title: Gaithersburg, Maryland

text: the population was spread out, with 25.0% under the age of 18, 9.0% from 18 to 24, 37.7% from 25 to 44, 20.0% from 45 to 64, and 8.2% who were 65 years of age or older. The median age was 34 years. For every 100 females, there were 95.1 males. For every 100 females age 18 and over, there were 92.4 males.


title: Gaithersburg, Maryland

text: According to the city's 2020 Comprehensive Annual Financial Report, the top employers in the city are:


title: Gaithersburg, Maryland

text: Gaithersburg also receives significant income from its conference organization platform including prominent conferences such as the CHI 84 conference.


title: Gaithersburg, Maryland

text: RIO Washingtonian Center a.k.a. rio a.k.a. Rio Lakefront, is a 760,000-square-foot complex of retail, restaurant and entertainment including AMC/Loews rio Cinemas, Target, LOFT, Barnes & Noble, Dave & Buster's, and Dick's Sporting Goods, built along an artificial lake.


title: Gaithersburg, Maryland

text: Gaithersburg has an elected, five-member City Council, which serves as the legislative body of the city. The mayor, who is also elected, serves as president of the council. The day-to-day administration of the city is overseen by a career city manager. Gaithersburg is also the location of the United States Army Reserve Legal Command.


title: Gaithersburg, Maryland

text: The city's current mayor is Jud Ashman, who has held the office since 2014. On October 6, 2014, the Gaithersburg City Council selected city council member Jud Ashman to serve as mayor until the next City of Gaithersburg election in November 2015, replacing resigning mayor Sidney Katz. Ashman was re-elected in November 2015 and would be re-elected to full terms in 2017 and 2021.


title: Gaithersburg, Maryland

text: The most prominent highways serving Gaithersburg are Interstate 270 and Interstate 370. I-270 is the main highway leading northwest out of metropolitan Washington, D.C., beginning at Interstate 495 (the Capital Beltway) and proceeding northwestward to Interstate 70 in Frederick. I-370 is a short spur, starting just west of I-270 in Gaithersburg and heading east to its junction with Maryland Route 200. Via MD 200, I-370 connects Gaithersburg with Interstate 95 near Laurel.


title: Gaithersburg, Maryland

text: Maryland Route 355 was the precursor to I-270 and follows a parallel route. It now serves as the main commercial roadway through Gaithersburg and neighboring communities. Other state highways serving Gaithersburg include Maryland Route 117, Maryland Route 119 and Maryland Route 124. Maryland Route 28 passes just outside the Gaithersburg corporate limits.


title: Gaithersburg, Maryland

text: Gaithersburg is connected to the Washington Metro via Shady Grove station, which is located just outside the city limits and is the north-western terminus of the Red Line.


title: Gaithersburg, Maryland

text: The Corridor Cities Transitway is a proposed bus rapid transit line that would have 8 stops in Gaithersburg, generally in the western half of the city.


title: Gaithersburg, Maryland

text: Maryland's MARC system operates commuter rail services connecting Gaithersburg to Washington, D.C. with two stations in the city, at Old Town Gaithersburg and Metropolitan Grove, and a third station — Washington Grove — just outside city limits.


title: Gaithersburg, Maryland

text: Bus service in Gaithersburg consists of Metrobus routes operated by WMATA and Ride-On routes operated by Montgomery County, as well as paratransit service provided by MetroAccess.


title: Biologist

text: A biologist is a scientist who conducts research in biology. Biologists are interested in studying life on Earth, whether it is an individual cell, a multicellular organism, or a community of interacting populations. They usually specialize in a particular branch (e.g., molecular biology, zoology, and evolutionary biology) of biology and have a specific research focus (e.g., studying malaria or cancer).


title: Biologist

text: Biologists who are involved in basic research have the aim of advancing knowledge about the natural world. They conduct their research using the scientific method, which is an empirical method for testing hypotheses. Their discoveries may have applications for some specific purpose such as in biotechnology, which has the goal of developing medically useful products for humans.


title: Biologist

text: In modern times, most biologists have one or more academic degrees such as a bachelor's degree plus an advanced degree like a master's degree or a doctorate. Like other scientists, biologists can be found working in different sectors of the economy such as in academia, nonprofits, private industry, or government.


title: Biologist

text: Francesco Redi, the founder of biology, is recognized to be one of the greatest biologists of all time. Robert Hooke, an English natural philosopher, coined the term "cell", suggesting plant structure's resemblance to honeycomb cells.


title: Biologist

text: Charles Darwin and Alfred Wallace independently formulated the theory of evolution by natural selection, which was described in detail in Darwin's book "On the Origin of Species", which was published in 1859. In it, Darwin proposed that the features of all living things, including humans, were shaped by natural processes of descent with accumulated modification leading to divergence over long periods of time. The theory of evolution in its current form affects almost all areas of biology. Separately, Gregor Mendel formulated in the principles of inheritance in 1866, which became the basis of modern genetics.


title: Biologist

text: In 1953, James D. Watson and Francis Crick described the basic structure of DNA, the genetic material for expressing life in all its forms, building on the work of Maurice Wilkins and Rosalind Franklin, suggested that the structure of DNA was a double helix.


title: Biologist

text: Ian Wilmut led a research group that in 1996 first cloned a mammal from an adult somatic cell, a Finnish Dorset lamb named Dolly.


title: Biologist

text: An undergraduate degree in biology typically requires coursework in molecular and cellular biology, development, ecology, genetics, microbiology, anatomy, physiology, botany, and zoology. Additional requirements may include physics, chemistry (general, organic, and biochemistry), calculus, and statistics.


title: Biologist

text: Students who aspire to a research-oriented career usually pursue a graduate degree such as a master’s or a doctorate (e.g., PhD) whereby they would receive training from a research head based on an apprenticeship model that has been in existence since the 1800s. Students in these graduate programs often receive specialized training in a particular subdiscipline of biology.


title: Biologist

text: Biologists who work in basic research formulate theories and devise experiments to advance human knowledge on life including topics such as evolution, biochemistry, molecular biology, neuroscience and cell biology.


title: Biologist

text: Biologists typically conduct laboratory experiments involving animals, plants, microorganisms or biomolecules. However, a small part of biological research also occurs outside the laboratory and may involve natural observation rather than experimentation. For example, a botanist may investigate the plant species present in a particular environment, while an ecologist might study how a forest area recovers after a fire.


title: Biologist

text: Biologists who work in applied research use instead the accomplishments gained by basic research to further knowledge in particular fields or applications. For example, this applied research may be used to develop new pharmaceutical drugs, treatments and medical diagnostic tests. Biological scientists conducting applied research and product development in private industry may be required to describe their research plans or results to non-scientists who are in a position to veto or approve their ideas. These scientists must consider the business effects of their work.


title: Biologist

text: Swift advances in knowledge of genetics and organic molecules spurred growth in the field of biotechnology, transforming the industries in which biological scientists work. Biological scientists can now manipulate the genetic material of animals and plants, attempting to make organisms (including humans) more productive or resistant to disease. Basic and applied research on biotechnological processes, such as recombining DNA, has led to the production of important substances, including human insulin and growth hormone. Many other substances not previously available in large quantities are now produced by biotechnological means. Some of these substances are useful in treating diseases.


title: Biologist

text: Those working on various genome (chromosomes with their associated genes) projects isolate genes and determine their function. This work continues to lead to the discovery of genes associated with specific diseases and inherited health risks, such as sickle cell anemia. Advances in biotechnology have created research opportunities in almost all areas of biology, with commercial applications in areas such as medicine, agriculture, and environmental remediation.


title: Biologist

text: Most biological scientists specialize in the study of a certain type of organism or in a specific activity, although recent advances have blurred some traditional classifications.


title: Biologist

text: Biologists typically work regular hours but longer hours are not uncommon. Researchers may be required to work odd hours in laboratories or other locations (especially while in the field), depending on the nature of their research.


title: Biologist

text: Many biologists depend on grant money to fund their research. They may be under pressure to meet deadlines and to conform to rigid grant-writing specifications when preparing proposals to seek new or extended funding.


title: Biologist

text: Marine biologists encounter a variety of working conditions. Some work in laboratories; others work on research ships, and those who work underwater must practice safe diving while working around sharp coral reefs and hazardous marine life. Although some marine biologists obtain their specimens from the sea, many still spend a good deal of their time in laboratories and offices, conducting tests, running experiments, recording results, and compiling data.


title: Biologist

text: Biologists are not usually exposed to unsafe or unhealthy conditions. Those who work with dangerous organisms or toxic substances in the laboratory must follow strict safety procedures to avoid contamination. Many biological scientists, such as botanists, ecologists, and zoologists, conduct field studies that involve strenuous physical activity and primitive living conditions. Biological scientists in the field may work in warm or cold climates, in all kinds of weather.


title: Biologist

text: The highest honor awarded to biologists is the Nobel Prize in Physiology or Medicine, awarded since 1901, by the Royal Swedish Academy of Sciences. Another significant award is the Crafoord Prize in Biosciences; established in 1980.


title: Travis Hunter (American football)

text: Travis Hunter Jr. (born May 18, 2003) is an American football cornerback and wide receiver who previously played for the Jackson State Tigers. A consensus five-star recruit in high school, he made headlines in December 2021 when he committed to Jackson State, becoming the highest-ranked prospect to ever commit to an HBCU or FCS program.


title: Travis Hunter (American football)

text: A native of West Palm Beach, Florida, Hunter moved to Georgia as an eighth grader. He attended Collins Hill High School in Suwanee, Georgia, where he played cornerback and wide receiver for the Eagles. As a sophomore, he led the county with seven interceptions while recording 49 receptions for 919 yards and 12 touchdowns. As a junior, Hunter made eight interceptions and 51 tackles in addition to catching 137 passes for 1,746 yards and 24 touchdowns, earning MaxPreps Georgia Player of the Year honors. He also set Gwinnett County single-season records in receptions, receiving yards and receiving touchdowns, leading the Eagles to a 12–3 record and an appearance in the Class 7A state title game.


title: Travis Hunter (American football)

text: As a senior, Hunter recorded 76 receptions for 1,128 yards and 10 touchdowns on offense and 23 tackles, four interceptions and a forced fumble on defense, even after missing five games due to a mid-season ankle injury. In the state title game, he made 10 catches for 153 yards and a touchdown, as well as a forced fumble, to cap off a perfect 15–0 record and Collins Hill's first state championship in school history. In his final high school game, Hunter made 10 catches for 178 yards and two touchdowns in their 40–36 defeat to Washington state champs Graham-Kapowsin in the GEICO State Championship Bowl Series. He also broke the Georgia state record in career receiving touchdowns, previously held by Braxton Hicks, with 48.


title: Travis Hunter (American football)

text: Hunter was selected to play in the 2022 Polynesian Bowl, where he won Offensive MVP honors after recording five receptions for 54 yards, in addition to an interception on defense.


title: Travis Hunter (American football)

text: Hunter was considered the number one overall recruit by both 247Sports.com and Rivals.com, as well as number two by ESPN (behind Walter Nolen). After having received a scholarship from Florida State in November 2019, he committed to the Seminoles on March 3, 2020, one day after making his first visit to the school. He made his choice over offers from Alabama, Auburn, Clemson, Florida, Georgia, Michigan and Oklahoma, among others.


title: Travis Hunter (American football)

text: On December 15, 2021, Hunter flipped his commitment to the Jackson State Tigers coached by Deion Sanders, a Florida State alum, in an unprecedented move. He had visited the school the month before, attending their sellout Soul Bowl victory on November 20. As the number one recruit in the nation, Hunter became the first five-star recruit to ever sign with a HBCU or an FCS program. Dustin Lewis of "Sports Illustrated" called it "probably the most shocking decision in the history of college football recruiting" while Steve Wiltfong, the national director of recruiting for 247Sports.com, said it was "the biggest signing day moment in the history of college football".


title: Travis Hunter (American football)

text: Hunter recorded two receiving touchdowns and two interceptions in the Jackson State spring game, which was the first HBCU spring game to ever be nationally televised.


title: Travis Hunter (American football)

text: Hunter recorded his first collegiate touchdown as well as his first interception against Alabama A&M on November 12, 2022.


title: Travis Hunter (American football)

text: At the conclusion of the 2022 season, Hunter accumulated 15 total tackles and 2 interceptions and a defensive touchdown in 7 games played. On offense, he contributed 14 receptions for 141 yards with 2 receiving touchdowns.


title: Travis Hunter (American football)

text: With Deion Sanders departing Jackson State for the head coaching position at the University of Colorado, it has been widely reported that Hunter will likely follow Sanders to the program. On December 19, Hunter entered the transfer portal.


title: Travis Hunter (American football)

text: Hunter's father, Travis Sr., starred on the football and track teams at Boynton Beach Community High School. He went on to play semipro football in the Florida Football Alliance and the Southern States Football League, winning the latter's Offensive Rookie of the Year award in 2007.


title: Travis Hunter (American football)

text: On February 16, 2022, it was announced that Hunter signed a Name, Image and Likeness (NIL) deal with J5 Caffe, a Black-owned coffee company based in Columbus, Mississippi.


title: Travis Hunter (American football)

text: On July 27, 2022, Hunter signed a multiyear endorsement deal with Greenwood, designed to create awareness around all that fintech has to offer to his community. He is the first NIL deal for Greenwood.


title: Michael Dowd (police officer)

text: Michael F. Dowd (born January 10, 1961) is a former New York City Police Department (NYPD) officer, drug distributor, and criminal associate of the Diaz organization who was arrested in 1992 for running a drug ring out of Suffolk County, Long Island, New York. He is the subject of the 2014 documentary film "The Seven Five" directed by Tiller Russell and produced by Eli Holzman. The Tiller Russell TV documentary version, "Precinct Seven Five" (2015), aired on Film4 on June 19, 2020 and featured Dowd's co-conspirator and "dirty cop" friend Kenneth (Kenny) Eurell, who turned States Evidence and agreed to be wired, in order to entrap Dowd and corroborate Eurell's testimony in exchange for a lenient sentence at trial.


title: Michael Dowd (police officer)

text: Dowd was born on January 10, 1961 in Brooklyn, New York City, the third of seven children in an Irish Catholic family. He grew up in Brentwood, Long Island, on a block mostly populated by the families of police officers and firefighters. According to Dowd, he was a good student in high school.


title: Michael Dowd (police officer)

text: In the course of his career, Dowd committed a host of crimes, including conspiring with drug traffickers to distribute cocaine, warning drug dealers about upcoming raids, providing them with guns and badges, allegedly planning to abduct a woman in Queens, and stealing food meant for the needy at a church. Dowd located a man who robbed the Diaz drug cartel and instead of arresting him turned him over to Diaz. He pocketed several thousand dollars a week as a result of corrupt arrangements.


title: Michael Dowd (police officer)

text: Dowd was arrested in 1992. After investigations by the Suffolk County Police, the DEA, and NYPD's internal affairs, Dowd was convicted of racketeering and conspiracy to distribute narcotics and sent to prison for his crimes. He cooperated with the Mollen Commission, which investigated allegations of corruption in the NYPD.


title: Michael Dowd (police officer)

text: Sentenced to 16 years in prison, Dowd served 12 years and 5 months. While he was in prison, Dowd claimed he worked as a peer counselor, worked out, read, and ran the addiction and suicide prevention programs.


title: Michael Dowd (police officer)

text: Dowd has been featured on podcasts, periodicals, broadcast radio and television programs, and was the subject of the documentary film "The Seven Five". A narrative feature adaptation by Sony Pictures is being produced by John Lesher and Megan Ellison.


title: Nile (band)

text: Nile is an American death metal band from Greenville, South Carolina, formed in 1993. The band's current lineup consists of Karl Sanders (guitars, vocals), George Kollias (drums), Brian Kingsland (guitars, vocals), and Scott Eames (guitars, vocals). The band is known for their dedicated lyrical and musical styles inspired by Egyptology and works of H.P. Lovecraft. Since its formation, the band has released nine studio albums, two compilations, three extended plays, two demos, three singles, and five music videos. Their latest studio album, "Vile Nilotic Rites", was released on November 1, 2019, through Nuclear Blast.


title: Nile (band)

text: Nile formed in Greenville, South Carolina in 1993 from a previous band called Morriah, in which founding member Karl Sanders performed in since the 1980s. The name of this band was taken from the Nile river. Nile released a self-titled thrash-influenced demo in 1994 (re-issued in 2011 under the title "Worship the Animal - 1994: The Lost Recordings"). Sanders, vocalist/bassist Chief Spires and drummer Pete Hammoura released the "Festivals of Atonement" EP in 1995, leading Nile to a southwestern tour in support of other metal acts such as Obituary, Deicide, and Broken Hope.


title: Nile (band)

text: The EP "Ramses Bringer of War" was released by Visceral Productions in 1997. The label was due to release the band's first full-length album "Amongst the Catacombs of Nephren-Ka", but went out of business later that year. However, Relapse Records was open to releasing it early on, giving Nile a wider distribution and a chance to tour with Incantation and Morbid Angel.


title: Nile (band)

text: The follow-up, "Black Seeds of Vengeance", was released in September 2000. Prior to the release, drummer Pete Hammoura left the band due to injuries sustained while touring with Catacomb. Derek Roddy stepped in as a session musician to perform on all of the tracks (except "To Dream of Ur"). Also, prior to the album, Dallas Toler-Wade joined as a second guitarist/vocalist. Chief Spires left the band a few months after the release of "Black Seeds of Vengeance", citing personal and professional differences. He was replaced by Jon Vesano. The band also found new drummer Tony Laureano.


title: Nile (band)

text: In 2002, "In Their Darkened Shrines" was released. One reviewer noted that the album gave "death metal a cinematic, symphonic twist". Two videos were shot with director Darren Doane, "Sarcophagus" and "Execration Text", respectively.


title: Nile (band)

text: The follow-up album "Annihilation of the Wicked" was released in 2005. Greek drummer George Kollias stepped in to replace Laureano, who left before the recording sessions began. After recording the bass parts for the album, Jon Vesano also left the band. He was temporarily replaced by Joe Payne on the following tour. Another Darren Doane directed video was shot for the song "Sacrifice Unto Sebek".


title: Nile (band)

text: In May 2006, Nile signed a recording contract with Nuclear Blast. On February 26, 2007, Nile entered the studio and began recording their fifth full-length album entitled "Ithyphallic", which was their first release with the label. Karl Sanders had confirmed a release date of June 29, 2007, via a posting on the band's message board, which was later pushed back to July 20, 2007. As a special bonus, Nuclear Blast gave several fans the opportunity to listen to the new album two weeks before its release if they could decipher a hieroglyphic message, to which "papyrus" was the answer. Upon the album's release, it sold 4,600 copies, which enabled Nile to get onto the "Billboard" 200 for the first time, reaching No. 162 on that chart.


title: Nile (band)

text: On March 21, 2007, Nile was confirmed as a "Second Stage" act for Ozzfest 2007. A message from Karl Sanders via Nile's MySpace profile indicated that they were excited to finally bring true death metal to an Ozzfest audience. From March to April 2008, Nile went on a headlining tour in the United States with Suicide Silence, The Faceless, Unexpect, and Warbringer. From September to October 2008, Nile set out on a European headlining tour with Grave, Omnium Gatherum and Severe Torture. In October 2008, the band planned to take their "Ithyphallic" tour to South Africa for a booking of three shows, but were canceled for unknown reasons.


title: Nile (band)

text: In June 2009, Nile entered the recording studio to begin work on their sixth full-length album entitled "Those Whom the Gods Detest". The album was released on November 3, 2009, and was critically well-received, with one reviewer suggesting "pretty much the only thing standing in Nile's way at this point is the fact that we as listeners have become so accustomed to the band's excellence that another excellent album seems almost par for the course".


title: Nile (band)

text: On October 21, 2010, bassist Chris Lollis was confirmed as the new member for Nile, after four years of session work with the band. On December 6, 2011, Nile announced work in progress, and the completion of drum tracking, for their upcoming album in 2012.


title: Nile (band)

text: On February 4, 2012, Karl Sanders announced on his Facebook page that they had a new bassist, Todd Ellis, who would contribute vocals to the new record. The next day, he explained on the band's official forum that he had lost contact with Chris Lollis during the writing and recording of the upcoming album, and after three months of silence from him they hired a replacement. The seventh album "At the Gate of Sethu" was released on June 29, 2012.


title: Nile (band)

text: On February 9, 2015, Todd Ellis left the band, citing "unforeseen circumstances beyond my control". Nile's eighth studio album "What Should Not Be Unearthed" was released on August 28, 2015.


title: Nile (band)

text: In October 2016, after the Summer Slaughter tour, Dallas Toler-Wade left the band, prioritizing his duties with Narcotic Wasteland, and was replaced by Enthean vocalist/guitarist Brian Kingsland. Nile made the announcement in February 2017, In May 2017, the band entered the studio to begin work on their ninth album "Vile Nilotic Rites", which was released on November 1, 2019.


title: Nile (band)

text: In July 2022, Sanders confirmed to "Guitar World" magazine that Nile has been working on their tenth studio album: "So far, we've got 10 songs written. We're probably going to do three more, and hopefully start recording by the end of the summer. Then we've got a European tour coming up in November/December, and after that, we've got to finish the record and hopefully get it out sometime early next year."


title: Nile (band)

text: In October 2022 Nile announced Julian Guillen would be joining the band on bass for the upcoming November/December European tour.


title: Nile (band)

text: The majority of Nile's music is written by vocalist/guitarist Karl Sanders. It combines traditional and technical death metal. Nile's music is characterized by the technical death metal genre's comparative complexity and speed combined with a riffing style consistent with brutal death metal. Nile's guitar and bass tunings are typically Dropped A tuning.


title: Nile (band)

text: Much of the lyrical content is based on Sanders' interest in Egyptology (mainly inspired by Ancient Egyptian and Near Eastern mysticism, history, religion, and ancient art) and other ancient Middle Eastern cultures, such as Mesopotamia. Works of H.P. Lovecraft also seems to have an enduring influence upon the band's lyrical themes; the word "Nile" and the title of the band's first full-length album ("Amongst the Catacombs of Nephren-Ka") can both be found in the same sentence of Lovecraft's short story "The Outsider". Extensive sleeve notes are included on all Nile's full-length releases from "Black Seeds of Vengeance" onwards, with the exception of "Ithyphallic", explaining the inspiration or source for the lyrics of each song.


title: Planetarium

text: A planetarium ( planetariums or "planetaria") is a theatre built primarily for presenting educational and entertaining shows about astronomy and the night sky, or for training in celestial navigation.


title: Planetarium

text: A dominant feature of most planetariums is the large dome-shaped projection screen onto which scenes of stars, planets, and other celestial objects can be made to appear and move realistically to simulate their motion. The projection can be created in various ways, such as a star ball, slide projector, video, fulldome projector systems, and lasers. Typical systems can be set to simulate the sky at any point in time, past or present, and often to depict the night sky as it would appear from any point of latitude on Earth.


title: Planetarium

text: Planetaria range in size from the 37 meter dome in St. Petersburg, Russia (called “Planetarium No 1”) to three-meter inflatable portable domes where attendees sit on the floor. The largest planetarium in the Western Hemisphere is the Jennifer Chalsty Planetarium at Liberty Science Center in New Jersey, its dome measuring 27 meters in diameter. The Birla Planetarium in Kolkata, India is the largest by seating capacity, having 630 seats. In North America, the Hayden Planetarium at the American Museum of Natural History in New York City has the greatest number of seats, at 423.


title: Planetarium

text: The term "planetarium" is sometimes used generically to describe other devices which illustrate the Solar System, such as a computer simulation or an orrery. "Planetarium software" refers to a software application that renders a three-dimensional image of the sky onto a two-dimensional computer screen, or in a virtual reality headset for a 3D representation. The term "planetarian" is used to describe a member of the professional staff of a planetarium.


title: Planetarium

text: The ancient Greek polymath Archimedes is attributed with creating a primitive planetarium device that could predict the movements of the Sun and the Moon and the planets. The discovery of the Antikythera mechanism proved that such devices already existed during antiquity, though likely after Archimedes' lifetime. Campanus of Novara described a planetary equatorium in his "Theorica Planetarum", and included instructions on how to build one. The Globe of Gottorf built around 1650 had constellations painted on the inside. These devices would today usually be referred to as orreries (named for the Earl of Orrery). In fact, many planetariums today have projection orreries, which project onto the dome the Solar System (including the Sun and planets up to Saturn) in their regular orbital paths.


title: Planetarium

text: In 1229, following the conclusion of the Fifth Crusade, Holy Roman Emperor Frederick II of Hohenstaufen brought back a tent with scattered holes representing stars or planets. The device was operated internally with a spinnable table that rotated the tent.


title: Planetarium

text: The small size of typical 18th century orreries limited their impact, and towards the end of that century a number of educators attempted to create a larger sized version. The efforts of Adam Walker (1730–1821) and his sons are noteworthy in their attempts to fuse theatrical illusions with education. Walker's Eidouranion was the heart of his public lectures or theatrical presentations. Walker's son describes this "Elaborate Machine" as "twenty feet high, and twenty-seven in diameter: it stands vertically before the spectators, and its globes are so large, that they are distinctly seen in the most distant parts of the Theatre. Every Planet and Satellite seems suspended in space, without any support; performing their annual and diurnal revolutions without any apparent cause". Other lecturers promoted their own devices: R E Lloyd advertised his Dioastrodoxon, or Grand Transparent Orrery, and by 1825 William Kitchener was offering his Ouranologia, which was in diameter. These devices most probably sacrificed astronomical accuracy for crowd-pleasing spectacle and sensational and awe-provoking imagery.


title: Planetarium

text: The oldest, still working planetarium can be found in the Dutch town Franeker. It was built by Eise Eisinga (1744–1828) in the living room of his house. It took Eisinga seven years to build his planetarium, which was completed in 1781.


title: Planetarium

text: In 1905 Oskar von Miller (1855–1934) of the Deutsches Museum in Munich commissioned updated versions of a geared orrery and planetarium from M Sendtner, and later worked with Franz Meyer, chief engineer at the Carl Zeiss optical works in Jena, on the largest mechanical planetarium ever constructed, capable of displaying both heliocentric and geocentric motion. This was displayed at the Deutsches Museum in 1924, construction work having been interrupted by the war. The planets travelled along overhead rails, powered by electric motors: the orbit of Saturn was 11.25 m in diameter. 180 stars were projected onto the wall by electric bulbs.


title: Planetarium

text: While this was being constructed, von Miller was also working at the Zeiss factory with German astronomer Max Wolf, director of the Landessternwarte Heidelberg-Königstuhl observatory of the University of Heidelberg, on a new and novel design, inspired by Wallace W. Atwood's work at the Chicago Academy of Sciences and by the ideas of Walther Bauersfeld and Rudolf Straubel at Zeiss. The result was a planetarium design which would generate all the necessary movements of the stars and planets inside the optical projector, and would be mounted centrally in a room, projecting images onto the white surface of a hemisphere. In August 1923, the first (Model I) Zeiss planetarium projected images of the night sky onto the white plaster lining of a 16 m hemispherical concrete dome, erected on the roof of the Zeiss works. The first official public showing was at the Deutsches Museum in Munich on October 21, 1923.


title: Planetarium

text: When Germany was divided into East and West Germany after the war, the Zeiss firm was also split. Part remained in its traditional headquarters at Jena, in East Germany, and part migrated to West Germany. The designer of the first planetariums for Zeiss, Walther Bauersfeld, also migrated to West Germany with the other members of the Zeiss management team. There he remained on the Zeiss West management team until his death in 1959.


title: Planetarium

text: The West German firm resumed making large planetariums in 1954, and the East German firm started making small planetariums a few years later. Meanwhile, the lack of planetarium manufacturers had led to several attempts at construction of unique models, such as one built by the California Academy of Sciences in Golden Gate Park, San Francisco, which operated 1952–2003. The Korkosz brothers built a large projector for the Boston Museum of Science, which was unique in being the first (and for a very long time only) planetarium to project the planet Uranus. Most planetariums ignore Uranus as being at best marginally visible to the naked eye.


title: Planetarium

text: A great boost to the popularity of the planetarium worldwide was provided by the Space Race of the 1950s and 60s when fears that the United States might miss out on the opportunities of the new frontier in space stimulated a massive program to install over 1,200 planetariums in U.S. high schools.


title: Planetarium

text: Armand Spitz recognized that there was a viable market for small inexpensive planetaria. His first model, the Spitz A, was designed to project stars from a dodecahedron, thus reducing machining expenses in creating a globe. Planets were not mechanized, but could be shifted by hand. Several models followed with various upgraded capabilities, until the A3P, which projected well over a thousand stars, had motorized motions for latitude change, daily motion, and annual motion for Sun, Moon (including phases), and planets. This model was installed in hundreds of high schools, colleges, and even small museums from 1964 to the 1980s.


title: Planetarium

text: Japan entered the planetarium manufacturing business in the 1960s, with Goto and Minolta both successfully marketing a number of different models. Goto was particularly successful when the Japanese Ministry of Education put one of their smallest models, the E-3 or E-5 (the numbers refer to the metric diameter of the dome) in every elementary school in Japan.


title: Planetarium

text: Phillip Stern, as former lecturer at New York City's Hayden Planetarium, had the idea of creating a small planetarium which could be programmed. His Apollo model was introduced in 1967 with a plastic program board, recorded lecture, and film strip. Unable to pay for this himself, Stern became the head of the planetarium division of Viewlex, a mid-size audio-visual firm on Long Island. About thirty canned programs were created for various grade levels and the public, while operators could create their own or run the planetarium live. Purchasers of the Apollo were given their choice of two canned shows, and could purchase more. A few hundred were sold, but in the late 1970s Viewlex went bankrupt for reasons unrelated to the planetarium business.


title: Planetarium

text: During the 1970s, the OmniMax movie system (now known as IMAX Dome) was conceived to operate on planetarium screens. More recently, some planetariums have re-branded themselves as "dome theaters", with broader offerings including wide-screen or "wraparound" films, fulldome video, and laser shows that combine music with laser-drawn patterns.


title: Planetarium

text: Learning Technologies Inc. in Massachusetts offered the first easily portable planetarium in 1977. Philip Sadler designed this patented system which projected stars, constellation figures from many mythologies, celestial coordinate systems, and much else, from removable cylinders (Viewlex and others followed with their own portable versions).


title: Planetarium

text: When Germany reunified in 1989, the two Zeiss firms did likewise, and expanded their offerings to cover many different size domes.


title: Planetarium

text: In 1983, Evans & Sutherland installed the first digital planetarium projector displaying computer graphics (Hansen planetarium, Salt Lake City, Utah)—the Digistar I projector used a vector graphics system to display starfields as well as line art. This gives the operator great flexibility in showing not only the modern night sky as visible from Earth, but as visible from points far distant in space and time. The newest generations of planetarium projectors, beginning with Digistar 3, offer fulldome video technology. This allows for the projection of any image.


title: Planetarium

text: Planetarium domes range in size from 3 to 35 m in diameter, accommodating from 1 to 500 people. They can be permanent or portable, depending on the application.


title: Planetarium

text: The realism of the viewing experience in a planetarium depends significantly on the dynamic range of the image, i.e., the contrast between dark and light. This can be a challenge in any domed projection environment, because a bright image projected on one side of the dome will tend to reflect light across to the opposite side, "lifting" the black level there and so making the whole image look less realistic. Since traditional planetarium shows consisted mainly of small points of light (i.e., stars) on a black background, this was not a significant issue, but it became an issue as digital projection systems started to fill large portions of the dome with bright objects (e.g., large images of the sun in context). For this reason, modern planetarium domes are often not painted white but rather a mid grey colour, reducing reflection to perhaps 35-50%. This increases the perceived level of contrast.


title: Planetarium

text: A major challenge in dome construction is to make seams as invisible as possible. Painting a dome after installation is a major task, and if done properly, the seams can be made almost to disappear.


title: Planetarium

text: Traditionally, planetarium domes were mounted horizontally, matching the natural horizon of the real night sky. However, because that configuration requires highly inclined chairs for comfortable viewing "straight up", increasingly domes are being built tilted from the horizontal by between 5 and 30 degrees to provide greater comfort. Tilted domes tend to create a favoured "sweet spot" for optimum viewing, centrally about a third of the way up the dome from the lowest point. Tilted domes generally have seating arranged stadium-style in straight, tiered rows; horizontal domes usually have seats in circular rows, arranged in concentric (facing center) or epicentric (facing front) arrays.


title: Planetarium

text: planetaria occasionally include controls such as buttons or joysticks in the arm rests of seats to allow audience feedback that influences the show in real time.


title: Planetarium

text: Traditionally, planetariums needed many incandescent lamps around the cove of the dome to help audience entry and exit, to simulate sunrise and sunset, and to provide working light for dome cleaning. More recently, solid-state LED lighting has become available that significantly decreases power consumption and reduces the maintenance requirement as lamps no longer have to be changed on a regular basis.


title: Planetarium

text: The world's largest mechanical planetarium is located in Monico, Wisconsin. The "Kovac Planetarium". It is 22 feet in diameter and weighs two tons. The globe is made of wood and is driven with a variable speed motor controller. This is the largest mechanical planetarium in the world, larger than the "Atwood Globe" in Chicago (15 feet in diameter) and one third the size of the Hayden.


title: Planetarium

text: Some new planetariums now feature a glass floor, which allows spectators to stand near the center of a sphere surrounded by projected images in all directions, giving the impression of floating in outer space. For example, a small planetarium at AHHAA in Tartu, Estonia features such an installation, with special projectors for images below the feet of the audience, as well as above their heads.


title: Planetarium

text: Traditional planetarium projection apparatus use a hollow ball with a light inside, and a pinhole for each star, hence the name "star ball". With some of the brightest stars (e.g. Sirius, Canopus, Vega), the hole must be so big to let enough light through that there must be a small lens in the hole to focus the light to a sharp point on the dome. In later and modern planetarium star balls, the individual bright stars often have individual projectors, shaped like small hand-held torches, with focusing lenses for individual bright stars. Contact breakers prevent the projectors from projecting below the "horizon".


title: Planetarium

text: The star ball is usually mounted so it can rotate as a whole to simulate the Earth's daily rotation, and to change the simulated latitude on Earth. There is also usually a means of rotating to produce the effect of precession of the equinoxes. Often, one such ball is attached at its south ecliptic pole. In that case, the view cannot go so far south that any of the resulting blank area at the south is projected on the dome. Some star projectors have two balls at opposite ends of the projector like a dumbbell. In that case all stars can be shown and the view can go to either pole or anywhere between. But care must be taken that the projection fields of the two balls match where they meet or overlap.


title: Planetarium

text: Smaller planetarium projectors include a set of fixed stars, Sun, Moon, and planets, and various nebulae. Larger projectors also include comets and a far greater selection of stars. Additional projectors can be added to show twilight around the outside of the screen (complete with city or country scenes) as well as the Milky Way. Others add coordinate lines and constellations, photographic slides, laser displays, and other images.


title: Planetarium

text: Each planet is projected by a sharply focused spotlight that makes a spot of light on the dome. Planet projectors must have gearing to move their positioning and thereby simulate the planets' movements. These can be of these types:-


title: Planetarium

text: Despite offering a good viewer experience, traditional star ball projectors suffer several inherent limitations. From a practical point of view, the low light levels require several minutes for the audience to "dark adapt" its eyesight. "Star ball" projection is limited in education terms by its inability to move beyond an earth-bound view of the night sky. Finally, in most traditional projectors the various overlaid projection systems are incapable of proper occultation. This means that a planet image projected on top of a star field (for example) will still show the stars shining through the planet image, degrading the quality of the viewing experience. For related reasons, some planetariums show stars below the horizon projecting on the walls below the dome or on the floor, or (with a bright star or a planet) shining in the eyes of someone in the audience.


title: Planetarium

text: However, the new breed of Optical-Mechanical projectors using fiber-optic technology to display the stars show a much more realistic view of the sky.


title: Planetarium

text: An increasing number of planetariums are using digital technology to replace the entire system of interlinked projectors traditionally employed around a star ball to address some of their limitations. Digital planetarium manufacturers claim reduced maintenance costs and increased reliability from such systems compared with traditional "star balls" on the grounds that they employ few moving parts and do not generally require synchronisation of movement across the dome between several separate systems. Some planetariums mix both traditional opto-mechanical projection and digital technologies on the same dome.


title: Planetarium

text: In a fully digital planetarium, the dome image is generated by a computer and then projected onto the dome using a variety of technologies including cathode ray tube, LCD, DLP, or laser projectors. Sometimes a single projector mounted near the centre of the dome is employed with a fisheye lens to spread the light over the whole dome surface, while in other configurations several projectors around the horizon of the dome are arranged to blend together seamlessly.


title: Planetarium

text: Digital projection systems all work by creating the image of the night sky as a large array of pixels. Generally speaking, the more pixels a system can display, the better the viewing experience. While the first generation of digital projectors were unable to generate enough pixels to match the image quality of the best traditional "star ball" projectors, high-end systems now offer a resolution that approaches the limit of human visual acuity.


title: Planetarium

text: LCD projectors have fundamental limits on their ability to project true black as well as light, which has tended to limit their use in planetaria. LCOS and modified LCOS projectors have improved on LCD contrast ratios while also eliminating the “screen door” effect of small gaps between LCD pixels. “Dark chip” DLP projectors improve on the standard DLP design and can offer relatively inexpensive solution with bright images, but the black level requires physical baffling of the projectors. As the technology matures and reduces in price, laser projection looks promising for dome projection as it offers bright images, large dynamic range and a very wide color space.


title: Planetarium

text: Worldwide, most planetariums provide shows to the general public. Traditionally, shows for these audiences with themes such as "What's in the sky tonight?", or shows which pick up on topical issues such as a religious festival (often the Christmas star) linked to the night sky, have been popular. Live format is preferred by many venues as a live speaker or presenter can answer questions raised by the audience.


title: Planetarium

text: Since the early 1990s, fully featured 3-D digital planetariums have added an extra degree of freedom to a presenter giving a show because they allow simulation of the view from any point in space, not only the Earth-bound view which we are most familiar with. This new virtual reality capability to travel through the universe provides important educational benefits because it vividly conveys that space has depth, helping audiences to leave behind the ancient misconception that the stars are stuck on the inside of a giant celestial sphere and instead to understand the true layout of the Solar System and beyond. For example, a planetarium can now 'fly' the audience towards one of the familiar constellations such as Orion, revealing that the stars which appear to make up a co-ordinated shape from our earth-bound viewpoint are at vastly different distances from Earth and so not connected, except in human imagination and mythology. For especially visual or spatially aware people, this experience can be more educationally beneficial than other demonstrations.


title: Jorge Soler

text: Jorge Carlos Soler Castillo (born February 25, 1992) is a Cuban professional baseball outfielder for the Miami Marlins of Major League Baseball (MLB). Soler has previously played in MLB for the Chicago Cubs, Kansas City Royals, and Atlanta Braves.


title: Jorge Soler

text: Soler played for the Cuban national baseball team in international competition. He defected from Cuba in 2011, seeking a career in MLB. After establishing residency in Haiti, Soler signed a nine-year contract with the Cubs. He made his MLB debut in 2014 and won the 2016 World Series with the Cubs. The Cubs traded Soler to the Royals after the 2016 season. He led the American League in home runs in 2019. Traded to Atlanta in 2021, Soler won the 2021 World Series and earned the World Series Most Valuable Player Award. Soler signed with the Marlins before the 2022 season.


title: Jorge Soler

text: Soler played for the Cuban national baseball team in the 2010 World Junior Baseball Championship, where he had a .304 batting average, .500 on-base percentage, and .522 slugging percentage. His nine walks were the second most in the tournament. Cuba won the bronze medal. Soler also played briefly with the Industriales in the Cuban National Series.


title: Jorge Soler

text: Soler defected from Cuba in 2011 to pursue his career in Major League Baseball (MLB). He established residency in Haiti. Soler was unblocked by the Office of Foreign Assets Control on June 2, 2012, making him an MLB free agent. As a free agent, many teams were involved in bidding on Soler.


title: Jorge Soler

text: Soler is tall and weighs . He was described as a power-hitting outfielder who would likely play right field. Jim Callis of "Baseball America" described Soler in 2011 as "a 19-year-old athlete with five-tool potential." According to Callis, Soler likely would have been a top-five pick in the 2010 draft had he been eligible. Kevin Goldstein of "Baseball Prospectus" did not rank Soler in his list of the top baseball prospects prior to the 2012 season, but said he would have ranked Soler as the 38th or 39th best prospect if he were eligible. Some teams preferred Soler to higher profile Cuban defector Yoenis Céspedes. Keith Law of ESPN.com indicated that Soler had the talent of a top-five draft choice in the 2012 Major League Baseball Draft, had he been eligible to be drafted. Writing for Fox News, Mauricio Rubio wrote that "Early in his career he was benched for not hustling, and in a separate incident he ran toward an opposing dugout with a bat." Writing for the Sporting News, Jeff Mans noted that: "The biggest issue with Soler aside from the hamstring injuries is his temper... He started out on the wrong foot with the Cubs after failing to report to minor league camp shortly after signing his nine-year, $30 million deal, feeling that he should have been in Chicago immediately. The other scare for the Cubs brass was his relative inability to hit righthanded pitching."


title: Jorge Soler

text: On June 11, 2012, Soler reportedly agreed to a nine-year $30 million contract with the Chicago Cubs. The Cubs had reportedly agreed to a deal with Soler before he was declared a free agent, though Cubs general manager Jed Hoyer denied this. He made his professional debut that same season with the AZL Cubs and was promoted to the Peoria Chiefs in August. In 34 games between the two teams he batted .299 with five home runs and 25 RBIs.


title: Jorge Soler

text: On April 10, 2013, while playing on the Daytona Cubs, immediately following a bench-clearing incident, Soler charged the opposing Clearwater Threshers' dugout while brandishing a baseball bat. He was ejected from the game, was fined, and received a five-game suspension. Soler spent all of 2013 with Daytona, slashing .281/.343/.467 with eight home runs and 35 RBIs in 55 games.


title: Jorge Soler

text: Soler began the 2014 season with the Tennessee Smokies. On July 22, 2014, after batting .415/.494/.862 with six home runs and 22 RBIS in 22 games, Soler was promoted to the Iowa Cubs.


title: Jorge Soler

text: On August 25, 2014, Soler was called up to the Chicago Cubs for the first time. In 32 games for Iowa prior to his call up he was batting .282 with eight home runs and 29 RBIs in 32 games. In his major league debut on August 27, facing Cincinnati Reds pitcher Mat Latos, Soler hit a home run in his first major league at-bat, becoming the 117th player in MLB history to do so. On September 1 Soler had two doubles in his home debut for the Cubs to become just the third major league player in the last 100 years to have at least one extra-base hit in each of his first five games in the majors. Two days later Soler became the second player in Cubs history with as many as 10 RBIs in his first seven games as a major leaguer. Soler was the starting right fielder for the Cubs 2015 season until an ankle injury sidelined him in early June. He returned to the starting lineup on July 5 after spending time in rehab. He finished the regular season with a .268 batting average, 15 home runs and 67 RBIs.


title: Jorge Soler

text: In 2015, Soler's postseason debut, he walked as a pinch-hitter in the ninth inning in Game 1 of the Division Series and followed up with a double, two-run home run to straightaway center and two more walks in Game 2, and another home run, a single and two walks in Game 3. Record setting Soler started his postseason career by reaching base nine times in a row, in which he recorded five walks and hit two home runs, a double and a single. In Game 4 Soler ended a game-tying St. Louis Cardinals sixth inning rally with an outfield assist on a game-saving inning-ending put out of Tony Cruz at home plate. The Cubs won the game 6–4 and beat the rival St. Louis Cardinals in four games to advance to the National League Championship Series.


title: Jorge Soler

text: Soler's playing time with Chicago dipped in 2016, playing in 86 games compared to 101 the previous year. The Cubs were the most dominant team for the entirety of the regular season, entering the postseason as the favorites. Through 13 at-bats in the playoffs, Soler totaled 4 strikeouts, 3 walks, and two hits. Both of his hits came in the World Series. The more notable of the two was a triple in Game 3 off of Bryan Shaw. The Cubs went on to win the 2016 World Series over the Cleveland Indians in seven games.


title: Jorge Soler

text: On December 7, 2016, the Cubs traded Soler to the Kansas City Royals for Wade Davis. After a string of injuries and inconsistency at the plate, Soler was demoted to the Omaha Storm Chasers on June 2. In 74 games for Omaha he batted .267 with 24 home runs and 59 RBIs, and in 35 games for Kansas City, he compiled a .144 batting average with two home runs and six RBIs.


title: Jorge Soler

text: Soler began the 2018 season as Kansas City's starting right fielder. However, after suffering a toe fracture in mid-June, he was sidelined for the remainder of the year. Over 61 games, he hit .265 with nine home runs. Soler returned from the injury in 2019, splitting time between right field and designated hitter. On September 3, 2019, he hit his 39th home run of the season, becoming Kansas City's record holder for most home runs in a single season. The very next night, Soler became the first Royals player in history to record at least 40 home runs in a single season.


title: Jorge Soler

text: In 2019, he batted .265, and led the American League with 48 home runs and 178 strikeouts in 589 at bats. His home run total was the most ever in a single season by a Cuban-born player.


title: Jorge Soler

text: In the 2020 pandemic-shortened season, Soler batted .228 with eight home runs and 24 RBIs in 43 games. At the beginning of the 2021 season with Kansas City, he batted .192/.288/.370 in 308 at bats.


title: Jorge Soler

text: In 2021, he batted .223/.316/.432 with 27 home runs and 70 RBIs in 516 at bats between the Royals and the Braves. With the Braves, he hit .269/.358/.524. His home runs averaged 423 feet, the longest average home run distance of all major league hitters.


title: Jorge Soler

text: In his first start after spending 10 days on the COVID-19 injured list, Soler hit a leadoff home run in Game 1 of the 2021 World Series off of Framber Valdez, becoming the first player to hit a home run in the first plate appearance of a World Series. In Game 4, Soler hit a pinch-hit go-ahead home run in the seventh inning to give the Braves a 3–2 lead, which led them to a 3–1 lead in the World Series. In Game 6, he hit his third home run of the series, a 446-foot, three run shot to give the Braves a 3–0 lead.


title: Andreas Cornelius

text: Andreas Evald Cornelius (born 16 March 1993) is a Danish professional footballer who plays as a striker for Copenhagen and the Denmark national team.


title: Andreas Cornelius

text: Cornelius had joined the FC Copenhagen first team squad for its training camps in July 2011 and January 2012.


title: Andreas Cornelius

text: On 9 April 2012, he played his first Superliga match, coming on as a substitute for César Santin against AGF at NRGi Park in Aarhus. In the previous Superliga match (against Brøndby at Parken Stadium on 5 April 2012), he had been an unused substitute. On 20 May 2012, it was announced that Cornelius would join the first team squad on a permanent basis at the beginning of the upcoming season, together with Christoffer Remmer and Jakob Busk.


title: Andreas Cornelius

text: Cornelius scored his first goal for Copenhagen against Midtjylland in the first game of the new 2012–13 season. He scored the opening goal against Odense in a 2–2 draw. He also scored the winning goal against Molde FK in the Europa League on 20 September.


title: Andreas Cornelius

text: Cornelius was awarded player of the month in the Danish Superliga on 7 October 2012. On 15 April, he scored the final goal in a 3–2 win against FC Nordsjælland. This gave FCK a 10-point lead in the league to number 2, and was crucial for Copenhagen winning the title.


title: Andreas Cornelius

text: Even though Cornelius only played one full season with the first team, he quickly became one of the most popular players of the club, as the print with his name and number on the official shirts being the most popular, selling around 40% of all prints.


title: Andreas Cornelius

text: On 27 June 2013, Cardiff City announced they had signed Cornelius, for a club record fee. Reports from Denmark suggested a fee of 75 million Danish kroner (approximately £8 million). On 1 July, Cornelius completed his medical and signed a five-year deal with the club, saying "I've been given a great opportunity and this is a very exciting project that's taking place at Cardiff City." Twenty-three days later, he made his debut in a friendly against Forest Green Rovers and scored a goal in a 4–3 victory. However, he sustained a minor injury to his ankle during training the following week, and consequently missed out on the team's first ever Premier League game, against West Ham United on 17 August.


title: Andreas Cornelius

text: Cornelius made his competitive debut for Cardiff on 25 August against Manchester City, coming on as a stoppage-time replacement for Fraizer Campbell who had scored twice in the 3–2 home victory. Three days later, in a League Cup match against League Two club Accrington Stanley, he made his first competitive start for the club in a 2–0 win; however he re-aggravated the ankle injury he had picked up in pre-season following a heavy tackle and was carried off on a stretcher just before half-time. The injury would prove worse than first feared; and left Cornelius on the sidelines for the next three months, meaning he missed 10 games for Cardiff, as well as the chance to compete for his country Denmark in their final qualification games for the 2014 FIFA World Cup in Brazil.


title: Andreas Cornelius

text: During the final international break of 2013, Cornelius once again began training with the first team squad, in order to be accessed by the medical staff. They deemed that he was fit to return to action, and therefore arranged a behind closed door friendly with Championship club Yeovil Town on 14 November. He played 60 minutes as planned in a 2–1 defeat and told the media that he was fit and ready to return to action.


title: Andreas Cornelius

text: On 31 January 2014, Cornelius returned to F.C. Copenhagen for an undisclosed fee, but Cardiff claim they lost most of the money they paid for him. He scored a hat-trick in his first game back at Copenhagen in a friendly match against Slovan Liberec. On 23 February 2014, Cornelius made his first competitive start after re-signing with the club and crowned his comeback with a header to put his team 1–0 in front against AGF at NRGi Park. The match ended in a 1-1 draw.


title: Andreas Cornelius

text: Cornelius ended his first half season with 5 goals in 13 appearances, including a brace against Randers FC in Parken where he dribbled past four men before sending the ball into the back of the net. In April 2015, he suffered a "horrific" ankle injury during Copenhagen's match against Silkeborg IF.


title: Andreas Cornelius

text: On 2 May 2017, it was announced that Cornelius would join Serie A side Atalanta for a fee of around €3.5 million.


title: Andreas Cornelius

text: On 31 August 2018, hours before the closing of the 2018 summer transfer window, Cornelius joined Ligue 1 side FC Girondins de Bordeaux on loan for the season. The deal reportedly included an option for Bordeaux to sign him permanently for a fee between €7 and 8 million.


title: Andreas Cornelius

text: On 18 July 2019, Cornelius signed to Serie A club Parma a 2-years loan with an obligation to buy. On 20 October 2019, Cornelius came off the bench in the 11th to substitute the injured Roberto Inglese, and scored a hat trick in a 5–1 home win against Genoa within a span of eight minutes. Cornelius scored a second hat-trick again against Genoa later on in the season in a 4–1 away win.


title: Andreas Cornelius

text: On 8 August 2021, Cornelius joined Süper Lig side Trabzonspor in a permanent deal. On 30 April 2022, Trabzonspor were crowned Turkish league champions for the first time in 38 years, clinching the title with a 2–2 draw against Antalyaspor.


title: Andreas Cornelius

text: On deadline day, 31 August 2022, Cornelius returned to his former club F.C. Copenhagen for the third time, signing a deal until June 2027.


title: Andreas Cornelius

text: In May 2018 he was named in Denmark’s preliminary 35-man squad for the 2018 World Cup in Russia. He played in 3 of Denmark's 4 games, starting in the final group match vs France and the round of 16 match against Croatia. Cornelius did not score any goals in the World Cup.


title: Andreas Cornelius

text: In June 2021, he was included in the national team's bid for 2020 UEFA Euro, where the team reached the semi-finals.


I Like It When You Sleep, for You Are So Beautiful yet So Unaware of It (stylised in sentence case) is the second studio album by English rock band the 1975, released on 26 February 2016 through Dirty Hit and Polydor. In 2014, frontman Matty Healy released a series of cryptic tweets containing lyrics from the album, revealing its title the following year. After their social accounts were deleted and reinstated with a new visual identity, the band officially confirmed the album in September 2015, a month before "Love Me" was released as the lead single. Over the course of five months, "Ugh!", "Somebody Else" and "The Sound" were released as singles, with "A Change of Heart" released four days prior to release. "She's American" and "Loving Someone" were later released in November 2016 and February 2017 as the final singles.


title: I Like It When You Sleep, for You Are So Beautiful yet So Unaware of It

text: Upon its release, the album received positive reviews from critics. Several publications, including "Pitchfork", "Rolling Stone", and "The Guardian", listed it as one of the best albums of 2016. It was also a commercial success, topping the charts in the United Kingdom and the United States, and its box set received a nomination for Best Boxed or Special Limited Edition Package at the 2017 Grammy Awards.


text: "NME" later placed the album sixth on their list of the Best Albums of The Decade. Additionally, "Stereogum", "Pitchfork", and "Billboard" placed it 61st, 161st, and 82nd, respectively.


text: After the band released their self-titled debut, in 2013, they spent 2015 mostly recording a follow-up.


text: On 1 June, the band's Twitter accounts were deleted, which caused mass speculation from both fans and media alike that the band had broken up. On 2 June, Healy reactivated the accounts and tweeted again, but revealed a cryptic and symbolic comic strip containing the message that the band had gone on hiatus. A blurred Instagram picture from Healy titled "The 1975-2" set anticipation for release. The same day, the social media accounts were reinstated.


text: The tweet by Healy was verified as the name for their second album later in October 2015 with "Love Me" (a "very funky new single", according to "Spin") released on 8 October.


text: The band cited D'Angelo, Christina Aguilera, Jimmy Jam, Terry Lewis, Roberta Flack, My Bloody Valentine's "Loveless", Boards of Canada, Kim Carnes, Scritti Politti, and Sigur Rós as inspirations for the record's 17 tracks. The album's sound has been described as pop, new wave, dance-rock, indie rock, pop punk and soul. It also incorporates synth-pop, jazz, post-rock, dance-pop and R&B elements.


text: On 8 October 2015, the 1975 released "Love Me" as a single from the album after announcement and was first played on BBC. On 10 December, they first played "Ugh!", the second single from the album, on Apple Music's Beats 1.


text: "The Sound" was premiered on Radio 1 on 14 January 2016, and was released as a radio single on 19 February. The music video premiered 6 days after. "Somebody Else" debuted on Beats 1 with Zane Lowe on 15 February, and released on iTunes and Spotify on 16 February. The music video debuted on 7 July. The next single, "A Change of Heart" debuted on BBC Radio 1 with Annie Mac on 22 February. A video for "She's American" was filmed but never released. "Loving Someone" was released as the seventh single.


text: A tour for the album began on 9 November 2015 in Liverpool. The band played the United Kingdom in November and the United States in December 2015, Asia and Oceania in January 2016, Europe in March to April 2016, and the US from April to May 2016. They played nine festivals over the summer of 2016, including Firefly Music Festival in June and the Reading and Leeds Festivals in August.


text: On 25 July 2016 the band announced a North American Tour for the Fall, beginning on 1 and 2 October at The Meadows Music & Arts Festival, and then playing three dates in Mexico, the first ever in the country. The band concluded the tour headlining Latitude Festival in Henham Park, United Kingdom on 14 July 2017, stating that it was "the end an of an era, but the start of a new era, called 'Music for Cars'". It is estimated that as of July 2017, the band had done over 150 concerts for this album-cycle.


text: The album's artwork and design was created by Samuel Burgess-Johnson and photographed by David Drake.


text: For each song on the album, a pink neon sign was created and put against various locations to create nostalgia for the song, but to also detail the thematic material and complexity of each song through the photo's atmosphere. Burgess-Johnson worked closely with Healy to help with the placement. The box set version of the album gained a nomination for the Grammy Award for Best Boxed or Special Limited Edition Package.


text: The album was released on 26 February 2016. The US Target edition of the album includes two bonus tracks: a demo of "A Change of Heart" and the song "How to Draw".


text: The record received mostly positive reviews from music critics. At Metacritic, the album has an average score of 75 out of 100, which indicates "generally favorable reviews" based on 24 reviews. Writing for "Exclaim!", Ian Gormely noted that the band's ambition was perhaps their sole stumbling block, though pursuing all musical avenues makes the result "overstuffed, awkwardly titled and frequently brilliant." In "Drowned in Sound"s review of the album, they praised the album's eclecticism and lyricism, concluding, "What they've made is a bold body of work that sounds effortless and odd and sophisticated. What they do next is likely to be stadium-filling and bonkers and brilliant, but it matters little when what they're doing now is so sensational."


text: "NME", who had previously been highly critical of the band, also praised the album for its scope and ambition, writing, "Any record that burrows as deep into your psyche as 'I Like It...' should be considered essential. It's hugely clever and wryly funny, too." Although music journalist Alexis Petridis noted that parts of the album were over-ambitious, he went on to claim that "[i]ncredibly, though, most of the time Healy gets away with it. That's sometimes because his observations are sharp – as a skewering of celebrity #squad culture, "you look famous, let's be friends / And portray we possess something important / And do the things we like" is pretty acute – but more usually because they come loaded with witty self-awareness and deprecation: the endless depictions of vacuous, coke-numbed girls he has met would get wearying were it not for the fact that he keeps turning the lyrical crosshair on himself." In a more mixed review, "Rolling Stone" criticised tracks like 'Lostmyhead' and 'Please Be Naked' for being 'boring-melty' but praised songs such as 'Somebody Else', 'Loving Someone' and 'Love Me'.


text: The album became the group's second number one in the United Kingdom, debuting atop the UK Albums Chart, with combined sales of over 58,000. It became the group's first number one on the US "Billboard" 200, with 98,000 pure album sales in its debut week and 108,000 sps, while also setting the record for longest title of a "Billboard" number-one album with 71 characters. The next week, it fell to number 26, tying with Amos Lee's 2011 album "Mission Bell" for the fourth largest drop from number 1 as of January 2017. It also debuted at number one in Australia, New Zealand and Canada. The album was certified platinum by the British Phonographic Industry (BPI) for sales of over 300,000 copies in the United Kingdom. On 10 April 2017 the album was certified gold by the Recording Industry Association of America (RIAA) for combined sales and album-equivalent units of over 500,000 units.


title: Glen Affric

text: Glen Affric () is a glen south-west of the village of Cannich in the Highland region of Scotland, some west of Loch Ness. The River Affric runs along its length, passing through Loch Affric and Loch Beinn a' Mheadhoin. A minor public road reaches as far as the end of Loch Beinn a' Mheadhoin, but beyond that point only rough tracks and footpaths continue along the glen.


title: Glen Affric

text: Often described as the most beautiful glen in Scotland, Glen Affric contains the third largest area of ancient Caledonian pinewoods in Scotland, as well as lochs, moorland and mountains.


title: Glen Affric

text: The area is a Caledonian Forest Reserve, a national scenic area and a national nature reserve, as well as holding several other conservation designations.


title: Glen Affric

text: The forests and open landscapes of the glen, and the mountains on either side, are a popular destination for hikers, climbers and mountain bikers.


title: Glen Affric

text: Glen Affric is listed in the Caledonian Pinewood Inventory, and contains the third largest area of ancient Caledonian pinewoods in Scotland. Due to the importance of this woodland it has been classified as a national nature reserve since 2002, and holds several other conservation designations. The pinewood consists predominantly of Scots pine, but also includes broadleaved species such as birch, rowan, aspen, willows and alder. The forest floor hosts many plant species typically found in Scotland's pinewoods, including creeping ladies tresses, lesser twayblade, twinflower, and four species of wintergreen. Many nationally rare or scarce species of lichens grow on the trees of Glen Affric.


title: Glen Affric

text: Scots pine trees first colonised the area after the last Ice Age, 10,000 to 8,000 years ago. Currently the oldest trees in the area are the gnarled "granny" pines that are the survivors of generations of felling. Although felling ceased many years ago, regrowth was hampered by unnaturally high populations of sheep and deer, and in the early 1950s the Forestry Commission found that very few of the remaining pines were less than 100 years old. Initially the Commission, which was tasked with increasing the total amount of tree cover without reference to the species used, reforested the area with non-native species such as Sitka spruce and lodgepole pine, as well as Scots pine from local seed stocks. Since the 1980s management priorities have changed, and non-native conifers have been felled and removed from the glen, alongside the removal of other non-native trees such as rhododendron. Some commercial forestry continues in order to maintain forest cover and to benefit local people financially, and considerable areas of non-native conifers are expected to remain in the forest over the next few decades.


title: Glen Affric

text: Forestry and Land Scotland (successor body to the Forestry Commission) aims to encourage regrowth of the pinewood by reducing deer numbers, thus minimising the use of fencing, which can injure black grouse and capercaillie which collide with the wires. The long-term aim is to provide a network of forest habitats, with corridors of new forest linking existing woodland, interspersed with open areas. Management of the reserve also seeks to establish a 'treeline transition zone', in which there is a more gradual transition between woodland and mountain heath with an intermediate zone of shorter, more twisted trees and low-growing shrubs. At the western end of the glen the National Trust for Scotland aims to encourage the growth of other tree species such as birch and rowan to complement the pinewood lower down the glen. The charity Trees for Life have also planted extensively in Glen Affric, in areas once grazed by deer. They own a bothy at Athnamulloch, by the edge of Loch Affric, that is used to house volunteers.


title: Glen Affric

text: After nearly seventy years of management to encourage restoration of the area, biodiversity has improved and Glen Affric now supports birds such as black grouse, capercaillie, crested tit and Scottish crossbill, as well as raptor species such as ospreys and golden eagles. Glen Affric is also home to Scottish wildcats and otters. The bogs and lochs of the glen provide a habitat for many species of dragonfly, including the rare brilliant emerald.


title: Glen Affric

text: In 2019 an elm tree in Glen Affric, christened the "Last Ent of Affric" was named Scotland's Tree of the Year by the Woodland Trust.


title: Glen Affric

text: Glen Affric, also written Glenaffric, was part of the lands of the Clan Chisholm and the Clan Fraser of Lovat from the 15th to the mid 19th centuries. By the early 15th century, Lord Lovat had passed the lands to his son Thomas who in turn passed it on to his son, William, who was recorded in "Burke's Landed Gentry Scotland" as William Fraser, first Laird of Guisachan. In 1579, Thomas Chisholm, Laird of Strathglass, was imprisoned for being a Catholic. By the 18th century, the title deeds of Glen Affric had been a source of feuding, with the Battle of Glen Affric taking place in 1721.


title: Glen Affric

text: There exists in the Scottish Register of Tartans a "Glenaffric Fragment" that possibly dates from the late 17th century.


title: Glen Affric

text: Dudley Marjoribanks, later Lord Tweedmouth was a rich Liberal MP who took a long lease on shooting rights over much of Glen Affric in 1846 and, by 1856, had acquired ownership of the Glen Affric Estate from "Laird Fraser" whose family had built the original Guisachan Georgian manor house around 1755. The estate held over at the time of its transference from the Clan Chisholm to Lord Tweedmouth. By the 1860s, Lord Tweedmouth, as the new laird, had much enlarged the house, using Scottish architect Alexander Reid who designed many buildings on Tweedmouth's vast Glen Affric Estate, including an entire village—Tomich—and the Glen Affric Hunting Lodge, described in appearance as "castle-like". Tweedmouth had enjoyed shooting rights over much of Glen Affric since 1846, and, following his acquisition of the estate he initiated the first breed of golden retrievers at kennels near Guisachan House. He put the retrievers to good use at the shooting parties he hosted when at Glen Affric Lodge. The retrievers were sent to other estates when, for some months of the years 1870–71, he leased the Glen Affric Estate to Lord Grosvenor.


title: Glen Affric

text: In 1894 Edward Marjoribanks, 2nd Baron Tweedmouth had inherited the Glenaffric and Guisachan estates from his father. His wife, the Baroness Tweedmouth, was born Lady Fanny Spencer-Churchill, the daughter of the 7th Duke of Marlborough, and died at Glen Affric Lodge in 1904. Known in the highlands as the Lady of Glenaffric and Guisachan, she was reported to be a "lover of the golden retriever dog".


title: Glen Affric

text: The Duke and Duchess of York are reported in "The Graphic", 25 September 1897 to have visited the Guisachan Estate in Strathglass, including Glen Affric Lodge. Lady Tweedmouth's nephew Winston Churchill also came to visit the estate in 1901, and amused himself learning how to drive a car in the grounds.


title: Glen Affric

text: Clan Marjoribanks' ownership ended with Edward’s son, Dudley Churchill Marjoribanks, who became 3rd Lord Tweedmouth in 1909. He and his wife had two daughters, but no male heir. For the next few years, until 1918, the estate was owned by the family of Newton Wallop, 6th Earl of Portsmouth (1856–1917). Marmaduke Furness, 1st Viscount Furness owned the estate throughout the 1920s and 30s. The entire property, then consisting of , had been sold by 1936 to a Mr Hunter. It was he who resold the Glen Affric deer forest to the west and a large area of grazing land to the Forestry Commission.


title: Glen Affric

text: Lady Islington acquired the Guisachan portion of the estate in 1939 but let the property go to ruin. In 1962 the Guisachan estate was bought by a descendant of the Frasers of Gortuleg. In 1990, this later generation laird wrote a booklet concerning his Fraser ancestors who had once owned Guisachan—"Guisachan, A History by Donald Fraser".


title: Glen Affric

text: Provost Robert Wotherspoon was recorded as owning Glen Affric Estate in 1951, having purchased it in 1944 and selling the "majority of its ground to the Forestry Commission" in 1948. His son, Iain Wotherspoon was listed as living at Glen Affric Lodge in 1958.


title: Glen Affric

text: Most of the glen was bought by the Forestry Commission in 1951. Although the House of Commons recorded that the Commission was considering returning at least some of its Glen Affric landholdings to private ownership in the early 1980s, the majority of the glen continues to form part of Scotland's National Forest Estate. The Forestry


title: Glen Affric

text: Commission's successor body, Forestry and Land Scotland (FLS), is the largest single landowner in Glen Affric, holding of the lower and central parts of the glen.


title: Glen Affric

text: The National Trust for Scotland has owned the West Affric Estate, which covers the upper part of the glen, since 1993.


title: Glen Affric

text: As of 2019 the main private landowner is the North Affric Estate with of land on the north side of Loch Affric centred on the baronial Affric Lodge. Since 2008 this land has been held by David Matthews, whose eldest son James Matthews is married to Pippa Middleton, sister to Catherine, Princess of Wales.


title: Glen Affric

text: The Guisachan area of Glen Affric, which lies to the south of the main glen, is also in private hands, now forming three separate estates. Wester Guisachan Estate covers of land to the south of Loch Affric, whilst the Hilton & Guisachan Estates, owned by Alexander Grigg, lies further east and covers . The final portion of the Guisachan Estate, which is in the ownership of Nigel Fraser, consists of at the very east of the glen. In this area lies the Conservation village of Tomich.


title: Glen Affric

text: As with all land in Scotland, there is a right of responsible access to most of the land in the glen for pursuits such as walking, cycling, horse-riding and wild camping. These rights apply regardless of whether the land is in public or private ownership, provided access is exercised in accordance with the Scottish Outdoor Access Code.


title: Glen Affric

text: Glen Affric is popular with hillwalkers, as it provides access to many Munros and Corbetts. The north side of the glen forms a ridge with eight Munro summits, including the highest peak north of the Great Glen, Càrn Eige . The three Munros at the western end of this ridge, Sgùrr nan Ceathreamhnan , Mullach na Dheireagain and An Socach , are amongst the remotest hills in Scotland, and are often climbed from the Scottish Youth Hostels Association hostel at Alltbeithe. The hostel is only open in the summer, and can only be reached by foot or by mountain bike via routes of between starting from lower down Glen Affric or from the A87 at Loch Cluanie or Morvich. The dormitories are unheated and hostellers are required to bring a sleeping bag, and to carry out all rubbish. east of the hostel is the Strawberry Cottage mountaineering hut, maintained by the An Teallach Mountaineering Club, described by The Scotsman as "one of the best-equipped huts in the country". Glen Affric is also the starting point for routes to the summits of Munros to the south and west of the glen, although these can also be accessed from the Kintail area. Corbetts accessible from Glen Affric include Sgùrr Gaorsaic, Càrn a' Choire Ghairbh and Aonach Shasuinn.


title: Glen Affric

text: The Affric Kintail Way is a long route from Drumnadrochit on the shore of Loch Ness to Morvich in Kintail via Glen Urquhart and Glen Affric. The route is suitable for both walkers and mountain bikers, and can usually be walked in four days.


title: Glen Affric

text: Shorter waymarked trails are provided in the lower parts of the glen, taking walkers to viewpoints and attractions such as the waterfalls at Plodda and the Dog Falls.


title: Glen Affric

text: The glen is part of the Affric/Beauly hydroelectric scheme, constructed by the North of Scotland Hydro-Electric Board. Loch Mullardoch, in the neighbouring Glen Cannich, is dammed, and a tunnel carries water to Loch Beinn a' Mheadhoin, which has also been dammed. From there, another tunnel takes water to Fasnakyle power station, near Cannich. As the rivers in this scheme are important for Atlantic salmon, flow in the rivers is kept above agreed levels. The dam at Loch Beinn a' Mheadhoin has a Borland fish ladder to allow salmon to pass.


title: Glen Affric

text: In addition to being a national nature reserve, Glen Affric is a Caledonian Forest Reserve, a National Scenic Area, and a Site of Special Scientific Interest (SSSI). The NNR is classified as a Category II protected area by the International Union for Conservation of Nature. Much of the area forms part of a European Union Special Protection Area for golden eagles, and is also classified as an EU Special Area of Conservation.


title: Glen Affric

text: Glen Affric was proposed for inclusion in a national park by the Ramsay committee, set up following the Second World War to consider the issue of national parks in Scotland, and in 2013 the Scottish Campaign for National Parks listed the area as one of seven deemed suitable for national park status, however as of 2019 no national park designation has occurred. In September 2016 Roseanna Cunningham, the Cabinet Secretary for Environment, Climate Change and Land Reform, told the Scottish Parliament that the Scottish Government had no plans to designate new national parks in Scotland and instead planned to focus on the two existing national parks.


title: Sawtooth wave

text: The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle. A single sawtooth, or an intermittently triggered sawtooth, is called a ramp waveform.


title: Sawtooth wave

text: The convention is that a sawtooth wave ramps upward and then sharply drops. In a reverse (or inverse) sawtooth wave, the wave ramps downward and then sharply rises. It can also be considered the extreme case of an asymmetric triangle wave.


title: Sawtooth wave

text: While a square wave is constructed from only odd harmonics, a sawtooth wave's sound is harsh and clear and its spectrum contains both even and odd harmonics of the fundamental frequency. Because it contains all the integer harmonics, it is one of the best waveforms to use for subtractive synthesis of musical sounds, particularly bowed string instruments like violins and cellos, since the slip-stick behavior of the bow drives the strings with a sawtooth-like motion.


title: Sawtooth wave

text: A sawtooth can be constructed using additive synthesis. For period "p" and amplitude "a", the following infinite Fourier series converge to a sawtooth and a reverse (inverse) sawtooth wave:


title: Sawtooth wave

text: In digital synthesis, these series are only summed over "k" such that the highest harmonic, "N", is less than the Nyquist frequency (half the sampling frequency). This summation can generally be more efficiently calculated with a fast Fourier transform. If the waveform is digitally created directly in the time domain using a non-bandlimited form, such as "y" = "x" − floor("x"), infinite harmonics are sampled and the resulting tone contains aliasing distortion.


title: Sawtooth wave

text: An audio demonstration of a sawtooth played at 440 Hz (A) and 880 Hz (A) and 1,760 Hz (A) is available below. Both bandlimited (non-aliased) and aliased tones are presented.


title: Creation and annihilation operators

text: Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually denoted formula_1) lowers the number of particles in a given state by one. A creation operator (usually denoted formula_2) increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization. They were introduced by Paul Dirac.


title: Creation and annihilation operators

text: Creation and annihilation operators can act on states of various types of particles. For example, in quantum chemistry and many-body theory the creation and annihilation operators often act on electron states. They can also refer specifically to the ladder operators for the quantum harmonic oscillator. In the latter case, the raising operator is interpreted as a creation operator, adding a quantum of energy to the oscillator system (similarly for the lowering operator). They can be used to represent phonons. Constructing Hamiltonians using these operators has the advantage that the theory automatically satisfies the cluster decomposition theorem.


title: Creation and annihilation operators

text: The mathematics for the creation and annihilation operators for bosons is the same as for the ladder operators of the quantum harmonic oscillator. For example, the commutator of the creation and annihilation operators that are associated with the same boson state equals one, while all other commutators vanish. However, for fermions the mathematics is different, involving anticommutators instead of commutators.


title: Creation and annihilation operators

text: In the context of the quantum harmonic oscillator, one reinterprets the ladder operators as creation and annihilation operators, adding or subtracting fixed quanta of energy to the oscillator system.


title: Creation and annihilation operators

text: Creation/annihilation operators are different for bosons (integer spin) and fermions (half-integer spin). This is because their wavefunctions have different symmetry properties.


title: Creation and annihilation operators

text: Start with the Schrödinger equation for the one-dimensional time independent quantum harmonic oscillator,


title: Creation and annihilation operators

text: Note that the quantity formula_6 is the same energy as that found for light quanta and that the parenthesis in the Hamiltonian can be written as


title: Creation and annihilation operators

text: The last two terms can be simplified by considering their effect on an arbitrary differentiable function formula_8


title: Creation and annihilation operators

text: coinciding with the usual canonical commutation relation formula_11, in position space representation: formula_12.


title: Creation and annihilation operators

text: and the Schrödinger equation for the oscillator becomes, with substitution of the above and rearrangement of the factor of 1/2,


title: Creation and annihilation operators

text: as the "annihilation operator" or the "lowering operator", the Schrödinger equation for the oscillator reduces to


title: Creation and annihilation operators

text: This is significantly simpler than the original form. Further simplifications of this equation enable one to derive all the properties listed above thus far.


title: Creation and annihilation operators

text: The operators formula_24 and formula_25 may be contrasted to normal operators, which commute with their adjoints.


title: Creation and annihilation operators

text: One may compute the commutation relations between the formula_24 and formula_25 operators and the Hamiltonian:


title: Creation and annihilation operators

text: These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator as follows.


title: Creation and annihilation operators

text: Assuming that formula_31 is an eigenstate of the Hamiltonian formula_32. Using these commutation relations, it follows that


title: Creation and annihilation operators

text: This shows that formula_35 and formula_36 are also eigenstates of the Hamiltonian, with eigenvalues formula_37 and formula_38 respectively. This identifies the operators formula_39 and formula_40 as "lowering" and "raising" operators between adjacent eigenstates. The energy difference between adjacent eigenstates is formula_41.


title: Creation and annihilation operators

text: The ground state can be found by assuming that the lowering operator possesses a nontrivial kernel: formula_42 with formula_43. Applying the Hamiltonian to the ground state,


title: Creation and annihilation operators

text: This gives the ground state energy formula_46, which allows one to identify the energy eigenvalue of any eigenstate formula_31 as


title: Creation and annihilation operators

text: Furthermore, it turns out that the first-mentioned operator in (*), the number operator formula_49 plays the most important role in applications, while the second one, formula_50 can simply be replaced by formula_51.


title: Creation and annihilation operators

text: The ground state formula_55 of the quantum harmonic oscillator can be found by imposing the condition that


title: Creation and annihilation operators

text: The normalization constant is found to be formula_59 from formula_60,  using the Gaussian integral. Explicit formulas for all the eigenfunctions can now be found by repeated application of formula_61 to formula_62.


title: Creation and annihilation operators

text: The matrix expression of the creation and annihilation operators of the quantum harmonic oscillator with respect to the above orthonormal basis is


title: Creation and annihilation operators

text: These can be obtained via the relationships formula_65 and formula_66. The eigenvectors formula_67 are those of the quantum harmonic oscillator, and are sometimes called the "number basis".


title: Creation and annihilation operators

text: The operators derived above are actually a specific instance of a more generalized notion of creation and annihilation operators. The more abstract form of the operators are constructed as follows. Let formula_68 be a one-particle Hilbert space (that is, any Hilbert space, viewed as representing the state of a single particle).


title: Creation and annihilation operators

text: The (bosonic) CCR algebra over formula_68 is the algebra-with-conjugation-operator (called "*") abstractly generated by elements formula_70, where formula_71ranges freely over formula_68, subject to the relations


title: Creation and annihilation operators

text: The map formula_75 from formula_68 to the bosonic CCR algebra is required to be complex antilinear (this adds more relations). Its adjoint is formula_77, and the map formula_78 is complex linear in . Thus formula_68 embeds as a complex vector subspace of its own CCR algebra. In a representation of this algebra, the element formula_70 will be realized as an annihilation operator, and formula_77 as a creation operator.


title: Creation and annihilation operators

text: In general, the CCR algebra is infinite dimensional. If we take a Banach space completion, it becomes a C*-algebra. The CCR algebra over formula_68 is closely related to, but not identical to, a Weyl algebra.


title: Creation and annihilation operators

text: For fermions, the (fermionic) CAR algebra over "formula_68" is constructed similarly, but using anticommutator relations instead, namely


title: Creation and annihilation operators

text: The CAR algebra is finite dimensional only if formula_68 is finite dimensional. If we take a Banach space completion (only necessary in the infinite dimensional case), it becomes a formula_87 algebra. The CAR algebra is closely related to, but not identical to, a Clifford algebra.


title: Creation and annihilation operators

text: Physically speaking, formula_70 removes (i.e. annihilates) a particle in the state formula_89 whereas formula_77 creates a particle in the state formula_89.


title: Creation and annihilation operators

text: If formula_89 is normalized so that formula_94, then formula_95 gives the number of particles in the state formula_89.


title: Creation and annihilation operators

text: The annihilation and creation operator description has also been useful to analyze classical reaction diffusion equations, such as the situation when a gas of molecules "formula_97" diffuse and interact on contact, forming an inert product: "formula_98". To see how this kind of reaction can be described by the annihilation and creation operator formalism, consider formula_99 particles at a site on a one dimensional lattice. Each particle moves to the right or left with a certain probability, and each pair of particles at the same site annihilates each other with a certain other probability.


title: Creation and annihilation operators

text: The probability that one particle leaves the site during the short time period ' is proportional to formula_100, let us say a probability formula_101 to hop left and formula_102 to hop right. All formula_103 particles will stay put with a probability formula_104. (Since ' is so short, the probability that two or more will leave during is very small and will be ignored.)


title: Creation and annihilation operators

text: formula_105. It represents the juxtaposition (or conjunction, or tensor product) of the number states formula_106 formula_107, formula_108 located at the individual sites of the lattice. Recall that


title: Creation and annihilation operators

text: This definition of the operators will now be changed to accommodate the "non-quantum" nature of this problem and we shall use the following definition:


title: Creation and annihilation operators

text: note that even though the behavior of the operators on the kets has been modified, these operators still obey the commutation relation


title: Creation and annihilation operators

text: Now define "formula_114" so that it applies "formula_115" to formula_116. Correspondingly, define formula_117 as applying formula_61 to formula_116. Thus, for example, the net effect of formula_120 is to move a particle from the "formula_121" to the "formula_122"site while multiplying with the appropriate factor.


title: Creation and annihilation operators

text: The reaction term can be deduced by noting that formula_125 particles can interact in formula_126 different ways, so that the probability that a pair annihilates is formula_127, yielding a term


title: Creation and annihilation operators

text: This kind of notation allows the use of quantum field theoretic techniques to be used in the analysis of reaction diffusion systems.


title: Creation and annihilation operators

text: In quantum field theories and many-body problems one works with creation and annihilation operators of quantum states, formula_131 and formula_132. These operators change the eigenvalues of the number operator,


title: Creation and annihilation operators

text: by one, in analogy to the harmonic oscillator. The indices (such as formula_124) represent quantum numbers that label the single-particle states of the system; hence, they are not necessarily single numbers. For example, a tuple of quantum numbers formula_135 is used to label states in the hydrogen atom.


title: Creation and annihilation operators

text: Therefore, exchanging disjoint (i.e. formula_143) operators in a product of creation or annihilation operators will reverse the sign in fermion systems, but not in boson systems.


title: Creation and annihilation operators

text: If the states labelled by "i" are an orthonormal basis of a Hilbert space "H", then the result of this construction coincides with the CCR algebra and CAR algebra construction in the previous section but one. If they represent "eigenvectors" corresponding to the continuous spectrum of some operator, as for unbound particles in QFT, then the interpretation is more subtle.


title: Creation and annihilation operators

text: While Zee obtains the momentum space normalization formula_144 via the symmetric convention for Fourier transforms, Tong and Peskin & Schroeder use the common asymmetric convention to obtain formula_145. Each derives formula_146.


title: Creation and annihilation operators

text: Srednicki additionally merges the Lorentz-invariant measure into his asymmetric Fourier measure, formula_147, yielding formula_148.