Definition.

The ISM bands are defined by the ITU Radio Regulations (article 5) in footnotes 5.138, 5.150, and 5.280 of the Radio Regulations. Individual countries' use of the bands designated in these sections may differ due to variations in national radio regulations. Because communication devices using the ISM bands must tolerate any interference from ISM equipment, unlicensed operations are typically permitted to use these bands, since unlicensed operation typically needs to be tolerant of interference from other devices anyway. The ISM bands share allocations with unlicensed and licensed operations; however, due to the high likelihood of harmful interference, licensed use of the bands is typically low. In the United States, uses of the ISM bands are governed by Part 18 of the Federal Communications Commission (FCC) rules, while Part 15 contains the rules for unlicensed communication devices, even those that share ISM frequencies. In Europe, the ETSI is responsible for regulating the use of Short Range Devices, some of which operate in ISM bands.

Frequency allocation.

The allocation of radio frequencies is provided according to "Article 5" of the ITU Radio Regulations (edition 2012).  In order to improve harmonisation in spectrum utilisation, the majority of service-allocations stipulated in this document were incorporated in national Tables of Frequency Allocations and Utilisations which is within the responsibility of the appropriate national administration. The allocation might be primary, secondary, exclusive, and shared.  Type A (footnote 5.138) = frequency bands are designated for "ISM applications". The use of these frequency bands for ISM applications shall be subject to special authorization by the administration concerned, in agreement with other administrations whose radiocommunication services might be affected. In applying this provision, administrations shall have due regard to the latest relevant ITU-R Recommendations.  Type B (footnote 5.150) = frequency bands are also designated for ISM applications. Radiocommunication services operating within these bands must accept harmful interference which may be caused by these applications.  ITU RR, (Footnote 5.280) = In Germany, Austria, Bosnia and Herzegovina, Croatia, Macedonia, Liechtenstein, Montenegro, Portugal, Serbia, Slovenia and Switzerland, the band 433.05-434.79 MHz (center frequency 433.92 MHz) is designated for "ISM applications". Radio communication services of these countries operating within this band must accept harmful interference which may be caused by these applications.  Footnote AU = Australia is part of ITU Region 3. The band 433.05 to 434.79 MHz is not a designated ISM band in Australia, however the operation of low powered devices in the radio frequency band 433.05 to 434.79 MHz is supported through Radio communications class licence for low interference potential devices (LIPDs).

History.

The ISM bands were first established at the International Telecommunications Conference of the ITU in Atlantic City, 1947. The American delegation specifically proposed several bands, including the now commonplace 2.4 GHz band, to accommodate the then nascent process of microwave heating; however, FCC annual reports of that time suggest that much preparation was done ahead of these presentations.  The report of the August 9th 1947 meeting of the Allocation of Frequencies committee includes the remark:  "The delegate of the United States, referring to his request that the frequency 2450 Mc/s be allocated for I.S.M., indicated that there was in existence in the United States, and working on this frequency a diathermy machine and an electronic cooker, and that the latter might eventually be installed in transatlantic ships and airplanes. There was therefore some point in attempting to reach world agreement on this subject."  Radio frequencies in the ISM bands have been used for communication purposes, although such devices may experience interference from non-communication sources. In the United States, as early as 1958 Class D Citizens Band, a Part 95 service, was allocated to frequencies that are also allocated to ISM. [1]  In the U.S., the FCC first made unlicensed spread spectrum available in the ISM bands in rules adopted on May 9, 1985.  Many other countries later developed similar regulations, enabling use of this technology. The FCC action was proposed by Michael Marcus of the FCC staff in 1980 and the subsequent regulatory action took five more years. It was part of a broader proposal to allow civil use of spread spectrum technology and was opposed at the time by mainstream equipment manufacturers and many radio system operators.

Applications.

Industrial, scientific and medical (ISM) applications (of radio frequency energy) (short: ISM applications) are – according to "article 1.15" of the International Telecommunication Union ́s (ITU) ITU Radio Regulations (RR) – defined as «"Operation of equipment or appliances designed to generate and use locally radio frequency energy for industrial, scientific, medical, domestic or similar purposes, excluding applications in the field of telecommunications".»  The original ISM specifications envisioned that the bands would be used primarily for noncommunication purposes, such as heating. The bands are still widely used for these purposes. For many people, the most commonly encountered ISM device is the home microwave oven operating at 2.45 GHz which uses microwaves to cook food. Industrial heating is another big application area; such as induction heating, microwave heat treating, plastic softening, and plastic welding processes. In medical settings, shortwave and microwave diathermy machines use radio waves in the ISM bands to apply deep heating to the body for relaxation and healing. More recently hyperthermia therapy uses microwaves to heat tissue to kill cancer cells.  However, as detailed below, the increasing congestion of the radio spectrum, the increasing sophistication of microelectronics, and the attraction of unlicensed use, has in recent decades led to an explosion of uses of these bands for short range communication systems for wireless devices, which are now by far the largest uses of these bands. These are sometimes called "non ISM" uses since they do not fall under the originally envisioned "industrial", "scientific", and "medical" application areas. One of the largest applications has been wireless networking (WiFi). The IEEE 802.11 wireless networking protocols, the standards on which almost all wireless systems are based, use the ISM bands. Virtually all laptops, tablet computers, computer printers and cellphones now have 802.11 wireless modems using the 2.4 and 5.7 GHz ISM bands. Bluetooth is another networking technology using the 2.4 GHz band, which can be problematic given the probability of interference. Near field communication devices such as proximity cards and contactless smart cards use the lower frequency 13 and 27 MHz ISM bands. Other short range devices using the ISM bands are: wireless microphones, baby monitors, garage door openers, wireless doorbells, keyless entry systems for vehicles, radio control channels for UAVs (drones), wireless surveillance systems, RFID systems for merchandise, and wild animal tracking systems.  Some electrodeless lamp designs are ISM devices, which use RF emissions to excite fluorescent tubes. Sulfur lamps are commercially available plasma lamps, which use a 2.45 GHz magnetron to heat sulfur into a brightly glowing plasma.  Long-distance wireless power systems have been proposed and experimented with which would use high-power transmitters and rectennas, in lieu of overhead transmission lines and underground cables, to send power to remote locations. NASA has studied using microwave power transmission on 2.45 GHz to send energy collected by solar power satellites back to the ground.  Also in space applications, a Helicon Double Layer ion thruster is a prototype spacecraft propulsion engine which uses a 13.56 MHz transmission to break down and heat gas into plasma.

Common non-ISM uses.

In recent years ISM bands have also been shared with (non-ISM) license-free error-tolerant communications applications such as wireless sensor networks in the 915 MHz and 2.450 GHz bands, as well as wireless LANs and cordless phones in the 915 MHz, 2.450 GHz, and 5.800 GHz bands. Because unlicensed devices are required to be tolerant of ISM emissions in these bands, unlicensed low power users are generally able to operate in these bands without causing problems for ISM users. ISM equipment does not necessarily include a radio receiver in the ISM band (e.g. a microwave oven does not have a receiver).  In the United States, according to 47 CFR Part 15.5, low power communication devices must accept interference from licensed users of that frequency band, and the Part 15 device must not cause interference to licensed users. Note that the 915 MHz band should not be used in countries outside Region 2, except those that specifically allow it, such as Australia and Israel, especially those that use the GSM-900 band for cellphones. The ISM bands are also widely used for Radio-frequency identification (RFID) applications with the most commonly used band being the 13.56 MHz band used by systems compliant with ISO/IEC 14443 including those used by biometric passports and contactless smart cards.  In Europe, the use of the ISM band is covered by Short Range Device regulations issued by European Commission, based on technical recommendations by CEPT and standards by ETSI. In most of Europe, LPD433 band is allowed for license-free voice communication in addition to PMR446.  Wireless LAN devices use wavebands as follows:  IEEE 802.15.4, ZigBee and other personal area networks may use the and ISM bands because of frequency sharing between different allocations.  Wireless LANs and cordless phones can also use bands other than those shared with ISM, but such uses require approval on a country by country basis. DECT phones use allocated spectrum outside the ISM bands that differs in Europe and North America. Ultra-wideband LANs require more spectrum than the ISM bands can provide, so the relevant standards such as IEEE 802.15.4a are designed to make use of spectrum outside the ISM bands. Despite the fact that these additional bands are outside the official ITU-R ISM bands, because they are used for the same types of low power personal communications, they are sometimes incorrectly referred to as ISM bands as well.  Several brands of radio control equipment use the band range for low power remote control of toys, from gas powered cars to miniature aircraft.  Worldwide Digital Cordless Telecommunications or WDCT is a technology that uses the radio spectrum.  Google's Project Loon uses ISM bands (specifically 2.4 and 5.8 GHz bands) for balloon-to-balloon and balloon-to-ground communications.  Pursuant to 47 CFR Part 97 some ISM bands are used by licensed amateur radio operators for communication - including amateur television.

The ISM radio bands are portions of the radio spectrum reserved internationally for industrial, scientific and medical (ISM) purposes other than telecommunications. Examples of applications for the use of radio frequency (RF) energy in these bands include radio-frequency process heating, microwave ovens, and medical diathermy machines. The powerful emissions of these devices can create electromagnetic interference and disrupt radio communication using the same frequency, so these devices are limited to certain bands of frequencies. In general, communications equipment operating in these bands must tolerate any interference generated by ISM applications, and users have no regulatory protection from ISM device operation.

Pásma ISM (industrial, scientific and medical) jsou pásma pro rádiové vysílání v oborech průmyslovém, vědeckém a zdravotnickém. Jsou to pásma volná, což znamená že je v nich při použití homologovaného (schváleného) zařízení dovolen provoz bez licenčních poplatků, avšak bez garance proti rušení. Zároveň je provoz takovýchto zařízení regulován všeobecnými oprávněními, které vydává Český telekomunikační úřad. Pásma ISM jsou součástí podmínek pro vysílací zařízení krátkého dosahu (SRD, short range devices), pro něž platí všeobecné oprávnění VO-R/10.


History.

Formation and early years.

David Monks (born January 21, 1987), Graham Wright (born February 16, 1987), Josh Hook (born May 11, 1987), and Greg Alsop (born March 20, 1985) grew up and attended school together in Newmarket, Ontario. In high school, the four played in a band called "Suburbia" which eventually disbanded. In 2004, the group informally came together again to form Tokyo Police Club; the name of the band comes from a 2000s-era online Band Name Generator.  Early on, the band played some small shows in the Toronto area. Tokyo Police Club were asked to play in the Pop Montreal festival, and soon after they signed with Toronto label Paper Bag Records; Monks and Alsop dropped out of college to become professional musicians.  The band released its debut EP, "A Lesson in Crime", in 2006 on Paper Bag Records. The recording was less than seventeen minutes long, and some of the songs were quite short. That year they appeared at Edgefest and the inaugural Osheaga Festival.  The "Smith EP" was released in 2007. During a July show in Omaha, Nebraska, the band signed a deal with Saddle Creek Records which would release their debut LP. The single "Your English Is Good" was released in July and later included on their album "Elephant Shell". That year the band performed at Coachella, Lollapalooza, Bumbershoot, the Glastonbury Festival, and the Reading and Leeds Festival.

Debut record and successful follow-up.

In 2008, Tokyo Police Club released their first album, "Elephant Shell", first via the US iTunes Store on March 25, 2008, and then on their new label Saddle Creek in April in North America, and in May in the U.K. and Continental Europe via Memphis Industries. In the same year, the band played the Roskilde Festival, the Rock am Ring-Festival, and Street Scene. In the fall, they played on the last six weeks of Weezer's Troublemaker Tour along with Angels and Airwaves.  In 2010, the band played the Bonnaroo Music Festival. They also played at San Francisco's 2010 Outside Lands Music and Arts Festival. Their second full-length album, "Champ", was released in June; it was produced by Rob Schnapf. That fall the album appeared on the!Earshot National Top 50 Chart.  On March 27, 2011, Tokyo Police Club performed their single "Bambi" at the 40th Juno Awards in Toronto. In June, the video for "Wait Up (Boots of Danger)" was nominated as best indie video of the year by MuchMusic. The video was directed by Mike Juneau and Kyle McCreight, and produced by Jesse Ewles and Chris Cunningham. In 2011, the band also released "10 Days. 10 Covers. 10 Years.", a project in which they recorded and released a new cover song from each of the previous 10 years, from 2001 to 2010.  In December, 2011, the band performed an unreleased song "Beaches" at a live show. In 2012, they joined Foster the People on the North American portion of their Torches tour along with Kimbra. While continuing to work with the band, Wright also hosts programming on CBC Radio 3.

New music.

In December, 2013, the band debuted a nine-minute lyric video for "Argentina (Pts. I, II, and III)" on YouTube. Both "Beaches" and "Argentina" are on their album "Forcefield" which was released March 24, 2014. In July the band performed at the 2014 Calgary Stampede in Calgary, Alberta. The band also recorded two songs for the "Daytrotter Sessions", which were released online.  Tokyo Police Club continued to record in 2018 in preparation for a fourth album. They released "TPC" on October 5, 2018 and received a Juno nomination for Alternative Album of the Year.

Television.

On April 19, 2007, Tokyo Police Club made their first US TV performance on the "Late Show with David Letterman". They played their single "Nature of the Experiment", along with a tambourine accompaniment by the CBS Orchestra. A year later the band made a second appearance on the "Late Show with David Letterman", performing the lead single "Tessellate" off their debut LP "Elephant Shell".  On November 16, 2008 they appeared on the television show "Desperate Housewives" in the episode "City on Fire" as "Cold Splash", a band competing in a battle-of-the-bands contest. They performed "In A Cave" from their album "Elephant Shell". A month later they played "Your English is Good" on The Late Late Show with Craig Ferguson on CBS. On June 28, 2010 they appeared on the "Late Show with David Letterman", performing the single "Wait Up (Boots of Danger)" off of their second album, "Champ".

Discography.

Solo projects.

In November 2008, Tokyo Police Club keyboardist Graham Wright released a solo EP called "The Lakes of Alberta" online only, available as a free download. He has also participated in a project called "Novels" with musicians Luke Lalonde (of Born Ruffians), Will Currie (of Will Currie & The Country French), Dean Marino (of Papermaps), and Jay Sadlowski (of Jay Sad) in which these musicians wrote and recorded an entire EP in 24 hours. The CDs were not released in stores, nor on the internet, but were given out to random people or placed in random places. His debut solo album, Shirts vs Skins, was released on June 27, 2010. Greg Alsop has posted comedy sketch videos online, Drumsters and Novelty T-Shirt College as well as a song titled'Losing Our Heads' in 2013 for the "Versus Valerie" Season One soundtrack. In June 2015, lead singer Dave Monks released his first solo EP on iTunes, Spotify and Google Play called "All Signs Point to Yes", with two singles having been released prior to the release of the album. The album "On a Wave" followed in 2019.. As of 2018 Dave Monks is also a member of the Canadian Supergroup Anyway Gang, releasing its self titled album in November 2019.

Tokyo Police Club is an indie rock band from Newmarket, Ontario, Canada. It consists of vocalist and bassist David Monks, keyboardist Graham Wright, guitarist Josh Hook, and drummer Greg Alsop. The band has been nominated for multiple Juno awards, most recently in 2019 for Alternative Album of the Year.

Tokyo Police Club je kanadská indie rocková skupina. Mezi její členy patří Dave Monks (basová kytara,zpěv), Graham Wright (klávesy), Josh Hook (kytara), Greg Alsop (bicí).


History.

The company was started in 1979 and took its name from its founder, Roger Sadowsky, known for "being one of the industry's master craftsmen". As of 2011 the company has 10 employees involved in instrument manufacturing excluding Sadowsky himself. Sadowsky started out modifying vintage Fender basses, which were at that time inexpensive, to improve their sound by adding more noise reduction technology and replacing the passive electronics with an active preamp, which increases the bass's signal-to-noise ratio. Once the price of vintage bass guitars began to increase, Sadowsky started making new bass guitars, and then signature models.

Models.

NYC Range.

The Sadowsky "NYC" line of instruments consists of four- and five-string bass guitars and six-string guitars. The bass guitars have a 34-inch scale length with no current models offered in longer or shorter scale lengths. All Sadowsky basses are available only with bolt-on necks. A weight-reducing chambered body is unique to the NYC range (including Satin series and Deluxe Satin series). The NYC series has a wide range of customisation options. The electronics of the NYC series are all hand wired.

NYC Satin.

The Satin range of NYC basses are made in the same workshop as the NYC basses, with a reduction in colour options with a satin finish. There are no customisation options available. Sadowsky introduced the Verdine White signature bass in the Satin line of instruments. This bass was built as a tribute to Verdine's late brother, Maurice White. Two instruments are being donated to the Verdine White Performing Arts Center with proceeds from sales of this model being donated in addition to the instruments. The electronics of the Satin range are pre-wired, and the tuners are Korean made.  The Deluxe Satin range of NYC basses follow the same cost-saving principals as the regular Satin range, but with the addition of flame maple tops at extra cost. The electronics of the Satin range are pre-wired, and the tuners are Korean made.

MetroLine.

The Sadowsky Metro line are the basic models of Sadowsky NYC basses made by Sadowsky Tokyo and offers most of the same features, minus the custom options, of the NYC line at a more affordable price. These instruments are made by hand by Yoshi Kikuchi and his team.  The Sadowsky Metro Express line of basses are factory-made instruments made in Japan. These models have a very limited colour choice.  Sadowsky, as of 2003, also offers a line of archtop guitars. There are currently five models available, the Jim Hall model, the Jimmy Bruno model, the semi-hollow model, the LS-17 (long scale/17" lower bout) and the SS-15 (short scale/15" lower bout).

Serial Numbering.

NYC Instruments.

Instruments manufactured in New York City are assigned as follows:

Japan (Metro) Instruments.

Instruments manufactured in Japan at assigned as follows:

Players and endorsers.

The various archtop guitar models are played and endorsed by many professional guitarists.  Sadowsky bass models are used by high-profile musicians

Sadowsky Guitars Limited is an American high-end guitar, bass guitar, and preamps manufacturer based in Long Island City, New York.

Sadowsky Guitars Limited je americká firma vyrábějící high-endové kytary, baskytary, kytarové zesilovače a předzesilovače. Byla založena v Brooklynu ve státě New York. Jméno společnosti nese příjmení jeho zakladatele Rogera Sadowsky.


Background and early life.

Yousuf Karsh was born to Armenian parents Amsih Karsh, a merchant, and Bahai Nakash, on December 23, 1908 in Mardin, Diyarbekir Vilayet, Ottoman Empire. The city's Armenian population was largely Arabic-speaking. His Armenian name, Hovsep, is a variant of Joseph, while Yousuf is the Arabic version of the same name. He grew up during the Armenian Genocide, during which some of his family were murdered. Karsh and his family escaped to a refugee camp in Aleppo, Syria in 1922 in a month-long journey with a Kurdish caravan. "The Economist" noted in their obituary of Karsh that he "thought of himself as an Armenian" and, according to Vartan Gregorian, "Although he was proud to be Canadian, Karsh was equally proud to be Armenian."  Karsh was sent to Canada by his family. He arrived in Halifax, Nova Scotia on December 31, 1923 by ship from Beirut. He immediately moved to Sherbrooke, Quebec to live with his maternal uncle George Nakashian (Nakash), a portrait photographer. Karsh worked for, and was taught photography by his uncle. He gave Karsh a Box Brownie camera. From 1928 to 1931, Karsh apprenticed in Boston, Massachusetts for John H. Garo, the most prominent Armenian photographer in America at the time who had made a name for himself photographing Boston celebrities.

Career.

Karsh settled in Ottawa and opened his first studio in 1932. It was located on the second floor of a building at 130 Sparks Street, which was later named the Hardy Arcade. He remained there until 1972, when he moved to Château Laurier. He was known professionally as "Karsh of Ottawa", which was also his signature. He achieved initial success by capturing the attention of Canadian Prime Minister Mackenzie King, who helped Karsh arrange photography sessions with visiting dignitaries.  Throughout his life, Karsh photographed "anyone who was anyone." When asked why he almost exclusively captured famous people, he replied, "I am working with the world's most remarkable cross-section of people. I do believe it's the minority who make the world go around, not the majority." He once also jokingly remarked, "I do it for my own immortality." By the time he retired in 1992, more than 20 of his photos had appeared on the cover of "Life" magazine. Karsh's photos were known for their use of dramatic lighting, which became the hallmark of his portrait style. He had studied it with both Garo in Boston and at the Ottawa Little Theatre, of which he was a member.  His 1941 photo of Winston Churchill, the British Prime Minister, brought him prominence. The photo was taken on December 30, 1941 in the chamber of the Speaker of the House of Commons in the Canadian Parliament in Ottawa after Churchill delivered a speech on World War II to the Canadian members of the parliament. It was arranged by Canadian Prime Minister Mackenzie King. Churchill is particularly noted for his posture and facial expression, which have been compared to the wartime feelings that prevailed in the UK: persistence in the face of an all-conquering enemy. The photo session was short and, just before exposure, Karsh moved towards Churchill and removed the cigar which he had in his hand. Churchill was miffed and showed his displeasure in the portrait. The photo, which according to "The Economist" is the "most reproduced portrait in the history of photography", has been described as one of the "most iconic portraits ever shot". USC Fisher Museum of Art described it as a "defiant and scowling portrait [which] became an instant icon of Britain's stand against fascism." It appeared on the cover of the May 21, 1945 issue of "Life", which bought it for $100. It now hangs on the wall of the Speaker's chamber.  During World War II, Karsh photographed political and military leaders and began capturing photos of writers, actors, artists, musicians, scientists, and celebrities in the post-war period. His 1957 portrait of the American novelist Ernest Hemingway, which was shot at Hemingway's Cuban home Finca Vigia, is another well-known photo by Karsh. (Finca Vigia is located in the small, working-class town of San Francisco de Paula, about 10 miles east of Havana.) According to Amanda Hopkinson it made Hemingway look like the hero of his novel "The Old Man and the Sea". His other notable portraits include George Bernard Shaw at an old age (1943), Dwight D. Eisenhower as a five-star general and Supreme Commander of the Allied Expeditionary Force (1946), American artist Georgia O'Keeffe in her New Mexico studio (1956), Soviet leader Nikita Khrushchev swathed in fur (1963).  Besides portraits of the famous, Karsh photographed assembly line workers in Windsor, Ontario, commissioned by the Ford Motor Company of Canada. He also photographed landscapes of Rome and the Holy Land to be included in books in collaboration with Bishop Fulton J. Sheen, an annual poster for the Muscular Dystrophy Association, and other works.  He was a visiting professor at Ohio University and at Emerson College in Boston.

Marriages.

Karsh's first marriage was to Solange Gauthier (1902−1961) in 1939. He met her at the Ottawa Little Theatre in 1933, where she was a star. Gauthier was born in Tours, France and migrated to Canada as a young girl. They initially moved into her apartment and in 1940, into an Art Deco home called Little Wings on the Rideau River just outside Ottawa. She died in January 1961 of cancer.  His second marriage was to Estrellita Maria Nachbar, a medical writer 21 years his junior, in August 1962. Their wedding was officiated by Fulton J. Sheen, Auxiliary Bishop of the Catholic Archdiocese of New York. From 1972 to 1992 they lived in a third-floor suite at Château Laurier, Ottawa and maintained Little Wings and an apartment and studio in Manhattan. They had no children.

Retirement and death.

Karsh retired in July 1993, aged 84. His penultimate sittings in May 1993 were with President Bill Clinton and First Lady Hillary. He closed his studio at Château Laurier and moved to Boston in 1997. He died on July 13, 2002 at Brigham and Women's Hospital in Boston after complications following surgery. A private funeral was held in Ottawa. He was interred in Notre-Dame Cemetery in Ottawa.

Recognition.

Karsh has been recognized as Canada's leading portrait photographer. In general, he is recognized as one of the best-known and great portrait photographers of the 20th century. "The Economist" wrote upon his death that Karsh was "for half a century perhaps the greatest portrait photographer in the monumental manner." The website of the Governor General of Canada describes him as the "pre-eminent portrait photographer of the twentieth century." The Metropolitan Museum of Art described him as "one of the greatest portrait photographers of the twentieth century [who] achieved a distinct style in his theatrical lighting." "The Canadian Encyclopedia" noted that his portraits "have come to represent the public images of major international figures of politics, science, and culture in the twentieth century."  By the time he died, his work was on display permanently at the Metropolitan Museum of Art, Museum of Modern Art (both in New York), National Gallery of Canada, National Portrait Gallery in London, National Museum of Modern Art in Tokyo, Art Institute of Chicago, Saint Louis Art Museum, George Eastman Museum, and elsewhere. In 1987 the National Archives of Canada acquired the complete collection of Karsh items, including the negatives, prints and transparencies produced and retained by Karsh since 1933. The current Library and Archives Canada collection has 355,000 items in its Karsh collection, including all of his 150,000 negatives, kept at a facility in Gatineau, Quebec.  He was a member of the Royal Canadian Academy of Arts and an honorary fellow of the Royal Photographic Society (UK).  On June 9, 2017 a bust of Karsh by Canadian-Armenian sculptor Megerditch Tarakdjian was unveiled before Château Laurier, Ottawa. It depicts Karsh with his famous camera and is a gift to Canada from the people of Armenia on the occasion of the 25th anniversary of the establishment of diplomatic relations between the two countries and the 150th anniversary of Canada. Among attendees were George Furey, the Speaker of the Senate, and Arif Virani, the Parliamentary Secretary to the Minister of Canadian Heritage.  The Karsh Award, dedicated to Yousuf and his brother Malak Karsh, is awarded by the City of Ottawa every two years to an established professional artist for outstanding artistic work in a photo-based medium.

Awards.

Karsh was awarded honorary degrees from Dartmouth College (1961), Ohio University (1968), Tufts University (D.F.A., 1981), Syracuse University (D.F.A., 1986), Ohio State University (Doctor of Humane Letters, 1996), and other universities.

Yousuf Karsh (December 23, 1908 – July 13, 2002), was an Armenian-Canadian photographer known for his portraits of notable individuals. He has been described as one of the greatest portrait photographers of the 20th century.

Yousuf Karsh (23. prosince 1908 Mardin - 13. července 2002, Boston), držitel řádu Order of Canada, byl kanadský fotograf původem z Arménie, jeden z nejznámějších portrétních fotografů. Proslul svými portréty politiků, umělců, hudebníků, spisovatelů, vědců a dalších známých osobností. Jeho snímky jsou součástí stálých expozic Metropolitního muzea umění a Muzea moderního umění v New Yorku stejně jako kanadské Národní galerie.


Life and career.

August was born on November 9, 1948 in Brede, Denmark. After attending local schools, he graduated from the Foto- og Dokumentarskolan in Stockholm, Sweden, and later from the National Film School of Denmark. He began his career as a cinematographer. He made his feature directorial debut in 1978 with "In My Life", which won the Bodil Award for Best Danish Film.  After decades of writing, directing and producing notable films in Denmark, on 23 September 2011, Bille August announced that he had opened a studio in Hangzhou, China and taken a position as Tianpeng Media's Art Director, in order to produce Chinese films for Tianpeng Media over the next few years. Tianpeng Media is a new media company established in 2010. The company produced two films, "The Women Knight of Mirror" (竞雄女侠秋瑾) and "The Years of Qi Xiao Fu" (七小福之燃情岁月), to be released later in 2011. Bille is the first foreign director to be hired by the Chinese film company. In 2011 August also accepted the invitation from the Hangzhou government to serve as a "culture consultant" for the city.  His film "Night Train to Lisbon" (2013) premiered out of competition at the 63rd Berlin International Film Festival. He had planned to direct a "Gianni Versace" biopic with Antonio Banderas as Versace, but this project was cancelled.

Personal life.

August has been married four times, and had four children by two of his wives.  He was first married to Annie Munksgård Lauritzen. His second wife was Masja Dessau. Together they had two children, Anders Frithiof August (born June 15, 1978) and Adam August (born in 1983). They have both become screenwriters. Thirdly he was married to Swedish actress Pernilla August from 1991 to 1997. Together they had daughters Asta Kamma August (born November 5, 1991) and Alba Adéle August (born June 6, 1993). In 2012 he married his current wife, Sara-Marie Maltha.  In addition, August fathered four other children by two women outside marriage.

Bille August (born 9 November 1948) is a Danish director, screenwriter, and cinematographer of film and television. In a career spanning over four decades, he has been the recipient of numerous accolades, making him one of the most acclaimed contemporary Danish filmmakers.

Bille August (9. listopadu 1948 v Brede, Dánsko) je dánský filmový a televizní režisér, kameraman a scenárista. Vzdělání získal v roce 1976 na Danske Filmskole. Celosvětový ohlas si získal po natočení filmu "Pelle Dobyvatel" podle románové předlohy spisovatele Martina Andersena Nexø. Film byl oceněn v roce 1988 Oscarem za nejlepší cizojazyčný film a režisér za něj dostal Zlatou palmu z MFF v Cannes.


Algebraic notation.

Algebraic notation describes the rules and conventions for writing mathematical expressions, as well as the terminology used for talking about parts of expressions. For example, the expression formula_1 has the following components:  A "coefficient" is a numerical value, or letter representing a numerical constant, that multiplies a variable (the operator is omitted). A "term" is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators. Letters represent variables and constants. By convention, letters at the beginning of the alphabet (e.g. formula_2) are typically used to represent constants, and those toward the end of the alphabet (e.g. formula_3 and ) are used to represent variables. They are usually written in italics.  Algebraic operations work in the same way as arithmetic operations, such as addition, subtraction, multiplication, division and exponentiation. and are applied to algebraic variables and terms. Multiplication symbols are usually omitted, and implied when there is no space between two variables or terms, or when a coefficient is used. For example, formula_4 is written as formula_5, and formula_6 may be written formula_7.  Usually terms with the highest power (exponent), are written on the left, for example, formula_8 is written to the left of. When a coefficient is one, it is usually omitted (e.g. formula_9 is written formula_8). Likewise when the exponent (power) is one, (e.g. formula_11 is written formula_12). When the exponent is zero, the result is always 1 (e.g. formula_13 is always rewritten to ). However formula_14, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents.

Alternative notation.

Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters and symbols are available. As an illustration of this, while exponents are usually formatted using superscripts, e.g., formula_8, in plain text, and in the TeX mark-up language, the caret symbol "^" represents exponentiation, so formula_8 is written as "x^2"., as well as some programming languages such as Lua. In programming languages such as Ada, Fortran, Perl, Python and Ruby, a double asterisk is used, so formula_8 is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol, and it must be explicitly used, for example, formula_12 is written "3*x".

Concepts.

Variables.

Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons.

Simplifying expressions.

Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition, subtraction, multiplication, division and exponentiation). For example,

Equations.

An equation states that two expressions are equal using the symbol for equality, (the equals sign). One of the best-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle:  This equation states that formula_39, representing the square of the length of the side that is the hypotenuse, the side opposite the right angle. That is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by and.  An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as formula_40); such equations are called identities. Conditional equations are true for only some values of the involved variables, e.g. formula_41 is true only for formula_42 and formula_43. The values of the variables which make the equation true are the solutions of the equation and can be found through equation solving.  Another type of equation is inequality. Inequalities are used to show that one side of the equation is greater, or less, than the other. The symbols used for this are: formula_44 where formula_45 represents 'greater than', and formula_46 where formula_47 represents 'less than'. Just like standard equality equations, numbers can be added, subtracted, multiplied or divided. The only exception is that when multiplying or dividing by a negative number, the inequality symbol must be flipped.

Properties of equality.

By definition, equality is an equivalence relation, meaning it has the properties (a) reflexive (i.e. formula_48), (b) symmetric (i.e. if formula_49 then formula_50) (c) transitive (i.e. if formula_49 and formula_52 then formula_53). It also satisfies the important property that if two symbols are used for equal things, then one symbol can be substituted for the other in any true statement about the first and the statement will remain true. This implies the following properties:

Properties of inequality.

The relations "less than" formula_47 and greater than formula_45 have the property of transitivity: By reversing the inequation, formula_47 and formula_45 can be swapped, for example:

Substitution.

Substitution is replacing the terms in an expression to create a new expression. Substituting 3 for in the expression makes a new expression with meaning. Substituting the terms of a statement makes a new statement. When the original statement is true independently of the values of the terms, the statement created by substitutions is also true. Hence, definitions can be made in symbolic terms and interpreted through substitution: if formula_81 is meant as the definition of formula_82 as the product of with itself, substituting for informs the reader of this statement that formula_83 means. Often it's not known whether the statement is true independently of the values of the terms. And, substitution allows one to derive restrictions on the possible values, or show what conditions the statement holds under. For example, taking the statement, if is substituted with, this implies, which is false, which implies that if then cannot be.  If and are integers, rationals, or real numbers, then implies or. Consider. Then, substituting for and for, we learn or. Then we can substitute again, letting and, to show that if then or. Therefore, if, then or ( or ), so implies or or.  If the original fact were stated as " implies or ", then when saying "consider," we would have a conflict of terms when substituting. Yet the above logic is still valid to show that if then or or if, instead of letting and, one substitutes for and for (and with, substituting for and for ). This shows that substituting for the terms in a statement isn't always the same as letting the terms from the statement equal the substituted terms. In this situation it's clear that if we substitute an expression into the term of the original equation, the substituted does not refer to the in the statement " implies or."

Solving algebraic equations.

The following sections lay out examples of some of the types of algebraic equations that may be encountered.

Linear equations with one variable.

Linear equations are so-called, because when they are plotted, they describe a straight line. The simplest equations to solve are linear equations that have only one variable. They contain only constant numbers and a single variable without an exponent. As an example, consider:  To solve this kind of equation, the technique is add, subtract, multiply, or divide both sides of the equation by the same number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other side of the equation is the value of the variable. This problem and its solution are as follows: In words: the child is 4 years old.  The general form of a linear equation with one variable, can be written as: formula_85  Following the same procedure (i.e. subtract from both sides, and then divide by ), the general solution is given by formula_86

Linear equations with two variables.

A linear equation with two variables has many (i.e. an infinite number of) solutions. For example:  That cannot be worked out by itself. If the son's age was made known, then there would no longer be two unknowns (variables). The problem then becomes a linear equation with just one variable, that can be solved as described above.  To solve a linear equation with two variables (unknowns), requires two related equations. For example, if it was also revealed that:  Now there are two related linear equations, each with two unknowns, which enables the production of a linear equation with just one variable, by subtracting one from the other (called the elimination method):  In other words, the son is aged 12, and since the father 22 years older, he must be 34. In 10 years, the son will be 22, and the father will be twice his age, 44. This problem is illustrated on the associated plot of the equations.  For other ways to solve this kind of equations, see below, System of linear equations.

Quadratic equations.

A quadratic equation is one which includes a term with an exponent of 2, for example, formula_30, and no term with higher exponent. The name derives from the Latin "quadrus", meaning square. In general, a quadratic equation can be expressed in the form formula_92, where is not zero (if it were zero, then the equation would not be quadratic but linear). Because of this a quadratic equation must contain the term formula_93, which is known as the quadratic term. Hence formula_94, and so we may divide by and rearrange the equation into the standard form  where formula_96 and formula_97. Solving this, by a process known as completing the square, leads to the quadratic formula  where the symbol "±" indicates that both  are solutions of the quadratic equation.  Quadratic equations can also be solved using factorization (the reverse process of which is expansion, but for two linear terms is sometimes denoted foiling). As an example of factoring:  which is the same thing as  It follows from the zero-product property that either formula_102 or formula_103 are the solutions, since precisely one of the factors must be equal to zero. All quadratic equations will have two solutions in the complex number system, but need not have any in the real number system. For example,  has no real number solution since no real number squared equals −1. Sometimes a quadratic equation has a root of multiplicity 2, such as:  For this equation, −1 is a root of multiplicity 2. This means −1 appears twice, since the equation can be rewritten in factored form as

Complex numbers.

All quadratic equations have exactly two solutions in complex numbers (but they may be equal to each other), a category that includes real numbers, imaginary numbers, and sums of real and imaginary numbers. Complex numbers first arise in the teaching of quadratic equations and the quadratic formula. For example, the quadratic equation  has solutions  Since formula_109 is not any real number, both of these solutions for "x" are complex numbers.

Exponential and logarithmic equations.

An exponential equation is one which has the form formula_110 for formula_111, which has solution  when formula_113. Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. For example, if  then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain  whence  or  A logarithmic equation is an equation of the form formula_118 for formula_111, which has solution  For example, if  then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get  whence  from which we obtain

Radical equations.

A radical equation is one that includes a radical sign, which includes square roots, formula_125 cube roots, formula_126, and "n"th roots, formula_127. Recall that an "n"th root can be rewritten in exponential format, so that formula_127 is equivalent to formula_129. Combined with regular exponents (powers), then formula_130 (the square root of cubed), can be rewritten as formula_131. So a common form of a radical equation is formula_132 (equivalent to formula_133) where and are integers. It has real solution(s): For example, if:  then  and thus

System of linear equations.

There are different methods to solve a system of linear equations with two variables.

Elimination method.

An example of solving a system of linear equations is by using the elimination method:  Multiplying the terms in the second equation by 2:  Adding the two equations together to get:  which simplifies to  Since the fact that formula_102 is known, it is then possible to deduce that formula_143 by either of the original two equations (by using "2" instead of ) The full solution to this problem is then  This is not the only way to solve this specific system; could have been resolved before.

Substitution method.

Another way of solving the same system of linear equations is by substitution.  An equivalent for can be deduced by using one of the two equations. Using the second equation:  Subtracting formula_147 from each side of the equation:  and multiplying by −1:  Using this value in the first equation in the original system:  Adding "2" on each side of the equation:  which simplifies to  Using this value in one of the equations, the same solution as in the previous method is obtained.  This is not the only way to solve this specific system; in this case as well, could have been solved before.

Other types of systems of linear equations.

Inconsistent systems.

In the above example, a solution exists. However, there are also systems of equations which do not have any solution. Such a system is called inconsistent. An obvious example is  As 0≠2, the second equation in the system has no solution. Therefore, the system has no solution. However, not all inconsistent systems are recognized at first sight. As an example, consider the system  Multiplying by 2 both sides of the second equation, and adding it to the first one results in which clearly has no solution.

Undetermined systems.

There are also systems which have infinitely many solutions, in contrast to a system with a unique solution (meaning, a unique pair of values for and ) For example:  Isolating in the second equation:  And using this value in the first equation in the system:  The equality is true, but it does not provide a value for. Indeed, one can easily verify (by just filling in some values of ) that for any there is a solution as long as formula_160. There is an infinite number of solutions for this system.

Over- and underdetermined systems.

Systems with more variables than the number of linear equations are called underdetermined. Such a system, if it has any solutions, does not have a unique one but rather an infinitude of them. An example of such a system is  When trying to solve it, one is led to express some variables as functions of the other ones if any solutions exist, but cannot express "all" solutions numerically because there are an infinite number of them if there are any.  A system with a higher number of equations than variables is called overdetermined. If an overdetermined system has any solutions, necessarily some equations are linear combinations of the others.

Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values, known as variables. This use of variables entails use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.

Elementární algebra je nejzákladnější forma algebraické teorie a navazuje na aritmetiku. Hlavní rozdíl mezi aritmetikou a algebrou je v používání proměnných. Zatímco v aritmetice jsou použita pouze čísla a jejich základní aritmetické operace, v algebře je možné použít např. symboly jako x či y k označení proměnných.


Biography.

Early years.

Bossuet was born at Dijon. He came from a family of prosperous Burgundian lawyers – on both his paternal and maternal side, his ancestors had held legal posts for at least a century. He was the fifth son born to Beneigne Bossuet, a judge of the "parlement" (a provincial high court) at Dijon, and Marguerite Mouchet. His parents decided on a career in the church for their fifth son, so he was tonsured at age eight.  The boy was sent to school at the Collège des Godrans, a classical school run by the Jesuits of Dijon. When his father was appointed to the "parlement" at Metz, Bossuet was left in Dijon under the care of his uncle Claude Bossuet d'Aiseray, a renowned scholar. At the Collège des Godrans, he gained a reputation for hard work: fellow students nicknamed him "Bos suetus aratro", an "ox broken in to the plough". His father's influence at Metz allowed him to obtain for the young Bossuet a canonicate in the cathedral of Metz when the boy was just 13 years old.  In 1642, Bossuet enrolled in the Collège de Navarre in Paris to finish his classical studies and to begin the study of philosophy and theology. His mentor at Navarre was the college's president, Nicolas Cornet, the theologian whose denunciation of Antoine Arnauld at the Sorbonne in 1649 was a major episode in the Jansenist controversy.  For the time being, however, Cornet and Arnaud were still on good terms. In 1643, Arnaud introduced Bossuet to the Hôtel de Rambouillet, a great centre of aristocratic culture and the original home of the "Précieuses". Bossuet was already showing signs of the oratorical brilliance which served him so well throughout his life. On one celebrated occasion at the Hôtel de Rambouillet, during a dispute about extempore preaching, the 16-year-old Bossuet was called on to deliver an impromptu sermon at 11 pm. Voiture famously quipped: "I never heard anybody preach so early nor so late".

Early clerical career.

Bossuet became a Master of Arts in 1643. He held his first thesis ("tentativa") in theology on 25 January 1648, in the presence of the Prince de Condé. Later in 1648, he became a sub-deacon at Metz. He became a deacon in 1649. During this period, he preached his first sermons.  He held his second thesis ("sorbonica") on November 9, 1650. Then, in preparation for the priesthood, he spent the next two years in retirement under the spiritual direction of Vincent de Paul.

Priest at Metz.

In January 1652, Bossuet re-entered public life, being named Archdeacon of Sarrebourg. He was ordained a priest on 18 March 1652. A few weeks later, he defended his brilliant doctoral work and became a Doctor of Divinity.  He spent the next seven years at Metz, where his father's influence had got him a canonry at age 13 and where he now also had the office of archdeacon. He was plunged at once into the thick of controversy; for nearly half of Metz was Protestant, and Bossuet's first appearance in print was a refutation of the Huguenot pastor Paul Ferry (1655), and he frequently engaged in religious controversies with Protestants (and, less regularly, with Jews) during his time at Metz. To reconcile the Protestants with the Roman Catholic Church became the great object of his dreams; and for this purpose, he began to train himself carefully for the pulpit, an all-important centre of influence in a land where political assemblies were unknown and novels and newspapers scarcely born. His youthful imagination was unbridled, and his ideas ran easily into a kind of paradoxical subtlety, redolent of the divinity school. Nevertheless, his time at Metz was an important time for developing his pulpit oratory and for allowing him to continue his studies of Scripture and the Fathers. He also gained political experience through his participation in the local Assembly of the Three Orders.  In 1657, in Metz, Bossuet preached before Anne of Austria, mother of Louis XIV. As a result, he received the honorific title of "Counselor and Preacher to the King".

Early career in Paris.

In 1657, St. Vincent de Paul convinced Bossuet to move to Paris and give himself entirely to preaching. (He did not entirely sever his connections with the cathedral of Metz, though: he continued to hold his benefice, and in 1664, when his widower father was ordained as a priest and became a canon at the cathedral at Metz, Bossuet was named the dean of the cathedral.)  Bossuet quickly gained a reputation as a great preacher, and by 1660, he was preaching regularly before the court in the Chapel Royal. In 1662, he preached his famous sermon "On the Duties of Kings" to Louis XIV at the Louvre.  In Paris, the congregations had no mercy on purely clerical logic or clerical taste; if a preacher wished to catch their ear, he had to manage to address them in terms they would agree to consider sensible and well bred. Having very stern ideas of the dignity of a priest, Bossuet refused to descend to the usual devices for arousing popular interest.  The narrative element in Bossuet's sermons grew shorter with each year. He never drew satirical pictures like his great rival Louis Bourdaloue. He would not write out his discourses in full, much less learn them off by heart: of the two hundred printed in his works, all but a fraction are rough drafts. Ladies such as Mme de Sévigné forsook him when Bourdaloue dawned on the Paris horizon in 1669, though Fénelon and La Bruyère, two much sounder critics, refused to follow their example.  Bossuet possessed the full equipment of the orator, voice, language, flexibility and strength. He never needed to strain for effect; his genius struck out at a single blow the thought, the feeling and the word. What he said of Martin Luther applies peculiarly to himself: he could fling his fury into theses and thus unite the dry light of argument with the fire and heat of passion. These qualities reached their highest point in the "Oraisons funèbres" ("Funeral Orations").  Bossuet was always best when at work on a large canvas; besides, here no conscientious scruples intervened to prevent him giving much time and thought to the artistic side of his subject. The "Oraison", as its name betokened, stood midway between the sermon proper and what would nowadays be called a biographical sketch. At least that was what Bossuet made it; for on this field, he stood not merely first, but alone.  One hundred and thirty-seven of Bossuet's sermons preached in the period from 1659 to 1669 are extant, and it is estimated that he preached more than a hundred more that have since been lost. Apart from state occasions, Bossuet seldom appeared in a Paris pulpit after 1669.

Tutor to the Dauphin, 1670–1681.

A favourite of the court, in 1669, Bossuet was gazetted bishop of Condom in Gascony, without being obliged to reside there. He was consecrated as a bishop on September 21, 1670, but he resigned the bishopric when he was elected to the "Académie française" in 1671.  On 18 September 1670 he was appointed tutor to the nine-year-old Dauphin, oldest child of Louis XIV. The choice was scarcely fortunate. Bossuet unbent as far as he could, but his genius was by no means fitted to enter into the feelings of a child; and the dauphin was a cross, ungainly, sullen lad. Probably no one was happier than the tutor when his charge turned sixteen and was married off to a Bavarian princess. Still, the nine years at court were by no means wasted.  Bossuet's tutorial functions involved composing all the necessary books of instruction, including not just handwriting samples, but also manuals of philosophy, history, and religion fit for a future king of France. Among the books written by Bossuet during this period are three classics. First came the "Traité de la connaissance de Dieu et de soi-même" ("Treatise on the Knowledge of God and of One's Self") (1677), then the "Discours sur l'histoire universelle" ("Speech of Universal History") (1679, published 1682), and lastly the "Politique tirée de l'Écriture Sainte" ("Politics Drawn from Holy Scripture") (1679, published 1709). The three books fit into each other. The "Traité" is a general sketch of the nature of God and the nature of man. The "Discours" is a history of God's dealings with humanity in the past. The "Politique" is a code of rights and duties drawn up in the light thrown by those dealings. Bossuet's conclusions are only drawn from Holy Scripture because he wished to gain the highest possible sanction for the institutions of his country and to hallow the France of Louis XIV by proving its astonishing likeness to the Israel of Solomon. Then, too, the veil of Holy Scripture enabled him to speak out more boldly than court etiquette would have otherwise allowed, to remind the son of Louis XIV that kings have duties as well as rights.  The Grand Dauphin had often forgotten these duties, but his son, the Petit Dauphin, would bear them in mind. The tutor's imagination looked forward to a time when France would blossom into Utopia, with a Christian philosopher on the throne. That is what made him so stalwart a champion of authority in all its forms: ""le roi, Jesus-Christ et l'Eglise, Dieu en ces trois noms"" ("the king, Jesus Christ, and the Church, God in His three names"), he says in a characteristic letter. The object of his books is to provide authority with a rational basis. Bossuet's worship of authority by no means killed his confidence in reason; what it did was make him doubt the honesty of those who reasoned otherwise than himself.  The whole chain of argument seemed to him so clear and simple. Philosophy proves that God exists and that He shapes and governs the course of human affairs. History shows that this governance is, for the most part, indirect, exercised through certain venerable corporations, as well civil and ecclesiastical, all of which demand implicit obedience as the immediate representatives of God. Thus all revolt, whether civil or religious, is a direct defiance of the Almighty.  Oliver Cromwell becomes a moral monster, and the revocation of the Edict of Nantes was the greatest achievement of the second Constantine. The France of his youth had known the misery of divided counsels and civil war; the France of his manhood, brought together under an absolute sovereign, had suddenly shot up into a splendour only comparable with ancient Rome. Why not, then, strain every nerve to hold innovation at bay and prolong that splendour for all time? Bossuet's own "Discours sur l'histoire universelle" might have furnished an answer, for there the fall of many empires is detailed; but then the "Discours" was composed under a single preoccupation.  To Bossuet, the establishment of Christianity was the one point of real importance in the whole history of the world. He totally ignores the history of Islam and Asia; on Greece and Rome, he only touched insofar as they formed part of the Praeparatio Evangelica. Yet his "Discours" is far more than a theological pamphlet. While Pascal might refer the rise and fall of empires to Providence or chance or a little grain of sand in the English lord protectors' veins, Bossuet held fast to his principle that God works through secondary causes. It is His will that every great change should have its roots in the ages that went before it. Bossuet, accordingly, made a heroic attempt to grapple with origins and causes, and in this way, his book deserves its place as one of the very first of philosophic histories.

Bishop of Meaux, 1681–1704.

With the period of the Dauphin's formal education ending in 1681, Bossuet was appointed Bishop of Meaux by the King on 2 May 1681, which was approved by Pope Innocent XI on 17 November. But before he could take possession of his see, he was drawn into a violent quarrel between Louis XIV and Pope Innocent XI. Here he found himself in a quandary: to support the Pope meant supporting the Jesuits; and he hated their supposed casuistry and "dévotion aisée" almost as much as Pascal; to oppose the Pope was to play into the hands of Louis XIV, who was eager to subject the Church to the will of the State. Bossuet therefore attempted to steer a middle course. In 1682, before the general Assembly of the French Clergy, he preached a great sermon on the unity of the Church and made it a magnificent plea for compromise. As Louis XIV insisted on his clergy making an anti-papal declaration, Bossuet got leave to draw it up and made it as moderate as he could, and when the Pope declared it null and void, he set to work on a gigantic "Defensio Cleri Gallicani", only published after his death. Throughout this controversy, unlike the court bishops, Bossuet constantly resided in his diocese and took an active interest in its administration.

Efforts to Combat Protestantism.

The Gallican storm a little abated, he turned back to a project very near his heart. Ever since the early days at Metz, he had been busy with schemes for uniting the Huguenots to the Catholic Church. In 1668, he converted Turenne; in 1670, he published an "Exposition de la foi catholique" ("Exposition of the Catholic Faith"), so moderate in tone that adversaries were driven to accuse him of having fraudulently watered down the Catholic dogmas to suit Protestant taste.  Finally, in 1688, his great "Histoire des variations des Églises protestantes" ("History of the Variations of the Protestant Churches"), perhaps the most brilliant of all his works, appeared. Few writers could have made the Justification controversy interesting or even intelligible. His argument is simple enough. Without rules, an organized society cannot hold together, and rules require an authorized interpreter. The Protestant churches had thrown over this interpreter; and Bossuet had small trouble in showing that, the longer they lived, the more they varied on increasingly important points.  For the moment, the Protestants were pulverized; but before long, they began to ask whether variation was necessarily so great an evil. Between 1691 and 1701, Bossuet corresponded with Leibniz with a view to reunion, but negotiations broke down precisely at this point. Leibniz thought his countrymen might accept individual Roman doctrines, but he flatly refused to guarantee that they would necessarily believe tomorrow what they believe today. We prefer, he said, a church eternally variable and for ever moving forwards.  Next, Protestant writers began to accumulate some alleged proofs of Rome's own variations; and here, they were backed up by Richard Simon, a priest of the Paris Oratory and the father of biblical criticism in France. He accused St Augustine, Bossuet's own special master, of having corrupted the primitive doctrine of grace.  Bossuet set to work on a "Defense de la tradition", but Simon calmly went on to raise issues graver still. Under a veil of politely ironic circumlocutions, such as did not deceive the Bishop of Meaux, he claimed his right to interpret the Bible like any other book. Bossuet denounced him again and again; Simon told his friends he would wait until the old fellow was no more. Another Oratorian proved more dangerous still. Simon had endangered miracles by applying to them lay rules of evidence, but Malebranche abrogated miracles altogether. It was blasphemous, he argued, to suppose that the Author of nature would violate the law He had Himself established. Bossuet might scribble "nova, mira, falsa" in the margins of his book and urge Fénelon to attack them; Malebranche politely met his threats by saying that to be refuted by such a pen would do him too much honor. These repeated checks soured Bossuet's temper.  In his earlier controversies, he had borne himself with great magnanimity, and the Huguenot ministers he refuted had found him a kindly advocate at court. His approval of the revocation of the Edict of Nantes stopped far short of approving dragonnades within his Diocese of Meaux, but now his patience was waning. A dissertation by one Father Caffaro, an obscure Italian monk, became his excuse for writing certain, violent "Maximes sur la comédie" (1694), wherein he made an attack on the memory of Molière, dead more than twenty years.

Controversy with Fénelon.

Three years later, he was battling with Bishop François Fénelon over the love of God. Fénelon, 24 years his junior, was an old pupil who had suddenly become a rival; like Bossuet, Fénelon was a bishop who served as a royal tutor.  The controversy concerned their different reactions to the opinions of Jeanne Guyon: her ideas were similar to the Quietism of Molinos, which was condemned by Pope Innocent XI in 1687. When Mme de Maintenon began questioning the orthodoxy of Mme Guyon's opinions, an ecclesiastical commission of three members, including Bossuet, was appointed to report on the matter. The commission issued 34 articles known as the "Articles d'Issy", which condemned Mme Guyon's ideas very briefly and provided a short treatise on the orthodox, Catholic conception of prayer. Fénelon, who had been attracted to Mme Guyon's ideas, signed off on the Articles, and Mme Guyon submitted to the judgment.  Bossuet now composed "Instructions sur les états d'oraison", a work that explained the "Articles d'Issy" in greater depth. Fénelon refused to endorse this treatise, however, and instead composed his own explanation as to the meaning of the "Articles d'Issy", his "Explication des Maximes des Saints". He explained his view that the goal of human life should be to have love of God as its perfect object, with neither fear of punishment nor desire for the reward of eternal life having anything to do with this pure love of God. King Louis XIV reproached Bossuet for failing to warn him that his grandsons' tutor had such unorthodox opinions and instructed Bossuet and other bishops to respond to the "Maximes des Saints".  Bossuet and Fénelon thus spent the years 1697–1699 battling each other in pamphlets and letters until the Inquisition finally condemned the "Maximes des Saints" on 12 March 1699. Pope Innocent XII selected 23 specific passages for condemnation. Bossuet triumphed in the controversy and Fénelon submitted to Rome's determination of the matter.

Death.

Until he was over 70 years, Bossuet enjoyed good health, but in 1702 he developed chronic kidney stones. Two years later he was a hopeless invalid, and on 12 April 1704 he died quietly. His funeral oration was given by Charles de la Rue, SJ. He was buried at Meaux Cathedral.

Preaching.

Bossuet is widely considered to be one of the most influential homilists of all time. He is one of the preachers, along with John Tillotson and Louis Bourdaloue, who began the transition from Baroque to Neoclassical preaching. He preached with a simple eloquence that eschewed the grandiose extravagances of earlier preaching. He focused on ethical rather than doctrinal messages, often drawing from the lives of saints or saintly contemporaries as examples. He preached, for example, on St. Francis de Sales as well as funeral orations on Queen Henrietta Maria of France and Henrietta Anne of England. Bossuet's funeral orations in particular had lasting importance and were translated early into many languages, including English. Such was their power that even Voltaire, normally so antagonistic toward clergy, praised his oratorical excellence.

Works.

An edition of Bossuet's sermons was edited by Abbé Lebarq in 6 vols. (Paris, 1890, 1896), as the "Œuvres oratoires de Bossuet". His complete works were edited by Lachat in 31 vols. (Paris, 1862–1864).

"Politics Drawn from the Very Words of Holy Scripture".

When Bossuet was chosen to be the tutor of the Dauphin, oldest child of Louis XIV, he wrote several works for the edification of his pupil, one of which was "Politics Derived from the Words of Holy Scripture", a discourse on the principles of royal absolutism. The work was published posthumously in 1709.  The work consists of several books which are divided into articles and propositions which lay out the nature, characteristics, duties, and resources of royalty. To justify his propositions, Bossuet quotes liberally from the Bible and various psalms.  Throughout his essay, Bossuet emphasizes the fact that royal authority comes directly from God and that the person of the king is sacred. In the third book, Bossuet asserts that "God establishes kings as his ministers, and reigns through them over the people." He also states that "the prince must be obeyed on principle, as a matter of religion and of conscience." While he declares the absolute authority of rulers, he emphasizes the fact that kings must use their power only for the public good and that the king is not above the law "for if he sins, he destroys the laws by his example."  In books six and seven, Bossuet describes the duties of the subjects to the prince and the special duties of royalty. For Bossuet, the prince was synonymous with the state, which is why, according to him, the subjects of the prince owe the prince the same duties that they owe their country. He also states that "only public enemies make a separation between the interest of the prince and the interest of the state." As far as the duties of royalty, the primary goal is the preservation of the state. Bossuet describes three ways that this can be achieved: by maintaining a good constitution, making good use of the state's resources, and protecting the state from the dangers and difficulties that threaten it.  In books nine and ten, Bossuet outlines the various resources of royalty (arms, wealth, and counsel) and how they should be used. In regards to arms, Bossuet explains that there are just and unjust grounds for war. Unjust causes include ambitious conquest, pillage, and jealousy. As far as wealth is concerned, he then lays out the types of expenditures that a king has and the various sources of wealth for the kingdom. He emphasizes that the true wealth of a kingdom is its men and says that it is important to improve the people's lot and that there would be no more poor.

Trivia.

"The Catholic Encyclopedia" (1913) calls Bossuet the greatest pulpit orator of all time, ranking him even ahead of Augustine and Chrysostom.  The exterior of Harvard's Sanders Theater includes busts of the eight greatest orators of all time – they include a bust of Bossuet alongside such giants of oratory as Demosthenes, Cicero, and Chrysostom.  A character in "Les Misérables", being from Meaux and an orator, is nicknamed Bossuet by his friends.  Bossuet was one of several co-editors on the Delphin Classics collection.  Bossuet was the uncle of Louis Bossuet.

References.

Attribution:

Jacques-Bénigne Lignel Bossuet (; 27 September 1627 – 12 April 1704) was a French bishop and theologian, renowned for his sermons and other addresses. He has been considered by many to be one of the most brilliant orators of all time and a masterly French stylist.

Jacques-Bénigne Bossuet (27. září 1627 – 12. dubna 1704) byl francouzský teolog, kazatel, politik a spisovatel, od roku 1681 biskup v Meaux.


Motivation and notation.

Consider, for example, the rational number, which is around 4.4624. As a first approximation, start with 4, which is the integer part;. The fractional part is the reciprocal of which is about 2.1628. Use the integer part, 2, as an approximation for the reciprocal to obtain a second approximation of. The remaining fractional part,, is the reciprocal of, and is around 6.1429. Use 6 as an approximation for this to obtain as an approximation for and, about 4.4615, as the third approximation;. Finally, the fractional part,, is the reciprocal of 7, so its approximation in this scheme, 7, is exact () and produces the exact expression for.  The expression is called the continued fraction representation of. This can be represented by the abbreviated notation = [4; 2, 6, 7]. (It is customary to replace only the "first" comma by a semicolon.) Some older textbooks use all commas in the -tuple, for example, [4, 2, 6, 7].  If the starting number is rational, then this process exactly parallels the Euclidean algorithm. In particular, it must terminate and produce a finite continued fraction representation of the number. If the starting number is irrational, then the process continues indefinitely. This produces a sequence of approximations, all of which are rational numbers, and these converge to the starting number as a limit. This is the (infinite) continued fraction representation of the number. Examples of continued fraction representations of irrational numbers are:  Continued fractions are, in some ways, more "mathematically natural" representations of a real number than other representations such as decimal representations, and they have several desirable properties:

Basic formula.

A continued fraction is an expression of the form  where a and b can be any complex numbers. Usually they are required to be integers. If b = 1 for all "i" the expression is called a "simple" continued fraction. If the expression contains a finite number of terms, it is called a "finite" continued fraction. If the expression contains an infinite number of terms, it is called an "infinite" continued fraction.  Thus, all of the following illustrate valid finite simple continued fractions:

Calculating continued fraction representations.

Consider a real number. Let formula_10 be the integer part of and let formula_11 be the fractional part of. Then the continued fraction representation of is formula_12, where formula_13 is the continued fraction representation of formula_14.  To calculate a continued fraction representation of a number, write down the integer part (technically the floor) of. Subtract this integer part from. If the difference is 0, stop; otherwise find the reciprocal of the difference and repeat. The procedure will halt if and only if is rational. This process can be efficiently implemented using the Euclidean algorithm when the number is rational. The table below shows an implementation of this procedure for the number 3.245, resulting in the continued fraction expansion [3; 4,12,4].

Notations.

The integers formula_2, formula_16 etc., are called the "coefficients" or "terms" of the continued fraction. One can abbreviate the continued fraction  in the notation of Carl Friedrich Gauss  or as  or in the notation of Pringsheim as  or in another related notation as  Sometimes angle brackets are used, like this:  The semicolon in the square and angle bracket notations is sometimes replaced by a comma.  One may also define "infinite simple continued fractions" as limits:  This limit exists for any choice of formula_2 and positive integers formula_25

Finite continued fractions.

Every finite continued fraction represents a rational number, and every rational number can be represented in precisely two different ways as a finite continued fraction, with the conditions that the first coefficient is an integer and the other coefficients are positive integers. These two representations agree except in their final terms. In the longer representation the final term in the continued fraction is 1; the shorter representation drops the final 1, but increases the new final term by 1. The final element in the short representation is therefore always greater than 1, if present. In symbols:

Of reciprocals.

The continued fraction representations of a positive rational number and its reciprocal are identical except for a shift one place left or right depending on whether the number is less than or greater than one respectively. In other words, the numbers represented by formula_26 and formula_27 are reciprocals.  For instance if formula_28 is an integer and formula_29 then  If formula_32 then  The last number that generates the remainder of the continued fraction is the same for both formula_35 and its reciprocal.  For example,

Infinite continued fractions and convergents.

Every infinite continued fraction is irrational, and every irrational number can be represented in precisely one way as an infinite continued fraction.  An infinite continued fraction representation for an irrational number is useful because its initial segments provide rational approximations to the number. These rational numbers are called the convergents of the continued fraction. The larger a term is in the continued fraction, the closer the corresponding convergent is to the irrational number being approximated. Numbers like π have occasional large terms in their continued fraction, which makes them easy to approximate with rational numbers. Other numbers like "e" have only small terms early in their continued fraction, which makes them more difficult to approximate rationally. The golden ratio φ has terms equal to 1 everywhere—the smallest values possible—which makes φ the most difficult number to approximate rationally. In this sense, therefore, it is the "most irrational" of all irrational numbers. Even-numbered convergents are smaller than the original number, while odd-numbered ones are larger.  For a continued fraction, the first four convergents (numbered 0 through 3) are  The numerator of the third convergent is formed by multiplying the numerator of the second convergent by the third quotient, and adding the numerator of the first convergent. The denominators are formed similarly. Therefore, each convergent can be expressed explicitly in terms of the continued fraction as the ratio of certain multivariate polynomials called "continuants".  If successive convergents are found, with numerators,,... and denominators,,... then the relevant recursive relation is:  The successive convergents are given by the formula  Thus to incorporate a new term into a rational approximation, only the two previous convergents are necessary. The initial "convergents" (required for the first two terms) are ⁄ and ⁄. For example, here are the convergents for [0;1,5,2,2].  When using the Babylonian method to generate successive approximations to the square root of an integer, if one starts with the lowest integer as first approximant, the rationals generated all appear in the list of convergents for the continued fraction. Specifically, the approximants will appear on the convergents list in positions 0, 1, 3, 7, 15,...,,... For example, the continued fraction expansion for is [1;1,2,1,2,1,2,1,2...]. Comparing the convergents with the approximants derived from the Babylonian method:

Properties.

A Baire space is a topological space on infinite sequences of natural numbers. The infinite continued fraction provides a homeomorphism from the Baire space to the space of irrational real numbers (with the subspace topology inherited from the usual topology on the reals). The infinite continued fraction also provides a map between the quadratic irrationals and the dyadic rationals, and from other irrationals to the set of infinite strings of binary numbers (i.e. the Cantor set); this map is called the Minkowski question mark function. The mapping has interesting self-similar fractal properties; these are given by the modular group, which is the subgroup of Möbius transformations having integer values in the transform. Roughly speaking, continued fraction convergents can be taken to be Möbius transformations acting on the (hyperbolic) upper half-plane; this is what leads to the fractal self-symmetry.

Some useful theorems.

If formula_2, formula_16, formula_40, formula_41 is an infinite sequence of positive integers, define the sequences formula_42 and formula_43 recursively: Theorem 1. For any positive real number formula_44  Theorem 2. The convergents of [formula_2; formula_16, formula_40, formula_41] are given by  Theorem 3. If the formula_1th convergent to a continued fraction is formula_42/formula_43, then  Corollary 1: Each convergent is in its lowest terms (for if formula_42 and formula_43 had a nontrivial common divisor it would divide formula_57, which is impossible).  Corollary 2: The difference between successive convergents is a fraction whose numerator is unity:  Corollary 3: The continued fraction is equivalent to a series of alternating terms:  Corollary 4: The matrix has determinant plus or minus one, and thus belongs to the group of formula_61 unimodular matrices formula_62.  Theorem 4. Each (formula_63th) convergent is nearer to a subsequent (formula_1th) convergent than any preceding (formula_65th) convergent is. In symbols, if the formula_1th convergent is taken to be formula_67, then for all formula_69.  Corollary 1: The even convergents (before the formula_1th) continually increase, but are always less than formula_71.  Corollary 2: The odd convergents (before the formula_1th) continually decrease, but are always greater than formula_71.  Theorem 5.  Corollary 1: A convergent is nearer to the limit of the continued fraction than any fraction whose denominator is less than that of the convergent.  Corollary 2: A convergent obtained by terminating the continued fraction just before a large term is a close approximation to the limit of the continued fraction.

Semiconvergents.

If  are consecutive convergents, then any fractions of the form  where formula_77 is an integer such that formula_78, are called "semiconvergents", "secondary convergents", or "intermediate fractions". The formula_79-st semiconvergent equals the mediant of the formula_77-th one and the convergent formula_81. It follows that semiconvergents represent a monotonic sequence of fractions between the convergents formula_82 (corresponding to formula_83) and formula_84 (corresponding to formula_85). Sometimes the term is taken to mean that being a semiconvergent excludes the possibility of being a convergent (i.e., formula_86), rather than that a convergent is a kind of semiconvergent.  The semiconvergents to the continued fraction expansion of a real number formula_35 include all the rational approximations that are better than any approximation with a smaller denominator. Another useful property is that consecutive semiconvergents formula_88 and formula_89 are such that formula_90.

Best rational approximations.

One can choose to define a "best rational approximation" to a real number as a rational number,, that is closer to than any approximation with a smaller or equal denominator. The simple continued fraction for can be used to generate "all" of the best rational approximations for by applying these three rules:  For example, 0.84375 has continued fraction [0;1,5,2,2]. Here are all of its best rational approximations.  The strictly monotonic increase in the denominators as additional terms are included permits an algorithm to impose a limit, either on size of denominator or closeness of approximation.  The "half rule" mentioned above requires that when is even, the halved term /2 is admissible if and only if This is equivalent to:  The convergents to are "best approximations" in a much stronger sense than the one defined above. Namely, / is a convergent for if and only if has the smallest value among the analogous expressions for all rational approximations / with ; that is, we have so long as. (Note also that as.)

Best rational within an interval.

A rational that falls within the interval, for, can be found with the continued fractions for and. When both and are irrational and where and have identical continued fraction expansions up through, a rational that falls within the interval is given by the finite continued fraction, This rational will be best in the sense that no other rational in will have a smaller numerator or a smaller denominator.  If is rational, it will have "two" continued fraction representations that are "finite", and, and similarly a rational will have two representations, and. The coefficients beyond the last in any of these representations should be interpreted as ; and the best rational will be one of,,, or.  For example, the decimal representation 3.1416 could be rounded from any number in the interval. The continued fraction representations of 3.14155 and 3.14165 are and the best rational between these two is Thus, is the best rational number corresponding to the rounded decimal number 3.1416, in the sense that no other rational number that would be rounded to 3.1416 will have a smaller numerator or a smaller denominator.

Interval for a convergent.

A rational number, which can be expressed as finite continued fraction in two ways, will be one of the convergents for the continued fraction expansion of a number, if and only if the number is strictly between The numbers and are formed by incrementing the last coefficient in the two representations for. It is the case that when is even, and when is odd.  For example, the number has the continued fraction representations and thus is a convergent of any number strictly between

Comparison.

Consider and. If is the smallest index for which is unequal to then if and otherwise.  If there is no such, but one expansion is shorter than the other, say and with for, then if is even and if is odd.

Continued fraction expansions of.

To calculate the convergents of we may set, define and, and,. Continuing like this, one can determine the infinite continued fraction of as The fourth convergent of is [3;7,15,1] = = 3.14159292035..., sometimes called Milü, which is fairly close to the true value of.  Let us suppose that the quotients found are, as above, [3;7,15,1]. The following is a rule by which we can write down at once the convergent fractions which result from these quotients without developing the continued fraction.  The first quotient, supposed divided by unity, will give the first fraction, which will be too small, namely,. Then, multiplying the numerator and denominator of this fraction by the second quotient and adding unity to the numerator, we shall have the second fraction,, which will be too large. Multiplying in like manner the numerator and denominator of this fraction by the third quotient, and adding to the numerator the numerator of the preceding fraction, and to the denominator the denominator of the preceding fraction, we shall have the third fraction, which will be too small. Thus, the third quotient being 15, we have for our numerator, and for our denominator,. The third convergent, therefore, is. We proceed in the same manner for the fourth convergent. The fourth quotient being 1, we say 333 times 1 is 333, and this plus 22, the numerator of the fraction preceding, is 355; similarly, 106 times 1 is 106, and this plus 7 is 113.  In this manner, by employing the four quotients [3;7,15,1], we obtain the four fractions:  These convergents are alternately smaller and larger than the true value of, and approach nearer and nearer to. The difference between a given convergent and is less than the reciprocal of the product of the denominators of that convergent and the next convergent. For example, the fraction is greater than, but − is less than = (in fact, − is just more than = ).  The demonstration of the foregoing properties is deduced from the fact that if we seek the difference between one of the convergent fractions and the next adjacent to it we shall obtain a fraction of which the numerator is always unity and the denominator the product of the two denominators. Thus the difference between and is, in excess; between and,, in deficit; between and,, in excess; and so on. The result being, that by employing this series of differences we can express in another and very simple manner the fractions with which we are here concerned, by means of a second series of fractions of which the numerators are all unity and the denominators successively be the product of every two adjacent denominators. Instead of the fractions written above, we have thus the series:  The first term, as we see, is the first fraction; the first and second together give the second fraction, ; the first, the second and the third give the third fraction, and so on with the rest; the result being that the series entire is equivalent to the original value.

Generalized continued fraction.

A generalized continued fraction is an expression of the form  where the "a" ("n" > 0) are the partial numerators, the "b" are the partial denominators, and the leading term "b" is called the "integer" part of the continued fraction.  To illustrate the use of generalized continued fractions, consider the following example. The sequence of partial denominators of the simple continued fraction of does not show any obvious pattern:  or  However, several generalized continued fractions for have a perfectly regular structure, such as:  The first two of these are special cases of the function with = 4 arctan (1).  The continued fraction of formula_98 above consisting of cubes uses the Nilakantha series and an exploit from Leonhard Euler.

Other continued fraction expansions.

Periodic continued fractions.

The numbers with periodic continued fraction expansion are precisely the irrational solutions of quadratic equations with rational coefficients; rational solutions have finite continued fraction expansions as previously stated. The simplest examples are the golden ratio φ = [1;1,1,1,1,1...] and = [1;2,2,2,2...], while = [3;1,2,1,6,1,2,1,6...] and = [6;2,12,2,12,2,12...]. All irrational square roots of integers have a special form for the period; a symmetrical string, like the empty string (for ) or 1,2,1 (for ), followed by the double of the leading integer.

A property of the golden ratio φ.

Because the continued fraction expansion for φ doesn't use any integers greater than 1, φ is one of the most "difficult" real numbers to approximate with rational numbers. Hurwitz's theorem states that any irrational number can be approximated by infinitely many rational with  While virtually all real numbers will eventually have infinitely many convergents whose distance from is significantly smaller than this limit, the convergents for φ (i.e., the numbers,,,, etc.) consistently "toe the boundary", keeping a distance of almost exactly formula_100 away from φ, thus never producing an approximation nearly as impressive as, for example, for. It can also be shown that every real number of the form, where,,, and are integers such that, shares this property with the golden ratio φ; and that all other real numbers can be more closely approximated.

Regular patterns in continued fractions.

While there is no discernable pattern in the simple continued fraction expansion of, there is one for, the base of the natural logarithm:  which is a special case of this general expression for positive integer :  Another, more complex pattern appears in this continued fraction expansion for positive odd :  with a special case for :  Other continued fractions of this sort are  where is a positive integer; also, for integer :  with a special case for :  If is the modified, or hyperbolic, Bessel function of the first kind, we may define a function on the rationals by  which is defined for all rational numbers, with and in lowest terms. Then for all nonnegative rationals, we have  with similar formulas for negative rationals; in particular we have  Many of the formulas can be proved using Gauss's continued fraction.

Typical continued fractions.

Most irrational numbers do not have any periodic or regular behavior in their continued fraction expansion. Nevertheless, Khinchin proved that for almost all real numbers, the (for ) have an astonishing property: their geometric mean tends to a constant (known as Khinchin's constant, ) independent of the value of. Paul Lévy showed that the th root of the denominator of the th convergent of the continued fraction expansion of almost all real numbers approaches an asymptotic limit, approximately 3.27582, which is known as Lévy's constant. Lochs' theorem states that th convergent of the continued fraction expansion of almost all real numbers determines the number to an average accuracy of just over decimal places.

Applications.

Square roots.

Generalized continued fractions are used in a method for computing square roots.  The identity leads via recursion to the generalized continued fraction for any square root:

Pell's equation.

Continued fractions play an essential role in the solution of Pell's equation. For example, for positive integers and, and non-square, it is true that if, then is a convergent of the regular continued fraction for. The converse holds if the period of the regular continued fraction for is 1, and in general the period describes which convergents give solutions to Pell's equation.

Dynamical systems.

Continued fractions also play a role in the study of dynamical systems, where they tie together the Farey fractions which are seen in the Mandelbrot set with Minkowski's question mark function and the modular group Gamma.  The backwards shift operator for continued fractions is the map called the Gauss map, which lops off digits of a continued fraction expansion:. The transfer operator of this map is called the Gauss–Kuzmin–Wirsing operator. The distribution of the digits in continued fractions is given by the zero'th eigenvector of this operator, and is called the Gauss–Kuzmin distribution.

Eigenvalues and eigenvectors.

The Lanczos algorithm uses a continued fraction expansion to iteratively approximate the eigenvalues and eigenvectors of a large sparse matrix.

Networking applications.

Continued fractions have also been used in modelling optimization problems for wireless network virtualization to find a route between a source and a destination.

} In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers formula_3 are called the coefficients or terms of the continued fraction.

Řetězový zlomek je výraz typu kde "a0" je celé číslo a čísla "a" jsou kladná přirozená čísla. Pokud je dána pouze konečná posloupnost ("a", "a", "a"...), pak mluvíme o konečném řetězovém zlomku, pokud je tato posloupnost nekonečná, pak mluvíme o nekonečném řetězovém zlomku, který bývá také značen: