecewise function. For example, if two different power laws fit better in different regions, one can use abroken power lawfor the overall relation; the word frequency in English Wikipedia (above) also demonstrates this.
TheYule–Simon distributionthat results frompreferential attachment(intuitively, "the rich get richer" and "success breeds success") simulates a broken power law and has been shown to "very well capture" word frequency versus rank distributions.[7]It originated from trying to explain the population versus rank in different species. It has also been shown to fit city population versus rank better.[8]
Therank-size rule(orlaw) describes the remarkable regularity in many phenomena, including the distribution of city sizes, the sizes of businesses, the sizes of particles (such as sand), the lengths of rivers, the frequencies of word usage, and wealth among individuals.
All are real-world observations that followpower laws, such asZipf's law, theYule distribution, or thePareto distribution. If one ranks the population size of cities in a given country or in the entire world and calculates thenatural logarithmof the rank and of the city population, the resulting graph will show a linear pattern. This is the rank-size distribution.[9]
While Zipf's law works well in many cases, it tends to not fit the largest cities in many countries; one type of deviation is known as theKing effect. A 2002 study found that Zipf's law was rejected in 53 of 73 countries, far more than would be expected based on random chance.[10]The study also found that variations of the Pareto exponent are better explained by political variables than by economic geography variables like proxies foreconomies of scaleor transportation costs.[11]A 2004 study showed that Zipf's law did not work well for the five largest cities in six countries.[12]In the richer countries, the distribution was flatter than predicted. For instance, in theUnited States, although its largest city,New York City, has more than twice the population of second-placeLos Angeles, the two cities' metropolitan areas (also the two largest in the country) are much closer in population. In metropolitan-area population, New York City is only 1.3 times larger than Los Angeles. In other countries, the largest city would dominate much more than expected. For instance, in theDemocratic Republic of the Congo, the capital,Kinshasa, is more than eight times larger than the second-largest city,Lubumbashi. When considering the entire distribution of cities, including the smallest ones, the rank-size rule does not hold. Instead, the distribution islog-normal. This follows fromGibrat's lawof proportionate growth.
Because exceptions are so easy to find, the function of the rule for analyzing cities today is to compare the city systems in different countries. The rank-size rule is a common standard by which urban primacy is established. A distribution such as that in the United States or China does not exhibit a pattern of primacy, but countries with a dominant "primate city" clearly vary from the rank-size rule in the opposite manner. Therefore, the rule helps to classify national (or regional) city systems according to the degree of dominance exhibited by the largest city. Countries with a primate city, for example, have typically had a colonial history that accounts for that city pattern. If a normal city distribution pattern is expected to follow the rank-size rule (i.e. if the rank-size principle correlates with central place theory), then it suggests that those countries or regions with distributions that do not follow the rule have experienced some conditions that have altered the normal distribution pattern. For example, the presence of multiple regions within large nations such as China and the United States tends to favor a pattern in which more large cities appear than would be predicted by the rule. By contrast, small countries that had been connected (e.g. colonially/economically) to much larger areas will exhibit a distribution in which the largest city is much larger than would fit the rule, compared with the other cities—the excessive size of the city theoretically stems from its connection with a larger system rather than the natural hierarchy that central place theory would predict within that one country or region alone.
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Letter frequencyis the number of timeslettersof thealphabetappear on average inwritten language. Letterfrequency analysisdates back to theArabmathematicianAl-Kindi(c. AD 801–873), who formally developed the method to breakciphers. Letter frequency analysis gained importance inEuropewith the development ofmovable typein AD 1450, wherein one must estimate the amount of type required for eachletterform. Linguists use letter frequency analysis as a rudimentary technique forlanguage identification, where it is particularly effective as an indication of whether an unknownwriting systemis alphabetic,syllabic, orideographic.
The use of letter frequencies andfrequency analysisplays a fundamental role incryptogramsand several word puzzle games, includinghangman,Scrabble,Wordle[2]and the television game showWheel of Fortune. One of the earliest descriptions in classical literature of applying the knowledge of English letter frequency to solving a cryptogram is found inEdgar Allan Poe's famous story "The Gold-Bug", where the method is successfully applied to decipher a message giving the location of a treasure hidden byCaptain Kidd.[3][citation needed]
Herbert S. Zim, in his classic introductory cryptography textCodes and Secret Writing, gives the English letter frequency sequence as "ETAON RISHD LFCMU GYPWB VKJXZQ", the most common letter pairs as "TH HE AN RE ER IN ON AT ND ST ES EN OF TE ED OR TI HI AS TO", and the most commondoubled lettersas "LL EE SS OO TT FF RR NN PP CC".[4]Different ways of counting can produce somewhat different orders.
Letter frequencies also have a strong effect on the design of somekeyboard layouts. The most frequent letters are placed on thehome rowof theBlickensderfer typewriter, theDvorak keyboard layout,Colemakand other optimized layouts.
The frequency of letters in text has been studied for use incryptanalysis, andfrequency analysisin particular, dating back to the Arab mathematicianal-Kindi(c. AD 801–873 ), who formally developed the method (the ciphers breakable by this technique go back at least to theCaesar cipherused byJulius Caesar,[citation needed]so this method could have been explored in classical times). Letter frequency analysis gained additional importance in Europe with the development of movable type in AD 1450, wherein one must estimate the amount of type required for each letterform, as evidenced by the variations in letter compartment size in typographer's type cases.
No exact letter frequency distribution underlies a given language, since all writers write slightly differently. However, most languages have a characteristic distribution which is strongly apparent in longer texts. Even language changes as extreme as fromOld Englishto modern English (regarded as mutually unintelligible) show strong trends in related letter frequencies: over a small sample of Biblical passages, from most frequent to least frequent,enaid sorhm tgþlwu æcfy ðbpxzof Old English compares toeotha sinrd luymw fgcbp kvjqxzof modern English, with the most extreme differences concerning letterforms not shared.[5]
Linotype machinesfor the English language assumed the letter order, from most to least common, to beetaoin shrdlucmfwyp vbgkqj xzbased on the experience and custom of manual compositors. The equivalent for the French language waselaoin sdrétu cmfhyp vbgwqj xz.
Arranging the alphabet in Morse into groups of letters that require equal amounts of time to transmit, and then sorting these groups in increasing order, yieldse it san hurdm wgvlfbk opxcz jyq.[a]Letter frequency was used by other telegraph systems, such as theMurray Code.
Similar ideas are used in moderndata-compressiontechniques such asHuffman coding.
Letter frequencies, likeword frequencies, tend to vary, both by writer and by subject. For instance,⟨d⟩occurs with greater frequency in fiction, as most fiction is written in past tense and thus most verbs will end in the inflectional suffix-ed / -d. One cannot write an essay about x-rays without using⟨x⟩frequently. Different authors have habits which can be reflected in their use of letters.Hemingway's writing style, for example, is visibly different fromFaulkner's. Letter,bigram,trigram, word frequencies, word length, and sentence length can be calculated for specific authors and used to prove or disprove authorship of texts, even for authors whose styles are not so divergent.
Accurate average letter frequencies can only be gleaned by analyzing a large amount of representative text. With the availability of modern computing and collections of largetext corpora, such calculations are easily made. Examples can be drawn from a variety of sources (press reporting, religious texts, scientific texts and general fiction) and there are differences especially for general fiction with the position of⟨h⟩and⟨i⟩, with⟨h⟩becoming more common.
Different dialects of a language will also affect a letter's frequency. For example, an author in the United States would produce something in which⟨z⟩is more common than an author in the United Kingdom writing on the same topic: words like "analyze", "apologize", and "recognize" contain the letter in American English, whereas the same words are spelled "analyse", "apologise", and "recognise" in British English. This would highly affect the frequency of the letter⟨z⟩, as it is rarely used by British writers in the English language.[6]
The "top twelve" letters constitute about 80% of the total usage. The "top eight" letters constitute about 65% of the total usage. Letter frequency as a function of rank can be fitted well by several rank functions, with the two-parameterCocho/Beta rank functionbeing the best.[7]Another rank function with no adjustable free parameter also fits the letter frequency distribution reasonably well[8](the same function has been used to fit the amino acid frequency in protein sequences.[9]) A spy using theVIC cipheror some other cipher based on astraddling checkerboardtypically uses a mnemonic such as "a sin to err" (dropping the second "r")[10][11]or "at one sir"[12]to remember the top eight characters.
There are three ways to count letter frequency that result in very different charts for common letters. The first method, used in the chart below, is to count letter frequency inlemmasof a dictionary. The lemma is the word in its canonical form. The second method is to include all word variants when counting, such as "abstracts", "abstracted" and "abstracting" and not just the lemma of "abstract". This second method results in letters like⟨s⟩appearing much more frequently, such as when counting letters from lists of the most used English words on the Internet.⟨s⟩is especially common in inflected words (non-lemma forms) because it is added to form plurals and third person singular present tense verbs. A final method is to count letters based on their frequency of use in actual texts, resulting in certain letter combinations like⟨th⟩becoming more common due to the frequent use of common words like "the", "then", "both", "this", etc. Absolute usage frequency measures like this are used when creating keyboard layouts or letter frequencies in old fashioned printing presses.
An analysis of entries in the Concise Oxford dictionary, ignoring frequency of word use, gives an order of "EARIOTNSLCUDPMHGBFYWKVXZJQ".[13]
The letter-frequency table above is taken from Pavel Mička's website, which cites Robert Lewand'sCryptological Mathematics.[14]
According to Lewand, arranged from most to least common in appearance, the letters are:etaoinshrdlcumwfgypbvkjxqz. Lewand's ordering differs slightly from others, such as Cornell University Math Explorer's Project, which produced a table after measuring 40,000 words.[15]
In English, the space character occurs almost twice as frequently as the top letter (⟨e⟩)[16]and the non-alphabetic characters (digits, punctuation, etc.) collectively occupy the fourth position (having already included the space) between⟨t⟩and⟨a⟩.[17]
The frequency of the first letters of words or names is helpful in pre-assigning space in physical files and indexes.[18]Given 26filing cabinetdrawers, rather than a 1:1 assignment of one drawer to one letter of the alphabet, it is often useful to use a more equal-frequency-letter code by assigning several low-frequency letters to the same drawer (often one drawer is labeled VWXYZ), and to split up the most-frequent initial letters (⟨s, a, c⟩) into several drawers (often 6 drawers Aa-An, Ao-Az, Ca-Cj, Ck-Cz, Sa-Si, Sj-Sz). The same system is used in some multi-volume works such as someencyclopedias.Cutter numbers, another mapping of names to a more equal-frequency code, are used in some libraries.
Both the overall letter distribution and the word-initial letter distribution approximately match theZipf distributionand even more closely match theYule distribution.[19]
Often the frequency distribution of the first digit in each datum is significantly different from the overall frequency of all the digits in a set of numeric data, an observation known asBenford's law.
An analysis byPeter Norvigon words that appear 100,000 times or more inGoogle Books datatranscribed usingoptical character recognition(OCR) determined the frequency of first letters of English words, among other things.[20]
*Seeİanddotless I.
The figure below illustrates the frequency distributions of the 26 most common Latin letters across some languages. All of these languages use a similar 25+ character alphabet.
Based on these tables, the 'etaoin shrdlu' equivalent for each language is as follows:
Useful tables for single letter, digram, trigram, tetragram, and pentagram frequencies based on 20,000 words that take into account word-length and letter-position combinations for words 3 to 7 letters in length:
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Studies that estimate and rank themost common words in Englishexamine texts written in English. Perhaps the most comprehensive such analysis is one that was conducted against theOxford English Corpus(OEC), a massivetext corpusthat is written in the English language.
In total, the texts in the Oxford English Corpus contain more than 2 billion words.[1]The OEC includes a wide variety of writing samples, such as literary works, novels, academic journals, newspapers, magazines,Hansard's Parliamentary Debates,blogs,chat logs, and emails.[2]
Another English corpus that has been used to study word frequency is theBrown Corpus, which was compiled by researchers atBrown Universityin the 1960s. The researchers published their analysis of the Brown Corpus in 1967. Their findings were similar, but not identical, to the findings of the OEC analysis.
According toThe Reading Teacher's Book of Lists, the first 25 words in the OEC make up about one-third of all printed material in English, and the first 100 words make up about half of all written English.[3]According to a study cited byRobert McCruminThe Story of English,all of the first hundred of the most common words in English are of eitherOld EnglishorOld Norseorigin,[4]except for "just", ultimately from Latin "iustus", "people", ultimately from Latin "populus", "use", ultimately from Latin "usare", and "because", in part from Latin "causa".
Some lists of common words distinguish betweenword forms, while others rank all forms of a word as a singlelexeme(the form of the word as it would appear in a dictionary). For example, the lexemebe(as into be) comprises all its conjugations (am,are,is,was,were, etc.), andcontractionsof those conjugations.[5]These top 100lemmaslisted below account for 50% of all the words in the Oxford English Corpus.[1]
A list of 100 words that occur most frequently in written English is given below, based on an analysis of theOxford English Corpus(a collection of texts in the English language, comprising over 2 billion words).[1]Apart of speechis provided for most of the words, but part-of-speech categories vary between analyses, and not all possibilities are listed. For example, "I" may be a pronoun or a Roman numeral; "to" may be a preposition or an infinitive marker; "time" may be a noun or a verb. Also, a single spelling can represent more than oneroot word. For example, "singer" may be a form of either "sing" or "singe". Different corpora may treat such difference differently.
The number of distinct senses that are listed inWiktionaryis shown in thepolysemycolumn. For example, "out" can refer to an escape, a removal from play in baseball, or any of 36 other concepts. On average, each word in the list has 15.38 senses. The sense count does not include the use of terms inphrasal verbssuch as "put out" (as in "inconvenienced") and othermultiword expressionssuch as the interjection "get out!", where the word "out" does not have an individual meaning.[6]As an example, "out" occurs in at least 560 phrasal verbs[7]and appears in nearly 1700 multiword expressions.[8]
The table also includes frequencies from other corpora. As well as usage differences,lemmatisationmay differ from corpus to corpus – for example splitting the prepositional use of "to" from the use as a particle. Also, theCorpus of Contemporary American English(COCA) list includes dispersion as well as frequency to calculate rank.
The following is a very similar list, also from the OEC, subdivided bypart of speech.[1]The list labeled "Others" includespronouns,possessives,articles,modal verbs,adverbs, andconjunctions.
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"The quick brown fox jumps over the lazy dog" is an English-languagepangram– asentencethat contains all the letters of thealphabet. The phrase is commonly used fortouch-typingpractice, testingtypewritersandcomputer keyboards, displaying examples offonts, and other applications involving text where the use of all letters in the alphabet is desired.
The earliest known appearance of the phrase was inThe Boston Journal. In an article titled "Current Notes" in the February 9, 1885, edition, the phrase is mentioned as a good practicesentencefor writing students: "A favorite copy set by writing teachers for their pupils is the following, because it contains every letter of the alphabet: 'A quick brown fox jumps over the lazy dog.'"[1]Dozens of other newspapers published the phrase over the next few months, all using the version of the sentence starting with "A" rather than "The".[2]The earliest known use of the phrase starting with "The" is from the 1888 bookIllustrative Shorthandby Linda Bronson.[3]The modern form (starting with "The") became more common even though it is two letters longer than the original (starting with "A").
A 1908 edition of theLos Angeles Herald Sunday Magazinerecords that when theNew York Heraldwas equipping an office with typewriters "a few years ago", staff found that the common practice sentence of "now is the time for all good men to come to the aid of the party" did not familiarize typists with the entire alphabet, and ran onto two lines in a newspaper column. They write that a staff member named Arthur F. Curtis invented the "quick brown fox" pangram to address this.[4]
As the use of typewriters grew in the late 19th century, the phrase began appearing in typing lesson books as a practice sentence. Early examples includeHow to Become Expert in Typewriting: A Complete Instructor Designed Especially for the Remington Typewriter(1890),[6]andTypewriting Instructor and Stenographer's Hand-book(1892). By the turn of the 20th century, the phrase had become widely known. In the January 10, 1903, issue ofPitman's Phonetic Journal, it is referred to as "the well known memorized typing line embracing all the letters of the alphabet".[7]Robert Baden-Powell's bookScouting for Boys(1908) uses the phrase as a practice sentence for signaling.[5]
The first message sent on theMoscow–Washington hotlineon August 30, 1963, was the test phrase "THE QUICK BROWN FOX JUMPED OVER THE LAZY DOG'S BACK 1234567890".[8]Later, during testing, the Russian translators sent a message asking their American counterparts, "What does it mean when your people say 'The quick brown fox jumped over the lazy dog'?"[9]
During the 20th century, technicians tested typewriters andteleprintersby typing the sentence.[10]
It is the sentence used in the annual Zaner-Bloser National Handwriting Competition, acursive writingcompetition which has been held in the U.S. since 1991.[11][12]
In the age of computers, this pangram is commonly used to display font samples and for testingcomputer keyboards. Incryptography, it is commonly used as a test vector for hash and encryption algorithms to verify their implementation, as well as to ensure alphabetic character set compatibility.[citation needed]
Microsoft Wordhas a command to auto-type the sentence, in versions up to Word 2003, using the command=rand(), and in Microsoft Office Word 2007 and later using the command=rand.old().[13]
Numerous references to the phrase have occurred in movies, television, books, video games, advertising, websites, and graphic arts.
ThelipogrammaticnovelElla Minnow PeabyMark Dunnis built entirely around the "quick brown fox" pangram and its inventor. It depicts a fictional island off theSouth Carolinacoast that idealizes the pangram, chronicling the effects on literature and social structure as various letters are banned from daily use by government dictum.[14]
With 35 letters, this is not the shortest pangram. Shorter examples include:
If abbreviations and non-dictionary words are allowed, it is possible to create a perfect pangram that uses each letter only once, such as "Mr. Jock, TV quiz PhD, bags few lynx".
TheNASASpace Shuttleflew ateleprinterthat used the phrase "THE LAZY YELLOW DOG WAS CAUGHT BY THE SLOW RED FOX AS HE LAY SLEEPING IN THE SUN", a reference to the eponymous phrase, as part of its self-test program. While the phrase is not apangram, as it lacks J, K, M, Q, and V, it was selected to be exactly 80 characters wide to match the length of the teleprinter's drum.[15]
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Apalindrome(/ˈpæl.ɪn.droʊm/) is a word,number, phrase, or other sequence of symbols that reads the same backwards as forwards, such asmadamorracecar, the date "22/02/2022" and the sentence: "A man, a plan, a canal –Panama". The 19-letterFinnishwordsaippuakivikauppias(asoapstonevendor) is the longest single-word palindrome in everyday use, while the 12-letter termtattarrattat(fromJames JoyceinUlysses) is the longest in English.
The wordpalindromewas introduced by English poet and writerHenry Peachamin 1638.[1]The concept of a palindrome can be dated to the 3rd-century BCE, although no examples survive. The earliest known examples are the 1st-century CE Latinacrosticword square, theSator Square(which contains both word and sentence palindromes), and the 4th-century Greek Byzantine sentence palindromenipson anomemata me monan opsin.[2][3]
Palindromes are also found in music (thetable canonandcrab canon) and biological structures (mostgenomesincludepalindromic gene sequences). Inautomata theory, the set of all palindromes over analphabetis acontext-freelanguage, but it is notregular.
The wordpalindromewas introduced by English poet and writerHenry Peachamin 1638.[1]It is derived from the Greek rootsπάλιν'again' andδρóμος'way, direction'; a different word is used in Greek, καρκινικός 'carcinic' (lit.'crab-like') to refer to letter-by-letter reversible writing.[2][3]
The ancient Greek poetSotades(3rd-century BC) invented a form ofIonic metercalled Sotadic orSotadeanverse, which is sometimes said to have been palindromic,[4]since it is sometimes possible to make a sotadean line by reversing a dactylic hexameter.[5][6][7]
A 1st-century Latin palindrome was found as a graffito atPompeii. This palindrome, known as theSator Square, consists of a sentence written in Latin:sator arepo tenet opera rotas'The sower Arepo holds with effort the wheels'. It is also anacrosticwhere the first letters of each word form the first word, the second letters form the second word, and so forth. Hence, it can be arranged into aword squarethat reads in four different ways: horizontally or vertically from either top left to bottom right or bottom right to top left. Other palindromes found at Pompeii include "Roma-Olim-Milo-Amor", which is also written as an acrostic square.[8][9]Indeed, composing palindromes was "a pastime of Roman landed gentry".[10]
Byzantinebaptismal fontswere often inscribed with the 4th-century Greek palindrome,ΝΙΨΟΝ ΑΝΟΜΗΜΑΤΑ(orΑΝΟΜΗΜΑ)ΜΗ ΜΟΝΑΝ ΟΨΙΝ("Nipson anomēmata mē monan opsin") 'Wash [your] sin(s), not only [your] face', attributed toGregory of Nazianzus;[11]most notably in the basilica ofHagia SophiainConstantinople. The inscription is found on fonts in many churches in Western Europe:Orléans(St. Menin's Abbey);Dulwich College;Nottingham(St. Mary's);Worlingworth;Harlow;Knapton;London(St Martin, Ludgate); andHadleigh (Suffolk).[12]
A 12th-century palindrome with the same square property is theHebrewpalindrome,פרשנו רעבתן שבדבש נתבער ונשרףperashnu: ra`avtan shebad'vash nitba`er venisraf'We explained the glutton who is in the honey was burned and incinerated', credited in 1924 to the medieval Jewish philosopherAbraham ibn Ezra,[13][unreliable fringe source?]and referring to thehalachicquestion as to whether a fly landing in honey makes the honeytreif(non-kosher).
The palindromic Latin riddle "In girum imus nocte et consumimur igni" 'we go in a circle at night and are consumed by fire' describes the behavior of moths. It is likely that this palindrome is from medieval rather than ancient times. The second word, borrowed from Greek, should properly be spelledgyrum.
In English, there are many palindromewordssuch aseye,madam, anddeified, but English writers generally cited Latin and Greek palindromic sentences in the early 19th century;[14]thoughJohn Taylorhad coined one in 1614: "Lewd did I live, & evil I did dwel" (with theampersandbeing something of a "fudge"[15]). This is generally considered the first English-language palindrome sentence and was long reputed, notably by the grammarianJames "Hermes" Harris, to be theonlyone, despite many efforts to find others.[16][17](Taylor had also composed two other, "rather indifferent", palindromic lines of poetry: "Deer Madam, Reed", "Deem if I meed".[4]) Then in 1848, a certain "J.T.R." coined "Able was I ere I saw Elba", which became famous after it was (implausibly) attributed toNapoleon(alluding to his exile on Elba).[18][17][19]Otherwell-known English palindromesare: "A man, a plan, a canal – Panama" (1948),[20]"Madam, I'm Adam" (1861),[21]and "Never odd or even" (1930).[22]
The most familiar palindromes in English are character-unit palindromes, where the characters read the same backward as forward. Examples arecivic,radar,level,rotor,kayak,madam, andrefer. The longest common ones arerotator, deified, racecar, andreviver; longer examples such asredivider,kinnikinnik, andtattarrattatare orders of magnitude rarer.[23]
There are also word-unit palindromes in which the unit of reversal is the word ("Is it crazy how saying sentences backwards creates backwards sentences saying how crazy it is?"). Word-unit palindromes were made popular in therecreational linguisticscommunity byJ. A. Lindonin the 1960s. Occasional examples in English were created in the 19th century. Several in French and Latin date to theMiddle Ages.[24]
There are also line-unit palindromes, most oftenpoems. These possess an initial set of lines which, precisely halfway through, is repeated in reverse order, without alteration to word order within each line, and in a way that the second half continues the "story" related in the first half in a way that makes sense, this last being key.[25]
Palindromes often consist of a sentence or phrase, e.g., "A man, a plan, a canal, Panama", "Mr. Owl ate my metal worm",
"Do geese see God?", or "Was it a car or a cat I saw?". Punctuation, capitalization, and spaces are usually ignored. Some, such as "Rats live on no evil star", "Live on time, emit no evil", and "Step on no pets", include the spaces.
Some names are palindromes, such as thegiven namesHannah, Ava, Aviva, Anna, Eve, Bob, and Otto, or thesurnamesHarrah, Renner, Salas, and Nenonen.Lon Nol(1913–1985) was Prime Minister of Cambodia.Nisio Isinis a Japanese novelist andmangawriter, whose pseudonym (西尾 維新,Nishio Ishin) is a palindrome when romanized using theKunrei-shikior theNihon-shikisystems, and is often written as NisiOisiN to emphasize this. Some people have changed their name in order to make it palindromic (including as the actorRobert Treborand rock-vocalistOla Salo), while others were given a palindromic name at birth (such as the philologistRevilo P. Oliver, the flamenco dancerSara Baras, the runnerAnuța Cătună, the creator of theEden ProjectTim Smit, and the Mexican racing driverNoel León).
Savas (Emil M. Savas) Painter fromDenmarkchristened with the palindrome Savas. The family name is ofKurdishorigin and derived from Savaş written with Ş /ʃ/ (Hawar alphabet,Bedirxan 1932). The spelling was changed in accordance with theDano-Norwegian alphabetwhen Savas’ father was granted danish citizenship.
There are also palindromic names in fictional media. "Stanley Yelnats" is the name of the main character inHoles, a 1998 novel and2003 film. Five of the fictionalPokémonspecieshave palindromic names in English (Eevee, Girafarig, Farigiraf, Ho-Oh, and Alomomola), as does the region Alola.
The 1970s pop bandABBAis a palindrome using the starting letter of the first name of each of the four band members.
The digits of a palindromic number are the same read backwards as forwards, for example, 91019;decimalrepresentation is usually assumed. Inrecreational mathematics, palindromic numbers with special properties are sought. For example, 191 and 313 arepalindromic primes.
WhetherLychrel numbersexist is an unsolved problem in mathematics about whether all numbers become palindromes when they are continuously reversed and added. For example, 56 is not a Lychrel number as 56 + 65 = 121, and 121 is a palindrome. The number 59 becomes a palindrome after three iterations: 59 + 95 = 154; 154 + 451 = 605; 605 + 506 = 1111, so 59 is not a Lychrel number either. Numbers such as 196 are thought to never become palindromes when this reversal process is carried out and are therefore suspected of being Lychrel numbers. If a number is not a Lychrel number, it is called a "delayed palindrome" (56 has a delay of 1 and 59 has a delay of 3). In January 2017 the number 1,999,291,987,030,606,810 was published in OEIS asA281509, and described as "The Largest Known Most Delayed Palindrome", with a delay of 261. Several smaller 261-delay palindromes were published separately asA281508.
Every positive integer can be written as the sum of three palindromic numbers in every number system with base 5 or greater.[26]
A day or timestamp is a palindrome when its digits are the same when reversed. Only the digits are considered in this determination and the component separators (hyphens, slashes, and dots) are ignored. Short digits may be used as in11/11/1111:11or long digits as in2 February 2020.
A notable palindrome day is this century's 2 February 2020 because this date is a palindrome regardless of thedate format by country(yyyy-mm-dd, dd-mm-yyyy, or mm-dd-yyyy) used in various countries. For this reason, this date has also been termed as a "Universal Palindrome Day".[27][28]Other universal palindrome days include, almost a millennium previously,11/11/1111, the future12/12/2121, and in a millennium03/03/3030.[29]
A phonetic palindrome is a portion ofspeechthat is identical or roughly identical when reversed. It can arise in context where language is played with, for example in slang dialects likeverlan.[30]In theFrench language, there is the phraseune Slave valse nue("a Slavic woman waltzes naked"), phonemically/ynslavvalsny/.[31]John Oswalddiscussed his experience of phonetic palindromes while working on audio tape versions of thecut-up techniqueusing recorded readings byWilliam S. Burroughs.[32][33]A list of phonetic palindromes discussed byword puzzlecolumnist O.V. Michaelsen (Ove Ofteness) include "crew work"/"work crew", "dry yard", "easy", "Funny enough", "Let Bob tell", "new moon", "selfless", "Sorry, Ross", "Talk, Scott", "to boot", "top spot" (also an orthographic palindrome), "Y'all lie", "You're caught. Talk, Roy", and "You're damn mad, Roy".[34]
The longest single-word palindrome in theOxford English Dictionaryis the 12-letteronomatopoeicwordtattarrattat, coined byJames JoyceinUlysses(1922) for a knock on the door.[35][36][37]TheGuinness Book of Recordsgives the title to the 11-letterdetartrated, thepreteriteand past participle ofdetartrate, a chemical term meaning to removetartrates. The 9-letter wordRotavator, a trademarked name for an agricultural machine, is listed in dictionaries as being the longest single-word palindrome. The 9-letter termredivideris used by some writers, but appears to be an invented or derived term; onlyredivideandredivisionappear in the Shorter Oxford English Dictionary; the 9-letter wordMalayalam, a language of southern India, is also of equal length.
According toGuinness World Records, theFinnish19-letter wordsaippuakivikauppias(asoapstonevendor), is the world's longest palindromic word in everyday use.[12]
English palindrome sentences of notable length include mathematicianPeter Hilton's "Doc, note: I dissent. A fast never prevents a fatness. I diet on cod",[38]and Scottish poetAlastair Reid's "T. Eliot, top bard, notes putrid tang emanating, is sad; I'd assign it a name: gnat dirt upset on drab pot toilet."[39]
In English, two palindromic novels have been published:Satire: Veritasby David Stephens (1980, 58,795 letters), andDr Awkward & Olson in Osloby Lawrence Levine (1986, 31,954 words).[40]Another palindromic English work is a 224-word long poem, "Dammit I'm Mad", written byDemetri Martin.[41]"Weird Al" Yankovic's song "Bob" is composed entirely of palindromes.[42]
Joseph Haydn'sSymphony No. 47in G is nicknamed "the Palindrome". In the third movement, aminuetandtrio, the second half of the minuet is the same as the first but backwards, the second half of the ensuing trio similarly reflects the first half, and then the minuet is repeated.
The interlude fromAlban Berg's operaLuluis a palindrome,[43]as are sections and pieces, inarch form, by many other composers, includingJames Tenney, and most famouslyBéla Bartók.George Crumbalso used musical palindrome to text paint theFederico García Lorcapoem "¿Por qué nací?", the first movement of three in his fourth book ofMadrigals.Igor Stravinsky's final composition,The Owl and the Pussy Cat, is a palindrome.[44][unreliable source?]
The first movement fromConstant Lambert'sballetHoroscope(1938) is entitled "Palindromic Prelude". Lambert claimed that the theme was dictated to him by the ghost ofBernard van Dieren, who had died in 1936.[45]
British composerRobert Simpsonalso composed music in the palindrome or based on palindromic themes; the slow movement of hisSymphony No. 2is a palindrome, as is the slow movement of hisString Quartet No. 1. His hour-longString Quartet No. 9consists of thirty-two variations and a fugue on a palindromic theme of Haydn (from the minuet of his Symphony No. 47). All of Simpson's thirty-two variations are themselves palindromic.
Hin und Zurück("There and Back": 1927) is an operatic 'sketch' (Op. 45a) in one scene by Paul Hindemith, with a German libretto by Marcellus Schiffer. It is essentially a dramatic palindrome. Through the first half, a tragedy unfolds between two lovers, involving jealousy, murder and suicide. Then, in the reversing second half, this is replayed with the lines sung in reverse order to produce a happy ending.
The music ofAnton Webernis often palindromic. Webern, who had studied the music of the Renaissance composerHeinrich Isaac, was extremely interested in symmetries in music, be they horizontal or vertical. An example of horizontal or linear symmetry in Webern's music is the first phrase in the second movement of thesymphony, Op. 21. A striking example of vertical symmetry is the second movement of thePiano Variations, Op. 27, in which Webern arranges every pitch of thisdodecaphonicwork around the central pitch axis of A4. From this, each downward reaching interval is replicated exactly in the opposite direction. For example, a G♯3—13 half-steps down from A4 is replicated as a B♭5—13 half-steps above.
Just as the letters of a verbal palindrome are not reversed, so are the elements of a musical palindrome usually presented in the same form in both halves. Although these elements are usually single notes, palindromes may be made using more complex elements. For example,Karlheinz Stockhausen's compositionMixtur, originally written in 1964, consists of twenty sections, called "moments", which may bepermutedin several different ways, including retrograde presentation, and two versions may be made in a single program. When the composer revised the work in 2003, he prescribed such a palindromic performance, with the twenty moments first played in a "forwards" version, and then "backwards". Each moment is a complex musical unit and is played in the same direction in each half of the program.[46]By contrast,Karel Goeyvaerts's 1953 electronic composition,Nummer 5(met zuivere tonen)is anexactpalindrome: not only does each event in the second half of the piece occur according to an axis of symmetry at the centre of the work, but each event itself is reversed, so that the note attacks in the first half become note decays in the second, and vice versa. It is a perfect example of Goeyvaerts's aesthetics, the perfect example of the imperfection of perfection.[47]
Inclassical music, acrab canonis acanonin which one line of the melody is reversed in time and pitch from the other.
A large-scale musical palindrome covering more than one movement is called "chiastic", referring to the cross-shaped Greek letter "χ" (pronounced /ˈkaɪ/.) This is usually a form of reference to the crucifixion; for example, theCrucifixusmovement of Bach'sMass in B minor. The purpose of such palindromic balancing is to focus the listener on the central movement, much as one would focus on the centre of the cross in the crucifixion. Other examples are found in Bach's cantata BWV 4,Christ lag in Todes Banden, Handel'sMessiahand Fauré'sRequiem.[48]
Atable canonis a rectangular piece of sheet music intended to be played by two musicians facing each other across a table with the music between them, with one musician viewing the music upside down compared to the other. The result is somewhat like two speakers simultaneously reading theSator Squarefrom opposite sides, except that it is typically in two-part polyphony rather than in unison.[49]
Palindromic motifs are found in mostgenomesor sets ofgeneticinstructions. The meaning of palindrome in the context of genetics is slightly different, from the definition used for words and sentences. Since theDNAis formed by two paired strands ofnucleotides, and the nucleotides always pair in the same way (Adenine(A) withThymine(T),Cytosine(C) withGuanine(G)), a (single-stranded) sequence of DNA is said to be a palindrome if it is equal to its complementary sequence read backward. For example, the sequenceACCTAGGTis palindromic because its complement isTGGATCCA, which is equal to the original sequence in reverse complement.
A palindromicDNAsequence may form ahairpin. Palindromic motifs are made by the order of thenucleotidesthat specify the complex chemicals (proteins) that, as a result of thosegeneticinstructions, thecellis to produce. They have been specially researched inbacterialchromosomes and in the so-called Bacterial Interspersed Mosaic Elements (BIMEs) scattered over them. In 2003, a research genome sequencing project discovered that many of the bases on theY-chromosomeare arranged as palindromes.[50]A palindrome structure allows the Y-chromosome to repair itself by bending over at the middle if one side is damaged.
It is believed that palindromes are also found in proteins,[51][52]but their role in the protein function is not clearly known. It has been suggested in 2008[53]that the prevalent existence of palindromes in peptides might be related to the prevalence of low-complexity regions in proteins, as palindromes frequently are associated with low-complexity sequences. Their prevalence might also be related to analpha helicalformation propensity of these sequences,[53]or in formation of protein/protein complexes.[54]
Inautomata theory, asetof all palindromes in a givenalphabetis a typical example of alanguagethat iscontext-free, but notregular. This means that it is impossible for afinite automatonto reliably test for palindromes.
In addition, the set of palindromes may not be reliably tested by adeterministic pushdown automatonwhich also means that they are notLR(k)-parsableorLL(k)-parsable. When reading a palindrome from left to right, it is, in essence, impossible to locate the "middle" until the entire word has been read completely.
It is possible to find thelongest palindromic substringof a given input string inlinear time.[55][56]
Thepalindromic densityof an infinite wordwover an alphabetAis defined to be zero if only finitely many prefixes are palindromes; otherwise, letting the palindromic prefixes be of lengthsnkfork=1,2,... we define the density to be
Among aperiodic words, the largest possible palindromic density is achieved by theFibonacci word, which has density 1/φ, where φ is theGolden ratio.[57]
Apalstaris aconcatenationof palindromic strings, excluding the trivial one-letter palindromes – otherwise all strings would be palstars.[55]
February 2, 2020, was the most recent palindromic date which was can perfectly fit to any date formats in 8-digit FIGURES. And it happens very rare in any Millennium. The next of it will occur on December 12, 2121, which will be the last in this 3rd millennium.
3rd Millennium: February 2, 2020, and December 12, 2121.
4th Millennium: March 3, 3030
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Anapostolic nunciatureis a top-leveldiplomatic missionof theHoly Seethat is equivalent to anembassy. However, it neither issues visas nor hasconsulates.
The head of the apostolic nunciature is called anuncio, an ecclesiastical diplomatic title. A papal nuncio (officially known as an apostolic nuncio) is a permanent diplomatic representative (head of diplomatic mission) of the Holy See to a state or to one of two international intergovernmental organizations, theEuropean UnionorASEAN, having the rank of anambassadorextraordinary and plenipotentiary, and the ecclesiastical rank oftitulararchbishop. Papal representatives to other intergovernmental organizations are known as "permanent observers" or "delegates".
In several countries that have diplomatic relations with the Holy See, the apostolic nuncio isipso factothedean of the diplomatic corps. The nuncio is, in such a country, first in theorder of precedenceamong all the diplomats accredited to the country, and he speaks for the diplomatic corps in matters of diplomatic privilege and protocol. Most countries that concede priority to the nuncio are officially Catholic, but some are not.
In addition, the nuncio serves as the liaison between the Holy See and the Church in that particular nation. The nuncio has an important role in the selection of bishops.
The pope accredits diplomats with the following states and other subjects of international law (list as per January 2010):[2]
Algeria,Angola,Benin,Burkina Faso,Burundi,Botswana,Cameroon,Cape Verde,Central African Republic,Chad,Congo (Republic of),Congo (Democratic Republic of),Côte d'Ivoire,Djibouti,Egypt,Equatorial Guinea,Eritrea,Ethiopia,Gabon,Gambia,Ghana,Guinea,Guinea-Bissau,Kenya,Lesotho,Liberia,Libya,Madagascar,Malawi,Mali,Mauritius,Morocco,Mozambique,Namibia,Niger,Nigeria,Rwanda,São Tomé and Príncipe,Sénégal,Seychelles,Sierra Leone,South Africa,Sudan,Swaziland,Tanzania,Togo,Tunisia,Uganda,Zambia,Zimbabwe
Antigua and Barbuda,Argentina, Bahamas, Barbados, Belize,Bolivia,Brazil,Canada,Chile,Colombia,Costa Rica, Cuba, Dominica,Dominican Republic, Ecuador, El Salvador, Grenada, Guatemala, Guyana, Haiti, Honduras, Jamaica,México, Nicaragua, Panama, Paraguay,Peru, Saint Kitts and Nevis, Saint Lucia, Saint Vincent and Grenadines, Suriname,Trinidad and Tobago,United States of America, Uruguay,Venezuela
Bahrain,Bangladesh, Cambodia,Republic of China (Taiwan), East Timor,India,Indonesia,Iran,Iraq,Israel,Japan, Jordan, Kazakhstan, Korea[which?], Kuwait,Kyrgyzstan,Lebanon, Malaysia, Mongolia,Nepal,Pakistan,Philippines, Qatar, Singapore, Sri Lanka, Syria, Tajikistan,Thailand, Turkmenistan, United Arab Emirates, Uzbekistan, Vietnam (Resident), Yemen.
Albania, Andorra, Armenia,Austria, Azerbaijan, Belarus,Belgium, Bosnia-Herzegovina, Bulgaria, Croatia, Cyprus, Czech Republic, Estonia, European Union,France, Georgia,Germany,Great Britain, Greece, Hungary,Ireland,Italy, Latvia, Liechtenstein, Lithuania, Luxembourg, Macedonia, Malta, Moldova, Monaco, Montenegro,The Netherlands,Nordic Countries,Poland,Portugal, Romania,Russia, San Marino, Serbia, Slovakia, Slovenia,Spain, Switzerland, Turkey,Ukraine
Australia, the Cook Islands, Fiji, Guam, Kiribati, Marshall Islands, Micronesia (Federated States of), Nauru, New Zealand, Palau, Papua New Guinea, Samoa, Solomon Islands, Tonga, Vanuatu.
An apostolic delegate may be sent to liaison between the Catholic Church and a country with which the Holy See has no diplomatic ties, though not accredited to the government of the country. Apostolic delegates have no formal diplomatic status, though in some countries they have some diplomatic privileges.
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Apapal legateorapostolic legate(from theancient Romantitlelegatus) is a personal representative of thePopeto foreign nations, to some other part of theCatholic Church, or to representatives of a state or monarchy. A legate is empowered in matters ofCatholic faithand for the settlement ofecclesiasticalmatters.
The legate is appointed directly by the Pope—the Bishop of Rome and head of the Catholic Church. Hence a legate is usually sent to a government, to a sovereign, to a large body of believers (such as a national church), or to take charge of a major religious effort, such as anecumenical council, acrusadeto the Holy Land, or even against aheresysuch as theCathars.
The termlegationis applied both to a legate's mandate and to the territory concerned (such as a state, or an ecclesiastical province). The relevant adjective islegatine.
In theHigh Middle Ages, papal legates were often used to strengthen the links between Rome and the many parts ofChristendom. More often than not, legates were learned men and skilled diplomats who were not from the country they were accredited to. For example, the Italian-bornGuala Bicchieriserved as papal legate to England in the early 13th century and played a major role in both the English government and church at the time. By theLate Middle Agesit had become more common to appoint native clerics to the position of legate within their own country, such asCardinal Wolseyacting as legate to the court ofHenry VIII of England. The reason for this switch in policy could be attributed to a change in attitude on the eve of theReformation; by this point, foreign men representing the papacy would be more likely to reinforce dissent than bring Christendom closer together.[1][non sequitur]
Papal legates often summonedlegatine councils, which dealt with church government and other ecclesiastical issues.[2]According to PopeGregory VII, writing in theDictatus papae, a papal legate "presides over all bishops in a council, even if he is inferior in rank, and he can pronounce sentence of deposition against them".[3]During theMiddle Ages, a legatine council was the usual means that a papal legate imposed his directives.[3]
There are several ranks of papal legates in diplomacy, some of which are no longer used.
The most common form of papal legate today is theapostolic nuncio, whose task it is to strengthen relations between theHoly Seeand the Catholic Church in a particular country and at the same time to act as the diplomatic representative of the Holy See to the government of that country.[4]An apostolic nuncio is generally equivalent in rank to that ofambassadorextraordinary andplenipotentiary, although inCatholic countriesthe nuncio often ranks above ambassadors in diplomatic protocol. A nuncio performs the same functions as an ambassador and has the same diplomatic privileges. Under the 1961Vienna Convention on Diplomatic Relations, to which the Holy See is a party, a nuncio is an ambassador like those from any other country. The Vienna Convention allows the host state to grant seniority of precedence to the nuncio over others of ambassadorial rank accredited to the same country, and may grant thedeanship of that country's diplomatic corpsto the nuncio regardless of seniority.[5]
Pro-nuncio was a term used from 1965 to 1991 for a papal diplomatic representative of full ambassadorial rank accredited to a country that did not accord him precedence over other ambassadors andex officiodeanship of the diplomatic corps. In those countries, the papal representative's precedence within the corps is exactly on a par with that of the other members of ambassadorial rank, so that he becomes dean only on becoming the senior member of the corps.[6]
For countries with which the Holy See has no diplomatic relations, an apostolic delegate is sent to serve as a liaison with the Catholic Church in that country, though not accredited to its government.[4]
This highest rank (literally "from the (pope's) side", i.e. "intimately" trusted) is normally awarded to a priest ofcardinal rank. It is an exceptional investiture and can either be focused or broad in scope. The legatea latereis the alter ego of the Pope, and as such, possesses full plenipotentiary powers.[7][8]
Literally "born legate", i.e. not nominated individually butex officio, namely a bishop holding this rank as a privilege of his see, e.g.archbishopsofCanterbury(pre-Reformation),Prague,Esztergom,Udine,Salzburg,GnieznoandCologne.[7][8]Thelegatus natuswould act as the Pope's representative in his province, with alegatus a latereonly being sent in extraordinary circumstances. Although limited in their jurisdiction compared tolegati a latere, alegatus natuswas not subordinate to them.[9]
Literally "sent legate", possessing limited powers for the purpose of completing a specific mission. This commission is normally focused in scope and of short duration.[7][8]
Some administrative (temporal) provinces of thePapal Statesin (mostly central) Italy were governed by a papal legate. This has been the case inBenevento, inPontecorvo(of Campagna e Marittima/of Frosinone) and inViterbo. In four cases, includingBologna, this post was awarded exclusively tocardinals; theVelletripost was created forBartolomeo Pacca.
The title could be changed toApostolic Delegate, as happened in Frosinone (for Pontecorvo) in 1827.
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The following is a sortablelist of the heads of the diplomatic mission of the Holy See. An apostolic nuncio (also known as a papal nuncio or simply as a nuncio) is anecclesiasticaldiplomat, serving as an envoy or a permanent diplomatic representative of theHoly Seeto astateor to an international organization. A nuncio is appointed by and represents the Holy See, and is the head of thediplomatic mission, called anApostolic Nunciature, which is the equivalent of anembassy. The Holy See is legally distinct from theVatican Cityor theCatholic Church. A nuncio is usually anarchbishop.
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https://en.wikipedia.org/wiki/List_of_heads_of_the_diplomatic_missions_of_the_Holy_See
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AGooglewhackwas a contest to find aGoogle Searchquery that returns a single result. A Googlewhack must consist of two words found in a dictionary and was only considered legitimate if both of the search terms appear in the result. Published googlewhacks were short-lived since when published to a website, the new number of hits would become at least two: one to the original hit found, and one to the publishing site, unless a screenshot was provided.[1]Googlewhacks generally no longer exist due to changes in Google search indexing.
The termgooglewhack, coined by Gary Stock, first appeared on the web at UnBlinking on 8 January 2002.[2]Subsequently, Stock created The Whack Stack, at googlewhack.com, to allow the verification and collection of user-submitted Googlewhacks.
Googlewhacks were the basis of British comedianDave Gorman's comedy tourDave Gorman's Googlewhack Adventureand book of the same name.[3]In these Gorman tells the true story of how, while attempting to write a novel for his publisher, he became obsessed with Googlewhacks and travelled across the world finding people who had authored them. Although he never completed his original novel,Dave Gorman's Googlewhack Adventurewent on to be aSunday TimesNo. 1 best seller in the UK.
Participants at Googlewhack.com discovered the sporadic "cleaner girl" bug in Google's search algorithm where "results 1–1 of thousands" were returned for two relatively common words[4]such as Anxiousness Scheduler[5]or Italianate Tablesides.[6]
Googlewhack went offline in November 2009 after Google stopped providing definition links.[definition needed]Gary Stock stated on the game's web page soon afterward that he was pursuing solutions for Googlewhack to remain viable.[citation needed]
Some people propose the googlewhack "score", which is the product of the hits of the individual words.[7]Thus a googlewhack score is highest when the individual words produce a large number of hits.
New Scientisthas discussed the idea of aGooglewhackblatt, which is similar to a Googlewhack except that it involves finding asingle wordthat produces only one Google result. Lists of these have become available, but as with Googlewhacks, they result in the Googlewhackblatt status of the word being destroyed—unless it is blocked byrobots.txtor the word does not produce any Google results before it is added to the list, thus forming the Googlewhackblatt Paradox. Those words that do not produce any Google search results at all are known asAntegooglewhackblattsbefore they are listed—and subsequently elevated to Googlewhackblatt status if it is not blocked by robots.txt.
Feedback stories are also available on theNew Scientistwebsite, thus resulting in the destruction of any existing Googlewhackblatts that are ever printed in the magazine. Antegooglewhackblatts that are posted on the Feedback website become known asFeedbackgooglewhackblattsas their Googlewhackblatt status is created. In addition,New Scientisthas more recently discovered another way to obtain a Googlewhackblatt without falling into the Googlewhackblatt Paradox. One can write the Googlewhackblatt on a website, but backward, and then search onelgooGto view the list properly while still keeping the Googlewhackblatt's status as a Googlewhackblatt.
In contrast to Googlewhacks, many Googlewhackblatts and Antegooglewhackblatts are nonsense words or uncommon misspellings that are not in dictionaries and probably never will be.
Practical use of specially constructed Googlewhackblatts was proposed byLeslie Lamport(although he did not use the term).[9]
The probabilities of internet search result values for multi-word queries was studied in 2008 with the help of Googlewhacks.[10][11][12]Based on data from 351 Googlewhacks from the "WhackStack" a list of previously documented Googlewhacks,[13]theHeaps' lawβ{\displaystyle \beta }coefficient for the indexedWorld Wide Web(about 8 billion pages in 2008) was measured to beβ=0.52{\displaystyle \beta =0.52}. This result is in line with previous studies which used under 20,000 pages.[14]The googlewhacks were a key in calibrating the model so that it could be extended automatically to analyse the relatedness of word pairs.
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Inlinguistics, aprotologismis a newly used or coinedword, anonce word, that has been repeated but has not gained acceptance beyond its original users or been published independently of the coiners.[1][2]The word may be proposed, may be extremely new, or may be established only within a very limited group of people.[3][4]
A protologism becomes aneologismas soon as it appears in published press, on a website, or in a book, independently of the coiner[5]—though, most definitively, in a dictionary.[6]A word whose developmental stage is between that of a protologism (freshly coined) and a neologism (a new word) is aprelogism.[7]
Protologisms constitute one stage in the development of neologisms. A protologism is coined to fill a gap in the language, with the hope of its becoming an accepted word.[8][9]As an example, when the wordprotologismitself was coined—in 2003[10]by the Americanliterary theoristMikhail Epstein—it wasautological: an example of the thing it describes.[11]
About the concept and his name for it, Epstein wrote:
I suggest calling such brand new words 'protologisms' (from Greekprotos, meaning 'first, original' and Greeklogos, meaning 'word'; cf.prototype,protoplasm). The protologism is a freshly minted word not yet widely accepted. It is a verbal prototype, which may eventually be adopted for public service or remain a whim of linguo-poetic imagination.[12]
According to Epstein, every word in use started out as a protologism, subsequently became a neologism, and then gradually grew to be part of the language.[12]
There is no fixed rule determining when a protologism becomes a stable neologism,[13]and according to Kerry Maxwell, author ofBrave New Words:
[A] protologism is unlikely to make the leap to neologism status unless society connects with the word or identifies a genuine need for it [...] there's no guarantee that simple exposure to these creations will be effective in getting them used, as discovered by British inventorSir James Dysonwhen he fruitlessly attempted to promote a verbdyson(by analogy withhoover) in the early 2000s.[14]
It has been suggestedprotologismsare needed in scientific fields, particularly in thelife sciences, where very complex interactions between partially understood components produce higher order phenomena.[15]Nevertheless, until the unappreciated concept in question has been thoroughly investigated and shown to be a real phenomenon, it is improbable that the term would be used by anyone other than its creator[15]and achieve the status ofneologism.
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Aword listis a list of words in alexicon, generally sorted by frequency of occurrence (either bygraded levels, or as a ranked list). A word list is compiled bylexical frequency analysiswithin a giventext corpus, and is used incorpus linguisticsto investigate genealogies and evolution of languages and texts. A word which appears only once in the corpus is called ahapax legomena. Inpedagogy, word lists are used incurriculum designforvocabulary acquisition. A lexicon sorted by frequency "provides a rational basis for making sure that learners get the best return for their vocabulary learning effort" (Nation 1997), but is mainly intended forcourse writers, not directly for learners. Frequency lists are also made for lexicographical purposes, serving as a sort ofchecklistto ensure that common words are not left out. Some major pitfalls are the corpus content, the corpusregister, and the definition of "word". While word counting is a thousand years old, with still gigantic analysis done by hand in the mid-20th century,natural language electronic processingof large corpora such as movie subtitles (SUBTLEX megastudy) has accelerated the research field.
Incomputational linguistics, afrequency listis a sorted list ofwords(word types) together with theirfrequency, where frequency here usually means the number of occurrences in a givencorpus, from which the rank can be derived as the position in the list.
Nation (Nation 1997) noted the incredible help provided by computing capabilities, making corpus analysis much easier. He cited several key issues which influence the construction of frequency lists:
Most of currently available studies are based on writtentext corpus, more easily available and easy to process.
However,New et al. 2007proposed to tap into the large number of subtitles available online to analyse large numbers of speeches.Brysbaert & New 2009made a long critical evaluation of the traditional textual analysis approach, and support a move toward speech analysis and analysis of film subtitles available online. The initial research saw a handful of follow-up studies,[1]providing valuable frequency count analysis for various languages. In depth SUBTLEX researches over cleaned up open subtitles were produce for French (New et al. 2007), American English (Brysbaert & New 2009;Brysbaert, New & Keuleers 2012), Dutch (Keuleers & New 2010), Chinese (Cai & Brysbaert 2010), Spanish (Cuetos et al. 2011), Greek (Dimitropoulou et al. 2010), Vietnamese (Pham, Bolger & Baayen 2011), Brazil Portuguese (Tang 2012) and Portugal Portuguese (Soares et al. 2015), Albanian (Avdyli & Cuetos 2013), Polish (Mandera et al. 2014) and Catalan (2019[2]), Welsh (Van Veuhen et al. 2024[3]). SUBTLEX-IT (2015) provides raw data only.[4]
In any case, the basic "word" unit should be defined. For Latin scripts, words are usually one or several characters separated either by spaces or punctuation. But exceptions can arise : English "can't" and French "aujourd'hui" include punctuations while French "chateau d'eau" designs a concept different from the simple addition of its components while including a space. It may also be preferable to group words of aword familyunder the representation of itsbase word. Thus,possible, impossible, possibilityare words of the same word family, represented by the base word*possib*. For statistical purpose, all these words are summed up under the base word form *possib*, allowing the ranking of a concept and form occurrence. Moreover, other languages may present specific difficulties. Such is the case of Chinese, which does not use spaces between words, and where a specified chain of several characters can be interpreted as either a phrase of unique-character words, or as a multi-character word.
It seems thatZipf's lawholds for frequency lists drawn from longer texts of any natural language. Frequency lists are a useful tool when building an electronic dictionary, which is a prerequisite for a wide range of applications incomputational linguistics.
German linguists define theHäufigkeitsklasse(frequency class)N{\displaystyle N}of an item in the list using thebase 2 logarithmof the ratio between its frequency and the frequency of the most frequent item. The most common item belongs to frequency class 0 (zero) and any item that is approximately half as frequent belongs in class 1. In the example list above, the misspelled wordoutragioushas a ratio of 76/3789654 and belongs in class 16.
where⌊…⌋{\displaystyle \lfloor \ldots \rfloor }is thefloor function.
Frequency lists, together withsemantic networks, are used to identify the least common, specialized terms to be replaced by theirhypernymsin a process ofsemantic compression.
Those lists are not intended to be given directly to students, but rather to serve as a guideline for teachers and textbook authors (Nation 1997).Paul Nation's modern language teaching summary encourages first to "move from high frequency vocabulary and special purposes [thematic] vocabulary to low frequency vocabulary, then to teach learners strategies to sustain autonomous vocabulary expansion" (Nation 2006).
Word frequency is known to have various effects (Brysbaert et al. 2011;Rudell 1993). Memorization is positively affected by higher word frequency, likely because the learner is subject to more exposures (Laufer 1997). Lexical access is positively influenced by high word frequency, a phenomenon calledword frequency effect(Segui et al.). The effect of word frequency is related to the effect ofage-of-acquisition, the age at which the word was learned.
Below is a review of available resources.
Word counting is an ancient field,[5]with known discussion back toHellenistictime. In 1944,Edward Thorndike,Irvin Lorgeand colleagues[6]hand-counted 18,000,000 running words to provide the first large-scale English language frequency list, before modern computers made such projects far easier (Nation 1997). 20th century's works all suffer from their age. In particular, words relating to technology, such as "blog," which, in 2014, was #7665 in frequency[7]in the Corpus of Contemporary American English,[8]was first attested to in 1999,[9][10][11]and does not appear in any of these three lists.
The Teacher Word Book contains 30,000 lemmas or ~13,000 word families (Goulden, Nation and Read, 1990). A corpus of 18 million written words was hand analysed. The size of its source corpus increased its usefulness, but its age, and language changes, have reduced its applicability (Nation 1997).
The General Service List contains 2,000 headwords divided into two sets of 1,000 words. A corpus of 5 million written words was analyzed in the 1940s. The rate of occurrence (%) for different meanings, and parts of speech, of the headword are provided. Various criteria, other than frequence and range, were carefully applied to the corpus. Thus, despite its age, some errors, and its corpus being entirely written text, it is still an excellent database of word frequency, frequency of meanings, and reduction of noise (Nation 1997). This list was updated in 2013 by Dr. Charles Browne, Dr. Brent Culligan and Joseph Phillips as theNew General Service List.
A corpus of 5 million running words, from written texts used in United States schools (various grades, various subject areas). Its value is in its focus on school teaching materials, and its tagging of words by the frequency of each word, in each of the school grade, and in each of the subject areas (Nation 1997).
These now contain 1 million words from a written corpus representing different dialects of English. These sources are used to produce frequency lists (Nation 1997).
A review has been made byNew & Pallier.
An attempt was made in the 1950s–60s with theFrançais fondamental. It includes the F.F.1 list with 1,500 high-frequency words, completed by a later F.F.2 list with 1,700 mid-frequency words, and the most used syntax rules.[12]It is claimed that 70 grammatical words constitute 50% of the communicatives sentence,[13][14]while 3,680 words make about 95~98% of coverage.[15]A list of 3,000 frequent words is available.[16]
The French Ministry of the Education also provide a ranked list of the 1,500 most frequentword families, provided by the lexicologueÉtienne Brunet.[17]Jean Baudot made a study on the model of the American Brown study, entitled "Fréquences d'utilisation des mots en français écrit contemporain".[18]
More recently, the projectLexique3provides 142,000 French words, withorthography,phonetic,syllabation,part of speech,gender, number of occurrence in the source corpus, frequency rank, associatedlexemes, etc., available under an open licenseCC-by-sa-4.0.[19]
This Lexique3 is a continuous study from which originate theSubtlex movementcited above.New et al. 2007made a completely new counting based on online film subtitles.
There have been several studies of Spanish word frequency (Cuetos et al. 2011).[20]
Chinese corpora have long been studied from the perspective of frequency lists. The historical way to learn Chinese vocabulary is based on characters frequency (Allanic 2003). American sinologistJohn DeFrancismentioned its importance for Chinese as a foreign language learning and teaching inWhy Johnny Can't Read Chinese(DeFrancis 1966). As a frequency toolkit, Da (Da 1998) and the Taiwanese Ministry of Education (TME 1997) provided large databases with frequency ranks for characters and words. TheHSKlist of 8,848 high and medium frequency words in thePeople's Republic of China, and theRepublic of China (Taiwan)'sTOPlist of about 8,600 common traditional Chinese words are two other lists displaying common Chinese words and characters. Following the SUBTLEX movement,Cai & Brysbaert 2010recently made a rich study of Chinese word and character frequencies.
Wiktionarycontains frequency lists in more languages.[21]
Most frequently used words in different languages based on Wikipedia or combined corpora.[22]
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Inabstract algebra, analternative algebrais analgebrain which multiplication need not beassociative, onlyalternative. That is, one must have
for allxandyin the algebra.
Everyassociative algebrais obviously alternative, but so too are some strictlynon-associative algebrassuch as theoctonions.
Alternative algebras are so named because they are the algebras for which theassociatorisalternating. The associator is atrilinear mapgiven by
By definition, amultilinear mapis alternating if itvanisheswhenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to[1]
Both of these identities together imply that:
for allx{\displaystyle x}andy{\displaystyle y}. This is equivalent to theflexible identity[2]
The associator of an alternative algebra is therefore alternating.Conversely, any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of:
is alternative and therefore satisfies all three identities.
An alternating associator is always totally skew-symmetric. That is,
for anypermutationσ{\displaystyle \sigma }. The converse holds so long as thecharacteristicof the basefieldis not 2.
Artin's theoremstates that in an alternative algebra thesubalgebragenerated by any two elements isassociative.[4]Conversely, any algebra for which this is true is clearly alternative. It follows that expressions involving only two variables can be written unambiguously without parentheses in an alternative algebra. A generalization of Artin's theorem states that whenever three elementsx,y,z{\displaystyle x,y,z}in an alternative algebra associate (i.e.,[x,y,z]=0{\displaystyle [x,y,z]=0}), the subalgebra generated by those elements is associative.
Acorollaryof Artin's theorem is that alternative algebras arepower-associative, that is, the subalgebra generated by a single element is associative.[5]The converse need not hold: the sedenions are power-associative but not alternative.
TheMoufang identities
hold in any alternative algebra.[2]
In a unital alternative algebra, multiplicativeinversesare unique whenever they exist. Moreover, for any invertible elementx{\displaystyle x}and ally{\displaystyle y}one has
This is equivalent to saying the associator[x−1,x,y]{\displaystyle [x^{-1},x,y]}vanishes for all suchx{\displaystyle x}andy{\displaystyle y}.
Ifx{\displaystyle x}andy{\displaystyle y}are invertible thenxy{\displaystyle xy}is also invertible with inverse(xy)−1=y−1x−1{\displaystyle (xy)^{-1}=y^{-1}x^{-1}}. The set of all invertible elements is therefore closed under multiplication and forms aMoufang loop. Thisloop of unitsin an alternative ring or algebra is analogous to thegroup of unitsin anassociative ringor algebra.
Kleinfeld's theorem states that any simple non-associative alternative ring is a generalized octonion algebra over itscenter.[6]The structure theory of alternative rings is presented in the bookRings That Are Nearly Associativeby Zhevlakov, Slin'ko, Shestakov, and Shirshov.[7]
Theprojective planeover any alternativedivision ringis aMoufang plane.
Everycomposition algebrais an alternative algebra, as shown by Guy Roos in 2008:[8]A composition algebraAover a fieldKhas anorm nthat is a multiplicativehomomorphism:n(a×b)=n(a)×n(b){\displaystyle n(a\times b)=n(a)\times n(b)}connecting (A, ×) and (K, ×).
Define the form ( _ : _ ):A×A→Kby(a:b)=n(a+b)−n(a)−n(b).{\displaystyle (a:b)=n(a+b)-n(a)-n(b).}Then the trace ofais given by (a:1) and the conjugate bya* = (a:1)e –awhere e is the basis element for 1. A series of exercises prove that a composition algebra is always an alternative algebra.[9]
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Ring homomorphisms
Algebraic structures
Related structures
Algebraic number theory
Noncommutative algebraic geometry
Free algebra
Clifford algebra
Inmathematics, aClifford algebra[a]is analgebragenerated by avector spacewith aquadratic form, and is aunitalassociative algebrawith the additional structure of a distinguished subspace. AsK-algebras, they generalize thereal numbers,complex numbers,quaternionsand several otherhypercomplex numbersystems.[1][2]The theory of Clifford algebras is intimately connected with the theory ofquadratic formsandorthogonal transformations. Clifford algebras have important applications in a variety of fields includinggeometry,theoretical physicsanddigital image processing. They are named after the English mathematicianWilliam Kingdon Clifford(1845–1879).
The most familiar Clifford algebras, theorthogonal Clifford algebras, are also referred to as (pseudo-)Riemannian Clifford algebras, as distinct fromsymplectic Clifford algebras.[b]
A Clifford algebra is aunitalassociative algebrathat contains and is generated by avector spaceVover afieldK, whereVis equipped with aquadratic formQ:V→K. The Clifford algebraCl(V,Q)is the "freest" unital associative algebra generated byVsubject to the condition[c]v2=Q(v)1for allv∈V,{\displaystyle v^{2}=Q(v)1\ {\text{ for all }}v\in V,}where the product on the left is that of the algebra, and the1on the right is the algebra'smultiplicative identity(not to be confused with the multiplicative identity ofK). The idea of being the "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of auniversal property, as donebelow.
WhenVis a finite-dimensional real vector space andQisnondegenerate,Cl(V,Q)may be identified by the labelClp,q(R), indicating thatVhas an orthogonal basis withpelements withei2= +1,qwithei2= −1, and whereRindicates that this is a Clifford algebra over the reals; i.e. coefficients of elements of the algebra are real numbers. This basis may be found byorthogonal diagonalization.
Thefree algebragenerated byVmay be written as thetensor algebra⨁n≥0V⊗ ⋯ ⊗V, that is, thedirect sumof thetensor productofncopies ofVover alln. Therefore one obtains a Clifford algebra as thequotientof this tensor algebra by the two-sidedidealgenerated by elements of the formv⊗v−Q(v)1for all elementsv∈V. The product induced by the tensor product in the quotient algebra is written using juxtaposition (e.g.uv). Its associativity follows from the associativity of the tensor product.
The Clifford algebra has a distinguishedsubspaceV, being theimageof theembeddingmap. Such a subspace cannot in general be uniquely determined given only aK-algebra that isisomorphicto the Clifford algebra.
If2isinvertiblein the ground fieldK, then one can rewrite the fundamental identity above in the formuv+vu=2⟨u,v⟩1for allu,v∈V,{\displaystyle uv+vu=2\langle u,v\rangle 1\ {\text{ for all }}u,v\in V,}where⟨u,v⟩=12(Q(u+v)−Q(u)−Q(v)){\displaystyle \langle u,v\rangle ={\frac {1}{2}}\left(Q(u+v)-Q(u)-Q(v)\right)}is thesymmetric bilinear formassociated withQ, via thepolarization identity.
Quadratic forms and Clifford algebras in characteristic2form an exceptional case in this respect. In particular, ifchar(K) = 2it is not true that a quadratic form necessarily or uniquely determines a symmetric bilinear form that satisfiesQ(v) =⟨v,v⟩,[3]Many of the statements in this article include the condition that the characteristic is not2, and are false if this condition is removed.
Clifford algebras are closely related toexterior algebras. Indeed, ifQ= 0then the Clifford algebraCl(V,Q)is just the exterior algebra⋀V. Whenever2is invertible in the ground fieldK, there exists a canonicallinearisomorphism between⋀VandCl(V,Q). That is, they arenaturally isomorphicas vector spaces, but with different multiplications (in the case of characteristic two, they are still isomorphic as vector spaces, just not naturally). Clifford multiplication together with the distinguished subspace is strictly richer than theexterior productsince it makes use of the extra information provided byQ.
The Clifford algebra is afiltered algebra; theassociated graded algebrais the exterior algebra.
More precisely, Clifford algebras may be thought of asquantizations(cf.quantum group) of the exterior algebra, in the same way that theWeyl algebrais a quantization of thesymmetric algebra.
Weyl algebras and Clifford algebras admit a further structure of a*-algebra, and can be unified as even and odd terms of asuperalgebra, as discussed inCCR and CAR algebras.
LetVbe avector spaceover afieldK, and letQ:V→Kbe aquadratic formonV. In most cases of interest the fieldKis either the field ofreal numbersR, or the field ofcomplex numbersC, or afinite field.
A Clifford algebraCl(V,Q)is a pair(B,i),[d][4]whereBis aunitalassociative algebraoverKandiis alinear mapi:V→Bthat satisfiesi(v)2=Q(v)1Bfor allvinV, defined by the followinguniversal property: given any unital associative algebraAoverKand any linear mapj:V→Asuch thatj(v)2=Q(v)1Afor allv∈V{\displaystyle j(v)^{2}=Q(v)1_{A}{\text{ for all }}v\in V}(where1Adenotes the multiplicative identity ofA), there is a uniquealgebra homomorphismf:B→Asuch that the following diagramcommutes(i.e. such thatf∘i=j):
The quadratic formQmay be replaced by a (not necessarily symmetric[5])bilinear form⟨⋅,⋅⟩that has the property⟨v,v⟩=Q(v),v∈V, in which case an equivalent requirement onjisj(v)j(v)=⟨v,v⟩1Afor allv∈V.{\displaystyle j(v)j(v)=\langle v,v\rangle 1_{A}\quad {\text{ for all }}v\in V.}
When the characteristic of the field is not2, this may be replaced by what is then an equivalent requirement,j(v)j(w)+j(w)j(v)=(⟨v,w⟩+⟨w,v⟩)1Afor allv,w∈V,{\displaystyle j(v)j(w)+j(w)j(v)=(\langle v,w\rangle +\langle w,v\rangle )1_{A}\quad {\text{ for all }}v,w\in V,}where the bilinear form may additionally be restricted to being symmetric without loss of generality.
A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that containsV, namely thetensor algebraT(V), and then enforce the fundamental identity by taking a suitablequotient. In our case we want to take the two-sidedidealIQinT(V)generated by all elements of the formv⊗v−Q(v)1{\displaystyle v\otimes v-Q(v)1}for allv∈V{\displaystyle v\in V}and defineCl(V,Q)as the quotient algebraCl(V,Q)=T(V)/IQ.{\displaystyle \operatorname {Cl} (V,Q)=T(V)/I_{Q}.}
Theringproduct inherited by this quotient is sometimes referred to as theClifford product[6]to distinguish it from the exterior product and the scalar product.
It is then straightforward to show thatCl(V,Q)containsVand satisfies the above universal property, so thatClis unique up to a unique isomorphism; thus one speaks of "the" Clifford algebraCl(V,Q). It also follows from this construction thatiisinjective. One usually drops theiand considersVas alinear subspaceofCl(V,Q).
The universal characterization of the Clifford algebra shows that the construction ofCl(V,Q)isfunctorialin nature. Namely,Clcan be considered as afunctorfrom thecategoryof vector spaces with quadratic forms (whosemorphismsare linear maps that preserve the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (that preserve the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras.
SinceVcomes equipped with a quadratic formQ, in characteristic not equal to2there existbasesforVthat areorthogonal. Anorthogonal basisis one such that for a symmetric bilinear form⟨ei,ej⟩=0{\displaystyle \langle e_{i},e_{j}\rangle =0}fori≠j{\displaystyle i\neq j}, and⟨ei,ei⟩=Q(ei).{\displaystyle \langle e_{i},e_{i}\rangle =Q(e_{i}).}
The fundamental Clifford identity implies that for an orthogonal basiseiej=−ejei{\displaystyle e_{i}e_{j}=-e_{j}e_{i}}fori≠j{\displaystyle i\neq j}, andei2=Q(ei).{\displaystyle e_{i}^{2}=Q(e_{i}).}
This makes manipulation of orthogonal basis vectors quite simple. Given a productei1ei2⋯eik{\displaystyle e_{i_{1}}e_{i_{2}}\cdots e_{i_{k}}}ofdistinctorthogonal basis vectors ofV, one can put them into a standard order while including an overall sign determined by the number ofpairwise swapsneeded to do so (i.e. thesignatureof the orderingpermutation).
If thedimensionofVoverKisnand{e1, ...,en}is an orthogonal basis of(V,Q), thenCl(V,Q)isfree overKwith a basis{ei1ei2⋯eik∣1≤i1<i2<⋯<ik≤nand0≤k≤n}.{\displaystyle \{e_{i_{1}}e_{i_{2}}\cdots e_{i_{k}}\mid 1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n{\text{ and }}0\leq k\leq n\}.}
The empty product (k= 0) is defined as being the multiplicativeidentity element. For each value ofkthere arenchoosekbasis elements, so the total dimension of the Clifford algebra isdimCl(V,Q)=∑k=0n(nk)=2n.{\displaystyle \dim \operatorname {Cl} (V,Q)=\sum _{k=0}^{n}{\binom {n}{k}}=2^{n}.}
The most important Clifford algebras are those overrealandcomplexvector spaces equipped withnondegenerate quadratic forms.
Each of the algebrasClp,q(R)andCln(C)is isomorphic toAorA⊕A, whereAis afull matrix ringwith entries fromR,C, orH. For a complete classification of these algebras seeClassification of Clifford algebras.
Clifford algebras are also sometimes referred to asgeometric algebras, most often over the real numbers.
Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:Q(v)=v12+⋯+vp2−vp+12−⋯−vp+q2,{\displaystyle Q(v)=v_{1}^{2}+\dots +v_{p}^{2}-v_{p+1}^{2}-\dots -v_{p+q}^{2},}wheren=p+qis the dimension of the vector space. The pair of integers(p,q)is called thesignatureof the quadratic form. The real vector space with this quadratic form is often denotedRp,q.The Clifford algebra onRp,qis denotedClp,q(R).The symbolCln(R)means eitherCln,0(R)orCl0,n(R), depending on whether the author prefers positive-definite or negative-definite spaces.
A standardbasis{e1, ...,en}forRp,qconsists ofn=p+qmutually orthogonal vectors,pof which square to+1andqof which square to−1. Of such a basis, the algebraClp,q(R)will therefore havepvectors that square to+1andqvectors that square to−1.
A few low-dimensional cases are:
One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space of dimensionnis equivalent to the standard diagonal formQ(z)=z12+z22+⋯+zn2.{\displaystyle Q(z)=z_{1}^{2}+z_{2}^{2}+\dots +z_{n}^{2}.}Thus, for each dimensionn, up to isomorphism there is only one Clifford algebra of a complex vector space with a nondegenerate quadratic form. We will denote the Clifford algebra onCnwith the standard quadratic form byCln(C).
For the first few cases one finds that
whereMn(C)denotes the algebra ofn×nmatrices overC.
In this section, Hamilton'squaternionsare constructed as the even subalgebra of the Clifford algebraCl3,0(R).
Let the vector spaceVbe real three-dimensional spaceR3, and the quadratic form be the usual quadratic form. Then, forv,winR3we have the bilinear form (or scalar product)v⋅w=v1w1+v2w2+v3w3.{\displaystyle v\cdot w=v_{1}w_{1}+v_{2}w_{2}+v_{3}w_{3}.}Now introduce the Clifford product of vectorsvandwgiven byvw+wv=2(v⋅w).{\displaystyle vw+wv=2(v\cdot w).}
Denote a set of orthogonal unit vectors ofR3as{e1,e2,e3}, then the Clifford product yields the relationse2e3=−e3e2,e1e3=−e3e1,e1e2=−e2e1,{\displaystyle e_{2}e_{3}=-e_{3}e_{2},\,\,\,e_{1}e_{3}=-e_{3}e_{1},\,\,\,e_{1}e_{2}=-e_{2}e_{1},}ande12=e22=e32=1.{\displaystyle e_{1}^{2}=e_{2}^{2}=e_{3}^{2}=1.}The general element of the Clifford algebraCl3,0(R)is given byA=a0+a1e1+a2e2+a3e3+a4e2e3+a5e1e3+a6e1e2+a7e1e2e3.{\displaystyle A=a_{0}+a_{1}e_{1}+a_{2}e_{2}+a_{3}e_{3}+a_{4}e_{2}e_{3}+a_{5}e_{1}e_{3}+a_{6}e_{1}e_{2}+a_{7}e_{1}e_{2}e_{3}.}
The linear combination of the even degree elements ofCl3,0(R)defines the even subalgebraCl[0]3,0(R)with the general elementq=q0+q1e2e3+q2e1e3+q3e1e2.{\displaystyle q=q_{0}+q_{1}e_{2}e_{3}+q_{2}e_{1}e_{3}+q_{3}e_{1}e_{2}.}The basis elements can be identified with the quaternion basis elementsi,j,kasi=e2e3,j=e1e3,k=e1e2,{\displaystyle i=e_{2}e_{3},j=e_{1}e_{3},k=e_{1}e_{2},}which shows that the even subalgebraCl[0]3,0(R)is Hamilton's realquaternionalgebra.
To see this, computei2=(e2e3)2=e2e3e2e3=−e2e2e3e3=−1,{\displaystyle i^{2}=(e_{2}e_{3})^{2}=e_{2}e_{3}e_{2}e_{3}=-e_{2}e_{2}e_{3}e_{3}=-1,}andij=e2e3e1e3=−e2e3e3e1=−e2e1=e1e2=k.{\displaystyle ij=e_{2}e_{3}e_{1}e_{3}=-e_{2}e_{3}e_{3}e_{1}=-e_{2}e_{1}=e_{1}e_{2}=k.}Finally,ijk=e2e3e1e3e1e2=−1.{\displaystyle ijk=e_{2}e_{3}e_{1}e_{3}e_{1}e_{2}=-1.}
In this section,dual quaternionsare constructed as the even subalgebra of a Clifford algebra of real four-dimensional space with a degenerate quadratic form.[9][10]
Let the vector spaceVbe real four-dimensional spaceR4,and let the quadratic formQbe a degenerate form derived from the Euclidean metric onR3.Forv,winR4introduce the degenerate bilinear formd(v,w)=v1w1+v2w2+v3w3.{\displaystyle d(v,w)=v_{1}w_{1}+v_{2}w_{2}+v_{3}w_{3}.}This degenerate scalar product projects distance measurements inR4onto theR3hyperplane.
The Clifford product of vectorsvandwis given byvw+wv=−2d(v,w).{\displaystyle vw+wv=-2\,d(v,w).}Note the negative sign is introduced to simplify the correspondence with quaternions.
Denote a set of mutually orthogonal unit vectors ofR4as{e1,e2,e3,e4}, then the Clifford product yields the relationsemen=−enem,m≠n,{\displaystyle e_{m}e_{n}=-e_{n}e_{m},\,\,\,m\neq n,}ande12=e22=e32=−1,e42=0.{\displaystyle e_{1}^{2}=e_{2}^{2}=e_{3}^{2}=-1,\,\,e_{4}^{2}=0.}
The general element of the Clifford algebraCl(R4,d)has 16 components. The linear combination of the even degree elements defines the even subalgebraCl[0](R4,d)with the general elementH=h0+h1e2e3+h2e3e1+h3e1e2+h4e4e1+h5e4e2+h6e4e3+h7e1e2e3e4.{\displaystyle H=h_{0}+h_{1}e_{2}e_{3}+h_{2}e_{3}e_{1}+h_{3}e_{1}e_{2}+h_{4}e_{4}e_{1}+h_{5}e_{4}e_{2}+h_{6}e_{4}e_{3}+h_{7}e_{1}e_{2}e_{3}e_{4}.}
The basis elements can be identified with the quaternion basis elementsi,j,kand the dual unitεasi=e2e3,j=e3e1,k=e1e2,ε=e1e2e3e4.{\displaystyle i=e_{2}e_{3},j=e_{3}e_{1},k=e_{1}e_{2},\,\,\varepsilon =e_{1}e_{2}e_{3}e_{4}.}This provides the correspondence ofCl[0]0,3,1(R)withdual quaternionalgebra.
To see this, computeε2=(e1e2e3e4)2=e1e2e3e4e1e2e3e4=−e1e2e3(e4e4)e1e2e3=0,{\displaystyle \varepsilon ^{2}=(e_{1}e_{2}e_{3}e_{4})^{2}=e_{1}e_{2}e_{3}e_{4}e_{1}e_{2}e_{3}e_{4}=-e_{1}e_{2}e_{3}(e_{4}e_{4})e_{1}e_{2}e_{3}=0,}andεi=(e1e2e3e4)e2e3=e1e2e3e4e2e3=e2e3(e1e2e3e4)=iε.{\displaystyle \varepsilon i=(e_{1}e_{2}e_{3}e_{4})e_{2}e_{3}=e_{1}e_{2}e_{3}e_{4}e_{2}e_{3}=e_{2}e_{3}(e_{1}e_{2}e_{3}e_{4})=i\varepsilon .}The exchanges ofe1ande4alternate signs an even number of times, and show the dual unitεcommutes with the quaternion basis elementsi,j,k.
LetKbe any field of characteristic not2.
FordimV= 1, ifQhas diagonalizationdiag(a), that is there is a non-zero vectorxsuch thatQ(x) =a, thenCl(V,Q)is algebra-isomorphic to aK-algebra generated by an elementxthat satisfiesx2=a, the quadratic algebraK[X] / (X2−a).
In particular, ifa= 0(that is,Qis the zero quadratic form) thenCl(V,Q)is algebra-isomorphic to thedual numbersalgebra overK.
Ifais a non-zero square inK, thenCl(V,Q) ≃K⊕K.
Otherwise,Cl(V,Q)is isomorphic to the quadratic field extensionK(√a)ofK.
FordimV= 2, ifQhas diagonalizationdiag(a,b)with non-zeroaandb(which always exists ifQis non-degenerate), thenCl(V,Q)is isomorphic to aK-algebra generated by elementsxandythat satisfiesx2=a,y2=bandxy= −yx.
ThusCl(V,Q)is isomorphic to the (generalized)quaternion algebra(a,b)K. We retrieve Hamilton's quaternions whena=b= −1, sinceH= (−1, −1)R.
As a special case, if somexinVsatisfiesQ(x) = 1, thenCl(V,Q) ≃ M2(K).
Given a vector spaceV, one can construct theexterior algebra⋀V, whose definition is independent of any quadratic form onV. It turns out that ifKdoes not have characteristic2then there is anatural isomorphismbetween⋀VandCl(V,Q)considered as vector spaces (and there exists an isomorphism in characteristic two, which may not be natural). This is an algebra isomorphism if and only ifQ= 0. One can thus consider the Clifford algebraCl(V,Q)as an enrichment (or more precisely, a quantization, cf. the Introduction) of the exterior algebra onVwith a multiplication that depends onQ(one can still define the exterior product independently ofQ).
The easiest way to establish the isomorphism is to choose anorthogonalbasis{e1, ...,en}forVand extend it to a basis forCl(V,Q)as describedabove. The mapCl(V,Q) → ⋀Vis determined byei1ei2⋯eik↦ei1∧ei2∧⋯∧eik.{\displaystyle e_{i_{1}}e_{i_{2}}\cdots e_{i_{k}}\mapsto e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}.}Note that this works only if the basis{e1, ...,en}is orthogonal. One can show that this map is independent of the choice of orthogonal basis and so gives a natural isomorphism.
If thecharacteristicofKis0, one can also establish the isomorphism by antisymmetrizing. Define functionsfk:V× ⋯ ×V→ Cl(V,Q)byfk(v1,…,vk)=1k!∑σ∈Sksgn(σ)vσ(1)⋯vσ(k){\displaystyle f_{k}(v_{1},\ldots ,v_{k})={\frac {1}{k!}}\sum _{\sigma \in \mathrm {S} _{k}}\operatorname {sgn}(\sigma )\,v_{\sigma (1)}\cdots v_{\sigma (k)}}where the sum is taken over thesymmetric grouponkelements,Sk. Sincefkisalternating, it induces a unique linear map⋀kV→ Cl(V,Q). Thedirect sumof these maps gives a linear map between⋀VandCl(V,Q). This map can be shown to be a linear isomorphism, and it is natural.
A more sophisticated way to view the relationship is to construct afiltrationonCl(V,Q). Recall that thetensor algebraT(V)has a natural filtration:F0⊂F1⊂F2⊂ ⋯, whereFkcontains sums of tensors withorder≤k. Projecting this down to the Clifford algebra gives a filtration onCl(V,Q). Theassociated graded algebraGrFCl(V,Q)=⨁kFk/Fk−1{\displaystyle \operatorname {Gr} _{F}\operatorname {Cl} (V,Q)=\bigoplus _{k}F^{k}/F^{k-1}}is naturally isomorphic to the exterior algebra⋀V. Since the associated graded algebra of a filtered algebra is always isomorphic to the filtered algebra as filtered vector spaces (by choosing complements ofFkinFk+1for allk), this provides an isomorphism (although not a natural one) in any characteristic, even two.
In the following, assume that the characteristic is not2.[e]
Clifford algebras areZ2-graded algebras(also known assuperalgebras). Indeed, the linear map onVdefined byv↦ −v(reflection through the origin) preserves the quadratic formQand so by the universal property of Clifford algebras extends to an algebraautomorphismα:Cl(V,Q)→Cl(V,Q).{\displaystyle \alpha :\operatorname {Cl} (V,Q)\to \operatorname {Cl} (V,Q).}
Sinceαis aninvolution(i.e. it squares to theidentity) one can decomposeCl(V,Q)into positive and negative eigenspaces ofαCl(V,Q)=Cl[0](V,Q)⊕Cl[1](V,Q){\displaystyle \operatorname {Cl} (V,Q)=\operatorname {Cl} ^{[0]}(V,Q)\oplus \operatorname {Cl} ^{[1]}(V,Q)}whereCl[i](V,Q)={x∈Cl(V,Q)∣α(x)=(−1)ix}.{\displaystyle \operatorname {Cl} ^{[i]}(V,Q)=\left\{x\in \operatorname {Cl} (V,Q)\mid \alpha (x)=(-1)^{i}x\right\}.}
Sinceαis an automorphism it follows that:Cl[i](V,Q)Cl[j](V,Q)=Cl[i+j](V,Q){\displaystyle \operatorname {Cl} ^{[i]}(V,Q)\operatorname {Cl} ^{[j]}(V,Q)=\operatorname {Cl} ^{[i+j]}(V,Q)}where the bracketed superscripts are read modulo 2. This givesCl(V,Q)the structure of aZ2-graded algebra. The subspaceCl[0](V,Q)forms asubalgebraofCl(V,Q), called theeven subalgebra. The subspaceCl[1](V,Q)is called theodd partofCl(V,Q)(it is not a subalgebra).ThisZ2-grading plays an important role in the analysis and application of Clifford algebras. The automorphismαis called themaininvolutionorgrade involution. Elements that are pure in thisZ2-grading are simply said to be even or odd.
Remark. The Clifford algebra is not aZ-graded algebra, but isZ-filtered, whereCl≤i(V,Q)is the subspace spanned by all products of at mostielements ofV.Cl⩽i(V,Q)⋅Cl⩽j(V,Q)⊂Cl⩽i+j(V,Q).{\displaystyle \operatorname {Cl} ^{\leqslant i}(V,Q)\cdot \operatorname {Cl} ^{\leqslant j}(V,Q)\subset \operatorname {Cl} ^{\leqslant i+j}(V,Q).}
Thedegreeof a Clifford number usually refers to the degree in theZ-grading.
The even subalgebraCl[0](V,Q)of a Clifford algebra is itself isomorphic to a Clifford algebra.[f][g]IfVis theorthogonal direct sumof a vectoraof nonzero normQ(a)and a subspaceU, thenCl[0](V,Q)is isomorphic toCl(U, −Q(a)Q|U), whereQ|Uis the formQrestricted toU. In particular over the reals this implies that:Clp,q[0](R)≅{Clp,q−1(R)q>0Clq,p−1(R)p>0{\displaystyle \operatorname {Cl} _{p,q}^{[0]}(\mathbf {R} )\cong {\begin{cases}\operatorname {Cl} _{p,q-1}(\mathbf {R} )&q>0\\\operatorname {Cl} _{q,p-1}(\mathbf {R} )&p>0\end{cases}}}
In the negative-definite case this gives an inclusionCl0,n− 1(R) ⊂ Cl0,n(R), which extends the sequence
Likewise, in the complex case, one can show that the even subalgebra ofCln(C)is isomorphic toCln−1(C).
In addition to the automorphismα, there are twoantiautomorphismsthat play an important role in the analysis of Clifford algebras. Recall that thetensor algebraT(V)comes with an antiautomorphism that reverses the order in all products of vectors:v1⊗v2⊗⋯⊗vk↦vk⊗⋯⊗v2⊗v1.{\displaystyle v_{1}\otimes v_{2}\otimes \cdots \otimes v_{k}\mapsto v_{k}\otimes \cdots \otimes v_{2}\otimes v_{1}.}Since the idealIQis invariant under this reversal, this operation descends to an antiautomorphism ofCl(V,Q)called thetransposeorreversaloperation, denoted byxt. The transpose is an antiautomorphism:(xy)t=ytxt. The transpose operation makes no use of theZ2-grading so we define a second antiautomorphism by composingαand the transpose. We call this operationClifford conjugationdenotedx¯{\displaystyle {\bar {x}}}x¯=α(xt)=α(x)t.{\displaystyle {\bar {x}}=\alpha (x^{\mathrm {t} })=\alpha (x)^{\mathrm {t} }.}Of the two antiautomorphisms, the transpose is the more fundamental.[h]
Note that all of these operations areinvolutions. One can show that they act as±1on elements that are pure in theZ-grading. In fact, all three operations depend on only the degree modulo4. That is, ifxis pure with degreekthenα(x)=±xxt=±xx¯=±x{\displaystyle \alpha (x)=\pm x\qquad x^{\mathrm {t} }=\pm x\qquad {\bar {x}}=\pm x}where the signs are given by the following table:
When the characteristic is not2, the quadratic formQonVcan be extended to a quadratic form on all ofCl(V,Q)(which we also denoted byQ). A basis-independent definition of one such extension isQ(x)=⟨xtx⟩0{\displaystyle Q(x)=\left\langle x^{\mathrm {t} }x\right\rangle _{0}}where⟨a⟩0denotes the scalar part ofa(the degree-0part in theZ-grading). One can show thatQ(v1v2⋯vk)=Q(v1)Q(v2)⋯Q(vk){\displaystyle Q(v_{1}v_{2}\cdots v_{k})=Q(v_{1})Q(v_{2})\cdots Q(v_{k})}where theviare elements ofV– this identity isnottrue for arbitrary elements ofCl(V,Q).
The associated symmetric bilinear form onCl(V,Q)is given by⟨x,y⟩=⟨xty⟩0.{\displaystyle \langle x,y\rangle =\left\langle x^{\mathrm {t} }y\right\rangle _{0}.}One can check that this reduces to the original bilinear form when restricted toV. The bilinear form on all ofCl(V,Q)isnondegenerateif and only if it is nondegenerate onV.
The operator of left (respectively right) Clifford multiplication by the transposeatof an elementais theadjointof left (respectively right) Clifford multiplication byawith respect to this inner product. That is,⟨ax,y⟩=⟨x,aty⟩,{\displaystyle \langle ax,y\rangle =\left\langle x,a^{\mathrm {t} }y\right\rangle ,}and⟨xa,y⟩=⟨x,yat⟩.{\displaystyle \langle xa,y\rangle =\left\langle x,ya^{\mathrm {t} }\right\rangle .}
In this section we assume that characteristic is not2, the vector spaceVis finite-dimensional and that the associated symmetric bilinear form ofQis nondegenerate.
Acentral simple algebraoverKis a matrix algebra over a (finite-dimensional) division algebra with centerK. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions.
The structure of Clifford algebras can be worked out explicitly using the following result. Suppose thatUhas even dimension and a non-singular bilinear form withdiscriminantd, and suppose thatVis another vector space with a quadratic form. The Clifford algebra ofU+Vis isomorphic to the tensor product of the Clifford algebras ofUand(−1)dim(U)/2dV, which is the spaceVwith its quadratic form multiplied by(−1)dim(U)/2d. Over the reals, this implies in particular thatClp+2,q(R)=M2(R)⊗Clq,p(R){\displaystyle \operatorname {Cl} _{p+2,q}(\mathbf {R} )=\mathrm {M} _{2}(\mathbf {R} )\otimes \operatorname {Cl} _{q,p}(\mathbf {R} )}Clp+1,q+1(R)=M2(R)⊗Clp,q(R){\displaystyle \operatorname {Cl} _{p+1,q+1}(\mathbf {R} )=\mathrm {M} _{2}(\mathbf {R} )\otimes \operatorname {Cl} _{p,q}(\mathbf {R} )}Clp,q+2(R)=H⊗Clq,p(R).{\displaystyle \operatorname {Cl} _{p,q+2}(\mathbf {R} )=\mathbf {H} \otimes \operatorname {Cl} _{q,p}(\mathbf {R} ).}These formulas can be used to find the structure of all real Clifford algebras and all complex Clifford algebras; see theclassification of Clifford algebras.
Notably, theMorita equivalenceclass of a Clifford algebra (its representation theory: the equivalence class of the category of modules over it) depends on only the signature(p−q) mod 8. This is an algebraic form ofBott periodicity.
The class of Lipschitz groups (a.k.a.[11]Clifford groupsor Clifford–Lipschitz groups) was discovered byRudolf Lipschitz.[12]
In this section we assume thatVis finite-dimensional and the quadratic formQisnondegenerate.
An action on the elements of a Clifford algebra by itsgroup of unitsmay be defined in terms of a twisted conjugation: twisted conjugation byxmapsy↦α(x)yx−1, whereαis themain involutiondefinedabove.
The Lipschitz groupΓis defined to be the set of invertible elementsxthatstabilize the set of vectorsunder this action,[13]meaning that for allvinVwe have:α(x)vx−1∈V.{\displaystyle \alpha (x)vx^{-1}\in V.}
This formula also defines an action of the Lipschitz group on the vector spaceVthat preserves the quadratic formQ, and so gives a homomorphism from the Lipschitz group to the orthogonal group. The Lipschitz group contains all elementsrofVfor whichQ(r)is invertible inK, and these act onVby the corresponding reflections that takevtov− (⟨r,v⟩+⟨v,r⟩)r/Q(r). (In characteristic2these are called orthogonal transvections rather than reflections.)
IfVis a finite-dimensional real vector space with anon-degeneratequadratic form then the Lipschitz group maps onto the orthogonal group ofVwith respect to the form (by theCartan–Dieudonné theorem) and the kernel consists of the nonzero elements of the fieldK. This leads to exact sequences1→K×→Γ→OV(K)→1,{\displaystyle 1\rightarrow K^{\times }\rightarrow \Gamma \rightarrow \operatorname {O} _{V}(K)\rightarrow 1,}1→K×→Γ0→SOV(K)→1.{\displaystyle 1\rightarrow K^{\times }\rightarrow \Gamma ^{0}\rightarrow \operatorname {SO} _{V}(K)\rightarrow 1.}
Over other fields or with indefinite forms, the map is not in general onto, and the failure is captured by the spinor norm.
In arbitrary characteristic, thespinor normQis defined on the Lipschitz group byQ(x)=xtx.{\displaystyle Q(x)=x^{\mathrm {t} }x.}It is a homomorphism from the Lipschitz group to the groupK×of non-zero elements ofK. It coincides with the quadratic formQofVwhenVis identified with a subspace of the Clifford algebra. Several authors define the spinor norm slightly differently, so that it differs from the one here by a factor of−1,2, or−2onΓ1. The difference is not very important in characteristic other than 2.
The nonzero elements ofKhave spinor norm in the group (K×)2of squares of nonzero elements of the fieldK. So whenVis finite-dimensional and non-singular we get an induced map from the orthogonal group ofVto the groupK×/(K×)2, also called the spinor norm. The spinor norm of the reflection aboutr⊥, for any vectorr, has imageQ(r)inK×/(K×)2, and this property uniquely defines it on the orthogonal group. This gives exact sequences:1→{±1}→PinV(K)→OV(K)→K×/(K×)2,1→{±1}→SpinV(K)→SOV(K)→K×/(K×)2.{\displaystyle {\begin{aligned}1\to \{\pm 1\}\to \operatorname {Pin} _{V}(K)&\to \operatorname {O} _{V}(K)\to K^{\times }/\left(K^{\times }\right)^{2},\\1\to \{\pm 1\}\to \operatorname {Spin} _{V}(K)&\to \operatorname {SO} _{V}(K)\to K^{\times }/\left(K^{\times }\right)^{2}.\end{aligned}}}
Note that in characteristic2the group{±1}has just one element.
From the point of view ofGalois cohomologyofalgebraic groups, the spinor norm is aconnecting homomorphismon cohomology. Writingμ2for thealgebraic group of square roots of 1(over a field of characteristic not2it is roughly the same as a two-element group with trivial Galois action), the short exact sequence1→μ2→PinV→OV→1{\displaystyle 1\to \mu _{2}\rightarrow \operatorname {Pin} _{V}\rightarrow \operatorname {O} _{V}\rightarrow 1}yields a long exact sequence on cohomology, which begins1→H0(μ2;K)→H0(PinV;K)→H0(OV;K)→H1(μ2;K).{\displaystyle 1\to H^{0}(\mu _{2};K)\to H^{0}(\operatorname {Pin} _{V};K)\to H^{0}(\operatorname {O} _{V};K)\to H^{1}(\mu _{2};K).}
The 0th Galois cohomology group of an algebraic group with coefficients inKis just the group ofK-valued points:H0(G;K) =G(K), andH1(μ2;K) ≅K×/(K×)2, which recovers the previous sequence1→{±1}→PinV(K)→OV(K)→K×/(K×)2,{\displaystyle 1\to \{\pm 1\}\to \operatorname {Pin} _{V}(K)\to \operatorname {O} _{V}(K)\to K^{\times }/\left(K^{\times }\right)^{2},}where the spinor norm is the connecting homomorphismH0(OV;K) →H1(μ2;K).
In this section we assume thatVis finite-dimensional and its bilinear form is non-singular.
Thepin groupPinV(K)is the subgroup of the Lipschitz groupΓof elements of spinor norm1, and similarly thespin groupSpinV(K)is the subgroup of elements ofDickson invariant0inPinV(K). When the characteristic is not2, these are the elements of determinant1. The spin group usually has index2in the pin group.
Recall from the previous section that there is a homomorphism from the Lipschitz group onto the orthogonal group. We define thespecial orthogonal groupto be the image ofΓ0. IfKdoes not have characteristic2this is just the group of elements of the orthogonal group of determinant1. IfKdoes have characteristic2, then all elements of the orthogonal group have determinant1, and the special orthogonal group is the set of elements of Dickson invariant0.
There is a homomorphism from the pin group to the orthogonal group. The image consists of the elements of spinor norm1 ∈K×/(K×)2. The kernel consists of the elements+1and−1, and has order2unlessKhas characteristic2. Similarly there is a homomorphism from the Spin group to the special orthogonal group ofV.
In the common case whenVis a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected whenVhas dimension at least3. Further the kernel of this homomorphism consists of1and−1. So in this case the spin group,Spin(n), is a double cover ofSO(n). Note, however, that the simple connectedness of the spin group is not true in general: ifVisRp,qforpandqboth at least2then the spin group is not simply connected. In this case the algebraic groupSpinp,qis simply connected as an algebraic group, even though its group of real valued pointsSpinp,q(R)is not simply connected. This is a rather subtle point, which completely confused the authors of at least one standard book about spin groups.[which?]
Clifford algebrasClp,q(C), withp+q= 2neven, are matrix algebras that have a complex representation of dimension2n. By restricting to the groupPinp,q(R)we get a complex representation of the Pin group of the same dimension, called thespin representation. If we restrict this to the spin groupSpinp,q(R)then it splits as the sum of twohalf spin representations(orWeyl representations) of dimension2n−1.
Ifp+q= 2n+ 1is odd then the Clifford algebraClp,q(C)is a sum of two matrix algebras, each of which has a representation of dimension2n, and these are also both representations of the pin groupPinp,q(R). On restriction to the spin groupSpinp,q(R)these become isomorphic, so the spin group has a complex spinor representation of dimension2n.
More generally, spinor groups and pin groups over any field have similar representations whose exact structure depends on thestructure of the corresponding Clifford algebras: whenever a Clifford algebra has a factor that is a matrix algebra over some division algebra, we get a corresponding representation of the pin and spin groups over that division algebra.
For examples over the reals see the article onspinors.
To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. Thepin group,Pinp,qis the set of invertible elements inClp,qthat can be written as a product of unit vectors:Pinp,q={v1v2⋯vr∣∀i‖vi‖=±1}.{\displaystyle \mathrm {Pin} _{p,q}=\left\{v_{1}v_{2}\cdots v_{r}\mid \forall i\,\|v_{i}\|=\pm 1\right\}.}Comparing with the above concrete realizations of the Clifford algebras, the pin group corresponds to the products of arbitrarily many reflections: it is a cover of the full orthogonal groupO(p,q). Thespin groupconsists of those elements ofPinp,qthat are products of an even number of unit vectors. Thus by theCartan–Dieudonné theoremSpin is a cover of the group of proper rotationsSO(p,q).
Letα: Cl → Clbe the automorphism that is given by the mappingv↦ −vacting on pure vectors. Then in particular,Spinp,qis the subgroup ofPinp,qwhose elements are fixed byα. LetClp,q[0]={x∈Clp,q∣α(x)=x}.{\displaystyle \operatorname {Cl} _{p,q}^{[0]}=\{x\in \operatorname {Cl} _{p,q}\mid \alpha (x)=x\}.}(These are precisely the elements of even degree inClp,q.) Then the spin group lies withinCl[0]p,q.
The irreducible representations ofClp,qrestrict to give representations of the pin group. Conversely, since the pin group is generated by unit vectors, all of its irreducible representation are induced in this manner. Thus the two representations coincide. For the same reasons, the irreducible representations of the spin coincide with the irreducible representations ofCl[0]p,q.
To classify the pin representations, one need only appeal to theclassification of Clifford algebras. To find the spin representations (which are representations of the even subalgebra), one can first make use of either of the isomorphisms (see above)Clp,q[0]≈Clp,q−1,forq>0{\displaystyle \operatorname {Cl} _{p,q}^{[0]}\approx \operatorname {Cl} _{p,q-1},{\text{ for }}q>0}Clp,q[0]≈Clq,p−1,forp>0{\displaystyle \operatorname {Cl} _{p,q}^{[0]}\approx \operatorname {Cl} _{q,p-1},{\text{ for }}p>0}and realize a spin representation in signature(p,q)as a pin representation in either signature(p,q− 1)or(q,p− 1).
One of the principal applications of the exterior algebra is indifferential geometrywhere it is used to define thebundleofdifferential formson asmooth manifold. In the case of a (pseudo-)Riemannian manifold, thetangent spacescome equipped with a natural quadratic form induced by themetric. Thus, one can define aClifford bundlein analogy with theexterior bundle. This has a number of important applications inRiemannian geometry. Perhaps more important is the link to aspin manifold, its associatedspinor bundleandspincmanifolds.
Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra that has a basis that is generated by the matricesγ0, ...,γ3, calledDirac matrices, which have the property thatγiγj+γjγi=2ηij,{\displaystyle \gamma _{i}\gamma _{j}+\gamma _{j}\gamma _{i}=2\eta _{ij},}whereηis the matrix of a quadratic form of signature(1, 3)(or(3, 1)corresponding to the two equivalent choices of metric signature). These are exactly the defining relations for the Clifford algebraCl1,3(R), whosecomplexificationisCl1,3(R)C, which, by theclassification of Clifford algebras, is isomorphic to the algebra of4 × 4complex matricesCl4(C) ≈ M4(C). However, it is best to retain the notationCl1,3(R)C, since any transformation that takes the bilinear form to the canonical form isnota Lorentz transformation of the underlying spacetime.
The Clifford algebra of spacetime used in physics thus has more structure thanCl4(C). It has in addition a set of preferred transformations – Lorentz transformations. Whether complexification is necessary to begin with depends in part on conventions used and in part on how much one wants to incorporate straightforwardly, but complexification is most often necessary in quantum mechanics where the spin representation of the Lie algebraso(1, 3)sitting inside the Clifford algebra conventionally requires a complex Clifford algebra. For reference, the spin Lie algebra is given byσμν=−i4[γμ,γν],[σμν,σρτ]=i(ητμσρν+ηντσμρ−ηρμστν−ηνρσμτ).{\displaystyle {\begin{aligned}\sigma ^{\mu \nu }&=-{\frac {i}{4}}\left[\gamma ^{\mu },\,\gamma ^{\nu }\right],\\\left[\sigma ^{\mu \nu },\,\sigma ^{\rho \tau }\right]&=i\left(\eta ^{\tau \mu }\sigma ^{\rho \nu }+\eta ^{\nu \tau }\sigma ^{\mu \rho }-\eta ^{\rho \mu }\sigma ^{\tau \nu }-\eta ^{\nu \rho }\sigma ^{\mu \tau }\right).\end{aligned}}}
This is in the(3, 1)convention, hence fits inCl3,1(R)C.[14]
The Dirac matrices were first written down byPaul Diracwhen he was trying to write a relativistic first-order wave equation for theelectron, and give an explicit isomorphism from the Clifford algebra to the algebra of complex matrices. The result was used to define theDirac equationand introduce theDirac operator. The entire Clifford algebra shows up inquantum field theoryin the form ofDirac field bilinears.
The use of Clifford algebras to describe quantum theory has been advanced among others byMario Schönberg,[i]byDavid Hestenesin terms ofgeometric calculus, byDavid BohmandBasil Hileyand co-workers in form of ahierarchy of Clifford algebras, and by Elio Conte et al.[15][16]
Clifford algebras have been applied in the problem of action recognition and classification incomputer vision. Rodriguez et al[17]propose a Clifford embedding to generalize traditional MACH filters to video (3D spatiotemporal volume), and vector-valued data such asoptical flow. Vector-valued data is analyzed using theClifford Fourier Transform. Based on these vectors action filters are synthesized in the Clifford Fourier domain and recognition of actions is performed using Clifford correlation. The authors demonstrate the effectiveness of the Clifford embedding by recognizing actions typically performed in classic feature films and sports broadcast television.
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Inmathematics, acomposition algebraAover afieldKis anot necessarily associativealgebraoverKtogether with anondegeneratequadratic formNthat satisfies
for allxandyinA.
A composition algebra includes aninvolutioncalled aconjugation:x↦x∗.{\displaystyle x\mapsto x^{*}.}The quadratic formN(x)=xx∗{\displaystyle N(x)=xx^{*}}is called thenormof the algebra.
A composition algebra (A, ∗,N) is either adivision algebraor asplit algebra, depending on the existence of a non-zerovinAsuch thatN(v) = 0, called anull vector.[1]Whenxisnota null vector, themultiplicative inverseofxisx∗N(x){\textstyle {\frac {x^{*}}{N(x)}}}.When there is a non-zero null vector,Nis anisotropic quadratic form, and "the algebra splits".
Everyunitalcomposition algebra over a fieldKcan be obtained by repeated application of theCayley–Dickson constructionstarting fromK(if thecharacteristicofKis different from2) or a 2-dimensional composition subalgebra (ifchar(K) = 2). The possible dimensions of a composition algebra are1,2,4, and8.[2][3][4]
For consistent terminology, algebras of dimension 1 have been calledunarion, and those of dimension 2binarion.[5]
Every composition algebra is analternative algebra.[3]
Using the doubled form ( _ : _ ):A×A→Kby(a:b)=n(a+b)−n(a)−n(b),{\displaystyle (a:b)=n(a+b)-n(a)-n(b),}then the trace ofais given by (a:1) and the conjugate bya* = (a:1)e –awhere e is the basis element for 1. A series of exercises proves that a composition algebra is always an alternative algebra.[6]
When the fieldKis taken to becomplex numbersCand the quadratic formz2, then four composition algebras overCareCitself, thebicomplex numbers, thebiquaternions(isomorphic to the2×2complexmatrix ringM(2,C)), and thebioctonionsC⊗O, which are also called complex octonions.
The matrix ringM(2,C)has long been an object of interest, first asbiquaternionsbyHamilton(1853), later in the isomorphic matrix form, and especially asPauli algebra.
Thesquaring functionN(x) =x2on thereal numberfield forms the primordial composition algebra.
When the fieldKis taken to be real numbersR, then there are just six other real composition algebras.[3]: 166In two, four, and eight dimensions there are both adivision algebraand asplit algebra:
Every composition algebra has an associatedbilinear formB(x,y) constructed with the norm N and apolarization identity:
The composition of sums of squares was noted by several early authors.Diophantuswas aware of the identity involving the sum of two squares, now called theBrahmagupta–Fibonacci identity, which is also articulated as a property of Euclidean norms of complex numbers when multiplied.Leonhard Eulerdiscussed thefour-square identityin 1748, and it ledW. R. Hamiltonto construct his four-dimensional algebra ofquaternions.[5]: 62In 1848tessarineswere described giving first light to bicomplex numbers.
About 1818 Danish scholar Ferdinand Degen displayed theDegen's eight-square identity, which was later connected with norms of elements of theoctonionalgebra:
In 1919Leonard Dicksonadvanced the study of theHurwitz problemwith a survey of efforts to that date, and by exhibiting the method of doubling the quaternions to obtainCayley numbers. He introduced a newimaginary unite, and for quaternionsqandQwrites a Cayley numberq+Qe. Denoting the quaternion conjugate byq′, the product of two Cayley numbers is[8]
The conjugate of a Cayley number isq'–Qe, and the quadratic form isqq′ +QQ′, obtained by multiplying the number by its conjugate. The doubling method has come to be called theCayley–Dickson construction.
In 1923 the case of real algebras withpositive definite formswas delimited by theHurwitz's theorem (composition algebras).
In 1931Max Zornintroduced a gamma (γ) into the multiplication rule in the Dickson construction to generatesplit-octonions.[9]Adrian Albertalso used the gamma in 1942 when he showed that Dickson doubling could be applied to anyfieldwith thesquaring functionto construct binarion, quaternion, and octonion algebras with their quadratic forms.[10]Nathan Jacobsondescribed theautomorphismsof composition algebras in 1958.[2]
The classical composition algebras overRandCareunital algebras. Composition algebraswithoutamultiplicative identitywere found by H.P. Petersson (Petersson algebras) and Susumu Okubo (Okubo algebras) and others.[11]: 463–81
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Inmathematics,differential algebrais, broadly speaking, the area of mathematics consisting in the study ofdifferential equationsanddifferential operatorsasalgebraic objectsin view of deriving properties of differential equations and operators without computing the solutions, similarly aspolynomial algebrasare used for the study ofalgebraic varieties, which are solution sets ofsystems of polynomial equations.Weyl algebrasandLie algebrasmay be considered as belonging to differential algebra.
More specifically,differential algebrarefers to the theory introduced byJoseph Rittin 1950, in whichdifferential rings,differential fields, anddifferential algebrasarerings,fields, andalgebrasequipped with finitely manyderivations.[1][2][3]
A natural example of a differential field is the field ofrational functionsin one variable over thecomplex numbers,C(t),{\displaystyle \mathbb {C} (t),}where the derivation is differentiation with respect tot.{\displaystyle t.}More generally, every differential equation may be viewed as an element of a differential algebra over the differential field generated by the (known) functions appearing in the equation.
Joseph Rittdeveloped differential algebra because he viewed attempts to reduce systems of differential equations to various canonical forms as an unsatisfactory approach. However, the success of algebraic elimination methods and algebraic manifold theory motivated Ritt to consider a similar approach for differential equations.[4]His efforts led to an initial paperManifolds Of Functions Defined By Systems Of Algebraic Differential Equationsand 2 books,Differential Equations From The Algebraic StandpointandDifferential Algebra.[5][6][2]Ellis Kolchin, Ritt's student, advanced this field and publishedDifferential Algebra And Algebraic Groups.[1]
Aderivation∂{\textstyle \partial }on a ringR{\textstyle R}is afunction∂:R→R{\displaystyle \partial :R\to R\,}such that∂(r1+r2)=∂r1+∂r2{\displaystyle \partial (r_{1}+r_{2})=\partial r_{1}+\partial r_{2}}and
for everyr1{\displaystyle r_{1}}andr2{\displaystyle r_{2}}inR.{\displaystyle R.}
A derivation islinearover the integers since these identities imply∂(0)=∂(1)=0{\displaystyle \partial (0)=\partial (1)=0}and∂(−r)=−∂(r).{\displaystyle \partial (-r)=-\partial (r).}
Adifferential ringis acommutative ringR{\displaystyle R}equipped with one or more derivations that commute pairwise; that is,∂1(∂2(r))=∂2(∂1(r)){\displaystyle \partial _{1}(\partial _{2}(r))=\partial _{2}(\partial _{1}(r))}for every pair of derivations and everyr∈R.{\displaystyle r\in R.}[7]When there is only one derivation one talks often of anordinary differential ring; otherwise, one talks of apartial differential ring.
Adifferential fieldis a differential ring that is also a field. Adifferential algebraA{\displaystyle A}over a differential fieldK{\displaystyle K}is a differential ring that containsK{\displaystyle K}as a subring such that the restriction toK{\displaystyle K}of the derivations ofA{\displaystyle A}equal the derivations ofK.{\displaystyle K.}(A more general definition is given below, which covers the case whereK{\displaystyle K}is not a field, and is essentially equivalent whenK{\displaystyle K}is a field.)
AWitt algebrais a differential ring that contains the fieldQ{\displaystyle \mathbb {Q} }of the rational numbers. Equivalently, this is a differential algebra overQ,{\displaystyle \mathbb {Q} ,}sinceQ{\displaystyle \mathbb {Q} }can be considered as a differential field on which every derivation is thezero function.
Theconstantsof a differential ring are the elementsr{\displaystyle r}such that∂r=0{\displaystyle \partial r=0}for every derivation∂.{\displaystyle \partial .}The constants of a differential ring form asubringand the constants of a differentiable field form a subfield.[8]This meaning of "constant" generalizes the concept of aconstant function, and must not be confused with the common meaning of aconstant.
In the followingidentities,δ{\displaystyle \delta }is a derivation of a differential ringR.{\displaystyle R.}[9]
Aderivation operatororhigher-order derivation[citation needed]is thecompositionof several derivations. As the derivations of a differential ring are supposed to commute, the order of the derivations does not matter, and a derivation operator may be written asδ1e1∘⋯∘δnen,{\displaystyle \delta _{1}^{e_{1}}\circ \cdots \circ \delta _{n}^{e_{n}},}whereδ1,…,δn{\displaystyle \delta _{1},\ldots ,\delta _{n}}are the derivations under consideration,e1,…,en{\displaystyle e_{1},\ldots ,e_{n}}are nonnegative integers, and the exponent of a derivation denotes the number of times this derivation is composed in the operator.
The sumo=e1+⋯+en{\displaystyle o=e_{1}+\cdots +e_{n}}is called theorderof derivation. Ifo=1{\displaystyle o=1}the derivation operator is one of the original derivations. Ifo=0{\displaystyle o=0}, one has theidentity function, which is generally considered as the unique derivation operator of order zero. With these conventions, the derivation operators form afree commutative monoidon the set of derivations under consideration.
Aderivativeof an elementx{\displaystyle x}of a differential ring is the application of a derivation operator tox,{\displaystyle x,}that is, with the above notation,δ1e1∘⋯∘δnen(x).{\displaystyle \delta _{1}^{e_{1}}\circ \cdots \circ \delta _{n}^{e_{n}}(x).}Aproper derivativeis a derivative of positive order.[7]
Adifferential idealI{\displaystyle I}of a differential ringR{\displaystyle R}is anidealof the ringR{\displaystyle R}that isclosed(stable) under the derivations of the ring; that is,∂x∈I,{\textstyle \partial x\in I,}for every derivation∂{\displaystyle \partial }and everyx∈I.{\displaystyle x\in I.}A differential ideal is said to beproperif it is not the whole ring. For avoiding confusion, an ideal that is not a differential ideal is sometimes called analgebraic ideal.
Theradicalof a differential ideal is the same as itsradicalas an algebraic ideal, that is, the set of the ring elements that have a power in the ideal. The radical of a differential ideal is also a differential ideal. Aradicalorperfectdifferential ideal is a differential ideal that equals its radical.[10]A prime differential ideal is a differential ideal that isprimein the usual sense; that is, if a product belongs to the ideal, at least one of the factors belongs to the ideal. A prime differential ideal is always a radical differential ideal.
A discovery of Ritt is that, although the classical theory of algebraic ideals does not work for differential ideals, a large part of it can be extended to radical differential ideals, and this makes them fundamental in differential algebra.
The intersection of any family of differential ideals is a differential ideal, and the intersection of any family of radical differential ideals is a radical differential ideal.[11]It follows that, given a subsetS{\displaystyle S}of a differential ring, there are three ideals generated by it, which are the intersections of, respectively, all algebraic ideals, all differential ideals, and all radical differential ideals that contain it.[11][12]
The algebraic ideal generated byS{\displaystyle S}is the set of finite linear combinations of elements ofS,{\displaystyle S,}and is commonly denoted as(S){\displaystyle (S)}or⟨S⟩.{\displaystyle \langle S\rangle .}
The differential ideal generated byS{\displaystyle S}is the set of the finite linear combinations of elements ofS{\displaystyle S}and of the derivatives of any order of these elements; it is commonly denoted as[S].{\displaystyle [S].}WhenS{\displaystyle S}is finite,[S]{\displaystyle [S]}is generally notfinitely generatedas an algebraic ideal.
The radical differential ideal generated byS{\displaystyle S}is commonly denoted as{S}.{\displaystyle \{S\}.}There is no known way to characterize its element in a similar way as for the two other cases.
A differential polynomial over a differential fieldK{\displaystyle K}is a formalization of the concept ofdifferential equationsuch that the known functions appearing in the equation belong toK,{\displaystyle K,}and the indeterminates are symbols for the unknown functions.
So, letK{\displaystyle K}be a differential field, which is typically (but not necessarily) a field ofrational fractionsK(X)=K(x1,…,xn){\displaystyle K(X)=K(x_{1},\ldots ,x_{n})}(fractions of multivariate polynomials), equipped with derivations∂i{\displaystyle \partial _{i}}such that∂ixi=1{\displaystyle \partial _{i}x_{i}=1}and∂ixj=0{\displaystyle \partial _{i}x_{j}=0}ifi≠j{\displaystyle i\neq j}(the usual partial derivatives).
For defining the ringK{Y}=K{y1,…,yn}{\textstyle K\{Y\}=K\{y_{1},\ldots ,y_{n}\}}of differential polynomials overK{\displaystyle K}with indeterminates inY={y1,…,yn}{\displaystyle Y=\{y_{1},\ldots ,y_{n}\}}with derivations∂1,…,∂n,{\displaystyle \partial _{1},\ldots ,\partial _{n},}one introduces an infinity of new indeterminates of the formΔyi,{\displaystyle \Delta y_{i},}whereΔ{\displaystyle \Delta }is any derivation operator of order higher than1. With this notation,K{Y}{\displaystyle K\{Y\}}is the set of polynomials in all these indeterminates, with the natural derivations (each polynomial involves only a finite number of indeterminates). In particular, ifn=1,{\displaystyle n=1,}one has
Even whenn=1,{\displaystyle n=1,}a ring of differential polynomials is notNoetherian. This makes the theory of this generalization of polynomial rings difficult. However, two facts allow such a generalization.
Firstly, a finite number of differential polynomials involves together a finite number of indeterminates. Its follows that every property of polynomials that involves a finite number of polynomials remains true for differential polynomials. In particular,greatest common divisorsexist, and a ring of differential polynomials is aunique factorization domain.
The second fact is that, if the fieldK{\displaystyle K}contains the field of rational numbers, the rings of differential polynomials overK{\displaystyle K}satisfy theascending chain conditionon radical differential ideals. This Ritt’s theorem is implied by its generalization, sometimes called theRitt-Raudenbush basis theoremwhich asserts that ifR{\displaystyle R}is aRitt Algebra(that, is a differential ring containing the field of rational numbers),[13]that satisfies the ascending chain condition on radical differential ideals, then the ring of differential polynomialsR{y}{\displaystyle R\{y\}}satisfies the same property (one passes from the univariate to the multivariate case by applying the theorem iteratively).[14][15]
This Noetherian property implies that, in a ring of differential polynomials, every radical differential idealIis finitely generated as a radical differential ideal; this means that there exists a finite setSof differential polynomials such thatIis the smallest radical differential ideal containingS.[16]This allows representing a radical differential ideal by such a finite set of generators, and computing with these ideals. However, some usual computations of the algebraic case cannot be extended. In particular no algorithm is known for testing membership of an element in a radical differential ideal or the equality of two radical differential ideals.
Another consequence of the Noetherian property is that a radical differential ideal can be uniquely expressed as the intersection of a finite number of prime differential ideals, calledessential prime componentsof the ideal.[17]
Elimination methodsare algorithms that preferentially eliminate a specified set of derivatives from a set of differential equations, commonly done to better understand and solve sets of differential equations.
Categories of elimination methods includecharacteristic setmethods, differentialGröbner basesmethods andresultantbased methods.[1][18][19][20][21][22][23]
Common operations used in elimination algorithms include 1) ranking derivatives, polynomials, and polynomial sets, 2) identifying a polynomial's leading derivative, initial and separant, 3) polynomial reduction, and 4) creating special polynomial sets.
Therankingof derivatives is atotal orderand anadmisible order, defined as:[24][25][26]
Each derivative has an integer tuple, and amonomial orderranks the derivative by ranking the derivative's integer tuple. The integer tuple identifies the differential indeterminate, the derivative's multi-index and may identify the derivative's order. Types of ranking include:[27]
In this example, the integer tuple identifies the differential indeterminate and derivative's multi-index, andlexicographic monomial order,≥lex{\textstyle \geq _{\text{lex}}}, determines the derivative's rank.[28]
This is the standard polynomial form:p=ad⋅upd+ad−1⋅upd−1+⋯+a1⋅up+a0{\displaystyle p=a_{d}\cdot u_{p}^{d}+a_{d-1}\cdot u_{p}^{d-1}+\cdots +a_{1}\cdot u_{p}+a_{0}}.[24][28]
Separant set isSA={Sp∣p∈A}{\displaystyle S_{A}=\{S_{p}\mid p\in A\}}, initial set isIA={Ip∣p∈A}{\displaystyle I_{A}=\{I_{p}\mid p\in A\}}and combined set isHA=SA∪IA{\textstyle H_{A}=S_{A}\cup I_{A}}.[29]
Partially reduced(partial normal form) polynomialq{\textstyle q}with respect to polynomialp{\textstyle p}indicates these polynomials are non-ground field elements,p,q∈K{Y}∖K{\textstyle p,q\in {\mathcal {K}}\{Y\}\setminus {\mathcal {K}}}, andq{\displaystyle q}contains no proper derivative ofup{\displaystyle u_{p}}.[30][31][29]
Partially reduced polynomialq{\textstyle q}with respect to polynomialp{\textstyle p}becomesreduced(normal form) polynomialq{\textstyle q}with respect top{\textstyle p}if the degree ofup{\textstyle u_{p}}inq{\textstyle q}is less than the degree ofup{\textstyle u_{p}}inp{\textstyle p}.[30][31][29]
Anautoreducedpolynomial set has every polynomial reduced with respect to every other polynomial of the set. Every autoreduced set is finite. An autoreduced set istriangularmeaning each polynomial element has a distinct leading derivative.[32][30]
Ritt's reduction algorithmidentifies integersiAk,sAk{\textstyle i_{A_{k}},s_{A_{k}}}and transforms a differential polynomialf{\textstyle f}usingpseudodivisionto a lower or equally ranked remainder polynomialfred{\textstyle f_{red}}that is reduced with respect to the autoreduced polynomial setA{\textstyle A}. The algorithm's first step partially reduces the input polynomial and the algorithm's second step fully reduces the polynomial. The formula for reduction is:[30]
SetA{\textstyle A}is adifferential chainif the rank of the leading derivatives isuA1<⋯<uAm{\textstyle u_{A_{1}}<\dots <u_{A_{m}}}and∀i,Ai{\textstyle \forall i,\ A_{i}}is reduced with respect toAi+1{\textstyle A_{i+1}}[33]
Autoreduced setsA{\textstyle A}andB{\textstyle B}each contain ranked polynomial elements. This procedure ranks two autoreduced sets by comparing pairs of identically indexed
polynomials from both autoreduced sets.[34]
Acharacteristic setC{\textstyle C}is thelowest rankedautoreduced subset among all the ideal's autoreduced subsets whose subset polynomial separants are non-members of the idealI{\textstyle {\mathcal {I}}}.[35]
Thedelta polynomialapplies to polynomial pairp,q{\textstyle p,q}whose leaders share a common derivative,θαup=θβuq{\textstyle \theta _{\alpha }u_{p}=\theta _{\beta }u_{q}}. The least common derivative operator for the polynomial pair's leading derivatives isθpq{\textstyle \theta _{pq}}, and the delta polynomial is:[36][37]
Acoherent setis a polynomial set that reduces its delta polynomial pairs to zero.[36][37]
Aregular systemΩ{\textstyle \Omega }contains a autoreduced and coherent set of differential equationsA{\textstyle A}and a inequation setHΩ⊇HA{\textstyle H_{\Omega }\supseteq H_{A}}with setHΩ{\textstyle H_{\Omega }}reduced with respect to the equation set.[37]
Regular differential idealIdif{\textstyle {\mathcal {I}}_{\text{dif}}}and regular algebraic idealIalg{\textstyle {\mathcal {I}}_{\text{alg}}}aresaturation idealsthat arise from a regular system.[37]Lazard's lemmastates that the regular differential and regular algebraic ideals are radical ideals.[38]
TheRosenfeld–Gröbner algorithmdecomposes the radical differential ideal as a finite intersection of regular radical differential ideals. These regular differential radical ideals, represented by characteristic sets, are not necessarily prime ideals and the representation is not necessarilyminimal.[39]
Themembership problemis to determine if a differential polynomialp{\textstyle p}is a member of an ideal generated from a set of differential polynomialsS{\textstyle S}. The Rosenfeld–Gröbner algorithm generates sets of Gröbner bases. The algorithm determines that a polynomial is a member of the ideal if and only if the partially reduced remainder polynomial is a member of the algebraic ideal generated by the Gröbner bases.[40]
The Rosenfeld–Gröbner algorithm facilitates creatingTaylor seriesexpansions of solutions to the differential equations.[41]
Example 1:(Mer(f(y),∂y)){\textstyle (\operatorname {Mer} (\operatorname {f} (y),\partial _{y}))}is the differentialmeromorphic functionfield with a singlestandard derivation.
Example 2:(C{y},p(y)⋅∂y){\textstyle (\mathbb {C} \{y\},p(y)\cdot \partial _{y})}is a differential field with alinear differential operatoras the derivation, for any polynomialp(y){\displaystyle p(y)}.
DefineEa(p(y))=p(y+a){\textstyle E^{a}(p(y))=p(y+a)}asshift operatorEa{\textstyle E^{a}}for polynomialp(y){\textstyle p(y)}.
A shift-invariant operatorT{\textstyle T}commutes with the shift operator:Ea∘T=T∘Ea{\textstyle E^{a}\circ T=T\circ E^{a}}.
ThePincherle derivative, a derivation of shift-invariant operatorT{\textstyle T}, isT′=T∘y−y∘T{\textstyle T^{\prime }=T\circ y-y\circ T}.[42]
Ring of integers is(Z.δ){\displaystyle (\mathbb {Z} .\delta )}, and every integer is a constant.
Field of rational numbers is(Q.δ){\displaystyle (\mathbb {Q} .\delta )}, and every rational number is a constant.
Constants form thesubring of constants(C,∂y)⊂(C{y},∂y){\textstyle (\mathbb {C} ,\partial _{y})\subset (\mathbb {C} \{y\},\partial _{y})}.[43]
Elementexp(y){\textstyle \exp(y)}simply generates differential ideal[exp(y)]{\textstyle [\exp(y)]}in the differential ring(C{y,exp(y)},∂y){\textstyle (\mathbb {C} \{y,\exp(y)\},\partial _{y})}.[44]
Any ring with identity is aZ-{\textstyle \operatorname {{\mathcal {Z}}-} }algebra.[45]Thus a differential ring is aZ-{\textstyle \operatorname {{\mathcal {Z}}-} }algebra.
If ringR{\textstyle {\mathcal {R}}}is a subring of the center of unital ringM{\textstyle {\mathcal {M}}}, thenM{\textstyle {\mathcal {M}}}is anR-{\textstyle \operatorname {{\mathcal {R}}-} }algebra.[45]Thus, a differential ring is an algebra over its differential subring. This is thenatural structureof an algebra over its subring.[30]
Ring(Q{y,z},∂y){\textstyle (\mathbb {Q} \{y,z\},\partial _{y})}has irreducible polynomials,p{\textstyle p}(normal, squarefree) andq{\textstyle q}(special, ideal generator).
Ring(Q{y1,y2},δ){\textstyle (\mathbb {Q} \{y_{1},y_{2}\},\delta )}has derivativesδ(y1)=y1′{\textstyle \delta (y_{1})=y_{1}^{\prime }}andδ(y2)=y2′{\textstyle \delta (y_{2})=y_{2}^{\prime }}
Theleading derivatives, andinitialsare:
Symbolic integration uses algorithms involving polynomials and their derivatives such as Hermite reduction, Czichowski algorithm, Lazard-Rioboo-Trager algorithm, Horowitz-Ostrogradsky algorithm, squarefree factorization and splitting factorization to special and normal polynomials.[46]
Differential algebra can determine if a set of differential polynomial equations has a solution. A total order ranking may identify algebraic constraints. An elimination ranking may determine if one or a selected group of independent variables can express the differential equations. Using triangular decomposition and elimination order, it may be possible to solve the differential equations one differential indeterminate at a time in a step-wise method. Another approach is to create a class of differential equations with a known solution form; matching a differential equation to its class identifies the equation's solution. Methods are available to facilitate the numerical integration of adifferential-algebraicsystem of equations.[47]
In a study of non-linear dynamical systems withchaos, researchers used differential elimination to reduce differential equations to ordinary differential equations involving a single state variable. They were successful in most cases, and this facilitated developing approximate solutions, efficiently evaluating chaos, and constructingLyapunov functions.[48]Researchers have applied differential elimination to understandingcellular biology,compartmental biochemical models,parameterestimation andquasi-steady stateapproximation (QSSA) for biochemical reactions.[49][50]Using differential Gröbner bases, researchers have investigated non-classicalsymmetryproperties ofnon-linear differential equations.[51]Other applications include control theory, model theory, and algebraic geometry.[52][16][53]Differential algebra also applies to differential-difference equations.[54]
AZ-graded{\textstyle \operatorname {\mathbb {Z} -graded} }vector spaceV∙{\textstyle V_{\bullet }}is a collection of vector spacesVm{\textstyle V_{m}}with integerdegree|v|=m{\textstyle |v|=m}forv∈Vm{\textstyle v\in V_{m}}. Adirect sumcan represent this graded vector space:[55]
Adifferential graded vector spaceorchain complex, is a graded vector spaceV∙{\textstyle V_{\bullet }}with adifferential maporboundary mapdm:Vm→Vm−1{\textstyle d_{m}:V_{m}\to V_{m-1}}withdm∘dm+1=0{\displaystyle d_{m}\circ d_{m+1}=0}.[56]
Acochain complexis a graded vector spaceV∙{\textstyle V^{\bullet }}with adifferential maporcoboundary mapdm:Vm→Vm+1{\textstyle d_{m}:V_{m}\to V_{m+1}}withdm+1∘dm=0{\displaystyle d_{m+1}\circ d_{m}=0}.[56]
Adifferential graded algebrais a graded algebraA{\textstyle A}with a linear derivationd:A→A{\textstyle d:A\to A}withd∘d=0{\displaystyle d\circ d=0}that follows the graded Leibniz product rule.[57]
ALie algebrais a finite-dimensional real or complex vector spaceg{\textstyle {\mathcal {g}}}with abilinearbracket operator[,]:g×g→g{\textstyle [,]:{\mathcal {g}}\times {\mathcal {g}}\to {\mathcal {g}}}withSkew symmetryand theJacobi identityproperty.[58]
for allX,Y,Z∈g{\displaystyle X,Y,Z\in {\mathcal {g}}}.
Theadjointoperator,adX(Y)=[Y,X]{\textstyle \operatorname {ad_{X}} (Y)=[Y,X]}is aderivation of the bracketbecause the adjoint's effect on the binary bracket operation is analogous to the derivation's effect on the binary product operation. This is theinner derivationdetermined byX{\textstyle X}.[59][60]
Theuniversal enveloping algebraU(g){\textstyle U({\mathcal {g}})}of Lie algebrag{\textstyle {\mathcal {g}}}is a maximal associative algebra with identity, generated by Lie algebra elementsg{\textstyle {\mathcal {g}}}and containing products defined by the bracket operation. Maximal means that a linear homomorphism maps the universal algebra to any other algebra that otherwise has these properties. The adjoint operator is a derivation following the Leibniz product rule.[61]
for allX,Y,Z∈U(g){\displaystyle X,Y,Z\in U({\mathcal {g}})}.
TheWeyl algebrais an algebraAn(K){\textstyle A_{n}(K)}over a ringK[p1,q1,…,pn,qn]{\textstyle K[p_{1},q_{1},\dots ,p_{n},q_{n}]}with a specific noncommutative product:[62]
All other indeterminate products are commutative fori,j∈{1,…,n}{\textstyle i,j\in \{1,\dots ,n\}}:
A Weyl algebra can represent the derivations for a commutative ring's polynomialsf∈K[y1,…,yn]{\textstyle f\in K[y_{1},\ldots ,y_{n}]}. The Weyl algebra's elements areendomorphisms, the elementsp1,…,pn{\textstyle p_{1},\ldots ,p_{n}}function as standard derivations, and map compositions generatelinear differential operators.D-moduleis a related approach for understanding differential operators. The endomorphisms are:[62]
The associative, possibly noncommutative ringA{\textstyle A}has derivationd:A→A{\textstyle d:A\to A}.[63]
Thepseudo-differential operatorringA((∂−1)){\textstyle A((\partial ^{-1}))}is a leftA-module{\textstyle \operatorname {A-module} }containing ring elementsL{\textstyle L}:[63][64][65]
The derivative operator isd(a)=∂∘a−a∘∂{\textstyle d(a)=\partial \circ a-a\circ \partial }.[63]
Thebinomial coefficientis(ik){\displaystyle {\Bigl (}{i \atop k}{\Bigr )}}.
Pseudo-differential operator multiplication is:[63]
TheRitt problemasks is there an algorithm that determines if one prime differential ideal contains a second prime differential ideal when characteristic sets identify both ideals.[66]
TheKolchin catenary conjecturestates given ad>0{\textstyle d>0}dimensional irreducible differential algebraic varietyV{\textstyle V}and an arbitrary pointp∈V{\textstyle p\in V}, a long gap chain of irreducible differential algebraic subvarieties occurs fromp{\textstyle p}to V.[67]
TheJacobi bound conjectureconcerns the upper bound for the order of an differential variety's irreducible component. The polynomial's orders determine a Jacobi number, and the conjecture is the Jacobi number determines this bound.[68]
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Ring homomorphisms
Algebraic structures
Related structures
Algebraic number theory
Noncommutative algebraic geometry
Free algebra
Clifford algebra
Inmathematics, especially in the area ofabstract algebraknown asring theory, afree algebrais the noncommutative analogue of apolynomial ringsince its elements may be described as "polynomials" with non-commuting variables. Likewise, thepolynomial ringmay be regarded as afree commutative algebra.
ForRacommutative ring, the free (associative,unital)algebraonnindeterminates{X1,...,Xn} is thefreeR-modulewith a basis consisting of allwordsover the alphabet {X1,...,Xn} (including the empty word, which is the unit of the free algebra). ThisR-module becomes anR-algebraby defining a multiplication as follows: the product of two basis elements is theconcatenationof the corresponding words:
and the product of two arbitraryR-module elements is thus uniquely determined (because the multiplication in anR-algebra must beR-bilinear). ThisR-algebra is denotedR⟨X1,...,Xn⟩. This construction can easily be generalized to an arbitrary setXof indeterminates.
In short, for an arbitrary setX={Xi;i∈I}{\displaystyle X=\{X_{i}\,;\;i\in I\}}, thefree (associative,unital)R-algebraonXis
with theR-bilinear multiplication that is concatenation on words, whereX* denotes thefree monoidonX(i.e. words on the lettersXi),⊕{\displaystyle \oplus }denotes the externaldirect sum, andRwdenotes thefreeR-moduleon 1 element, the wordw.
For example, inR⟨X1,X2,X3,X4⟩, for scalarsα, β, γ, δ∈R, a concrete example of a product of two elements is
The non-commutative polynomial ring may be identified with themonoid ringoverRof thefree monoidof all finite words in theXi.
Since the words over the alphabet {X1, ...,Xn} form a basis ofR⟨X1,...,Xn⟩, it is clear that any element ofR⟨X1, ...,Xn⟩ can be written uniquely in the form:
whereai1,i2,...,ik{\displaystyle a_{i_{1},i_{2},...,i_{k}}}are elements ofRand all but finitely many of these elements are zero. This explains why the elements ofR⟨X1,...,Xn⟩ are often denoted as "non-commutative polynomials" in the "variables" (or "indeterminates")X1,...,Xn; the elementsai1,i2,...,ik{\displaystyle a_{i_{1},i_{2},...,i_{k}}}are said to be "coefficients" of these polynomials, and theR-algebraR⟨X1,...,Xn⟩ is called the "non-commutative polynomial algebra overRinnindeterminates". Note that unlike in an actualpolynomial ring, the variables do notcommute. For example,X1X2does not equalX2X1.
More generally, one can construct the free algebraR⟨E⟩ on any setEofgenerators. Since rings may be regarded asZ-algebras, afree ringonEcan be defined as the free algebraZ⟨E⟩.
Over afield, the free algebra onnindeterminates can be constructed as thetensor algebraon ann-dimensionalvector space. For a more general coefficient ring, the same construction works if we take thefree moduleonngenerators.
The construction of the free algebra onEisfunctorialin nature and satisfies an appropriateuniversal property. The free algebra functor isleft adjointto theforgetful functorfrom the category ofR-algebras to thecategory of sets.
Free algebras overdivision ringsarefree ideal rings.
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Inmathematics, ageometric algebra(also known as aClifford algebra) is analgebrathat can represent and manipulate geometrical objects such asvectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects calledmultivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division (though generally not by all elements) and addition of objects of different dimensions.
The geometric product was first briefly mentioned byHermann Grassmann,[1]who was chiefly interested in developing the closely relatedexterior algebra. In 1878,William Kingdon Cliffordgreatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). Clifford defined the Clifford algebra and its product as a unification of theGrassmann algebraand Hamilton'squaternion algebra. Adding thedualof the Grassmann exterior product allows the use of theGrassmann–Cayley algebra. In the late 1990s,plane-based geometric algebraandconformal geometric algebra(CGA) respectively provided a framework for euclidean geometry andclassical geometries.[2]In practice, these and several derived operations allow a correspondence of elements,subspacesand operations of the algebra with geometric interpretations. For several decades, geometric algebras went somewhat ignored, greatly eclipsed by thevector calculusthen newly developed to describe electromagnetism. The term "geometric algebra" was repopularized in the 1960s byDavid Hestenes, who advocated its importance to relativistic physics.[3]
The scalars and vectors have their usual interpretation and make up distinct subspaces of a geometric algebra.Bivectorsprovide a more natural representation of thepseudovectorquantities of 3D vector calculus that are derived as across product, such as oriented area, oriented angle of rotation, torque, angular momentum and themagnetic field. Atrivectorcan represent an oriented volume, and so on. An element called ablademay be used to represent a subspace andorthogonal projectionsonto that subspace. Rotations and reflections are represented as elements. Unlike a vector algebra, a geometric algebra naturally accommodates any number of dimensions and any quadratic form such as inrelativity.
Examples of geometric algebras applied in physics include thespacetime algebra(and the less commonalgebra of physical space).Geometric calculus, an extension of GA that incorporatesdifferentiationandintegration, can be used to formulate other theories such ascomplex analysisanddifferential geometry, e.g. by using the Clifford algebra instead ofdifferential forms. Geometric algebra has been advocated, most notably byDavid Hestenes[4]andChris Doran,[5]as the preferred mathematical framework forphysics. Proponents claim that it provides compact and intuitive descriptions in many areas includingclassicalandquantum mechanics,electromagnetic theory, andrelativity.[6]GA has also found use as a computational tool incomputer graphics[7]androbotics.
There are a number of different ways to define a geometric algebra. Hestenes's original approach was axiomatic,[8]"full of geometric significance" and equivalent to the universal[a]Clifford algebra.[9]Given a finite-dimensional vector spaceV{\displaystyle V}over afieldF{\displaystyle F}with a symmetric bilinear form (theinner product,[b]e.g., the Euclidean orLorentzian metric)g:V×V→F{\displaystyle g:V\times V\to F}, thegeometric algebraof thequadratic space(V,g){\displaystyle (V,g)}is theClifford algebraCl(V,g){\displaystyle \operatorname {Cl} (V,g)}, an element of which is called a multivector. The Clifford algebra is commonly defined as aquotient algebraof thetensor algebra, though this definition is abstract, so the following definition is presented without requiringabstract algebra.
To cover degenerate symmetric bilinear forms, the last condition must be modified.[c]It can be shown that these conditions uniquely characterize the geometric product.
For the remainder of this article, only therealcase,F=R{\displaystyle F=\mathbb {R} }, will be considered. The notationG(p,q){\displaystyle {\mathcal {G}}(p,q)}(respectivelyG(p,q,r){\displaystyle {\mathcal {G}}(p,q,r)}) will be used to denote a geometric algebra for which the bilinear formg{\displaystyle g}has thesignature(p,q){\displaystyle (p,q)}(respectively(p,q,r){\displaystyle (p,q,r)}).
The product in the algebra is called thegeometric product, and the product in the contained exterior algebra is called theexterior product(frequently called thewedge productor theouter product[d]). It is standard to denote these respectively by juxtaposition (i.e., suppressing any explicit multiplication symbol) and the symbol∧{\displaystyle \wedge }.
The above definition of the geometric algebra is still somewhat abstract, so we summarize the properties of the geometric product here. For multivectorsA,B,C∈G(p,q){\displaystyle A,B,C\in {\mathcal {G}}(p,q)}:
The exterior product has the same properties, except that the last property above is replaced bya∧a=0{\displaystyle a\wedge a=0}fora∈V{\displaystyle a\in V}.
Note that in the last property above, the real numberg(a,a){\displaystyle g(a,a)}need not be nonnegative ifg{\displaystyle g}is not positive-definite. An important property of the geometric product is the existence of elements that have a multiplicative inverse. For a vectora{\displaystyle a}, ifa2≠0{\displaystyle a^{2}\neq 0}thena−1{\displaystyle a^{-1}}exists and is equal tog(a,a)−1a{\displaystyle g(a,a)^{-1}a}. A nonzero element of the algebra does not necessarily have a multiplicative inverse. For example, ifu{\displaystyle u}is a vector inV{\displaystyle V}such thatu2=1{\displaystyle u^{2}=1}, the element12(1+u){\displaystyle \textstyle {\frac {1}{2}}(1+u)}is both a nontrivialidempotent elementand a nonzerozero divisor, and thus has no inverse.[e]
It is usual to identifyR{\displaystyle \mathbb {R} }andV{\displaystyle V}with their images under the naturalembeddingsR→G(p,q){\displaystyle \mathbb {R} \to {\mathcal {G}}(p,q)}andV→G(p,q){\displaystyle V\to {\mathcal {G}}(p,q)}. In this article, this identification is assumed. Throughout, the termsscalarandvectorrefer to elements ofR{\displaystyle \mathbb {R} }andV{\displaystyle V}respectively (and of their images under this embedding).
For vectorsa{\displaystyle a}andb{\displaystyle b}, we may write the geometric product of any two vectorsa{\displaystyle a}andb{\displaystyle b}as the sum of a symmetric product and an antisymmetric product:
Thus we can define theinner productof vectors as
so that the symmetric product can be written as
Conversely,g{\displaystyle g}is completely determined by the algebra. The antisymmetric part is the exterior product of the two vectors, the product of the containedexterior algebra:
Then by simple addition:
The inner and exterior products are associated with familiar concepts from standard vector algebra. Geometrically,a{\displaystyle a}andb{\displaystyle b}areparallelif their geometric product is equal to their inner product, whereasa{\displaystyle a}andb{\displaystyle b}areperpendicularif their geometric product is equal to their exterior product. In a geometric algebra for which the square of any nonzero vector is positive, the inner product of two vectors can be identified with thedot productof standard vector algebra. The exterior product of two vectors can be identified with thesigned areaenclosed by aparallelogramthe sides of which are the vectors. Thecross productof two vectors in3{\displaystyle 3}dimensions with positive-definite quadratic form is closely related to their exterior product.
Most instances of geometric algebras of interest have a nondegenerate quadratic form. If the quadratic form is fullydegenerate, the inner product of any two vectors is always zero, and the geometric algebra is then simply an exterior algebra. Unless otherwise stated, this article will treat only nondegenerate geometric algebras.
The exterior product is naturally extended as an associative bilinear binary operator between any two elements of the algebra, satisfying the identities
where the sum is over all permutations of the indices, withsgn(σ){\displaystyle \operatorname {sgn} (\sigma )}thesign of the permutation, andai{\displaystyle a_{i}}are vectors (not general elements of the algebra). Since every element of the algebra can be expressed as the sum of products of this form, this defines the exterior product for every pair of elements of the algebra. It follows from the definition that the exterior product forms analternating algebra.
The equivalent structure equation for Clifford algebra is[16][17]
wherePf(A){\displaystyle \operatorname {Pf} (A)}is thePfaffianofA{\displaystyle A}andC=(n2i){\textstyle {\mathcal {C}}={\binom {n}{2i}}}providescombinations,μ{\displaystyle \mu }, ofn{\displaystyle n}indices divided into2i{\displaystyle 2i}andn−2i{\displaystyle n-2i}parts andk{\displaystyle k}is theparityof thecombination.
The Pfaffian provides a metric for the exterior algebra and, as pointed out by Claude Chevalley, Clifford algebra reduces to the exterior algebra with a zero quadratic form.[18]The role the Pfaffian plays can be understood from a geometric viewpoint by developing Clifford algebra fromsimplices.[19]This derivation provides a better connection betweenPascal's triangleandsimplicesbecause it provides an interpretation of the first column of ones.
A multivector that is the exterior product ofr{\displaystyle r}linearly independent vectors is called ablade, and is said to be of grader{\displaystyle r}.[f]A multivector that is the sum of blades of grader{\displaystyle r}is called a (homogeneous) multivector of grader{\displaystyle r}. From the axioms, with closure, every multivector of the geometric algebra is a sum of blades.
Consider a set ofr{\displaystyle r}linearly independent vectors{a1,…,ar}{\displaystyle \{a_{1},\ldots ,a_{r}\}}spanning anr{\displaystyle r}-dimensional subspace of the vector space. With these, we can define a realsymmetric matrix(in the same way as aGramian matrix)
By thespectral theorem,A{\displaystyle \mathbf {A} }can be diagonalized todiagonal matrixD{\displaystyle \mathbf {D} }by anorthogonal matrixO{\displaystyle \mathbf {O} }via
Define a new set of vectors{e1,…,er}{\displaystyle \{e_{1},\ldots ,e_{r}\}}, known as orthogonal basis vectors, to be those transformed by the orthogonal matrix:
Since orthogonal transformations preserve inner products, it follows thatei⋅ej=[D]ij{\displaystyle e_{i}\cdot e_{j}=[\mathbf {D} ]_{ij}}and thus the{e1,…,er}{\displaystyle \{e_{1},\ldots ,e_{r}\}}are perpendicular. In other words, the geometric product of two distinct vectorsei≠ej{\displaystyle e_{i}\neq e_{j}}is completely specified by their exterior product, or more generally
Therefore, every blade of grader{\displaystyle r}can be written as the exterior product ofr{\displaystyle r}vectors. More generally, if a degenerate geometric algebra is allowed, then the orthogonal matrix is replaced by ablock matrixthat is orthogonal in the nondegenerate block, and the diagonal matrix has zero-valued entries along the degenerate dimensions. If the new vectors of the nondegenerate subspace arenormalizedaccording to
then these normalized vectors must square to+1{\displaystyle +1}or−1{\displaystyle -1}. BySylvester's law of inertia, the total number of+1{\displaystyle +1}and the total number of−1{\displaystyle -1}s along the diagonal matrix is invariant. By extension, the total numberp{\displaystyle p}of these vectors that square to+1{\displaystyle +1}and the total numberq{\displaystyle q}that square to−1{\displaystyle -1}is invariant. (The total number of basis vectors that square to zero is also invariant, and may be nonzero if the degenerate case is allowed.) We denote this algebraG(p,q){\displaystyle {\mathcal {G}}(p,q)}. For example,G(3,0){\displaystyle {\mathcal {G}}(3,0)}models three-dimensionalEuclidean space,G(1,3){\displaystyle {\mathcal {G}}(1,3)}relativisticspacetimeandG(4,1){\displaystyle {\mathcal {G}}(4,1)}aconformal geometric algebraof a three-dimensional space.
The set of all possible products ofn{\displaystyle n}orthogonal basis vectors with indices in increasing order, including1{\displaystyle 1}as the empty product, forms a basis for the entire geometric algebra (an analogue of thePBW theorem). For example, the following is a basis for the geometric algebraG(3,0){\displaystyle {\mathcal {G}}(3,0)}:
A basis formed this way is called astandard basisfor the geometric algebra, and any other orthogonal basis forV{\displaystyle V}will produce another standard basis. Each standard basis consists of2n{\displaystyle 2^{n}}elements. Every multivector of the geometric algebra can be expressed as a linear combination of the standard basis elements. If the standard basis elements are{Bi∣i∈S}{\displaystyle \{B_{i}\mid i\in S\}}withS{\displaystyle S}being an index set, then the geometric product of any two multivectors is
The terminology "k{\displaystyle k}-vector" is often encountered to describe multivectors containing elements of only one grade. In higher dimensional space, some such multivectors are not blades (cannot be factored into the exterior product ofk{\displaystyle k}vectors). By way of example,e1∧e2+e3∧e4{\displaystyle e_{1}\wedge e_{2}+e_{3}\wedge e_{4}}inG(4,0){\displaystyle {\mathcal {G}}(4,0)}cannot be factored; typically, however, such elements of the algebra do not yield to geometric interpretation as objects, although they may represent geometric quantities such as rotations. Only0{\displaystyle 0}-,1{\displaystyle 1}-,(n−1){\displaystyle (n-1)}- andn{\displaystyle n}-vectors are always blades inn{\displaystyle n}-space.
Ak{\displaystyle k}-versor is a multivector that can be expressed as the geometric product ofk{\displaystyle k}invertible vectors.[g][21]Unit quaternions (originally called versors by Hamilton) may be identified with rotors in 3D space in much the same way as real 2D rotors subsume complex numbers; for the details refer to Dorst.[22]
Some authors use the term "versor product" to refer to the frequently occurring case where an operand is "sandwiched" between operators. The descriptions for rotations and reflections, including their outermorphisms, are examples of such sandwiching. These outermorphisms have a particularly simple algebraic form.[h]Specifically, a mapping of vectors of the form
Since both operators and operand are versors there is potential for alternative examples such as rotating a rotor or reflecting a spinor always provided that some geometrical or physical significance can be attached to such operations.
By theCartan–Dieudonné theoremwe have that every isometry can be given as reflections in hyperplanes and since composed reflections provide rotations then we have that orthogonal transformations are versors.
In group terms, for a real, non-degenerateG(p,q){\displaystyle {\mathcal {G}}(p,q)}, having identified the groupG×{\displaystyle {\mathcal {G}}^{\times }}as the group of all invertible elements ofG{\displaystyle {\mathcal {G}}}, Lundholm gives a proof that the "versor group"{v1v2⋯vk∈G∣vi∈V×}{\displaystyle \{v_{1}v_{2}\cdots v_{k}\in {\mathcal {G}}\mid v_{i}\in V^{\times }\}}(the set of invertible versors) is equal to the Lipschitz groupΓ{\displaystyle \Gamma }(a.k.a.Clifford group, although Lundholm deprecates this usage).[23]
We denote the grade involution asS^{\displaystyle {\widehat {S}}}and reversion asS~{\displaystyle {\widetilde {S}}}.
Although the Lipschitz group (defined as{S∈G×∣S^VS−1⊆V}{\displaystyle \{S\in {\mathcal {G}}^{\times }\mid {\widehat {S}}VS^{-1}\subseteq V\}}) and the versor group (defined as{∏i=0kvi∣vi∈V×,k∈N}{\displaystyle \textstyle \{\prod _{i=0}^{k}v_{i}\mid v_{i}\in V^{\times },k\in \mathbb {N} \}}) have divergent definitions, they are the same group. Lundholm defines thePin{\displaystyle \operatorname {Pin} },Spin{\displaystyle \operatorname {Spin} }, andSpin+{\displaystyle \operatorname {Spin} ^{+}}subgroups of the Lipschitz group.[24]
Multiple analyses of spinors use GA as a representation.[25]
AZ{\displaystyle \mathbb {Z} }-graded vector spacestructure can be established on a geometric algebra by use of the exterior product that is naturally induced by the geometric product.
Since the geometric product and the exterior product are equal on orthogonal vectors, this grading can be conveniently constructed by using an orthogonal basis{e1,…,en}{\displaystyle \{e_{1},\ldots ,e_{n}\}}.
Elements of the geometric algebra that are scalar multiples of1{\displaystyle 1}are of grade0{\displaystyle 0}and are calledscalars. Elements that are in the span of{e1,…,en}{\displaystyle \{e_{1},\ldots ,e_{n}\}}are of grade1{\displaystyle 1}and are the ordinary vectors. Elements in the span of{eiej∣1≤i<j≤n}{\displaystyle \{e_{i}e_{j}\mid 1\leq i<j\leq n\}}are of grade2{\displaystyle 2}and are the bivectors. This terminology continues through to the last grade ofn{\displaystyle n}-vectors. Alternatively,n{\displaystyle n}-vectors are calledpseudoscalars,(n−1){\displaystyle (n-1)}-vectors are called pseudovectors, etc. Many of the elements of the algebra are not graded by this scheme since they are sums of elements of differing grade. Such elements are said to be ofmixed grade. The grading of multivectors is independent of the basis chosen originally.
This is a grading as a vector space, but not as an algebra. Because the product of anr{\displaystyle r}-blade and ans{\displaystyle s}-blade is contained in the span of0{\displaystyle 0}throughr+s{\displaystyle r+s}-blades, the geometric algebra is afiltered algebra.
A multivectorA{\displaystyle A}may be decomposed with thegrade-projection operator⟨A⟩r{\displaystyle \langle A\rangle _{r}}, which outputs the grade-r{\displaystyle r}portion ofA{\displaystyle A}. As a result:
As an example, the geometric product of two vectorsab=a⋅b+a∧b=⟨ab⟩0+⟨ab⟩2{\displaystyle ab=a\cdot b+a\wedge b=\langle ab\rangle _{0}+\langle ab\rangle _{2}}since⟨ab⟩0=a⋅b{\displaystyle \langle ab\rangle _{0}=a\cdot b}and⟨ab⟩2=a∧b{\displaystyle \langle ab\rangle _{2}=a\wedge b}and⟨ab⟩i=0{\displaystyle \langle ab\rangle _{i}=0}, fori{\displaystyle i}other than0{\displaystyle 0}and2{\displaystyle 2}.
A multivectorA{\displaystyle A}may also be decomposed into even and odd components, which may respectively be expressed as the sum of the even and the sum of the odd grade components above:
This is the result of forgetting structure from aZ{\displaystyle \mathrm {Z} }-graded vector spacetoZ2{\displaystyle \mathrm {Z} _{2}}-graded vector space. The geometric product respects this coarser grading. Thus in addition to being aZ2{\displaystyle \mathrm {Z} _{2}}-graded vector space, the geometric algebra is aZ2{\displaystyle \mathrm {Z} _{2}}-graded algebra,a.k.a.asuperalgebra.
Restricting to the even part, the product of two even elements is also even. This means that the even multivectors defines aneven subalgebra. The even subalgebra of ann{\displaystyle n}-dimensional geometric algebra isalgebra-isomorphic(without preserving either filtration or grading) to a full geometric algebra of(n−1){\displaystyle (n-1)}dimensions. Examples includeG[0](2,0)≅G(0,1){\displaystyle {\mathcal {G}}^{[0]}(2,0)\cong {\mathcal {G}}(0,1)}andG[0](1,3)≅G(3,0){\displaystyle {\mathcal {G}}^{[0]}(1,3)\cong {\mathcal {G}}(3,0)}.
Geometric algebra represents subspaces ofV{\displaystyle V}as blades, and so they coexist in the same algebra with vectors fromV{\displaystyle V}. Ak{\displaystyle k}-dimensional subspaceW{\displaystyle W}ofV{\displaystyle V}is represented by taking an orthogonal basis{b1,b2,…,bk}{\displaystyle \{b_{1},b_{2},\ldots ,b_{k}\}}and using the geometric product to form thebladeD=b1b2⋯bk{\displaystyle D=b_{1}b_{2}\cdots b_{k}}. There are multiple blades representingW{\displaystyle W}; all those representingW{\displaystyle W}are scalar multiples ofD{\displaystyle D}. These blades can be separated into two sets: positive multiples ofD{\displaystyle D}and negative multiples ofD{\displaystyle D}. The positive multiples ofD{\displaystyle D}are said to havethe sameorientationasD{\displaystyle D}, and the negative multiples theopposite orientation.
Blades are important since geometric operations such as projections, rotations and reflections depend on the factorability via the exterior product that (the restricted class of)n{\displaystyle n}-blades provide but that (the generalized class of) grade-n{\displaystyle n}multivectors do not whenn≥4{\displaystyle n\geq 4}.
Unit pseudoscalars are blades that play important roles in GA. Aunit pseudoscalarfor a non-degenerate subspaceW{\displaystyle W}ofV{\displaystyle V}is a blade that is the product of the members of an orthonormal basis forW{\displaystyle W}. It can be shown that ifI{\displaystyle I}andI′{\displaystyle I'}are both unit pseudoscalars forW{\displaystyle W}, thenI=±I′{\displaystyle I=\pm I'}andI2=±1{\displaystyle I^{2}=\pm 1}. If one doesn't choose an orthonormal basis forW{\displaystyle W}, then thePlücker embeddinggives a vector in the exterior algebra but only up to scaling. Using the vector space isomorphism between the geometric algebra and exterior algebra, this gives the equivalence class ofαI{\displaystyle \alpha I}for allα≠0{\displaystyle \alpha \neq 0}. Orthonormality gets rid of this ambiguity except for the signs above.
Suppose the geometric algebraG(n,0){\displaystyle {\mathcal {G}}(n,0)}with the familiar positive definite inner product onRn{\displaystyle \mathbb {R} ^{n}}is formed. Given a plane (two-dimensional subspace) ofRn{\displaystyle \mathbb {R} ^{n}}, one can find an orthonormal basis{b1,b2}{\displaystyle \{b_{1},b_{2}\}}spanning the plane, and thus find a unit pseudoscalarI=b1b2{\displaystyle I=b_{1}b_{2}}representing this plane. The geometric product of any two vectors in the span ofb1{\displaystyle b_{1}}andb2{\displaystyle b_{2}}lies in{α0+α1I∣αi∈R}{\displaystyle \{\alpha _{0}+\alpha _{1}I\mid \alpha _{i}\in \mathbb {R} \}}, that is, it is the sum of a0{\displaystyle 0}-vector and a2{\displaystyle 2}-vector.
By the properties of the geometric product,I2=b1b2b1b2=−b1b2b2b1=−1{\displaystyle I^{2}=b_{1}b_{2}b_{1}b_{2}=-b_{1}b_{2}b_{2}b_{1}=-1}. The resemblance to theimaginary unitis not incidental: the subspace{α0+α1I∣αi∈R}{\displaystyle \{\alpha _{0}+\alpha _{1}I\mid \alpha _{i}\in \mathbb {R} \}}isR{\displaystyle \mathbb {R} }-algebra isomorphic to thecomplex numbers. In this way, a copy of the complex numbers is embedded in the geometric algebra for each two-dimensional subspace ofV{\displaystyle V}on which the quadratic form is definite.
It is sometimes possible to identify the presence of an imaginary unit in a physical equation. Such units arise from one of the many quantities in the real algebra that square to−1{\displaystyle -1}, and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces.
InG(3,0){\displaystyle {\mathcal {G}}(3,0)}, a further familiar case occurs. Given a standard basis consisting of orthonormal vectorsei{\displaystyle e_{i}}ofV{\displaystyle V}, the set ofall2{\displaystyle 2}-vectors is spanned by
Labelling thesei{\displaystyle i},j{\displaystyle j}andk{\displaystyle k}(momentarily deviating from our uppercase convention), the subspace generated by0{\displaystyle 0}-vectors and2{\displaystyle 2}-vectors is exactly{α0+iα1+jα2+kα3∣αi∈R}{\displaystyle \{\alpha _{0}+i\alpha _{1}+j\alpha _{2}+k\alpha _{3}\mid \alpha _{i}\in \mathbb {R} \}}. This set is seen to be the even subalgebra ofG(3,0){\displaystyle {\mathcal {G}}(3,0)}, and furthermore is isomorphic as anR{\displaystyle \mathbb {R} }-algebra to thequaternions, another important algebraic system.
It is common practice to extend the exterior product on vectors to the entire algebra. This may be done through the use of the above-mentionedgrade projectionoperator:
This generalization is consistent with the above definition involving antisymmetrization. Another generalization related to the exterior product is the commutator product:
The regressive product is the dual of the exterior product (respectively corresponding to the "meet" and "join" in this context).[i]The dual specification of elements permits, for bladesC{\displaystyle C}andD{\displaystyle D}, the intersection (or meet) where the duality is to be taken relative to the a blade containing bothC{\displaystyle C}andD{\displaystyle D}(the smallest such blade being the join).[27]
withI{\displaystyle I}the unit pseudoscalar of the algebra. The regressive product, like the exterior product, is associative.[28]
The inner product on vectors can also be generalized, but in more than one non-equivalent way. The paper (Dorst 2002) gives a full treatment of several different inner products developed for geometric algebras and their interrelationships, and the notation is taken from there. Many authors use the same symbol as for the inner product of vectors for their chosen extension (e.g. Hestenes and Perwass). No consistent notation has emerged.
Among these several different generalizations of the inner product on vectors are:
Dorst (2002)makes an argument for the use of contractions in preference to Hestenes's inner product; they are algebraically more regular and have cleaner geometric interpretations.
A number of identities incorporating the contractions are valid without restriction of their inputs.
For example,
Benefits of using the left contraction as an extension of the inner product on vectors include that the identityab=a⋅b+a∧b{\displaystyle ab=a\cdot b+a\wedge b}is extended toaB=a⌋B+a∧B{\displaystyle aB=a\;\rfloor \;B+a\wedge B}for any vectora{\displaystyle a}and multivectorB{\displaystyle B}, and that theprojectionoperationPb(a)=(a⋅b−1)b{\displaystyle {\mathcal {P}}_{b}(a)=(a\cdot b^{-1})b}is extended toPB(A)=(A⌋B−1)⌋B{\displaystyle {\mathcal {P}}_{B}(A)=(A\;\rfloor \;B^{-1})\;\rfloor \;B}for any bladeB{\displaystyle B}and any multivectorA{\displaystyle A}(with a minor modification to accommodate nullB{\displaystyle B}, givenbelow).
Let{e1,…,en}{\displaystyle \{e_{1},\ldots ,e_{n}\}}be a basis ofV{\displaystyle V}, i.e. a set ofn{\displaystyle n}linearly independent vectors that span then{\displaystyle n}-dimensional vector spaceV{\displaystyle V}. The basis that is dual to{e1,…,en}{\displaystyle \{e_{1},\ldots ,e_{n}\}}is the set of elements of thedual vector spaceV∗{\displaystyle V^{*}}that forms abiorthogonal systemwith this basis, thus being the elements denoted{e1,…,en}{\displaystyle \{e^{1},\ldots ,e^{n}\}}satisfying
whereδ{\displaystyle \delta }is theKronecker delta.
Given a nondegenerate quadratic form onV{\displaystyle V},V∗{\displaystyle V^{*}}becomes naturally identified withV{\displaystyle V}, and the dual basis may be regarded as elements ofV{\displaystyle V}, but are not in general the same set as the original basis.
Given further a GA ofV{\displaystyle V}, let
be the pseudoscalar (which does not necessarily square to±1{\displaystyle \pm 1}) formed from the basis{e1,…,en}{\displaystyle \{e_{1},\ldots ,e_{n}\}}. The dual basis vectors may be constructed as
where theeˇi{\displaystyle {\check {e}}_{i}}denotes that thei{\displaystyle i}th basis vector is omitted from the product.
A dual basis is also known as areciprocal basisor reciprocal frame.
A major usage of a dual basis is to separate vectors into components. Given a vectora{\displaystyle a}, scalar componentsai{\displaystyle a^{i}}can be defined as
in terms of whicha{\displaystyle a}can be separated into vector components as
We can also define scalar componentsai{\displaystyle a_{i}}as
in terms of whicha{\displaystyle a}can be separated into vector components in terms of the dual basis as
A dual basis as defined above for the vector subspace of a geometric algebra can be extended to cover the entire algebra.[29]For compactness, we'll use a single capital letter to represent an ordered set of vector indices. I.e., writing
wherej1<j2<⋯<jn{\displaystyle j_{1}<j_{2}<\dots <j_{n}},
we can write a basis blade as
The corresponding reciprocal blade has the indices in opposite order:
Similar to the case above with vectors, it can be shown that
where∗{\displaystyle *}is the scalar product.
WithA{\displaystyle A}a multivector, we can define scalar components as[30]
in terms of whichA{\displaystyle A}can be separated into component blades as
We can alternatively define scalar components
in terms of whichA{\displaystyle A}can be separated into component blades as
Although a versor is easier to work with because it can be directly represented in the algebra as a multivector, versors are a subgroup oflinear functionson multivectors, which can still be used when necessary. The geometric algebra of ann{\displaystyle n}-dimensional vector space is spanned by a basis of2n{\displaystyle 2^{n}}elements. If a multivector is represented by a2n×1{\displaystyle 2^{n}\times 1}realcolumn matrixof coefficients of a basis of the algebra, then all linear transformations of the multivector can be expressed as thematrix multiplicationby a2n×2n{\displaystyle 2^{n}\times 2^{n}}real matrix. However, such a general linear transformation allows arbitrary exchanges among grades, such as a "rotation" of a scalar into a vector, which has no evident geometric interpretation.
A general linear transformation from vectors to vectors is of interest. With the natural restriction to preserving the induced exterior algebra, theoutermorphismof the linear transformation is the unique[k]extension of the versor. Iff{\displaystyle f}is a linear function that maps vectors to vectors, then its outermorphism is the function that obeys the rule
for a blade, extended to the whole algebra through linearity.
Although a lot of attention has been placed on CGA, it is to be noted that GA is not just one algebra, it is one of a family of algebras with the same essential structure.[31]
Theeven subalgebraofG(2,0){\displaystyle {\mathcal {G}}(2,0)}is isomorphic to thecomplex numbers, as may be seen by writing a vectorP{\displaystyle P}in terms of its components in an orthonormal basis and left multiplying by the basis vectore1{\displaystyle e_{1}}, yielding
where we identifyi↦e1e2{\displaystyle i\mapsto e_{1}e_{2}}since
Similarly, the even subalgebra ofG(3,0){\displaystyle {\mathcal {G}}(3,0)}with basis{1,e2e3,e3e1,e1e2}{\displaystyle \{1,e_{2}e_{3},e_{3}e_{1},e_{1}e_{2}\}}is isomorphic to thequaternionsas may be seen by identifyingi↦−e2e3{\displaystyle i\mapsto -e_{2}e_{3}},j↦−e3e1{\displaystyle j\mapsto -e_{3}e_{1}}andk↦−e1e2{\displaystyle k\mapsto -e_{1}e_{2}}.
Everyassociative algebrahas a matrix representation; replacing the three Cartesian basis vectors by thePauli matricesgives a representation ofG(3,0){\displaystyle {\mathcal {G}}(3,0)}:
Dotting the "Pauli vector" (adyad):
In physics, the main applications are the geometric algebra ofMinkowski 3+1 spacetime,G(1,3){\displaystyle {\mathcal {G}}(1,3)}, calledspacetime algebra(STA),[3]or less commonly,G(3,0){\displaystyle {\mathcal {G}}(3,0)}, interpreted thealgebra of physical space(APS).
While in STA, points of spacetime are represented simply by vectors, in APS, points of(3+1){\displaystyle (3+1)}-dimensional spacetime are instead represented byparavectors, a three-dimensional vector (space) plus a one-dimensional scalar (time).
In spacetime algebra the electromagnetic field tensor has a bivector representationF=(E+icB)γ0{\displaystyle {F}=({E}+ic{B})\gamma _{0}}.[32]Here, thei=γ0γ1γ2γ3{\displaystyle i=\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}}is the unit pseudoscalar (or four-dimensional volume element),γ0{\displaystyle \gamma _{0}}is the unit vector in time direction, andE{\displaystyle E}andB{\displaystyle B}are the classic electric and magnetic field vectors (with a zero time component). Using thefour-currentJ{\displaystyle {J}},Maxwell's equationsthen become
D⌋A=0{\displaystyle D~\rfloor ~A=0}
In geometric calculus, juxtaposition of vectors such as inDF{\displaystyle DF}indicate the geometric product and can be decomposed into parts asDF=D⌋F+D∧F{\displaystyle DF=D~\rfloor ~F+D\wedge F}. HereD{\displaystyle D}is the covector derivative in any spacetime and reduces to∇{\displaystyle \nabla }in flat spacetime. Where▽{\displaystyle \bigtriangledown }plays a role in Minkowski4{\displaystyle 4}-spacetime which is synonymous to the role of∇{\displaystyle \nabla }in Euclidean3{\displaystyle 3}-space and is related to thed'Alembertianby◻=▽2{\displaystyle \Box =\bigtriangledown ^{2}}. Indeed, given an observer represented by a future pointing timelike vectorγ0{\displaystyle \gamma _{0}}we have
Boostsin this Lorentzian metric space have the same expressioneβ{\displaystyle e^{\beta }}as rotation in Euclidean space, whereβ{\displaystyle {\beta }}is the bivector generated by the time and the space directions involved, whereas in the Euclidean case it is the bivector generated by the two space directions, strengthening the "analogy" to almost identity.
TheDirac matricesare a representation ofG(1,3){\displaystyle {\mathcal {G}}(1,3)}, showing the equivalence with matrix representations used by physicists.
Homogeneous models generally refer to a projective representation in which the elements of the one-dimensional subspaces of a vector space represent points of a geometry.
In a geometric algebra of a space ofn{\displaystyle n}dimensions, the rotors represent a set of transformations withn(n−1)/2{\displaystyle n(n-1)/2}degrees of freedom, corresponding to rotations – for example,3{\displaystyle 3}whenn=3{\displaystyle n=3}and6{\displaystyle 6}whenn=4{\displaystyle n=4}. Geometric algebra is often used to model aprojective space, i.e. as ahomogeneous model: a point, line, plane, etc. is represented by an equivalence class of elements of the algebra that differ by an invertible scalar factor.
The rotors in a space of dimensionn+1{\displaystyle n+1}haven(n−1)/2+nn(n-1)/2+ndegrees of freedom, the same as the number of degrees of freedom in the rotations and translations combined for ann{\displaystyle n}-dimensional space.
This is the case inProjective Geometric Algebra(PGA), which is used[33][34][35]to representEuclidean isometriesin Euclidean geometry (thereby covering the large majority of engineering applications of geometry). In this model, a degenerate dimension is added to the three Euclidean dimensions to form the algebraG(3,0,1){\displaystyle {\mathcal {G}}(3,0,1)}. With a suitable identification of subspaces to represent points, lines and planes, the versors of this algebra represent all proper Euclidean isometries, which are alwaysscrew motionsin 3-dimensional space, along with all improper Euclidean isometries, which includes reflections, rotoreflections, transflections, and point reflections. PGA allows projection, meet, and angle formulas to be derived fromG(3,0,1){\displaystyle {\mathcal {G}}(3,0,1)}- with a very minor extension to the algebra it is also possible to derive distances and joins.
PGA is a widely used system that combines geometric algebra with homogeneous representations in geometry, but there exist several other such systems. The conformal model discussed below is homogeneous, as is "Conic Geometric Algebra",[36]and seePlane-based geometric algebrafor discussion of homogeneous models of elliptic and hyperbolic geometry compared with the Euclidean geometry derived from PGA.
Working within GA, Euclidean spaceE3{\displaystyle \mathbb {E} ^{3}}(along with a conformal point at infinity) is embedded projectively in the CGAG(4,1){\displaystyle {\mathcal {G}}(4,1)}via the identification of Euclidean points with 1D subspaces in the 4D null cone of the 5D CGA vector subspace. This allows all conformal transformations to be performed as rotations and reflections and iscovariant, extending incidence relations of projective geometry to rounds objects such as circles and spheres.
Specifically, we add orthogonal basis vectorse+{\displaystyle e_{+}}ande−{\displaystyle e_{-}}such thate+2=+1{\displaystyle e_{+}^{2}=+1}ande−2=−1{\displaystyle e_{-}^{2}=-1}to the basis of the vector space that generatesG(3,0){\displaystyle {\mathcal {G}}(3,0)}and identifynull vectors
(Some authors sete4=no{\displaystyle e_{4}=n_{\text{o}}}ande5=n∞{\displaystyle e_{5}=n_{\infty }}.[37]) This procedure has some similarities to the procedure for working withhomogeneous coordinatesin projective geometry, and in this case allows the modeling ofEuclidean transformationsofR3{\displaystyle \mathbb {R} ^{3}}asorthogonal transformationsof a subset ofR4,1{\displaystyle \mathbf {R} ^{4,1}}.
A fast changing and fluid area of GA, CGA is also being investigated for applications to relativistic physics.
Note in this list thatp{\displaystyle p}andq{\displaystyle q}can be swapped and the same name applies; for example, withrelativelylittle change occurring, seesign convention. For example,G(3,1,0){\displaystyle {\mathcal {G}}(3,1,0)}andG(1,3,0){\displaystyle {\mathcal {G}}(1,3,0)}arebothreferred to as Spacetime Algebra.[38]
Algebra of Physical Space, APS
Quadric Conformal 2D GA QC2GA[42][36]
For any vectora{\displaystyle a}and any invertible vectorm{\displaystyle m},
where theprojectionofa{\displaystyle a}ontom{\displaystyle m}(or the parallel part) is
and therejectionofa{\displaystyle a}fromm{\displaystyle m}(or the orthogonal part) is
Using the concept of ak{\displaystyle k}-bladeB{\displaystyle B}as representing a subspace ofV{\displaystyle V}and every multivector ultimately being expressed in terms of vectors, this generalizes to projection of a general multivector onto any invertiblek{\displaystyle k}-bladeB{\displaystyle B}as[l]
with the rejection being defined as
The projection and rejection generalize to null bladesB{\displaystyle B}by replacing the inverseB−1{\displaystyle B^{-1}}with the pseudoinverseB+{\displaystyle B^{+}}with respect to the contractive product.[m]The outcome of the projection coincides in both cases for non-null blades.[45][46]For null bladesB{\displaystyle B}, the definition of the projection given here with the first contraction rather than the second being onto the pseudoinverse should be used,[n]as only then is the result necessarily in the subspace represented byB{\displaystyle B}.[45]The projection generalizes through linearity to general multivectorsA{\displaystyle A}.[o]The projection is not linear inB{\displaystyle B}and does not generalize to objectsB{\displaystyle B}that are not blades.
Simple reflectionsin a hyperplane are readily expressed in the algebra through conjugation with a single vector. These serve to generate the group of generalrotoreflectionsandrotations.
The reflectionc′{\displaystyle c'}of a vectorc{\displaystyle c}along a vectorm{\displaystyle m}, or equivalently in the hyperplane orthogonal tom{\displaystyle m}, is the same as negating the component of a vector parallel tom{\displaystyle m}. The result of the reflection will be
This is not the most general operation that may be regarded as a reflection when the dimensionn≥4{\displaystyle n\geq 4}. A general reflection may be expressed as the composite of any odd number of single-axis reflections. Thus, a general reflectiona′{\displaystyle a'}of a vectora{\displaystyle a}may be written
where
If we define the reflection along a non-null vectorm{\displaystyle m}of the product of vectors as the reflection of every vector in the product along the same vector, we get for any product of an odd number of vectors that, by way of example,
and for the product of an even number of vectors that
Using the concept of every multivector ultimately being expressed in terms of vectors, the reflection of a general multivectorA{\displaystyle A}using any reflection versorM{\displaystyle M}may be written
whereα{\displaystyle \alpha }is theautomorphismofreflection through the originof the vector space (v↦−v{\displaystyle v\mapsto -v}) extended through linearity to the whole algebra.
If we have a product of vectorsR=a1a2⋯ar{\displaystyle R=a_{1}a_{2}\cdots a_{r}}then we denote the reverse as
As an example, assume thatR=ab{\displaystyle R=ab}we get
ScalingR{\displaystyle R}so thatRR~=1{\displaystyle R{\widetilde {R}}=1}then
soRvR~{\displaystyle Rv{\widetilde {R}}}leaves the length ofv{\displaystyle v}unchanged. We can also show that
so the transformationRvR~{\displaystyle Rv{\widetilde {R}}}preserves both length and angle. It therefore can be identified as a rotation or rotoreflection;R{\displaystyle R}is called arotorif it is aproper rotation(as it is if it can be expressed as a product of an even number of vectors) and is an instance of what is known in GA as aversor.
There is a general method for rotating a vector involving the formation of a multivector of the formR=e−Bθ/2{\displaystyle R=e^{-B\theta /2}}that produces a rotationθ{\displaystyle \theta }in theplaneand with the orientation defined by a2{\displaystyle 2}-bladeB{\displaystyle B}.
Rotors are a generalization of quaternions ton{\displaystyle n}-dimensional spaces.
For vectorsa{\displaystyle a}andb{\displaystyle b}spanning a parallelogram we have
with the result thata∧b{\displaystyle a\wedge b}is linear in the product of the "altitude" and the "base" of the parallelogram, that is, its area.
Similar interpretations are true for any number of vectors spanning ann{\displaystyle n}-dimensionalparallelotope; the exterior product of vectorsa1,a2,…,an{\displaystyle a_{1},a_{2},\ldots ,a_{n}}, that is⋀i=1nai{\displaystyle \textstyle \bigwedge _{i=1}^{n}a_{i}}, has a magnitude equal to the volume of then{\displaystyle n}-parallelotope. Ann{\displaystyle n}-vector does not necessarily have a shape of a parallelotope – this is a convenient visualization. It could be any shape, although the volume equals that of the parallelotope.
We may define the line parametrically byp=t+αv{\displaystyle p=t+\alpha \ v}, wherep{\displaystyle p}andt{\displaystyle t}are position vectors for points P and T andv{\displaystyle v}is the direction vector for the line.
Then
so
and
A rotational quantity such astorqueorangular momentumis described in geometric algebra as a bivector. Suppose a circular path in an arbitrary plane containing orthonormal vectorsu^{\displaystyle {\widehat {u}}}andv^{\displaystyle {\widehat {\ \!v}}}is parameterized by angle.
By designating the unit bivector of this plane as the imaginary number
this path vector can be conveniently written in complex exponential form
and the derivative with respect to angle is
For example, torque is generally defined as the magnitude of the perpendicular force component times distance, or work per unit angle. Thus the torque, the rate of change of workW{\displaystyle W}with respect to angle, due to a forceF{\displaystyle F}, is
Rotational quantities are represented invector calculusin three dimensions using thecross product. Together with a choice of an oriented volume formI{\displaystyle I}, these can be related to the exterior product with its more natural geometric interpretation of such quantities as a bivectors by using thedualrelationship
Unlike the cross product description of torque,τ=r×F{\displaystyle \tau =\mathbf {r} \times F}, the geometric algebra description does not introduce a vector in the normal direction; a vector that does not exist in two and that is not unique in greater than three dimensions. The unit bivector describes the plane and the orientation of the rotation, and the sense of the rotation is relative to the angle between the vectorsu^{\displaystyle {\widehat {u}}}andv^{\displaystyle {\widehat {\ \!v}}}.
Geometric calculus extends the formalism to include differentiation and integration including differential geometry anddifferential forms.[47]
Essentially, the vector derivative is defined so that the GA version ofGreen's theoremis true,
and then one can write
as a geometric product, effectively generalizingStokes' theorem(including the differential form version of it).
In 1D whenA{\displaystyle A}is a curve with endpointsa{\displaystyle a}andb{\displaystyle b}, then
reduces to
or the fundamental theorem of integral calculus.
Also developed are the concept ofvector manifoldand geometric integration theory (which generalizes differential forms).
Although the connection of geometry with algebra dates as far back at least toEuclid'sElementsin the third century B.C. (seeGreek geometric algebra), GA in the sense used in this article was not developed until 1844, when it was used in asystematic wayto describe the geometrical properties andtransformationsof a space. In that year,Hermann Grassmannintroduced the idea of a geometrical algebra in full generality as a certain calculus (analogous to thepropositional calculus) that encoded all of the geometrical information of a space.[48]Grassmann's algebraic system could be applied to a number of different kinds of spaces, the chief among them beingEuclidean space,affine space, andprojective space. Following Grassmann, in 1878William Kingdon Cliffordexamined Grassmann's algebraic system alongside thequaternionsofWilliam Rowan Hamiltonin (Clifford 1878). From his point of view, the quaternions described certaintransformations(which he calledrotors), whereas Grassmann's algebra described certainproperties(orStreckensuch as length, area, and volume). His contribution was to define a new product – thegeometric product– on an existing Grassmann algebra, which realized the quaternions as living within that algebra. Subsequently,Rudolf Lipschitzin 1886 generalized Clifford's interpretation of the quaternions and applied them to the geometry of rotations inn{\displaystyle n}dimensions. Later these developments would lead other 20th-century mathematicians to formalize and explore the properties of the Clifford algebra.
Nevertheless, another revolutionary development of the 19th-century would completely overshadow the geometric algebras: that ofvector analysis, developed independently byJosiah Willard GibbsandOliver Heaviside. Vector analysis was motivated byJames Clerk Maxwell's studies ofelectromagnetism, and specifically the need to express and manipulate conveniently certaindifferential equations. Vector analysis had a certain intuitive appeal compared to the rigors of the new algebras. Physicists and mathematicians alike readily adopted it as their geometrical toolkit of choice, particularly following the influential 1901 textbookVector AnalysisbyEdwin Bidwell Wilson, following lectures of Gibbs.
In more detail, there have been three approaches to geometric algebra:quaternionicanalysis, initiated by Hamilton in 1843 and geometrized as rotors by Clifford in 1878; geometric algebra, initiated by Grassmann in 1844; and vector analysis, developed out of quaternionic analysis in the late 19th century by Gibbs and Heaviside. The legacy of quaternionic analysis in vector analysis can be seen in the use ofi{\displaystyle i},j{\displaystyle j},k{\displaystyle k}to indicate the basis vectors ofR3{\displaystyle \mathbf {R} ^{3}}: it is being thought of as the purely imaginary quaternions. From the perspective of geometric algebra, the even subalgebra of the Space Time Algebra is isomorphic to the GA of 3D Euclidean space and quaternions are isomorphic to the even subalgebra of the GA of 3D Euclidean space, which unifies the three approaches.
Progress on the study of Clifford algebras quietly advanced through the twentieth century, although largely due to the work ofabstract algebraistssuch asÉlie Cartan,Hermann WeylandClaude Chevalley. Thegeometricalapproach to geometric algebras has seen a number of 20th-century revivals. In mathematics,Emil Artin'sGeometric Algebra[49]discusses the algebra associated with each of a number of geometries, includingaffine geometry,projective geometry,symplectic geometry, andorthogonal geometry. In physics, geometric algebras have been revived as a "new" way to do classical mechanics and electromagnetism, together with more advanced topics such as quantum mechanics and gauge theory.[5]David Hestenesreinterpreted thePauliandDiracmatrices as vectors in ordinary space and spacetime, respectively, and has been a primary contemporary advocate for the use of geometric algebra.
Incomputer graphicsand robotics, geometric algebras have been revived in order to efficiently represent rotations and other transformations. For applications of GA in robotics (screw theory, kinematics and dynamics using versors), computer vision, control and neural computing (geometric learning) see Bayro (2010).
English translations of early books and papers
Research groups
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Inidempotent analysis, thetropical semiringis asemiringofextended real numberswith the operations ofminimum(ormaximum) and addition replacing the usual ("classical") operations of addition and multiplication, respectively.
The tropical semiring has various applications (seetropical analysis), and forms the basis oftropical geometry. The nametropicalis a reference to the Hungarian-born computer scientistImre Simon, so named because he lived and worked in Brazil.[1]
Themin tropical semiring(ormin-plus semiringormin-plus algebra) is thesemiring(R∪{+∞}{\displaystyle \mathbb {R} \cup \{+\infty \}},⊕{\displaystyle \oplus },⊗{\displaystyle \otimes }), with the operations:
The operations⊕{\displaystyle \oplus }and⊗{\displaystyle \otimes }are referred to astropical additionandtropical multiplicationrespectively. The identity element for⊕{\displaystyle \oplus }is+∞{\displaystyle +\infty }, and the identity element for⊗{\displaystyle \otimes }is 0.
Similarly, themax tropical semiring(ormax-plus semiringormax-plus algebraorArctic semiring[citation needed]) is the semiring (R∪{−∞}{\displaystyle \mathbb {R} \cup \{-\infty \}},⊕{\displaystyle \oplus },⊗{\displaystyle \otimes }), with operations:
The identity element unit for⊕{\displaystyle \oplus }is−∞{\displaystyle -\infty }, and the identity element unit for⊗{\displaystyle \otimes }is 0.
The two semirings are isomorphic under negationx↦−x{\displaystyle x\mapsto -x}, and generally one of these is chosen and referred to simply as thetropical semiring. Conventions differ between authors and subfields: some use theminconvention, some use themaxconvention.
The two tropical semirings are the limit ("tropicalization", "dequantization") of thelog semiringas the base goes to infinityb→∞{\displaystyle b\to \infty }(max-plus semiring) or to zerob→0{\displaystyle b\to 0}(min-plus semiring).
Tropical addition isidempotent, thus a tropical semiring is an example of anidempotent semiring.
A tropical semiring is also referred to as atropical algebra,[2]though this should not be confused with anassociative algebraover a tropical semiring.
Tropicalexponentiationis defined in the usual way as iterated tropical products.
The tropical semiring operations model howvaluationsbehave under addition and multiplication in avalued field. A real-valued fieldK{\displaystyle K}is a field equipped with a function
which satisfies the following properties for alla{\displaystyle a},b{\displaystyle b}inK{\displaystyle K}:
Therefore the 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.
Some common valued fields:
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In the theory ofalgebras over a field,mutationis a construction of a newbinary operationrelated to the multiplication of the algebra. In specific cases the resulting algebra may be referred to as ahomotopeor anisotopeof the original.
LetAbe an algebra over afieldFwith multiplication (not assumed to beassociative) denoted by juxtaposition. For an elementaofA, define thelefta-homotopeA(a){\displaystyle A(a)}to be the algebra with multiplication
Similarly define theleft (a,b) mutationA(a,b){\displaystyle A(a,b)}
Right homotope and mutation are defined analogously. Since the right (p,q) mutation ofAis the left (−q, −p) mutation of theopposite algebratoA, it suffices to study left mutations.[1]
IfAis aunital algebraandais invertible, we refer to theisotopebya.
AJordan algebrais a commutative algebra satisfying theJordan identity(xy)(xx)=x(y(xx)){\displaystyle (xy)(xx)=x(y(xx))}. TheJordan triple productis defined by
ForyinAthemutation[3]orhomotope[4]Ayis defined as the vector spaceAwith multiplication
and ifyis invertible this is referred to as anisotope. A homotope of a Jordan algebra is again a Jordan algebra: isotopy defines an equivalence relation.[5]Ifyisnuclearthen the isotope byyis isomorphic to the original.[6]
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Ring homomorphisms
Algebraic structures
Related structures
Algebraic number theory
Noncommutative algebraic geometry
Free algebra
Clifford algebra
Infunctional analysis, a branch ofmathematics, anoperator algebrais analgebraofcontinuouslinear operatorson atopological vector space, with the multiplication given by thecomposition of mappings.
The results obtained in the study of operator algebras are often phrased inalgebraicterms, while the techniques used are often highlyanalytic.[1]Although the study of operator algebras is usually classified as a branch of functional analysis, it has direct applications torepresentation theory,differential geometry,quantum statistical mechanics,quantum information, andquantum field theory.
Operator algebras can be used to study arbitrary sets of operators with little algebraic relationsimultaneously. From this point of view, operator algebras can be regarded as a generalization ofspectral theoryof a single operator. In general, operator algebras arenon-commutativerings.
An operator algebra is typically required to beclosedin a specified operatortopologyinside the whole algebra of continuous linear operators. In particular, it is a set of operators with both algebraic and topological closure properties. In some disciplines such properties areaxiomatizedand algebras with certain topological structure become the subject of the research.
Though algebras of operators are studied in various contexts (for example, algebras ofpseudo-differential operatorsacting on spaces ofdistributions), the termoperator algebrais usually used in reference to algebras ofbounded operatorson aBanach spaceor, even more specially in reference to algebras of operators on aseparableHilbert space, endowed with theoperator normtopology.
In the case of operators on a Hilbert space, theHermitian adjointmap on operators gives a naturalinvolution, which provides an additional algebraic structure that can be imposed on the algebra. In this context, the best studied examples areself-adjointoperator algebras, meaning that they are closed under taking adjoints. These includeC*-algebras,von Neumann algebras, andAW*-algebras. C*-algebras can be easily characterized abstractly by a condition relating the norm, involution and multiplication. Such abstractly defined C*-algebras can be identified to a certain closedsubalgebraof the algebra of the continuous linear operators on a suitable Hilbert space. A similar result holds for von Neumann algebras.
Commutativeself-adjoint operator algebras can be regarded as the algebra ofcomplex-valued continuous functions on alocally compact space, or that ofmeasurable functionson astandard measurable space. Thus, general operator algebras are often regarded as a noncommutative generalizations of these algebras, or the structure of thebase spaceon which the functions are defined. This point of view is elaborated as the philosophy ofnoncommutative geometry, which tries to study various non-classical and/or pathological objects by noncommutative operator algebras.
Examples of operator algebras that are not self-adjoint include:
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Inalgebra,Zariski's lemma, proved byOscar Zariski(1947), states that, if afieldKisfinitely generatedas anassociative algebraover another fieldk, thenKis afinite field extensionofk(that is, it is also finitely generated as avector space).
An important application of the lemma is a proof of the weak form ofHilbert's Nullstellensatz:[1]ifIis a properidealofk[t1,...,tn]{\displaystyle k[t_{1},...,t_{n}]}(kanalgebraically closed field), thenIhas a zero; i.e., there is a pointxinkn{\displaystyle k^{n}}such thatf(x)=0{\displaystyle f(x)=0}for allfinI. (Proof: replacingIby amaximal idealm{\displaystyle {\mathfrak {m}}}, we can assumeI=m{\displaystyle I={\mathfrak {m}}}is maximal. LetA=k[t1,...,tn]{\displaystyle A=k[t_{1},...,t_{n}]}andϕ:A→A/m{\displaystyle \phi :A\to A/{\mathfrak {m}}}be the natural surjection. By the lemmaA/m{\displaystyle A/{\mathfrak {m}}}is a finite extension. Sincekis algebraically closed that extension must bek. Then for anyf∈m{\displaystyle f\in {\mathfrak {m}}},
that is to say,x=(ϕ(t1),⋯,ϕ(tn)){\displaystyle x=(\phi (t_{1}),\cdots ,\phi (t_{n}))}is a zero ofm{\displaystyle {\mathfrak {m}}}.)
The lemma may also be understood from the following perspective. In general, a ringRis aJacobson ringif and only if every finitely generatedR-algebra that is a field is finite overR.[2]Thus, the lemma follows from the fact that a field is a Jacobson ring.
Two direct proofs are given in Atiyah–MacDonald;[3][4]the one is due to Zariski and the other uses theArtin–Tate lemma. For Zariski's original proof, see the original paper.[5]Another direct proof in the language ofJacobson ringsis given below. The lemma is also a consequence of theNoether normalization lemma. Indeed, by the normalization lemma,Kis afinite moduleover the polynomial ringk[x1,…,xd]{\displaystyle k[x_{1},\ldots ,x_{d}]}wherex1,…,xd{\displaystyle x_{1},\ldots ,x_{d}}are elements ofKthat are algebraically independent overk. But sinceKhas Krull dimension zero and since anintegral ring extension(e.g., a finite ring extension) preserves Krull dimensions, the polynomial ring must have dimension zero; i.e.,d=0{\displaystyle d=0}.
The following characterization of a Jacobson ring contains Zariski's lemma as a special case. Recall that a ring is a Jacobson ring if every prime ideal is an intersection of maximal ideals. (WhenAis a field,Ais a Jacobson ring and the theorem below is precisely Zariski's lemma.)
Theorem—[2]LetAbe a ring. Then the following are equivalent.
Proof: 2.⇒{\displaystyle \Rightarrow }1.: Letp{\displaystyle {\mathfrak {p}}}be a prime ideal ofAand setB=A/p{\displaystyle B=A/{\mathfrak {p}}}. We need to show theJacobson radicalofBis zero. For that end, letfbe a nonzero element ofB. Letm{\displaystyle {\mathfrak {m}}}be a maximal ideal of the localizationB[f−1]{\displaystyle B[f^{-1}]}. ThenB[f−1]/m{\displaystyle B[f^{-1}]/{\mathfrak {m}}}is a field that is a finitely generatedA-algebra and so is finite overAby assumption; thus it is finite overB=A/p{\displaystyle B=A/{\mathfrak {p}}}and so is finite over the subringB/q{\displaystyle B/{\mathfrak {q}}}whereq=m∩B{\displaystyle {\mathfrak {q}}={\mathfrak {m}}\cap B}. By integrality,q{\displaystyle {\mathfrak {q}}}is a maximal ideal not containingf.
1.⇒{\displaystyle \Rightarrow }2.: Since a factor ring of a Jacobson ring is Jacobson, we can assumeBcontainsAas a subring. Then the assertion is a consequence of the next algebraic fact:
Indeed, choose a maximal idealm{\displaystyle {\mathfrak {m}}}ofAnot containinga. WritingKfor some algebraic closure ofA/m{\displaystyle A/{\mathfrak {m}}}, the canonical mapϕ:A→A/m↪K{\displaystyle \phi :A\to A/{\mathfrak {m}}\hookrightarrow K}extends toϕ~:B→K{\displaystyle {\widetilde {\phi }}:B\to K}. SinceBis a field,ϕ~{\displaystyle {\widetilde {\phi }}}is injective and soBis algebraic (thus finite algebraic) overA/m{\displaystyle A/{\mathfrak {m}}}. We now prove (*). IfBcontains an element that is transcendental overA, then it contains a polynomial ring overAto whichφextends (without a requirement ona) and so we can assumeBis algebraic overA(by Zorn's lemma, say). Letx1,…,xr{\displaystyle x_{1},\dots ,x_{r}}be the generators ofBasA-algebra. Then eachxi{\displaystyle x_{i}}satisfies the relation
wherendepends oniandai0≠0{\displaystyle a_{i0}\neq 0}. Seta=a10a20…ar0{\displaystyle a=a_{10}a_{20}\dots a_{r0}}. ThenB[a−1]{\displaystyle B[a^{-1}]}is integral overA[a−1]{\displaystyle A[a^{-1}]}. Now givenϕ:A→K{\displaystyle \phi :A\to K}, we first extend it toϕ~:A[a−1]→K{\displaystyle {\widetilde {\phi }}:A[a^{-1}]\to K}by settingϕ~(a−1)=ϕ(a)−1{\displaystyle {\widetilde {\phi }}(a^{-1})=\phi (a)^{-1}}. Next, letm=kerϕ~{\displaystyle {\mathfrak {m}}=\operatorname {ker} {\widetilde {\phi }}}. By integrality,m=n∩A[a−1]{\displaystyle {\mathfrak {m}}={\mathfrak {n}}\cap A[a^{-1}]}for some maximal idealn{\displaystyle {\mathfrak {n}}}ofB[a−1]{\displaystyle B[a^{-1}]}. Thenϕ~:A[a−1]→A[a−1]/m→K{\displaystyle {\widetilde {\phi }}:A[a^{-1}]\to A[a^{-1}]/{\mathfrak {m}}\to K}extends toB[a−1]→B[a−1]/n→K{\displaystyle B[a^{-1}]\to B[a^{-1}]/{\mathfrak {n}}\to K}. Restrict the last map toBto finish the proof.◻{\displaystyle \square }
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Inmathematics,categorificationis the process of replacingset-theoretictheoremswithcategory-theoreticanalogues. Categorification, when done successfully, replacessetswithcategories,functionswithfunctors, andequationswithnatural isomorphismsof functors satisfying additional properties. The term was coined byLouis Crane.[1][2]
The reverse of categorification is the process ofdecategorification. Decategorification is a systematic process by whichisomorphicobjects in a category are identified asequal. Whereas decategorification is a straightforward process, categorification is usually much less straightforward. In therepresentation theoryofLie algebras,modulesover specific algebras are the principal objects of study, and there are several frameworks for what a categorification of such a module should be, e.g., so called (weak) abelian categorifications.[3]
Categorification and decategorification are not precise mathematical procedures, but rather a class of possible analogues. They are used in a similar way to the words like 'generalization', and not like 'sheafification'.[4]
One form of categorification takes a structure described in terms of sets, and interprets the sets asisomorphism classesof objects in a category. For example, the set ofnatural numberscan be seen as the set ofcardinalitiesoffinite sets(and any two sets with the same cardinality are isomorphic). In this case, operations on the set of natural numbers, such as addition and multiplication, can be seen as carrying information aboutcoproductsandproductsof thecategory of finite sets. Less abstractly, the idea here is that manipulating sets of actual objects, and taking coproducts (combining two sets in a union) or products (building arrays of things to keep track of large numbers of them) came first. Later, the concrete structure of sets was abstracted away – taken "only up to isomorphism", to produce the abstract theory of arithmetic. This is a "decategorification" – categorification reverses this step.
Other examples includehomology theoriesintopology.Emmy Noethergave the modern formulation of homology as therankof certainfree abelian groupsby categorifying the notion of aBetti number.[5]See alsoKhovanov homologyas aknot invariantinknot theory.
An example infinite group theoryis that thering of symmetric functionsis categorified by thecategory of representationsof thesymmetric group. The decategorification map sends theSpecht moduleindexed by partitionλ{\displaystyle \lambda }to theSchur functionindexed by the same partition,
essentially following thecharactermap from a favorite basis of the associatedGrothendieck groupto a representation-theoretic favorite basis of the ring ofsymmetric functions. This map reflects how the structures are similar; for example
have the same decomposition numbers over their respective bases, both given byLittlewood–Richardson coefficients.
For a categoryB{\displaystyle {\mathcal {B}}}, letK(B){\displaystyle K({\mathcal {B}})}be theGrothendieck groupofB{\displaystyle {\mathcal {B}}}.
LetA{\displaystyle A}be aringwhich isfree as an abelian group, and leta={ai}i∈I{\displaystyle \mathbf {a} =\{a_{i}\}_{i\in I}}be a basis ofA{\displaystyle A}such that the multiplication is positive ina{\displaystyle \mathbf {a} }, i.e.
LetB{\displaystyle B}be anA{\displaystyle A}-module. Then a (weak) abelian categorification of(A,a,B){\displaystyle (A,\mathbf {a} ,B)}consists of anabelian categoryB{\displaystyle {\mathcal {B}}}, an isomorphismϕ:K(B)→B{\displaystyle \phi :K({\mathcal {B}})\to B}, and exactendofunctorsFi:B→B{\displaystyle F_{i}:{\mathcal {B}}\to {\mathcal {B}}}such that
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Inmathematics, especially (higher)category theory,higher-dimensional algebrais the study ofcategorifiedstructures. It has applications in nonabelianalgebraic topology, and generalizesabstract algebra.
A first step towards defining higher dimensional algebras is the concept of2-categoryofhigher category theory, followed by the more 'geometric' concept of double category.[1][2][3]
A higher level concept is thus defined as acategoryof categories, or super-category, which generalises to higher dimensions the notion ofcategory– regarded as any structure which is an interpretation ofLawvere's axioms of the elementary theory of abstract categories (ETAC).[4][5][6][7]Thus, a supercategory and also asuper-category, can be regarded as natural extensions of the concepts ofmeta-category,[8]multicategory, andmulti-graph,k-partite graph, orcolored graph(see acolor figure, and also its definition ingraph theory).
Supercategories were first introduced in 1970,[9]and were subsequently developed for applications intheoretical physics(especiallyquantum field theoryandtopological quantum field theory) andmathematical biologyormathematical biophysics.[10]
Other pathways in higher-dimensional algebra involve:bicategories, homomorphisms of bicategories,variable categories(also known as indexed orparametrized categories),topoi, effective descent, andenrichedandinternal categories.
Inhigher-dimensional algebra(HDA), adouble groupoidis a generalisation of a one-dimensionalgroupoidto two dimensions,[11]and the latter groupoid can be considered as a special case of a category with all invertible arrows, ormorphisms.
Double groupoidsare often used to capture information aboutgeometricalobjects such ashigher-dimensional manifolds(orn-dimensional manifolds).[11]In general, ann-dimensional manifoldis a space that locally looks like ann-dimensional Euclidean space, but whose global structure may benon-Euclidean.
Double groupoids were first introduced byRonald BrowninDouble groupoids and crossed modules(1976),[11]and were further developed towards applications innonabelianalgebraic topology.[12][13][14][15]A related, 'dual' concept is that of a doublealgebroid, and the more general concept ofR-algebroid.
SeeNonabelian algebraic topology
Inquantum field theory, there existquantum categories.[16][17][18]andquantum double groupoids.[18]One can consider quantum double groupoids to befundamental groupoidsdefined via a2-functor, which allows one to think about the physically interesting case ofquantum fundamental groupoids(QFGs) in terms of thebicategorySpan(Groupoids), and then constructing 2-Hilbert spacesand 2-linear mapsfor manifolds andcobordisms. At the next step, one obtainscobordismswith corners vianatural transformationsof such 2-functors. A claim was then made that, with thegauge groupSU(2), "the extendedTQFT, or ETQFT, gives a theory equivalent to thePonzano–Regge modelofquantum gravity";[18]similarly, theTuraev–Viro modelwould be then obtained withrepresentationsof SUq(2). Therefore, one can describe thestate spaceof a gauge theory – or many kinds ofquantum field theories(QFTs) and local quantum physics, in terms of thetransformation groupoidsgiven by symmetries, as for example in the case of a gauge theory, by thegauge transformationsacting on states that are, in this case, connections. In the case of symmetries related toquantum groups, one would obtain structures that are representation categories ofquantum groupoids,[16]instead of the 2-vector spacesthat are representation categories of groupoids.
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In mathematics, aLien-algebrais a generalization of aLie algebra, a vector space with a bracket, to higher order operations. For example, in the case of a Lie 2-algebra, the Jacobi identity is replaced by anisomorphismcalled aJacobiator.[1]
Thisalgebra-related article is astub. You can help Wikipedia byexpanding it.
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Module theoryis the branch of mathematics in whichmodulesare studied. This is a glossary of some terms of the subject.
See also:Glossary of linear algebra,Glossary of ring theory,Glossary of representation theory.
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Inmathematics, anon-emptycollectionofsetsR{\displaystyle {\mathcal {R}}}is called aδ-ring(pronounced "delta-ring") if it isclosedunderunion,relative complementation, and countableintersection. The name "delta-ring" originates from the German word for intersection, "Durchschnitt", which is meant to highlight the ring's closure under countable intersection, in contrast to a𝜎-ringwhich is closed under countable unions.
Afamily of setsR{\displaystyle {\mathcal {R}}}is called aδ-ringif it has all of the following properties:
If only the first two properties are satisfied, thenR{\displaystyle {\mathcal {R}}}is aring of setsbut not aδ-ring. Every𝜎-ringis aδ-ring, but not everyδ-ring is a𝜎-ring.
δ-rings can be used instead ofσ-algebrasin the development ofmeasure theoryif one does not wish to allow sets of infinite measure.
The familyK={S⊆R:Sis bounded}{\displaystyle {\mathcal {K}}=\{S\subseteq \mathbb {R} :S{\text{ is bounded}}\}}is aδ-ring but not a𝜎-ringbecause⋃n=1∞[0,n]{\textstyle \bigcup _{n=1}^{\infty }[0,n]}is not bounded.
Additionally, asemiringis aπ-systemwhere every complementB∖A{\displaystyle B\setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}Asemialgebrais a semiring where every complementΩ∖A{\displaystyle \Omega \setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}A,B,A1,A2,…{\displaystyle A,B,A_{1},A_{2},\ldots }are arbitrary elements ofF{\displaystyle {\mathcal {F}}}and it is assumed thatF≠∅.{\displaystyle {\mathcal {F}}\neq \varnothing .}
Thismathematical analysis–related article is astub. You can help Wikipedia byexpanding it.
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Inmathematics, afield of setsis amathematical structureconsisting of a pair(X,F){\displaystyle (X,{\mathcal {F}})}consisting of asetX{\displaystyle X}and afamilyF{\displaystyle {\mathcal {F}}}ofsubsetsofX{\displaystyle X}called analgebra overX{\displaystyle X}that contains theempty setas an element, and is closed under the operations of takingcomplementsinX,{\displaystyle X,}finiteunions, and finiteintersections.
Fields of sets should not be confused withfieldsinring theorynor withfields in physics. Similarly the term "algebra overX{\displaystyle X}" is used in the sense of a Boolean algebra and should not be confused withalgebras over fields or ringsin ring theory.
Fields of sets play an essential role in therepresentation theoryof Boolean algebras. Every Boolean algebra can be represented as a field of sets.
A field of sets is a pair(X,F){\displaystyle (X,{\mathcal {F}})}consisting of asetX{\displaystyle X}and afamilyF{\displaystyle {\mathcal {F}}}ofsubsetsofX,{\displaystyle X,}called analgebra overX,{\displaystyle X,}that has the following properties:
In other words,F{\displaystyle {\mathcal {F}}}forms asubalgebraof the power setBoolean algebraofX{\displaystyle X}(with the same identity elementX∈F{\displaystyle X\in {\mathcal {F}}}).
Many authors refer toF{\displaystyle {\mathcal {F}}}itself as a field of sets.
Elements ofX{\displaystyle X}are calledpointswhile elements ofF{\displaystyle {\mathcal {F}}}are calledcomplexesand are said to be theadmissible setsofX.{\displaystyle X.}
A field of sets(X,F){\displaystyle (X,{\mathcal {F}})}is called aσ-field of setsand the algebraF{\displaystyle {\mathcal {F}}}is called aσ-algebraif the following additional condition (4) is satisfied:
For an arbitrary setY,{\displaystyle Y,}itspower set2Y{\displaystyle 2^{Y}}(or, somewhat pedantically, the pair(Y,2Y){\displaystyle (Y,2^{Y})}of this set and its power set) is a field of sets. IfY{\displaystyle Y}is finite (namely,n{\displaystyle n}-element), then2Y{\displaystyle 2^{Y}}is finite (namely,2n{\displaystyle 2^{n}}-element). It appears that every finite field of sets (it means,(X,F){\displaystyle (X,{\mathcal {F}})}withF{\displaystyle {\mathcal {F}}}finite, whileX{\displaystyle X}may be infinite) admits a representation of the form(Y,2Y){\displaystyle (Y,2^{Y})}with finiteY{\displaystyle Y}; it means a functionf:X→Y{\displaystyle f:X\to Y}that establishes a one-to-one correspondence betweenF{\displaystyle {\mathcal {F}}}and2Y{\displaystyle 2^{Y}}viainverse image:S=f−1[B]={x∈X∣f(x)∈B}{\displaystyle S=f^{-1}[B]=\{x\in X\mid f(x)\in B\}}whereS∈F{\displaystyle S\in {\mathcal {F}}}andB∈2Y{\displaystyle B\in 2^{Y}}(that is,B⊂Y{\displaystyle B\subset Y}). One notable consequence: the number of complexes, if finite, is always of the form2n.{\displaystyle 2^{n}.}
To this end one choosesY{\displaystyle Y}to be the set of allatomsof the given field of sets, and definesf{\displaystyle f}byf(x)=A{\displaystyle f(x)=A}wheneverx∈A{\displaystyle x\in A}for a pointx∈X{\displaystyle x\in X}and a complexA∈F{\displaystyle A\in {\mathcal {F}}}that is an atom; the latter means that a nonempty subset ofA{\displaystyle A}different fromA{\displaystyle A}cannot be a complex.
In other words: the atoms are a partition ofX{\displaystyle X};Y{\displaystyle Y}is the correspondingquotient set; andf{\displaystyle f}is the corresponding canonical surjection.
Similarly, every finiteBoolean algebracan be represented as a power set – the power set of its set ofatoms; each element of the Boolean algebra corresponds to the set of atoms below it (the join of which is the element). Thispower set representationcan be constructed more generally for anycompleteatomicBoolean algebra.
In the case of Boolean algebras which are not complete and atomic we can still generalize the power set representation by considering fields of sets instead of whole power sets. To do this we first observe that the atoms of a finite Boolean algebra correspond to itsultrafiltersand that an atom is below an element of a finite Boolean algebra if and only if that element is contained in the ultrafilter corresponding to the atom. This leads us to construct a representation of a Boolean algebra by taking its set of ultrafilters and forming complexes by associating with each element of the Boolean algebra the set of ultrafilters containing that element. This construction does indeed produce a representation of the Boolean algebra as a field of sets and is known as theStone representation. It is the basis ofStone's representation theorem for Boolean algebrasand an example of a completion procedure inorder theorybased onidealsorfilters, similar toDedekind cuts.
Alternatively one can consider the set ofhomomorphismsonto the two element Boolean algebra and form complexes by associating each element of the Boolean algebra with the set of such homomorphisms that map it to the top element. (The approach is equivalent as the ultrafilters of a Boolean algebra are precisely the pre-images of the top elements under these homomorphisms.) With this approach one sees that Stone representation can also be regarded as a generalization of the representation of finite Boolean algebras bytruth tables.
These definitions arise from considering thetopologygenerated by the complexes of a field of sets. (It is just one of notable topologies on the given set of points; it often happens that another topology is given, with quite different properties, in particular, not zero-dimensional). Given a field of setsX=(X,F){\displaystyle \mathbf {X} =(X,{\mathcal {F}})}the complexes form abasefor a topology. We denote byT(X){\displaystyle T(\mathbf {X} )}the corresponding topological space,(X,T){\displaystyle (X,{\mathcal {T}})}whereT{\displaystyle {\mathcal {T}}}is the topology formed by taking arbitrary unions of complexes. Then
The Stone representation of a Boolean algebra is always separative and compact; the corresponding Boolean space is known as theStone spaceof the Boolean algebra. The clopen sets of the Stone space are then precisely the complexes of the Stone representation. The area of mathematics known asStone dualityis founded on the fact that the Stone representation of a Boolean algebra can be recovered purely from the corresponding Stone space whence adualityexists between Boolean algebras and Boolean spaces.
If an algebra over a set is closed under countableunions(hence also undercountableintersections), it is called asigma algebraand the corresponding field of sets is called ameasurable space. The complexes of a measurable space are calledmeasurable sets. TheLoomis-Sikorskitheorem provides a Stone-type duality between countably complete Boolean algebras (which may be calledabstract sigma algebras) and measurable spaces.
Ameasure spaceis a triple(X,F,μ){\displaystyle (X,{\mathcal {F}},\mu )}where(X,F){\displaystyle (X,{\mathcal {F}})}is a measurable space andμ{\displaystyle \mu }is ameasuredefined on it. Ifμ{\displaystyle \mu }is in fact aprobability measurewe speak of aprobability spaceand call its underlying measurable space asample space. The points of a sample space are calledsample pointsand represent potential outcomes while the measurable sets (complexes) are calledeventsand represent properties of outcomes for which we wish to assign probabilities. (Many use the termsample spacesimply for the underlying set of a probability space, particularly in the case where every subset is an event.) Measure spaces and probability spaces play a foundational role inmeasure theoryandprobability theoryrespectively.
In applications toPhysicswe often deal with measure spaces and probability spaces derived from rich mathematical structures such asinner product spacesortopological groupswhich already have a topology associated with them - this should not be confused with the topology generated by taking arbitrary unions of complexes.
Atopological field of setsis a triple(X,T,F){\displaystyle (X,{\mathcal {T}},{\mathcal {F}})}where(X,T){\displaystyle (X,{\mathcal {T}})}is atopological spaceand(X,F){\displaystyle (X,{\mathcal {F}})}is a field of sets which is closed under theclosure operatorofT{\displaystyle {\mathcal {T}}}or equivalently under theinterior operatori.e. the closure and interior of every complex is also a complex. In other words,F{\displaystyle {\mathcal {F}}}forms a subalgebra of the power setinterior algebraon(X,T).{\displaystyle (X,{\mathcal {T}}).}
Topological fields of sets play a fundamental role in the representation theory of interior algebras andHeyting algebras. These two classes of algebraic structures provide thealgebraic semanticsfor themodal logicS4(a formal mathematical abstraction ofepistemic logic) andintuitionistic logicrespectively. Topological fields of sets representing these algebraic structures provide a related topologicalsemanticsfor these logics.
Every interior algebra can be represented as a topological field of sets with the underlying Boolean algebra of the interior algebra corresponding to the complexes of the topological field of sets and the interior and closure operators of the interior algebra corresponding to those of the topology. EveryHeyting algebracan be represented by a topological field of sets with the underlying lattice of the Heyting algebra corresponding to the lattice of complexes of the topological field of sets that are open in the topology. Moreover the topological field of sets representing a Heyting algebra may be chosen so that the open complexes generate all the complexes as a Boolean algebra. These related representations provide a well defined mathematical apparatus for studying the relationship between truth modalities (possibly true vs necessarily true, studied in modal logic) and notions of provability and refutability (studied in intuitionistic logic) and is thus deeply connected to the theory ofmodal companionsofintermediate logics.
Given a topological space theclopensets trivially form a topological field of sets as each clopen set is its own interior and closure. The Stone representation of a Boolean algebra can be regarded as such a topological field of sets, however in general the topology of a topological field of sets can differ from the topology generated by taking arbitrary unions of complexes and in general the complexes of a topological field of sets need not be open or closed in the topology.
A topological field of sets is calledalgebraicif and only if there is a base for its topology consisting of complexes.
If a topological field of sets is both compact and algebraic then its topology is compact and its compact open sets are precisely the open complexes. Moreover, the open complexes form a base for the topology.
Topological fields of sets that are separative, compact and algebraic are calledStone fieldsand provide a generalization of the Stone representation of Boolean algebras. Given an interior algebra we can form the Stone representation of its underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated by the complexes corresponding to theopen elementsof the interior algebra (which form a base for a topology). These complexes are then precisely the open complexes and the construction produces a Stone field representing the interior algebra - theStone representation. (The topology of the Stone representation is also known as theMcKinsey–Tarski Stone topologyafter the mathematicians who first generalized Stone's result for Boolean algebras to interior algebras and should not be confused with the Stone topology of the underlying Boolean algebra of the interior algebra which will be a finer topology).
Apreorder fieldis a triple(X,≤,F){\displaystyle (X,\leq ,{\mathcal {F}})}where(X,≤){\displaystyle (X,\leq )}is apreordered setand(X,F){\displaystyle (X,{\mathcal {F}})}is a field of sets.
Like the topological fields of sets, preorder fields play an important role in the representation theory of interior algebras. Every interior algebra can be represented as a preorder field with its interior and closure operators corresponding to those of theAlexandrov topologyinduced by the preorder. In other words, for allS∈F{\displaystyle S\in {\mathcal {F}}}:Int(S)={x∈X:there exists ay∈Swithy≤x}{\displaystyle \mathrm {Int} (S)=\{x\in X:{\text{ there exists a }}y\in S{\text{ with }}y\leq x\}}andCl(S)={x∈X:there exists ay∈Swithx≤y}{\displaystyle \mathrm {Cl} (S)=\{x\in X:{\text{ there exists a }}y\in S{\text{ with }}x\leq y\}}
Similarly to topological fields of sets, preorder fields arise naturally in modal logic where the points represent thepossible worldsin theKripke semanticsof a theory in the modal logicS4, the preorder represents the accessibility relation on these possible worlds in this semantics, and the complexes represent sets of possible worlds in which individual sentences in the theory hold, providing a representation of theLindenbaum–Tarski algebraof the theory. They are a special case of thegeneral modal frameswhich are fields of sets with an additional accessibility relation providing representations of modal algebras.
A preorder field is calledalgebraic(ortight) if and only if it has a set of complexesA{\displaystyle {\mathcal {A}}}which determines the preorder in the following manner:x≤y{\displaystyle x\leq y}if and only if for every complexS∈A{\displaystyle S\in {\mathcal {A}}},x∈S{\displaystyle x\in S}impliesy∈S{\displaystyle y\in S}. The preorder fields obtained fromS4theories are always algebraic, the complexes determining the preorder being the sets of possible worlds in which the sentences of the theory closed under necessity hold.
A separative compact algebraic preorder field is said to becanonical. Given an interior algebra, by replacing the topology of its Stone representation with the correspondingcanonical preorder(specialization preorder) we obtain a representation of the interior algebra as a canonical preorder field. By replacing the preorder by its correspondingAlexandrov topologywe obtain an alternative representation of the interior algebra as a topological field of sets. (The topology of this "Alexandrov representation" is just theAlexandrov bi-coreflectionof the topology of the Stone representation.) While representation of modal algebras by general modal frames is possible for any normal modal algebra, it is only in the case of interior algebras (which correspond to the modal logicS4) that the general modal frame corresponds to topological field of sets in this manner.
The representation of interior algebras by preorder fields can be generalized to a representation theorem for arbitrary (normal)Boolean algebras with operators. For this we consider structures(X,(Ri)I,F){\displaystyle (X,(R_{i})_{I},{\mathcal {F}})}where(X,(Ri)I){\displaystyle (X,(R_{i})_{I})}is arelational structurei.e. a set with an indexed family ofrelationsdefined on it, and(X,F){\displaystyle (X,{\mathcal {F}})}is a field of sets. Thecomplex algebra(oralgebra of complexes) determined by a field of setsX=(X,(Ri)I,F){\displaystyle \mathbf {X} =(X,\left(R_{i}\right)_{I},{\mathcal {F}})}on a relational structure, is the Boolean algebra with operatorsC(X)=(F,∩,∪,′,∅,X,(fi)I){\displaystyle {\mathcal {C}}(\mathbf {X} )=({\mathcal {F}},\cap ,\cup ,\prime ,\emptyset ,X,(f_{i})_{I})}where for alli∈I,{\displaystyle i\in I,}ifRi{\displaystyle R_{i}}is a relation of arityn+1,{\displaystyle n+1,}thenfi{\displaystyle f_{i}}is an operator of arityn{\displaystyle n}and for allS1,…,Sn∈F{\displaystyle S_{1},\ldots ,S_{n}\in {\mathcal {F}}}fi(S1,…,Sn)={x∈X:there existx1∈S1,…,xn∈Snsuch thatRi(x1,…,xn,x)}{\displaystyle f_{i}(S_{1},\ldots ,S_{n})=\left\{x\in X:{\text{ there exist }}x_{1}\in S_{1},\ldots ,x_{n}\in S_{n}{\text{ such that }}R_{i}(x_{1},\ldots ,x_{n},x)\right\}}
This construction can be generalized to fields of sets on arbitraryalgebraic structureshaving bothoperatorsand relations as operators can be viewed as a special case of relations. IfF{\displaystyle {\mathcal {F}}}is the whole power set ofX{\displaystyle X}thenC(X){\displaystyle {\mathcal {C}}(\mathbf {X} )}is called afull complex algebraorpower algebra.
Every (normal) Boolean algebra with operators can be represented as a field of sets on a relational structure in the sense that it isisomorphicto the complex algebra corresponding to the field.
(Historically the termcomplexwas first used in the case where the algebraic structure was agroupand has its origins in 19th centurygroup theorywhere a subset of a group was called acomplex.)
Additionally, asemiringis aπ-systemwhere every complementB∖A{\displaystyle B\setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}Asemialgebrais a semiring where every complementΩ∖A{\displaystyle \Omega \setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}A,B,A1,A2,…{\displaystyle A,B,A_{1},A_{2},\ldots }are arbitrary elements ofF{\displaystyle {\mathcal {F}}}and it is assumed thatF≠∅.{\displaystyle {\mathcal {F}}\neq \varnothing .}
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ADynkin system,[1]named afterEugene Dynkin, is acollectionofsubsetsof another universalsetΩ{\displaystyle \Omega }satisfying a set ofaxiomsweaker than those of𝜎-algebra. Dynkin systems are sometimes referred to as𝜆-systems(Dynkin himself used this term) ord-system.[2]These set families have applications inmeasure theoryandprobability.
A major application of 𝜆-systems is theπ-𝜆 theorem, see below.
LetΩ{\displaystyle \Omega }be anonemptyset, and letD{\displaystyle D}be acollection of subsetsofΩ{\displaystyle \Omega }(that is,D{\displaystyle D}is a subset of thepower setofΩ{\displaystyle \Omega }). ThenD{\displaystyle D}is a Dynkin system if
It is easy to check[note 2]that any Dynkin systemD{\displaystyle D}satisfies:
Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class.[note 3]For this reason, a small group of authors have adopted conditions 4-6 to define a Dynkin system.
An important fact is that any Dynkin system that is also aπ-system(that is, closed under finite intersections) is a𝜎-algebra. This can be verified by noting that conditions 2 and 3 together with closure under finite intersections imply closure under finite unions, which in turn implies closure under countable unions.
Given any collectionJ{\displaystyle {\mathcal {J}}}of subsets ofΩ,{\displaystyle \Omega ,}there exists a unique Dynkin system denotedD{J}{\displaystyle D\{{\mathcal {J}}\}}which is minimal with respect to containingJ.{\displaystyle {\mathcal {J}}.}That is, ifD~{\displaystyle {\tilde {D}}}is any Dynkin system containingJ,{\displaystyle {\mathcal {J}},}thenD{J}⊆D~.{\displaystyle D\{{\mathcal {J}}\}\subseteq {\tilde {D}}.}D{J}{\displaystyle D\{{\mathcal {J}}\}}is called theDynkin system generated byJ.{\displaystyle {\mathcal {J}}.}For instance,D{∅}={∅,Ω}.{\displaystyle D\{\varnothing \}=\{\varnothing ,\Omega \}.}For another example, letΩ={1,2,3,4}{\displaystyle \Omega =\{1,2,3,4\}}andJ={1}{\displaystyle {\mathcal {J}}=\{1\}}; thenD{J}={∅,{1},{2,3,4},Ω}.{\displaystyle D\{{\mathcal {J}}\}=\{\varnothing ,\{1\},\{2,3,4\},\Omega \}.}
Sierpiński-Dynkin'sπ-𝜆 theorem:[3]IfP{\displaystyle P}is aπ-systemandD{\displaystyle D}is a Dynkin system withP⊆D,{\displaystyle P\subseteq D,}thenσ{P}⊆D.{\displaystyle \sigma \{P\}\subseteq D.}
In other words, the 𝜎-algebra generated byP{\displaystyle P}is contained inD.{\displaystyle D.}Thus a Dynkin system contains aπ-system if and only if it contains the 𝜎-algebra generated by thatπ-system.
One application of Sierpiński-Dynkin'sπ-𝜆 theorem is the uniqueness of a measure that evaluates the length of an interval (known as theLebesgue measure):
Let(Ω,B,ℓ){\displaystyle (\Omega ,{\mathcal {B}},\ell )}be theunit interval[0,1] with the Lebesgue measure onBorel sets. Letm{\displaystyle m}be anothermeasureonΩ{\displaystyle \Omega }satisfyingm[(a,b)]=b−a,{\displaystyle m[(a,b)]=b-a,}and letD{\displaystyle D}be the family of setsS{\displaystyle S}such thatm[S]=ℓ[S].{\displaystyle m[S]=\ell [S].}LetI:={(a,b),[a,b),(a,b],[a,b]:0<a≤b<1},{\displaystyle I:=\{(a,b),[a,b),(a,b],[a,b]:0<a\leq b<1\},}and observe thatI{\displaystyle I}is closed under finite intersections, thatI⊆D,{\displaystyle I\subseteq D,}and thatB{\displaystyle {\mathcal {B}}}is the 𝜎-algebra generated byI.{\displaystyle I.}It may be shown thatD{\displaystyle D}satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin'sπ-𝜆 Theorem it follows thatD{\displaystyle D}in fact includes all ofB{\displaystyle {\mathcal {B}}}, which is equivalent to showing that the Lebesgue measure is unique onB{\displaystyle {\mathcal {B}}}.
Theπ-𝜆 theorem motivates the common definition of theprobability distributionof arandom variableX:(Ω,F,P)→R{\displaystyle X:(\Omega ,{\mathcal {F}},\operatorname {P} )\to \mathbb {R} }in terms of itscumulative distribution function. Recall that the cumulative distribution of a random variable is defined asFX(a)=P[X≤a],a∈R,{\displaystyle F_{X}(a)=\operatorname {P} [X\leq a],\qquad a\in \mathbb {R} ,}whereas the seemingly more generallawof the variable is the probability measureLX(B)=P[X−1(B)]for allB∈B(R),{\displaystyle {\mathcal {L}}_{X}(B)=\operatorname {P} \left[X^{-1}(B)\right]\quad {\text{ for all }}B\in {\mathcal {B}}(\mathbb {R} ),}whereB(R){\displaystyle {\mathcal {B}}(\mathbb {R} )}is the Borel 𝜎-algebra. The random variablesX:(Ω,F,P)→R{\displaystyle X:(\Omega ,{\mathcal {F}},\operatorname {P} )\to \mathbb {R} }andY:(Ω~,F~,P~)→R{\displaystyle Y:({\tilde {\Omega }},{\tilde {\mathcal {F}}},{\tilde {\operatorname {P} }})\to \mathbb {R} }(on two possibly differentprobability spaces) areequal in distribution(orlaw), denoted byX=DY,{\displaystyle X\,{\stackrel {\mathcal {D}}{=}}\,Y,}if they have the same cumulative distribution functions; that is, ifFX=FY.{\displaystyle F_{X}=F_{Y}.}The motivation for the definition stems from the observation that ifFX=FY,{\displaystyle F_{X}=F_{Y},}then that is exactly to say thatLX{\displaystyle {\mathcal {L}}_{X}}andLY{\displaystyle {\mathcal {L}}_{Y}}agree on theπ-system{(−∞,a]:a∈R}{\displaystyle \{(-\infty ,a]:a\in \mathbb {R} \}}which generatesB(R),{\displaystyle {\mathcal {B}}(\mathbb {R} ),}and so by theexample above:LX=LY.{\displaystyle {\mathcal {L}}_{X}={\mathcal {L}}_{Y}.}
A similar result holds for the joint distribution of a random vector. For example, supposeX{\displaystyle X}andY{\displaystyle Y}are two random variables defined on the same probability space(Ω,F,P),{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} ),}with respectively generatedπ-systemsIX{\displaystyle {\mathcal {I}}_{X}}andIY.{\displaystyle {\mathcal {I}}_{Y}.}The joint cumulative distribution function of(X,Y){\displaystyle (X,Y)}isFX,Y(a,b)=P[X≤a,Y≤b]=P[X−1((−∞,a])∩Y−1((−∞,b])],for alla,b∈R.{\displaystyle F_{X,Y}(a,b)=\operatorname {P} [X\leq a,Y\leq b]=\operatorname {P} \left[X^{-1}((-\infty ,a])\cap Y^{-1}((-\infty ,b])\right],\quad {\text{ for all }}a,b\in \mathbb {R} .}
However,A=X−1((−∞,a])∈IX{\displaystyle A=X^{-1}((-\infty ,a])\in {\mathcal {I}}_{X}}andB=Y−1((−∞,b])∈IY.{\displaystyle B=Y^{-1}((-\infty ,b])\in {\mathcal {I}}_{Y}.}BecauseIX,Y={A∩B:A∈IX,andB∈IY}{\displaystyle {\mathcal {I}}_{X,Y}=\left\{A\cap B:A\in {\mathcal {I}}_{X},{\text{ and }}B\in {\mathcal {I}}_{Y}\right\}}is aπ-system generated by the random pair(X,Y),{\displaystyle (X,Y),}theπ-𝜆 theorem is used to show that the joint cumulative distribution function suffices to determine the joint law of(X,Y).{\displaystyle (X,Y).}In other words,(X,Y){\displaystyle (X,Y)}and(W,Z){\displaystyle (W,Z)}have the same distribution if and only if they have the same joint cumulative distribution function.
In the theory ofstochastic processes, two processes(Xt)t∈T,(Yt)t∈T{\displaystyle (X_{t})_{t\in T},(Y_{t})_{t\in T}}are known to be equal in distribution if and only if they agree on all finite-dimensional distributions; that is, for allt1,…,tn∈T,n∈N,{\displaystyle t_{1},\ldots ,t_{n}\in T,\,n\in \mathbb {N} ,}(Xt1,…,Xtn)=D(Yt1,…,Ytn).{\displaystyle \left(X_{t_{1}},\ldots ,X_{t_{n}}\right)\,{\stackrel {\mathcal {D}}{=}}\,\left(Y_{t_{1}},\ldots ,Y_{t_{n}}\right).}
The proof of this is another application of theπ-𝜆 theorem.[4]
This article incorporates material from Dynkin system onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.
Additionally, asemiringis aπ-systemwhere every complementB∖A{\displaystyle B\setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}Asemialgebrais a semiring where every complementΩ∖A{\displaystyle \Omega \setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}A,B,A1,A2,…{\displaystyle A,B,A_{1},A_{2},\ldots }are arbitrary elements ofF{\displaystyle {\mathcal {F}}}and it is assumed thatF≠∅.{\displaystyle {\mathcal {F}}\neq \varnothing .}
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Inmeasure theoryandprobability, themonotone class theoremconnects monotone classes and𝜎-algebras. The theorem says that the smallestmonotone classcontaining analgebra of setsG{\displaystyle G}is precisely the smallest𝜎-algebracontainingG.{\displaystyle G.}It is used as a type oftransfinite inductionto prove many other theorems, such asFubini's theorem.
Amonotone classis afamily(i.e. class)M{\displaystyle M}of sets that isclosedunder countable monotone unions and also under countable monotone intersections. Explicitly, this meansM{\displaystyle M}has the following properties:
Monotone class theorem for sets—LetG{\displaystyle G}be analgebra of setsand defineM(G){\displaystyle M(G)}to be the smallest monotone class containingG.{\displaystyle G.}ThenM(G){\displaystyle M(G)}is precisely the𝜎-algebragenerated byG{\displaystyle G}; that isσ(G)=M(G).{\displaystyle \sigma (G)=M(G).}
Monotone class theorem for functions—LetA{\displaystyle {\mathcal {A}}}be aπ-systemthat containsΩ{\displaystyle \Omega \,}and letH{\displaystyle {\mathcal {H}}}be a collection of functions fromΩ{\displaystyle \Omega }toR{\displaystyle \mathbb {R} }with the following properties:
ThenH{\displaystyle {\mathcal {H}}}contains all bounded functions that are measurable with respect toσ(A),{\displaystyle \sigma ({\mathcal {A}}),}which is the 𝜎-algebra generated byA.{\displaystyle {\mathcal {A}}.}
The following argument originates inRick Durrett's Probability: Theory and Examples.[1]
The assumptionΩ∈A,{\displaystyle \Omega \,\in {\mathcal {A}},}(2), and (3) imply thatG={A:1A∈H}{\displaystyle {\mathcal {G}}=\left\{A:\mathbf {1} _{A}\in {\mathcal {H}}\right\}}is a 𝜆-system.
By (1) and theπ−𝜆 theorem,σ(A)⊆G.{\displaystyle \sigma ({\mathcal {A}})\subseteq {\mathcal {G}}.}Statement (2) implies thatH{\displaystyle {\mathcal {H}}}contains all simple functions, and then (3) implies thatH{\displaystyle {\mathcal {H}}}contains all bounded functions measurable with respect toσ(A).{\displaystyle \sigma ({\mathcal {A}}).}
As a corollary, ifG{\displaystyle G}is aring of sets, then the smallest monotone class containing it coincides with the𝜎-ringofG.{\displaystyle G.}
By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a𝜎-algebra.
The monotone class theorem for functions can be a powerful tool that allows statements about particularly simple classes of functions to be generalized to arbitrary bounded and measurable functions.
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Inmathematics, aπ-system(orpi-system) on asetΩ{\displaystyle \Omega }is acollectionP{\displaystyle P}of certainsubsetsofΩ,{\displaystyle \Omega ,}such that
That is,P{\displaystyle P}is a non-empty family of subsets ofΩ{\displaystyle \Omega }that isclosedunder non-empty finiteintersections.[nb 1]The importance ofπ-systems arises from the fact that if twoprobability measuresagree on aπ-system, then they agree on the𝜎-algebragenerated by thatπ-system. Moreover, if other properties, such as equality of integrals, hold for theπ-system, then they hold for the generated 𝜎-algebra as well. This is the case whenever the collection of subsets for which the property holds is a𝜆-system.π-systems are also useful for checking independence of random variables.
This is desirable because in practice,π-systems are often simpler to work with than 𝜎-algebras. For example, it may be awkward to work with 𝜎-algebras generated by infinitely many setsσ(E1,E2,…).{\displaystyle \sigma (E_{1},E_{2},\ldots ).}So instead we may examine the union of all 𝜎-algebras generated by finitely many sets⋃nσ(E1,…,En).{\textstyle \bigcup _{n}\sigma (E_{1},\ldots ,E_{n}).}This forms aπ-system that generates the desired 𝜎-algebra. Another example is the collection of allintervalsof thereal line, along with the empty set, which is aπ-system that generates the very importantBorel 𝜎-algebraof subsets of the real line.
Aπ-systemis a non-empty collection of setsP{\displaystyle P}that is closed under non-empty finite intersections, which is equivalent toP{\displaystyle P}containing the intersection of any two of its elements.
If every set in thisπ-system is a subset ofΩ{\displaystyle \Omega }then it is called aπ-system onΩ.{\displaystyle \Omega .}
For any non-emptyfamilyΣ{\displaystyle \Sigma }of subsets ofΩ,{\displaystyle \Omega ,}there exists aπ-systemIΣ,{\displaystyle {\mathcal {I}}_{\Sigma },}called theπ-system generated byΣ{\displaystyle {\boldsymbol {\varSigma }}}, that is the unique smallestπ-system ofΩ{\displaystyle \Omega }containing every element ofΣ.{\displaystyle \Sigma .}It is equal to the intersection of allπ-systems containingΣ,{\displaystyle \Sigma ,}and can be explicitly described as the set of all possible non-empty finite intersections of elements ofΣ:{\displaystyle \Sigma :}{E1∩⋯∩En:1≤n∈NandE1,…,En∈Σ}.{\displaystyle \left\{E_{1}\cap \cdots \cap E_{n}~:~1\leq n\in \mathbb {N} {\text{ and }}E_{1},\ldots ,E_{n}\in \Sigma \right\}.}
A non-empty family of sets has thefinite intersection propertyif and only if theπ-system it generates does not contain the empty set as an element.
A𝜆-systemonΩ{\displaystyle \Omega }is a setD{\displaystyle D}of subsets ofΩ,{\displaystyle \Omega ,}satisfying
Whilst it is true that any 𝜎-algebra satisfies the properties of being both aπ-system and a 𝜆-system, it is not true that anyπ-system is a 𝜆-system, and moreover it is not true that anyπ-system is a 𝜎-algebra. However, a useful classification is that any set system which is both a 𝜆-system and aπ-system is a 𝜎-algebra. This is used as a step in proving theπ-𝜆 theorem.
LetD{\displaystyle D}be a 𝜆-system, and letI⊆D{\displaystyle {\mathcal {I}}\subseteq D}be aπ-system contained inD.{\displaystyle D.}Theπ-𝜆 theorem[1]states that the 𝜎-algebraσ(I){\displaystyle \sigma ({\mathcal {I}})}generated byI{\displaystyle {\mathcal {I}}}is contained inD:{\displaystyle D~:~}σ(I)⊆D.{\displaystyle \sigma ({\mathcal {I}})\subseteq D.}
Theπ-𝜆 theorem can be used to prove many elementarymeasure theoreticresults. For instance, it is used in proving the uniqueness claim of theCarathéodory extension theoremfor 𝜎-finite measures.[2]
Theπ-𝜆 theorem is closely related to themonotone class theorem, which provides a similar relationship between monotone classes and algebras, and can be used to derive many of the same results. Sinceπ-systems are simpler classes than algebras, it can be easier to identify the sets that are in them while, on the other hand, checking whether the property under consideration determines a 𝜆-system is often relatively easy. Despite the difference between the two theorems, theπ-𝜆 theorem is sometimes referred to as the monotone class theorem.[1]
Letμ1,μ2:F→R{\displaystyle \mu _{1},\mu _{2}:F\to \mathbb {R} }be two measures on the 𝜎-algebraF,{\displaystyle F,}and suppose thatF=σ(I){\displaystyle F=\sigma (I)}is generated by aπ-systemI.{\displaystyle I.}If
thenμ1=μ2.{\displaystyle \mu _{1}=\mu _{2}.}This is the uniqueness statement of the Carathéodory extension theorem for finite measures. If this result does not seem very remarkable, consider the fact that it usually is very difficult or even impossible to fully describe every set in the 𝜎-algebra, and so the problem of equating measures would be completely hopeless without such a tool.
Idea of the proof[2]Define the collection of setsD={A∈σ(I):μ1(A)=μ2(A)}.{\displaystyle D=\left\{A\in \sigma (I)\colon \mu _{1}(A)=\mu _{2}(A)\right\}.}By the first assumption,μ1{\displaystyle \mu _{1}}andμ2{\displaystyle \mu _{2}}agree onI{\displaystyle I}and thusI⊆D.{\displaystyle I\subseteq D.}By the second assumption,Ω∈D,{\displaystyle \Omega \in D,}and it can further be shown thatD{\displaystyle D}is a 𝜆-system. It follows from theπ-𝜆 theorem thatσ(I)⊆D⊆σ(I),{\displaystyle \sigma (I)\subseteq D\subseteq \sigma (I),}and soD=σ(I).{\displaystyle D=\sigma (I).}That is to say, the measures agree onσ(I).{\displaystyle \sigma (I).}
π-systems are more commonly used in the study of probability theory than in the general field of measure theory. This is primarily due to probabilistic notions such asindependence, though it may also be a consequence of the fact that theπ-𝜆 theorem was proven by the probabilistEugene Dynkin. Standard measure theory texts typically prove the same results viamonotone classes, rather thanπ-systems.
Theπ-𝜆 theorem motivates the common definition of theprobability distributionof arandom variableX:(Ω,F,P)→R{\displaystyle X:(\Omega ,{\mathcal {F}},\operatorname {P} )\to \mathbb {R} }in terms of itscumulative distribution function. Recall that the cumulative distribution of a random variable is defined asFX(a)=P[X≤a],a∈R,{\displaystyle F_{X}(a)=\operatorname {P} [X\leq a],\qquad a\in \mathbb {R} ,}whereas the seemingly more generallawof the variable is the probability measureLX(B)=P[X−1(B)]for allB∈B(R),{\displaystyle {\mathcal {L}}_{X}(B)=\operatorname {P} \left[X^{-1}(B)\right]\quad {\text{ for all }}B\in {\mathcal {B}}(\mathbb {R} ),}whereB(R){\displaystyle {\mathcal {B}}(\mathbb {R} )}is the Borel 𝜎-algebra. The random variablesX:(Ω,F,P)→R{\displaystyle X:(\Omega ,{\mathcal {F}},\operatorname {P} )\to \mathbb {R} }andY:(Ω~,F~,P~)→R{\displaystyle Y:({\tilde {\Omega }},{\tilde {\mathcal {F}}},{\tilde {\operatorname {P} }})\to \mathbb {R} }(on two possibly differentprobability spaces) areequal in distribution(orlaw), denoted byX=DY,{\displaystyle X\,{\stackrel {\mathcal {D}}{=}}\,Y,}if they have the same cumulative distribution functions; that is, ifFX=FY.{\displaystyle F_{X}=F_{Y}.}The motivation for the definition stems from the observation that ifFX=FY,{\displaystyle F_{X}=F_{Y},}then that is exactly to say thatLX{\displaystyle {\mathcal {L}}_{X}}andLY{\displaystyle {\mathcal {L}}_{Y}}agree on theπ-system{(−∞,a]:a∈R}{\displaystyle \{(-\infty ,a]:a\in \mathbb {R} \}}which generatesB(R),{\displaystyle {\mathcal {B}}(\mathbb {R} ),}and so by theexample above:LX=LY.{\displaystyle {\mathcal {L}}_{X}={\mathcal {L}}_{Y}.}
A similar result holds for the joint distribution of a random vector. For example, supposeX{\displaystyle X}andY{\displaystyle Y}are two random variables defined on the same probability space(Ω,F,P),{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} ),}with respectively generatedπ-systemsIX{\displaystyle {\mathcal {I}}_{X}}andIY.{\displaystyle {\mathcal {I}}_{Y}.}The joint cumulative distribution function of(X,Y){\displaystyle (X,Y)}isFX,Y(a,b)=P[X≤a,Y≤b]=P[X−1((−∞,a])∩Y−1((−∞,b])],for alla,b∈R.{\displaystyle F_{X,Y}(a,b)=\operatorname {P} [X\leq a,Y\leq b]=\operatorname {P} \left[X^{-1}((-\infty ,a])\cap Y^{-1}((-\infty ,b])\right],\quad {\text{ for all }}a,b\in \mathbb {R} .}
However,A=X−1((−∞,a])∈IX{\displaystyle A=X^{-1}((-\infty ,a])\in {\mathcal {I}}_{X}}andB=Y−1((−∞,b])∈IY.{\displaystyle B=Y^{-1}((-\infty ,b])\in {\mathcal {I}}_{Y}.}BecauseIX,Y={A∩B:A∈IX,andB∈IY}{\displaystyle {\mathcal {I}}_{X,Y}=\left\{A\cap B:A\in {\mathcal {I}}_{X},{\text{ and }}B\in {\mathcal {I}}_{Y}\right\}}is aπ-system generated by the random pair(X,Y),{\displaystyle (X,Y),}theπ-𝜆 theorem is used to show that the joint cumulative distribution function suffices to determine the joint law of(X,Y).{\displaystyle (X,Y).}In other words,(X,Y){\displaystyle (X,Y)}and(W,Z){\displaystyle (W,Z)}have the same distribution if and only if they have the same joint cumulative distribution function.
In the theory ofstochastic processes, two processes(Xt)t∈T,(Yt)t∈T{\displaystyle (X_{t})_{t\in T},(Y_{t})_{t\in T}}are known to be equal in distribution if and only if they agree on all finite-dimensional distributions; that is, for allt1,…,tn∈T,n∈N,{\displaystyle t_{1},\ldots ,t_{n}\in T,\,n\in \mathbb {N} ,}(Xt1,…,Xtn)=D(Yt1,…,Ytn).{\displaystyle \left(X_{t_{1}},\ldots ,X_{t_{n}}\right)\,{\stackrel {\mathcal {D}}{=}}\,\left(Y_{t_{1}},\ldots ,Y_{t_{n}}\right).}
The proof of this is another application of theπ-𝜆 theorem.[3]
The theory ofπ-system plays an important role in the probabilistic notion ofindependence. IfX{\displaystyle X}andY{\displaystyle Y}are two random variables defined on the same probability space(Ω,F,P){\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}then the random variables are independent if and only if theirπ-systemsIX,IY{\displaystyle {\mathcal {I}}_{X},{\mathcal {I}}_{Y}}satisfy for allA∈IX{\displaystyle A\in {\mathcal {I}}_{X}}andB∈IY,{\displaystyle B\in {\mathcal {I}}_{Y},}P[A∩B]=P[A]P[B],{\displaystyle \operatorname {P} [A\cap B]~=~\operatorname {P} [A]\operatorname {P} [B],}which is to say thatIX,IY{\displaystyle {\mathcal {I}}_{X},{\mathcal {I}}_{Y}}are independent. This actually is a special case of the use ofπ-systems for determining the distribution of(X,Y).{\displaystyle (X,Y).}
LetZ=(Z1,Z2),{\displaystyle Z=\left(Z_{1},Z_{2}\right),}whereZ1,Z2∼N(0,1){\displaystyle Z_{1},Z_{2}\sim {\mathcal {N}}(0,1)}areiidstandard normal random variables. Define the radius and argument (arctan) variablesR=Z12+Z22,Θ=tan−1(Z2/Z1).{\displaystyle R={\sqrt {Z_{1}^{2}+Z_{2}^{2}}},\qquad \Theta =\tan ^{-1}\left(Z_{2}/Z_{1}\right).}
ThenR{\displaystyle R}andΘ{\displaystyle \Theta }are independent random variables.
To prove this, it is sufficient to show that theπ-systemsIR,IΘ{\displaystyle {\mathcal {I}}_{R},{\mathcal {I}}_{\Theta }}are independent: that is, for allρ∈[0,∞){\displaystyle \rho \in [0,\infty )}andθ∈[0,2π],{\displaystyle \theta \in [0,2\pi ],}P[R≤ρ,Θ≤θ]=P[R≤ρ]P[Θ≤θ].{\displaystyle \operatorname {P} [R\leq \rho ,\Theta \leq \theta ]=\operatorname {P} [R\leq \rho ]\operatorname {P} [\Theta \leq \theta ].}
Confirming that this is the case is an exercise in changing variables. Fixρ∈[0,∞){\displaystyle \rho \in [0,\infty )}andθ∈[0,2π],{\displaystyle \theta \in [0,2\pi ],}then the probability can be expressed as an integral of the probability density function ofZ.{\displaystyle Z.}P[R≤ρ,Θ≤θ]=∫R≤ρ,Θ≤θ12πexp(−12(z12+z22))dz1dz2=∫0θ∫0ρ12πe−r22rdrdθ~=(∫0θ12πdθ~)(∫0ρe−r22rdr)=P[Θ≤θ]P[R≤ρ].{\displaystyle {\begin{aligned}\operatorname {P} [R\leq \rho ,\Theta \leq \theta ]&=\int _{R\leq \rho ,\,\Theta \leq \theta }{\frac {1}{2\pi }}\exp \left({-{\frac {1}{2}}(z_{1}^{2}+z_{2}^{2})}\right)dz_{1}\,dz_{2}\\[5pt]&=\int _{0}^{\theta }\int _{0}^{\rho }{\frac {1}{2\pi }}e^{-{\frac {r^{2}}{2}}}\;r\,dr\,d{\tilde {\theta }}\\[5pt]&=\left(\int _{0}^{\theta }{\frac {1}{2\pi }}\,d{\tilde {\theta }}\right)\;\left(\int _{0}^{\rho }e^{-{\frac {r^{2}}{2}}}\;r\,dr\right)\\[5pt]&=\operatorname {P} [\Theta \leq \theta ]\operatorname {P} [R\leq \rho ].\end{aligned}}}
Additionally, asemiringis aπ-systemwhere every complementB∖A{\displaystyle B\setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}Asemialgebrais a semiring where every complementΩ∖A{\displaystyle \Omega \setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}A,B,A1,A2,…{\displaystyle A,B,A_{1},A_{2},\ldots }are arbitrary elements ofF{\displaystyle {\mathcal {F}}}and it is assumed thatF≠∅.{\displaystyle {\mathcal {F}}\neq \varnothing .}
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https://en.wikipedia.org/wiki/Pi-system
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Inmathematical analysisand inprobability theory, aσ-algebra("sigma algebra") is part of the formalism for definingsets that can be measured. Incalculusandanalysis, for example, σ-algebras are used to define the concept of sets withareaorvolume. In probability theory, they are used to define events with a well-defined probability. In this way, σ-algebras help to formalize the notion ofsize.
In formal terms, a σ-algebra (alsoσ-field, where the σ comes from theGerman"Summe",[1]meaning "sum") on a setXis a nonempty collection Σ ofsubsetsofXclosedundercomplement, countableunions, and countableintersections. The ordered pair(X,Σ){\displaystyle (X,\Sigma )}is called ameasurable space.
The setXis understood to be an ambient space (such as the 2D plane or the set of outcomes when rolling a six-sided die {1,2,3,4,5,6}), and the collection Σ is a choice of subsets declared to have a well-defined size. The closure requirements for σ-algebras are designed to capture our intuitive ideas about how sizes combine: if there is a well-defined probability that an event occurs, there should be a well-defined probability that it does not occur (closure under complements); if several sets have a well-defined size, so should their combination (countable unions); if several events have a well-defined probability of occurring, so should the event where they all occur simultaneously (countable intersections).
The definition of σ-algebra resembles other mathematical structures such as atopology(which is required to be closed under all unions but only finite intersections, and which doesn't necessarily contain all complements of its sets) or aset algebra(which is closed only underfiniteunions and intersections).
IfX={a,b,c,d}{\displaystyle X=\{a,b,c,d\}}one possible σ-algebra onX{\displaystyle X}isΣ={∅,{a,b},{c,d},{a,b,c,d}},{\displaystyle \Sigma =\{\varnothing ,\{a,b\},\{c,d\},\{a,b,c,d\}\},}where∅{\displaystyle \varnothing }is theempty set. In general, a finite algebra is always a σ-algebra.
If{A1,A2,A3,…},{\displaystyle \{A_{1},A_{2},A_{3},\ldots \},}is a countablepartitionofX{\displaystyle X}then the collection of all unions of sets in the partition (including the empty set) is a σ-algebra.
A more useful example is the set of subsets of thereal lineformed by starting with allopen intervalsand adding in all countable unions, countable intersections, and relative complements and continuing this process (bytransfinite iterationthrough allcountable ordinals) until the relevant closure properties are achieved (a construction known as theBorel hierarchy).
There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets, and managing partial information characterized by sets.
AmeasureonX{\displaystyle X}is afunctionthat assigns a non-negativereal numberto subsets ofX;{\displaystyle X;}this can be thought of as making precise a notion of "size" or "volume" for sets. We want the size of the union of disjoint sets to be the sum of their individual sizes, even for an infinite sequence ofdisjoint sets.
One would like to assign a size toeverysubset ofX,{\displaystyle X,}but in many natural settings, this is not possible. For example, theaxiom of choiceimplies that when the size under consideration is the ordinary notion of length for subsets of the real line, then there exist sets for which no size exists, for example, theVitali sets. For this reason, one considers instead a smaller collection of privileged subsets ofX.{\displaystyle X.}These subsets will be called the measurable sets. They are closed under operations that one would expect for measurable sets, that is, the complement of a measurable set is a measurable set and the countable union of measurable sets is a measurable set. Non-empty collections of sets with these properties are called σ-algebras.
Many uses of measure, such as the probability concept ofalmost sure convergence, involvelimits of sequences of sets. For this, closure under countable unions and intersections is paramount. Set limits are defined as follows on σ-algebras.
The inner limit is always a subset of the outer limit:lim infn→∞An⊆lim supn→∞An.{\displaystyle \liminf _{n\to \infty }A_{n}~\subseteq ~\limsup _{n\to \infty }A_{n}.}If these two sets are equal then their limitlimn→∞An{\displaystyle \lim _{n\to \infty }A_{n}}exists and is equal to this common set:limn→∞An:=lim infn→∞An=lim supn→∞An.{\displaystyle \lim _{n\to \infty }A_{n}:=\liminf _{n\to \infty }A_{n}=\limsup _{n\to \infty }A_{n}.}
In much of probability, especially whenconditional expectationis involved, one is concerned with sets that represent only part of all the possible information that can be observed. This partial information can be characterized with a smaller σ-algebra which is a subset of the principal σ-algebra; it consists of the collection of subsets relevant only to and determined only by the partial information. Formally, ifΣ,Σ′{\displaystyle \Sigma ,\Sigma '}are σ-algebras onX{\displaystyle X}, thenΣ′{\displaystyle \Sigma '}is a sub σ-algebra ofΣ{\displaystyle \Sigma }ifΣ′⊆Σ{\displaystyle \Sigma '\subseteq \Sigma }.
TheBernoulli processprovides a simple example. This consists of a sequence of random coin flips, coming up Heads (H{\displaystyle H}) or Tails (T{\displaystyle T}), of unbounded length. Thesample spaceΩ consists of all possible infinite sequences ofH{\displaystyle H}orT:{\displaystyle T:}Ω={H,T}∞={(x1,x2,x3,…):xi∈{H,T},i≥1}.{\displaystyle \Omega =\{H,T\}^{\infty }=\{(x_{1},x_{2},x_{3},\dots ):x_{i}\in \{H,T\},i\geq 1\}.}
The full sigma algebra can be generated from an ascending sequence of subalgebras, by considering the information that might be obtained after observing some or all of the firstn{\displaystyle n}coin flips. This sequence of subalgebras is given byGn={A×{Ω}:A⊆{H,T}n}{\displaystyle {\mathcal {G}}_{n}=\{A\times \{\Omega \}:A\subseteq \{H,T\}^{n}\}}Each of these is finer than the last, and so can be ordered as afiltration
G0⊆G1⊆G2⊆⋯⊆G∞{\displaystyle {\mathcal {G}}_{0}\subseteq {\mathcal {G}}_{1}\subseteq {\mathcal {G}}_{2}\subseteq \cdots \subseteq {\mathcal {G}}_{\infty }}
The first subalgebraG0={∅,Ω}{\displaystyle {\mathcal {G}}_{0}=\{\varnothing ,\Omega \}}is the trivial algebra: it has only two elements in it, the empty set and the total space. The second subalgebraG1{\displaystyle {\mathcal {G}}_{1}}has four elements: the two inG0{\displaystyle {\mathcal {G}}_{0}}plus two more: sequences that start withH{\displaystyle H}and sequences that start withT{\displaystyle T}. Each subalgebra is finer than the last. Then{\displaystyle n}'th subalgebra contains2n+1{\displaystyle 2^{n+1}}elements: it divides the total spaceΩ{\displaystyle \Omega }into all of the possible sequences that might have been observed aftern{\displaystyle n}flips, including the possible non-observation of some of the flips.
The limiting algebraG∞{\displaystyle {\mathcal {G}}_{\infty }}is the smallest σ-algebra containing all the others. It is the algebra generated by theproduct topologyorweak topologyon the product space{H,T}∞.{\displaystyle \{H,T\}^{\infty }.}
LetX{\displaystyle X}be some set, and letP(X){\displaystyle P(X)}represent itspower set, the set of all subsets ofX{\displaystyle X}. Then a subsetΣ⊆P(X){\displaystyle \Sigma \subseteq P(X)}is called aσ-algebraif and only if it satisfies the following three properties:[2]
From these properties, it follows that the σ-algebra is also closed under countableintersections(by applyingDe Morgan's laws).
It also follows that theempty set∅{\displaystyle \varnothing }is inΣ,{\displaystyle \Sigma ,}since by(1)X{\displaystyle X}is inΣ{\displaystyle \Sigma }and(2)asserts that its complement, the empty set, is also inΣ.{\displaystyle \Sigma .}Moreover, since{X,∅}{\displaystyle \{X,\varnothing \}}satisfies all 3 conditions, it follows that{X,∅}{\displaystyle \{X,\varnothing \}}is the smallest possible σ-algebra onX.{\displaystyle X.}The largest possible σ-algebra onX{\displaystyle X}isP(X).{\displaystyle P(X).}
Elements of the σ-algebra are calledmeasurable sets. An ordered pair(X,Σ),{\displaystyle (X,\Sigma ),}whereX{\displaystyle X}is a set andΣ{\displaystyle \Sigma }is a σ-algebra overX,{\displaystyle X,}is called ameasurable space. A function between two measurable spaces is called ameasurable functionif thepreimageof every measurable set is measurable. The collection of measurable spaces forms acategory, with themeasurable functionsasmorphisms.Measuresare defined as certain types of functions from a σ-algebra to[0,∞].{\displaystyle [0,\infty ].}
A σ-algebra is both aπ-systemand aDynkin system(λ-system). The converse is true as well, by Dynkin's theorem (see below).
This theorem (or the relatedmonotone class theorem) is an essential tool for proving many results about properties of specific σ-algebras. It capitalizes on the nature of two simpler classes of sets, namely the following.
Dynkin's π-λ theorem says, ifP{\displaystyle P}is a π-system andD{\displaystyle D}is a Dynkin system that containsP,{\displaystyle P,}then the σ-algebraσ(P){\displaystyle \sigma (P)}generatedbyP{\displaystyle P}is contained inD.{\displaystyle D.}Since certain π-systems are relatively simple classes, it may not be hard to verify that all sets inP{\displaystyle P}enjoy the property under consideration while, on the other hand, showing that the collectionD{\displaystyle D}of all subsets with the property is a Dynkin system can also be straightforward. Dynkin's π-λ Theorem then implies that all sets inσ(P){\displaystyle \sigma (P)}enjoy the property, avoiding the task of checking it for an arbitrary set inσ(P).{\displaystyle \sigma (P).}
One of the most fundamental uses of the π-λ theorem is to show equivalence of separately defined measures or integrals. For example, it is used to equate a probability for a random variableX{\displaystyle X}with theLebesgue-Stieltjes integraltypically associated with computing the probability:P(X∈A)=∫AF(dx){\displaystyle \mathbb {P} (X\in A)=\int _{A}\,F(dx)}for allA{\displaystyle A}in the Borel σ-algebra onR,{\displaystyle \mathbb {R} ,}whereF(x){\displaystyle F(x)}is thecumulative distribution functionforX,{\displaystyle X,}defined onR,{\displaystyle \mathbb {R} ,}whileP{\displaystyle \mathbb {P} }is aprobability measure, defined on a σ-algebraΣ{\displaystyle \Sigma }of subsets of somesample spaceΩ.{\displaystyle \Omega .}
Suppose{Σα:α∈A}{\displaystyle \textstyle \left\{\Sigma _{\alpha }:\alpha \in {\mathcal {A}}\right\}}is a collection of σ-algebras on a spaceX.{\displaystyle X.}
Meet
The intersection of a collection of σ-algebras is a σ-algebra. To emphasize its character as a σ-algebra, it often is denoted by:⋀α∈AΣα.{\displaystyle \bigwedge _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }.}
Sketch of Proof:LetΣ∗{\displaystyle \Sigma ^{*}}denote the intersection. SinceX{\displaystyle X}is in everyΣα,Σ∗{\displaystyle \Sigma _{\alpha },\Sigma ^{*}}is not empty. Closure under complement and countable unions for everyΣα{\displaystyle \Sigma _{\alpha }}implies the same must be true forΣ∗.{\displaystyle \Sigma ^{*}.}Therefore,Σ∗{\displaystyle \Sigma ^{*}}is a σ-algebra.
Join
The union of a collection of σ-algebras is not generally a σ-algebra, or even an algebra, but itgeneratesa σ-algebra known as the join which typically is denoted⋁α∈AΣα=σ(⋃α∈AΣα).{\displaystyle \bigvee _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }=\sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right).}A π-system that generates the join isP={⋂i=1nAi:Ai∈Σαi,αi∈A,n≥1}.{\displaystyle {\mathcal {P}}=\left\{\bigcap _{i=1}^{n}A_{i}:A_{i}\in \Sigma _{\alpha _{i}},\alpha _{i}\in {\mathcal {A}},\ n\geq 1\right\}.}Sketch of Proof:By the casen=1,{\displaystyle n=1,}it is seen that eachΣα⊂P,{\displaystyle \Sigma _{\alpha }\subset {\mathcal {P}},}so⋃α∈AΣα⊆P.{\displaystyle \bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\subseteq {\mathcal {P}}.}This impliesσ(⋃α∈AΣα)⊆σ(P){\displaystyle \sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right)\subseteq \sigma ({\mathcal {P}})}by the definition of a σ-algebrageneratedby a collection of subsets. On the other hand,P⊆σ(⋃α∈AΣα){\displaystyle {\mathcal {P}}\subseteq \sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right)}which, by Dynkin's π-λ theorem, impliesσ(P)⊆σ(⋃α∈AΣα).{\displaystyle \sigma ({\mathcal {P}})\subseteq \sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right).}
SupposeY{\displaystyle Y}is a subset ofX{\displaystyle X}and let(X,Σ){\displaystyle (X,\Sigma )}be a measurable space.
Aσ-algebraΣ{\displaystyle \Sigma }is just aσ-ringthat contains the universal setX.{\displaystyle X.}[3]Aσ-ring need not be aσ-algebra, as for example measurable subsets of zero Lebesgue measure in the real line are aσ-ring, but not aσ-algebra since the real line has infinite measure and thus cannot be obtained by their countable union. If, instead of zero measure, one takes measurable subsets of finite Lebesgue measure, those are aringbut not aσ-ring, since the real line can be obtained by their countable union yet its measure is not finite.
σ-algebras are sometimes denoted usingcalligraphiccapital letters, or theFraktur typeface. Thus(X,Σ){\displaystyle (X,\Sigma )}may be denoted as(X,F){\displaystyle \scriptstyle (X,\,{\mathcal {F}})}or(X,F).{\displaystyle \scriptstyle (X,\,{\mathfrak {F}}).}
Aseparableσ{\displaystyle \sigma }-algebra(orseparableσ{\displaystyle \sigma }-field) is aσ{\displaystyle \sigma }-algebraF{\displaystyle {\mathcal {F}}}that is aseparable spacewhen considered as ametric spacewithmetricρ(A,B)=μ(A△B){\displaystyle \rho (A,B)=\mu (A{\mathbin {\triangle }}B)}forA,B∈F{\displaystyle A,B\in {\mathcal {F}}}and a given finitemeasureμ{\displaystyle \mu }(and with△{\displaystyle \triangle }being thesymmetric differenceoperator).[4]Anyσ{\displaystyle \sigma }-algebra generated by acountablecollection ofsetsis separable, but the converse need not hold. For example, the Lebesgueσ{\displaystyle \sigma }-algebra is separable (since every Lebesgue measurable set is equivalent to some Borel set) but not countably generated (since its cardinality is higher than continuum).
A separable measure space has a naturalpseudometricthat renders itseparableas apseudometric space. The distance between two sets is defined as the measure of thesymmetric differenceof the two sets. The symmetric difference of two distinct sets can have measure zero; hence the pseudometric as defined above need not to be a true metric. However, if sets whose symmetric difference has measure zero are identified into a singleequivalence class, the resultingquotient setcan be properly metrized by the induced metric. If the measure space is separable, it can be shown that the corresponding metric space is, too.
LetX{\displaystyle X}be any set.
Astopping timeτ{\displaystyle \tau }can define aσ{\displaystyle \sigma }-algebraFτ,{\displaystyle {\mathcal {F}}_{\tau },}the
so-calledstopping time sigma-algebra, which in afiltered probability spacedescribes the information up to the random timeτ{\displaystyle \tau }in the sense that, if the filtered probability space is interpreted as a random experiment, the maximum information that can be found out about the experiment from arbitrarily often repeating it until the timeτ{\displaystyle \tau }isFτ.{\displaystyle {\mathcal {F}}_{\tau }.}[5]
LetF{\displaystyle F}be an arbitrary family of subsets ofX.{\displaystyle X.}Then there exists a unique smallest σ-algebra which contains every set inF{\displaystyle F}(even thoughF{\displaystyle F}may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containingF.{\displaystyle F.}(See intersections of σ-algebras above.) This σ-algebra is denotedσ(F){\displaystyle \sigma (F)}and is calledthe σ-algebra generated byF.{\displaystyle F.}
IfF{\displaystyle F}is empty, thenσ(∅)={∅,X}.{\displaystyle \sigma (\varnothing )=\{\varnothing ,X\}.}Otherwiseσ(F){\displaystyle \sigma (F)}consists of all the subsets ofX{\displaystyle X}that can be made from elements ofF{\displaystyle F}by a countable number of complement, union and intersection operations.
For a simple example, consider the setX={1,2,3}.{\displaystyle X=\{1,2,3\}.}Then the σ-algebra generated by the single subset{1}{\displaystyle \{1\}}isσ({1})={∅,{1},{2,3},{1,2,3}}.{\displaystyle \sigma (\{1\})=\{\varnothing ,\{1\},\{2,3\},\{1,2,3\}\}.}By anabuse of notation, when a collection of subsets contains only one element,A,{\displaystyle A,}σ(A){\displaystyle \sigma (A)}may be written instead ofσ({A});{\displaystyle \sigma (\{A\});}in the prior exampleσ({1}){\displaystyle \sigma (\{1\})}instead ofσ({{1}}).{\displaystyle \sigma (\{\{1\}\}).}Indeed, usingσ(A1,A2,…){\displaystyle \sigma \left(A_{1},A_{2},\ldots \right)}to meanσ({A1,A2,…}){\displaystyle \sigma \left(\left\{A_{1},A_{2},\ldots \right\}\right)}is also quite common.
There are many families of subsets that generate useful σ-algebras. Some of these are presented here.
Iff{\displaystyle f}is a function from a setX{\displaystyle X}to a setY{\displaystyle Y}andB{\displaystyle B}is aσ{\displaystyle \sigma }-algebra of subsets ofY,{\displaystyle Y,}then theσ{\displaystyle \sigma }-algebra generated by the functionf,{\displaystyle f,}denoted byσ(f),{\displaystyle \sigma (f),}is the collection of all inverse imagesf−1(S){\displaystyle f^{-1}(S)}of the setsS{\displaystyle S}inB.{\displaystyle B.}That is,σ(f)={f−1(S):S∈B}.{\displaystyle \sigma (f)=\left\{f^{-1}(S)\,:\,S\in B\right\}.}
A functionf{\displaystyle f}from a setX{\displaystyle X}to a setY{\displaystyle Y}ismeasurablewith respect to a σ-algebraΣ{\displaystyle \Sigma }of subsets ofX{\displaystyle X}if and only ifσ(f){\displaystyle \sigma (f)}is a subset ofΣ.{\displaystyle \Sigma .}
One common situation, and understood by default ifB{\displaystyle B}is not specified explicitly, is whenY{\displaystyle Y}is ametricortopological spaceandB{\displaystyle B}is the collection ofBorel setsonY.{\displaystyle Y.}
Iff{\displaystyle f}is a function fromX{\displaystyle X}toRn{\displaystyle \mathbb {R} ^{n}}thenσ(f){\displaystyle \sigma (f)}is generated by the family of subsets which are inverse images of intervals/rectangles inRn:{\displaystyle \mathbb {R} ^{n}:}σ(f)=σ({f−1([a1,b1]×⋯×[an,bn]):ai,bi∈R}).{\displaystyle \sigma (f)=\sigma \left(\left\{f^{-1}(\left[a_{1},b_{1}\right]\times \cdots \times \left[a_{n},b_{n}\right]):a_{i},b_{i}\in \mathbb {R} \right\}\right).}
A useful property is the following. Assumef{\displaystyle f}is a measurable map from(X,ΣX){\displaystyle \left(X,\Sigma _{X}\right)}to(S,ΣS){\displaystyle \left(S,\Sigma _{S}\right)}andg{\displaystyle g}is a measurable map from(X,ΣX){\displaystyle \left(X,\Sigma _{X}\right)}to(T,ΣT).{\displaystyle \left(T,\Sigma _{T}\right).}If there exists a measurable maph{\displaystyle h}from(T,ΣT){\displaystyle \left(T,\Sigma _{T}\right)}to(S,ΣS){\displaystyle \left(S,\Sigma _{S}\right)}such thatf(x)=h(g(x)){\displaystyle f(x)=h(g(x))}for allx,{\displaystyle x,}thenσ(f)⊆σ(g).{\displaystyle \sigma (f)\subseteq \sigma (g).}IfS{\displaystyle S}is finite or countably infinite or, more generally,(S,ΣS){\displaystyle \left(S,\Sigma _{S}\right)}is astandard Borel space(for example, a separable complete metric space with its associated Borel sets), then the converse is also true.[6]Examples of standard Borel spaces includeRn{\displaystyle \mathbb {R} ^{n}}with its Borel sets andR∞{\displaystyle \mathbb {R} ^{\infty }}with the cylinder σ-algebra described below.
An important example is theBorel algebraover anytopological space: the σ-algebra generated by theopen sets(or, equivalently, by theclosed sets). This σ-algebra is not, in general, the whole power set. For a non-trivial example that is not a Borel set, see theVitali setorNon-Borel sets.
On theEuclidean spaceRn,{\displaystyle \mathbb {R} ^{n},}another σ-algebra is of importance: that of allLebesgue measurablesets. This σ-algebra contains more sets than the Borel σ-algebra onRn{\displaystyle \mathbb {R} ^{n}}and is preferred inintegrationtheory, as it gives acomplete measure space.
Let(X1,Σ1){\displaystyle \left(X_{1},\Sigma _{1}\right)}and(X2,Σ2){\displaystyle \left(X_{2},\Sigma _{2}\right)}be two measurable spaces. The σ-algebra for the correspondingproduct spaceX1×X2{\displaystyle X_{1}\times X_{2}}is called theproduct σ-algebraand is defined byΣ1×Σ2=σ({B1×B2:B1∈Σ1,B2∈Σ2}).{\displaystyle \Sigma _{1}\times \Sigma _{2}=\sigma \left(\left\{B_{1}\times B_{2}:B_{1}\in \Sigma _{1},B_{2}\in \Sigma _{2}\right\}\right).}
Observe that{B1×B2:B1∈Σ1,B2∈Σ2}{\displaystyle \{B_{1}\times B_{2}:B_{1}\in \Sigma _{1},B_{2}\in \Sigma _{2}\}}is a π-system.
The Borel σ-algebra forRn{\displaystyle \mathbb {R} ^{n}}is generated by half-infinite rectangles and by finite rectangles. For example,B(Rn)=σ({(−∞,b1]×⋯×(−∞,bn]:bi∈R})=σ({(a1,b1]×⋯×(an,bn]:ai,bi∈R}).{\displaystyle {\mathcal {B}}(\mathbb {R} ^{n})=\sigma \left(\left\{(-\infty ,b_{1}]\times \cdots \times (-\infty ,b_{n}]:b_{i}\in \mathbb {R} \right\}\right)=\sigma \left(\left\{\left(a_{1},b_{1}\right]\times \cdots \times \left(a_{n},b_{n}\right]:a_{i},b_{i}\in \mathbb {R} \right\}\right).}
For each of these two examples, the generating family is a π-system.
SupposeX⊆RT={f:f(t)∈R,t∈T}{\displaystyle X\subseteq \mathbb {R} ^{\mathbb {T} }=\{f:f(t)\in \mathbb {R} ,\ t\in \mathbb {T} \}}
is a set of real-valued functions. LetB(R){\displaystyle {\mathcal {B}}(\mathbb {R} )}denote the Borel subsets ofR.{\displaystyle \mathbb {R} .}Acylinder subsetofX{\displaystyle X}is a finitely restricted set defined asCt1,…,tn(B1,…,Bn)={f∈X:f(ti)∈Bi,1≤i≤n}.{\displaystyle C_{t_{1},\dots ,t_{n}}(B_{1},\dots ,B_{n})=\left\{f\in X:f(t_{i})\in B_{i},1\leq i\leq n\right\}.}
Each{Ct1,…,tn(B1,…,Bn):Bi∈B(R),1≤i≤n}{\displaystyle \left\{C_{t_{1},\dots ,t_{n}}\left(B_{1},\dots ,B_{n}\right):B_{i}\in {\mathcal {B}}(\mathbb {R} ),1\leq i\leq n\right\}}is a π-system that generates a σ-algebraΣt1,…,tn.{\displaystyle \textstyle \Sigma _{t_{1},\dots ,t_{n}}.}Then the family of subsetsFX=⋃n=1∞⋃ti∈T,i≤nΣt1,…,tn{\displaystyle {\mathcal {F}}_{X}=\bigcup _{n=1}^{\infty }\bigcup _{t_{i}\in \mathbb {T} ,i\leq n}\Sigma _{t_{1},\dots ,t_{n}}}is an algebra that generates thecylinder σ-algebraforX.{\displaystyle X.}This σ-algebra is a subalgebra of the Borel σ-algebra determined by theproduct topologyofRT{\displaystyle \mathbb {R} ^{\mathbb {T} }}restricted toX.{\displaystyle X.}
An important special case is whenT{\displaystyle \mathbb {T} }is the set of natural numbers andX{\displaystyle X}is a set of real-valued sequences. In this case, it suffices to consider the cylinder setsCn(B1,…,Bn)=(B1×⋯×Bn×R∞)∩X={(x1,x2,…,xn,xn+1,…)∈X:xi∈Bi,1≤i≤n},{\displaystyle C_{n}\left(B_{1},\dots ,B_{n}\right)=\left(B_{1}\times \cdots \times B_{n}\times \mathbb {R} ^{\infty }\right)\cap X=\left\{\left(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots \right)\in X:x_{i}\in B_{i},1\leq i\leq n\right\},}for whichΣn=σ({Cn(B1,…,Bn):Bi∈B(R),1≤i≤n}){\displaystyle \Sigma _{n}=\sigma \left(\{C_{n}\left(B_{1},\dots ,B_{n}\right):B_{i}\in {\mathcal {B}}(\mathbb {R} ),1\leq i\leq n\}\right)}is a non-decreasing sequence of σ-algebras.
The ball σ-algebra is the smallest σ-algebra containing all the open (and/or closed) balls. This is never larger than theBorel σ-algebra. Note that the two σ-algebra are equal for separable spaces. For some nonseparable spaces, some maps are ball measurable even though they are not Borel measurable, making use of the ball σ-algebra useful in the analysis of such maps.[7]
Suppose(Ω,Σ,P){\displaystyle (\Omega ,\Sigma ,\mathbb {P} )}is aprobability space. IfY:Ω→Rn{\displaystyle \textstyle Y:\Omega \to \mathbb {R} ^{n}}is measurable with respect to the Borel σ-algebra onRn{\displaystyle \mathbb {R} ^{n}}thenY{\displaystyle Y}is called arandom variable(n=1{\displaystyle n=1}) orrandom vector(n>1{\displaystyle n>1}). The σ-algebra generated byY{\displaystyle Y}isσ(Y)={Y−1(A):A∈B(Rn)}.{\displaystyle \sigma (Y)=\left\{Y^{-1}(A):A\in {\mathcal {B}}\left(\mathbb {R} ^{n}\right)\right\}.}
Suppose(Ω,Σ,P){\displaystyle (\Omega ,\Sigma ,\mathbb {P} )}is aprobability spaceandRT{\displaystyle \mathbb {R} ^{\mathbb {T} }}is the set of real-valued functions onT.{\displaystyle \mathbb {T} .}IfY:Ω→X⊆RT{\displaystyle \textstyle Y:\Omega \to X\subseteq \mathbb {R} ^{\mathbb {T} }}is measurable with respect to the cylinder σ-algebraσ(FX){\displaystyle \sigma \left({\mathcal {F}}_{X}\right)}(see above) forX{\displaystyle X}thenY{\displaystyle Y}is called astochastic processorrandom process. The σ-algebra generated byY{\displaystyle Y}isσ(Y)={Y−1(A):A∈σ(FX)}=σ({Y−1(A):A∈FX}),{\displaystyle \sigma (Y)=\left\{Y^{-1}(A):A\in \sigma \left({\mathcal {F}}_{X}\right)\right\}=\sigma \left(\left\{Y^{-1}(A):A\in {\mathcal {F}}_{X}\right\}\right),}the σ-algebra generated by the inverse images of cylinder sets.
Additionally, asemiringis aπ-systemwhere every complementB∖A{\displaystyle B\setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}Asemialgebrais a semiring where every complementΩ∖A{\displaystyle \Omega \setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}A,B,A1,A2,…{\displaystyle A,B,A_{1},A_{2},\ldots }are arbitrary elements ofF{\displaystyle {\mathcal {F}}}and it is assumed thatF≠∅.{\displaystyle {\mathcal {F}}\neq \varnothing .}
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Inmathematics, particularlymeasure theory, a𝜎-ideal, orsigma ideal, of aσ-algebra(𝜎, read "sigma") is asubsetwith certain desirableclosureproperties. It is a special type ofideal. Its most frequent application is inprobability theory.[citation needed]
Let(X,Σ){\displaystyle (X,\Sigma )}be ameasurable space(meaningΣ{\displaystyle \Sigma }is a 𝜎-algebra of subsets ofX{\displaystyle X}). A subsetN{\displaystyle N}ofΣ{\displaystyle \Sigma }is a 𝜎-ideal if the following properties are satisfied:
Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. The concept of 𝜎-ideal isdualto that of acountablycomplete(𝜎-)filter.
If ameasureμ{\displaystyle \mu }is given on(X,Σ),{\displaystyle (X,\Sigma ),}the set ofμ{\displaystyle \mu }-negligible sets(S∈Σ{\displaystyle S\in \Sigma }such thatμ(S)=0{\displaystyle \mu (S)=0}) is a 𝜎-ideal.
The notion can be generalized topreorders(P,≤,0){\displaystyle (P,\leq ,0)}with a bottom element0{\displaystyle 0}as follows:I{\displaystyle I}is a 𝜎-ideal ofP{\displaystyle P}just when
(i')0∈I,{\displaystyle 0\in I,}
(ii')x≤yandy∈I{\displaystyle x\leq y{\text{ and }}y\in I}impliesx∈I,{\displaystyle x\in I,}and
(iii') given a sequencex1,x2,…∈I,{\displaystyle x_{1},x_{2},\ldots \in I,}there exists somey∈I{\displaystyle y\in I}such thatxn≤y{\displaystyle x_{n}\leq y}for eachn.{\displaystyle n.}
ThusI{\displaystyle I}contains the bottom element, is downward closed, and satisfies a countable analogue of the property of beingupwards directed.
A𝜎-idealof a setX{\displaystyle X}is a 𝜎-ideal of the power set ofX.{\displaystyle X.}That is, when no 𝜎-algebra is specified, then one simply takes the full power set of the underlying set. For example, themeager subsetsof a topological space are those in the 𝜎-ideal generated by the collection of closed subsets with empty interior.
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https://en.wikipedia.org/wiki/Sigma-ideal
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Inmathematics, a nonempty collection ofsetsis called a𝜎-ring(pronouncedsigma-ring) if it isclosedunder countableunionandrelative complementation.
LetR{\displaystyle {\mathcal {R}}}be a nonemptycollection of sets. ThenR{\displaystyle {\mathcal {R}}}is a𝜎-ringif:
These two properties imply:⋂n=1∞An∈R{\displaystyle \bigcap _{n=1}^{\infty }A_{n}\in {\mathcal {R}}}wheneverA1,A2,…{\displaystyle A_{1},A_{2},\ldots }are elements ofR.{\displaystyle {\mathcal {R}}.}
This is because⋂n=1∞An=A1∖⋃n=2∞(A1∖An).{\displaystyle \bigcap _{n=1}^{\infty }A_{n}=A_{1}\setminus \bigcup _{n=2}^{\infty }\left(A_{1}\setminus A_{n}\right).}
Every 𝜎-ring is aδ-ringbut there exist δ-rings that are not 𝜎-rings.
If the first property is weakened to closure under finite union (that is,A∪B∈R{\displaystyle A\cup B\in {\mathcal {R}}}wheneverA,B∈R{\displaystyle A,B\in {\mathcal {R}}}) but not countable union, thenR{\displaystyle {\mathcal {R}}}is aringbut not a 𝜎-ring.
𝜎-rings can be used instead of𝜎-fields(𝜎-algebras) in the development ofmeasureandintegrationtheory, if one does not wish to require that theuniversal setbe measurable. Every 𝜎-field is also a 𝜎-ring, but a 𝜎-ring need not be a 𝜎-field.
A 𝜎-ringR{\displaystyle {\mathcal {R}}}that is a collection of subsets ofX{\displaystyle X}induces a𝜎-fieldforX.{\displaystyle X.}DefineA={E⊆X:E∈RorEc∈R}.{\displaystyle {\mathcal {A}}=\{E\subseteq X:E\in {\mathcal {R}}\ {\text{or}}\ E^{c}\in {\mathcal {R}}\}.}ThenA{\displaystyle {\mathcal {A}}}is a 𝜎-field over the setX{\displaystyle X}- to check closure under countable union, recall aσ{\displaystyle \sigma }-ring is closed under countable intersections. In factA{\displaystyle {\mathcal {A}}}is the minimal 𝜎-field containingR{\displaystyle {\mathcal {R}}}since it must be contained in every 𝜎-field containingR.{\displaystyle {\mathcal {R}}.}
Additionally, asemiringis aπ-systemwhere every complementB∖A{\displaystyle B\setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}Asemialgebrais a semiring where every complementΩ∖A{\displaystyle \Omega \setminus A}is equal to a finitedisjoint unionof sets inF.{\displaystyle {\mathcal {F}}.}A,B,A1,A2,…{\displaystyle A,B,A_{1},A_{2},\ldots }are arbitrary elements ofF{\displaystyle {\mathcal {F}}}and it is assumed thatF≠∅.{\displaystyle {\mathcal {F}}\neq \varnothing .}
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The term "information algebra" refers to mathematical techniques ofinformation processing. Classicalinformation theorygoes back toClaude Shannon. It is a theory of information transmission, looking at communication and storage. However, it has not been considered so far that information comes from different sources and that it is therefore usually combined. It has furthermore been neglected in classical information theory that one wants to extract those parts out of a piece of information that are relevant to specific questions.
A mathematical phrasing of these operations leads to analgebra of information, describing basic modes of information processing. Such an algebra involves several formalisms ofcomputer science, which seem to be different on the surface: relational databases, multiple systems of formal logic or numerical problems of linear algebra. It allows the development of generic procedures of information processing and thus a unification of basic methods of computer science, in particular ofdistributed information processing.
Information relates to precise questions, comes from different sources, must be aggregated, and can be focused on questions of interest. Starting from these considerations, information algebras (Kohlas 2003) aretwo-sortedalgebras(Φ,D){\displaystyle (\Phi ,D)}:
WhereΦ{\displaystyle \Phi }is asemigroup, representing combination or aggregation of information, and
D{\displaystyle D}is alatticeofdomains(related to questions) whosepartial orderreflects the granularity of the domain or the question, and amixed operationrepresenting focusing or extraction of information.
More precisely, in the two-sorted algebra(Φ,D){\displaystyle (\Phi ,D)}, the following operations are defined
Additionally, inD{\displaystyle D}the usual lattice operations (meet and join) are defined.
The axioms of the two-sorted algebra(Φ,D){\displaystyle (\Phi ,D)}, in addition to the axioms of the latticeD{\displaystyle D}:
To focus an information onx{\displaystyle x}combined with another information to domainx{\displaystyle x}, one may as well first focus the second information tox{\displaystyle x}and then combine.
To focus an information onx{\displaystyle x}andy{\displaystyle y}, one may focus it tox∧y{\displaystyle x\wedge y}.
An information combined with a part of itself gives nothing new.
Each information refers to at least one domain (question).
A two-sorted algebra(Φ,D){\displaystyle (\Phi ,D)}satisfying these axioms is called anInformation Algebra.
A partial order of information can be introduced by definingϕ≤ψ{\displaystyle \phi \leq \psi }ifϕ⊗ψ=ψ{\displaystyle \phi \otimes \psi =\psi }. This means thatϕ{\displaystyle \phi }is less informative thanψ{\displaystyle \psi }if it adds no new information toψ{\displaystyle \psi }. The semigroupΦ{\displaystyle \Phi }is a semilattice relative to this order, i.e.ϕ⊗ψ=ϕ∨ψ{\displaystyle \phi \otimes \psi =\phi \vee \psi }. Relative to any domain (question)x∈D{\displaystyle x\in D}a partial order can be introduced by definingϕ≤xψ{\displaystyle \phi \leq _{x}\psi }ifϕ⇒x≤ψ⇒x{\displaystyle \phi ^{\Rightarrow x}\leq \psi ^{\Rightarrow x}}. It represents the order of information content ofϕ{\displaystyle \phi }andψ{\displaystyle \psi }relative to the domain (question)x{\displaystyle x}.
The pairs(ϕ,x){\displaystyle (\phi ,x)\ }, whereϕ∈Φ{\displaystyle \phi \in \Phi }andx∈D{\displaystyle x\in D}such thatϕ⇒x=ϕ{\displaystyle \phi ^{\Rightarrow x}=\phi }form alabeled Information Algebra. More precisely, in the two-sorted algebra(Φ,D){\displaystyle (\Phi ,D)\ }, the following operations are defined
Here follows an incomplete list of instances of information algebras:
LetA{\displaystyle {\mathcal {A}}}be a set of symbols, calledattributes(orcolumn names). For eachα∈A{\displaystyle \alpha \in {\mathcal {A}}}letUα{\displaystyle U_{\alpha }}be a non-empty set, the set of all possible values of the attributeα{\displaystyle \alpha }. For example, ifA={name,age,income}{\displaystyle {\mathcal {A}}=\{{\texttt {name}},{\texttt {age}},{\texttt {income}}\}}, thenUname{\displaystyle U_{\texttt {name}}}could
be the set of strings, whereasUage{\displaystyle U_{\texttt {age}}}andUincome{\displaystyle U_{\texttt {income}}}are both the set of non-negative integers.
Letx⊆A{\displaystyle x\subseteq {\mathcal {A}}}. Anx{\displaystyle x}-tupleis a functionf{\displaystyle f}so thatdom(f)=x{\displaystyle {\hbox{dom}}(f)=x}andf(α)∈Uα{\displaystyle f(\alpha )\in U_{\alpha }}for eachα∈x{\displaystyle \alpha \in x}The set
of allx{\displaystyle x}-tuples is denoted byEx{\displaystyle E_{x}}. For anx{\displaystyle x}-tuplef{\displaystyle f}and a subsety⊆x{\displaystyle y\subseteq x}the restrictionf[y]{\displaystyle f[y]}is defined to be they{\displaystyle y}-tupleg{\displaystyle g}so thatg(α)=f(α){\displaystyle g(\alpha )=f(\alpha )}for allα∈y{\displaystyle \alpha \in y}.
ArelationR{\displaystyle R}overx{\displaystyle x}is a set ofx{\displaystyle x}-tuples, i.e. a subset ofEx{\displaystyle E_{x}}.
The set of attributesx{\displaystyle x}is called thedomainofR{\displaystyle R}and denoted byd(R){\displaystyle d(R)}. Fory⊆d(R){\displaystyle y\subseteq d(R)}theprojectionofR{\displaystyle R}ontoy{\displaystyle y}is defined
as follows:
Thejoinof a relationR{\displaystyle R}overx{\displaystyle x}and a relationS{\displaystyle S}overy{\displaystyle y}is
defined as follows:
As an example, letR{\displaystyle R}andS{\displaystyle S}be the following relations:
Then the join ofR{\displaystyle R}andS{\displaystyle S}is:
A relational database with natural join⋈{\displaystyle \bowtie }as combination and the usual projectionπ{\displaystyle \pi }is an information algebra.
The operations are well defined since
It is easy to see that relational databases satisfy the axioms of a labeled
information algebra:
The axioms for information algebras are derived from
the axiom system proposed in (Shenoy and Shafer, 1990), see also (Shafer, 1991).
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Inmathematics, specificallyalgebraic geometry, aschemeis astructurethat enlarges the notion ofalgebraic varietyin several ways, such as taking account ofmultiplicities(the equationsx= 0andx2= 0define the same algebraic variety but different schemes) and allowing "varieties" defined over anycommutative ring(for example,Fermat curvesare defined over theintegers).
Scheme theorywas introduced byAlexander Grothendieckin 1960 in his treatiseÉléments de géométrie algébrique(EGA); one of its aims was developing the formalism needed to solve deep problems ofalgebraic geometry, such as theWeil conjectures(the last of which was proved byPierre Deligne).[1]Strongly based oncommutative algebra, scheme theory allows a systematic use of methods oftopologyandhomological algebra. Scheme theory also unifies algebraic geometry with much ofnumber theory, which eventually led toWiles's proof of Fermat's Last Theorem.
Schemes elaborate the fundamental idea that an algebraic variety is best analyzed through thecoordinate ringof regular algebraic functions defined on it (or on its subsets), and each subvariety corresponds to theidealof functions which vanish on the subvariety. Intuitively, a scheme is atopological spaceconsisting of closed points which correspond to geometric points, together with non-closed points which aregeneric pointsof irreducible subvarieties. The space is covered by anatlasof open sets, each endowed with a coordinate ring of regular functions, with specified coordinate changes between the functions over intersecting open sets. Such a structure is called aringed spaceor asheafof rings. The cases of main interest are theNoetherian schemes, in which the coordinate rings areNoetherian rings.
Formally, a scheme is a ringed space covered by affine schemes. An affine scheme is thespectrumof a commutative ring; its points are theprime idealsof the ring, and its closed points aremaximal ideals. The coordinate ring of an affine scheme is the ring itself, and the coordinate rings of open subsets arerings of fractions.
Therelative point of viewis that much of algebraic geometry should be developed for a morphismX→Yof schemes (called a schemeXover the baseY), rather than for an individual scheme. For example, in studyingalgebraic surfaces, it can be useful to consider families of algebraic surfaces over any schemeY. In many cases, the family of all varieties of a given type can itself be viewed as a variety or scheme, known as amoduli space.
For some of the detailed definitions in the theory of schemes, see theglossary of scheme theory.
The origins of algebraic geometry mostly lie in the study ofpolynomialequations over thereal numbers. By the 19th century, it became clear (notably in the work ofJean-Victor PonceletandBernhard Riemann) that algebraic geometry over the real numbers is simplified by working over thefieldofcomplex numbers, which has the advantage of beingalgebraically closed.[2]The early 20th century saw analogies between algebraic geometry and number theory, suggesting the question: can algebraic geometry be developed over other fields, such as those with positivecharacteristic, and more generally overnumber ringslike the integers, where the tools of topology andcomplex analysisused to study complex varieties do not seem to apply?
Hilbert's Nullstellensatzsuggests an approach to algebraic geometry over any algebraically closed fieldk: themaximal idealsin thepolynomial ringk[x1, ... ,xn]are in one-to-one correspondence with the setknofn-tuples of elements ofk, and theprime idealscorrespond to the irreducible algebraic sets inkn, known as affine varieties. Motivated by these ideas,Emmy NoetherandWolfgang Krulldeveloped commutative algebra in the 1920s and 1930s.[3]Their work generalizes algebraic geometry in a purely algebraic direction, generalizing the study of points (maximal ideals in a polynomial ring) to the study of prime ideals in any commutative ring. For example, Krull defined thedimensionof a commutative ring in terms of prime ideals and, at least when the ring isNoetherian, he proved that this definition satisfies many of the intuitive properties of geometric dimension.
Noether and Krull's commutative algebra can be viewed as an algebraic approach toaffinealgebraic varieties. However, many arguments in algebraic geometry work better forprojective varieties, essentially because they arecompact. From the 1920s to the 1940s,B. L. van der Waerden,André WeilandOscar Zariskiapplied commutative algebra as a new foundation for algebraic geometry in the richer setting of projective (orquasi-projective) varieties.[4]In particular, theZariski topologyis a useful topology on a variety over any algebraically closed field, replacing to some extent the classical topology on a complex variety (based on themetric topologyof the complex numbers).
For applications to number theory, van der Waerden and Weil formulated algebraic geometry over any field, not necessarily algebraically closed. Weil was the first to define anabstract variety(not embedded inprojective space), by gluing affine varieties along open subsets, on the model of abstractmanifoldsin topology. He needed this generality for his construction of theJacobian varietyof a curve over any field. (Later, Jacobians were shown to be projective varieties by Weil,ChowandMatsusaka.)
The algebraic geometers of theItalian schoolhad often used the somewhat foggy concept of thegeneric pointof an algebraic variety. What is true for the generic point is true for "most" points of the variety. In Weil'sFoundations of Algebraic Geometry(1946), generic points are constructed by taking points in a very large algebraically closed field, called auniversal domain.[4]This worked awkwardly: there were many different generic points for the same variety. (In the later theory of schemes, each algebraic variety has a single generic point.)
In the 1950s,Claude Chevalley,Masayoshi NagataandJean-Pierre Serre, motivated in part by theWeil conjecturesrelating number theory and algebraic geometry, further extended the objects of algebraic geometry, for example by generalizing the base rings allowed. The wordschemewas first used in the 1956 Chevalley Seminar, in which Chevalley pursued Zariski's ideas.[5]According toPierre Cartier, it wasAndré Martineauwho suggested to Serre the possibility of using the spectrum of an arbitrary commutative ring as a foundation for algebraic geometry.[6]
The theory took its definitive form in Grothendieck'sÉléments de géométrie algébrique(EGA) and the laterSéminaire de géométrie algébrique(SGA), bringing to a conclusion a generation of experimental suggestions and partial developments.[7]Grothendieck defined thespectrumX{\displaystyle X}of acommutative ringR{\displaystyle R}as the space ofprime idealsofR{\displaystyle R}with a natural topology (known as the Zariski topology), but augmented it with asheafof rings: to every open subsetU{\displaystyle U}he assigned a commutative ringOX(U){\displaystyle {\mathcal {O}}_{X}(U)}, which may be thought of as the coordinate ring of regular functions onU{\displaystyle U}. These objectsSpec(R){\displaystyle \operatorname {Spec} (R)}are the affine schemes; a general scheme is then obtained by "gluing together" affine schemes.
Much of algebraic geometry focuses on projective or quasi-projective varieties over a fieldk{\displaystyle k}, most often over the complex numbers. Grothendieck developed a large body of theory for arbitrary schemes extending much of the geometric intuition for varieties. For example, it is common to construct a moduli space first as a scheme, and only later study whether it is a more concrete object such as a projective variety. Applying Grothendieck's theory to schemes over the integers and other number fields led to powerful new perspectives in number theory.
Anaffine schemeis alocally ringed spaceisomorphic to thespectrumSpec(R){\displaystyle \operatorname {Spec} (R)}of a commutative ringR{\displaystyle R}. Aschemeis a locally ringed spaceX{\displaystyle X}admitting a covering by open setsUi{\displaystyle U_{i}}, such that eachUi{\displaystyle U_{i}}(as a locally ringed space) is an affine scheme.[8]In particular,X{\displaystyle X}comes with a sheafOX{\displaystyle {\mathcal {O}}_{X}}, which assigns to every open subsetU{\displaystyle U}a commutative ringOX(U){\displaystyle {\mathcal {O}}_{X}(U)}called thering of regular functionsonU{\displaystyle U}. One can think of a scheme as being covered by "coordinate charts" that are affine schemes. The definition means exactly that schemes are obtained by gluing together affine schemes using the Zariski topology.
In the early days, this was called aprescheme, and a scheme was defined to be aseparatedprescheme. The term prescheme has fallen out of use, but can still be found in older books, such as Grothendieck's "Éléments de géométrie algébrique" andMumford's "Red Book".[9]The sheaf properties ofOX(U){\displaystyle {\mathcal {O}}_{X}(U)}mean that its elements,which are not necessarily functions, can neverthess be patched together from their restrictions in the same way as functions.
A basic example of an affine scheme isaffinen{\displaystyle n}-spaceover a fieldk{\displaystyle k}, for anatural numbern{\displaystyle n}. By definition,Akn{\displaystyle A_{k}^{n}}is the spectrum of the polynomial ringk[x1,…,xn]{\displaystyle k[x_{1},\dots ,x_{n}]}. In the spirit of scheme theory, affinen{\displaystyle n}-space can in fact be defined over any commutative ringR{\displaystyle R}, meaningSpec(R[x1,…,xn]){\displaystyle \operatorname {Spec} (R[x_{1},\dots ,x_{n}])}.
Schemes form acategory, with morphisms defined as morphisms of locally ringed spaces. (See also:morphism of schemes.) For a schemeY, a schemeXoverY(or aY-scheme) means a morphismX→Yof schemes. A schemeXovera commutative ringRmeans a morphismX→ Spec(R).
An algebraic variety over a fieldkcan be defined as a scheme overkwith certain properties. There are different conventions about exactly which schemes should be called varieties. One standard choice is that avarietyoverkmeans anintegral separatedscheme offinite typeoverk.[10]
A morphismf:X→Yof schemes determines apullback homomorphismon the rings of regular functions,f*:O(Y) →O(X). In the case of affine schemes, this construction gives a one-to-one correspondence between morphisms Spec(A) → Spec(B) of schemes and ring homomorphismsB→A.[11]In this sense, scheme theory completely subsumes the theory of commutative rings.
SinceZis aninitial objectin thecategory of commutative rings, the category of schemes has Spec(Z) as aterminal object.
For a schemeXover a commutative ringR, anR-pointofXmeans asectionof the morphismX→ Spec(R). One writesX(R) for the set ofR-points ofX. In examples, this definition reconstructs the old notion of the set of solutions of the defining equations ofXwith values inR. WhenRis a fieldk,X(k) is also called the set ofk-rational pointsofX.
More generally, for a schemeXover a commutative ringRand any commutativeR-algebraS, anS-pointofXmeans a morphism Spec(S) →XoverR. One writesX(S) for the set ofS-points ofX. (This generalizes the old observation that given some equations over a fieldk, one can consider the set of solutions of the equations in anyfield extensionEofk.) For a schemeXoverR, the assignmentS↦X(S) is afunctorfrom commutativeR-algebras to sets. It is an important observation that a schemeXoverRis determined by thisfunctor of points.[12]
Thefiber product of schemesalways exists. That is, for any schemesXandZwith morphisms to a schemeY, thecategorical fiber productX×YZ{\displaystyle X\times _{Y}Z}exists in the category of schemes. IfXandZare schemes over a fieldk, their fiber product over Spec(k) may be called theproductX×Zin the category ofk-schemes. For example, the product of affine spacesAm{\displaystyle \mathbb {A} ^{m}}andAn{\displaystyle \mathbb {A} ^{n}}overkis affine spaceAm+n{\displaystyle \mathbb {A} ^{m+n}}overk.
Since the category of schemes has fiber products and also a terminal object Spec(Z), it has all finitelimits.
Here and below, all the rings considered are commutative.
Letkbe an algebraically closed field. The affine spaceX¯=Akn{\displaystyle {\bar {X}}=\mathbb {A} _{k}^{n}}is the algebraic variety of all pointsa=(a1,…,an){\displaystyle a=(a_{1},\ldots ,a_{n})}with coordinates ink; its coordinate ring is the polynomial ringR=k[x1,…,xn]{\displaystyle R=k[x_{1},\ldots ,x_{n}]}. The corresponding schemeX=Spec(R){\displaystyle X=\mathrm {Spec} (R)}is a topological space with the Zariski topology, whose closed points are the maximal idealsma=(x1−a1,…,xn−an){\displaystyle {\mathfrak {m}}_{a}=(x_{1}-a_{1},\ldots ,x_{n}-a_{n})}, the set of polynomials vanishing ata{\displaystyle a}. The scheme also contains a non-closed point for each non-maximal prime idealp⊂R{\displaystyle {\mathfrak {p}}\subset R}, whose vanishing defines an irreducible subvarietyV¯=V¯(p)⊂X¯{\displaystyle {\bar {V}}={\bar {V}}({\mathfrak {p}})\subset {\bar {X}}}; the topological closure of the scheme pointp{\displaystyle {\mathfrak {p}}}is the subschemeV(p)={q∈Xwithp⊂q}{\displaystyle V({\mathfrak {p}})=\{{\mathfrak {q}}\in X\ \ {\text{with}}\ \ {\mathfrak {p}}\subset {\mathfrak {q}}\}}, specially including all the closed points of the subvariety, i.e.ma{\displaystyle {\mathfrak {m}}_{a}}witha∈V¯{\displaystyle a\in {\bar {V}}}, or equivalentlyp⊂ma{\displaystyle {\mathfrak {p}}\subset {\mathfrak {m}}_{a}}.
The schemeX{\displaystyle X}has a basis of open subsets given by the complements of hypersurfaces,Uf=X∖V(f)={p∈Xwithf∉p}{\displaystyle U_{f}=X\setminus V(f)=\{{\mathfrak {p}}\in X\ \ {\text{with}}\ \ f\notin {\mathfrak {p}}\}}for irreducible polynomialsf∈R{\displaystyle f\in R}. This set is endowed with its coordinate ring of regular functionsOX(Uf)=R[f−1]={rfmforr∈R,m∈Z≥0}.{\displaystyle {\mathcal {O}}_{X}(U_{f})=R[f^{-1}]=\left\{{\tfrac {r}{f^{m}}}\ \ {\text{for}}\ \ r\in R,\ m\in \mathbb {Z} _{\geq 0}\right\}.}This induces a unique sheafOX{\displaystyle {\mathcal {O}}_{X}}which gives the usual ring of rational functions regular on a given open setU{\displaystyle U}.
Each ring elementr=r(x1,…,xn)∈R{\displaystyle r=r(x_{1},\ldots ,x_{n})\in R}, a polynomial function onX¯{\displaystyle {\bar {X}}}, also defines a function on the points of the schemeX{\displaystyle X}whose value atp{\displaystyle {\mathfrak {p}}}lies in the quotient ringR/p{\displaystyle R/{\mathfrak {p}}}, theresidue ring. We definer(p){\displaystyle r({\mathfrak {p}})}as the image ofr{\displaystyle r}under the natural mapR→R/p{\displaystyle R\to R/{\mathfrak {p}}}. A maximal idealma{\displaystyle {\mathfrak {m}}_{a}}gives theresidue fieldk(ma)=R/ma≅k{\displaystyle k({\mathfrak {m}}_{a})=R/{\mathfrak {m}}_{a}\cong k}, with the natural isomorphismxi↦ai{\displaystyle x_{i}\mapsto a_{i}}, so thatr(ma){\displaystyle r({\mathfrak {m}}_{a})}corresponds to the original valuer(a){\displaystyle r(a)}.
The vanishing locus of a polynomialf=f(x1,…,xn){\displaystyle f=f(x_{1},\ldots ,x_{n})}is ahypersurfacesubvarietyV¯(f)⊂Akn{\displaystyle {\bar {V}}(f)\subset \mathbb {A} _{k}^{n}}, corresponding to theprincipal ideal(f)⊂R{\displaystyle (f)\subset R}. The corresponding scheme isV(f)=Spec(R/(f)){\textstyle V(f)=\operatorname {Spec} (R/(f))}, a closed subscheme of affine space. For example, takingkto be the complex or real numbers, the equationx2=y2(y+1){\displaystyle x^{2}=y^{2}(y+1)}defines anodal cubic curvein the affine planeAk2{\displaystyle \mathbb {A} _{k}^{2}}, corresponding to the schemeV=Speck[x,y]/(x2−y2(y+1)){\displaystyle V=\operatorname {Spec} k[x,y]/(x^{2}-y^{2}(y+1))}.
The ring of integersZ{\displaystyle \mathbb {Z} }can be considered as the coordinate ring of the schemeZ=Spec(Z){\displaystyle Z=\operatorname {Spec} (\mathbb {Z} )}. The Zariski topology has closed pointsmp=(p){\displaystyle {\mathfrak {m}}_{p}=(p)}, the principal ideals of the prime numbersp∈Z{\displaystyle p\in \mathbb {Z} }; as well as the generic pointp0=(0){\displaystyle {\mathfrak {p}}_{0}=(0)}, the zero ideal, whoseclosure is the whole scheme. Closed sets are finite sets, and open sets are their complements, the cofinite sets; any infinite set of points is dense.
The basis open set corresponding to the irreducible elementp∈Z{\displaystyle p\in \mathbb {Z} }isUp=Z∖{mp}{\displaystyle U_{p}=Z\smallsetminus \{{\mathfrak {m}}_{p}\}}, with coordinate ringOZ(Up)=Z[p−1]={npmforn∈Z,m≥0}{\displaystyle {\mathcal {O}}_{Z}(U_{p})=\mathbb {Z} [p^{-1}]=\{{\tfrac {n}{p^{m}}}\ {\text{for}}\ n\in \mathbb {Z} ,\ m\geq 0\}}. For the open setU=Z∖{mp1,…,mpℓ}{\displaystyle U=Z\smallsetminus \{{\mathfrak {m}}_{p_{1}},\ldots ,{\mathfrak {m}}_{p_{\ell }}\}}, this inducesOZ(U)=Z[p1−1,…,pℓ−1]{\displaystyle {\mathcal {O}}_{Z}(U)=\mathbb {Z} [p_{1}^{-1},\ldots ,p_{\ell }^{-1}]}.
A numbern∈Z{\displaystyle n\in \mathbb {Z} }corresponds to a function on the schemeZ{\displaystyle Z}, a function whose value atmp{\displaystyle {\mathfrak {m}}_{p}}lies in the residue fieldk(mp)=Z/(p)=Fp{\displaystyle k({\mathfrak {m}}_{p})=\mathbb {Z} /(p)=\mathbb {F} _{p}}, thefinite fieldof integers modulop{\displaystyle p}:the function is defined byn(mp)=nmodp{\displaystyle n({\mathfrak {m}}_{p})=n\ {\text{mod}}\ p}, and alson(p0)=n{\displaystyle n({\mathfrak {p}}_{0})=n}in the generic residue ringZ/(0)=Z{\displaystyle \mathbb {Z} /(0)=\mathbb {Z} }. The functionn{\displaystyle n}is determined by its values at the pointsmp{\displaystyle {\mathfrak {m}}_{p}}only, so we can think ofn{\displaystyle n}as a kind of "regular function" on the closed points, a very special type among the arbitrary functionsf{\displaystyle f}withf(mp)∈Fp{\displaystyle f({\mathfrak {m}}_{p})\in \mathbb {F} _{p}}.
Note that the pointmp{\displaystyle {\mathfrak {m}}_{p}}is the vanishing locus of the functionn=p{\displaystyle n=p}, the point where the value ofp{\displaystyle p}is equal to zero in the residue field. The field of "rational functions" onZ{\displaystyle Z}is the fraction field of the generic residue ring,k(p0)=Frac(Z)=Q{\displaystyle k({\mathfrak {p}}_{0})=\operatorname {Frac} (\mathbb {Z} )=\mathbb {Q} }. A fractiona/b{\displaystyle a/b}has "poles" at the pointsmp{\displaystyle {\mathfrak {m}}_{p}}corresponding to prime divisors of the denominator.
This also gives a geometric interpretaton ofBezout's lemmastating that if the integersn1,…,nr{\displaystyle n_{1},\ldots ,n_{r}}have no common prime factor, then there are integersa1,…,ar{\displaystyle a_{1},\ldots ,a_{r}}witha1n1+⋯+arnr=1{\displaystyle a_{1}n_{1}+\cdots +a_{r}n_{r}=1}. Geometrically, this is a version of the weakHilbert Nullstellensatzfor the schemeZ{\displaystyle Z}: if the functionsn1,…,nr{\displaystyle n_{1},\ldots ,n_{r}}have no common vanishing pointsmp{\displaystyle {\mathfrak {m}}_{p}}inZ{\displaystyle Z}, then they generate the unit ideal(n1,…,nr)=(1){\displaystyle (n_{1},\ldots ,n_{r})=(1)}in the coordinate ringZ{\displaystyle \mathbb {Z} }. Indeed, we may consider the termsρi=aini{\displaystyle \rho _{i}=a_{i}n_{i}}as forming a kind ofpartition of unitysubordinate to the covering ofZ{\displaystyle Z}by the open setsUi=Z∖V(ni){\displaystyle U_{i}=Z\smallsetminus V(n_{i})}.
The affine spaceAZ1={afora∈Z}{\displaystyle \mathbb {A} _{\mathbb {Z} }^{1}=\{a\ {\text{for}}\ a\in \mathbb {Z} \}}is a variety with coordinate ringZ[x]{\displaystyle \mathbb {Z} [x]}, the polynomials with integer coefficients. The corresponding scheme isY=Spec(Z[x]){\displaystyle Y=\operatorname {Spec} (\mathbb {Z} [x])}, whose points are all of the prime idealsp⊂Z[x]{\displaystyle {\mathfrak {p}}\subset \mathbb {Z} [x]}. The closed points are maximal ideals of the formm=(p,f(x)){\displaystyle {\mathfrak {m}}=(p,f(x))}, wherep{\displaystyle p}is a prime number, andf(x){\displaystyle f(x)}is a non-constant polynomial with no integer factor and which is irreducible modulop{\displaystyle p}. Thus, we may pictureY{\displaystyle Y}as two-dimensional, with a "characteristic direction" measured by the coordinatep{\displaystyle p}, and a "spatial direction" with coordinatex{\displaystyle x}.
A given prime numberp{\displaystyle p}defines a "vertical line", the subschemeV(p){\displaystyle V(p)}of the prime idealp=(p){\displaystyle {\mathfrak {p}}=(p)}: this containsm=(p,f(x)){\displaystyle {\mathfrak {m}}=(p,f(x))}for allf(x){\displaystyle f(x)}, the "characteristicp{\displaystyle p}points" of the scheme. Fixing thex{\displaystyle x}-coordinate, we have the "horizontal line"x=a{\displaystyle x=a}, the subschemeV(x−a){\displaystyle V(x-a)}of the prime idealp=(x−a){\displaystyle {\mathfrak {p}}=(x-a)}. We also have the lineV(bx−a){\displaystyle V(bx-a)}corresponding to the rational coordinatex=a/b{\displaystyle x=a/b}, which does not intersectV(p){\displaystyle V(p)}for thosep{\displaystyle p}which divideb{\displaystyle b}.
A higher degree "horizontal" subscheme likeV(x2+1){\displaystyle V(x^{2}+1)}corresponds tox{\displaystyle x}-values which are roots ofx2+1{\displaystyle x^{2}+1}, namelyx=±−1{\displaystyle x=\pm {\sqrt {-1}}}. This behaves differently under differentp{\displaystyle p}-coordinates. Atp=5{\displaystyle p=5}, we get two pointsx=±2mod5{\displaystyle x=\pm 2\ {\text{mod}}\ 5}, since(5,x2+1)=(5,x−2)∩(5,x+2){\displaystyle (5,x^{2}+1)=(5,x-2)\cap (5,x+2)}. Atp=2{\displaystyle p=2}, we get oneramifieddouble-pointx=1mod2{\displaystyle x=1\ {\text{mod}}\ 2}, since(2,x2+1)=(2,(x−1)2){\displaystyle (2,x^{2}+1)=(2,(x-1)^{2})}. And atp=3{\displaystyle p=3}, we get thatm=(3,x2+1){\displaystyle {\mathfrak {m}}=(3,x^{2}+1)}is a prime ideal corresponding tox=±−1{\displaystyle x=\pm {\sqrt {-1}}}in an extension field ofF3{\displaystyle \mathbb {F} _{3}}; since we cannot distinguish between these values (they are symmetric under theGalois group), we should pictureV(3,x2+1){\displaystyle V(3,x^{2}+1)}as two fused points. Overall,V(x2+1){\displaystyle V(x^{2}+1)}is a kind of fusion of two Galois-symmetric horizonal lines, a curve of degree 2.
The residue field atm=(p,f(x)){\displaystyle {\mathfrak {m}}=(p,f(x))}isk(m)=Z[x]/m=Fp[x]/(f(x))≅Fp(α){\displaystyle k({\mathfrak {m}})=\mathbb {Z} [x]/{\mathfrak {m}}=\mathbb {F} _{p}[x]/(f(x))\cong \mathbb {F} _{p}(\alpha )}, a field extension ofFp{\displaystyle \mathbb {F} _{p}}adjoining a rootx=α{\displaystyle x=\alpha }off(x){\displaystyle f(x)}; this is a finite field withpd{\displaystyle p^{d}}elements,d=deg(f){\displaystyle d=\operatorname {deg} (f)}. A polynomialr(x)∈Z[x]{\displaystyle r(x)\in \mathbb {Z} [x]}corresponds to a function on the schemeY{\displaystyle Y}with valuesr(m)=rmodm{\displaystyle r({\mathfrak {m}})=r\ \mathrm {mod} \ {\mathfrak {m}}}, that isr(m)=r(α)∈Fp(α){\displaystyle r({\mathfrak {m}})=r(\alpha )\in \mathbb {F} _{p}(\alpha )}. Again eachr(x)∈Z[x]{\displaystyle r(x)\in \mathbb {Z} [x]}is determined by its valuesr(m){\displaystyle r({\mathfrak {m}})}at closed points;V(p){\displaystyle V(p)}is the vanishing locus of the constant polynomialr(x)=p{\displaystyle r(x)=p}; andV(f(x)){\displaystyle V(f(x))}contains the points in each characteristicp{\displaystyle p}corresponding to Galois orbits of roots off(x){\displaystyle f(x)}in the algebraic closureF¯p{\displaystyle {\overline {\mathbb {F} }}_{p}}.
The schemeY{\displaystyle Y}is notproper, so that pairs of curves may fail tointersect with the expected multiplicity. This is a major obstacle to analyzingDiophantine equationswithgeometric tools.Arakelov theoryovercomes this obstacle by compactifying affine arithmetic schemes, adding points at infinity corresponding tovaluations.
If we consider a polynomialf∈Z[x,y]{\displaystyle f\in \mathbb {Z} [x,y]}then the affine schemeX=Spec(Z[x,y]/(f)){\displaystyle X=\operatorname {Spec} (\mathbb {Z} [x,y]/(f))}has a canonical morphism toSpecZ{\displaystyle \operatorname {Spec} \mathbb {Z} }and is called anarithmetic surface. The fibersXp=X×Spec(Z)Spec(Fp){\displaystyle X_{p}=X\times _{\operatorname {Spec} (\mathbb {Z} )}\operatorname {Spec} (\mathbb {F} _{p})}are then algebraic curves over the finite fieldsFp{\displaystyle \mathbb {F} _{p}}. Iff(x,y)=y2−x3+ax2+bx+c{\displaystyle f(x,y)=y^{2}-x^{3}+ax^{2}+bx+c}is anelliptic curve, then the fibers over its discriminant locus, whereΔf=−4a3c+a2b2+18abc−4b3−27c2=0modp,{\displaystyle \Delta _{f}=-4a^{3}c+a^{2}b^{2}+18abc-4b^{3}-27c^{2}=0\ {\text{mod}}\ p,}are all singular schemes.[13]For example, ifp{\displaystyle p}is a prime number andX=SpecZ[x,y](y2−x3−p){\displaystyle X=\operatorname {Spec} {\frac {\mathbb {Z} [x,y]}{(y^{2}-x^{3}-p)}}}then its discriminant is−27p2{\displaystyle -27p^{2}}. This curve is singular over the prime numbers3,p{\displaystyle 3,p}.
It is also fruitful to consider examples of morphisms as examples of schemes since they demonstrate their technical effectiveness for encapsulating many objects of study in algebraic and arithmetic geometry.
Here are some of the ways in which schemes go beyond older notions of algebraic varieties, and their significance.
A central part of scheme theory is the notion ofcoherent sheaves, generalizing the notion of (algebraic)vector bundles. For a schemeX, one starts by considering theabelian categoryofOX-modules, which are sheaves of abelian groups onXthat form amoduleover the sheaf of regular functionsOX. In particular, a moduleMover a commutative ringRdetermines anassociatedOX-module~MonX= Spec(R). Aquasi-coherent sheafon a schemeXmeans anOX-module that is the sheaf associated to a module on each affine open subset ofX. Finally, acoherent sheaf(on a Noetherian schemeX, say) is anOX-module that is the sheaf associated to afinitely generated moduleon each affine open subset ofX.
Coherent sheaves include the important class ofvector bundles, which are the sheaves that locally come from finitely generatedfree modules. An example is thetangent bundleof a smooth variety over a field. However, coherent sheaves are richer; for example, a vector bundle on a closed subschemeYofXcan be viewed as a coherent sheaf onXthat is zero outsideY(by thedirect imageconstruction). In this way, coherent sheaves on a schemeXinclude information about all closed subschemes ofX. Moreover,sheaf cohomologyhas good properties for coherent (and quasi-coherent) sheaves. The resulting theory ofcoherent sheaf cohomologyis perhaps the main technical tool in algebraic geometry.[18][19]
Considered as its functor of points, a scheme is a functor that is a sheaf of sets for the Zariski topology on the category of commutative rings, and that, locally in the Zariski topology, is an affine scheme. This can be generalized in several ways. One is to use theétale topology.Michael Artindefined analgebraic spaceas a functor that is a sheaf in the étale topology and that, locally in the étale topology, is an affine scheme. Equivalently, an algebraic space is the quotient of a scheme by an étale equivalence relation. A powerful result, theArtin representability theorem, gives simple conditions for a functor to be represented by an algebraic space.[20]
A further generalization is the idea of astack. Crudely speaking,algebraic stacksgeneralize algebraic spaces by having analgebraic groupattached to each point, which is viewed as the automorphism group of that point. For example, anyactionof an algebraic groupGon an algebraic varietyXdetermines aquotient stack[X/G], which remembers thestabilizer subgroupsfor the action ofG. More generally, moduli spaces in algebraic geometry are often best viewed as stacks, thereby keeping track of the automorphism groups of the objects being classified.
Grothendieck originally introduced stacks as a tool for the theory ofdescent. In that formulation, stacks are (informally speaking) sheaves of categories.[21]From this general notion, Artin defined the narrower class of algebraic stacks (or "Artin stacks"), which can be considered geometric objects. These includeDeligne–Mumford stacks(similar toorbifoldsin topology), for which the stabilizer groups are finite, and algebraic spaces, for which the stabilizer groups are trivial. TheKeel–Mori theoremsays that an algebraic stack with finite stabilizer groups has acoarse moduli spacethat is an algebraic space.
Another type of generalization is to enrich the structure sheaf, bringing algebraic geometry closer tohomotopy theory. In this setting, known asderived algebraic geometryor "spectral algebraic geometry", the structure sheaf is replaced by a homotopical analog of a sheaf of commutative rings (for example, a sheaf ofE-infinity ring spectra). These sheaves admit algebraic operations that are associative and commutative only up to an equivalence relation. Taking the quotient by this equivalence relation yields the structure sheaf of an ordinary scheme. Not taking the quotient, however, leads to a theory that can remember higher information, in the same way thatderived functorsin homological algebra yield higher information about operations such astensor productand theHom functoron modules.
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Inalgebraic geometry, aprojective varietyis analgebraic varietythat is a closedsubvarietyof aprojective space. That is, it is the zero-locus inPn{\displaystyle \mathbb {P} ^{n}}of some finite family ofhomogeneous polynomialsthat generate aprime ideal, the defining ideal of the variety.
A projective variety is aprojective curveif its dimension is one; it is aprojective surfaceif its dimension is two; it is aprojective hypersurfaceif its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a singlehomogeneous polynomial.
IfXis a projective variety defined by a homogeneous prime idealI, then thequotient ring
is called thehomogeneous coordinate ringofX. Basic invariants ofXsuch as thedegreeand thedimensioncan be read off theHilbert polynomialof thisgraded ring.
Projective varieties arise in many ways. They arecomplete, which roughly can be expressed by saying that there are no points "missing". The converse is not true in general, butChow's lemmadescribes the close relation of these two notions. Showing that a variety is projective is done by studyingline bundlesordivisorsonX.
A salient feature of projective varieties are the finiteness constraints on sheaf cohomology. For smooth projective varieties,Serre dualitycan be viewed as an analog ofPoincaré duality. It also leads to theRiemann–Roch theoremfor projective curves, i.e., projective varieties ofdimension1. The theory of projective curves is particularly rich, including a classification by thegenusof the curve. The classification program for higher-dimensional projective varieties naturally leads to the construction of moduli of projective varieties.[1]Hilbert schemesparametrize closed subschemes ofPn{\displaystyle \mathbb {P} ^{n}}with prescribed Hilbert polynomial. Hilbert schemes, of whichGrassmanniansare special cases, are also projective schemes in their own right.Geometric invariant theoryoffers another approach. The classical approaches include theTeichmüller spaceandChow varieties.
A particularly rich theory, reaching back to the classics, is available for complex projective varieties, i.e., when the polynomials definingXhavecomplexcoefficients. Broadly, theGAGA principlesays that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties. For example, the theory ofholomorphic vector bundles(more generallycoherent analytic sheaves) onXcoincide with that of algebraic vector bundles.Chow's theoremsays that a subset of projective space is the zero-locus of a family of holomorphic functions if and only if it is the zero-locus of homogeneous polynomials. The combination of analytic and algebraic methods for complex projective varieties lead to areas such asHodge theory.
Letkbe an algebraically closed field. The basis of the definition of projective varieties is projective spacePn{\displaystyle \mathbb {P} ^{n}}, which can be defined in different, but equivalent ways:
Aprojective varietyis, by definition, a closed subvariety ofPn{\displaystyle \mathbb {P} ^{n}}, where closed refers to theZariski topology.[2]In general, closed subsets of the Zariski topology are defined to be the common zero-locus of a finite collection of homogeneous polynomial functions. Given a polynomialf∈k[x0,…,xn]{\displaystyle f\in k[x_{0},\dots ,x_{n}]}, the condition
does not make sense for arbitrary polynomials, but only iffishomogeneous, i.e., the degrees of all themonomials(whose sum isf) are the same. In this case, the vanishing of
is independent of the choice ofλ≠0{\displaystyle \lambda \neq 0}.
Therefore, projective varieties arise from homogeneousprime idealsIofk[x0,…,xn]{\displaystyle k[x_{0},\dots ,x_{n}]}, and setting
Moreover, the projective varietyXis an algebraic variety, meaning that it is covered by open affine subvarieties and satisfies the separation axiom. Thus, the local study ofX(e.g., singularity) reduces to that of an affine variety. The explicit structure is as follows. The projective spacePn{\displaystyle \mathbb {P} ^{n}}is covered by the standard open affine charts
which themselves are affinen-spaces with the coordinate ring
Sayi= 0 for the notational simplicity and drop the superscript (0). ThenX∩U0{\displaystyle X\cap U_{0}}is a closed subvariety ofU0≃An{\displaystyle U_{0}\simeq \mathbb {A} ^{n}}defined by the ideal ofk[y1,…,yn]{\displaystyle k[y_{1},\dots ,y_{n}]}generated by
for allfinI. Thus,Xis an algebraic variety covered by (n+1) open affine chartsX∩Ui{\displaystyle X\cap U_{i}}.
Note thatXis the closure of the affine varietyX∩U0{\displaystyle X\cap U_{0}}inPn{\displaystyle \mathbb {P} ^{n}}. Conversely, starting from some closed (affine) varietyV⊂U0≃An{\displaystyle V\subset U_{0}\simeq \mathbb {A} ^{n}}, the closure ofVinPn{\displaystyle \mathbb {P} ^{n}}is the projective variety called theprojective completionofV. IfI⊂k[y1,…,yn]{\displaystyle I\subset k[y_{1},\dots ,y_{n}]}definesV, then the defining ideal of this closure is the homogeneous ideal[3]ofk[x0,…,xn]{\displaystyle k[x_{0},\dots ,x_{n}]}generated by
for allfinI.
For example, ifVis an affine curve given by, say,y2=x3+ax+b{\displaystyle y^{2}=x^{3}+ax+b}in the affine plane, then its projective completion in the projective plane is given byy2z=x3+axz2+bz3.{\displaystyle y^{2}z=x^{3}+axz^{2}+bz^{3}.}
For various applications, it is necessary to consider more general algebro-geometric objects than projective varieties, namely projective schemes. The first step towards projective schemes is to endow projective space with a scheme structure, in a way refining the above description of projective space as an algebraic variety, i.e.,Pn(k){\displaystyle \mathbb {P} ^{n}(k)}is a scheme which it is a union of (n+ 1) copies of the affinen-spacekn. More generally,[4]projective space over a ringAis the union of theaffine schemes
in such a way the variables match up as expected. The set ofclosed pointsofPkn{\displaystyle \mathbb {P} _{k}^{n}}, for algebraically closed fieldsk, is then the projective spacePn(k){\displaystyle \mathbb {P} ^{n}(k)}in the usual sense.
An equivalent but streamlined construction is given by theProj construction, which is an analog of thespectrum of a ring, denoted "Spec", which defines an affine scheme.[5]For example, ifAis a ring, then
IfRis aquotientofk[x0,…,xn]{\displaystyle k[x_{0},\ldots ,x_{n}]}by a homogeneous idealI, then the canonical surjection induces theclosed immersion
Compared to projective varieties, the condition that the idealIbe a prime ideal was dropped. This leads to a much more flexible notion: on the one hand thetopological spaceX=ProjR{\displaystyle X=\operatorname {Proj} R}may have multipleirreducible components. Moreover, there may benilpotentfunctions onX.
Closed subschemes ofPkn{\displaystyle \mathbb {P} _{k}^{n}}correspond bijectively to the homogeneous idealsIofk[x0,…,xn]{\displaystyle k[x_{0},\ldots ,x_{n}]}that aresaturated; i.e.,I:(x0,…,xn)=I.{\displaystyle I:(x_{0},\dots ,x_{n})=I.}[6]This fact may be considered as a refined version ofprojective Nullstellensatz.
We can give a coordinate-free analog of the above. Namely, given a finite-dimensional vector spaceVoverk, we let
wherek[V]=Sym(V∗){\displaystyle k[V]=\operatorname {Sym} (V^{*})}is thesymmetric algebraofV∗{\displaystyle V^{*}}.[7]It is theprojectivizationofV; i.e., it parametrizes lines inV. There is a canonical surjective mapπ:V∖{0}→P(V){\displaystyle \pi :V\setminus \{0\}\to \mathbb {P} (V)}, which is defined using the chart described above.[8]One important use of the construction is this (cf.,§ Duality and linear system). A divisorDon a projective varietyXcorresponds to a line bundleL. One then set
it is called thecomplete linear systemofD.
Projective space over anyschemeScan be defined as afiber product of schemes
IfO(1){\displaystyle {\mathcal {O}}(1)}is thetwisting sheaf of SerreonPZn{\displaystyle \mathbb {P} _{\mathbb {Z} }^{n}}, we letO(1){\displaystyle {\mathcal {O}}(1)}denote thepullbackofO(1){\displaystyle {\mathcal {O}}(1)}toPSn{\displaystyle \mathbb {P} _{S}^{n}}; that is,O(1)=g∗(O(1)){\displaystyle {\mathcal {O}}(1)=g^{*}({\mathcal {O}}(1))}for the canonical mapg:PSn→PZn.{\displaystyle g:\mathbb {P} _{S}^{n}\to \mathbb {P} _{\mathbb {Z} }^{n}.}
A schemeX→Sis calledprojectiveoverSif it factors as a closed immersion
followed by the projection toS.
A line bundle (or invertible sheaf)L{\displaystyle {\mathcal {L}}}on a schemeXoverSis said to bevery ample relative toSif there is animmersion(i.e., an open immersion followed by a closed immersion)
for somenso thatO(1){\displaystyle {\mathcal {O}}(1)}pullbacks toL{\displaystyle {\mathcal {L}}}. Then aS-schemeXis projective if and only if it isproperand there exists a very ample sheaf onXrelative toS. Indeed, ifXis proper, then an immersion corresponding to the very ample line bundle is necessarily closed. Conversely, ifXis projective, then the pullback ofO(1){\displaystyle {\mathcal {O}}(1)}under the closed immersion ofXinto a projective space is very ample. That "projective" implies "proper" is deeper: themain theorem of elimination theory.
By definition, a variety iscomplete, if it isproperoverk. Thevaluative criterion of propernessexpresses the intuition that in a proper variety, there are no points "missing".
There is a close relation between complete and projective varieties: on the one hand, projective space and therefore any projective variety is complete. The converse is not true in general. However:
Some properties of a projective variety follow from completeness. For example,
for any projective varietyXoverk.[10]This fact is an algebraic analogue ofLiouville's theorem(any holomorphic function on a connected compact complex manifold is constant). In fact, the similarity between complex analytic geometry and algebraic geometry on complex projective varieties goes much further than this, as is explained below.
Quasi-projective varietiesare, by definition, those which are open subvarieties of projective varieties. This class of varieties includesaffine varieties. Affine varieties are almost never complete (or projective). In fact, a projective subvariety of an affine variety must have dimension zero. This is because only the constants are globallyregular functionson a projective variety.
By definition, any homogeneous ideal in a polynomial ring yields a projective scheme (required to be prime ideal to give a variety). In this sense, examples of projective varieties abound. The following list mentions various classes of projective varieties which are noteworthy since they have been studied particularly intensely. The important class of complex projective varieties, i.e., the casek=C{\displaystyle k=\mathbb {C} }, is discussed further below.
The product of two projective spaces is projective. In fact, there is the explicit immersion (calledSegre embedding)
As a consequence, theproductof projective varieties overkis again projective. ThePlücker embeddingexhibits aGrassmannianas a projective variety.Flag varietiessuch as the quotient of thegeneral linear groupGLn(k){\displaystyle \mathrm {GL} _{n}(k)}modulo the subgroup of uppertriangular matrices, are also projective, which is an important fact in the theory ofalgebraic groups.[11]
As the prime idealPdefining a projective varietyXis homogeneous, thehomogeneous coordinate ring
is agraded ring, i.e., can be expressed as thedirect sumof its graded components:
There exists a polynomialPsuch thatdimRn=P(n){\displaystyle \dim R_{n}=P(n)}for all sufficiently largen; it is called theHilbert polynomialofX. It is a numerical invariant encoding some extrinsic geometry ofX. The degree ofPis thedimensionrofXand its leading coefficient timesr!is thedegreeof the varietyX. Thearithmetic genusofXis (−1)r(P(0) − 1) whenXis smooth.
For example, the homogeneous coordinate ring ofPn{\displaystyle \mathbb {P} ^{n}}isk[x0,…,xn]{\displaystyle k[x_{0},\ldots ,x_{n}]}and its Hilbert polynomial isP(z)=(z+nn){\displaystyle P(z)={\binom {z+n}{n}}}; its arithmetic genus is zero.
If the homogeneous coordinate ringRis anintegrally closed domain, then the projective varietyXis said to beprojectively normal. Note, unlikenormality, projective normality depends onR, the embedding ofXinto a projective space. The normalization of a projective variety is projective; in fact, it's the Proj of the integral closure of some homogeneous coordinate ring ofX.
LetX⊂PN{\displaystyle X\subset \mathbb {P} ^{N}}be a projective variety. There are at least two equivalent ways to define the degree ofXrelative to its embedding. The first way is to define it as the cardinality of the finite set
wheredis the dimension ofXandHi's are hyperplanes in "general positions". This definition corresponds to an intuitive idea of a degree. Indeed, ifXis a hypersurface, then the degree ofXis the degree of the homogeneous polynomial definingX. The "general positions" can be made precise, for example, byintersection theory; one requires that the intersection isproperand that the multiplicities of irreducible components are all one.
The other definition, which is mentioned in the previous section, is that the degree ofXis the leading coefficient of theHilbert polynomialofXtimes (dimX)!. Geometrically, this definition means that the degree ofXis the multiplicity of the vertex of the affine cone overX.[12]
LetV1,…,Vr⊂PN{\displaystyle V_{1},\dots ,V_{r}\subset \mathbb {P} ^{N}}be closed subschemes of pure dimensions that intersect properly (they are in general position). Ifmidenotes the multiplicity of an irreducible componentZiin the intersection (i.e.,intersection multiplicity), then the generalization ofBézout's theoremsays:[13]
The intersection multiplicitymican be defined as the coefficient ofZiin the intersection productV1⋅⋯⋅Vr{\displaystyle V_{1}\cdot \cdots \cdot V_{r}}in theChow ringofPN{\displaystyle \mathbb {P} ^{N}}.
In particular, ifH⊂PN{\displaystyle H\subset \mathbb {P} ^{N}}is a hypersurface not containingX, then
whereZiare the irreducible components of thescheme-theoretic intersectionofXandHwith multiplicity (length of the local ring)mi.
A complex projective variety can be viewed as acompact complex manifold; the degree of the variety (relative to the embedding) is then the volume of the variety as a manifold with respect to the metric inherited from the ambientcomplex projective space. A complex projective variety can be characterized as a minimizer of the volume (in a sense).
LetXbe a projective variety andLa line bundle on it. Then the graded ring
is called thering of sectionsofL. IfLisample, then Proj of this ring isX. Moreover, ifXis normal andLis very ample, thenR(X,L){\displaystyle R(X,L)}is the integral closure of the homogeneous coordinate ring ofXdetermined byL; i.e.,X↪PN{\displaystyle X\hookrightarrow \mathbb {P} ^{N}}so thatOPN(1){\displaystyle {\mathcal {O}}_{\mathbb {P} ^{N}}(1)}pulls-back toL.[14]
For applications, it is useful to allow fordivisors(orQ{\displaystyle \mathbb {Q} }-divisors) not just line bundles; assumingXis normal, the resulting ring is then called a generalized ring of sections. IfKX{\displaystyle K_{X}}is acanonical divisoronX, then the generalized ring of sections
is called thecanonical ringofX. If the canonical ring is finitely generated, then Proj of the ring is called thecanonical modelofX. The canonical ring or model can then be used to define theKodaira dimensionofX.
Projective schemes of dimension one are calledprojective curves. Much of the theory of projective curves is about smooth projective curves, since thesingularitiesof curves can be resolved bynormalization, which consists in taking locally theintegral closureof the ring of regular functions. Smooth projective curves are isomorphic if and only if theirfunction fieldsare isomorphic. The study of finite extensions of
or equivalently smooth projective curves overFp{\displaystyle \mathbb {F} _{p}}is an important branch inalgebraic number theory.[15]
A smooth projective curve of genus one is called anelliptic curve. As a consequence of theRiemann–Roch theorem, such a curve can be embedded as a closed subvariety inP2{\displaystyle \mathbb {P} ^{2}}. In general, any (smooth) projective curve can be embedded inP3{\displaystyle \mathbb {P} ^{3}}(for a proof, seeSecant variety#Examples). Conversely, any smooth closed curve inP2{\displaystyle \mathbb {P} ^{2}}of degree three has genus one by thegenus formulaand is thus an elliptic curve.
A smooth complete curve of genus greater than or equal to two is called ahyperelliptic curveif there is a finite morphismC→P1{\displaystyle C\to \mathbb {P} ^{1}}of degree two.[16]
Every irreducible closed subset ofPn{\displaystyle \mathbb {P} ^{n}}of codimension one is ahypersurface; i.e., the zero set of some homogeneous irreducible polynomial.[17]
Another important invariant of a projective varietyXis thePicard groupPic(X){\displaystyle \operatorname {Pic} (X)}ofX, the set of isomorphism classes of line bundles onX. It is isomorphic toH1(X,OX∗){\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})}and therefore an intrinsic notion (independent of embedding). For example, the Picard group ofPn{\displaystyle \mathbb {P} ^{n}}is isomorphic toZ{\displaystyle \mathbb {Z} }via the degree map. The kernel ofdeg:Pic(X)→Z{\displaystyle \deg :\operatorname {Pic} (X)\to \mathbb {Z} }is not only an abstract abelian group, but there is a variety called theJacobian varietyofX, Jac(X), whose points equal this group. The Jacobian of a (smooth) curve plays an important role in the study of the curve. For example, the Jacobian of an elliptic curveEisEitself. For a curveXof genusg, Jac(X) has dimensiong.
Varieties, such as the Jacobian variety, which are complete and have a group structure are known asabelian varieties, in honor ofNiels Abel. In marked contrast toaffine algebraic groupssuch asGLn(k){\displaystyle GL_{n}(k)}, such groups are always commutative, whence the name. Moreover, they admit an ampleline bundleand are thus projective. On the other hand, anabelian schememay not be projective. Examples of abelian varieties are elliptic curves, Jacobian varieties andK3 surfaces.
LetE⊂Pn{\displaystyle E\subset \mathbb {P} ^{n}}be a linear subspace; i.e.,E={s0=s1=⋯=sr=0}{\displaystyle E=\{s_{0}=s_{1}=\cdots =s_{r}=0\}}for some linearly independent linear functionalssi. Then theprojection fromEis the (well-defined) morphism
The geometric description of this map is as follows:[18]
Projections can be used to cut down the dimension in which a projective variety is embedded, up tofinite morphisms. Start with some projective varietyX⊂Pn.{\displaystyle X\subset \mathbb {P} ^{n}.}Ifn>dimX,{\displaystyle n>\dim X,}the projection from a point not onXgivesϕ:X→Pn−1.{\displaystyle \phi :X\to \mathbb {P} ^{n-1}.}Moreover,ϕ{\displaystyle \phi }is a finite map to its image. Thus, iterating the procedure, one sees there is a finite map
This result is the projective analog ofNoether's normalization lemma. (In fact, it yields a geometric proof of the normalization lemma.)
The same procedure can be used to show the following slightly more precise result: given a projective varietyXover a perfect field, there is a finite birational morphism fromXto a hypersurfaceHinPd+1.{\displaystyle \mathbb {P} ^{d+1}.}[20]In particular, ifXis normal, then it is the normalization ofH.
While a projectiven-spacePn{\displaystyle \mathbb {P} ^{n}}parameterizes the lines in an affinen-space, thedualof it parametrizes the hyperplanes on the projective space, as follows. Fix a fieldk. ByP˘kn{\displaystyle {\breve {\mathbb {P} }}_{k}^{n}}, we mean a projectiven-space
equipped with the construction:
wheref:SpecL→P˘kn{\displaystyle f:\operatorname {Spec} L\to {\breve {\mathbb {P} }}_{k}^{n}}is anL-pointofP˘kn{\displaystyle {\breve {\mathbb {P} }}_{k}^{n}}for a field extensionLofkandαi=f∗(ui)∈L.{\displaystyle \alpha _{i}=f^{*}(u_{i})\in L.}
For eachL, the construction is a bijection between the set ofL-points ofP˘kn{\displaystyle {\breve {\mathbb {P} }}_{k}^{n}}and the set of hyperplanes onPLn{\displaystyle \mathbb {P} _{L}^{n}}. Because of this, the dual projective spaceP˘kn{\displaystyle {\breve {\mathbb {P} }}_{k}^{n}}is said to be themoduli spaceof hyperplanes onPkn{\displaystyle \mathbb {P} _{k}^{n}}.
A line inP˘kn{\displaystyle {\breve {\mathbb {P} }}_{k}^{n}}is called apencil: it is a family of hyperplanes onPkn{\displaystyle \mathbb {P} _{k}^{n}}parametrized byPk1{\displaystyle \mathbb {P} _{k}^{1}}.
IfVis a finite-dimensional vector space overk, then, for the same reason as above,P(V∗)=Proj(Sym(V)){\displaystyle \mathbb {P} (V^{*})=\operatorname {Proj} (\operatorname {Sym} (V))}is the space of hyperplanes onP(V){\displaystyle \mathbb {P} (V)}. An important case is whenVconsists of sections of a line bundle. Namely, letXbe an algebraic variety,La line bundle onXandV⊂Γ(X,L){\displaystyle V\subset \Gamma (X,L)}a vector subspace of finite positive dimension. Then there is a map:[21]
determined by the linear systemV, whereB, called thebase locus, is theintersectionof the divisors of zero of nonzero sections inV(seeLinear system of divisors#A map determined by a linear systemfor the construction of the map).
LetXbe a projective scheme over a field (or, more generally over a Noetherian ringA).Cohomology of coherent sheavesF{\displaystyle {\mathcal {F}}}onXsatisfies the following important theorems due to Serre:
These results are proven reducing to the caseX=Pn{\displaystyle X=\mathbb {P} ^{n}}using the isomorphism
where in the right-hand sideF{\displaystyle {\mathcal {F}}}is viewed as a sheaf on the projective space by extension by zero.[22]The result then follows by a direct computation forF=OPr(n),{\displaystyle {\mathcal {F}}={\mathcal {O}}_{\mathbb {P} ^{r}}(n),}nany integer, and for arbitraryF{\displaystyle {\mathcal {F}}}reduces to this case without much difficulty.[23]
As a corollary to 1. above, iffis a projective morphism from a noetherian scheme to a noetherian ring, then the higher direct imageRpf∗F{\displaystyle R^{p}f_{*}{\mathcal {F}}}is coherent. The same result holds for proper morphismsf, as can be shown with the aid ofChow's lemma.
Sheaf cohomologygroupsHion a noetherian topological space vanish foristrictly greater than the dimension of the space. Thus the quantity, called theEuler characteristicofF{\displaystyle {\mathcal {F}}},
is a well-defined integer (forXprojective). One can then showχ(F(n))=P(n){\displaystyle \chi ({\mathcal {F}}(n))=P(n)}for some polynomialPover rational numbers.[24]Applying this procedure to the structure sheafOX{\displaystyle {\mathcal {O}}_{X}}, one recovers the Hilbert polynomial ofX. In particular, ifXis irreducible and has dimensionr, the arithmetic genus ofXis given by
which is manifestly intrinsic; i.e., independent of the embedding.
The arithmetic genus of a hypersurface of degreedis(d−1n){\displaystyle {\binom {d-1}{n}}}inPn{\displaystyle \mathbb {P} ^{n}}. In particular, a smooth curve of degreedinP2{\displaystyle \mathbb {P} ^{2}}has arithmetic genus(d−1)(d−2)/2{\displaystyle (d-1)(d-2)/2}. This is thegenus formula.
LetXbe a smooth projective variety where all of its irreducible components have dimensionn. In this situation, thecanonical sheafωX, defined as the sheaf ofKähler differentialsof top degree (i.e., algebraicn-forms), is a line bundle.
Serre dualitystates that for any locally free sheafF{\displaystyle {\mathcal {F}}}onX,
where the superscript prime refers to the dual space andF∨{\displaystyle {\mathcal {F}}^{\vee }}is the dual sheaf ofF{\displaystyle {\mathcal {F}}}.
A generalization to projective, but not necessarily smooth schemes is known asVerdier duality.
For a (smooth projective) curveX,H2and higher vanish for dimensional reason and the space of the global sections of the structure sheaf is one-dimensional. Thus the arithmetic genus ofXis the dimension ofH1(X,OX){\displaystyle H^{1}(X,{\mathcal {O}}_{X})}. By definition, thegeometric genusofXis the dimension ofH0(X,ωX). Serre duality thus implies that the arithmetic genus and the geometric genus coincide. They will simply be called the genus ofX.
Serre duality is also a key ingredient in the proof of theRiemann–Roch theorem. SinceXis smooth, there is an isomorphism of groups
from the group of(Weil) divisorsmodulo principal divisors to the group of isomorphism classes of line bundles. A divisor corresponding to ωXis called the canonical divisor and is denoted byK. Letl(D) be the dimension ofH0(X,O(D)){\displaystyle H^{0}(X,{\mathcal {O}}(D))}. Then the Riemann–Roch theorem states: ifgis a genus ofX,
for any divisorDonX. By the Serre duality, this is the same as:
which can be readily proved.[25]A generalization of the Riemann–Roch theorem to higher dimension is theHirzebruch–Riemann–Roch theorem, as well as the far-reachingGrothendieck–Riemann–Roch theorem.
Hilbert schemesparametrize all closed subvarieties of a projective schemeXin the sense that the points (in the functorial sense) ofHcorrespond to the closed subschemes ofX. As such, the Hilbert scheme is an example of amoduli space, i.e., a geometric object whose points parametrize other geometric objects. More precisely, the Hilbert scheme parametrizes closed subvarieties whoseHilbert polynomialequals a prescribed polynomialP.[26]It is a deep theorem of Grothendieck that there is a scheme[27]HXP{\displaystyle H_{X}^{P}}overksuch that, for anyk-schemeT, there is a bijection
The closed subscheme ofX×HXP{\displaystyle X\times H_{X}^{P}}that corresponds to the identity mapHXP→HXP{\displaystyle H_{X}^{P}\to H_{X}^{P}}is called theuniversal family.
ForP(z)=(z+rr){\displaystyle P(z)={\binom {z+r}{r}}}, the Hilbert schemeHPnP{\displaystyle H_{\mathbb {P} ^{n}}^{P}}is called theGrassmannianofr-planes inPn{\displaystyle \mathbb {P} ^{n}}and, ifXis a projective scheme,HXP{\displaystyle H_{X}^{P}}is called theFano schemeofr-planes onX.[28]
In this section, all algebraic varieties arecomplexalgebraic varieties. A key feature of the theory of complex projective varieties is the combination of algebraic and analytic methods. The transition between these theories is provided by the following link: since any complex polynomial is also a holomorphic function, any complex varietyXyields a complexanalytic space, denotedX(C){\displaystyle X(\mathbb {C} )}. Moreover, geometric properties ofXare reflected by the ones ofX(C){\displaystyle X(\mathbb {C} )}. For example, the latter is acomplex manifoldif and only ifXis smooth; it is compact if and only ifXis proper overC{\displaystyle \mathbb {C} }.
Complex projective space is aKähler manifold. This implies that, for any projective algebraic varietyX,X(C){\displaystyle X(\mathbb {C} )}is a compact Kähler manifold. The converse is not in general true, but theKodaira embedding theoremgives a criterion for a Kähler manifold to be projective.
In low dimensions, there are the following results:
Chow's theoremprovides a striking way to go the other way, from analytic to algebraic geometry. It states that every analytic subvariety of a complex projective space is algebraic. The theorem may be interpreted to saying that aholomorphic functionsatisfying certain growth condition is necessarily algebraic: "projective" provides this growth condition. One can deduce from the theorem the following:
Chow's theorem can be shown via Serre'sGAGA principle. Its main theorem states:
The complex manifold associated to an abelian varietyAoverC{\displaystyle \mathbb {C} }is a compactcomplex Lie group. These can be shown to be of the form
and are also referred to ascomplex tori. Here,gis the dimension of the torus andLis a lattice (also referred to asperiod lattice).
According to theuniformization theoremalready mentioned above, any torus of dimension 1 arises from an abelian variety of dimension 1, i.e., from anelliptic curve. In fact, theWeierstrass's elliptic function℘{\displaystyle \wp }attached toLsatisfies a certain differential equation and as a consequence it defines a closed immersion:[33]
There is ap-adic analog, thep-adic uniformizationtheorem.
For higher dimensions, the notions of complex abelian varieties and complex tori differ: onlypolarizedcomplex tori come from abelian varieties.
The fundamentalKodaira vanishing theoremstates that for an ample line bundleL{\displaystyle {\mathcal {L}}}on a smooth projective varietyXover a field of characteristic zero,
fori> 0, or, equivalently by Serre dualityHi(X,L−1)=0{\displaystyle H^{i}(X,{\mathcal {L}}^{-1})=0}fori<n.[34]The first proof of this theorem used analytic methods of Kähler geometry, but a purely algebraic proof was found later. The Kodaira vanishing in general fails for a smooth projective variety in positive characteristic. Kodaira's theorem is one of various vanishing theorems, which give criteria for higher sheaf cohomologies to vanish. Since the Euler characteristic of a sheaf (see above) is often more manageable than individual cohomology groups, this often has important consequences about the geometry of projective varieties.[35]
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In themathematicaldiscipline ofalgebraic geometry,Serre's theorem on affineness(also calledSerre's cohomological characterization of affinenessorSerre's criterion on affineness) is a theorem due toJean-Pierre Serrewhich gives sufficient conditions for aschemeto beaffine, stated in terms ofsheaf cohomology.[1]The theorem was first published by Serre in 1957.[2]
LetXbe a scheme withstructure sheafOX.If:
thenXisaffine.[3]
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Inmathematics, the (right)Ziegler spectrumof aringRis atopological spacewhose points are (isomorphism classes of)indecomposablepure-injectiverightR-modules. Itsclosed subsetscorrespond to theories of modules closed under arbitrary products and direct summands. Ziegler spectra are named after Martin Ziegler, who first defined and studied them in 1984.[1]
LetRbe a ring (associative, with 1, not necessarily commutative). A (right)pp-n-formulais a formula in the language of (right)R-modules of the form
whereℓ,n,m{\displaystyle \ell ,n,m}are natural numbers,A{\displaystyle A}is an(ℓ+n)×m{\displaystyle (\ell +n)\times m}matrix with entries fromR, andy¯{\displaystyle {\overline {y}}}is anℓ{\displaystyle \ell }-tuple of variables andx¯{\displaystyle {\overline {x}}}is ann{\displaystyle n}-tuple of variables.
The (right) Ziegler spectrum,ZgR{\displaystyle \operatorname {Zg} _{R}}, ofRis the topological space whose points are isomorphism classes of indecomposable pure-injective right modules, denoted bypinjR{\displaystyle \operatorname {pinj} _{R}}, and the topology
has the sets
assubbasisof open sets, whereφ,ψ{\displaystyle \varphi ,\psi }range over
(right) pp-1-formulae andφ(N){\displaystyle \varphi (N)}denotes the subgroup ofN{\displaystyle N}consisting of all elements that satisfy the one-variable formulaφ{\displaystyle \varphi }. One can show that these sets form a basis.
Ziegler spectra are rarelyHausdorffand often fail to have theT0{\displaystyle T_{0}}-property. However they are alwayscompactand have a basis of compact open sets given by the sets(φ/ψ){\displaystyle (\varphi /\psi )}whereφ,ψ{\displaystyle \varphi ,\psi }are pp-1-formulae.
When the ringRis countableZgR{\displaystyle \operatorname {Zg} _{R}}issober.[2]It is not currently known if all Ziegler spectra are sober.
Ivo Herzog showed in 1997 how to define the Ziegler spectrum of a locally coherentGrothendieck category, which generalizes the construction above.[3]
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Inmathematics, specificallyring theory, a leftprimitive idealis theannihilatorof a (nonzero)simpleleftmodule. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals.
Primitive ideals areprime. Thequotientof aringby a left primitive ideal is a leftprimitive ring. Forcommutative ringsthe primitive ideals aremaximal, and so commutative primitive rings are allfields.
Theprimitive spectrumof a ring is a non-commutative analog[note 1]of theprime spectrumof a commutative ring.
LetAbe a ring andPrim(A){\displaystyle \operatorname {Prim} (A)}thesetof all primitive ideals ofA. Then there is atopologyonPrim(A){\displaystyle \operatorname {Prim} (A)}, called theJacobson topology, defined so that theclosureof asubsetTis the set of primitive ideals ofAcontaining theintersectionof elements ofT.
Now, supposeAis anassociative algebraover a field. Then, by definition, a primitive ideal is the kernel of anirreducible representationπ{\displaystyle \pi }ofAand thus there is asurjection
Example: thespectrum of a unital C*-algebra.
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Inmathematics, there is an ample supply ofcategorical dualitiesbetween certaincategoriesoftopological spacesand categories ofpartially ordered sets. Today, these dualities are usually collected under the labelStone duality, since they form a natural generalization ofStone's representation theorem for Boolean algebras. These concepts are named in honor ofMarshall Stone. Stone-type dualities also provide the foundation forpointless topologyand are exploited intheoretical computer sciencefor the study offormal semantics.
This article gives pointers to special cases of Stone duality and explains a very general instance thereof in detail.
Probably the most general duality that is classically referred to as "Stone duality" is the duality between the categorySobofsober spaceswithcontinuous functionsand the categorySFrmof spatialframeswith appropriate frame homomorphisms. Thedual categoryofSFrmis the category of spatiallocalesdenoted bySLoc. Thecategorical equivalenceofSobandSLocis the basis for the mathematical area ofpointless topology, which is devoted to the study ofLoc—the category of all locales, of whichSLocis afull subcategory. The involved constructions are characteristic for this kind of duality, and are detailed below.
Now one can easily obtain a number of other dualities by restricting to certain special classes of sober spaces:
Many other Stone-type dualities could be added to these basic dualities.
The starting point for the theory is the fact that every topological space is characterized by a set of pointsXand a system Ω(X) ofopen setsof elements fromX, i.e. a subset of thepowersetofX. It is known that Ω(X) has certain special properties: it is acomplete latticewithin whichsupremaand finiteinfimaare given by set unions and finite set intersections, respectively. Furthermore, it contains bothXand theempty set. Since theembeddingof Ω(X) into the powerset lattice ofXpreserves finite infima and arbitrary suprema, Ω(X) inherits the following distributivity law:
for every element (open set)xand every subsetSof Ω(X). Hence Ω(X) is not an arbitrary complete lattice but acomplete Heyting algebra(also calledframeorlocale– the various names are primarily used to distinguish several categories that have the same class of objects but different morphisms: frame morphisms, locale morphisms and homomorphisms of complete Heyting algebras). Now an obvious question is: To what extent is a topological space characterized by its locale of open sets?
As already hinted at above, one can go even further. The categoryTopof topological spaces has as morphisms the continuous functions, where a functionfis continuous if theinverse imagef−1(O) of any open set in thecodomainoffis open in thedomainoff. Thus any continuous functionffrom a spaceXto a spaceYdefines an inverse mappingf−1from Ω(Y) to Ω(X). Furthermore, it is easy to check thatf−1(like any inverse image map) preserves finite intersections and arbitrary unions and therefore is amorphism of frames. If we define Ω(f) =f−1then Ω becomes acontravariant functorfrom the categoryTopto the categoryFrmof frames and frame morphisms. Using the tools of category theory, the task of finding a characterization of topological spaces in terms of their open set lattices is equivalent to finding a functor fromFrmtoTopwhich isadjointto Ω.
The goal of this section is to define a functor pt fromFrmtoTopthat in a certain sense "inverts" the operation of Ω by assigning to each localeLa set of points pt(L) (hence the notation pt) with a suitable topology. But how can we recover the set of points just from the locale, though it is not given as a lattice of sets? It is certain that one cannot expect in general that pt can reproduce all of the original elements of a topological space just from its lattice of open sets – for example all sets with theindiscrete topologyyield (up to isomorphism) the same locale, such that the information on the specific set is no longer present. However, there is still a reasonable technique for obtaining "points" from a locale, which indeed gives an example of a central construction for Stone-type duality theorems.
Let us first look at the points of a topological spaceX. One is usually tempted to consider a point ofXas an elementxof the setX, but there is in fact a more useful description for our current investigation. Any pointxgives rise to a continuous functionpxfrom the one element topological space 1 (all subsets of which are open) to the spaceXby definingpx(1) =x. Conversely, any function from 1 toXclearly determines one point: the element that it "points" to. Therefore, the set of points of a topological space is equivalently characterized as the set of functions from 1 toX.
When using the functor Ω to pass fromToptoFrm, all set-theoretic elements of a space are lost, but – using a fundamental idea of category theory – one can as well work on thefunction spaces. Indeed, any "point"px: 1 →XinTopis mapped to a morphism Ω(px): Ω(X) → Ω(1). The open set lattice of the one-element topological space Ω(1) is just (isomorphic to) the two-element locale 2 = { 0, 1 } with 0 < 1. After these observations it appears reasonable to define the set of points of a localeLto be the set of frame morphisms fromLto 2. Yet, there is no guarantee that every point of the locale Ω(X) is in one-to-one correspondence to a point of the topological spaceX(consider again the indiscrete topology, for which the open set lattice has only one "point").
Before defining the required topology on pt(X), it is worthwhile to clarify the concept of a point of a locale further. The perspective motivated above suggests to consider a point of a localeLas a frame morphismpfromLto 2. But these morphisms are characterized equivalently by the inverse images of the two elements of 2. From the properties of frame morphisms, one can derive thatp−1(0) is a lower set (sincepismonotone), which contains a greatest elementap= Vp−1(0) (sinceppreserves arbitrary suprema). In addition, theprincipal idealp−1(0) is aprime idealsinceppreserves finite infima and thus the principalapis ameet-prime element. Now the set-inverse ofp−1(0) given byp−1(1) is acompletely prime filterbecausep−1(0) is a principal prime ideal. It turns out that all of these descriptions uniquely determine the initial frame morphism. We sum up:
All of these descriptions have their place within the theory and it is convenient to switch between them as needed.
Now that a set of points is available for any locale, it remains to equip this set with an appropriate topology in order to define the object part of the functor pt. This is done by defining the open sets of pt(L) as
for every elementaofL. Here we viewed the points ofLas morphisms, but one can of course state a similar definition for all of the other equivalent characterizations. It can be shown that setting Ω(pt(L)) = {φ(a) |a∈L} does really yield a topological space (pt(L), Ω(pt(L))). It is common to abbreviate this space as pt(L).
Finally pt can be defined on morphisms ofFrmrather canonically by defining, for a frame morphismgfromLtoM, pt(g): pt(M) → pt(L) as pt(g)(p) =pog. In words, we obtain a morphism fromLto 2 (a point ofL) by applying the morphismgto get fromLtoMbefore applying the morphismpthat maps fromMto 2. Again, this can be formalized using the other descriptions of points of a locale as well – for example just calculate (pog)−1(0).
As noted several times before, pt and Ω usually are not inverses. In general neither isXhomeomorphicto pt(Ω(X)) nor isLorder-isomorphicto Ω(pt(L)). However, when introducing the topology of pt(L) above, a mapping φ fromLto Ω(pt(L)) was applied. This mapping is indeed a frame morphism. Conversely, we can define a continuous function ψ fromXto pt(Ω(X)) by setting ψ(x) = Ω(px), wherepxis just the characteristic function for the pointxfrom 1 toXas described above. Another convenient description is given by viewing points of a locale as meet-prime elements. In this case we have ψ(x) =X\ Cl{x}, where Cl{x} denotes the topological closure of the set {x} and \ is just set-difference.
At this point we already have more than enough data to obtain the desired result: the functors Ω and pt define an adjunction between the categoriesTopandLoc=Frmop, where pt is right adjoint to Ω and thenatural transformationsψ and φopprovide the required unit and counit, respectively.
The above adjunction is not an equivalence of the categoriesTopandLoc(or, equivalently, a duality ofTopandFrm). For this it is necessary that both ψ and φ are isomorphisms in their respective categories.
For a spaceX, ψ:X→ pt(Ω(X)) is a homeomorphismif and only ifit isbijective. Using the characterization via meet-prime elements of the open set lattice, one sees that this is the case if and only if every meet-prime open set is of the formX\ Cl{x} for a uniquex. Alternatively, every join-prime closed set is the closure of a unique point, where "join-prime" can be replaced by(join-) irreduciblesince we are in a distributive lattice. Spaces with this property are calledsober.
Conversely, for a localeL, φ:L→ Ω(pt(L)) is always surjective. It is additionally injective if and only if any two elementsaandbofLfor whichais not less-or-equal tobcan be separated by points of the locale, formally:
If this condition is satisfied for all elements of the locale, then the locale isspatial, or said to have enough points. (See alsowell-pointed categoryfor a similar condition in more general categories.)
Finally, one can verify that for every spaceX, Ω(X) is spatial and for every localeL, pt(L) is sober. Hence, it follows that the above adjunction ofTopandLocrestricts to an equivalence of the full subcategoriesSobof sober spaces andSLocof spatial locales. This main result is completed by the observation that for the functor pt o Ω, sending each space to the points of its open set lattice is left adjoint to theinclusion functorfromSobtoTop. For a spaceX, pt(Ω(X)) is called itssoberification. The case of the functor Ω o pt is symmetric but a special name for this operation is not commonly used.
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Inmathematics, anEn{\displaystyle {\mathcal {E}}_{n}}-algebrain asymmetric monoidalinfinity categoryCconsists of the following data:
subject to the requirements that the multiplication maps are compatible with composition, and thatμ{\displaystyle \mu }is an equivalence ifm=1{\displaystyle m=1}. An equivalent definition is thatAis analgebrainCover the littlen-disksoperad.
Thisalgebra-related article is astub. You can help Wikipedia byexpanding it.
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https://en.wikipedia.org/wiki/E_n-ring
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TheQ-codeis a standardised collection of three-letter codes that each start with the letter "Q". It is anoperating signalinitially developed for commercialradiotelegraphcommunication and later adopted by other radio services, especiallyamateur radio. To distinguish the use of a Q-code transmitted as a question from the same Q-code transmitted as a statement, operators either prefixed it with the military network question marker "INT" (▄ ▄ ▄▄▄ ▄ ▄▄▄) or suffixed it with the standard Morse question markUD(▄ ▄ ▄▄▄ ▄▄▄ ▄ ▄).
Although Q-codes were created when radio usedMorse codeexclusively, they continued to be employed after the introduction of voice transmissions. To avoid confusion, transmittercall signsare restricted; countries can be issued unused Q-Codes as theirITU prefixe.g.Qatar is QAT.
Codes in the range QAA–QNZ are reserved for aeronautical use; QOA–QQZ for maritime use and QRA–QUZ for all services.
"Q" has no official meaning, but it is sometimes assigned a word withmnemonicvalue, such as "question" or "query", for example in QFE: "query field elevation".[1]
The original Q-codes were created,circa1909, by the British government as a "List of abbreviations ... prepared for the use of British ships and coast stations licensed by thePostmaster General".[2]The Q-codes facilitated communication between maritime radio operators speaking different languages, so they were soon adopted internationally. A total of forty-five Q-codes appeared in the "List of Abbreviations to be used in Radio Communications", which was included in the Service Regulations affixed to theSecond International Radiotelegraph Convention in London(The convention was signed on July 5, 1912, and became effective July 1, 1913.)
The following table reviews a sample of the all-services Q-codes adopted by the 1912 convention:
Over the years the original Q-codes were modified to reflect changes in radio practice. For example, QSW / QSX originally stood for, "Shall I increase / decrease my spark frequency?", but in the 1920sspark-gap transmitterswere gradually being banned from land stations, making that meaning obsolete.[3]By the 1970s, thePost Office Handbook for Radio Operatorslisted over a hundred Q-codes,[4]covering a wide range of subjects including radio procedures, meteorology, radio direction finding, and search and rescue.
Some Q-codes are also used inaviation, in particular QNE, QNH and QFE, referring to certainaltimeter settings. These codes are used in radiotelephone conversations withair traffic controlas unambiguous shorthand, where safety and efficiency are of vital importance. A subset of Q-codes is used by theMiami-Dade County, Floridalocal government for law enforcement and fire rescue communications, one of the few instances where Q-codes are used in ground voice communication.[5]
The QAA–QNZ code range includes phrases applicable primarily to the aeronautical service,[6]as defined by theInternational Civil Aviation Organization.[7]The QOA–QQZ code range is reserved for the maritime service. The QRA–QUZ code range includes phrases applicable to all services and is allocated to theInternational Telecommunication Union.[8]QVA–QZZ are not allocated.[9]Many codes have no immediate applicability outside one individual service, such as maritime operation (many QO or QU series codes) orradioteletypeoperation (the QJ series).[10]
Many military and other organisations that use Morse code have adopted additional codes, including theZ codeused by most European andNATOcountries. The Z code adds commands and questions adapted for military radio transmissions, for example, "ZBW 2", which means "change to backup frequency number 2", and "ZNB abc", which means "my checksum is abc, what is yours?"[11]
Used in their formal question / answer sense, the meaning of a Q-code varies depending on whether the individual Q-code is sent as a question or an answer. For example, the message "QRP?" means "Shall I decrease transmitter power?", and a reply of "QRP" means "Yes, decrease your transmitter power", whereas an unprompted statement "QRP" means "Please decrease your transmitter power". This structured use of Q-codes is fairly rare and now mainly limited to amateur radio and militaryMorse code(CW) traffic networks.
Under U.S. F.C.C. Regulations, CFR 47 97.113(a)(4), Amateurs are not permitted to 'transmit codes or ciphers' (historical description), allowing only that 'plain- language communications' may be transmitted. The term 'Q-Signal' has been historically used in that context, not 'Q-Code,' - the use of 'Code' is technically a violation of F.C.C. Regulations. This essentially examples, in the U.S. the historic separation between Amateur and Commercial/Military radio services
First defined in ICAO publication "Doc 6100-COM/504/1" and in "ICAO Procedures for Air Navigation Services, Abbreviations and Codes (PANS-ABC)" [Doc8400-4] (4th edition 1989), the majority of the Q-codes have fallen out of common use; for example today reports such as QAU ("I am about to jettison fuel") and QAZ ("I am flying in a storm") would be voice or computerised transmissions. But several remain part of the standardICAOradiotelephony phraseology in aviation. These are also part ofACP131, which lists all ITU-R Q-codes, without grouping them by aeronautical/marine/general use.[12]
orI am arranging my flight in order to arrive over ____ (place) at ____ hours.
orArrange your flight so as to reach flight level / altitude ____ at ____ (hours or place).
orHas aircraft ____ landed at ____ (place)?
or(You may) land at ____ (place).orAircraft ____ landed at ____ (place).
orAm I near area ____ (identification of area)?
orJettison fuel in ____ (area).
orMaintain a vertical distance of ____ (figures and units) above clouds, smoke, haze or fog levels.
orMaintain a vertical distance of ____ (figures and units) below cloud.
orReport reaching flight level/altitude ____ [or____ (area or place)].
orReport leaving flight level/altitude ____ [or____ (area or place)].
orI am changing my flight level/altitude from ____ to ____.
orNo delay expected.
orI am making a 360-degree turn immediately (turning to the ____).
orUse your full call sign until further notice.
orWork on a trailing aerial.
orWhat is the D-Value at ____ (place or position) (at ____ hours) for the ____ millibar level?
orThe D-Value at ____ (place or position) at ____ hours for the ____ millibar level is (D-Value figures and units) ____ (specify plus or minus).[c]
orYou are cleared subject to maintaining own separation and visual meteorological conditions.
orIFRflight cancelled at ____ (time).
orHave you reached your parking area?
orI have reached my parking area.
orHave you left the parking area?
orI have left the parking area.
orHave you moved to the holding position for runway number ____ ?
orI have moved to the holding position for runway number ____.
orHave you assumed position for take-off?
orI am assuming take-off position for runway number ____ and am holding.
orHave you cleared the runway (orlanding area)?
orI have cleared the runway (orlanding area).
orPlease light the aerodrome lights.
1. Maintain (orfly at) flight level / altitude ____.2. I am maintaining flight level / altitude ____3. I intend cruising at flight level/altitude ____.
orPlease light the approach and runway lights.
orPlease have the ____ radio facility at ____ (place) put in operation.
orPlease switch on the floodlights.
orWork on a fixed aerial.
orWhat track are you making good?
orI am making good a track from ____ (place) on ____ degrees ____ (trueormagnetic).
orCan you see the aerodrome?orCan you see ____ (aircraft)?
orI can see the aerodrome.orI can see ____ (aircraft).
orEmergency landing being made at ____ (place). All aircraft below flight level/altitude ____ and within a distance of ____ (figures and units) leave ____ (place or headings).
orAre you making a ____ approach?
orI am making a ____ approach.
orI will establish communication with ____ radio station on ____ kHz (orMHz) [noworat ____ hours].
orIn the parallel sweep (track) search being (orto be) conducted, what is (are) ____.1. the direction of sweeps,2. the separation between sweeps,3. the flight level/altitude ____ employed in the search pattern?
1. with direction of sweeps ____ degrees ____ (trueormagnetic).2. with ____ (distance figures and units) separation between sweeps.3. at flight level/altitude ____.
or____ (name) unit is taking part in operation [____ (identification] (with effect from ____ hours).
lowest layer observed* ____ eights (____ type) with base of ____ (figures and units) and tops of ____ (figures and units)[*and similarly in sequence for each of the layers observed.]height above ____ (datum).
This assignment is specified inRECOMMENDATION ITU-R M.1172.[13]
Q signals are not substantially used in the maritime service. Morse code is now very rarely used for maritime communications, but in isolated maritime regions like Antarctica and the South Pacific the use of Q-codes continues. Q-codes still work when HF voice circuits are not possible due to atmospherics and the nearest vessel is one ionospheric hop away.
First defined by the Washington 1927 ITU Radio Regulations. Later defined by ITU-R in Appendix 9 to the Radio Regulations Annex to theInternational Telecommunications Convention(Atlantic City, 1947). The current callsign table is found in ITU-R Appendix 42. Current interpretation of the Q-code can be found in ITU-R Appendices 14 and 15.
ITU Radio Regulations 1990, Appendix 13: Miscellaneous Abbreviations and Signals to be Used in Radiotelegraphy Communications Except in the Maritime Mobile Service:[14]
orReturn to ____ (place).
orIs my transmission being interfered with?[15]
orYour transmission is being interfered with ____.[15]
orAre you a low traffic ship?[15]
orI am a low traffic ship.[15]
orAre my signals mutilated?[15]
orYour signals are mutilated.[15]
orWill you inform ____ (call sign) that I have been unable to break in on his transmission (on ____ kHz (orMHz)).
orWill you listen to ____ (call sign(s)) on ____ kHz (orMHz), or in the bands ____ / channels ____ ?[15]
orI am listening to ____ (call sign(s)) on ____ kHz (orMHz), or in the bands ____ / channels ____.[15]
orWhat is my TRUE bearing from ____ (call sign)?orWhat is the TRUE bearing of ____ (call sign) from ____ (call sign)?
orYour TRUE bearing from ____ (call sign) was ____ degrees at ____ hours.orThe TRUE bearing of ____ (call sign) from ____ (call sign) was ____ degrees at ____ hours.
orWill you request ____ to send two dashes of ten seconds followed by his call sign (repeated ____ times) on ____ kHz (orMHz)?
orI have requested ____ to send two dashes of ten seconds followed by his call sign (repeated ____ times) on ____ kHz (orMHz).
(Requests the speed of a ship or aircraft through the water or air respectively).
(Indicates the speed of a ship or aircraft through the water or air respectively).
orAre you airborne?
orI am airborne.
orAre you going to alight (orland)?
orI am going to alight (orland).
orWill you send your call sign (and/orname) for ____ seconds?[15]
orI will send my call sign (and/orname) for ____ seconds.[15]
orCan you speak in ____ (language), – with interpreter if necessary; if so, on what frequencies?[15]
orI can speak in ____ (language) on ____ kHz (orMHz).[15]
orI shall be forced to alight (orland) at ____ (position or place) at ____ hours.[14]
or
or
or2. When directed to a single station:[15]
in the vicinity of ____ latitude, ____ longitude (or according to any other indication)?
in the vicinity of ____ latitude, ____ longitude (or according to any other indication).
Amateur radio has adapted two different sets of Q-codes for use in amateur communications. The first set comes from the ITU civil series QRA through QUZ. Most of the meanings are identical to the ITU definitions, however, they must be looked at in the context of amateur communications. For example, QSJ? asks what the charges are for sending the telegraph. Since by regulation amateur communications are without charge, this Q-code couldn't make sense.
The second set is the set ofQN Signals, used only in ARRLNTSnets. These operating signals generally have no equivalent in the ACP 131 publication or ITU publications, and are specifically defined only for use in ARRL NTS nets. They are not used in casual amateur radio communications.[16][17]
Selected Q-codes were soon adopted byamateur radio operators. In December 1915, theAmerican Radio Relay Leaguebegan publication of a magazine titledQST, named after the Q-code for "General call to all stations". In amateur radio, the Q-codes were originally used in Morse code transmissions to shorten lengthy phrases and were followed by a Morse code question mark (▄ ▄ ▄▄▄ ▄▄▄ ▄ ▄) if the phrase was a question.
Q-codes are commonly used in voice communications as shorthand nouns, verbs, and adjectives making up phrases. For example, an amateur radio operator will complain about QRM (man-made interference), or tell another operator that there is "QSB on the signal" (fading); "to QSY" is to change your operating frequency, or to break in on a conversation QSK is often used even on VHF and UHF frequencies. (See alsoInformal usage, below.)
Often heard colloquially as:I am suspending operation / shutting off the radio.
Responses to a radiotelegraph Q-code query or a Q-code assertion may vary depending upon the code. For Q-code assertions or queries which only need to be acknowledged as received, the usual practice is to respond with the letter "R" for "Roger" which means "Received correctly". Sending an "R" merely means the code has been correctly received and does not necessarily mean that the receiving operator has taken any other action.
For Q-code queries that need to be answered in the affirmative, the usual practice is to respond with the letter "C" (Sounds like the Spanish word "Si"). For Q-code queries that need to be answered in the negative, the usual practice it to respond with the letter "N" for "no". For those Q-code assertions that merely need to be acknowledged as understood, the usual practice is to respond with theprosignSN(orVE) which means "understood". On telegraph cable networks "KK" was often used at the end of a reply to a Q-code to mean "OK" or "Acknowledged". This practice predates amateur radio as telegraph operators in the late 19th century are known to have used it.
QAC- Taken from the Articles of Association of the South Hampshire International Telegraphy Society, para 9: "...and amongst themselves they shall promote the Use of the Code QAC, which shall be taken as implying "All Compliments" and shall include:- VY 73 73 OM CUL BCNU & mni tnx fer nice/FB/rotten QSO GL GB hpe cuagn wid gud/btr/wrse condx mri Xms Hpi Nw Yr mni hpi rtrns gtgs fer Rosh Hoshanah/Id el Fitr/May Day/Tksgvg 88 to XYL/YL/Widow Ciao Cheerio & gud/FB/best DX or any Part or Parts thereof in any Permutation or Combination.[20]
QLF– "Are you sending with your left foot? Try sending with your left foot!" A humorously derogatory comment about the quality of a person's sending.[21][22]
QNB– QNB? “How many buttons on your radio?” “QNB 100/5” Means there are 100 and I know what 5 of them do.
QSK– "I can hear you during my transmission" – refers to a particular mode of Morse code operating often calledQSK operation (full break-in)in which the receiver is quickly enabled during the spaces between the dits and dahs, which allows another operator to interrupt transmissions. Many moderntransceiversincorporate this function, sometimes referred to asfull break-inas againstsemi-break-inin which there is a short delay before the transceiver goes to receive.[23]
QSY– "Change to transmission on another frequency"; colloquially, "move [=change address]". E.g., "When didGKBQSY from Northolt to Portishead?"[24]
QTH– "My location is ____"; colloquially in voice or writing, "location". E.g., "The OCF[antenna type]is an interesting build but at my QTH a disappointing performer."[25]
QTHR– "At the registered location ____"; chiefly British use. Historically, the location in the printed Callbook; modernly, "as given in online government records for my callsign". E.g., "You can contact me QTHR".[26]
QBL– “Quit Bein' a Lid” QBL is used among amateur radio operators to indicate humour in their CW transmission. While QBL is generally used by a small subsection of operators who can properly decode, it is available to anyone.
During World War II, according toBletchley Park’sGeneral Report on Tunny,[27]German radio teleprinter networks used Q-codes to establish and maintain circuit connections.
In particular:QEPwas to indicate theLorenz ciphermachine setting for each message and,QZZto indicate that the daily key change was about to take place at the sender's station.
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Z Code(likeQ CodeandX Code) is a set of operating signals used inCW,TTYandRTTYradio communication.
There are at least three sets of Z codes.
Many of the old C&W codes are derived frommnemonics
The old C&W Z codes are not widely used today.
There are other sets of codes internally used byRussia's military and other operating agencies.
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Acontinuous waveorcontinuous waveform(CW) is anelectromagnetic waveof constantamplitudeandfrequency, typically asine wave, that formathematical analysisis considered to be of infinite duration.[1]It may refer to e.g. alaserorparticle acceleratorhaving a continuous output, as opposed to apulsedoutput.
By extension, the termcontinuous wavealso refers to an early method ofradiotransmissionin which a sinusoidalcarrier waveis switched on and off. This is more precisely calledinterrupted continuous wave(ICW).[2]Informationis carried in the varying duration of theon and off periodsof the signal, for example byMorse codein early radio. In earlywireless telegraphyradio transmission, CW waves were also known as "undamped waves", to distinguish this method fromdamped wavesignals produced by earlierspark gaptype transmitters.
Very early radio transmitters used aspark gapto produce radio-frequency oscillations in the transmitting antenna. The signals produced by thesespark-gap transmittersconsisted of strings of brief pulses ofsinusoidalradio frequency oscillations which died out rapidly to zero, calleddamped waves. The disadvantage of damped waves was that their energy was spread over an extremely wide band offrequencies; they had widebandwidth. As a result, they producedelectromagnetic interference(RFI) that spread over the transmissions of stations at other frequencies.
This motivated efforts to produce radio frequency oscillations that decayed more slowly; had less damping. There is an inverse relation between the rate of decay (thetime constant) of a damped wave and its bandwidth; the longer the damped waves take to decay toward zero, the narrower the frequency band the radio signal occupies, so the less it interferes with other transmissions. As more transmitters began crowding the radio spectrum, reducing the frequency spacing between transmissions, government regulations began to limit the maximum damping or "decrement" a radio transmitter could have. Manufacturers produced spark transmitters which generated long "ringing" waves with minimal damping.
It was realized that the ideal radio wave forradiotelegraphiccommunication would be a sine wave with zero damping, acontinuous wave. An unbroken continuous sine wave theoretically has no bandwidth; all its energy is concentrated at a single frequency, so it doesn't interfere with transmissions on other frequencies. Continuous waves could not be produced with an electric spark, but were achieved with thevacuum tubeelectronic oscillator, invented around 1913 byEdwin ArmstrongandAlexander Meissner. AfterWorld War I, transmitters capable of producing continuous wave, theAlexanderson alternatorandvacuum tubeoscillators, became widely available.
Damped wave spark transmitters were replaced by continuous wave vacuum tube transmitters around 1920, and damped wave transmissions were finally outlawed in 1934.
In order to transmit information, the continuous wave must be turned off and on with atelegraph keyto produce the different length pulses, "dots" and "dashes", that spell out text messages inMorse code, so a "continuous wave" radiotelegraphy signal consists of pulses of sine waves with a constant amplitude interspersed with gaps of no signal.
In on-off carrier keying, if the carrier wave is turned on or off abruptly,communications theorycan show that thebandwidthwill be large; if the carrier turns on and off more gradually, the bandwidth will be smaller. The bandwidth of an on-off keyed signal is related to the data transmission rate as:Bn=BK{\displaystyle B_{n}=BK}whereBn{\displaystyle B_{n}}is the necessary bandwidth in hertz,B{\displaystyle B}is the keying rate in signal changes per second (baudrate),
andK{\displaystyle K}is a constant related to the expected radio propagation conditions; K=1 is difficult for a human ear to decode, K=3 or K=5 is used when fading ormultipath propagationis expected.[3]
The spurious noise emitted by atransmitterwhich abruptly switches a carrier on and off is calledkey clicks. The noise occurs in the part of the signal bandwidth further above and below the carrier than required for normal, less abrupt switching. The solution to the problem for CW is to make the transition between on and off to be more gradual, making the edges of pulsessoft, appearing more rounded, or to use other modulation methods (e.g.phase modulation). Certain types of power amplifiers used in transmission may aggravate the effect of key clicks.
Early radio transmitters could not bemodulatedto transmit speech, and so CW radio telegraphy was the only form of communication available. CW still remains a viable form of radio communication many years after voice transmission was perfected, because simple, robust transmitters can be used, and because its signals are the simplest of the forms ofmodulationable to penetrate interference. The low bandwidth of the code signal, due in part to low information transmission rate, allows very selective filters to be used in the receiver, which block out much of the radio noise that would otherwise reduce the intelligibility of the signal.
Continuous-wave radio was calledradiotelegraphybecause like thetelegraph, it worked by means of a simple switch to transmitMorse code. However, instead of controlling the electricity in a cross-country wire, the switch controlled the power sent to a radiotransmitter. This mode is still in common use byamateur radiooperators due to its narrow bandwidth and highsignal-to-noise ratiocompared to other modes of communication.
In military communications andamateur radiothe terms "CW" and "Morse code" are often used interchangeably, despite the distinctions between the two. Aside from radio signals, Morse code may be sent usingdirect currentin wires, sound, or light, for example. For radio signals, a carrier wave is keyed on and off to represent the dots and dashes of the code elements. The carrier's amplitude and frequency remainconstantduring each code element. At the receiver, the received signal is mixed with aheterodynesignal from a BFO (beat frequency oscillator) to change the radio frequency impulses to sound. Almost all commercial traffic has now ceased operation using Morse, but it is still used by amateur radio operators.Non-directional beacons (NDB)andVHF omnidirectional radio range (VOR)used in air navigation use Morse to transmit their identifier.
Morse code is all but extinct outside the amateur service, so in non-amateur contexts the term CW usually refers to acontinuous-wave radarsystem, as opposed to one transmitting short pulses. Somemonostatic (single antenna) CW radarstransmit and receive a single (non-swept) frequency, often using the transmitted signal as thelocal oscillatorfor the return; examples include police speed radars and microwave-type motion detectors and automatic door openers. This type of radar is effectively "blinded" by its own transmitted signal to stationary targets; they must move toward or away from the radar quickly enough to create a Doppler shift sufficient to allow the radar to isolate the outbound and return signal frequencies. This kind of CW radar can measurerange ratebut notrange(distance).
Other CW radars linearly or pseudo-randomly "chirp" (frequency modulate) their transmitters rapidly enough to avoid self-interference with returns from objects beyond some minimum distance; this kind of radar can detect and range static targets. This approach is commonly used inradar altimeters, inmeteorologyand in oceanic and atmospheric research. Thelanding radaron theApollo Lunar Modulecombined both CW radar types.
CWbistatic radarsuse physically separate transmit and receive antennas to lessen the self-interference problems inherent in monostatic CW radars.
Inlaser physicsand engineering, "continuous wave" or "CW" refers to alaserthat produces a continuous output beam, sometimes referred to as "free-running," as opposed to aq-switched,gain-switchedormodelockedlaser, which has a pulsed output beam.
The continuous wavesemiconductor laserwas invented by Japanese physicistIzuo Hayashiin 1970.[citation needed]It led directly to the light sources infiber-optic communication,laser printers,barcode readers, andoptical disc drives, commercialized by Japanese entrepreneurs,[4]and opened up the field ofoptical communication, playing an important role in futurecommunication networks.[5]Optical communication in turn provided the hardware basis forinternettechnology, laying the foundations for theDigital RevolutionandInformation Age.[6]
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Radiois the technology ofcommunicatingusingradio waves.[1][2][3]Radio waves areelectromagnetic wavesoffrequencybetween 3hertz(Hz) and 300gigahertz(GHz). They are generated by anelectronic devicecalled atransmitterconnected to anantennawhich radiates the waves. They can be received by other antennas connected to aradio receiver; this is the fundamental principle of radio communication. In addition to communication, radio is used forradar,radio navigation,remote control,remote sensing, and other applications.
Inradio communication, used in radio andtelevision broadcasting, cell phones,two-way radios,wireless networking, andsatellite communication, among numerous other uses, radio waves are used to carry information across space from a transmitter to a receiver, bymodulatingthe radio signal (impressing an information signal on the radio wave by varying some aspect of the wave) in the transmitter. In radar, used to locate and track objects like aircraft, ships, spacecraft and missiles, a beam of radio waves emitted by a radar transmitter reflects off the target object, and the reflected waves reveal the object's location to a receiver that is typically colocated with the transmitter. In radio navigation systems such asGPSandVOR, a mobile navigation instrument receives radio signals from multiplenavigational radio beaconswhose position is known, and by precisely measuring the arrival time of the radio waves the receiver can calculate its position on Earth. In wirelessradio remote controldevices likedrones,garage door openers, andkeyless entry systems, radio signals transmitted from a controller device control the actions of a remote device.
The existence of radio waves was first proven by German physicistHeinrich Hertzon 11 November 1886.[4]In the mid-1890s, building on techniques physicists were using to study electromagnetic waves, Italian physicistGuglielmo Marconideveloped the first apparatus for long-distance radio communication,[5]sending a wirelessMorse Codemessage to a recipient over a kilometer away in 1895,[6]and the first transatlantic signal on 12 December 1901.[7]The first commercial radio broadcast was transmitted on 2 November 1920, when the live returns of theHarding-Cox presidential electionwere broadcast by Westinghouse Electric and Manufacturing Company in Pittsburgh, under the call signKDKA.[8]
The emission of radio waves is regulated by law, coordinated by theInternational Telecommunication Union(ITU), which allocates frequency bands in theradio spectrumfor various uses.
The wordradiois derived from the Latin wordradius, meaning "spoke of a wheel, beam of light, ray." It was first applied to communications in 1881 when, at the suggestion of French scientistErnest Mercadier[fr],Alexander Graham Belladoptedradiophone(meaning "radiated sound") as an alternate name for hisphotophoneoptical transmission system.[9][10]
Following Hertz's discovery of the existence ofradio wavesin 1886, the termHertzian waveswas initially used for this radiation.[11]The first practical radio communication systems, developed byMarconiin 1894–1895, transmittedtelegraphsignals by radio waves,[4]so radio communication was first calledwireless telegraphy. Up until about 1910 the termwireless telegraphyalso included a variety of other experimental systems for transmitting telegraph signals without wires, includingelectrostatic induction,electromagnetic inductionandaquatic and earth conduction, so there was a need for a more precise term referring exclusively to electromagnetic radiation.[12][13]
The French physicistÉdouard Branly, who in 1890 developed the radio wave detectingcoherer, called it in French aradio-conducteur.[14][15]Theradio-prefix was later used to form additional descriptive compound and hyphenated words, especially in Europe. For example, in early 1898 the British publicationThe Practical Engineerincluded a reference tothe radiotelegraphandradiotelegraphy.[14][16]
The use ofradioas a standalone word dates back to at least 30 December 1904, when instructions issued by the British Post Office for transmitting telegrams specified that "The word 'Radio'... is sent in the Service Instructions."[14][17]This practice was universally adopted, and the word "radio" introduced internationally, by the 1906 Berlin Radiotelegraphic Convention, which included a Service Regulation specifying that "Radiotelegrams shall show in the preamble that the service is 'Radio'".[14]
The switch toradioin place ofwirelesstook place slowly and unevenly in the English-speaking world.Lee de Foresthelped popularize the new word in the United States—in early 1907, he founded the DeForest Radio Telephone Company, and his letter in the 22 June 1907Electrical Worldabout the need for legal restrictions warned that "Radio chaos will certainly be the result until such stringent regulation is enforced."[18]The United States Navy would also play a role. Although its translation of the 1906 Berlin Convention used the termswireless telegraphandwireless telegram, by 1912 it began to promote the use ofradioinstead. The term started to become preferred by the general public in the 1920s with the introduction of broadcasting.
Electromagnetic waveswere predicted byJames Clerk Maxwellin his 1873 theory ofelectromagnetism, now calledMaxwell's equations, who proposed that a coupled oscillatingelectric fieldandmagnetic fieldcould travel through space as a wave, and proposed that light consisted of electromagnetic waves of shortwavelength. On 11 November 1886, German physicistHeinrich Hertz, attempting to confirm Maxwell's theory, first observed radio waves he generated using a primitivespark-gap transmitter.[4]Experiments by Hertz and physicistsJagadish Chandra Bose,Oliver Lodge,Lord Rayleigh, andAugusto Righi, among others, showed that radio waves like light demonstrated reflection,refraction,diffraction,polarization,standing waves, and traveled at the same speed as light, confirming that both light and radio waves were electromagnetic waves, differing only infrequency.[19]In 1895,Guglielmo Marconideveloped the first radio communication system, using a spark-gap transmitter to sendMorse codeover long distances. By December 1901, he had transmitted across the Atlantic Ocean.[4][5][6][7]Marconi andKarl Ferdinand Braunshared the 1909 Nobel Prize in Physics "for their contributions to the development of wireless telegraphy".[20]
During radio's first two decades, called theradiotelegraphyera, the primitive radio transmitters could only transmit pulses of radio waves, not the continuous waves which were needed for audiomodulation, so radio was used for person-to-person commercial, diplomatic and military text messaging. Starting around 1908 industrial countries built worldwide networks of powerful transoceanic transmitters to exchangetelegramtraffic between continents and communicate with their colonies and naval fleets. DuringWorld War Ithe development ofcontinuous waveradio transmitters,rectifyingelectrolytic, and crystalradio receiver detectorsenabledamplitude modulation(AM)radiotelephonyto be achieved byReginald Fessendenand others, allowingaudioto be transmitted. On 2 November 1920, the first commercial radio broadcast was transmitted by Westinghouse Electric and Manufacturing Company in Pittsburgh, under the call signKDKAfeaturing live coverage of theHarding-Cox presidential election.[8]
Radio waves are radiated byelectric chargesundergoingacceleration.[21][22]They are generated artificially by time-varyingelectric currents, consisting ofelectronsflowing back and forth in a metal conductor called anantenna.[23][24]
As they travel farther from the transmitting antenna, radio waves spread out so theirsignal strength(intensityin watts per square meter) decreases (seeInverse-square law), so radio transmissions can only be received within a limited range of the transmitter, the distance depending on the transmitter power, the antennaradiation pattern, receiver sensitivity,background noiselevel, and presence ofobstructions between transmitter and receiver. Anomnidirectional antennatransmits or receives radio waves in all directions, while adirectional antennatransmits radio waves in a beam in a particular direction, or receives waves from only one direction.[25][26][27][28]
Radio waves travel at thespeed of lightin vacuum[29]and at slightly lower velocity in air.[30]
The other types ofelectromagnetic wavesbesides radio waves,infrared,visible light,ultraviolet,X-raysandgamma rays, can also carry information and be used for communication. The wide use of radio waves for telecommunication is mainly due to their desirablepropagationproperties stemming from their longer wavelength.[24]Radio waves have the ability to pass through the atmosphere in any weather, foliage, and at longer wavelengths through most building materials. Bydiffraction, longer wavelengths can bend around obstructions, and unlike other electromagnetic waves they tend to be scattered rather than absorbed by objects larger than their wavelength.
In radio communication systems, information is carried across space using radio waves. At the sending end, the information to be sent is converted by some type oftransducerto a time-varyingelectrical signalcalled the modulation signal.[24][31]The modulation signal may be anaudio signalrepresenting sound from amicrophone, avideo signalrepresenting moving images from avideo camera, or adigital signalconsisting of a sequence ofbitsrepresenting binary data from a computer. The modulation signal is applied to aradio transmitter. In the transmitter, anelectronic oscillatorgenerates analternating currentoscillating at aradio frequency, called thecarrier wavebecause it serves to generate the radio waves thatcarrythe information through the air. The modulation signal is used tomodulatethe carrier, varying some aspect of the carrier wave, impressing the information in the modulation signal onto the carrier. Different radio systems use different modulation methods:[32]
Many other types of modulation are also used. In some types, the carrier wave is suppressed, and only one or both modulationsidebandsare transmitted.[34]
The modulated carrier isamplifiedin the transmitter and applied to a transmittingantennawhich radiates the energy as radio waves. The radio waves carry the information to the receiver location.[35]At the receiver, the radio wave induces a tiny oscillatingvoltagein the receiving antenna – a weaker replica of the current in the transmitting antenna.[24][31]This voltage is applied to theradio receiver, whichamplifiesthe weak radio signal so it is stronger, thendemodulatesit, extracting the original modulation signal from the modulated carrier wave. The modulation signal is converted by atransducerback to a human-usable form: an audio signal is converted tosound wavesby a loudspeaker or earphones, avideo signalis converted to images by adisplay, while a digital signal is applied to a computer or microprocessor, which interacts with human users.[32]
The radio waves from many transmitters pass through the air simultaneously without interfering with each other because each transmitter's radio waves oscillate at a different frequency, measured inhertz(Hz),kilohertz(kHz),megahertz(MHz) orgigahertz(GHz). The receiving antenna typically picks up the radio signals of many transmitters. The receiver usestuned circuitsto select the radio signal desired out of all the signals picked up by the antenna and reject the others. A tuned circuit acts like aresonator, similar to atuning fork.[31]It has a naturalresonant frequencyat which it oscillates. The resonant frequency of the receiver's tuned circuit is adjusted by the user to the frequency of the desired radio station; this is calledtuning. The oscillating radio signal from the desired station causes the tuned circuit to oscillate in sympathy, and it passes the signal on to the rest of the receiver. Radio signals at other frequencies are blocked by the tuned circuit and not passed on.[36]
A modulated radio wave, carrying an information signal, occupies a range of frequencies. The information in a radio signal is usually concentrated in narrow frequency bands calledsidebands(SB) just above and below thecarrierfrequency. The width inhertzof the frequency range that the radio signal occupies, the highest frequency minus the lowest frequency, is called itsbandwidth(BW).[32][37]For any givensignal-to-noise ratio, a given bandwidth can carry the same amount of information regardless of where in the radio frequency spectrum it is located; bandwidth is a measure ofinformation-carrying capacity. The bandwidth required by a radio transmission depends on the data rate of the information being sent, and thespectral efficiencyof the modulation method used; how much data it can transmit in each unit of bandwidth. Different types of information signals carried by radio have different data rates. For example, a television signal has a greater data rate than anaudio signal.[32][38]
Theradio spectrum, the total range of radio frequencies that can be used for communication in a given area, is a limited resource.[37][3]Each radio transmission occupies a portion of the total bandwidth available. Radio bandwidth is regarded as aneconomic goodwhich has a monetary cost and is in increasing demand. In some parts of the radio spectrum, the right to use a frequency band or even a single radio channel is bought and sold for millions of dollars. So there is an incentive to employ technology to minimize the bandwidth used by radio services.[38]
A slow transition fromanalogtodigitalradio transmission technologies began in the late 1990s.[39][40]Part of the reason for this is thatdigital modulationcan often transmit more information (a greater data rate) in a given bandwidth thananalog modulation, by usingdata compressionalgorithms, which reduce redundancy in the data to be sent, and more efficient modulation. Other reasons for the transition is that digital modulation has greaternoise immunitythan analog,digital signal processingchips have more power and flexibility than analog circuits, and a wide variety of types of information can be transmitted using the same digital modulation.[32]
Because it is a fixed resource which is in demand by an increasing number of users, theradio spectrumhas become increasingly congested in recent decades, and the need to use it more effectively is driving many additional radio innovations such astrunked radio systems,spread spectrum(ultra-wideband) transmission,frequency reuse,dynamic spectrum management, frequency pooling, andcognitive radio.[38]
TheITUarbitrarily divides theradio spectruminto 12 bands, each beginning at a wavelength which is a power of ten (10n) metres, with corresponding frequency of 3 times a power of ten, and each covering a decade of frequency or wavelength.[3][41]Each of these bands has a traditional name:[42]
It can be seen that thebandwidth, the range of frequencies, contained in each band is not equal but increases exponentially as the frequency increases; each band contains ten times the bandwidth of the preceding band.[43]
The term "tremendously low frequency" (TLF) has been used for wavelengths from 1–3 Hz (300,000–100,000 km),[44]though the term has not been defined by the ITU.[42]
The airwaves are a resource shared by many users. Two radio transmitters in the same area that attempt to transmit on the same frequency will interfere with each other, causing garbled reception, so neither transmission may be received clearly.[37]Interferencewith radio transmissions can not only have a large economic cost, but it can also be life-threatening (for example, in the case of interference with emergency communications orair traffic control).[45][46]
To prevent interference between different users, the emission of radio waves is strictly regulated by national laws, coordinated by an international body, theInternational Telecommunication Union(ITU), which allocates bands in theradio spectrumfor different uses.[37][3]Radio transmitters must be licensed by governments, under a variety of license classes depending on use, and are restricted to certain frequencies and power levels. In some classes, such as radio and television broadcasting stations, the transmitter is given a unique identifier consisting of a string of letters and numbers called acall sign, which must be used in all transmissions.[47]In order to adjust, maintain, or internally repair radiotelephone transmitters, individuals must hold a government license, such as thegeneral radiotelephone operator licensein the US, obtained by taking a test demonstrating adequate technical and legal knowledge of safe radio operation.[48]
Exceptions to the above rules allow the unlicensed operation by the public of low power short-range transmitters in consumer products such as cell phones,cordless phones,wireless devices,walkie-talkies,citizens band radios,wireless microphones,garage door openers, andbaby monitors. In the US, these fall underPart 15of theFederal Communications Commission(FCC) regulations. Many of these devices use theISM bands, a series of frequency bands throughout the radio spectrum reserved for unlicensed use. Although they can be operated without a license, like all radio equipment these devices generally must betype-approvedbefore the sale.[49]
Below are some of the most important uses of radio, organized by function.
Broadcasting is the one-way transmission of information from a transmitter to receivers belonging to a public audience.[50]Since the radio waves become weaker with distance, abroadcasting stationcan only be received within a limited distance of its transmitter.[51]Systems that broadcast fromsatellitescan generally be received over an entire country or continent. Older terrestrial radio and television are paid for bycommercial advertisingor governments. In subscription systems like satellite television andsatellite radiothe customer pays a monthly fee. In these systems, the radio signal isencryptedand can only be decrypted by the receiver, which is controlled by the company and can be deactivated if the customer does not pay.[52]
Broadcasting uses several parts of the radio spectrum, depending on the type of signals transmitted and the desired target audience.Longwaveandmedium wavesignals can give reliable coverage of areas several hundred kilometers across, but have a more limited information-carrying capacity and so work best with audio signals (speech and music), and the sound quality can be degraded byradio noisefrom natural and artificial sources. Theshortwavebands have a greater potential range but are more subject to interference by distant stations and varying atmospheric conditions that affect reception.[53][54]
In thevery high frequencyband, greater than 30 megahertz, the Earth's atmosphere has less of an effect on the range of signals, andline-of-sight propagationbecomes the principal mode. These higher frequencies permit the great bandwidth required for television broadcasting. Since natural and artificial noise sources are less present at these frequencies, high-quality audio transmission is possible, usingfrequency modulation.[55][56]
Radio broadcastingmeans transmission ofaudio(sound) toradio receiversbelonging to a public audience. Analog audio is the earliest form of radio broadcast.AM broadcastingbegan around 1920.FM broadcastingwas introduced in the late 1930s with improvedfidelity. A broadcast radio receiver is called aradio. Most radios can receive both AM and FM.[57]
Television broadcastingis the transmission of moving images along with a synchronized audio (sound) channel by radio. The sequence of still images is displayed on a screen on atelevision receiver(a "television" or TV), which includes aloudspeaker. Television (video) signals occupy a widerbandwidththan broadcast radio (audio) signals.Analog television, the original television technology, required 6 MHz, so the television frequency bands are divided into 6 MHz channels, now called "RF channels".[75]
The current television standard, introduced beginning in 2006, is a digital format calledhigh-definition television(HDTV), which transmits pictures at higher resolution, typically 1080pixelshigh by 1920 pixels wide, at a rate of 25 or 30 frames per second.Digital television(DTV) transmission systems, which replaced older analog television in atransitionbeginning in 2006, useimage compressionand high-efficiency digital modulation such asOFDMand8VSBto transmit HDTV video within a smaller bandwidth than the old analog channels, saving scarceradio spectrumspace. Therefore, each of the 6 MHz analog RF channels now carries up to 7 DTV channels – these are called "virtual channels". Digital television receivers have different behavior in the presence of poor reception or noise than analog television, called the "digital cliff" effect. Unlike analog television, in which increasingly poor reception causes the picture quality to gradually degrade, in digital television picture quality is not affected by poor reception until, at a certain point, the receiver stops working and the screen goes black.[76][77]
Governmentstandard frequency and time signal servicesoperate time radio stations which continuously broadcast extremely accurate time signals produced byatomic clocks, as a reference to synchronize other clocks.[84]Examples areBPC,DCF77,JJY,MSF,RTZ,TDF,WWV, andYVTO.[85]One use is inradio clocksand watches, which include an automated receiver that periodically (usually weekly) receives and decodes the time signal and resets the watch's internalquartz clockto the correct time, thus allowing a small watch or desk clock to have the same accuracy as an atomic clock. Government time stations are declining in number becauseGPSsatellites and the InternetNetwork Time Protocol(NTP) provide equally accurate time standards.[86]
Atwo-way radiois anaudiotransceiver, areceiverandtransmitterin the same device, used for bidirectional person-to-person voice communication with other users with similar radios. An older term for this mode of communication isradiotelephony. The radio link may behalf-duplex, as in awalkie-talkie, using a single radio channel in which only one radio can transmit at a time, so different users take turns talking, pressing a "push to talk" button on their radio which switches off the receiver and switches on the transmitter. Or the radio link may befull duplex, a bidirectional link using two radio channels so both people can talk at the same time, as in a cell phone.[87]
One way, unidirectional radio transmission is calledsimplex.
This is radio communication between aspacecraftand an Earth-based ground station, or another spacecraft. Communication with spacecraft involves the longest transmission distances of any radio links, up to billions of kilometers forinterplanetary spacecraft. In order to receive the weak signals from distant spacecraft,satellite ground stationsuse largeparabolic "dish" antennasup to 25 metres (82 ft) in diameter and extremely sensitive receivers. High frequencies in themicrowaveband are used, since microwaves pass through theionospherewithoutrefraction, and at microwave frequencies thehigh-gain antennasneeded to focus the radio energy into a narrow beam pointed at the receiver are small and take up a minimum of space in a satellite. Portions of theUHF,L,C,S,kuandkabandare allocated for space communication. A radio link that transmits data from the Earth's surface to a spacecraft is called anuplink, while a link that transmits data from the spacecraft to the ground is called a downlink.[126]
Radaris aradiolocationmethod used to locate and track aircraft, spacecraft, missiles, ships, vehicles, and also to map weather patterns and terrain. A radar set consists of a transmitter and receiver.[130][131]The transmitter emits a narrow beam of radio waves which is swept around the surrounding space. When the beam strikes a target object, radio waves are reflected back to the receiver. The direction of the beam reveals the object's location. Since radio waves travel at a constant speed close to thespeed of light, by measuring the brief time delay between the outgoing pulse and the received "echo", the range to the target can be calculated. The targets are often displayed graphically on a map display called aradar screen.Doppler radarcan measure a moving object's velocity, by measuring the change in frequency of the return radio waves due to theDoppler effect.[132]
Radar sets mainly use high frequencies in themicrowavebands, because these frequencies create strong reflections from objects the size of vehicles and can be focused into narrow beams with compact antennas.[131]Parabolic (dish) antennasare widely used. In most radars the transmitting antenna also serves as the receiving antenna; this is called amonostatic radar. A radar which uses separate transmitting and receiving antennas is called abistatic radar.[133]
Radiolocation is a generic term covering a variety of techniques that use radio waves to find the location of objects, or for navigation.[144]
Radio remote control is the use of electroniccontrol signalssent by radio waves from a transmitter to control the actions of a device at a remote location. Remote control systems may also include telemetry channels in the other direction, used to transmit real-time information on the state of the device back to the control station. Uncrewedspacecraftare an example of remote-controlled machines, controlled by commands transmitted bysatellite ground stations. Most handheldremote controlsused to controlconsumer electronicsproducts like televisions or DVD players actually operate byinfrared lightrather than radio waves, so are not examples of radio remote control. A security concern with remote control systems isspoofing, in which an unauthorized person transmits an imitation of the control signal to take control of the device.[162]Examples of radio remote control:
Radio jammingis the deliberate radiation of radio signals designed to interfere with the reception of other radio signals. Jamming devices are called "signal suppressors" or "interference generators" or just jammers.[174]
During wartime, militaries use jamming to interfere with enemies' tactical radio communication. Since radio waves can pass beyond national borders, sometotalitariancountries which practice censorship use jamming to prevent their citizens from listening to broadcasts from radio stations in other countries. Jamming is usually accomplished by a powerful transmitter which generates noise on the same frequency as the target transmitter.[175][176]
US Federal law prohibits the nonmilitary operation or sale of any type of jamming devices, including ones that interfere with GPS, cellular, Wi-Fi and police radars.[177]
ELF3 Hz/100 Mm30 Hz/10 Mm
SLF30 Hz/10 Mm300 Hz/1 Mm
ULF300 Hz/1 Mm3 kHz/100 km
VLF3 kHz/100 km30 kHz/10 km
LF30 kHz/10 km300 kHz/1 km
MF300 kHz/1 km3 MHz/100 m
HF3 MHz/100 m30 MHz/10 m
VHF30 MHz/10 m300 MHz/1 m
UHF300 MHz/1 m3 GHz/100 mm
SHF3 GHz/100 mm30 GHz/10 mm
EHF30 GHz/10 mm300 GHz/1 mm
THF300 GHz/1 mm3 THz/0.1 mm
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From early in the 20th century, the radio frequency of500 kilohertz(500 kHz) was aninternational calling and distress frequencyforMorse codemaritime communication. For much of its early history, this frequency was referred to by its equivalentwavelength,600 meters, or, using the earlier frequency unit name, 500kilocycles(per second) or 500 kc.
Maritime authorities of many nations, including theMaritime and Coastguard Agencyand theUnited States Coast Guard, once maintained 24 hour watches on this frequency, staffed by skilled radio operators. ManySOScalls and medical emergencies at sea were handled via this frequency. However, as the use of Morse code over radio is now obsolete in commercial shipping, 500 kHz is obsolete as a Morse distress frequency. Beginning in the late 1990s, most nations ended monitoring of transmissions on 500 kHz and emergency traffic on 500 kHz has been replaced by theGlobal Maritime Distress Safety System(GMDSS).
The 500 kHz frequency has now been allocated to the maritime Navigational Data orNAVDATbroadcast system.
The nearby frequencies of 518 kHz and 490 kHz are used for theNAVTEXcomponent ofGMDSS. Proposals to allocate frequencies at or near 500 kHz toamateur radiouse resulted in the international allocation of 472–479 kHz to the630-meter amateur radio band, now implemented in many countries.
International standards for the use of 500 kHz first appeared in the firstInternational Radiotelegraph ConventioninBerlin, which was signed 3 November 1906, and became effective 1 July 1908.
The second service regulation affixed to this Convention designated 500 kHz as one of the standard frequencies to be employed by shore stations, specifying that
These regulations also specified that ship stations normally used 1 MHz.[1][2]
International standards for the use of 500 kHz were expanded by the secondInternational Radiotelegraph Convention, which was held in London after thesinking of the RMSTitanic. This meeting produced an agreement which was signed on 5 July 1912, and became effective 1 July 1913.
The Service Regulations, affixed to the 1912 convention, established 500 kHz as the primary frequency for seagoing communication, and the standard ship frequency was changed from 1,000 kHz to 500 kHz, to match the coastal station standard. Communication was generally conducted inMorse code, initially usingspark-gap transmitters. Most two-way radio contacts were to be initiated on this frequency, although once established, the participating stations could shift to another frequency to avoid the congestion on 500 kHz. To facilitate communication between operators speaking different languages, standardized abbreviations were used, including a set of "Q codes" specified by the 1912 Service Regulations.
Article XXI of the Service Regulations required that whenever anSOSdistress call was heard, all transmissions unrelated to the emergency had to immediately cease until the emergency was declared over.
There was a potential problem if a ship transmitted a distress call: The use of 500 kHz as a common frequency often led to heavy congestion, especially around major ports and shipping lanes, and it was possible the distress message would be drowned out by the bedlam of ongoing commercial traffic. To help address this problem, the Service Regulation's Article XXXII specified that
During distress working all non-distress traffic was banned from 500 kHz and adjacent coast stations then monitored 512 kHz as an additional calling frequency for ordinary traffic.
The silent and monitoring periods were soon expanded and standardized. For example, Regulation 44, from the 27 July 1914, edition ofRadio Communication Laws of the United States, stated: "The international standard wave length is600meters, and the operators of allcoast stationsare required, during the hours the station is in operation, to 'listen in' at intervals of not more than 15 minutes and for a period not less than 2 minutes, with the receiving apparatus tuned to receive this wave length, for the purpose of determining if any distress signals or messages are being sent and to determine if the transmitting operations of the 'listening station' are causing interference with other radio communication."
International refinements for the use of 500 kHz were specified in later agreements, including the 1932 Madrid Radio Conference. In later years, except for distress traffic, stations shifted to nearby "working frequencies" to exchange messages once contact was established: 425, 454, 468, 480, and 512 kHz were used by ships while the coast stations had their own individual working frequencies.
Twice each hour, all stations operating on 500 kHz were required to maintain a strictly enforced three-minute silent period, starting at 15 and 45 minutes past the hour.
As a visual memory aid, a typical clock in a ship's radio room would have the silence periods marked by shading thesectorsbetween h+15ᵐ to h+18ᵐ and h+45ᵐ to h+48ᵐ in RED. Similar sectors between h+00ᵐ to h+03ᵐ and h+30ᵐ to h+33ᵐ are marked in GREEN which is the corresponding silence period for the2182 kHzvoice communications distress signals.
In addition, during this silent period all coastal and ship stations were required to monitor the frequency, listening for any distress signals.[3][full citation needed]All large ships at sea had to monitor 500 kHz at all times, either with a licensed radio operator or with equipment (called an auto-alarm) that detected an automatically sent distress signal consisting of long dashes.
Shore stations throughout the world operated on this frequency to exchange messages with ships and to issue warning about weather and other navigational warnings.
At night, transmission ranges of 3,000–4,000 miles (4,500–6,500 kilometers) were typical. Daytime ranges were much shorter, on the order of 300–1,500 miles (500–2,500 kilometers).Terman's Radio Engineering Handbook(1948) shows the maximum distance for 1 kW over salt water to be 1,500 miles, and this distance was routinely covered by ships at sea, where signals from ships and nearby coastal stations would cause congestion, covering up distant and weaker signals. During the silence, a distress signal could more easily be heard at great distances.
Following the adoption ofGMDSSin 1999 and the subsequent obsolescence of 500 kHz as a Morse distress frequency, the 2019World Radiocommunication Conference(WRC-19) allocated500 ± 5 kHzto the maritime NAVDAT service.[4]
NAVDAT is intended for the broadcast of data from shore-to-ship and may thus be compared toNAVTEX. However, NAVDAT usesQAM modulation(in comparison to theSITORused by NAVTEX) and is therefore capable of much higher data throughput. This allows NAVDAT broadcasts to carry images and other data as well as plain text, further allowing this data to be presented directly on anElectronic Chart Display and Information System (ECDIS).[5]This presents a significant improvement over the text-only NAVTEX system.
As of February 2023, no maritime authorities have begunNAVDATbroadcasts.
Maritime traffic currently displaced from the 500 kHz band in most countries, and with theITU472–479 kHz amateur allocation, most countries no longer using it have allocated frequencies near 500 kHz toamateur radiouse on a secondary basis, although the primary allocation of the band remains with the maritime mobile service.
Full details of these allocations can be found under the article on the630 metreamateur radio band.
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The earlyhistory of radiois thehistory of technologythat produces and usesradio instrumentsthat useradio waves. Within thetimeline of radio, many people contributed theory and inventions in what becameradio. Radio development began as "wireless telegraphy". Later radio history increasingly involves matters ofbroadcasting.
In an 1864 presentation, published in 1865,James Clerk Maxwellproposed theories ofelectromagnetismand mathematical proofs demonstrating that light, radio and x-rays were all types of electromagnetic waves propagating throughfree space.[1][2][3][4][5]
Between 1886 and 1888Heinrich Rudolf Hertzpublished the results of experiments wherein he was able to transmit electromagnetic waves (radio waves) through the air, proving Maxwell's electromagnetic theory.[6][7]
After their discovery many scientists and inventors experimented with transmitting and detecting "Hertzian waves" (it would take almost 20 years for the term "radio" to be universally adopted for this type of electromagnetic radiation).[8]Maxwell's theory showing that light and Hertzian electromagnetic waves were the same phenomenon at different wavelengths led "Maxwellian" scientists such as John Perry,Frederick Thomas Troutonand Alexander Trotter to assume they would be analogous to optical light.[9][10]
Following Hertz' untimely death in 1894, British physicist and writerOliver Lodgepresented a widely covered lecture on Hertzian waves at theRoyal Institutionon June 1 of the same year.[11]Lodge focused on the optical qualities of the waves and demonstrated how to transmit and detect them (using an improved variation of French physicistÉdouard Branly's detector Lodge named the "coherer").[12]Lodge further expanded on Hertz' experiments showing how these new waves exhibited like lightrefraction,diffraction,polarization,interferenceandstanding waves,[13]confirming that Hertz' waves and light waves were both forms of Maxwell'selectromagnetic waves. During part of the demonstration the waves were sent from the neighboringClarendon Laboratorybuilding, and received by apparatus in the lecture theater.[14]
After Lodge's demonstrations researchers pushed their experiments further down the electromagnetic spectrum towards visible light to further explore thequasiopticalnature at these wavelengths.[15]Oliver LodgeandAugusto Righiexperimented with 1.5 and 12 GHz microwaves respectively, generated by small metal ball spark resonators.[13]Russian physicistPyotr Lebedevin 1895 conducted experiments in the 50 GHz (6 millimeter) range.[13]Bengali Indian physicistJagadish Chandra Boseconducted experiments at wavelengths of 60 GHz (5 millimeter) and inventedwaveguides,horn antennas, andsemiconductorcrystal detectorsfor use in his experiments.[16]He would later write an essay, "Adrisya Alok" ("Invisible Light") on how in November 1895 he conducted a public demonstration at the Town Hall ofKolkata,Indiausing millimeter-range-wavelength microwaves to trigger detectors that ignited gunpowder and rang a bell at a distance.[17]
Between 1890 and 1892 physicists such as John Perry,Frederick Thomas TroutonandWilliam Crookesproposed electromagnetic or Hertzian waves as a navigation aid or means of communication, with Crookes writing on the possibilities of wirelesstelegraphybased on Hertzian waves in 1892.[18]Among physicists, what were perceived as technical limitations to using these new waves, such as delicate equipment, the need for large amounts of power to transmit over limited ranges, and its similarity to already existent optical light transmitting devices, lead them to a belief that applications were very limited. The Serbian American engineerNikola Teslaconsidered Hertzian waves relatively useless for long range transmission since "light" could not transmit further thanline of sight.[19]There was speculation that this fog and stormy weather penetrating "invisible light" could be used in maritime applications such as lighthouses.[18]The London journalThe Electrician(December 1895) commented on Bose's achievements, saying "we may in time see the whole system of coast lighting throughout the navigable world revolutionized by an Indian Bengali scientist working single handed[ly] in our Presidency College Laboratory."[20]
In 1895, adapting the techniques presented in Lodge's published lectures, Russian physicistAlexander Stepanovich Popovbuilt alightning detectorthat used a coherer based radio receiver.[21]He presented it to the Russian Physical and Chemical Society on May 7, 1895.
In 1894, the young Italian inventorGuglielmo Marconibegan working on the idea of building long-distance wireless transmission systems based on the use of Hertzian waves (radio waves), a line of inquiry that he noted other inventors did not seem to be pursuing.[22]Marconi read through the literature and used the ideas of others who were experimenting with radio waves but did a great deal to develop devices such as portable transmitters and receiver systems that could work over long distances,[22]turning what was essentially a laboratory experiment into a useful communication system.[23]By August 1895, Marconi was field testing his system but even with improvements he was only able to transmit signals up to one-half mile, a distance Oliver Lodge had predicted in 1894 as the maximum transmission distance for radio waves. Marconi raised the height of his antenna and hit upon the idea of grounding his transmitter and receiver. With these improvements the system was capable of transmitting signals up to 2 miles (3.2 km) and over hills.[24]This apparatus proved to be the first engineering-complete, commercially successfulradio transmissionsystem[25][26][27]and Marconi went on to file British patent GB189612039A,Improvements in transmitting electrical impulses and signals and in apparatus there-for, in 1896. This patent was granted in the UK on 2 July 1897.[28]
In 1897, Marconi established a radio station on theIsle of Wight, England and opened his "wireless" factory in the formersilk-works at Hall Street,Chelmsford, England, in 1898, employing around 60 people.
On 12 December 1901, using a 500-foot (150 m) kite-supported antenna for reception—signals transmitted by the company's new high-power station atPoldhu, Cornwall, Marconi transmitted a message across the Atlantic Ocean toSignal HillinSt. John's,Newfoundland.[29][30][31][32]
Marconi began to build high-powered stations on both sides of the Atlantic to communicate with ships at sea. In 1904, he established a commercial service to transmit nightly news summaries to subscribing ships, which could incorporate them into their on-board newspapers. A regular transatlantic radio-telegraph service was finally begun on 17 October 1907[33][34]betweenClifden, Ireland, andGlace Bay, but even after this the company struggled for many years to provide reliable communication to others.
Marconi's apparatus is also credited with saving the 700 people who survived the tragicTitanicdisaster.[35]
In the late 1890s, Canadian-American inventorReginald Fessendencame to the conclusion that he could develop a far more efficient system than the spark-gap transmitter and coherer receiver combination.[36][37]To this end he worked on developing a high-speed alternator (referred to as "an alternating-current dynamo") that generated "pure sine waves" and produced "a continuous train of radiant waves of substantially uniform strength", or, in modern terminology, acontinuous-wave(CW) transmitter.[38]While working for theUnited States Weather BureauonCobb Island, Maryland, Fessenden researched using this setup for audio transmissions via radio. By fall of 1900, he successfully transmitted speech over a distance of about 1.6 kilometers (one mile),[39]which appears to have been the first successful audio transmission using radio signals.[40][41]Although successful, the sound transmitted was far too distorted to be commercially practical.[42]According to some sources, notably Fessenden's wife Helen's biography, onChristmas Eve1906,Reginald Fessendenused anAlexanderson alternatorand rotaryspark-gap transmitterto make the first radio audio broadcast, fromBrant Rock, Massachusetts. Ships at sea heard a broadcast that included Fessenden playingO Holy Nighton theviolinand reading a passage from theBible.[43][44]
Around the same time American inventorLee de Forestexperimented with anarc transmitter, which unlike the discontinuous pulses produced by spark transmitters, created steady "continuous wave" signal that could be used foramplitude modulated(AM) audio transmissions. In February 1907 he transmitted electronictelharmoniummusic from his laboratory station in New York City.[45]This was followed by tests that included, in the fall,Eugenia Farrarsinging "I Love You Truly".[46]In July 1907 he made ship-to-shore transmissions by radiotelephone—race reports for the Annual Inter-Lakes Yachting Association (I-LYA) Regatta held onLake Erie—which were sent from the steam yachtThelmato his assistant, Frank E. Butler, located in the Fox's Dock Pavilion onSouth Bass Island.[47]
The Dutch companyNederlandsche Radio-Industrieand its owner-engineer,Hanso Idzerda, made its first regular entertainment radio broadcast over stationPCGGfrom its workshop inThe Hagueon 6 November 1919. The company manufactured both transmitters and receivers. Its popular program was broadcast four nights per week using narrow-band FM transmissions on 670 metres (448 kHz),[48]until 1924 when the company ran into financial trouble.
Regular entertainment broadcasts began inArgentina, pioneered byEnrique Telémaco Susiniand his associates. At 9 pm on August 27, 1920, Sociedad Radio Argentina aired a live performance of Richard Wagner's operaParsifalfrom the Coliseo Theater in downtownBuenos Aires. Only about twenty homes in the city had receivers to tune in this program.
On 31 August 1920 theDetroit Newsbegan publicized daily news and entertainment "Detroit News Radiophone" broadcasts, originally as licensed amateur station 8MK, then later as WBL andWWJinDetroit, Michigan.
Union College in Schenectady,New Yorkbegan broadcasting on October 14, 1920, over2ADD, an amateur station licensed to Wendell King, anAfrican-Americanstudent at the school.[49]Broadcasts included a series of Thursday night concerts initially heard within a 100-mile (160 km) radius and later for a 1,000-mile (1,600 km) radius.[49][50]
In 1922 regular audio broadcasts for entertainment began in the UK from theMarconiResearch Centre2MTatWrittlenearChelmsford, England.
In early radio, and to a limited extent much later, the transmission signal of the radio station was specified in meters, referring to thewavelength, the length of the radio wave. This is the origin of the termslong wave,medium wave, andshort waveradio.[51]Portions of the radio spectrum reserved for specific purposes were often referred to by wavelength: the40-meter band, used foramateur radio, for example. The relation between wavelength and frequency is reciprocal: the higher the frequency, the shorter the wave, and vice versa.
As equipment progressed, precise frequency control became possible; early stations often did not have a precise frequency, as it was affected by the temperature of the equipment, among other factors. Identifying a radio signal by its frequency rather than its length proved much more practical and useful, and starting in the 1920s this became the usual method of identifying a signal, especially in the United States. Frequencies specified in number of cycles per second (kilocycles, megacycles) were replaced by the more specific designation ofhertz(cycles per second) about 1965.
Using variouspatents, theBritish Marconicompany was established in 1897 by Guglielmo Marconi and began communication betweencoast radio stationsand ships at sea.[52]A year after, in 1898, they successfully introduced their first radio station in Chelmsford. This company, along with its subsidiariesCanadian MarconiandAmerican Marconi, had a stranglehold on ship-to-shore communication. It operated much the wayAmerican Telephone and Telegraphoperated until 1983, owning all of its equipment and refusing to communicate with non-Marconi equipped ships. Many inventions improved the quality of radio, and amateurs experimented with uses of radio, thus planting the first seeds of broadcasting.
The companyTelefunkenwas founded on May 27, 1903, as "Telefunken society for wireless telefon" ofSiemens & Halske(S & H) and theAllgemeine Elektrizitäts-Gesellschaft (General Electricity Company)as joint undertakings for radio engineering in Berlin.[53]It continued as a joint venture ofAEGandSiemens AG, until Siemens left in 1941. In 1911,Kaiser Wilhelm IIsent Telefunken engineers toWest Sayville,New Yorkto erect three 600-foot (180-m) radio towers there. Nikola Tesla assisted in the construction. A similar station was erected inNauen, creating the only wireless communication between North America and Europe.
The invention of amplitude-modulated (AM) radio, which allows more closely spaced stations to simultaneously send signals (as opposed to spark-gap radio, where each transmission occupies a wide bandwidth) is attributed toReginald Fessenden,Valdemar PoulsenandLee de Forest.
The most common type of receiver before vacuum tubes was thecrystal set, although some early radios used some type of amplification through electric current or battery. Inventions of thetriode amplifier,motor-generator, anddetectorenabled audio radio. The use ofamplitude modulation(AM), by which soundwaves can be transmitted over a continuous-wave radio signal of narrow bandwidth (as opposed to spark-gap radio, which sent rapid strings of damped-wave pulses that consumed much bandwidth and were only suitable for Morse-code telegraphy) was pioneered by Fessenden, Poulsen and Lee de Forest.[54]
The art and science of crystal sets is still pursued as a hobby in the form of simple un-amplified radios that 'runs on nothing, forever'. They are used as a teaching tool by groups such as theBoy Scouts of Americato introduce youngsters to electronics and radio. As the only energy available is that gathered by the antenna system, loudness is necessarily limited.
During the mid-1920s, amplifyingvacuum tubesrevolutionizedradio receiversandtransmitters.John Ambrose Flemingdeveloped a vacuum tubediode.Lee de Forestplaced a screen, added a"grid" electrode, creating thetriode.[55]
Early radios ran the entire power of the transmitter through acarbon microphone. In the 1920s, theWestinghouse companybought Lee de Forest's andEdwin Armstrong's patent. During the mid-1920s, Amplifyingvacuum tubesrevolutionizedradio receiversand transmitters. Westinghouse engineers developed a more modern vacuum tube.
The first radios still required batteries, but in 1926 the "battery eliminator" was introduced to the market. This tube technology allowed radios to be powered through the grid instead. They still required batteries to heat up the vacuum-tube filaments, but after the invention ofindirectly heated vacuum tubes, the first completely battery free radios became available in 1927.[56]
In 1929 a new screen grid tube called UY-224 was introduced, an amplifier designed to operate directly on alternating current.[57]
A problem with the early radios was fading stations and fluctuating volume. The invention of thesuperheterodyne receiversolved this problem, and the first radios with a heterodyne radio receiver went for sale in 1924. But it was costly, and the technology was shelved while waiting for the technology to mature, and in 1929 the Radiola 66 and Radiola 67 went for sale.[58][59][60]
In the early days one had to use headphones to listen to radio. Later loudspeakers in the form of a horn of the type used by phonographs, equipped with a telephone receiver, became available. But the sound quality was poor. In 1926 the first radios with electrodynamic loudspeakers went for sale, which improved the quality significantly. At first the loudspeakers were separated from the radio, but soon radios would come with a built-in loudspeaker.[61]
Other inventions related to sound included the automatic volume control (AVC), first commercially available in 1928.[62]In 1930 a tone control knob was added to the radios. This allowed listeners to improve imperfect broadcasting.[63]
Themagnetic cartridge, which was introduced in the mid 20's, greatly improved the broadcasting of music. When playing music from a phonograph before the magnetic cartridge, a microphone had to be placed close to a horn loudspeaker. The invention allowed the electric signals to be amplified and then fed directly to thebroadcast transmitter.[64]
Following development oftransistortechnology,bipolar junction transistorsled to the development of thetransistor radio. In 1954, the Regency company introduced a pocket transistor radio, theTR-1, powered by a "standard 22.5 V Battery." In 1955, the newly formedSonycompany introduced its first transistorized radio, theTR-55.[65]It was small enough to fit in avestpocket, powered by a small battery. It was durable, because it had no vacuum tubes to burn out. In 1957, Sony introduced the TR-63, the first mass-produced transistor radio, leading to the mass-market penetration of transistor radios.[66]Over the next 20 years, transistors replaced tubes almost completely except for high-powertransmitters.
By the mid-1960s, theRadio Corporation of America(RCA) were usingmetal–oxide–semiconductor field-effect transistors(MOSFETs) in their consumer products, includingFM radio, television andamplifiers.[67]Metal–oxide–semiconductor(MOS)large-scale integration(LSI) provided a practical and economic solution for radio technology, and was used inmobile radiosystems by the early 1970s.[68]
The first integrated circuit (IC) radio, P1740 byGeneral Electric, became available in 1966.[69]
The first car radio was introduced in 1922, but it was so large that it took up too much space in the car.[70]The first commercial car radio that could easily be installed in most cars went for sale in 1930.[71][72]
Telegraphydid not go away on radio. Instead, the degree of automation increased. On land-lines in the 1930s,teletypewritersautomated encoding, and were adapted to pulse-code dialing to automate routing, a service calledtelex. For thirty years, telex was the cheapest form of long-distance communication, because up to 25 telex channels could occupy the same bandwidth as one voice channel. For business and government, it was an advantage that telex directly produced written documents.
Telex systems were adapted to short-wave radio by sending tones oversingle sideband.CCITTR.44 (the most advanced pure-telex standard) incorporated character-level error detection and retransmission as well as automated encoding and routing. For many years, telex-on-radio (TOR) was the only reliable way to reach some third-world countries. TOR remains reliable, though less-expensive forms of e-mail are displacing it. Many national telecom companies historically ran nearly pure telex networks for their governments, and they ran many of these links over short wave radio.
Documents including maps and photographs went byradiofax, or wireless photoradiogram, invented in 1924 byRichard H. RangerofRadio Corporation of America(RCA). This method prospered in the mid-20th century and faded late in the century.
One of the first developments in the early 20th century was that aircraft used commercial AM radio stations for navigation, AM stations are still marked on U.S. aviation charts.Radio navigationplayed an important role during war time, especially in World War II. Before the discovery of the crystal oscillator, radio navigation had many limits.[73]However, as radio technology expanding, navigation is easier to use, and it provides a better position. Although there are many advantages, the radio navigation systems often comes with complex equipment such as the radio compass receiver, compass indicator, or the radar plan position indicator. All of these require users to obtain certain knowledge.
In the 1960sVORsystems became widespread. In the 1970s,LORANbecame the premier radio navigation system. Soon, the US Navy experimented withsatellite navigation. In 1987, theGlobal Positioning System(GPS) constellation ofsatelliteswas launched; it was followed by otherGNSSsystems likeGlonass,BeiDouandGalileo.
In 1933,FM radiowas patented by inventorEdwin H. Armstrong.[74]FM usesfrequency modulationof the radio wave to reducestaticandinterferencefrom electrical equipment and the atmosphere. In 1937,W1XOJ, the first experimental FM radio station after Armstrong'sW2XMNin Alpine, New Jersey, was granted a construction permit by the USFederal Communications Commission(FCC).
After World War II,FM radiobroadcasting was introduced in Germany. At a meeting inCopenhagenin 1948, a newwavelength planwas set up for Europe. Because of the recent war, Germany (which did not exist as a state and so was not invited) was only given a small number ofmedium-wavefrequencies, which were not very good for broadcasting. For this reason Germany began broadcasting on UKW ("Ultrakurzwelle", i.e. ultra short wave, nowadays calledVHF) which was not covered by the Copenhagen plan. After someamplitude modulationexperience with VHF, it was realized that FM radio was a much better alternative for VHF radio than AM. Because of this history, FM radio is still referred to as "UKW Radio" in Germany. Other European nations followed a bit later, when the superior sound quality of FM and the ability to run many more local stations because of the more limited range of VHF broadcasts were realized.
In the 1930s, regularanalog televisionbroadcasting began in some parts of Europe and North America. By the end of the decade there were roughly 25,000 all-electronic television receivers in existence worldwide, the majority of them in the UK. In the US, Armstrong's FM system was designated by the FCC to transmit and receive television sound.
By 1963,color televisionwas being broadcast commercially (though not all broadcasts or programs were in color), and the first (radio)communication satellite,Telstar, was launched. In the 1970s,
In 1947 AT&T commercialized theMobile Telephone Service. From its start in St. Louis in 1946, AT&T then introduced Mobile Telephone Service to one hundred towns and highway corridors by 1948. Mobile Telephone Service was a rarity with only 5,000 customers placing about 30,000 calls each week. Because only three radio channels were available, only three customers in any given city could make mobile telephone calls at one time.[76]Mobile Telephone Service was expensive, costing US$15 per month, plus $0.30–0.40 per local call, equivalent to (in 2012 US dollars) about $176 per month and $3.50–4.75 per call.[77]TheAdvanced Mobile Phone Systemanalog mobile phone system, developed byBell Labs, was introduced in the Americas in 1978,[78][79][80]gave much more capacity. It was the primary analog mobile phone system in North America (and other locales) through the 1980s and into the 2000s.
The development ofmetal–oxide–semiconductor(MOS)large-scale integration(LSI) technology,information theoryandcellular networkingled to the development of affordablemobile communications.[81]TheAdvanced Mobile Phone Systemanalog mobile phone system, developed byBell Labsand introduced in theAmericasin 1978,[78][79][80]gave much more capacity. It was the primary analog mobile phone system inNorth America(and other locales) through the 1980s and into the 2000s.
The British government and the state-owned postal services found themselves under massive pressure from the wireless industry (including telegraphy) and early radio adopters to open up to the new medium. In an internal confidential report from February 25, 1924, theImperial Wireless Telegraphy Committeestated:
When radio was introduced in the early 1920s, many predicted it would kill thephonograph recordindustry. Radio was a free medium for the public to hear music for which they would normally pay. While some companies saw radio as a new avenue for promotion, others feared it would cut into profits from record sales and live performances. Many record companies would not license their records to be played over the radio, and had their major stars sign agreements that they would not perform on radio broadcasts.[83][84]
Indeed, the music recording industry had a severe drop in profits after the introduction of the radio. For a while, it appeared as though radio was a definite threat to the record industry. Radio ownership grew from two out of five homes in 1931 to four out of five homes in 1938. Meanwhile, record sales fell from $75 million in 1929 to $26 million in 1938 (with a low point of $5 million in 1933), though the economics of the situation were also affected by theGreat Depression.[85]
The copyright owners were concerned that they would see no gain from the popularity of radio and the 'free' music it provided. What they needed to make this new medium work for them already existed in previous copyright law. The copyright holder for a song had control over all public performances 'for profit.' The problem now was proving that the radio industry, which was just figuring out for itself how to make money from advertising and currently offered free music to anyone with a receiver, was making a profit from the songs.
Thetest casewas againstBamberger'sDepartment Store inNewark, New Jerseyin 1922. The store was broadcasting music from its store on the radio station WOR. No advertisements were heard, except at the beginning of the broadcast which announced "L. Bamberger and Co., One of America's Great Stores, Newark, New Jersey." It was determined through this and previous cases (such as the lawsuit against Shanley's Restaurant) that Bamberger was using the songs for commercial gain, thus making it a public performance for profit, which meant the copyright owners were due payment.
With this ruling theAmerican Society of Composers, Authors and Publishers(ASCAP) began collecting licensing fees from radio stations in 1923. The beginning sum was $250 for all music protected under ASCAP, but for larger stations the price soon ballooned to $5,000. Edward Samuels reports in his bookThe Illustrated Story of Copyrightthat "radio and TV licensing represents the single greatest source of revenue for ASCAP and its composers […] and [a]n average member of ASCAP gets about $150–$200 per work per year, or about $5,000-$6,000 for all of a member's compositions." Not long after the Bamberger ruling, ASCAP had to once again defend their right to charge fees, in 1924. The Dill Radio Bill would have allowed radio stations to play music without paying and licensing fees to ASCAP or any other music-licensing corporations. The bill did not pass.[86]
Radio technology was first used for ships to communicate at sea. To ensure safety, theWireless Ship Act of 1910marks the first time the U.S. government implies regulations on radio systems on ships.[87]This act requires ships to have a radio system with a professional operator if they want to travel more than 200 miles offshore or have more than 50 people on board. However, this act had many flaws including the competition ofradio operatorsincluding the two majors company (British and American Marconi). They tended to delay communication for ships that used their competitor's system. This contributed to the tragic incident of the sinking of theTitanicin 1912.
In 1912, distress calls to aid the sinkingTitanicwere met with a large amount of interfering radio traffic, severely hampering the rescue effort. Subsequently, the US government passed theRadio Act of 1912to help mitigate the repeat of such a tragedy. The act helps distinguish between normal radio traffic and (primarily maritime) emergency communication, and specifies the role of government during such an emergency.[88]
TheRadio Act of 1927gave theFederal Radio Commissionthe power to grant and deny licenses, and to assign frequencies and power levels for each licensee. In 1928 it began requiring licenses of existing stations and setting controls on who could broadcast from where on what frequency and at what power. Some stations could not obtain a license and ceased operations. In section 29, the Radio Act of 1927 mentioned that the content of the broadcast should be freely present, and the government cannot interfere with this.[89]
The introduction of theCommunications Act of 1934led to the establishment of the Federal Communications Commissions (FCC). The FCC's responsibility is to control the industry including "telephone, telegraph, and radio communications."[90]Under this Act, all carriers have to keep records of authorized interference and unauthorized interference. This Act also supports the President in time of war. If the government needs to use the communication facilities in time of war, they are allowed to.
TheTelecommunications Act of 1996was the first significant overhaul in over 60 years amending the work of the Communications Act of 1934. Coming only two dozen years after the breakup of AT&T, the act sets out to move telecommunications into a state of competition with their markets and the networks they are a part of.[91]Up to this point the effects of the Telecommunications Act of 1996 have been seen, but some of the changes the Act set out to fix are still ongoing problems, such as being unable to create an open competitive market.
The question of the 'first' publicly targeted licensed radio station in the U.S. has more than one answer and depends on semantics. Settlement of this 'first' question may hang largely upon what constitutes 'regular' programming
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https://en.wikipedia.org/wiki/History_of_radio
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Sir Jagadish Chandra Bose[1](/boʊs/;[2]IPA:[d͡ʒɔɡod̪iʃt͡ʃɔn̪d̪roboʃu]; 30 November 1858 – 23 November 1937)[3]was apolymathwith interests inbiology,physicsand writing science fiction.[4]He was a pioneer in the investigation of radiomicrowaveoptics, made significant contributions to botany, and was a major force behind the expansion of experimental science on theIndian subcontinent.[5]Bose is considered the father ofBengali science fiction.A crater on the Moonwas named in his honour.[6]He founded theBose Institute, a premier research institute in India and also one of its oldest. Established in 1917, the institute was the first interdisciplinary research centre in Asia.[7]He served as the Director of Bose Institute from its inception until his death.
Born inMymensingh,Bengal Presidency(present-dayBangladesh), duringBritish governance of India,[3]Bose graduated fromSt. Xavier's College, Calcutta(nowKolkata, West Bengal, India). Prior to his enrollment at St. Xavier's College, Calcutta, Bose attendedPabna Zilla SchoolandDhaka Collegiate School, where he began his educational journey. He attended theUniversity of Londonto study medicine, but had to give it up due to health problems. Instead, he conducted research withNobel Laureate,Lord Rayleighat theUniversity of Cambridge. Bose returned to India to join thePresidency Collegeof theUniversity of Calcuttaas a professor of physics. There, despiteracial discriminationand a lack of funding and equipment, Bose carried on his scientific research. He made progress in his research into radio waves in the microwave spectrum and was the first to usesemiconductorjunctions to detect radio waves.
Bose made pioneering discoveries in plant physiology. He used his own invention, thecrescograph, to measure plant response to variousstimuliand proved parallelism between animal and plant tissues. Bose filed for a patent for one of his inventions because of peer pressure, but he was generallycritical of the patent system. To facilitate his research, he constructed automatic recorders capable of registering extremely slight movements; these instruments produced some striking results, such as quivering of injured plants, which Bose interpreted as apower of feeling in plants. His books includeResponse in the Living and Non-Living(1902) andThe Nervous Mechanism of Plants(1926). In a 2004BBCpoll to name theGreatest Bengali of All Time, Bose placed seventh.[8]
Jagadish Chandra Bose was born on 30 November 1858 to aBengali Kayasthafamily ofBrahmosinMymensingh,Bengal Presidency(now part ofBangladesh).[3][9]His family were originally from the village ofRarhikhalinMunshiganj,Dacca district.[10]His father was a leading member of theBrahmo Samajand worked as a civil servant with the title Deputy Magistrate and Assistant Commissioner of Police (ACP) in several places, includingFaridpurandBardhaman.[11][12]
Bose's father sent Bose to aBengali-languageschool for his early education, as it was important to him that his son should study in hisnative languageand culture before studying in English. Speaking at theBikrampurConference in 1915, Bose described the effect this early education had on him:
At that time, sending children to English schools was an aristocratic status symbol. In the vernacular school, to which I was sent, the son of the Muslim attendant of my father sat on my right side, and the son of a fisherman sat on my left. They were my playmates. I listened spellbound to their stories of birds, animals, and aquatic creatures. Perhaps these stories created in my mind a keen interest in investigating the workings of Nature. When I returned home from school accompanied by my school fellows, my mother welcomed and fed all of us without discrimination. Although she was an orthodox old-fashioned lady, she never considered herself guilty of impiety by treating these 'untouchables' as her own children. It was because of my childhood friendship with them that I could never feel that there were 'creatures' who might be labeled 'low-caste', I never realized that there existed a 'problem' common to the two communities, Hindus and Muslims.[12]
Bose joined theHare Schoolin Kolkata in 1869, followed bySFX Greenherald International School, also in Dhaka. In 1875, he passed the entrance examination of theUniversity of Dhakaand was admitted toSt Xavier's College, Mohamudpur. There, he metJesuitFatherEugene Lafont, who played a significant role in developing his interest in natural sciences.[12][13]He received a BA from the University of Dhaka in 1879.[11]
Bose wanted to follow his father into theIndian Civil Service, but his father forbade it, saying his son should be a scholar who would "rule nobody but himself."[14]Bose went to England to study medicine at theUniversity of London, but had to quit because of ill health, possibly worsened by the chemicals used in the dissection rooms.[11]
Through the recommendation ofAnandamohan Bose, his brother-in-law and the first IndianWranglerat theUniversity of Cambridge, Bose secured admission inChrist's College, Cambridge to studynatural sciences. In 1884 he received a BA (Natural Sciences Tripos) from the University of Cambridge[13]as well as a BSc from theUniversity College Londonaffiliated under University of London in 1883.[15][16]
Among Bose's teachers at Cambridge wereLord Rayleigh,Michael Foster,James Dewar,Francis Darwin,Francis Balfour, and Sidney Vines. While at Cambridge, he metUniversity of EdinburghstudentPrafulla Chandra Roy, with whom he became close friends.[11][12]In 1887, Bose married feminist and social workerAbala Bose.[17]
After obtaining a degree from the University of Cambridge Bose returned to India.Henry Fawcetthad given Bose an introduction toLord Ripon, theViceroy of India, who recommended him for a post to the Director of Public Instruction in Kolkata. In those days such posts in the Imperial Education Service were usually reserved for Europeans. Bose was appointed as an officiating professor of physics atPresidency College. Although the principalCharles Henry Tawneyand Director of EducationAlfred Woodley Croftwere reluctant to appoint him, Bose took up his post in January 1885.[15][18]
At that time, an Indian professor was paid two thirds the salary of a European and since his appointment was considered temporary, his salary was further halved, making his salary one-third that of his European peers. As a protest, Bose did not accept his salary and worked without remuneration for the first three years at Presidency College.
He was popular among the students for his teaching style and demonstration of experiments. He got rid of the roll call. After three years in this temporary post, the value of his professorial work was recognized by Tawney and Croft, who made Bose's appointment permanent with retrospective effect.[19]Bose received his full pay for the last three years in a lump sum. However, another source states that his appointment was made permanent on 21 September 1903, some 8 years after his joining the college.[20]
Bose used his own money to fund his research projects as well as receiving funding and support from the social activist nunSister Nivedita.[21]
Bose became interested in radio following the 1894 publication of British physicistOliver Lodge's demonstrations on how to transmit and detect radio waves.[22]He began his own research in the new field in November 1894, setting up his equipment in small 20 ft sq room at Presidency College.[18]Wanting to study the light-like properties of radio waves which were hard to study usinglong radio waves, he managed to reduce the waves to the millimetre level (in the microwave range of about 5 mm wavelength).[22]
Bose's research was not initially appreciated by his department at the college. They felt he should focus only on teaching and that research involved neglect of his duties as a teacher, in spite of Bose giving 26 hours of weekly lectures. Later, when interest was generated in the wider scientific community, the Lieutenant-Governor of Bengal proposed a research post to help Bose. But this scheme was withdrawn when Bose voted against the government's stance during a university meeting. The Lieutenant-Governor persevered to have a Rs.2500 annual grant issued. Despite this, Bose struggled to find time for research due to his teaching duties.[citation needed]
Bose submitted hisfirst scientific paper, "On polarisation of electric rays by double-refracting crystals," to theAsiatic Society of Bengalin May 1895. He submittedhis second paper, "On a new electro-polariscope," to theRoyal Society of Londonin October 1895, and it was published byThe Electricianin December 1895. This may have been the first paper to be published by an Indian in Western scientific periodicals.[23]The paper described Bose's plans for acoherer, a term coined by Lodge referring toradio wavereceivers, which he intended to "perfect" but never patented. The paper was well received byThe ElectricianandThe Englishman, which in January 1896 (commenting on how this new type of wall and fog penetrating "invisible light" could be used inlighthouses) wrote:[22]
Should Professor Bose succeed in perfecting and patenting his 'Coherer', we may in time see the whole system of coast lighting throughout the navigable world revolutionised by a Bengali scientist working single handed in our Presidency College Laboratory.
In November 1895 at a public demonstration at theTown Hallof Kolkata, Bose showed how the millimetre range wavelength microwaves could travel through the human body (of Lieutenant Governor Sir William Mackenzie), and over a distance of 23 metres (75') through two intervening walls to a trigger apparatus he had set up to ring a bell and ignite gunpowder in a closed room.[24][18][25]
Wanting to meet other scientists in Europe, Bose was given a six month scientific deputation in 1896.[26]Bose went to London on a lecture tour and met Italian inventorGuglielmo Marconi, who had been developing a radio wavewireless telegraphysystem for over a year and was trying to market it to the British post service. He was also congratulated byWilliam Thomson, 1st Baron Kelvinand received an honorary Doctor of Science ( DSc) from the University of London.[23][13]In an interview, Bose expressed his disinterest in commercial telegraphy and suggested others use his research work.
In 1899, Bose announced the development of an "iron-mercury-ironcohererwith telephone detector" in apaperpresented at theRoyal Society, London.[27]
In 1900, he presented his research at the firstInternational Congress of Physicsin Paris.[28]
Bose's work in radio microwave optics was specifically directed towards studying the nature of the phenomenon and was not an attempt to develop radio into a communication medium.[29]His experiments took place during the same period (from late 1894 on) when Marconi was making breakthroughs on a radio system specifically designed for wireless telegraphy[30]and others were finding practical applications for radio waves, such as Russian physicistAlexander Stepanovich Popov's radio wave based lightning detector, also inspired by Lodge's experiment.[31]Although Bose's work was not related to communication he, like Lodge and other laboratory experimenters, probably had an influence on other inventors trying to develop radio as communications medium.[31][32][33]Bose was not interested in patenting his work, and openly revealed the operation of his galena crystal detector in his lectures. A friend in the US persuaded him to take out a US patent on his detector, but he did not actively pursue it and allowed it to lapse."[11]
Bose was the first to use a semiconductor junction to detect radio waves, and he invented various now-commonplace microwave components.[31]In 1954, Pearson and Brattain gave priority to Bose for the use of a semi-conducting crystal as a detector of radio waves.[31]In fact, further work at millimetre wavelengths was almost non-existent for the following 50 years. In 1897, Bose described to the Royal Institution in London his research carried out in Kolkata at millimetre wavelengths. He used waveguides, horn antennas, dielectric lenses, various polarisers and even semiconductors at frequencies as high as 60 GHz.[31]Much of his original equipment is still in existence, especially at theBose Institutein Kolkata. A 1.3 mm multi-beam receiver now in use on the NRAO 12 Metre Telescope, Arizona, US, incorporates concepts from his original 1897 papers.[31]
Sir Nevill Mott, Nobel Laureate in 1977 for his own contributions to solid-state electronics, remarked that "J.C. Bose was at least 60 years ahead of his time. In fact, he had anticipated the existence ofP-typeandN-typesemiconductors."[31]
Bose's 1898 experiment on theoptical rotationof microwaves in a twistedjutestructure[34]pioneered the study ofchiral media, and preceded the fields ofartificial dielectricsandmetamaterialsby decades and a century, respectively.[35][36][37]
Bose conducted most of his studies in plant research onMimosa pudicaandDesmodium gyransplants. His major contribution in the field of biophysics was the demonstration of the electrical nature of the conduction of various stimuli (e.g., wounds, chemical agents) in plants, which were earlier thought to be of a chemical nature.[citation needed]In order to understand theheliotropicmovements of plants (the movement of a plant towards a light source), Bose invented a torsional recorder. He found that light applied to one side of the sunflower caused turgor to increase on the opposite side.[38]He was also the first to study the action of microwaves in plant tissues and corresponding changes in the cell membrane potential. He researched the mechanism of the seasonal effect on plants, the effect of chemical inhibitors on plant stimuli and the effect of temperature.[citation needed]
Bose performed a comparative study of the fatigue response of various metals and organic tissue in plants. He subjected metals to a combination of mechanical, thermal, chemical, and electrical stimuli and noted the similarities between metals and cells. Bose's experiments demonstrated a cyclical fatigue response in both stimulated cells and metals, as well as a distinctive cyclical fatigue and recovery response across multiple types of stimuli in both living cells and metals.[citation needed]
Bose documented a characteristic electrical response curve of plant cells to electrical stimulus, as well as the decrease and eventual absence of this response in plants treated with anaesthetics or poison. The response was also absent in metal treated withoxalic acid.[39]
In 1896, Bose wroteNiruddesher Kahini (The Story of the Missing One), a short story that was later expanded and added toAbyakta (অব্যক্ত)collection in 1921 with the new titlePalatak Tuphan (Runaway Sea-Storm). It was one of the first works ofBengali science fiction.[40][41]
In 1917 Bose established the Bose Institute inKolkata, West Bengal, India. Bose served as its director for its first twenty years until his death. Today it is a public research institute of India and also one of its oldest. Bose in his inaugural address on 30 November 1917 dedicated the institute to the nation saying:
I dedicate today this Institute—not merely a Laboratory but a Temple. The power of physical methods applies to the establishment of that truth which can be realised directly through our senses, or through the vast expansion of the perceptive range by means of artificially created organs... Thirty-two years ago I chose the teaching of science as my vocation. It was held that by its very peculiar constitution, the Indian mind would always turn away from the study of Nature to metaphysical speculations. Even had the capacity for inquiry and accurate observation been assumed to be present, there were no opportunities for their employment; there were neither well-equipped laboratories nor skilled mechanicians. This was all too true. It is not for man to complain of circumstances, but bravely to accept, to confront and to dominate them; and we belong to that race which has accomplished great things with simple means.[42][43]
He spent the last years of his life inGiridih. Here he lived in the house located near Jhanda Maidan. This building was named Jagdish Chandra Bose Smriti Vigyan Bhavan. It was inaugurated on 28 February 1997 by then Governor of BiharAkhlaqur Rahman Kidwai.[citation needed]
Jatras, which were popular ancient plays, sparked his interest in the stories of theMahabharataandRamayana. In the latter, he was particularly impressed by the character ofRamaand even more so by the soldierly devotion of his brotherLakshmana.[44]However, he found that most of the characters in these stories seemed too good and perfect. It was the elderly warriors of theMahabharata, with their flaws and qualities that were both human and superhuman, who appealed more to his imagination as a boy.
Impressed byKarna, Bose said:
Always in struggle for the uplift of the people, yet with so little success, such frequent failures, that to most he seemed a failure. All this too gave me a lower and lower idea of all worldly success - how small its so-called victories are! - and higher and higher idea of conflict and defeat; and of true success born of defeat. In such ways I have come to feel one with the highest spirit of my race; with every fibre thrilling with the emotion of the past. That is its noblest teaching - that the only real and spiritual advantage is to fight fair, never to take crooked ways, but keep to the straight path, whatever be in the way.[45]
Bose's place in history is now being re-evaluated. His work may have contributed to the development of radio communication.[27]He is also credited with discovering millimetre length electromagnetic waves and being a pioneer in the field of biophysics.[47]
Many of his instruments are still on display and remain largely usable over 100 years later. They include various antennas, polarisers, and waveguides.
To commemorate his birth centenary in 1958, theJBNSTSscholarship programme was started inWest Bengal. In the same year, India issued a postage stamp bearing his portrait.[48]The same yearAcharya Jagdish Chandra Bose, a documentary film directed by Pijush Bose, was released. It was produced by theGovernment of India'sFilms Division.[49][50]Films Division also produced another documentary film, again titledAcharya Jagdish Chandra Bose, this time directed by the prominent Indian filmmakerTapan Sinha.[51]
On 14 September 2012, Bose's experimental work in millimetre-band radio was recognised as an IEEE Milestone in Electrical and Computer Engineering, the first such recognition of a discovery in India.[52]
On 30 November 2016, Bose was celebrated in a Google Doodle on the 158th anniversary of his birth.[53]
In 2018, theBank of Englanddecided to redesign the50 pound notewith a prominent scientist. Jagadish Chandra Bose was featured in that nomination list for his pioneering work on technology that would enable later development ofWi-Fi.[54][55][56]However, he was not shortlisted.
Journals
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This is alist of people on stamps of Ireland, including the years when they appeared on a stamp.
Because no Irishstampswere designed prior to 1929, the first Irish stamps issued by theProvisional Government of Irelandwere the then-current Britishdefinitivepostage stamps bearing a portrait ofGeorge Vthat wereoverprintedRialtas Sealadaċ na hÉireann 1922(translates as Provisional Government of Ireland 1922) and issued on 17 February 1922. The overprint was later changed toSaorstát Éireann 1922(Irish Free State 1922).[1]: 8
The Irish Free State issued the first commemorative stamps depicting a person on 22 June 1929 whenOifig an Phoist, the Irish Post Office, a section of theDepartment of Posts and Telegraphs, issued a set of three stamps showingDaniel O'Connell.[1]: 21O'Connell is one of a small number of people shown in two issues, includingWolfe ToneandArthur Guinness. The 2009 Guinness issue included postmarks with histrade marksignature, a first inphilately.[2]
The Department of Posts and Telegraphs and, after 1984,An Post, designed stamps showing statesmen, religious, literary and cultural figures, athletes, etc. Until the mid-1990s it was usual policy not to issue stamps showing living persons, the only exceptions being Douglas Hyde (stamp 1943, d. 1949,illustrated below)[1]: 23andLouis le Brocquy(stamp 1977, d. 2012,illustrated below),[1]: 56but this policy has been put aside and there have recently been several issues showing living persons. For themillennium, 30millennium stampswere issued showing living Irish sportsmen.
During the release of the 2022 Irish Oscar Winners stamps, An Post's retain managing director stated thatThe Irish Oscar Winners stamps celebrate the best in the business and serve as a reminder of what we, as Irish people, can achieve.[3]
[30]
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https://en.wikipedia.org/wiki/List_of_people_on_the_postage_stamps_of_Ireland
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This is a list of people and other topics appearing on the cover ofTimemagazine in the 1920s.Timewas first published in 1923. AsTimebecame established as one of the United States' leadingnews magazines, an appearance on the cover ofTimebecame an indicator of notability, fame or notoriety. Such features were accompanied by articles.
For other decades, seeList of covers ofTimemagazine.
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https://en.wikipedia.org/wiki/List_of_covers_of_Time_magazine_(1920s)
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TheFriendship Radiosport Games (FRG)is an internationalmulti-sport eventthat includes competitions in the various sports collectively referred to asradiosport. The Friendship Radiosport Games began in 1989 as a result of asister cityagreement betweenKhabarovsk, RussiaandPortland, Oregon, United States. Since then, participation has been extended to othersister citiesin thePacific Rim. The Friendship Radiosport Games are generally held in the month of August.
The most recent Friendship Radiosport Games were held on August 19–21, 2016, in Portland, Oregon. Planning for the next games in Khabarovsk is starting with a target date of 2018.
The first Friendship Radiosport Games were held in 1989 inKhabarovsk, Russia, which was then still a part of theSoviet Union. The games were organized as a result of the signing of asister cityagreement between the Far Eastern Russia city of Khabarovsk and the city ofPortland, Oregon, on the west coast of the United States. The origination of the idea for a friendlyradiosportcompetition between the two cities can be credited to Yevgeny Stavitsky UAØCA, an activeamateur radiooperator in Khabarovsk. Participants from Portland traveled to Khabarovsk to participate in the games, an event that would not have been possible only a few years before, as the two nations squared off against one another in theCold War. In 1991, the second Friendship Radiosport Games were held in Portland, hosted by the Friendship Amateur Radio Society, and participants from Khabarovsk traveled to Oregon to attend the event. This would start a tradition of holding the event in August of every odd-numbered year.
Extending the event to additionalsister cities, the host for the 1993 Friendship Radiosport Games wasVictoria, British Columbia, Canada. In addition to competitors from Canada, Russia, and the United States, competitors from thesister cityofNiigata, Japan also came to the event in 1993. The 1995 Friendship Radiosport Games were held inKhabarovsk, Russiafor the second time, and representatives from all four cities were in attendance.Tokyo, Japan became the fourth host city for the Friendship Radiosport Games when the event has held there in 1997. The 1999 games returned toPortland, Oregon, United States, where theARDFevent was also designated theIARURegion II Championships, the first suchIARUsanctioned championships in theAmericas. The event returned toVictoria, British Columbia, Canada in 2001, where for the first time competitors fromMelbourne, Victoria, Australia were also in attendance. Breaking with the established pattern, the Friendship Radiosport Games were not held in 2003, but were instead held in 2004, again inKhabarovsk, Russia. The invitation to participation was further extended to radio clubs in thePacific Rimsister citiesofHarbin, China, andBucheon,Korea.
The Friendship Radiosport Games have traditionally included events from all of the three activities collectively known asradiosport. This includesHF contesting,Amateur Radio Direction Finding, andHigh Speed Telegraphy. Some competitors participate in only one of these activities, while others have been competitive in multiple events.
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Procedural signsorprosignsare shorthand signals used inMorse codetelegraphy, for the purpose of simplifying and standardizing procedural protocols for landline and radio communication. The procedural signs are distinct from conventionalMorse code abbreviations, which consist mainly ofbrevity codesthat convey messages to other parties with greaterspeedandaccuracy. However, some codes are usedbothas prosigns and as single letters or punctuation marks, and for those, the distinction between a prosign and abbreviation is ambiguous, even in context.
In the broader sense prosigns are just standardised parts of short form radio protocol, and can include any abbreviation. Examples would beKfor "okay, heard you, continue" orRfor "message, received".[1][2]In a more restricted sense, "prosign" refers to something analogous to the nonprintingcontrol charactersinteleprinterand computercharacter sets, such asBaudotandASCII. Different from abbreviations, those are universally recognizable across language barriers as distinct and well-definedsymbols.
At the coding level, prosigns admit any form the Morse code can take, unlike abbreviations which have to be sent as a sequence of individual letters, like ordinary text. On the other hand, most prosigns codes are much longer than typical codes for letters and numbers. They are individual and indivisiblecode pointswithin the broader Morse code, fully at par with basic letters and numbers.
The development of prosigns began in the 1860s for wired telegraphy. Since telegraphy preceded voice communications by several decades, many of the much older Morse prosigns have acquired precisely equivalentprowordsfor use in more recentvoice protocols.
Not all prosigns used by telegraphers are standard: There are regional and community-specific variations of the coding convention used in certain radio networks to manage transmission and formatting of messages, and many unofficial prosign conventions exist; some of which might be redundant or ambiguous. One typical example of something which is not an officially recognized prosign, but is yet fairly often used in Europe, is one or two freely timed dits at the end of a message,IIor▄ ▄▄ ▄; it is equivalent to the prowordOUT, meaning "I'm done; go ahead". However theofficialprosign with the same meaning isAR, or▄ ▄▄▄ ▄ ▄▄▄ ▄, which takes a little longer to send.[3][2]
Even though represented as strings of letters, prosigns are rendered without the intercharactercommas or pausesthat would occur between the letters shown, if the representation were (mistakenly) sent as a sequence of letters: In printed material describing their meaning and use, prosigns are shown either as a sequence of dots and dashes for the sound of a telegraph, or by an overlined sequence of letters from theInternational Morse Code, which when sentwithoutthe usual spacing, sounds like the prosign symbol.
The best-known example of the convention is the standarddistress callpreamble:SOS. As a prosign it is not really composed of the three separate lettersS,O, andS, (in International Morse:▄ ▄ ▄▄▄▄ ▄▄▄ ▄▄▄▄ ▄ ▄) but is run together as a single symbol▄ ▄ ▄ ▄▄▄ ▄▄▄ ▄▄▄ ▄ ▄ ▄, which is asignin its own right.
In the early decades of telegraphy, many efficiency improvements were incorporated into operations. Each of the early versions of Morse code was an example of that: With only one glaring exception (Intl. MorseO), they all encoded more common characters into shorter keying sequences, and the rare ones into longer, thus effecting onlinedata compression. The introduction of Morse symbols calledprocedural signsorprosignswas then just a logical progression. They were not defined by the developers of Morse code, but were gradually introduced by telegraph operators to improve the speed and accuracy of high-volume message handling, especially those sent over that era's problematic long distance communication channels, such astransoceanic cablesand laterlongwavewireless telegraphy.
Among other prosign uses, improvement in the legibility of written messages sent by telegraph (telegrams) using white space formatting was supported by the procedural symbols. To become an efficienttelegraph operatorit was important to master the Morse code prosigns, as well as themany standard abbreviationsused to facilitate checking and re-sending sections of text.
There are at least three methods used to represent Morse prosign symbols:
Although some of the prosigns as-written appear to be simply two adjacent letters,mostprosigns are transmitted asdigraphsthat havenopauses between the patterns that represent the "combined" letters, and are most commonly written with a single bar over the merged letters (if more than one single character) to indicate this.[4]The only difference between what is transmitted for the Morse code prosign vs. the separate letter signs is the presence or absence of an inter-letter space between the two "dit" / "dah" sequences.
Although the difference in the transmission is subtle, the difference in meaning is gross:
Because no letter boundaries are transmitted with the codes counted asprosigns, their representation by two letters is usually arbitrary, and may be done in multiple equivalent ways. Normally, one particular form is used by convention, but some prosigns have multiple forms in common use:
Many Morse code prosigns do not have written or printed textual character representations in the original source information, even if they do represent characters in other contexts. For example, when embedded in text the Morse code sequence▄▄▄ ▄ ▄ ▄ ▄▄▄represents the "double hyphen" character (normally "=", but also"– –").[1]When the same code appears alone it indicates the action of spacing down two lines on a page in order to create the white space indicating the start of a new paragraph[2]or new section in a message heading.[1]When used as a prosign, there is no actual written or printed character representation or symbol for a new paragraph (i.e. no symbol corresponding to "¶"), other than the two-line white space itself.
Some prosigns are in unofficial use for special characters in languages other thanEnglish, for exampleAAis used unofficially for both the "next line" prosign[b]and for "Ä",[6][7]neither of which is in the international standard.[1]Other prosigns are officially designated for both characters and prosigns, such asARequiv. "+", which marks the end of a message.[d][1]Some genuinely have only one use, such asCTor the equivalentKA(▄▄▄ ▄ ▄▄▄ ▄ ▄▄▄), the International Morse prosign that marks the start of a new transmission[1]or new message.[2]
The procedure signs below are compiled from the official specification for Morse Code, ITU-R M.1677, International Morse Code,[1]while others are defined theInternational Radio Regulations for Mobile Maritime Service, including ITU-R M.1170,[8]ITU-R M.1172,[4]and the MaritimeInternational Code of Signals,[5]with a few details of their use appearing inACP 131,[9]which otherwise definesoperating signals, not procedure signals.
The following table of prosigns includesKandR, which could be considered either abbreviations (for "okay, go ahead", and for "received") or prosigns that are also letters. All of the rest of the symbols are not letters, but in some cases are also used as punctuation.
The following table lists standard abbreviations used for organizing radiotelegraph traffic, however none of them are actual prosigns, despite their similar purpose. All are strictly used as normal strings of one to several letters, never asdigraphsymbols, and have standard meanings used for the management of sending and receiving messages. Dots following indicate that in use, the abbreviation is always followed by more information.
(in-tur-ko)
For the special purpose of exchangingARRL RadiogramsduringNational Traffic Systemnets, the following prosigns and signals can be used, most of which are an exact match withITU-RandCombined Communications Electronics Board(military) standards; a few have no equivalent in any other definition of Morse code procedure signals or abbreviations.
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Telegraphyis the long-distance transmission of messages where the sender uses symbolic codes, known to the recipient, rather than a physical exchange of an object bearing the message. Thusflag semaphoreis a method of telegraphy, whereaspigeon postis not. Ancientsignallingsystems, although sometimes quite extensive and sophisticated as in China, were generally not capable of transmitting arbitrary text messages. Possible messages were fixed and predetermined, so such systems are thus not true telegraphs.
The earliest true telegraph put into widespread use was theChappe telegraph, anoptical telegraphinvented byClaude Chappein the late 18th century. The system was used extensively in France, and European nations occupied by France, during theNapoleonic era. Theelectric telegraphstarted to replace the optical telegraph in the mid-19th century. It was first taken up in Britain in the form of theCooke and Wheatstone telegraph, initially used mostly as an aid torailway signalling. This was quickly followed by a different system developed in the United States bySamuel Morse. The electric telegraph was slower to develop in France due to the established optical telegraph system, but an electrical telegraph was put into use with a code compatible with the Chappe optical telegraph. The Morse system was adopted as the international standard in 1865, using a modifiedMorse codedeveloped in Germany in 1848.[1]
Theheliographis a telegraph system using reflected sunlight for signalling. It was mainly used in areas where the electrical telegraph had not been established and generally used the same code. The most extensive heliograph network established was in Arizona and New Mexico during theApache Wars. The heliograph was standard military equipment as late asWorld War II.Wireless telegraphydeveloped in the early 20th century became important for maritime use, and was a competitor to electrical telegraphy usingsubmarine telegraph cablesin international communications.
Telegrams became a popular means of sending messages once telegraph prices had fallen sufficiently. Traffic became high enough to spur the development of automated systems—teleprintersandpunched tapetransmission. These systems led to newtelegraph codes, starting with theBaudot code. However, telegrams were never able to compete with the letter post on price, and competition from thetelephone, which removed their speed advantage, drove the telegraph into decline from 1920 onwards. The few remaining telegraph applications were largely taken over by alternatives on theinternettowards the end of the 20th century.
The wordtelegraph(fromAncient Greek:τῆλε(têle) 'at a distance' andγράφειν(gráphein) 'to write') was coined by the French inventor of thesemaphore telegraph,Claude Chappe, who also coined the wordsemaphore.[2]
A telegraph is a device for transmitting and receiving messages over long distances, i.e., for telegraphy. The wordtelegraphalone generally refers to anelectrical telegraph. Wireless telegraphy is transmission of messages over radio with telegraphic codes.
Contrary to the extensive definition used by Chappe, Morse argued that the termtelegraphcan strictly be applied only to systems that transmitandrecord messages at a distance. This is to be distinguished fromsemaphore, which merely transmits messages. Smoke signals, for instance, are to be considered semaphore, not telegraph. According to Morse, telegraph dates only from 1832 whenPavel Schillinginvented one of the earliest electrical telegraphs.[3]
A telegraph message sent by anelectrical telegraphoperator or telegrapher usingMorse code(or aprinting telegraphoperator using plain text) was known as a telegram. A cablegram was a message sent by a submarine telegraph cable,[4]often shortened to "cable" or "wire". The suffix -gram is derived from ancient Greek:γραμμα(gramma), meaning something written, i.e. telegram means something written at a distance and cablegram means something written via a cable, whereas telegraph implies the process of writing at a distance.
Later, a Telex was a message sent by aTelexnetwork, a switched network ofteleprinterssimilar to a telephone network.
Awirephotoor wire picture was a newspaper picture that was sent from a remote location by afacsimile telegraph. A diplomatic telegram, also known as adiplomatic cable, is a confidential communication between adiplomatic missionand theforeign ministryof its parent country.[5][6]These continue to be called telegrams or cables regardless of the method used for transmission.
Passing messages by signalling over distance is an ancient practice. One of the oldest examples is the signal towers of theGreat Wall of China. By 400 BC, signals could be sent bybeacon firesordrum beats, and by 200 BC complex flag signalling had developed. During theHan dynasty(202 BC – 220 AD), signallers mainly used flags and wood fires—via the light of the flames swung high into the air at night, and via dark smoke produced by the addition of wolf dung during the day—to send signals.[7]By theTang dynasty(618–907) a message could be sent 1,100 kilometres (700 mi) in 24 hours. TheMing dynasty(1368–1644) usedartilleryas another possible signalling method. While the signalling was complex (for instance, flags of different colours could be used to indicate enemy strength), only predetermined messages could be sent.[8]The Chinese signalling system extended well beyond the Great Wall. Signal towers away from the wall were used to give early warning of an attack. Others were built even further out as part of the protection of trade routes, especially theSilk Road.[9]
Signal fires were widely used in Europe and elsewhere for military purposes. The Roman army made frequent use of them, as did their enemies, and the remains of some of the stations still exist. Few details have been recorded of European/Mediterranean signalling systems and the possible messages. One of the few for which details are known is a system invented byAeneas Tacticus(4th century BC). Tacticus's system had water filled pots at the two signal stations which were drained in synchronisation. Annotation on a floating scale indicated which message was being sent or received. Signals sent by means oftorchesindicated when to start and stop draining to keep the synchronisation.[10]
None of the signalling systems discussed above are true telegraphs in the sense of a system that can transmit arbitrary messages over arbitrary distances. Lines of signallingrelaystations can send messages to any required distance, but all these systems are limited to one extent or another in the range of messages that they can send. A system likeflag semaphore, with an alphabetic code, can certainly send any given message, but the system is designed for short-range communication between two persons. Anengine order telegraph, used to send instructions from the bridge of a ship to the engine room, fails to meet both criteria; it has a limited distance and very simple message set. There was only one ancient signalling system described thatdoesmeet these criteria. That was a system using thePolybius squareto encode an alphabet.Polybius(2nd century BC) suggested using two successive groups of torches to identify the coordinates of the letter of the alphabet being transmitted. The number of said torches held up signalled the grid square that contained the letter. There is no definite record of the system ever being used, but there are several passages in ancient texts that some think are suggestive. Holzmann and Pehrson, for instance, suggest thatLivyis describing its use byPhilip V of Macedonin 207 BC during theFirst Macedonian War. Nothing else that could be described as a true telegraph existed until the 17th century.[10][11]: 26–29Possibly the first alphabetictelegraph codein the modern era is due toFranz Kesslerwho published his work in 1616. Kessler used a lamp placed inside a barrel with a moveable shutter operated by the signaller. The signals were observed at a distance with the newly invented telescope.[11]: 32–34
Anoptical telegraphis a telegraph consisting of a line of stations in towers or natural high points which signal to each other by means of shutters or paddles. Signalling by means of indicator pointers was calledsemaphore. Early proposals for an optical telegraph system were made to theRoyal SocietybyRobert Hookein 1684[12]and were first implemented on an experimental level by SirRichard Lovell Edgeworthin 1767.[13]The first successful optical telegraph network was invented byClaude Chappeand operated in France from 1793.[14]The two most extensive systems were Chappe's in France, with branches into neighbouring countries, and the system ofAbraham Niclas Edelcrantzin Sweden.[11]: ix–x, 47
During 1790–1795, at the height of theFrench Revolution, France needed a swift and reliable communication system to thwart the war efforts of its enemies. In 1790, the Chappe brothers set about devising a system of communication that would allow the central government to receive intelligence and to transmit orders in the shortest possible time. On 2 March 1791, at 11 am, they sent the message "si vous réussissez, vous serez bientôt couverts de gloire" (If you succeed, you will soon bask in glory) between Brulon and Parce, a distance of 16 kilometres (10 mi). The first means used a combination of black and white panels, clocks, telescopes, and codebooks to send their message.
In 1792, Claude was appointedIngénieur-Télégraphisteand charged with establishing a line of stations between Paris andLille, a distance of 230 kilometres (140 mi). It was used to carry dispatches for the war between France and Austria. In 1794, it brought news of a French capture ofCondé-sur-l'Escautfrom the Austrians less than an hour after it occurred.[15]A decision to replace the system with an electric telegraph was made in 1846, but it took a decade before it was fully taken out of service. Thefall of Sevastopolwas reported by Chappe telegraph in 1855.[11]: 92–94
ThePrussian systemwas put into effect in the 1830s. However, they were highly dependent on good weather and daylight to work and even then could accommodate only about two words per minute. The last commercial semaphore link ceased operation in Sweden in 1880. As of 1895, France still operated coastal commercial semaphore telegraph stations, for ship-to-shore communication.[16]
The early ideas for an electric telegraph included in 1753 usingelectrostaticdeflections ofpithballs,[17]proposals forelectrochemicalbubbles in acid byCampilloin 1804 andvon Sömmeringin 1809.[18][19]The first experimental system over a substantial distance was byRonaldsin 1816 using anelectrostatic generator. Ronalds offered his invention to theBritish Admiralty, but it was rejected as unnecessary,[20]the existing optical telegraph connecting the Admiralty in London to their main fleet base inPortsmouthbeing deemed adequate for their purposes. As late as 1844, after the electrical telegraph had come into use, the Admiralty's optical telegraph was still used, although it was accepted that poor weather ruled it out on many days of the year.[21]: 16, 37France had an extensive optical telegraph system dating from Napoleonic times and was even slower to take up electrical systems.[22]: 217–218
Eventually, electrostatic telegraphs were abandoned in favour ofelectromagneticsystems. An early experimental system (Schilling, 1832) led to a proposal to establish a telegraph betweenSt PetersburgandKronstadt, but it was never completed.[23]The first operative electric telegraph (GaussandWeber, 1833) connectedGöttingen Observatoryto the Institute of Physics about 1 km away during experimental investigations of the geomagnetic field.[24]
The first commercial telegraph was byCookeandWheatstonefollowing their English patent of 10 June 1837. It was demonstrated on theLondon and Birmingham Railwayin July of the same year.[25]In July 1839, a five-needle, five-wire system was installed to provide signalling over a record distance of 21 km on a section of theGreat Western RailwaybetweenLondon Paddington stationand West Drayton.[26][27]However, in trying to get railway companies to take up his telegraph more widely forrailway signalling, Cooke was rejected several times in favour of the more familiar, but shorter range, steam-powered pneumatic signalling. Even when his telegraph was taken up, it was considered experimental and the company backed out of a plan to finance extending the telegraph line out toSlough. However, this led to a breakthrough for the electric telegraph, as up to this point the Great Western had insisted on exclusive use and refused Cooke permission to open public telegraph offices. Cooke extended the line at his own expense and agreed that the railway could have free use of it in exchange for the right to open it up to the public.[21]: 19–20
Most of the early electrical systems required multiple wires (Ronalds' system was an exception), but the system developed in the United States byMorseandVailwas a single-wire system. This was the system that first used the soon-to-become-ubiquitousMorse code.[25]By 1844, the Morse system connectedBaltimore to Washington, and by 1861 the west coast of the continent was connected to the east coast.[28][29]TheCooke and Wheatstone telegraph, in a series of improvements, also ended up with a one-wire system, but still using their own code andneedle displays.[26]
The electric telegraph quickly became a means of more general communication. The Morse system was officially adopted as the standard for continental European telegraphy in 1851 with a revised code, which later became the basis ofInternational Morse Code.[30]However, Great Britain and theBritish Empirecontinued to use the Cooke and Wheatstone system, in some places as late as the 1930s.[26]Likewise, the United States continued to useAmerican Morse codeinternally, requiring translation operators skilled in both codes for international messages.[30]
Railway signal telegraphy was developed in Britain from the 1840s onward. It was used to manage railway traffic and to prevent accidents as part of the railway signalling system. On 12 June 1837 Cooke and Wheatstone were awarded a patent for an electric telegraph.[31]This was demonstrated betweenEuston railway station—where Wheatstone was located—and the engine house at Camden Town—where Cooke was stationed, together withRobert Stephenson, theLondon and Birmingham Railwayline's chief engineer. The messages were for the operation of the rope-haulage system for pulling trains up the 1 in 77 bank. The world's first permanent railway telegraph was completed in July 1839 between London Paddington and West Drayton on theGreat Western Railwaywith an electric telegraph using a four-needle system.
The concept of asignalling "block" systemwas proposed by Cooke in 1842. Railway signal telegraphy did not change in essence from Cooke's initial concept for more than a century. In this system each line of railway was divided into sections or blocks of varying length. Entry to and exit from the block was to be authorised by electric telegraph and signalled by the line-side semaphore signals, so that only a single train could occupy the rails. In Cooke's original system, a single-needle telegraph was adapted to indicate just two messages: "Line Clear" and "Line Blocked". Thesignallerwould adjust his line-side signals accordingly. As first implemented in 1844 each station had as many needles as there were stations on the line, giving a complete picture of the traffic. As lines expanded, a sequence of pairs of single-needle instruments were adopted, one pair for each block in each direction.[32]
Wigwag is a form offlag signallingusing a single flag. Unlike most forms of flag signalling, which are used over relatively short distances, wigwag is designed to maximise the distance covered—up to 32 km (20 mi) in some cases. Wigwag achieved this by using a large flag—a single flag can be held with both hands unlike flag semaphore which has a flag in each hand—and using motions rather than positions as its symbols since motions are more easily seen. It was invented by US Army surgeonAlbert J. Myerin the 1850s who later became the first head of theSignal Corps. Wigwag was used extensively during theAmerican Civil Warwhere it filled a gap left by the electrical telegraph. Although the electrical telegraph had been in use for more than a decade, the network did not yet reach everywhere and portable, ruggedized equipment suitable for military use was not immediately available. Permanent or semi-permanent stations were established during the war, some of them towers of enormous height and the system was extensive enough to be described as a communications network.[33][34]
Aheliographis a telegraph that transmits messages by flashing sunlight with a mirror, usually using Morse code. The idea for a telegraph of this type was first proposed as a modification of surveying equipment (Gauss, 1821). Various uses of mirrors were made for communication in the following years, mostly for military purposes, but the first device to become widely used was a heliograph with a moveable mirror (Mance, 1869). The system was used by the French during the1870–71 siege of Paris, with night-time signalling usingkerosene lampsas the source of light. An improved version (Begbie, 1870) was used by British military in many colonial wars, including theAnglo-Zulu War(1879). At some point, a morse key was added to the apparatus to give the operator the same degree of control as in the electric telegraph.[35]
Another type of heliograph was theheliostatorheliotropefitted with a Colomb shutter. The heliostat was essentially a surveying instrument with a fixed mirror and so could not transmit a code by itself. The termheliostatis sometimes used as a synonym forheliographbecause of this origin. The Colomb shutter (BoltonandColomb, 1862) was originally invented to enable the transmission of morse code bysignal lampbetweenRoyal Navyships at sea.[35]
The heliograph was heavily used byNelson A. MilesinArizonaandNew Mexicoafter he took over command (1886) of the fight againstGeronimoand otherApachebands in theApache Wars. Miles had previously set up the first heliograph line in the US betweenFort KeoghandFort CusterinMontana. He used the heliograph to fill in vast, thinly populated areas that were not covered by the electric telegraph. Twenty-six stations covered an area 320 by 480 km (200 by 300 mi). In a test of the system, a message was relayed 640 km (400 mi) in four hours. Miles' enemies usedsmoke signalsand flashes of sunlight from metal, but lacked a sophisticated telegraph code.[36]The heliograph was ideal for use in the American Southwest due to its clear air and mountainous terrain on which stations could be located. It was found necessary to lengthen the morse dash (which is much shorter in American Morse code than in the modern International Morse code) to aid differentiating from the morse dot.[35]
Use of the heliograph declined from 1915 onwards, but remained in service in Britain andBritish Commonwealthcountries for some time. Australian forces used the heliograph as late as 1942 in theWestern Desert CampaignofWorld War II. Some form of heliograph was used by themujahideenin theSoviet–Afghan War(1979–1989).[35]
A teleprinter is a telegraph machine that can send messages from a typewriter-like keyboard and print incoming messages in readable text with no need for the operators to be trained in the telegraph code used on the line. It developed from various earlier printing telegraphs and resulted in improved transmission speeds.[37]TheMorse telegraph(1837) was originally conceived as a system marking indentations on paper tape. A chemical telegraph making blue marks improved the speed of recording (Bain, 1846), but was delayed by a patent challenge from Morse. The first true printing telegraph (that is printing in plain text) used a spinning wheel oftypesin the manner of adaisy wheel printer(House, 1846, improved byHughes, 1855). The system was adopted byWestern Union.[38]
Early teleprinters used theBaudot code, a five-bit sequential binary code. This was a telegraph code developed for use on the French telegraph using a five-key keyboard (Baudot, 1874). Teleprinters generated the same code from a full alphanumeric keyboard. A feature of the Baudot code, and subsequent telegraph codes, was that, unlike Morse code, every character has a code of the same length making it more machine friendly.[39]The Baudot code was used on the earliestticker tapemachines (Calahan, 1867), a system for mass distributing information on current price of publicly listed companies.[40]
In apunched-tapesystem, the message is first typed onto punched tape using the code of the telegraph system—Morse code for instance. It is then, either immediately or at some later time, run through a transmission machine which sends the message to the telegraph network. Multiple messages can be sequentially recorded on the same run of tape. The advantage of doing this is that messages can be sent at a steady, fast rate making maximum use of the available telegraph lines. The economic advantage of doing this is greatest on long, busy routes where the cost of the extra step of preparing the tape is outweighed by the cost of providing more telegraph lines. The first machine to use punched tape was Bain's teleprinter (Bain, 1843), but the system saw only limited use. Later versions of Bain's system achieved speeds up to 1000 words per minute, far faster than a human operator could achieve.[41]
The first widely used system (Wheatstone, 1858) was first put into service with the BritishGeneral Post Officein 1867. A novel feature of the Wheatstone system was the use ofbipolar encoding. That is, both positive and negative polarity voltages were used.[42]Bipolar encoding has several advantages, one of which is that it permitsduplexcommunication.[43]The Wheatstone tape reader was capable of a speed of 400 words per minute.[44]: 190
A worldwide communication network meant that telegraph cables would have to be laid across oceans. On land cables could be run uninsulated suspended from poles. Underwater, a good insulator that was both flexible and capable of resisting the ingress of seawater was required. A solution presented itself withgutta-percha, a natural rubber from thePalaquium guttatree, afterWilliam Montgomeriesent samples to London from Singapore in 1843. The new material was tested byMichael Faradayand in 1845 Wheatstone suggested that it should be used on the cable planned betweenDoverandCalaisbyJohn Watkins Brett. The idea was proved viable when theSouth Eastern Railwaycompany successfully tested a three-kilometre (two-mile) gutta-percha insulated cable with telegraph messages to a ship off the coast ofFolkestone.[45]The cable to France was laid in 1850 but was almost immediately severed by a French fishing vessel.[46]It was relaid the next year[46]and connections to Ireland and theLow Countriessoon followed.
Getting a cable across the Atlantic Ocean proved much more difficult. TheAtlantic Telegraph Company, formed inLondonin 1856, had several failed attempts. A cable laid in 1858 worked poorly for a few days, sometimes taking all day to send a message despite the use of the highly sensitivemirror galvanometerdeveloped by William Thomson (the futureLord Kelvin) before being destroyed by applying too high a voltage. Its failure and slow speed of transmission prompted Thomson andOliver Heavisideto find better mathematical descriptions of longtransmission lines.[47]The company finally succeeded in 1866 with an improved cable laid bySSGreat Eastern, the largest ship of its day, designed byIsambard Kingdom Brunel.[48][47]
An overland telegraph from Britain to India was first connected in 1866 but was unreliable so a submarine telegraph cable was connected in 1870.[49]Several telegraph companies were combined to form theEastern Telegraph Companyin 1872. Australia was first linked to the rest of the world in October 1872 by a submarine telegraph cable atDarwin.[50]
From the 1850s until well into the 20th century, British submarine cable systems dominated the world system. This was set out as a formal strategic goal, which became known as theAll Red Line.[51]In 1896, there were thirty cable-laying ships in the world and twenty-four of them were owned by British companies. In 1892, British companies owned and operated two-thirds of the world's cables and by 1923, their share was still 42.7 percent.[52]DuringWorld War I, Britain's telegraph communications were almost completely uninterrupted while it was able to quickly cut Germany's cables worldwide.[51]
In 1843, Scottish inventorAlexander Baininvented a device that could be considered the firstfacsimile machine. He called his invention a "recording telegraph". Bain's telegraph was able to transmit images by electrical wires.Frederick Bakewellmade several improvements on Bain's design and demonstrated a telefax machine. In 1855, an Italian priest,Giovanni Caselli, also created an electric telegraph that could transmit images. Caselli called his invention "Pantelegraph". Pantelegraph was successfully tested and approved for a telegraph line betweenParisandLyon.[53][54]
In 1881, English inventorShelford Bidwellconstructed thescanning phototelegraphthat was the first telefax machine to scan any two-dimensional original, not requiring manual plotting or drawing. Around 1900, German physicistArthur Korninvented theBildtelegraphwidespread in continental Europe especially since a widely noticed transmission of a wanted-person photograph from Paris to London in 1908 used until the wider distribution of the radiofax. Its main competitors were theBélinographebyÉdouard Belinfirst, then since the 1930s, theHellschreiber, invented in 1929 by German inventorRudolf Hell, a pioneer in mechanical image scanning and transmission.
The late 1880s through to the 1890s saw the discovery and then development of a newly understood phenomenon into a form ofwireless telegraphy, calledHertzian wavewireless telegraphy, radiotelegraphy, or (later) simply "radio". Between 1886 and 1888,Heinrich Rudolf Hertzpublished the results of his experiments where he was able to transmitelectromagnetic waves(radio waves) through the air, provingJames Clerk Maxwell's 1873 theory ofelectromagnetic radiation. Many scientists and inventors experimented with this new phenomenon but the consensus was that these new waves (similar to light) would be just as short range as light, and, therefore, useless for long range communication.[56]
At the end of 1894, the young Italian inventorGuglielmo Marconibegan working on the idea of building a commercial wireless telegraphy system based on the use of Hertzian waves (radio waves), a line of inquiry that he noted other inventors did not seem to be pursuing.[57]Building on the ideas of previous scientists and inventors Marconi re-engineered their apparatus by trial and error attempting to build a radio-based wireless telegraphic system that would function the same as wired telegraphy. He would work on the system through 1895 in his lab and then in field tests making improvements to extend its range. After many breakthroughs, including applying the wired telegraphy concept of grounding the transmitter and receiver, Marconi was able, by early 1896, to transmit radio far beyond the short ranges that had been predicted.[58]Having failed to interest the Italian government, the 22-year-old inventor brought his telegraphy system to Britain in 1896 and metWilliam Preece, a Welshman, who was a major figure in the field and Chief Engineer of theGeneral Post Office. A series of demonstrations for the British government followed—by March 1897, Marconi had transmitted Morse code signals over a distance of about6 km (3+1⁄2mi) acrossSalisbury Plain.
On 13 May 1897, Marconi, assisted by George Kemp, aCardiffPost Office engineer, transmitted the first wireless signals over water toLavernock(near Penarth in Wales) fromFlat Holm.[59]His star rising, he was soon sending signals across theEnglish Channel(1899), from shore to ship (1899) and finally across the Atlantic (1901).[60]A study of these demonstrations of radio, with scientists trying to work out how a phenomenon predicted to have a short range could transmit "over the horizon", led to the discovery of a radio reflecting layer in the Earth's atmosphere in 1902, later called theionosphere.[61]
Radiotelegraphy proved effective for rescue work in seadisastersby enabling effective communication between ships and from ship to shore. In 1904, Marconi began the first commercial service to transmit nightly news summaries to subscribing ships, which could incorporate them into their on-board newspapers. A regular transatlantic radio-telegraph service was finally begun on 17 October 1907.[62][63]Notably, Marconi's apparatus was used to help rescue efforts after the sinking ofRMSTitanic. Britain's postmaster-general summed up, referring to theTitanicdisaster, "Those who have been saved, have been saved through one man, Mr. Marconi...and his marvellous invention."
The successful development of radiotelegraphy was preceded by a 50-year history of ingenious but ultimately unsuccessful experiments by inventors to achieve wireless telegraphy by other means.[citation needed]
Several wireless electrical signaling schemes based on the (sometimes erroneous) idea that electric currents could be conducted long-range through water, ground, and air were investigated for telegraphy before practical radio systems became available.
The original telegraph lines used two wires between the two stations to form a completeelectrical circuitor "loop". In 1837, however,Carl August von SteinheilofMunich,Germany, found that by connecting one leg of the apparatus at each station to metal plates buried in the ground, he could eliminate one wire and use a single wire for telegraphic communication. This led to speculation that it might be possible to eliminate both wires and therefore transmit telegraph signals through the ground without any wires connecting the stations. Other attempts were made to send the electric current through bodies of water, to span rivers, for example. Prominent experimenters along these lines includedSamuel F. B. Morsein the United States andJames Bowman Lindsayin Great Britain, who in August 1854, was able to demonstrate transmission across a mill dam at a distance of 500 yards (457 metres).[64]
US inventorsWilliam Henry Ward(1871) andMahlon Loomis(1872) developed electrical conduction systems based on the erroneous belief that there was an electrified atmospheric stratum accessible at low altitude.[65][66]They thought atmosphere current, connected with a return path using "Earth currents" would allow for wireless telegraphy as well as supply power for the telegraph, doing away with artificial batteries.[67][68]A more practical demonstration of wireless transmission via conduction came inAmos Dolbear's 1879 magneto electric telephone that used ground conduction to transmit over a distance of a quarter of a mile.[69]
In the 1890s inventorNikola Teslaworked on an air and ground conductionwireless electric power transmission system, similar to Loomis',[70][71][72]which he planned to include wireless telegraphy. Tesla's experiments had led him to incorrectly conclude that he could use the entire globe of the Earth to conduct electrical energy[73][69]and his 1901 large scale application of his ideas, a high-voltage wireless power station, now calledWardenclyffe Tower, lost funding and was abandoned after a few years.
Telegraphic communication using earth conductivity was eventually found to be limited to impractically short distances, as was communication conducted through water, or between trenches during World War I.
Both electrostatic and electromagnetic induction were used to develop wireless telegraph systems that saw limited commercial application. In the United States,Thomas Edison, in the mid-1880s, patented an electromagnetic induction system he called "grasshopper telegraphy", which allowed telegraphic signals to jump the short distance between a running train and telegraph wires running parallel to the tracks.[74]This system was successful technically but not economically, as there turned out to be little interest by train travelers in the use of an on-board telegraph service. During theGreat Blizzard of 1888, this system was used to send and receive wireless messages fromtrainsburied in snowdrifts. The disabled trains were able to maintain communications via their Edison induction wireless telegraph systems,[75]perhaps the first successful use of wireless telegraphy to send distress calls. Edison would also help to patent a ship-to-shore communication system based on electrostatic induction.[76]
The most successful creator of an electromagnetic induction telegraph system wasWilliam Preece, chief engineer of Post Office Telegraphs of theGeneral Post Office(GPO) in theUnited Kingdom. Preece first noticed the effect in 1884 when overhead telegraph wires inGrays Inn Roadwere accidentally carrying messages sent on buried cables. Tests inNewcastlesucceeded in sending a quarter of a mile using parallel rectangles of wire.[21]: 243In tests across theBristol Channelin 1892, Preece was able to telegraph across gaps of about 5 kilometres (3.1 miles). However, his induction system required extensive lengths ofantenna wires, many kilometers long, at both the sending and receiving ends. The length of those sending and receiving wires needed to be about the same length as the width of the water or land to be spanned. For example, for Preece's station to span theEnglish ChannelfromDover, England, to the coast ofFrancewould require sending and receiving wires of about 30 miles (48 kilometres) along the two coasts. These facts made the system impractical on ships, boats, and ordinary islands, which are much smaller thanGreat BritainorGreenland. Also, the relatively short distances that a practical Preece system could span meant that it had few advantages overunderwater telegraph cables.
A telegram service is a company or public entity that delivers telegraphed messages directly to the recipient. Telegram services were not inaugurated untilelectric telegraphybecame available. Earlier optical systems were largely limited to official government and military purposes.
Historically, telegrams were sent between a network of interconnected telegraph offices. A person visiting a local telegraph office paid by the word to have a message telegraphed to another office and delivered to the addressee on a paper form.[77]: 276Messages (i.e. telegrams) sent by telegraph could be delivered bytelegraph messengerfaster than mail,[40]and even in the telephone age, the telegram remained popular for social and business correspondence. At their peak in 1929, an estimated 200 million telegrams were sent.[77]: 274
In 1919, the Central Bureau for Registered Addresses was established in thefinancial districtofNew York City. The bureau was created to ease the growing problem of messages being delivered to the wrong recipients. To combat this issue, the bureau offered telegraph customers the option to register unique code names for their telegraph addresses. Customers were charged $2.50 per year per code. By 1934, 28,000 codes had been registered.[78]
Telegram services still operate in much of the world (seeworldwide use of telegrams by country), but e-mail andtext messaginghave rendered telegrams obsolete in many countries, and the number of telegrams sent annually has been declining rapidly since the 1980s.[79]Where telegram services still exist, the transmission method between offices is no longer by telegraph, but bytelexorIPlink.[80]
As telegrams have been traditionally charged by the word, messages were often abbreviated to pack information into the smallest possible number of words, in what came to be called "telegram style".
The average length of a telegram in the 1900s in the US was 11.93 words; more than half of the messages were 10 words or fewer.[81]According to another study, the mean length of the telegrams sent in the UK before 1950 was 14.6 words or 78.8 characters.[82]For German telegrams, the mean length is 11.5 words or 72.4 characters.[82]At the end of the 19th century, the average length of a German telegram was calculated as 14.2 words.[82]
Telex(telegraph exchange) was a public switched network of teleprinters. It used rotary-telephone-stylepulse diallingfor automatic routing through the network. It initially used theBaudot codefor messages. Telex development began in Germany in 1926, becoming an operational service in 1933 run by theReichspost(the German imperial postal service). It had a speed of 50 baud—approximately 66 words per minute. Up to 25 telex channels could share a single long-distance telephone channel by usingvoice frequency telegraphymultiplexing, making telex the least expensive method of reliable long-distance communication.[83]Telex was introduced into Canada in July 1957, and the United States in 1958.[84]A new code,ASCII, was introduced in 1963 by theAmerican Standards Association. ASCII was a seven-bit code and could thus support a larger number of characters than Baudot. In particular, ASCII supported upper and lower case whereas Baudot was upper case only.
Telegraph use began to permanently decline around 1920.[21]: 248The decline began with the growth of the use of thetelephone.[21]: 253Ironically, the invention of the telephone grew out of the development of theharmonic telegraph, a device which was supposed to increase the efficiency of telegraph transmission and improve the profits of telegraph companies. Western Union gave up its patent battle withAlexander Graham Bellbecause it believed the telephone was not a threat to its telegraph business. TheBell Telephone Companywas formed in 1877 and had 230 subscribers which grew to 30,000 by 1880. By 1886 there were a quarter of a million phones worldwide,[77]: 276–277and nearly 2 million by 1900.[44]: 204The decline was briefly postponed by the rise of special occasion congratulatory telegrams. Traffic continued to grow between 1867 and 1893 despite the introduction of the telephone in this period,[77]: 274but by 1900 the telegraph was definitely in decline.[77]: 277
There was a brief resurgence in telegraphy duringWorld War Ibut the decline continued as the world entered theGreat Depressionyears of the 1930s.[77]: 277After theSecond World Warnew technology improved communication in the telegraph industry.[85]Telegraph lines continued to be an important means of distributing news feeds fromnews agenciesby teleprinter machine until the rise of the internet in the 1990s. For Western Union, one service remained highly profitable—thewire transferof money. This service kept Western Union in business long after the telegraph had ceased to be important.[77]: 277In the modern era, the telegraph that began in 1837 has been gradually replaced bydigital datatransmission based oncomputerinformation systems.[85]
Optical telegraph lines were installed by governments, often for a military purpose, and reserved for official use only. In many countries, this situation continued after the introduction of the electric telegraph. Starting in Germany and the UK, electric telegraph lines were installed by railway companies. Railway use quickly led to private telegraph companies in the UK and the US offering a telegraph service to the public using telegraph along railway lines. The availability of this new form of communication brought on widespread social and economic changes.
The electric telegraph freed communication from the time constraints of postal mail and revolutionized the global economy and society.[86][87]By the end of the 19th century, the telegraph was becoming an increasingly common medium of communication for ordinary people. The telegraph isolated the message (information) from the physical movement of objects or the process.[88]
There was some fear of the new technology. According to authorAllan J. Kimmel, some people "feared that the telegraph would erode the quality of public discourse through the transmission of irrelevant, context-free information."Henry David Thoreauthought of the Transatlantic cable "...perchance the first news that will leak through into the broad flapping American ear will be that Princess Adelaide has the whooping cough." Kimmel says these fears anticipate many of the characteristics of the modern internet age.[89]
Initially, the telegraph was expensive, but it had an enormous effect on three industries: finance, newspapers, and railways. Telegraphy facilitated the growth of organizations "in the railroads, consolidated financial and commodity markets, and reduced information costs within and between firms".[87]In the US, there were 200 to 300 stock exchanges before the telegraph, but most of these were unnecessary and unprofitable once the telegraph made financial transactions at a distance easy and drove down transaction costs.[77]: 274–75This immense growth in the business sectors influenced society to embrace the use of telegrams once the cost had fallen.
Worldwide telegraphy changed the gathering of information for news reporting. Journalists were using the telegraph for war reporting as early as 1846 when theMexican–American Warbroke out. News agencies were formed, such as theAssociated Press, for the purpose of reporting news by telegraph.[77]: 274–75Messages and information would now travel far and wide, and the telegraph demanded a language "stripped of the local, the regional; and colloquial", to better facilitate a worldwide media language.[88]Media language had to be standardized, which led to the gradual disappearance of different forms of speech and styles ofjournalismand storytelling.
The spread of the railways created a need for an accuratestandard timeto replace local standards based on localnoon. The means of achieving this synchronisation was the telegraph. This emphasis on precise time has led to major societal changes such as the concept of thetime value of money.[77]: 273–74
During the telegraph era there was widespread employment ofwomen in telegraphy. The shortage of men to work as telegraph operators in theAmerican Civil Waropened up the opportunity for women of a well-paid skilled job.[77]: 274In the UK, there was widespread employment of women as telegraph operators even earlier – from the 1850s by all the major companies. The attraction of women for the telegraph companies was that they could pay them less than men. Nevertheless, the jobs were popular with women for the same reason as in the US; most other work available for women was very poorly paid.[39]: 77[21]: 85
The economic impact of the telegraph was not much studied by economic historians until parallels started to be drawn with the rise of the internet. In fact, the electric telegraph was as important as the invention of printing in this respect. According to economist Ronnie J. Phillips, the reason for this may be thatinstitutional economistspaid more attention to advances that required greater capital investment. The investment required to build railways, for instance, is orders of magnitude greater than that for the telegraph.[77]: 269–70
The optical telegraph was quickly forgotten once it went out of service. While it was in operation, it was very familiar to the public across Europe. Examples appear in many paintings of the period. Poems include "Le Telégraphe" byVictor Hugo, and the collectionTelegrafen: Optisk kalender för 1858byElias Sehlstedt[sv][90]is dedicated to the telegraph. In novels, the telegraph is a major component inLucien LeuwenbyStendhal, and it features inThe Count of Monte Cristo, byAlexandre Dumas.[11]: vii–ixJoseph Chudy's 1796 opera,Der Telegraph oder die Fernschreibmaschine, was written to publicise Chudy's telegraph (a binary code with five lamps) when it became clear that Chappe's design was being taken up.[11]: 42–43
Rudyard Kiplingwrote a poem in praise of submarine telegraph cables; "And a new Word runs between: whispering, 'Let us be one!'"[91][92]Kipling's poem represented a widespread idea in the late nineteenth century that international telegraphy (and new technology in general)[93]would bring peace and mutual understanding to the world.[94]When a submarine telegraph cable first connected America and Britain, theNew York Postdeclared:
It is the harbinger of an age when international difficulties will not have time to ripen into bloody results, and when, in spite of the fatuity and perveseness of rulers, war will be impossible.[95]
Numerous newspapers and news outlets in various countries, such asThe Daily Telegraphin Britain,The Telegraphin India,De Telegraafin the Netherlands, and theJewish Telegraphic Agencyin the US, were given names which include the word "telegraph" due to their having received news by means of electric telegraphy. Some of these names are retained even though different means of news acquisition are now used.
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Atelegraphist(British English),telegrapher(American English), ortelegraph operatoris a person who uses atelegraph keyto send and receiveMorse codemessages in atelegraphysystem. These messages, also called telegrams, can be transmittedelectronically by land lines, orwirelesslybyradio.
During theFirst World War, theRoyal Navyenlisted many volunteers as radio telegraphists. Telegraphists were indispensable at sea in the early days ofwireless telegraphy, and many young men were called to sea as professional radiotelegraph operators who were always accorded high-paying officer status at sea. Subsequent to theTitanicdisasterand theRadio Act of 1912, the International Safety of Life at Sea (SOLAS) conventions established the500kHzmaritime distress frequency monitoring and mandated that all passenger-carrying ships carry licensed radio telegraph operators.[1]
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Awire signalis abrevity codeused by telegraphers to save time and cost when sending long messages. The best-known code was the92 Codeadopted byWestern Unionin 1859. The code was designed to reducebandwidthconsumption overtelegraph lines, thus speeding transmissions by utilizing a numerical code system for frequently used phrases.[1]
Several of the codes are taken fromThe Telegraph Instructorby G.M. Dodge.[2]Dodge notes:
Codes that are not listed in the 1901 edition of Dodge are marked with an asterisk (*).
In the above list, the numbers 19 and 31 refer totrain order operationswhereby messages from the dispatcher about changes in railroad routing and scheduling were written on paper forms. Form 19 was designed to be passed to the train as it went through a station at speed. Form 31 required hand delivery for confirmation.
Today,amateur radiooperators still use codes 73 and 88 regularly, and-30-is used in journalism, as it was shorthand for "No more - the end". TheYoung Ladies Radio Leagueuses code 33 to mean "love sealed with friendship and mutual respect between one YL [young lady] and another YL"[3]or simply "hugs." A once-used but unofficial code 99 meant "go to hell." The other codes have mostly fallen into disuse.[1]
The following code was taken from 1873 telegraph rulebook of theLakeshore and Tuscarawas Valley Railway CompanyofCleveland, Ohio.[4]
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https://en.wikipedia.org/wiki/92_Code
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ThePhillips Codeis abrevity code(shorthand) compiled and expanded in 1879 byWalter P. Phillips(then of theAssociated Press) for the rapid transmission oftelegraphmessages, including press reports.
It was compiled in 1879 byWalter P. Phillips, who explained that he was in large part putting down the collective experience of generations of telegraph operators. In the introduction to the 1907 edition of his book, "The Phillips Code: A Thoroughly Tested Method of Shorthand Arranged for Telegraphic Purposes. And Contemplating the Rapid Transmission of Press Reports; Also Intended to be Used as an Easily Acquired Method for General Newspaper and Court Reporting," Phillips wrote, "Research suggests that at one time, commercial telegraphs and railroads had numerical codes that contained at least 100 groupings. Few survived beyond the turn of the century. The compilation in this book represents the consensus of many whose duties brought them into close contact with this subject."[1]
His code defined hundreds of abbreviations and initialisms for commonly used words that news authors and copy desk staff would commonly use. There were subcodes for commodities and stocks called the Market Code, a Baseball Supplement, and single-letter codes for Option Months. The last official edition was published in 1925, but there was also a Market supplement last published in 1909 that was separate.
The code consists of a dictionary of common words or phrases and their associated abbreviations. Extremely common terms are represented by a single letter (C: See; Y: Year); those less frequently used gain successively longer abbreviations (Ab: About; Abb: Abbreviate; Abty: Ability; Acmpd: Accompanied).
Later, The Evans Basic English Code[2]expanded the 1,760 abbreviations in the Phillips Code to 3,848 abbreviations.
Using the Phillips Code, this ten-word telegraphic transmission:
ABBG LG WORDS CAN SAVE XB AMTS MON AVOG FAPIB
expands to this:
Abbreviating long words can save exorbitant amounts of money, avoiding filing a petition in bankruptcy.
In 1910, an article explaining the basic structure and purpose of the Phillips Code appeared in various US newspapers and magazines.[3]One example given is:
T tri o HKT ft mu o SW on Ms roof garden, nw in pg,etc.
which the article translates as:
Thetrial of Harry K. Thawfor the murder of Stanford White on Madison Square Roof Garden, now in progress,etc.
The termsPOTUSandSCOTUSoriginated in telegraph code, and are included in the Phillips code.[4][5][6]SCOTUS appeared in the very first edition of 1879[7]and POTUS was in use by 1895,[4]and was officially included in the 1923 edition. These abbreviations entered common parlance when news gathering services, in particular, theAssociated Press, adopted the terminology.
Telegraph operators would often interleave Phillips Code with numericwire signalsthat had been developed during theAmerican Civil Warera, such as the92 Code. These codes were used byrailroadtelegraphers to indicatelogisticsinstructions and they proved to be useful when describing an article's priority or confirming its transmission and receipt. Thismeta-datawould occasionally appear in print whentypesettersincluded the codes in newspapers,[8][failed verification]especially the code for "No more—the end", abbreviated as"- 30 -"on atypewriter.
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https://en.wikipedia.org/wiki/Phillips_Code
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TheR-S-T systemis used byamateur radio operators,shortwave listeners, and other radio hobbyists to exchange information about the quality of a radio signal being received. The code is a three digit number, with one digit each for conveying an assessment of the signal's readability, strength, and tone.[1][2]The code was developed in 1934 by Amateur radio operator Arthur W. Braaten, W2BSR,[3][4][5][6]and was similar to that codified in the ITU Radio Regulations, Cairo, 1938.[7]
TheRstands for "Readability". Readability is a qualitative assessment of how easy or difficult it is to correctly copy the information being sent during thetransmission. In aMorse codetelegraphytransmission, readability refers to how easy or difficult it is to distinguish each of the characters in the text of the message being sent; in avoicetransmission, readability refers to how easy or difficult it is for each spoken word to be understood correctly. Readability is measured on a scale of 1 to 5.[8]
TheSstands for "Strength". Strength is an assessment of how powerful the received signal is at the receiving location. Although an accurate signal strength meter can determine a quantitative value for signal strength, in practice this portion of the RST code is a qualitative assessment, often made based on theS meterof the radioreceiverat the location of signal reception. "Strength" is measured on a scale of 1 to 9.[8]
For a quantitative assessment, quality HF receivers are calibrated so that S9 on theS-metercorresponds to a signal of 50 μV at the antenna standard terminal impedance 50 ohms.[9]One "S" difference should correspond to 6 dB at signal strength (2x voltage = 4x power). On VHF and UHF receivers used for weak signal communications, S9 often corresponds to 5 μV at the antenna terminal 50 ohms. Amateur radio (ham) operators may also use a signal strength of "20 to 60 over 9", or "+20 to +60 over 9." This is in reference to a signal that exceeds S9 on a signal meter on a HF receiver.
TheTstands for "Tone" and is measured on a scale of 1 to 9. Tone only pertains to Morse code and other digital transmission modes and is therefore omitted during voice operations. With modern technology, imperfections in the quality of transmitters’ digital modulation severe enough to be detected by human ears are rare.[8]
Suffixes were historically added to indicate other signal properties, and might be sent as599Kto indicate a clear, strong signal but with bothersomekey clicks.
An example RST report for a voice transmission is "59", usually pronounced "five nine" or "five by nine", a report that indicates a perfectly readable and very strong signal. Exceptionally strong signals are designated by the quantitative number ofdecibels, in excess of "S9", displayed on the receiver's S meter. Example: "Your signal is 30 dB over S9," or more simply, "Your signal is 30 over 9."
Because the N character in Morse code requires less time to send than the 9, during amateur radiocontestswhere the competingamateur radio stationsare all using Morse code, the nines in the RST are typically abbreviated to N to read 5NN.[11]In general, this practice is referred to as abbreviated or "cut" numbers.[12][13][14]
The RSQ system has also been proposed for digital modes as an alternative to the RST system. The Q replaces "Tone" with "Quality" on a similar 1-9 scale indicating presence or number of unwanted 'sidebar pairs' in a narrow-band digital mode, such asPSK31orRTTY.[citation needed]
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https://en.wikipedia.org/wiki/R-S-T_system
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TheAllied military phonetic spelling alphabetsprescribed the words that are used to represent each letter of the alphabet, when spelling other words out loud, letter-by-letter, and how the spelling words should be pronounced for use by theAllies of World War II. They are not a "phonetic alphabet" in the sense in which that term is used inphonetics, i.e. they are not a system for transcribing speech sounds.
The Allied militaries – primarily the US and the UK – had their own radiotelephone spelling alphabets which had origins back toWorld War Iand had evolved separately in the different services in the two countries. For communication between the different countries and different services specific alphabets were mandated.
The last WWII spelling alphabet continued to be used through theKorean War, being replaced in 1956 as a result of both countries adopting theICAO/ITURadiotelephony Spelling Alphabet, with theNATO memberscalling their usage the "NATO Phonetic Alphabet".
During WWII, theAllieshad defined terminology to describe the scope of communications procedures among different services and nations. A summary of the terms used was published in a post-WWIINATOmemo:[1]
Thus, theCombined Communications Board(CCB), created in 1941, derived a spelling alphabet that was mandated for use when any US military branch was communicating with any British military branch; when operating without any British forces, the Joint Army/Navy spelling alphabet was mandated for use whenever the US Army and US Navy were communicating in joint operations; if the US Army was operating on its own, it would use its own spelling alphabet, in which some of the letters were identical to the other spelling alphabets and some completely different.
The US and UK began to coordinate calling alphabets by the military during World War II and by 1943 they had settled on a streamline communications that became known as the CCB. Both nations had previous independently developed alphabet naming system dating back to World War I. Subsequently, this second world war era letter naming became accepted as standard by the ICAO in 1947.
After the creation of NATO in 1949, modifications began to take place. An alternative name for the ICAO spelling alphabet, "NATO phonetic alphabet", exists because it appears in Allied Tactical Publication ATP-1, Volume II:Allied Maritime Signal and Maneuvering Bookused by all navies of NATO, which adopted a modified form of theInternational Code of Signals. Because the latter allows messages to be spelled via flags orMorse code, it naturally named the code words used to spell out messages byvoiceits "phonetic alphabet". The nameNATO phonetic alphabetbecame widespread because the signals used to facilitate the naval communications and tactics of NATO have become global.[2]However, ATP-1 is markedNATO Confidential(or the lowerNATO Restricted) so it is not available publicly. Nevertheless, a NATO unclassified version of the document is provided to foreign, even hostile, militaries, even though they are not allowed to make it available publicly. The spelling alphabet is now also defined in other unclassified international military documents.[3]The NATO alphabet appeared in some United States Air Force Europe publications during the Cold War. A particular example was the Ramstein Air Base Telephone Directory, published between 1969 and 1973 (currently out of print). The US and NATO versions had differences, and the translation was provided as a convenience. Differences included Alfa, Bravo and Able, Baker for the first two letters.
The NATO phonetic spelling alphabet was first adopted on January 1, 1956, while the ICAO radiotelephony spelling alphabet was still undergoing final changes.[4]
TheRAF radiotelephony spelling alphabet, sometimes referred to as the "RAF Phonetic Alphabet", was used by the BritishRoyal Air Force(RAF) to aid communication after the take-up of radio, especially to spell out aircraft identification letters, e.g. "H for Harry", "G for George", etc. Several alphabets were used, before being superseded by the adoption of the NATO/ICAO radiotelephony alphabet.
DuringWorld War Ibattle lines were often static and forces were commonly linked by wired telephone networks. Signals were weak on long wire runs andfield telephonesystems often used a single wire withearth return, which made them subject to inadvertent and deliberate interference. Spelling alphabets were introduced for wire telephony as well as on the newer radio voice equipment.[14]
The British Army and the Royal Navy had developed their own quite separate spelling alphabets. The Navy system was a full alphabet, starting:Apples, Butter, Charlie, Duff, Edward, but the RAF alphabet was based on that of the "signalese" of the army signallers. This was not a full alphabet, but differentiated only the letters most frequently misunderstood:Ack (originally "Ak"), Beer (or Bar), C, D, E, F, G, H, I, J, K, L, eMma, N, O, Pip, Q, R, eSses, Toc, U, Vic, W, X, Y, Z.
By 1921, the RAF "Telephony Spelling Alphabet" had been adopted by all three armed services, and was then made mandatory for UK civil aviation, as announced inNotice to AirmenNumber 107.[15]
In 1956, theNATO phonetic alphabetwas adopted due to the RAF's wide commitments with NATO and worldwide sharing of civil aviation facilities.[16]
aThe choice of Nuts following Monkey is probably[citation needed]from "monkey nuts" (peanuts); likewise Orange and Pip can be similarly paired, as in "orange pip".b"Vic" subsequently entered the English language as the standard"Vee"-shaped flight patternof three aircraft.
1916–1939
1939–1941
1941–1943
1943–1955
1956–present[9]
†'Interrogatory' was used in place of 'Inter' in joint Army/Navy Operations.
The US Navy's first phonetic spelling alphabet was not used for radio, but was instead used on the deck of ships "in calling out flags to be hoisted in a signal". There were two alternative alphabets used, which were almost completely different from each other, with only the code word "Xray" in common.[22]
The US Navy's firstradiotelephonyphonetic spelling alphabet was published in 1913, in the Naval Radio Service's Handbook of Regulations developed by CaptainWilliam H. G. Bullard. The Handbook's procedures were described in the November 1917 edition ofPopular Science Monthly.[23]
1956–present[9]
The Joint Army/Navy (JAN) spelling alphabet was developed by the Joint Board on November 13, 1940, and it took effect on March 1, 1941.[28][29]It was reformulated by theCCBfollowing the entrance of the US intoWorld War IIby the CCB "Methods and Procedures" committee,[29]and was used by all branches of theUnited States Armed Forcesuntil the promulgation of its replacement, theICAO spelling alphabet(Alfa, Bravo, etc.), in 1956. Before the JAN phonetic alphabet, each branch of the armed forces had used its own radio alphabet, leading to difficulties in interbranch communication.
TheUS Armyused this alphabet in modified form, along with theBritish ArmyandCanadian Armyfrom 1943 onward, with "Sugar" replacing "Sail".
The JAN spelling alphabet was used to name Atlantic basin storms duringhurricane seasonfrom1947to1952, before being replaced with a new system of using female names.
Vestiges of the JAN spelling system remain in use in theUS Navy, in the form ofMaterial Conditions of Readiness, used in damage control. Dog, William, X-Ray, Yoke, and Zebra all reference designations of fittings, hatches, or doors.[30]The response "Roger" for "· – ·" or "R", to mean "received", also derives from this alphabet.
The namesAbletoFoxwere also widely used in the early days ofhexadecimaldigital encoding of text, for speaking the hexadecimal digits A to F (equivalent to decimal 10 to 15), although the written form was simply the capital letters A to F.
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https://en.wikipedia.org/wiki/Allied_military_phonetic_spelling_alphabets
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TheAPCO phonetic alphabet, a.k.a.LAPD radio alphabet,is the term for an old competingspelling alphabetto theICAO radiotelephony alphabet, defined by theAssociation of Public-Safety Communications Officials-International[1]from 1941 to 1974, that is used by theLos Angeles Police Department(LAPD) and other local and statelaw enforcement agenciesacross the state ofCaliforniaand elsewhere in the United States. It is the "over the air" communication used for properly understanding a broadcast of letters in the form of easily understood words. Despite often being called a "phonetic alphabet", it is not aphonetic alphabetfor transcribing phonetics.
In 1974, APCO adopted the ICAO Radiotelephony Spelling Alphabet, making the APCO alphabet officially obsolete; however, it is still widely used, and relatively few police departments in the U.S. use the ICAO alphabet.[citation needed]
The APCO first suggested that its Procedure and Signals Committee work out a system for a "standard set of words representing the alphabet should be used by all stations" in its April 1940 newsletter.[2][3]By this point, APCO President Herb Wareing "came out in favor of a standard list of words for alphabet letters, preferably suitable for both radiophone and radiotelegraph use."[4]
The list was based on the results of questionnaires sent out by the Procedures Committee to all zone and interzone police radio stations. The questionnaire solicited suggestions, but also included the existing Western Union and Bell Telephone word lists, plus another list then in general use by a number of police stations. Lists used by military services were excluded because of a lack of permission to reproduce. The resulting final list differs from the Bell Telephone word list by only five words, and from the Western Union word list by only eight words.[5]
In 1974, APCO adopted the ICAO International Radiotelephony Spelling Alphabet,[6]replacing the Adam-Boy-Charlie alphabet APCO first published in 1940. However, most police departments nationwide have kept using the 1940 APCO spelling alphabet, with those using the 1974 APCO spelling alphabet being the exception, rather than the rule. A partial list of police departments using the modern APCO/ICAO spelling alphabet includes:
At some point in the early history of emergency service mobile radio systems,[when?]theLAPDadopted the APCOradio spelling alphabetfor relaying precise information on individual letters. For example, the license plate "8QXG518" might be read by a civilian as "eight cue ex gee five eighteen" but with accuracy being paramount, the police dispatcher would say "eight queen x-ray george five one eight." Despite the development in 1941 of theJoint Army/Navy Phonetic Alphabetand its replacement, circa 1956, by theNATO phonetic alphabet(currently used by all NATO armed forces, civil aviation,amateur radio, telecommunications, and some law enforcement agencies), the LAPD and other law enforcement and emergency service agencies throughout theUnited Statescontinue to use their traditional system.[citation needed]
APCO's Project 14 updated the definition ofTen-codes, and also adopted the international radiotelephony spelling alphabet for use by law enforcement nationwide.[8]
The APCO radiotelephony spelling alphabet and its variations represent the letters of the English alphabet using words as follows:
1967[9]
[16][17]
There are several local variations of this system in use. TheMetropolitan Police Department (Washington DC), uses the APCO alphabet,[19]however theCalifornia Highway Patrol,Las Vegas Metropolitan Police Department,Los Angeles County Sheriff's Department,[citation needed]San Jose Police Department,[citation needed]San Francisco Police Department,[citation needed]and other agencies across theWest CoastandSouthwestern United States,[citation needed]use versions that allocateYellowto "Y" and other agencies' versions allocateBakerorBravoto "B", or use variations that includeNancyinstead ofNorafor "N",Easyinstead ofEdwardfor "E", orYesterdayfor "Y".
With the ultimate goal of clarity, especially in circumstances where signals can be garbled, the use of the wordOceanseems to be advantageous in the radio communication of the letter "O" because it begins with the long, clear vowel "O". The phonetic wordsIdaandUnionfeature this same advantage. However, spelling alphabets seem to rarely use initial long vowels. With the exception ofUniform, none of the initial vowels in the NATO alphabet is like this. In an earlier U.S. military alphabet, "A" was indicated byAble, which does start with a long "A", but has since been changed toAlpha(also spelledAlfa, particularly outside the English-speaking countries). In like manner, for clarity, the use of "niner" instead of "nine" for the numeral 9 prevents confusion with the numeral 5, which can sound similar, especially when communications are garbled.[citation needed]
The origin of the nameAdam-12from thetelevisionseries of the same title comes from this alphabet. The LAPD still calls its basic two-man patrol car an "A" unit, and the letter "A" is spoken as "Adam" in the spelling alphabet. The entire callsign "1-Adam-12" translates to [Division] One (LAPD Central Division) Two Man Patrol Car (Adam unit) in patrol car 12. The 12 refers to what is called "The Basic Car Plan". That is, the patrol area within the precinct. Specialized units use the last numbers as designating the officers. An example would be 6U2, Hollywood Division report writing unit. The patrol car, in LAPD jargon, is called a "black-and-white", owing to the colors. The number that is on the car is called the shop number and is only used for identifying the vehicle.
In the American television seriesCHiPsfrom 1977 to 1983, motorcycle units are identified with the letter "M", such as 7M4 (Seven Mary Four) for Officer Frank Poncherello (portrayed by Erik Estrada). His partner, Officer Jon Baker (portrayed by actor Larry Wilcox), is identified as 7M3 (Seven Mary Three). In these callsigns, "7" designates the patrol beat, "M" designates a motorcycle unit, and "3" is the unit number.
Hunterfrom 1984 to 1991 had actorFred Dryeras Rick Hunter identify as "1 William 1 Paul 156" as his call sign where W is "William" and P is "Paul" when he was with the LAPD.
Also, since many police, fire department, and rescue squad TV programs and movies are set inLos Angeles, the words of the LAPD phonetic alphabet have become familiar in the United States,Canadaand English-speaking countries around the world[citation needed]due to the wide reach of American entertainment media. When used by workers such as telephone operators speaking to "civilians" who may be unfamiliar with the use of a phonetic alphabet, both the everyday letter and its phonetic alphabet equivalent are spoken, such as "B as in boy", "V as in Victor", etc.
On early seasons ofWheel of Fortune, a close variant of the LAPD phonetic alphabet was used. Players would be encouraged to say things like "I'll have B as in boy" when choosing letters.
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TheInternational Code of Signals(INTERCO) is an international system of signals and codes for use byvesselsto communicate important messages regarding safety of navigation and related matters. Signals can be sent byflaghoist,signal lamp("blinker"),flag semaphore, radiotelegraphy, and radiotelephony. The International Code is the most recent evolution of a wide variety ofmaritime flag signallingsystems.
The International Code of Signals was preceded by a variety of naval signals and private signals, most notablyMarryat's Code, the most widely used code flags prior to 1857. What is now the International Code of Signals was drafted in 1855 by the BritishBoard of Tradeand published in 1857 as the Commercial Code. It came in two parts: the first containing universal and international signals, and the second British signals only. Eighteen separate signal flags (see chart) were used to make over 70,000 possible messages. Vowels were omitted from the set to avoid spelling out any word that might be objectionable in any language, and some little-used letters were also omitted. It was revised by the Board of Trade in 1887, and was modified at the International Conference of 1889 in Washington, D.C.[1]The new international code of signals officially came into worldwide operation on 1 January 1901. At first it was used concurrently with the old system until 1 January 1902, and then used exclusively after 1 January 1903. In this new edition, the number of flags was increased from 18 flags plus a code pennant to 26 flags and a code pennant. The eight new flags represented the vowels A E I O U and the letters X Y Z.[2]
A slightly different version was published in Brown's Signalling, 18th Edition, February, 1916, pages 9-28. Charlie, Delta, Echo, Foxtrot and Golf were pennants corresponding to more modern numeral pennants 1, 2, 3, 4 and 5. Otherwise the letters appear to correspond to the more modern formats.[3]
The code was severely tested during World War I, and it was found that, "when coding signals, word by word, the occasions upon which signaling failed were more numerous than those when the result was successful."[4]A 1920 meeting of the five Principal Allied and Associated Powers met in Paris and proposed forming the Universal Electrical Communications Union on October 8, 1920 in Washington, D.C.[5]The group suggested revisions to the International Code of Signals, and adopted a phonetic spelling alphabet, but the creation of the organization was not agreed upon.
The 1927 International Radiotelegraph Conference in Washington[6]considered proposals for a new revision of the Code, including preparation in seven languages: English, French, Italian, German, Japanese, Spanish, and Norwegian. This new edition was completed in 1930 and was adopted by the International Radiotelegraph Conference held in Madrid in 1932.[7]The Madrid Conference also set up a standing committee for continual revision of the code. The new version introduced vocabulary for aviation and a complete medical section with the assistance and by the advice of theOffice International d'Hygiène Publique. A certain number of signals were also inserted for communications between vessels and shipowners, agents, repair yards, and other maritime stakeholders. The new international code of signals was officially brought into force worldwide on 1 January 1934. Thirteen new flags were introduced, whereby the triangular pennants used for letters, C, D, E, F, and G were replaced with new square flags, and became the numerals 1, 2, 3, 4, and 5. The numerals 6, 7, 8, 9, and 0 were introduced by five new flags, and there were three new substitute flags added.[8]
After World War II, the 1947 International Radio Conference of theInternational Telecommunication Unionsuggested in that the International Code of Signals should fall within the competence of the Inter-Governmental Maritime Consultative Organization (IMCO), which became the IMO.[9]In January 1959, the First Assembly of IMCO decided that the organization should assume all the functions then being performed by the Standing Committee of the International Code of Signals.
The Second Assembly of IMCO 1961 endorsed plans for a comprehensive review of the International Code of Signals to meet the needs of mariners. The revisions were prepared in the previous seven languages plus Russian and Greek.
The code was revised in 1964 taking into account recommendations from the 1960Conference on Safety of Life at Sea(SOLAS) and the 1959 Administrative Radio Conference.[10]Changes included a shift in focus from general communications to safety of navigation, abandonment of the "vocabulary" method of spelling out messages word by word, adaptation to all forms of communication, and elimination of the separate radiotelegraph and geographical sections. It was adopted in 1965. The 1969 English-language version of the code (United States edition, revised 2020) is available online through theNational Geospatial-Intelligence Agency(NGA, formerly the National Imagery and Mapping Agency) and can be foundhere.
The International Code of Signals is currently maintained by theInternational Maritime Organization(IMO), which published an edition in 2005.[11]
"The purpose of the International Code of Signals is to provide ways and means of communication in situations related essentially to safety of navigation and persons, especially when language difficulties arise."[12]It has done this by first establishing a standardized alphabet (the letters A to Z and the ten digits), along with a spoken form of each letter (to avoid confusing similar-sounding letters, such asb,p, andv), and associating this alphabet with standardized flags. (See chart to the right.)
Combinations of these alphanumeric characters are assigned as codes for various standardized messages. For instance, the master of a ship may wish to communicate with another ship, where their own radio may not be working or the other ship's call sign is not known or the other ship may not be maintaining a radio watch. One simply raises the Kilo flag (see diagram at the top), or sends the Morse Code equivalent (dash-dot-dash) by flashing light; this has the assigned message of "I wish to communicate with you."
One practical benefit of using the ICS is that all of the standardized messages come in nine languages (English, French, Italian, German, Japanese, Spanish, Norwegian, and, since 1969, Russian and Greek). This solves the problems which may potentially arise when the sender and the receiver(s) of a message are fluent only in differing languages; each language has a book with equivalent messages keyed to the same code. This is also useful in radiotelephony, or even when ships are within hailing distance, if there is no common language. A crew member on a burning ship could yell, "Yuliet alfa vore", to a ship which has come to offer aid, in order to communicate exactly what the distressed ship needs—in this case, "material [foaming agent] for use in foam fire extinguishers". (Seede:Flaggenalphabetfor the German version of single-letter signals.)
The code also covers procedural aspects (how to initiate a call, the format of a message, how to format date and time, etc.), how naval ships (which usually use their own codes) indicate that they are using the ICS (by flying the code pennant), use in radiotelephony (use of the spoken word "Interco"), and various other matters (such as how an aircraft directs a vessel to another vessel in distress and how to order unidentified submarines to surface).
Prior to 1969, the code was much more extensive, covering a wider range of messages and including a list of five-letter codes for every prominent maritime location in the world. Since 1969, it has been reduced to focus on navigation and safety, including a medical section. Signals can be sorted into three groups:
In some cases, additional characters are added to indicate quantities, bearing, course, distance, date, time, latitude, or longitude. There is also provision for spelling words and for indicating use of other codes. Several of the most common single-letter signals are shown at the right. Two-letter signals cover a broad gamut of situations.
Repeated characters can be a problem in flaghoist. To avoid having to carry multiple sets of signal flags, the Code uses three "substitute" (or "repeater") flags. These repeat the flag at the indicated position. For instance, to signal MAA ("I request urgent medical advice") the Mike, Alfa, and 2nd substitute flags would be flown, the substitute indicating a repeat of the second character.
The Medical Signal Code[13](incorporated in the International Code of Signals since 1930) is a means of providing assistance when medical personnel are not present. Plain language is generally preferred in such cases (presumably via radiotelephone), but the various codes provide a succinct method of communicating to a doctor the nature of the problem where there are language or communication difficulties, and in return the recommended treatment. Even where there are no language problems, the Medical Signal Code is useful in providing a standard method of case description and treatment. There is also a standard list of medicaments (medicines), keyed to a standard ships medicine chest carried by all merchant ships. The Medical signals all begin with the letter "M" (Mike) followed by two more letters, and sometimes with additional numerals or letters.
Prior to 1969:"The way is off my ship; you may feel your way past me."
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TheFinnish Defence Forcesswitched over to theNATO phonetic alphabetin 2005, but the Finnish one is used for Å, Ä, Ö and digits.[1]International operations use only the NATO alphabet.
On the Finnish rail network the Finnish Armed Forces spelling alphabet was used until May 31, 2020 and starting on July 1 the railways switched to NATO phonetic alphabet, but still retained Finnish spelling words for Å, Ä, Ö and numbers.[2]
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https://en.wikipedia.org/wiki/Finnish_Armed_Forces_radio_alphabet
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German orthographyis theorthographyused inwritingtheGerman language, which is largelyphonemic. However, it shows many instances of spellings that are historic or analogous to other spellings rather than phonemic. The pronunciation of almost every word can be derived from its spelling once the spelling rules are known, but the opposite is not generally the case.
Today,Standard High Germanorthography is regulated by theRat für deutsche Rechtschreibung(Council for German Orthography), composed of representatives from mostGerman-speaking countries.
The modern Germanalphabetconsists of the twenty-sixlettersof theISO basic Latin alphabetplus four special letters.
German has four special letters; three arevowelsaccentedwith anumlaut sign(⟨ä,ö,ü⟩) and one is derived from aligatureof⟨ſ⟩(long s) and⟨z⟩(⟨ß⟩; calledEszett"ess-zed/zee" orscharfes S"sharp s"). They have their own names separate from the letters they are based on.
While the Council for German Orthography considers⟨ä, ö, ü, ß⟩distinct letters,[4]disagreement on how to categorize and count them has led to a dispute over the exact number of letters the German alphabet has, the number ranging between 26 (considering special letters as variants of⟨a, o, u, s⟩) and 30 (counting all special letters separately).[5]
Theaccentedletters⟨ä,ö,ü⟩are used to indicate the presence ofumlauts(frontingof back vowels). Before the introduction of theprinting press, frontalization was indicated by placing an⟨e⟩after the back vowel to be modified, but German printers developed the space-saving typographical convention of replacing the full⟨e⟩with a small version placed above the vowel to be modified. In GermanKurrentwriting, the superscripted⟨e⟩was simplified to two vertical dashes (as the Kurrent⟨e⟩consists largely of two short vertical strokes), which have further been reduced to dots in both handwriting and German typesetting. Although the two dots of umlaut look like those in thediaeresis(trema), the two have different origins and functions.
When it is not possible to use the umlauts (for example, when using a restricted character set) the characters⟨Ä, Ö, Ü, ä, ö, ü⟩should be transcribed as⟨Ae, Oe, Ue, ae, oe, ue⟩respectively, following the earlier postvocalic-⟨e⟩convention; simply using the base vowel (e.g.⟨u⟩instead of⟨ü⟩) would be wrong and misleading. However, such transcription should be avoided if possible, especially with names. Names often exist in different variants, such asMüllerandMueller, and with such transcriptions in use one could not work out the correct spelling of the name.
Automatic back-transcribing is wrong not only for names. Consider, for example,das neue Buch("the new book"). This should never be changed todas neü Buch, as the second⟨e⟩is completely separate from the⟨u⟩and does not even belong in the same syllable;neue([ˈnɔʏ.ə]) isneu(the root for "new") followed by⟨e⟩, an inflection. The word⟨neü⟩does not exist in German.
Furthermore, in northern and western Germany, there are family names and place names in which⟨e⟩lengthens the preceding vowel (by acting as aDehnungs-e), as in the former Dutch orthography, such asStraelen, which is pronounced with a long⟨a⟩, not an⟨ä⟩. Similar cases areCoesfeldandBernkastel-Kues.
In proper names and ethnonyms, there may also appear a rare⟨ë⟩and⟨ï⟩, which are not letters with an umlaut, but adiaeresis, used as in French and English to distinguish what could be adigraph, for example,⟨ai⟩inKaraïmen,⟨eu⟩inAlëuten,⟨ie⟩inPiëch,⟨oe⟩invon LoëandHoëcker(although Hoëcker added the diaeresis himself), and⟨ue⟩inNiuë.[6]Occasionally, a diaeresis may be used in some well-known names, i.e.:Italiën[7](usually written asItalien).
Swiss keyboardsand typewriters do not allow easy input of uppercase letters with umlauts (nor⟨ß⟩) because their positions are taken by the most frequent French diacritics. Uppercase umlauts were dropped because they are less common than lowercase ones (especially in Switzerland). Geographical names in particular are supposed to be written with⟨a, o, u⟩plus⟨e⟩, exceptÖsterreich. The omission can cause some inconvenience, since the first letter of everynounis capitalized in German.
Unlike inHungarian, the exact shape of the umlaut diacritics – especially when handwritten – is not important, because they are the only ones in the language (not counting thetittleon⟨i⟩and⟨j⟩). They will be understood whether they look like dots (⟨¨⟩),acute accents(⟨ ˝ ⟩) orvertical bars(⟨‖⟩). A horizontal bar (macron,⟨¯⟩), abreve(⟨˘⟩), a tiny⟨N⟩or⟨e⟩, atilde(⟨˜⟩), and such variations are often used in stylized writing (e.g. logos). However, the breve – or thering(⟨°⟩) – was traditionally used in some scripts to distinguish a⟨u⟩from an⟨n⟩. In rare cases, the⟨n⟩was underlined. The breved⟨u⟩was common in someKurrent-derived handwritings; it was mandatory inSütterlin.
Eszettorscharfes S(⟨ß⟩) represents the“s”sound. In the current orthography, the letter is used only after long vowels and diphthongs. Prior to theGerman spelling reform of 1996, it was used additionally whenever the letter combination⟨ss⟩occurred at the end of a syllable or word. It is not used inSwitzerlandandLiechtenstein.
As⟨ß⟩derives from a ligature of lowercase letters, it is exclusively used in the middle or at the end of a word. The proper transcription when it cannot be used is⟨ss⟩(⟨sz⟩and⟨SZ⟩in earlier times). This transcription can give rise to ambiguities, albeit rarely; one such case isin Maßen"in moderation" vs.in Massen"en masse". In all-caps,⟨ß⟩is replaced by⟨SS⟩or, optionally, by theuppercase⟨ß⟩.[8]The uppercase⟨ß⟩was included inUnicode 5.1as U+1E9E in 2008. Since 2010 its use is mandatory in official documentation in Germany when writing geographical names in all-caps.[9]The option of using the uppercase⟨ẞ⟩in all-caps was officially added to the German orthography in 2017.[10]
There are three ways to deal with the umlauts inalphabetic sorting.
Microsoft Windowsin German versions offers the choice between the first two variants in its internationalization settings.
A sort of combination of nos. 1 and 2 also exists, in use in a couple of lexica: The umlaut is sorted with the base character, but an⟨ae, oe, ue⟩in proper names is sorted with the umlaut if it is actually spoken that way (with the umlaut getting immediate precedence). A possible sequence of names then would beMukovic; Muller; Müller; Mueller; Multmannin this order.
Eszettis sorted as though it were⟨ss⟩. Occasionally it is treated as⟨s⟩, but this is generally considered incorrect. Words distinguished only by⟨ß⟩vs.⟨ss⟩are rare. The word with⟨ß⟩gets precedence, andGeschoß(story of a building; South German pronunciation) would be sorted beforeGeschoss(projectile).[citation needed]
Accents in Frenchloanwordsare always ignored in collation.
In rare contexts (e.g. in older indices)⟨sch⟩(phonetic value equal to English⟨sh⟩) and likewise⟨st⟩and⟨ch⟩are treated as single letters, but the vocalicdigraphs⟨ai, ei⟩(historically⟨ay, ey⟩),⟨au, äu, eu⟩and the historic⟨ui, oi⟩never are.
German names containing umlauts (⟨ä, ö, ü⟩) and/or⟨ß⟩are spelled in the correct way in the non-machine-readable zone of the passport, but with⟨AE, OE, UE⟩and/or⟨SS⟩in themachine-readable zone, e.g.⟨Müller⟩becomes⟨MUELLER⟩,⟨Weiß⟩becomes⟨WEISS⟩, and⟨Gößmann⟩becomes⟨GOESSMANN⟩. The transcription mentioned above is generally used for aircraft tickets et cetera, but sometimes (like in US visas) simple vowels are used (MULLER, GOSSMANN). As a result, passport, visa, and aircraft ticket may display different spellings of the same name. The three possible spelling variants of the same name (e.g.Müller/Mueller/Muller) in different documents sometimes lead to confusion, and the use of two different spellings within the same document may give persons unfamiliar with German orthography the impression that the document is a forgery.
Even before the introduction of the capital⟨ẞ⟩, it was recommended to use the minuscule⟨ß⟩as a capital letter in family names in documents (e.g.HEINZ GROßE, today's spelling:HEINZ GROẞE).
German naming law accepts umlauts and/or⟨ß⟩in family names as a reason for an official name change. Even a spelling change, e.g. fromMüllertoMuelleror fromWeißtoWeissis regarded as a name change.
A typical feature of German spelling is the generalcapitalizationof nouns and of mostnominalizedwords. In addition, capital letters are used: at the beginning of sentences (may be used after a colon, when the part of a sentence after the colon can be treated as a sentence); in the formal pronounSie'you' and the determinerIhr'your' (optionally in other second-person pronouns in letters); in adjectives at the beginning of proper names (e.g.der Stille Ozean'the Pacific Ocean'); in adjectives with the suffix '-er' from geographical names (e.g.Berliner); in adjectives with the suffix '-sch' from proper names if written with the apostrophe before the suffix (e.g.Ohm'sches Gesetz'Ohm's law', also writtenohmsches Gesetz).
Compound words, including nouns, are usually written together, e.g.Haustür(Haus+Tür; 'house door'),Tischlampe(Tisch+Lampe; 'table lamp'),Kaltwasserhahn(Kalt+Wasser+Hahn; 'cold water tap/faucet). This can lead to long words: the longest word in regular use,Rechtsschutzversicherungsgesellschaften[11]('legal protection insurance companies'), consists of 39 letters.
Compounds involving letters, abbreviations, or numbers (written in figures, even with added suffixes) are hyphenated:A-Dur'A major',US-Botschaft'US embassy',10-prozentig'with 10 percent',10er-Gruppe'group of ten'. The hyphen is used when adding suffixes to letters:n-te'nth'. It is used in substantivated compounds such asEntweder-oder'alternative' (literally 'either-or'); in phrase-word compounds such asTag-und-Nacht-Gleiche'equinox',Auf-die-lange-Bank-Schieben'postponing' (substantivation ofauf die lange Bank schieben'to postpone'); in compounds of words containing hyphen with other words:A-Dur-Tonleiter'A major scale'; in coordinated adjectives:deutsch-englisches Wörterbuch'German-English dictionary'. Compound adjectives meaning colours are written with a hyphen if they mean two colours:rot-braun'red and brown', but without a hyphen if they mean an intermediate colour:rotbraun'reddish brown' (from the spelling reform of 1996 to the 2024 revision of the orthographic rules, both variants could be used in both meanings). Optionally the hyphen can be used to emphasize individual components, to clarify the meaning of complicated compounds, to avoid misunderstandings or when three identical letters occur together (in practice, in this case it is mostly used when writing nouns with triple vowels, e.g.See-Elefant'elephant seal').
The hyphen is used in compounds where the second part or both parts are proper names, e.g.Foto-Hansen'the photographer Hansen',Müller-Lüdenscheid'Lüdenscheid, the city of millers', double-barrelled surnames such asMeyer-Schmidt; geographical names such asBaden-Württemberg. Double given names are variously written asAnna-Maria, Anna Maria, Annamaria. Some compound geographical names are written as one word (e.g.Nordkorea'North Korea') or as two words (e.g. geographical names beginning withSanktorBad). The hyphen is not used when compounds with a proper name in the second part are used as common nouns, e.g.Heulsuse'crybaby'; also in the name of the fountainGänseliesel. The hyphen is used in words derived from proper names with hyphen, from proper names of more than one word, or from more than one proper name (optional in derivations with the suffix-erfrom geographical names from more than one word). Optionally the hyphen can be used in compounds where the first part is a proper name. Compounds of the type "geographical name+specification" are written with a hyphen or as two words:München-OstorMünchen Ost.
Even thoughvowel lengthisphonemicin German, it is not consistently represented. However, there are different ways of identifying long vowels:
Even though German does not have phonemicconsonant length, there are many instances of doubled or even tripled consonants in the spelling. A single consonant following achecked vowelis doubled if another vowel follows, for instanceimmer'always',lassen'let'. These consonants are analyzed asambisyllabicbecause they constitute not only thesyllable onsetof the second syllable but also thesyllable codaof the first syllable, which must not be empty because thesyllable nucleusis a checked vowel.
By analogy, if a word has one form with a doubled consonant, all forms of that word are written with a doubled consonant, even if they do not fulfill the conditions for consonant doubling; for instance,rennen'to run' →er rennt'he runs';Küsse'kisses' →Kuss'kiss'.
Doubled consonants can occur in composite words when the first part ends in the same consonant the second part starts with, e.g. in the wordSchaffell('sheepskin', composed ofSchaf'sheep' andFell'skin, fur, pelt').
Composite words can also have tripled letters. While this is usually a sign that the consonant is actually spoken long, it does not affect the pronunciation per se: the⟨fff⟩inSauerstoffflasche('oxygen bottle', composed ofSauerstoff'oxygen' andFlasche'bottle') is exactly as long as the ff inSchaffell. According to the spelling before 1996, the three consonants would be shortened before vowels, but retained before consonants and in hyphenation, so the wordSchifffahrt('navigation, shipping', composed ofSchiff'ship' andFahrt'drive, trip, tour') was then writtenSchiffahrt, whereasSauerstoffflaschealready had a triple⟨fff⟩. With the aforementioned change in⟨ß⟩spelling, even a new source of triple consonants⟨sss⟩, which in pre-1996 spelling could not occur as it was rendered⟨ßs⟩, was introduced, e.g.Mussspiel('compulsory round' in certain card games, composed ofmuss'must' andSpiel'game').
For technical terms, the foreign spelling is often retained such as⟨ph⟩/f/or⟨y⟩/yː/in the wordPhysik(physics) of Greek origin. For some common affixes however, like-graphieorPhoto-, it is allowed to use-grafieorFoto-instead.[12]BothPhotographieandFotografieare correct, but the mixed variants*Fotographieor*Photografieare not.[12]
For other foreign words, both the foreign spelling and a revised German spelling are correct such asDelphin/Delfin[13]orPortemonnaie/Portmonee, though in the latter case the revised one does not usually occur.[14]
For some words for which the Germanized form was common even before the reform of 1996, the foreign version is no longer allowed. A notable example is the wordFoto"photograph", which may no longer be spelled asPhoto.[15]Other examples areTelephon(telephone) which was already Germanized asTelefonsome decades ago orBureau(office) which got replaced by the Germanized versionBüroeven earlier.
Except for the common sequencessch(/ʃ/),ch([x]or[ç]) andck(/k/), the letter⟨c⟩appears only inloanwordsor inproper nouns. In many loanwords, including most words of Latin origin, the letter⟨c⟩pronounced (/k/) has been replaced by⟨k⟩. Alternatively, German words which come from Latin words with⟨c⟩before⟨e, i, y, ae, oe⟩are usually pronounced with (/ts/) and spelled with⟨z⟩. However, certain older spellings occasionally remain, mostly for decorative reasons, such asCircusinstead ofZirkus.
The letter⟨q⟩in German appears only in the sequence⟨qu⟩(/kv/) except for loanwords such asCoq au vinorQigong(the latter is also writtenChigong).
The letter⟨x⟩(Ix,/ɪks/) occurs almost exclusively in loanwords such asXylofon(xylophone) and names, e.g.AlexanderandXanthippe. Native German words now pronounced with a/ks/sound are usually written using⟨chs⟩or⟨(c)ks⟩, as withFuchs(fox). Some exceptions occur such asHexe(witch),Nixe(mermaid),Axt(axe) andXanten.
The letter⟨y⟩(Ypsilon,/ˈʏpsilɔn/) occurs almost exclusively in loanwords, especially words of Greek origin, but some such words (such asTyp) have become so common that they are no longer perceived as foreign. It used to be more common in earlier centuries, and traces of this earlier usage persist in proper names. It is used either as an alternative letter for⟨i⟩, for instance inMayer/Meyer(a commonfamily namethat occurs also in the spellingsMaier/Meier), or especially in the Southwest, as a representation of[iː]that goes back to an oldIJ (digraph), for instance inSchwyzorSchnyder(anAlemannicvariant of the nameSchneider).[citation needed]Another notable exception isBayern("Bavaria") and derived words likebayrisch("Bavarian"); this actually used to be spelt with an⟨i⟩until the King of Bavaria introduced the⟨y⟩as a sign of hisphilhellenism(his son would become King of Greece later).
The Latin and Ancient Greek diphthongs⟨ae (αι)⟩and⟨oe (οι)⟩are normally rendered as⟨ä⟩and⟨ö⟩in German, whereas English usually uses a simple⟨e⟩(but seeList of English words that may be spelled with a ligature):Präsens'present tense' (Latintempus praesens),Föderation'federation' (Latinfoederatio).
The etymological spelling⟨-ti-⟩for the sounds[tsɪ̯]before vowels is used in many words of Latin origin, mostly ending in⟨-tion⟩, but also⟨-tiell, -tiös⟩, etc. Latin⟨-tia⟩in feminine nouns is typically simplified to⟨-z⟩in German; in related words, both⟨-ti-⟩and⟨-zi-⟩are allowed:Potenz'power' (from Latinpotentia),Potential/Potenzial'potential' (noun),potentiell/potenziell'potential' (adj.). Latin⟨-tia⟩in neuter plural nouns may be retained, but is also Germanized orthographically and morphologically to⟨-zien⟩:Ingrediens'ingredient', pluralIngredienzien;Solvens'expectorant', pluralSolventiaorSolvenzien.
In loan words from theFrench language, spelling and accents are usually preserved. For instance,caféin the sense of "coffeehouse" is always writtenCaféin German; accentlessCafewould be considered erroneous, and the word cannot be writtenKaffee, which means "coffee". (Caféis normally pronounced/kaˈfeː/;Kaffeeis mostly pronounced/ˈkafe/in Germany but/kaˈfeː/in Austria.) Thus, Germantypewritersand computer keyboards offer twodead keys: one for theacuteandgrave accentsand one forcircumflex. Other letters occur less often such as⟨ç⟩in loan words from French or Portuguese, and⟨ñ⟩in loan words from Spanish.
A number of loanwords from French are spelled in a partially adapted way:Quarantäne/kaʁanˈtɛːnə/(quarantine),Kommuniqué/kɔmyniˈkeː,kɔmuniˈkeː/(communiqué),Ouvertüre/u.vɛʁˈtyː.ʁə/(overture) from Frenchquarantaine, communiqué, ouverture. In Switzerland, where French is one of the official languages, people are less prone to use adapted and especially partially adapted spellings of loanwords from French and more often use original spellings, e.g.Communiqué.
In one curious instance, the wordSki('ski') is pronounced as if it were*Schiall over the German-speaking areas (reflecting its pronunciation in its source languageNorwegian), but only written that way in Austria.[16]
This section lists German letters and letter combinations, and how to pronounce them transliterated into theInternational Phonetic Alphabet. This is the pronunciation ofStandard German. Note that the pronunciation of standard German varies slightly from region to region. In fact, it is possible to tell where most German speakers come from by their accent in standard German (not to be confused with the differentGerman dialects).
Foreign words are usually pronounced approximately as they are in the original language.
Doubleconsonantsare pronounced as single consonants, except in compound words.
[ɐ]in other cases
Consonants are often doubled in writing to indicate the preceding vowel is to be pronounced as a short vowel, mostly when the vowel is stressed. Only consonants written by single letters can be doubled; compareWasser'water'towaschen'wash', not *waschschen. Hence, short and long vowels before the digraph⟨ch⟩are not distinguished in writing:Drache/ˈdʁaxə/'dragon',Sprache/ˈʃpʁaːxə/'language'.
Most one-syllable words that end in a single consonant are pronounced with long vowels, but there are some exceptions such asan, das, es, in, mit, andvon. The suffixes -in, -nisand the word endings -as, -is, -os, -uscontain short unstressed vowels, but duplicate the final consonants in the plurals:Leserin'female reader'—Leserinnen'female readers',Kürbis'pumpkin'—Kürbisse'pumpkins'.
The⟨e⟩in the ending -enis often silent, as inbitten'to ask, request'. The ending -eris often pronounced[ɐ], but in some regions, it is[ʀ̩]or[r̩]. The⟨e⟩in the endings -el([əl~l̩], e.g.Tunnel,Mörtel'mortar') and -em([əm~m̩]in the dative case of adjectives, e.g.kleinemfromklein'small') is pronounced as aschwa.
In the following cases, the vowel letter always represents a long vowel:
Also, the vowel letter usually represents a long vowel:
The Germandefinite articleis pronounced with long vowels in the formsder, dem, den, die, but with short vowels in the formsdasanddes.
A vowel before two or more different consonants is usually pronounced short, but there are some words where it is pronounced long, e.g.Mond"moon".
Long vowels are generally pronounced with greatertensenessthan short vowels.
The long vowels map as follows:
A pre-stress long vowel shortens, mostly in the unstressed position:
A vowel bearing secondary stress may also shorten, as inMonolog'monologue'[ˌmonoˈloːk]. Phonemically, they are typically analysed as allophones of the long/iː,yː,uː,eː,øː,oː/(thus/ˌmoːnoːˈloːk/etc.) and are mostly restricted to loanwords.
In some German proper names, unusual spellings occur, e. g.⟨ui⟩[yː]:Duisburg/dyːsbʊʁk/;⟨ow⟩[oː]:Treptow/ˈtʁeːptoː/.
Theperiod (full stop)is used at the end of sentences, for abbreviations, and for ordinal numbers, such asder 1.forder erste(the first). It is omitted before a full stop at the end of a sentence.
Thecommais used between for enumerations (but theserial commais not used), before adversative conjunctions, after vocative phrases, for clarifying words such as appositions, before and after infinitive and participle constructions, and between clauses in a sentence. A comma may link two independent clauses without a conjunction. The comma is not used before the direct speech; in this case, the colon is used. Using the comma in infinitive phrases was optional before 2024, when the revision of the orthographic rules made it mandatory.
Theexclamation markand thequestion markare used for exclamative and interrogative sentences. It is not preceded by a space, in contrast with languages like French. The exclamation mark may be used for addressing people in letters.
Thesemicolonis used for divisions of a sentence greater than that with the comma.
Thecolonis used before direct speech and quotes, after a generalizing word before enumerations (but not when the wordsdas ist, das heißt, nämlich, zum Beispielare inserted), before explanations and generalizations, and after words in questionnaires, timetables, etc. (e.g.Vater: Franz Müller).
Theem dashis used for marking a sharp transition from one thought to another one, between remarks of a dialogue (as aquotation dash), between keywords in a review, between commands, for contrasting, for marking unexpected changes, for marking an unfinished direct speech, and sometimes instead of parentheses in parenthetical constructions.
Theellipsisis used for unfinished thoughts and incomplete citations.
Theparenthesesare used for parenthetical information.
Thesquare bracketsare used instead of parentheses inside parentheses and for editor's words inside quotations.
Thequotation marksare written as »…« or „…“. They are used for direct speech, quotes, names of books, periodicals, films, etc., and for words in unusual meaning. Quotation inside a quotation is written in single quotation marks: ›…‹ or ‚…‘. If a quotation is followed by a period or a comma, it is placed outside the quotation marks.
Theapostropheis used for contracted forms (such as’sfores) except forms with omitted final⟨e⟩(was sometimes used in this case in the past) and preposition+article contractions. It is also used for genitive of proper names ending in⟨s, ß, x, z, ce⟩, but not if preceded by the definite article.
The oldest knownGermantexts date back to the 8th century. They were written mainly inmonasteriesin different local dialects ofOld High German. In these texts,⟨z⟩along with combinations such as⟨tz, cz, zz, sz, zs⟩was chosen to transcribe the sounds/ts/and/s(ː)/, which is ultimately the origin of the modern German letters⟨z, tz⟩and⟨ß⟩(an old⟨sz⟩ligature). After theCarolingian Renaissance, however, during the reigns of theOttonianandSaliandynasties in the 10th century and 11th century, German was rarely written, the literary language being almost exclusivelyLatin.
Notker the Germanis a notable exception in his period: not only are his German compositions of high stylistic value, but his orthography is also the first to follow a strictly coherent system.
Significant production of German texts only resumed during the reign of theHohenstaufendynasty (in theHigh Middle Ages). Around the year 1200, there was a tendency towards a standardizedMiddle High Germanlanguage and spelling for the first time, based on theFranconian-Swabianlanguage of the Hohenstaufen court. However, that language was used only in theepic poetryandminnesanglyric of theknightculture. These early tendencies of standardization ceased in theinterregnumafter the death of the last Hohenstaufen king in 1254. Certain features of today's Germanorthographystill date back to Middle High German: the use of thetrigraph⟨sch⟩for/ʃ/and the occasional use of⟨v⟩for/f/because around the 12th and 13th century, the prevocalic/f/was voiced.
In the following centuries, the only variety that showed a marked tendency to be used across regions was theMiddle Low Germanof theHanseatic League, based on the variety ofLübeckand used in many areas of northern Germany and indeed northern Europe in general.
By the 16th century, a new interregional standard developed on the basis of theEast Central GermanandAustro-Bavarianvarieties. This was influenced by several factors:
Mid-16th centuryCounter-ReformationreintroducedCatholicismto Austria and Bavaria, prompting a rejection of the Lutheran language. Instead, a specific southern interregional language was used, based on the language of the Habsburg chancellery.
In northern Germany, the Lutheran East Central German replaced theLow Germanwritten language until the mid-17th century. In the early 18th century, the Lutheran standard was also introduced in the southern states and countries, Austria, Bavaria and Switzerland, due to the influence of northern German writers, grammarians such asJohann Christoph Gottschedor language cultivation societies such as theFruitbearing Society.
Though, by the mid-18th century, one norm was generally established, there was no institutionalized standardization. Only with the introduction ofcompulsory educationin late 18th and early 19th century was the spelling further standardized, though at first independently in each state because of the political fragmentation of Germany. Only the foundation of theGerman Empirein 1871 allowed for further standardization.
In 1876, the Prussian government instituted theFirst Orthographic Conference[de]to achieve a standardization for the entire German Empire. However, its results were rejected, notably byPrime Minister of PrussiaOtto von Bismarck.
In 1880,GymnasiumdirectorKonrad Dudenpublished theVollständiges Orthographisches Wörterbuch der deutschen Sprache('Complete Orthographic Dictionary of the German Language'), known simply as the "Duden". In the same year, the Duden was declared to be authoritative in Prussia.[citation needed]Since Prussia was, by far, the largest state in the German Empire, its regulations also influenced spelling elsewhere, for instance, in 1894, whenSwitzerlandrecognized the Duden.[citation needed]
In 1901, theinterior ministerof the German Empire instituted theSecond Orthographic Conference. It declared the Duden to be authoritative, with a few innovations. In 1902, its results were approved by the governments of the German Empire, Austria and Switzerland.
In 1944, theNazi Germangovernment planned areform of the orthography, but because ofWorld War II, it was never implemented.
After 1902, German spelling was essentially decidedde factoby the editors of the Duden dictionaries. AfterWorld War II, this tradition was followed with two different centers:MannheiminWest GermanyandLeipziginEast Germany. By the early 1950s, a few other publishing houses had begun to attack the Duden monopoly in the West by putting out their own dictionaries, which did not always hold to the "official" spellings prescribed by Duden. In response, the Ministers of Culture of the federal states in West Germany officially declared the Duden spellings to be binding as of November 1955.
The Duden editors used their power cautiously because they considered their primary task to be the documentation of usage, not the creation of rules. At the same time, however, they found themselves forced to make finer and finer distinctions in the production of German spelling rules, and each newprint runintroduced a few reformed spellings.
German spelling and punctuation was changed in 1996 (Reform der deutschen Rechtschreibung von 1996) with the intent to simplify German orthography, and thus to make the language easier to learn,[18]without substantially changing the rules familiar to users of the language. The rules of the new spelling concern correspondence between sounds and written letters (including rules for spellingloan words), capitalization, joined and separate words, hyphenated spellings, punctuation, and hyphenation at the end of a line. Place names and family names were excluded from the reform.
The reform was adopted initially by Germany, Austria, Liechtenstein and Switzerland, and later by Luxembourg as well.
The new orthography is mandatory only in schools. A 1998 decision of theFederal Constitutional Court of Germanyconfirmed that there is no law on the spelling people use in daily life, so they can use the old or the new spelling.[19]While the reform is not very popular in opinion polls, it has been adopted by all major dictionaries and the majority of publishing houses.
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https://en.wikipedia.org/wiki/German_spelling_alphabet
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TheGreek spelling alphabetis aspelling alphabet(or "phonetic alphabet") forGreek, i.e. an accepted set of easily differentiated names given to the letters of the alphabet for the purpose of spelling out words. It is used mostly on radio voice channels by theGreek army, thenavyand thepolice. The names for some Greek letters are easily confused in noisy conditions.
Similar sounding Greek letters:
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https://en.wikipedia.org/wiki/Greek_spelling_alphabet
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TheJapanese radiotelephony alphabet(和文通話表,wabuntsūwahyō, literally "Japanese character telecommunication chart")is a radiotelephonyspelling alphabet, similar in purpose to the NATO/ICAO radiotelephony alphabet, but designed to communicate the Japanesekanasyllables rather thanLatin letters. The alphabet was sponsored by the now-defunct Ministry for Posts and Telecommunications.
Each kana is assigned a code word, so that critical combinations of kana (andnumbers) can be pronounced and clearly understood by those who transmit and receive voice messages by radio or telephone, especially when the safety of navigation or persons is essential.
There are specific names for kana, numerals, and special characters (i.e. vowel extender, comma, quotation mark, and parentheses).
Every kana name takes the form of aX no Y(X の Y). For example,ringo no ri(りんごのリ)means "riofringo". Voiced kana do not have special names of their own. Instead, one simply states the unvoiced form, followed by "nidakuten". /p/ sounds are named similarly, with "nihandakuten". Thus, to conveyba(ば), one would say "hagaki no ha ni dakuten(はがきのハに濁点)". To conveypa(ぱ), one would say "hagaki no ha ni handakuten(はがきのハに半濁点)". As no word begins with the syllabic n, the wordoshimai(おしまい), meaning end, is used forn(ん).
Digits are identified with "数字の..." (sūji no.../Number X) followed by the name of the number, analogous to English phrases such asthe number five.
When a number can be named in multiple ways, the most distinctive pronunciation is used. Thus 1, 7, 4 are pronouncedhito, nana, yonrather thanichi, shichi, shiwhich could easily be confused with each other.
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https://en.wikipedia.org/wiki/Japanese_radiotelephony_alphabet
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TheKorean spelling alphabet(Korean:한국어 표준 음성 기호;RR:hangugeo pyojun eumseong giho; also한글 통화표;hangeul tonghwapyo) is a spelling alphabet for theKorean language, similar to theNATO phonetic alphabet.
ThisHangul-related article is astub. You can help Wikipedia byexpanding it.
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https://en.wikipedia.org/wiki/Korean_spelling_alphabet
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TheRussian spelling alphabetis aspelling alphabet(or "phonetic alphabet") forRussian,i.e.a set of names given to the alphabet letters for the purpose of unambiguous verbal spelling. It is used primarily by theRussian army,navyand thepolice. The large majority of the identifiers are common individualfirst names, with a handful of ordinary nouns and grammatical identifiers also. A good portion of the letters also have an accepted alternative name.
The letter words are as follows:[1]
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https://en.wikipedia.org/wiki/Russian_spelling_alphabet
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TheSwedish Armed Forces' radio alphabetwas aradiotelephony alphabetmade up of Swedish two-syllable male names with the exception of Z which is just the name of the letter as pronounced in Swedish.
TheSwedish Armed Forcesare since 2006 instructed to use theNATO alphabetinstead of the original Swedish alphabet, along with and adaptation of theNATOvoice procedures to communicate, since most activity is in various internationalUNandNATOmissions. This has been changed back again since the administrative authorities are required to use the Swedish language according to Swedish law even the Swedish Armed Forces.[clarification needed]
The alphabet is also used for civil communications in Sweden, one example being local flights operating underVFR.
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https://en.wikipedia.org/wiki/Swedish_Armed_Forces_radio_alphabet
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Themilitary time zonesare a standardized, uniform set oftime zonesfor expressing time across different regions of the world, named after theNATO phonetic alphabet. The Zulu time zone (Z) is equivalent toCoordinated Universal Time(UTC) and is often referred to asthemilitary time zone. The military time zone system ensures clear communication in a concise manner, and avoids confusion when coordinating across time zones. TheCCEB, representing thearmed forcesof Australia, Canada, New Zealand, the United Kingdom, and the United States, publishes the military time zone system as the ACP 121 standard.[1]The armed forces of Austria and many nations inNATOuse it.[citation needed]
Going east from theprime meridianat Greenwich, letters "Alfa"[a]to "Mike" (skipping "J", see below) represent the 12 time zones with positiveUTC offsetsuntil reaching theinternational Date Line. Going west from Greenwich, letters "November" to "Yankee" represent zones with negative offsets.
The letters are typically used in conjunction withmilitary time. For example, 6:00 a.m. in zoneUTC−5is written "0600R" and spoken "zero six hundred Romeo".
The numeric zone description or "plus and minus system" indicates the correction which must be applied to the time as expressed in order to convert to UTC. For example, the zone description for the Romeo time zone is +5. Therefore, adding 5 hours to 0600R produces the time in UTC, 1100Z.[1]
The letter "J" ("Juliet"), originally skipped, may be used to indicate the observer's local time.[2]The letter 'L' was previously misidentified in some editions of U.S. Army publications, such as FM 5-0,[3]as representing 'Local' time, which conflicted with its established use for the Lima time zone (UTC+11). This error has been rectified in the latest edition of FM 5-0, released in May 2022,[4]which no longer includes this incorrect usage. "LT" may instead be used to denote local time.
The letter "N" is also used to designate zone −13; this is to provide for a ship in zone −12 keeping Daylight Saving Time.[1]
The letter "Z" ("Zulu") indicatesCoordinated Universal Time(UTC).
The ACP 121 standard actually refers toGreenwich Mean Time(GMT) as the base time standard,[1]but UTC has superseded GMT as a more precise time standard,[5]so the time offsets are commonly understood as UTC.[6][7]
Sandford Flemingdevised a system assigning the letters A–Y excluding J to 1-hour time zones, which may have been the inspiration for the system.[8]
The standard was first distributed byNATOas a note in 1950. The note states "This method is based on the systems in use in the Armed Forces of these countries and the United States".[9]The British used a system of lettered zones, which was likely the direct influence.[10][better source needed]
RFC 733 published in 1977 allowed using military time zones in theDate:field ofemails.[11]RFC 1233 in 1989 noted that the signs of the offsets were specified as opposite the common convention (e.g. A=UTC−1 instead of A=UTC+1),[12]and the use of military time zones in emails was deprecated in RFC 2822 in 2001. It is recommended to ignore such designations and treat all such time designations as UTC unless out-of-band information is present.[13]
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https://en.wikipedia.org/wiki/List_of_military_time_zones
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This is alist of heritage NATO country codes. Up to and including the seventh edition ofSTANAG1059, these were two-letter codes (digrams). The eighth edition, promulgated 19 February 2004, and effective 1 April 2004, replaced all codes with new ones based on theISO 3166-1 alpha-2codes. Additional codes cover gaps in the ISO coverage, deal with imaginary countries used for exercise purposes, and designate large geographical groupings and water bodies (ranging from oceans to rivers). It consists of two-letter codes for geographical entities, four-letter codes for subdivisions, and lists the ISO three-letter codes for reference. The digrams match theFIPS 10-4 codeswith a few exceptions.
The ninth edition's ratification draft was published on 6 July 2005, with a reply deadline of 6 October 2005. It replaces all two- and four-letter codes with ISO or ISO-like three- and six-letter codes. It is intended as a transitional standard: once allNATOnations have updated their information systems, a tenth edition will be published.
For diplomatic reasons, the country that is now known asNorth Macedoniawas designated as theFormer Yugoslav Republic of Macedoniafor a number of years and received a temporary code, FY/FYR, explicitly different from the ISO one, which was 3166 MKD. Since its name change following the 2018Prespa agreementwithGreece, the country is identified with the MK digram and the MKD trigram,[1]but on car license plates, they must be changed to NM or NMK.[2]
The Republic of Palau is also often indicated (at least in the United States) as PW.
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https://en.wikipedia.org/wiki/List_of_NATO_country_codes
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Radiotelephony procedure(alsoon-air protocolandvoice procedure) includes various techniques used to clarify, simplify and standardize spoken communications overtwo-way radios, in use by the armed forces, incivil aviation, police and fire dispatching systems,citizens' band radio(CB), andamateur radio.
Voice procedure communications are intended to maximize clarity of spoken communication and reduce errors in the verbal message by use of an accepted nomenclature. It consists of a signalling protocol such as the use of abbreviated codes like the CB radioten-code,Q codesin amateur radio and aviation, police codes, etc., and jargon.
Some elements of voice procedure are understood across many applications, but significant variations exist. The armed forces of theNATOcountries have similar procedures in order to make cooperation easier.
The impacts of having radio operators who are not well-trained in standard procedures can cause significant operational problems and delays, as exemplified by one case of amateur radio operators duringHurricane Katrina, in which:
...many of the operators who were deployed had excellent go-kits and technical ability, but were seriously wanting in traffic handling skill. In one case it took almost 15 minutes to pass one 25 word message.[1]
Radiotelephony procedures encompass international regulations, official procedures, technical standards, and commonly understood conventions intended to ensure efficient, reliable, and inter-operable communications via all modes of radio communications. The most well-developed and public procedures are contained in theCombined Communications Electronics Board's Allied Communications ProcedureACP 125(G):Communications Instructions Radiotelephone Procedures.[2]
These procedures consist of many different components. The three most important ones are:
These procedures have been developed, tested under the most difficult of conditions, then revised to implement the lessons learned, many times since the early 1900s[clarification needed]. According to ACP 125(G)[2]and the Virginia Defense Force Signal Operating Instructions:[4]
Voice procedure is designed to provide the fastest and most accurate method of speech transmission. All messages should be pre-planned, brief and straightforward. Ideally, messages should be written down: even brief notes reduce the risk of error. Messages should be constructed clearly and logically in order not to confuse the recipient.
Voice procedure is necessary because:
Radio operators must talk differently because two-way radios reduce the quality of human speech in such a way that it becomes harder to understand. A large part of the radio-specific procedures is the specialized language that has been refined over more than 100 years.
There are several main methods of communication over radio, and they should be used in this order of preference:
All radio communications on the planet operate under regulations created by theITU-R, which prescribes most of the basic voice radio procedures, and these are further codified by each individual country.
In the U.S., radio communications are regulated by the NTIA and the FCC. Regulations created by the FCC are codified inTitle 47 of the Code of Federal Regulations:
Radio call signs are a globally unique identifier assigned to all stations that are required to obtain a license in order to emit RF energy. The identifiers consist of from 3 to 9 letters and digits, and while the basic format of the call signs are specified by the ITU-R Radio Regulations, Article 19, Identification of stations,[5]the details are left up to each country's radio licensing organizations.
Each country is assigned a range of prefixes, and the radiotelecommunications agencies within each country then responsible for allocating call signs, within the format defined by the RR, as they see fit. The Radio Regulations require most radio stations to regularly identify themselves by means of their official station call sign or other unique identifier.
Because official radio call signs have no inherent meaning outside of the above-described patterns, and other than individually licensed Amateur radio stations, do not serve to identify the person using the radio, they are not usually desirable as the primary means of identifying which person, department, or function is transmitting or is being contacted.
For this reason, functionaldesignators(a.k.a.tactical call signs) are frequently used to provide such identification. Such designators are not sufficient to meet the FCC requirements that stations regularly identify the license they are operating under, typically every x number of minutes and at the end of each transmission, where x ranges from 10 to 30 minutes (longer for broadcast stations).
For the some radio services, the FCC authorizes alternate station IDs,[6]typically in situations where the alternate station ID serves the purposes of identifying the transmitting station better than the standard ITU format. These include:
The United States has been assigned all call signs with the prefixes K, N, and W, as well as AAA–ALZ. Allocating call signs within these groups is the responsibility of theNational Telecommunications and Information Administration(almost all government stations) or theFederal Communications Commission(all other stations), and they subdivide the radio call signs into the following groups:
Ham station call signs begin with A, K, N or W, and have a single digit from 0 to 9 that separates the 1 or 2 letter prefix from the 1 to 3 letter suffix (special event stations have only three characters: the prefix, the digit, and a one-letter suffix).[7]
Maritime call signs have a much more complex structure, and are sometimes replaced with the name of the vessel or aMaritime Mobile Service Identity(MMSI) number.
Microphones are imperfect reproducers of the human voice, and will distort the human voice in ways that make it unintelligible unless a set of techniques are used to avoid the problems. The recommended techniques vary, but generally align with the following guidelines, which are extracted from the IARU Emergency Telecommunications Guide[8]
Similarly, the U.S. military radio procedures recommend headsets with noise-cancelling microphones:
Use of Audio Equipment. In many situations, particularly in noisy or difficult conditions, the use of headsets fitted with a noise cancelling microphone is preferable to loudspeakers as a headset will aid concentration and the audibility of the incoming signal. The double-sided, noise cancelling microphone is designed to cancel out surrounding noise, for example engine noise or gunfire, allowing speech entering on one side to pass freely. The microphone should be as close to the mouth as possible.[9]
The U.S. Navy radio operator training manuals contain similar guidelines, including NAVPERS 10228-B, Radioman 3 & 2 training course (1957 edition):[10]
Dos:
Do Nots:
Many radio systems also require the operator to wait a few seconds after depressing the PTT button before speaking, and so this is a recommended practice on all systems. The California Statewide EMS Operations and Communications Resource Manual explains why:
Key your transmitter before engaging in speech. The complexities in communications system design often introduce delay in the time it takes to turn on the various components comprising the system. Transmitters take time to come up to full power output, tone squelch decoding equipment requires time to open receivers and receiver voting systems take time to select the best receiver. While these events generally are accomplished in less than one second's time, there are many voice transmissions that could be missed in their entirety if the operator did not delay slightly before beginning his/her voice message. Pausing one second after depressing the push-to-talk button on the microphone or handset is sufficient in most cases to prevent missed words or responses.[11]
Further, transmissions should be kept as short as possible; a maximum limit of 20 or 30 seconds is typically suggested:
Transmissions should generally be kept to less than 20 seconds, or within the time specifically allocated by the system. Most radio systems limit transmissions to less than 30 seconds to prevent malfunctioning transmitters or accidentally keyed microphones from dominating a system, and will automatically stop transmitting at the expiration of the allowed time cutting off additional audio.[11]
Communicating by voice over two-way radios is more difficult than talking with other people face-to-face or over the telephone. The human voice is changed dramatically by two-way radio circuits. In addition to cutting off important audio bandwidth at both the low and high ends of the human speech spectrum (reducing the bandwidth by at least half), other distortions of the voice occur in the microphone, transmitter, receiver, and speaker—and the radio signal itself is subject to fading, interruptions, and other interference. All of these make human speech more difficult to recognize; in particular, momentary disruptions or distortions of the signal are likely to block the transmission of entire syllables.
The best way to overcome these problems is by greatly reducing the number of single-syllable words used. This is very much counter to the human nature of taking shortcuts, and so takes training, discipline, and having all operators using the same language, techniques, and procedures.[12]
Several radio operation procedures manuals, including ACP 125(G) teach the same mnemonic of Rhythm, Speed, Volume, and Pitch (RSVP):[13]
According to the UK's Radiotelephony Manual, CAP 413, radio operators should talk at a speed of fewer than 100 words per minute.
Communicating over a half-duplex, shared circuit with multiple parties requires a large amount of discipline in following the established procedures and conventions, because whenever one particular radio operator is transmitting, that operator can not hear any other station on the channel being used.[15][16][17][18]
The initialism ABC is commonly used as a memory aid to reinforce the three most important rules about what to transmit.[19]
Whenever a report or a request is transmitted over a two-way radio, the operator should consider including the standard Five Ws in the transmission, so as to eliminate additional requests for information that may occur and thereby delay the request (and other communications).
The procedures described in this section can be viewed as the base of all voice radio communications procedures.
However, the international aviation and maritime industries, because their global expansion in the 20th century coincided with, and were heavily integral to the development of voice procedures and other aspects in the development of two-way radio technology, gradually developed their own variations on these procedures.
Voice communications procedures for international air traffic control and communications among airplanes are defined by the followingInternational Civil Aviation Organizationdocuments:
Refinements and localization of these procedures can be done by each member country of ICAO.
Voice procedures for use on ships and boats are defined by theInternational Telecommunication Unionand theInternational Maritime Organizationbodies of the United Nations, and by international treaties such as theSafety of Life a Sea Convention(a.k.a. SOLAS 74), and by other documents, such as theInternational Code of Signals.
In the U.S., the organization chartered with devising police communications procedures is APCO International, the Association of Police Communications Officers, which was founded in 1935. For the most part, APCO's procedures have been developed independently of the worldwide standard operating procedures, leading to most police departments using a different spelling alphabet, and the reverse order of calling procedure (e.g. 1-Adam-12 calling Dispatch).
However, APCO occasionally follows the international procedure standards, having adopted the U.S. Navy's Morse code procedure signs in the 1930s, and adopting the ICAO radiotelephony spelling alphabet in 1974, replacing its own Adam-Boy-Charles alphabet adopted in 1940, although very few U.S. police departments made the change.
APCO has also specified Standard Description Forms, a standard order of reporting information describing people and vehicles.
The Standard Description of Persons format first appeared in the April 1950 edition of the APCO Bulletin.[29]It starts with a description of the person themself and finishes with a description of what they are wearing at the time.
APCO promotes the mnemonic CYMBALS for reporting vehicle descriptions:[29]
The voice calling procedure (sometimes referred to as "method of calling" or "communications order model") is the standardized method of establishing communications. The order of transmitting the called station's call sign, followed by the calling station's call sign, was first specified for voice communications in the International Radiotelegraph Convention of Washington, 1927,[30]however it matches the order used for the radiotelegraph calling procedure that had already existed since at least 1912.[citation needed]In the United States, the radiotelegraph calling procedure is legally defined in FCC regulations Part 80.97 (47 CFR 80.97(c)), which specifies that the method of calling begins with the call sign of the station called, not more than twice, [THIS IS] and the call sign of the calling station, not more than twice".[31]This order is also specified by theICAOfor international aviation radio procedures (Annex 10 to the Convention on. International Civil Aviation: Aeronautical Telecommunications.[32]), the FAA (Aeronautical Information Manual[33]) and by theITU-Rfor the Maritime Mobile Service (ITU-R M.1171),[34]and the U.S. Coast Guard (Radiotelephone Handbook[35]). The March, 1940 issue of The APCO Bulletin explains the origin of this order was found to have better results than other methods,[36]
Stations needing to interrupt other communications in-progress shall use the most appropriate of the belowprocedure words, followed by their call sign.
The use of these emergency signals is governed by the International Radio Regulations that have the force of law in most countries, and were originally defined in theInternational Code of Signalsand theInternational Convention for the Safety of Life at Sea, so the rules for their use emanate from that document.
All of these break-in procedure words must be followed by your call sign, because that information will help the NCS determine the relevant importance when dealing with multiple break-ins of the same precedence, and to determine the relevance when multiple calls offering a CORRECTION or INFO are received.
The priority levels described below are derived from Article 44 of the ITU Radio Regulations, Chapter VIII, and were codified as early as theInternational Telecommunication Convention, Atlantic City, 1947(but probably existed much earlier).
Procedure words are a direct voice replacement forprocedure signs(prosigns) andoperating signals(such asQ codes), and must always be used on radiotelephone channels in their place. Prosigns/operating signals may only be used with Morse Code (as well as semaphore flags, light signals, etc.) and TTY (including all forms of landline and radio teletype, and Amateur radio digital interactive modes). The most complete set of procedure words is defined in the U.S. Military's Allied Communications Publication ACP 125(G).[9]
NIL
(civilian)
This usage comes from the Morse code prosign "R", which means "received": from 1943 to early 1956, the code word for R was Roger in theAllied Military phonetic spelling alphabetsin use by the armed forces, including theJoint Army/Navy Phonetic AlphabetandRAF phonetic alphabet.[40][41]This use was officially continued even after the spelling word for R was changed to ROMEO.[42]Contrary to popular belief, Roger does not mean or imply both "received" and "I will comply." That distinction goes to the contractionwilco(from "will comply"), which is used exclusively if the speaker intends to say "received and will comply". The phrase "Roger Wilco" is procedurally incorrect, as it is redundant with respect to the intent to say "received".[43][9]
An error has been made in this transmission (or message indicated). The correct version is...... That which follows is a corrected version in answer to your request for verification"
N (civilian)
QTRIMI(civilian)
(military) QTR
(civilian)
Whenever an operator is transmitting and uncertain of how good their radio and/or voice signal are, they can use the following procedure words to ask for asignal strength and readability report. This is the modern method of signal reporting that replaced the old 1 to 5 scale reports for the two aspects of a radio signal, and as with the procedure words, are defined in ACP 125(G):
The prowords listed below are for use when initiating and answering queries concerning signal strength and readability.
In the tables below, the mappings of the QSA and QRK Morse code prosigns is interpreted because there is not a 1:1 correlation. SeeQSA and QRK codefor the full procedure specification.
The reporting format is one of the signal strength prowords followed by an appropriate conjunction, with that followed by one of the readability prowords:
LOUD AND CLEARmeans Excellent copy with no noise
GOOD AND READABLEmeans Good copy with slight noise
FAIR BUT READABLEmeans Fair copy, occasional fills are needed
WEAK WITH INTERFERENCEmeans Weak copy, frequent fills are needed because of interference from other radio signals.
WEAK AND UNREADABLEmeans Unable to copy, a relay is required
According to military usage, if the response would beLOUD AND CLEAR, you may also respond simply with the prowordROGER. However, because this reporting format is not currently used widely outside of military organizations, it is better to always use the full format, so that there is no doubt about the response by parties unfamiliar with minimization and other shorthand radio operating procedures.
TheInternational Civil Aviation Organization(ICAO),International Telecommunication Union, and theInternational Maritime Organization(all agencies of the United Nations), plusNATO, all specify the use of the ICAO Radiotelephony Spelling Alphabet for use when it is necessary to spell out words, callsigns, and other letter/number sequences. It was developed with international cooperation and ratified in 1956, and has been in use unmodified ever since.
Spelling is necessary when difficult radio conditions prevent the reception of an obscure word, or of a word or group, which is unpronounceable. Such words or groups within the text of plain language messages may be spelt using the phonetic alphabet; they are preceded by the proword "I SPELL". If the word is pronounceable and it is advantageous to do so, then it should be spoken before and after the spelling to help identify the word.[44]
When radio conditions are satisfactory and confusion will not arise, numbers in the text of a message may be spoken as in normal speech. During difficult conditions, or when extra care is necessary to avoid misunderstanding, numbers are sent figure by figure preceded by the proword FIGURES. This proword warns that figures follow immediately, to help distinguish them from other similarly pronounced words.[44]
Ending a two-way radio call has its own set of procedures:
Nets operate either on schedule or continuously (continuous watch). Nets operating on schedule handle traffic only at definite, prearranged times and in accordance with a prearranged schedule of intercommunication. Nets operating continuously are prepared to handle traffic at any time; they maintain operators on duty at all stations in the net at all times. When practicable, messages relating to schedules will be transmitted by a means of signal communication other than radio.[45]
A net manager is the person who supervises the creation and operation of a net over multiple sessions. This person will specify the format, date, time, participants, and the net control script. The net manager will also choose the Net Control Station for each net, and may occasionally take on that function, especially in smaller organizations.
Radio nets are like conference calls in that both have a moderator who initiates the group communication, who ensures all participants follow the standard procedures, and who determines and directs when each other station may talk. The moderator in a radio net is called the Net Control Station, formally abbreviated NCS, and has the following duties:[46]
The Net Control Station will, for each net, appoint at least one Alternate Net Control Station, formally abbreviated ANCS (abbreviated NC2 in WWII procedures), who has the following duties:[46]
Nets can be described as always having a net opening and a net closing, with a roll call normally following the net opening, itself followed by regular net business, which may include announcements, official business, and message passing. Military nets will follow a very abbreviated and opaque version of the structure outlined below, but will still have the critical elements of opening, roll call, late check-ins, and closing.
A net should always operate on the same principle as theinverted pyramidused in journalism—the most important communications always come first, followed by content in ever lower levels of priority.
Each net will typically have a main purpose, which varies according to the organization conducting the net, which occurs during the net business phase. For amateur radio nets, it's typically for the purpose of allowing stations to discuss their recent operating activities (stations worked, antennas built, etc.) or to swap equipment. ForMilitary Auxiliary Radio SystemandNational Traffic Systemnets, net business will involve mainly the passing of formal messages, known asradiograms.
Stations without the ability to acquire a time signal accurate to at least one second should request a time check at the start of every shift, or once a day minimum. Stations may ask the NCS for a time check by waiting for an appropriate pause, keying up and stating your call sign, and then using the prowords "REQUEST TIME CHECK, OVER" when the NCS calls on you. Otherwise, you may ask any station that has access to any of the above time signals for a time check.
Once requested, the sending station will state the current UTC time plus one minute, followed by a countdown as follows:
This is Net Control, TIME CHECK WUN AIT ZERO TOO ZULU (pause) WUN FIFE SECONDS…WUN ZERO SECONDS…FIFE FOWER TREE TOO WUN…TIME WUN AIT ZERO TOO ZULU…OVER
The receiving station will then use the proword "TIME" as the synch mark, indicating zero seconds. If the local time is desired instead of UTC, substitute the time zone code "JULIETT" for "ZULU".
Instead of providing time checks on an individual basis, the NCS should give advance notice of a time check by stating, for example, "TIME CHECK AT 0900 JULIETT", giving all stations sufficient time to prepare their clocks and watches for adjustment. A period of at least five minutes is suggested.
When calling stations who are part of a net, a variety of types of calls can be used:
TheCivil Air PatrolandInternational Amateur Radio Uniondefine a number of different nets which represent the typical type and range used in civilian radio communications:
U.S. Army Field Manual ACP 125(G) has the most complete set of procedure words used in radio nets:[50]
TheFederal Aviation Administrationuses the termphraseologyto describe voice procedure or communications protocols used over telecommunications circuits. An example is air traffic control radio communications. Standardised wording is used and the person receiving the message may repeat critical parts of the message back to the sender. This is especially true of safety-critical messages.[51]Consider this example of an exchange between a controller and an aircraft:
Aircraft:Boston Tower, Warrior three five foxtrot (35F), holding short of two two right.Tower:Warrior three five foxtrot, Boston Tower, runway two two right, cleared for immediate takeoff.Aircraft:Roger, cleared for immediate takeoff, two two right, Warrier three five foxtrot.
On telecommunications circuits, disambiguation is a critical function of voice procedure. Due to any number of variables, including radio static, a busy or loud environment, or similarity in the phonetics of different words, a critical piece of information can be misheard or misunderstood; for instance, a pilot being ordered toeleventhousand as opposed toseventhousand (by hearing "even"). To reduce ambiguity, critical information may be broken down and read as separate letters and numbers. To avoid error or misunderstanding, pilots will often read back altitudes in the tens of thousands using both separate numbers and the single word (example: given a climb to 10,000 ft, the pilot replies "[Callsign] climbing to One zero, Ten Thousand"). However, this is usually only used to differentiate between 10,000 and 11,000 ft since these are the most common altitude deviations. The runway number read visually as eighteen, when read over a voice circuit as part of an instruction, becomesone eight. In some cases aspelling alphabetis used (also called aradio alphabetor aphonetic alphabet). Instead of the letters AB, the wordsAlpha Bravoare used.MainStreet becomesMike Alpha India November street, clearly separating it from Drain Street and Wayne Street. The numbers 5 and 9 are pronounced "fife" and "niner" respectively, since "five" and "nine" can sound the same over the radio. The use of 'niner' in place of 'nine' is due to German-speaking NATO allies for whom the spoken word 'nine' could be confused with the German word 'nein' or 'no'.
Over fire service radios, phraseology may include words that indicate the priority of a message, for example:[52]
Forty Four Truck to the Bronx, Urgent!
or
San Diego, Engine Forty, Emergency traffic!
Words may be repeated to modify them from traditional use in order to describe a critical message:[53]
Evacuate! Evacuate! Evacuate!
A similar technique may be used in aviation for critical messages. For example, this transmission might be sent to an aircraft that has just landed and has not yet cleared the runway.
Echo-Foxtrot-Charlie, Tower. I have engine out traffic on short final. Exit runway at next taxiway. Expedite! Expedite!
Police Radios[where?]also use this technique to escalate a call that is quickly becoming an emergency.
Code 3! Code 3! Code 3!
Railroads have similar processes. Wheninstructions are read to a locomotive engineer, they are preceded by the train or locomotive number, direction of travel and the engineer's name. This reduces the possibility that a set of instructions will be acted on by the wrong locomotive engineer:
Five Sixty Six West, Engineer Jones, okay to proceed two blocks west to Ravendale.
Phraseology on telecommunications circuits may employ special phrases liketen codes,Sigalert,Quick Alert!or road service towing abbreviations such asT6. This jargon may abbreviate critical data and alert listeners by identifying the priority of a message. It may also reduce errors caused by ambiguities involving rhyming, or similar-sounding, words.
(Done on VHF Ch 16)
Boat "Albacore" talking to Boat "Bronwyn"
Albacore:Bronwyn, Bronwyn, Bronwyn* this is Albacore, OVER. (*3×1, repeating the receiver's callsign up to 3 times, and the sender's once, is proper procedure and should be used when first establishing contact, especially over a long distance. A 1×1, i.e. 'Bronwyn this is Albacore,' or 2×1, i.e. 'Bronwyn, Bronwyn, this is Albacore,' is less proper, but acceptable especially for a subsequent contact.)[54]
Bronwyn:Albacore, this is Bronwyn, OVER. (** At this point switch to a working channel as 16 is for distress and hailing only**)
Albacore:This is Albacore. Want a tow and are you OK for tea at Osbourne Bay? OVER.
Bronwyn:This is Bronwyn. Negative, got engine running, 1600 at clubhouse fine with us. OVER.
Albacore:This is Albacore, ROGER, OUT.
"Copy that" is incorrect. COPY is used when a message has been intercepted by another station, i.e. a third station would respond:
Nonesuch:Bronwyn, this is Nonesuch. Copied your previous, will also see you there, OUT.
One should always use one's own callsign when transmitting.
Station C21A (charlie-two-one Alpha) talking to C33B (charlie-three-three Bravo):[55]
C21A:C33B, this is C21A, message, OVER.
C33B:C33B, send, OVER.
C21A:Have you got C1ØD Sunray at your location?, OVER.
C33B:Negative, I think he is with C3ØC, OVER.
C21A:Roger, OUT.
The advantage of this sequence is that the recipient always knows who sent the message.
The downside is that the listener only knows the intended recipient from the context of the conversation. Requires moderate signal quality for the radio operator to keep track of the conversations.
However a broadcast message and response is fairly efficient.
Sunray (Lead)Charlie Charlie (Collective Call - everyone), this is Sunray. Radio check, OVER.
C-E-5-9:Sunray, this is Charlie Echo five niner, LOUD AND CLEAR, OVER.
Y-S-7-2Sunray, this is Yankee Sierra Seven Two, reading three by four. OVER.
B-G-5-2:Sunray, this is Bravo Golf Five Two, Say again. OVER.
E-F-2-0:Sunray, this is Echo Foxtrot Two Zero, reading Five by Four OVER.
Sunray:Charlie Charlie this is Sunray, OUT.
The "Say again" response from B-G-5-2 tells Sunray that the radio signal is not good and possibly unreadable. Sunray can then re-initiate a Call onto B-G-5-2 and start another R/C or instruct them to relocate, change settings, etc.
So it could carry on with:
Sunray:Bravo Golf Five Two this is sunray, RADIO CHECK OVER.
B-G-5-2:Sunray this is Bravo Golf Five Two, unclear, read you 2 by 3 OVER.
Sunray:Sunray copies, Relocate to Grid One Niner Zero Three Three Two for a better signal OVER.
B-G-5-2:Bravo Golf Five Two copies and is Oscar Mike, Bravo Golf Five Two OUT.
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https://en.wikipedia.org/wiki/Radiotelephony_procedure
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Procedure words(abbreviated toprowords) are words or phrases limited toradiotelephony procedureused to facilitatecommunicationby conveying information in a condensed standard verbal format.[1]Prowords are voice versions of the much olderprocedural signs for Morse codewhich were first developed in the 1860s forMorse telegraphy, and their meaning is identical.
The NATO communications manual ACP-125[2]contains the most formal and perhaps earliest modern (post-World War II) glossary of prowords, but its definitions have been adopted by many other organizations, including theUnited Nations Development Programme,[3]theU.S. Coast Guard,[4]USCivil Air Patrol,[5]USMilitary Auxiliary Radio System,[6]and others.
Prowords are one of several structured parts of radio voice procedures, including brevity codes andplain language radio checks.
According to theU.S. Marine Corpstraining document FMSO 108, "understanding the following PROWORDS and their respective definitions is the key to clear and concise communication procedures".
Thistransmissionis from thestationwhose designator immediately follows. For clarity, the stationcalledshould be named before the stationcalling. So, "Victor Juliet zero, THIS IS Golf Mike Oscar three...", or for brevity, "Victor Juliet zero, Golf Mike Oscar three, ...". One can never say, "This is GMO3 calling VJ0".[citation needed]
"This is the end of my transmission to you and a responseisnecessary. Go ahead: transmit."
"Over" and "Out" areneverused at the same time, since their meanings are mutually exclusive. With spring-loadedPush to talk(PTT) buttons on modern combinedtransceivers, the same meaning can be communicated with just "OUT", as in "Ops, Alpha, ETA five minutes. OUT."[clarification needed]
"This is the end of my transmission to you and no answer is required or expected."[citation needed]
A question about whether the receiver can hear and understand the transmission.
Example: "Bob, you read me? What is the situation from your position?"
Example:
"ROGER" may be used to mean "yes" with regard to confirming a command; however, in Air Traffic Control phraseology, it does not signify that a clearance has been given.[citation needed]
The term originates from the practice oftelegrapherssending an "R" to stand for "received" after successfully getting a message. This was extended into spoken radio during World War II, with the "R" changed to thespelling alphabetequivalent word "Roger".[8][9][10]The modernNATO spelling alphabetuses the word "Romeo" for "R" instead of "Roger", and "Romeo" is sometimes used for the same purpose as "Roger", mainly in Australian maritime operations.[citation needed]
"I understand and will comply." It is used on receipt of an order. "Roger" and "Wilco" used together (e.g. "Roger, Wilco") are redundant, since "Wilco" includes the acknowledgement element of "Roger".[11]
"I have not understood your message, please SAY AGAIN". Usually used with prowords "ALL AFTER" or "ALL BEFORE". Example: radio working betweenSolentCoastguard and a motor vessel, call-sign EG 93, where part of the initial transmission is unintelligible.
Example:
At this juncture, Solent Coastguard would reply, preceding the message with the prowords "I SAY AGAIN":
The word "REPEAT" should not be used in place of "SAY AGAIN", especially in the vicinity of naval or other firing ranges, as "REPEAT" is anartillery proworddefined in ACP 125 U.S. Supp-2(A) with the wholly different meaning of "request for the same volume of fire to be fired again with or without corrections or changes" (e.g., at the same coordinates as the previous round).[12]
"Please repeat the message you just sent me beginning after the word or phrase said after this proword."[citation needed]
Example:
At this juncture, Solent Coastguard would reply, preceding the message after "position" with the prowords "I SAY AGAIN":
"Please repeat the message you just sent me ending before the word or phrase said after this proword."[citation needed]
"Wait for some time."
"I must pause for a few seconds."[citation needed]
"I must pause for longer than a few seconds.."
"Please repeat my entire transmission back to me."[citation needed]
"The following is my response to your READ BACK proword."[citation needed]
"I made an error in this transmission. Transmission will continue with the last word correctly sent."[citation needed]
"What is mysignal strength and readability; how do you hear me?"[citation needed]
The sender requests a response indicating the strength and readability of their transmission, according to plain language radio check standards:
"5 by 5" is an older term used to assess radio signals, as in 5 out of 5 units for both signal strength and readability. Other terms similar to 5x5 are "LOUD AND CLEAR" or "Lima and Charlie". Example:
Similar example in shorter form:
If the initiating station (Alpha 12 in the example) cannot hear the responding station (X-ray 23 above), then the initiator attempts a radio-check again, or if the responder's signal was not heard, the initiator replies to the responder with "Negative contact, Alpha 12 OUT".
The following readability scale is used: 1 = bad (unreadable); 2 = poor (readable now and then); 3 = fair (readable, but with difficulty); 4 = good (readable); 5 = excellent (perfectly readable).
Example of correct US Army radio check, for receiver A-11 (Alpha 11) and sender D-12 (Delta 12):
International Telecommunication Union (ITU) Radio Regulations and theInternational Civil Aviation Organization(ICAO) Convention and Procedures for Air Navigation Services set out "distress, urgency and safety procedures".
On the radio, distress (emergency) and rescue usage takes precedence above all other usage, and the radio stations at the scene of the disaster (on land, in a plane, or on a boat) are authorized to commandeer the frequency and prohibit all transmissions that are not involved in assisting them. These procedure words originate in theInternational Radio Regulations.
TheCombined Communications-Electronics Board(representing military use by Australia, Canada, New Zealand, United Kingdom and United States) sets out their usage in the Allied Communications Publications "ACP 135(F) Communications instructions Distress and Rescue Procedures".[13]
Mayday is used internationally as the official SOS/distress call for voice. It means that the caller, their vessel or a person aboard the vessel is in grave and imminent danger, send immediate assistance. This call takes priority over all other calls.[14]
The correct format for a Mayday call is as follows:
[The first part of the signal is known as the "call"]
Mayday, Mayday, Mayday,This is (vessel name repeated three times, followed bycall signif available)[The subsequent part of the signal is known as the "message"]
Mayday (vessel name)My position is (position as a lat-long position or bearing and distance from a fixed point)I am (type of distress, e.g. on fire and sinking)I require immediate assistanceI have (number of people on board and their condition)(Any other information e.g. "I am abandoning to life rafts")
Over
VHF instructors, specifically those working for theRoyal Yachting Association, often suggest themnemonicMIPDANIO for learning the message of a Mayday signal: mayday, identify, position, distress, assistance, number-of-crew, information, over.[citation needed]
In aviation a different format is used:
[First part of the message] Mayday, Mayday, Mayday
[Second part of the message] Callsign
[Third part of the message] Nature of the emergency
For example: "Mayday, Mayday, Mayday, Wiki Air 999, we have lost both of our engines due to a bird strike, we are gliding now."
After that pilot can give, or the controller can ask for, additional information, such as, fuel and number of passengers on board.
Pan-pan (pronounced/ˈpænˈpæn/)[15]is the official urgency voice call.
Meaning "I, my vessel or a person aboard my vessel requires assistance but is not in distress." This overrides all but a mayday call, and is used, as an example, for calling for medical assistance or if the station has no means of propulsion. The correct usage is:
Pan-Pan, Pan-Pan, Pan-Pan
All stations, all stations, all stationsThis is [vessel name repeated three times]My position is [position as a lat-long position or bearing and distance from a fixed point]I am [type of urgency, e.g. drifting without power in a shipping lane]I require [type of assistance required][Any other information e.g. size of vessel, which may be important for towing]
Over[citation needed]
Pronounced/seɪˈkjuːrɪteɪ/say-KURE-i-tay, this is the official safety voice call.
"I have important meteorological, navigational or safety information to pass on."
This call is normally broadcast on a defined channel (channel 16 for maritime VHF) and then moved onto another channel to pass the message. Example:
[On channel 16]
SÉCURITÉ, SÉCURITÉ, SÉCURITÉ
All stations, all stations, all stations.
This is Echo Golf niner three, Echo Golf niner three, Echo Golf niner three.
For urgent navigational warning, listen on channel six-seven.
OUT
[Then on channel 67]
SÉCURITÉ, SÉCURITÉ, SÉCURITÉ
All stations, all stations, all stations.
This is Echo Golf niner tree (three), Echo Golf niner tree, Echo Golf niner tree.
Floating debris sighted offCalshot Spit.
Considered a danger to surface navigation.
OUT
"Seelonce" is an approximation rendition of the French wordsilence. Indicates that your vessel has an emergency and that you are requiring radio silence from all other stations not assisting you.
Indicates that you are relaying or assisting a station that has placed a MAYDAY call and you are requiring radio silence from all other stations not assisting you or the station in distress.
When the emergency issue is winding down and then has been resolved, these prowords are used to open up the frequency for use by stations not involved in the emergency:
From the French wordprudence. Indicates that complete radio silence is no longer required and restricted (limited) use of the frequency may resume, but immediately giving way to all further distress communications.
From the French wordsilenceandfini (ended). Indicates that emergency communications have ceased and normal use of the frequency may resume.
More formally known as "Aeronautical Mobile communications", radio communications from and to aircraft are governed by rules created by theInternational Civil Aviation Organization. ICAO defines a very similar but shorter list of prowords in Annex 10 of its Radiotelephony Procedures (to theConvention on International Civil Aviation). Material in the following table is quoted from their list.[16]ICAO also defines "ICAO Radio Telephony Phraseology".[17]
Marine radio procedure words follow from the ACP-125 definition, and those in theInternational Radio Regulationspublished by theITU, and should be used by small vessels as their standard radio procedure. Beginning in 2001, large vessels, defined as being 500gross tonnageor greater, theInternational Convention on Standards of Training, Certification and Watchkeeping for Seafarershas required that a restricted and simplified English vocabulary with pre-set phrases, calledStandard Marine Communication Phrases(SMCP), be used and understood by all officers in charge of a navigational watch. These rules are enforced by theInternational Maritime Organization(IMO).[18]The IMO describes the purpose of SMCP, explaining "The IMO SMCP includes phrases which have been developed to cover the most important safety-related fields of verbal shore-to-ship (and vice-versa), ship-to-ship and on-board communications. The aim is to get round the problem of language barriers at sea and avoid misunderstandings which can cause accidents."[19]
The SMCP language is not free-form like the standard radio voice procedures and procedure words. Instead, it consists of entire pre-formed phrases carefully designed for each situation, and watch officers must pass a test of their usage in order to be certified under international maritime regulations. For example, ships in their own territorial waters might be allowed to use their native language, but when navigating at sea or communicating with foreign vessels in their own territorial waters, they should switch to SMCP, and will state the switch over the radio before using the procedures. When it is necessary to indicate that the SMCP are to be used, the following message may be sent: "Please use Standard Marine Communication Phrases." "I will use Standard Marine Communication Phrases."
"Clear" is sometimes heard inamateur radiotransmissions to indicate the sending station is done transmitting and leaving the airways, i.e. turning off the radio, but theClearproword is reserved for a different purpose, that of specifying the classification of a16-line formatradio message as one which can be sent 'in clear [language]' (without encryption), as well as being reserved for use in responding to theRadio Checkproword to indicate the readability of the radio transmission.
"Confirm" or "yes" and sometimes shortened toAffirmis heard in several radio services, but is not listed in ACP-125 as a proword. In poor radio conditionsAffirmativecan be confused withNegative. Instead, the prowordCorrectis used.
Means "no", and can be abbreviated toNegat. Because over a poor quality connection the words "affirmative" and "negative" can be mistaken for one another (for example over asound-powered telephonecircuit), United States Navy instruction omits the use of either as prowords.[20]Sailors are instructed to instead use "yes" and "no".
Two helicopters, call signs "Swiss 610" and "Swiss 613", are flying in formation :
Anytime a radio call is made (excepting "standby", where the correct response is silence), there is some kind of response indicating that the original call was heard. 613's "Roger" confirms to 610 that the information was heard. In the second radio call from 610, direction was given. 613's "Wilco" means "will comply."
Reading back an instruction confirms that it was heard correctly. For example, if all 613 says is "Wilco", 610 cannot be certain that he correctly heard the heading as 090. If 613 replies with a read backandthe word "Wilco" ("Turn right zero-niner-zero, Wilco") then 610 knows that the heading was correctly understood, and that 613 intends to comply.
The following is the example of working between two stations, EG93 and VJ50 demonstrating how to confirm information:
The following is the example of working betweenMACV-SOGoperator and agunshipdemonstrating how to confirm information:
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https://en.wikipedia.org/wiki/Procedure_word
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Aspelling alphabet(also called by various other names) is a set of words used to represent thelettersof analphabetinoral communication, especially over atwo-way radioortelephone. The words chosen to represent the letters sound sufficiently different from each other to clearly differentiate them. This avoids any confusion that could easily otherwise result from the names of letters that sound similar, except for some small difference easily missed or easily degraded by the imperfect sound quality of the apparatus. For example, in the Latin alphabet, the letters B, P, and D ("bee", "pee" and "dee") sound similar and could easily be confused, but the words "bravo", "papa" and "delta" sound completely different, making confusion unlikely.
Any suitable words can be used in the moment, making this form of communication easy even for people not trained on any particular standardized spelling alphabet. For example, it is common to hear anonceform like "A as in 'apple', D as in 'dog', P as in 'paper'" over the telephone incustomer supportcontexts. However, to gain the advantages ofstandardizationin contexts involving trained persons, a standard version can be convened by an organization. Many (loosely or strictly) standardized spelling alphabets exist, mostly owing to historicalsiloization, where each organization simply created its own. International air travel created a need for a worldwide standard.
Today the most widely known spelling alphabet is theICAOInternational Radiotelephony Spelling Alphabet, also known as the NATO phonetic alphabet, which is used for Roman letters. Spelling alphabets also existfor Greekandfor Russian.
Spelling alphabets are called by various names, according to context. These synonyms includespelling alphabet,word-spelling alphabet,voice procedure alphabet,radio alphabet,radiotelephony alphabet,telephone alphabet, andtelephony alphabet. A spelling alphabet is also often called aphonetic alphabet, especially byamateur radioenthusiasts,[1]recreational sailors in the US and Australia,[2]and NATO military organizations,[3]despite this usage of the term producing anaming collisionwith the usage of the same phrase inphoneticsto mean a notation used forphonetic transcriptionorphonetic spelling, such as theInternational Phonetic Alphabet, which is used to indicate the sounds of human speech.
The names of the letters of theEnglish alphabetare "a", "bee", "cee", "dee", "e", etc. These can be difficult to discriminate, particularly over a limited-bandwidth[further explanation needed]and noisy communications channel, hence the use in aviation and by armed services of unambiguous substitute names for use in electrical voice communication such as telephone and radio.
A large number of spelling alphabets have been developed over the past century, with the first ones being used to overcome problems with the early wired telephone networks, and the later ones being focused on wireless two-way radio (radiotelephony) links. Often, each communications company and each branch of each country's military developed its own spelling alphabet, with the result that one 1959 research effort documented a full 203 different spelling alphabets, comprising 1600 different words, leading the author of the report to ask:
Should an efficient American secretary, for example, know several alphabets—one for use on the telephone, another to talk to the telegraph operator, another to call the police, and still another for civil defense?[4]
Each word in the spelling alphabet typically replaces the name of the letter with which it starts (acrophony). It is used to spell out words when speaking to someone not able to see the speaker, or when the audio channel is not clear. The lack of high frequencies on standardtelephonesmakes it hard to distinguish an 'F' from an 'S' for example. Also, the lack of visual cues during oral communication can cause confusion. For example, lips are closed at the start of saying the letter "B" but open at the beginning of the letter "D" making these otherwise similar-sounding letters more easily discriminated when looking at the speaker. Without these visual cues, such as during announcements of airline gate numbers "B1" and "D1" at an airport, "B" may be confused with "D" by the listener. Spelling out one's name, a password or aticker symbolover the telephone are other scenarios where a spelling alphabet is useful.
British Armysignallersbegan using a partial spelling alphabet in the late 19th century. Recorded in the 1898 "Signalling Instruction" issued by the War Office and followed by the 1904 Signalling Regulations[5]this system differentiated only the letters most frequently misunderstood: Ack (originally "Ak") Beer (or Bar) C D E F G H I J K L eMma N O Pip Q R eSses Toc U Vic W X Y Z. This alphabet was the origin of phrases such as "ack-ack" (A.A. foranti-aircraft), "pip-emma" forpmandToc Hfor an ex-servicemen's association. It was developed on theWestern Frontof the First World War. The RAF developed their "telephony spelling alphabet", which was adopted by all three services and civil aviation in the UK from 1921.
It was later formally codified to provide a word for all 26 letters (seecomparative tabulationof Western military alphabets).
For civilian users, in particular in the field of finance, alternative alphabets arose. Common personal names were a popular choice, and the First Name Alphabet came into common use.
Spelling alphabets are especially useful when speaking in a noisy environment when clarity and promptness of communication is essential, for example duringtwo-way radiocommunication between an aircraft pilot andair traffic control, or in military operations. Whereas the names of many letters sound alike, the set of replacement words can be selected to be as distinct from each other as possible, to minimise the likelihood of ambiguity or mistaking one letter for another. For example, if a burst ofstaticcuts off the start of anEnglish-languageutterance of the letterJ, it may be mistaken forAorK. In the international radiotelephony spelling alphabet known as theICAO (or NATO) phonetic alphabet, the sequenceJ–A–Kwould be pronouncedJuliett–Alfa–Kilo. Some voice procedure standards requirenumbersto be spelled outdigitby digit, so some spelling alphabets replace confusable digit names with more distinct alternatives; for example, the NATO alphabet has “niner” for 9 to distinguish it better from 5 (pronounced as “fife”) and the German word “nein”.
Although no radio or traditional telephone communications are involved in communicating flag signals among ships, the instructions for which flags to hoist are relayed by voice on each ship displaying flags, and whether this is done by shouting between decks, sound tubes, orsound-powered telephones, some of the same distortions that make a spelling alphabet for radiotelephony also make a spelling alphabet desirable for directing seamen in which flags to hoist. The first documented use of this were two different alphabets used by U.S. Navy circa 1908. By 1942, the U.S. Army's radiotelephony spelling alphabet was associated with theInternational Code of Signals(ICS) flags.[6]
(proposed)[8]
While spelling alphabets today are mostly used over two-way radio voice circuits (radiotelephony), early on in telecommunications there were also telephone-specific spelling alphabets, which were developed to deal with the noisy conditions on long-distance circuits. Their development was loosely intertwined with radiotelephony spelling alphabets, but were developed by different organizations; for example, AT&T developed a spelling alphabet for its long-distance operators, another for its international operators; Western Union developed one for the public to use when dictating telegrams over the telephone;[10]and ITU-T developed a spelling alphabet for telephone networks, while ITU-R was involved in the development of radiotelephony spelling alphabets. Even though both of these groups were part of the same ITU, and thus part of the UN, their alphabets often differed from each other.
Uniquely, the 1908 Tasmanian telegraph operator's code was designed to be memorized as follows:[11]
Englishmen Invariably Support High Authority Unless Vindictive.The Managing Owners Never Destroy Bills.Remarks When Loose Play Jangling. Fractious Galloping Zigzag Knights eXpeditely Capture Your Queen.
InWorld War Ibattle lines were relatively static and forces were commonly linked by wired telephones. Signals could be weak on long wire runs andfield telephonesystems often used a single wire withearth return, which made them subject to inadvertent and deliberate interference. Spelling alphabets were introduced for wire telephony as well as on the newer radio voice equipment.[15]
Commercial and international telephone and radiotelephone spelling alphabets.
(Harald prior 1960)
(Etta prior 1960)
(Sjua prior 1960)
The laterNATO phonetic alphabetevolved from theprocedures of several different Allied nationsduring World War II, including:
For the 1938 and 1947 alphabets, each transmission of figures is preceded and followed by the words "as a number" spoken twice.
The ITU adopted theInternational Maritime Organization's phonetic spelling alphabet in 1959,[23]and in 1969 specified that it be "for application in the maritime mobile service only".[24]
During the late 1940s and early 1950s, there were two international aviation radio spelling alphabets, the "Able Baker" was used by most Western countries, while the "Ana Brazil" alphabet was used by South American and Caribbean regions.[25][26]
Pronunciation was not defined prior to 1959. From 1959 to present, the underlined syllable of each code word[whose?]for the letters should be stressed, and from 1969 to present, each syllable of the code words for the digits should be equally stressed, with the exceptions of the unstressed second syllables of fower, seven, niner, hundred.
After WWII, the major work in producing a better spelling alphabet was conducted by the ICAO, which was subsequently adopted in modified form by the ITU and IMO. Its development is related to these various international conventions on radio, including:
TheICAO Radiotelephony Alphabetis defined by theInternational Civil Aviation Organizationfor international aircraft communications.[36][37]
Defined by theAssociation of Public-Safety Communications Officials-International.[39]
The APCO first suggested that its Procedure and Signals Committee work out a system for a "standard set of words representing the alphabet should be used by all stations" in its April 1940 newsletter.[40][41]
Note: The old APCO alphabet has wide usage among Public Safety agencies nationwide,[clarification needed]even though APCO itself deprecated the alphabet in 1974, replacing it with the ICAO spelling alphabet. Seehttps://www.apcointl.organdAPCO radiotelephony spelling alphabet.
1967[42]
The FCC regulations for Amateur radio state that "Use of a phonetic alphabet as an aid for correct station identification is encouraged" (47 C.F.R. § 97.119(b)(2)[44]), but does not state which set of words should be used. Officially the same as used by ICAO, but there are significant variations commonly used by stations participating in HF contests and DX (especially in international HF communications).[45][46]
The official ARRL alphabet changed over the years, sometimes to reflect the current norms, and sometimes by the force of law. In rules made effective beginning April 1, 1946, the FCC forbade using the names of cities, states, or countries in spelling alphabets.[47]
Certain languages' standard alphabets have letters, or letters with diacritics (e.g.,umlauts,rings,tildes), that do not exist in the English alphabet. If these letters have two-letterASCIIsubstitutes, the ICAO/ITU code words for the two letters are used.
InDanishandNorwegianthe letters "æ", "ø" and "å" have their own code words. In DanishÆgir,ØdisandÅserepresent the three letters,[50]while in Norwegian the three code words areÆgir,ØrnulfandÅgotfor civilians andÆrlig,ØstenandÅsefor military personnel.[51]
Estonianhas four special letters,õ,ä,öandü.Õnnerepresentsõ,Ärniforä,ÖöbikforöandÜlleforü.[citation needed]
InFinnishthere are special code words for the letterså,äandö.Åkeis used to representå,Äitiis used foräandÖljyforö. These code words are used only in national operations, the last remnants of theFinnish radio alphabet.[52]
InGerman,Alfa-Echo(ae) may be used for "ä",Oscar-Echo(oe) for "ö",Sierra-Sierra(ss) for "ß", andUniform-Echo(ue) for "ü".
TheGreek spelling alphabetis a spelling alphabet for the Greek language, i.e. a set of names used in lieu of alphabet letters for the purpose of spelling out words. It is used by the Greek armed and emergency services.
Malay(includingIndonesian) represents the letter "L" with "London", since the wordlimameans "five" in this language.[53][54][55]
TheRussian spelling alphabetis a spelling alphabet for the Russian version of theCyrillic alphabet.
InSpanishthe wordñoño([ˈɲo.ɲo], 'dull') is used forñ.[56][57]
Åkeis used for "å"Ärligfor "ä" andÖstenfor "ö" in theSwedishspelling alphabet, though the two-letter substitutesaa,aeandoerespectively may be used in absence of the specific letters.[58][17]
[68]
/Julius
/Quotiënt
ThePGP word list, theBubble Babblewordlist used byssh-keygen, and theS/KEYdictionary, are spelling alphabets forpublic keyfingerprints (or other binary data) – a set of names given to data bytes for the purpose of spelling out binary data in a clear and unambiguous way via a voice channel.
Many unofficial spelling alphabets are in use that are not based on a standard, but are based on words the transmitter can remember easily, includingfirst names, states, or cities. TheLAPD phonetic alphabethas many first names. TheGerman spelling alphabet("Deutsches Funkalphabet" (literally "German Radio Alphabet")) also uses first names. Also, during the Vietnam war, soldiers used 'Cain' instead of 'Charlie' because 'Charlie' meant Viet Cong (Charlie being short for Victor Charlie, the International alphabet spelling of the initials VC).
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Incommunicationsandinformation processing,codeis a system of rules to convertinformation—such as aletter,word, sound, image, orgesture—into another form, sometimesshortenedorsecret, for communication through acommunication channelor storage in astorage medium. An early example is an invention oflanguage, which enabled a person, throughspeech, to communicate what they thought, saw, heard, or felt to others. But speech limits the range of communication to the distance a voice can carry and limits the audience to those present when the speech is uttered. The invention ofwriting, which converted spoken language intovisualsymbols, extended the range of communication across space andtime.
The process ofencodingconverts information from asourceinto symbols for communication or storage.Decodingis the reverse process, converting code symbols back into a form that the recipient understands, such as English, Spanish, etc.
One reason for coding is to enable communication in places where ordinaryplain language, spoken or written, is difficult or impossible. For example,semaphore, where the configuration offlagsheld by a signaler or the arms of asemaphore towerencodes parts of the message, typically individual letters, and numbers. Another person standing a great distance away can interpret the flags and reproduce the words sent.
Ininformation theoryandcomputer science, a code is usually considered as analgorithmthat uniquely representssymbolsfrom some sourcealphabet, byencodedstrings, which may be in some other target alphabet. An extension of the code for representing sequences of symbols over the source alphabet is obtained by concatenating the encoded strings.
Before giving a mathematically precise definition, this is a brief example. The mapping
is a code, whose source alphabet is the set{a,b,c}{\displaystyle \{a,b,c\}}and whose target alphabet is the set{0,1}{\displaystyle \{0,1\}}. Using the extension of the code, the encoded string 0011001 can be grouped into codewords as 0 011 0 01, and these in turn can be decoded to the sequence of source symbolsacab.
Using terms fromformal language theory, the precise mathematical definition of this concept is as follows: let S and T be two finite sets, called the source and targetalphabets, respectively. AcodeC:S→T∗{\displaystyle C:\,S\to T^{*}}is atotal functionmapping each symbol from S to asequence of symbolsover T. TheextensionC′{\displaystyle C'}ofC{\displaystyle C}, is ahomomorphismofS∗{\displaystyle S^{*}}intoT∗{\displaystyle T^{*}}, which naturally maps each sequence of source symbols to a sequence of target symbols.
In this section, we consider codes that encode each source (clear text) character by acode wordfrom some dictionary, andconcatenationof such code words give us an encoded string. Variable-length codes are especially useful when clear text characters have different probabilities; see alsoentropy encoding.
Aprefix codeis a code with the "prefix property": there is no valid code word in the system that is aprefix(start) of any other valid code word in the set.Huffman codingis the most known algorithm for deriving prefix codes. Prefix codes are widely referred to as "Huffman codes" even when the code was not produced by a Huffman algorithm. Other examples of prefix codes aretelephone country codes, the country and publisher parts ofISBNs, and the Secondary Synchronization Codes used in theUMTSWCDMA3G Wireless Standard.
Kraft's inequalitycharacterizes the sets of codeword lengths that are possible in a prefix code. Virtually any uniquely decodable one-to-many code, not necessarily a prefix one, must satisfy Kraft's inequality.
Codes may also be used to represent data in a way more resistant to errors in transmission or storage. This so-callederror-correcting codeworks by including carefully crafted redundancy with the stored (or transmitted) data. Examples includeHamming codes,Reed–Solomon,Reed–Muller,Walsh–Hadamard,Bose–Chaudhuri–Hochquenghem,Turbo,Golay,algebraic geometry codes,low-density parity-check codes, andspace–time codes.
Error detecting codes can be optimised to detectburst errors, orrandom errors.
A cable code replaces words (e.g.shiporinvoice) with shorter words, allowing the same information to be sent with fewercharacters, more quickly, and less expensively.
Codes can be used for brevity. Whentelegraphmessages were the state of the art in rapid long-distance communication, elaborate systems ofcommercial codesthat encoded complete phrases into single mouths (commonly five-minute groups) were developed, so that telegraphers became conversant with such "words" asBYOXO("Are you trying to weasel out of our deal?"),LIOUY("Why do you not answer my question?"),BMULD("You're a skunk!"), orAYYLU("Not clearly coded, repeat more clearly.").Code wordswere chosen for various reasons:length,pronounceability, etc. Meanings were chosen to fit perceived needs: commercial negotiations, military terms for military codes, diplomatic terms for diplomatic codes, any and all of the preceding for espionage codes. Codebooks and codebook publishers proliferated, including one run as a front for the AmericanBlack Chamberrun byHerbert Yardleybetween the First and Second World Wars. The purpose of most of these codes was to save on cable costs. The use of data coding fordata compressionpredates the computer era; an early example is the telegraphMorse codewhere more-frequently used characters have shorter representations. Techniques such asHuffman codingare now used by computer-basedalgorithmsto compress large data files into a more compact form for storage or transmission.
Character encodings are representations of textual data. A given character encoding may be associated with a specific character set (the collection of characters which it can represent), though some character sets have multiple character encodings and vice versa. Character encodings may be broadly grouped according to the number of bytes required to represent a single character: there are single-byte encodings,multibyte(also called wide) encodings, andvariable-width(also called variable-length) encodings. The earliest character encodings were single-byte, the best-known example of which isASCII. ASCII remains in use today, for example inHTTP headers. However, single-byte encodings cannot model character sets with more than 256 characters. Scripts that require large character sets such asChinese, Japanese and Koreanmust be represented with multibyte encodings. Early multibyte encodings were fixed-length, meaning that although each character was represented by more than one byte, all characters used the same number of bytes ("word length"), making them suitable for decoding with a lookup table. The final group, variable-width encodings, is a subset of multibyte encodings. These use more complex encoding and decoding logic to efficiently represent large character sets while keeping the representations of more commonly used characters shorter or maintaining backward compatibility properties. This group includesUTF-8, an encoding of theUnicodecharacter set; UTF-8 is the most common encoding of text media on the Internet.
Biologicalorganisms contain genetic material that is used to control their function and development. This isDNA, which contains units namedgenesfrom whichmessenger RNAis derived. This in turn producesproteinsthrough agenetic codein which a series of triplets (codons) of four possiblenucleotidescan be translated into one of twenty possibleamino acids. A sequence of codons results in a corresponding sequence of amino acids that form a protein molecule; a type of codon called astop codonsignals the end of the sequence.
Inmathematics, aGödel codeis the basis for the proof ofGödel'sincompleteness theorem. Here, the idea is to mapmathematical notationto anatural number(using aGödel numbering).
There are codes using colors, liketraffic lights, thecolor codeemployed to mark the nominal value of theelectrical resistorsor that of the trashcans devoted to specific types of garbage (paper, glass, organic, etc.).
Inmarketing,couponcodes can be used for a financial discount or rebate when purchasing a product from a (usual internet) retailer.
In military environments, specific sounds with thecornetare used for different uses: to mark some moments of the day, to command the infantry on the battlefield, etc.
Communication systems for sensory impairments, such assign languagefor deaf people andbraillefor blind people, are based on movement or tactile codes.
Musical scoresare the most common way to encodemusic.
Specific games have their own code systems to record the matches, e.g.chess notation.
In thehistory of cryptography,codeswere once common for ensuring the confidentiality of communications, althoughciphersare now used instead.
Secret codes intended to obscure the real messages, ranging from serious (mainlyespionagein military, diplomacy, business, etc.) to trivial (romance, games) can be any kind of imaginative encoding:flowers, game cards, clothes, fans, hats, melodies, birds, etc., in which the sole requirement is the pre-agreement on the meaning by both the sender and the receiver.
Other examples of encoding include:
Other examples of decoding include:
Acronymsand abbreviations can be considered codes, and in a sense, alllanguagesandwriting systemsare codes for human thought.
International Air Transport Association airport codesare three-letter codes used to designate airports and used forbag tags.Station codesare similarly used on railways but are usually national, so the same code can be used for different stations if they are in different countries.
Occasionally, a code word achieves an independent existence (and meaning) while the original equivalent phrase is forgotten or at least no longer has the precise meaning attributed to the code word. For example, '30' was widely used injournalismto mean "end of story", and has been used inother contextsto signify "the end".[1][2]
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InCWMorse codeoperations,QSKorfull break-in operationdescribes an operating mode in which the transmitting station can detect signals from other stations between the elements (dots and dashes) or letters of the Morse transmission. This allows other stations to interrupt the transmitting station between individual coding elements, and such allows for a conversational style of communication.
"QSK" is one of theQ-codesignals established forradiotelegraphoperators in the first decade of the 1900s. The three letter code "QSK" literally means "I can hear you between my signals; you may break in on my transmission." AlthoughMorse codeis no longer used for commercial or professional purposes, it continues to be used inamateur radio.
InQSKorfull break-inoperation the silent periods between the Morse code dits and dahs enable operators to listen between their transmitted signals for signals from the other operator, thus enabling a conversational style of communication. This is especially useful inhigh-speed telegraphy.
Morse code has silent periods between its symbol elements (dots and dashes), letters, words, and sentences. These silent periods provide the sending operator with opportunities to listen for interruptions from receiving stations. Whereas in usual CW operation the sendingcarrier waveis always on, and only gated to the antenna, in QSK operation the antenna is switched to receiver status on the off-time between dits and dahs, and then switched right back.
This fast switching between sending and receiving on the transceiver sending the currently message segment lets the sending side of the conversation hear that the receiving side has switched to transmitting, in between the off-times of Morse code. The intermixed signal is normally too garbled for either station to decode the other.[a]But when the QSK sender does hear the other station's sound filling in the normally silent gaps between the sender's own sent code elements, that informs the sender that the other station wants to break in, and the sender then has the option to politely stop and let the other operator back into the conversation early, before the sender has finished his / her current message segment.
QSK operationis a technical challenge: It requires very fast T/RRF switchesat the high power and voltage side of the radio transceiver. Such switches must be controlled automatically by thetelegraph key, and as such they must be rapid enough to be perceptually undetectable by the telegraph operator. In QSK operation, the T/R switches must be capable of automatically and rapidly switching the radio antenna or antennas between the transmitter and receiver during the short (dot duration) silent periods between Morse code signals, sometimes hundreds or even thousands of times per minute, for years on end. As such, the T/R switches used in QSK operation generally have the most stringent timing, reliability, and power handling specifications of all, and are quite expensive. Typically switching of this kind is taken on by vacuum reed switches, maybe aided by high voltagePIN diodes, and sometimes augmented by various circuit improvements that passively take the load off the active elements.
Full break-inorQSK operation,[1][2][3]is a hardware supportedMorse codecommunications channelturn overcommunications protocol. So,full break-inis also aduplexing protocol, which facilitates a style of two-way Morse code communications on traditionalhalf-duplexradiotelegraphchannels that closely simulatesfull-duplexchannel operations similar to the way normal human voice communications proceeds. It is not just a technology, but also a communications protocol and culture revolving around said technology.
With full break-in operation, the receiving operator can interrupt a sending operator in mid-character, similar to the way in which normal human voice conversations allow mid-syllable interruption of speakers by listeners.[4]This makes it possible to creatively break the normal Morse or other radiotelegraphy protocols, and as such, in QSK-operation, many informal idioms appear which are not in use in more codified forms of telegraphy.
For instance, to take turn in conversation, a held note of indefinite length and often varying by discussion/discussee, can be used. This is only possible because the other participant can hear a held note between his keying his own tone. Or as another break from convention, not only do high speed telegraphers often sign off with two non-spaced dits. Many of those whom go with HST and QSK sign off with rather elaborate combinations of dits, dahs, bad spacing, and a "click" here and there. As such, QSK, in its communicational style, also leads to lots of conversational style, which cannot be codified for real.
Semi break-inis a technique used by stations where slow (T/R) antenna switches are controlled indirectly by the telegraph key which lack the faster switching offull break-instations. Semi break-in hardware T/R switches are not required to switch as fast or to have the same long term reliability as their more expensive full break-in counterparts. Instead of using the telegraph key to directly control antenna switching, semi break-in radio transceiver equipment typically uses the telegraph key to control T/R switches indirectly, but still automatically, by passing the telegraph key information (usually in the form of a keyed audio tone) through a radio transceiver'sVoice-operated switch or VOXcircuitry.
In this technique, the relatively slow actingVOX circuitryis used to control the T/R switches.Voice-operated switch (VOX) circuitryis designed to be normally activated by human voice audio picked up by the transceiver microphone during voice communications in order to effect antenna change over at a rate no faster than the typicalhuman voice syllabic rateor slower. VOX circuitry usually has a front panel adjustable delay that can be used to control the length of time it takes for T/R switches to operate but generally the delay adjustment range is limited to that of human voice syllables and, although automatic, is generally not fast enough to act in the short periods between Morse code dots and dashes. Receiving stations thus cannot break-in or interruptsemi break-inorVOX controlledMorse code stations in mid-symbol or mid-word, during Morse code operation because the semi break-in sending station simply cannot hear in the short duration between the Morse code signals and words or code groups.
Receiving stations wishing to break-in on semi break-in stations must wait for the longer silent periods between the sending station's words or sentences before attempting to interrupt or break-in. At worst, receiving stations must wait until semi break-in stations explicitly turn over the channel to the receiving station by sending a breakprosign. Unlike full break-in operation, semi break-in operation is not fast enough to provide a fluid Morse code conversational capability approximating that of normal human voice conversation.
Although not as fluid and efficient as full break-in, semi break-in or VOX controlled break-in is a better Morse code channel turn over technique than puremanual break-in operation[citation needed]as described in the following paragraph.
Manual break-inis a technique used in a rudimentary Morse code radio station set up where antenna change over (T/R) switches are not controlled by the telegraph key. Instead antenna change over is accomplished manually by mechanical switches separate from the telegraph key on which the operator sends the Morse code. With such a simple manual turn over system there is no possibility of the sending operator listening between signals or symbols and therefore no possibility for the receiving operator to interrupt the sending operator. Instead the receiving operator must wait until a transmitting operator has indicated the end of transmission by means of a turn overprosignand has manually changed the antenna over from transmitter to receiver. Suchmanual break-inoperation leads to a very slow and stilted style of Morse code conversations.
QSK operationcomprises a hardware switch technology andprotocol, wherein participating Morse code stations are equipped with very fast analog radio frequency T/R switches connecting the transmitter, receiver, and antenna. This fast analog hardware switching capability enables a receiving station to interrupt or break-in on a transmitting station in mid-symbol (mid-character), a process known asfull break-in. The ability to hear between transmitted signals conferred by fast radio frequency hardware switching only requires Morse code operators to make use of simplecommunications protocolsto manage the channel turn over process. The typical QSK protocol technique is quite simple to learn and to master.
Since not all Morse code radio stations are equipped for QSK operation, sending stations equipped for QSK operation will often open a Morse conversation by sending the three letter groupQSK(e.g. the operator will assertQSK) during an initial (opening) Morse code transmission to alert receiving stations that the sending station has the ability to listen between signals and that the receiving station can interrupt, or break-in, on the sending station at will. Conversely a station may query another Morse code station's QSK capability by sending the QSK signal followed by a question mark. The queryQSK?asks if the receiving station has full break-in capability. If a receiving station is equipped for QSK operation the receiving operator will respond to the queryQSK?with the assertionQSKindicating that the station is operating with QSK enabled. Subsequently, the two stations can then use the fluid and efficient Morse code conversational QSK protocols outlined in the following paragraphs.
In practice, many skilled operators do not bother to open a conversation with the preliminary opening QSK assertion or QSK query protocols, instead merely attempting to interrupt a sending station by tapping their telegraph key while listening between the signals (dots and dashes) for what happens next. If the sending station pauses when interrupted each party automatically knows the other is using QSK operation and then the two stations immediately start using the following QSK interrupt and turn over protocol with no further ado.
Interruptions or break-ins are initiated by receiving stations momentarily depressing theirtelegraph keywhile the sending station is actively sending Morse code, thus generating a short interrupting signal which is heard by the sending station between its own signals. In practice usuallyonly a single dot is required to initiate a break-in.
Upon hearing the break-in signal between the dots and dashes being sent, the interrupted station stops sending immediately and
either way turning over the channel to the interrupter, and subsequently listens for the other station during the momentary pause. Skilledtelegraphistsseldom bother to send theKprosign when interrupted, instead simply letting the interrupter take over the channel during the pause.
The interrupting station, recognizing the momentary sending pause by the sender, immediately begins sending its own information to the interrupted station. Meanwhile, the interrupting station continues listening between its own transmitted signals in case of interruption in the reverse direction by the original sender.
These simple full break-in channel turn over protocols literally mimic the conversational style in which people interrupt each other mid-syllable during normal voice conversations. Full break-in QSK T/R switch hardware together with use of the simple QSK protocols enables a fast, efficient, fluid conversational style of Morse code communication.
Enormous signal level ranges must be accommodated by radio transceiver equipment. Transmitter output power for amateur radio stations might typically be 100 watts (+50dBm) or more, while received power atradio receiverantenna input terminals might typically be as low as −130dBm. This range of signal power that must be handled by various components of the T/R switching hardware encompasses an enormous total power handling range of up to 180 dBm (−130 to 50 dBm). This logarithmic measure of range encompasses a signal power ratio of 1 to 1 followed by 18 zeros (1:1,000,000,000,000,000,000).
Depending upon the Engineering set up, radiotelegraph stations may use either a single antenna for both transmit and receive or, separate transmit and receive antennas. In either case, when receivers are operating on the same or nearby radio frequencies as used by their associated transmitters, while using the same or nearby antennas, the typical radio receiver is thus exposed to extremely large signals from the nearby transmitter. This situation would generally result in the destruction or degradation of the receiver front end circuitry and would be problematic at best and destructive at worst. As of this writing there has yet apparently been no receiver technology developed that can operate with full sensitivity over such a huge range of received signal levels whilst also safely withstanding the high power levels presented by the associated nearby transmitter. And so receiver inputs cannot simply be bridged across transmitting antenna terminals! Receivers must be isolated from the powerful transmitter signals by some means. These means are provided by the so-called T/R switches.
The low level analog front end (AFE) amplifier circuitry of receivers sensitive enough to detect signals at −130 dBm levels and below are invariably extremely sensitive to high power levels. Typically, without the protection and isolation provided by T/R switches, the receiver AFE would be overwhelmed or destroyed by the normal transmitter power levels which are in the +50 dBm or more range. Consequently, receiver AFE antenna input terminals must be protected. With QSK operation this receiver protection is provided by well designed robust analog hardware T/R switches placed between the receiver AFE circuitry and the radio antenna.
The result of extreme receiver AFE sensitivity to high power levels is that, for most practical purposes, signal reception is impossible during periods when the associated transmitter is actually transmitting the dot and dash signals. Consequently, radiotelegraph operators cannot hear interruptions from remote receiving stations during normal signal transmission periods when the full transmitter power is applied to the antenna.
To protect receiver circuitry, radiotelegraph channels on nearby frequencies and antennas must operate in so-calledhalf-duplexmode wherein the stations at either end alternate between transmitting and receiving (because e.g. simultaneous transmit and receive is simply not possible). To support two way conversations on half duplex channels, analog radio frequency hardware antenna switches must be provided at each station location to connect and disconnect the transmitters and receivers from their antennas whenever the channel transmission control is turned over from one station to the other.
The aforementioned considerations are the prime motivations and considerations driving the development of radiotelegraph channelfull break-inQSK technologies:
Not all radio receivers are amenable to QSK operation.
Adding fast robust T/R switching externally to a transmitter/receiver combination (transceiver) will not necessarily result in good QSK operation. Adding such fast switching externally to a transceiver may create transients within receiver circuitry that makes signal copy: very noisy at best, and difficult, or impossible at worst.
Apart from the requirement for fast robust T/R switches, the main factor affecting good QSK operation is the ability for the radio receiver to recover its sensitivity quickly whilst operating quietly (without popping noises) during and after the fast transient signals created by the fast T/R switch operation. Many receivers haveautomatic gain control (AGC)circuitswith time constants that take many milliseconds to recover their sensitivity and volume level after a strong transient signal is presented to their antenna input port. Without modifications or AGC circuit re-design such receivers are not suitable for QSK operation. In cases of slow responding AGC circuitry operators may accept the thumping noise and loss of AGC functionality and choose to turn their receiver AGC function off, instead operating their receivers using only manual gain control during QSK operation.
Morse code operators aspiring to the convenience and conversational fluency of Morse code QSK operation who plan to add external QSK T/R switches to their existing or planned radio transceiver setups should ensure that their receiver AGC circuitry has recovery times commensurate with the T/R switching transients to be expected and that the AGC circuits can operate quickly in the sub-millisecond range without creating noisy pops and static at the receiver audio output (speaker or headphones). Of particular note is that many of the modern so-calledsoftware defined radio(SDR) transceivers have particularly slow AGC functions because of the latency created by the extensivedigital signal processing(A/D conversion, D/A conversion, digital filtering, digital modulation and digital demodulation) used for the SDR implementation. For these reasons, generally most SDR radios will not have the capability to operate QSK at the higher speed Morse code rates.
Expensive high end radio transceiver equipment that has been designed and manufactured withintegrated QSK capabilitywill generally meet such fast AGC recovery time requirements. Receiver recovery times may however be a potential issue for QSK operators who plan to addexternal QSK switchingto an existing radio equipment set up.
Full break-in hardware capability requires fast, robust, high power, analog, radio frequency (RF) transmit/receive (T/R) switches orRF switchescapable of operating in sub-millisecond response times over long periods of continuous operation while handling the high radio frequency power of the transmitter. Some high-end manufactured radio transceiver equipment contains integrated (factory installed) QSK switching hardware while in other cases external QSK switching hardware or commercial switching products may be added to existing non-QSK capable equipment.[5]
As an example illustrating switching speeds or timing requirements consider that when sendingMorse codeat a 20 word per minute rate the typicaldot signal durationis a mere 50 milliseconds. To enable good quality QSK operation the switching hardware must switch the radio antenna from receiver to transmitter in much less than one tenth of thedot duration. At 20 word per minute code speed this means that QSK T/R switching times must be in the range of 1 to ½ millisecond or below. Even smaller sub-millisecond times are required with higher speed Morse code transmissions.
Thedotting rateof Morse code is the reciprocal of the dot duration, e.g. at twenty words per minute based upon the standard word PARIS with a dot duration of 50 milliseconds, the dotting rate is 20 times per second (20 = 1.0 / 0.050). The dotting rate is even faster for higher speed Morse code. For long time reliability QSK T/R switches must be robust enough to open and close at least at a dotting rate of twenty times per second or even higher over thousands of hours of operation.
T/R switches must operate reliably at high dotting rates over many thousands of hours enabling the reception of extremely low level signals between dots and dashes while handling very high radio transmitter power levels of hundreds to thousands of Watts. Such robust high power analog radio frequency high speed switches are not inexpensive.
Examples of radio frequency analog hardware switch orRF switchtechnologies are high voltagevacuum relays[6]or high power semiconductorPIN diodeswitches. In recent times, as PIN diode power handling capabilities have been improved by the semiconductor industry, PIN diodes have largely supplanted vacuum relays in the QSK switch function because the absence of moving parts in PIN diode semiconductor devices results in: higher speeds, higher reliability and longer lifetimes.[7][8]An alternative approach uses power relays for QSK operation by adding a few milliseconds of delay in the keying line.[9]
Switching hardware technologies that can handle the radio frequency currents of high power transmitters and also switch quietly at these high Morse code rates over long periods of time are difficult to design and quite expensive to manufacture. Mechanical switches or relays are most problematic and least reliable and must be protected from arcs (sparking) usually by operating in a vacuum enclosure with elaborate timing circuitry. Not all radio transceiver equipment provides the costly high speed analog transmit/receive (T/R) radio frequency switching hardware support necessary for QSK full break-in operation. Generally full break-in is available only on more expensive radio transceivers. Radiotelegraphers who aspire to the fluency of Morse code QSK operation must ensure that their radio equipment includes the hardware capability for radio frequency antenna switching that operates rapidly enough to allow listening between signals, at the appropriate Morse code sending speeds, with appropriate lifetimes and reliability.
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Words per minute, commonly abbreviated asWPM(sometimes lowercased aswpm), is a measure ofwordsprocessed in a minute, often used as a measurement of the speed of typing,readingorMorse codesending and receiving.
Since words vary in length, for the purpose of measurement of text entry the definition of each "word" is often standardized to be five characters orkeystrokeslong in English,[1]including spaces and punctuation. For example, under such a method applied to plain English text the phrase "I run" counts as one word, but "rhinoceros" and "let's talk" would both count as two.
Karat et al. found in one study of averagecomputer usersin 1999 that the average rate for transcription was 32.5 words per minute, and 19.0 words per minute for composition.[2]In the same study, when the group was divided into "fast", "moderate", and "slow" groups, the average speeds were 40 wpm, 35 wpm, and 23 wpm, respectively.
With the onset of the era ofdesktop computersandsmartphones, fast typing skills became much more widespread. As of 2019, the average typing speed on a mobile phone was 36.2 wpm with 2.3% uncorrected errors—there were significant correlations with age, level of English proficiency, and number of fingers used to type.[3]Some typists have sustained speeds over 200 wpm for a 15-second typing test with simple English words.[4]
Typically, professionaltypiststype at speeds of 43 to 80 wpm, while some positions can require 80 to 95 (usually the minimum required fordispatchpositions and other time-sensitive typing jobs), and some advanced typists work at speeds above 120 wpm.[5]Two-finger typists, sometimes also referred to as "hunt and peck" typists, commonly reach sustained speeds of about 37 wpm for memorized text and 27 wpm when copying text, but in bursts may be able to reach much higher speeds.[6]From the 1920s through the 1970s, typing speed (along withshorthandspeed) was an important secretarial qualification, and typing contests were popular and often publicized by typewriter companies as promotional tools.
Stenotype keyboards enable the trained user to input text as fast as 360 wpm at very highaccuracyfor an extended period, which is sufficient for real-time activities such as court reporting or closed captioning. While training dropout rates are very high — in some cases only 10% or even fewer graduate — stenotype students are usually able to reach speeds of 100–120 wpm within six months, which is faster than most alphanumeric typists. Guinness World Records gives 360 wpm with 97.23% accuracy as the highest achieved speed using a stenotype.[7]
The numeric entry or 10-key speed is a measure of one's ability to manipulate thenumeric keypadfound on most modern separate computer keyboards. It is used to measure speed for jobs such asdata entryof number information on items such asremittance advice, bills, or checks, as deposited tolock boxes. It is measured in keystrokes per hour (KPH). Many jobs require a certain KPH, often 8,000 or 10,000.[8]
For an adult population (age range 18–64) the average speed of copying is 68 letters per minute (approximately 13 wpm), with the range from a minimum of 26 to a maximum of 113 letters per minute (approximately 5 to 20 wpm).[9]
A study of police interview records showed that the highest speed fell in the range 120–155 characters per minute, the highest possible limit being 190 characters per minute.[10]
According to various studies the speed of handwriting of 3–7 graders varies from 25 to 94 letters per minute.[11]
Usingstenography(shorthand) methods, this rate increases greatly. Handwriting speeds up to 350 words per minute have been achieved in shorthand competitions.[12]
Words per minute is a common metric for assessingreadingspeed and is often used in the context of remedial skills evaluation, as well as in the context ofspeed reading, where it is a controversial measure of reading performance.
A word in this context is the same as in the context of speech.
Research done in 2012[13]measured the speed at which subjects read a text aloud, and found the typical range of speeds across 17 different languages to be 184±29 wpm or 863±234 characters per minute. However, the number of wpm varied between languages, even for languages that use the Latin orCyrillicalphabets: as low as 161±18 forFinnishand as high as 228±30 for English. This was because different languages have different average word lengths (longer words in such languages as Finnish and shorter words in English). However, the number of characters per minute tends to be around 1000 for all the tested languages. For the tested Asian languages that use particular writing systems (Arabic, Hebrew, Chinese, Japanese) these numbers are lower.
Scientific studies have demonstrated that reading—defined here as capturing and decoding all the words on every page—faster than 900 wpm is not feasible given the limits set by the anatomy of the eye.[14]
Whileproofreadingmaterials, people are able to read English at 200 wpm on paper, and 180 wpm on a monitor.[15][Those numbers from Ziefle, 1998, are for studies that used monitors prior to 1992. See Noyes & Garland 2008 for a modern tech view of equivalence.]
Audiobooksare recommended to be 150–160 words per minute, which is the range that people comfortably hear and vocalize words.[16]
Slide presentationstend to be closer to 100–125 wpm for a comfortable pace,[17]auctioneerscan speak at about 250 wpm,[18]and the fastest speakingpolicy debatersspeak from 350[19]to over 500 words per minute.[20]Internet speech calculators show that various things influence words per minute including nervousness.[18]
An example of anagglutinative language, the average rate ofTurkishspeech is reported to be about 220 syllables per minute. When the time spent for the silent parts of speech are removed, the so-called average articulation rate reaches 310 syllables per minute.[21]The average number of syllables per (written) word has been measured as 2.6.[22][23]For a comparison,Fleschhas suggested that the conversational English for consumers aims 1.5 syllables per word,[24]although these measures are dependent on corpus.
John Moschitta Jr.was listed inGuinness World Records, for a time, as the world's fastest speaker, being able to talk at 586 wpm.[25]He has since been surpassed bySteve Woodmore, who achieved a rate of 637 wpm.[26]
In the realm ofAmerican Sign Language, the American Sign Language University (ASLU) specifies a cutoff proficiency for students who clock a signing speed of 110-130 wpm.[27]
Morse codeuses variable length sequences of short and long durationsignals(dits and dahs, colloquially called dots and dashes) to represent source information[28]e.g., sequences for the letter "K" and numeral "2" are respectively (▄▄▄ ▄ ▄▄▄) and (▄ ▄ ▄▄▄ ▄▄▄ ▄▄▄). This variability complicates the measurement of Morse code speed rated in words per minute. Using telegram messages, the average English word length is about five characters, each averaging 5.124 dot durations orbaud. Spacing between words should also be considered, being seven dot durations in the USA and five in British territories. So the average British telegraph word was 30.67 dot times.[29]So the baud rate of a Morse code is50⁄60× word per minute rate.
It is standard practice to use two different suchstandard wordsto measure Morse code speeds in words per minute. The standard words are: "PARIS" and "CODEX". In Morse code "PARIS" has a dot duration of 50, while "CODEX" has 60.
Although many countries no longer require it for licensing, Morse is still widely used byamateur radio("ham") operators. Experienced hams routinely send Morse at 20 words per minute, using manually operated handtelegraph keys; enthusiasts such as members ofThe CW Operators' Clubroutinely send and receive Morse code at speeds up to 60 wpm. The upper limit for Morse operators attempting to write down Morse code received by ear using paper and pencil is roughly 20 wpm. Many skilled Morse code operators can receive Morse code by ear mentally without writing down the information at speeds up to 70 wpm.[30]To write down the Morse code information manually at speeds higher than 20 wpm it is usual for the operators to use a typewriter or computer keyboard to enable higher speed copying.
In the United States a commercial radiotelegraph operator's license is still issued, although there is almost no demand for it, since for long distance communication ships now use the satellite-basedGlobal Maritime Distress and Safety System. Besides a written examination, proficiency at receiving Morse at 20 wpm plain language and 16 wpm in code groups must be demonstrated.[31]
High-speed telegraphycontests are still held. The fastest Morse code operator wasTheodore Roosevelt McElroycopying at 75.6 wpm using a typewriter at the 1939 world championship.[32]
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AT&T Corporation, an abbreviation for its former name, theAmerican Telephone and Telegraph Company, was an Americantelecommunicationscompany that provided voice, video, data, and Internet telecommunications and professional services to businesses, consumers, and government agencies.
During theBell System's long history, AT&T was at times the world's largest telecommunications company, the world's largest cable television operator, and a regulated monopoly. At its peak in the 1950s and 1960s, it employed one million people and its revenue ranged between US$3 billion in 1950[4]($42.6 billion in present-day terms[5]) and $12 billion in 1966[6]($120 billion in present-day terms[5]).
In 2005, AT&T was acquired by "Baby Bell" and former subsidiary SBC Communications for more than $16 billion ($25.8 billion in present-day terms[5]). SBC then changed its name toAT&T Inc., with AT&T Corporation continuing to exist as along-distance callingsubsidiaryuntil its dissolution on May 1, 2024.[7]
AT&T started withBell Patent Association, a legal entity established in 1874 to protect the patent rights ofAlexander Graham Bellafter he invented the telephone system. Originally a verbal agreement, it was formalized in writing in 1875 asBell Telephone Company.[8][9]
In 1880 the management of American Bell created what would becomeAT&T Long Lines. The project was the first of its kind to create a nationwide long-distance network with a commercially viable cost-structure. The project was formally incorporated in New York as a separate company namedAmerican Telephone and Telegraph Companyon March 3, 1885. Originating in New York City, its long-distance telephone network reachedChicago, Illinois, in 1892,[10]with its multitudes of local exchanges continuing to stretch further and further yearly, eventually creating a continent-wide telephone system. On December 30, 1899, the assets of American Bell were transferred into its subsidiary American Telephone and Telegraph Company (formerlyAT&T Long Lines); this was becauseMassachusettscorporate laws were very restrictive, and limited capitalization to ten million dollars, forestalling American Bell's further growth. With this assets transfer at the very end of the 19th century, AT&T became the parent of both American Bell and theBell System.[11]
AT&T was involved mainly in the telephone business and, although it was a partner withRCA, was reluctant to see radio grow because such growth might diminish the demand for wired services. It established stationWEAFin New York as what was termed atoll station. AT&T could provide no programming, but anyone who wished to broadcast a message could pay a "toll" to AT&T and then air the message publicly. The original studio was the size of a telephone booth. The idea, however, did not take hold, because people would pay to broadcast messages only if they were sure that someone was listening. As a result, WEAF began broadcasting entertainment material, drawing amateur talent found among its employees. Opposition to AT&T's expansion into radio and an agreement with theNational Broadcasting Companyto lease long-distance lines for their broadcasts resulted in the sale of the station and its developing network of affiliates to NBC.[12]
On April 30, 1907,Theodore Newton Vailbecame president of AT&T.[13][14]Vail believed in the superiority of one phone system and AT&T adopted the slogan "One Policy, One System, Universal Service."[13][14]This would be the company's philosophy for the next 70 years.[14]
Under Vail, AT&T began buying up many of the smaller telephone companies includingWestern Union telegraph.[13][14]These actions brought unwanted attention from antitrust regulators. Anxious to avoid action from governmentantitrustsuits, AT&T and the federal government entered into an agreement known as theKingsbury Commitment.[13][14]In the Kingsbury Commitment, AT&T and the government reached an agreement that allowed AT&T to continue operating as a telephone monopoly, subject to certain conditions, including divesting its interest in Western Union. While AT&T periodically faced scrutiny from regulators, this state of affairs continued until the company's breakup in 1984. Throughout most of the 20th century, AT&T held a semi-monopoly on phone service in the United States and Canada through a network of companies called theBell System. At this time, the company was nicknamedMa Bell.
AT&T had a domestic and global presence in laying the infrastructure of undersea routes for telecommunications. In 1950, the U.S. Navy commissioned a network of undersea surveillance cables for foreign submarine detection. AT&T was probably, according to internal employees, involved in this Sound Surveillance System (SOSUS). After completion, AT&T began commercial operations in cable laying for communications in 1955.[15]The implementation of cables assured local and long-distance telephone or data services would provide revenue for the company.[16]AT&T Long Lineswas one of the divisions responsible for the cable-laying and maintaining of Long Lines' undersea cables.[17]Western Electricwas the manufacturing company responsible for production and supply of undersea coaxial equipment and later, fiber cables. Equipment such as repeaters was manufactured in Clark, New Jersey and coaxial cable was manufactured in Baltimore, Maryland .[18]Also,Bell Labswas responsible for the innovations of products[19]or technologies in cabling in transmission by undersea systems.[20]In 1955, the first trans-Atlantic telephone undersea cable,TAT-1, from North America to Europe was installed by AT&T. This installation allowed 48 telephone circuits to be used for long-distance calling.[21]When partnering with other global Telecommunications companies, such as the French Cables de Lyon and German Felten & Guilleaume, Bell Labs provided the specification and inspection of non Bell System cable for networks such as theTAT-2.[18]By the continuous undersea network installations, AT&T was a globally technology leader with the 1970 installedTAT-5and the 1975 installedTAT-6, achieving 720 channels and then 4000 channels for transmitting voice or data.[18]Prior to 1963, AT&T had to charter oceanic ships, such as theCS Monarch (1945)for installations. AT&T purchasedCS Long Linesin 1961 and operated it with several cable laying ships that would provide, either the laying or repair of cabling under the subsidiary, Transoceanic Cable Ship Company. After the break-up, AT&T operated their ships under a subsidiary called AT&T Submarine Systems Inc, based in Morristown, New Jersey, until they sold six ships to Tyco International Ltd in 1997 for $850 million.[22]AT&T continued to maintain their communication building facilities. Here is a list of the cable laying-ship fleet:
Between 1951 and 2000, AT&T was listed 73 times in cable laying operations for specific routes deployed.[31]The Cable ShipLong Lineshad 23 cable runs from 1963 to 1992, with the first deep-sea trial of optical fiber cable in 1982 leading to the consortium of countries and locations for theTAT-8fiber cable implementation of 1988.[32]
The United States Justice Department opened the caseUnited States v. AT&Tin 1974. This was prompted by suspicion that AT&T was using monopoly profits from itsWestern Electricsubsidiary to subsidize the cost of its network, a violation of antitrust law.[33]A settlement to this case was finalized in 1982, leading to thedivision of the companyon January 1, 1984, into sevenRegional Bell Operating Companies, commonly known as Baby Bells. These companies were:
Post-breakup, the former parent company's main business was nowAT&T Communications Inc., which focused on long-distance services, and with other non-RBOC activities.
AT&T acquiredNCR Corporationin 1991. AT&T announced in 1995 that it would split into three companies: a manufacturing/R&D company, a computer company, and a services company. NCR,Bell Labsand AT&T Technologies were to be spun off by 1997. In preparation for its spin-off, AT&T Technologies was nowLucent Technologies. Lucent was completely spun off from AT&T in 1996.
On January 31, 2005, the "Baby Bell" company SBC Communications announced its plans to acquire "Ma Bell" AT&T Corp. for $16 billion. SBC announced in October 2005 that it would shed the "SBC" brand and take the more recognizable AT&T brand, along with the old AT&T's "T"NYSEticker symbol.
Merger approval concluded on November 18, 2005; SBC Communications began rebranding the following Monday, November 21 as "the new AT&T" and began trading under the "T" symbol on December 1. Present-day AT&T Inc. claims AT&T Corp.'s history as its own, but retains SBC's pre-2005 stock price history and corporate structure. As well, all SEC filings before 2005 are under SBC, not AT&T.
From 1885 to 1910, AT&T was headquartered at 125 Milk Street in Boston. With its expansion it moved to New York City, to a headquarters on195 Broadway(close to what is now theWorld Trade Center site). The property originally belonged toWestern Union, of which AT&T held a controlling interest until 1913 when AT&T divested its interest as part of theKingsbury Commitment.[14]Construction of the current building began in 1912. Designed byWilliam Welles Bosworth, who played a significant role in designingKykuit, the Rockefeller mansion north ofTarrytown, New York, it was a modern steel structure clad top to bottom in a Greek-styled exterior, the three-story-high Ionic columns of Vermont granite forming eight registers over a Doric base.[34]The lobby of the AT&T Building was one of the most unusual ones of the era. Instead of a large double-high space, similar to the nearbyWoolworth Building, Bosworth designed what is called a "hypostyle hall", with full-bodied Doric columns modeled on the Parthenon, marking out a grid. Bosworth was seeking to coordinate the classical tradition with the requirements of a modern building. Columns were not merely the decorative elements they had become in the hands of other architects but created all the illusion of being real supports. Bosworth also designed the campus ofMITas well asTheodore N. Vail's mansion inMorristown, New Jersey.
In 1978, AT&T commissioned a new building at550 Madison Avenue. This new AT&T Building was designed byPhilip Johnsonand quickly became an icon of the newPostmodernarchitectural style. The building was completed in 1984, the very year of the divestiture of the Bell System. The building proved to be too large for the post-divestiture corporation and in 1993, AT&T leased the building toSony, who then subsequently owned the building until it was sold in 2013.[35][36]
In 1969, AT&T began plans to construct an administration corporate complex in the suburbs. In early 1970, AT&T began purchases of land in the suburbs of New Jersey for this office complex and began construction in 1974. The award-winning architect,Vincent Kling, designed aFordism[37]style, luxurious "Pagoda"[38]campus layout and the construction firms: New York–based Walter Kidde andNewark, New Jersey–based Frank Briscoe, managed this joint venture construction project with Vollers Construction ofBranchburg, New Jersey, as the subcontractor. The 295 North Maple Avenue and Interstate 287 location ofBasking RidgeinBernards Township, New Jersey was completed in 1975 for the AT&T General Department offices. Employees began moving, in November 1975, to the seven inter-connected building complex using 28 acre of the property. The property had a 15-acre underground parking garage with spaces for 3,900 vehicles, and included a Class 1 licensed privatehelipad, a two-story cafeteria, a wood-burning fireplace, an indoor waterfall at the entrance lobby, and a seven-acre created lake for flood control. The entire property was 130 acre and cost $219 million to construct. Later, across the street from the complex, AT&T purchased additional land and established its Learning Center in 1985, at 300 North Maple Avenue, to become a 171 conference room inn. The AT&T Learning Center won the commercial property known as Somerset County's Land Development Award that year. In 1992, Basking Ridge location would become a corporate headquarters just before AT&T leased the New York City, 550 Madison Avenue building to Sony in 1993. The corporate statue, known as "Golden Boy" was moved in 1992, from the former New York City headquarters to this current New Jersey headquarters. In 1992, a corporate art consultant approached, artist sculptor,Elyn Zimmerman, to commission a 30-foot diameter project with fountain and seating area for the conference center courtyard gardens. In 1994, the project was completed and had one 34 ton granite boulder centered on top of the other boulders, which flowed water from the fountain designed by fountain engineer, Dr. Gerald Palevsky.[39]AT&T occupancy at the location peaked to 6,000 employees in its heyday before AT&T experienced competition and downsizing.
In October 2001, the Basking Ridge property was 140 acre with 2.6 million square feet and was placed for sale.[40]Basking Ridge employee occupancy, prior to the sale were approximately 3,200 employees. In April 2002,Pharmacia Corporationpurchased the complex for $210 million for their corporate headquarters from existingPeapack-Gladstone, New Jerseyheadquarters.[41]A short time afterwards, in 2005,Verizonpurchased the former complex, excluding the hotel/conference room building,[42]fromPfizerfor Verizon Wireless Headquarters and consolidation of employees from Manhattan as well as other nearby New Jersey building locations.[43]In 2007, Pfizer placed the North Maple Inn for sale. At the time, it was a four-diamond, certified hotel and conference center under IACC ("International Association of Conference Centers") designation.[44]In 2015, Verizon performed a sale-leaseback agreement valued at $650.3 million on the complex with the address previously known as One Verizon Way.[45]In 2017, the 35 acre hotel/conference center was known as the Dolce Basking Ridge Hotel and sold for $30 million.[46]
On February 15, 2024, AT&T Inc. filed notice with the Kentucky Public Service Commission that it intends to make an internal structural change and merge AT&T Corp. into AT&T Enterprises, Inc., which will become alimited liability company. In a filing with theSouth Dakota Secretary of Statedated January 30, 2024, the reason given for the merger is thatNew Yorkstate law does not allow AT&T Corp. to be directly converted into an LLC.[47]Although acquired by SBC in 2005, AT&T Corp. has remained a separate entity within the corporate structure of AT&T Inc. The merger, said to create “greater operational efficiencies”, will end the existence of the nearly 140-year-old entity. The internal merger took effect on May 1, 2024.[48]
AT&T, prior to its merger withSBC Communications, had three core companies:
AT&T Alascomsold service in Alaska.AT&T Communicationswas renamedAT&T Communications – East, Inc.and sold long-distance telephone service and operated as aCLECoutside of the borders of theBell Operating CompaniesthatAT&Towned. It has now been absorbed into AT&T Corp. and all but 4 of the original 22 subsidiaries that formed AT&T Communications continue to exist.AT&T Laboratorieshas been integrated intoAT&T Labs, formerly namedSBC Laboratories.
AT&T was formerly known as "Ma Bell" and affectionately called "Mother" by phonephreaks. During some strikes by its employees, picketers would wear T-shirts reading, "Ma Bell is a real mother." Before the break-up, there was greater consumer recognition of the "Bell System" name, in comparison to the name AT&T. This prompted the company to launch an advertising campaign after the break-up to increase its name recognition. Spinoffs like theRegional Bell Operating CompaniesorRBOCs were often called "Baby Bells". Ironically, "Ma Bell" was acquired by one of its "Baby Bells",SBC Communications, in 2005.
The AT&T Globe Symbol,[49]the corporate logo designed bySaul Bassin 1983 and originally used byAT&T Information Systems, was created because part of theUnited States v. AT&Tsettlementrequired AT&T to relinquish all claims to the use ofBell Systemtrademarks. It has been nicknamed the "Death Star" in reference to the Death Star space station inStar Warswhich the logo resembles. In 1999 it was changed from the 12-line design to the 8-line design. Again in 2005 it was changed to the 3D transparent "marble" design created by Interbrand for use by the parent companyAT&T Inc.This name was also given to the iconicBell Labs facilityinHolmdel, New Jersey, now a multi-tenant office facility.[citation needed]
The following is a list of the 16 CEOs of AT&T Corporation, from its incorporation in 1885 until its purchase by SBC Communications in 2005.[50]
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Electrical telegraphyispoint-to-pointdistance communicating via sending electric signals over wire, a system primarily used from the 1840s until the late 20th century. It was the first electricaltelecommunicationssystem and the most widely used of a number of early messaging systems calledtelegraphs, that were devised to send text messages more quickly than physically carrying them.[1][2]Electrical telegraphy can be considered the first example ofelectrical engineering.[3]
Electrical telegraphy consisted of two or more geographically separated stations, calledtelegraph offices. The offices were connected by wires, usually supported overhead onutility poles. Many electrical telegraph systems were invented that operated in different ways, but the ones that became widespread fit into two broad categories. First are the needle telegraphs, in which electric current sent down the telegraph line produces electromagnetic force to move a needle-shaped pointer into position over a printed list. Early needle telegraph models used multiple needles, thus requiring multiple wires to be installed between stations. The first commercial needle telegraph system and the most widely used of its type was theCooke and Wheatstone telegraph, invented in 1837. The second category are armature systems, in which the current activates atelegraph sounderthat makes a click; communication on this type of system relies on sending clicks in coded rhythmic patterns. The archetype of this category was the Morse system and the code associated with it, both invented bySamuel Morsein 1838. In 1865, the Morse system became the standard for international communication, using a modified form of Morse's code that had been developed for German railways.
Electrical telegraphs were used by the emerging railway companies to provide signals for train control systems, minimizing the chances of trains colliding with each other.[4]This was built around thesignalling block systemin which signal boxes along the line communicate with neighbouring boxes by telegraphic sounding ofsingle-stroke bellsand three-positionneedle telegraphinstruments.
In the 1840s, the electrical telegraph supersededoptical telegraphsystems such as semaphores, becoming the standard way to send urgent messages. By the latter half of the century, mostdeveloped nationshad commercial telegraph networks with local telegraph offices in most cities and towns, allowing the public to send messages (calledtelegrams) addressed to any person in the country, for a fee.
Beginning in 1850,submarine telegraph cablesallowed for the first rapid communication between people on different continents. The telegraph's nearly-instant transmission of messages across continents – and between continents – had widespread social and economic impacts. The electric telegraph led toGuglielmo Marconi's invention ofwireless telegraphy, the first means ofradiowavetelecommunication, which he began in 1894.[5]
In the early 20th century, manual operation of telegraph machines was slowly replaced byteleprinternetworks. Increasing use of thetelephonepushed telegraphy into only a few specialist uses; its use by the general public dwindled to greetings for special occasions. The rise of theInternetandemailin the 1990s largely made dedicated telegraphy networks obsolete.
Prior to the electric telegraph, visual systems were used, includingbeacons,smoke signals,flag semaphore, andoptical telegraphsfor visual signals to communicate over distances of land.[6]
An auditory predecessor was West Africantalking drums. In the 19th century,Yorubadrummers used talking drums to mimic humantonallanguage[7][8]to communicate complex messages – usually regarding news of birth, ceremonies, and military conflict – over 4–5 mile distances.[9]
Possibly the earliest design and conceptualization for a telegraph system was by the BritishpolymathRobert Hooke, who gave a vivid and comprehensive outline of visual telegraphy to theRoyal Societyin a 1684 submission in which he outlined many practical details. The system was largely motivated by military concerns, following the Battle of Vienna in 1683.[10][11]
The first official optical telegraph was invented in France in the 18th century byClaude Chappeand his brothers. The Chappe system would stretch nearly 5,000 km with 556 stations and was used until the 1850s.[12]
Fromearly studies of electricity, electrical phenomena were known to travel with great speed, and many experimenters worked on the application of electricity tocommunicationsat a distance. All the known effects of electricity – such assparks,electrostatic attraction,chemical changes,electric shocks, and laterelectromagnetism– were applied to the problems of detecting controlled transmissions of electricity at various distances.[13]
In 1753, an anonymous writer in theScots Magazinesuggested an electrostatic telegraph. Using one wire for each letter of the alphabet, a message could be transmitted by connecting the wire terminals in turn to an electrostatic machine, and observing the deflection ofpithballs at the far end.[14]The writer has never been positively identified, but the letter was signed C.M. and posted fromRenfrewleading to a Charles Marshall of Renfrew being suggested.[15]Telegraphs employing electrostatic attraction were the basis of early experiments in electrical telegraphy in Europe, but were abandoned as being impractical and were never developed into a useful communication system.[16]
In 1774,Georges-Louis Le Sagerealised an early electric telegraph. The telegraph had a separate wire for each of the 26 letters of thealphabetand its range was only between two rooms of his home.[17]
In 1800,Alessandro Voltainvented thevoltaic pile, providing acontinuous currentofelectricityfor experimentation. This became a source of a low-voltage current that could be used to produce more distinct effects, and which was far less limited than the momentary discharge of anelectrostatic machine, which withLeyden jarswere the only previously known human-made sources of electricity.
Another very early experiment in electrical telegraphy was an "electrochemical telegraph" created by theGermanphysician, anatomist and inventorSamuel Thomas von Sömmeringin 1809, based on an earlier 1804 design by Spanishpolymathand scientistFrancisco Salva Campillo.[18]Both their designs employed multiple wires (up to 35) to represent almost all Latin letters and numerals. Thus, messages could be conveyed electrically up to a few kilometers (in von Sömmering's design), with each of the telegraph receiver's wires immersed in a separate glass tube of acid. An electric current was sequentially applied by the sender through the various wires representing each letter of a message; at the recipient's end, the currents electrolysed the acid in the tubes in sequence, releasing streams of hydrogen bubbles next to each associated letter or numeral. The telegraph receiver's operator would watch the bubbles and could then record the transmitted message.[18]This is in contrast to later telegraphs that used a single wire (with ground return).
Hans Christian Ørsteddiscovered in 1820 that an electric current produces a magnetic field that will deflect a compass needle. In the same yearJohann Schweiggerinvented thegalvanometer, with a coil of wire around a compass, that could be used as a sensitive indicator for an electric current.[19]Also that year,André-Marie Ampèresuggested that telegraphy could be achieved by placing small magnets under the ends of a set of wires, one pair of wires for each letter of the alphabet. He was apparently unaware of Schweigger's invention at the time, which would have made his system much more sensitive. In 1825,Peter Barlowtried Ampère's idea but only got it to work over 200 feet (61 m) and declared it impractical. In 1830William Ritchieimproved on Ampère's design by placing the magnetic needles inside a coil of wire connected to each pair of conductors. He successfully demonstrated it, showing the feasibility of the electromagnetic telegraph, but only within a lecture hall.[20]
In 1825,William Sturgeoninvented theelectromagnet, with a single winding of uninsulated wire on a piece of varnishediron, which increased the magnetic force produced by electric current.Joseph Henryimproved it in 1828 by placing several windings of insulated wire around the bar, creating a much more powerful electromagnet which could operate a telegraph through the high resistance of long telegraph wires.[21]During his tenure atThe Albany Academyfrom 1826 to 1832, Henry first demonstrated the theory of the 'magnetic telegraph' by ringing a bell through one-mile (1.6 km) of wire strung around the room in 1831.[22]
In 1835,Joseph HenryandEdward Davyindependently invented the mercury dippingelectrical relay, in which a magnetic needle is dipped into a pot of mercury when an electric current passes through the surrounding coil.[23][24][25]In 1837, Davy invented the much more practical metallic make-and-break relay which became the relay of choice in telegraph systems and a key component for periodically renewing weak signals.[26]Davy demonstrated his telegraph system inRegent's Parkin 1837 and was granted a patent on 4 July 1838.[27]Davy also invented a printing telegraph which used the electric current from the telegraph signal to mark a ribbon of calico infused withpotassium iodideandcalcium hypochlorite.[28]
The first working telegraph was built by the English inventorFrancis Ronaldsin 1816 and used static electricity.[29]At the family home onHammersmith Mall, he set up a complete subterranean system in a 175-yard (160 m) long trench as well as an eight-mile (13 km) long overhead telegraph. The lines were connected at both ends to revolving dials marked with the letters of the alphabet and electrical impulses sent along the wire were used to transmit messages. Offering his invention to theAdmiraltyin July 1816, it was rejected as "wholly unnecessary".[30]His account of the scheme and the possibilities of rapid global communication inDescriptions of an Electrical Telegraph and of some other Electrical Apparatus[31]was the first published work on electric telegraphy and even described the risk ofsignal retardationdue to induction.[32]Elements of Ronalds' design were utilised in the subsequent commercialisation of the telegraph over 20 years later.[33]
TheSchilling telegraph, invented byBaron Schillingvon Canstatt in 1832, was an earlyneedle telegraph. It had a transmitting device that consisted of a keyboard with 16 black-and-white keys.[34]These served for switching the electric current. The receiving instrument consisted of sixgalvanometerswith magnetic needles, suspended fromsilkthreads. The two stations of Schilling's telegraph were connected by eight wires; six were connected with the galvanometers, one served for the return current and one for a signal bell. When at the starting station the operator pressed a key, the corresponding pointer was deflected at the receiving station. Different positions of black and white flags on different disks gave combinations which corresponded to the letters or numbers. Pavel Schilling subsequently improved its apparatus by reducing the number of connecting wires from eight to two.
On 21 October 1832, Schilling managed a short-distance transmission of signals between two telegraphs in different rooms of his apartment. In 1836, the British government attempted to buy the design but Schilling instead accepted overtures fromNicholas I of Russia. Schilling's telegraph was tested on a 5-kilometre-long (3.1 mi) experimental underground and underwater cable, laid around the building of the main Admiralty in Saint Petersburg and was approved for a telegraph between the imperial palace atPeterhofand the naval base atKronstadt. However, the project was cancelled following Schilling's death in 1837.[35]Schilling was also one of the first to put into practice the idea of thebinarysystem of signal transmission.[34]His work was taken over and developed byMoritz von Jacobiwho invented telegraph equipment that was used by TsarAlexander IIIto connect the Imperial palace atTsarskoye SeloandKronstadt Naval Base.
In 1833,Carl Friedrich Gauss, together with the physics professorWilhelm WeberinGöttingen, installed a 1,200-metre-long (3,900 ft) wire above the town's roofs. Gauss combined thePoggendorff-Schweigger multiplicatorwith his magnetometer to build a more sensitive device, thegalvanometer. To change the direction of the electric current, he constructed acommutatorof his own. As a result, he was able to make the distant needle move in the direction set by the commutator on the other end of the line.
At first, Gauss and Weber used the telegraph to coordinate time, but soon they developed other signals and finally, their own alphabet. The alphabet was encoded in a binary code that was transmitted by positive or negative voltage pulses which were generated by means of moving an induction coil up and down over a permanent magnet and connecting the coil with the transmission wires by means of the commutator. The page of Gauss's laboratory notebook containing both his code and the first message transmitted, as well as a replica of the telegraph made in the 1850s under the instructions of Weber are kept in the faculty of physics at theUniversity of Göttingen, in Germany.
Gauss was convinced that this communication would be of help to his kingdom's towns. Later in the same year, instead of avoltaic pile, Gauss used aninductionpulse, enabling him to transmit seven letters a minute instead of two. The inventors and university did not have the funds to develop the telegraph on their own, but they received funding fromAlexander von Humboldt.Carl August SteinheilinMunichwas able to build a telegraph network within the city in 1835–1836. In 1838, Steinheil installed a telegraph along theNuremberg–Fürth railway line, built in 1835 as the first German railroad, which was the firstearth-return telegraphput into service.
By 1837,William Fothergill CookeandCharles Wheatstonehad co-developed atelegraph systemwhich used a number of needles on a board that could be moved to point to letters of the alphabet. Any number of needles could be used, depending on the number of characters it was required to code. In May 1837 they patented their system. The patent recommended five needles, which coded twenty of the alphabet's 26 letters.
Samuel Morseindependently developed and patented a recording electric telegraph in 1837. Morse's assistantAlfred Vaildeveloped an instrument that was called the register for recording the received messages. It embossed dots and dashes on a moving paper tape by a stylus which was operated by an electromagnet.[36]Morse and Vail developed theMorse codesignallingalphabet.
On 24 May 1844, Morse sent to Vail the historic first message “WHAT HATH GOD WROUGHT" from theCapitolin Washington to theold Mt. Clare DepotinBaltimore.[37][38]
The first commercial electrical telegraph was theCooke and Wheatstone system. A demonstration four-needle system was installed on theEustontoCamden Townsection ofRobert Stephenson'sLondon and Birmingham Railwayin 1837 for signalling rope-hauling of locomotives.[39]It was rejected in favour of pneumatic whistles.[40]Cooke and Wheatstone had their first commercial success with a system installed on theGreat Western Railwayover the 13 miles (21 km) fromPaddington stationtoWest Draytonin 1838.[41]This was a five-needle, six-wire[40]system, and had the major advantage of displaying the letter being sent so operators did not need to learn a code. The insulation failed on the underground cables between Paddington and West Drayton,[42][43]and when the line was extended toSloughin 1843, the system was converted to a one-needle, two-wire configuration with uninsulated wires on poles.[44]The cost of installing wires was ultimately more economically significant than the cost of training operators. The one-needle telegraph proved highly successful on British railways, and 15,000 sets were in use at the end of the nineteenth century; some remained in service in the 1930s.[45]TheElectric Telegraph Company, the world's first public telegraphy company, was formed in 1845 by financierJohn Lewis Ricardoand Cooke.[46][47]
Wheatstone developed a practical alphabetical system in 1840 called the A.B.C. System, used mostly on private wires. This consisted of a "communicator" at the sending end and an "indicator" at the receiving end. The communicator consisted of a circular dial with a pointer and the 26 letters of the alphabet (and four punctuation marks) around its circumference. Against each letter was a key that could be pressed. A transmission would begin with the pointers on the dials at both ends set to the start position. The transmitting operator would then press down the key corresponding to the letter to be transmitted. In the base of the communicator was amagnetoactuated by a handle on the front. This would be turned to apply an alternating voltage to the line. Each half cycle of the current would advance the pointers at both ends by one position. When the pointer reached the position of the depressed key, it would stop and the magneto would be disconnected from the line. The communicator's pointer was geared to the magneto mechanism. The indicator's pointer was moved by a polarised electromagnet whosearmaturewas coupled to it through anescapement. Thus the alternating line voltage moved the indicator's pointer on to the position of the depressed key on the communicator. Pressing another key would then release the pointer and the previous key, and re-connect the magneto to the line.[48]These machines were very robust and simple to operate, and they stayed in use in Britain until well into the 20th century.[49][50]
The Morse system uses a single wire between offices. At the sending station, an operator taps on a switch called atelegraph key, spelling out text messages inMorse code. Originally, the armature was intended to make marks on paper tape, but operators learned to interpret the clicks and it was more efficient to write down the message directly.
In 1851, a conference in Vienna of countries in the German-Austrian Telegraph Union (which included many central European countries) adopted the Morse telegraph as the system for international communications.[51]Theinternational Morse codeadopted was considerably modified from the originalAmerican Morse code, and was based on a code used on Hamburg railways (Gerke, 1848).[52]A common code was a necessary step to allow direct telegraph connection between countries. With different codes, additional operators were required to translate and retransmit the message. In 1865, a conference in Paris adopted Gerke's code as the International Morse code and was henceforth the international standard. The US, however, continued to use American Morse code internally for some time, hence international messages required retransmission in both directions.[53]
In the United States, the Morse/Vail telegraph wasquickly deployed in the two decades following the first demonstrationin 1844. Theoverland telegraphconnected the west coast of the continent to the east coast by 24 October 1861, bringing an end to thePony Express.[54]
France was slow to adopt the electrical telegraph, because of the extensiveoptical telegraphsystem built during theNapoleonic era. There was also serious concern that an electrical telegraph could be quickly put out of action by enemy saboteurs, something that was much more difficult to do with optical telegraphs which had no exposed hardware between stations. TheFoy-Breguet telegraphwas eventually adopted. This was a two-needle system using two signal wires but displayed in a uniquely different way to other needle telegraphs. The needles made symbols similar to theChappeoptical system symbols, making it more familiar to the telegraph operators. The optical system was decommissioned starting in 1846, but not completely until 1855. In that year the Foy-Breguet system was replaced with the Morse system.[55]
As well as the rapid expansion of the use of the telegraphs along the railways, they soon spread into the field of mass communication with the instruments being installed inpost offices. The era of mass personal communication had begun. Telegraph networks were expensive to build, but financing was readily available, especially from London bankers. By 1852, National systems were in operation in major countries:[56][57]
The New York and Mississippi Valley Printing Telegraph Company, for example, was created in 1852 in Rochester, New York and eventually became theWestern Union Telegraph Company.[60]Although many countries had telegraph networks, there was noworldwideinterconnection. Message by post was still the primary means of communication to countries outside Europe.
Telegraphy was introduced inCentral Asiaduring the 1870s.[62]
A continuing goal in telegraphy was to reduce the cost per message by reducing hand-work, or increasing the sending rate. There were many experiments with moving pointers, and various electrical encodings. However, most systems were too complicated and unreliable. A successful expedient to reduce the cost per message was the development oftelegraphese.
The first system that did not require skilled technicians to operate was Charles Wheatstone's ABC system in 1840 in which the letters of the alphabet were arranged around a clock-face, and the signal caused a needle to indicate the letter. This early system required the receiver to be present in real time to record the message and it reached speeds of up to 15 words a minute.
In 1846,Alexander Bainpatented a chemical telegraph in Edinburgh. The signal current moved an iron pen across a moving paper tape soaked in a mixture of ammonium nitrate and potassium ferrocyanide, decomposing the chemical and producing readable blue marks in Morse code. The speed of the printing telegraph was 16 and a half words per minute, but messages still required translation into English by live copyists. Chemical telegraphy came to an end in the US in 1851, when the Morse group defeated the Bain patent in the US District Court.[63]
For a brief period, starting with the New York–Boston line in 1848, some telegraph networks began to employ sound operators, who were trained to understand Morse code aurally. Gradually, the use of sound operators eliminated the need for telegraph receivers to include register and tape. Instead, the receiving instrument was developed into a "sounder", an electromagnet that was energized by a current and attracted a small iron lever. When the sounding key was opened or closed, the sounder lever struck an anvil. The Morse operator distinguished a dot and a dash by the short or long interval between the two clicks. The message was then written out in long-hand.[64]
Royal Earl Housedeveloped and patented a letter-printing telegraph system in 1846 which employed an alphabetic keyboard for the transmitter and automatically printed the letters on paper at the receiver,[65]and followed this up with a steam-powered version in 1852.[66]Advocates of printing telegraphy said it would eliminate Morse operators' errors. The House machine was used on four main American telegraph lines by 1852. The speed of the House machine was announced as 2600 words an hour.[67]
David Edward Hughesinvented the printing telegraph in 1855; it used a keyboard of 26 keys for the alphabet and a spinning type wheel that determined the letter being transmitted by the length of time that had elapsed since the previous transmission. The system allowed for automatic recording on the receiving end. The system was very stable and accurate and became accepted around the world.[68]
The next improvement was theBaudot codeof 1874. French engineerÉmile Baudotpatented a printing telegraph in which the signals were translated automatically into typographic characters. Each character was assigned a five-bit code, mechanically interpreted from the state of five on/off switches. Operators had to maintain a steady rhythm, and the usual speed of operation was 30 words per minute.[69]
By this point, reception had been automated, but the speed and accuracy of the transmission were still limited to the skill of the human operator. The first practical automated system was patented by Charles Wheatstone. The message (inMorse code) was typed onto a piece of perforated tape using a keyboard-like device called the 'Stick Punch'. The transmitter automatically ran the tape through and transmitted the message at the then exceptionally high speed of 70 words per minute.
An early successfulteleprinterwas invented byFrederick G. Creed. InGlasgowhe created his first keyboard perforator, which used compressed air to punch the holes. He also created a reperforator (receiving perforator) and a printer. The reperforator punched incoming Morse signals onto paper tape and the printer decoded this tape to produce alphanumeric characters on plain paper. This was the origin of the Creed High Speed Automatic Printing System, which could run at an unprecedented 200 words per minute. His system was adopted by theDaily Mailfor daily transmission of the newspaper contents.
With the invention of theteletypewriter, telegraphic encoding became fully automated. Early teletypewriters used the ITA-1Baudot code, a five-bit code. This yielded only thirty-two codes, so it was over-defined into two "shifts", "letters" and "figures". An explicit, unshared shift code prefaced each set of letters and figures. In 1901, Baudot's code was modified byDonald Murray.
In the 1930s, teleprinters were produced byTeletypein the US,Creedin Britain andSiemensin Germany.
By 1935, message routing was the last great barrier to full automation. Large telegraphy providers began to develop systems that usedtelephone-like rotary diallingto connect teletypewriters. These resulting systems were called "Telex" (TELegraph EXchange). Telex machines first performed rotary-telephone-stylepulse diallingforcircuit switching, and then sent data byITA2. This "type A" Telex routing functionally automated message routing.
The first wide-coverage Telex network was implemented in Germany during the 1930s[70]as a network used to communicate within the government.
At the rate of 45.45 (±0.5%)baud– considered speedy at the time – up to 25 telex channels could share a single long-distance telephone channel by usingvoice frequency telegraphymultiplexing, making telex the least expensive method of reliable long-distance communication.
Automatic teleprinter exchange service was introduced into Canada byCPR TelegraphsandCN Telegraphin July 1957 and in 1958,Western Unionstarted to build a Telex network in the United States.[71]
The most expensive aspect of a telegraph system was the installation – the laying of the wire, which was often very long. The costs would be better covered by finding a way to send more than one message at a time through the single wire, thus increasing revenue per wire. Early devices included theduplexand thequadruplexwhich allowed, respectively, one or two telegraph transmissions in each direction. However, an even greater number of channels was desired on the busiest lines. In the latter half of the 1800s, several inventors worked towards creating a method for doing just that, includingCharles Bourseul,Thomas Edison,Elisha Gray, andAlexander Graham Bell.
One approach was to have resonators of several different frequencies act as carriers of a modulated on-off signal. This was the harmonic telegraph, a form offrequency-division multiplexing. These various frequencies, referred to as harmonics, could then be combined into one complex signal and sent down the single wire. On the receiving end, the frequencies would be separated with a matching set of resonators.
With a set of frequencies being carried down a single wire, it was realized that the human voice itself could be transmitted electrically through the wire. This effort led to theinvention of the telephone. (While the work toward packing multiple telegraph signals onto one wire led to telephony, later advances would pack multiple voice signals onto one wire by increasing the bandwidth by modulating frequencies much higher than human hearing. Eventually, the bandwidth was widened much further by using laser light signals sent through fiber optic cables. Fiber optic transmission can carry 25,000 telephone signals simultaneously down a single fiber.[72])
Soon after the first successful telegraph systems were operational, the possibility of transmitting messages across the sea by way ofsubmarine communications cableswas first proposed. One of the primary technical challenges was to sufficiently insulate the submarine cable to prevent the electric current from leaking out into the water. In 1842, a Scottish surgeonWilliam Montgomerie[73]introducedgutta-percha, the adhesive juice of thePalaquium guttatree, to Europe.Michael Faradayand Wheatstone soon discovered the merits of gutta-percha as an insulator, and in 1845, the latter suggested that it should be employed to cover the wire which was proposed to be laid fromDovertoCalais. Gutta-percha was used as insulation on a wire laid across theRhinebetweenDeutzandCologne.[74]In 1849,C. V. Walker, electrician to theSouth Eastern Railway, submerged a 2 miles (3.2 km) wire coated with gutta-percha off the coast from Folkestone, which was tested successfully.[73]
John Watkins Brett, an engineer fromBristol, sought and obtained permission fromLouis-Philippein 1847 to establishtelegraphic communicationbetween France and England. The first undersea cable was laid in 1850, connecting the two countries and was followed by connections to Ireland and the Low Countries.
TheAtlantic Telegraph Companywas formed inLondonin 1856 to undertake to construct a commercial telegraph cable across the Atlantic Ocean. It was successfully completed on 18 July 1866 by the shipSSGreat Eastern, captained bySir James Anderson, after many mishaps along the way.[75]John Pender, one of the men on the Great Eastern, later founded several telecommunications companies primarily laying cables between Britain and Southeast Asia.[76]Earlier transatlanticsubmarine cablesinstallations were attempted in 1857, 1858 and 1865. The 1857 cable only operated intermittently for a few days or weeks before it failed. The study of underwater telegraph cables accelerated interest in mathematical analysis of very longtransmission lines. The telegraph lines from Britain to India were connected in 1870. (Those several companies combined to form theEastern Telegraph Companyin 1872.) The HMSChallengerexpedition in 1873–1876 mapped the ocean floor for future underwater telegraph cables.[77]
Australia was first linked to the rest of the world in October 1872 by a submarine telegraph cable at Darwin.[78]This brought news reports from the rest of the world.[79]The telegraph across the Pacific was completed in 1902, finally encircling the world.
From the 1850s until well into the 20th century, British submarine cable systems dominated the world system. This was set out as a formal strategic goal, which became known as theAll Red Line.[80]In 1896, there were thirty cable laying ships in the world and twenty-four of them were owned by British companies. In 1892, British companies owned and operated two-thirds of the world's cables and by 1923, their share was still 42.7 percent.[81]
Cable & Wirelesswas a British telecommunications company that traced its origins back to the 1860s, with SirJohn Penderas the founder,[82]although the name was only adopted in 1934. It was formed from successive mergers including:
Main article § Section:History of longitude § Land surveying and telegraphy.
The telegraph was very important for sending time signals to determine longitude, providing greater accuracy than previously available. Longitude was measured by comparing local time (for example local noon occurs when the sun is at its highest above the horizon) with absolute time (a time that is the same for an observer anywhere on earth). If the local times of two places differ by one hour, the difference in longitude between them is 15° (360°/24h). Before telegraphy, absolute time could be obtained from astronomical events, such aseclipses,occultationsorlunar distances, or by transporting an accurate clock (achronometer) from one location to the other.
The idea of using the telegraph to transmit a time signal for longitude determination was suggested byFrançois AragotoSamuel Morsein 1837,[85]and the first test of this idea was made byCapt. Wilkesof the U.S. Navy in 1844, over Morse's line between Washington and Baltimore.[86]The method was soon in practical use for longitude determination, in particular by the U.S. Coast Survey, and over longer and longer distances as the telegraph network spread across North America and the world, and as technical developments improved accuracy and productivity[87]: 318–330[88]: 98–107
The "telegraphic longitude net"[89]soon became worldwide. Transatlantic links between Europe and North America were established in 1866 and 1870. The US Navy extended observations into the West Indies and Central and South America with an additional transatlantic link from South America to Lisbon between 1874 and 1890.[90][91][92][93]British, Russian and US observations created a chain from Europe through Suez, Aden, Madras, Singapore, China and Japan, to Vladivostok, thence to Saint Petersburg and back to Western Europe.[94]
Australia's telegraph network was linked to Singapore's via Java in 1871,[95]and the net circled the globe in 1902 with the connection of the Australia and New Zealand networks to Canada's via theAll Red Line. The two determinations of longitudes, one transmitted from east to west and the other from west to east, agreed within one second of arc (1⁄15second of time – less than 30 metres).[96]
The ability to send telegrams brought obvious advantages to those conducting war. Secret messages were encoded, so interception alone would not be sufficient for the opposing side to gain an advantage. There were also geographical constraints on intercepting the telegraph cables that improved security, however once radio telegraphy was developed interception became far more widespread.
TheCrimean Warwas one of the first conflicts to usetelegraphsand was one of the first to be documented extensively. In 1854, the government in London created a military Telegraph Detachment for the Army commanded by an officer of theRoyal Engineers. It was to comprise twenty-five men from the Royal Corps of Sappers & Miners trained by the Electric Telegraph Company to construct and work the first field electric telegraph.[97]
Journalistic recording of the war was provided byWilliam Howard Russell(writing forThe Timesnewspaper) with photographs byRoger Fenton.[98]News from war correspondents kept the public of the nations involved in the war informed of the day-to-day events in a way that had not been possible in any previous war. After the French extended their telegraph lines to the coast of the Black Sea in late 1854, war news began reachingLondonin two days. When the British laid an underwater cable to the Crimean peninsula in April 1855, news reached London in a few hours. These prompt daily news reports energised British public opinion on the war, which brought down the government and led to Lord Palmerston becoming prime minister.[99]
During theAmerican Civil Warthe telegraph proved its value as a tactical, operational, and strategic communication medium and an important contributor to Union victory.[100]By contrast the Confederacy failed to make effective use of the South's much smaller telegraph network. Prior to the War the telegraph systems were primarily used in the commercial sector. Government buildings were not inter-connected with telegraph lines, but relied on runners to carry messages back and forth.[101]Before the war the Government saw no need to connect lines within city limits, however, they did see the use in connections between cities. Washington D.C. being the hub of government, it had the most connections, but there were only a few lines running north and south out of the city.[101]It was not until the Civil War that the government saw the true potential of the telegraph system. Soon after the shelling ofFort Sumter, the South cut telegraph lines running into D.C., which put the city in a state of panic because they feared an immediate Southern invasion.[102][101]
Within 6 months of the start of the war, theU.S. Military Telegraph Corps(USMT) had laid approximately 300 miles (480 km) of line. By war's end they had laid approximately 15,000 miles (24,000 km) of line, 8,000 for military and 5,000 for commercial use, and had handled approximately 6.5 million messages. The telegraph was not only important for communication within the armed forces, but also in the civilian sector, helping political leaders to maintain control over their districts.[102]
Even before the war, theAmerican Telegraph Companycensored suspect messages informally to block aid to the secession movement. During the war,Secretary of WarSimon Cameron, and laterEdwin Stanton, wanted control over the telegraph lines to maintain the flow of information. Early in the war, one of Stanton's first acts as Secretary of War was to move telegraph lines from ending atMcClellan'sheadquarters to terminating at the War Department. Stanton himself said "[telegraphy] is my right arm". Telegraphy assisted Northern victories, including theBattle of Antietam(1862), theBattle of Chickamauga(1863), andSherman's March to the Sea(1864).[102]
The telegraph system still had its flaws. The USMT, while the main source of telegraphers and cable, was still a civilian agency. Most operators were first hired by the telegraph companies and then contracted out to the War Department. This created tension between generals and their operators. One source of irritation was that USMT operators did not have to follow military authority. Usually they performed without hesitation, but they were not required to, soAlbert Myercreated aU.S. Army Signal Corpsin February 1863. As the new head of the Signal Corps, Myer tried to get all telegraph and flag signaling under his command, and therefore subject to military discipline. After creating the Signal Corps, Myer pushed to further develop new telegraph systems. While the USMT relied primarily on civilian lines and operators, the Signal Corp's new field telegraph could be deployed and dismantled faster than USMT's system.[102]
DuringWorld War I, Britain's telegraph communications were almost completely uninterrupted, while it was able to quickly cut Germany's cables worldwide.[103]The British government censored telegraph cable companies in an effort to root out espionage and restrict financial transactions with Central Powers nations.[104]British access to transatlantic cables and its codebreaking expertise led to theZimmermann Telegramincident that contributed to theUS joining the war.[105]Despite British acquisition of German colonies and expansion into the Middle East, debt from the war led to Britain's control over telegraph cables to weaken while US control grew.[106]
World War IIrevived the 'cable war' of 1914–1918. In 1939, German-owned cables across the Atlantic were cut once again, and, in 1940, Italian cables to South America and Spain were cut in retaliation for Italian action against two of the five British cables linking Gibraltar and Malta.Electra House, Cable & Wireless's head office and central cable station, was damaged by German bombing in 1941.
Resistance movementsin occupied Europe sabotaged communications facilities such as telegraph lines,[107]forcing the Germans to usewireless telegraphy, which could then beinterceptedby Britain.
The Germans developed a highly complex teleprinter attachment (German:Schlüssel-Zusatz, "cipher attachment") that was used for enciphering telegrams, using theLorenz cipher, between German High Command (OKW) and the army groups in the field. These contained situation reports, battle plans, and discussions of strategy and tactics. Britain intercepted these signals, diagnosed how the encrypting machine worked, anddecrypteda large amount of teleprinter traffic.[108]
In America, the end of the telegraph era can be associated with the fall of theWestern Union Telegraph Company. Western Union was the leading telegraph provider for America and was seen as the best competition for theNational Bell Telephone Company. Western Union and Bell were both invested in telegraphy and telephone technology. Western Union's decision to allow Bell to gain the advantage in telephone technology was the result of Western Union's upper management's failure to foresee the surpassing of the telephone over the, at the time, dominant telegraph system. Western Union soon lost the legal battle for the rights to their telephone copyrights. This led to Western Union agreeing to a lesser position in the telephone competition, which in turn led to the lessening of the telegraph.[102]
While the telegraph was not the focus of the legal battles that occurred around 1878, the companies that were affected by the effects of the battle were the main powers of telegraphy at the time. Western Union thought that the agreement of 1878 would solidify telegraphy as the long-range communication of choice. However, due to the underestimates of telegraph's future[further explanation needed]and poor contracts, Western Union found itself declining.[102]AT&Tacquired working control of Western Union in 1909 but relinquished it in 1914 under threat of antitrust action. AT&T bought Western Union's electronic mail andTelexbusinesses in 1990.
Although commercial "telegraph" services are still available inmany countries, transmission is usually done via acomputer networkrather than a dedicated wired connection.
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TheImperial Wireless Chainwas a strategic international communications network of powerful long rangeradiotelegraphystations, created by theBritish governmentto link the countries of theBritish Empire. The stations exchanged commercial and diplomatic text message traffic transmitted at high speed byMorse codeusingpaper tapemachines. Although the idea was conceived prior toWorld War I, the United Kingdom was the last of the world'sgreat powersto implement an operational system.[1]The first link in the chain, betweenLeafieldin Oxfordshire andCairo, Egypt, eventually opened on 24 April 1922,[2]with the final link, between Australia and Canada, opening on 16 June 1928.[3]
Guglielmo Marconiinvented the first practicalradio transmittersandreceivers, and radio began to be used for practical ship-to-shore communication around 1900. His company, theMarconi Wireless Telegraph Company, dominated early radio. In the period leading up toWorld War I, long distanceradiotelegraphybecame a strategic defense technology, as it was realized that a nation without radio could be isolated by an enemy cutting itssubmarine telegraph cables, as indeed happened during the war. Starting around 1908, industrialized nations built global networks of powerful transoceanic wireless telegraphy stations to exchangeMorse codetelegramtraffic with their overseas colonies.[4][5]
In 1910 theColonial Officereceived a formal proposal from theMarconi Companyto construct a series of wireless telegraphy stations to link the British Empire within three years.[1]While not then accepted, the Marconi proposal created serious interest in the concept.[6]
A dilemma faced by Britain throughout the negotiations to establish the chain was that Britain owned the largest network ofsubmarine telegraph cables. The proposed stations would directly compete with cables for a fixed amount of transoceanic telegram traffic, reducing the revenue of the cable companies and possibly bankrupting them.[citation needed]
Parliament ruled out the creation of a privatemonopolyto provide the service and concluded that no government department was in a position to do so, and the Treasury were reluctant to fund the creation of a new department. Contracting the construction to a commercial "wireless company" was the favoured option,[6]and a contract was signed with Marconi's Wireless Telegraph Company in March 1912. The government then found itself facing severe criticism and appointed aselect committeeto examine the topic.[7]After hearing evidence from theAdmiralty,War Office,India Office, and representatives fromSouth Africa, the committee unanimously concluded that a "chain of Imperial wireless stations" should be established as a matter of urgency.[6]An expert committee also advised that Marconi were the only company with technology that was proven to operate reliably over the distances required (in excess of 2,000 miles (3,200 km)) "if rapid installation and immediate and trustworthy communication be desired".[6]
After further negotiations prompted by Treasury pressure, a modified contract wasratifiedby Parliament on 8 August 1913, with 221Members of Parliamentvoting in favour, 140 against.[6]The course of these events was disrupted somewhat by theMarconi scandal, when it was alleged that highly placed members of the governingLiberalparty had used their knowledge of the negotiations to indulge ininsider tradingin Marconi shares. The outbreak ofWorld War Iled to the suspension of the contract by the government.[8]Meanwhile Germany successfully constructed its own wireless chain before the war, at a cost equivalent to two millionpounds sterling, and was able to use it to its advantage during the conflict.[9]
With the end of the war and theDominionscontinuing to apply pressure on the government to provide an "Imperial wireless system",[8]theHouse of Commonsagreed in 1919 that £170,000 should be spent constructing the first two radio stations in the chain, in Oxfordshire (atLeafield) and Egypt (in Cairo), to be completed in early 1920[10]– although in the event the link opened on 24 April 1922,[11]two months after the UKdeclared Egypt independent.
Parliament's decision came shortly after legal action initiated by Marconi in June 1919, claiming £7,182,000 in damages from the British government forbreach of their July 1912 contract, and in which they were awarded £590,000 by the court.[12]The government also commissioned the "Imperial Wireless Telegraphy Committee" chaired bySir Henry Norman(the Norman Committee), which reported in 1920. The Norman Report recommended that transmitters should have a range of 2,000 miles, which required relay stations,[13]and that Britain should be connected toCanada,Australia,South Africa, Egypt,India,East Africa,Singapore, andHong Kong.[14]However, the report was not acted upon.[15]While British politicians procrastinated, Marconi constructed stations for other nations, linking North and South America, as well as China and Japan, in 1922.[16]In January 1922 theBritish Chambers of Commerceadded their voice to the demands for action, adopting a resolution urging the government to urgently resolve the matter,[17]as did other organisations such as theEmpire Press Union, which claimed that the Empire was suffering "incalculable loss" in its absence.[18]
Under this pressure, after the1922 General Election, theConservativegovernment commissioned the Empire Wireless Committee, chaired bySir Robert Donald, to "consider and advise upon the policy to be adopted as regards an Imperial wireless service so as to protect and facilitate public interest." Its report was presented to the Postmaster-General on 23 February 1924[19]The committee's recommendations were similar to those of the Norman Committee – that any stations in the United Kingdom used to communicate with the Empire should be in the hands of the state, that they should be operated by the Post Office, and that eight high-powerlongwavestations should be used, as well as land-lines.[8][20]The scheme was estimated at £500,000.[20]At the time the committee was unaware of Marconi's 1923 experiments intoshortwaveradio transmissions, which offered a much cheaper alternative – although not a commercially proven one – to high-power long-wave transmission system.[8]
Following the Donald Report and discussions with the Dominions, it was decided that the high-powerRugby longwave station(announced on 13 July 1922 by the previous government)[21]would be completed since it used proven technology, in addition to which a number of shortwave "beam stations" would be built (so called because adirectional antennaconcentrated the radio transmission into a narrow directional beam). The beam stations would communicate with those Dominions that chose the new shortwave technology. Parliament finally approved an agreement between the Post Office and Marconi to build beam stations to communicate with Canada, South Africa, India and Australia, on 1 August 1924.[8]
From when the Post Office began operating the "Post Office Beam" services, through to 31 March 1929, they had earned gross receipts of £813,100 at a cost of £538,850, leaving a net surplus of £274,250.[22]
Even before the final link became operational between Australia and Canada, it was apparent that the commercial success of the Wireless Chain was threatening the viability of thecable telegraphycompanies. An "Imperial Wireless and Cable Conference" was therefore held in London in January 1928, with delegates from the United Kingdom, the self-governing Dominions, India, the Crown Colonies and Protectorates, to "examine the situation which arose as a result of the competition of the Imperial Beam Wireless Services with the cable services of various parts of the empire, to report upon it and to make recommendations with a view to a common policy being adopted by the various governments concerned."[23]It concluded that the cable companies would not be able to compete in an unrestricted market, but that the cable links remained of both commercial and strategic value. It therefore recommended that the cable and wireless interests of theEastern Telegraph Company, theEastern Extension, Australasia and China Telegraph Company,Western Telegraph Companyand Marconi's Wireless Telegraph Company should be merged to form a single organisation holding amonopolisticposition. The merged company would be overseen by an Imperial Advisory Committee, would purchase the government-owned cables in the Pacific, West Indies and Atlantic, and would also be given a lease on the beam stations for a period of 25 years, for the sum of £250,000 per year.[24][25]
The conference's recommendations were incorporated into theImperial Telegraphs Act 1929(5 & 6 Eliz. 2. c. 62), leading to the creation of two new companies on 8 April 1929; an operating company Imperial and International Communications, in turn owned by a holding company namedCable & Wireless Limited. In 1934 Imperial and International Communications was renamed asCable & WirelessLimited, with Cable and Wireless Limited being renamed as Cable and Wireless (Holding) Limited.[citation needed]From the beginning of April 1928 the beam services were operated by the Post Office asagentfor Imperial and International Communications Limited.[22]
The 1930s saw the arrival of theGreat Depression, as well as competition from theInternational Telephone and Telegraph Corporationand affordableairmail. Due to such factors Cable and Wireless were never able to earn the revenue which had been forecast, resulting in low dividends and an inability to reduce the rates charged to customers as much as had been expected.[26]To ease the financial pressure, the British Government finally decided to transfer the beam stations to Cable and Wireless, in exchange for 2,600,000 of the 30,000,000sharesin the company, under the provisions of theImperial Telegraphs Act 1938.[26]The ownership of the beam stations was reversed in 1947, when theLabourGovernment nationalised Cable and Wireless, integrating its UK assets with those of the Post Office.[27]By this stage, however, three of the original stations had been closed, after the service was centralised during 1939–1940 at Dorchester and Somerton.[28]The longwave Rugby radio station continued to remain under Post Office ownership throughout.[citation needed]
The shortwave Imperial Wireless Chain "beam stations" operated in pairs; one transmitting and one receiving. Pairs of stations were sited at (transmitters first):[28]
At Bodmin and Bridgwater, each aerial stretched to nearly half a mile (800 m) long, and consisted of a row of five 277 feet (84 m) highlattice masts, erected in a line at 640 feet (200 m) intervals and at right angles to the overseas receiving station. These were topped by cross-arm measuring 10 feet (3.0 m) high by 90 feet (27 m) wide, from which the vertical wires of the aerial were hung, forming a "curtain antenna".[29]At Tetney the antenna for India was similar to those at Bodmin and Bridgwater, while the Australian aerial was carried on three 275 feet (84 m) high masts.[28]
Electronic components for the system were built at Marconi's New Street wireless factory inChelmsford.[31]
Devizeswas home to a receiving station until the outbreak ofWorld War I.[citation needed]
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Radioteletype(RTTY)[a]is atelecommunicationssystem consisting originally of two or moreelectromechanicalteleprintersin different locations connected byradiorather than a wired link. Radioteletype evolved from earlier landline teleprinter operations that began in the mid-1800s.[1]TheUS Navy Departmentsuccessfully tested printing telegraphy between an airplane and ground radio station in 1922. Later that year, the Radio Corporation of America successfully tested printing telegraphy via theirChatham, Massachusetts, radio station to theRMSMajestic. Commercial RTTY systems were in active service betweenSan FranciscoandHonoluluas early as April 1932 and between San Francisco andNew York Cityby 1934. TheUS militaryused radioteletype in the 1930s and expanded this usage duringWorld War II. From the 1980s, teleprinters were replaced bypersonal computers(PCs) runningsoftware to emulate teleprinters.
The term radioteletype is used to describe both the original radioteletype system, sometimes described as "Baudot", as well as the entire family of systems connecting two or more teleprinters or PCs using software to emulate teleprinters, over radio, regardless of alphabet, link system or modulation.
In some applications, notably military and government, radioteletype is known by the acronym RATT (Radio Automatic Teletype).[2]
Landline teleprinter operations began in 1849 when a circuit was put in service betweenPhiladelphiaand New York City.[3]Émile Baudotdesigned a system using a five unit code in 1874 that is still in use today. Teleprinter system design was gradually improved until, at the beginning of World War II, it represented the principal distribution method used by the news services.
Radioteletype evolved from these earlier landline teleprinter operations. The US Department of the Navy successfully tested printing telegraphy between an airplane and ground radio station in August 1922.[4][5][6]Later that year, the Radio Corporation of America successfully tested printing telegraphy via their Chatham, MA radio station to the RMSMajestic.[7]An early implementation of the Radioteletype was the Watsongraph,[8]named afterDetroitinventor Glenn Watson in March 1931.[9]Commercial RTTY systems were in active service between San Francisco and Honolulu as early as April 1932[10][11]and between San Francisco and New York City by 1934.[12]The US Military used radioteletype in the 1930s and expanded this usage during World War II.[13]The Navy called radioteletypeRATT(Radio Automatic Teletype) and the Army Signal Corps called radioteletypeSCRT, an abbreviation of Single-Channel Radio Teletype. The military usedfrequency shift keying(FSK) technology and this technology proved very reliable even over long distances.
A radioteletype station consists of three distinct parts: the Teletype or teleprinter, themodemand theradio.
The Teletype or teleprinter is anelectromechanicalorelectronicdevice. The wordTeletypewas a trademark of the Teletype Corporation, so the terms "TTY", "RTTY", "RATT" and "teleprinter" are usually used to describe a generic device without reference to a particular manufacturer.
Electromechanical teleprinters are heavy, complex and noisy, and have largely been replaced with electronic units. The teleprinter includes a keyboard, which is the main means of entering text, and a printer orvisual display unit(VDU). An alternative input device is aperforated tapereader and, more recently, computerstorage media(such as floppy disks). Alternative output devices are tape perforators and computer storage media.
The line output of a teleprinter can be at eitherdigital logiclevels (+5 V signifies a logical "1" ormarkand 0 V signifies a logical "0" orspace) orline levels(−80 V signifies a "1" and +80 V a "0"). When no traffic is passed, the line idles at the "mark" state.
When a key of the teleprinter keyboard is pressed, a5-bit characteris generated. The teleprinter converts it toserial formatand transmits a sequence of astart bit(a logical 0 or space), then one after the other the 5 data bits, finishing with astop bit(a logical 1 or mark, lasting 1, 1.5 or 2 bits). When a sequence of start bit, 5 data bits and stop bit arrives at the input of the teleprinter, it is converted to a 5-bit word and passed to the printer or VDU. With electromechanical teleprinters, these functions required complicated electromechanical devices, but they are easily implemented with standard digital electronics usingshift registers. Specialintegrated circuitshave been developed for this function, for example theIntersil6402 and 6403.[14]These are stand-aloneUARTdevices, similar to computer serial port peripherals.
The 5 data bits allow for only 32 different codes, which cannot accommodate the 26 letters, 10 figures, space, a fewpunctuationmarks and the requiredcontrol codes, such as carriage return, new line, bell, etc. To overcome this limitation, the teleprinter has twostates, theunshiftedorlettersstate and theshiftedornumbersorfiguresstate. The change from one state to the other takes place when the special control codesLETTERSandFIGURESare sent from the keyboard or received from the line. In thelettersstate the teleprinter prints the letters and space while in the shifted state it prints the numerals and punctuation marks. Teleprinters for languages using otheralphabetsalso use an additionalthird shiftstate, in which they print letters in the alternative alphabet.
The modem is sometimes called the terminal unit and is an electronic device which is connected between the teleprinter and the radiotransceiver. The transmitting part of the modem converts the digital signal transmitted by the teleprinter or tape reader to one or the other of a pair ofaudio frequencytones, traditionally 2295/2125 Hz (US) or 2125/1955 Hz (Europe). One of the tones corresponds to themarkcondition and the other to thespacecondition. These audio tones, then,modulateanSSBtransmitter to produce the final audio-frequency shift keying (AFSK) radio frequency signal. Some transmitters are capable of directfrequency-shift keying(FSK) as they can directly accept the digital signal and change their transmitting frequency according to themarkorspaceinput state. In this case the transmitting part of the modem is bypassed.
On reception, the FSK signal is converted to the original tones by mixing the FSK signal with a local oscillator called the BFO orbeat frequency oscillator. These tones are fed to the demodulator part of the modem, which processes them through a series of filters and detectors to recreate the original digital signal. The FSK signals are audible on a communications radio receiver equipped with a BFO, and have a distinctive "beedle-eeeedle-eedle-eee" sound, usually starting and ending on one of the two tones ("idle on mark").
The transmission speed is a characteristic of the teleprinter while the shift (the difference between the tones representing mark and space) is a characteristic of the modem. These two parameters are therefore independent, provided they have satisfied theminimum shift sizefor a given transmission speed. Electronic teleprinters can readily operate in a variety of speeds, but mechanical teleprinters require the change of gears in order to operate at different speeds.
Today, both functions can be performed with modern computers equipped with digital signal processors orsound cards. The sound card performs the functions of the modem and theCPUperforms the processing of the digital bits. This approach is very common inamateur radio, using specialized computer programs likefldigi, MMTTY or MixW.
Before the computer mass storage era, most RTTY stations stored text on paper tape using paper tape punchers and readers. The operator would type the message on the TTY keyboard and punch the code onto the tape. The tape could then be transmitted at a steady, high rate, without typing errors. A tape could be reused, and in some cases - especially for use with ASCII on NC Machines - might be made of plastic or even very thin metal material in order to be reused many times.
The most common test signal is a series of "RYRYRY" characters, as these form an alternating tone pattern exercising all bits and are easily recognized.Pangramsare also transmitted on RTTY circuits as test messages, the most common one being "The quick brown fox jumps over the lazy dog", and in French circuits, "Voyez le brick géant que j'examine près du wharf"
The original (or "Baudot") radioteletype system is based almost invariably on theBaudot codeor ITA-2 5 bit alphabet. The link is based on character asynchronous transmission with 1 start bit and 1, 1.5 or 2 stop bits. Transmitter modulation is normallyFSK(F1B). Occasionally, an AFSK signal modulating an RF carrier (A2B, F2B) is used on VHF or UHF frequencies. Standard transmission speeds are 45.45, 50, 75, 100, 150 and 300 baud.
Common carrier shifts are 85 Hz (used on LF and VLF frequencies), 170 Hz, 425 Hz, 450 Hz and 850 Hz, although some stations use non-standard shifts. There are variations of the standard Baudot alphabet to cover languages written in Cyrillic, Arabic, Greek etc., using special techniques.[15][16]
Some combinations of speed and shift are standardized for specific services using the original radioteletype system:
After World War II,amateur radio operatorsin the U.S. started to receive obsolete but usable Teletype Model 26 equipment from commercial operators with the understanding that this equipment would not be used for or returned to commercial service. "The Amateur Radioteletype and VHF Society" was founded in 1946 in Woodside, NY. This organization soon changed its name to "The VHF Teletype Society" and started US amateur radio operations on2 metersusingaudio frequency shift keying(AFSK). The first two-way amateur radio teletypecontact(QSO) of record took place in May 1946 between Dave Winters, W2AUF, Brooklyn, NY, and W2BFD, John Evans Williams, Woodside Long Island, NY.[21]On the west coast, amateur RTTY also started on 2 meters. Operation on 80 meters, 40 meters and the otherHigh Frequency(HF) amateur radio bands was initially accomplished using make and break keying sincefrequency shift keying(FSK) was not yet authorized.
In early 1949, the first American transcontinental two-way RTTYcontactwas accomplished on11 metersusingAFSKbetween Tom McMullen (W1QVF) operating atW1AWand Johnny Agalsoff, W6PSW.[22]The stations effected partial contact on January 30, 1949, and repeated more successfully on January 31. On February 1, 1949, the stations exchanged solid print congratulatory message traffic andrag-chewed. Earlier, on January 23, 1949, William T. Knott, W2QGH, Larchmont, NY, had been able to make rough copy of W6PSW's test transmissions. Whilecontactscould be accomplished, it was quickly realized thatFSKwas technically superior to make and break keying. Due to the efforts of Merrill Swan, W6AEE, of "The RTTY Society of Southern California" publisher ofRTTYand Wayne Green, W2NSD, ofCQ Magazine, amateur radio operators successfully petitioned the U.S.Federal Communications Commission(FCC) to amend Part 12 of the Regulations, which was effective on February 20, 1953.[23]The amended Regulations permittedFSKin the non-voice parts of the80,40, and20 meter bandsand also specified the use of single channel 60 words-per-minute five unit code corresponding toITA2. A shift of850 ± 50Hzwas specified. Amateur radio operators also had to identify their station callsign at the beginning and the end of each transmission and at ten-minute intervals usingInternational Morse code. Use of this wide shift proved to be a problem for amateur radio operations. Commercial operators had already discovered that narrow shift worked best on theHF bands. After investigation and a petition to the FCC, Part 12 was amended, in March 1956, to allow amateur radio operators to use any shift that was 900 Hz or less.
The FCCNotice of Proposed Rule Making(NPRM) that resulted in the authorization ofFSKin the amateur high frequency (HF) bands responded to petitions by theAmerican Radio Relay League(ARRL), the National Amateur Radio Council, and a Mr. Robert Weinstein. The NPRM specifically states this, and this information may be found in its entirety in the December 1951 issue ofQST Magazine. WhileThe New RTTY Handbook[23]gives ARRL no credit, it was published byCQ Magazineand its author was aCQcolumnist (CQwas generally hostile to the ARRL at that time).
The first RTTY Contest was held by the RTTY Society of Southern California from October 31 to November 1, 1953.[24]Named the RTTY Sweepstakes Contest, twenty nine participants exchanged messages that contained a serial number, originating station call, check or RST report of two or three numbers, ARRL section of originator, local time (0000-2400 preferred) and date. Example: NR 23 W0BP CK MINN 1325 FEB 15. By the late 1950s, the contest exchange was expanded to include band used. Example: NR 23 W0BP CK MINN 1325 FEB 15 FORTY METERS. The contest was scored as follows: One point for each message sent and received entirely by RTTY and one point for each message received and acknowledged by RTTY. The final score was computed by multiplying the total number of message points by the number of ARRL sections worked. Two stations could exchange messages again on a different band for added points, but the section multiplier did not increase when the same section was reworked on a different band. Each DXCC entity was counted as an additional ARRL section for RTTY multiplier credit.
A new magazine namedRTTY, later renamedRTTY Journal, also published the first listing of stations, mostly located in the continental US, that were interested in RTTY in 1956.[25]Amateur radio operators used this callbook information to contact other operators both inside and outside the United States. For example, the first recorded USA to New Zealand two-way RTTYcontacttook place in 1956 between W0BP and ZL1WB.
By the late 1950s, new organizations focused on amateur radioteletype started to appear. The "British Amateur Radio Teletype Group", BARTG, now known as the "British Amateur Radio Teledata Group"[26]was formed in June 1959. The Florida RTTY Society was formed in September 1959.[27]Amateur radio operators outside of Canada and the U.S. began to acquire surplus teleprinter and receive permission to get on the air. The first recorded RTTYcontactin the U.K. occurred in September 1959 between G2UK and G3CQE. A few weeks later, G3CQE had the first G/VE RTTY QSO with VE7KX.[28]This was quickly followed up by G3CQE QSOs with VK3KF and ZL3HJ.[29]Information on how to acquire surplus teleprinter equipment continued to spread and before long it was possible to work all continents on RTTY.
Amateur radio operators used various equipment designs to get on the air using RTTY in the 1950s and 1960s. Amateurs used their existing receivers for RTTY operation but needed to add a terminal unit, sometimes called a demodulator, to convert the received audio signals to DC signals for the teleprinter.
Most of the terminal unit equipment used for receiving RTTY signals was home built, using designs published in amateur radio publications. These original designs can be divided into two classes of terminal units: audio-type and intermediate frequency converters. The audio-type converters proved to be more popular with amateur radio operators. The Twin City, W2JAV and W2PAT designs were examples of typical terminal units that were used into the middle 1960s. The late 1960s and early 1970s saw the emergence of terminal units designed by W6FFC, such as the TT/L, ST-3, ST-5, and ST-6. These designs were first published inRTTY Journalstarting in September 1967 and ending in 1970.
An adaptation of the W6FFC TT/L terminal unit was developed by Keith Petersen, W8SDZ, and it was first published in theRTTY Journalin September 1967. The drafting of the schematic in the article was done by Ralph Leland, W8DLT.
Amateur radio operators needed to modify their transmitters to allow for HF RTTY operation. This was accomplished by adding a frequency shift keyer that used a diode to switch a capacitor in and out of the circuit, shifting the transmitter’s frequency in synchronism with the teleprinter signal changing from mark to space to mark. A very stable transmitter was required for RTTY. The typical frequency multiplication type transmitter that was popular in the 1950s and 1960s would be relatively stable on80 metersbut become progressively less stable on40 meters,20 meters, and15 meters. By the middle 1960s, transmitter designs were updated, mixing a crystal-controlled high frequency oscillator with a variable low frequency oscillator, resulting in better frequency stability across all amateur radio HF bands.
During the early days of Amateur RTTY, the RTTYWorked All ContinentsAward was conceived by the RTTY Society of Southern California and issued by RTTY Journal.[30]The first amateur radio station to achieve this WAC – RTTY Award was VE7KX.[31]The first stations recognized as having achieved single band WAC RTTY were W1MX (3.5 MHz); DL0TD (7.0 MHz); K3SWZ (14.0 MHz); W0MT (21.0 MHz) and FG7XT (28.0 MHz).[32]The ARRL began issuingWACRTTY certificates in 1969.
By the early 1970s, amateur radio RTTY had spread around the world and it was finally possible to work more than 100 countries via RTTY. FG7XT was the first amateur radio station to claim to achieve this honor. However, Jean did not submit hisQSLcards for independent review. ON4BX, in 1971, was the first amateur radio station to submit his cards to the DX editor ofRTTY Journaland to achieve this honor.[33]The ARRL began issuingDXCCRTTY Awards on November 1, 1976.[34]Prior to that date, an award for working 100 countries on RTTY was only available via RTTY Journal.
In the 1950s through the 1970s, "RTTY art" was a popular on-air activity. This consisted of (sometimes very elaborate and artistic) pictures sent over RTTY through the use of lengthy punched tape transmissions and then printed by the receiving station on paper.
On January 7, 1972, the FCC amended Part 97 to allow faster RTTY speeds. Four standard RTTY speeds were authorized, namely, 60words per minute(WPM) (45baud), 67WPM(50 baud), 75WPM(56.25 baud), and 100WPM(75 baud). Many amateur radio operators had equipment that was capable of being upgraded to 75 and 100 words per minute by changing teleprinter gears. While there was an initial interest in 100WPMoperation, many amateur radio operators moved back to 60WPM. Some of the reasons for the failure of 100WPMHF RTTY included poor operation of improperly maintained mechanical teleprinters, narrow bandwidth terminal units, continued use of 170 Hz shift at 100WPM, and excessive error rates due to multipath distortion and the nature of ionospheric propagation.
The FCC approved the use ofASCIIby amateur radio stations on March 17, 1980 with speeds up to 300baudfrom3.5 MHzto21.25 MHzand 1200 baud between28 MHzand225 MHz. Speeds up to 19.2 kilobaud was authorized on amateur frequencies above420 MHz.[35]
These symbol rates were later modified:[36]
The requirement for amateur radio operators in the U.S. to identify their stationcallsignat the beginning and the end of each digital transmission, and at ten-minute intervals using International Morse code, was finally lifted by the FCC on June 15, 1983.
RTTY has a typicalbaud ratefor Amateur operation of 45.45 baud (approximately 60 words per minute). It remains popular as a "keyboard to keyboard" mode in Amateur Radio.[37]RTTY has declined in commercial popularity as faster, more reliable alternative data modes have become available, using satellite or other connections.
For its transmission speed, RTTY has lowspectral efficiency. The typical RTTY signal with 170 Hz shift at 45.45 baud requires around 250 Hz receiver bandwidth, more than double that required byPSK31. In theory, at this baud rate, the shift size can be decreased to 22.725 Hz, reducing the overall band footprint substantially. Because RTTY, using eitherAFSKorFSKmodulation, produces a waveform with constant power, a transmitter does not need to use alinear amplifier, which is required for many digital transmission modes. A more efficientClass C amplifiermay be used.
RTTY, using either AFSK or FSK modulation, is moderately resistant to vagaries of HF propagation and interference, however modern digital modes, such asMFSK, useForward Error Correctionto provide much better data reliability.
Principally, the primary users are those who need robust shortwave communications. Examples are:
One regular service transmitting RTTY meteorological information is theGerman Meteorological Service(Deutscher Wetterdienst or DWD). The DWD regularly transmit two programs on various frequencies onLFandHFin standard RTTY (ITA-2 alphabet). The list of callsigns, frequencies,baudrates and shifts are as follows:[19]
The DWD signals can be easily received in Europe, North Africa and parts of North America.
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Inmathematical analysis,asymptotic analysis, also known asasymptotics, is a method of describinglimitingbehavior.
As an illustration, suppose that we are interested in the properties of a functionf(n)asnbecomes very large. Iff(n) =n2+ 3n, then asnbecomes very large, the term3nbecomes insignificant compared ton2. The functionf(n)is said to be "asymptotically equivalentton2, asn→ ∞". This is often written symbolically asf(n) ~n2, which is read as "f(n)is asymptotic ton2".
An example of an important asymptotic result is theprime number theorem. Letπ(x)denote theprime-counting function(which is not directly related to the constantpi), i.e.π(x)is the number ofprime numbersthat are less than or equal tox. Then the theorem states thatπ(x)∼xlnx.{\displaystyle \pi (x)\sim {\frac {x}{\ln x}}.}
Asymptotic analysis is commonly used incomputer scienceas part of theanalysis of algorithmsand is often expressed there in terms ofbig O notation.
Formally, given functionsf(x)andg(x), we define a binary relationf(x)∼g(x)(asx→∞){\displaystyle f(x)\sim g(x)\quad ({\text{as }}x\to \infty )}if and only if (de Bruijn 1981, §1.4)limx→∞f(x)g(x)=1.{\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=1.}
The symbol~is thetilde. The relation is anequivalence relationon the set of functions ofx; the functionsfandgare said to beasymptotically equivalent. Thedomainoffandgcan be any set for which the limit is defined: e.g. real numbers, complex numbers, positive integers.
The same notation is also used for other ways of passing to a limit: e.g.x→ 0,x↓ 0,|x| → 0. The way of passing to the limit is often not stated explicitly, if it is clear from the context.
Although the above definition is common in the literature, it is problematic ifg(x)is zero infinitely often asxgoes to the limiting value. For that reason, some authors use an alternative definition. The alternative definition, inlittle-o notation, is thatf~gif and only iff(x)=g(x)(1+o(1)).{\displaystyle f(x)=g(x)(1+o(1)).}
This definition is equivalent to the prior definition ifg(x)is not zero in someneighbourhoodof the limiting value.[1][2]
Iff∼g{\displaystyle f\sim g}anda∼b{\displaystyle a\sim b}, then, under some mild conditions,[further explanation needed]the following hold:
Such properties allow asymptotically equivalent functions to be freely exchanged in many algebraic expressions.
Also, if we further haveg∼h{\displaystyle g\sim h}, then, because the asymptote is atransitive relation, then we also havef∼h{\displaystyle f\sim h}.
Anasymptotic expansionof a functionf(x)is in practice an expression of that function in terms of aseries, thepartial sumsof which do not necessarily converge, but such that taking any initial partial sum provides an asymptotic formula forf. The idea is that successive terms provide an increasingly accurate description of the order of growth off.
In symbols, it means we havef∼g1,{\displaystyle f\sim g_{1},}but alsof−g1∼g2{\displaystyle f-g_{1}\sim g_{2}}andf−g1−⋯−gk−1∼gk{\displaystyle f-g_{1}-\cdots -g_{k-1}\sim g_{k}}for each fixedk. In view of the definition of the∼{\displaystyle \sim }symbol, the last equation meansf−(g1+⋯+gk)=o(gk){\displaystyle f-(g_{1}+\cdots +g_{k})=o(g_{k})}in thelittle o notation, i.e.,f−(g1+⋯+gk){\displaystyle f-(g_{1}+\cdots +g_{k})}is much smaller thangk.{\displaystyle g_{k}.}
The relationf−g1−⋯−gk−1∼gk{\displaystyle f-g_{1}-\cdots -g_{k-1}\sim g_{k}}takes its full meaning ifgk+1=o(gk){\displaystyle g_{k+1}=o(g_{k})}for allk, which means thegk{\displaystyle g_{k}}form anasymptotic scale. In that case, some authors mayabusivelywritef∼g1+⋯+gk{\displaystyle f\sim g_{1}+\cdots +g_{k}}to denote the statementf−(g1+⋯+gk)=o(gk).{\displaystyle f-(g_{1}+\cdots +g_{k})=o(g_{k}).}One should however be careful that this is not a standard use of the∼{\displaystyle \sim }symbol, and that it does not correspond to the definition given in§ Definition.
In the present situation, this relationgk=o(gk−1){\displaystyle g_{k}=o(g_{k-1})}actually follows from combining stepskandk−1; by subtractingf−g1−⋯−gk−2=gk−1+o(gk−1){\displaystyle f-g_{1}-\cdots -g_{k-2}=g_{k-1}+o(g_{k-1})}fromf−g1−⋯−gk−2−gk−1=gk+o(gk),{\displaystyle f-g_{1}-\cdots -g_{k-2}-g_{k-1}=g_{k}+o(g_{k}),}one getsgk+o(gk)=o(gk−1),{\displaystyle g_{k}+o(g_{k})=o(g_{k-1}),}i.e.gk=o(gk−1).{\displaystyle g_{k}=o(g_{k-1}).}
In case the asymptotic expansion does not converge, for any particular value of the argument there will be a particular partial sum which provides the best approximation and adding additional terms will decrease the accuracy. This optimal partial sum will usually have more terms as the argument approaches the limit value.
Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. For example, we might start with the ordinary series11−w=∑n=0∞wn{\displaystyle {\frac {1}{1-w}}=\sum _{n=0}^{\infty }w^{n}}
The expression on the left is valid on the entire complex planew≠1{\displaystyle w\neq 1}, while the right hand side converges only for|w|<1{\displaystyle |w|<1}. Multiplying bye−w/t{\displaystyle e^{-w/t}}and integrating both sides yields∫0∞e−wt1−wdw=∑n=0∞tn+1∫0∞e−uundu{\displaystyle \int _{0}^{\infty }{\frac {e^{-{\frac {w}{t}}}}{1-w}}\,dw=\sum _{n=0}^{\infty }t^{n+1}\int _{0}^{\infty }e^{-u}u^{n}\,du}
The integral on the left hand side can be expressed in terms of theexponential integral. The integral on the right hand side, after the substitutionu=w/t{\displaystyle u=w/t}, may be recognized as thegamma function. Evaluating both, one obtains the asymptotic expansione−1tEi(1t)=∑n=0∞n!tn+1{\displaystyle e^{-{\frac {1}{t}}}\operatorname {Ei} \left({\frac {1}{t}}\right)=\sum _{n=0}^{\infty }n!\;t^{n+1}}
Here, the right hand side is clearly not convergent for any non-zero value oft. However, by keepingtsmall, and truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value ofEi(1/t){\displaystyle \operatorname {Ei} (1/t)}. Substitutingx=−1/t{\displaystyle x=-1/t}and noting thatEi(x)=−E1(−x){\displaystyle \operatorname {Ei} (x)=-E_{1}(-x)}results in the asymptotic expansion given earlier in this article.
Inmathematical statistics, anasymptotic distributionis a hypothetical distribution that is in a sense the "limiting" distribution of a sequence of distributions. A distribution is an ordered set of random variablesZifori= 1, …,n, for some positive integern. An asymptotic distribution allowsito range without bound, that is,nis infinite.
A special case of an asymptotic distribution is when the late entries go to zero—that is, theZigo to 0 asigoes to infinity. Some instances of "asymptotic distribution" refer only to this special case.
This is based on the notion of anasymptoticfunction which cleanly approaches a constant value (theasymptote) as the independent variable goes to infinity; "clean" in this sense meaning that for any desired closeness epsilon there is some value of the independent variable after which the function never differs from the constant by more than epsilon.
Anasymptoteis a straight line that a curve approaches but never meets or crosses. Informally, one may speak of the curve meeting the asymptote "at infinity" although this is not a precise definition. In the equationy=1x,{\displaystyle y={\frac {1}{x}},}ybecomes arbitrarily small in magnitude asxincreases.
Asymptotic analysis is used in severalmathematical sciences. Instatistics, asymptotic theory provides limiting approximations of theprobability distributionofsample statistics, such as thelikelihood ratiostatisticand theexpected valueof thedeviance. Asymptotic theory does not provide a method of evaluating the finite-sample distributions of sample statistics, however. Non-asymptotic bounds are provided by methods ofapproximation theory.
Examples of applications are the following.
Asymptotic analysis is a key tool for exploring theordinaryandpartialdifferential equations which arise in themathematical modellingof real-world phenomena.[3]An illustrative example is the derivation of theboundary layer equationsfrom the fullNavier-Stokes equationsgoverning fluid flow. In many cases, the asymptotic expansion is in power of a small parameter,ε: in the boundary layer case, this is thenondimensionalratio of the boundary layer thickness to a typical length scale of the problem. Indeed, applications of asymptotic analysis in mathematical modelling often[3]center around a nondimensional parameter which has been shown, or assumed, to be small through a consideration of the scales of the problem at hand.
Asymptotic expansions typically arise in the approximation of certain integrals (Laplace's method,saddle-point method,method of steepest descent) or in the approximation of probability distributions (Edgeworth series). TheFeynman graphsinquantum field theoryare another example of asymptotic expansions which often do not converge.
De Bruijn illustrates the use of asymptotics in the following dialog between Dr. N.A., a Numerical Analyst, and Dr. A.A., an Asymptotic Analyst:
N.A.: I want to evaluate my functionf(x){\displaystyle f(x)}for large values ofx{\displaystyle x}, with a relative error of at most 1%.
A.A.:f(x)=x−1+O(x−2)(x→∞){\displaystyle f(x)=x^{-1}+\mathrm {O} (x^{-2})\qquad (x\to \infty )}.
N.A.: I am sorry, I don't understand.
A.A.:|f(x)−x−1|<8x−2(x>104).{\displaystyle |f(x)-x^{-1}|<8x^{-2}\qquad (x>10^{4}).}
N.A.: But my value ofx{\displaystyle x}is only 100.
A.A.: Why did you not say so? My evaluations give
|f(x)−x−1|<57000x−2(x>100).{\displaystyle |f(x)-x^{-1}|<57000x^{-2}\qquad (x>100).}
N.A.: This is no news to me. I know already that0<f(100)<1{\displaystyle 0<f(100)<1}.
A.A.: I can gain a little on some of my estimates. Now I find that
|f(x)−x−1|<20x−2(x>100).{\displaystyle |f(x)-x^{-1}|<20x^{-2}\qquad (x>100).}
N.A.: I asked for 1%, not for 20%.
A.A.: It is almost the best thing I possibly can get. Why don't you take larger values ofx{\displaystyle x}?
N.A.: !!! I think it's better to ask my electronic computing machine.
Machine: f(100) = 0.01137 42259 34008 67153
A.A.: Haven't I told you so? My estimate of 20% was not far off from the 14% of the real error.
N.A.: !!! . . . !
Some days later, Miss N.A. wants to know the value of f(1000), but her machine would take a month of computation to give the answer. She returns to her Asymptotic Colleague, and gets a fully satisfactory reply.[4]
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https://en.wikipedia.org/wiki/Asymptotic_analysis
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Inmathematics, asingular perturbationproblem is a problem containing a small parameter that cannot be approximated by setting the parameter value to zero. More precisely, the solution cannot be uniformly approximated by anasymptotic expansion
asε→0{\displaystyle \varepsilon \to 0}. Hereε{\displaystyle \varepsilon }is the small parameter of the problem andδn(ε){\displaystyle \delta _{n}(\varepsilon )}are a sequence of functions ofε{\displaystyle \varepsilon }of increasing order, such asδn(ε)=εn{\displaystyle \delta _{n}(\varepsilon )=\varepsilon ^{n}}. This is in contrast toregular perturbationproblems, for which a uniform approximation of this form can be obtained. Singularly perturbed problems are generally characterized by dynamics operating on multiple scales. Several classes of singular perturbations are outlined below.
The term "singular perturbation" was
coined in the 1940s byKurt Otto FriedrichsandWolfgang R. Wasow.[1]
A perturbed problem whose solution can be approximated on the whole problem domain, whether space or time, by a singleasymptotic expansionhas aregular perturbation. Most often in applications, an acceptable approximation to a regularly perturbed problem is found by simply replacing the small parameterε{\displaystyle \varepsilon }by zero everywhere in the problem statement. This corresponds to taking only the first term of the expansion, yielding an approximation that converges, perhaps slowly, to the true solution asε{\displaystyle \varepsilon }decreases. The solution to a singularly perturbed problem cannot be approximated in this way: As seen in the examples below, a singular perturbation generally occurs when a problem's small parameter multiplies its highest operator. Thus naively taking the parameter to be zero changes the very nature of the problem. In the case of differential equations, boundary conditions cannot be satisfied; in algebraic equations, the possible number of solutions is decreased.
Singular perturbation theory is a rich and ongoing area of exploration for mathematicians, physicists, and other researchers. The methods used to tackle problems in this field are many. The more basic of these include themethod of matched asymptotic expansionsandWKB approximationfor spatial problems, and in time, thePoincaré–Lindstedt method, themethod of multiple scalesandperiodic averaging.
The numerical methods for solving singular perturbation problems are also very popular.[2]
For books on singular perturbation in ODE and PDE's, see for example Holmes,Introduction to Perturbation Methods,[3]Hinch,Perturbation methods[4]orBenderandOrszag,Advanced Mathematical Methods for Scientists and Engineers.[5]
Each of the examples described below shows how a naive perturbation analysis, which assumes that the problem is regular instead of singular, will fail. Some show how the problem may be solved by more sophisticated singular methods.
Differential equations that contain a small parameter that premultiplies the highest order term typically exhibit boundary layers, so that the solution evolves in two different scales. For example, consider the boundary value problem
Its solution whenε=0.1{\displaystyle \varepsilon =0.1}is the solid curve shown below. Note that the solution changes rapidly near the origin. If we naively setε=0{\displaystyle \varepsilon =0}, we would get the solution labelled "outer" below which does not model the boundary layer, for whichxis close to zero. For more details that show how to obtain the uniformly valid approximation, seemethod of matched asymptotic expansions.
An electrically driven robot manipulator can have slower mechanical dynamics and faster electrical dynamics, thus exhibiting two time scales. In such cases, we can divide the system into two subsystems, one corresponding to faster dynamics and other corresponding to slower dynamics, and then design controllers for each one of them separately. Through a singular perturbation technique, we can make these two subsystems independent of each other, thereby simplifying the control problem.
Consider a class of system described by the following set of equations:
with0<ε≪1{\displaystyle 0<\varepsilon \ll 1}. The second equation indicates that the dynamics ofx2{\displaystyle x_{2}}is much faster than that ofx1{\displaystyle x_{1}}. A theorem due toTikhonov[6]states that, with the correct conditions on the system, it will initially and very quickly approximate the solution to the equations
on some interval of time and that, asε{\displaystyle \varepsilon }decreases toward zero, the system will approach the solution more closely in that same interval.[7]
Influid mechanics, the properties of a slightly viscous fluid are dramatically different outside and inside a narrowboundary layer. Thus the fluid exhibits multiple spatial scales.
Reaction–diffusion systemsin which one reagent diffuses much more slowly than another can formspatial patternsmarked by areas where a reagent exists, and areas where it does not, with sharp transitions between them. Inecology, predator-prey models such as
whereu{\displaystyle u}is the prey andv{\displaystyle v}is the predator, have been shown to exhibit such patterns.[8]
Consider the problem of finding allrootsof the polynomialp(x)=εx3−x2+1{\displaystyle p(x)=\varepsilon x^{3}-x^{2}+1}. In the limitε→0{\displaystyle \varepsilon \to 0}, thiscubicdegenerates into thequadratic1−x2{\displaystyle 1-x^{2}}with roots atx=±1{\displaystyle x=\pm 1}. Substituting a regular perturbation series
in the equation and equating equal powers ofε{\displaystyle \varepsilon }only yields corrections to these two roots:
To find the other root, singular perturbation analysis must be used. We must then deal with the fact that the equation degenerates into a quadratic when we letε{\displaystyle \varepsilon }tend to zero, in that limit one of the roots escapes to infinity. To prevent this root from becoming invisible to the perturbative analysis, we must rescalex{\displaystyle x}to keep track with this escaping root so that in terms of the rescaled variables, it doesn't escape. We define a rescaled variabley=xεν{\displaystyle y=x\varepsilon ^{\nu }}where the exponentν{\displaystyle \nu }will be chosen such that we rescale just fast enough so that the root is at a finite value ofy{\displaystyle y}in the limit ofε{\displaystyle \varepsilon }to zero, but such that it doesn't collapse to zero where the other two roots will end up. In terms ofy{\displaystyle y}we have
We can see that forν<1{\displaystyle \nu <1}they3{\displaystyle y^{3}}is dominated by the lower degree terms, while atν=1{\displaystyle \nu =1}it becomes as dominant as they2{\displaystyle y^{2}}term while they both dominate the remaining term. This point where the highest order term will no longer vanish in the limitε{\displaystyle \varepsilon }to zero by becoming equally dominant to another term, is called significant degeneration; this yields the correct rescaling to make the remaining root visible. This choice yields
Substituting the perturbation series
yields
We are then interested in the root aty0=1{\displaystyle y_{0}=1}; the double root aty0=0{\displaystyle y_{0}=0}are the two roots that we've found above that collapse to zero in the limit of an infinite rescaling. Calculating the first few terms of the series then yields
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Incomputational complexity theory, theelement distinctness problemorelement uniqueness problemis the problem of determining whether all the elements of a list are distinct.
It is a well studied problem in many different models of computation. The problem may be solved bysortingthe list and then checking if there are any consecutive equal elements; it may also be solved in linear expected time by arandomized algorithmthat inserts each item into ahash tableand compares only those elements that are placed in the same hash table cell.[1]
Several lower bounds in computational complexity are proved by reducing the element distinctness problem to the problem in question, i.e., by demonstrating that the solution of the element uniqueness problem may be quickly found after solving the problem in question.
The number of comparisons needed to solve the problem of sizen{\displaystyle n}, in a comparison-based model of computation such as adecision treeoralgebraic decision tree, isΘ(nlogn){\displaystyle \Theta (n\log n)}. Here,Θ{\displaystyle \Theta }invokesbig theta notation, meaning that the problem can be solved in a number of comparisons proportional tonlogn{\displaystyle n\log n}(alinearithmic function) and that all solutions require this many comparisons.[2]In these models of computation, the input numbers may not be used to index the computer's memory (as in the hash table solution) but may only be accessed by computing and comparing simple algebraic functions of their values. For these models, an algorithm based oncomparison sortsolves the problem within a constant factor of the best possible number of comparisons. The same lower bound applies as well to theexpected numberof comparisons in therandomizedalgebraic decision treemodel.[3][4]
If the elements in the problem arereal numbers, the decision-tree lower bound extends to thereal random-access machinemodel with an instruction set that includes addition, subtraction and multiplication of real numbers, as well as comparison and either division or remaindering ("floor").[5]It follows that the problem's complexity in this model is alsoΘ(nlogn){\displaystyle \Theta (n\log n)}. This RAM model covers more algorithms than the algebraic decision-tree model, as it encompasses algorithms that use indexing into tables. However, in this model all program steps are counted, not just decisions.
A single-tape deterministicTuring machinecan solve the problem, fornelements ofm≥ lognbits each, in timeO(n2m(m+2–logn)),
while on a nondeterministic machine the time complexity isO(nm(n+ logm)).[6]
Quantum algorithmscan solve this problem faster, inΘ(n2/3){\textstyle \Theta (n^{2/3})}queries. The optimal algorithm is byAndris Ambainis.[7]Yaoyun Shifirst proved a tight lower bound when the size of the range is sufficiently large.[8]Ambainis[9]and Kutin[10]independently (and via different proofs) extended his work to obtain the lower bound for all functions.
Elements that occur more thann/k{\displaystyle n/k}times in a multiset of sizen{\displaystyle n}may be found by a comparison-based algorithm, theMisra–Gries heavy hitters algorithm, in timeO(nlogk){\displaystyle O(n\log k)}. The element distinctness problem is a special case of this problem wherek=n{\displaystyle k=n}. This time is optimal under thedecision tree modelof computation.[11]
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Inmathematics, the concepts ofessential infimumandessential supremumare related to the notions ofinfimum and supremum, but adapted tomeasure theoryandfunctional analysis, where one often deals with statements that are not valid forallelements in aset, but ratheralmost everywhere, that is, except on aset of measure zero.
While the exact definition is not immediately straightforward, intuitively the essential supremum of a function is the smallest value that is greater than or equal to the function values everywhere while ignoring what the function does at a set of points of measure zero. For example, if one takes the functionf(x){\displaystyle f(x)}that is equal to zero everywhere except atx=0{\displaystyle x=0}wheref(0)=1,{\displaystyle f(0)=1,}then the supremum of the function equals one. However, its essential supremum is zero since (under theLebesgue measure) one can ignore what the function does at the single point wheref{\displaystyle f}is peculiar. The essential infimum is defined in a similar way.
As is often the case in measure-theoretic questions, the definition of essential supremum and infimum does not start by asking what a functionf{\displaystyle f}does at pointsx{\displaystyle x}(that is, theimageoff{\displaystyle f}), but rather by asking for the set of pointsx{\displaystyle x}wheref{\displaystyle f}equals a specific valuey{\displaystyle y}(that is, thepreimageofy{\displaystyle y}underf{\displaystyle f}).
Letf:X→R{\displaystyle f:X\to \mathbb {R} }be arealvaluedfunctiondefined on a setX.{\displaystyle X.}Thesupremumof a functionf{\displaystyle f}is characterized by the following property:f(x)≤supf≤∞{\displaystyle f(x)\leq \sup f\leq \infty }forallx∈X{\displaystyle x\in X}and if for somea∈R∪{+∞}{\displaystyle a\in \mathbb {R} \cup \{+\infty \}}we havef(x)≤a{\displaystyle f(x)\leq a}forallx∈X{\displaystyle x\in X}thensupf≤a.{\displaystyle \sup f\leq a.}More concretely, a real numbera{\displaystyle a}is called anupper boundforf{\displaystyle f}iff(x)≤a{\displaystyle f(x)\leq a}for allx∈X;{\displaystyle x\in X;}that is, if the setf−1(a,∞)={x∈X:f(x)>a}{\displaystyle f^{-1}(a,\infty )=\{x\in X:f(x)>a\}}isempty. LetUf={a∈R:f−1(a,∞)=∅}{\displaystyle U_{f}=\{a\in \mathbb {R} :f^{-1}(a,\infty )=\varnothing \}\,}be the set of upper bounds off{\displaystyle f}and define theinfimumof the empty set byinf∅=+∞.{\displaystyle \inf \varnothing =+\infty .}Then the supremum off{\displaystyle f}issupf=infUf{\displaystyle \sup f=\inf U_{f}}if the set of upper boundsUf{\displaystyle U_{f}}is nonempty, andsupf=+∞{\displaystyle \sup f=+\infty }otherwise.
Now assume in addition that(X,Σ,μ){\displaystyle (X,\Sigma ,\mu )}is ameasure spaceand, for simplicity, assume that the functionf{\displaystyle f}ismeasurable. Similar to the supremum, the essential supremum of a function is characterised by the following property:f(x)≤esssupf≤∞{\displaystyle f(x)\leq \operatorname {ess} \sup f\leq \infty }forμ{\displaystyle \mu }-almost allx∈X{\displaystyle x\in X}and if for somea∈R∪{+∞}{\displaystyle a\in \mathbb {R} \cup \{+\infty \}}we havef(x)≤a{\displaystyle f(x)\leq a}forμ{\displaystyle \mu }-almost allx∈X{\displaystyle x\in X}thenesssupf≤a.{\displaystyle \operatorname {ess} \sup f\leq a.}More concretely, a numbera{\displaystyle a}is called anessential upper boundoff{\displaystyle f}if the measurable setf−1(a,∞){\displaystyle f^{-1}(a,\infty )}is a set ofμ{\displaystyle \mu }-measure zero,[a]That is, iff(x)≤a{\displaystyle f(x)\leq a}forμ{\displaystyle \mu }-almost allx{\displaystyle x}inX.{\displaystyle X.}LetUfess={a∈R:μ(f−1(a,∞))=0}{\displaystyle U_{f}^{\operatorname {ess} }=\{a\in \mathbb {R} :\mu (f^{-1}(a,\infty ))=0\}}be the set of essential upper bounds. Then theessential supremumis defined similarly asesssupf=infUfess{\displaystyle \operatorname {ess} \sup f=\inf U_{f}^{\mathrm {ess} }}ifUfess≠∅,{\displaystyle U_{f}^{\operatorname {ess} }\neq \varnothing ,}andesssupf=+∞{\displaystyle \operatorname {ess} \sup f=+\infty }otherwise.
Exactly in the same way one defines theessential infimumas the supremum of theessential lower bounds, that is,essinff=sup{b∈R:μ({x:f(x)<b})=0}{\displaystyle \operatorname {ess} \inf f=\sup\{b\in \mathbb {R} :\mu (\{x:f(x)<b\})=0\}}if the set of essential lower bounds is nonempty, and as−∞{\displaystyle -\infty }otherwise; again there is an alternative expression asessinff=sup{a∈R:f(x)≥afor almost allx∈X}{\displaystyle \operatorname {ess} \inf f=\sup\{a\in \mathbb {R} :f(x)\geq a{\text{ for almost all }}x\in X\}}(with this being−∞{\displaystyle -\infty }if the set is empty).
On the real line consider theLebesgue measureand its corresponding𝜎-algebraΣ.{\displaystyle \Sigma .}Define a functionf{\displaystyle f}by the formulaf(x)={5,ifx=1−4,ifx=−12,otherwise.{\displaystyle f(x)={\begin{cases}5,&{\text{if }}x=1\\-4,&{\text{if }}x=-1\\2,&{\text{otherwise.}}\end{cases}}}
The supremum of this function (largest value) is 5, and the infimum (smallest value) is −4. However, the function takes these values only on the sets{1}{\displaystyle \{1\}}and{−1},{\displaystyle \{-1\},}respectively, which are of measure zero. Everywhere else, the function takes the value 2. Thus, the essential supremum and the essential infimum of this function are both 2.
As another example, consider the functionf(x)={x3,ifx∈Qarctanx,ifx∈R∖Q{\displaystyle f(x)={\begin{cases}x^{3},&{\text{if }}x\in \mathbb {Q} \\\arctan x,&{\text{if }}x\in \mathbb {R} \smallsetminus \mathbb {Q} \\\end{cases}}}whereQ{\displaystyle \mathbb {Q} }denotes therational numbers. This function is unbounded both from above and from below, so its supremum and infimum are∞{\displaystyle \infty }and−∞,{\displaystyle -\infty ,}respectively. However, from the point of view of the Lebesgue measure, the set of rational numbers is of measure zero; thus, what really matters is what happens in the complement of this set, where the function is given asarctanx.{\displaystyle \arctan x.}It follows that the essential supremum isπ/2{\displaystyle \pi /2}while the essential infimum is−π/2.{\displaystyle -\pi /2.}
On the other hand, consider the functionf(x)=x3{\displaystyle f(x)=x^{3}}defined for all realx.{\displaystyle x.}Its essential supremum is+∞,{\displaystyle +\infty ,}and its essential infimum is−∞.{\displaystyle -\infty .}
Lastly, consider the functionf(x)={1/x,ifx≠00,ifx=0.{\displaystyle f(x)={\begin{cases}1/x,&{\text{if }}x\neq 0\\0,&{\text{if }}x=0.\\\end{cases}}}Then for anya∈R,{\displaystyle a\in \mathbb {R} ,}μ({x∈R:1/x>a})≥1|a|{\displaystyle \mu (\{x\in \mathbb {R} :1/x>a\})\geq {\tfrac {1}{|a|}}}and soUfess=∅{\displaystyle U_{f}^{\operatorname {ess} }=\varnothing }andesssupf=+∞.{\displaystyle \operatorname {ess} \sup f=+\infty .}
Ifμ(X)>0{\displaystyle \mu (X)>0}theninff≤essinff≤esssupf≤supf.{\displaystyle \inf f~\leq ~\operatorname {ess} \inf f~\leq ~\operatorname {ess} \sup f~\leq ~\sup f.}and otherwise, ifX{\displaystyle X}has measure zero then[1]+∞=essinff≥esssupf=−∞.{\displaystyle +\infty ~=~\operatorname {ess} \inf f~\geq ~\operatorname {ess} \sup f~=~-\infty .}
Iff{\displaystyle f}andg{\displaystyle g}are measurable, then
and
Iff{\displaystyle f}andg{\displaystyle g}are measurable and iff≤g{\displaystyle f\leq g}almost everywhere, then
and
If the essential supremums of two functionsf{\displaystyle f}andg{\displaystyle g}are both nonnegative, thenesssup(fg)≤(esssupf)(esssupg).{\displaystyle \operatorname {ess} \sup(fg)~\leq ~(\operatorname {ess} \sup f)\,(\operatorname {ess} \sup g).}
The essential supremum of a function is not just the infimum of the essential lower bounds, but also their minimum. A similar result holds for the essential infimum.
Given ameasure space(S,Σ,μ),{\displaystyle (S,\Sigma ,\mu ),}thespaceL∞(S,μ){\displaystyle {\mathcal {L}}^{\infty }(S,\mu )}consisting of all of measurable functions that are bounded almost everywhere is aseminormed spacewhoseseminorm‖f‖∞=inf{C∈R≥0:|f(x)|≤Cfor almost everyx}={esssup|f|if0<μ(S),0if0=μ(S),{\displaystyle \|f\|_{\infty }=\inf\{C\in \mathbb {R} _{\geq 0}:|f(x)|\leq C{\text{ for almost every }}x\}={\begin{cases}\operatorname {ess} \sup |f|&{\text{ if }}0<\mu (S),\\0&{\text{ if }}0=\mu (S),\end{cases}}}is the essential supremum of a function's absolute value whenμ(S)≠0.{\displaystyle \mu (S)\neq 0.}[b]
This article incorporates material fromEssential supremumonPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.
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https://en.wikipedia.org/wiki/Essential_infimum_and_essential_supremum
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Inphysicsandengineering, theenvelopeof anoscillatingsignalis asmooth curveoutlining its extremes.[1]The envelope thus generalizes the concept of a constantamplitudeinto aninstantaneous amplitude. The figure illustrates a modulatedsine wavevarying between anupper envelopeand alower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.
A common situation resulting in an envelope function in both spacexand timetis the superposition of two waves of almost the same wavelength and frequency:[2]
which uses the trigonometric formula for theaddition of two sine waves, and the approximation Δλ≪λ:
Here themodulation wavelengthλmodis given by:[2][3]
The modulation wavelength is double that of the envelope itself because each half-wavelength of the modulating cosine wave governs both positive and negative values of the modulated sine wave. Likewise thebeat frequencyis that of the envelope, twice that of the modulating wave, or 2Δf.[4]
If this wave is a sound wave, the ear hears the frequency associated withfand the amplitude of this sound varies with the beat frequency.[4]
The argument of the sinusoids above apart from a factor 2πare:
with subscriptsCandEreferring to thecarrierand theenvelope. The same amplitudeFof the wave results from the same values of ξCand ξE, each of which may itself return to the same value over different but properly related choices ofxandt. This invariance means that one can trace these waveforms in space to find the speed of a position of fixed amplitude as it propagates in time; for the argument of thecarrier waveto stay the same, the condition is:
which shows to keep a constant amplitude the distance Δxis related to the time interval Δtby the so-calledphase velocityvp
On the other hand, the same considerations show the envelope propagates at the so-calledgroup velocityvg:[5]
A more common expression for the group velocity is obtained by introducing thewavevectork:
We notice that for small changes Δλ, the magnitude of the corresponding small change in wavevector, say Δk, is:
so the group velocity can be rewritten as:
whereωis the frequency in radians/s:ω= 2πf. In all media, frequency and wavevector are related by adispersion relation,ω=ω(k), and the group velocity can be written:
In a medium such asclassical vacuumthe dispersion relation for electromagnetic waves is:
wherec0is thespeed of lightin classical vacuum. For this case, the phase and group velocities both arec0.
In so-calleddispersive mediathedispersion relationcan be a complicated function of wavevector, and the phase and group velocities are not the same. For example, for several types of waves exhibited by atomic vibrations (phonons) inGaAs, the dispersion relations are shown in the figure forvarious directionsof wavevectork. In the general case, the phase and group velocities may have different directions.[7]
Incondensed matter physicsan energyeigenfunctionfor a mobilecharge carrierin a crystal can be expressed as aBloch wave:
wherenis the index for the band (for example, conduction or valence band)ris a spatial location, andkis awavevector. The exponential is a sinusoidally varying function corresponding to a slowly varying envelope modulating the rapidly varying part of thewave functionun,kdescribing the behavior of the wave function close to the cores of the atoms of the lattice. The envelope is restricted tok-values within a range limited by theBrillouin zoneof the crystal, and that limits how rapidly it can vary with locationr.
In determining the behavior of the carriers usingquantum mechanics, theenvelope approximationusually is used in which theSchrödinger equationis simplified to refer only to the behavior of the envelope, and boundary conditions are applied to the envelope function directly, rather than to the complete wave function.[9]For example, the wave function of a carrier trapped near an impurity is governed by an envelope functionFthat governs a superposition of Bloch functions:
where the Fourier components of the envelopeF(k) are found from the approximate Schrödinger equation.[10]In some applications, the periodic partukis replaced by its value near the band edge, sayk=k0, and then:[9]
Diffraction patternsfrom multiple slits have envelopes determined by the single slit diffraction pattern. For a single slit the pattern is given by:[11]
where α is the diffraction angle,dis the slit width, and λ is the wavelength. For multiple slits, the pattern is[11]
whereqis the number of slits, andgis the grating constant. The first factor, the single-slit resultI1, modulates the more rapidly varying second factor that depends upon the number of slits and their spacing.
Anenvelope detectoris acircuitthat attempts to extract the envelope from ananalog signal.
Indigital signal processing, the envelope may be estimated employing theHilbert transformor amovingRMS amplitude.[12]
This article incorporates material from theCitizendiumarticle "Envelope function", which is licensed under theCreative Commons Attribution-ShareAlike 3.0 Unported Licensebut not under theGFDL.
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https://en.wikipedia.org/wiki/Envelope_(waves)
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Incalculus, aone-sided limitrefers to either one of the twolimitsof afunctionf(x){\displaystyle f(x)}of arealvariablex{\displaystyle x}asx{\displaystyle x}approaches a specified point either from the left or from the right.[1][2]
The limit asx{\displaystyle x}decreases in value approachinga{\displaystyle a}(x{\displaystyle x}approachesa{\displaystyle a}"from the right"[3]or "from above") can be denoted:[1][2]
limx→a+f(x)orlimx↓af(x)orlimx↘af(x)orf(x+){\displaystyle \lim _{x\to a^{+}}f(x)\quad {\text{ or }}\quad \lim _{x\,\downarrow \,a}\,f(x)\quad {\text{ or }}\quad \lim _{x\searrow a}\,f(x)\quad {\text{ or }}\quad f(x+)}
The limit asx{\displaystyle x}increases in value approachinga{\displaystyle a}(x{\displaystyle x}approachesa{\displaystyle a}"from the left"[4][5]or "from below") can be denoted:[1][2]
limx→a−f(x)orlimx↑af(x)orlimx↗af(x)orf(x−){\displaystyle \lim _{x\to a^{-}}f(x)\quad {\text{ or }}\quad \lim _{x\,\uparrow \,a}\,f(x)\quad {\text{ or }}\quad \lim _{x\nearrow a}\,f(x)\quad {\text{ or }}\quad f(x-)}
If the limit off(x){\displaystyle f(x)}asx{\displaystyle x}approachesa{\displaystyle a}exists then the limits from the left and from the right both exist and are equal. In some cases in which the limitlimx→af(x){\displaystyle \lim _{x\to a}f(x)}does not exist, the two one-sided limits nonetheless exist. Consequently, the limit asx{\displaystyle x}approachesa{\displaystyle a}is sometimes called a "two-sided limit".[citation needed]
It is possible for exactly one of the two one-sided limits to exist (while the other does not exist). It is also possible for neither of the two one-sided limits to exist.
IfI{\displaystyle I}represents someintervalthat is contained in thedomainoff{\displaystyle f}and ifa{\displaystyle a}is a point inI{\displaystyle I}then the right-sided limit asx{\displaystyle x}approachesa{\displaystyle a}can be rigorously defined as the valueR{\displaystyle R}that satisfies:[6][verification needed]for allε>0there exists someδ>0such that for allx∈I,if0<x−a<δthen|f(x)−R|<ε,{\displaystyle {\text{for all }}\varepsilon >0\;{\text{ there exists some }}\delta >0\;{\text{ such that for all }}x\in I,{\text{ if }}\;0<x-a<\delta {\text{ then }}|f(x)-R|<\varepsilon ,}and the left-sided limit asx{\displaystyle x}approachesa{\displaystyle a}can be rigorously defined as the valueL{\displaystyle L}that satisfies:for allε>0there exists someδ>0such that for allx∈I,if0<a−x<δthen|f(x)−L|<ε.{\displaystyle {\text{for all }}\varepsilon >0\;{\text{ there exists some }}\delta >0\;{\text{ such that for all }}x\in I,{\text{ if }}\;0<a-x<\delta {\text{ then }}|f(x)-L|<\varepsilon .}
We can represent the same thing more symbolically, as follows.
LetI{\displaystyle I}represent an interval, whereI⊆domain(f){\displaystyle I\subseteq \mathrm {domain} (f)}, anda∈I{\displaystyle a\in I}.
In comparison to the formal definition for thelimit of a functionat a point, the one-sided limit (as the name would suggest) only deals with input values to one side of the approached input value.
For reference, the formal definition for the limit of a function at a point is as follows:
To define a one-sided limit, we must modify this inequality. Note that the absolute distance betweenx{\displaystyle x}anda{\displaystyle a}is
|x−a|=|(−1)(−x+a)|=|(−1)(a−x)|=|(−1)||a−x|=|a−x|.{\displaystyle |x-a|=|(-1)(-x+a)|=|(-1)(a-x)|=|(-1)||a-x|=|a-x|.}
For the limit from the right, we wantx{\displaystyle x}to be to the right ofa{\displaystyle a}, which means thata<x{\displaystyle a<x}, sox−a{\displaystyle x-a}is positive. From above,x−a{\displaystyle x-a}is the distance betweenx{\displaystyle x}anda{\displaystyle a}. We want to bound this distance by our value ofδ{\displaystyle \delta }, giving the inequalityx−a<δ{\displaystyle x-a<\delta }. Putting together the inequalities0<x−a{\displaystyle 0<x-a}andx−a<δ{\displaystyle x-a<\delta }and using thetransitivityproperty of inequalities, we have the compound inequality0<x−a<δ{\displaystyle 0<x-a<\delta }.
Similarly, for the limit from the left, we wantx{\displaystyle x}to be to the left ofa{\displaystyle a}, which means thatx<a{\displaystyle x<a}. In this case, it isa−x{\displaystyle a-x}that is positive and represents the distance betweenx{\displaystyle x}anda{\displaystyle a}. Again, we want to bound this distance by our value ofδ{\displaystyle \delta }, leading to the compound inequality0<a−x<δ{\displaystyle 0<a-x<\delta }.
Now, when our value ofx{\displaystyle x}is in its desired interval, we expect that the value off(x){\displaystyle f(x)}is also within its desired interval. The distance betweenf(x){\displaystyle f(x)}andL{\displaystyle L}, the limiting value of the left sided limit, is|f(x)−L|{\displaystyle |f(x)-L|}. Similarly, the distance betweenf(x){\displaystyle f(x)}andR{\displaystyle R}, the limiting value of the right sided limit, is|f(x)−R|{\displaystyle |f(x)-R|}. In both cases, we want to bound this distance byε{\displaystyle \varepsilon }, so we get the following:|f(x)−L|<ε{\displaystyle |f(x)-L|<\varepsilon }for the left sided limit, and|f(x)−R|<ε{\displaystyle |f(x)-R|<\varepsilon }for the right sided limit.
Example 1:
The limits from the left and from the right ofg(x):=−1x{\displaystyle g(x):=-{\frac {1}{x}}}asx{\displaystyle x}approachesa:=0{\displaystyle a:=0}arelimx→0−−1/x=+∞andlimx→0+−1/x=−∞{\displaystyle \lim _{x\to 0^{-}}{-1/x}=+\infty \qquad {\text{ and }}\qquad \lim _{x\to 0^{+}}{-1/x}=-\infty }The reason whylimx→0−−1/x=+∞{\displaystyle \lim _{x\to 0^{-}}{-1/x}=+\infty }is becausex{\displaystyle x}is always negative (sincex→0−{\displaystyle x\to 0^{-}}means thatx→0{\displaystyle x\to 0}with all values ofx{\displaystyle x}satisfyingx<0{\displaystyle x<0}), which implies that−1/x{\displaystyle -1/x}is always positive so thatlimx→0−−1/x{\displaystyle \lim _{x\to 0^{-}}{-1/x}}diverges[note 1]to+∞{\displaystyle +\infty }(and not to−∞{\displaystyle -\infty }) asx{\displaystyle x}approaches0{\displaystyle 0}from the left.
Similarly,limx→0+−1/x=−∞{\displaystyle \lim _{x\to 0^{+}}{-1/x}=-\infty }since all values ofx{\displaystyle x}satisfyx>0{\displaystyle x>0}(said differently,x{\displaystyle x}is always positive) asx{\displaystyle x}approaches0{\displaystyle 0}from the right, which implies that−1/x{\displaystyle -1/x}is always negative so thatlimx→0+−1/x{\displaystyle \lim _{x\to 0^{+}}{-1/x}}diverges to−∞.{\displaystyle -\infty .}
Example 2:
One example of a function with different one-sided limits isf(x)=11+2−1/x,{\displaystyle f(x)={\frac {1}{1+2^{-1/x}}},}(cf. picture) where the limit from the left islimx→0−f(x)=0{\displaystyle \lim _{x\to 0^{-}}f(x)=0}and the limit from the right islimx→0+f(x)=1.{\displaystyle \lim _{x\to 0^{+}}f(x)=1.}To calculate these limits, first show thatlimx→0−2−1/x=∞andlimx→0+2−1/x=0{\displaystyle \lim _{x\to 0^{-}}2^{-1/x}=\infty \qquad {\text{ and }}\qquad \lim _{x\to 0^{+}}2^{-1/x}=0}(which is true becauselimx→0−−1/x=+∞andlimx→0+−1/x=−∞{\displaystyle \lim _{x\to 0^{-}}{-1/x}=+\infty {\text{ and }}\lim _{x\to 0^{+}}{-1/x}=-\infty })
so that consequently,limx→0+11+2−1/x=11+limx→0+2−1/x=11+0=1{\displaystyle \lim _{x\to 0^{+}}{\frac {1}{1+2^{-1/x}}}={\frac {1}{1+\displaystyle \lim _{x\to 0^{+}}2^{-1/x}}}={\frac {1}{1+0}}=1}whereaslimx→0−11+2−1/x=0{\displaystyle \lim _{x\to 0^{-}}{\frac {1}{1+2^{-1/x}}}=0}because the denominator diverges to infinity; that is, becauselimx→0−1+2−1/x=∞.{\displaystyle \lim _{x\to 0^{-}}1+2^{-1/x}=\infty .}Sincelimx→0−f(x)≠limx→0+f(x),{\displaystyle \lim _{x\to 0^{-}}f(x)\neq \lim _{x\to 0^{+}}f(x),}the limitlimx→0f(x){\displaystyle \lim _{x\to 0}f(x)}does not exist.
The one-sided limit to a pointp{\displaystyle p}corresponds to thegeneral definition of limit, with the domain of the function restricted to one side, by either allowing that the function domain is a subset of the topological space, or by considering a one-sided subspace, includingp.{\displaystyle p.}[1][verification needed]Alternatively, one may consider the domain with ahalf-open interval topology.[citation needed]
A noteworthy theorem treating one-sided limits of certainpower seriesat the boundaries of theirintervals of convergenceisAbel's theorem.[citation needed]
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https://en.wikipedia.org/wiki/One-sided_limit
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Inmathematicsand, specifically,real analysis, theDini derivatives(orDini derivates) are a class of generalizations of thederivative. They were introduced byUlisse Dini, who studied continuous but nondifferentiable functions.
Theupper Dini derivative, which is also called anupper right-hand derivative,[1]of acontinuous function
is denoted byf′+and defined by
wherelim supis thesupremum limitand the limit is aone-sided limit. Thelower Dini derivative,f′−, is defined by
wherelim infis theinfimum limit.
Iffis defined on avector space, then the upper Dini derivative attin the directiondis defined by
IffislocallyLipschitz, thenf′+is finite. Iffisdifferentiableatt, then the Dini derivative attis the usualderivativeatt.
and
and
which are the same as the first pair, but with thesupremumand theinfimumreversed. For only moderately ill-behaved functions, the two extra Dini derivatives aren't needed. For particularly badly behaved functions, if all four Dini derivatives have the same value (D+f(t)=D+f(t)=D−f(t)=D−f(t){\displaystyle D^{+}f(t)=D_{+}f(t)=D^{-}f(t)=D_{-}f(t)}) then the functionfis differentiable in the usual sense at the pointt.
This article incorporates material from Dini derivative onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.
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https://en.wikipedia.org/wiki/Dini_derivative
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Inmathematics, thelimitof asequenceofsetsA1,A2,…{\displaystyle A_{1},A_{2},\ldots }(subsetsof a common setX{\displaystyle X}) is a set whose elements are determined by the sequence in either of two equivalent ways:(1)by upper and lower bounds on the sequence that converge monotonically to the same set (analogous toconvergence of real-valued sequences) and(2)by convergence of a sequence ofindicator functionswhich are themselvesreal-valued. As is the case with sequences of other objects, convergence is not necessary or even usual.
More generally, again analogous to real-valued sequences, the less restrictivelimit infimumandlimit supremumof a set sequence always exist and can be used to determine convergence: the limit exists if the limit infimum and limit supremum are identical. (See below). Such set limits are essential inmeasure theoryandprobability.
It is a common misconception that the limits infimum and supremum described here involve sets of accumulation points, that is, sets ofx=limk→∞xk,{\displaystyle x=\lim _{k\to \infty }x_{k},}where eachxk{\displaystyle x_{k}}is in someAnk.{\displaystyle A_{n_{k}}.}This is only true if convergence is determined by thediscrete metric(that is,xn→x{\displaystyle x_{n}\to x}if there isN{\displaystyle N}such thatxn=x{\displaystyle x_{n}=x}for alln≥N{\displaystyle n\geq N}). This article is restricted to that situation as it is the only one relevant for measure theory and probability. See the examples below. (On the other hand, there are more generaltopological notions of set convergencethat do involve accumulation points under differentmetricsortopologies.)
Suppose that(An)n=1∞{\displaystyle \left(A_{n}\right)_{n=1}^{\infty }}is a sequence of sets. The two equivalent definitions are as follows.
To see the equivalence of the definitions, consider the limit infimum. The use ofDe Morgan's lawbelow explains why this suffices for the limit supremum. Since indicator functions take only values0{\displaystyle 0}and1,{\displaystyle 1,}lim infn→∞1An(x)=1{\displaystyle \liminf _{n\to \infty }\mathbb {1} _{A_{n}}(x)=1}if and only if1An(x){\displaystyle \mathbb {1} _{A_{n}}(x)}takes value0{\displaystyle 0}only finitely many times. Equivalently,x∈⋃n≥1⋂j≥nAj{\textstyle x\in \bigcup _{n\geq 1}\bigcap _{j\geq n}A_{j}}if and only if there existsn{\displaystyle n}such that the element is inAm{\displaystyle A_{m}}for everym≥n,{\displaystyle m\geq n,}which is to say if and only ifx∉An{\displaystyle x\not \in A_{n}}for only finitely manyn.{\displaystyle n.}Therefore,x{\displaystyle x}is in thelim infn→∞An{\displaystyle \liminf _{n\to \infty }A_{n}}if and only ifx{\displaystyle x}is in all but finitely manyAn.{\displaystyle A_{n}.}For this reason, a shorthand phrase for the limit infimum is "x{\displaystyle x}is inAn{\displaystyle A_{n}}all but finitely often", typically expressed by writing "An{\displaystyle A_{n}}a.b.f.o.".
Similarly, an elementx{\displaystyle x}is in the limit supremum if, no matter how largen{\displaystyle n}is, there existsm≥n{\displaystyle m\geq n}such that the element is inAm.{\displaystyle A_{m}.}That is,x{\displaystyle x}is in the limit supremum if and only ifx{\displaystyle x}is in infinitely manyAn.{\displaystyle A_{n}.}For this reason, a shorthand phrase for the limit supremum is "x{\displaystyle x}is inAn{\displaystyle A_{n}}infinitely often", typically expressed by writing "An{\displaystyle A_{n}}i.o.".
To put it another way, the limit infimum consists of elements that "eventually stay forever" (are ineachset aftersomen{\displaystyle n}), while the limit supremum consists of elements that "never leave forever" (are insomeset aftereachn{\displaystyle n}). Or more formally:
The sequence(An){\displaystyle \left(A_{n}\right)}is said to benonincreasingifAn+1⊆An{\displaystyle A_{n+1}\subseteq A_{n}}for eachn,{\displaystyle n,}andnondecreasingifAn⊆An+1{\displaystyle A_{n}\subseteq A_{n+1}}for eachn.{\displaystyle n.}In each of these cases the set limit exists. Consider, for example, a nonincreasing sequence(An).{\displaystyle \left(A_{n}\right).}Then⋂j≥nAj=⋂j≥1Ajand⋃j≥nAj=An.{\displaystyle \bigcap _{j\geq n}A_{j}=\bigcap _{j\geq 1}A_{j}{\text{ and }}\bigcup _{j\geq n}A_{j}=A_{n}.}From these it follows thatlim infn→∞An=⋃n≥1⋂j≥nAj=⋂j≥1Aj=⋂n≥1⋃j≥nAj=lim supn→∞An.{\displaystyle \liminf _{n\to \infty }A_{n}=\bigcup _{n\geq 1}\bigcap _{j\geq n}A_{j}=\bigcap _{j\geq 1}A_{j}=\bigcap _{n\geq 1}\bigcup _{j\geq n}A_{j}=\limsup _{n\to \infty }A_{n}.}Similarly, if(An){\displaystyle \left(A_{n}\right)}is nondecreasing thenlimn→∞An=⋃j≥1Aj.{\displaystyle \lim _{n\to \infty }A_{n}=\bigcup _{j\geq 1}A_{j}.}
TheCantor setis defined this way.
lim infn→∞An=⋃n⋂j≥n(−1j,1−1j]=⋃n[0,1−1n]=[0,1){\displaystyle \liminf _{n\to \infty }A_{n}=\bigcup _{n}\bigcap _{j\geq n}\left(-{\tfrac {1}{j}},1-{\tfrac {1}{j}}\right]=\bigcup _{n}\left[0,1-{\tfrac {1}{n}}\right]=[0,1)}andlim supn→∞An=⋂n⋃j≥n(−1j,1−1j]=⋂n(−1n,1)=[0,1){\displaystyle \limsup _{n\to \infty }A_{n}=\bigcap _{n}\bigcup _{j\geq n}\left(-{\tfrac {1}{j}},1-{\tfrac {1}{j}}\right]=\bigcap _{n}\left(-{\tfrac {1}{n}},1\right)=[0,1)}solimn→∞An=[0,1){\displaystyle \lim _{n\to \infty }A_{n}=[0,1)}exists.
lim infn→∞An=⋃n⋂j≥n((−1)jj,1−(−1)jj]=⋃n(12n,1−12n]=(0,1){\displaystyle \liminf _{n\to \infty }A_{n}=\bigcup _{n}\bigcap _{j\geq n}\left({\tfrac {(-1)^{j}}{j}},1-{\tfrac {(-1)^{j}}{j}}\right]=\bigcup _{n}\left({\tfrac {1}{2n}},1-{\tfrac {1}{2n}}\right]=(0,1)}andlim supn→∞An=⋂n⋃j≥n((−1)jj,1−(−1)jj]=⋂n(−12n−1,1+12n−1]=[0,1],{\displaystyle \limsup _{n\to \infty }A_{n}=\bigcap _{n}\bigcup _{j\geq n}\left({\tfrac {(-1)^{j}}{j}},1-{\tfrac {(-1)^{j}}{j}}\right]=\bigcap _{n}\left(-{\tfrac {1}{2n-1}},1+{\tfrac {1}{2n-1}}\right]=[0,1],}solimn→∞An{\displaystyle \lim _{n\to \infty }A_{n}}does not exist, despite the fact that the left and right endpoints of theintervalsconverge to 0 and 1, respectively.
⋃j≥nAj=Q∩[0,1]{\displaystyle \bigcup _{j\geq n}A_{j}=\mathbb {Q} \cap [0,1]}is the set of allrational numbersbetween 0 and 1 (inclusive), since even forj<n{\displaystyle j<n}and0≤k≤j,{\displaystyle 0\leq k\leq j,}kj=nknj{\displaystyle {\tfrac {k}{j}}={\tfrac {nk}{nj}}}is an element of the above. Therefore,lim supn→∞An=Q∩[0,1].{\displaystyle \limsup _{n\to \infty }A_{n}=\mathbb {Q} \cap [0,1].}On the other hand,⋂j≥nAj={0,1},{\displaystyle \bigcap _{j\geq n}A_{j}=\{0,1\},}which implieslim infn→∞An={0,1}.{\displaystyle \liminf _{n\to \infty }A_{n}=\{0,1\}.}In this case, the sequenceA1,A2,…{\displaystyle A_{1},A_{2},\ldots }does not have a limit. Note thatlimn→∞An{\displaystyle \lim _{n\to \infty }A_{n}}is not the set of accumulation points, which would be the entire interval[0,1]{\displaystyle [0,1]}(according to the usualEuclidean metric).
Set limits, particularly the limit infimum and the limit supremum, are essential forprobabilityandmeasure theory. Such limits are used to calculate (or prove) the probabilities and measures of other, more purposeful, sets. For the following,(X,F,P){\displaystyle (X,{\mathcal {F}},\mathbb {P} )}is aprobability space, which meansF{\displaystyle {\mathcal {F}}}is aσ-algebraof subsets ofX{\displaystyle X}andP{\displaystyle \mathbb {P} }is aprobability measuredefined on that σ-algebra. Sets in the σ-algebra are known asevents.
IfA1,A2,…{\displaystyle A_{1},A_{2},\ldots }is amonotone sequenceof events inF{\displaystyle {\mathcal {F}}}thenlimn→∞An{\displaystyle \lim _{n\to \infty }A_{n}}exists andP(limn→∞An)=limn→∞P(An).{\displaystyle \mathbb {P} \left(\lim _{n\to \infty }A_{n}\right)=\lim _{n\to \infty }\mathbb {P} \left(A_{n}\right).}
In probability, the two Borel–Cantelli lemmas can be useful for showing that the limsup of a sequence of events has probability equal to 1 or to 0. The statement of the first (original) Borel–Cantelli lemma is
First Borel–Cantelli lemma—If∑n=1∞P(An)<∞{\displaystyle \sum _{n=1}^{\infty }\mathbb {P} \left(A_{n}\right)<\infty }thenP(lim supn→∞An)=0.{\displaystyle \mathbb {P} \left(\limsup _{n\to \infty }A_{n}\right)=0.}
The second Borel–Cantelli lemma is a partial converse:
Second Borel–Cantelli lemma—IfA1,A2,…{\displaystyle A_{1},A_{2},\ldots }are independent events and∑n=1∞P(An)=∞{\displaystyle \sum _{n=1}^{\infty }\mathbb {P} \left(A_{n}\right)=\infty }thenP(lim supn→∞An)=1.{\displaystyle \mathbb {P} \left(\limsup _{n\to \infty }A_{n}\right)=1.}
One of the most important applications toprobabilityis for demonstrating thealmost sure convergenceof a sequence ofrandom variables. The event that a sequence of random variablesY1,Y2,…{\displaystyle Y_{1},Y_{2},\ldots }converges to another random variableY{\displaystyle Y}is formally expressed as{lim supn→∞|Yn−Y|=0}.{\textstyle \left\{\limsup _{n\to \infty }\left|Y_{n}-Y\right|=0\right\}.}It would be a mistake, however, to write this simply as a limsup of events. That is, thisis notthe eventlim supn→∞{|Yn−Y|=0}{\textstyle \limsup _{n\to \infty }\left\{\left|Y_{n}-Y\right|=0\right\}}! Instead, thecomplementof the event is{lim supn→∞|Yn−Y|≠0}={lim supn→∞|Yn−Y|>1kfor somek}=⋃k≥1⋂n≥1⋃j≥n{|Yj−Y|>1k}=limk→∞lim supn→∞{|Yn−Y|>1k}.{\displaystyle {\begin{aligned}\left\{\limsup _{n\to \infty }\left|Y_{n}-Y\right|\neq 0\right\}&=\left\{\limsup _{n\to \infty }\left|Y_{n}-Y\right|>{\frac {1}{k}}{\text{ for some }}k\right\}\\&=\bigcup _{k\geq 1}\bigcap _{n\geq 1}\bigcup _{j\geq n}\left\{\left|Y_{j}-Y\right|>{\tfrac {1}{k}}\right\}\\&=\lim _{k\to \infty }\limsup _{n\to \infty }\left\{\left|Y_{n}-Y\right|>{\tfrac {1}{k}}\right\}.\end{aligned}}}Therefore,P({lim supn→∞|Yn−Y|≠0})=limk→∞P(lim supn→∞{|Yn−Y|>1k}).{\displaystyle \mathbb {P} \left(\left\{\limsup _{n\to \infty }\left|Y_{n}-Y\right|\neq 0\right\}\right)=\lim _{k\to \infty }\mathbb {P} \left(\limsup _{n\to \infty }\left\{\left|Y_{n}-Y\right|>{\tfrac {1}{k}}\right\}\right).}
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Incomputer science, theAkra–Bazzi method, orAkra–Bazzi theorem, is used to analyze the asymptotic behavior of the mathematicalrecurrencesthat appear in theanalysisofdivide and conquer algorithmswhere the sub-problems have substantially different sizes. It is a generalization of themaster theorem for divide-and-conquer recurrences, which assumes that the sub-problems have equal size. It is named after mathematicians Mohamad Akra and Louay Bazzi.[1]
The Akra–Bazzi method applies to recurrence formulas of the form:[1]
The conditions for usage are:
The asymptotic behavior ofT(x){\displaystyle T(x)}is found by determining the value ofp{\displaystyle p}for which∑i=1kaibip=1{\displaystyle \sum _{i=1}^{k}a_{i}b_{i}^{p}=1}and plugging that value into the equation:[2]
(seeΘ). Intuitively,hi(x){\displaystyle h_{i}(x)}represents a small perturbation in the index ofT{\displaystyle T}. By noting that⌊bix⌋=bix+(⌊bix⌋−bix){\displaystyle \lfloor b_{i}x\rfloor =b_{i}x+(\lfloor b_{i}x\rfloor -b_{i}x)}and that the absolute value of⌊bix⌋−bix{\displaystyle \lfloor b_{i}x\rfloor -b_{i}x}is always between 0 and 1,hi(x){\displaystyle h_{i}(x)}can be used to ignore thefloor functionin the index. Similarly, one can also ignore theceiling function. For example,T(n)=n+T(12n){\displaystyle T(n)=n+T\left({\frac {1}{2}}n\right)}andT(n)=n+T(⌊12n⌋){\displaystyle T(n)=n+T\left(\left\lfloor {\frac {1}{2}}n\right\rfloor \right)}will, as per the Akra–Bazzi theorem, have the same asymptotic behavior.
SupposeT(n){\displaystyle T(n)}is defined as 1 for integers0≤n≤3{\displaystyle 0\leq n\leq 3}andn2+74T(⌊12n⌋)+T(⌈34n⌉){\displaystyle n^{2}+{\frac {7}{4}}T\left(\left\lfloor {\frac {1}{2}}n\right\rfloor \right)+T\left(\left\lceil {\frac {3}{4}}n\right\rceil \right)}for integersn>3{\displaystyle n>3}. In applying the Akra–Bazzi method, the first step is to find the value ofp{\displaystyle p}for which74(12)p+(34)p=1{\displaystyle {\frac {7}{4}}\left({\frac {1}{2}}\right)^{p}+\left({\frac {3}{4}}\right)^{p}=1}. In this example,p=2{\displaystyle p=2}. Then, using the formula, the asymptotic behavior can be determined as follows:[3]
The Akra–Bazzi method is more useful than most other techniques for determining asymptotic behavior because it covers such a wide variety of cases. Its primary application is the approximation of the running time of many divide-and-conquer algorithms. For example, in themerge sort, the number of comparisons required in the worst case, which is roughly proportional to its runtime, is given recursively asT(1)=0{\displaystyle T(1)=0}and
for integersn>0{\displaystyle n>0}, and can thus be computed using the Akra–Bazzi method to beΘ(nlogn){\displaystyle \Theta (n\log n)}.
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https://en.wikipedia.org/wiki/Akra%E2%80%93Bazzi_method
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Incomputer science, thecomputational complexityor simplycomplexityof analgorithmis the amount of resources required to run it.[1]Particular focus is given tocomputation time(generally measured by the number of needed elementary operations) andmemory storagerequirements. The complexity of aproblemis the complexity of the best algorithms that allow solving the problem.
The study of the complexity of explicitly given algorithms is calledanalysis of algorithms, while the study of the complexity of problems is calledcomputational complexity theory. Both areas are highly related, as the complexity of an algorithm is always anupper boundon the complexity of the problem solved by this algorithm. Moreover, for designing efficient algorithms, it is often fundamental to compare the complexity of a specific algorithm to the complexity of the problem to be solved. Also, in most cases, the only thing that is known about the complexity of a problem is that it is lower than the complexity of the most efficient known algorithms. Therefore, there is a large overlap between analysis of algorithms and complexity theory.
As the amount of resources required to run an algorithm generally varies with the size of the input, the complexity is typically expressed as a functionn→f(n), wherenis the size of the input andf(n)is either theworst-case complexity(the maximum of the amount of resources that are needed over all inputs of sizen) or theaverage-case complexity(the average of the amount of resources over all inputs of sizen).Time complexityis generally expressed as the number of required elementary operations on an input of sizen, where elementary operations are assumed to take a constant amount of time on a given computer and change only by a constant factor when run on a different computer.Space complexityis generally expressed as the amount ofmemoryrequired by an algorithm on an input of sizen.
The resource that is most commonly considered is time. When "complexity" is used without qualification, this generally means time complexity.
The usual units of time (seconds, minutes etc.) are not used incomplexity theorybecause they are too dependent on the choice of a specific computer and on the evolution of technology. For instance, a computer today can execute an algorithm significantly faster than a computer from the 1960s; however, this is not an intrinsic feature of the algorithm but rather a consequence of technological advances incomputer hardware. Complexity theory seeks to quantify the intrinsic time requirements of algorithms, that is, the basic time constraints an algorithm would place onanycomputer. This is achieved by counting the number ofelementary operationsthat are executed during the computation. These operations are assumed to take constant time (that is, not affected by the size of the input) on a given machine, and are often calledsteps.
Formally, thebit complexityrefers to the number of operations onbitsthat are needed for running an algorithm. With mostmodels of computation, it equals the time complexity up to a constant factor. Oncomputers, the number of operations onmachine wordsthat are needed is also proportional to the bit complexity. So, thetime complexityand thebit complexityare equivalent for realistic models of computation.
Another important resource is the size ofcomputer memorythat is needed for running algorithms.
For the class ofdistributed algorithmsthat are commonly executed by multiple, interacting parties, the resource that is of most interest is the communication complexity. It is the necessary amount of communication between the executing parties.
The number ofarithmetic operationsis another resource that is commonly used. In this case, one talks ofarithmetic complexity. If one knows anupper boundon the size of thebinary representationof the numbers that occur during a computation, the time complexity is generally the product of the arithmetic complexity by a constant factor.
For many algorithms the size of the integers that are used during a computation is not bounded, and it is not realistic to consider that arithmetic operations take a constant time. Therefore, the time complexity, generally calledbit complexityin this context, may be much larger than the arithmetic complexity. For example, the arithmetic complexity of the computation of thedeterminantof an×ninteger matrixisO(n3){\displaystyle O(n^{3})}for the usual algorithms (Gaussian elimination). The bit complexity of the same algorithms isexponentialinn, because the size of the coefficients may grow exponentially during the computation. On the other hand, if these algorithms are coupled withmulti-modular arithmetic, the bit complexity may be reduced toO~(n4).
Insortingandsearching, the resource that is generally considered is the number of entry comparisons. This is generally a good measure of the time complexity if data are suitably organized.
It is impossible to count the number of steps of an algorithm on all possible inputs. As the complexity generally increases with the size of the input, the complexity is typically expressed as a function of the sizen(inbits) of the input, and therefore, the complexity is a function ofn. However, the complexity of an algorithm may vary dramatically for different inputs of the same size. Therefore, several complexity functions are commonly used.
Theworst-case complexityis the maximum of the complexity over all inputs of sizen, and theaverage-case complexityis the average of the complexity over all inputs of sizen(this makes sense, as the number of possible inputs of a given size is finite). Generally, when "complexity" is used without being further specified, this is the worst-case time complexity that is considered.
It is generally difficult to compute precisely the worst-case and the average-case complexity. In addition, these exact values provide little practical application, as any change of computer or of model of computation would change the complexity somewhat. Moreover, the resource use is not critical for small values ofn, and this makes that, for smalln, the ease of implementation is generally more interesting than a low complexity.
For these reasons, one generally focuses on the behavior of the complexity for largen, that is on itsasymptotic behaviorwhenntends to the infinity. Therefore, the complexity is generally expressed by usingbig O notation.
For example, the usual algorithm for integermultiplicationhas a complexity ofO(n2),{\displaystyle O(n^{2}),}this means that there is a constantcu{\displaystyle c_{u}}such that the multiplication of two integers of at mostndigits may be done in a time less thancun2.{\displaystyle c_{u}n^{2}.}This bound issharpin the sense that the worst-case complexity and the average-case complexity areΩ(n2),{\displaystyle \Omega (n^{2}),}which means that there is a constantcl{\displaystyle c_{l}}such that these complexities are larger thancln2.{\displaystyle c_{l}n^{2}.}Theradixdoes not appear in these complexity, as changing of radix changes only the constantscu{\displaystyle c_{u}}andcl.{\displaystyle c_{l}.}
The evaluation of the complexity relies on the choice of amodel of computation, which consists in defining the basic operations that are done in a unit of time. When the model of computation is not explicitly specified, it is generally implicitely assumed as being amultitape Turing machine, since several more realistic models of computation, such asrandom-access machinesare asymptotically equivalent for most problems. It is only for very specific and difficult problems, such asinteger multiplicationin timeO(nlogn),{\displaystyle O(n\log n),}that the explicit definition of the model of computation is required for proofs.
Adeterministic modelof computation is a model of computation such that the successive states of the machine and the operations to be performed are completely determined by the preceding state. Historically, the first deterministic models wererecursive functions,lambda calculus, andTuring machines. The model ofrandom-access machines(also called RAM-machines) is also widely used, as a closer counterpart to realcomputers.
When the model of computation is not specified, it is generally assumed to be amultitape Turing machine. For most algorithms, the time complexity is the same on multitape Turing machines as on RAM-machines, although some care may be needed in how data is stored in memory to get this equivalence.
In anon-deterministic model of computation, such asnon-deterministic Turing machines, some choices may be done at some steps of the computation. In complexity theory, one considers all possible choices simultaneously, and the non-deterministic time complexity is the time needed, when the best choices are always done. In other words, one considers that the computation is done simultaneously on as many (identical) processors as needed, and the non-deterministic computation time is the time spent by the first processor that finishes the computation. This parallelism is partly amenable toquantum computingvia superposedentangled statesin running specificquantum algorithms, like e.g.Shor's factorizationof yet only small integers (as of March 2018[update]: 21 = 3 × 7).
Even when such a computation model is not realistic yet, it has theoretical importance, mostly related to theP = NPproblem, which questions the identity of the complexity classes formed by taking "polynomial time" and "non-deterministic polynomial time" as least upper bounds. Simulating an NP-algorithm on a deterministic computer usually takes "exponential time". A problem is in the complexity classNP, if it may be solved inpolynomial timeon a non-deterministic machine. A problem isNP-completeif, roughly speaking, it is in NP and is not easier than any other NP problem. Manycombinatorialproblems, such as theKnapsack problem, thetravelling salesman problem, and theBoolean satisfiability problemare NP-complete. For all these problems, the best known algorithm has exponential complexity. If any one of these problems could be solved in polynomial time on a deterministic machine, then all NP problems could also be solved in polynomial time, and one would have P = NP. As of 2017[update]it is generally conjectured thatP ≠ NP,with the practical implication that the worst cases of NP problems are intrinsically difficult to solve, i.e., take longer than any reasonable time span (decades!) for interesting lengths of input.
Parallel and distributed computing consist of splitting computation on several processors, which work simultaneously. The difference between the different model lies mainly in the way of transmitting information between processors. Typically, in parallel computing the data transmission between processors is very fast, while, in distributed computing, the data transmission is done through anetworkand is therefore much slower.
The time needed for a computation onNprocessors is at least the quotient byNof the time needed by a single processor. In fact this theoretically optimal bound can never be reached, because some subtasks cannot be parallelized, and some processors may have to wait a result from another processor.
The main complexity problem is thus to design algorithms such that the product of the computation time by the number of processors is as close as possible to the time needed for the same computation on a single processor.
Aquantum computeris a computer whose model of computation is based onquantum mechanics. TheChurch–Turing thesisapplies to quantum computers; that is, every problem that can be solved by a quantum computer can also be solved by a Turing machine. However, some problems may theoretically be solved with a much lowertime complexityusing a quantum computer rather than a classical computer. This is, for the moment, purely theoretical, as no one knows how to build an efficient quantum computer.
Quantum complexity theoryhas been developed to study thecomplexity classesof problems solved using quantum computers. It is used inpost-quantum cryptography, which consists of designingcryptographic protocolsthat are resistant to attacks by quantum computers.
The complexity of a problem is theinfimumof the complexities of the algorithms that may solve the problem[citation needed], including unknown algorithms. Thus the complexity of a problem is not greater than the complexity of any algorithm that solves the problems.
It follows that every complexity of an algorithm, that is expressed withbig O notation, is also an upper bound on the complexity of the corresponding problem.
On the other hand, it is generally hard to obtain nontrivial lower bounds for problem complexity, and there are few methods for obtaining such lower bounds.
For solving most problems, it is required to read all input data, which, normally, needs a time proportional to the size of the data. Thus, such problems have a complexity that is at leastlinear, that is, usingbig omega notation, a complexityΩ(n).{\displaystyle \Omega (n).}
The solution of some problems, typically incomputer algebraandcomputational algebraic geometry, may be very large. In such a case, the complexity is lower bounded by the maximal size of the output, since the output must be written. For example, asystem ofnpolynomial equations of degreedinnindeterminatesmay have up todn{\displaystyle d^{n}}complexsolutions, if the number of solutions is finite (this isBézout's theorem). As these solutions must be written down, the complexity of this problem isΩ(dn).{\displaystyle \Omega (d^{n}).}For this problem, an algorithm of complexitydO(n){\displaystyle d^{O(n)}}is known, which may thus be considered as asymptotically quasi-optimal.
A nonlinear lower bound ofΩ(nlogn){\displaystyle \Omega (n\log n)}is known for the number of comparisons needed for asorting algorithm. Thus the best sorting algorithms are optimal, as their complexity isO(nlogn).{\displaystyle O(n\log n).}This lower bound results from the fact that there aren!ways of orderingnobjects. As each comparison splits in two parts this set ofn!orders, the number ofNof comparisons that are needed for distinguishing all orders must verify2N>n!,{\displaystyle 2^{N}>n!,}which impliesN=Ω(nlogn),{\displaystyle N=\Omega (n\log n),}byStirling's formula.
A standard method for getting lower bounds of complexity consists ofreducinga problem to another problem. More precisely, suppose that one may encode a problemAof sizeninto a subproblem of sizef(n)of a problemB, and that the complexity ofAisΩ(g(n)).{\displaystyle \Omega (g(n)).}Without loss of generality, one may suppose that the functionfincreases withnand has aninverse functionh. Then the complexity of the problemBisΩ(g(h(n))).{\displaystyle \Omega (g(h(n))).}This is the method that is used to prove that, ifP ≠ NP(an unsolved conjecture), the complexity of everyNP-complete problemisΩ(nk),{\displaystyle \Omega (n^{k}),}for every positive integerk.
Evaluating the complexity of an algorithm is an important part ofalgorithm design, as this gives useful information on the performance that may be expected.
It is a common misconception that the evaluation of the complexity of algorithms will become less important as a result ofMoore's law, which posits theexponential growthof the power of moderncomputers. This is wrong because this power increase allows working with large input data (big data). For example, when one wants to sort alphabetically a list of a few hundreds of entries, such as thebibliographyof a book, any algorithm should work well in less than a second. On the other hand, for a list of a million of entries (the phone numbers of a large town, for example), the elementary algorithms that requireO(n2){\displaystyle O(n^{2})}comparisons would have to do a trillion of comparisons, which would need around three hours at the speed of 10 million of comparisons per second. On the other hand, thequicksortandmerge sortrequire onlynlog2n{\displaystyle n\log _{2}n}comparisons (as average-case complexity for the former, as worst-case complexity for the latter). Forn= 1,000,000, this gives approximately 30,000,000 comparisons, which would only take 3 seconds at 10 million comparisons per second.
Thus the evaluation of the complexity may allow eliminating many inefficient algorithms before any implementation. This may also be used for tuning complex algorithms without testing all variants. By determining the most costly steps of a complex algorithm, the study of complexity allows also focusing on these steps the effort for improving the efficiency of an implementation.
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https://en.wikipedia.org/wiki/Asymptotic_complexity
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Borel, then an unknown young man, discovered that his summation method gave the 'right' answer for many classical divergent series. He decided to make a pilgrimage to Stockholm to seeMittag-Leffler, who was the recognized lord of complex analysis. Mittag-Leffler listened politely to what Borel had to say and then, placing his hand upon the complete works byWeierstrass, his teacher, he said in Latin, 'The Master forbids it'.
In mathematics,Borel summationis asummation methodfordivergent series, introduced byÉmile Borel(1899). It is particularly useful for summingdivergent asymptotic series, and in some sense gives the best possible sum for such series. There are several variations of this method that are also called Borel summation, and a generalization of it calledMittag-Leffler summation.
There are (at least) three slightly different methods called Borel summation. They differ in which series they can sum, but are consistent, meaning that if two of the methods sum the same series they give the same answer.
Throughout letA(z)denote aformal power series
and define theBorel transformofAto be its corresponding exponential series
LetAn(z)denote the partial sum
A weak form of Borel's summation method defines the Borel sum ofAto be
If this converges atz∈Cto some functiona(z), we say that the weak Borel sum ofAconverges atz, and write∑akzk=a(z)(wB){\displaystyle {\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {wB}})}.
Suppose that the Borel transform converges for all positive real numbers to a function growing sufficiently slowly that the following integral is well defined (as an improper integral), theBorel sumofAis given by
representingLaplace transformofBA(z){\displaystyle {\mathcal {B}}A(z)}.
If the integral converges atz∈Cto somea(z), we say that the Borel sum ofAconverges atz, and write∑akzk=a(z)(B){\displaystyle {\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {B}})}.
This is similar to Borel's integral summation method, except that the Borel transform need not converge for allt, but converges to ananalytic functionoftnear 0 that can beanalytically continuedalong thepositive real axis.
The methods(B)and(wB)are bothregularsummation methods, meaning that wheneverA(z)converges (in the standard sense), then the Borel sum and weak Borel sum also converge, and do so to the same value. i.e.
Regularity of(B)is easily seen by a change in order of integration, which is valid due toabsolute convergence: ifA(z)is convergent atz, then
where the rightmost expression is exactly the Borel sum atz.
Regularity of(B)and(wB)imply that these methods provide analytic extensions toA(z).
Any seriesA(z)that is weak Borel summable atz∈Cis also Borel summable atz. However, one can constructexamplesof series which are divergent under weak Borel summation, but which are Borel summable. The following theorem characterises the equivalence of the two methods.
There are always many different functions with any given asymptotic expansion. However, there is sometimes a best possible function, in the sense that the errors in the finite-dimensional approximations are as small as possible in some region. Watson's theorem and Carleman's theorem show that Borel summation produces such a best possible sum of the series.
Watson's theorem gives conditions for a function to be the Borel sum of its asymptotic series. Suppose thatfis a function satisfying the following conditions:
is bounded by
for allzin the region (for some positive constantC).
Then Watson's theorem says that in this regionfis given by the Borel sum of its asymptotic series. More precisely, the series for the Borel transform converges in a neighborhood of the origin, and can be analytically continued to the positive real axis, and the integral defining the Borel sum converges tof(z)forzin the region above.
Carleman's theorem shows that a function is uniquely determined by an asymptotic series in a sector provided the errors in the finite order approximations do not grow too fast. More precisely it states that iffis analytic in the interior of the sector|z| <C,Re(z) > 0and|f(z)| < |bnz|nin this region for alln, thenfis zero provided that the series1/b0+ 1/b1+ ...diverges.
Carleman's theorem gives a summation method for any asymptotic series whose terms do not grow too fast, as the sum can be defined to be the unique function with this asymptotic series in a suitable sector if it exists. Borel summation is slightly weaker than special case of this whenbn=cnfor some constantc. More generally one can define summation methods slightly stronger than Borel's by taking the numbersbnto be slightly larger, for examplebn=cnlognorbn=cnlognlog logn. In practice this generalization is of little use, as there are almost no natural examples of series summable by this method that cannot also be summed by Borel's method.
The functionf(z) = exp(–1/z)has the asymptotic series0 + 0z+ ...with an error bound of the form above in the region|arg(z)| <θfor anyθ<π/2, but is not given by the Borel sum of its asymptotic series. This shows that the numberπ/2in Watson's theorem cannot be replaced by any smaller number (unless the bound on the error is made smaller).
Consider thegeometric series
which converges (in the standard sense) to1/(1 −z)for|z| < 1. The Borel transform is
from which we obtain the Borel sum
which converges in the larger regionRe(z) < 1, giving ananalytic continuationof the original series.
Considering instead the weak Borel transform, the partial sums are given byAN(z) = (1 − zN+1)/(1 −z), and so the weak Borel sum is
where, again, convergence is onRe(z) < 1. Alternatively this can be seen by appealing to part 2 of the equivalence theorem, since forRe(z) < 1,
Consider the series
thenA(z)does not converge for any nonzeroz∈C. The Borel transform is
for|t| < 1, which can be analytically continued to allt≥ 0. So the Borel sum is
(whereΓis theincomplete gamma function).
This integral converges for allz≥ 0, so the original divergent series is Borel summable for all suchz. This function has anasymptotic expansionasztends to 0 that is given by the original divergent series. This is a typical example of the fact that Borel summation will sometimes "correctly" sum divergent asymptotic expansions.
Again, since
for allz, the equivalence theorem ensures that weak Borel summation has the same domain of convergence,z≥ 0.
The following example extends on that given in (Hardy 1992, 8.5). Consider
After changing the order of summation, the Borel transform is given by
Atz= 2the Borel sum is given by
whereS(x)is theFresnel integral. Via theconvergence theoremalong chords, the Borel integral converges for allz≤ 2(the integral diverges forz> 2).
For the weak Borel sum we note that
holds only forz< 1, and so the weak Borel sum converges on this smaller domain.
If a formal seriesA(z)is Borel summable atz0∈C, then it is also Borel summable at all points on the chordOz0connectingz0to the origin. Moreover, there exists a functiona(z)analytic throughout the disk with radiusOz0such that
for allz= θz0, θ ∈ [0,1].
An immediate consequence is that the domain of convergence of the Borel sum is astar domaininC. More can be said about the domain of convergence of the Borel sum, than that it is a star domain, which is referred to as the Borel polygon, and is determined by the singularities of the seriesA(z).
Suppose thatA(z)has strictly positive radius of convergence, so that it is analytic in a non-trivial region containing the origin, and letSAdenote the set of singularities ofA. This means thatP∈SAif and only ifAcan be continued analytically along the open chord from 0 toP, but not toPitself. ForP∈SA, letLPdenote the line passing throughPwhich is perpendicular to the chordOP. Define the sets
the set of points which lie on the same side ofLPas the origin. The Borel polygon ofAis the set
An alternative definition was used by Borel and Phragmén (Sansone & Gerretsen 1960, 8.3). LetS⊂C{\displaystyle S\subset \mathbb {C} }denote the largest star domain on which there is an analytic extension ofA, thenΠA{\displaystyle \Pi _{A}}is the largest subset ofS{\displaystyle S}such that for allP∈ΠA{\displaystyle P\in \Pi _{A}}the interior of the circle with diameterOPis contained inS{\displaystyle S}. Referring to the setΠA{\displaystyle \Pi _{A}}as a polygon is something of a misnomer, since the set need not be polygonal at all; if, however,A(z)has only finitely many singularities thenΠA{\displaystyle \Pi _{A}}will in fact be a polygon.
The following theorem, due to Borel andPhragménprovides convergence criteria for Borel summation.
Note that(B)summability forz∈∂ΠA{\displaystyle z\in \partial \Pi _{A}}depends on the nature of the point.
Letωi∈Cdenote them-th roots of unity,i= 1, ...,m, and consider
which converges onB(0,1) ⊂C. Seen as a function onC,A(z)has singularities atSA= {ωi:i= 1, ...,m}, and consequently the Borel polygonΠA{\displaystyle \Pi _{A}}is given by the regularm-goncentred at the origin, and such that1 ∈Cis a midpoint of an edge.
The formal series
converges for all|z|<1{\displaystyle |z|<1}(for instance, by thecomparison testwith the geometric series). It can however be shown[2]thatAdoes not converge for any pointz∈Csuch thatz2n= 1for somen. Since the set of suchzis dense in theunit circle, there can be no analytic extension ofAoutside ofB(0,1). Subsequently the largest star domain to whichAcan be analytically extended isS=B(0,1)from which (via the second definition) one obtainsΠA=B(0,1){\displaystyle \Pi _{A}=B(0,1)}. In particular one sees that the Borel polygon is not polygonal.
ATauberian theoremprovides conditions under which convergence of one summation method implies convergence under another method. The principal Tauberian theorem[1]for Borel summation provides conditions under which the weak Borel method implies convergence of the series.
Borel summation finds application inperturbation expansionsin quantum field theory. In particular in 2-dimensional Euclidean field theory the Schwinger functions can often be recovered from their perturbation series using Borel summation (Glimm & Jaffe 1987, p. 461). Some of the singularities of the Borel transform are related toinstantonsandrenormalonsin quantum field theory (Weinberg 2005, 20.7).
Borel summation requires that the coefficients do not grow too fast: more precisely,anhas to be bounded byn!Cn+1for someC. There is a variation of Borel summation that replacesfactorialsn!with(kn)!for some positiveintegerk, which allows the summation of some series withanbounded by(kn)!Cn+1for someC. This generalization is given byMittag-Leffler summation.
In the most general case, Borel summation is generalized byNachbin resummation, which can be used when the bounding function is of some general type (psi-type), instead of beingexponential type.
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https://en.wikipedia.org/wiki/Borel_summation
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In the mathematics ofconvergentanddivergent series,Euler summationis a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σan, if itsEuler transformconverges to a sum, then that sum is called theEuler sumof the original series. As well as being used to define values for divergent series, Euler summation can be used to speed the convergence of series.
Euler summation can be generalized into a family of methods denoted (E,q), whereq≥ 0. The (E, 1) sum is the ordinary Euler sum. All of these methods are strictly weaker thanBorel summation; forq> 0 they are incomparable withAbel summation.
For some valueywe may define the Euler sum (if it converges for that value ofy) corresponding to a particular formal summation as:
If all the formal sums actually converge, the Euler sum will equal the left hand side. However, using Euler summation canaccelerate the convergence(this is especially useful for alternating series); sometimes it can also give a useful meaning to divergent sums.
To justify the approach notice that for interchanged sum, Euler's summation reduces to the initial series, because
This method itself cannot be improved by iterated application, as
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https://en.wikipedia.org/wiki/Euler_summation
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Inmathematical analysis,Cesàro summation(also known as theCesàro mean[1][2]orCesàro limit[3]) assigns values to someinfinite sumsthat arenot necessarily convergentin the usual sense. The Cesàro sum is defined as the limit, asntends to infinity, of the sequence of arithmetic means of the firstnpartial sums of the series.
This special case of amatrix summability methodis named for the Italian analystErnesto Cesàro(1859–1906).
The termsummationcan be misleading, as some statements and proofs regarding Cesàro summation can be said to implicate theEilenberg–Mazur swindle. For example, it is commonly applied toGrandi's serieswith the conclusion that thesumof that series is 1/2.
Let(an)n=1∞{\displaystyle (a_{n})_{n=1}^{\infty }}be asequence, and let
be itskthpartial sum.
The sequence(an)is calledCesàro summable, with Cesàro sumA∈R{\displaystyle \mathbb {R} }, if, asntends to infinity, thearithmetic meanof its firstnpartial sumss1,s2, ...,sntends toA:
The value of the resulting limit is called the Cesàro sum of the series∑n=1∞an.{\displaystyle \textstyle \sum _{n=1}^{\infty }a_{n}.}If this series is convergent, then it is Cesàro summable and its Cesàro sum is the usual sum.
Letan= (−1)nforn≥ 0. That is,(an)n=0∞{\displaystyle (a_{n})_{n=0}^{\infty }}is the sequence
LetGdenote the series
The seriesGis known asGrandi's series.
Let(sk)k=0∞{\displaystyle (s_{k})_{k=0}^{\infty }}denote the sequence of partial sums ofG:
This sequence of partial sums does not converge, so the seriesGis divergent. However,GisCesàro summable. Let(tn)n=1∞{\displaystyle (t_{n})_{n=1}^{\infty }}be the sequence of arithmetic means of the firstnpartial sums:
Then
and therefore, the Cesàro sum of the seriesGis1/2.
As another example, letan=nforn≥ 1. That is,(an)n=1∞{\displaystyle (a_{n})_{n=1}^{\infty }}is the sequence
LetGnow denote the series
Then the sequence of partial sums(sk)k=1∞{\displaystyle (s_{k})_{k=1}^{\infty }}is
Since the sequence of partial sums grows without bound, the seriesGdiverges to infinity. The sequence(tn)of means of partial sums of G is
This sequence diverges to infinity as well, soGisnotCesàro summable. In fact, for the series of any sequence which diverges to (positive or negative) infinity, the Cesàro method also leads to the series of a sequence that diverges likewise, and hence such a series is not Cesàro summable.
In 1890, Ernesto Cesàro stated a broader family of summation methods which have since been called(C,α)for non-negative integersα. The(C, 0)method is just ordinary summation, and(C, 1)is Cesàro summation as described above.
The higher-order methods can be described as follows: given a seriesΣan, define the quantities
(where the upper indices do not denote exponents) and defineEαnto beAαnfor the series1 + 0 + 0 + 0 + .... Then the(C,α)sum ofΣanis denoted by(C,α)-Σanand has the value
if it exists (Shawyer & Watson 1994, pp.16-17). This description represents anα-times iterated application of the initial summation method and can be restated as
Even more generally, forα∈R{\displaystyle \mathbb {R} }\Z{\displaystyle \mathbb {Z} }−, letAαnbe implicitly given by the coefficients of the series
andEαnas above. In particular,Eαnare thebinomial coefficientsof power−1 −α. Then the(C,α)sum ofΣanis defined as above.
IfΣanhas a(C,α)sum, then it also has a(C,β)sum for everyβ>α, and the sums agree; furthermore we havean=o(nα)ifα> −1(seelittle-onotation).
Letα≥ 0. Theintegral∫0∞f(x)dx{\displaystyle \textstyle \int _{0}^{\infty }f(x)\,dx}is(C,α)summable if
exists and is finite (Titchmarsh 1948, §1.15). The value of this limit, should it exist, is the(C,α)sum of the integral. Analogously to the case of the sum of a series, ifα= 0, the result is convergence of theimproper integral. In the caseα= 1,(C, 1)convergence is equivalent to the existence of the limit
which is the limit of means of the partial integrals.
As is the case with series, if an integral is(C,α)summable for some value ofα≥ 0, then it is also(C,β)summable for allβ>α, and the value of the resulting limit is the same.
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https://en.wikipedia.org/wiki/Ces%C3%A0ro_summation
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Inmathematical analysisandanalytic number theory,Lambert summationis a summability method for summing infinite series related toLambert seriesspecially relevant in analytic number theory.
Define the Lambert kernel byL(x)=log(1/x)x1−x{\displaystyle L(x)=\log(1/x){\frac {x}{1-x}}}withL(1)=1{\displaystyle L(1)=1}. Note thatL(xn)>0{\displaystyle L(x^{n})>0}is decreasing as a function ofn{\displaystyle n}when0<x<1{\displaystyle 0<x<1}. A sum∑n=0∞an{\displaystyle \sum _{n=0}^{\infty }a_{n}}is Lambert summable toA{\displaystyle A}iflimx→1−∑n=0∞anL(xn)=A{\displaystyle \lim _{x\to 1^{-}}\sum _{n=0}^{\infty }a_{n}L(x^{n})=A}, written∑n=0∞an=A(L){\displaystyle \sum _{n=0}^{\infty }a_{n}=A\,\,(\mathrm {L} )}.
Abelian theorem: If a series is convergent toA{\displaystyle A}then it is Lambert summable toA{\displaystyle A}.
Tauberian theorem: Suppose that∑n=1∞an{\displaystyle \sum _{n=1}^{\infty }a_{n}}is Lambert summable toA{\displaystyle A}. Then it is Abel summable toA{\displaystyle A}. In particular, if∑n=0∞an{\displaystyle \sum _{n=0}^{\infty }a_{n}}is Lambert summable toA{\displaystyle A}andnan≥−C{\displaystyle na_{n}\geq -C}then∑n=0∞an{\displaystyle \sum _{n=0}^{\infty }a_{n}}converges toA{\displaystyle A}.
The Tauberian theorem was first proven byG. H. HardyandJohn Edensor Littlewoodbut was not independent of number theory, in fact they used a number-theoretic estimate which is somewhat stronger than the prime number theorem itself. The unsatisfactory situation around the Lambert Tauberian theorem was resolved byNorbert Wiener.
Thismathematical analysis–related article is astub. You can help Wikipedia byexpanding it.
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https://en.wikipedia.org/wiki/Lambert_summation
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