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During an event on March 21, 2016, Apple provided a status update on its environmental initiative to be 100% renewable in all of its worldwide operations. Lisa P. Jackson, Apple's vice president of Environment, Policy and Social Initiatives who reports directly to CEO, Tim Cook, announced that , 93% of Apple's worldwide operations are powered with renewable energy. Also featured was the company's efforts to use sustainable paper in their product packaging; 99% of all paper used by Apple in the product packaging comes from post-consumer recycled paper or sustainably managed forests, as the company continues its move to all paper packaging for all of its products. Apple working in partnership with Conservation Fund, have preserved 36,000 acres of working forests in Maine and North Carolina. Another partnership announced is with the World Wildlife Fund to preserve up to of forests in China. Featured was the company's installation of a 40 MW solar power plant in the Sichuan province of China that was tailor-made to coexist with the indigenous yaks that eat hay produced on the land, by raising the panels to be several feet off of the ground so the yaks and their feed would be unharmed grazing beneath the array. This installation alone compensates for more than all of the energy used in Apple's Stores and Offices in the whole of China, negating the company's energy carbon footprint in the country. In Singapore, Apple has worked with the Singaporean government to cover the rooftops of 800 buildings in the city-state with solar panels allowing Apple's Singapore operations to be run on 100% renewable energy. | [
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Liam was introduced to the world, an advanced robotic disassembler and sorter designed by Apple Engineers in California specifically for recycling outdated or broken iPhones. Reuses and recycles parts from traded in products. Apple announced on August 16, 2016, that Lens Technology, one of its major suppliers in China, has committed to power all its glass production for Apple with 100 percent renewable energy by 2018. The commitment is a large step in Apple's efforts to help manufacturers lower their carbon footprint in China. Apple also announced that all 14 of its final assembly sites in China are now compliant with UL's Zero Waste to Landfill validation. The standard, which started in January 2015, certifies that all manufacturing waste is reused, recycled, composted, or converted into energy (when necessary). Since the program began, nearly, 140,000 metric tons of waste have been diverted from landfills. On July 21, 2020, Apple announced its plan to become carbon neutral across its entire business, manufacturing supply chain, and product life cycle by 2030. In the next 10 years, Apple will try to lower emissions with a series of innovative actions, including: low carbon product design, expanding energy efficiency, renewable energy, process and material innovations, and carbon removal. In April 2021, Apple said that it had started a $200 million fund in order to combat climate change by removing 1 million metric tons of carbon dioxide from the atmosphere each year. | [
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Toxins
Following further campaigns by Greenpeace, in 2008, Apple became the first electronics manufacturer to fully eliminate all polyvinyl chloride (PVC) and brominated flame retardants (BFRs) in its complete product line. In June 2007, Apple began replacing the cold cathode fluorescent lamp (CCFL) backlit LCD displays in its computers with mercury-free LED-backlit LCD displays and arsenic-free glass, starting with the upgraded MacBook Pro. Apple offers comprehensive and transparent information about the CO2e, emissions, materials, and electrical usage concerning every product they currently produce or have sold in the past (and which they have enough data needed to produce the report), in their portfolio on their homepage. Allowing consumers to make informed purchasing decisions on the products they offer for sale. In June 2009, Apple's iPhone 3GS was free of PVC, arsenic, and BFRs. All Apple products now have mercury-free LED-backlit LCD displays, arsenic-free glass, and non-PVC cables. All Apple products have EPEAT Gold status and beat the latest Energy Star guidelines in each product's respective regulatory category. In November 2011, Apple was featured in Greenpeace's Guide to Greener Electronics, which ranks electronics manufacturers on sustainability, climate and energy policy, and how "green" their products are. The company ranked fourth of fifteen electronics companies (moving up five places from the previous year) with a score of 4.6/10. Greenpeace praises Apple's sustainability, noting that the company exceeded its 70% global recycling goal in 2010. It continues to score well on the products rating with all Apple products now being free of PVC plastic and BFRs. | [
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However, the guide criticizes Apple on the Energy criteria for not seeking external verification of its greenhouse gas emissions data and for not setting out any targets to reduce emissions. In January 2012, Apple requested that its cable maker, Volex, begin producing halogen-free USB and power cables. Green bonds
In February 2016, Apple issued a US$1.5 billion green bond (climate bond), the first ever of its kind by a U.S. tech company. The green bond proceeds are dedicated to the financing of environmental projects. Racial Justice and Equality Initiatives
In June 2020, Apple committed $100 million for its Racial Equity and Justice initiative (REJI) and in Jan 2021 announced various projects as part of the initiative. Finance
Apple is the world's largest information technology company by revenue, the world's largest technology company by total assets, and the world's second-largest mobile phone manufacturer after Samsung. In its fiscal year ending in September 2011, Apple Inc. reported a total of $108 billion in annual revenues—a significant increase from its 2010 revenues of $65 billion—and nearly $82 billion in cash reserves. On March 19, 2012, Apple announced plans for a $2.65-per-share dividend beginning in fourth quarter of 2012, per approval by their board of directors. The company's worldwide annual revenue in 2013 totaled $170 billion. In May 2013, Apple entered the top ten of the Fortune 500 list of companies for the first time, rising 11 places above its 2012 ranking to take the sixth position. | [
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, Apple has around US$234 billion of cash and marketable securities, of which 90% is located outside the United States for tax purposes. Apple amassed 65% of all profits made by the eight largest worldwide smartphone manufacturers in quarter one of 2014, according to a report by Canaccord Genuity. In the first quarter of 2015, the company garnered 92% of all earnings. On April 30, 2017, The Wall Street Journal reported that Apple had cash reserves of $250 billion, officially confirmed by Apple as specifically $256.8 billion a few days later. , Apple was the largest publicly traded corporation in the world by market capitalization. On August 2, 2018, Apple became the first publicly traded U.S. company to reach a $1 trillion market value. Apple was ranked No. 4 on the 2018 Fortune 500 rankings of the largest United States corporations by total revenue. Tax practices
Apple has created subsidiaries in low-tax places such as Ireland, the Netherlands, Luxembourg, and the British Virgin Islands to cut the taxes it pays around the world. According to The New York Times, in the 1980s Apple was among the first tech companies to designate overseas salespeople in high-tax countries in a manner that allowed the company to sell on behalf of low-tax subsidiaries on other continents, sidestepping income taxes. In the late 1980s, Apple was a pioneer of an accounting technique known as the "Double Irish with a Dutch sandwich," which reduces taxes by routing profits through Irish subsidiaries and the Netherlands and then to the Caribbean. | [
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British Conservative Party Member of Parliament Charlie Elphicke published research on October 30, 2012, which showed that some multinational companies, including Apple Inc., were making billions of pounds of profit in the UK, but were paying an effective tax rate to the UK Treasury of only 3 percent, well below standard corporation tax. He followed this research by calling on the Chancellor of the Exchequer George Osborne to force these multinationals, which also included Google and The Coca-Cola Company, to state the effective rate of tax they pay on their UK revenues. Elphicke also said that government contracts should be withheld from multinationals who do not pay their fair share of UK tax. Apple Inc. claims to be the single largest taxpayer to the Department of the Treasury of the United States of America with an effective tax rate of approximately of 26% as of the second quarter of the Apple fiscal year 2016. In an interview with the German newspaper FAZ in October 2017, Tim Cook stated, that Apple is the biggest taxpayer worldwide. In 2015, Reuters reported that Apple had earnings abroad of $54.4 billion which were untaxed by the IRS of the United States. Under U.S. tax law governed by the IRC, corporations don't pay income tax on overseas profits unless the profits are repatriated into the United States and as such Apple argues that to benefit its shareholders it will leave it overseas until a repatriation holiday or comprehensive tax reform takes place in the United States. | [
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On July 12, 2016, the Central Statistics Office of Ireland announced that 2015 Irish GDP had grown by 26.3%, and 2015 Irish GNP had grown by 18.7%. The figures attracted international scorn, and were labelled by Nobel-prize winning economist, Paul Krugman, as leprechaun economics. It was not until 2018 that Irish economists could definitively prove that the 2015 growth was due to Apple restructuring its controversial double Irish subsidiaries (Apple Sales International), which Apple converted into a new Irish capital allowances for intangible assets tax scheme (expires in January 2020). The affair required the Central Bank of Ireland to create a new measure of Irish economic growth, Modified GNI* to replace Irish GDP, given the distortion of Apple's tax schemes. Irish GDP is 143% of Irish Modified GNI*. On August 30, 2016, after a two-year investigation, the EU Competition Commissioner concluded Apple received "illegal state aid" from Ireland. The EU ordered Apple to pay 13 billion euros ($14.5 billion), plus interest, in unpaid Irish taxes for 2004–2014. It is the largest tax fine in history. The Commission found that Apple had benefited from a private Irish Revenue Commissioners tax ruling regarding its double Irish tax structure, Apple Sales International (ASI). Instead of using two companies for its double Irish structure, Apple was given a ruling to split ASI into two internal "branches". | [
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The Chancellor of Austria, Christian Kern, put this decision into perspective by stating that "every Viennese cafe, every sausage stand pays more tax in Austria than a multinational corporation". , Apple agreed to start paying €13 billion in back taxes to the Irish government, the repayments will be held in an escrow account while Apple and the Irish government continue their appeals in EU courts. On July 15, 2020, the EU General Court annuls the European Commission's decision in Apple State aid case: Apple will not have to repay €13 billion to Ireland. Board of directors
the following individuals sit on the board of Apple Inc.
Arthur D. Levinson (chairman)
Tim Cook (executive director and CEO)
James A. | [
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Bell (non-executive director)
Al Gore (non-executive director)
Andrea Jung (non-executive director)
Ronald Sugar (non-executive director)
Susan Wagner (non-executive director)
Executive management
the management of Apple Inc. includes:
Tim Cook (chief executive officer)
Jeff Williams (chief operating officer)
Luca Maestri (senior vice president and chief financial officer)
Katherine L. Adams (senior vice president and general counsel)
Eddy Cue (senior vice president – Internet Software and Services)
Craig Federighi (senior vice president – Software Engineering)
John Giannandrea (senior vice president – Machine Learning and AI Strategy)
Deirdre O'Brien (senior vice president – Retail + People)
John Ternus (senior vice president – Hardware Engineering)
Greg Josiwak (senior vice president – Worldwide Marketing)
Johny Srouji (senior vice president – Hardware Technologies)
Sabih Khan (senior vice president – Operations)
Lisa P. Jackson (vice president – Environment, Policy, and Social Initiatives)
Isabel Ge Mahe (vice president and managing director – Greater China)
Tor Myhren (vice president – Marketing Communications)
Adrian Perica (vice president – Corporate Development)
List of chief executives
Michael Scott (1977–1981)
Mike Markkula (1981–1983)
John Sculley (1983–1993)
Michael Spindler (1993–1996)
Gil Amelio (1996–1997)
Steve Jobs (1997–2011)
Tim Cook (2011–present)
List of chairmen
The role of chairman of the Board has not always been in use; notably, between 1981 to 1985, and 1997 to 2011. | [
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Mike Markkula (1977–1981)
Steve Jobs (1985)
Mike Markkula (1985–1993); second term
John Sculley (1993)
Mike Markkula (1993–1997); third term
Steve Jobs (2011); second term
Arthur D. Levinson (2011–present)
Litigation
Apple has been a participant in various legal proceedings and claims since it began operation. In particular, Apple is known for and promotes itself as actively and aggressively enforcing its intellectual property interests. Some litigation examples include Apple v. Samsung, Apple v. Microsoft, Motorola Mobility v. Apple Inc., and Apple Corps v. Apple Computer. Apple has also had to defend itself against charges on numerous occasions of violating intellectual property rights. Most have been dismissed in the courts as shell companies known as patent trolls, with no evidence of actual use of patents in question. On December 21, 2016, Nokia announced that in the U.S. and Germany, it has filed a suit against Apple, claiming that the latter's products infringe on Nokia's patents. Most recently, in November 2017, the United States International Trade Commission announced an investigation into allegations of patent infringement in regards to Apple's remote desktop technology; Aqua Connect, a company that builds remote desktop software, has claimed that Apple infringed on two of its patents. Privacy stance
Apple has a notable pro-privacy stance, actively making privacy-conscious features and settings part of its conferences, promotional campaigns, and public image. | [
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With its iOS 8 mobile operating system in 2014, the company started encrypting all contents of iOS devices through users' passcodes, making it impossible at the time for the company to provide customer data to law enforcement requests seeking such information. With the popularity rise of cloud storage solutions, Apple began a technique in 2016 to do deep learning scans for facial data in photos on the user's local device and encrypting the content before uploading it to Apple's iCloud storage system. It also introduced "differential privacy", a way to collect crowdsourced data from many users, while keeping individual users anonymous, in a system that Wired described as "trying to learn as much as possible about a group while learning as little as possible about any individual in it". Users are explicitly asked if they want to participate, and can actively opt-in or opt-out. With Apple's release of an update to iOS 14, Apple required all developers of iPhone, iPad, and iPod touch applications to directly ask iPhone users permission to track them. The feature, titled "App Tracking Transparency", received heavy criticism from Facebook, whose primary business model revolves around the tracking of users' data and sharing such data with advertisers so users can see more relevant ads, a technique commonly known as targeted advertising. Despite Facebook's measures, including purchasing full-page newspaper advertisements protesting App Tracking Transparency, Apple released the update in mid-spring 2021. A study by Verizon subsidiary Flurry Analytics reported only 4% of iOS users in the United States and 12% worldwide have opted into tracking. | [
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However, Apple aids law enforcement in criminal investigations by providing iCloud backups of users' devices, and the company's commitment to privacy has been questioned by its efforts to promote biometric authentication technology in its newer iPhone models, which don't have the same level of constitutional privacy as a passcode in the United States. Prior to the release of iOS 15, Apple announced new efforts at combating child sexual abuse material on iOS and Mac platforms. Parents of minor iMessage users can now be alerted if their child sends or receives nude photographs. Additionally, on-device hashing would take place on media destined for upload to iCloud, and hashes would be compared to a list of known abusive images provided by law enforcement; if enough matches were found, Apple would be alerted and authorities informed. The new features received praise from law enforcement and victims rights advocates, however privacy advocates, including the Electronic Frontier Foundation, condemned the new features as invasive and highly prone to abuse by authoritarian governments. Charitable causes
Apple is a partner of (PRODUCT)RED, a fundraising campaign for AIDS charity. In November 2014, Apple arranged for all App Store revenue in a two-week period to go to the fundraiser, generating more than US$20 million, and in March 2017, it released an iPhone 7 with a red color finish. Apple contributes financially to fundraisers in times of natural disasters. | [
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In November 2012, it donated $2.5 million to the American Red Cross to aid relief efforts after Hurricane Sandy, and in 2017 it donated $5 million to relief efforts for both Hurricane Irma and Hurricane Harvey, as well as for the 2017 Central Mexico earthquake. The company has also used its iTunes platform to encourage donations in the wake of environmental disasters and humanitarian crises, such as the 2010 Haiti earthquake, the 2011 Japan earthquake, Typhoon Haiyan in the Philippines in November 2013, and the 2015 European migrant crisis. Apple emphasizes that it does not incur any processing or other fees for iTunes donations, sending 100% of the payments directly to relief efforts, though it also acknowledges that the Red Cross does not receive any personal information on the users donating and that the payments may not be tax deductible. On April 14, 2016, Apple and the World Wide Fund for Nature (WWF) announced that they have engaged in a partnership to, "help protect life on our planet." Apple released a special page in the iTunes App Store, Apps for Earth. In the arrangement, Apple has committed that through April 24, WWF will receive 100% of the proceeds from the applications participating in the App Store via both the purchases of any paid apps and the In-App Purchases. Apple and WWF's Apps for Earth campaign raised more than $8 million in total proceeds to support WWF's conservation work. WWF announced the results at WWDC 2016 in San Francisco. | [
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During the COVID-19 pandemic, Apple's CEO Cook announced that the company will be donating "millions" of masks to health workers in the United States and Europe. On January 13, 2021, Apple announced a $100 million "Racial Equity and Justice Initiative" to help combat institutional racism worldwide. Criticism and controversies
Apple has been criticized for alleged unethical business practices such as anti-competitive behavior, rash litigation, dubious tax tactics, production methods involving the use of sweatshop labor, customer service issues involving allegedly misleading warranties and insufficient data security, and its products' environmental footprint. Apple has also received criticism for its willingness to work and conduct business with nations such as China and Russia, engaging in practices that have been criticized by human rights groups. Critics have claimed that Apple products combine stolen or purchased designs that Apple claims are its original creations. It has been criticized for its alleged collaboration with the U.S. surveillance program PRISM. The company denied any collaboration. Products and services
Apple's issues regarding music over the years include those with the European Union regarding iTunes, trouble over updating the Spotify app on Apple devices and collusion with record labels. In 2018–19, Apple faced criticism for its failure to approve NVIDIA web drivers for GPUs installed on legacy Mac Pro machines (up to mid 2012 5,1 running macOS Mojave 10.14). Without access to Apple-approved NVIDIA web drivers, Apple users faced replacing their NVIDIA cards with graphic cards produced by supported brands (such as the AMD Radeon), from a list of recommendations provided by Apple to its consumers. | [
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In June 2019, Apple issued a recall for its 2015 MacBook Pro Retina 15" following reports of batteries catching fire. The recall affected 432,000 units, and Apple was criticized for the long waiting periods consumers experienced, sometimes extending up to 3 weeks for replacements to arrive; the company also did not provide alternative replacements or repair options. In July 2019, following a campaign by the "right to repair" movement, challenging Apple's tech repair restrictions on devices, the FTC held a workshop to establish the framework of a future nationwide Right to Repair rule. The movement argues Apple is preventing consumers from legitimately fixing their devices at local repair shops which is having a negative impact on consumers. On November 19, 2020, it was announced that Apple will be paying out $113 million related to lawsuits stemming from their iPhone's battery problems and subsequent performance slow-downs. Apple continues to face litigation related to the performance throttling of iPhone 6 and 7 devices, an action that Apple argued was done in order to balance the functionality of the software with the impacts of a chemically aged battery. On January 25, 2021, Apple was hit with another lawsuit from an Italian consumer group, with more groups to follow, despite the rationale for the throttling. On November 30, 2020, the Italian antitrust authority AGCM fined Apple $12 million for misleading trade practices. AGCM stated that Apple's claims of the iPhone's water resistance weren't true as the phones could only resist water up to 4 meters deep in ideal laboratory conditions and not in regular circumstances. | [
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The authority added that Apple provided no assistance to customers with water-damaged phones, which it said constituted an aggressive trade practice. Privacy
Ireland's Data Protection Commission also launched a privacy investigation to examine whether Apple complied with the EU's GDPR law following an investigation into how the company processes personal data with targeted ads on its platform. In December 2019, a report found that the iPhone 11 Pro continues tracking location and collecting user data even after users have disabled location services. In response, an Apple engineer said the Location Services icon "appears for system services that do not have a switch in settings." Antitrust
The United States Department of Justice also began a review of Big Tech firms to establish whether they could be unlawfully stifling competition in a broad antitrust probe in 2019. On March 16, 2020, France fined Apple €1.1 billion for colluding with two wholesalers to stifle competition and keep prices high by handicapping independent resellers. The arrangement created aligned prices for Apple products such as iPads and personal computers for about half the French retail market. According to the French regulators, the abuses occurred between 2005 and 2017 but were first discovered after a complaint by an independent reseller, eBizcuss, in 2012. On August 13, 2020, Epic Games, the maker of the popular game Fortnite, sued Apple and Google after its hugely popular video game was removed from Apple and Google's App Store. The suits come after both Apple and Google blocked the game after it introduced a direct payment system, effectively shutting out the tech titans from collecting fees. | [
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In September 2020 Epic Games founded the Coalition for App Fairness together with other thirteen companies, which aims for better conditions for the inclusion of apps in the app stores. Later in December 2020, Facebook agreed to assist Epic in their legal game against Apple, planning to support the company by providing materials and documents to Epic. Facebook had, however, stated that the company will not participate directly with the lawsuit, although did commit to helping with the discovery of evidence relating to the trial of 2021. In the months prior to their agreement, Facebook had been dealing with feuds against Apple relating to the prices of paid apps as well as privacy rule changes. Head of ad products for Facebook Dan Levy commented, saying that "this is not really about privacy for them, this is about an attack on personalized ads and the consequences it's going to have on small-business owners," commenting on the full-page ads placed by Facebook in various newspapers in December 2020. Politics
In January 2020, US President Donald Trump and attorney general William P. Barr criticized Apple for refusing to unlock two iPhones of a Saudi national, Mohammed Saeed Alshamrani, who shot and killed three American sailors and injured eight others in the Naval Air Station Pensacola. The shooting was declared an "act of terrorism" by the FBI, but Apple denied the request to crack the phones to reveal possible terrorist information citing its data privacy policy. | [
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Apple Inc., shareholders increased pressure on the company to publicly commit “to respect freedom of expression as a human right”, upon which Apple committed to freedom of expression and information in its human rights policy document. It said that the policy is based on the guidelines of the United Nations on business and human rights, in early September 2020. In 2021, Apple complied with a request by the Chinese government to ban a Quran app from its devices and platforms. The request occurred in the context of the Chinese government's ongoing mass repression of Muslims, particularly Uyghurs, in Xinjiang, which some have labeled a genocide. In December 2021, The Information reported that CEO Tim Cook had negotiated in 2016 a five-year agreement with the Chinese government, motivated in part to allay regulatory issues that had harmed the company's business in China. The agreement entailed promised investments totaling $275 billion. In September 2021, Apple removed an app from its App Store created by Alexei Navalny meant to coordinate protest voting during the 2021 Russian legislative election. The Russian government had threatened to arrest individual Apple employees working in the country unless Apple complied. Patents
In January 2022, Ericsson sued Apple over payment of royalty of 5G technology. | [
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] |
See also
List of Apple Inc. media events
Pixar
References
Bibliography
Further reading
External links
1976 establishments in California
1980s initial public offerings
American brands
Companies based in Cupertino, California
Companies in the Dow Jones Industrial Average
Companies in the PRISM network
Companies listed on the Nasdaq
Computer companies established in 1976
Computer companies of the United States
Display technology companies
Electronics companies of the United States
Home computer hardware companies
Mobile phone manufacturers
Multinational companies headquartered in the United States
Networking hardware companies
Portable audio player manufacturers
Retail companies of the United States
Software companies based in the San Francisco Bay Area
Software companies established in 1976
Steve Jobs
Technology companies based in the San Francisco Bay Area
Technology companies established in 1976
Technology companies of the United States | [
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An author is the creator or originator of any written work such as a book or play, and is also considered a writer or poet. More broadly defined, an author is "the person who originated or gave existence to anything" and whose authorship determines responsibility for what was created. Legal significance of authorship
Typically, the first owner of a copyright is the person who created the work, i.e. the author. If more than one person created the work, then a case of joint authorship can be made provided some criteria are met. In the copyright laws of various jurisdictions, there is a necessity for little flexibility regarding what constitutes authorship. The United States Copyright Office, for example, defines copyright as "a form of protection provided by the laws of the United States (title 17, U.S. Code) to authors of 'original works of authorship.'" Holding the title of "author" over any "literary, dramatic, musical, artistic, [or] certain other intellectual works" gives rights to this person, the owner of the copyright, especially the exclusive right to engage in or authorize any production or distribution of their work. Any person or entity wishing to use intellectual property held under copyright must receive permission from the copyright holder to use this work, and often will be asked to pay for the use of copyrighted material. After a fixed amount of time, the copyright expires on intellectual work and it enters the public domain, where it can be used without limit. | [
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Copyright laws in many jurisdictions – mostly following the lead of the United States, in which the entertainment and publishing industries have very strong lobbying power – have been amended repeatedly since their inception, to extend the length of this fixed period where the work is exclusively controlled by the copyright holder. However, copyright is merely the legal reassurance that one owns their work. Technically, someone owns their work from the time it's created. A notable aspect of authorship emerges with copyright in that, in many jurisdictions, it can be passed down to another upon one's death. The person who inherits the copyright is not the author, but enjoys the same legal benefits. Questions arise as to the application of copyright law. How does it, for example, apply to the complex issue of fan fiction? If the media agency responsible for the authorized production allows material from fans, what is the limit before legal constraints from actors, music, and other considerations, come into play? Additionally, how does copyright apply to fan-generated stories for books? What powers do the original authors, as well as the publishers, have in regulating or even stopping the fan fiction? This particular sort of case also illustrates how complex intellectual property law can be, since such fiction may also involved trademark law (e.g. for names of characters in media franchises), likeness rights (such as for actors, or even entirely fictional entities), fair use rights held by the public (including the right to parody or satirize), and many other interacting complications. | [
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Authors may portion out different rights they hold to different parties, at different times, and for different purposes or uses, such as the right to adapt a plot into a film, but only with different character names, because the characters have already been optioned by another company for a television series or a video game. An author may also not have rights when working under contract that they would otherwise have, such as when creating a work for hire (e.g., hired to write a city tour guide by a municipal government that totally owns the copyright to the finished work), or when writing material using intellectual property owned by others (such as when writing a novel or screenplay that is a new installment in an already established media franchise). Philosophical views of the nature of authorship
In literary theory, critics find complications in the term author beyond what constitutes authorship in a legal setting. In the wake of postmodern literature, critics such as Roland Barthes and Michel Foucault have examined the role and relevance of authorship to the meaning or interpretation of a text. Barthes challenges the idea that a text can be attributed to any single author. He writes, in his essay "Death of the Author" (1968), that "it is language which speaks, not the author." The words and language of a text itself determine and expose meaning for Barthes, and not someone possessing legal responsibility for the process of its production. | [
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Every line of written text is a mere reflection of references from any of a multitude of traditions, or, as Barthes puts it, "the text is a tissue of quotations drawn from the innumerable centres of culture"; it is never original. With this, the perspective of the author is removed from the text, and the limits formerly imposed by the idea of one authorial voice, one ultimate and universal meaning, are destroyed. The explanation and meaning of a work does not have to be sought in the one who produced it, "as if it were always in the end, through the more or less transparent allegory of the fiction, the voice of a single person, the author 'confiding' in us." The psyche, culture, fanaticism of an author can be disregarded when interpreting a text, because the words are rich enough themselves with all of the traditions of language. To expose meanings in a written work without appealing to the celebrity of an author, their tastes, passions, vices, is, to Barthes, to allow language to speak, rather than author. Michel Foucault argues in his essay "What is an author?" (1969) that all authors are writers, but not all writers are authors. He states that "a private letter may have a signatory—it does not have an author." For a reader to assign the title of author upon any written work is to attribute certain standards upon the text which, for Foucault, are working in conjunction with the idea of "the author function." | [
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Foucault's author function is the idea that an author exists only as a function of a written work, a part of its structure, but not necessarily part of the interpretive process. The author's name "indicates the status of the discourse within a society and culture," and at one time was used as an anchor for interpreting a text, a practice which Barthes would argue is not a particularly relevant or valid endeavour. Expanding upon Foucault's position, Alexander Nehamas writes that Foucault suggests "an author [...] is whoever can be understood to have produced a particular text as we interpret it," not necessarily who penned the text. It is this distinction between producing a written work and producing the interpretation or meaning in a written work that both Barthes and Foucault are interested in. Foucault warns of the risks of keeping the author's name in mind during interpretation, because it could affect the value and meaning with which one handles an interpretation. Literary critics Barthes and Foucault suggest that readers should not rely on or look for the notion of one overarching voice when interpreting a written work, because of the complications inherent with a writer's title of "author." They warn of the dangers interpretations could suffer from when associating the subject of inherently meaningful words and language with the personality of one authorial voice. Instead, readers should allow a text to be interpreted in terms of the language as "author." | [
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Relationship with publisher
Self-publishing
Self-publishing, self-publishing, independent publishing, or artisanal publishing is the "publication of any book, album or other media by its author without the involvement of a traditional publisher. It is the modern equivalent to traditional publishing." Types
Unless a book is to be sold directly from the author to the public, an ISBN is required to uniquely identify the title. ISBN is a global standard used for all titles worldwide. Most self-publishing companies either provide their own ISBN to a title or can provide direction; it may be in the best interest of the self-published author to retain ownership of ISBN and copyright instead of using a number owned by a vanity press. A separate ISBN is needed for each edition of the book. Electronic (e-book) publishing
There are a variety of book formats and tools that can be used to create them. Because it is possible to create e-books with no up-front or per-book costs, this is a popular option for self-publishers. E-book publishing platforms include Pronoun, Smashwords, Blurb, Amazon Kindle Direct Publishing, CinnamonTeal Publishing, Papyrus Editor, ebook leap, Bookbaby, Pubit, Lulu, Llumina Press, and CreateSpace. E-book formats include epub, mobi, and PDF, among others. Print-on-demand
Print-on-demand (POD) publishing refers to the ability to print high-quality books as needed. For self-published books, this is often a more economical option than conducting a print run of hundreds or thousands of books. | [
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Many companies, such as Createspace (owned by Amazon.com), Outskirts Press, Blurb, Lulu, Llumina Press, ReadersMagnet, and iUniverse, allow printing single books at per-book costs not much higher than those paid by publishing companies for large print runs. Traditional publishing
With commissioned publishing, the publisher makes all the publication arrangements and the author covers all expenses. The author of a work may receive a percentage calculated on a wholesale or a specific price or a fixed amount on each book sold. Publishers, at times, reduced the risk of this type of arrangement, by agreeing only to pay this after a certain number of copies had sold. In Canada, this practice occurred during the 1890s, but was not commonplace until the 1920s. Established and successful authors may receive advance payments, set against future royalties, but this is no longer common practice. Most independent publishers pay royalties as a percentage of net receipts – how net receipts are calculated varies from publisher to publisher. Under this arrangement, the author does not pay anything towards the expense of publication. The costs and financial risk are all carried by the publisher, who will then take the greatest percentage of the receipts. See Compensation for more. Vanity publishing
This type of publisher normally charges a flat fee for arranging publication, offers a platform for selling, and then takes a percentage of the sale of every copy of a book. The author receives the rest of the money made. | [
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Relationship with editor
The relationship between the author and the editor, often the author's only liaison to the publishing company, is often characterized as the site of tension. For the author to reach their audience, often through publication, the work usually must attract the attention of the editor. The idea of the author as the sole meaning-maker of necessity changes to include the influences of the editor and the publisher in order to engage the audience in writing as a social act. There are three principal areas covered by editors – Proofing (checking the Grammar and spelling, looking for typing errors), Story (potentially an area of deep angst for both author and publisher), and Layout (the setting of the final proof ready for publishing often requires minor text changes so a layout editor is required to ensure that these do not alter the sense of the text). Pierre Bourdieu's essay "The Field of Cultural Production" depicts the publishing industry as a "space of literary or artistic position-takings," also called the "field of struggles," which is defined by the tension and movement inherent among the various positions in the field. Bourdieu claims that the "field of position-takings [...] is not the product of coherence-seeking intention or objective consensus," meaning that an industry characterized by position-takings is not one of harmony and neutrality. In particular for the writer, their authorship in their work makes their work part of their identity, and there is much at stake personally over the negotiation of authority over that identity. | [
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However, it is the editor who has "the power to impose the dominant definition of the writer and therefore to delimit the population of those entitled to take part in the struggle to define the writer". As "cultural investors," publishers rely on the editor position to identify a good investment in "cultural capital" which may grow to yield economic capital across all positions. According to the studies of James Curran, the system of shared values among editors in Britain has generated a pressure among authors to write to fit the editors' expectations, removing the focus from the reader-audience and putting a strain on the relationship between authors and editors and on writing as a social act. Even the book review by the editors has more significance than the readership's reception. Compensation
Authors rely on advance fees, royalty payments, adaptation of work to a screenplay, and fees collected from giving speeches. A standard contract for an author will usually include provision for payment in the form of an advance and royalties. An advance is a lump sum paid in advance of publication. An advance must be earned out before royalties are payable. An advance may be paid in two lump sums: the first payment on contract signing, and the second on delivery of the completed manuscript or on publication. Royalty payment is the sum paid to authors for each copy of a book sold and is traditionally around 10-12%, but self-published authors can earn about 40% – 60% royalties per each book sale. | [
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An author's contract may specify, for example, that they will earn 10% of the retail price of each book sold. Some contracts specify a scale of royalties payable (for example, where royalties start at 10% for the first 10,000 sales, but then increase to a higher percentage rate at higher sale thresholds). An author's book must earn the advance before any further royalties are paid. For example, if an author is paid a modest advance of $2000, and their royalty rate is 10% of a book priced at $20 – that is, $2 per book – the book will need to sell 1000 copies before any further payment will be made. Publishers typically withhold payment of a percentage of royalties earned against returns. In some countries, authors also earn income from a government scheme such as the ELR (educational lending right) and PLR (public lending right) schemes in Australia. Under these schemes, authors are paid a fee for the number of copies of their books in educational and/or public libraries. These days, many authors supplement their income from book sales with public speaking engagements, school visits, residencies, grants, and teaching positions. Ghostwriters, technical writers, and textbooks writers are typically paid in a different way: usually a set fee or a per word rate rather than on a percentage of sales. In the year 2016, according to the U.S. Bureau of Labor Statistics, nearly 130,000 people worked in the U.S. as authors making an average of $61,240 per year. | [
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] |
See also
Academic authorship
Auteur
Authors' editor
Distributive writing
Lead author
List of novelists
Lists of poets
Lists of writers
Novelist
Professional writing
References
Writing occupations
Literary criticism | [
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Andrey Andreyevich Markov (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research later became known as Markov chains or Markov processes. Markov and his younger brother Vladimir Andreevich Markov (1871–1897) proved the Markov brothers' inequality. His son, another Andrey Andreyevich Markov (1903–1979), was also a notable mathematician, making contributions to constructive mathematics and recursive function theory. Biography
Andrey Markov was born on 14 June 1856 in Russia. He attended the St. Petersburg Grammar School, where some teachers saw him as a rebellious student. In his academics he performed poorly in most subjects other than mathematics. Later in life he attended Saint Petersburg Imperial University (now Saint Petersburg State University). among his teachers were Yulian Sokhotski (differential calculus, higher algebra), Konstantin Posse (analytic geometry), Yegor Zolotarev (integral calculus), Pafnuty Chebyshev (number theory and probability theory), Aleksandr Korkin (ordinary and partial differential equations), Mikhail Okatov (mechanism theory), Osip Somov (mechanics), and Nikolai Budajev (descriptive and higher geometry). He completed his studies at the university and was later asked if he would like to stay and have a career as a Mathematician. He later taught at high schools and continued his own mathematical studies. In this time he found a practical use for his mathematical skills. He figured out that he could use chains to model the alliteration of vowels and consonants in Russian literature. He also contributed to many other mathematical aspects in his time. | [
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He died at age 66 on 20 July 1922. Timeline
In 1877, Markov was awarded a gold medal for his outstanding solution of the problem
About Integration of Differential Equations by Continued Fractions with an Application to the Equation . During the following year, he passed the candidate's examinations, and he remained at the university to prepare for a lecturer's position. In April 1880, Markov defended his master's thesis "On the Binary Square Forms with Positive Determinant", which was directed by Aleksandr Korkin and Yegor Zolotarev. Four years later in 1884, he defended his doctoral thesis titled "On Certain Applications of the Algebraic Continuous Fractions". His pedagogical work began after the defense of his master's thesis in autumn 1880. As a privatdozent he lectured on differential and integral calculus. Later he lectured alternately on "introduction to analysis", probability theory (succeeding Chebyshev, who had left the university in 1882) and the calculus of differences. From 1895 through 1905 he also lectured in differential calculus. One year after the defense of his doctoral thesis, Markov was appointed extraordinary professor (1886) and in the same year he was elected adjunct to the Academy of Sciences. In 1890, after the death of Viktor Bunyakovsky, Markov became an extraordinary member of the academy. His promotion to an ordinary professor of St. Petersburg University followed in the fall of 1894. In 1896, Markov was elected an ordinary member of the academy as the successor of Chebyshev. In 1905, he was appointed merited professor and was granted the right to retire, which he did immediately. | [
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Until 1910, however, he continued to lecture in the calculus of differences. In connection with student riots in 1908, professors and lecturers of St. Petersburg University were ordered to monitor their students. Markov refused to accept this decree, and he wrote an explanation in which he declined to be an "agent of the governance". Markov was removed from further teaching duties at St. Petersburg University, and hence he decided to retire from the university. Markov was an atheist. In 1912, he protested Leo Tolstoy's excommunication from the Russian Orthodox Church by requesting his own excommunication. The Church complied with his request. In 1913, the council of St. Petersburg elected nine scientists honorary members of the university. Markov was among them, but his election was not affirmed by the minister of education. The affirmation only occurred four years later, after the February Revolution in 1917. Markov then resumed his teaching activities and lectured on probability theory and the calculus of differences until his death in 1922. See also
List of things named after Andrey Markov
Chebyshev–Markov–Stieltjes inequalities
Gauss–Markov theorem
Gauss–Markov process
Hidden Markov model
Markov blanket
Markov chain
Markov decision process
Markov's inequality
Markov brothers' inequality
Markov information source
Markov network
Markov number
Markov property
Markov process
Stochastic matrix (also known as Markov matrix)
Subjunctive possibility
Notes
References
Further reading
А. А. Марков. "Распространение закона больших чисел на величины, зависящие друг от друга". "Известия Физико-математического общества при Казанском университете", 2-я серия, том 15, с. 135–156, 1906. A. A. Markov. | [
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"Extension of the limit theorems of probability theory to a sum of variables connected in a chain". reprinted in Appendix B of: R. Howard. Dynamic Probabilistic Systems, volume 1: Markov Chains. John Wiley and Sons, 1971. External links
Markov, Andrei Andreyevich
Markov, Andrei Andreyevich
19th-century Russian mathematicians
20th-century Russian mathematicians
Russian atheists
Former Russian Orthodox Christians
Probability theorists
Saint Petersburg State University alumni
Full members of the Saint Petersburg Academy of Sciences
Full Members of the Russian Academy of Sciences (1917–1925)
People from Ryazan
Russian statisticians | [
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An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. As used in modern logic, an axiom is a premise or starting point for reasoning. As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (A and B) implies A), while non-logical axioms (e.g., ) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., parallel postulate in Euclidean geometry). | [
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To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there may be multiple ways to axiomatize a given mathematical domain. Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the philosophy of mathematics. Etymology
The word axiom comes from the Greek word (axíōma), a verbal noun from the verb (axioein), meaning "to deem worthy", but also "to require", which in turn comes from (áxios), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the ancient Greek philosophers an axiom was a claim which could be seen to be self-evidently true without any need for proof. The root meaning of the word postulate is to "demand"; for instance, Euclid demands that one agree that some things can be done (e.g., any two points can be joined by a straight line). Ancient geometers maintained some distinction between axioms and postulates. While commenting on Euclid's books, Proclus remarks that "Geminus held that this [4th] Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property." | [
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Boethius translated 'postulate' as petitio and called the axioms notiones communes but in later manuscripts this usage was not always strictly kept. Historical development
Early Greeks
The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (syllogisms, rules of inference) was developed by the ancient Greeks, and has become the core principle of modern mathematics. Tautologies excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are thus the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions (theorems, in the case of mathematics) must be proven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms axiom and postulate hold a slightly different meaning for the present day mathematician, than they did for Aristotle and Euclid. The ancient Greeks considered geometry as just one of several sciences, and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle's posterior analytics is a definitive exposition of the classical view. An "axiom", in classical terminology, referred to a self-evident assumption common to many branches of science. A good example would be the assertion that When an equal amount is taken from equals, an equal amount results. At the foundation of the various sciences lay certain additional hypotheses that were accepted without proof. | [
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Such a hypothesis was termed a postulate. While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Aristotle warns that the content of a science cannot be successfully communicated if the learner is in doubt about the truth of the postulates. The classical approach is well-illustrated by Euclid's Elements, where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions). Postulates
It is possible to draw a straight line from any point to any other point. It is possible to extend a line segment continuously in both directions. It is possible to describe a circle with any center and any radius. It is true that all right angles are equal to one another. ("Parallel postulate") It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles. Common notions
Things which are equal to the same thing are also equal to one another. If equals are added to equals, the wholes are equal. If equals are subtracted from equals, the remainders are equal. Things which coincide with one another are equal to one another. The whole is greater than the part. | [
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Modern development
A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions, theorems) and definitions. One must concede the need for primitive notions, or undefined terms or concepts, in any study. Such abstraction or formalization makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts. Alessandro Padoa, Mario Pieri, and Giuseppe Peano were pioneers in this movement. Structuralist mathematics goes further, and develops theories and axioms (e.g. field theory, group theory, topology, vector spaces) without any particular application in mind. The distinction between an "axiom" and a "postulate" disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate, one can get theories that have meaning in wider contexts (e.g., hyperbolic geometry). As such, one must simply be prepared to use labels such as "line" and "parallel" with greater flexibility. The development of hyperbolic geometry taught mathematicians that it is useful to regard postulates as purely formal statements, and not as facts based on experience. When mathematicians employ the field axioms, the intentions are even more abstract. The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction. | [
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There are many examples of fields; field theory gives correct knowledge about them all. It is not correct to say that the axioms of field theory are "propositions that are regarded as true without proof." Rather, the field axioms are a set of constraints. If any given system of addition and multiplication satisfies these constraints, then one is in a position to instantly know a great deal of extra information about this system. Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and mathematics itself can be regarded as a branch of logic. Frege, Russell, Poincaré, Hilbert, and Gödel are some of the key figures in this development. Another lesson learned in modern mathematics is to examine purported proofs carefully for hidden assumptions. In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formally stated assertions follow – by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be consistent; it should be impossible to derive a contradiction from the axioms. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom. It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbert's formalization of Euclidean geometry, and the related demonstration of the consistency of those axioms. | [
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In a wider context, there was an attempt to base all of mathematics on Cantor's set theory. Here, the emergence of Russell's paradox and similar antinomies of naïve set theory raised the possibility that any such system could turn out to be inconsistent. The formalist project suffered a decisive setback, when in 1931 Gödel showed that it is possible, for any sufficiently large set of axioms (Peano's axioms, for example) to construct a statement whose truth is independent of that set of axioms. As a corollary, Gödel proved that the consistency of a theory like Peano arithmetic is an unprovable assertion within the scope of that theory. It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of natural numbers, an infinite but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern Zermelo–Fraenkel axioms for set theory. Furthermore, using techniques of forcing (Cohen) one can show that the continuum hypothesis (Cantor) is independent of the Zermelo–Fraenkel axioms. Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics. Other sciences
Experimental sciences - as opposed to mathematics and logic - also have general founding assertions from which a deductive reasoning can be built so as to express propositions that predict properties - either still general or much more specialized to a specific experimental context. | [
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For instance, Newton's laws in classical mechanics, Maxwell's equations in classical electromagnetism, Einstein's equation in general relativity, Mandel's laws of genetics, Darwin's Natural selection law, etc. These founding assertions are usually called principles or postulates so as to distinguish from mathematical axioms. As a matter of facts, the role of axioms in mathematics and postulates in experimental sciences is different. In mathematics one neither "proves" nor "disproves" an axiom. A set of mathematical axioms gives a set of rules that fix a conceptual realm, in which the theorems logically follow. In contrast, in experimental sciences, a set of postulates shall allow deducing results that match or do not match experimental results. If postulates do not allow deducing experimental predictions, they do not set a scientific conceptual framework and have to be completed or made more accurate. If the postulates allow deducing predictions of experimental results, the comparison with experiments allows falsifying (falsified) the theory that the postulates install. A theory is considered valid as long as it has not been falsified. Now, the transition between the mathematical axioms and scientific postulates is always slightly blurred, especially in physics. This is due to the heavy use of mathematical tools to support the physical theories. For instance, the introduction of Newton's laws rarely establishes as a prerequisite neither Euclidian geometry or differential calculus that they imply. | [
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It became more apparent when Albert Einstein first introduced special relativity where the invariant quantity is no more the Euclidian length (defined as ) > but the Minkowski spacetime interval (defined as ), and then general relativity where flat Minkowskian geometry is replaced with pseudo-Riemannian geometry on curved manifolds. In quantum physics, two sets of postulates have coexisted for some time, which provide a very nice example of falsification. The 'Copenhagen school' (Niels Bohr, Werner Heisenberg, Max Born) developed an operational approach with a complete mathematical formalism that involves the description of quantum system by vectors ('states') in a separable Hilbert space, and physical quantities as linear operators that act in this Hilbert space. This approach is fully falsifiable and has so far produced the most accurate predictions in physics. But it has the unsatisfactory aspect of not allowing answers to questions one would naturally ask. For this reason, another 'hidden variables' approach was developed for some time by Albert Einstein, Erwin Schrödinger, David Bohm. It was created so as to try to give deterministic explanation to phenomena such as entanglement. This approach assumed that the Copenhagen school description was not complete, and postulated that some yet unknown variable was to be added to the theory so as to allow answering some of the questions it does not answer (the founding elements of which were discussed as the EPR paradox in 1935). Taking this ideas seriously, John Bell derived in 1964 a prediction that would lead to different experimental results (Bell's inequalities) in the Copenhagen and the Hidden variable case. | [
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The experiment was conducted first by Alain Aspect in the early 1980's, and the result excluded the simple hidden variable approach (sophisticated hidden variables could still exist but their properties would still be more disturbing than the problems they try to solve). This does not mean that the conceptual framework of quantum physics can be considered as complete now, since some open questions still exist (the limit between the quantum and classical realms, what happens during a quantum measurement, what happens in a completely closed quantum system such as the universe itself, etc). Mathematical logic
In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical and non-logical (somewhat similar to the ancient distinction between "axioms" and "postulates" respectively). Logical axioms
These are certain formulas in a formal language that are universally valid, that is, formulas that are satisfied by every assignment of values. Usually one takes as logical axioms at least some minimal set of tautologies that is sufficient for proving all tautologies in the language; in the case of predicate logic more logical axioms than that are required, in order to prove logical truths that are not tautologies in the strict sense. | [
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Examples
Propositional logic
In propositional logic it is common to take as logical axioms all formulae of the following forms, where , , and can be any formulae of the language and where the included primitive connectives are only "" for negation of the immediately following proposition and "" for implication from antecedent to consequent propositions:
Each of these patterns is an axiom schema, a rule for generating an infinite number of axioms. For example, if , , and are propositional variables, then and are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and modus ponens, one can prove all tautologies of the propositional calculus. It can also be shown that no pair of these schemata is sufficient for proving all tautologies with modus ponens. Other axiom schemata involving the same or different sets of primitive connectives can be alternatively constructed. These axiom schemata are also used in the predicate calculus, but additional logical axioms are needed to include a quantifier in the calculus. First-order logic
Axiom of Equality. Let be a first-order language. For each variable , the formula
is universally valid. This means that, for any variable symbol the formula can be regarded as an axiom. | [
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Also, in this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by (or, for that matter, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol has to be enforced, only regarding it as a string and only a string of symbols, and mathematical logic does indeed do that. Another, more interesting example axiom scheme, is that which provides us with what is known as Universal Instantiation:
Axiom scheme for Universal Instantiation. Given a formula in a first-order language , a variable and a term that is substitutable for in , the formula
is universally valid. Where the symbol stands for the formula with the term substituted for . (See Substitution of variables.) In informal terms, this example allows us to state that, if we know that a certain property holds for every and that stands for a particular object in our structure, then we should be able to claim . Again, we are claiming that the formula is valid, that is, we must be able to give a "proof" of this fact, or more properly speaking, a metaproof. These examples are metatheorems of our theory of mathematical logic since we are dealing with the very concept of proof itself. Aside from this, we can also have Existential Generalization:
Axiom scheme for Existential Generalization. | [
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Given a formula in a first-order language , a variable and a term that is substitutable for in , the formula
is universally valid. Non-logical axioms
Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example, the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as groups). Thus non-logical axioms, unlike logical axioms, are not tautologies. Another name for a non-logical axiom is postulate. Almost every modern mathematical theory starts from a given set of non-logical axioms, and it was thought that in principle every theory could be axiomatized in this way and formalized down to the bare language of logical formulas. Non-logical axioms are often simply referred to as axioms in mathematical discourse. This does not mean that it is claimed that they are true in some absolute sense. For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom, we can do quite well developing (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups. Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system. Examples
This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). | [
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A rigorous treatment of any of these topics begins with a specification of these axioms. Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo–Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann–Bernays–Gödel set theory, a conservative extension of ZFC. Sometimes slightly stronger theories such as Morse–Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe is used, but in fact, most mathematicians can actually prove all they need in systems weaker than ZFC, such as second-order arithmetic. The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory. The development of abstract algebra brought with itself group theory, rings, fields, and Galois theory. This list could be expanded to include most fields of mathematics, including measure theory, ergodic theory, probability, representation theory, and differential geometry. Arithmetic
The Peano axioms are the most widely used axiomatization of first-order arithmetic. They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem. We have a language where is a constant symbol and is a unary function and the following axioms:
for any formula with one free variable. | [
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The standard structure is where is the set of natural numbers, is the successor function and is naturally interpreted as the number 0. Euclidean geometry
Probably the oldest, and most famous, list of axioms are the 4 + 1 Euclid's postulates of plane geometry. The axioms are referred to as "4 + 1" because for nearly two millennia the fifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. One can assume that exactly one parallel through a point outside a line exists, or that infinitely many exist. This choice gives us two alternative forms of geometry in which the interior angles of a triangle add up to exactly 180 degrees or less, respectively, and are known as Euclidean and hyperbolic geometries. If one also removes the second postulate ("a line can be extended indefinitely") then elliptic geometry arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees. Real analysis
The objectives of the study are within the domain of real numbers. The real numbers are uniquely picked out (up to isomorphism) by the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound. However, expressing these properties as axioms requires the use of second-order logic. | [
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The Löwenheim–Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger. Some of the latter are studied in non-standard analysis. Role in mathematical logic
Deductive systems and completeness
A deductive system consists of a set of logical axioms, a set of non-logical axioms, and a set of rules of inference. A desirable property of a deductive system is that it be complete. A system is said to be complete if, for all formulas ,
that is, for any statement that is a logical consequence of there actually exists a deduction of the statement from . This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation". Gödel's completeness theorem establishes the completeness of a certain commonly used type of deductive system. Note that "completeness" has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms. There is thus, on the one hand, the notion of completeness of a deductive system and on the other hand that of completeness of a set of non-logical axioms. | [
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The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another. Further discussion
Early mathematicians regarded axiomatic geometry as a model of physical space, and obviously, there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic. Galois showed just before his untimely death that these efforts were largely wasted. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details, and modern algebra was born. In the modern view, axioms may be any set of formulas, as long as they are not known to be inconsistent. See also
Axiomatic system
Dogma
First principle, axiom in science and philosophy
List of axioms
Model theory
Regulæ Juris
Theorem
Presupposition
Physical law
Principle
Notes
References
Further reading
Mendelson, Elliot (1987). Introduction to mathematical logic. Belmont, California: Wadsworth & Brooks. External links
Metamath axioms page
Ancient Greek philosophy
Concepts in ancient Greek metaphysics
Concepts in epistemology
Concepts in ethics
Concepts in logic
Concepts in metaphysics
Concepts in the philosophy of science
Deductive reasoning
Formal systems
History of logic
History of mathematics
History of philosophy
History of science
Intellectual history
Logic
Mathematical logic
Mathematical terminology
Philosophical terminology
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Augusta Ada King, Countess of Lovelace (née Byron; 10 December 1815 – 27 November 1852) was an English mathematician and writer, chiefly known for her work on Charles Babbage's proposed mechanical general-purpose computer, the Analytical Engine. She was the first to recognise that the machine had applications beyond pure calculation, and to have published the first algorithm intended to be carried out by such a machine. As a result, she is often regarded as the first computer programmer. Ada Byron was the only child of poet Lord Byron and mathematician Lady Byron. All of Byron's other children were born out of wedlock to other women. Byron separated from his wife a month after Ada was born and left England forever. Four months later, he commemorated the parting in a poem that begins, "Is thy face like thy mother's my fair child! ADA! sole daughter of my house and heart?". He died in Greece when Ada was eight years old. Her mother remained bitter and promoted Ada's interest in mathematics and logic in an effort to prevent her from developing her father's perceived insanity. Despite this, Ada remained interested in him, naming her two sons Byron and Gordon. Upon her death, she was buried next to him at her request. Although often ill in her childhood, Ada pursued her studies assiduously. She married William King in 1835. King was made Earl of Lovelace in 1838, Ada thereby becoming Countess of Lovelace. | [
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Her educational and social exploits brought her into contact with scientists such as Andrew Crosse, Charles Babbage, Sir David Brewster, Charles Wheatstone, Michael Faraday and the author Charles Dickens, contacts which she used to further her education. Ada described her approach as "poetical science" and herself as an "Analyst (& Metaphysician)". When she was a teenager (18), her mathematical talents led her to a long working relationship and friendship with fellow British mathematician Charles Babbage, who is known as "the father of computers". She was in particular interested in Babbage's work on the Analytical Engine. Lovelace first met him in June 1833, through their mutual friend, and her private tutor, Mary Somerville. Between 1842 and 1843, Ada translated an article by Italian military engineer Luigi Menabrea about the Analytical Engine, supplementing it with an elaborate set of notes, simply called "Notes". Lovelace's notes are important in the early history of computers, containing what many consider to be the first computer program—that is, an algorithm designed to be carried out by a machine. Other historians reject this perspective and point out that Babbage's personal notes from the years 1836/1837 contain the first programs for the engine. She also developed a vision of the capability of computers to go beyond mere calculating or number-crunching, while many others, including Babbage himself, focused only on those capabilities. | [
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Her mindset of "poetical science" led her to ask questions about the Analytical Engine (as shown in her notes) examining how individuals and society relate to technology as a collaborative tool. She died of uterine cancer in 1852 at the age of 36, the same age at which her father died. Biography
Childhood
Lord Byron expected his child to be a "glorious boy" and was disappointed when Lady Byron gave birth to a girl. The child was named after Byron's half-sister, Augusta Leigh, and was called "Ada" by Byron himself. On 16 January 1816, at Lord Byron's command, Lady Byron left for her parents' home at Kirkby Mallory, taking their five-week-old daughter with her. Although English law at the time granted full custody of children to the father in cases of separation, Lord Byron made no attempt to claim his parental rights, but did request that his sister keep him informed of Ada's welfare. On 21 April, Lord Byron signed the deed of separation, although very reluctantly, and left England for good a few days later. Aside from an acrimonious separation, Lady Byron continued throughout her life to make allegations about her husband's immoral behaviour. This set of events made Lovelace infamous in Victorian society. Ada did not have a relationship with her father. He died in 1824 when she was eight years old. Her mother was the only significant parental figure in her life. Lovelace was not shown the family portrait of her father until her 20th birthday. Lovelace did not have a close relationship with her mother. | [
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She was often left in the care of her maternal grandmother Judith, Hon. Lady Milbanke, who doted on her. However, because of societal attitudes of the time—which favoured the husband in any separation, with the welfare of any child acting as mitigation—Lady Byron had to present herself as a loving mother to the rest of society. This included writing anxious letters to Lady Milbanke about her daughter's welfare, with a cover note saying to retain the letters in case she had to use them to show maternal concern. In one letter to Lady Milbanke, she referred to her daughter as "it": "I talk to it for your satisfaction, not my own, and shall be very glad when you have it under your own." Lady Byron had her teenage daughter watched by close friends for any sign of moral deviation. Lovelace dubbed these observers the "Furies" and later complained they exaggerated and invented stories about her. Lovelace was often ill, beginning in early childhood. At the age of eight, she experienced headaches that obscured her vision. In June 1829, she was paralyzed after a bout of measles. She was subjected to continuous bed rest for nearly a year, something which may have extended her period of disability. By 1831, she was able to walk with crutches. Despite the illnesses, she developed her mathematical and technological skills. Ada Byron had an affair with a tutor in early 1833. She tried to elope with him after she was caught, but the tutor's relatives recognised her and contacted her mother. | [
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Lady Byron and her friends covered the incident up to prevent a public scandal. Lovelace never met her younger half-sister, Allegra, the daughter of Lord Byron and Claire Clairmont. Allegra died in 1822 at the age of five. Lovelace did have some contact with Elizabeth Medora Leigh, the daughter of Byron's half-sister Augusta Leigh, who purposely avoided Lovelace as much as possible when introduced at court. Adult years
Lovelace became close friends with her tutor Mary Somerville, who introduced her to Charles Babbage in 1833. She had a strong respect and affection for Somerville, and they corresponded for many years. Other acquaintances included the scientists Andrew Crosse, Sir David Brewster, Charles Wheatstone, Michael Faraday and the author Charles Dickens. She was presented at Court at the age of seventeen "and became a popular belle of the season" in part because of her "brilliant mind." By 1834 Ada was a regular at Court and started attending various events. She danced often and was able to charm many people, and was described by most people as being dainty, although John Hobhouse, Byron's friend, described her as "a large, coarse-skinned young woman but with something of my friend's features, particularly the mouth". This description followed their meeting on 24 February 1834 in which Ada made it clear to Hobhouse that she did not like him, probably due to her mother's influence, which led her to dislike all of her father's friends. This first impression was not to last, and they later became friends. | [
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On 8 July 1835, she married William, 8th Baron King, becoming Lady King. They had three homes: Ockham Park, Surrey; a Scottish estate on Loch Torridon in Ross-shire; and a house in London. They spent their honeymoon at Worthy Manor in Ashley Combe near Porlock Weir, Somerset. The Manor had been built as a hunting lodge in 1799 and was improved by King in preparation for their honeymoon. It later became their summer retreat and was further improved during this time. From 1845, the family's main house was Horsley Towers, built in the Tudorbethan fashion by the architect of the Houses of Parliament, Charles Barry, and later greatly enlarged to Lovelace's own designs. They had three children: Byron (born 1836); Anne Isabella (called Annabella, born 1837); and Ralph Gordon (born 1839). Immediately after the birth of Annabella, Lady King experienced "a tedious and suffering illness, which took months to cure." Ada was a descendant of the extinct Barons Lovelace and in 1838, her husband was made Earl of Lovelace and Viscount Ockham, meaning Ada became the Countess of Lovelace. In 1843–44, Ada's mother assigned William Benjamin Carpenter to teach Ada's children and to act as a "moral" instructor for Ada. He quickly fell for her and encouraged her to express any frustrated affections, claiming that his marriage meant he would never act in an "unbecoming" manner. When it became clear that Carpenter was trying to start an affair, Ada cut it off. | [
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In 1841, Lovelace and Medora Leigh (the daughter of Lord Byron's half-sister Augusta Leigh) were told by Ada's mother that Ada's father was also Medora's father. On 27 February 1841, Ada wrote to her mother: "I am not in the least astonished. In fact, you merely confirm what I have for years and years felt scarcely a doubt about, but should have considered it most improper in me to hint to you that I in any way suspected." She did not blame the incestuous relationship on Byron, but instead blamed Augusta Leigh: "I fear she is more inherently wicked than he ever was." In the 1840s, Ada flirted with scandals: firstly, from a relaxed approach to extra-marital relationships with men, leading to rumours of affairs; and secondly, from her love of gambling. She apparently lost more than £3,000 on the horses during the later 1840s. The gambling led to her forming a syndicate with male friends, and an ambitious attempt in 1851 to create a mathematical model for successful large bets. This went disastrously wrong, leaving her thousands of pounds in debt to the syndicate, forcing her to admit it all to her husband. She had a shadowy relationship with Andrew Crosse's son John from 1844 onwards. John Crosse destroyed most of their correspondence after her death as part of a legal agreement. She bequeathed him the only heirlooms her father had personally left to her. During her final illness, she would panic at the idea of the younger Crosse being kept from visiting her. | [
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Education
From 1832, when she was seventeen, her mathematical abilities began to emerge, and her interest in mathematics dominated the majority of her adult life. Her mother's obsession with rooting out any of the insanity of which she accused Byron was one of the reasons that Ada was taught mathematics from an early age. She was privately educated in mathematics and science by William Frend, William King, and Mary Somerville, the noted 19th-century researcher and scientific author. In the 1840s, the mathematician Augustus De Morgan extended her "much help in her mathematical studies" including study of advanced calculus topics including the "numbers of Bernoulli" (that formed her celebrated algorithm for Babbage's Analytical Engine). In a letter to Lady Byron, De Morgan suggested that Ada's skill in mathematics might lead her to become "an original mathematical investigator, perhaps of first-rate eminence." Lovelace often questioned basic assumptions through integrating poetry and science. Whilst studying differential calculus, she wrote to De Morgan:
I may remark that the curious transformations many formulae can undergo, the unsuspected and to a beginner apparently impossible identity of forms exceedingly dissimilar at first sight, is I think one of the chief difficulties in the early part of mathematical studies. I am often reminded of certain sprites and fairies one reads of, who are at one's elbows in one shape now, and the next minute in a form most dissimilar. Lovelace believed that intuition and imagination were critical to effectively applying mathematical and scientific concepts. | [
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She valued metaphysics as much as mathematics, viewing both as tools for exploring "the unseen worlds around us." Death
Lovelace died at the age of 36 on 27 November 1852, from uterine cancer. The illness lasted several months, in which time Annabella took command over whom Ada saw, and excluded all of her friends and confidants. Under her mother's influence, Ada had a religious transformation and was coaxed into repenting of her previous conduct and making Annabella her executor. She lost contact with her husband after confessing something to him on 30 August which caused him to abandon her bedside. It is not known what she told him. She was buried, at her request, next to her father at the Church of St. Mary Magdalene in Hucknall, Nottinghamshire. A memorial plaque, written in Latin, to her and her father is in the chapel attached to Horsley Towers. Work
Throughout her life, Lovelace was strongly interested in scientific developments and fads of the day, including phrenology and mesmerism. After her work with Babbage, Lovelace continued to work on other projects. In 1844, she commented to a friend Woronzow Greig about her desire to create a mathematical model for how the brain gives rise to thoughts and nerves to feelings ("a calculus of the nervous system"). She never achieved this, however. In part, her interest in the brain came from a long-running pre-occupation, inherited from her mother, about her "potential" madness. | [
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As part of her research into this project, she visited the electrical engineer Andrew Crosse in 1844 to learn how to carry out electrical experiments. In the same year, she wrote a review of a paper by Baron Karl von Reichenbach, Researches on Magnetism, but this was not published and does not appear to have progressed past the first draft. In 1851, the year before her cancer struck, she wrote to her mother mentioning "certain productions" she was working on regarding the relation of maths and music. Lovelace first met Charles Babbage in June 1833, through their mutual friend Mary Somerville. Later that month, Babbage invited Lovelace to see the prototype for his difference engine. She became fascinated with the machine and used her relationship with Somerville to visit Babbage as often as she could. Babbage was impressed by Lovelace's intellect and analytic skills. He called her "The Enchantress of Number." In 1843, he wrote to her:
During a nine-month period in 1842–43, Lovelace translated the Italian mathematician Luigi Menabrea's article on Babbage's newest proposed machine, the Analytical Engine. With the article, she appended a set of notes. Explaining the Analytical Engine's function was a difficult task, as many other scientists did not really grasp the concept and the British establishment had shown little interest in it. Lovelace's notes even had to explain how the Analytical Engine differed from the original Difference Engine. Her work was well received at the time; the scientist Michael Faraday described himself as a supporter of her writing. | [
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The notes are around three times longer than the article itself and include (in Note G), in complete detail, a method for calculating a sequence of Bernoulli numbers using the Analytical Engine, which might have run correctly had it ever been built (only Babbage's Difference Engine has been built, completed in London in 2002). Based on this work, Lovelace is now considered by many to be the first computer programmer and her method has been called the world's first computer program. Others dispute this because some of Charles Babbage's earlier writings could be considered computer programs. Note G also contains Lovelace's dismissal of artificial intelligence. She wrote that "The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform. It can follow analysis; but it has no power of anticipating any analytical relations or truths." This objection has been the subject of much debate and rebuttal, for example by Alan Turing in his paper "Computing Machinery and Intelligence". Lovelace and Babbage had a minor falling out when the papers were published, when he tried to leave his own statement (criticising the government's treatment of his Engine) as an unsigned preface, which could have been mistakenly interpreted as a joint declaration. When Taylor's Scientific Memoirs ruled that the statement should be signed, Babbage wrote to Lovelace asking her to withdraw the paper. This was the first that she knew he was leaving it unsigned, and she wrote back refusing to withdraw the paper. | [
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The historian Benjamin Woolley theorised that "His actions suggested he had so enthusiastically sought Ada's involvement, and so happily indulged her ... because of her 'celebrated name'." Their friendship recovered, and they continued to correspond. On 12 August 1851, when she was dying of cancer, Lovelace wrote to him asking him to be her executor, though this letter did not give him the necessary legal authority. Part of the terrace at Worthy Manor was known as Philosopher's Walk, as it was there that Lovelace and Babbage were reputed to have walked while discussing mathematical principles. First computer program
In 1840, Babbage was invited to give a seminar at the University of Turin about his Analytical Engine. Luigi Menabrea, a young Italian engineer and the future Prime Minister of Italy, transcribed Babbage's lecture into French, and this transcript was subsequently published in the Bibliothèque universelle de Genève in October 1842. Babbage's friend Charles Wheatstone commissioned Ada Lovelace to translate Menabrea's paper into English. She then augmented the paper with notes, which were added to the translation. Ada Lovelace spent the better part of a year doing this, assisted with input from Babbage. These notes, which are more extensive than Menabrea's paper, were then published in the September 1843 edition of Taylor's Scientific Memoirs under the initialism AAL. Ada Lovelace's notes were labelled alphabetically from A to G. In note G, she describes an algorithm for the Analytical Engine to compute Bernoulli numbers. | [
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It is considered to be the first published algorithm ever specifically tailored for implementation on a computer, and Ada Lovelace has often been cited as the first computer programmer for this reason. The engine was never completed so her program was never tested. In 1953, more than a century after her death, Ada Lovelace's notes on Babbage's Analytical Engine were republished as an appendix to B. V. Bowden's Faster than Thought: A Symposium on Digital Computing Machines. The engine has now been recognised as an early model for a computer and her notes as a description of a computer and software. Insight into potential of computing devices
In her notes, Ada Lovelace emphasised the difference between the Analytical Engine and previous calculating machines, particularly its ability to be programmed to solve problems of any complexity. She realised the potential of the device extended far beyond mere number crunching. In her notes, she wrote:
This analysis was an important development from previous ideas about the capabilities of computing devices and anticipated the implications of modern computing one hundred years before they were realised. Walter Isaacson ascribes Ada's insight regarding the application of computing to any process based on logical symbols to an observation about textiles: "When she saw some mechanical looms that used punchcards to direct the weaving of beautiful patterns, it reminded her of how Babbage's engine used punched cards to make calculations." This insight is seen as significant by writers such as Betty Toole and Benjamin Woolley, as well as the programmer John Graham-Cumming, whose project Plan 28 has the aim of constructing the first complete Analytical Engine. | [
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According to the historian of computing and Babbage specialist Doron Swade: Ada saw something that Babbage in some sense failed to see. In Babbage's world his engines were bound by number...What Lovelace saw...was that number could represent entities other than quantity. So once you had a machine for manipulating numbers, if those numbers represented other things, letters, musical notes, then the machine could manipulate symbols of which number was one instance, according to rules. It is this fundamental transition from a machine which is a number cruncher to a machine for manipulating symbols according to rules that is the fundamental transition from calculation to computation—to general-purpose computation—and looking back from the present high ground of modern computing, if we are looking and sifting history for that transition, then that transition was made explicitly by Ada in that 1843 paper. Controversy over contribution
Though Lovelace is often referred to as the first computer programmer, some biographers, computer scientists and historians of computing claim otherwise. Allan G. Bromley, in the 1990 article Difference and Analytical Engines:
Bruce Collier, who later wrote a biography of Babbage, wrote in his 1970 Harvard University PhD thesis that Lovelace "made a considerable contribution to publicizing the Analytical Engine, but there is no evidence that she advanced the design or theory of it in any way". Eugene Eric Kim and Betty Alexandra Toole consider it "incorrect" to regard Lovelace as the first computer programmer, as Babbage wrote the initial programs for his Analytical Engine, although the majority were never published. | [
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Bromley notes several dozen sample programs prepared by Babbage between 1837 and 1840, all substantially predating Lovelace's notes. Dorothy K. Stein regards Lovelace's notes as "more a reflection of the mathematical uncertainty of the author, the political purposes of the inventor, and, above all, of the social and cultural context in which it was written, than a blueprint for a scientific development." Doron Swade, a specialist on history of computing known for his work on Babbage, discussed Lovelace during a lecture on Babbage's analytical engine. He explained that Ada was only a "promising beginner" instead of genius in mathematics, that she began studying basic concepts of mathematics five years after Babbage conceived the analytical engine so she could not have made important contributions to it, and that she only published the first computer program instead of actually writing it. But he agrees that Ada was the only person to see the potential of the analytical engine as a machine capable of expressing entities other than quantities. In his self-published book, Idea Makers, Stephen Wolfram defends Lovelace's contributions. While acknowledging that Babbage wrote several unpublished algorithms for the Analytical Engine prior to Lovelace's notes, Wolfram argues that "there's nothing as sophisticated—or as clean—as Ada's computation of the Bernoulli numbers. Babbage certainly helped and commented on Ada's work, but she was definitely the driver of it." Wolfram then suggests that Lovelace's main achievement was to distill from Babbage's correspondence "a clear exposition of the abstract operation of the machine—something which Babbage never did." | [
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In popular culture
1810s
Lord Byron wrote the poem "Fare Thee Well" to his wife Lady Byron in 1816, following their separation after the birth of Ada Lovelace. In the poem he writes:
And when thou would'st solace gather—
When our child's first accents flow—
Wilt thou teach her to say "Father!" Though his care she must forego? When her little hands shall press thee—
When her lip to thine is pressed—
Think of him whose prayer shall bless thee—
Think of him thy love had blessed! Should her lineaments resemble
Those thou never more may'st see,
Then thy heart will softly tremble
With a pulse yet true to me. 1970s
Lovelace is portrayed in Romulus Linney's 1977 play Childe Byron. 1990s
In the 1990 steampunk novel The Difference Engine by William Gibson and Bruce Sterling, Lovelace delivers a lecture on the "punched cards" programme which proves Gödel's incompleteness theorems decades before their actual discovery. In the 1997 film Conceiving Ada, a computer scientist obsessed with Ada finds a way of communicating with her in the past by means of "undying information waves". In Tom Stoppard's 1993 play Arcadia, the precocious teenage genius Thomasina Coverly—a character "apparently based" on Ada Lovelace (the play also involves Lord Byron)—comes to understand chaos theory, and theorises the second law of thermodynamics, before either is officially recognised. 2000s
Lovelace features in John Crowley's 2005 novel, Lord Byron's Novel: The Evening Land, as an unseen character whose personality is forcefully depicted in her annotations and anti-heroic efforts to archive her father's lost novel. | [
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2010s
The 2015 play Ada and the Engine by Lauren Gunderson portrays Lovelace and Charles Babbage in unrequited love, and it imagines a post-death meeting between Lovelace and her father. Lovelace and Babbage are the main characters in Sydney Padua's webcomic and graphic novel The Thrilling Adventures of Lovelace and Babbage. The comic features extensive footnotes on the history of Ada Lovelace, and many lines of dialogue are drawn from actual correspondence. Lovelace and Mary Shelley as teenagers are the central characters in Jordan Stratford's steampunk series, The Wollstonecraft Detective Agency. Lovelace, identified as Ada Augusta Byron, is portrayed by Lily Lesser in the second season of The Frankenstein Chronicles. She is employed as an "analyst" to provide the workings of a life-sized humanoid automaton. The brass workings of the machine are reminiscent of Babbage's analytical engine. Her employment is described as keeping her occupied until she returns to her studies in advanced mathematics. Lovelace and Babbage appear as characters in the second season of the ITV series Victoria (2017). Emerald Fennell portrays Lovelace in the episode, "The Green-Eyed Monster." The Cardano cryptocurrency platform, which was launched in 2017, uses Ada as the name for their cryptocurrency and Lovelace as the smallest sub-unit of an Ada. "Lovelace" is the name given to the operating system designed by the character Cameron Howe in Halt and Catch Fire. Lovelace is a primary character in the 2019 Big Finish Doctor Who audio play The Enchantress of Numbers, starring Tom Baker as the Fourth Doctor and Jane Slavin as his current companion, WPC Ann Kelso. | [
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Lovelace is played by Finty Williams. In 2019, Lovelace is a featured character in the play STEM FEMMES by Philadelphia theater company Applied Mechanics. 2020s
Lovelace features as a character in "Spyfall, Part 2", the second episode of Doctor Who, series 12, which first aired on BBC One on 5 January 2020. The character was portrayed by Sylvie Briggs, alongside characterisations of Charles Babbage and Noor Inayat Khan. In 2021, Nvidia named their upcoming GPU architecture (to be released in 2022), "Ada Lovelace", after her. Commemoration
The computer language Ada, created on behalf of the United States Department of Defense, was named after Lovelace. The reference manual for the language was approved on 10 December 1980 and the Department of Defense Military Standard for the language, MIL-STD-1815, was given the number of the year of her birth. In 1981, the Association for Women in Computing inaugurated its Ada Lovelace Award. Since 1998, the British Computer Society (BCS) has awarded the Lovelace Medal, and in 2008 initiated an annual competition for women students. BCSWomen sponsors the Lovelace Colloquium, an annual conference for women undergraduates. Ada College is a further-education college in Tottenham Hale, London, focused on digital skills. Ada Lovelace Day is an annual event celebrated on the second Tuesday of October, which began in 2009. Its goal is to "... raise the profile of women in science, technology, engineering, and maths," and to "create new role models for girls and women" in these fields. | [
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Events have included Wikipedia edit-a-thons with the aim of improving the representation of women on Wikipedia in terms of articles and editors to reduce unintended gender bias on Wikipedia. The Ada Initiative was a non-profit organisation dedicated to increasing the involvement of women in the free culture and open source movements. The Engineering in Computer Science and Telecommunications College building in Zaragoza University is called the Ada Byron Building. The computer centre in the village of Porlock, near where Lovelace lived, is named after her. Ada Lovelace House is a council-owned building in Kirkby-in-Ashfield, Nottinghamshire, near where Lovelace spent her infancy. In 2012, a Google Doodle and blog post honoured her on her birthday. In 2013, Ada Developers Academy was founded and named after her. The mission of Ada Developers Academy is to diversify tech by providing women and gender diverse people the skills, experience, and community support to become professional software developers to change the face of tech. On 17 September 2013, an episode of Great Lives about Ada Lovelace aired. As of November 2015, all new British passports have included an illustration of Lovelace and Babbage. In 2017, a Google Doodle honoured her with other women on International Women's Day. On 2 February 2018, Satellogic, a high-resolution Earth observation imaging and analytics company, launched a ÑuSat type micro-satellite named in honour of Ada Lovelace. In March 2018, The New York Times published a belated obituary for Ada Lovelace. | [
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On 27 July 2018, Senator Ron Wyden submitted, in the United States Senate, the designation of 9 October 2018 as National Ada Lovelace Day: "To honor the life and contributions of Ada Lovelace as a leading woman in science and mathematics". The resolution (S.Res.592) was considered, and agreed to without amendment and with a preamble by unanimous consent. In November 2020 it was announced that Trinity College Dublin whose library had previously held forty busts, all of them of men, was commissioning four new busts of women, one of whom was to be Lovelace. Bicentenary
The bicentenary of Ada Lovelace's birth was celebrated with a number of events, including:
The Ada Lovelace Bicentenary Lectures on Computability, Israel Institute for Advanced Studies, 20 December 2015 – 31 January 2016. Ada Lovelace Symposium, University of Oxford, 13–14 October 2015. Ada.Ada.Ada, a one-woman show about the life and work of Ada Lovelace (using an LED dress), premiered at Edinburgh International Science Festival on 11 April 2015, and continues to touring internationally to promote diversity on STEM at technology conferences, businesses, government and educational organisations. Special exhibitions were displayed by the Science Museum in London, England and the Weston Library (part of the Bodleian Library) in Oxford, England. Publications
Lovelace, Ada King. Ada, the Enchantress of Numbers: A Selection from the Letters of Lord Byron's Daughter and her Description of the First Computer. Mill Valley, CA: Strawberry Press, 1992. . | [
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Publication history
Six copies of the 1843 first edition of Sketch of the Analytical Engine with Ada Lovelace's "Notes" have been located. Three are held at Harvard University, one at the University of Oklahoma, and one at the United States Air Force Academy. On 20 July 2018, the sixth copy was sold at auction to an anonymous buyer for £95,000. A digital facsimile of one of the copies in the Harvard University Library is available online. In December 2016, a letter written by Ada Lovelace was forfeited by Martin Shkreli to the New York State Department of Taxation and Finance for unpaid taxes owed by Shkreli. See also
Ai-Da (robot)
Code: Debugging the Gender Gap
List of pioneers in computer science
Timeline of women in science
Women in computing
Women in STEM fields
Explanatory notes
References
General sources
. . . . . . . With notes upon the memoir by the translator. Miller, Clair Cain. "Ada Lovelace, 1815–1852," New York Times, 8 March 2018.
. . . . . . . Further reading
Miranda Seymour, In Byron's Wake: The Turbulent Lives of Byron's Wife and Daughter: Annabella Milbanke and Ada Lovelace, Pegasus, 2018, 547 pp. Christopher Hollings, Ursula Martin, and Adrian Rice, Ada Lovelace: The Making of a Computer Scientist, Bodleian Library, 2018, 114 pp. Jenny Uglow, "Stepping Out of Byron's Shadow", The New York Review of Books, vol. LXV, no. 18 (22 November 2018), pp. 30–32. | [
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] |
Jennifer Chiaverini, Enchantress of Numbers, Dutton, 2017, 426 pp. External links
"Ada's Army gets set to rewrite history at Inspirefest 2018" by Luke Maxwell, 4 August 2018
"Untangling the Tale of Ada Lovelace" by Stephen Wolfram, December 2015
1815 births
1852 deaths
19th-century British women scientists
19th-century British writers
19th-century English mathematicians
19th-century English women writers
19th-century British inventors
19th-century English nobility
Ada (programming language)
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Burials in Nottinghamshire
Ada
Women computer scientists
Computer designers
Daughters of barons
Deaths from cancer in England
Deaths from uterine cancer
English computer programmers
English people of Scottish descent
English women poets
Lord Byron
Mathematicians from London
Women of the Victorian era
Burials at the Church of St Mary Magdalene, Hucknall | [
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In mathematics, the absolute value or modulus of a real number , is the non-negative value without regard to its sign. Namely, if is a positive number, and if is negative (in which case negating makes positive), and For example, the absolute value of 3 and the absolute value of −3 is The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts. Terminology and notation
In 1806, Jean-Robert Argand introduced the term module, meaning unit of measure in French, specifically for the complex absolute value, and it was borrowed into English in 1866 as the Latin equivalent modulus. The term absolute value has been used in this sense from at least 1806 in French and 1857 in English. The notation , with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical value and magnitude. In programming languages and computational software packages, the absolute value of x is generally represented by abs(x), or a similar expression. The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes its cardinality; when applied to a matrix, it denotes its determinant. | [
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Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably an element of a normed division algebra, for example a real number, a complex number, or a quaternion. A closely related but distinct notation is the use of vertical bars for either the Euclidean norm or sup norm of a vector although double vertical bars with subscripts respectively) are a more common and less ambiguous notation. Definition and properties
Real numbers
For any the absolute value or modulus is denoted , with a vertical bar on each side of the quantity, and is defined as
The absolute value is thus always either a positive number or zero, but never negative. When itself is negative then its absolute value is necessarily positive
From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line, and more generally the absolute value of the difference of two real numbers is the distance between them. The notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference (see "Distance" below). Since the square root symbol represents the unique positive square root, when applied to a positive number, it follows that
This is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers. | [
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The absolute value has the following four fundamental properties (a, b are real numbers), that are used for generalization of this notion to other domains:
Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that with its sign chosen to make the result positive. Now, since it follows that, whichever of is the value one has for all Consequently, , as desired. Some additional useful properties are given below. These are either immediate consequences of the definition or implied by the four fundamental properties above. Two other useful properties concerning inequalities are:
These relations may be used to solve inequalities involving absolute values. For example:
The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standard metric on the real numbers. Complex numbers
Since the complex numbers are not ordered, the definition given at the top for the real absolute value cannot be directly applied to complex numbers. However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the origin. This can be computed using the Pythagorean theorem: for any complex number
where and are real numbers, the absolute value or modulus is and is defined by
the Pythagorean addition of and , where and denote the real and imaginary parts respectively. | [
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When the is zero, this coincides with the definition of the absolute value of the
When a complex number is expressed in its polar form its absolute value
Since the product of any complex number and its with the same absolute value, is always the non-negative real number the absolute value of a complex number is the square root which is therefore called the absolute square or squared modulus
This generalizes the alternative definition for reals:
The complex absolute value shares the four fundamental properties given above for the real absolute value. Absolute value function
The real absolute value function is continuous everywhere. It is differentiable everywhere except for . It is monotonically decreasing on the interval and monotonically increasing on the interval . Since a real number and its opposite have the same absolute value, it is an even function, and is hence not invertible. The real absolute value function is a piecewise linear, convex function. For both real and complex numbers the absolute value function is idempotent (meaning that the absolute value of any absolute value is itself). Relationship to the sign function
The absolute value function of a real number returns its value irrespective of its sign, whereas the sign (or signum) function returns a number's sign irrespective of its value. The following equations show the relationship between these two functions:
or
and for ,
Derivative
The real absolute value function has a derivative for every , but is not differentiable at . Its derivative for is given by the step function:
The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist. | [
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The subdifferential of at is the interval . The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann equations. The second derivative of with respect to is zero everywhere except zero, where it does not exist. As a generalised function, the second derivative may be taken as two times the Dirac delta function. Antiderivative
The antiderivative (indefinite integral) of the real absolute value function is
where is an arbitrary constant of integration. This is not a complex antiderivative because complex antiderivatives can only exist for complex-differentiable (holomorphic) functions, which the complex absolute value function is not. Distance
The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them. The standard Euclidean distance between two points
and
in Euclidean -space is defined as:
This can be seen as a generalisation, since for and real, i.e. in a 1-space, according to the alternative definition of the absolute value,
and for and complex numbers, i.e. | [
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in a 2-space,
{|
|-
|
|
|-
|
|
|-
|
|
|}
The above shows that the "absolute value"-distance, for real and complex numbers, agrees with the standard Euclidean distance, which they inherit as a result of considering them as one and two-dimensional Euclidean spaces, respectively. The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a distance function as follows:
A real valued function on a set is called a metric (or a distance function) on , if it satisfies the following four axioms:
{|
|-
|style="width:250px" |
|Non-negativity
|-
|
|Identity of indiscernibles
|-
|
|Symmetry
|-
|
|Triangle inequality
|}
Generalizations
Ordered rings
The definition of absolute value given for real numbers above can be extended to any ordered ring. That is, if is an element of an ordered ring R, then the absolute value of , denoted by , is defined to be:
where is the additive inverse of , 0 is the additive identity, and < and ≥ have the usual meaning with respect to the ordering in the ring. Fields
The four fundamental properties of the absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows. | [
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A real-valued function on a field is called an absolute value (also a modulus, magnitude, value, or valuation) if it satisfies the following four axioms:
{| cellpadding=10
|-
|
|Non-negativity
|-
|
|Positive-definiteness
|-
|
|Multiplicativity
|-
|
|Subadditivity or the triangle inequality
|}
Where 0 denotes the additive identity of . It follows from positive-definiteness and multiplicativity that , where 1 denotes the multiplicative identity of . The real and complex absolute values defined above are examples of absolute values for an arbitrary field. If is an absolute value on , then the function on , defined by , is a metric and the following are equivalent:
satisfies the ultrametric inequality for all , , in . is bounded in R.
for every . for all
for all . An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwise it is said to be Archimedean. Vector spaces
Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space. A real-valued function on a vector space over a field , represented as , is called an absolute value, but more usually a norm, if it satisfies the following axioms:
For all in , and , in ,
{| cellpadding=10
|-
|
|Non-negativity
|-
|
|Positive-definiteness
|-
|
|Positive homogeneity or positive scalability
|-
|
|Subadditivity or the triangle inequality
|}
The norm of a vector is also called its length or magnitude. In the case of Euclidean space , the function defined by
is a norm called the Euclidean norm. | [
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When the real numbers are considered as the one-dimensional vector space , the absolute value is a norm, and is the -norm (see Lp space) for any . In fact the absolute value is the "only" norm on , in the sense that, for every norm on , . The complex absolute value is a special case of the norm in an inner product space, which is identical to the Euclidean norm when the complex plane is identified as the Euclidean plane . Composition algebras
Every composition algebra A has an involution x → x* called its conjugation. The product in A of an element x and its conjugate x* is written N(x) = x x* and called the norm of x. The real numbers , complex numbers , and quaternions are all composition algebras with norms given by definite quadratic forms. The absolute value in these division algebras is given by the square root of the composition algebra norm. In general the norm of a composition algebra may be a quadratic form that is not definite and has null vectors. However, as in the case of division algebras, when an element x has a non-zero norm, then x has a multiplicative inverse given by x*/N(x). Notes
References
Bartle; Sherbert; Introduction to real analysis (4th ed. ), John Wiley & Sons, 2011 . Nahin, Paul J.; An Imaginary Tale; Princeton University Press; (hardcover, 1998). . Mac Lane, Saunders, Garrett Birkhoff, Algebra, American Mathematical Soc., 1999. . | [
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Mendelson, Elliott, Schaum's Outline of Beginning Calculus, McGraw-Hill Professional, 2008. . O'Connor, J.J. and Robertson, E.F.; "Jean Robert Argand". Schechter, Eric; Handbook of Analysis and Its Foundations, pp. 259–263, "Absolute Values", Academic Press (1997) . External links
Special functions
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An analog signal is any continuous signal representing some other quantity, i.e., "analogous" to another quantity. For example, in an analog audio signal, the instantaneous signal voltage varies continuously with the pressure of the sound waves. In contrast, a digital signal represents the original time-varying quantity as a sampled sequence of quantized values. This imposes some bandwidth and dynamic range constraints on the representation but, unlike the analog signal, can avoid further electronic noise and distortion. The term analog signal usually refers to electrical signals; however, mechanical, pneumatic, hydraulic, and other systems may also convey or be considered analog signals. Representation
An analog signal uses some property of the medium to convey the signal's information. For example, an aneroid barometer uses rotary position as the signal to convey pressure information. In an electrical signal, the voltage, current, or frequency of the signal may be varied to represent the information. Any information may be conveyed by an analog signal; such a signal may be a measured response to changes in a physical variable, such as sound, light, temperature, position, or pressure. The physical variable is converted to an analog signal by a transducer. For example, sound striking the diaphragm of a microphone induces corresponding fluctuations in the current produced by a coil in an electromagnetic microphone or the voltage produced by a condenser microphone. The voltage or the current is said to be an analog of the sound. | [
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Noise
An analog signal is subject to electronic noise and distortion introduced by communication channels, recording and signal processing operations, which can progressively degrade the signal-to-noise ratio (SNR). As the signal is transmitted, copied, or processed, the unavoidable noise introduced in the signal path will accumulate as a generation loss, progressively and irreversibly degrading the SNR, until in extreme cases, the signal can be overwhelmed. Noise can show up as hiss and intermodulation distortion in audio signals, or snow in video signals. Generation loss is irreversible as there is no reliable method to distinguish the noise from the signal. In contrast, although converting an analog signal to digital form introduces a low-level quantization noise into the signal due to finite resolution of digital systems, once in digital form, the signal can be transmitted, stored, or processed without introducing significant additional noise or distortion. Noise accumulation in analog systems can be minimized by electromagnetic shielding, balanced lines, low-noise amplifiers and high-quality electrical components. In analog systems, it is difficult to detect when such degradation occurs. However, in digital systems, degradation can not only be detected but corrected as well. See also
Amplifier
Analog computer
Analog device
Analog signal processing
Magnetic tape
Preamplifier
References
Analog circuits
Electronic design
Television terminology
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The aspect ratio of a geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter sidethe ratio of width to height, when the rectangle is oriented as a "landscape". The aspect ratio is most often expressed as two integer numbers separated by a colon (x:y), less commonly as a simple or decimal fraction. The values x and y do not represent actual widths and heights but, rather, the proportion between width and height. As an example, 8:5, 16:10, 1.6:1, and 1.6 are all ways of representing the same aspect ratio. In objects of more than two dimensions, such as hyperrectangles, the aspect ratio can still be defined as the ratio of the longest side to the shortest side. Applications and uses
The term is most commonly used with reference to:
Graphic / image
Image aspect ratio
Display aspect ratio
Paper size
Standard photographic print sizes
Motion picture film formats
Standard ad size
Pixel aspect ratio
Photolithography: the aspect ratio of an etched, or deposited structure is the ratio of the height of its vertical side wall to its width. HARMST High Aspect Ratios allow the construction of tall microstructures without slant
Tire code
Tire sizing
Turbocharger impeller sizing
Wing aspect ratio of an aircraft or bird
Astigmatism of an optical lens
Nanorod dimensions
Shape factor (image analysis and microscopy)
Aspect ratios of simple shapes
Rectangles
For a rectangle, the aspect ratio denotes the ratio of the width to the height of the rectangle. | [
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A square has the smallest possible aspect ratio of 1:1. Examples:
4:3 = 1.: Some (not all) 20th century computer monitors (VGA, XGA, etc. ), standard-definition television
: international paper sizes (ISO 216)
3:2 = 1.5: 35mm still camera film, iPhone (until iPhone 5) displays
16:10 = 1.6: commonly used widescreen computer displays (WXGA)
Φ:1 = 1.618...: golden ratio, close to 16:10
5:3 = 1.: super 16 mm, a standard film gauge in many European countries
16:9 = 1.: widescreen TV and most laptops
2:1 = 2: dominoes
64:27 = 2.: ultra-widescreen, 21:9
32:9 = 3.: super ultra-widescreen
Ellipses
For an ellipse, the aspect ratio denotes the ratio of the major axis to the minor axis. An ellipse with an aspect ratio of 1:1 is a circle. Aspect ratios of general shapes
In geometry, there are several alternative definitions to aspect ratios of general compact sets in a d-dimensional space:
The diameter-width aspect ratio (DWAR) of a compact set is the ratio of its diameter to its width. A circle has the minimal DWAR which is 1. A square has a DWAR of . The cube-volume aspect ratio (CVAR) of a compact set is the d-th root of the ratio of the d-volume of the smallest enclosing axes-parallel d-cube, to the set's own d-volume. A square has the minimal CVAR which is 1. A circle has a CVAR of . An axis-parallel rectangle of width W and height H, where W>H, has a CVAR of . | [
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] |
If the dimension d is fixed, then all reasonable definitions of aspect ratio are equivalent to within constant factors. Notations
Aspect ratios are mathematically expressed as x:y (pronounced "x-to-y"). Cinematographic aspect ratios are usually denoted as a (rounded) decimal multiple of width vs unit height, while photographic and videographic aspect ratios are usually defined and denoted by whole number ratios of width to height. In digital images there is a subtle distinction between the display aspect ratio (the image as displayed) and the storage aspect ratio (the ratio of pixel dimensions); see Distinctions. See also
Axial ratio
Ratio
Equidimensional ratios in 3D
List of film formats
Squeeze mapping
Scale (ratio)
Vertical orientation
References
Ratios | [
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Algorithms for calculating variance play a major role in computational statistics. A key difficulty in the design of good algorithms for this problem is that formulas for the variance may involve sums of squares, which can lead to numerical instability as well as to arithmetic overflow when dealing with large values. Naïve algorithm
A formula for calculating the variance of an entire population of size N is:
Using Bessel's correction to calculate an unbiased estimate of the population variance from a finite sample of n observations, the formula is:
Therefore, a naïve algorithm to calculate the estimated variance is given by the following:
Let
For each datum :
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by N instead of n − 1 on the last line. Because and can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation. Thus this algorithm should not be used in practice, and several alternate, numerically stable, algorithms have been proposed. This is particularly bad if the standard deviation is small relative to the mean. Computing shifted data
The variance is invariant with respect to changes in a location parameter, a property which can be used to avoid the catastrophic cancellation in this formula. with any constant, which leads to the new formula
the closer is to the mean value the more accurate the result will be, but just choosing a value inside the
samples range will guarantee the desired stability. | [
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If just the first sample is taken as the algorithm can be written in Python programming language as
def shifted_data_variance(data):
if len(data) < 2:
return 0.0
K = data[0]
n = Ex = Ex2 = 0.0
for x in data:
n = n + 1
Ex += x - K
Ex2 += (x - K) * (x - K)
variance = (Ex2 - (Ex * Ex) / n) / (n - 1)
# use n instead of (n-1) if want to compute the exact variance of the given data
# use (n-1) if data are samples of a larger population
return variance
This formula also facilitates the incremental computation that can be expressed as
K = n = Ex = Ex2 = 0.0
def add_variable(x):
global K, n, Ex, Ex2
if n == 0:
K = x
n += 1
Ex += x - K
Ex2 += (x - K) * (x - K)
def remove_variable(x):
global K, n, Ex, Ex2
n -= 1
Ex -= x - K
Ex2 -= (x - K) * (x - K)
def get_mean():
global K, n, Ex
return K + Ex / n
def get_variance():
global n, Ex, Ex2
return (Ex2 - (Ex * Ex) / n) / (n - 1)
Two-pass algorithm
An alternative approach, using a different formula for the variance, first computes the sample mean,
and then computes the sum of the squares of the differences from the mean,
where s is the standard deviation. | [
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This is given by the following code:
def two_pass_variance(data):
n = sum1 = sum2 = 0
for x in data:
n += 1
sum1 += x
mean = sum1 / n
for x in data:
sum2 += (x - mean) * (x - mean)
variance = sum2 / (n - 1)
return variance
This algorithm is numerically stable if n is small. However, the results of both of these simple algorithms ("naïve" and "two-pass") can depend inordinately on the ordering of the data and can give poor results for very large data sets due to repeated roundoff error in the accumulation of the sums. Techniques such as compensated summation can be used to combat this error to a degree. Welford's online algorithm
It is often useful to be able to compute the variance in a single pass, inspecting each value only once; for example, when the data is being collected without enough storage to keep all the values, or when costs of memory access dominate those of computation. For such an online algorithm, a recurrence relation is required between quantities from which the required statistics can be calculated in a numerically stable fashion. The following formulas can be used to update the mean and (estimated) variance of the sequence, for an additional element xn. Here, denotes the sample mean of the first n samples , their biased sample variance, and their unbiased sample variance. | [
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These formulas suffer from numerical instability, as they repeatedly subtract a small number from a big number which scales with n. A better quantity for updating is the sum of squares of differences from the current mean, , here denoted :
This algorithm was found by Welford, and it has been thoroughly analyzed. It is also common to denote and . An example Python implementation for Welford's algorithm is given below. # For a new value newValue, compute the new count, new mean, the new M2. # mean accumulates the mean of the entire dataset
# M2 aggregates the squared distance from the mean
# count aggregates the number of samples seen so far
def update(existingAggregate, newValue):
(count, mean, M2) = existingAggregate
count += 1
delta = newValue - mean
mean += delta / count
delta2 = newValue - mean
M2 += delta * delta2
return (count, mean, M2)
# Retrieve the mean, variance and sample variance from an aggregate
def finalize(existingAggregate):
(count, mean, M2) = existingAggregate
if count < 2:
return float("nan")
else:
(mean, variance, sampleVariance) = (mean, M2 / count, M2 / (count - 1))
return (mean, variance, sampleVariance)
This algorithm is much less prone to loss of precision due to catastrophic cancellation, but might not be as efficient because of the division operation inside the loop. | [
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For a particularly robust two-pass algorithm for computing the variance, one can first compute and subtract an estimate of the mean, and then use this algorithm on the residuals. The parallel algorithm below illustrates how to merge multiple sets of statistics calculated online. Weighted incremental algorithm
The algorithm can be extended to handle unequal sample weights, replacing the simple counter n with the sum of weights seen so far. West (1979) suggests this incremental algorithm:
def weighted_incremental_variance(data_weight_pairs):
w_sum = w_sum2 = mean = S = 0
for x, w in data_weight_pairs:
w_sum = w_sum + w
w_sum2 = w_sum2 + w * w
mean_old = mean
mean = mean_old + (w / w_sum) * (x - mean_old)
S = S + w * (x - mean_old) * (x - mean)
population_variance = S / w_sum
# Bessel's correction for weighted samples
# Frequency weights
sample_frequency_variance = S / (w_sum - 1)
# Reliability weights
sample_reliability_variance = S / (w_sum - w_sum2 / w_sum)
Parallel algorithm
Chan et al. note that Welford's online algorithm detailed above is a special case of an algorithm that works for combining arbitrary sets and :
. This may be useful when, for example, multiple processing units may be assigned to discrete parts of the input. Chan's method for estimating the mean is numerically unstable when and both are large, because the numerical error in is not scaled down in the way that it is in the case. In such cases, prefer . | [
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def parallel_variance(n_a, avg_a, M2_a, n_b, avg_b, M2_b):
n = n_a + n_b
delta = avg_b - avg_a
M2 = M2_a + M2_b + delta ** 2 * n_a * n_b / n
var_ab = M2 / (n - 1)
return var_ab
This can be generalized to allow parallelization with AVX, with GPUs, and computer clusters, and to covariance. Example
Assume that all floating point operations use standard IEEE 754 double-precision arithmetic. Consider the sample (4, 7, 13, 16) from an infinite population. Based on this sample, the estimated population mean is 10, and the unbiased estimate of population variance is 30. Both the naïve algorithm and two-pass algorithm compute these values correctly. Next consider the sample (, , , ), which gives rise to the same estimated variance as the first sample. The two-pass algorithm computes this variance estimate correctly, but the naïve algorithm returns 29.333333333333332 instead of 30. While this loss of precision may be tolerable and viewed as a minor flaw of the naïve algorithm, further increasing the offset makes the error catastrophic. Consider the sample (, , , ). Again the estimated population variance of 30 is computed correctly by the two-pass algorithm, but the naïve algorithm now computes it as −170.66666666666666. This is a serious problem with naïve algorithm and is due to catastrophic cancellation in the subtraction of two similar numbers at the final stage of the algorithm. | [
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Higher-order statistics
Terriberry extends Chan's formulae to calculating the third and fourth central moments, needed for example when estimating skewness and kurtosis:
Here the are again the sums of powers of differences from the mean , giving
For the incremental case (i.e., ), this simplifies to:
By preserving the value , only one division operation is needed and the higher-order statistics can thus be calculated for little incremental cost. An example of the online algorithm for kurtosis implemented as described is:
def online_kurtosis(data):
n = mean = M2 = M3 = M4 = 0
for x in data:
n1 = n
n = n + 1
delta = x - mean
delta_n = delta / n
delta_n2 = delta_n * delta_n
term1 = delta * delta_n * n1
mean = mean + delta_n
M4 = M4 + term1 * delta_n2 * (n*n - 3*n + 3) + 6 * delta_n2 * M2 - 4 * delta_n * M3
M3 = M3 + term1 * delta_n * (n - 2) - 3 * delta_n * M2
M2 = M2 + term1
# Note, you may also calculate variance using M2, and skewness using M3
# Caution: If all the inputs are the same, M2 will be 0, resulting in a division by 0.
kurtosis = (n * M4) / (M2 * M2) - 3
return kurtosis
Pébaÿ
further extends these results to arbitrary-order central moments, for the incremental and the pairwise cases, and subsequently Pébaÿ et al. for weighted and compound moments. One can also find there similar formulas for covariance. | [
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Choi and Sweetman
offer two alternative methods to compute the skewness and kurtosis, each of which can save substantial computer memory requirements and CPU time in certain applications. The first approach is to compute the statistical moments by separating the data into bins and then computing the moments from the geometry of the resulting histogram, which effectively becomes a one-pass algorithm for higher moments. One benefit is that the statistical moment calculations can be carried out to arbitrary accuracy such that the computations can be tuned to the precision of, e.g., the data storage format or the original measurement hardware. A relative histogram of a random variable can be constructed in the conventional way: the range of potential values is
divided into bins and the number of occurrences within each bin are counted and plotted such that the area of each rectangle equals the portion of the sample values within that bin:
where and represent the frequency and the relative frequency at bin and is the total area of the histogram. After this normalization, the raw moments and central moments of can be calculated from the relative histogram:
where the superscript indicates the moments are calculated from the histogram. For constant bin width these two expressions can be simplified using :
The second approach from Choi and Sweetman is an analytical methodology to combine statistical moments from individual segments of a time-history such that the resulting overall moments are those of the complete time-history. This methodology could be used for parallel computation of statistical moments with subsequent combination of those moments, or for combination of statistical moments computed at sequential times. | [
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If sets of statistical moments are known:
for , then each can
be expressed in terms of the equivalent raw moments:
where is generally taken to be the duration of the time-history, or the number of points if is constant. The benefit of expressing the statistical moments in terms of is that the sets can be combined by addition, and there is no upper limit on the value of . where the subscript represents the concatenated time-history or combined . These combined values of can then be inversely transformed into raw moments representing the complete concatenated time-history
Known relationships between the raw moments () and the central moments ()
are then used to compute the central moments of the concatenated time-history. Finally, the statistical moments of the concatenated history are computed from the central moments:
Covariance
Very similar algorithms can be used to compute the covariance. Naïve algorithm
The naïve algorithm is
For the algorithm above, one could use the following Python code:
def naive_covariance(data1, data2):
n = len(data1)
sum12 = 0
sum1 = sum(data1)
sum2 = sum(data2)
for i1, i2 in zip(data1, data2):
sum12 += i1 * i2
covariance = (sum12 - sum1 * sum2 / n) / n
return covariance
With estimate of the mean
As for the variance, the covariance of two random variables is also shift-invariant, so given any two constant values and it can be written:
and again choosing a value inside the range of values will stabilize the formula against catastrophic cancellation as well as make it more robust against big sums. | [
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Taking the first value of each data set, the algorithm can be written as:
def shifted_data_covariance(data_x, data_y):
n = len(data_x)
if n < 2:
return 0
kx = data_x[0]
ky = data_y[0]
Ex = Ey = Exy = 0
for ix, iy in zip(data_x, data_y):
Ex += ix - kx
Ey += iy - ky
Exy += (ix - kx) * (iy - ky)
return (Exy - Ex * Ey / n) / n
Two-pass
The two-pass algorithm first computes the sample means, and then the covariance:
The two-pass algorithm may be written as:
def two_pass_covariance(data1, data2):
n = len(data1)
mean1 = sum(data1) / n
mean2 = sum(data2) / n
covariance = 0
for i1, i2 in zip(data1, data2):
a = i1 - mean1
b = i2 - mean2
covariance += a * b / n
return covariance
A slightly more accurate compensated version performs the full naive algorithm on the residuals. The final sums and should be zero, but the second pass compensates for any small error. Online
A stable one-pass algorithm exists, similar to the online algorithm for computing the variance, that computes co-moment :
The apparent asymmetry in that last equation is due to the fact that , so both update terms are equal to . Even greater accuracy can be achieved by first computing the means, then using the stable one-pass algorithm on the residuals. | [
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] |
Thus the covariance can be computed as
def online_covariance(data1, data2):
meanx = meany = C = n = 0
for x, y in zip(data1, data2):
n += 1
dx = x - meanx
meanx += dx / n
meany += (y - meany) / n
C += dx * (y - meany)
population_covar = C / n
# Bessel's correction for sample variance
sample_covar = C / (n - 1)
A small modification can also be made to compute the weighted covariance:
def online_weighted_covariance(data1, data2, data3):
meanx = meany = 0
wsum = wsum2 = 0
C = 0
for x, y, w in zip(data1, data2, data3):
wsum += w
wsum2 += w * w
dx = x - meanx
meanx += (w / wsum) * dx
meany += (w / wsum) * (y - meany)
C += w * dx * (y - meany)
population_covar = C / wsum
# Bessel's correction for sample variance
# Frequency weights
sample_frequency_covar = C / (wsum - 1)
# Reliability weights
sample_reliability_covar = C / (wsum - wsum2 / wsum)
Likewise, there is a formula for combining the covariances of two sets that can be used to parallelize the computation:
Weighted batched version
A version of the weighted online algorithm that does batched updated also exists: let
denote the weights, and write
The covariance can then be computed as
See also
Kahan summation algorithm
Squared deviations from the mean
Yamartino method
References
External links
Statistical algorithms
Statistical deviation and dispersion
Articles with example pseudocode
Articles with example Python (programming language) code | [
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Ibn Sina (), also known as Abu Ali Sina (), Pour Sina (), and often known in the West as Avicenna (; – June 1037), was a Persian polymath who is regarded as one of the most significant physicians, astronomers, thinkers and writers of the Islamic Golden Age, and the father of early modern medicine. Sajjad H. Rizvi has called Avicenna "arguably the most influential philosopher of the pre-modern era". He was a Muslim Peripatetic philosopher influenced by Greek Aristotelian philosophy. Of the 450 works he is believed to have written, around 240 have survived, including 150 on philosophy and 40 on medicine. His most famous works are The Book of Healing, a philosophical and scientific encyclopedia, and The Canon of Medicine, a medical encyclopedia which became a standard medical text at many medieval universities and remained in use as late as 1650. Besides philosophy and medicine, Avicenna's corpus includes writings on astronomy, alchemy, geography and geology, psychology, Islamic theology, logic, mathematics, physics and works of poetry. Name
is a Latin corruption of the Arabic patronym Ibn Sīnā (), meaning "Son of Sina". However, Avicenna was not the son but the great-great-grandson of a man named Sina. His formal Arabic name was Abū ʿAlī al-Ḥusayn bin ʿAbdullāh ibn al-Ḥasan bin ʿAlī bin Sīnā al-Balkhi al-Bukhari (). Circumstances
Avicenna created an extensive corpus of works during what is commonly known as the Islamic Golden Age, in which the translations of Byzantine Greco-Roman, Persian and Indian texts were studied extensively. | [
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