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# Bluetongue disease
**Bluetongue (BT) disease** is a noncontagious, arthropod-borne viral disease affecting ruminants, primarily sheep and other domestic or wild ruminants, including cattle, yaks, goats, buffalo, deer, dromedaries, and antelope. It is caused by Bluetongue virus (**BTV**), a non-enveloped, double-stranded RNA virus belongs to the genus *Orbivirus* within the family *Sedoreoviridae*. The virus is mainly transmitted by biting midges, specifically *Culicoides* species (e.g. *Culicoides imicola*, *Culicoides oxystoma,* and *Culicoides variipennis*). BTV has a widespread geographical distribution, encompassing numerous continents and regions, including Africa, Asia, Australia, Europe, North America, and various tropical and subtropical regions. At present, there are more than 28 recognized serotypes of BTV. Bluetongue outbreaks have had a significant economic impact, with estimated global losses reaching approximately US\$3 billion.
## Clinical signs {#clinical_signs}
In sheep, BTV causes an acute disease with high morbidity and mortality. BTV also infects goats, cattle, and other domestic animals, as well as wild ruminants (for example, blesbuck, white-tailed deer, elk, and pronghorn antelope). The clinical signs are summarized under the term FFF (fever, face, feet).
Major signs are high fever, excessive salivation, swelling of the face and tongue, and cyanosis (in severe conditions) of the tongue. Swelling of the lips and tongue gives the tongue its typical blue appearance, though this sign is confined to a minority of the animals. Nasal signs may be prominent, with nasal discharge and stertorous respiration.
Some animals also develop foot lesions, beginning with coronitis, with consequent lameness. In sheep, this can lead to knee-walking. In cattle, constant changing of position of the feet gives bluetongue the nickname **the dancing disease**. Torsion of the neck (opisthotonos or torticollis) is observed in severely affected animals.
Not all animals develop signs, but all those that do lose condition rapidly, and the sickest die within a week. For affected animals that do not die, recovery is very slow, lasting several months.
The incubation period is 5--20 days, and all signs usually develop within a month. The mortality rate is normally low, but it is high in susceptible breeds of sheep. In Africa, local breeds of sheep may have no mortality, but in imported breeds, it may be up to 90%.
The manifestation of clinical signs in cattle is contingent upon the strain of virus. BTV-8 has been documented to cause a severe disease state and mortality in cattle. The current circulation of BTV-3 in Northern Europe is epidemiologically noteworthy due to the presentation of clinical signs in cattle and a higher sheep mortality rate than that observed with BTV-8. Other ruminants, such as goats, typically exhibit minimal or no clinical signs despite high virus levels in blood. Therefore, they could serve as potential virus reservoirs of BTV. Red deer are an exception, and in them the disease may be as acute as in sheep.
Lamb infected in utero can develop congenital hydranencephaly. This abnormality is a condition in which the brain\'s cerebral hemispheres are like Swiss cheese, or absent, and replaced by sacs filled with cerebrospinal fluid. Ewes infected with bluetongue virus while pregnant can have lambs with this defect, as well as giving birth to lambs who are small, weak, deformed or blind. These affected lambs die within a few days of birth, or are born dead.
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# Bluetongue disease
## Microbiology
Bluetongue is caused by the pathogenic vector-borne RNA virus, Bluetongue virus (BTV), of the genus *Orbivirus* within the *Sedoreoviridae* family. The virus particle consists of 10 strands of double-stranded RNA surrounded by two protein shells. Unlike other arboviruses, BTV lacks a lipid envelope. The virus exhibits icosahedral symmetry, with a diameter of approximately 80--90 nm. The structure of the 70 nm core was determined in 1998 and was at the time the largest atomic structure to be solved.
The 10 viral genome segments have been found to encode 7 structural (VP1--VP7) and 5 non-structural (NS1, NS2, NS3/NS3A, NS4 and NS5) proteins. There are currently more than 28 known serotypes of BTV. The sequence of genome Seg-2 and its translated protein VP2, as well as that of Seg-6 and its translated protein VP5, exhibit variations that determine the serotypes.
The two outer capsid proteins, VP2 and VP5, mediate attachment and penetration of BTV into the target cell. VP2 and VP5 are the primary antigenic targets for antibody targeting by the host immune system. The virus makes initial contact with the cell with VP2, triggering receptor-mediated endocytosis of the virus. The low pH within the endosome then triggers BTV\'s membrane penetration protein VP5 to undergo a conformational change that disrupts the endosomal membrane. Uncoating yields a transcriptionally active 470S core particle which is composed of two major proteins VP7 and VP3, and the three minor proteins VP1, VP4 and VP6 in addition to the dsRNA genome. There is no evidence that any trace of the outer capsid remains associated with these cores, as has been described for reovirus. The cores may be further uncoated to form 390S subcore particles that lack VP7, also in contrast to reovirus. Subviral particles are probably akin to cores derived *in vitro* from virions by physical or proteolytic treatments that remove the outer capsid and causes activation of the BTV transcriptase. In addition to the seven structural proteins, three non-structural (NS) proteins, NS1, NS2, NS3 (and a related NS3A) are synthesised in BTV-infected cells. Of these, NS3/NS3A is involved in the egress of the progeny virus. The two remaining non-structural proteins, NS1 and NS2, are produced at high levels in the cytoplasm and are believed to be involved in virus replication, assembly and morphogenesis.
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# Bluetongue disease
## Evolution
The viral genome is replicated via structural protein VP1, an RNA-dependent RNA polymerase. The lack of proof-reading abilities results in high levels of transcription errors, resulting in single nucleotide mutations. Despite this, the BTV genome is quite stable, exhibiting a low rate of variants arising in populations. Evidence suggests this is due to purifying selection across the genome as the virus is transmitted alternately through its insect and animal hosts. However, individual gene segments undergo different selective pressures and some, particularly segments 4 and 5, are subject to positive selection.
The BTV genome exhibits rapid evolution through genetic drift, reassortment of genome segments (genetic shift), and intragenic recombination. This evolutionary process, in conjunction with the random fixation of quasispecies variants during transmission between susceptible animals and vectors, is postulated to be the primary driver of the genetic diversity observed in BTV field strains. Reassortment can lead to a rapid shift in phenotypes independent of the slow rate of mutation. During this process, gene segments are not randomly reassorted. Rather, there appears to be a mechanism for selecting for or against certain segments from the parental serotypes present. However, this selective mechanism is still poorly understood.
To date, BTV serotypes 25 and above have been identified as the causative agents of infection in small ruminants. The infection is subclinical, which likely explains why these serotypes, which are less or non-virulent, have not been identified earlier through laboratory diagnosis studies. It is noteworthy that BTV serotypes 25 and higher are transmitted without midges, indicating that direct contact between sheep or goats may be a potential vector.
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# Bluetongue disease
## Epidemiology
The presence of the insect vectors determines the bluetongue disease\'s global distribution, with regions in Africa, Asia, Australia, Europe, North America, and other tropical/subtropical area being most affected.
The virus persists in areas where climatic conditions support the survival of *Culicoides* midges during winter. This adaptability allows the disease to establish itself in new regions when conditions become favorable.
An outline of the transmission cycle of BTV is illustrated in article Parasitic flies of domestic animals.
Its occurrence is seasonal in the affected Mediterranean countries, subsiding when temperatures drop and hard frosts kill the adult midge vectors. Viral survival and vector longevity is seen during milder winters. A significant contribution to the northward spread of bluetongue disease has been the ability of *C. obsoletus* and *C.pulicaris* to acquire and transmit the pathogen, both of which are spread widely throughout Europe. This is in contrast to the original *C.imicola* vector, which is limited to North Africa and the Mediterranean. The relatively recent novel vector has facilitated a far more rapid spread than the simple expansion of habitats north through global warming.
In August 2006, cases of bluetongue were found in the Netherlands, then Belgium, Germany, and Luxembourg. In 2007, the first case of bluetongue in the Czech Republic was detected in one bull near Cheb at the Czech-German border. In September 2007, the UK reported its first ever suspected case of the disease, in a Highland cow on a rare-breeds farm near Ipswich, Suffolk. Since then, the virus has spread from cattle to sheep in Britain. By October 2007, bluetongue had become a serious threat in Scandinavia and Switzerland and the first outbreak in Denmark was reported. In autumn 2008, several cases were reported in the southern Swedish provinces of Småland, Halland, and Skåne, as well as in areas of the Netherlands bordering Germany, prompting veterinary authorities in Germany to intensify controls. Norway had its first finding in February 2009, when cows at two farms in Vest-Agder in the south of Norway showed an immune response to bluetongue. A number of countries, including Norway and Finland, were certified as free of the disease in 2011 and 2021, respectively.
In 2023, Europe witnessed a series of notable epizootic occurrences at higher latitudes, partially attributable to the emergence of a novel serotype, BTV-3. The serotype was first identified in the Netherlands in September 2023 and has since been documented in numerous European countries, including Belgium, Germany, the Netherlands, France, Spain, the UK, Norway, and Sweden.
Although the disease is not a threat to humans, the most vulnerable common domestic ruminants are cattle, goats, and especially, sheep.
### Overwintering
A puzzling aspect of BTV is its survival between midge seasons in temperate regions. Adults of *Culicoides* are killed by cold winter temperatures, and BTV infections typically do not last for more than 60 days, which is not long enough for BTV to survive until the next spring. It is believed that the virus somehow survives in overwintering midges or animals. Multiple mechanisms have been proposed. A few adult *Culicoides* midges infected with BTV may survive the mild winters of the temperate zone. Some midges may even move indoors to avoid the cold temperature of the winter. Additionally, BTV could cause a chronic or latent infection in some animals, providing another means for BTV to survive the winter. BTV can also be transmitted from mother to fetus. The outcome is abortion or stillbirth if fetal infection occurs early in gestation and survival if infection occurs late. However infection at an intermediate stage, before the fetal immune system is fully developed, may result in a chronic infection that lingers until the first months after birth of the lamb. Midges then spread the pathogen from the calves to other animals, starting a new season of infection.
### Climate change {#climate_change}
The spread of bluetongue to Southern, Central, and Northern Europe provides an illustrative example of the complex interactions between climate change, vector habitat suitability, animal population density, distribution, and movement, which collectively influence the patterns of disease emergence and transmission.`{{Excerpt|Culicoides imicola#Role of climate change}}`{=mediawiki}
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# Bluetongue disease
## Treatment and prevention {#treatment_and_prevention}
There are currently no antiviral medications that have been approved for the treatment of bluetongue disease. The standard of care involves the administration of anti-inflammatory drugs and supportive nursing care to alleviate the clinical signs and symptoms. Prevention is effected via quarantine, vaccination, and control of the midges vector, including inspection of aircraft. The recurrent emergence of novel strains and the occurrence of new outbreaks with significant socio-economic impacts highlight the urgent need for effective antiviral strategies. The current vaccines for bluetongue virus (BTV) are serotype-specific, which limits their utility and has led to interest in host-targeted antiviral strategies that offer broader activity against multiple serotypes and a reduced risk of resistance development.
### Livestock management and insect control {#livestock_management_and_insect_control}
Some available key measures include vector control, such as the use of insecticides, insect-proof nets, and improved housing to reduce exposure to biting midges. Additionally, the removal of infected animals helps prevent further transmission by reducing the number of viremic hosts, while movement restrictions---including quarantines and health certifications---prevent the introduction of the virus to uninfected regions.
### Vaccines
Vaccination still represents an effective strategy for protecting ruminants against bluetongue. However, this is only possible with a vaccine that is effective against the relevant serotype. The most prevalent vaccines are live attenuated vaccines and killed or inactivated vaccines. Other potential vaccines include subunit vaccines, virus-like particles, DNA vaccines, disabled unfectious single animal vaccines (DISA), and disabled infectious single-cycle vaccines (DISC).
Protection by live attenuated vaccines (LAVs) are serotype specific. Multiserotype LAV cocktails can induce neutralizing antibodies against unincluded serotypes, and subsequent vaccinations with three different pentavalent LAV cocktails induce broad protection. These pentavalent cocktails contain 15 different serotypes in total: serotypes 1 through 14, as well as 19.
Immunization with any of the available vaccines, though, precludes later serological monitoring of affected cattle populations, a problem that could be resolved using next-generation subunit vaccines.
In January 2015, the vaccine Raksha Blu was launched in India. It is designed to protect livestock against five strains of the bluetongue virus.
The vaccine Syvazul BTV was authorized for veterinary use in the European Union in January 2019.
In January 2025, the Committee for Veterinary Medicinal Products (CVMP) of the European Medicines Agency adopted a positive opinion, recommending the granting of a marketing authorization for the veterinary medicinal product Bluevac-3, suspension for injection, intended for cattle and sheep. The applicant for this veterinary medicinal product is CZ Vaccines S.A.U. Bluevac-3 is a vaccine containing inactivated bluetongue virus, serotype 3, BTV-3/NET2023 as active substance. The vaccine is intended to stimulate the active immunity of sheep and cattle against bluetongue virus serotype 3. The CVMP also adopted a positive opinion, recommending the granting of a marketing authorization for the veterinary medicinal product Syvazul BTV 3, suspension for injection, intended for sheep. The applicant for this veterinary medicinal product is Laboratorios Syva S.A. Syvazul BTV 3 is a vaccine containing Bluetongue virus, serotype 3, BTV-3/NET2023, inactivated as active substance. It is intended for the active immunization of sheep against bluetongue virus serotype 3.
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# Bluetongue disease
## History
In the early stages of its identification, BT was referred to by a number of different names, including \"epizootic catarrh,\" \"fever,\" \"malarial catarrhal fever of sheep,\" and \"epizootic malignant catarrhal fever of sheep.\" This was due to the prevailing belief at the time that BT was caused by an intraerythrocytic parasite. The English translation \"Bluetongue\" was initially proposed by Spreull and derived from the Afrikaans term \"bloutong,\" which refers to the condition of cyanosis of the tongue in clinically affected sheep. Although bluetongue disease was already recognized in South Africa in the early 19th century, a comprehensive description of the disease was not published until the first decade of the 20th century. In 1906, Arnold Theiler showed that bluetongue was caused by a filterable agent. He also created the first bluetongue vaccine, which was developed from an attenuated BT V strain. For many decades, bluetongue was thought to be confined to Africa. The first confirmed outbreak outside of Africa occurred in Cyprus in 1943. In 2021, a vessel owned by Khalifeh Livestock Trading and managed by Talia Shipping Line, both based in Lebanon, has been denied right to dock in Spain, as it has about 895 male calves suspected to be infected by bluetongue disease.
## Related diseases {#related_diseases}
African horse sickness is related to bluetongue and is spread by the same midges (*Culicoides* species). It can kill the horses it infects and mortality may go as high as 90% of the infected horses during an epidemic.
Epizootic hemorrhagic disease virus is closely related and crossreacts with Bluetongue virus on many blood tests
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# Bruce Perens
**Bruce Perens** (born around 1958) is an American computer programmer and advocate in the free software movement. He created *The Open Source Definition* and published the first formal announcement and manifesto of open source. He co-founded the Open Source Initiative (OSI) with Eric S. Raymond.
In 2005, Perens represented Open Source at the United Nations World Summit on the Information Society, at the invitation of the United Nations Development Programme. He has appeared before national legislatures and is often quoted in the press, advocating for open source and the reform of national and international technology policy.
Perens is also an amateur radio operator, with call sign K6BP. He promotes open radio communications standards and open-source hardware.
In 2016 Perens, along with Boalt Hall (Berkeley Law) professor Lothar Determann, co-authored \"Open Cars\" which appeared in the Berkeley Technology Law Journal.
In 2018 Perens founded the Open Research Institute (ORI), a non-profit research and development organization to address technologies involving Open Source, Open Hardware, Open Standards, Open Content, and Open Access to Research. In April 2022 he divorced himself from the organization and reported he was starting a new charity, HamOpen.org, to redirect his focus, and align with the ARRL organization for their liability insurance benefit. HamOpen has been most visible supporting the convention exhibitions of projects Perens supports, including M17 and FreeDV.
## Companies
Perens operates two companies: Algoram is a start-up which is creating a web-based control system for radio transmitters and other devices. Legal Engineering is a legal-technical consultancy which specializes in resolving copyright infringement in relation to open source software.
## Early life {#early_life}
Perens grew up in Long Island, New York. He was born with cerebral palsy, which caused him to have slurred speech as a child, a condition that led to a misdiagnosis of him as developmentally disabled in school and led the school to fail to teach him to read. He developed an interest in technology at an early age: besides his interest in amateur radio, he ran a pirate radio station in the town of Lido Beach and briefly engaged in phone phreaking.
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# Bruce Perens
## Career
### Computer graphics {#computer_graphics}
Perens worked for seven years at the New York Institute of Technology Computer Graphics Lab. After that, he worked at Pixar for 12 years, from 1987 to 1999. He is credited as a studio tools engineer on the Pixar films *A Bug\'s Life* (1998) and *Toy Story 2* (1999).
### No-Code International {#no_code_international}
Perens founded No-Code International in 1998 with the goal of ending the Morse Code test then required for an amateur radio license. His rationale was that amateur radio should be a tool for young people to learn advanced technology and networking, rather than something that preserved antiquity and required new hams to master outmoded technology before they were allowed on the air.
Perens lobbied intensively on the Internet, at amateur radio events in the United States, and during visits to other nations. One of his visits was to Iceland, where he had half of that nation\'s radio amateurs in the room, and their vote in the International Amateur Radio Union was equivalent to that of the entire United States.
### Debian Social Contract {#debian_social_contract}
In 1997, Perens was carbon-copied on an email conversation between Donnie Barnes of Red Hat and Ean Schuessler, who was then working on Debian. Schuessler bemoaned that Red Hat had never stated its social contract with the developer community. Perens took this as inspiration to create a formal social contract for Debian. In a blog posting, Perens claims not to have made use of the Three Freedoms (later the Four Freedoms) published by the Free Software Foundation in composing his document. Perens proposed a draft of the Debian Social Contract to the Debian developers on the debian-private mailing list early in June 1997. Debian developers contributed discussion and changes for the rest of the month while Perens edited, and the completed document was then announced as Debian project policy. Part of the Debian Social Contract was the Debian Free Software Guidelines, a set of 10 guidelines for determining whether a set of software can be described as \"free software\", and thus whether it could be included in Debian.
### Open Source Definition and The Open Source Initiative {#open_source_definition_and_the_open_source_initiative}
On February 3, 1998, a group of people (not including Perens) met at VA Linux Systems to discuss the promotion of Free Software to business in pragmatic terms, rather than the moral terms preferred by Richard Stallman. Christine Petersen of the nanotechnology organization Foresight Institute, who was present because Foresight took an early interest in Free Software, suggested the term \"Open Source\". The next day, Eric S. Raymond recruited Perens to work with him on the formation of Open Source. Perens modified the Debian Free Software Guidelines into the Open Source Definition by removing Debian references and replacing them with \"Open Source\".
The original announcement of The Open Source Definition was made on February 9, 1998, on Slashdot and elsewhere; the definition was given in Linux Gazette on February 10, 1998.
Concurrently, Perens and Raymond established the Open Source Initiative, an organization intended to promote open source software.
Perens left OSI in 1999, a year after co-founding it. In February 1999 in an email to the Debian developers mailing list he explained his decision and stated that, though \"most hackers know that Free Software and Open Source are just two words for the same thing\", the success of \"open source\" as a marketing term had \"de-emphasized the importance of the freedoms involved in Free Software\"; he added, \"It\'s time for us to fix that.\" He stated his regret that OSI co-founder Eric Raymond \"seems to be losing his free software focus.\" But in the following 2000s he spoke about Open source again. Perens presently volunteers as the Open Source Initiative\'s representative to the European Technical Standards Institute (\"ETSI\"), and is a frequent participant in review of license texts submitted to OSI for certification as Open Source licenses.
### Linux Capital Group {#linux_capital_group}
In 1999, Perens left Pixar and became the president of Linux Capital Group, a business incubator and venture capital firm focusing on Linux-based businesses. Their major investment was in Progeny Linux Systems, a company headed by Debian founder Ian Murdock. In 2000, as a result of the economic downturn, Perens shut down Linux Capital Group. (Progeny Linux Systems would end operations in 2007.)
### Hewlett-Packard {#hewlett_packard}
From December 2000 to September 2002, Perens served as \"Senior Global Strategist for Linux and Open Source\" at Hewlett-Packard, internally evangelizing for the use of Linux and other open-source software. He was fired as a result of his anti-Microsoft statements, which especially became an issue after HP acquired Compaq, a major manufacturer of Microsoft Windows-based PCs, in 2002.
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# Bruce Perens
## Career
### UserLinux
In 2003 Perens created UserLinux, a Debian-based distribution whose stated goal was, \"Provide businesses with freely available, high quality Linux operating systems accompanied by certifications, service, and support options designed to encourage productivity and security while reducing overall costs.\" UserLinux was eventually overtaken in popularity by Ubuntu, another Debian-based distribution, which was started in 2004, and UserLinux became unmaintained in 2006.
### SourceLabs
Perens was an employee of SourceLabs, a Seattle-based open source software and services company, from June 2005 until December 2007. He produced a video commercial, *Impending Security Breach*, for SourceLabs in 2007. (SourceLabs was acquired by EMC in 2009.)
### University faculty {#university_faculty}
Between 1981 and 1986, Perens was on the staff of the New York Institute of Technology Computer Graphics Lab as a Unix kernel programmer.
In 2002, Perens was a remote Senior Scientist for Open Source with the Cyber Security Policy Laboratory of George Washington University under the direction of Tony Stanco. Stanco was director of the laboratory for a year, while its regular director was on sabbatical.
Between 2006 and 2007, Perens was a visiting lecturer and researcher for the University of Agder under a three-year grant from the Competence Fund of Southern Norway. During this time he consulted the Norwegian Government and other entities on government policy issues related to computers and software. After this time Perens worked remotely on Agder programs, mainly concerning the European Internet Accessibility Observatory.
### Other activities {#other_activities}
In 2007, some of Perens\'s government advisory roles included a meeting with the President of the Chamber of Deputies (the lower house of parliament) in Italy and testimony to the Culture Committee of the Chamber of Deputies; a keynote speech at the foundation of Norway\'s Open Source Center, following Norway\'s Minister of Governmental Reform (Perens is on the advisory board of the center); he provided input on the revision of the European Interoperability Framework; and he was keynote speaker at a European Commission conference on *Digital Business Ecosystems at the Centre Borschette, Brussels, on November 7*.
In 2009, Perens acted as an expert witness on open source in the Jacobsen v. Katzer U.S. Federal lawsuit. His report, which was made publicly available by Jacobsen, presented the culture and impact of open-source software development to the federal courts.
Perens delivered one of the keynote addresses at the 2012 linux.conf.au conference in Ballarat, Australia. He discussed the need for open source software to market itself better to non-technical users. He also discussed some of the latest developments in open-source hardware, such as Papilio and Bus Pirate.
In 2013, Perens spoke in South America, as the closing keynote at Latinoware 2013. He was the keynote of CISL -- Conferencia Internacional de Software Libre, in Buenos Aires, Argentina, and keynoted a special event along with the Minister of software and innovation of Chubut Province, in Puerto Madrin, Patagonia, Argentina. He keynoted the Festival de Software Libre 2013, in Puerto Vallarta, Mexico.
In 2014--2015, Perens took a break from Open Source conferences, having spoken at them often since 1996. In 2016, he returned to the conference circuit, keynoting the Open Source Insight conference in Seoul, sponsored by the Copyright Commission of South Korea. Perens web site presently advertises his availability to keynote conferences as long as travel and lodging expenses are compensated.
In 2020, Perens delivered the talk, \"What Comes After Open Source?\" for DebConf 2020. He discussed the future of open source licensing and the need to develop alternative licensing structures so that open source developers could get paid for their work.
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# Bruce Perens
## Views
Perens poses \"Open Source\" as a means of marketing the free and open-source software idea to business people and mainstream who might be more interested in the practical benefits of an open source development model and ecosystem than abstract ethics. He states that open source and free software are only two ways of talking about the same phenomenon, a point of view not shared by Stallman and his free software movement. Perens postulated in 2004 an economic theory for business use of Open Source in his paper *The Emerging Economic Paradigm of Open Source* and his speech *Innovation Goes Public*. This differs from Raymond\'s theory in *The Cathedral and the Bazaar*, which having been written before there was much business involvement in open source, explains open source as a consequence of programmer motivation and leisure.
In February 2008, for the 10th anniversary of the phrase \"open source\", Perens published a message to the community called \"State of Open Source Message: A New Decade For Open Source\". Around the same time the ezine RegDeveloper published an interview with Perens where he spoke of the successes of open source, but also warned of dangers, including a proliferation of OSI-approved licenses which had not undergone legal scrutiny. He advocated the use of the GPLv3 license, especially noting Linus Torvalds\' refusal to switch away from GPLv2 for the Linux kernel.
Bruce Perens supported Bernie Sanders for President and he claims that his experience with the open source movement influenced that decision. On July 13, 2016, following Sanders\'s endorsement of Hillary Clinton for president, Perens endorsed Clinton.
In January 2013, Perens advocated for abolishment of the Second Amendment to the U.S. constitution, stating that he does \"not believe in private ownership of firearms\" and that he would \"take away guns currently held by individuals, without compensation for their value.\" He reiterated this view in a June 2014 interview in Slashdot, and in November 2017 on his Twitter account.
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# Bruce Perens
## Amateur radio and other activities {#amateur_radio_and_other_activities}
Perens is an avid amateur radio enthusiast (call sign K6BP) and maintained technocrat.net, which he closed in late 2008, because its revenues did not cover its costs.
## Media appearances {#media_appearances}
Perens is featured in the 2001 documentary film *Revolution OS* and the 2006 BBC television documentary *The Code-Breakers*.
From 2002 to 2006, Prentice Hall PTR published the Bruce Perens\' Open Source Series, a set of 24 books covering various open source software tools, for which Perens served as the series editor. It was the first book series to be published under an open license.
## Personal life {#personal_life}
Perens lives in Berkeley, California with his wife, Valerie, and son, Stanley, born in 2000
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# Bundle theory
**Bundle theory**, originated by the 18th century Scottish philosopher David Hume, is the ontological theory about objecthood in which an object consists only of a collection (*bundle*) of properties, relations or tropes.
According to bundle theory, an object consists of its properties and nothing more; thus, there cannot be an object without properties and one cannot *conceive* of such an object. For example, when we think of an apple, we think of its properties: redness, roundness, being a type of fruit, *etc*. There is nothing above and beyond these properties; the apple is nothing more than the collection of its properties. In particular, there is no *substance* in which the properties are *inherent*.
Bundle theory has been contrasted with the *ego theory* of the self, which views the egoic self as a soul-like substance existing in the same manner as the corporeal self.
## Arguments in favor {#arguments_in_favor}
The difficulty in conceiving and or describing an object without also conceiving and or describing its properties is a common justification for bundle theory, especially among current philosophers in the Anglo-American tradition.
The inability to comprehend any aspect of the thing other than its properties implies, this argument maintains, that one cannot conceive of a *bare particular* (a *substance* without properties), an implication that directly opposes substance theory. The conceptual difficulty of *bare particulars* was illustrated by John Locke when he described a *substance* by itself, apart from its properties as \"something, I know not what. \[\...\] The idea then we have, to which we give the general name substance, being nothing but the supposed, but unknown, support of those qualities we find existing, which we imagine cannot subsist sine re substante, without something to support them, we call that support substantia; which, according to the true import of the word, is, in plain English, standing under or upholding.\"
Whether a *relation* of an object is one of its properties may complicate such an argument. However, the argument concludes that the conceptual challenge of *bare particulars* leaves a bundle of properties and nothing more as the only possible conception of an object, thus justifying bundle theory.
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# Bundle theory
## Objections
Bundle theory maintains that properties are *bundled* together in a collection without describing how they are tied together. For example, bundle theory regards an apple as red, 4 in wide, and juicy but lacking an underlying *substance*. The apple is said to be a *bundle of properties* including redness, being 4 in wide, and juiciness. Hume used the term \"bundle\" in this sense, also referring to the personal identity, in his main work: \"I may venture to affirm of the rest of mankind, that they are nothing but a bundle or collection of different perceptions, which succeed each other with inconceivable rapidity, and are in a perpetual flux and movement\".
Critics question how bundle theory accounts for the properties\' *compresence* (the *togetherness* relation between those properties) without an underlying *substance*. Critics also question how any two given properties are determined to be properties of the same object if there is no *substance* in which they both *inhere*. This argument is done away with if one considers spatio-temporal location to be a property as well.
Traditional bundle theory explains the *compresence* of properties by defining an object as a collection of properties *bound* together. Thus, different combinations of properties and relations produce different objects. Redness and juiciness, for example, may be found together on top of the table because they are part of a bundle of properties located on the table, one of which is the \"looks like an apple\" property.
By contrast, substance theory explains the *compresence* of properties by asserting that the properties are found together because it is the *substance* that has those properties. In substance theory, a *substance* is the thing in which properties *inhere*. For example, redness and juiciness are found on top of the table because redness and juiciness *inhere* in an apple, making the apple red and juicy.
The *bundle theory of substance* explains *compresence*. Specifically, it maintains that properties\' compresence itself engenders a *substance*. Thus, it determines *substancehood* empirically by the *togetherness* of properties rather than by a *bare particular* or by any other non-empirical underlying strata. The *bundle theory of substance* thus rejects the substance theories of Aristotle, Descartes, Leibniz, and more recently, J. P. Moreland, Jia Hou, Joseph Bridgman, Quentin Smith, and others.
## Buddhism
The Indian Madhyamaka philosopher, Chandrakirti, used the aggregate nature of objects to demonstrate the lack of essence in what is known as the sevenfold reasoning. In his work, *Guide to the Middle Way* (Sanskrit: *Madhyamakāvatāra*), he says:
He goes on to explain what is meant by each of these seven assertions, but briefly in a subsequent commentary he explains that the conventions of the world do not exist essentially when closely analyzed, but exist only through being taken for granted, without being subject to scrutiny that searches for an essence within them.
Another view of the Buddhist theory of the self, especially in early Buddhism, is that the Buddhist theory is essentially an eliminativist theory. According to this understanding, the self can not be reduced to a bundle because there is nothing that answers to the concept of a self. Consequently, the idea of a self must be eliminated
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# List of major biblical figures
The Bible is a collection of canonical sacred texts of Judaism and Christianity. Different religious groups include different books within their canons, in different orders, and sometimes divide or combine books, or incorporate additional material into canonical books. Christian Bibles range from the sixty-six books of the Protestant canon to the eighty-one books of the Ethiopian Orthodox Church canon.
## Hebrew Bible {#hebrew_bible}
- Tubal-cain
### Prophets
- Samuel
- Enoch
### Kings
- David
- Solomon
### Priests
- Aaron
- Eleazar
- Eli
- Phinehas
### Tribes of Israel {#tribes_of_israel}
According to the Book of Genesis, the Israelites were descendants of the sons of Jacob, who was renamed Israel after wrestling with an angel. His twelve male children become the ancestors of the Twelve Tribes of Israel.
- Asher
- Benjamin
- Dan
- Gad
- Issachar
- Joseph, which was split into two tribes descended from his sons:
- Tribe of Ephraim
- Tribe of Manasseh
- Judah
- Levi
- Naphtali
- Reuben
- Simeon
- Zebulun
## Deuterocanon
### Maccabees
- Eleazar Avaran
- John Gaddi
- John Hyrcanus
- Jonathan Apphus
- Judas Maccabeus
- Mattathias
- Simon Thassi
### Greek rulers {#greek_rulers}
- Alexander the Great
- Antiochus III the Great
- Antiochus IV Epiphanes
- Philip II of Macedon
### Persian rulers {#persian_rulers}
- Astyages
- Darius III
### Others
- Baruch
- Tobit
- Judith
- Susanna
## New Testament {#new_testament}
### Jesus and his relatives {#jesus_and_his_relatives}
- Jesus Christ
- Mary, mother of Jesus
- Joseph
- Brothers of Jesus
- James (often identified with James, son of Alphaeus)
- Joseph (Joses)
- Judas (Jude) (often identified with Thaddeus)
- Simon
- Mary of Clopas
- Cleopas (often identified with Alphaeus and Clopas)
### Apostles of Jesus {#apostles_of_jesus}
The Thirteen:
- Peter (a.k.a. Simon or Cephas)
- Andrew (Simon Peter\'s brother)
- James, son of Zebedee
- John, son of Zebedee
- Philip
- Bartholomew also known as \"Nathanael\"
- Thomas also known as \"Doubting Thomas\"
- Matthew also known as \"Levi\"
- James, son of Alphaeus
- Judas, son of James (a.k.a. Thaddeus or Lebbaeus)
- Simon the Zealot
- Judas Iscariot (the traitor)
- Matthias
Others:
- Paul
- Barnabas
- Mary Magdalene (the one who discovered Jesus\' empty tomb)
### Priests {#priests_1}
- Caiaphas, high priest
- Annas, first high priest of Roman Judea
- Zechariah, father of John the Baptist
### Prophets {#prophets_1}
- Agabus
- Anna
- Simeon
- John the Baptist
### Other believers {#other_believers}
- Apollos
- Aquila
- Dionysius the Areopagite
- Epaphras, fellow prisoner of Paul, fellow worker
- John Mark (often identified with Mark)
- Joseph of Arimathea
- Lazarus
- Luke
- Mark
- Martha
- Mary Magdalene
- Mary, sister of Martha
- Nicodemus
- Onesimus
- Philemon
- Priscilla
- Silas
- Sopater
- Stephen, first martyr
- Timothy
- Titus
### Secular rulers {#secular_rulers}
Herod}}
- Herod Agrippa I, called \"King Herod\" or \"Herod\" in Acts 12
- Felix governor of Judea who was present at the trial of Paul, and his wife Drusilla in Acts 24:24
- Herod Agrippa II, king over several territories, before whom Paul made his defense in Acts 26
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# Basketball
**Basketball** is a team sport in which two teams, most commonly of five players each, opposing one another on a rectangular court, compete with the primary objective of shooting a basketball (approximately 9.4 in in diameter) through the defender\'s hoop (a basket 18 in in diameter mounted 10 ft high to a backboard at each end of the court), while preventing the opposing team from shooting through their own hoop. A field goal is worth two points, unless made from behind the three-point line, when it is worth three. After a foul, timed play stops and the player fouled or designated to shoot a technical foul is given one, two or three one-point free throws. The team with the most points at the end of the game wins, but if regulation play expires with the score tied, an additional period of play (overtime) is mandated.
Players advance the ball by bouncing it while walking or running (dribbling) or by passing it to a teammate, both of which require considerable skill. On offense, players may use a variety of shots`{{snd}}`{=mediawiki}the layup, the jump shot, or a dunk; on defense, they may steal the ball from a dribbler, intercept passes, or block shots; either offense or defense may collect a rebound, that is, a missed shot that bounces from rim or backboard. It is a violation to lift or drag one\'s pivot foot without dribbling the ball, to carry it, or to hold the ball with both hands then resume dribbling.
The five players on each side fall into five playing positions. The tallest player is usually the center, the second-tallest and strongest is the power forward, a slightly shorter but more agile player is the small forward, and the shortest players or the best ball handlers are the shooting guard and the point guard, who implement the coach\'s game plan by managing the execution of offensive and defensive plays (player positioning). Informally, players may play three-on-three, two-on-two, and one-on-one.
Invented in 1891 by Canadian-American gym teacher James Naismith in Springfield, Massachusetts, in the United States, basketball has evolved to become one of the world\'s most popular and widely viewed sports. The National Basketball Association (NBA) is the most significant professional basketball league in the world in terms of popularity, salaries, talent, and level of competition (drawing most of its talent from U.S. college basketball). Outside North America, the top clubs from national leagues qualify to continental championships such as the EuroLeague and the Basketball Champions League Americas. The FIBA Basketball World Cup and Men\'s Olympic Basketball Tournament are the major international events of the sport and attract top national teams from around the world. Each continent hosts regional competitions for national teams, like EuroBasket and FIBA AmeriCup.
The FIBA Women\'s Basketball World Cup and women\'s Olympic basketball tournament feature top national teams from continental championships. The main North American league is the WNBA (NCAA Women\'s Division I Basketball Championship is also popular), whereas the strongest European clubs participate in the EuroLeague Women.
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# Basketball
## History
### Early history {#early_history}
A game similar to basketball is mentioned in a 1591 book published in Frankfurt am Main that reports on the lifestyles and customs of coastal North American residents, *Wahrhafftige Abconterfaytung der Wilden* (German; translates as *Truthful Depictions of the Savages*: \"Among other things, a game of skill is described in which balls must be thrown against a target woven from twigs, mounted high on a pole. There\'s a small reward for the player if the target is being hit.\"
### Creation
In December 1891, James Naismith, a Canadian-American professor of physical education and instructor at the International Young Men\'s Christian Association Training School (now Springfield College) in Springfield, Massachusetts, was trying to keep his gym class active on a rainy day. He sought a vigorous indoor game to keep his students occupied and at proper levels of fitness during the long New England winters. After rejecting other ideas as either too rough or poorly suited to walled-in gymnasiums, he invented a new game in which players would pass a ball to teammates and try to score points by tossing the ball into a basket mounted on a wall.
Naismith wrote the basic rules and nailed a peach basket onto an elevated track. Naismith initially set up the peach basket with its bottom intact, which meant that the ball had to be retrieved manually after each \"basket\" or point scored. This quickly proved tedious, so Naismith removed the bottom of the basket to allow the balls to be poked out with a long dowel after each scored basket.
Shortly after, Senda Berenson, instructor of physical culture at the nearby Smith College, went to Naismith to learn more about the game. Fascinated by the new sport and the values it could teach, she started to organize games with her pupils, following adjusted rules. The first official women\'s interinstitutional game was played barely 11 months later, between the University of California and Miss Head\'s School. In 1899, a committee was established at the Conference of Physical Training in Springfield to draw up general rules for women\'s basketball. Thus, the sport quickly spread throughout America\'s schools, colleges and universities with uniform rules for both sexes.
Basketball was originally played with a soccer ball. These round balls from \"association football\" were made, at the time, with a set of laces to close off the hole needed for inserting the inflatable bladder after the other sewn-together segments of the ball\'s cover had been flipped outside-in. These laces could cause bounce passes and dribbling to be unpredictable. Eventually a lace-free ball construction method was invented, and this change to the game was endorsed by Naismith (whereas in American football, the lace construction proved to be advantageous for gripping and remains to this day). The first balls made specifically for basketball were brown, and it was only in the late 1950s that Tony Hinkle, searching for a ball that would be more visible to players and spectators alike, introduced the orange ball that is now in common use. Dribbling was not part of the original game except for the \"bounce pass\" to teammates. Passing the ball was the primary means of ball movement. Dribbling was eventually introduced but limited by the asymmetric shape of early balls.`{{dubious|date=January 2019}}`{=mediawiki} Dribbling was common by 1896, with a rule against the double dribble by 1898.
The peach baskets were used until 1906 when they were finally replaced by metal hoops with backboards. A further change was soon made, so the ball merely passed through. Whenever a person got the ball in the basket, their team would gain a point. Whichever team got the most points won the game. The baskets were originally nailed to the mezzanine balcony of the playing court, but this proved impractical when spectators in the balcony began to interfere with shots. The backboard was introduced to prevent this interference; it had the additional effect of allowing rebound shots. Naismith\'s handwritten diaries, discovered by his granddaughter in early 2006, indicate that he was nervous about the new game he had invented, which incorporated rules from a children\'s game called duck on a rock, as many had failed before it.
Frank Mahan, one of the players from the original first game, approached Naismith after the Christmas break, in early 1892, asking him what he intended to call his new game. Naismith replied that he had not thought of it because he had been focused on just getting the game started. Mahan suggested that it be called \"Naismith ball\", at which he laughed, saying that a name like that would kill any game. Mahan then said, \"Why not call it basketball?\" Naismith replied, \"We have a basket and a ball, and it seems to me that would be a good name for it.\" The first official game was played in the YMCA gymnasium in Albany, New York, on January 20, 1892, with nine players. The game ended at 1--0; the shot was made from 25 ft, on a court just half the size of a present-day Streetball or National Basketball Association (NBA) court.
At the time, soccer was being played with 10 to a team (which was increased to 11). When winter weather got too icy to play soccer, teams were taken indoors, and it was convenient to have them split in half and play basketball with five on each side. By 1897--98, teams of five became standard.
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# Basketball
## History
### College basketball {#college_basketball}
Basketball\'s early adherents were dispatched to YMCAs throughout the United States, and it quickly spread through the United States and Canada. By 1895, it was well established at several women\'s high schools. While YMCA was responsible for initially developing and spreading the game, within a decade it discouraged the new sport, as rough play and rowdy crowds began to detract from YMCA\'s primary mission. However, other amateur sports clubs, colleges, and professional clubs quickly filled the void. In the years before World War I, the Amateur Athletic Union and the Intercollegiate Athletic Association of the United States (forerunner of the NCAA) vied for control over the rules for the game. The first pro league, the National Basketball League, was formed in 1898 to protect players from exploitation and to promote a less rough game. This league only lasted five years.
James Naismith was instrumental in establishing college basketball. His colleague C. O. Beamis fielded the first college basketball team just a year after the Springfield YMCA game at the suburban Pittsburgh Geneva College. Naismith himself later coached at the University of Kansas for six years, before handing the reins to renowned coach Forrest \"Phog\" Allen. Naismith\'s disciple Amos Alonzo Stagg brought basketball to the University of Chicago, while Adolph Rupp, a student of Naismith\'s at Kansas, enjoyed great success as coach at the University of Kentucky. On February 9, 1895, the first intercollegiate 5-on-5 game was played at Hamline University between Hamline and the School of Agriculture, which was affiliated with the University of Minnesota. The School of Agriculture won in a 9--3 game.
In 1901, colleges, including the University of Chicago, Columbia University, Cornell University, Dartmouth College, the University of Minnesota, the U.S. Naval Academy, the University of Colorado and Yale University began sponsoring men\'s games. In 1905, frequent injuries on the football field prompted President Theodore Roosevelt to suggest that colleges form a governing body, resulting in the creation of the Intercollegiate Athletic Association of the United States (IAAUS). In 1910, that body changed its name to the National Collegiate Athletic Association (NCAA). The first Canadian interuniversity basketball game was played at YMCA in Kingston, Ontario on February 6, 1904, when McGill University`{{snd}}`{=mediawiki}Naismith\'s alma mater`{{snd}}`{=mediawiki}visited Queen\'s University. McGill won 9--7 in overtime; the score was 7--7 at the end of regulation play, and a ten-minute overtime period settled the outcome. A good turnout of spectators watched the game.
The first men\'s national championship tournament, the National Association of Intercollegiate Basketball tournament, which still exists as the National Association of Intercollegiate Athletics (NAIA) tournament, was organized in 1937. The first national championship for NCAA teams, the National Invitation Tournament (NIT) in New York, was organized in 1938; the NCAA national tournament began one year later. College basketball was rocked by gambling scandals from 1948 to 1951, when dozens of players from top teams were implicated in game-fixing and point shaving. Partially spurred by an association with cheating, the NIT lost support to the NCAA tournament.
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# Basketball
## History
### High school basketball {#high_school_basketball}
Before widespread school district consolidation, most American high schools were far smaller than their present-day counterparts. During the first decades of the 20th century, basketball quickly became the ideal interscholastic sport due to its modest equipment and personnel requirements. In the days before widespread television coverage of professional and college sports, the popularity of high school basketball was unrivaled in many parts of America. Perhaps the most legendary of high school teams was Indiana\'s Franklin Wonder Five, which took the nation by storm during the 1920s, dominating Indiana basketball and earning national recognition.
Today virtually every high school in the United States fields a basketball team in varsity competition. Basketball\'s popularity remains high, both in rural areas where they carry the identification of the entire community, as well as at some larger schools known for their basketball teams where many players go on to participate at higher levels of competition after graduation. In the 2016--17 season, 980,673 boys and girls represented their schools in interscholastic basketball competition, according to the National Federation of State High School Associations. The states of Illinois, Indiana and Kentucky are particularly well known for their residents\' devotion to high school basketball, commonly called Hoosier Hysteria in Indiana; the critically acclaimed film *Hoosiers* shows high school basketball\'s depth of meaning to these communities.
There is currently no tournament to determine a national high school champion. The most serious effort was the National Interscholastic Basketball Tournament at the University of Chicago from 1917 to 1930. The event was organized by Amos Alonzo Stagg and sent invitations to state champion teams. The tournament started out as a mostly Midwest affair but grew. In 1929 it had 29 state champions. Faced with opposition from the National Federation of State High School Associations and North Central Association of Colleges and Schools that bore a threat of the schools losing their accreditation the last tournament was in 1930. The organizations said they were concerned that the tournament was being used to recruit professional players from the prep ranks. The tournament did not invite minority schools or private/parochial schools.
The National Catholic Interscholastic Basketball Tournament ran from 1924 to 1941 at Loyola University. The National Catholic Invitational Basketball Tournament from 1954 to 1978 played at a series of venues, including Catholic University, Georgetown and George Mason. The National Interscholastic Basketball Tournament for Black High Schools was held from 1929 to 1942 at Hampton Institute. The National Invitational Interscholastic Basketball Tournament was held from 1941 to 1967 starting out at Tuskegee Institute. Following a pause during World War II it resumed at Tennessee State College in Nashville. The basis for the champion dwindled after 1954 when *Brown v. Board of Education* began an integration of schools. The last tournaments were held at Alabama State College from 1964 to 1967.
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# Basketball
## History
### Professional basketball {#professional_basketball}
Teams abounded throughout the 1920s. There were hundreds of men\'s professional basketball teams in towns and cities all over the United States, and little organization of the professional game. Players jumped from team to team and teams played in armories and smoky dance halls. Leagues came and went. Barnstorming squads such as the Original Celtics and two all-African American teams, the New York Renaissance Five (\"Rens\") and the (still existing) Harlem Globetrotters played up to two hundred games a year on their national tours.
In 1946, the Basketball Association of America (BAA) was formed. The first game was played in Toronto, Ontario, Canada between the Toronto Huskies and New York Knickerbockers on November 1, 1946. Three seasons later, in 1949, the BAA merged with the National Basketball League (NBL) to form the National Basketball Association (NBA). By the 1950s, basketball had become a major college sport, thus paving the way for a growth of interest in professional basketball. In 1959, a basketball hall of fame was founded in Springfield, Massachusetts, site of the first game. Its rosters include the names of great players, coaches, referees and people who have contributed significantly to the development of the game. The hall of fame has people who have accomplished many goals in their career in basketball. An upstart organization, the American Basketball Association, emerged in 1967 and briefly threatened the NBA\'s dominance until the ABA-NBA merger in 1976. Today the NBA is the top professional basketball league in the world in terms of popularity, salaries, talent, and level of competition.
The NBA has featured many famous players, including George Mikan, the first dominating \"big man\"; ball-handling wizard Bob Cousy and defensive genius Bill Russell of the Boston Celtics; charismatic center Wilt Chamberlain, who originally played for the barnstorming Harlem Globetrotters; all-around stars Oscar Robertson and Jerry West; more recent big men Kareem Abdul-Jabbar, Shaquille O\'Neal, Hakeem Olajuwon and Karl Malone; playmakers John Stockton, Isiah Thomas and Steve Nash; crowd-pleasing forwards Julius Erving and Charles Barkley; European stars Dirk Nowitzki, Pau Gasol, Nikola Jokić and Tony Parker; Latin American stars Manu Ginobili, more recent superstars, Allen Iverson, Kobe Bryant, Tim Duncan, LeBron James, Stephen Curry, Giannis Antetokounmpo, etc.; and the three players who many credit with ushering the professional game to its highest level of popularity during the 1980s and 1990s: Larry Bird, Earvin \"Magic\" Johnson, and Michael Jordan.
In 2001, the NBA formed a developmental league, the National Basketball Development League (later known as the NBA D-League and then the NBA G League after a branding deal with Gatorade). As of the 2023--24 season, the G League has 31 teams.
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# Basketball
## History
### International basketball {#international_basketball}
FIBA (International Basketball Federation) was formed in 1932 by eight founding nations: Argentina, Czechoslovakia, Greece, Italy, Latvia, Portugal, Romania and Switzerland. At this time, the organization only oversaw amateur players. Its acronym, derived from the French *Fédération Internationale de Basket-ball Amateur*, was thus \"FIBA\". Men\'s basketball was first included at the Berlin 1936 Summer Olympics, although a demonstration tournament was held in 1904. The United States defeated Canada in the first final, played outdoors. This competition has usually been dominated by the United States, whose team has won all but three titles. The first of these came in a controversial final game in Munich in 1972 against the Soviet Union, in which the ending of the game was replayed three times until the Soviet Union finally came out on top. In 1950 the first FIBA World Championship for men, now known as the FIBA Basketball World Cup, was held in Argentina. Three years later, the first FIBA World Championship for women, now known as the FIBA Women\'s Basketball World Cup, was held in Chile. Women\'s basketball was added to the Olympics in 1976, which were held in Montreal, Quebec, Canada with teams such as the Soviet Union, Brazil and Australia rivaling the American squads.
In 1989, FIBA allowed professional NBA players to participate in the Olympics for the first time. Prior to the 1992 Summer Olympics, only European and South American teams were allowed to field professionals in the Olympics. The United States\' dominance continued with the introduction of the original Dream Team. In the 2004 Athens Olympics, the United States suffered its first Olympic loss while using professional players, falling to Puerto Rico (in a 19-point loss) and Lithuania in group games, and being eliminated in the semifinals by Argentina. It eventually won the bronze medal defeating Lithuania, finishing behind Argentina and Italy. The Redeem Team, won gold at the 2008 Olympics, and the B-Team, won gold at the 2010 FIBA World Championship in Turkey despite featuring no players from the 2008 squad. The United States continued its dominance as they won gold at the 2012 Olympics, 2014 FIBA World Cup and the 2016 Olympics.
Worldwide, basketball tournaments are held for boys and girls of all age levels. The global popularity of the sport is reflected in the nationalities represented in the NBA. Players from all six inhabited continents currently play in the NBA. Top international players began coming into the NBA in the mid-1990s, including Croatians Dražen Petrović and Toni Kukoč, Serbian Vlade Divac, Lithuanians Arvydas Sabonis and Šarūnas Marčiulionis, Dutchman Rik Smits and German Detlef Schrempf.
In the Philippines, the Philippine Basketball Association\'s first game was played on April 9, 1975, at the Araneta Coliseum in Cubao, Quezon City, Philippines. It was founded as a \"rebellion\" of several teams from the now-defunct Manila Industrial and Commercial Athletic Association, which was tightly controlled by the Basketball Association of the Philippines (now defunct), the then-FIBA recognized national association. Nine teams from the MICAA participated in the league\'s first season that opened on April 9, 1975. The NBL is Australia\'s pre-eminent men\'s professional basketball league. The league commenced in 1979, playing a winter season (April--September) and did so until the completion of the 20th season in 1998. The 1998--99 season, which commenced only months later, was the first season after the shift to the current summer season format (October--April). This shift was an attempt to avoid competing directly against Australia\'s various football codes. It features 8 teams from around Australia and one in New Zealand. A few players including Luc Longley, Andrew Gaze, Shane Heal, Chris Anstey and Andrew Bogut made it big internationally, becoming poster figures for the sport in Australia. The Women\'s National Basketball League began in 1981.
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# Basketball
## History
### Women\'s basketball {#womens_basketball}
upright=1.8\|thumb\|The Australian women\'s basketball team on winning the 2006 FIBA World Championship for Women
Women began to play basketball in the fall of 1892 at Smith College through Senda Berenson, substitute director of the newly opened gymnasium and physical education teacher, after having modified the rules for women. Shortly after Berenson was hired at Smith, she visited Naismith to learn more about the game. Fascinated by the new sport and the values it could teach, she instantly introduced the game as a class exercise and soon after teams were organized. The first women\'s collegiate basketball game was played on March 21, 1893, when her Smith freshmen and sophomores played against one another. The first official women\'s interinstitutional game was played later that year between the University of California and the Miss Head\'s School. In 1899, a committee was established at the Conference of Physical Training in Springfield to draw up general rules for women\'s basketball. These rules, designed by Berenson, were published in 1899. In 1902 Berenson became the editor of A. G. Spalding\'s first Women\'s Basketball Guide. The same year women of Mount Holyoke and Sophie Newcomb College (coached by Clara Gregory Baer), began playing basketball. By 1895, the game had spread to colleges across the country, including Wellesley, Vassar, and Bryn Mawr. The first intercollegiate women\'s game was on April 4, 1896. Stanford women played Berkeley, 9-on-9, ending in a 2--1 Stanford victory.
Women\'s basketball development was more structured than that for men in the early years. In 1905, the executive committee on Basket Ball Rules (National Women\'s Basketball Committee) was created by the American Physical Education Association. These rules called for six to nine players per team and 11 officials. The International Women\'s Sports Federation (1924) included a women\'s basketball competition. 37 women\'s high school varsity basketball or state tournaments were held by 1925. And in 1926, the Amateur Athletic Union backed the first national women\'s basketball championship, complete with men\'s rules. The Edmonton Grads, a touring Canadian women\'s team based in Edmonton, Alberta, operated between 1915 and 1940. The Grads toured all over North America, and were exceptionally successful. They posted a record of 522 wins and only 20 losses over that span, as they met any team that wanted to challenge them, funding their tours from gate receipts. The Grads also shone on several exhibition trips to Europe, and won four consecutive exhibition Olympics tournaments, in 1924, 1928, 1932, and 1936; however, women\'s basketball was not an official Olympic sport until 1976. The Grads\' players were unpaid, and had to remain single. The Grads\' style focused on team play, without overly emphasizing skills of individual players. The first women\'s AAU All-America team was chosen in 1929. Women\'s industrial leagues sprang up throughout the United States, producing famous athletes, including Babe Didrikson of the Golden Cyclones, and the All American Red Heads Team, which competed against men\'s teams, using men\'s rules. By 1938, the women\'s national championship changed from a three-court game to two-court game with six players per team. The NBA-backed Women\'s National Basketball Association (WNBA) began in 1997. Though it had shaky attendance figures, several marquee players (Lisa Leslie, Diana Taurasi, and Candace Parker among others) have helped the league\'s popularity and level of competition. Other professional women\'s basketball leagues in the United States, such as the American Basketball League (1996--98), have folded in part because of the popularity of the WNBA. The WNBA has been looked at by many as a niche league. However, the league has recently taken steps forward. In June 2007, the WNBA signed a contract extension with ESPN. The new television deal ran from 2009 to 2016. Along with this deal, came the first-ever rights fees to be paid to a women\'s professional sports league. Over the eight years of the contract, \"millions and millions of dollars\" were \"dispersed to the league\'s teams.\" In a March 12, 2009, article, NBA commissioner David Stern said that in the bad economy, \"the NBA is far less profitable than the WNBA. We\'re losing a lot of money among a large number of teams. We\'re budgeting the WNBA to break even this year.\"
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# Basketball
## Rules and regulations {#rules_and_regulations}
*Main article: Rules of basketball*
Measurements and time limits discussed in this section often vary among tournaments and organizations; international and NBA rules are used in this section.
The object of the game is to outscore one\'s opponents by throwing the ball through the opponents\' basket from above while preventing the opponents from doing so on their own. An attempt to score in this way is called a shot. A successful shot is worth two points, or three points if it is taken from beyond the three-point arc 6.75 m from the basket in international games and 23 ft in NBA games. A one-point shot can be earned when shooting from the foul line after a foul is made. After a team has scored from a field goal or free throw, play is resumed with a **throw-in** awarded to the non-scoring team taken from a point beyond the endline of the court where the points were scored.
### Playing regulations {#playing_regulations}
Games are played in four quarters of 10 (FIBA) or 12 minutes (NBA). College men\'s games use two 20-minute halves, college women\'s games use 10-minute quarters, and most United States high school varsity games use 8-minute quarters; however, this varies from state to state. 15 minutes are allowed for a half-time break under FIBA, NBA, and NCAA rules and 10 minutes in United States high schools. Overtime periods are five minutes in length except for high school, which is four minutes in length. Teams exchange baskets for the second half. The time allowed is actual playing time; the clock is stopped while the play is not active. Therefore, games generally take much longer to complete than the allotted game time, typically about two hours.
Five players from each team may be on the court at one time. Substitutions are unlimited but can only be done when play is stopped. Teams also have a coach, who oversees the development and strategies of the team, and other team personnel such as assistant coaches, managers, statisticians, doctors and trainers.
For both men\'s and women\'s teams, a standard uniform consists of a pair of shorts and a jersey with a clearly visible number, unique within the team, printed on both the front and back. Players wear high-top sneakers that provide extra ankle support. Typically, team names, players\' names and, outside of North America, sponsors are printed on the uniforms.
A limited number of time-outs, clock stoppages requested by a coach (or sometimes mandated in the NBA) for a short meeting with the players, are allowed. They generally last no longer than one minute (100 seconds in the NBA) unless, for televised games, a commercial break is needed.
The game is controlled by the officials consisting of the referee (referred to as crew chief in the NBA), one or two umpires (referred to as referees in the NBA) and the table officials. For college, the NBA, and many high schools, there are a total of three referees on the court. The table officials are responsible for keeping track of each team\'s scoring, timekeeping, individual and team fouls, player substitutions, team possession arrow, and the shot clock.
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# Basketball
## Rules and regulations {#rules_and_regulations}
### Equipment
The only essential equipment in a basketball game is the ball and the court: a flat, rectangular surface with baskets at opposite ends. Competitive levels require the use of more equipment such as clocks, score sheets, scoreboards, alternating possession arrows, and whistle-operated stop-clock systems.
A regulation basketball court in international games is 28 m long and 15 m wide. In the NBA and NCAA the court is 94 by. Most courts have wood flooring, usually constructed from maple planks running in the same direction as the longer court dimension. The name and logo of the home team is usually painted on or around the center circle.
The basket is a steel rim 18 in diameter with an attached net affixed to a backboard that measures 6 by and one basket is at each end of the court. The white outlined box on the backboard is 18 in high and 2 ft wide. At almost all levels of competition, the top of the rim is exactly 10 ft above the court and 4 ft inside the baseline. While variation is possible in the dimensions of the court and backboard, it is considered important for the basket to be of the correct height -- a rim that is off by just a few inches can have an adverse effect on shooting. The net must \"check the ball momentarily as it passes through the basket\" to aid the visual confirmation that the ball went through. The act of checking the ball has the further advantage of slowing down the ball so the rebound does not go as far.
The size of the basketball is also regulated. For men, the official ball is 29.5 in in circumference (size 7, or a \"295 ball\") and weighs 22 oz. If women are playing, the official basketball size is 28.5 in in circumference (size 6, or a \"285 ball\") with a weight of 20 oz. In 3x3, a formalized version of the halfcourt 3-on-3 game, a dedicated ball with the circumference of a size 6 ball but the weight of a size 7 ball is used in all competitions (men\'s, women\'s, and mixed teams).
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# Basketball
## Rules and regulations {#rules_and_regulations}
### Violations
The ball may be advanced toward the basket by being shot, passed between players, thrown, tapped, rolled or dribbled (bouncing the ball while running).
The ball must stay within the court; the last team to touch the ball before it travels out of bounds forfeits possession. The ball is out of bounds if it touches a boundary line, or touches any player or object that is out of bounds.
There are limits placed on the steps a player may take without dribbling, which commonly results in an infraction known as traveling. Nor may a player stop their dribble and then resume dribbling. A dribble that touches both hands is considered stopping the dribble, giving this infraction the name double dribble. Within a dribble, the player cannot carry the ball by placing their hand on the bottom of the ball; doing so is known as carrying the ball. A team, once having established ball control in the front half of their court, may not return the ball to the backcourt and be the first to touch it. A violation of these rules results in loss of possession.
The ball may not be kicked, nor be struck with the fist. For the offense, a violation of these rules results in loss of possession; for the defense, most leagues reset the shot clock and the offensive team is given possession of the ball out of bounds.
There are limits imposed on the time taken before progressing the ball past halfway (8 seconds in FIBA and the NBA; 10 seconds in NCAA and high school for both sexes), before attempting a shot (24 seconds in FIBA, the NBA, and U Sports (Canadian universities) play for both sexes, and 30 seconds in NCAA play for both sexes), holding the ball while closely guarded (5 seconds), and remaining in the restricted area known as the free-throw lane, (or the \"key\") (3 seconds). These rules are designed to promote more offense.
There are also limits on how players may block an opponent\'s field goal attempt or help a teammate\'s field goal attempt. Goaltending is a defender\'s touching of a ball that is on a downward flight toward the basket, while the related violation of basket interference is the touching of a ball that is on the rim or above the basket, or by a player reaching through the basket from below. Goaltending and basket interference committed by a defender result in awarding the basket to the offense, while basket interference committed by an offensive player results in cancelling the basket if one is scored. The defense gains possession in all cases of goaltending or basket interference.
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# Basketball
## Rules and regulations {#rules_and_regulations}
### Fouls
*Main article: Personal foul (basketball), Technical foul* An attempt to unfairly disadvantage an opponent through certain types of physical contact is illegal and is called a personal foul. These are most commonly committed by defensive players; however, they can be committed by offensive players as well. Players who are fouled either receive the ball to pass inbounds again, or receive one or more free throws if they are fouled in the act of shooting, depending on whether the shot was successful. One point is awarded for making a free throw, which is attempted from a line 15 ft from the basket.
The referee is responsible for judging whether contact is illegal, sometimes resulting in controversy. The calling of fouls can vary between games, leagues and referees.
There is a second category of fouls called technical fouls, which may be charged for various rules violations including failure to properly record a player in the scorebook, or for unsportsmanlike conduct. These infractions result in one or two free throws, which may be taken by any of the five players on the court at the time. Repeated incidents can result in disqualification. A blatant foul involving physical contact that is either excessive or unnecessary is called an intentional foul (flagrant foul in the NBA). In FIBA and NCAA women\'s basketball, a foul resulting in ejection is called a disqualifying foul, while in leagues other than the NBA, such a foul is referred to as flagrant.
If a team exceeds a certain limit of team fouls in a given period (quarter or half) -- four for NBA, NCAA women\'s, and international games -- the opposing team is awarded one or two free throws on all subsequent non-shooting fouls for that period, the number depending on the league. In the US college men\'s game and high school games for both sexes, if a team reaches 7 fouls in a half, the opposing team is awarded one free throw, along with a second shot if the first is made. This is called shooting \"one-and-one\". If a team exceeds 10 fouls in the half, the opposing team is awarded two free throws on all subsequent fouls for the half.
When a team shoots foul shots, the opponents may not interfere with the shooter, nor may they try to regain possession until the last or potentially last free throw is in the air.
After a team has committed a specified number of fouls, the other team is said to be \"in the bonus\". On scoreboards, this is usually signified with an indicator light reading \"Bonus\" or \"Penalty\" with an illuminated directional arrow or dot indicating that team is to receive free throws when fouled by the opposing team. (Some scoreboards also indicate the number of fouls committed.)
If a team misses the first shot of a two-shot situation, the opposing team must wait for the completion of the second shot before attempting to reclaim possession of the ball and continuing play.
If a player is fouled while attempting a shot and the shot is unsuccessful, the player is awarded a number of free throws equal to the value of the attempted shot. A player fouled while attempting a regular two-point shot thus receives two shots, and a player fouled while attempting a three-point shot receives three shots.
If a player is fouled while attempting a shot and the shot is successful, typically the player will be awarded one additional free throw for one point. In combination with a regular shot, this is called a \"three-point play\" or \"four-point play\" (or more colloquially, an \"and one\") because of the basket made at the time of the foul (2 or 3 points) and the additional free throw (1 point).
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# Basketball
## Common techniques and practices {#common_techniques_and_practices}
### Positions
thumb\|upright=1.15\|Basketball positions in the offensive zone Although the rules do not specify any positions whatsoever, they have evolved as part of basketball. During the early years of basketball\'s evolution, two guards, two forwards, and one center were used. In more recent times specific positions evolved, but the current trend, advocated by many top coaches including Mike Krzyzewski, is towards positionless basketball, where big players are free to shoot from outside and dribble if their skill allows it. Popular descriptions of positions include:
Point guard (often called the \"**1**\") : usually the fastest player on the team, organizes the team\'s offense by controlling the ball and making sure that it gets to the right player at the right time.
Shooting guard (the \"**2**\") : creates a high volume of shots on offense, mainly long-ranged; and guards the opponent\'s best perimeter player on defense.
Small forward (the \"**3**\") : often primarily responsible for scoring points via cuts to the basket and dribble penetration; on defense seeks rebounds and steals, but sometimes plays more actively.
Power forward (the \"**4**\"): plays offensively often with their back to the basket; on defense, plays under the basket (in a zone defense) or against the opposing power forward (in man-to-man defense).
Center (the \"**5**\"): uses height and size to score (on offense), to protect the basket closely (on defense), or to rebound.
The above descriptions are flexible. For most teams today, the shooting guard and small forward have very similar responsibilities and are often called **the wings**, as do the power forward and center, who are often called **post players.** While most teams describe two players as guards, two as forwards, and one as a center, on some occasions teams choose to call them by different designations.
### Strategy
There are two main defensive strategies: *zone defense* and *man-to-man defense*. In a zone defense, each player is assigned to guard a specific area of the court. Zone defenses often allow the defense to double team the ball, a manoeuver known as a **trap**. In a man-to-man defense, each defensive player guards a specific opponent.
Offensive plays are more varied, normally involving planned passes and movement by players without the ball. A quick movement by an offensive player without the ball to gain an advantageous position is known as a *cut*. A legal attempt by an offensive player to stop an opponent from guarding a teammate, by standing in the defender\'s way such that the teammate cuts next to him, is a *screen* or *pick*. The two plays are combined in the *pick and roll*, in which a player sets a pick and then \"rolls\" away from the pick towards the basket. Screens and cuts are very important in offensive plays; these allow the quick passes and teamwork, which can lead to a successful basket. Teams almost always have several offensive plays planned to ensure their movement is not predictable. On court, the point guard is usually responsible for indicating which play will occur.
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# Basketball
## Common techniques and practices {#common_techniques_and_practices}
### Shooting
Shooting is the act of attempting to score points by throwing the ball through the basket, methods varying with players and situations.
Typically, a player faces the basket with both feet facing the basket. A player will rest the ball on the fingertips of the dominant hand (the shooting arm) slightly above the head, with the other hand supporting the side of the ball. The ball is usually shot by jumping (though not always) and extending the shooting arm. The shooting arm, fully extended with the wrist fully bent, is held stationary for a moment following the release of the ball, known as a *follow-through*. Players often try to put a steady backspin on the ball to absorb its impact with the rim. The ideal trajectory of the shot is somewhat controversial, but generally a proper arc is recommended. Players may shoot directly into the basket or may use the backboard to redirect the ball into the basket.
The two most common shots that use the above described setup are the *set shot* and the *jump shot*. Both are preceded by a crouching action which preloads the muscles and increases the power of the shot. In a set shot, the shooter straightens up and throws from a standing position with neither foot leaving the floor; this is typically used for free throws. For a jump shot, the throw is taken in mid-air with the ball being released near the top of the jump. This provides much greater power and range, and it also allows the player to elevate over the defender. Failure to release the ball before the feet return to the floor is considered a traveling violation.
Another common shot is called the *layup*. This shot requires the player to be in motion toward the basket, and to \"lay\" the ball \"up\" and into the basket, typically off the backboard (the backboard-free, underhand version is called a *finger roll*). The most crowd-pleasing and typically highest-percentage accuracy shot is the *slam dunk*, in which the player jumps very high and throws the ball downward, through the basket while touching it.
Another shot that is less common than the layup, is the \"circus shot\". The circus shot is a low-percentage shot that is flipped, heaved, scooped, or flung toward the hoop while the shooter is off-balance, airborne, falling down or facing away from the basket. A back-shot is a shot taken when the player is facing away from the basket, and may be shot with the dominant hand, or both; but there is a very low chance that the shot will be successful.
A shot that misses both the rim and the backboard completely is referred to as an *air ball*. A particularly bad shot, or one that only hits the backboard, is jocularly called a brick. The *hang time* is the length of time a player stays in the air after jumping, either to make a slam dunk, layup or jump shot.
### Rebounding
The objective of rebounding is to successfully gain possession of the basketball after a missed field goal or free throw, as it rebounds from the hoop or backboard. This plays a major role in the game, as most possessions end when a team misses a shot. There are two categories of rebounds: offensive rebounds, in which the ball is recovered by the offensive side and does not change possession, and defensive rebounds, in which the defending team gains possession of the loose ball. The majority of rebounds are defensive, as the team on defense tends to be in better position to recover missed shots; for example, about 75% of rebounds in the NBA are defensive.
### Passing
A pass is a method of moving the ball between players. Most passes are accompanied by a step forward to increase power and are followed through with the hands to ensure accuracy.
A staple pass is the *chest pass*. The ball is passed directly from the passer\'s chest to the receiver\'s chest. A proper chest pass involves an outward snap of the thumbs to add velocity and leaves the defence little time to react.
Another type of pass is the *bounce pass*. Here, the passer bounces the ball crisply about two-thirds of the way from his own chest to the receiver. The ball strikes the court and bounces up toward the receiver. The bounce pass takes longer to complete than the chest pass, but it is also harder for the opposing team to intercept (kicking the ball deliberately is a violation). Thus, players often use the bounce pass in crowded moments, or to pass around a defender.
The *overhead pass* is used to pass the ball over a defender. The ball is released while over the passer\'s head.
The *outlet pass* occurs after a team gets a defensive rebound. The next pass after the rebound is the *outlet pass*.
The crucial aspect of any good pass is it being difficult to intercept. Good passers can pass the ball with great accuracy and they know exactly where each of their other teammates prefers to receive the ball. A special way of doing this is passing the ball without looking at the receiving teammate. This is called a *no-look pass*.
Another advanced style of passing is the *behind-the-back pass*, which, as the description implies, involves throwing the ball behind the passer\'s back to a teammate. Although some players can perform such a pass effectively, many coaches discourage no-look or behind-the-back passes, believing them to be difficult to control and more likely to result in turnovers or violations.
### Dribbling
thumb\|right\|upright=1.45\|A demonstration of the basic types of dribbling in basketball *Main article: Dribble* Dribbling is the act of bouncing the ball continuously with one hand and is a requirement for a player to take steps with the ball. To dribble, a player pushes the ball down towards the ground with the fingertips rather than patting it; this ensures greater control.
When dribbling past an opponent, the dribbler should dribble with the hand farthest from the opponent, making it more difficult for the defensive player to get to the ball. It is therefore important for a player to be able to dribble competently with both hands.
Good dribblers (or \"ball handlers\") tend to keep their dribbling hand low to the ground, reducing the distance of travel of the ball from the floor to the hand, making it more difficult for the defender to \"steal\" the ball. Good ball handlers frequently dribble behind their backs, between their legs, and switch directions suddenly, making a less predictable dribbling pattern that is more difficult to defend against. This is called a crossover, which is the most effective way to move past defenders while dribbling.
A skilled player can dribble without watching the ball, using the dribbling motion or peripheral vision to keep track of the ball\'s location. By not having to focus on the ball, a player can look for teammates or scoring opportunities, as well as avoid the danger of having someone steal the ball away from him/her.
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# Basketball
## Common techniques and practices {#common_techniques_and_practices}
### Blocking
A block is performed when, after a shot is attempted, a defender succeeds in altering the shot by touching the ball. In almost all variants of play, it is illegal to touch the ball after it is in the downward path of its arc; this is known as *goaltending*. It is also illegal under NBA and Men\'s NCAA basketball to block a shot after it has touched the backboard, or when any part of the ball is directly above the rim. Under international rules it is illegal to block a shot that is in the downward path of its arc or one that has touched the backboard until the ball has hit the rim. After the ball hits the rim, it is again legal to touch it even though it is no longer considered as a block performed.
To block a shot, a player has to be able to reach a point higher than where the shot is released. Thus, height can be an advantage in blocking. Players who are taller and playing the power forward or center positions generally record more blocks than players who are shorter and playing the guard positions. However, with good timing and a sufficiently high vertical leap, even shorter players can be effective shot blockers.
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# Basketball
## Height
At the professional level, most male players are above 6 ft and most women above 5 ft. Guards, for whom physical coordination and ball-handling skills are crucial, tend to be the smallest players. Almost all forwards in the top men\'s pro leagues are 6 ft or taller. Most centers are over 6 ft tall. According to a survey given to all NBA teams,`{{when|date=July 2015}}`{=mediawiki} the average height of all NBA players is just under 6 ft, with the average weight being close to 222 lb. The tallest players ever in the NBA were Manute Bol and Gheorghe Mureșan, who were both 7 ft tall. At 7 ft, Margo Dydek was the tallest player in the history of the WNBA.
The shortest player ever to play in the NBA is Muggsy Bogues at 5 ft. Other average-height or relatively short players have thrived at the pro level, including Anthony \"Spud\" Webb, who was 5 ft tall, but had a 42 in vertical leap, giving him significant height when jumping, and Temeka Johnson, who won the WNBA Rookie of the Year Award and a championship with the Phoenix Mercury while standing only 5 ft. While shorter players are often at a disadvantage in certain aspects of the game, their ability to navigate quickly through crowded areas of the court and steal the ball by reaching low are strengths.
Players regularly inflate their height in high school or college. Many prospects exaggerate their height while in high school or college to make themselves more appealing to coaches and scouts, who prefer taller players. Charles Barkley stated; \"I\'ve been measured at 6--5, 6-`{{frac|4|3|4}}`{=mediawiki}. But I started in college at 6--6.\" Sam Smith, a former writer from the *Chicago Tribune*, said: \"We sort of know the heights, because after camp, the sheet comes out. But you use that height, and the player gets mad. And then you hear from his agent. Or you file your story with the right height, and the copy desk changes it because they have the \'official\' N.B.A. media guide, which is wrong. So you sort of go along with the joke.\"
Since the 2019--20 NBA season heights of NBA players are recorded definitively by measuring players with their shoes off.
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# Basketball
## Variations and similar games {#variations_and_similar_games}
Variations of basketball are activities based on the game of basketball, using common basketball skills and equipment (primarily the ball and basket). Some variations only have superficial rule changes, while others are distinct games with varying degrees of influence from basketball. Other variations include children\'s games, contests or activities meant to help players reinforce skills.
An earlier version of basketball, played primarily by women and girls, was six-on-six basketball. Horseball is a game played on horseback where a ball is handled and points are scored by shooting it through a high net (approximately 1.5m×1.5m). The sport is like a combination of polo, rugby, and basketball. There is even a form played on donkeys known as Donkey basketball, which has attracted criticism from animal rights groups.
### Half-court {#half_court}
Perhaps the single most common variation of basketball is the half-court game, played in informal settings without referees or strict rules. Only one basket is used, and the ball must be \"taken back\" or \"cleared\" -- passed or dribbled outside the three-point line each time possession of the ball changes from one team to the other. Half-court games require less cardiovascular stamina, since players need not run back and forth a full court. Half-court raises the number of players that can use a court or, conversely, can be played if there is an insufficient number to form full 5-on-5 teams.
Half-court basketball is usually played 1-on-1, 2-on-2 or 3-on-3. The last of these variations is gradually gaining official recognition as 3x3, originally known as FIBA 33. It was first tested at the 2007 Asian Indoor Games in Macau and the first official tournaments were held at the 2009 Asian Youth Games and the 2010 Youth Olympics, both in Singapore. The first FIBA 3x3 Youth World Championships were held in Rimini, Italy in 2011, with the first FIBA 3x3 World Championships for senior teams following a year later in Athens. The sport is highly tipped to become an Olympic sport as early as 2016. In the summer of 2017, the BIG3 basketball league, a professional 3x3 half court basketball league that features former NBA players, began. The BIG3 features several rule variants including a four-point field goal.
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# Basketball
## Variations and similar games {#variations_and_similar_games}
### Other variations {#other_variations}
Variations of basketball with their own page or subsection include:
- **21** (also known as **American**, **cutthroat** and **roughhouse**)
```{=html}
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```
- **42**
- **Around the World**
- **Bounce**
- **Firing Squad**
- **Fives**
- **H-O-R-S-E**
- **Hotshot**
- **Knockout**
- **One-shot conquer**
- **Steal The Bacon**
- **Tip-it**
- **Tips**
- **\"The One\"**
- **Basketball War**
- **Water basketball**
- **Beach basketball**
- **Streetball**
```{=html}
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```
- **One-on-one** is a variation in which two players will use only a small section of the court (often no more than a half of a court) and compete to play the ball into a single hoop. Such games tend to emphasize individual dribbling and ball stealing skills over shooting and team play.
- **Dunk Hoops** is a variation played on basketball hoops with lowered (under basketball regulation 10 feet) rims. It originated when the popularity of the slam dunk grew and was developed to create better chances for dunks with lowered rims and using altered goaltending rules.
- **Unicycle basketball** is played using a regulation basketball on a regular basketball court with the same rules, for example, one must dribble the ball while riding. There are a number of rules that are particular to unicycle basketball as well, for example, a player must have at least one foot on a pedal when in-bounding the ball. Unicycle basketball is usually played using 24\" or smaller unicycles, and using plastic pedals, both to preserve the court and the players\' shins. Popular unicycle basketball games are organized in North America.
Spin-offs from basketball that are now separate sports include:
- **Ringball**, a traditional South African sport that stems from basketball, has been played since 1907. The sport is now promoted in South Africa, Namibia, Botswana, Lesotho, India, and Mauritius to establish Ringball as an international sport.
- **Korfball** (Dutch: Korfbal, *korf* meaning \'basket\') started in the Netherlands and is now played worldwide as a mixed-gender team ball game, similar to mixed netball and basketball.
- **Netball** is a limited-contact team sport in which two teams of seven try to score points against one another by placing a ball through a high hoop. Australia New Zealand champions (so called ANZ Championship) is very famous in Australia and New Zealand as the premier netball league. Formerly played exclusively by women, netball today features mixed-gender competitions.
- **Slamball**, invented by television writer Mason Gordon, is a full-contact sport featuring trampolines. The main difference from basketball is the court; below the padded rim and backboard are four trampolines set into the floor, which serve to propel players to great heights for slam dunks. The rules also permit some physical contact between the members of the four-player teams. Professional games of Slamball aired on Spike TV in 2002, and the sport has since expanded to China and other countries.
<File:Dan> Hadani collection (990044347560205171).jpg\|A basketball player in Israel, 1969 <File:Girls> play basketball in Dharmsala, India.jpg\|Schoolgirls shooting hoops among the Himalayas in Dharamsala, India. <File:Sân> trường THPT Phan Đình Phùng, Hà Nội.JPG\|A basketball training course at the Phan Đình Phùng High School, Hanoi, Vietnam <File:MECVOLLEYBALL> GROUND.JPG\|A basketball court in Tamil Nadu, India <File:Kevyen> liikenteen väylä Baana - G8537 - hkm.HKMS000005-km0000n5j4.jpg\|A basketball court on Baana - Helsinki, Finland.
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# Basketball
## Social forms of basketball {#social_forms_of_basketball}
Basketball as a social and communal sport features environments, rules and demographics different from those seen in professional and televised basketball.
### Recreational basketball {#recreational_basketball}
Basketball is played widely as an extracurricular, intramural or amateur sport in schools and colleges. Notable institutions of recreational basketball include:
- **Basketball schools and academies**, where students are trained in developing basketball fundamentals, undergo fitness and endurance exercises and learn various basketball skills. Basketball students learn proper ways of passing, ball handling, dribbling, shooting from various distances, rebounding, offensive moves, defense, layups, screens, basketball rules and basketball ethics. Also popular are the basketball camps organized for various occasions, often to get prepared for basketball events, and basketball clinics for improving skills.
- **College and university basketball** played in educational institutions of higher learning. This includes National Collegiate Athletic Association (NCAA) intercollegiate basketball.
### Disabled basketball {#disabled_basketball}
- **Deaf basketball**: One of several deaf sports, deaf basketball relies on signing for communication. Any deaf sporting event that happens, its purpose is to serve as a catalyst for the socialization of a low-incidence and geographically dispersed population.
- **Wheelchair basketball**: A sport based on basketball but designed for disabled people in wheelchairs and considered one of the major disabled sports practiced. There is a functional classification system that is used to help determine if the wheelchair basketball player classification system reflects the existing differences in the performance of elite female players. This system gives an analysis of the players\' functional resources through field-testing and game observation. During this system\'s process, players are assigned a score of 1 to 4.5.
### Other forms {#other_forms}
- **Biddy basketball** played by minors, sometimes in formal tournaments, around the globe.
- **Midnight basketball**, an initiative to curb inner-city crime in the United States and elsewhere by engaging youth in urban areas with sports as an alternative to drugs and crime.
- **Rezball**, short for reservation ball, is the avid Native American following of basketball, particularly a style of play particular to Native American teams of some areas.
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# Basketball
## Fantasy basketball {#fantasy_basketball}
**Fantasy basketball** was popularized during the 1990s by ESPN Fantasy Sports, NBA.com, and Yahoo! Fantasy Sports. On the model of fantasy baseball and football, players create fictional teams, select professional basketball players to \"play\" on these teams through a mock draft or trades, then calculate points based on the players\' real-world performance.
### Basics of fantasy basketball {#basics_of_fantasy_basketball}
1. **League Setup**:
- You can join public leagues or create private leagues with friends.
- Popular platforms include ESPN, Yahoo Sports, Sleeper, and Fantrax.
2. **Draft**:
- A draft (snake or auction) is held at the beginning of the season.
- Participants select NBA players to form their teams.
3. **Scoring Formats**:
- **Points League**: Players earn points based on specific stats (e.g., 2 points per rebound, 1.5 points per assist).
- **Categories League**: Teams compete in specific categories (e.g., best in assists, steals).
- **Rotisserie (Roto)**: Teams rank in each category, and rankings are combined to determine the overall score.
4. **Roster Management**:
- Teams set lineups daily or weekly, determining which players\' stats will count.
- You can trade players, pick up free agents, or drop underperforming players.
5. **Playoffs**:
- At the end of the regular NBA season, fantasy leagues often have playoffs to determine the champion
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# Blowfish (cipher)
**Blowfish** is a symmetric-key block cipher, designed in 1993 by Bruce Schneier and included in many cipher suites and encryption products. Blowfish provides a good encryption rate in software, and no effective cryptanalysis of it has been found to date for smaller files. It is recommended Blowfish should not be used to encrypt files larger than 4GB in size, Twofish should be used instead.
Blowfish has a 64-bit block size and therefore it could be vulnerable to Sweet32 birthday attacks.
Schneier designed Blowfish as a general-purpose algorithm, intended as an alternative to the aging DES and free of the problems and constraints associated with other algorithms. At the time Blowfish was released, many other designs were proprietary, encumbered by patents, or were commercial or government secrets. Schneier has stated that \"Blowfish is unpatented, and will remain so in all countries. The algorithm is hereby placed in the public domain, and can be freely used by anyone.\"
Notable features of the design include key-dependent S-boxes and a highly complex key schedule.
## The algorithm {#the_algorithm}
Blowfish has a 64-bit block size and a variable key length from 32 bits up to 448 bits. It is a 16-round Feistel cipher and uses large key-dependent S-boxes. In structure it resembles CAST-128, which uses fixed S-boxes. The adjacent diagram shows Blowfish\'s encryption routine. Each line represents 32 bits. There are five subkey-arrays: one 18-entry P-array (denoted as K in the diagram, to avoid confusion with the Plaintext) and four 256-entry S-boxes (S0, S1, S2 and S3).
Every round *r* consists of 4 actions:
-------------- ------------------------------------------------------------------
**Action 1** XOR the left half (L) of the data with the *r* th P-array entry
**Action 2** Use the XORed data as input for Blowfish\'s F-function
**Action 3** XOR the F-function\'s output with the right half (R) of the data
**Action 4** Swap L and R
-------------- ------------------------------------------------------------------
The F-function splits the 32-bit input into four 8-bit quarters and uses the quarters as input to the S-boxes. The S-boxes accept 8-bit input and produce 32-bit output. The outputs are added modulo 2^32^ and XORed to produce the final 32-bit output (see image in the upper right corner).
After the 16th round, undo the last swap, and XOR L with K18 and R with K17 (output whitening).
Decryption is exactly the same as encryption, except that P1, P2, \..., P18 are used in the reverse order. This is not so obvious because xor is commutative and associative. A common misconception is to use inverse order of encryption as decryption algorithm (i.e. first XORing P17 and P18 to the ciphertext block, then using the P-entries in reverse order).
Blowfish\'s key schedule starts by initializing the P-array and S-boxes with values derived from the hexadecimal digits of pi, which contain no obvious pattern (see nothing up my sleeve number). The secret key is then, byte by byte, cycling the key if necessary, XORed with all the P-entries in order. A 64-bit all-zero block is then encrypted with the algorithm as it stands. The resultant ciphertext replaces P~1~ and P~2~. The same ciphertext is then encrypted again with the new subkeys, and the new ciphertext replaces P~3~ and P~4~. This continues, replacing the entire P-array and all the S-box entries. In all, the Blowfish encryption algorithm will run 521 times to generate all the subkeys`{{snd}}`{=mediawiki} about 4 KB of data is processed.
Because the P-array is 576 bits long, and the key bytes are XORed through all these 576 bits during the initialization, many implementations support key sizes up to 576 bits. The reason for that is a discrepancy between the original Blowfish description, which uses 448-bit keys, and its reference implementation, which uses 576-bit keys. The test vectors for verifying third-party implementations were also produced with 576-bit keys. When asked which Blowfish version is the correct one, Bruce Schneier answered: \"The test vectors should be used to determine the one true Blowfish\".
Another opinion is that the 448 bits limit is present to ensure that every bit of every subkey depends on every bit of the key, as the last four values of the P-array don\'t affect every bit of the ciphertext. This point should be taken in consideration for implementations with a different number of rounds, as even though it increases security against an exhaustive attack, it weakens the security guaranteed by the algorithm. And given the slow initialization of the cipher with each change of key, it is granted a natural protection against brute-force attacks, which doesn\'t really justify key sizes longer than 448 bits.
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# Blowfish (cipher)
## Blowfish in pseudocode {#blowfish_in_pseudocode}
`P[18] // `*`P-array of 18 elements`*\
`S[4][256] // `*`S-boxes: 4 arrays of 256 elements`*\
\
**`function`**` f(x):`\
` // `*`Calculates a function f on a 32-bit input x, using S-boxes and bit manipulation`*\
` high_byte := (x `**`shifted right by`**` 24 `**`bits`**`)`\
` second_byte := (x `**`shifted right by`**` 16 `**`bits`**`) `**`AND`**` 0xff`\
` third_byte := (x `**`shifted right by`**` 8 `**`bits`**`) `**`AND`**` 0xff`\
` low_byte := x `**`AND`**` 0xff`\
\
` h := S[0][high_byte] + S[1][second_byte]`\
` `**`return`**` (h `**`XOR`**` S[2][third_byte]) + S[3][low_byte]`\
\
**`procedure`**` blowfish_encrypt(L, R):`\
` // `*`Encrypts two 32-bit halves L and R using the P-array and function f over 16 rounds`*\
` `**`for`**` round := 0 `**`to`**` 15:`\
` L := L `**`XOR`**` P[round]`\
` R := f(L) `**`XOR`**` R`\
` `**`swap values of L and R`**\
` `**`swap values of L and R`**\
` R := R `**`XOR`**` P[16]`\
` L := L `**`XOR`**` P[17]`\
\
**`procedure`**` blowfish_decrypt(L, R):`\
` // `*`Decrypts two 32-bit halves L and R using the P-array and function f over 16 rounds in reverse`*\
` `**`for`**` round := 17 `**`down to`**` 2:`\
` L := L `**`XOR`**` P[round]`\
` R := f(L) `**`XOR`**` R`\
` `**`swap values of L and R`**\
` `**`swap values of L and R`**\
` R := R `**`XOR`**` P[1]`\
` L := L `**`XOR`**` P[0]`\
` `\
`// `*`Initializes the P-array and S-boxes using the provided key, followed by key expansion`*\
`//`*`Initialize P-array with the key values`*\
`key_position := 0`\
**`for`**` i := 0 `**`to`**` 17:`\
` k := 0`\
` `**`for`**` j := 0 `**`to`**` 3:`\
` k := (k `**`shifted left by`**` 8 `**`bits`**`) `**`OR`**` key[key_position]`\
` key_position := (key_position + 1) `**`mod`**` key_length`\
` P[i] := P[i] `**`XOR`**` k`\
\
`//`*`Blowfish key expansion (521 iterations)`*\
`L := 0, R := 0`\
**`for`**` i := 0 `**`to`**` 17 `**`by`**` 2:`\
` blowfish_encrypt(L, R)`\
` P[i] := L`\
` P[i + 1] := R`\
\
`//`*`Fill S-boxes by encrypting L and R`*\
**`for`**` i := 0 `**`to`**` 3:`\
` `**`for`**` j := 0 `**`to`**` 255 `**`by`**` 2:`\
` blowfish_encrypt(L, R)`\
` S[i][j] := L`\
` S[i][j + 1] := R`
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# Blowfish (cipher)
## Blowfish in practice {#blowfish_in_practice}
Blowfish is a fast block cipher, except when changing keys. Each new key requires the pre-processing equivalent of encrypting about 4 kilobytes of text, which is very slow compared to other block ciphers. This prevents its use in certain applications, but is not a problem in others.
Blowfish must be initialized with a key. It is good practice to have this key hashed with a hash function before use.
In one application Blowfish\'s slow key changing is actually a benefit: the password-hashing method (crypt \$2, i.e. bcrypt) used in OpenBSD uses an algorithm derived from Blowfish that makes use of the slow key schedule; the idea is that the extra computational effort required gives protection against dictionary attacks. *See* key stretching.
Blowfish has a memory footprint of just over 4 kilobytes of RAM. This constraint is not a problem even for older desktop and laptop computers, though it does prevent use in the smallest embedded systems such as early smartcards.
Blowfish was one of the first secure block ciphers not subject to any patents and therefore freely available for anyone to use. This benefit has contributed to its popularity in cryptographic software.
bcrypt is a password hashing function which, combined with a variable number of iterations (work \"cost\"), exploits the expensive key setup phase of Blowfish to increase the workload and duration of hash calculations, further reducing threats from brute force attacks.
bcrypt is also the name of a cross-platform file encryption utility developed in 2002 that implements Blowfish.
## Weakness and successors {#weakness_and_successors}
Blowfish\'s use of a 64-bit block size (as opposed to e.g. AES\'s 128-bit block size) makes it vulnerable to birthday attacks, particularly in contexts like HTTPS. In 2016, the SWEET32 attack demonstrated how to leverage birthday attacks to perform plaintext recovery (i.e. decrypting ciphertext) against ciphers with a 64-bit block size. The GnuPG project recommends that Blowfish not be used to encrypt files larger than 4 GB due to its small block size.
A reduced-round variant of Blowfish is known to be susceptible to known-plaintext attacks on reflectively weak keys. Blowfish implementations use 16 rounds of encryption, and are not susceptible to this attack.
Bruce Schneier has recommended migrating to his Blowfish successor, Twofish.
was released in 2005, developed by Alexander Pukall. It has exactly the same design but has twice as many S tables and uses 64-bit integers instead of 32-bit integers. It no longer works on 64-bit blocks but on 128-bit blocks like AES. Blowfish2 is used for example, in FreePascal
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# Bijection
`{{Functions}}`{=mediawiki}
In mathematics, a **bijection**, **bijective function**, or **one-to-one correspondence** is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set.
A function is bijective if it is invertible; that is, a function $f:X\to Y$ is bijective if and only if there is a function $g:Y\to X,$ the *inverse* of `{{mvar|f}}`{=mediawiki}, such that each of the two ways for composing the two functions produces an identity function: $g(f(x)) = x$ for each $x$ in $X$ and $f(g(y)) = y$ for each $y$ in $Y.$
For example, the *multiplication by two* defines a bijection from the integers to the even numbers, which has the *division by two* as its inverse function.
A function is bijective if and only if it is both injective (or *one-to-one*)---meaning that each element in the codomain is mapped from at most one element of the domain---and surjective (or *onto*)---meaning that each element of the codomain is mapped from at least one element of the domain. The term *one-to-one correspondence* must not be confused with *one-to-one function*, which means injective but not necessarily surjective.
The elementary operation of counting establishes a bijection from some finite set to the first natural numbers `{{math|(1, 2, 3, ...)}}`{=mediawiki}, up to the number of elements in the counted set. It results that two finite sets have the same number of elements if and only if there exists a bijection between them. More generally, two sets are said to have the same cardinal number if there exists a bijection between them.
A bijective function from a set to itself is also called a permutation, and the set of all permutations of a set forms its symmetric group.
Some bijections with further properties have received specific names, which include automorphisms, isomorphisms, homeomorphisms, diffeomorphisms, permutation groups, and most geometric transformations. Galois correspondences are bijections between sets of mathematical objects of apparently very different nature.
## Definition
For a binary relation pairing elements of set *X* with elements of set *Y* to be a bijection, four properties must hold:
1. each element of *X* must be paired with at least one element of *Y*,
2. no element of *X* may be paired with more than one element of *Y*,
3. each element of *Y* must be paired with at least one element of *X*, and
4. no element of *Y* may be paired with more than one element of *X*.
Satisfying properties (1) and (2) means that a pairing is a function with domain *X*. It is more common to see properties (1) and (2) written as a single statement: Every element of *X* is paired with exactly one element of *Y*. Functions which satisfy property (3) are said to be \"onto *Y* \" and are called surjections (or *surjective functions*). Functions which satisfy property (4) are said to be \"one-to-one functions\" and are called injections (or *injective functions*). With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both \"one-to-one\" and \"onto\".
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# Bijection
## Examples
### Batting line-up of a baseball or cricket team {#batting_line_up_of_a_baseball_or_cricket_team}
Consider the batting line-up of a baseball or cricket team (or any list of all the players of any sports team where every player holds a specific spot in a line-up). The set *X* will be the players on the team (of size nine in the case of baseball) and the set *Y* will be the positions in the batting order (1st, 2nd, 3rd, etc.) The \"pairing\" is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere in the list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says that for each position in the order, there is some player batting in that position and property (4) states that two or more players are never batting in the same position in the list.
### Seats and students of a classroom {#seats_and_students_of_a_classroom}
In a classroom there are a certain number of seats. A group of students enter the room and the instructor asks them to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. What the instructor observed in order to reach this conclusion was that:
1. Every student was in a seat (there was no one standing),
2. No student was in more than one seat,
3. Every seat had someone sitting there (there were no empty seats), and
4. No seat had more than one student in it.
The instructor was able to conclude that there were just as many seats as there were students, without having to count either set.
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# Bijection
## More mathematical examples {#more_mathematical_examples}
- For any set *X*, the identity function **1**~*X*~: *X* → *X*, **1**~*X*~(*x*) = *x* is bijective.
- The function *f*: **R** → **R**, *f*(*x*) = 2*x* + 1 is bijective, since for each *y* there is a unique *x* = (*y* − 1)/2 such that *f*(*x*) = *y*. More generally, any linear function over the reals, *f*: **R** → **R**, *f*(*x*) = *ax* + *b* (where *a* is non-zero) is a bijection. Each real number *y* is obtained from (or paired with) the real number *x* = (*y* − *b*)/*a*.
- The function *f*: **R** → (−π/2, π/2), given by *f*(*x*) = arctan(*x*) is bijective, since each real number *x* is paired with exactly one angle *y* in the interval (−π/2, π/2) so that tan(*y*) = *x* (that is, *y* = arctan(*x*)). If the codomain (−π/2, π/2) was made larger to include an integer multiple of π/2, then this function would no longer be onto (surjective), since there is no real number which could be paired with the multiple of π/2 by this arctan function.
- The exponential function, *g*: **R** → **R**, *g*(*x*) = e^*x*^, is not bijective: for instance, there is no *x* in **R** such that *g*(*x*) = −1, showing that *g* is not onto (surjective). However, if the codomain is restricted to the positive real numbers $\R^+ \equiv \left(0, \infty\right)$, then *g* would be bijective; its inverse (see below) is the natural logarithm function ln.
- The function *h*: **R** → **R**^+^, *h*(*x*) = *x*^2^ is not bijective: for instance, *h*(−1) = *h*(1) = 1, showing that *h* is not one-to-one (injective). However, if the domain is restricted to $\R^+_0 \equiv \left[0, \infty\right)$, then *h* would be bijective; its inverse is the positive square root function.
- By Schröder--Bernstein theorem, given any two sets *X* and *Y*, and two injective functions *f*: *X → Y* and *g*: *Y → X*, there exists a bijective function *h*: *X → Y*.
## Inverses
A bijection *f* with domain *X* (indicated by *f*: *X → Y* in functional notation) also defines a converse relation starting in *Y* and going to *X* (by turning the arrows around). The process of \"turning the arrows around\" for an arbitrary function does not, *in general*, yield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain *Y*. Moreover, properties (1) and (2) then say that this inverse *function* is a surjection and an injection, that is, the inverse function exists and is also a bijection. Functions that have inverse functions are said to be invertible. A function is invertible if and only if it is a bijection.
Stated in concise mathematical notation, a function *f*: *X → Y* is bijective if and only if it satisfies the condition
: for every *y* in *Y* there is a unique *x* in *X* with *y* = *f*(*x*).
Continuing with the baseball batting line-up example, the function that is being defined takes as input the name of one of the players and outputs the position of that player in the batting order. Since this function is a bijection, it has an inverse function which takes as input a position in the batting order and outputs the player who will be batting in that position.
## Composition
The composition $g \,\circ\, f$ of two bijections *f*: *X → Y* and *g*: *Y → Z* is a bijection, whose inverse is given by $g \,\circ\, f$ is $(g \,\circ\, f)^{-1} \;=\; (f^{-1}) \,\circ\, (g^{-1})$.
Conversely, if the composition $g \, \circ\, f$ of two functions is bijective, it only follows that *f* is injective and *g* is surjective.
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# Bijection
## Cardinality
If *X* and *Y* are finite sets, then there exists a bijection between the two sets *X* and *Y* if and only if *X* and *Y* have the same number of elements. Indeed, in axiomatic set theory, this is taken as the definition of \"same number of elements\" (equinumerosity), and generalising this definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets.
## Properties
- A function *f*: **R** → **R** is bijective if and only if its graph meets every horizontal and vertical line exactly once.
- If *X* is a set, then the bijective functions from *X* to itself, together with the operation of functional composition ($\circ$), form a group, the symmetric group of *X*, which is denoted variously by S(*X*), *S~X~*, or *X*! (*X* factorial).
- Bijections preserve cardinalities of sets: for a subset *A* of the domain with cardinality \|*A*\| and subset *B* of the codomain with cardinality \|*B*\|, one has the following equalities:
: \|*f*(*A*)\| = \|*A*\| and \|*f*^−1^(*B*)\| = \|*B*\|.
- If *X* and *Y* are finite sets with the same cardinality, and *f*: *X → Y*, then the following are equivalent:
1. *f* is a bijection.
2. *f* is a surjection.
3. *f* is an injection.
- For a finite set *S*, there is a bijection between the set of possible total orderings of the elements and the set of bijections from *S* to *S*. That is to say, the number of permutations of elements of *S* is the same as the number of total orderings of that set---namely, *n*!.
## Category theory {#category_theory}
Bijections are precisely the isomorphisms in the category *Set* of sets and set functions. However, the bijections are not always the isomorphisms for more complex categories. For example, in the category *Grp* of groups, the morphisms must be homomorphisms since they must preserve the group structure, so the isomorphisms are *group isomorphisms* which are bijective homomorphisms.
## Generalization to partial functions {#generalization_to_partial_functions}
The notion of one-to-one correspondence generalizes to partial functions, where they are called *partial bijections*, although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the symmetric inverse semigroup.
Another way of defining the same notion is to say that a partial bijection from *A* to *B* is any relation *R* (which turns out to be a partial function) with the property that *R* is the graph of a bijection *f*:*`{{prime|A}}`{=mediawiki}*→*`{{prime|B}}`{=mediawiki}*, where *`{{prime|A}}`{=mediawiki}* is a subset of *A* and *`{{prime|B}}`{=mediawiki}* is a subset of *B*.
When the partial bijection is on the same set, it is sometimes called a *one-to-one partial transformation*. An example is the Möbius transformation simply defined on the complex plane, rather than its completion to the extended complex plane
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# Binary function
In mathematics, a **binary function** (also called **bivariate function**, or **function of two variables**) is a function that takes two inputs.
Precisely stated, a function $f$ is binary if there exists sets $X, Y, Z$ such that
$$\,f \colon X \times Y \rightarrow Z$$ where $X \times Y$ is the Cartesian product of $X$ and $Y.$
## Alternative definitions {#alternative_definitions}
Set-theoretically, a binary function can be represented as a subset of the Cartesian product $X \times Y \times Z$, where $(x,y,z)$ belongs to the subset if and only if $f(x,y) = z$. Conversely, a subset $R$ defines a binary function if and only if for any $x \in X$ and $y \in Y$, there exists a unique $z \in Z$ such that $(x,y,z)$ belongs to $R$. $f(x,y)$ is then defined to be this $z$.
Alternatively, a binary function may be interpreted as simply a function from $X \times Y$ to $Z$. Even when thought of this way, however, one generally writes $f(x,y)$ instead of $f((x,y))$. (That is, the same pair of parentheses is used to indicate both function application and the formation of an ordered pair.)
## Examples
Division of whole numbers can be thought of as a function. If $\Z$ is the set of integers, $\N^+$ is the set of natural numbers (except for zero), and $\Q$ is the set of rational numbers, then division is a binary function $f:\Z \times \N^+ \to \Q$.
In a vector space *V* over a field *F*, scalar multiplication is a binary function. A scalar *a* ∈ *F* is combined with a vector *v* ∈ *V* to produce a new vector *av* ∈ *V*.
Another example is that of inner products, or more generally functions of the form $(x,y)\mapsto x^\mathrm{T}My$, where `{{mvar|x}}`{=mediawiki}, `{{mvar|y}}`{=mediawiki} are real-valued vectors of appropriate size and `{{mvar|M}}`{=mediawiki} is a matrix. If `{{mvar|M}}`{=mediawiki} is a positive definite matrix, this yields an inner product.
## Functions of two real variables {#functions_of_two_real_variables}
Functions whose domain is a subset of $\mathbb{R}^2$ are often also called functions of two variables even if their domain does not form a rectangle and thus the cartesian product of two sets.
## Restrictions to ordinary functions {#restrictions_to_ordinary_functions}
In turn, one can also derive ordinary functions of one variable from a binary function. Given any element $x \in X$, there is a function $f^x$, or $f(x,\cdot)$, from $Y$ to $Z$, given by $f^x(y) = f(x,y)$. Similarly, given any element $y \in Y$, there is a function $f_y$, or $f(\cdot,y)$, from $X$ to $Z$, given by $f_y(x) = f(x,y)$. In computer science, this identification between a function from $X \times Y$ to $Z$ and a function from $X$ to $Z^Y$, where $Z^Y$ is the set of all functions from $Y$ to $Z$, is called *currying*.
## Generalisations
The various concepts relating to functions can also be generalised to binary functions. For example, the division example above is *surjective* (or *onto*) because every rational number may be expressed as a quotient of an integer and a natural number. This example is *injective* in each input separately, because the functions *f* ^*x*^ and *f* ~*y*~ are always injective. However, it\'s not injective in both variables simultaneously, because (for example) *f* (2,4) = *f* (1,2).
One can also consider *partial* binary functions, which may be defined only for certain values of the inputs. For example, the division example above may also be interpreted as a partial binary function from **Z** and **N** to **Q**, where **N** is the set of all natural numbers, including zero. But this function is undefined when the second input is zero.
A binary operation is a binary function where the sets *X*, *Y*, and *Z* are all equal; binary operations are often used to define algebraic structures.
In linear algebra, a bilinear transformation is a binary function where the sets *X*, *Y*, and *Z* are all vector spaces and the derived functions *f* ^*x*^ and *f*~*y*~ are all linear transformations. A bilinear transformation, like any binary function, can be interpreted as a function from *X* × *Y* to *Z*, but this function in general won\'t be linear. However, the bilinear transformation can also be interpreted as a single linear transformation from the tensor product $X \otimes Y$ to *Z*.
## Generalisations to ternary and other functions {#generalisations_to_ternary_and_other_functions}
The concept of binary function generalises to *ternary* (or *3-ary*) *function*, *quaternary* (or *4-ary*) *function*, or more generally to *n-ary function* for any natural number *n*. A *0-ary function* to *Z* is simply given by an element of *Z*. One can also define an *A-ary function* where *A* is any set; there is one input for each element of *A*.
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# Binary function
## Category theory {#category_theory}
In category theory, *n*-ary functions generalise to *n*-ary morphisms in a multicategory. The interpretation of an *n*-ary morphism as an ordinary morphisms whose domain is some sort of product of the domains of the original *n*-ary morphism will work in a monoidal category. The construction of the derived morphisms of one variable will work in a closed monoidal category. The category of sets is closed monoidal, but so is the category of vector spaces, giving the notion of bilinear transformation above
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# Binary operation
In mathematics, a **binary operation** or **dyadic operation** is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, a **binary operation** on a set is a binary function that maps every pair of elements of the set to an element of the set. Examples include the familiar arithmetic operations like addition, subtraction, multiplication, set operations like union, complement, intersection. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups.
A binary function that involves several sets is sometimes also called a *binary operation*. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar.
Binary operations are the keystone of most structures that are studied in algebra, in particular in semigroups, monoids, groups, rings, fields, and vector spaces.
## Terminology
More precisely, a binary operation on a set $S$ is a mapping of the elements of the Cartesian product $S \times S$ to $S$:
$$\,f \colon S \times S \rightarrow S.$$
If $f$ is not a function but a partial function, then $f$ is called a **partial binary operation**. For instance, division is a partial binary operation on the set of all real numbers, because one cannot divide by zero: $\frac{a}{0}$ is undefined for every real number $a$. In both model theory and classical universal algebra, binary operations are required to be defined on all elements of $S \times S$. However, partial algebras generalize universal algebras to allow partial operations.
Sometimes, especially in computer science, the term binary operation is used for any binary function.
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# Binary operation
## Properties and examples {#properties_and_examples}
Typical examples of binary operations are the addition ($+$) and multiplication ($\times$) of numbers and matrices as well as composition of functions on a single set. For instance,
- On the set of real numbers $\mathbb R$, $f(a,b)=a+b$ is a binary operation since the sum of two real numbers is a real number.
- On the set of natural numbers $\mathbb N$, $f(a,b)=a+b$ is a binary operation since the sum of two natural numbers is a natural number. This is a different binary operation than the previous one since the sets are different.
- On the set $M(2,\mathbb R)$ of $2 \times 2$ matrices with real entries, $f(A,B)=A+B$ is a binary operation since the sum of two such matrices is a $2 \times 2$ matrix.
- On the set $M(2,\mathbb R)$ of $2 \times 2$ matrices with real entries, $f(A,B)=AB$ is a binary operation since the product of two such matrices is a $2 \times 2$ matrix.
- For a given set $C$, let $S$ be the set of all functions $h \colon C \rightarrow C$. Define $f \colon S \times S \rightarrow S$ by $f(h_1,h_2)(c)=(h_1 \circ h_2)(c)=h_1(h_2(c))$ for all $c \in C$, the composition of the two functions $h_1$ and $h_2$ in $S$. Then $f$ is a binary operation since the composition of the two functions is again a function on the set $C$ (that is, a member of $S$).
Many binary operations of interest in both algebra and formal logic are commutative, satisfying $f(a,b)=f(b,a)$ for all elements $a$ and $b$ in $S$, or associative, satisfying $f(f(a,b),c)=f(a,f(b,c))$ for all $a$, $b$, and $c$ in $S$. Many also have identity elements and inverse elements.
The first three examples above are commutative and all of the above examples are associative.
On the set of real numbers $\mathbb R$, subtraction, that is, $f(a,b)=a-b$, is a binary operation which is not commutative since, in general, $a-b \neq b-a$. It is also not associative, since, in general, $a-(b-c) \neq (a-b)-c$; for instance, $1-(2-3)=2$ but $(1-2)-3=-4$.
On the set of natural numbers $\mathbb N$, the binary operation exponentiation, $f(a,b)=a^b$, is not commutative since, $a^b \neq b^a$ (cf. Equation x^y^ = y^x^), and is also not associative since $f(f(a,b),c) \neq f(a,f(b,c))$. For instance, with $a=2$, $b=3$, and $c=2$, $f(2^3,2)=f(8,2)=8^2=64$, but $f(2,3^2)=f(2,9)=2^9=512$. By changing the set $\mathbb N$ to the set of integers $\mathbb Z$, this binary operation becomes a partial binary operation since it is now undefined when $a=0$ and $b$ is any negative integer. For either set, this operation has a *right identity* (which is $1$) since $f(a,1)=a$ for all $a$ in the set, which is not an *identity* (two sided identity) since $f(1,b) \neq b$ in general.
Division ($\div$), a partial binary operation on the set of real or rational numbers, is not commutative or associative. Tetration ($\uparrow\uparrow$), as a binary operation on the natural numbers, is not commutative or associative and has no identity element.
## Notation
Binary operations are often written using infix notation such as $a \ast b$, $a+b$, $a \cdot b$ or (by juxtaposition with no symbol) $ab$ rather than by functional notation of the form $f(a, b)$. Powers are usually also written without operator, but with the second argument as superscript.
Binary operations are sometimes written using prefix or (more frequently) postfix notation, both of which dispense with parentheses. They are also called, respectively, Polish notation $\ast a b$ and reverse Polish notation $a b \ast$.
## Binary operations as ternary relations {#binary_operations_as_ternary_relations}
A binary operation $f$ on a set $S$ may be viewed as a ternary relation on $S$, that is, the set of triples $(a, b, f(a,b))$ in $S \times S \times S$ for all $a$ and $b$ in $S$.
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# Binary operation
## Other binary operations {#other_binary_operations}
For example, scalar multiplication in linear algebra. Here $K$ is a field and $S$ is a vector space over that field.
Also the dot product of two vectors maps $S \times S$ to $K$, where $K$ is a field and $S$ is a vector space over $K$. It depends on authors whether it is considered as a binary operation
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# Bedrock Records
**Bedrock Records** is an English record label for trance, progressive house and techno started by John Digweed. Its name comes from a long running and successful club night held in Hastings and also at Heaven nightclub, London -- both also called Bedrock. Bedrock Records has released many singles from artists such as Astro & Glyde, Brancaccio & Aisher, Steve Lawler, Shmuel Flash, Steve Porter, Sahar Z, Guy J, Henry Saiz, Stelios Vassiloudis, Electric Rescue, The Japanese Popstars and Jerry Bonham. Bedrock is also the name that Digweed and Muir use as their production moniker.
Bedrock has had different imprints: Bedrock Breaks, B_Rock and Black (Bedrock). Currently it has Bedrock Digital and one called Lost & Found belonging to Guy J.
The first Bedrock album compiled and mixed by John Digweed was released in 1999, containing several tracks signed to the Bedrock label.
In 2018, Digweed marked the 20th anniversary of the label with the release of *Bedrock XX*
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# Boolean algebra (structure)
In abstract algebra, a **Boolean algebra** or **Boolean lattice** is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution).
Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle. \_\_TOC\_\_
## History
The term \"Boolean algebra\" honors George Boole (1815--1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, *The Mathematical Analysis of Logic*, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, *The Laws of Thought*, published in 1854. Boole\'s formulation differs from that described above in some important respects. For example, conjunction and disjunction in Boole were not a dual pair of operations. Boolean algebra emerged in the 1860s, in papers written by William Jevons and Charles Sanders Peirce. The first systematic presentation of Boolean algebra and distributive lattices is owed to the 1890 *Vorlesungen* of Ernst Schröder. The first extensive treatment of Boolean algebra in English is A. N. Whitehead\'s 1898 *Universal Algebra*. Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward V. Huntington. Boolean algebra came of age as serious mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoff\'s 1940 *Lattice Theory*. In the 1960s, Paul Cohen, Dana Scott, and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing and Boolean-valued models.
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# Boolean algebra (structure)
## Definition
A **Boolean algebra** is a set `{{math|1=''A''}}`{=mediawiki}, equipped with two binary operations `{{math|1=∧}}`{=mediawiki} (called \"meet\" or \"and\"), `{{math|1=∨}}`{=mediawiki} (called \"join\" or \"or\"), a unary operation `{{math|1=¬}}`{=mediawiki} (called \"complement\" or \"not\") and two elements `{{math|1=0}}`{=mediawiki} and `{{math|1=1}}`{=mediawiki} in `{{math|1=''A''}}`{=mediawiki} (called \"bottom\" and \"top\", or \"least\" and \"greatest\" element, also denoted by the symbols `{{math|1=⊥}}`{=mediawiki} and `{{math|1=⊤}}`{=mediawiki}, respectively), such that for all elements `{{math|''a''}}`{=mediawiki}, `{{math|''b''}}`{=mediawiki} and `{{math|''c''}}`{=mediawiki} of `{{math|''A''}}`{=mediawiki}, the following axioms hold:
:
: {\| cellpadding=5
\|`{{math|1=''a'' ∨ (''b'' ∨ ''c'') = (''a'' ∨ ''b'') ∨ ''c''}}`{=mediawiki} \|`{{math|1=''a'' ∧ (''b'' ∧ ''c'') = (''a'' ∧ ''b'') ∧ ''c''}}`{=mediawiki} \| associativity \|- \|`{{math|1=''a'' ∨ ''b'' = ''b'' ∨ ''a''}}`{=mediawiki} \|`{{math|1=''a'' ∧ ''b'' = ''b'' ∧ ''a''}}`{=mediawiki} \| commutativity \|- \|`{{math|1=''a'' ∨ (''a'' ∧ ''b'') = ''a''}}`{=mediawiki} \|`{{math|1=''a'' ∧ (''a'' ∨ ''b'') = ''a''}}`{=mediawiki} \| absorption \|- \|`{{math|1=''a'' ∨ 0 = ''a''}}`{=mediawiki} \|`{{math|1=''a'' ∧ 1 = ''a''}}`{=mediawiki} \| identity \|- \|`{{math|1=''a'' ∨ (''b'' ∧ ''c'') = (''a'' ∨ ''b'') ∧ (''a'' ∨ ''c'') }}`{=mediawiki} \|`{{math|1=''a'' ∧ (''b'' ∨ ''c'') = (''a'' ∧ ''b'') ∨ (''a'' ∧ ''c'') }}`{=mediawiki} \| distributivity \|- \|`{{math|1=''a'' ∨ ¬''a'' = 1}}`{=mediawiki} \|`{{math|1=''a'' ∧ ¬''a'' = 0}}`{=mediawiki} \| complements \|} Note, however, that the absorption law and even the associativity law can be excluded from the set of axioms as they can be derived from the other axioms (see Proven properties).
A Boolean algebra with only one element is called a **trivial Boolean algebra** or a **degenerate Boolean algebra**. (In older works, some authors required `{{math|0}}`{=mediawiki} and `{{math|1}}`{=mediawiki} to be *distinct* elements in order to exclude this case.)
It follows from the last three pairs of axioms above (identity, distributivity and complements), or from the absorption axiom, that
: if and only if `{{math|1=''a'' ∨ ''b'' = ''b''}}`{=mediawiki}.
The relation `{{math|≤}}`{=mediawiki} defined by `{{math|''a'' ≤ ''b''}}`{=mediawiki} if these equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet `{{math|1=''a'' ∧ ''b''}}`{=mediawiki} and the join `{{math|1=''a'' ∨ ''b''}}`{=mediawiki} of two elements coincide with their infimum and supremum, respectively, with respect to ≤.
The first four pairs of axioms constitute a definition of a bounded lattice.
It follows from the first five pairs of axioms that any complement is unique.
The set of axioms is self-dual in the sense that if one exchanges `{{math|1=∨}}`{=mediawiki} with `{{math|1=∧}}`{=mediawiki} and `{{math|0}}`{=mediawiki} with `{{math|1}}`{=mediawiki} in an axiom, the result is again an axiom. Therefore, by applying this operation to a Boolean algebra (or Boolean lattice), one obtains another Boolean algebra with the same elements; it is called its **dual**.
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# Boolean algebra (structure)
## Examples
- The simplest non-trivial Boolean algebra, the two-element Boolean algebra, has only two elements, `{{math|0}}`{=mediawiki} and `{{math|1}}`{=mediawiki}, and is defined by the rules:
+---+------------+---+---+------------+---+---+------------+
| | -- -- -- | | | -- -- -- | | | -- -- -- |
| | | | | | | | |
| | | | | | | | -- -- -- |
| | -- -- -- | | | -- -- -- | | | |
+---+------------+---+---+------------+---+---+------------+
:\* It has applications in logic, interpreting `{{math|0}}`{=mediawiki} as *false*, `{{math|1}}`{=mediawiki} as *true*, `{{math|1=∧}}`{=mediawiki} as *and*, `{{math|1=∨}}`{=mediawiki} as *or*, and `{{math|¬}}`{=mediawiki} as *not*. Expressions involving variables and the Boolean operations represent statement forms, and two such expressions can be shown to be equal using the above axioms if and only if the corresponding statement forms are logically equivalent.
:\* The two-element Boolean algebra is also used for circuit design in electrical engineering;`{{refn|group=note|Strictly, electrical engineers tend to use additional states to represent other circuit conditions such as high impedance - see [[IEEE 1164]] or [[IEEE 1364]].}}`{=mediawiki} here 0 and 1 represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if and only if the corresponding circuits have the same input--output behavior. Furthermore, every possible input--output behavior can be modeled by a suitable Boolean expression.
:\* The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can be checked by a trivial brute force algorithm for small numbers of variables). This can for example be used to show that the following laws (*Consensus theorems*) are generally valid in all Boolean algebras:
:\*\* `{{math|1=(''a'' ∨ ''b'') ∧ (¬''a'' ∨ ''c'') ∧ (''b'' ∨ ''c'') ≡ (''a'' ∨ ''b'') ∧ (¬''a'' ∨ ''c'')}}`{=mediawiki}
:\*\* `{{math|1=(''a'' ∧ ''b'') ∨ (¬''a'' ∧ ''c'') ∨ (''b'' ∧ ''c'') ≡ (''a'' ∧ ''b'') ∨ (¬''a'' ∧ ''c'')}}`{=mediawiki}
- The power set (set of all subsets) of any given nonempty set `{{math|''S''}}`{=mediawiki} forms a Boolean algebra, an algebra of sets, with the two operations `{{math|1=∨ := ∪}}`{=mediawiki} (union) and `{{math|1=∧ := ∩}}`{=mediawiki} (intersection). The smallest element 0 is the empty set and the largest element `{{math|1}}`{=mediawiki} is the set `{{math|''S''}}`{=mediawiki} itself.
:\* After the two-element Boolean algebra, the simplest Boolean algebra is that defined by the power set of two atoms:
+---+------------------+---+---+------------------+---+---+------------------+
| | -- -- -- -- -- | | | -- -- -- -- -- | | | -- -- -- -- -- |
| | | | | | | | |
| | | | | | | | -- -- -- -- -- |
| | | | | | | | |
| | | | | | | | |
| | -- -- -- -- -- | | | -- -- -- -- -- | | | |
+---+------------------+---+---+------------------+---+---+------------------+
- The set `{{mvar|A}}`{=mediawiki} of all subsets of `{{mvar|S}}`{=mediawiki} that are either finite or cofinite is a Boolean algebra and an algebra of sets called the finite--cofinite algebra. If `{{mvar|S}}`{=mediawiki} is infinite then the set of all cofinite subsets of `{{mvar|S}}`{=mediawiki}, which is called the Fréchet filter, is a free ultrafilter on `{{mvar|A}}`{=mediawiki}. However, the Fréchet filter is not an ultrafilter on the power set of `{{mvar|S}}`{=mediawiki}.
- Starting with the propositional calculus with `{{math|κ}}`{=mediawiki} sentence symbols, form the Lindenbaum algebra (that is, the set of sentences in the propositional calculus modulo logical equivalence). This construction yields a Boolean algebra. It is in fact the free Boolean algebra on `{{math|κ}}`{=mediawiki} generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to the two-element Boolean algebra.
- Given any linearly ordered set `{{math|''L''}}`{=mediawiki} with a least element, the interval algebra is the smallest Boolean algebra of subsets of `{{math|''L''}}`{=mediawiki} containing all of the half-open intervals `{{math|[''a'', ''b'')}}`{=mediawiki} such that `{{math|''a''}}`{=mediawiki} is in `{{math|''L''}}`{=mediawiki} and `{{math|''b''}}`{=mediawiki} is either in `{{math|''L''}}`{=mediawiki} or equal to `{{math|∞}}`{=mediawiki}. Interval algebras are useful in the study of Lindenbaum--Tarski algebras; every countable Boolean algebra is isomorphic to an interval algebra.
```{=html}
<!-- -->
```
- For any natural number `{{math|''n''}}`{=mediawiki}, the set of all positive divisors of `{{math|''n''}}`{=mediawiki}, defining `{{math|''a'' ≤ ''b''}}`{=mediawiki} if `{{math|''a''}}`{=mediawiki} divides `{{math|''b''}}`{=mediawiki}, forms a distributive lattice. This lattice is a Boolean algebra if and only if `{{math|''n''}}`{=mediawiki} is square-free. The bottom and the top elements of this Boolean algebra are the natural numbers `{{math|1}}`{=mediawiki} and `{{math|''n''}}`{=mediawiki}, respectively. The complement of `{{math|''a''}}`{=mediawiki} is given by `{{math|''n''/''a''}}`{=mediawiki}. The meet and the join of `{{math|''a''}}`{=mediawiki} and `{{math|''b''}}`{=mediawiki} are given by the greatest common divisor (`{{math|gcd}}`{=mediawiki}) and the least common multiple (`{{math|lcm}}`{=mediawiki}) of `{{math|''a''}}`{=mediawiki} and `{{math|''b''}}`{=mediawiki}, respectively. The ring addition `{{math|''a'' + ''b''}}`{=mediawiki} is given by `{{math|lcm(''a'', ''b'') / gcd(''a'', ''b'')}}`{=mediawiki}. The picture shows an example for `{{math|1=''n'' = 30}}`{=mediawiki}. As a counter-example, considering the non-square-free `{{math|1=''n'' = 60}}`{=mediawiki}, the greatest common divisor of 30 and its complement 2 would be 2, while it should be the bottom element 1.
- Other examples of Boolean algebras arise from topological spaces: if `{{math|''X''}}`{=mediawiki} is a topological space, then the collection of all subsets of `{{math|''X''}}`{=mediawiki} that are both open and closed forms a Boolean algebra with the operations `{{math|1=∨ := ∪}}`{=mediawiki} (union) and `{{math|1=∧ := ∩}}`{=mediawiki} (intersection).
- If `{{mvar|R}}`{=mediawiki} is an arbitrary ring then its set of *central idempotents*, which is the set
$A = \left\{e \in R : e^2 = e \text{ and } ex = xe \; \text{ for all } \; x \in R\right\},$ becomes a Boolean algebra when its operations are defined by `{{math|1=''e'' ∨ ''f'' := ''e'' + ''f'' − ''ef''}}`{=mediawiki} and `{{math|1=''e'' ∧ ''f'' := ''ef''}}`{=mediawiki}.
## Homomorphisms and isomorphisms {#homomorphisms_and_isomorphisms}
A *homomorphism* between two Boolean algebras `{{math|''A''}}`{=mediawiki} and `{{math|''B''}}`{=mediawiki} is a function `{{math|''f'' : ''A'' → ''B''}}`{=mediawiki} such that for all `{{math|''a''}}`{=mediawiki}, `{{math|''b''}}`{=mediawiki} in `{{math|''A''}}`{=mediawiki}:
: ,
: ,
: ,
: .
It then follows that `{{math|1=''f''(¬''a'') = ¬''f''(''a'')}}`{=mediawiki} for all `{{math|''a''}}`{=mediawiki} in `{{math|''A''}}`{=mediawiki}. The class of all Boolean algebras, together with this notion of morphism, forms a full subcategory of the category of lattices.
An *isomorphism* between two Boolean algebras `{{math|''A''}}`{=mediawiki} and `{{math|''B''}}`{=mediawiki} is a homomorphism `{{math|''f'' : ''A'' → ''B''}}`{=mediawiki} with an inverse homomorphism, that is, a homomorphism `{{math|''g'' : ''B'' → ''A''}}`{=mediawiki} such that the composition `{{math|''g'' ∘ ''f'' : ''A'' → ''A''}}`{=mediawiki} is the identity function on `{{math|''A''}}`{=mediawiki}, and the composition `{{math|''f'' ∘ ''g'' : ''B'' → ''B''}}`{=mediawiki} is the identity function on `{{math|''B''}}`{=mediawiki}. A homomorphism of Boolean algebras is an isomorphism if and only if it is bijective.
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# Boolean algebra (structure)
## Boolean rings {#boolean_rings}
Every Boolean algebra `{{math|1=(''A'', ∧, ∨)}}`{=mediawiki} gives rise to a ring `{{math|(''A'', +, ·)}}`{=mediawiki} by defining `{{math|1=''a'' + ''b'' := (''a'' ∧ ¬''b'') ∨ (''b'' ∧ ¬''a'') = (''a'' ∨ ''b'') ∧ ¬(''a'' ∧ ''b'')}}`{=mediawiki} (this operation is called symmetric difference in the case of sets and XOR in the case of logic) and `{{math|1=''a'' · ''b'' := ''a'' ∧ ''b''}}`{=mediawiki}. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the `{{math|1}}`{=mediawiki} of the Boolean algebra. This ring has the property that `{{math|1=''a'' · ''a'' = ''a''}}`{=mediawiki} for all `{{math|''a''}}`{=mediawiki} in `{{math|''A''}}`{=mediawiki}; rings with this property are called Boolean rings.
Conversely, if a Boolean ring `{{math|''A''}}`{=mediawiki} is given, we can turn it into a Boolean algebra by defining `{{math|1=''x'' ∨ ''y'' := ''x'' + ''y'' + (''x'' · ''y'')}}`{=mediawiki} and `{{math|1=''x'' ∧ ''y'' := ''x'' · ''y''}}`{=mediawiki}. Since these two constructions are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa. Furthermore, a map `{{math|''f'' : ''A'' → ''B''}}`{=mediawiki} is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The categories of Boolean rings and Boolean algebras are equivalent; in fact the categories are isomorphic.
Hsiang (1985) gave a rule-based algorithm to check whether two arbitrary expressions denote the same value in every Boolean ring. More generally, Boudet, Jouannaud, and Schmidt-Schauß (1989) gave an algorithm to solve equations between arbitrary Boolean-ring expressions. Employing the similarity of Boolean rings and Boolean algebras, both algorithms have applications in automated theorem proving.
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# Boolean algebra (structure)
## Ideals and filters {#ideals_and_filters}
An *ideal* of the Boolean algebra `{{mvar|A}}`{=mediawiki} is a nonempty subset `{{mvar|I}}`{=mediawiki} such that for all `{{mvar|x}}`{=mediawiki}, `{{mvar|y}}`{=mediawiki} in `{{mvar|I}}`{=mediawiki} we have `{{math|{{var|x}} ∨ {{var|y}}}}`{=mediawiki} in `{{mvar|I}}`{=mediawiki} and for all `{{mvar|a}}`{=mediawiki} in `{{mvar|A}}`{=mediawiki} we have `{{math|{{var|a}} ∧ {{var|x}}}}`{=mediawiki} in `{{mvar|I}}`{=mediawiki}. This notion of ideal coincides with the notion of ring ideal in the Boolean ring `{{mvar|A}}`{=mediawiki}. An ideal `{{mvar|I}}`{=mediawiki} of `{{mvar|A}}`{=mediawiki} is called *prime* if `{{math|{{var|I}} ≠ {{var|A}}}}`{=mediawiki} and if `{{math|{{var|a}} ∧ {{var|b}}}}`{=mediawiki} in `{{mvar|I}}`{=mediawiki} always implies `{{mvar|a}}`{=mediawiki} in `{{mvar|I}}`{=mediawiki} or `{{mvar|b}}`{=mediawiki} in `{{mvar|I}}`{=mediawiki}. Furthermore, for every `{{math|{{var|a}} ∈ {{var|A}}}}`{=mediawiki} we have that `{{math|{{var|a}} ∧ −{{var|a}} {{=}}`{=mediawiki} 0 ∈ `{{var|I}}`{=mediawiki}}}, and then if `{{mvar|I}}`{=mediawiki} is prime we have `{{math|{{var|a}} ∈ {{var|I}}}}`{=mediawiki} or `{{math|−{{var|a}} ∈ {{var|I}}}}`{=mediawiki} for every `{{math|{{var|a}} ∈ {{var|A}}}}`{=mediawiki}. An ideal `{{mvar|I}}`{=mediawiki} of `{{mvar|A}}`{=mediawiki} is called *maximal* if `{{math|{{var|I}} ≠ {{var|A}}}}`{=mediawiki} and if the only ideal properly containing `{{mvar|I}}`{=mediawiki} is `{{mvar|A}}`{=mediawiki} itself. For an ideal `{{mvar|I}}`{=mediawiki}, if `{{math|{{var|a}} ∉ {{var|I}}}}`{=mediawiki} and `{{math|−{{var|a}} ∉ {{var|I}}}}`{=mediawiki}, then `{{math|{{var|I}} ∪ {{mset|{{var|a}}}}}}`{=mediawiki} or `{{math|{{var|I}} ∪ {{mset|−{{var|a}}}}}}`{=mediawiki} is contained in another proper ideal `{{mvar|J}}`{=mediawiki}. Hence, such an `{{mvar|I}}`{=mediawiki} is not maximal, and therefore the notions of prime ideal and maximal ideal are equivalent in Boolean algebras. Moreover, these notions coincide with ring theoretic ones of prime ideal and maximal ideal in the Boolean ring `{{mvar|A}}`{=mediawiki}.
The dual of an *ideal* is a *filter*. A *filter* of the Boolean algebra `{{mvar|A}}`{=mediawiki} is a nonempty subset `{{mvar|p}}`{=mediawiki} such that for all `{{mvar|x}}`{=mediawiki}, `{{mvar|y}}`{=mediawiki} in `{{mvar|p}}`{=mediawiki} we have `{{math|{{var|x}} ∧ {{var|y}}}}`{=mediawiki} in `{{mvar|p}}`{=mediawiki} and for all `{{mvar|a}}`{=mediawiki} in `{{mvar|A}}`{=mediawiki} we have `{{math|{{var|a}} ∨ {{var|x}}}}`{=mediawiki} in `{{mvar|p}}`{=mediawiki}. The dual of a *maximal* (or *prime*) *ideal* in a Boolean algebra is *ultrafilter*. Ultrafilters can alternatively be described as 2-valued morphisms from `{{mvar|A}}`{=mediawiki} to the two-element Boolean algebra. The statement *every filter in a Boolean algebra can be extended to an ultrafilter* is called the *ultrafilter lemma* and cannot be proven in Zermelo--Fraenkel set theory (ZF), if ZF is consistent. Within ZF, the ultrafilter lemma is strictly weaker than the axiom of choice. The ultrafilter lemma has many equivalent formulations: *every Boolean algebra has an ultrafilter*, *every ideal in a Boolean algebra can be extended to a prime ideal*, etc.
## Representations
It can be shown that every *finite* Boolean algebra is isomorphic to the Boolean algebra of all subsets of a finite set. Therefore, the number of elements of every finite Boolean algebra is a power of two.
Stone\'s celebrated *representation theorem for Boolean algebras* states that *every* Boolean algebra `{{math|''A''}}`{=mediawiki} is isomorphic to the Boolean algebra of all clopen sets in some (compact totally disconnected Hausdorff) topological space.
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# Boolean algebra (structure)
## Axiomatics
+-----------------------------------------------------------------------------------------------------+
| **Proven properties** |
+=====================================================================================================+
| **UId~1~** If *x* ∨ *o* = *x* for all *x*, then *o* = 0 |
| ------------ --- ---------------------------------------------- |
| Proof: If *x* ∨ *o* = *x*, then |
| 0 |
| = 0 ∨ *o* |
| = *o* ∨ 0 |
| = *o* |
+-----------------------------------------------------------------------------------------------------+
| **Idm~1~** *x* ∨ *x* = *x* |
| ------------ --- ---------------------------- |
| Proof: *x* ∨ *x* |
| = (*x* ∨ *x*) ∧ 1 |
| = (*x* ∨ *x*) ∧ (*x* ∨ ¬*x*) |
| = *x* ∨ (*x* ∧ ¬*x*) |
| = *x* ∨ 0 |
| = *x* |
+-----------------------------------------------------------------------------------------------------+
| **Bnd~1~** *x* ∨ 1 = 1 |
| ------------ --- -------------------------- |
| Proof: *x* ∨ 1 |
| = (*x* ∨ 1) ∧ 1 |
| = 1 ∧ (*x* ∨ 1) |
| = (*x* ∨ ¬*x*) ∧ (*x* ∨ 1) |
| = *x* ∨ (¬*x* ∧ 1) |
| = *x* ∨ ¬*x* |
| = 1 |
+-----------------------------------------------------------------------------------------------------+
| **Abs~1~** *x* ∨ (*x* ∧ *y*) = *x* |
| ------------ --- ------------------------- |
| Proof: *x* ∨ (*x* ∧ *y*) |
| = (*x* ∧ 1) ∨ (*x* ∧ *y*) |
| = *x* ∧ (1 ∨ *y*) |
| = *x* ∧ (*y* ∨ 1) |
| = *x* ∧ 1 |
| = *x* |
+-----------------------------------------------------------------------------------------------------+
| **UNg** If *x* ∨ *x*~n~ = 1 and *x* ∧ *x*~n~ = 0, then *x*~n~ = ¬*x* |
| --------- --- -------------------------------------------------------------- |
| Proof: If *x* ∨ *x*~n~ = 1 and *x* ∧ *x*~n~ = 0, then |
| *x*~n~ |
| = *x*~n~ ∧ 1 |
| = *x*~n~ ∧ (*x* ∨ ¬*x*) |
| = (*x*~n~ ∧ *x*) ∨ (*x*~n~ ∧ ¬*x*) |
| = (*x* ∧ *x*~n~) ∨ (¬*x* ∧ *x*~n~) |
| = 0 ∨ (¬*x* ∧ *x*~n~) |
| = (*x* ∧ ¬*x*) ∨ (¬*x* ∧ *x*~n~) |
| = (¬*x* ∧ *x*) ∨ (¬*x* ∧ *x*~n~) |
| = ¬*x* ∧ (*x* ∨ *x*~n~) |
| = ¬*x* ∧ 1 |
| = ¬*x* |
+-----------------------------------------------------------------------------------------------------+
| **DNg** ¬¬*x* = *x* |
| --------- ------- ----------------------------- |
| Proof: ¬*x* ∨ *x* = *x* ∨ ¬*x* = 1 |
| and ¬*x* ∧ *x* = *x* ∧ ¬*x* = 0 |
| hence *x* = ¬¬*x* |
+-----------------------------------------------------------------------------------------------------+
| **A~1~** *x* ∨ (¬*x* ∨ *y*) = 1 |
| ---------- --- ------------------------------------- |
| Proof: *x* ∨ (¬*x* ∨ *y*) |
| = (*x* ∨ (¬*x* ∨ *y*)) ∧ 1 |
| = 1 ∧ (*x* ∨ (¬*x* ∨ *y*)) |
| = (*x* ∨ ¬*x*) ∧ (*x* ∨ (¬*x* ∨ *y*)) |
| = *x* ∨ (¬*x* ∧ (¬*x* ∨ *y*)) |
| = *x* ∨ ¬*x* |
| = 1 |
+-----------------------------------------------------------------------------------------------------+
| **B~1~** (*x* ∨ *y*) ∨ (¬*x* ∧ ¬*y*) = 1 |
| ---------- --- ------------------------------------------------- |
| Proof: (*x* ∨ *y*) ∨ (¬*x* ∧ ¬*y*) |
| = ((*x* ∨ *y*) ∨ ¬*x*) ∧ ((*x* ∨ *y*) ∨ ¬*y*) |
| = (¬*x* ∨ (*x* ∨ *y*)) ∧ (¬*y* ∨ (*y* ∨ *x*)) |
| = (¬*x* ∨ (¬¬*x* ∨ *y*)) ∧ (¬*y* ∨ (¬¬*y* ∨ *x*)) |
| = 1 ∧ 1 |
| = 1 |
+-----------------------------------------------------------------------------------------------------+
| **C~1~** (*x* ∨ *y*) ∧ (¬*x* ∧ ¬*y*) = 0 |
| ---------- --- ----------------------------------------------- |
| Proof: (*x* ∨ *y*) ∧ (¬*x* ∧ ¬*y*) |
| = (¬*x* ∧ ¬*y*) ∧ (*x* ∨ *y*) |
| = ((¬*x* ∧ ¬*y*) ∧ *x*) ∨ ((¬*x* ∧ ¬*y*) ∧ *y*) |
| = (*x* ∧ (¬*x* ∧ ¬*y*)) ∨ (*y* ∧ (¬*y* ∧ ¬*x*)) |
| = 0 ∨ 0 |
| = 0 |
+-----------------------------------------------------------------------------------------------------+
| **DMg~1~** ¬(*x* ∨ *y*) = ¬*x* ∧ ¬*y* |
| ------------ -- ------------------------------------ |
| Proof: by **B~1~**, **C~1~**, and **UNg** |
+-----------------------------------------------------------------------------------------------------+
| **D~1~** (*x*∨(*y*∨*z*)) ∨ ¬*x* = 1 |
| ---------- --- ------------------------------ |
| Proof: (*x* ∨ (*y* ∨ *z*)) ∨ ¬*x* |
| = ¬*x* ∨ (*x* ∨ (*y* ∨ *z*)) |
| = ¬*x* ∨ (¬¬*x* ∨ (*y* ∨ *z*)) |
| = 1 |
+-----------------------------------------------------------------------------------------------------+
| **E~1~** *y* ∧ (*x*∨(*y*∨*z*)) = *y* |
| ---------- --- ----------------------------------- |
| Proof: *y* ∧ (*x* ∨ (*y* ∨ *z*)) |
| = (*y* ∧ *x*) ∨ (*y* ∧ (*y* ∨ *z*)) |
| = (*y* ∧ *x*) ∨ *y* |
| = *y* ∨ (*y* ∧ *x*) |
| = *y* |
+-----------------------------------------------------------------------------------------------------+
| **F~1~** (*x*∨(*y*∨*z*)) ∨ ¬*y* = 1 |
| ---------- --- --------------------------------------------- |
| Proof: (*x* ∨ (*y* ∨ *z*)) ∨ ¬*y* |
| = ¬*y* ∨ (*x* ∨ (*y* ∨ *z*)) |
| = (¬*y* ∨ (*x* ∨ (*y* ∨ *z*))) ∧ 1 |
| = 1 ∧ (¬*y* ∨ (*x* ∨ (*y* ∨ *z*))) |
| = (*y* ∨ ¬*y*) ∧ (¬*y* ∨ (*x* ∨ (*y* ∨ *z*))) |
| = (¬*y* ∨ *y*) ∧ (¬*y* ∨ (*x* ∨ (*y* ∨ *z*))) |
| = ¬*y* ∨ (*y* ∧ (*x* ∨ (*y* ∨ *z*))) |
| = ¬*y* ∨ *y* |
| = *y* ∨ ¬*y* |
| = 1 |
+-----------------------------------------------------------------------------------------------------+
| **G~1~** (*x*∨(*y*∨*z*)) ∨ ¬*z* = 1 |
| ---------- --- ---------------------------- |
| Proof: (*x* ∨ (*y* ∨ *z*)) ∨ ¬*z* |
| = (*x* ∨ (*z* ∨ *y*)) ∨ ¬*z* |
| = 1 |
+-----------------------------------------------------------------------------------------------------+
| **H~1~** ¬((*x*∨*y*)∨*z*) ∧ *x* = 0 |
| ---------- --- ----------------------------------------------- |
| Proof: ¬((*x* ∨ *y*) ∨ *z*) ∧ *x* |
| = (¬(*x* ∨ *y*) ∧ ¬*z*) ∧ *x* |
| = ((¬*x* ∧ ¬*y*) ∧ ¬*z*) ∧ *x* |
| = *x* ∧ ((¬*x* ∧ ¬*y*) ∧ ¬*z*) |
| = (*x* ∧ ((¬*x* ∧ ¬*y*) ∧ ¬*z*)) ∨ 0 |
| = 0 ∨ (*x* ∧ ((¬*x* ∧ ¬*y*) ∧ ¬*z*)) |
| = (*x* ∧ ¬*x*) ∨ (*x* ∧ ((¬*x* ∧ ¬*y*) ∧ ¬*z*)) |
| = *x* ∧ (¬*x* ∨ ((¬*x* ∧ ¬*y*) ∧ ¬*z*)) |
| = *x* ∧ (¬*x* ∨ (¬*z* ∧ (¬*x* ∧ ¬*y*))) |
| = *x* ∧ ¬*x* |
| = 0 |
+-----------------------------------------------------------------------------------------------------+
| **I~1~** ¬((*x*∨*y*)∨*z*) ∧ *y* = 0 |
| ---------- --- ---------------------------- |
| Proof: ¬((*x* ∨ *y*) ∨ *z*) ∧ *y* |
| = ¬((*y* ∨ *x*) ∨ *z*) ∧ *y* |
| = 0 |
+-----------------------------------------------------------------------------------------------------+
| **J~1~** ¬((*x*∨*y*)∨*z*) ∧ *z* = 0 |
| ---------- --- ------------------------------ |
| Proof: ¬((*x* ∨ *y*) ∨ *z*) ∧ *z* |
| = (¬(*x* ∨ *y*) ∧ ¬*z*) ∧ *z* |
| = *z* ∧ (¬(*x* ∨ *y*) ∧ ¬*z*) |
| = *z* ∧ ( ¬*z* ∧ ¬(*x* ∨ *y*)) |
| = 0 |
+-----------------------------------------------------------------------------------------------------+
| **K~1~** (*x* ∨ (*y* ∨ *z*)) ∨ ¬((*x* ∨ *y*) ∨ *z*) = 1 |
| ---------- --- ---------------------------------------------------------------------------------- |
| Proof: (*x*∨(*y*∨*z*)) ∨ ¬((*x* ∨ *y*) ∨ *z*) |
| = (*x*∨(*y*∨*z*)) ∨ (¬(*x* ∨ *y*) ∧ ¬*z*) |
| = (*x*∨(*y*∨*z*)) ∨ ((¬*x* ∧ ¬*y*) ∧ ¬*z*) |
| = ((*x*∨(*y*∨*z*)) ∨ (¬*x* ∧ ¬*y*)) ∧ ((*x*∨(*y*∨*z*)) ∨ ¬*z*) |
| = (((*x*∨(*y*∨*z*)) ∨ ¬*x*) ∧ ((*x*∨(*y*∨*z*)) ∨ ¬*y*)) ∧ ((*x*∨(*y*∨*z*)) ∨ ¬*z*) |
| = (1 ∧ 1) ∧ 1 |
| = 1 |
+-----------------------------------------------------------------------------------------------------+
| **L~1~** (*x* ∨ (*y* ∨ *z*)) ∧ ¬((*x* ∨ *y*) ∨ *z*) = 0 |
| ---------- --- ---------------------------------------------------------------------------------- |
| Proof: (*x* ∨ (*y* ∨ *z*)) ∧ ¬((*x* ∨ *y*) ∨ *z*) |
| = ¬((*x*∨*y*)∨*z*) ∧ (*x* ∨ (*y* ∨ *z*)) |
| = (¬((*x*∨*y*)∨*z*) ∧ *x*) ∨ (¬((*x*∨*y*)∨*z*) ∧ (*y* ∨ *z*)) |
| = (¬((*x*∨*y*)∨*z*) ∧ *x*) ∨ ((¬((*x*∨*y*)∨*z*) ∧ *y*) ∨ (¬((*x*∨*y*)∨*z*) ∧ *z*)) |
| = 0 ∨ (0 ∨ 0) |
| = 0 |
+-----------------------------------------------------------------------------------------------------+
| **Ass~1~** *x* ∨ (*y* ∨ *z*) = (*x* ∨ *y*) ∨ *z* |
| ------------ -- ----------------------------------------- |
| Proof: by **K~1~**, **L~1~**, **UNg**, **DNg** |
+-----------------------------------------------------------------------------------------------------+
| Abbreviations |
| --------------- |
| **UId** |
| **Idm** |
| **Bnd** |
| **Abs** |
| **UNg** |
| **DNg** |
| **DMg** |
| **Ass** |
+-----------------------------------------------------------------------------------------------------+
+--------------------------------------------+
| **Huntington 1904 Boolean algebra axioms** |
+============================================+
| **Idn~1~** |
+--------------------------------------------+
| **Cmm~1~** |
+--------------------------------------------+
| **Dst~1~** |
+--------------------------------------------+
| **Cpl~1~** |
+--------------------------------------------+
| Abbreviations |
| --------------- |
| **Idn** |
| **Cmm** |
| **Dst** |
| **Cpl** |
+--------------------------------------------+
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# Boolean algebra (structure)
## Axiomatics
The first axiomatization of Boolean lattices/algebras in general was given by the English philosopher and mathematician Alfred North Whitehead in 1898. It included the above axioms and additionally `{{math|1=''x'' ∨ 1 = 1}}`{=mediawiki} and `{{math|1=''x'' ∧ 0 = 0}}`{=mediawiki}. In 1904, the American mathematician Edward V. Huntington (1874--1952) gave probably the most parsimonious axiomatization based on `{{math|1=∧}}`{=mediawiki}, `{{math|1=∨}}`{=mediawiki}, `{{math|1=¬}}`{=mediawiki}, even proving the associativity laws (see box). He also proved that these axioms are independent of each other. In 1933, Huntington set out the following elegant axiomatization for Boolean algebra. It requires just one binary operation `{{math|1=+}}`{=mediawiki} and a unary functional symbol `{{math|1=''n''}}`{=mediawiki}, to be read as \'complement\', which satisfy the following laws: `{{olist
|1= ''Commutativity'': {{math|1=''x'' + ''y'' = ''y'' + ''x''}}.
|2= ''Associativity'': {{math|1=(''x'' + ''y'') + ''z'' = ''x'' + (''y'' + ''z'')}}.
|3= ''Huntington equation'': {{math|1=''n''(''n''(''x'') + ''y'') + ''n''(''n''(''x'') + ''n''(''y'')) = ''x''}}.
}}`{=mediawiki} Herbert Robbins immediately asked: If the Huntington equation is replaced with its dual, to wit: `{{olist|start=4
|1= ''Robbins Equation'': {{math|1=''n''(''n''(''x'' + ''y'') + ''n''(''x'' + ''n''(''y''))) = ''x''}},
}}`{=mediawiki} do (1), (2), and (4) form a basis for Boolean algebra? Calling (1), (2), and (4) a *Robbins algebra*, the question then becomes: Is every Robbins algebra a Boolean algebra? This question (which came to be known as the Robbins conjecture) remained open for decades, and became a favorite question of Alfred Tarski and his students. In 1996, William McCune at Argonne National Laboratory, building on earlier work by Larry Wos, Steve Winker, and Bob Veroff, answered Robbins\'s question in the affirmative: Every Robbins algebra is a Boolean algebra. Crucial to McCune\'s proof was the computer program EQP he designed. For a simplification of McCune\'s proof, see Dahn (1998).
Further work has been done for reducing the number of axioms; see Minimal axioms for Boolean algebra.
## Generalizations
Removing the requirement of existence of a unit from the axioms of Boolean algebra yields \"generalized Boolean algebras\". Formally, a distributive lattice `{{math|1=''B''}}`{=mediawiki} is a generalized Boolean lattice, if it has a smallest element `{{math|1=0}}`{=mediawiki} and for any elements `{{math|1=''a''}}`{=mediawiki} and `{{math|1=''b''}}`{=mediawiki} in `{{math|1=''B''}}`{=mediawiki} such that `{{math|1=''a'' ≤ ''b''}}`{=mediawiki}, there exists an element `{{math|1=''x''}}`{=mediawiki} such that `{{math|1=''a'' ∧ ''x'' = 0}}`{=mediawiki} and `{{math|1=''a'' ∨ ''x'' = ''b''}}`{=mediawiki}. Defining `{{math|1=''a'' \ ''b''}}`{=mediawiki} as the unique `{{math|1=''x''}}`{=mediawiki} such that `{{math|1=(''a'' ∧ ''b'') ∨ ''x'' = ''a''}}`{=mediawiki} and `{{math|1=(''a'' ∧ ''b'') ∧ ''x'' = 0}}`{=mediawiki}, we say that the structure `{{math|(''B'', ∧, ∨, \, 0)}}`{=mediawiki} is a *generalized Boolean algebra*, while `{{math|(''B'', ∨, 0)}}`{=mediawiki} is a *generalized Boolean semilattice*. Generalized Boolean lattices are exactly the ideals of Boolean lattices.
A structure that satisfies all axioms for Boolean algebras except the two distributivity axioms is called an orthocomplemented lattice. Orthocomplemented lattices arise naturally in quantum logic as lattices of closed linear subspaces for separable Hilbert spaces
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# Bill Joy
vi{{•}}csh{{•}}chroot{{•}}TCP/IP driver{{•}}co-founder of Sun Microsystems{{•}}Java{{•}}SPARC{{•}}Solaris{{•}}NFS{{•}}*Why The Future Doesn\'t Need Us* \| author_abbrev_bot = \| author_abbrev_zoo = \| influences = \| influenced = \| spouse = \| children = 2 \| awards = \*ACM Grace Murray Hopper Award (1986)
- Elected to National Academy of Engineering (1999)
- Elected to American Academy of Arts and Sciences (1999)
- Fellow of the Computer History Museum (2011)
\| education = University of Michigan (BS)\
University of California, Berkeley (MS) }}
**William Nelson Joy** (born November 8, 1954) is an American computer engineer and venture capitalist. He co-founded Sun Microsystems in 1982 along with Scott McNealy, Vinod Khosla, and Andy Bechtolsheim, and served as Chief Scientist and CTO at the company until 2003.
He played an integral role in the early development of BSD UNIX while being a graduate student at Berkeley, and he is the original author of the vi text editor. He also wrote the 2000 essay \"Why The Future Doesn\'t Need Us\", in which he expressed deep concerns over the development of modern technologies.
Joy was elected a member of the National Academy of Engineering (1999) for contributions to operating systems and networking software.
## Early career {#early_career}
Joy was born in the Detroit suburb of Farmington Hills, Michigan, to William Joy, a school vice-principal and counselor, and Ruth Joy. He earned a Bachelor of Science in electrical engineering from the University of Michigan and a Master of Science in electrical engineering and computer science from the University of California, Berkeley, in 1979.
While a graduate student at Berkeley, he worked for Fabry\'s Computer Systems Research Group (CSRG) on the Berkeley Software Distribution (BSD) version of the Unix operating system. He initially worked on a Pascal compiler left at Berkeley by Ken Thompson, who had been visiting the university when Joy had just started his graduate work.
He later moved on to improving the Unix kernel, and also handled BSD distributions. Some of his most notable contributions were the ex and vi editors and the C shell. Joy\'s prowess as a computer programmer is legendary, with an oft-told anecdote that he wrote the vi editor in a weekend. Joy denies this assertion. A few of his other accomplishments have also been sometimes exaggerated; Eric Schmidt, CEO of Novell at the time, inaccurately reported during an interview in PBS\'s documentary *Nerds 2.0.1* that Joy had personally rewritten the BSD kernel in a weekend. In 1980, he also wrote `cat -v`, about which Rob Pike and Brian W. Kernighan wrote that it went against Unix philosophy.
According to a *Salon* article, during the early 1980s, DARPA had contracted the company Bolt, Beranek and Newman (BBN) to add TCP/IP to Berkeley UNIX. Joy had been instructed to plug BBN\'s stack into Berkeley Unix, but he refused to do so, as he had a low opinion of BBN\'s TCP/IP. So, Joy wrote his own high-performance TCP/IP stack. According to John Gage:
Rob Gurwitz, who was working at BBN at the time, disputes this version of events.
## Sun Microsystems {#sun_microsystems}
In 1982, after the firm had been going for six months, Joy, Sun\'s sixteenth employee, was brought in with full co-founder status at Sun Microsystems. At Sun, Joy was an inspiration for the development of NFS, the SPARC microprocessors, the Java programming language, Jini/JavaSpaces, and JXTA.
In 1986, Joy was awarded a Grace Murray Hopper Award by the ACM for his work on the Berkeley UNIX Operating System.
On September 9, 2003, Sun announced Joy was leaving the company and that he \"is taking time to consider his next move and has no definite plans\".
## Post-Sun activities {#post_sun_activities}
In 1999, Joy co-founded a venture capital firm, HighBAR Ventures, with two Sun colleagues, Andy Bechtolsheim and Roy Thiele-Sardiña. In January 2005 he was named a partner in venture capital firm Kleiner Perkins. There, Joy has made several investments in green energy industries, even though he does not have any credentials in the field. He once said, \"My method is to look at something that seems like a good idea and assume it\'s true\".
In 2011, he was inducted as a Fellow of the Computer History Museum for his work on the Berkeley Software Distribution (BSD) Unix system and the co-founding of Sun Microsystems.
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# Bill Joy
## Technology concerns {#technology_concerns}
In 2000, Joy gained notoriety with the publication of his article in *Wired* magazine, \"Why The Future Doesn\'t Need Us\", in which he declared, in what some have described as a \"neo-Luddite\" position, that he was convinced that growing advances in genetic engineering and nanotechnology would bring risks to humanity. He argued that intelligent robots would replace humanity, at the very least in intellectual and social dominance, in the relatively near future. He supports and promotes the idea of abandonment of GNR (genetics, nanotechnology, and robotics) technologies, instead of going into an arms race between negative uses of the technology and defense against those negative uses (good nano-machines patrolling and defending against Grey goo \"bad\" nano-machines). This stance of broad relinquishment was criticized by technologists such as technological-singularity thinker Ray Kurzweil, who instead advocates fine-grained relinquishment and ethical guidelines. Joy was also criticized by *The American Spectator*, which characterized Joy\'s essay as a (possibly unwitting) rationale for statism.
A bar-room discussion of these technologies with Ray Kurzweil started to set Joy\'s thinking along this path. He states in his essay that during the conversation, he became surprised that other serious scientists were considering such possibilities likely, and even more astounded at what he felt was a lack of consideration of the contingencies. After bringing the subject up with a few more acquaintances, he states that he was further alarmed by what he felt was that although many people considered these futures possible or probable, that very few of them shared as serious a concern for the dangers as he seemed to. This concern led to his in-depth examination of the issue and the positions of others in the scientific community on it, and eventually, to his current activities regarding it.
Despite this, he is a venture capitalist, investing in `{{Abbr|GNR|genetics, nanotechnology, and robotics}}`{=mediawiki} technology companies. He has also raised a specialty venture fund to address the dangers of pandemic diseases, such as the H5N1 avian influenza and biological weapons.
## Joy\'s law {#joys_law}
### Of management {#of_management}
In his 2013 book *Makers*, author Chris Anderson credited Joy with establishing \"Joy\'s law\" based on a quip: \"No matter who you are, most of the smartest people work for someone else \[other than you\].\" His argument was that companies use an inefficient process by not hiring the best employees, only those they are able to hire. His \"law\" was a continuation of Friedrich Hayek\'s \"The Use of Knowledge in Society\" and warned that the competition outside of a company would always have the potential to be greater than the company itself.
### Of computing {#of_computing}
Joy devised a formula in 1983, also called *Joy\'s law*, stating that the peak computer speed doubles each year and thus is given by a simple function of time. Specifically,
$$S = 2^{Y-1984},$$ in which `{{mvar|S}}`{=mediawiki} is the peak computer speed attained during year `{{mvar|Y}}`{=mediawiki}, expressed in MIPS
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# Bandwidth (signal processing)
**Bandwidth** is the difference between the upper and lower frequencies in a continuous band of frequencies. It is typically measured in unit of hertz (symbol Hz).
It may refer more specifically to two subcategories: *Passband bandwidth* is the difference between the upper and lower cutoff frequencies of, for example, a band-pass filter, a communication channel, or a signal spectrum. *Baseband bandwidth* is equal to the upper cutoff frequency of a low-pass filter or baseband signal, which includes a zero frequency.
Bandwidth in hertz is a central concept in many fields, including electronics, information theory, digital communications, radio communications, signal processing, and spectroscopy and is one of the determinants of the capacity of a given communication channel.
A key characteristic of bandwidth is that any band of a given width can carry the same amount of information, regardless of where that band is located in the frequency spectrum. For example, a 3 kHz band can carry a telephone conversation whether that band is at baseband (as in a POTS telephone line) or modulated to some higher frequency. However, wide bandwidths are easier to obtain and process at higher frequencies because the `{{section link|#Fractional bandwidth}}`{=mediawiki} is smaller.
## Overview
Bandwidth is a key concept in many telecommunications applications. In radio communications, for example, bandwidth is the frequency range occupied by a modulated carrier signal. An FM radio receiver\'s tuner spans a limited range of frequencies. A government agency (such as the Federal Communications Commission in the United States) may apportion the regionally available bandwidth to broadcast license holders so that their signals do not mutually interfere. In this context, bandwidth is also known as channel spacing.
For other applications, there are other definitions. One definition of bandwidth, for a system, could be the range of frequencies over which the system produces a specified level of performance. A less strict and more practically useful definition will refer to the frequencies beyond which performance is degraded. In the case of frequency response, degradation could, for example, mean more than 3 dB below the maximum value or it could mean below a certain absolute value. As with any definition of the *width* of a function, many definitions are suitable for different purposes.
In the context of, for example, the sampling theorem and Nyquist sampling rate, bandwidth typically refers to baseband bandwidth. In the context of Nyquist symbol rate or Shannon-Hartley channel capacity for communication systems it refers to passband bandwidth.
The **`{{vanchor|Rayleigh bandwidth}}`{=mediawiki}** of a simple radar pulse is defined as the inverse of its duration. For example, a one-microsecond pulse has a Rayleigh bandwidth of one megahertz.
The **`{{vanchor|essential bandwidth}}`{=mediawiki}** is defined as the portion of a signal spectrum in the frequency domain which contains most of the energy of the signal.
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# Bandwidth (signal processing)
## *x* dB bandwidth {#x_db_bandwidth}
In some contexts, the signal bandwidth in hertz refers to the frequency range in which the signal\'s spectral density (in W/Hz or V^2^/Hz) is nonzero or above a small threshold value. The threshold value is often defined relative to the maximum value, and is most commonly the `{{no wrap|[[3 dB point]]}}`{=mediawiki}, that is the point where the spectral density is half its maximum value (or the spectral amplitude, in $\mathrm{V}$ or $\mathrm{V/\sqrt{Hz}}$, is 70.7% of its maximum). This figure, with a lower threshold value, can be used in calculations of the lowest sampling rate that will satisfy the sampling theorem.
The bandwidth is also used to denote **system bandwidth**, for example in filter or communication channel systems. To say that a system has a certain bandwidth means that the system can process signals with that range of frequencies, or that the system reduces the bandwidth of a white noise input to that bandwidth.
The 3 dB bandwidth of an electronic filter or communication channel is the part of the system\'s frequency response that lies within 3 dB of the response at its peak, which, in the passband filter case, is typically at or near its center frequency, and in the low-pass filter is at or near its cutoff frequency. If the maximum gain is 0 dB, the 3 dB bandwidth is the frequency range where attenuation is less than 3 dB. 3 dB attenuation is also where power is half its maximum. This same *half-power gain* convention is also used in spectral width, and more generally for the extent of functions as full width at half maximum (FWHM).
In electronic filter design, a filter specification may require that within the filter passband, the gain is nominally 0 dB with a small variation, for example within the ±1 dB interval. In the stopband(s), the required attenuation in decibels is above a certain level, for example \>100 dB. In a transition band the gain is not specified. In this case, the filter bandwidth corresponds to the passband width, which in this example is the 1 dB-bandwidth. If the filter shows amplitude ripple within the passband, the *x* dB point refers to the point where the gain is *x* dB below the nominal passband gain rather than *x* dB below the maximum gain.
In signal processing and control theory the bandwidth is the frequency at which the closed-loop system gain drops 3 dB below peak.
In communication systems, in calculations of the Shannon--Hartley channel capacity, bandwidth refers to the 3 dB-bandwidth. In calculations of the maximum symbol rate, the Nyquist sampling rate, and maximum bit rate according to the Hartley\'s law, the bandwidth refers to the frequency range within which the gain is non-zero.
The fact that in equivalent baseband models of communication systems, the signal spectrum consists of both negative and positive frequencies, can lead to confusion about bandwidth since they are sometimes referred to only by the positive half, and one will occasionally see expressions such as $B = 2W$, where $B$ is the total bandwidth (i.e. the maximum passband bandwidth of the carrier-modulated RF signal and the minimum passband bandwidth of the physical passband channel), and $W$ is the positive bandwidth (the baseband bandwidth of the equivalent channel model). For instance, the baseband model of the signal would require a low-pass filter with cutoff frequency of at least $W$ to stay intact, and the physical passband channel would require a passband filter of at least $B$ to stay intact.
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# Bandwidth (signal processing)
## Relative bandwidth {#relative_bandwidth}
The absolute bandwidth is not always the most appropriate or useful measure of bandwidth. For instance, in the field of antennas the difficulty of constructing an antenna to meet a specified absolute bandwidth is easier at a higher frequency than at a lower frequency. For this reason, bandwidth is often quoted relative to the frequency of operation which gives a better indication of the structure and sophistication needed for the circuit or device under consideration.
There are two different measures of relative bandwidth in common use: *fractional bandwidth* ($B_\mathrm F$) and *ratio bandwidth* ($B_\mathrm R$). In the following, the absolute bandwidth is defined as follows, $B = \Delta f = f_\mathrm H - f_\mathrm L$ where $f_\mathrm H$ and $f_\mathrm L$ are the upper and lower frequency limits respectively of the band in question.
### Fractional bandwidth {#fractional_bandwidth}
Fractional bandwidth is defined as the absolute bandwidth divided by the center frequency ($f_\mathrm C$), $B_\mathrm F = \frac {\Delta f}{f_\mathrm C} \, .$
The center frequency is usually defined as the arithmetic mean of the upper and lower frequencies so that, $f_\mathrm C = \frac {f_\mathrm H + f_\mathrm L}{2} \$ and $B_\mathrm F = \frac {2 (f_\mathrm H - f_\mathrm L)}{f_\mathrm H + f_\mathrm L} \, .$
However, the center frequency is sometimes defined as the geometric mean of the upper and lower frequencies, $f_\mathrm C = \sqrt {f_\mathrm H f_\mathrm L}$ and $B_\mathrm F = \frac {f_\mathrm H - f_\mathrm L}{\sqrt {f_\mathrm H f_\mathrm L}} \, .$
While the geometric mean is more rarely used than the arithmetic mean (and the latter can be assumed if not stated explicitly) the former is considered more mathematically rigorous. It more properly reflects the logarithmic relationship of fractional bandwidth with increasing frequency. For narrowband applications, there is only marginal difference between the two definitions. The geometric mean version is inconsequentially larger. For wideband applications they diverge substantially with the arithmetic mean version approaching 2 in the limit and the geometric mean version approaching infinity.
Fractional bandwidth is sometimes expressed as a percentage of the center frequency (**percent bandwidth**, $\%B$), $\%B_\mathrm F = 100 \frac {\Delta f}{f_\mathrm C} \, .$
### Ratio bandwidth {#ratio_bandwidth}
Ratio bandwidth is defined as the ratio of the upper and lower limits of the band, $B_\mathrm R= \frac {f_\mathrm H}{f_\mathrm L} \, .$
Ratio bandwidth may be notated as $B_\mathrm R:1$. The relationship between ratio bandwidth and fractional bandwidth is given by, $B_\mathrm F = 2 \frac {B_\mathrm R - 1}{B_\mathrm R + 1}$ and $B_\mathrm R = \frac {2 + B_\mathrm F}{2 - B_\mathrm F} \, .$
Percent bandwidth is a less meaningful measure in wideband applications. A percent bandwidth of 100% corresponds to a ratio bandwidth of 3:1. All higher ratios up to infinity are compressed into the range 100--200%.
Ratio bandwidth is often expressed in octaves (i.e., as a frequency level) for wideband applications. An octave is a frequency ratio of 2:1 leading to this expression for the number of octaves, $\log_2 \left(B_\mathrm R\right) .$
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# Bandwidth (signal processing)
## Noise equivalent bandwidth {#noise_equivalent_bandwidth}
Further information: Spectral leakage#Noise bandwidth
The **noise equivalent bandwidth** (or **equivalent noise bandwidth (enbw)**) of a system of frequency response $H(f)$ is the bandwidth of an ideal filter with rectangular frequency response centered on the system\'s central frequency that produces the same average power outgoing $H(f)$ when both systems are excited with a white noise source. The value of the noise equivalent bandwidth depends on the ideal filter reference gain used. Typically, this gain equals $|H(f)|$ at its center frequency, but it can also equal the peak value of $|H(f)|$.
The noise equivalent bandwidth $B_n$ can be calculated in the frequency domain using $H(f)$ or in the time domain by exploiting the Parseval\'s theorem with the system impulse response $h(t)$. If $H(f)$ is a lowpass system with zero central frequency and the filter reference gain is referred to this frequency, then:
$B_n = \frac{\int_{-\infty}^{\infty} |H(f)|^2 df}{2|H(0)|^2} = \frac{\int_{-\infty}^{\infty} |h(t)|^2 dt}{2\left|\int_{-\infty}^{\infty} h(t)dt\right|^2} \, .$
The same expression can be applied to bandpass systems by substituting the equivalent baseband frequency response for $H(f)$.
The noise equivalent bandwidth is widely used to simplify the analysis of telecommunication systems in the presence of noise.
## Photonics
In photonics, the term *bandwidth* carries a variety of meanings:
- the bandwidth of the output of some light source, e.g., an ASE source or a laser; the bandwidth of ultrashort optical pulses can be particularly large
- the width of the frequency range that can be transmitted by some element, e.g. an optical fiber
- the gain bandwidth of an optical amplifier
- the width of the range of some other phenomenon, e.g., a reflection, the phase matching of a nonlinear process, or some resonance
- the maximum modulation frequency (or range of modulation frequencies) of an optical modulator
- the range of frequencies in which some measurement apparatus (e.g., a power meter) can operate
- the data rate (e.g., in Gbit/s) achieved in an optical communication system; see bandwidth (computing).
A related concept is the spectral linewidth of the radiation emitted by excited atoms
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# Bernie Federko
**Bernard Allan Federko** (born May 12, 1956) is a Canadian former professional ice hockey centre who played fourteen seasons in the National Hockey League from 1976 through 1990.
## Playing career {#playing_career}
Federko began playing hockey at a young age in his home town of Foam Lake, Saskatchewan. He was captain of the 1971 Bantam provincial champions. He also played Senior hockey with the local Foam Lake Flyers of the Fishing Lake Hockey League, winning the league scoring title as a bantam-aged player. Federko continued his career with the Saskatoon Blades of the WHL where he set and still holds the team record for assists. He played three seasons with the Blades, and in his final year with the club he led the league in assists and points in both the regular season *and* playoffs. Federko was drafted 7th overall by the St. Louis Blues in the 1976 NHL Amateur Draft. He started the next season with the Kansas City Blues of the Central Hockey League and was leading the league in points when he was called up mid-season to play 31 games with St. Louis. He scored three hat tricks in those 31 games. In the 1978--79 NHL season, Federko developed into a bona fide star, as he scored 95 points.
Federko scored 100 points in a season four times, and was a consistent and underrated performer for the Blues. Federko scored at least 90 points in seven of the eight seasons between 1978 and 1986, and became the first player in NHL history to record at least 50 assists in 10 consecutive seasons. However, in an era when Wayne Gretzky was scoring 200 points a season, Federko never got the attention many felt he deserved. In 1986, in a poll conducted by GOAL magazine, he was named the most overlooked talent in hockey. His General Manager Ron Caron said he was \"A great playmaker. He makes the average or above average player look like a star at times. He\'s such an unselfish player.\"
On March 19, 1988, Federko became the 22nd NHL player to record 1000 career points. After he had a poor season as a captain in 1988--89, he was traded to the Detroit Red Wings with Tony McKegney for future Blues star Adam Oates, and Paul MacLean. In Detroit, Federko re-united with former Blues head coach Jacques Demers, but he had to play behind Steve Yzerman and did not get his desired ice time. After his lowest point output since his rookie season, Federko decided to retire after the 1989--90 season, having played exactly 1,000 NHL games with his final game on April 1, 1990.
## Post-NHL career {#post_nhl_career}
Less than a year after retiring as a player, the Blues retired number **24** in his honour on March 16, 1991. Federko was eventually inducted into the Hockey Hall of Fame in 2002, the first Hall of Famer to earn his credentials primarily as a Blue.
Currently, Federko is a television colour commentator and studio analyst for Bally Sports Midwest during Blues broadcasts. He was the head coach/general manager of the St. Louis Vipers roller hockey team of the Roller Hockey International for the 1993 and 1994 seasons.
## Career statistics {#career_statistics}
### Regular season and playoffs {#regular_season_and_playoffs}
Regular season
------------ ------------------- -------- ------- ----------------
Season Team League GP G
1973--74 Saskatoon Blades WCHL 68 22
1974--75 Saskatoon Blades WCHL 66 39
1975--76 Saskatoon Blades WCHL 72 72
1976--77 Kansas City Blues CHL 42 30
1976--77 St. Louis Blues NHL 31 14
1977--78 St. Louis Blues NHL 72 17
1978--79 St. Louis Blues NHL 74 31
1979--80 St. Louis Blues NHL 79 38
1980--81 St. Louis Blues NHL 78 31
1981--82 St. Louis Blues NHL 74 30
1982--83 St. Louis Blues NHL 75 24
1983--84 St. Louis Blues NHL 79 41
1984--85 St. Louis Blues NHL 76 30
1985--86 St. Louis Blues NHL 80 34
1986--87 St. Louis Blues NHL 64 20
1987--88 St. Louis Blues NHL 79 20
1988--89 St. Louis Blues NHL 66 22
1989--90 Detroit Red Wings NHL 73 17
NHL totals 1,000 369
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# Bernie Federko
## Awards
- Bob Brownridge Memorial Trophy (WCHL leading scorer) - 1976
- Named to the WCHL First All-Star Team (1976)
- Named WCHL MVP (1976)
- Named to the CHL Second All-Star Team (1977)
- Won Ken McKenzie Trophy as CHL Rookie of the Year (1977)
- Played in the NHL All-Star Game (1980, 1981)
- Named NHL Player of the Week (For week ending December 3, 1984)
## Records
- St. Louis Blues team record for career games played (927)
- St. Louis Blues team record for career assists (721)
- St. Louis Blues team record for career points (1073)
- Shares St. Louis Blues team record for assists in one game (5 on February 27, 1988)
- St. Louis Blues team record for career playoff assists (66)
- First NHL player to get 50 assists in 10 consecutive seasons
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# Contract bridge
*Pandoc failed*: ```
Error at (line 83, column 2):
unexpected '+'
|+ {{center|'''Example auction'''}}
^
``
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# Basel Convention
The **Basel Convention on the Control of Transboundary Movements of Hazardous Wastes and Their Disposal**, usually known as the **Basel Convention**, is an international treaty that was designed to reduce the movements of hazardous waste between nations, and specifically to restrict the transfer of hazardous waste from developed to less developed countries. It does not address the movement of radioactive waste, controlled by the International Atomic Energy Agency. The Basel Convention is also intended to minimize the rate and toxicity of wastes generated, to ensure their environmentally sound management as closely as possible to the source of generation, and to assist developing countries in environmentally sound management of the hazardous and other wastes they generate.
The convention was opened for signature on 21 March 1989, and entered into force on 5 May 1992. As of June 2024, there are 191 parties to the convention. In addition, Haiti and the United States have signed the convention but did not ratify it.
Following a petition urging action on the issue signed by more than a million people around the world, most of the world\'s countries, but not the United States, agreed in May 2019 to an amendment of the Basel Convention to include plastic waste as regulated material. Although the United States is not a party to the treaty, export shipments of plastic waste from the United States are now \"criminal traffic as soon as the ships get on the high seas,\" according to the Basel Action Network (BAN), and carriers of such shipments may face liability, because the transportation of plastic waste is prohibited in just about every other country.
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# Basel Convention
## History
With the tightening of environmental laws (for example, RCRA) in developed nations in the 1970s, disposal costs for hazardous waste rose dramatically. At the same time, the globalization of shipping made cross-border movement of waste easier, and many less developed countries were desperate for foreign currency. Consequently, the trade in hazardous waste, particularly to poorer countries, grew rapidly. In 1990, OECD countries exported around 1.8 million tons of hazardous waste. Although most of this waste was shipped to other developed countries, a number of high-profile incidents of hazardous waste-dumping led to calls for regulation.
One of the incidents which led to the creation of the Basel Convention was the *Khian Sea* waste disposal incident, in which a ship carrying incinerator ash from the city of Philadelphia in the United States dumped half of its load on a beach in Haiti before being forced away. It sailed for many months, changing its name several times. Unable to unload the cargo in any port, the crew was believed to have dumped much of it at sea.
Another incident was a 1988 case in which five ships transported 8,000 barrels of hazardous waste from Italy to the small Nigerian town of Koko in exchange for \$100 monthly rent which was paid to a Nigerian for the use of his farmland.
At its meeting that took place from 27 November to 1 December 2006, the parties of the Basel Agreement focused on issues of electronic waste and the dismantling of ships.
Increased trade in recyclable materials has led to an increase in a market for used products such as computers. This market is valued in billions of dollars. At issue is the distinction when used computers stop being a \"commodity\" and become a \"waste\".
As of June 2023, there are 191 parties to the treaty, which includes 188 UN member states, the Cook Islands, the European Union, and the State of Palestine. The five UN member states that are not party to the treaty are East Timor, Fiji, Haiti, South Sudan, and United States.
## Definition of *hazardous waste* {#definition_of_hazardous_waste}
Waste falls under the scope of the convention if it is within the category of wastes listed in Annex I of the convention and it exhibits one of the hazardous characteristics contained in Annex III. In other words, it must both be listed and possess a characteristic such as being explosive, flammable, toxic, or corrosive. The other way that a waste may fall under the scope of the convention is if it is defined as or considered to be a hazardous waste under the laws of either the exporting country, the importing country, or any of the countries of transit.
The definition of the term disposal is made in Article 2 al 4 and just refers to annex IV, which gives a list of operations which are understood as disposal or recovery. Examples of disposal are broad, including recovery and recycling.
Alternatively, to fall under the scope of the convention, it is sufficient for waste to be included in Annex II, which lists other wastes, such as household wastes and residue that comes from incinerating household waste.
Radioactive waste that is covered under other international control systems and wastes from the normal operation of ships are not covered.
Annex IX attempts to define wastes which are not considered hazardous wastes and which would be excluded from the scope of the Basel Convention. If these wastes however are contaminated with hazardous materials to an extent causing them to exhibit an Annex III characteristic, they are not excluded.
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# Basel Convention
## Obligations
In addition to conditions on the import and export of the above wastes, there are stringent requirements for notice, consent and tracking for movement of wastes across national boundaries. The convention places a general prohibition on the exportation or importation of wastes between parties and non-parties. The exception to this rule is where the waste is subject to another treaty that does not take away from the Basel Convention. The United States is a notable non-party to the convention and has a number of such agreements for allowing the shipping of hazardous wastes to Basel Party countries.
The OECD Council also has its own control system that governs the transboundary movement of hazardous materials between OECD member countries. This allows, among other things, the OECD countries to continue trading in wastes with countries like the United States that have not ratified the Basel Convention.
Parties to the convention must honor import bans of other parties.
Article 4 of the Basel Convention calls for an overall reduction of waste generation. By encouraging countries to keep wastes within their boundaries and as close as possible to its source of generation, the internal pressures should provide incentives for waste reduction and pollution prevention. Parties are generally prohibited from exporting covered wastes to, or importing covered waste from, non-parties to the convention.
The convention states that illegal hazardous waste traffic is criminal but contains no enforcement provisions.
According to Article 12, parties are directed to adopt a protocol that establishes liability rules and procedures that are appropriate for damage that comes from the movement of hazardous waste across borders.
The current consensus is that as space is not classed as a \"country\" under the specific definition, export of e-waste to non-terrestrial locations would not be covered.
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# Basel Convention
## Basel Ban Amendment {#basel_ban_amendment}
After the initial adoption of the convention, some least developed countries and environmental organizations argued that it did not go far enough. Many nations and NGOs argued for a total ban on shipment of all hazardous waste to developing countries. In particular, the original convention did not prohibit waste exports to any location except Antarctica but merely required a notification and consent system known as \"prior informed consent\" or PIC. Further, many waste traders sought to exploit the good name of recycling and begin to justify all exports as moving to recycling destinations. Many believed a full ban was needed including exports for recycling. These concerns led to several regional waste trade bans, including the Bamako Convention.
Lobbying at 1995 Basel conference by developing countries, Greenpeace and several European countries such as Denmark, led to the adoption of an amendment to the convention in 1995 termed the Basel Ban Amendment to the Basel Convention. The amendment has been accepted by 86 countries and the European Union, but has not entered into force (as that requires ratification by three-fourths of the member states to the convention). On 6 September 2019, Croatia became the 97th country to ratify the amendment which will enter into force after 90 days on 5 December 2019. The amendment prohibits the export of hazardous waste from a list of developed (mostly OECD) countries to developing countries. The Basel Ban applies to export for any reason, including recycling. An area of special concern for advocates of the amendment was the sale of ships for salvage, shipbreaking. The Ban Amendment was strenuously opposed by a number of industry groups as well as nations including Australia and Canada. The number of ratification for the entry-into force of the Ban Amendment is under debate: Amendments to the convention enter into force after ratification of \"three-fourths of the Parties who accepted them\" \[Art. 17.5\]; so far, the parties of the Basel Convention could not yet agree whether this would be three-fourths of the parties that were party to the Basel Convention when the ban was adopted, or three-fourths of the current parties of the convention \[see Report of COP 9 of the Basel Convention\]. The status of the amendment ratifications can be found on the Basel Secretariat\'s web page. The European Union fully implemented the Basel Ban in its Waste Shipment Regulation (EWSR), making it legally binding in all EU member states. Norway and Switzerland have similarly fully implemented the Basel Ban in their legislation.
In the light of the blockage concerning the entry into force of the Ban Amendment, Switzerland and Indonesia have launched a \"Country-led Initiative\" (CLI) to discuss in an informal manner a way forward to ensure that the trans boundary movements of hazardous wastes, especially to developing countries and countries with economies in the transition, do not lead to an unsound management of hazardous wastes. This discussion aims at identifying and finding solutions to the reasons why hazardous wastes are still brought to countries that are not able to treat them in a safe manner. It is hoped that the CLI will contribute to the realization of the objectives of the Ban Amendment. The Basel Convention\'s website informs about the progress of this initiative.
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# Basel Convention
## Regulation of plastic waste {#regulation_of_plastic_waste}
In the wake of popular outcry, in May 2019 most of the world\'s countries, but not the United States, agreed to amend the Basel Convention to include plastic waste as a regulated material. The world\'s oceans are estimated to contain 100 million metric tons of plastic, with up to 90% of this quantity originating in land-based sources. The United States, which produces an annual 42 million metric tons of plastic waste, more than any other country in the world, opposed the amendment, but since it is not a party to the treaty it did not have an opportunity to vote on it to try to block it. Information about, and visual images of, wildlife, such as seabirds, ingesting plastic, and scientific findings that nanoparticles do penetrate through the blood--brain barrier were reported to have fueled public sentiment for coordinated international legally binding action. Over a million people worldwide signed a petition demanding official action. Although the United States is not a party to the treaty, export shipments of plastic waste from the United States are now \"criminal traffic as soon as the ships get on the high seas,\" according to the Basel Action Network (BAN), and carriers of such shipments may face liability, because the Basel Convention as amended in May 2019 prohibits the transportation of plastic waste to just about every other country.
The Basel Convention contains three main entries on plastic wastes in Annex II, VIII and IX of the convention. The Plastic Waste Amendments of the convention are now binding on 186 States. In addition to ensuring the trade in plastic waste is more transparent and better regulated, under the Basel Convention governments must take steps not only to ensure the environmentally sound management of plastic waste, but also to tackle plastic waste at its source.
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# Basel Convention
## Basel watchdog {#basel_watchdog}
The Basel Action Network (BAN) is a charitable civil society non-governmental organization that works as a consumer watchdog for implementation of the Basel Convention. BAN\'s principal aims is fighting exportation of toxic waste, including plastic waste, from industrialized societies to developing countries. BAN is based in Seattle, Washington, United States, with a partner office in the Philippines. BAN works to curb trans-border trade in hazardous electronic waste, land dumping, incineration, and the use of prison labor
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# Bar Kokhba (album)
***Bar Kokhba*** is a double album by John Zorn, recorded between 1994 and 1996. It features music from Zorn\'s *Masada* project, rearranged for small ensembles. It also features the original soundtrack from *[The Art of Remembrance -- Simon Wiesenthal](http://riverlightspictures.com/taor/index.html)*, a film by [Hannah Heer](http://hannahheer.com) and Werner Schmiedel (1994--95).
## Reception
The AllMusic review by Marc Gilman noted: \"While some compositions retain their original structure and sound, some are expanded and probed by Zorn\'s arrangements, and resemble avant-garde classical music more than jazz. But this is the beauty of the album; the ensembles provide a forum for Zorn to expand his compositions. The album consistently impresses.\" `{{Album ratings
| rev1 = [[AllMusic]]
| rev1Score = {{rating|4.5|5}}<ref name="Allmusic" />
|rev2 = ''[[The Penguin Guide to Jazz Recordings]]''
|rev2score = {{Rating|3.5|4}}<ref name="Penguin">{{cite book |last1=Cook |first1=Richard |authorlink1=Richard Cook (journalist) |last2=Morton |first2=Brian |authorlink2=Brian Morton (Scottish writer) |title=[[The Penguin Guide to Jazz|The Penguin Guide to Jazz Recordings]] |year=2008 |edition=9th |publisher=[[Penguin Books|Penguin]] |isbn=978-0-141-03401-0 |page=1544}}</ref>
}}`{=mediawiki}
## Track listing {#track_listing}
*All compositions by John Zorn*
Disc One
1. \"Gevurah\" -- 6:55
2. \"Nezikin\" -- 1:51
3. \"Mahshav\" -- 4:33
4. \"Rokhev\" -- 3:10
5. \"Abidan\" -- 5:19
6. \"Sheloshim\" -- 5:03
7. \"Hath-Arob\" -- 2:25
8. \"Paran\" -- 4:48
9. \"Mahlah\" -- 7:48
10. \"Socoh\" -- 4:07
11. \"Yechida\" -- 8:24
12. \"Bikkurim\" -- 3:25
13. \"Idalah-Abal\" -- 5:04
Disc Two
1. \"Tannaim\" -- 4:38
2. \"Nefesh\" -- 3:33
3. \"Abidan\" -- 3:13
4. \"Mo\'ed\" -- 4:59
5. \"Maskil\" -- 4:41
6. \"Mishpatim\" -- 6:46
7. \"Sansanah\" -- 6:56
8. \"Shear-Jashub\" -- 2:06
9. \"Mahshav\" -- 4:50
10. \"Sheloshim\" -- 6:45
11. \"Mochin\" -- 13:11
12
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# Brian Kernighan
\| birth_place = Toronto, Ontario, Canada \| residence = \| citizenship = Canada \| nationality = Canadian \| field = Computer science \| work_institution = Princeton University \| alma_mater = University of Toronto (BASc)\
Princeton University (PhD) \| thesis_title = Some Graph Partitioning Problems Related to Program Segmentation \| thesis_url = <http://www.worldcat.org/oclc/39166855> \| thesis_year = 1969 \| doctoral_advisor = Peter Weiner \| doctoral_students = \| known_for = `{{Plainlist|
* [[Unix]]
* [[AWK]]
* [[AMPL|A Mathematical Programming Language (AMPL)]]
* [[Kernighan–Lin algorithm]]
* [[Lin–Kernighan heuristic]]
* ''[[The C Programming Language]]'' (book)<ref name=numberphile>{{YouTube|id=de2Hsvxaf8M|title="C" Programming Language: Brian Kernighan - Computerphile}}</ref>}}`{=mediawiki} \| prizes = \| website = `{{URL|https://www.cs.princeton.edu/~bwk/}}`{=mediawiki} }}
**Brian Wilson Kernighan** (`{{IPAc-en|ˈ|k|ɜːr|n|ɪ|h|æ|n}}`{=mediawiki}; born January 30, 1942) is a Canadian computer scientist. He worked at Bell Labs and contributed to the development of Unix alongside Unix creators Ken Thompson and Dennis Ritchie. Kernighan\'s name became widely known through co-authorship of the first book on the C programming language (*The C Programming Language*) with Dennis Ritchie. Kernighan affirmed that he had no part in the design of the C language (\"it\'s entirely Dennis Ritchie\'s work\").
Kernighan authored many Unix programs, including ditroff. He is coauthor of the AWK and AMPL programming languages. The \"K\" of K&R C and of AWK both stand for \"Kernighan\".
In collaboration with Shen Lin he devised well-known heuristics for two NP-complete optimization problems: graph partitioning and the travelling salesman problem. In a display of authorial equity, the former is usually called the Kernighan--Lin algorithm, while the latter is known as the Lin--Kernighan heuristic.
Kernighan has been a professor of computer science at Princeton University since 2000 and is the director of undergraduate studies in the department of computer science. In 2015, he co-authored the book *The Go Programming Language*.
## Early life and education {#early_life_and_education}
Kernighan was born in Toronto. He attended the University of Toronto between 1960 and 1964, earning his bachelor\'s degree in engineering physics. He received his Ph.D. in electrical engineering from Princeton University in 1969, completing a doctoral dissertation titled \"Some graph partitioning problems related to program segmentation\" under the supervision of Peter G. Weiner.
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# Brian Kernighan
## Career and research {#career_and_research}
Kernighan has held a professorship in the department of computer science at Princeton since 2000. Each fall he teaches a course called \"Computers in Our World\", which introduces the fundamentals of computing to non-majors.
Kernighan was the software editor for Prentice Hall International. His \"Software Tools\" series spread the essence of \"C/Unix thinking\" with makeovers for BASIC, FORTRAN, and Pascal, and most notably his \"Ratfor\" (rational FORTRAN) was put in the public domain.
He has said that if stranded on an island with only one programming language it would have to be C.
Kernighan coined the term \"Unix\" and helped popularize Thompson\'s Unix philosophy. Kernighan is also known for coining the expression \"What You See Is All You Get\" (WYSIAYG), which is a sarcastic variant of the original \"What You See Is What You Get\" (WYSIWYG). Kernighan\'s term is used to indicate that WYSIWYG systems might throw away information in a document that could be useful in other contexts.
In 1972, Kernighan described memory management in strings using \"hello\" and \"world\", in the B programming language, which became the iconic example we know today. Kernighan\'s original 1978 implementation of `{{not_typo|[["Hello, World!" program|hello, world!]]}}`{=mediawiki} was sold at The Algorithm Auction, the world\'s first auction of computer algorithms.
In 1996, Kernighan taught CS50 which is the Harvard University introductory course in computer science. Kernighan was an influence on David J. Malan who subsequently taught the course and scaled it up to run at multiple universities and in multiple digital formats.
Kernighan was elected a member of the National Academy of Engineering in 2002 for contributions to software and to programming languages. He was also elected a member of the American Academy of Arts and Sciences in 2019.
In 2022, Kernighan stated that he was actively working on improvements to the AWK programming language, which he took part in creating in 1977.
### Books and reports {#books_and_reports}
- *The Elements of Programming Style*, with P. J. Plauger
- *Software Tools*, a book and set of tools for Ratfor, co-created in part with P. J. Plauger
- *Software Tools in Pascal*, a book and set of tools for Pascal, with P. J. Plauger
- *The C Programming Language*, with C creator Dennis Ritchie, the first book on C
- *The Practice of Programming*, with Rob Pike
- *The Unix Programming Environment*, a tutorial book, with Rob Pike
- [\"Why Pascal is Not My Favorite Programming Language\"](http://wiki.c2.com/?WhyPascalIsNotMyFavoriteProgrammingLanguage), a popular criticism of Niklaus Wirth\'s Pascal. Some parts of the criticism are obsolete due to ISO 7185 (Programming Languages - Pascal); the criticism was written before ISO 7185 was created. (AT&T Computing Science Technical Report #100)
- *UNIX: A History and a Memoir*, a historical account of the development of Unix from the perspective of his role at Bell Labs
### Programs
- 1972: The first documented \"Hello, world!\" program, in Kernighan\'s [\"A Tutorial Introduction to the Language B\"](https://web.archive.org/web/20150611114644/https://www.bell-labs.com/usr/dmr/www/btut.pdf)
- 1973: ditroff, or \"device independent troff\", which allowed troff to be used with any device
- 1974: The eqn typesetting language for troff, with Lorinda Cherry
- 1976: Ratfor
- 1977: The m4 macro processing language, with Dennis Ritchie
- 1977: The AWK programming language, with Alfred Aho and Peter J. Weinberger, and its book *The AWK Programming Language*
- 1985: The AMPL programming language
- 1988: The pic typesetting language for troff
## Publications
- *The Elements of Programming Style* (1974, 1978) with P. J. Plauger
- *Software Tools* (1976) with P. J. Plauger
- *The C Programming Language* (1978, 1988) with Dennis M. Ritchie
- *Software Tools in Pascal* (1981) with P. J. Plauger
- *The Unix Programming Environment* (1984) with Rob Pike
- *The AWK Programming Language* (1988, 2023) with Alfred Aho and Peter J. Weinberger
- *The Practice of Programming* (1999) with Rob Pike
- *AMPL: A Modeling Language for Mathematical Programming, 2nd ed
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# Battlecruiser
upright=1.5\|thumb\|right\|`{{HMS|Hood}}`{=mediawiki}, the largest battlecruiser ever built, in Australia on 17 March 1924
The **battlecruiser** (also written as **battle cruiser** or **battle-cruiser**) was a type of capital ship of the first half of the 20th century. These were similar in displacement, armament and cost to battleships, but differed in form and balance of attributes. Battlecruisers typically had thinner armour (to a varying degree) and a somewhat lighter main gun battery than contemporary battleships, installed on a longer hull with much higher engine power in order to attain greater speeds. The first battlecruisers were designed in the United Kingdom as a successor to the armoured cruiser, at the same time as the dreadnought succeeded the pre-dreadnought battleship. The goal of the battlecruiser concept was to outrun any ship with similar armament, and chase down any ship with lesser armament; they were intended to hunt down slower, older armoured cruisers and destroy them with heavy gunfire while avoiding combat with the more powerful but slower battleships. However, as more and more battlecruisers were built, they were increasingly used alongside the better-protected battleships.
Battlecruisers served in the navies of the United Kingdom, Germany, the Ottoman Empire, Australia and Japan during World War I, most notably at the Battle of the Falkland Islands and in the several raids and skirmishes in the North Sea which culminated in a pitched fleet battle, the Battle of Jutland. British battlecruisers in particular suffered heavy losses at Jutland, where poor fire safety and ammunition handling practices left them vulnerable to catastrophic magazine explosions following hits to their main turrets from large-calibre shells. This dismal showing led to a persistent general belief that battlecruisers were too thinly armoured to function successfully. By the end of the war, capital ship design had developed, with battleships becoming faster and battlecruisers becoming more heavily armoured, blurring the distinction between a battlecruiser and a fast battleship. The Washington Naval Treaty, which limited capital ship construction from 1922 onwards, treated battleships and battlecruisers identically, and the new generation of battlecruisers planned by the United States, Great Britain and Japan were scrapped or converted into aircraft carriers under the terms of the treaty.
Improvements in armour design and propulsion created the 1930s \"fast battleship\" with the speed of a battlecruiser and armour of a battleship, making the battlecruiser in the traditional sense effectively an obsolete concept. Thus from the 1930s on, only the Royal Navy continued to use \"battlecruiser\" as a classification for the World War I--era capital ships that remained in the fleet; while Japan\'s battlecruisers remained in service, they had been significantly reconstructed and were re-rated as full-fledged fast battleships. Some new vessels built during that decade, the German `{{sclass|Scharnhorst|battleship|1}}`{=mediawiki}s and `{{sclass|Deutschland|cruiser|1}}`{=mediawiki}s and the French `{{sclass|Dunkerque|battleship|1}}`{=mediawiki}s are all sometimes referred to as battlecruisers, although the owning navies referred to them as \"battleships\" (*Schlachtschiffe*), \"armoured ships\" (*Panzerschiffe*) and \"battleships\" (*Bâtiments de ligne*) respectively.`{{Refn|The German {{sclass|Scharnhorst|battleship|1}}s and {{sclass|Deutschland|cruiser|1}}s and the French {{sclass|Dunkerque|battleship|1}}s are all sometimes referred to as battlecruisers, although the owning navies referred to them as "battleships" ({{langx|de|Schlachtschiffe}}), "armoured ships" ({{langx|de|Panzerschiffe}}) and "battleships" ({{langx|fr|Bâtiments de ligne}}) respectively. Since neither their operators nor a significant number of naval historians classify them as such, they are not discussed in this article.<ref>Gröner, pp. 31, 60; Gille, p. 139; Koop & Schmolke, p. 4</ref><ref name="Gardiner & Chesneau, p. 259">Chesneau, p. 259</ref><ref>Bidlingmaier, pp. 73–74</ref>|group=Note}}`{=mediawiki}
Battlecruisers were put into action again during World War II, and only one survived to the end, `{{HMS|Renown|1916|2}}`{=mediawiki}. There was also renewed interest in large \"cruiser-killer\" type warships whose design was scaled-up from a heavy cruiser rather than a lighter/faster battleship derivative, but few were ever begun and only two members of the `{{sclass|Alaska|cruiser|0}}`{=mediawiki} were commissioned in time to see war service. Construction of large cruisers as well as fast battleships were curtailed in favor of more-needed aircraft carriers, convoy escorts, and cargo ships.
During (and after) the Cold War, the Soviet `{{sclass|Kirov|battlecruiser|4}}`{=mediawiki} of large guided missile cruisers have been the only ships termed \"battlecruisers\"; the class is also the only example of a nuclear-powered battlecruiser. As of 2024, Russia operates two units: the *Pyotr Velikiy* has remained in active service since its 1998 commissioning, while the *Admiral Nakhimov* has been inactive (in storage or refitting) since 1999.
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# Battlecruiser
## Background
The battlecruiser was developed by the Royal Navy in the first years of the 20th century as an evolution of the armoured cruiser. The first armoured cruisers had been built in the 1870s, as an attempt to give armour protection to ships fulfilling the typical cruiser roles of patrol, trade protection and power projection. However, the results were rarely satisfactory, as the weight of armour required for any meaningful protection usually meant that the ship became almost as slow as a battleship. As a result, navies preferred to build protected cruisers with an armoured deck protecting their engines, or simply no armour at all.
In the 1890s, new Krupp steel armour meant that it was now possible to give a cruiser side armour which would protect it against the quick-firing guns of enemy battleships and cruisers alike. In 1896--97 France and Russia, who were regarded as likely allies in the event of war, started to build large, fast armoured cruisers taking advantage of this. In the event of a war between Britain and France or Russia, or both, these cruisers threatened to cause serious difficulties for the British Empire\'s worldwide trade.
Britain, which had concluded in 1892 that it needed twice as many cruisers as any potential enemy to adequately protect its empire\'s sea lanes, responded to the perceived threat by laying down its own large armoured cruisers. Between 1899 and 1905, it completed or laid down seven classes of this type, a total of 35 ships. This building program, in turn, prompted the French and Russians to increase their own construction. The Imperial German Navy began to build large armoured cruisers for use on their overseas stations, laying down eight between 1897 and 1906. In the period 1889--1896, the Royal Navy spent £7.3 million on new large cruisers. From 1897 to 1904, it spent £26.9 million. Many armoured cruisers of the new kind were just as large and expensive as the equivalent battleship.
The increasing size and power of the armoured cruiser led to suggestions in British naval circles that cruisers should displace battleships entirely. The battleship\'s main advantage was its 12-inch heavy guns, and heavier armour designed to protect from shells of similar size. However, for a few years after 1900 it seemed that those advantages were of little practical value. The torpedo now had a range of 2,000 yards, and it seemed unlikely that a battleship would engage within torpedo range. However, at ranges of more than 2,000 yards it became increasingly unlikely that the heavy guns of a battleship would score any hits, as the heavy guns relied on primitive aiming techniques. The secondary batteries of 6-inch quick-firing guns, firing more plentiful shells, were more likely to hit the enemy. As naval expert Fred T. Jane wrote in June 1902,
> Is there anything outside of 2,000 yards that the big gun in its hundreds of tons of medieval castle can affect, that its weight in 6-inch guns without the castle could not affect equally well? And inside 2,000, what, in these days of gyros, is there that the torpedo cannot effect with far more certainty?
In 1904, Admiral John \"Jacky\" Fisher became First Sea Lord, the senior officer of the Royal Navy. He had for some time thought about the development of a new fast armoured ship. He was very fond of the \"second-class battleship\" `{{HMS|Renown|1895|2}}`{=mediawiki}, a faster, more lightly armoured battleship. As early as 1901, there is confusion in Fisher\'s writing about whether he saw the battleship or the cruiser as the model for future developments. This did not stop him from commissioning designs from naval architect W. H. Gard for an armoured cruiser with the heaviest possible armament for use with the fleet. The design Gard submitted was for a ship between 14000 -, capable of 25 kn, armed with four 9.2-inch and twelve 7.5 in guns in twin gun turrets and protected with six inches of armour along her belt and 9.2-inch turrets, 4 in on her 7.5-inch turrets, 10 inches on her conning tower and up to 2.5 in on her decks. However, mainstream British naval thinking between 1902 and 1904 was clearly in favour of heavily armoured battleships, rather than the fast ships that Fisher favoured.
The Battle of Tsushima proved the effectiveness of heavy guns over intermediate ones and the need for a uniform main caliber on a ship for fire control. Even before this, the Royal Navy had begun to consider a shift away from the mixed-calibre armament of the 1890s pre-dreadnought to an \"all-big-gun\" design, and preliminary designs circulated for battleships with all 12-inch or all 10-inch guns and armoured cruisers with all 9.2-inch guns. In late 1904, not long after the Royal Navy had decided to use 12-inch guns for its next generation of battleships because of their superior performance at long range, Fisher began to argue that big-gun cruisers could replace battleships altogether. The continuing improvement of the torpedo meant that submarines and destroyers would be able to destroy battleships; this in Fisher\'s view heralded the end of the battleship or at least compromised the validity of heavy armour protection. Nevertheless, armoured cruisers would remain vital for commerce protection.
Fisher\'s views were very controversial within the Royal Navy, and even given his position as First Sea Lord, he was not in a position to insist on his own approach. Thus he assembled a \"Committee on Designs\", consisting of a mixture of civilian and naval experts, to determine the approach to both battleship and armoured cruiser construction in the future. While the stated purpose of the committee was to investigate and report on future requirements of ships, Fisher and his associates had already made key decisions. The terms of reference for the committee were for a battleship capable of 21 kn with 12-inch guns and no intermediate calibres, capable of docking in existing drydocks; and a cruiser capable of 25.5 kn, also with 12-inch guns and no intermediate armament, armoured like `{{HMS|Minotaur|1906|2}}`{=mediawiki}, the most recent armoured cruiser, and also capable of using existing docks.
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# Battlecruiser
## First battlecruisers {#first_battlecruisers}
Under the Selborne plan of 1903,`{{Ambiguous|date=April 2025}}`{=mediawiki} the Royal Navy intended to start three new battleships and four armoured cruisers each year. However, in late 1904 it became clear that the 1905--1906 programme would have to be considerably smaller, because of lower than expected tax revenue and the need to buy out two Chilean battleships under construction in British yards, lest they be purchased by the Russians for use against the Japanese, Britain\'s ally. These economic realities meant that the 1905--1906 programme consisted only of one battleship, but three armoured cruisers. The battleship became the revolutionary battleship `{{HMS|Dreadnought|1906|2}}`{=mediawiki}, and the cruisers became the three ships of the `{{sclass|Invincible|battlecruiser|4}}`{=mediawiki}. Fisher later claimed, however, that he had argued during the committee for the cancellation of the remaining battleship.
The construction of the new class was begun in 1906 and completed in 1908, delayed perhaps to allow their designers to learn from any problems with *Dreadnought*. The ships fulfilled the design requirement quite closely. On a displacement similar to *Dreadnought*, the *Invincible*s were 40 ft longer to accommodate additional boilers and more powerful turbines to propel them at 25 kn. Moreover, the new ships could maintain this speed for days, whereas pre-dreadnought battleships could not generally do so for more than an hour. Armed with eight 12-inch Mk X guns, compared to ten on *Dreadnought*, they had 6 - of armour protecting the hull and the gun turrets. (*Dreadnought*{{\'}}s armour, by comparison, was 11 - at its thickest.) The class had a very marked increase in speed, displacement and firepower compared to the most recent armoured cruisers but no more armour.
While the *Invincible*s were to fill the same role as the armoured cruisers they succeeded, they were expected to do so more effectively. Specifically their roles were:
- **Heavy reconnaissance.** Because of their power, the *Invincible*s could sweep away the screen of enemy cruisers to close with and observe an enemy battlefleet before using their superior speed to retire.
- **Close support for the battle fleet.** They could be stationed at the ends of the battle line to stop enemy cruisers harassing the battleships, and to harass the enemy\'s battleships if they were busy fighting battleships. Also, the *Invincible*s could operate as the fast wing of the battlefleet and try to outmanoeuvre the enemy.
- **Pursuit.** If an enemy fleet ran, then the *Invincible*s would use their speed to pursue, and their guns to damage or slow enemy ships.
- **Commerce protection.** The new ships would hunt down enemy cruisers and commerce raiders.
Confusion about how to refer to these new battleship-size armoured cruisers set in almost immediately. Even in late 1905, before work was begun on the *Invincible*s, a Royal Navy memorandum refers to \"large armoured ships\" meaning both battleships and large cruisers. In October 1906, the Admiralty began to classify all post-Dreadnought battleships and armoured cruisers as \"capital ships\", while Fisher used the term \"dreadnought\" to refer either to his new battleships or the battleships and armoured cruisers together. At the same time, the *Invincible* class themselves were referred to as \"cruiser-battleships\", \"dreadnought cruisers\"; the term \"battlecruiser\" was first used by Fisher in 1908. Finally, on 24 November 1911, Admiralty Weekly Order No. 351 laid down that \"All cruisers of the \"Invincible\" and later types are for the future to be described and classified as \"battle cruisers\" to distinguish them from the armoured cruisers of earlier date.\"
Along with questions over the new ships\' nomenclature came uncertainty about their actual role due to their lack of protection. If they were primarily to act as scouts for the battle fleet and hunter-killers of enemy cruisers and commerce raiders, then the seven inches of belt armour with which they had been equipped would be adequate. If, on the other hand, they were expected to reinforce a battle line of dreadnoughts with their own heavy guns, they were too thin-skinned to be safe from an enemy\'s heavy guns. The *Invincible*s were essentially extremely large, heavily armed, fast armoured cruisers. However, the viability of the armoured cruiser was already in doubt. A cruiser that could have worked with the Fleet might have been a more viable option for taking over that role.
Because of the *Invincible*s{{\'}} size and armament, naval authorities considered them capital ships almost from their inception---an assumption that might have been inevitable. Complicating matters further was that many naval authorities, including Lord Fisher, had made overoptimistic assessments from the Battle of Tsushima in 1905 about the armoured cruiser\'s ability to survive in a battle line against enemy capital ships due to their superior speed. These assumptions had been made without taking into account the Russian Baltic Fleet\'s inefficiency and tactical ineptitude. By the time the term \"battlecruiser\" had been given to the *Invincible*s, the idea of their parity with battleships had been fixed in many people\'s minds.
Not everyone was so convinced. *Brassey{{\'}}s Naval Annual*, for instance, stated that with vessels as large and expensive as the *Invincible*s, an admiral \"will be certain to put them in the line of battle where their comparatively light protection will be a disadvantage and their high speed of no value.\" Those in favor of the battlecruiser countered with two points---first, since all capital ships were vulnerable to new weapons such as the torpedo, armour had lost some of its validity; and second, because of its greater speed, the battlecruiser could control the range at which it engaged an enemy.
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# Battlecruiser
## Battlecruisers in the dreadnought arms race {#battlecruisers_in_the_dreadnought_arms_race}
Between the launching of the *Invincible*s to just after the outbreak of the First World War, the battlecruiser played a junior role in the developing dreadnought arms race, as it was never wholeheartedly adopted as the key weapon in British imperial defence, as Fisher had presumably desired. The biggest factor for this lack of acceptance was the marked change in Britain\'s strategic circumstances between their conception and the commissioning of the first ships. The prospective enemy for Britain had shifted from a Franco-Russian alliance with many armoured cruisers to a resurgent and increasingly belligerent Germany. Diplomatically, Britain had entered the Entente cordiale in 1904 and the Anglo-Russian Entente. Neither France nor Russia posed a particular naval threat; the Russian navy had largely been sunk or captured in the Russo-Japanese War of 1904--1905, while the French were in no hurry to adopt the new dreadnought-type design. Britain also boasted very cordial relations with two of the significant new naval powers: Japan (bolstered by the Anglo-Japanese Alliance, signed in 1902 and renewed in 1905), and the US. These changed strategic circumstances, and the great success of the *Dreadnought* ensured that she rather than the *Invincible* became the new model capital ship. Nevertheless, battlecruiser construction played a part in the renewed naval arms race sparked by the *Dreadnought*.
For their first few years of service, the *Invincible*s entirely fulfilled Fisher\'s vision of being able to sink any ship fast enough to catch them, and run from any ship capable of sinking them. An *Invincible* would also, in many circumstances, be able to take on an enemy pre-dreadnought battleship. Naval circles concurred that the armoured cruiser in its current form had come to the logical end of its development and the *Invincible*s were so far ahead of any enemy armoured cruiser in firepower and speed that it proved difficult to justify building more or bigger cruisers. This lead was extended by the surprise both *Dreadnought* and *Invincible* produced by having been built in secret; this prompted most other navies to delay their building programmes and radically revise their designs. This was particularly true for cruisers, because the details of the *Invincible* class were kept secret for longer; this meant that the last German armoured cruiser, `{{SMS|Blücher||2}}`{=mediawiki}, was armed with only 21 cm guns, and was no match for the new battlecruisers.
The Royal Navy\'s early superiority in capital ships led to the rejection of a 1905--1906 design that would, essentially, have fused the battlecruiser and battleship concepts into what would eventually become the fast battleship. The \'X4\' design combined the full armour and armament of *Dreadnought* with the 25-knot speed of *Invincible*. The additional cost could not be justified given the existing British lead and the new Liberal government\'s need for economy; the slower and cheaper `{{HMS|Bellerophon|1907|2}}`{=mediawiki}, a relatively close copy of *Dreadnought*, was adopted instead. The X4 concept would eventually be fulfilled in the `{{sclass|Queen Elizabeth|battleship|4}}`{=mediawiki} and later by other navies.
The next British battlecruisers were the three `{{sclass|Indefatigable|battlecruiser|4}}`{=mediawiki}, slightly improved *Invincible*s built to fundamentally the same specification, partly due to political pressure to limit costs and partly due to the secrecy surrounding German battlecruiser construction, particularly about the heavy armour of `{{SMS|Von der Tann}}`{=mediawiki}. This class came to be widely seen as a mistake and the next generation of British battlecruisers were markedly more powerful. By 1909--1910 a sense of national crisis about rivalry with Germany outweighed cost-cutting, and a naval panic resulted in the approval of a total of eight capital ships in 1909--1910. Fisher pressed for all eight to be battlecruisers, but was unable to have his way; he had to settle for six battleships and two battlecruisers of the `{{sclass|Lion|battlecruiser|4}}`{=mediawiki}. The *Lion*s carried eight 13.5-inch guns, the now-standard caliber of the British \"super-dreadnought\" battleships. Speed increased to 27 kn and armour protection, while not as good as in German designs, was better than in previous British battlecruisers, with 9 in armour belt and barbettes. The two *Lion*s were followed by the very similar `{{HMS|Queen Mary||2}}`{=mediawiki}.
By 1911 Germany had built battlecruisers of her own, and the superiority of the British ships could no longer be assured. Moreover, the German Navy did not share Fisher\'s view of the battlecruiser. In contrast to the British focus on increasing speed and firepower, Germany progressively improved the armour and staying power of their ships to better the British battlecruisers. *Von der Tann*, begun in 1908 and completed in 1910, carried eight 11.1-inch guns, but with 11.1-inch (283 mm) armour she was far better protected than the *Invincible*s. The two `{{sclass|Moltke|battlecruiser|5}}`{=mediawiki}s were quite similar but carried ten 11.1-inch guns of an improved design. `{{SMS|Seydlitz||2}}`{=mediawiki}, designed in 1909 and finished in 1913, was a modified *Moltke*; speed increased by one knot to 26.5 kn, while her armour had a maximum thickness of 12 inches, equivalent to the `{{sclass|Helgoland|battleship|2}}`{=mediawiki}s of a few years earlier. *Seydlitz* was Germany\'s last battlecruiser completed before World War I.
The next step in battlecruiser design came from Japan. The Imperial Japanese Navy had been planning the `{{sclass|Kongō|battlecruiser|0}}`{=mediawiki} ships from 1909, and was determined that, since the Japanese economy could support relatively few ships, each would be more powerful than its likely competitors. Initially the class was planned with the *Invincible*s as the benchmark. On learning of the British plans for *Lion*, and the likelihood that new U.S. Navy battleships would be armed with 14 in guns, the Japanese decided to radically revise their plans and go one better. A new plan was drawn up, carrying eight 14-inch guns, and capable of 27.5 kn, thus marginally having the edge over the *Lion*s in speed and firepower. The heavy guns were also better-positioned, being superfiring both fore and aft with no turret amidships. The armour scheme was also marginally improved over the *Lion*s, with nine inches of armour on the turrets and 8 in on the barbettes. The first ship in the class was built in Britain, and a further three constructed in Japan. The Japanese also re-classified their powerful armoured cruisers of the *Tsukuba* and *Ibuki* classes, carrying four 12-inch guns, as battlecruisers; nonetheless, their armament was weaker and they were slower than any battlecruiser.
The next British battlecruiser, `{{HMS|Tiger|1913|2}}`{=mediawiki}, was intended initially as the fourth ship in the *Lion* class, but was substantially redesigned. She retained the eight 13.5-inch guns of her predecessors, but they were positioned like those of *Kongō* for better fields of fire. She was faster (making 29 kn on sea trials), and carried a heavier secondary armament. *Tiger* was also more heavily armoured on the whole; while the maximum thickness of armour was the same at nine inches, the height of the main armour belt was increased. Not all the desired improvements for this ship were approved, however. Her designer, Sir Eustace Tennyson d\'Eyncourt, had wanted small-bore water-tube boilers and geared turbines to give her a speed of 32 kn, but he received no support from the authorities and the engine makers refused his request.
1912 saw work begin on three more German battlecruisers of the `{{sclass|Derfflinger|battlecruiser|4}}`{=mediawiki}, the first German battlecruisers to mount 12-inch guns. These ships, like *Tiger* and the *Kongō*s, had their guns arranged in superfiring turrets for greater efficiency. Their armour and speed was similar to the previous *Seydlitz* class. In 1913, the Russian Empire also began the construction of the four-ship `{{sclass|Borodino|battlecruiser|4}}`{=mediawiki}, which were designed for service in the Baltic Sea. These ships were designed to carry twelve 14-inch guns, with armour up to 12 inches thick, and a speed of 26.6 kn. The heavy armour and relatively slow speed of these ships made them more similar to German designs than to British ships; construction of the *Borodino*s was halted by the First World War and all were scrapped after the end of the Russian Civil War.
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# Battlecruiser
## World War I {#world_war_i}
### Construction
For most of the combatants, capital ship construction was very limited during the war. Germany finished the *Derfflinger* class and began work on the `{{sclass|Mackensen|battlecruiser|4}}`{=mediawiki}. The *Mackensen*s were a development of the *Derfflinger* class, with 13.8-inch guns and a broadly similar armour scheme, designed for 28 kn.
In Britain, Jackie Fisher returned to the office of First Sea Lord in October 1914. His enthusiasm for big, fast ships was unabated, and he set designers to producing a design for a battlecruiser with 15-inch guns. Because Fisher expected the next German battlecruiser to steam at 28 knots, he required the new British design to be capable of 32 knots. He planned to reorder two `{{sclass|Revenge|battleship}}`{=mediawiki}s, which had been approved but not yet laid down, to a new design. Fisher finally received approval for this project on 28 December 1914 and they became the `{{sclass|Renown|battlecruiser|4}}`{=mediawiki}. With six 15-inch guns but only 6-inch armour they were a further step forward from *Tiger* in firepower and speed, but returned to the level of protection of the first British battlecruisers.
At the same time, Fisher resorted to subterfuge to obtain another three fast, lightly armoured ships that could use several spare 15 in gun turrets left over from battleship construction. These ships were essentially light battlecruisers, and Fisher occasionally referred to them as such, but officially they were classified as *large light cruisers*. This unusual designation was required because construction of new capital ships had been placed on hold, while there were no limits on light cruiser construction. They became `{{HMS|Courageous|50|2}}`{=mediawiki} and her sisters `{{HMS|Glorious||2}}`{=mediawiki} and `{{HMS|Furious|47|2}}`{=mediawiki}, and there was a bizarre imbalance between their main guns of 15 inches (or 18 in in *Furious*) and their armour, which at 3 in thickness was on the scale of a light cruiser. The design was generally regarded as a failure (nicknamed in the Fleet *Outrageous*, *Uproarious* and *Spurious*), though the later conversion of the ships to aircraft carriers was very successful. Fisher also speculated about a new mammoth, but lightly built battlecruiser, that would carry 20 in guns, which he termed `{{HMS|Incomparable}}`{=mediawiki}; this never got beyond the concept stage.
It is often held that the *Renown* and *Courageous* classes were designed for Fisher\'s plan to land troops (possibly Russian) on the German Baltic coast. Specifically, they were designed with a reduced draught, which might be important in the shallow Baltic. This is not clear-cut evidence that the ships were designed for the Baltic: it was considered that earlier ships had too much draught and not enough freeboard under operational conditions. Roberts argues that the focus on the Baltic was probably unimportant at the time the ships were designed, but was inflated later, after the disastrous Dardanelles Campaign.
The final British battlecruiser design of the war was the `{{sclass2|Admiral|battlecruiser|4}}`{=mediawiki}, which was born from a requirement for an improved version of the *Queen Elizabeth* battleship. The project began at the end of 1915, after Fisher\'s final departure from the Admiralty. While initially envisaged as a battleship, senior sea officers felt that Britain had enough battleships, but that new battlecruisers might be required to combat German ships being built (the British overestimated German progress on the *Mackensen* class as well as their likely capabilities). A battlecruiser design with eight 15-inch guns, 8 inches of armour and capable of 32 knots was decided on. The experience of battlecruisers at the Battle of Jutland meant that the design was radically revised and transformed again into a fast battleship with armour up to 12 inches thick, but still capable of 31.5 kn. The first ship in the class, `{{HMS|Hood|51|2}}`{=mediawiki}, was built according to this design to counter the possible completion of any of the Mackensen-class ship. The plans for her three sisters, on which little work had been done, were revised once more later in 1916 and in 1917 to improve protection.
The Admiral class would have been the only British ships capable of taking on the German *Mackensen* class; nevertheless, German shipbuilding was drastically slowed by the war, and while two *Mackensen*s were launched, none were ever completed. The Germans also worked briefly on a further three ships, of the `{{sclass|Ersatz Yorck|battlecruiser|4}}`{=mediawiki}, which were modified versions of the *Mackensen*s with 15-inch guns. Work on the three additional Admirals was suspended in March 1917 to enable more escorts and merchant ships to be built to deal with the new threat from U-boats to trade. They were finally cancelled in February 1919.
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# Battlecruiser
## World War I {#world_war_i}
### Battlecruisers in action {#battlecruisers_in_action}
The first combat involving battlecruisers during World War I was the Battle of Heligoland Bight in August 1914. A force of British light cruisers and destroyers entered the Heligoland Bight (the part of the North Sea closest to Hamburg) to attack German destroyer patrols. When they met opposition from light cruisers, Vice Admiral David Beatty took his squadron of five battlecruisers into the Bight and turned the tide of the battle, ultimately sinking three German light cruisers and killing their commander, Rear Admiral Leberecht Maass.
The German battlecruiser `{{SMS|Goeben||2}}`{=mediawiki} perhaps made the most impact early in the war. Stationed in the Mediterranean, she and the escorting light cruiser `{{SMS|Breslau}}`{=mediawiki} evaded British and French ships on the outbreak of war, and steamed to Constantinople (Istanbul) with two British battlecruisers in hot pursuit. The two German ships were handed over to the Ottoman Navy, and this was instrumental in bringing the Ottoman Empire into the war as one of the Central Powers. *Goeben* herself, renamed *Yavuz Sultan Selim*, fought engagements against the Imperial Russian Navy in the Black Sea before being knocked out of the action for the remainder of the war after the Battle of Imbros against British forces in the Aegean Sea in January 1918.
The original battlecruiser concept proved successful in December 1914 at the Battle of the Falkland Islands. The British battlecruisers `{{HMS|Inflexible|1907|2}}`{=mediawiki} and `{{HMS|Invincible|1907|2}}`{=mediawiki} did precisely the job for which they were intended when they chased down and annihilated the German East Asia Squadron, centered on the armoured cruisers `{{SMS|Scharnhorst||2}}`{=mediawiki} and `{{SMS|Gneisenau||2}}`{=mediawiki}, along with three light cruisers, commanded by Admiral Maximilian Graf Von Spee, in the South Atlantic Ocean. Prior to the battle, the Australian battlecruiser `{{HMAS|Australia|1911|2}}`{=mediawiki} had unsuccessfully searched for the German ships in the Pacific.
During the Battle of Dogger Bank in 1915, the aftermost barbette of the German flagship *Seydlitz* was struck by a British 13.5-inch shell from HMS *Lion*. The shell did not penetrate the barbette, but it dislodged a piece of the barbette armour that allowed the flame from the shell\'s detonation to enter the barbette. The propellant charges being hoisted upwards were ignited, and the fireball flashed up into the turret and down into the magazine, setting fire to charges removed from their brass cartridge cases. The gun crew tried to escape into the next turret, which allowed the flash to spread into that turret as well, killing the crews of both turrets. *Seydlitz* was saved from near-certain destruction only by emergency flooding of her after magazines, which had been effected by Wilhelm Heidkamp. This near-disaster was due to the way that ammunition handling was arranged and was common to both German and British battleships and battlecruisers, but the lighter protection on the latter made them more vulnerable to the turret or barbette being penetrated. The Germans learned from investigating the damaged *Seydlitz* and instituted measures to ensure that ammunition handling minimised any possible exposure to flash.
Apart from the cordite handling, the battle was mostly inconclusive, though both the British flagship *Lion* and *Seydlitz* were severely damaged. *Lion* lost speed, causing her to fall behind the rest of the battleline, and Beatty was unable to effectively command his ships for the remainder of the engagement. A British signalling error allowed the German battlecruisers to withdraw, as most of Beatty\'s squadron mistakenly concentrated on the crippled armoured cruiser *Blücher*, sinking her with great loss of life. The British blamed their failure to win a decisive victory on their poor gunnery and attempted to increase their rate of fire by stockpiling unprotected cordite charges in their ammunition hoists and barbettes.
At the Battle of Jutland on 31 May 1916, both British and German battlecruisers were employed as fleet units. The British battlecruisers became engaged with both their German counterparts, the battlecruisers, and then German battleships before the arrival of the battleships of the British Grand Fleet. The result was a disaster for the Royal Navy\'s battlecruiser squadrons: *Invincible*, *Queen Mary*, and `{{HMS|Indefatigable|1909|2}}`{=mediawiki} exploded with the loss of all but a handful of their crews. The exact reason why the ships\' magazines detonated is not known, but the abundance of exposed cordite charges stored in their turrets, ammunition hoists and working chambers in the quest to increase their rate of fire undoubtedly contributed to their loss. Beatty\'s flagship *Lion* herself was almost lost in a similar manner, save for the heroic actions of Major Francis Harvey.
The better-armoured German battlecruisers fared better, in part due to the poor performance of British fuzes (the British shells tended to explode or break up on impact with the German armour). `{{SMS|Lützow||2}}`{=mediawiki}---the only German battlecruiser lost at Jutland---had only 128 killed, for instance, despite receiving more than thirty hits. The other German battlecruisers, `{{SMS|Moltke||2}}`{=mediawiki}, *Von der Tann*, *Seydlitz*, and `{{SMS|Derfflinger||2}}`{=mediawiki}, were all heavily damaged and required extensive repairs after the battle, *Seydlitz* barely making it home, for they had been the focus of British fire for much of the battle.
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# Battlecruiser
## Interwar period {#interwar_period}
In the years immediately after World War I, Britain, Japan and the US all began design work on a new generation of ever more powerful battleships and battlecruisers. The new burst of shipbuilding that each nation\'s navy desired was politically controversial and potentially economically crippling. This nascent arms race was prevented by the Washington Naval Treaty of 1922, where the major naval powers agreed to limits on capital ship numbers. The German navy was not represented at the talks; under the terms of the Treaty of Versailles, Germany was not allowed any modern capital ships at all.
Through the 1920s and 1930s only Britain and Japan retained battlecruisers, often modified and rebuilt from their original designs. The line between the battlecruiser and the modern fast battleship became blurred; indeed, the Japanese *Kongō*s were formally redesignated as battleships after their very comprehensive reconstruction in the 1930s.
### Plans in the aftermath of World War I {#plans_in_the_aftermath_of_world_war_i}
*Hood*, launched in 1918, was the last World War I battlecruiser to be completed. Owing to lessons from Jutland, the ship was modified during construction; the thickness of her belt armour was increased by an average of 50 percent and extended substantially, she was given heavier deck armour, and the protection of her magazines was improved to guard against the ignition of ammunition. This was hoped to be capable of resisting her own weapons---the classic measure of a \"balanced\" battleship. *Hood* was the largest ship in the Royal Navy when completed; because of her great displacement, in theory she combined the firepower and armour of a battleship with the speed of a battlecruiser, causing some to refer to her as a fast battleship. However, her protection was markedly less than that of the British battleships built immediately after World War I, the `{{sclass|Nelson|battleship|4}}`{=mediawiki}.
The navies of Japan and the United States, not being affected immediately by the war, had time to develop new heavy 16 in guns for their latest designs and to refine their battlecruiser designs in light of combat experience in Europe. The Imperial Japanese Navy began four `{{sclass|Amagi|battlecruiser|2}}`{=mediawiki}s. These vessels would have been of unprecedented size and power, as fast and well armoured as *Hood* whilst carrying a main battery of ten 16-inch guns, the most powerful armament ever proposed for a battlecruiser. They were, for all intents and purposes, fast battleships---the only differences between them and the `{{sclass|Tosa|battleship|2}}`{=mediawiki}s which were to precede them were 1 in less side armour and a .25 kn increase in speed. The United States Navy, which had worked on its battlecruiser designs since 1913 and watched the latest developments in this class with great care, responded with the `{{sclass|Lexington|battlecruiser|4}}`{=mediawiki}. If completed as planned, they would have been exceptionally fast and well armed with eight 16-inch guns, but carried armour little better than the *Invincible*s---this after an 8000 LT increase in protection following Jutland. The final stage in the post-war battlecruiser race came with the British response to the *Amagi* and *Lexington* types: four 48000 LT G3 battlecruisers. Royal Navy documents of the period often described any battleship with a speed of over about 24 kn as a battlecruiser, regardless of the amount of protective armour, although the G3 was considered by most to be a well-balanced fast battleship.
The Washington Naval Treaty meant that none of these designs came to fruition. Ships that had been started were either broken up on the slipway or converted to aircraft carriers. In Japan, *Amagi* and `{{ship|Japanese aircraft carrier|Akagi||2}}`{=mediawiki} were selected for conversion. *Amagi* was damaged beyond repair by the 1923 Great Kantō earthquake and was broken up for scrap; the hull of one of the proposed *Tosa*-class battleships, `{{ship|Japanese aircraft carrier|Kaga||2}}`{=mediawiki}, was converted in her stead. The United States Navy also converted two battlecruiser hulls into aircraft carriers in the wake of the Washington Treaty: `{{USS|Lexington|CV-2|6}}`{=mediawiki} and `{{USS|Saratoga|CV-3|6}}`{=mediawiki}, although this was only considered marginally preferable to scrapping the hulls outright (the remaining four: *Constellation*, *Ranger*, *Constitution* and *United States* were scrapped). In Britain, Fisher\'s \"large light cruisers,\" were converted to carriers. *Furious* had already been partially converted during the war and *Glorious* and *Courageous* were similarly converted.
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# Battlecruiser
## Interwar period {#interwar_period}
### Rebuilding programmes {#rebuilding_programmes}
In total, nine battlecruisers survived the Washington Naval Treaty, although HMS *Tiger* later became a victim of the London Naval Conference 1930 and was scrapped. Because their high speed made them valuable surface units in spite of their weaknesses, most of these ships were significantly updated before World War II. `{{HMS|Renown|1916|2}}`{=mediawiki} and `{{HMS|Repulse|1916|2}}`{=mediawiki} were modernized significantly in the 1920s and 1930s. Between 1934 and 1936, *Repulse* was partially modernized and had her bridge modified, an aircraft hangar, catapult and new gunnery equipment added and her anti-aircraft armament increased. *Renown* underwent a more thorough reconstruction between 1937 and 1939. Her deck armour was increased, new turbines and boilers were fitted, an aircraft hangar and catapult added and she was completely rearmed aside from the main guns which had their elevation increased to +30 degrees. The bridge structure was also removed and a large bridge similar to that used in the `{{sclass|King George V|battleship (1939)|0}}`{=mediawiki} battleships installed in its place. While conversions of this kind generally added weight to the vessel, *Renown*{{\'}}s tonnage actually decreased due to a substantially lighter power plant. Similar thorough rebuildings planned for *Repulse* and *Hood* were cancelled due to the advent of World War II.
Unable to build new ships, the Imperial Japanese Navy also chose to improve its existing battlecruisers of the *Kongō* class (initially the `{{ship|Japanese battleship|Haruna||2}}`{=mediawiki}, `{{ship|Japanese battleship|Kirishima||2}}`{=mediawiki}, and `{{ship|Japanese battleship|Kongō||2}}`{=mediawiki}---the `{{ship|Japanese battleship|Hiei||2}}`{=mediawiki} only later as it had been disarmed under the terms of the Washington treaty) in two substantial reconstructions (one for *Hiei*). During the first of these, elevation of their main guns was increased to +40 degrees, anti-torpedo bulges and 3800 LT of horizontal armour added, and a \"pagoda\" mast with additional command positions built up. This reduced the ships\' speed to 25.9 kn. The second reconstruction focused on speed as they had been selected as fast escorts for aircraft carrier task forces. Completely new main engines, a reduced number of boilers and an increase in hull length by 26 ft allowed them to reach up to 30 knots once again. They were reclassified as \"fast battleships,\" although their armour and guns still fell short compared to surviving World War I--era battleships in the American or the British navies, with dire consequences during the Pacific War, when *Hiei* and *Kirishima* were easily crippled by US gunfire during actions off Guadalcanal, forcing their scuttling shortly afterwards. Perhaps most tellingly, *Hiei* was crippled by medium-caliber gunfire from heavy and light cruisers in a close-range night engagement.
There were two exceptions: Turkey\'s *Yavuz Sultan Selim* and the Royal Navy\'s *Hood*. The Turkish Navy made only minor improvements to the ship in the interwar period, which primarily focused on repairing wartime damage and the installation of new fire control systems and anti-aircraft batteries. *Hood* was in constant service with the fleet and could not be withdrawn for an extended reconstruction. She received minor improvements over the course of the 1930s, including modern fire control systems, increased numbers of anti-aircraft guns, and in March 1941, radar.
### Naval rearmament {#naval_rearmament}
In the late 1930s navies began to build capital ships again, and during this period a number of large commerce raiders and small, fast battleships were built that are sometimes referred to as battlecruisers, such as the `{{sclass|Scharnhorst|battleship|1}}`{=mediawiki}s and `{{sclass|Deutschland|cruiser|1}}`{=mediawiki}s and the French `{{sclass|Dunkerque|battleship|1}}`{=mediawiki}s.`{{refn|The German {{sclass|Scharnhorst|battleship|1}}s and {{sclass|Deutschland|cruiser|1}}s and the French {{sclass|Dunkerque|battleship|1}}s are all sometimes referred to as battlecruisers, although the owning navies referred to them as "battleships" ({{langx|de|Schlachtschiffe}}), "armoured ships" ({{langx|de|Panzerschiffe}}) and "battleships" ({{langx|fr|Bâtiments de ligne}}) respectively. Since neither their operators nor a significant number of naval historians classify them as such, they are not discussed in this article.<ref>Gröner, pp. 31, 60; Gille, p. 139; Koop & Schmolke, p. 4</ref><ref name="Gardiner & Chesneau, p. 259">Chesneau, p. 259</ref><ref>Bidlingmaier, pp. 73–74</ref>|group=Note}}`{=mediawiki} Germany and Russia designed new battlecruisers during this period, though only the latter laid down two of the 35,000-ton `{{sclass|Kronshtadt|battlecruiser|4}}`{=mediawiki}. They were still on the slipways when the Germans invaded in 1941 and construction was suspended. Both ships were scrapped after the war. The Germans planned three battlecruisers of the `{{sclass2|O|battlecruiser|4}}`{=mediawiki} as part of the expansion of the Kriegsmarine (Plan Z). With six 15-inch guns, high speed, excellent range, but very thin armour, they were intended as commerce raiders. Only one was ordered shortly before World War II; no work was ever done on it. No names were assigned, and they were known by their contract names: \'O\', \'P\', and \'Q\'. The new class was not universally welcomed in the Kriegsmarine. Their abnormally-light protection gained it the derogatory nickname *Ohne Panzer Quatsch* (without armour nonsense) within certain circles of the Navy.
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# Battlecruiser
## World War II {#world_war_ii}
The Royal Navy deployed some of its battlecruisers during the Norwegian Campaign in April 1940. The `{{ship|German battleship|Gneisenau||2}}`{=mediawiki} and the `{{ship|German battleship|Scharnhorst||2}}`{=mediawiki} were engaged during the action off Lofoten by *Renown* in very bad weather and disengaged after *Gneisenau* was damaged. One of *Renown*{{\'}}s 15-inch shells passed through *Gneisenau*{{\'}}s director-control tower without exploding, severing electrical and communication cables as it went and destroyed the rangefinders for the forward 150 mm (5.9 in) turrets. Main-battery fire control had to be shifted aft due to the loss of electrical power. Another shell from *Renown* knocked out *Gneisenau*{{\'}}s aft turret. The British ship was struck twice by German shells that failed to inflict any significant damage. She was the only pre-war battlecruiser to survive the war.
In the early years of the war various German ships had a measure of success hunting merchant ships in the Atlantic. Allied battlecruisers such as *Renown*, *Repulse*, and the fast battleships *Dunkerque* and `{{ship|French battleship|Strasbourg||2}}`{=mediawiki} were employed on operations to hunt down the commerce-raiding German ships. The one stand-up fight occurred when the battleship `{{ship|German battleship|Bismarck||2}}`{=mediawiki} and the heavy cruiser `{{ship|German cruiser|Prinz Eugen||2}}`{=mediawiki} sortied into the North Atlantic to attack British shipping and were intercepted by *Hood* and the battleship `{{HMS|Prince of Wales|53|2}}`{=mediawiki} in May 1941 in the Battle of the Denmark Strait. *Hood* was destroyed when the *Bismarck*{{\'}}s 15-inch shells caused a magazine explosion. Only three men survived.
The first battlecruiser to see action in the Pacific War was *Repulse* when she was sunk by Japanese torpedo bombers north of Singapore on 10 December 1941 whilst in company with *Prince of Wales*. She was lightly damaged by a single 250 kg bomb and near-missed by two others in the first Japanese attack. Her speed and agility enabled her to avoid the other attacks by level bombers and dodge 33 torpedoes. The last group of torpedo bombers attacked from multiple directions and *Repulse* was struck by five torpedoes. She quickly capsized with the loss of 27 officers and 486 crewmen; 42 officers and 754 enlisted men were rescued by the escorting destroyers. The loss of *Repulse* and *Prince of Wales* conclusively proved the vulnerability of capital ships to aircraft without air cover of their own.
The Japanese *Kongō*-class battlecruisers were extensively used as carrier escorts for most of their wartime career due to their high speed. Although classified as fast battleships by the Japanese, their World War I--era armament was weaker and their upgraded armour was still thin compared to contemporary battleships. On 13 November 1942, during the First Naval Battle of Guadalcanal, *Hiei* stumbled across American cruisers and destroyers at point-blank range. The ship was badly damaged in the encounter and had to be towed by her sister ship *Kirishima*. Both were spotted by American aircraft the following morning and *Kirishima* was forced to cast off her tow because of repeated aerial attacks. *Hiei*{{\'}}s captain ordered her crew to abandon ship after further damage and scuttled *Hiei* in the early evening of 14 November. On the night of 14/15 November during the Second Naval Battle of Guadalcanal, *Kirishima* returned to Ironbottom Sound, but encountered the American battleships `{{USS|South Dakota|BB-57|2}}`{=mediawiki} and `{{USS|Washington|BB-56|2}}`{=mediawiki}. While failing to detect *Washington*, *Kirishima* engaged *South Dakota* with some effect. *Washington* opened fire a few minutes later at short range and badly damaged *Kirishima*, knocking out her aft turrets, jamming her rudder, and hitting the ship below the waterline. The flooding proved to be uncontrollable and *Kirishima* capsized three and a half hours later.
Returning to Japan after the Battle of Leyte Gulf, *Kongō* was torpedoed and sunk by the American submarine `{{USS|Sealion II|SS-315|2}}`{=mediawiki} on 21 November 1944. *Haruna* was moored at Kure, Japan when the naval base was attacked by American carrier aircraft on 24 and 28 July 1945. The ship was only lightly damaged by a single bomb hit on 24 July, but was hit a dozen more times on 28 July and sank at her pier. She was refloated after the war and scrapped in early 1946.
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# Battlecruiser
## World War II {#world_war_ii}
### Large cruisers or \"cruiser killers\" {#large_cruisers_or_cruiser_killers}
A late renaissance in popularity of ships between battleships and cruisers in size occurred on the eve of World War II. Described by some as battlecruisers, but never classified as capital ships, they were variously described as \"super cruisers\", \"large cruisers\" or even \"unrestricted cruisers\". The Dutch, American, and Japanese navies all planned these new classes specifically to counter the heavy cruisers, or their counterparts, being built by their naval rivals.
The first such battlecruisers were the Dutch Design 1047, designed to protect their colonies in the East Indies in the face of Japanese aggression. Never officially assigned names, these ships were designed with German and Italian assistance. While they broadly resembled the German *Scharnhorst* class and had the same main battery, they would have been more lightly armoured and only protected against eight-inch gunfire. Although the design was mostly completed, work on the vessels never commenced as the Germans overran the Netherlands in May 1940. The first ship would have been laid down in June of that year.
The only class of these late battlecruisers actually built were the United States Navy\'s `{{sclass|Alaska|cruiser|0}}`{=mediawiki} \"large cruisers\". Two of them were completed, `{{USS|Alaska|CB-1|2}}`{=mediawiki} and `{{USS|Guam|CB-2|2}}`{=mediawiki}; a third, `{{USS|Hawaii|CB-3|2}}`{=mediawiki}, was cancelled while under construction and three others, to be named *Philippines*, *Puerto Rico* and *Samoa*, were cancelled before they were laid down. The USN classified them \"large cruisers\" instead of battlecruisers. These ships were named after territories or protectorates, while battleships were named after states and cruisers after cities. With a displacement of 27000 LT and a main armament of nine 12-inch guns in three triple turrets, they were twice the size of `{{sclass|Baltimore|cruiser|2}}`{=mediawiki}s and had guns some 50% larger in diameter. The *Alaska*s design was a scaled-up cruiser rather than a lighter/faster battleship derivative, as they lacked the thick armoured belt and intricate torpedo defence system of contemporary battleships. However, unlike World War I-era battlecruisers, the *Alaska*s were considered a balanced design according to cruiser standards as their protection could withstand fire from their own caliber of gun, albeit only in a very narrow range band. They were designed to hunt down Japanese heavy cruisers, though by the time they entered service most Japanese cruisers had been sunk by American aircraft or submarines. Like the contemporary `{{sclass|Iowa|battleship|0}}`{=mediawiki} fast battleships, their speed ultimately made them more useful as carrier escorts and bombardment ships than as the surface combatants they were developed to be.
The Japanese started designing the B64 class, which was similar to the *Alaska* but with 310 mm guns. News of the *Alaska*s led them to upgrade the design, creating Design B-65. Armed with 356 mm guns, the B65s would have been the best armed of the new breed of battlecruisers, but they still would have had only sufficient protection to keep out eight-inch shells. Much like the Dutch, the Japanese got as far as completing the design for the B65s, but never laid them down. By the time the designs were ready the Japanese Navy recognized that they had little use for the vessels and that their priority for construction should lie with aircraft carriers. Like the *Alaska*s, the Japanese did not call these ships battlecruisers, referring to them instead as super-heavy cruisers.
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# Battlecruiser
## Cold War--era designs {#cold_warera_designs}
In spite of the fact that most navies abandoned the battleship and battlecruiser concepts after World War II, Joseph Stalin\'s fondness for big-gun-armed warships caused the Soviet Union to plan a large cruiser class in the late 1940s. In the Soviet Navy, they were termed \"heavy cruisers\" (*tyazhelny kreyser*). The fruits of this program were the Project 82 (*Stalingrad*) cruisers, of 36500 t standard load, nine 305 mm guns and a speed of 35 kn. Three ships were laid down in 1951--1952, but they were cancelled in April 1953 after Stalin\'s death. Only the central armoured hull section of the first ship, *Stalingrad*, was launched in 1954 and then used as a target.
The Soviet `{{sclass|Kirov|battlecruiser|4}}`{=mediawiki} is sometimes referred to as a battlecruiser. This description arises from their over 24000 t displacement, which is roughly equal to that of a First World War battleship and more than twice the displacement of contemporary cruisers; upon entry into service, *Kirov* was the largest surface combatant to be built since World War II. The *Kirov* class lacks the armour that distinguishes battlecruisers from ordinary cruisers and they are classified as heavy nuclear-powered missile cruisers (*Тяжелый Атомный Ракетный Крейсер* (ТАРКР)) by Russia, with their primary surface armament consisting of twenty P-700 Granit surface to surface missiles. Four members of the class were completed during the 1980s and 1990s, but due to budget constraints only the `{{ship|Russian battlecruiser|Pyotr Velikiy||2}}`{=mediawiki} is operational with the Russian Navy, though plans were announced in 2010 to return the other three ships to service. As of 2021, `{{ship|Russian battlecruiser|Admiral Nakhimov||2}}`{=mediawiki} was being refitted, but the other two ships are reportedly beyond economical repair.
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# Battlecruiser
## Operators
- operates one `{{sclass|Kirov|battlecruiser|2}}`{=mediawiki} with one more being overhauled.
### Former operators {#former_operators}
- five surviving battlecruisers were all scuttled at Scapa Flow in 1919.
- decommissioned its only battlecruiser HMAS *Australia* in 1921.
- whose `{{sclass|Kongo|battlecruiser|2}}`{=mediawiki}s (upgraded and reclassified as fast battleships in the 1930s) all served in and were sunk in World War II, the last in 1945.
- last battlecruiser, HMS *Renown* was decommissioned in 1945, following World War II.
- two *Alaska*-class large cruisers were both decommissioned in 1947.
- decommissioned its only battlecruiser TCG *Yavuz* in 1950
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# Baldr
**Baldr** (Old Norse also **Balder**, **Baldur**) is a god in Germanic mythology. In Norse mythology, he is a son of the god Odin and the goddess Frigg, and has numerous brothers, such as Thor and Váli. In wider Germanic mythology, the god was known in Old English as ***Bældæġ***, and in Old High German as ***Balder***, all ultimately stemming from the Proto-Germanic theonym *\*\'\'\'Balðraz\'\'\'* (\'hero\' or \'prince\').
During the 12th century, Danish accounts by Saxo Grammaticus and other Danish Latin chroniclers recorded a euhemerized account of his story. Compiled in Iceland during the 13th century, but based on older Old Norse poetry, the *Poetic Edda* and the *Prose Edda* contain numerous references to the death of Baldr as both a great tragedy to the Æsir and a harbinger of Ragnarök.
According to *Gylfaginning*, a book of Snorri Sturluson\'s Prose Edda, Baldr\'s wife is Nanna and their son is Forseti. Baldr had the greatest ship ever built, *Hringhorni*, and there is no place more beautiful than his hall, Breidablik.
## Name
The Old Norse theonym *Baldr* (\'brave, defiant\'; also \'lord, prince\') and its various Germanic cognates -- including Old English *Bældæg* and Old High German *Balder* (or *Palter*) -- probably stems from Proto-Germanic *\*Balðraz* (\'Hero, Prince\'; cf. Old Norse *mann-baldr* \'great man\', Old English *bealdor* \'prince, hero\'), itself a derivative of *\*balþaz*, meaning \'brave\' (cf. Old Norse *ballr* \'hard, stubborn\', Gothic *balþa\** \'bold, frank\', Old English *beald* \'bold, brave, confident\', Old Saxon *bald* \'valiant, bold\', Old High German *bald* \'brave, courageous\').
This etymology was originally proposed by Jacob Grimm (1835), who also speculated on a comparison with the Lithuanian *báltas* (\'white\', also the name of a light-god) based on the semantic development from \'white\' to \'shining\' then \'strong\'. According to linguist Vladimir Orel, this could be linguistically tenable. Philologist Rudolf Simek also argues that the Old English *Bældæg* should be interpreted as meaning \'shining day\', from a Proto-Germanic root \**bēl*- (cf. Old English *bæl*, Old Norse *bál* \'fire\') attached to *dæg* (\'day\').
Old Norse also shows the usage of the word as an honorific in a few cases, as in *baldur î brynju* (Sæm. 272b) and *herbaldr* (Sæm. 218b), in general epithets of heroes. In continental Saxon and Anglo-Saxon tradition, the son of Woden is called not *Bealdor* but *Baldag* (Saxon) and *Bældæg, Beldeg* (Anglo-Saxon), which shows association with \"day\", possibly with Day personified as a deity. This, as Grimm points out, would agree with the meaning \"shining one, white one, a god\" derived from the meaning of Baltic *baltas*, further adducing Slavic *Belobog* and German *Berhta*.
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# Baldr
## Attestations
### Merseburg Incantation {#merseburg_incantation}
One of the two Merseburg Incantations names *Balder* (in the genitive singular *Balderes*), but also mentions a figure named *Phol*, considered to be a byname for Baldr (as in Scandinavian *Falr*, *Fjalarr*; (in Saxo) *Balderus* : *Fjallerus*). The incantation relates of *Phol ende Wotan* riding to the woods, where the foot of Baldr\'s foal is sprained. Sinthgunt (the sister of the sun), Frigg and Odin sing to the foot in order for it to heal. The identification with Balder is not conclusive. Modern scholarship suggests that the god Freyr might be meant.
### *Poetic Edda* {#poetic_edda}
Unlike the Prose Edda, in the Poetic Edda the tale of Baldr\'s death is referred to rather than recounted at length. Baldr is mentioned in *Völuspá*, in *Lokasenna*, and is the subject of the Eddic poem *Baldr\'s Dreams*.
Among the visions which the Völva sees and describes in Völuspá is Baldr\'s death. In stanza 32, the Völva says she saw the fate of Baldr \"the bleeding god\":
the bleeding god, The son of Othin, {{!}} his destiny set: Famous and fair {{!}} in the lofty fields, Full grown in strength {{!}} the mistletoe stood.}}
In the next two stanzas, the Völva refers to Baldr\'s killing, describes the birth of Váli for the slaying of Höðr and the weeping of Frigg: `{{poemquote|''Stanza 33:''
From the branch which seemed {{!}}`{=mediawiki} so slender and fair Came a harmful shaft {{!}} that Hoth should hurl; But the brother of Baldr {{!}} was born ere long, And one night old {{!}} fought Othin\'s son.
*Stanza 34:* His hands he washed not, {{!}} his hair he combed not, Till he bore to the bale-blaze {{!}} Baldr\'s foe. But in Fensalir {{!}} did Frigg weep sore For Valhall\'s need: {{!}} would you know yet more?}}
In stanza 62 of Völuspá, looking far into the future, the Völva says that Höðr and Baldr will come back, with the union, according to Bellows, being a symbol of the new age of peace:
bear ripened fruit, All ills grow better, {{!}} and Baldr comes back; Baldr and Hoth dwell {{!}} in Hropt\'s battle-hall, And the mighty gods: {{!}} would you know yet more?}}
Baldr is mentioned in two stanzas of Lokasenna, a poem which describes a flyting between the gods and the god Loki. In the first of the two stanzas, Frigg, Baldr\'s mother, tells Loki that if she had a son like Baldr, Loki would be killed:
In the next stanza, Loki responds to Frigg, and says that he is the reason Baldr \"will never ride home again\":
thumb\|right\|upright=1.36\|\"Odin rides to Hel\" (1908) by W. G. Collingwood
The Eddic poem *Baldr\'s Dreams* opens with the gods holding a council discussing why Baldr had had bad dreams: `{{poemquote|''Henry Adams Bellows translation:''
Once were the gods {{!}}`{=mediawiki} together met, And the goddesses came {{!}} and council held, And the far-famed ones {{!}} the truth would find, Why baleful dreams {{!}} to Baldr had come.}}
Odin then rides to Hel to a Völva\'s grave and awakens her using magic. The Völva asks Odin, who she does not recognize, who he is, and Odin answers that he is Vegtam (\"Wanderer\"). Odin asks the Völva for whom are the benches covered in rings and the floor covered in gold. The Völva tells him that in their location mead is brewed for Baldr, and that she spoke unwillingly, so she will speak no more:
the mead is brewed, The shining drink, {{!}} and a shield lies o\'er it; But their hope is gone {{!}} from the mighty gods. Unwilling I spake, {{!}} and now would be still.}}
Odin asks the Völva to not be silent and asks her who will kill Baldr. The Völva replies and says that Höðr will kill Baldr, and again says that she spoke unwillingly, and that she will speak no more:
the far-famed branch, He shall the bane {{!}} of Baldr become, And steal the life {{!}} from Othin\'s son. Unwilling I spake, {{!}} and now would be still.}}
Odin again asks the Völva to not be silent and asks her who will avenge Baldr\'s death. The Völva replies that Váli will, when he will be one night old. Once again, she says that she will speak no more:
in Vestrsalir, And one night old {{!}} fights Othin\'s son; His hands he shall wash not, {{!}} his hair he shall comb not, Till the slayer of Baldr {{!}} he brings to the flames. Unwilling I spake, {{!}} and now would be still.}}
Odin again asks the Völva to not be silent and says that he seeks to know who the women that will then weep be. The Völva realizes that Vegtam is Odin in disguise. Odin says that the Völva is not a Völva, and that she is the mother of three giants. The Völva tells Odin to ride back home proud, because she will speak to no more men until Loki escapes his bounds.
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# Baldr
## Attestations
### *Prose Edda* {#prose_edda}
In *Gylfaginning*, Baldr is described as follows:
Apart from this description, Baldr is known primarily for the story of his death, which is seen as the first in a chain of events that will ultimately lead to the destruction of the gods at Ragnarök.
Baldr had a dream of his own death and his mother, Frigg, had the same dream. Since dreams were usually prophetic, this depressed him, and so Frigg made every object on earth vow never to hurt Baldr. All objects made this vow, save for the mistletoe---a detail which has traditionally been explained with the idea that it was too unimportant and nonthreatening to bother asking it to make the vow, but which Merrill Kaplan has instead argued echoes the fact that young people were not eligible to swear legal oaths, which could make them a threat later in life.
When Loki, the mischief-maker, heard of this, he made a magical spear from this plant (in some later versions, an arrow). He hurried to the place where the gods were indulging in their new pastime of hurling objects at Baldr, which would bounce off without harming him. Loki gave the spear to Baldr\'s brother, the blind god Höðr, who then inadvertently killed his brother with it (other versions suggest that Loki guided the arrow himself). For this act, Odin and the *ásynja* Rindr gave birth to Váli, who grew to adulthood within a day and slew Höðr.
Baldr was ceremonially burnt upon his ship Hringhorni, the largest of all ships. On the pyre he was given the magical ring Draupnir. At first the gods were not able to push the ship out onto sea, and so they sent for Hyrrokin, a giantess, who came riding on a wolf and gave the ship such a push that fire flashed from the rollers and all the earth shook.
As he was carried to the ship, Odin whispered something in his ear. The import of this speech was held to be unknowable, and the question of what was said was thus used as an unanswerable riddle by Odin in other sources, namely against the giant Vafthrudnir in the Eddic poem *Vafthrudnismal* and in the riddles of Gestumblindi in *Hervarar saga*.
Upon seeing the corpse being carried to the ship, Nanna, his wife, died of grief. She was then placed on the funeral fire (perhaps a toned-down instance of Sati, also attested in the Arab traveller Ibn Fadlan\'s account of a funeral among the Rus\'), after which it was set on fire. Baldr\'s horse with all its trappings was also laid on the pyre.
As the pyre was set on fire, Thor blessed it with his hammer Mjǫllnir. As he did a small dwarf named Litr came running before his feet. Thor then kicked him into the pyre.
Upon Frigg\'s entreaties, delivered through the messenger Hermod, Hel promised to release Baldr from the underworld if all objects alive and dead would weep for him. All did, except a giantess, Þökk (often presumed to be the god Loki in disguise), who refused to mourn the slain god. Thus Baldr had to remain in the underworld, not to emerge until after Ragnarök, when he and his brother Höðr would be reconciled and rule the new earth together with Thor\'s sons.
Besides these descriptions of Baldr, the Prose Edda also explicitly links him to the Anglo-Saxon *Beldeg* in its prologue.
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# Baldr
## Attestations
### *Gesta Danorum* {#gesta_danorum}
Writing during the end of the 12th century, the Danish historian Saxo Grammaticus tells the story of Baldr (recorded as *Balderus*) in a form that professes to be historical. According to him, Balderus and Høtherus were rival suitors for the hand of Nanna, daughter of Gewar, King of Norway. Balderus was a demigod and common steel could not wound his sacred body. The two rivals encountered each other in a terrific battle. Though Odin and Thor and the other gods fought for Balderus, he was defeated and fled away, and Høtherus married the princess.
Nevertheless, Balderus took heart of grace and again met Høtherus in a stricken field. But he fared even worse than before. Høtherus dealt him a deadly wound with a magic sword which he had received from Mimir, the satyr of the woods; after lingering three days in pain Balderus died of his injury and was buried with royal honours in a barrow.
### Utrecht Inscription {#utrecht_inscription}
A Latin votive inscription from Utrecht, from the 3rd or 4th century C.E., has been theorized as containing the dative form *Baldruo*, pointing to a Latin nominative singular \**Baldruus*, which some have identified with the Norse/Germanic god, although both the reading and this interpretation have been questioned.
### *Anglo-Saxon Chronicle* {#anglo_saxon_chronicle}
In the Anglo-Saxon Chronicle Baldr is named as the ancestor of the monarchy of Kent, Bernicia, Deira, and Wessex through his supposed son Brond.
### Toponyms
There are a few old place names in Scandinavia that contain the name *Baldr*. The most certain and notable one is the (former) parish name Balleshol in Hedmark county, Norway: \"a Balldrshole\" 1356 (where the last element is *hóll* m \"mound; small hill\"). Others may be (in Norse forms) *Baldrsberg* in Vestfold county, *Baldrsheimr* in Hordaland county *Baldrsnes* in Sør-Trøndelag county---and (very uncertain) the Balsfjorden fjord and Balsfjord Municipality in Troms county.
In Copenhagen, there is also a Baldersgade, or \"Balder\'s Street\". A street in downtown Reykjavík is called Baldursgata (Baldur\'s Street).
In Sweden there is a Baldersgatan (Balder\'s Street) in Stockholm. There is also Baldersnäs (Balder\'s isthmus), Baldersvik (Balder\'s bay), Balders udde (Balder\'s headland) and Baldersberg (Balder\'s mountain) at various places
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# Breidablik
`{{About|the location in Nordic mythology|other uses|Breiðablik (disambiguation){{!}}`{=mediawiki}Breiðablik}} **Breiðablik** (sometimes anglicised as **Breithablik** or **Breidablik**) is the home of Baldr in Nordic mythology.
## Meaning
The word *Breiðablik* has been variously translated as \'broad sheen\', \'Broad gleam\', \'Broad-gleaming\' or \'the far-shining one\',
## Attestations
### Grímismál
The Eddic poem Grímnismál describes Breiðablik as the fair home of Baldr:
+----------------+-------------------------------------------------+
| Old Norse text | Bellows translation |
+================+=================================================+
| : | : The seventh is Breithablik; Baldr has there |
| | : For himself a dwelling set, |
| : | : In the land I know that lies so fair, |
| | : And from evil fate is free. |
| : | |
+----------------+-------------------------------------------------+
### Gylfaginning
In Snorri Sturluson\'s Gylfaginning, Breiðablik is described in a list of places in heaven, identified by some scholars as Asgard:
+----------------+-------------------------------------------------------------------------------------------------------------+
| Old Norse text | Brodeur translation |
+================+=============================================================================================================+
| | Then there is also in that place the abode called Breidablik, and there is not in heaven a fairer dwelling. |
+----------------+-------------------------------------------------------------------------------------------------------------+
Later in the work, when Snorri describes Baldr, he gives another description, citing *Grímnismál*, though he does not name the poem:
+----------------+----------------------------------------------------------------------------------------------------------------------+
| Old Norse text | Brodeur translation |
+================+======================================================================================================================+
| | : He \[Baldr\] dwells in the place called Breidablik, which is in heaven; in that place may nothing unclean be\... |
+----------------+----------------------------------------------------------------------------------------------------------------------+
## Interpretation and discussion {#interpretation_and_discussion}
The name of Breiðablik has been noted to link with Baldr\'s attributes of light and beauty.
Similarities have been drawn between the description of Breiðablik in Grímnismál and Heorot in Beowulf, which are both free of \'baleful runes\' (*feicnstafi* and *fācenstafas* respectively). In Beowulf, the lack of *fācenstafas* refers to the absence of crimes being committed, and therefore both halls have been proposed to be sanctuaries.
## In popular culture {#in_popular_culture}
- Breidablik is a sacred weapon in *Fire Emblem Heroes* that the Summoner uses to summon Heroes coming from different *Fire Emblem* games.
- In the PlayStation game *Xenogears*, Bledavik is the name of the capital city of the desert kingdom of Aveh on the Ignas continent
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# Bilskirnir
**Bilskirnir** (Old Norse \"lightning-crack\") is the hall of the god Thor in Norse mythology. Here he lives with his wife Sif and their children. According to *Grímnismál*, the hall is the greatest of buildings and contains 540 rooms, located in Asgard, as are all the dwellings of the gods, in the kingdom of Þrúðheimr (or Þrúðvangar according to *Gylfaginning* and *Ynglinga saga*).
## Modern influence {#modern_influence}
- The hall inspired the name of an Asgard starship commanded by Supreme Commander Thor, in the television series Stargate SG-1 named Beliskner.
- There is a NS / pagan black metal band from Hesse, Germany named Bilskirnir
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# Brísingamen
In Norse mythology, ***Brísingamen**\'\' (or***Brísinga men**\'\') is the torc or necklace of the goddess Freyja, of which little else is known for certain.
## Etymology
The name is an Old Norse compound *brísinga-men* whose second element is *men* \"(ornamental) neck-ring (of precious metal), torc\".`{{Refn|group="lower-alpha"|German sources use ''[[:de:Halsband|Halsband]]''<ref>e.g., Simek, Rudolf (1984) ''Lexikon der germanischen Mythologie'', s.v. "{{URL|1=https://archive.org/details/lexikondergerman0000sime/page/n551/mode/2up?q=Brisingamen |2=Brisingamen}}"</ref> lit. "neck-band", which differentiates from ''[[:de:Halskette|Halskette]]'' for 'necklace'.}}`{=mediawiki}
The etymology of the first element is uncertain. It has been derived from Old Norse *brísingr*, a poetic term for \"fire\" or \"amber\" mentioned in the anonymous versified word-lists (*þulur*) appended to many manuscripts of the Prose Edda, making Brísingamen \"gleaming torc\", \"sunny torc\", or the like. However, *Brísingr* can also be an ethnonym, in which case *Brísinga men* is \"torc of the Brísings\"; the Old English parallel in *Beowulf* supports this derivation, though who the Brísings (Old Norse *Brísingar*) may have been remains unknown.
## Attestations
### *Beowulf*
Brísingamen is referred to in the Anglo-Saxon epic *Beowulf* as *Brosinga mene*. The brief mention in *Beowulf* is as follows (trans. by Howell Chickering, 1977):
The *Beowulf* poet is clearly referring to the legends about Theoderic the Great. The *Þiðrekssaga* tells that the warrior Heime (*Háma* in Old English) takes sides against Ermanaric (\"Eormanric\"), king of the Goths, and has to flee his kingdom after robbing him; later in life, Hama enters a monastery and gives them all his stolen treasure. However, this saga makes no mention of the great necklace.
### *Poetic Edda* {#poetic_edda}
In the poem *Þrymskviða* of the *Poetic Edda*, Þrymr, the king of the jǫtnar, steals Thor\'s hammer, Mjölnir. Freyja lends Loki her falcon cloak to search for it; but upon returning, Loki tells Freyja that Þrymr has hidden the hammer and demanded to marry her in return. Freyja is so wrathful that all the Æsir's halls beneath her are shaken and the necklace Brísingamen breaks off from her neck. Later, Thor borrows Brísingamen when he dresses up as Freyja to go to the wedding at Jǫtunheimr.
### *Prose Edda* {#prose_edda}
*Húsdrápa*, a skaldic poem partially preserved in the *Prose Edda*, relates the story of the theft of Brísingamen by Loki. One day when Freyja wakes up and finds Brísingamen missing, she enlists the help of Heimdallr to help her search for it. Eventually they find the thief, who turns out to be Loki and who has transformed himself into a seal. Heimdallr turns into a seal as well and fights Loki (trans. Byock 2005): `{{Blockquote|...it was on this occasion that [Heimdall] and Loki came to blows over the ring of the Brisings. The skald Ulf Uggason devotes a lengthy passage to that story in his poem ''Husdrapa'', and it is stated there that Heimdall and Loki took on the shape of seals.}}`{=mediawiki} After a lengthy battle at Singasteinn, Heimdallr wins and returns Brísingamen to Freyja.
Snorri Sturluson quoted this old poem in *Skáldskaparmál*, saying that because of this legend Heimdallr is called \"Seeker of Freyja\'s Necklace\" (*Skáldskaparmál*, section 8) and Loki is called \"Thief of Brísingamen\" (*Skáldskaparmál*, section 16). A similar story appears in the later *Sörla þáttr*, where Heimdallr does not appear.
### *Sörla þáttr* {#sörla_þáttr}
Sörla þáttr is a short story in the later and extended version of the *Saga of Olaf Tryggvason* in the manuscript of the *Flateyjarbók*, which was written and compiled by two Christian priests, Jon Thordson and Magnus Thorhalson, in the late 14th century. In the end of the story, the arrival of Christianity dissolves the old curse that traditionally was to endure until Ragnarök.
The battle of Högni and Heðinn is recorded in several medieval sources, including the skaldic poem *Ragnarsdrápa*, *Skáldskaparmál* (section 49), and *Gesta Danorum*: king Högni\'s daughter, Hildr, is kidnapped by king Heðinn. When Högni comes to fight Heðinn on an island, Hildr comes to offer her father a necklace on behalf of Heðinn for peace; but the two kings still battle, and Hildr resurrects the fallen to make them fight until Ragnarök. None of these earlier sources mentions Freyja or king Olaf Tryggvason, the historical figure who Christianized Norway and Iceland in the 10th Century.
## Archaeological record {#archaeological_record}
A Völva was buried c. 1000 with considerable splendour in Hagebyhöga in Östergötland, Sweden. In addition to being buried with her wand, she had received great riches which included horses, a wagon and an Arabian bronze pitcher. There was also a silver pendant, which represents a woman with a broad necklace around her neck. This kind of necklace was only worn by the most prominent women during the Iron Age and some have interpreted it as Freyja\'s necklace Brísingamen. The pendant may represent Freyja herself.
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# Brísingamen
## Modern influence {#modern_influence}
Alan Garner wrote a children\'s fantasy novel called *The Weirdstone of Brisingamen*, published in 1960, about an enchanted teardrop bracelet.
Diana Paxson\'s novel *Brisingamen* features Freyja and her necklace.
Black Phoenix Alchemy Lab has a perfumed oil scent named Brisingamen.
Freyja\'s necklace Brisingamen features prominently in Betsy Tobin\'s novel *Iceland*, where the necklace is seen to have significant protective powers.
The Brisingamen feature as a major item in Joel Rosenberg\'s Keepers of the Hidden Ways series of books. In it, there are seven jewels that were created for the necklace by the Dwarfs and given to the Norse goddess. She in turn eventually split them up into the seven separate jewels and hid them throughout the realm, as together they hold the power to shape the universe by its holder. The book\'s plot is about discovering one of them and deciding what to do with the power they allow while avoiding Loki and other Norse characters.
In Christopher Paolini\'s *The Inheritance Cycle*, the word \"brisingr\" means fire. This is probably a distillation of the word *brisinga*.
Ursula Le Guin\'s short story *Semley\'s Necklace*, the first part of her novel *Rocannon\'s World*, is a retelling of the Brisingamen story on an alien planet.
Brisingamen is represented as a card in the *Yu-Gi-Oh!* Trading Card Game, \"Nordic Relic Brisingamen\".
Brisingamen was part of MMORPG *Ragnarok Online* lore, which is ranked as \"God item\". The game is heavily based from Norse mythology.
In the *Firefly Online* game, one of the planets of the Himinbjörg system (which features planets named after figures from Germanic mythology) is named Brisingamen. It is third from the star, and has moons named Freya, Beowulf, and Alberich.
The Brisingamen is an item that can be found and equipped in the video game, *Castlevania: Lament of Innocence*.
In the French comics *Freaks\' Squeele*, the character of Valkyrie accesses her costume change ability by touching a decorative torc necklace affixed to her forehead, named Brizingamen
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# Borsuk–Ulam theorem
In mathematics, the **Borsuk--Ulam theorem** states that every continuous function from an *n*-sphere into Euclidean *n*-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere\'s center.
Formally: if $f: S^n \to \R^n$ is continuous then there exists an $x\in S^n$ such that: $f(-x)=f(x)$.
The case $n=1$ can be illustrated by saying that there always exist a pair of opposite points on the Earth\'s equator with the same temperature. The same is true for any circle. This assumes the temperature varies continuously in space, which is, however, not always the case.
The case $n=2$ is often illustrated by saying that at any moment, there is always a pair of antipodal points on the Earth\'s surface with equal temperatures and equal barometric pressures, assuming that both parameters vary continuously in space.
The Borsuk--Ulam theorem has several equivalent statements in terms of odd functions. Recall that $S^n$ is the *n*-sphere and $B^n$ is the *n*-ball:
- If $g : S^n \to \R^n$ is a continuous odd function, then there exists an $x\in S^n$ such that: $g(x)=0$.
- If $g : B^n \to \R^n$ is a continuous function which is odd on $S^{n-1}$ (the boundary of $B^n$), then there exists an $x\in B^n$ such that: $g(x)=0$.
## History
According to `{{harvtxt|Matoušek|2003|p=25}}`{=mediawiki}, the first historical mention of the statement of the Borsuk--Ulam theorem appears in `{{harvtxt|Lyusternik|Shnirel'man|1930}}`{=mediawiki}. The first proof was given by `{{harvs|first=Karol|last=Borsuk|authorlink=Karol Borsuk|year=1933|txt}}`{=mediawiki}, where the formulation of the problem was attributed to Stanisław Ulam. Since then, many alternative proofs have been found by various authors, as collected by `{{harvtxt|Steinlein|1985}}`{=mediawiki}.
## Equivalent statements {#equivalent_statements}
The following statements are equivalent to the Borsuk--Ulam theorem.
### With odd functions {#with_odd_functions}
A function $g$ is called *odd* (aka *antipodal* or *antipode-preserving*) if for every $x$, $g(-x)=-g(x)$.
The Borsuk--Ulam theorem is equivalent to each of the following statements:
\(1\) Each continuous odd function $S^n\to \R^n$ has a zero.
\(2\) There is no continuous odd function $S^n \to S^{n-1}$.
Here is a proof that the Borsuk-Ulam theorem is equivalent to (1):
($\Longrightarrow$) If the theorem is correct, then it is specifically correct for odd functions, and for an odd function, $g(-x)=g(x)$ iff $g(x)=0$. Hence every odd continuous function has a zero.
($\Longleftarrow$) For every continuous function $f:S^n\to \R^n$, the following function is continuous and odd: $g(x)=f(x)-f(-x)$. If every odd continuous function has a zero, then $g$ has a zero, and therefore, $f(x)=f(-x)$.
To prove that (1) and (2) are equivalent, we use the following continuous odd maps:
- the obvious inclusion $i: S^{n-1}\to \R^n\setminus \{0\}$,
- and the radial projection map $p: \R^n\setminus \{0\} \to S^{n-1}$ given by $x \mapsto \frac{x}{|x|}$.
The proof now writes itself.
$((1) \Longrightarrow (2))$ We prove the contrapositive. If there exists a continuous odd function $f:S^n\to S^{n-1}$, then $i\circ f$ is a continuous odd function $S^n\to \R^n\setminus \{0\}$.
$((1) \Longleftarrow (2))$ Again we prove the contrapositive. If there exists a continuous odd function $f:S^n\to \R^{n}\setminus\{0\}$, then $p\circ f$ is a continuous odd function $S^n\to S^{n-1}$.
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# Borsuk–Ulam theorem
## Proofs
### 1-dimensional case {#dimensional_case}
The 1-dimensional case can easily be proved using the intermediate value theorem (IVT).
Let $g$ be the odd real-valued continuous function on a circle defined by $g(x)=f(x)-f(-x)$. Pick an arbitrary $x$. If $g(x)=0$ then we are done. Otherwise, without loss of generality, $g(x)>0.$ But $g(-x)<0.$ Hence, by the IVT, there is a point $y$ at which $g(y)=0$.
### General case {#general_case}
#### Algebraic topological proof {#algebraic_topological_proof}
Assume that $h: S^n \to S^{n-1}$ is an odd continuous function with $n > 2$ (the case $n = 1$ is treated above, the case $n = 2$ can be handled using basic covering theory). By passing to orbits under the antipodal action, we then get an induced continuous function $h': \mathbb{RP}^n \to \mathbb{RP}^{n-1}$ between real projective spaces, which induces an isomorphism on fundamental groups. By the Hurewicz theorem, the induced ring homomorphism on cohomology with $\mathbb F_2$ coefficients \[where $\mathbb F_2$ denotes the field with two elements\],
$$\mathbb F_2[a]/a^{n+1} = H^*\left(\mathbb{RP}^n; \mathbb{F}_2\right) \leftarrow H^*\left(\mathbb{RP}^{n-1}; \mathbb F_2\right) = \mathbb F_2[b]/b^{n},$$
sends $b$ to $a$. But then we get that $b^n = 0$ is sent to $a^n \neq 0$, a contradiction.
One can also show the stronger statement that any odd map $S^{n-1} \to S^{n-1}$ has odd degree and then deduce the theorem from this result.
#### Combinatorial proof {#combinatorial_proof}
The Borsuk--Ulam theorem can be proved from Tucker\'s lemma.
Let $g : S^n \to \R^n$ be a continuous odd function. Because *g* is continuous on a compact domain, it is uniformly continuous. Therefore, for every $\epsilon > 0$, there is a $\delta > 0$ such that, for every two points of $S_n$ which are within $\delta$ of each other, their images under *g* are within $\epsilon$ of each other.
Define a triangulation of $S_n$ with edges of length at most $\delta$. Label each vertex $v$ of the triangulation with a label $l(v)\in {\pm 1, \pm 2, \ldots, \pm n}$ in the following way:
- The absolute value of the label is the *index* of the coordinate with the highest absolute value of *g*: $|l(v)| = \arg\max_k (|g(v)_k|)$.
- The sign of the label is the sign of *g* at the above coordinate, so that: $l(v) = \sgn (g(v)_{|l(v)|}) |l(v)|$.
Because *g* is odd, the labeling is also odd: $l(-v) = -l(v)$. Hence, by Tucker\'s lemma, there are two adjacent vertices $u, v$ with opposite labels. Assume w.l.o.g. that the labels are $l(u)=1, l(v)=-1$. By the definition of *l*, this means that in both $g(u)$ and $g(v)$, coordinate #1 is the largest coordinate: in $g(u)$ this coordinate is positive while in $g(v)$ it is negative. By the construction of the triangulation, the distance between $g(u)$ and $g(v)$ is at most $\epsilon$, so in particular $|g(u)_1 - g(v)_1| = |g(u)_1| + |g(v)_1| \leq \epsilon$ (since $g(u)_1$ and $g(v)_1$ have opposite signs) and so $|g(u)_1| \leq \epsilon$. But since the largest coordinate of $g(u)$ is coordinate #1, this means that $|g(u)_k| \leq \epsilon$ for each $1 \leq k \leq n$. So $|g(u)| \leq c_n \epsilon$, where $c_n$ is some constant depending on $n$ and the norm $|\cdot|$ which you have chosen.
The above is true for every $\epsilon > 0$; since $S_n$ is compact there must hence be a point *u* in which $|g(u)|=0$.
## Corollaries
- No subset of $\R^n$ is homeomorphic to $S^n$
- The ham sandwich theorem: For any compact sets *A*~1~, \..., *A~n~* in $\R^n$ we can always find a hyperplane dividing each of them into two subsets of equal measure.
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# Borsuk–Ulam theorem
## Equivalent results {#equivalent_results}
Above we showed how to prove the Borsuk--Ulam theorem from Tucker\'s lemma. The converse is also true: it is possible to prove Tucker\'s lemma from the Borsuk--Ulam theorem. Therefore, these two theorems are equivalent. `{{Analogous fixed-point theorems}}`{=mediawiki}
## Generalizations
- In the original theorem, the domain of the function *f* is the unit *n*-sphere (the boundary of the unit *n*-ball). In general, it is true also when the domain of *f* is the boundary of any open bounded symmetric subset of $\R^n$ containing the origin (Here, symmetric means that if *x* is in the subset then -*x* is also in the subset).
- More generally, if $M$ is a compact *n*-dimensional Riemannian manifold, and $f: M \rightarrow \mathbb{R}^n$ is continuous, there exists a pair of points *x* and *y* in $M$ such that $f(x) = f(y)$ and *x* and *y* are joined by a geodesic of length $\delta$, for any prescribed $\delta > 0$.
- Consider the function *A* which maps a point to its antipodal point: $A(x) = -x.$ Note that $A(A(x))=x.$ The original theorem claims that there is a point *x* in which $f(A(x))=f(x).$ In general, this is true also for every function *A* for which $A(A(x))=x.$ However, in general this is not true for other functions *A*
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# Bragi
**Bragi** (Old Norse) is the skaldic god of poetry in Norse mythology.
## Etymology
The theonym Bragi probably stems from the masculine noun *bragr*, which can be translated in Old Norse as \'poetry\' (cf. Icelandic *bragur* \'poem, melody, wise\') or as \'the first, noblest\' (cf. poetic Old Norse *bragnar* \'chiefs, men\', *bragningr* \'king\'). It is unclear whether the theonym semantically derives from the first meaning or the second.
A connection has been also suggested with the Old Norse *bragarfull*, the cup drunk in solemn occasions with the taking of vows. The word is usually taken to semantically derive from the second meaning of *bragr* (\'first one, noblest\'). A relation with the Old English term *brego* (\'lord, prince\') remains uncertain.
*Bragi* regularly appears as a personal name in Old Norse and Old Swedish sources, which according to linguist Jan de Vries might indicate the secondary character of the god\'s name.
## Attestations
Snorri Sturluson writes in the *Gylfaginning* after describing Odin, Thor, and Baldr: `{{blockquote|One is called Bragi: he is renowned for wisdom, and most of all for fluency of speech and skill with words. He knows most of skaldship, and after him skaldship is called ''bragr'', and from his name that one is called ''bragr''-man or -woman, who possesses eloquence surpassing others, of women or of men. His wife is [[Iðunn]].}}`{=mediawiki}
In *Skáldskaparmál* Snorri writes: `{{blockquote|How should one periphrase Bragi? By calling him ''husband of Iðunn'', ''first maker of poetry'', and ''the long-bearded god'' (after his name, a man who has a great beard is called Beard-Bragi), and ''son of Odin''.}}`{=mediawiki}
That Bragi is Odin\'s son is clearly mentioned only here and in some versions of a list of the sons of Odin (see Sons of Odin). But \"wish-son\" in stanza 16 of the *Lokasenna* could mean \"Odin\'s son\" and is translated by Hollander as *Odin\'s kin*. Bragi\'s mother is possibly Frigg.
In that poem Bragi at first forbids Loki to enter the hall but is overruled by Odin. Loki then gives a greeting to all gods and goddesses who are in the hall save to Bragi. Bragi generously offers his sword, horse, and an arm ring as peace gift but Loki only responds by accusing Bragi of cowardice, of being the most afraid to fight of any of the Æsir and Elves within the hall. Bragi responds that if they were outside the hall, he would have Loki\'s head, but Loki only repeats the accusation. When Bragi\'s wife Iðunn attempts to calm Bragi, Loki accuses her of embracing her brother\'s slayer, a reference to matters that have not survived. It may be that Bragi had slain Iðunn\'s brother.
A passage in the *Poetic Edda* poem *Sigrdrífumál* describes runes being graven on the sun, on the ear of one of the sun-horses and on the hoofs of the other, on Sleipnir\'s teeth, on bear\'s paw, on eagle\'s beak, on wolf\'s claw, and on several other things including on Bragi\'s tongue. Then the runes are shaved off and the shavings are mixed with mead and sent abroad so that Æsir have some, Elves have some, Vanir have some, and Men have some, these being speech runes and birth runes, ale runes, and magic runes. The meaning of this is obscure.
The first part of Snorri Sturluson\'s *Skáldskaparmál* is a dialogue between Ægir and Bragi about the nature of poetry, particularly skaldic poetry. Bragi tells the origin of the mead of poetry from the blood of Kvasir and how Odin obtained this mead. He then goes on to discuss various poetic metaphors known as *kennings*.
Snorri Sturluson clearly distinguishes the god Bragi from the mortal skald Bragi Boddason, whom he often mentions separately. The appearance of Bragi in the *Lokasenna* indicates that if these two Bragis were originally the same, they have become separated for that author also, or that chronology has become very muddled and Bragi Boddason has been relocated to mythological time. Compare the appearance of the Welsh Taliesin in the second branch of the Mabinogi. Legendary chronology sometimes does become muddled. Whether Bragi the god originally arose as a deified version of Bragi Boddason was much debated in the 19th century, especially by the scholars Eugen Mogk and Sophus Bugge. The debate remains undecided.
In the poem *Eiríksmál* Odin, in Valhalla, hears the coming of the dead Norwegian king Eric Bloodaxe and his host, and bids the heroes Sigmund and Sinfjötli rise to greet him. Bragi is then mentioned, questioning how Odin knows that it is Eric and why Odin has let such a king die. In the poem *Hákonarmál*, Hákon the Good is taken to Valhalla by the valkyrie Göndul and Odin sends Hermóðr and Bragi to greet him. In these poems Bragi could be either a god or a dead hero in Valhalla. Attempting to decide is further confused because *Hermóðr* also seems to be sometimes the name of a god and sometimes the name of a hero. That Bragi was also the first to speak to Loki in the *Lokasenna* as Loki attempted to enter the hall might be a parallel. It might have been useful and customary that a man of great eloquence and versed in poetry should greet those entering a hall. He is also depicted in tenth-century court poetry of helping to prepare Valhalla for new arrivals and welcoming the kings who have been slain in battle to the hall of Odin.
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# Bragi
## Skalds named Bragi {#skalds_named_bragi}
### Bragi Boddason {#bragi_boddason}
In the *Prose Edda* Snorri Sturluson quotes many stanzas attributed to **Bragi Boddason** the old (*Bragi Boddason inn gamli*), a Norwegian court poet who served several Swedish kings, Ragnar Lodbrok, Östen Beli and Björn at Hauge who reigned in the first half of the 9th century. This Bragi was reckoned as the first skaldic poet, and was certainly the earliest skaldic poet then remembered by name whose verse survived in memory.
Snorri especially quotes passages from Bragi\'s *Ragnarsdrápa*, a poem supposedly composed in honor of the famous legendary Viking Ragnar Lodbrok (\'Hairy-breeches\') describing the images on a decorated shield which Ragnar had given to Bragi. The images included Thor\'s fishing for Jörmungandr, Gefjun\'s ploughing of Zealand from the soil of Sweden, the attack of Hamdir and Sorli against King Jörmunrekk, and the never-ending battle between Hedin and Högni.
### Bragi son of Hálfdan the Old {#bragi_son_of_hálfdan_the_old}
**Bragi son of Hálfdan the Old** is mentioned only in the *Skjáldskaparmál*. This Bragi is the sixth of the second of two groups of nine sons fathered by King Hálfdan the Old on Alvig the Wise, daughter of King Eymund of Hólmgard. This second group of sons are all eponymous ancestors of legendary families of the north. Snorri says:
> Bragi, from whom the Bragnings are sprung (that is the race of Hálfdan the Generous).
Of the Bragnings as a race and of Hálfdan the Generous nothing else is known. However, *Bragning* is often, like some others of these dynastic names, used in poetry as a general word for \'king\' or \'ruler\'.
### Bragi Högnason {#bragi_högnason}
In the eddic poem *Helgakviða Hundingsbana II*, **Bragi Högnason**, his brother Dag, and his sister Sigrún were children of Högne, the king of East Götaland. The poem relates how Sigmund\'s son Helgi Hundingsbane agreed to take Sigrún daughter of Högni as his wife against her unwilling betrothal to Hodbrodd son of Granmar the king of Södermanland. In the subsequent battle of Frekastein (probably one of the 300 hill forts of Södermanland, as *stein* meant \"hill fort\") against Högni and Granmar, all the chieftains on Granmar\'s side are slain, including Bragi, except for Bragi\'s brother Dag.
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# Bragi
## In popular culture {#in_popular_culture}
In the 2002 Ensemble Studios game *Age of Mythology*, Bragi is one of nine minor gods Norse players can worship
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# Bronski Beat
*Pandoc failed*: ```
Error at (line 216, column 1):
unexpected end of input
``
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# BPP (complexity)
In computational complexity theory, a branch of computer science, **bounded-error probabilistic polynomial time** (**BPP**) is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability bounded by 1/3 for all instances. **BPP** is one of the largest *practical* classes of problems, meaning most problems of interest in **BPP** have efficient probabilistic algorithms that can be run quickly on real modern machines. **BPP** also contains **P**, the class of problems solvable in polynomial time with a deterministic machine, since a deterministic machine is a special case of a probabilistic machine.
+---------------------------------+
| BPP algorithm (1 run) |
+=================================+
| \"padding-left:4em;\"\>produced |
| |
| }} |
+---------------------------------+
| |
+---------------------------------+
| |
+---------------------------------+
| BPP algorithm (*k* runs) |
+---------------------------------+
| \"padding-left:4em;\"\>Answer |
| |
| produced}} |
+---------------------------------+
| |
+---------------------------------+
| |
+---------------------------------+
| for some constant *c* \> 0 |
+---------------------------------+
Informally, a problem is in **BPP** if there is an algorithm for it that has the following properties:
- It is allowed to flip coins and make random decisions
- It is guaranteed to run in polynomial time
- On any given run of the algorithm, it has a probability of at most 1/3 of giving the wrong answer, whether the answer is YES or NO.
## Definition
A language *L* is in **BPP** if and only if there exists a probabilistic Turing machine *M*, such that
- *M* runs for polynomial time on all inputs
- For all *x* in *L*, *M* outputs 1 with probability greater than or equal to 2/3
- For all *x* not in *L*, *M* outputs 1 with probability less than or equal to 1/3
Unlike the complexity class **ZPP**, the machine *M* is required to run for polynomial time on all inputs, regardless of the outcome of the random coin flips.
Alternatively, **BPP** can be defined using only deterministic Turing machines. A language *L* is in **BPP** if and only if there exists a polynomial *p* and deterministic Turing machine *M*, such that
- *M* runs for polynomial time on all inputs
- For all *x* in *L*, the fraction of strings *y* of length *p*(\|*x*\|) which satisfy `{{tmath|1=M(x,y) = 1}}`{=mediawiki} is greater than or equal to 2/3
- For all *x* not in *L*, the fraction of strings *y* of length *p*(\|*x*\|) which satisfy `{{tmath|1=M(x,y) = 1}}`{=mediawiki} is less than or equal to 1/3
In this definition, the string *y* corresponds to the output of the random coin flips that the probabilistic Turing machine would have made. For some applications this definition is preferable since it does not mention probabilistic Turing machines.
In practice, an error probability of 1/3 might not be acceptable; however, the choice of 1/3 in the definition is arbitrary. Modifying the definition to use any constant between 0 and 1/2 (exclusive) in place of 1/3 would not change the resulting set **BPP**. For example, if one defined the class with the restriction that the algorithm can be wrong with probability at most 1/2^100^, this would result in the same class of problems. The error probability does not even have to be constant: the same class of problems is defined by allowing error as high as 1/2 − *n*^−*c*^ on the one hand, or requiring error as small as 2^−*n^c^*^ on the other hand, where *c* is any positive constant, and *n* is the length of input. This flexibility in the choice of error probability is based on the idea of running an error-prone algorithm many times, and using the majority result of the runs to obtain a more accurate algorithm. The chance that the majority of the runs are wrong drops off exponentially as a consequence of the Chernoff bound.
## Problems
All problems in **P** are obviously also in **BPP**. However, many problems have been known to be in **BPP** but not known to be in **P**. The number of such problems is decreasing, and it is conjectured that **P** = **BPP**.
For a long time, one of the most famous problems known to be in **BPP** but not known to be in **P** was the problem of determining whether a given number is prime. However, in the 2002 paper *PRIMES is in **P***, Manindra Agrawal and his students Neeraj Kayal and Nitin Saxena found a deterministic polynomial-time algorithm for this problem, thus showing that it is in **P**.
An important example of a problem in **BPP** (in fact in **co-RP**) still not known to be in **P** is polynomial identity testing, the problem of determining whether a polynomial is identically equal to the zero polynomial, when you have access to the value of the polynomial for any given input, but not to the coefficients. In other words, is there an assignment of values to the variables such that when a nonzero polynomial is evaluated on these values, the result is nonzero? It suffices to choose each variable\'s value uniformly at random from a finite subset of at least *d* values to achieve bounded error probability, where *d* is the total degree of the polynomial.
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# BPP (complexity)
## Related classes {#related_classes}
If the access to randomness is removed from the definition of **BPP**, we get the complexity class **P**. In the definition of the class, if we replace the ordinary Turing machine with a quantum computer, we get the class **BQP**.
Adding postselection to **BPP**, or allowing computation paths to have different lengths, gives the class **BPP**~path~. **BPP**~path~ is known to contain **NP**, and it is contained in its quantum counterpart **PostBQP**.
A Monte Carlo algorithm is a randomized algorithm which is likely to be correct. Problems in the class **BPP** have Monte Carlo algorithms with polynomial bounded running time. This is compared to a Las Vegas algorithm which is a randomized algorithm which either outputs the correct answer, or outputs \"fail\" with low probability. Las Vegas algorithms with polynomial bound running times are used to define the class **ZPP**. Alternatively, **ZPP** contains probabilistic algorithms that are always correct and have expected polynomial running time. This is weaker than saying it is a polynomial time algorithm, since it may run for super-polynomial time, but with very low probability.
## Complexity-theoretic properties {#complexity_theoretic_properties}
alt=Diagram of randomised complexity classes\|thumb\|upright=1.25\|BPP in relation to other probabilistic complexity classes (ZPP, RP, co-RP, BQP, PP), which generalise P within PSPACE. It is unknown if any of these containments are strict.
It is known that **BPP** is closed under complement; that is, **BPP** = **co-BPP**. **BPP** is low for itself, meaning that a **BPP** machine with the power to solve **BPP** problems instantly (a **BPP** oracle machine) is not any more powerful than the machine without this extra power. In symbols, **BPP**^**BPP**^ = **BPP**.
The relationship between **BPP** and **NP** is unknown: it is not known whether **BPP** is a subset of **NP**, **NP** is a subset of **BPP** or neither. If **NP** is contained in **BPP**, which is considered unlikely since it would imply practical solutions for NP-complete problems, then **NP** = **RP** and **PH** ⊆ **BPP**.
It is known that **RP** is a subset of **BPP**, and **BPP** is a subset of **PP**. It is not known whether those two are strict subsets, since we don\'t even know if **P** is a strict subset of **PSPACE**. **BPP** is contained in the second level of the polynomial hierarchy and therefore it is contained in **PH**. More precisely, the Sipser--Lautemann theorem states that $\mathsf{BPP} \subseteq \Sigma_2 \cap \Pi_2$. As a result, **P** = **NP** leads to **P** = **BPP** since **PH** collapses to **P** in this case. Thus either **P** = **BPP** or **P** ≠ **NP** or both.
Adleman\'s theorem states that membership in any language in **BPP** can be determined by a family of polynomial-size Boolean circuits, which means **BPP** is contained in **P/poly**. Indeed, as a consequence of the proof of this fact, every **BPP** algorithm operating on inputs of bounded length can be derandomized into a deterministic algorithm using a fixed string of random bits. Finding this string may be expensive, however. Some weak separation results for Monte Carlo time classes were proven by `{{harvtxt|Karpinski|Verbeek|1987a}}`{=mediawiki}, see also `{{harvtxt|Karpinski|Verbeek|1987b}}`{=mediawiki}.
### Closure properties {#closure_properties}
The class BPP is closed under complementation, union, intersection, and concatenation.
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