title: Topper Headon

text: In a February 2008 newspaper article Headon revealed that in 2003 he started to experience serious back pain, a frequent complaint of ageing rock drummers. Diagnosed with hyperkyphosis, a forward curvature of the back, he underwent intense posture adjustment treatment and continues to exercise daily. He notes that, on his recent appearance with Jones, he exhibited his new upright stance.


title: Topper Headon

text: At some point in the 1980s, Headon contracted Hepatitis C, which, along with his alcohol intake, led to severe liver problems. Headon successfully underwent interferon treatment for his hepatitis in 2007 and became a spokesman for the Hepatitis C Trust.


title: Topper Headon

text: The BBC featured Headon in a February 2009 feature on drumming as therapy. He shares some of his story in a brief video interview. In 2012 Headon was interviewed by fellow drummer Spike Webb, sharing stories from his years drumming for The Clash and his experience writing 'Rock The Casbah'.


title: Topper Headon

text: Actor Alex Gold portrayed Headon in the 2016 film "London Town", which tells the story of a Clash-obsessed teenager who crosses paths with Joe Strummer by happenstance in 1979 and finds his life changing as a result. Reviews were mostly negative.


title: Topper Headon

text: As a drummer, Headon often employed a distinctive style which emphasised a simple bass-snare up-down beat, accentuated with closed hi-hat flourishes. Such a method can be found in the songs "Janie Jones (song)","Clash City Rockers", "Clampdown", "Train in Vain", and "Lost in the Supermarket". His drumming on "Train in Vain" has been characterised as one of the most important and distinctive beats in rock music. Scott Kenemore of "PopMatters" writes, "his contribution to the music was tremendous, and his drumming remains an undiscovered treasure for too many."


title: Topper Headon

text: Topper Headon has released one studio album, one EP, and three singles as a solo artist and featured on several other artists' albums.


title: Renée Richards

text: Renée Richards (born August 19, 1934) is an American ophthalmologist and former tennis player who competed on the professional circuit in the 1970s, and became widely known following male-to-female sex reassignment surgery, when she fought to compete as a woman in the 1976 US Open.


title: Renée Richards

text: The United States Tennis Association began that year requiring genetic screening for female players. She challenged that policy, and the New York Supreme Court ruled in her favor, a landmark case in transgender rights. As one of the first professional athletes to identify as transgender, she became a spokesperson for transgender people in sports. After retiring as a player, she coached Martina Navratilova to two Wimbledon titles.


title: Renée Richards

text: Richards was born Richard Raskind on August 19, 1934, in New York City and raised, as she put it, as "a nice Jewish boy" in Forest Hills, Queens. Her father David Raskind was an orthopedic surgeon, and her mother was one of the first female psychiatrists in the United States, in addition to being a professor at Columbia University.


title: Renée Richards

text: Richards attended Horace Mann School and excelled as the wide receiver for the football team, the pitcher for the baseball team, and on the tennis and swim teams. Her baseball skills even led to an invitation to join the New York Yankees, but she decided to focus on tennis. After high school Richards attended Yale University and was captain of the men's tennis team, and was considered by some to be one of the best college tennis players in the country. After graduating from Yale, she went to the University of Rochester Medical Center and specialized in ophthalmology, graduating in 1959 and serving a two-year internship at Lenox Hill Hospital in New York. After an internship, she served two years of residency at the Manhattan Eye, Ear and Throat Hospital in New York. She played competitive tennis for a while and was ranked sixth out of the top 20 males over 35. After an internship and residency, she joined the United States Navy to continue medical training and played tennis in the Navy. While serving in the Navy, she won both the singles and doubles at the All Navy Championship, with a very effective left-hand serve. During this time she was ranked as high as fourth in the region.


title: Renée Richards

text: During college Richards began dressing as a woman, which at the time was considered to be a perversion, with transsexualism classified as a form of insanity. Richards named her female persona "Renée", which is French for "reborn". Her struggle with gender identity created sexual confusion, depression, and suicidal tendencies. She began seeing Dr. Charles Ihlenfeld, a disciple of Harry Benjamin who specialized in endocrinology, transsexualism, and sexual reassignment. Upon seeing Ihlenfeld she began getting hormone injections with the long-term hope for a life change. In the mid-1960s she traveled in Europe dressed as a woman, intending to go to North Africa to see Georges Burou, a famous gynecological surgeon at Clinique Parc in Casablanca, Morocco, regarding sex reassignment surgery; however, she ultimately decided against it and returned to New York. Richards married model Barbara Mole in June 1970, and together they had a son Nicholas in 1972. They were divorced in 1975.


title: Renée Richards

text: In the early 1970s, Richards resolved to undergo sex reassignment and was referred to surgeon Roberto C. Granato Sr. by Harry Benjamin, successfully transitioning in 1975. After surgery, Richards went to Newport Beach, California, and started working as an ophthalmologist in practice with another doctor.


title: Renée Richards

text: In 1976, Richards's gender reassignment was outed by local TV anchor Richard Carlson, the father of Tucker Carlson. Subsequently the United States Tennis Association (USTA), the Women's Tennis Association (WTA), and the United States Open Committee (USOC) required all female competitors to verify their sex with a Barr body test of their chromosomes. Richards applied to play in the US Open in 1976 as a woman, but refused to take the test, and thus was not allowed to compete in the Open, Wimbledon, or the Italian Open in the summer of 1976.


title: Renée Richards

text: Richards then sued the United States Tennis Association (USTA), which runs the US Open, in New York state court, alleging discrimination by gender in violation of the New York Human Rights Law. She asserted that participating in the tournament would constitute "an acceptance of her right to be a woman." Some USTA members felt that others would undergo sex change to enter women's tennis. "Sports Illustrated" called Richards an "extraordinary spectacle", and characterized reactions to her as "varying from astonishment to suspicion, sympathy, resentment, and more often than not, utter confusion." The USOC stated "there is competitive advantage for a male who has undergone a sex change surgery as a result of physical training and development as a male." Richards finally agreed to take the Barr body test. The test results were ambiguous. She refused to take it again and was barred from play.


title: Renée Richards

text: On August 16, 1977, Judge Alfred M. Ascione found in Richards's favor. He ruled: "This person is now a female" and that requiring Richards to pass the Barr body test was "grossly unfair, discriminatory and inequitable, and a violation of her rights." He further ruled that the USTA intentionally discriminated against Richards, and granted Richards an injunction against the USTA and the USOC, allowing her to play in the US Open. Richards lost to Virginia Wade in the first round of the singles competition, but made it to the finals in doubles.


title: Renée Richards

text: After moving to California, Richards played in regional competitions for her local club, the John Wayne Tennis Club, under the name Renée Clark. In the summer of 1976 she entered the La Jolla Tennis Tournament Championships, where she crushed the competition, and her unique left hand serve was recognized by Bob Perry, a tour player from UCLA. Her long-time friend Gene Scott then invited her to play in his professional tennis tournament, the Tennis Week Open in South Orange, New Jersey. The USTA and the WTA then withdrew their sanction for the Tennis Week Open, and organized another tournament; 25 of the 32 participants withdrew from the Tennis Week Open. This was just the beginning of the issues Richards would encounter in trying to play professional women's tennis, which eventually led to her suing the USTA and winning.


title: Renée Richards

text: Richards played professionally from 1977 to 1981 when she retired at age 47. She was ranked as high as 20th overall (in February 1979), and her highest ranking at the end of a year was 22nd (in 1977). Her first professional event as a female was the 1977 U.S. Open. Her greatest successes on court were reaching the doubles final at her first U.S. Open in 1977, with Betty Ann Grubb Stuart – the pair lost a close match to Martina Navratilova and Betty Stöve – and winning the 35-and-over women's singles. Richards was twice a semifinalist in mixed doubles, with Ilie Năstase, at the U.S. Open. In 1979, she defeated Nancy Richey for the 35-and-over singles title at the Open. Richards posted wins over Hana Mandlíková, Sylvia Hanika, Virginia Ruzici, and Pam Shriver. She later coached Navratilova to two Wimbledon wins.


title: Renée Richards

text: Richards was inducted into the USTA Eastern Tennis Hall of Fame in 2000. On August 2, 2013, Richards was among the first class of inductees into the National Gay and Lesbian Sports Hall of Fame.


title: Renée Richards

text: Richards has since expressed ambivalence about her legacy, and came to believe her past as a man provided her with advantages over her competitors, saying "Having lived for the past 30 years, I know if I'd had surgery at the age of 22, and then at 24 went on the tour, no genetic woman in the world would have been able to come close to me. And so I've reconsidered my opinion."


title: Renée Richards

text: After four years of playing tennis, she decided to return to her medical practice, which she moved to Park Avenue in New York. She then became the surgeon director of ophthalmology and head of the eye-muscle clinic at the Manhattan Eye, Ear and Throat Hospital. In addition she served on the editorial board of the "Journal of Pediatric Ophthalmology and Strabismus". She now lives in a small town north of New York City with her platonic companion Arleen Larzelere.


title: Renée Richards

text: In 2014 a wooden racket used by her was donated to the National Museum of American History, which is part of the Smithsonian.


title: Renée Richards

text: In 1983, Richards published an autobiography, "Second Serve", and in 2007, a second, "No Way Renée: The Second Half of My Notorious Life", in which she expresses regret over the type of fame that came with her transgenderism, although she said in 2007 that she did not regret undergoing the sex reassignment process in itself. In 2021, Richards continued her autobiography with "Diary 1999: An Eye-Opening Medical Memoir". Richards's first autobiography served as the basis for the film "Second Serve". "Renée" is a 2011 documentary film about Richards directed by Eric Drath. The film was one of the anchor films of the 2011 Tribeca Film Festival and the documentary premiered on ESPN on October 4, 2011.


title: Renée Richards

text: In 2017, Richards was interviewed by Katie Couric near the end of the documentary "Gender Revolution".


title: Julian Schnabel

text: Julian Schnabel (born October 26, 1951) is an American painter and filmmaker. In the 1980s, he received international attention for his "plate paintings" — with broken ceramic plates set onto large-scale paintings. Since the 1990s, he has been a proponent of independent arthouse cinema. Schnabel directed "Before Night Falls", which became Javier Bardem's breakthrough Academy Award-nominated role, and "The Diving Bell and the Butterfly", which was nominated for four Academy Awards. For the latter, he won the Cannes Film Festival Award for Best Director and the Golden Globe Award for Best Director, as well as receiving nominations for the Academy Award for Best Director and the César Award for Best Director.


title: Julian Schnabel

text: Schnabel was born in Brooklyn, New York to a Jewish family, the son of Esta (née Greenberg) and Jack Schnabel. He moved with his family to Brownsville, Texas in 1965. He received his B.F.A. at the University of Houston. After graduating, he sent an application to the Independent Study Program (ISP) at the Whitney Museum of American Art in New York. His application included slides of his work sandwiched between two pieces of bread. He was admitted into the program and studied there from 1973–1975.


title: Julian Schnabel

text: Schnabel returned to Houston in 1975 and rented a studio in the Heights neighborhood. After reported, repeated badgering by the artist, Jim Harithas, director of the Contemporary Arts Museum Houston, agreed to give him a show. The eponymously titled exhibit ran from February 20 to March 7, 1976 in the parallelogram building's lower gallery.


title: Julian Schnabel

text: On seeing the show, "ARTnews" critic Charlotte Moser wrote, “Though still formative, Schnabel’s paintings possess a palpable presence,” but found the work “clearly influenced by post-minimalist artists whose intellectual ideas he might share but whose technical expertise and clarity of vision he has yet to acquire.”


title: Julian Schnabel

text: It was with his first solo show, at the Mary Boone Gallery in 1979 that Schnabel had his breakthrough; all his works were sold in advance. He participated at the Venice Biennale in 1980 with Anselm Kiefer and Georg Baselitz. By the time he exhibited his work in a show jointly organized by Boone and Leo Castelli in 1981, he had become firmly established and was the youngest artist in the legendary exhibition 'A New Spirit in Painting' in the Royal Academy of Arts. His now famous "plate paintings" — large-scale paintings set on broken ceramic plates—received a boisterous and critical reception from the art world. His wild and expressive works were classed as neo-expressionism by art critics. In the years to follow Schnabel's success on the art market would above all be criticised.


title: Julian Schnabel

text: Schnabel's style is characterised by very large scale paintings. He uses diverse materials such as plaster, wax, photographs, antlers, velvet and ceramics. His paintings make use of canvas, wood, muslin and even surfboards. His paintings often combine abstract and figurative elements. Due to the size, weight and depth of his works, they are often given sculptural properties.


title: Julian Schnabel

text: In 2002, Schnabel painted the cover artwork for the Red Hot Chili Peppers' eighth studio album, "By the Way". The woman featured on the cover of "By the Way" is Julian's daughter, Stella Schnabel, who was band member John Frusciante's then-girlfriend. Regarding the artwork, Frusciante noted: "My girlfriend's father offered to do the album art, so we sent him rough mixes of eight songs, and he just got the vibe of the album from that. He said that he wouldn't be offended if we didn't like it, but we loved what he did. He's also given us great covers for all the singles. He's a true artist."


title: Julian Schnabel

text: Schnabel had an exhibition at the Art Gallery of Ontario in Toronto, which ran from September 1, 2010 to January 2, 2011 and occupied the entirety of the gallery's fifth floor. It examined "the rich interplay between Schnabel's paintings and films". In 2011 Museo Correr exhibited "Julian Schnabel: Permanently Becoming and the Architecture of Seeing", a selected survey show of Schnabel's career curated by Norman Rosenthal.


title: Julian Schnabel

text: Art critic Robert Hughes was one of the most outspoken critics of his work; he once stated that "Schnabel's work is to painting what Stallone's is to acting: a lurching display of oily pectorals." In the 2017 Swedish film "The Square", set in a museum of modern art, Dominic West plays a character modeled on Schnabel.


title: Julian Schnabel

text: Schnabel began his film career in the 1990s with the film "Basquiat", a biopic on the painter Jean-Michel Basquiat (1996), followed by "Before Night Falls" (2000), an adaptation of Reinaldo Arenas' autobiographical novel, which he also produced, and which won the Grand Jury Prize at the Venice Film Festival. He directed "The Diving Bell and the Butterfly" (2007), an adaptation (with a screenplay by Ronald Harwood) of a French memoir by Jean-Dominique Bauby. "The Diving Bell and the Butterfly" earned him the award for best director at the 2007 Cannes Film Festival, the Golden Globe for best director, the Independent Spirit Award for best director, and a nomination for the Academy Award for Best Director.


title: Julian Schnabel

text: Despite the fact that producing "The Diving Bell and the Butterfly" might seem like a commission to do someone else's work, Schnabel took on the film. According to Schnabel,


title: Julian Schnabel

text: In 2007, Schnabel designed Lou Reed's critically acclaimed "Berlin" Tour and released "". In 2010, Schnabel then directed the film "Miral". In May 2017, Schnabel announced plans for a film about the painter Vincent van Gogh during his time in Arles and Auvers-sur-Oise, France. The film "At Eternity's Gate" was released in 2018 and the script was written by Schnabel, French screenwriter Jean-Claude Carrière, and Louise Kugelberg. The film stars Willem Dafoe as Van Gogh. Other actors include Mathieu Amalric, Mads Mikkelsen, Niels Arestrup, Oscar Isaac as Paul Gauguin and Emmanuelle Seigner as "the woman from Arles" or L'Arlésienne.


title: Julian Schnabel

text: Schnabel published his autobiography, "CVJ: Nicknames of Maitre D's & Other Excerpts From Life" (Random House, New York), in 1987 and released the album "Every Silver Lining Has a Cloud" on Island Records (Catalog #314-524 111-2) in 1995.


title: Julian Schnabel

text: Recorded in Brooklyn, New York, in 1993, the album features guest musicians including Bill Laswell, Bernie Worrell, Buckethead, and Nicky Skopelitis.


title: Julian Schnabel

text: In 1980, he married Belgian clothing designer Jacqueline Beaurang. They have three children: two daughters – Lola, a painter and filmmaker; and Stella, a poet and actress – and a son, Vito, an art dealer.


title: Julian Schnabel

text: His collaboration with Palestinian journalist Rula Jebreal, who penned the screenplay and original source novel for Schnabel's film "Miral", extended beyond the movie. Schnabel was in a relationship with her from 2007 to 2011.


title: Julian Schnabel

text: Schnabel dated Danish model May Andersen, from whom he parted ways in 2014. They have a son, who was born in June 2013.


title: Julian Schnabel

text: Schnabel lives in New York City with his current wife Louise Kugelberg, a Swedish interior designer. She is also the co-editor and co-writer of "At Eternity's Gate". Schnabel maintains studios in New York City and in Montauk at the east end of Long Island. Schnabel resides in a former West Village horse stable that he purchased and converted for residential use, adding five luxury condominiums in the style of a Northern Italian palazzo. It is named the Palazzo Chupi, and it is easy to spot because it is painted pink.


title: Julian Schnabel

text: In 2009, Schnabel signed a petition in support of film director Roman Polanski, calling for his release after Polanski was arrested in Switzerland in relation to his 1977 charge for drugging and raping a 13-year-old girl.


title: Julian Schnabel

text: His works are exhibited in the collections of various museums throughout the world, among them the Metropolitan Museum of Art; Museum of Modern Art in New York; the Whitney Museum of American Art; the Museum of Contemporary Art (MOCA), Los Angeles; Los Angeles County Museum of Art; Reina Sofia in Madrid; Tate Modern in London and the Centre Georges Pompidou in Paris.


title: Meg Turney

text: Megan LeeAnn Turney (born March 12, 1987) is an American streamer, internet personality, cosplayer, glamour model, and vlogger. She became an internet personality through her hosting work, formerly with "SourceFed" and Rooster Teeth's "The Know". She also maintains a personal YouTube channel where she posts vlogs and Let's Play videos and live streams. In addition to her work with online media, Turney also cosplays at various conventions, such as Comic-Con.


title: Meg Turney

text: Through a friend's invitation to Ushicon 1, Turney was introduced to cosplaying and anime conventions in 2002. Turney's first cosplay was of Sailor Heavy Metal Papillon from the anime, "Sailor Moon". On her first cosplay, Turney recalled, "I failed terribly. I couldn’t make half the costume, it was incredibly difficult, but I was as happy as a clam." Since then, Turney became recognized for her cosplaying and was able to generate a following large enough to turn cosplaying into her full-time career. The "Vancouver Sun" wrote that "Turney is an internet mainstay. She has her own YouTube channel and her Patreon fan site is an earner." When she began to receive cover in online publications, Turney's Faye Valentine, Princess Leia, and Psycho cosplays were particularly recognized. Turney's notability as a cosplayer led her to appear on the Syfy series "Heroes of Cosplay" and a commercial for "Guild Wars 2". She hosted a cosplay contest in 2013 at SXSW. Turney was also a special guest at the 2016 edition of Fan Expo Vancouver.


title: Meg Turney

text: Her cosplaying has been noted by various media publications to be racy and erotic. Turney's online content majorly includes images of her cosplaying and modeling, with Inquisitr writing that "she has a tendency to upload photos that push the boundaries of Instagram's community guidelines regarding nudity." On sexuality in cosplay, Turney has stated, "I think there is an element of sexuality for cosplay and for some people there is no sexuality whatsoever. I think it is all about how you approach it. I think that is true of any community really." Turney also maintains that there is more work involved in her cosplaying than showing off her body, such as costume designing, and stated "I think some people just go, 'She [fellow cosplayer Jessica Nigri] has boobs, that's why she's popular.' But that's not really the case." In January 2015, "Playboy" ranked Turney at number 10 on its "25 Hottest YouTube Stars" listing. In April 2015, "FHM" placed Turney at number 20 on its annual "100 Sexiest Women in the World" list.


title: Meg Turney

text: Having worked with various photographers, Turney also has experience modeling outside of cosplay. In June 2015, "Playboy" conducted a photoshoot with Turney.


title: Meg Turney

text: One of her cosplays was of her boyfriend Gavin's character from Rooster Teeth's show "X-Ray and Vav", with Ashley Jenkins joining her as X-Ray at RTX 2014. Ray and Gavin themselves had attended the Achievement Hunter panel of the previous year's RTX in costume as the duo, which had originally started out as an intro flub on Gavin's part for a "Halo 4" achievement guide.


title: Meg Turney

text: Turney registered her YouTube channel in November 2009 but began posting vlogs in 2012. During the early years of her online media career, Turney hosted event coverage for deviantART and appeared as a featured host on CraveOnline's "TechKnow" show. In July 2012, Turney would break through as an online media host, making her debut appearance on "SourceFed", alongside Steve Zaragoza, in which the two discussed "Star Wars", Comic-Con, and cosplaying. Later that year Turney was alongside Elliott Morgan and Philip DeFranco during SourceFed's videos for YouTube's "Election Hub." In February 2013, Turney and her "SourceFed" co-hosts won an Audience Choice Streamy Award for Series of the Year. Along with being a co-host of the channel, Turney was also a writer for the "SourceFed" website. Turney would remain a host of the channel, as well as its "Nerd" spinoff, until April 2014, when she announced she would be leaving "SourceFed".


title: Meg Turney

text: During her time with "SourceFed", Turney appeared as a host of other series including the AMA's "On Demand", as well as "Nerdist News". Additionally, in August 2013, Turney discussed Hollywood portrayals about life on Mars, alongside Bobak Ferdowsi, Matt Mira, and Kevin Lieber.


title: Meg Turney

text: After departing from "SourceFed", Turney was announced as a new host for Rooster Teeth's news channel, "The Know", on May 30, 2014. On joining Rooster Teeth, Turney claimed, "They have been pioneers in the industry for many years, and their level of originality and just plain fun in their content is unmatched in the space." On June 1, 2014, Turney made her first appearance on "The Know". In July 2014, Rooster Teeth distributed the music video for "Did I Say That Out Loud?" by Barenaked Ladies, in which Turney portrays the exasperated girlfriend of Gavin Free's character. In August 2014, Turney began co-hosting "The Know It All: Leaderboard", a collaborative series between Rooster Teeth and "The Daily Dot". Turney also voiced Neon Katt in the animated series "RWBY", and was a regular host on The Patch, Rooster Teeth's weekly gaming podcast. On June 27, 2016, Turney announced that she had left Rooster Teeth on good terms.


title: Meg Turney

text: In October 2014, Turney was involved in a controversy when live streaming video website Twitch updated its rules of conduct telling users to not shoot videos in "lingerie, swimsuits, pasties, and undergarments". Some news websites reported that Turney was the first user banned for violating the updated rules of conduct. Turney clarified on Twitter, stating, "Dear everyone writing articles saying I was banned from Twitch, I wasn't. I've never been banned. I was told to change my picture," adding "while we're on the subject I've never done any 'sexy broadcasts'. I've never even streamed from that account." Additionally, Turney also provided her opinion that the rule was not unfair to women, but just unfair in general, stating, "If someone has a big push-up bra and a low-cut shirt while they play ["League of Legends"] or a guy streams shirtless, who cares?"


title: Meg Turney

text: In June 2016, Turney began streaming on Twitch. In the years since leaving "SourceFed" and Rooster Teeth, Turney has continued to upload video game commentaries and streams, as well as cosplay-related vlogs on her personal YouTube channel.


title: Meg Turney

text: In April 2019, Turney signed with talent agency Abrams Artists, joining its gaming roster. In June 2020, Turney moved from Abrams to TalentX Gaming, where she is currently represented. In October, she participated in a "Dead by Daylight" tournament hosted by Twitch Rivals.


title: Meg Turney

text: Turney currently streams Dead by Daylight regularly, along with a variety of other games, and celebrated 5 years of Twitch partnership in June 2021. Turney's YouTube channel also became primarily used as a selection of highlights from her Twitch streams.


title: Meg Turney

text: In March 2020, Turney created an OnlyFans account to supplement her Patreon account. At the time, Patreon was her primary venue for releasing her modeling work while OnlyFans was there primarily to share older photosets that were no longer available on Patreon, but in October 2020, Turney announced on her Twitter that she was moving all of her content to OnlyFans and that the site would be her new primary content hub. After releasing her book of topless photos on her personal website, she began creating boudoir content full time on her OnlyFans. Turney still releases Cosplay photosets on her OnlyFans, but most of her content is boudoir photography.


title: Meg Turney

text: Turney has openly spoken about her bisexuality with her audience. In a video interview with the YouTube channel "VlogBrothers", she further discussed being bisexual, and spoke about how she was teased for her sexual orientation. At VidCon 2013, Turney spoke during the convention's "Being LGBT on YouTube" panel.


title: Meg Turney

text: On January 26, 2018, a man armed with a handgun broke into Turney's and Free's home, breaking a window to enter and firing one round as he did. Turney and Free hid in a closet and called police. When the man could not find the couple, he left but was confronted by police outside. The suspect fired one shot and police returned fire; when officers approached the suspect's vehicle, they found him dead. Turney and Free were physically unharmed during the incident. It has been assessed the suspect was a fan who had developed an obsession with Turney, and he was attempting to cause harm to Free, against whom he felt resentment. Turney ceased updating her YouTube channel for nearly half a year following the incident, citing concern for her safety in sharing her life with an audience.


title: Categorical variable

text: In statistics, a categorical variable (also called qualitative variable) is a variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or nominal category on the basis of some qualitative property. In computer science and some branches of mathematics, categorical variables are referred to as enumerations or enumerated types. Commonly (though not in this article), each of the possible values of a categorical variable is referred to as a level. The probability distribution associated with a random categorical variable is called a categorical distribution.


title: Categorical variable

text: Categorical data is the statistical data type consisting of categorical variables or of data that has been converted into that form, for example as grouped data. More specifically, categorical data may derive from observations made of qualitative data that are summarised as counts or cross tabulations, or from observations of quantitative data grouped within given intervals. Often, purely categorical data are summarised in the form of a contingency table. However, particularly when considering data analysis, it is common to use the term "categorical data" to apply to data sets that, while containing some categorical variables, may also contain non-categorical variables.


title: Categorical variable

text: A categorical variable that can take on exactly two values is termed a "binary variable" or a dichotomous variable; an important special case is the Bernoulli variable. Categorical variables with more than two possible values are called polytomous variables; categorical variables are often assumed to be polytomous unless otherwise specified. Discretization is treating continuous data as if it were categorical. Dichotomization is treating continuous data or polytomous variables as if they were binary variables. Regression analysis often treats category membership with one or more quantitative dummy variables.


title: Categorical variable

text: For ease in statistical processing, categorical variables may be assigned numeric indices, e.g. 1 through "K" for a "K"-way categorical variable (i.e. a variable that can express exactly "K" possible values). In general, however, the numbers are arbitrary, and have no significance beyond simply providing a convenient label for a particular value. In other words, the values in a categorical variable exist on a nominal scale: they each represent a logically separate concept, cannot necessarily be meaningfully ordered, and cannot be otherwise manipulated as numbers could be. Instead, valid operations are equivalence, set membership, and other set-related operations.


title: Categorical variable

text: As a result, the central tendency of a set of categorical variables is given by its mode; neither the mean nor the median can be defined. As an example, given a set of people, we can consider the set of categorical variables corresponding to their last names. We can consider operations such as equivalence (whether two people have the same last name), set membership (whether a person has a name in a given list), counting (how many people have a given last name), or finding the mode (which name occurs most often). However, we cannot meaningfully compute the "sum" of Smith + Johnson, or ask whether Smith is "less than" or "greater than" Johnson. As a result, we cannot meaningfully ask what the "average name" (the mean) or the "middle-most name" (the median) is in a set of names.


title: Categorical variable

text: Note that this ignores the concept of alphabetical order, which is a property that is not inherent in the names themselves, but in the way we construct the labels. For example, if we write the names in Cyrillic and consider the Cyrillic ordering of letters, we might get a different result of evaluating "Smith < Johnson" than if we write the names in the standard Latin alphabet; and if we write the names in Chinese characters, we cannot meaningfully evaluate "Smith < Johnson" at all, because no consistent ordering is defined for such characters. However, if we do consider the names as written, e.g., in the Latin alphabet, and define an ordering corresponding to standard alphabetical order, then we have effectively converted them into ordinal variables defined on an ordinal scale.


title: Categorical variable

text: Categorical random variables are normally described statistically by a categorical distribution, which allows an arbitrary "K"-way categorical variable to be expressed with separate probabilities specified for each of the "K" possible outcomes. Such multiple-category categorical variables are often analyzed using a multinomial distribution, which counts the frequency of each possible combination of numbers of occurrences of the various categories. Regression analysis on categorical outcomes is accomplished through multinomial logistic regression, multinomial probit or a related type of discrete choice model.


title: Categorical variable

text: Categorical variables that have only two possible outcomes (e.g., "yes" vs. "no" or "success" vs. "failure") are known as "binary variables" (or "Bernoulli variables"). Because of their importance, these variables are often considered a separate category, with a separate distribution (the Bernoulli distribution) and separate regression models (logistic regression, probit regression, etc.). As a result, the term "categorical variable" is often reserved for cases with 3 or more outcomes, sometimes termed a "multi-way" variable in opposition to a binary variable.


title: Categorical variable

text: It is also possible to consider categorical variables where the number of categories is not fixed in advance. As an example, for a categorical variable describing a particular word, we might not know in advance the size of the vocabulary, and we would like to allow for the possibility of encountering words that we haven't already seen. Standard statistical models, such as those involving the categorical distribution and multinomial logistic regression, assume that the number of categories is known in advance, and changing the number of categories on the fly is tricky. In such cases, more advanced techniques must be used. An example is the Dirichlet process, which falls in the realm of nonparametric statistics. In such a case, it is logically assumed that an infinite number of categories exist, but at any one time most of them (in fact, all but a finite number) have never been seen. All formulas are phrased in terms of the number of categories actually seen so far rather than the (infinite) total number of potential categories in existence, and methods are created for incremental updating of statistical distributions, including adding "new" categories.


title: Categorical variable

text: Categorical variables represent a qualitative method of scoring data (i.e. represents categories or group membership). These can be included as independent variables in a regression analysis or as dependent variables in logistic regression or probit regression, but must be converted to quantitative data in order to be able to analyze the data. One does so through the use of coding systems. Analyses are conducted such that only "g" -1 ("g" being the number of groups) are coded. This minimizes redundancy while still representing the complete data set as no additional information would be gained from coding the total "g" groups: for example, when coding gender (where "g" = 2: male and female), if we only code females everyone left over would necessarily be males. In general, the group that one does not code for is the group of least interest.


title: Categorical variable

text: There are three main coding systems typically used in the analysis of categorical variables in regression: dummy coding, effects coding, and contrast coding. The regression equation takes the form of Y = bX + a, where "b" is the slope and gives the weight empirically assigned to an explanator, "X" is the explanatory variable, and "a" is the "Y"-intercept, and these values take on different meanings based on the coding system used. The choice of coding system does not affect the "F" or "R" statistics. However, one chooses a coding system based on the comparison of interest since the interpretation of "b" values will vary.


title: Categorical variable

text: Dummy coding is used when there is a control or comparison group in mind. One is therefore analyzing the data of one group in relation to the comparison group: "a" represents the mean of the control group and "b" is the difference between the mean of the experimental group and the mean of the control group. It is suggested that three criteria be met for specifying a suitable control group: the group should be a well-established group (e.g. should not be an “other” category), there should be a logical reason for selecting this group as a comparison (e.g. the group is anticipated to score highest on the dependent variable), and finally, the group's sample size should be substantive and not small compared to the other groups.


title: Categorical variable

text: In dummy coding, the reference group is assigned a value of 0 for each code variable, the group of interest for comparison to the reference group is assigned a value of 1 for its specified code variable, while all other groups are assigned 0 for that particular code variable.


title: Categorical variable

text: The "b" values should be interpreted such that the experimental group is being compared against the control group. Therefore, yielding a negative b value would entail the experimental group have scored less than the control group on the dependent variable. To illustrate this, suppose that we are measuring optimism among several nationalities and we have decided that French people would serve as a useful control. If we are comparing them against Italians, and we observe a negative "b" value, this would suggest Italians obtain lower optimism scores on average.


title: Categorical variable

text: The following table is an example of dummy coding with "French" as the control group and C1, C2, and C3 respectively being the codes for "Italian", "German", and "Other" (neither French nor Italian nor German):


title: Categorical variable

text: In the effects coding system, data are analyzed through comparing one group to all other groups. Unlike dummy coding, there is no control group. Rather, the comparison is being made at the mean of all groups combined ("a" is now the grand mean). Therefore, one is not looking for data in relation to another group but rather, one is seeking data in relation to the grand mean.


title: Categorical variable

text: Effects coding can either be weighted or unweighted. Weighted effects coding is simply calculating a weighted grand mean, thus taking into account the sample size in each variable. This is most appropriate in situations where the sample is representative of the population in question. Unweighted effects coding is most appropriate in situations where differences in sample size are the result of incidental factors. The interpretation of "b" is different for each: in unweighted effects coding "b" is the difference between the mean of the experimental group and the grand mean, whereas in the weighted situation it is the mean of the experimental group minus the weighted grand mean.


title: Categorical variable

text: In effects coding, we code the group of interest with a 1, just as we would for dummy coding. The principal difference is that we code −1 for the group we are least interested in. Since we continue to use a "g" - 1 coding scheme, it is in fact the −1 coded group that will not produce data, hence the fact that we are least interested in that group. A code of 0 is assigned to all other groups.


title: Categorical variable

text: The "b" values should be interpreted such that the experimental group is being compared against the mean of all groups combined (or weighted grand mean in the case of weighted effects coding). Therefore, yielding a negative "b" value would entail the coded group as having scored less than the mean of all groups on the dependent variable. Using our previous example of optimism scores among nationalities, if the group of interest is Italians, observing a negative "b" value suggest they obtain a lower optimism score.


title: Categorical variable

text: The contrast coding system allows a researcher to directly ask specific questions. Rather than having the coding system dictate the comparison being made (i.e., against a control group as in dummy coding, or against all groups as in effects coding) one can design a unique comparison catering to one's specific research question. This tailored hypothesis is generally based on previous theory and/or research. The hypotheses proposed are generally as follows: first, there is the central hypothesis which postulates a large difference between two sets of groups; the second hypothesis suggests that within each set, the differences among the groups are small. Through its a priori focused hypotheses, contrast coding may yield an increase in power of the statistical test when compared with the less directed previous coding systems.


title: Categorical variable

text: Certain differences emerge when we compare our a priori coefficients between ANOVA and regression. Unlike when used in ANOVA, where it is at the researcher's discretion whether they choose coefficient values that are either orthogonal or non-orthogonal, in regression, it is essential that the coefficient values assigned in contrast coding be orthogonal. Furthermore, in regression, coefficient values must be either in fractional or decimal form. They cannot take on interval values.


title: Categorical variable

text: Violating rule 2 produces accurate "R" and "F" values, indicating that we would reach the same conclusions about whether or not there is a significant difference; however, we can no longer interpret the "b" values as a mean difference.


title: Categorical variable

text: To illustrate the construction of contrast codes consider the following table. Coefficients were chosen to illustrate our a priori hypotheses: Hypothesis 1: French and Italian persons will score higher on optimism than Germans (French = +0.33, Italian = +0.33, German = −0.66). This is illustrated through assigning the same coefficient to the French and Italian categories and a different one to the Germans. The signs assigned indicate the direction of the relationship (hence giving Germans a negative sign is indicative of their lower hypothesized optimism scores). Hypothesis 2: French and Italians are expected to differ on their optimism scores (French = +0.50, Italian = −0.50, German = 0). Here, assigning a zero value to Germans demonstrates their non-inclusion in the analysis of this hypothesis. Again, the signs assigned are indicative of the proposed relationship.


title: Categorical variable

text: Nonsense coding occurs when one uses arbitrary values in place of the designated “0”s “1”s and “-1”s seen in the previous coding systems. Although it produces correct mean values for the variables, the use of nonsense coding is not recommended as it will lead to uninterpretable statistical results.


title: Categorical variable

text: "Embeddings" are codings of categorical values into high-dimensional real-valued (sometimes complex-valued) vector spaces, usually in such a way that ‘similar’ values are assigned ‘similar’ vectors, or with respect to some other kind of criterion making the vectors useful for the respective application. A common special case are word embeddings, where the possible values of the categorical variable are the words in a language and words with similar meanings are to be assigned similar vectors.


title: Categorical variable

text: An interaction may arise when considering the relationship among three or more variables, and describes a situation in which the simultaneous influence of two variables on a third is not additive. Interactions may arise with categorical variables in two ways: either categorical by categorical variable interactions, or categorical by continuous variable interactions.


title: Categorical variable

text: This type of interaction arises when we have two categorical variables. In order to probe this type of interaction, one would code using the system that addresses the researcher's hypothesis most appropriately. The product of the codes yields the interaction. One may then calculate the "b" value and determine whether the interaction is significant.


title: Categorical variable

text: Simple slopes analysis is a common post hoc test used in regression which is similar to the simple effects analysis in ANOVA, used to analyze interactions. In this test, we are examining the simple slopes of one independent variable at specific values of the other independent variable. Such a test is not limited to use with continuous variables, but may also be employed when the independent variable is categorical. We cannot simply choose values to probe the interaction as we would in the continuous variable case because of the nominal nature of the data (i.e., in the continuous case, one could analyze the data at high, moderate, and low levels assigning 1 standard deviation above the mean, at the mean, and at one standard deviation below the mean respectively). In our categorical case we would use a simple regression equation for each group to investigate the simple slopes. It is common practice to standardize or center variables to make the data more interpretable in simple slopes analysis; however, categorical variables should never be standardized or centered. This test can be used with all coding systems.


title: Roche limit

text: In celestial mechanics, the Roche limit, also called Roche radius, is the distance from a celestial body within which a second celestial body, held together only by its own force of gravity, will disintegrate because the first body's tidal forces exceed the second body's gravitational self-attraction. Inside the Roche limit, orbiting material disperses and forms rings, whereas outside the limit, material tends to coalesce. The Roche radius depends on the radius of the first body and on the ratio of the bodies' densities.


title: Roche limit

text: The term is named after Édouard Roche (, ), the French astronomer who first calculated this theoretical limit in 1848.


title: Roche limit

text: The Roche limit typically applies to a satellite's disintegrating due to tidal forces induced by its "primary", the body around which it orbits. Parts of the satellite that are closer to the primary are attracted more strongly by gravity from the primary than parts that are farther away; this disparity effectively pulls the near and far parts of the satellite apart from each other, and if the disparity (combined with any centrifugal effects due to the object's spin) is larger than the force of gravity holding the satellite together, it can pull the satellite apart. Some real satellites, both natural and artificial, can orbit within their Roche limits because they are held together by forces other than gravitation. Objects resting on the surface of such a satellite would be lifted away by tidal forces. A weaker satellite, such as a comet, could be broken up when it passes within its Roche limit.


title: Roche limit

text: Since, within the Roche limit, tidal forces overwhelm the gravitational forces that might otherwise hold the satellite together, no satellite can gravitationally coalesce out of smaller particles within that limit. Indeed, almost all known planetary rings are located within their Roche limit. (Notable exceptions are Saturn's E-Ring and Phoebe ring. These two rings could possibly be remnants from the planet's proto-planetary accretion disc that failed to coalesce into moonlets, or conversely have formed when a moon passed within its Roche limit and broke apart.)


title: Roche limit

text: The Roche limit is not the only factor that causes comets to break apart. Splitting by thermal stress, internal gas pressure and rotational splitting are other ways for a comet to split under stress.


title: Roche limit

text: The limiting distance to which a satellite can approach without breaking up depends on the rigidity of the satellite. At one extreme, a completely rigid satellite will maintain its shape until tidal forces break it apart. At the other extreme, a highly fluid satellite gradually deforms leading to increased tidal forces, causing the satellite to elongate, further compounding the tidal forces and causing it to break apart more readily.


title: Roche limit

text: Most real satellites would lie somewhere between these two extremes, with tensile strength rendering the satellite neither perfectly rigid nor perfectly fluid. For example, a rubble-pile asteroid will behave more like a fluid than a solid rocky one; an icy body will behave quite rigidly at first but become more fluid as tidal heating accumulates and its ices begin to melt.


title: Roche limit

text: But note that, as defined above, the Roche limit refers to a body held together solely by the gravitational forces which cause otherwise unconnected particles to coalesce, thus forming the body in question. The Roche limit is also usually calculated for the case of a circular orbit, although it is straightforward to modify the calculation to apply to the case (for example) of a body passing the primary on a parabolic or hyperbolic trajectory.


title: Roche limit

text: The "rigid-body" Roche limit is a simplified calculation for a spherical satellite. Irregular shapes such as those of tidal deformation on the body or the primary it orbits are neglected. It is assumed to be in hydrostatic equilibrium. These assumptions, although unrealistic, greatly simplify calculations.


title: Roche limit

text: The Roche limit for a rigid spherical satellite is the distance, formula_1, from the primary at which the gravitational force on a test mass at the surface of the object is exactly equal to the tidal force pulling the mass away from the object:


title: Roche limit

text: where formula_3 is the radius of the primary, formula_4 is the density of the primary, and formula_5 is the density of the satellite. This can be equivalently written as


title: Roche limit

text: where formula_7 is the radius of the secondary, formula_8 is the mass of the primary, and formula_9 is the mass of the secondary.


title: Roche limit

text: This does not depend on the size of the objects, but on the ratio of densities. This is the orbital distance inside of which loose material (e.g. regolith) on the surface of the satellite closest to the primary would be pulled away, and likewise material on the side opposite the primary will also go away from, rather than toward, the satellite.


title: Roche limit

text: A more accurate approach for calculating the Roche limit takes the deformation of the satellite into account. An extreme example would be a tidally locked liquid satellite orbiting a planet, where any force acting upon the satellite would deform it into a prolate spheroid.


title: Roche limit

text: The calculation is complex and its result cannot be represented in an exact algebraic formula. Roche himself derived the following approximate solution for the Roche limit:


title: Roche limit

text: However, a better approximation that takes into account the primary's oblateness and the satellite's mass is:


title: Roche limit

text: where formula_12 is the oblateness of the primary. The numerical factor is calculated with the aid of a computer.


title: Roche limit

text: The fluid solution is appropriate for bodies that are only loosely held together, such as a comet. For instance, comet Shoemaker–Levy 9's decaying orbit around Jupiter passed within its Roche limit in July 1992, causing it to fragment into a number of smaller pieces. On its next approach in 1994 the fragments crashed into the planet. Shoemaker–Levy 9 was first observed in 1993, but its orbit indicated that it had been captured by Jupiter a few decades prior.


title: Metric tensor

text: In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point of is a bilinear form defined on the tangent space at (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric tensor on consists of a metric tensor at each point of that varies smoothly with .


title: Metric tensor

text: A metric tensor is "positive-definite" if for every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying "infinitesimal" distance on the manifold. On a Riemannian manifold , the length of a smooth curve between two points and can be defined by integration, and the distance between and can be defined as the infimum of the lengths of all such curves; this makes a metric space. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner).


title: Metric tensor

text: While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita, who first codified the notion of a tensor. The metric tensor is an example of a tensor field.


title: Metric tensor

text: The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent point of view, a metric tensor field is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point.


title: Metric tensor

text: Carl Friedrich Gauss in his 1827 "Disquisitiones generales circa superficies curvas" ("General investigations of curved surfaces") considered a surface parametrically, with the Cartesian coordinates , , and of points on the surface depending on two auxiliary variables and . Thus a parametric surface is (in today's terms) a vector-valued function


title: Metric tensor

text: depending on an ordered pair of real variables , and defined in an open set in the -plane. One of the chief aims of Gauss's investigations was to deduce those features of the surface which could be described by a function which would remain unchanged if the surface underwent a transformation in space (such as bending the surface without stretching it), or a change in the particular parametric form of the same geometrical surface.


title: Metric tensor

text: One natural such invariant quantity is the length of a curve drawn along the surface. Another is the angle between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is the area of a piece of the surface. The study of these invariants of a surface led Gauss to introduce the predecessor of the modern notion of the metric tensor.


title: Metric tensor

text: The metric tensor is formula_2 in the description below; E, F, and G in the matrix can contain any number as long as the matrix is positive definite.


title: Metric tensor

text: If the variables and are taken to depend on a third variable, , taking values in an interval , then will trace out a parametric curve in parametric surface . The arc length of that curve is given by the integral


title: Metric tensor

text: where formula_4 represents the Euclidean norm. Here the chain rule has been applied, and the subscripts denote partial derivatives:


title: Metric tensor

text: The quantity in () is called the line element, while is called the first fundamental form of . Intuitively, it represents the principal part of the square of the displacement undergone by when is increased by units, and is increased by units.


title: Metric tensor

text: Suppose now that a different parameterization is selected, by allowing and to depend on another pair of variables and . Then the analog of () for the new variables is


title: Metric tensor

text: where the superscript T denotes the matrix transpose. The matrix with the coefficients , , and arranged in this way therefore transforms by the Jacobian matrix of the coordinate change


title: Metric tensor

text: first observed the significance of a system of coefficients , , and , that transformed in this way on passing from one system of coordinates to another. The upshot is that the first fundamental form () is "invariant" under changes in the coordinate system, and that this follows exclusively from the transformation properties of , , and . Indeed, by the chain rule,


title: Metric tensor

text: Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of tangent vectors to the surface, as well as the angle between two tangent vectors. In contemporary terms, the metric tensor allows one to compute the dot product(non-euclidean geometry) of tangent vectors in a manner independent of the parametric description of the surface. Any tangent vector at a point of the parametric surface can be written in the form


title: Metric tensor

text: This is plainly a function of the four variables , , , and . It is more profitably viewed, however, as a function that takes a pair of arguments and which are vectors in the -plane. That is, put


title: Metric tensor

text: The surface area is another numerical quantity which should depend only on the surface itself, and not on how it is parameterized. If the surface is parameterized by the function over the domain in the -plane, then the surface area of is given by the integral


title: Metric tensor

text: where denotes the cross product, and the absolute value denotes the length of a vector in Euclidean space. By Lagrange's identity for the cross product, the integral can be written


title: Metric tensor

text: Let be a smooth manifold of dimension ; for instance a surface (in the case ) or hypersurface in the Cartesian space formula_21. At each point there is a vector space , called the tangent space, consisting of all tangent vectors to the manifold at the point . A metric tensor at is a function which takes as inputs a pair of tangent vectors and at , and produces as an output a real number (scalar), so that the following conditions are satisfied:


title: Metric tensor

text: A metric tensor field on assigns to each point of a metric tensor in the tangent space at in a way that varies smoothly with . More precisely, given any open subset of manifold and any (smooth) vector fields and on , the real function


title: Metric tensor

text: are two vectors at , then the value of the metric applied to and is determined by the coefficients () by bilinearity:


title: Metric tensor

text: where and denote the transpose of the vectors and , respectively. Under a change of basis of the form


title: Metric tensor

text: For this reason, the system of quantities is said to transform covariantly with respect to changes in the frame .


title: Metric tensor

text: A system of real-valued functions , giving a local coordinate system on an open set in , determines a basis of vector fields on


title: Metric tensor

text: If is positive for all non-zero , then the metric is positive-definite at . If the metric is positive-definite at every , then is called a Riemannian metric. More generally, if the quadratic forms have constant signature independent of , then the signature of is this signature, and is called a pseudo-Riemannian metric. If is connected, then the signature of does not depend on .


title: Metric tensor

text: By Sylvester's law of inertia, a basis of tangent vectors can be chosen locally so that the quadratic form diagonalizes in the following manner


title: Metric tensor

text: for some between 1 and . Any two such expressions of (at the same point of ) will have the same number of positive signs. The signature of is the pair of integers , signifying that there are positive signs and negative signs in any such expression. Equivalently, the metric has signature if the matrix of the metric has positive and negative eigenvalues.


title: Metric tensor

text: One can consider the inverse matrix , which is identified with the inverse metric (or "conjugate" or "dual metric"). The inverse metric satisfies a transformation law when the frame is changed by a matrix via


title: Metric tensor

text: The inverse metric transforms "contravariantly", or with respect to the inverse of the change of basis matrix . Whereas the metric itself provides a way to measure the length of (or angle between) vector fields, the inverse metric supplies a means of measuring the length of (or angle between) covector fields; that is, fields of linear functionals.


title: Metric tensor

text: To see this, suppose that is a covector field. To wit, for each point , determines a function defined on tangent vectors at so that the following linearity condition holds for all tangent vectors and , and all real numbers and :


title: Metric tensor

text: The resulting definition, although it involves the choice of basis , does not actually depend on in an essential way. Indeed, changing basis to gives


title: Metric tensor

text: So that the right-hand side of equation () is unaffected by changing the basis to any other basis whatsoever. Consequently, the equation may be assigned a meaning independently of the choice of basis. The entries of the matrix are denoted by , where the indices and have been raised to indicate the transformation law ().


title: Metric tensor

text: for some uniquely determined smooth functions . Upon changing the basis by a nonsingular matrix , the coefficients change in such a way that equation () remains true. That is,


title: Metric tensor

text: Consequently, . In other words, the components of a vector transform "contravariantly" (that is, inversely or in the opposite way) under a change of basis by the nonsingular matrix . The contravariance of the components of is notationally designated by placing the indices of in the upper position.


title: Metric tensor

text: A frame also allows covectors to be expressed in terms of their components. For the basis of vector fields define the dual basis to be the linear functionals such that


title: Metric tensor

text: where denotes the row vector . The components transform when the basis is replaced by in such a way that equation () continues to hold. That is,


title: Metric tensor

text: whence, because , it follows that . That is, the components transform "covariantly" (by the matrix rather than its inverse). The covariance of the components of is notationally designated by placing the indices of in the lower position.


title: Metric tensor

text: Now, the metric tensor gives a means to identify vectors and covectors as follows. Holding fixed, the function


title: Metric tensor

text: of tangent vector defines a linear functional on the tangent space at . This operation takes a vector at a point and produces a covector . In a basis of vector fields , if a vector field has components , then the components of the covector field in the dual basis are given by the entries of the row vector


title: Metric tensor

text: so that : transforms covariantly. The operation of associating to the (contravariant) components of a vector field the (covariant) components of the covector field , where


title: Metric tensor

text: To "raise the index", one applies the same construction but with the inverse metric instead of the metric. If are the components of a covector in the dual basis , then the column vector


title: Metric tensor

text: Consequently, the quantity does not depend on the choice of basis in an essential way, and thus defines a vector field on . The operation () associating to the (covariant) components of a covector the (contravariant) components of a vector given is called raising the index. In components, () is


title: Metric tensor

text: Let be an open set in , and let be a continuously differentiable function from into the Euclidean space , where . The mapping is called an immersion if its differential is injective at every point of . The image of is called an immersed submanifold. More specifically, for , which means that the ambient Euclidean space is , the induced metric tensor is called the first fundamental form.


title: Metric tensor

text: Suppose that is an immersion onto the submanifold . The usual Euclidean dot product in is a metric which, when restricted to vectors tangent to , gives a means for taking the dot product of these tangent vectors. This is called the induced metric.


title: Metric tensor

text: where are the standard coordinate vectors in . When is applied to , the vector goes over to the vector tangent to given by


title: Metric tensor

text: (This is called the pushforward of along .) Given two such vectors, and , the induced metric is defined by


title: Metric tensor

text: It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields is given by


title: Metric tensor

text: The notion of a metric can be defined intrinsically using the language of fiber bundles and vector bundles. In these terms, a metric tensor is a function


title: Metric tensor

text: from the fiber product of the tangent bundle of with itself to such that the restriction of to each fiber is a nondegenerate bilinear mapping


title: Metric tensor

text: The mapping () is required to be continuous, and often continuously differentiable, smooth, or real analytic, depending on the case of interest, and whether can support such a structure.


title: Metric tensor

text: By the universal property of the tensor product, any bilinear mapping () gives rise naturally to a section of the dual of the tensor product bundle of with itself


title: Metric tensor

text: and is defined on arbitrary elements of by extending linearly to linear combinations of simple elements. The original bilinear form is symmetric if and only if


title: Metric tensor

text: so that is regarded also as a section of the bundle of the cotangent bundle with itself. Since is symmetric as a bilinear mapping, it follows that is a symmetric tensor.


title: Metric tensor

text: More generally, one may speak of a metric in a vector bundle. If is a vector bundle over a manifold , then a metric is a mapping


title: Metric tensor

text: Using duality as above, a metric is often identified with a section of the tensor product bundle . (See metric (vector bundle).)


title: Metric tensor

text: The metric tensor gives a natural isomorphism from the tangent bundle to the cotangent bundle, sometimes called the musical isomorphism. This isomorphism is obtained by setting, for each tangent vector ,


title: Metric tensor

text: the linear functional on which sends a tangent vector at to . That is, in terms of the pairing between and its dual space ,


title: Metric tensor

text: for all tangent vectors and . The mapping is a linear transformation from to . It follows from the definition of non-degeneracy that the kernel of is reduced to zero, and so by the rank–nullity theorem, is a linear isomorphism. Furthermore, is a symmetric linear transformation in the sense that


title: Metric tensor

text: This bilinear form is symmetric if and only if is symmetric. There is thus a natural one-to-one correspondence between symmetric bilinear forms on and symmetric linear isomorphisms of to the dual .


title: Metric tensor

text: As varies over , defines a section of the bundle of vector bundle isomorphisms of the tangent bundle to the cotangent bundle. This section has the same smoothness as : it is continuous, differentiable, smooth, or real-analytic according as . The mapping , which associates to every vector field on a covector field on gives an abstract formulation of "lowering the index" on a vector field. The inverse of is a mapping which, analogously, gives an abstract formulation of "raising the index" on a covector field.


title: Metric tensor

text: for all covectors , . Such a nonsingular symmetric mapping gives rise (by the tensor-hom adjunction) to a map


title: Metric tensor

text: Suppose that is a Riemannian metric on . In a local coordinate system , , the metric tensor appears as a matrix, denoted here by , whose entries are the components of the metric tensor relative to the coordinate vector fields.


title: Metric tensor

text: Let be a piecewise-differentiable parametric curve in , for . The arclength of the curve is defined by


title: Metric tensor

text: is called the first fundamental form associated to the metric, while is the line element. When is pulled back to the image of a curve in , it represents the square of the differential with respect to arclength.


title: Metric tensor

text: For a pseudo-Riemannian metric, the length formula above is not always defined, because the term under the square root may become negative. We generally only define the length of a curve when the quantity under the square root is always of one sign or the other. In this case, define


title: Metric tensor

text: Note that, while these formulas use coordinate expressions, they are in fact independent of the coordinates chosen; they depend only on the metric, and the curve along which the formula is integrated.


title: Metric tensor

text: Given a segment of a curve, another frequently defined quantity is the (kinetic) energy of the curve:


title: Metric tensor

text: This usage comes from physics, specifically, classical mechanics, where the integral can be seen to directly correspond to the kinetic energy of a point particle moving on the surface of a manifold. Thus, for example, in Jacobi's formulation of Maupertuis' principle, the metric tensor can be seen to correspond to the mass tensor of a moving particle.


title: Metric tensor

text: In many cases, whenever a calculation calls for the length to be used, a similar calculation using the energy may be done as well. This often leads to simpler formulas by avoiding the need for the square-root. Thus, for example, the geodesic equations may be obtained by applying variational principles to either the length or the energy. In the latter case, the geodesic equations are seen to arise from the principle of least action: they describe the motion of a "free particle" (a particle feeling no forces) that is confined to move on the manifold, but otherwise moves freely, with constant momentum, within the manifold.


title: Metric tensor

text: In analogy with the case of surfaces, a metric tensor on an -dimensional paracompact manifold gives rise to a natural way to measure the -dimensional volume of subsets of the manifold. The resulting natural positive Borel measure allows one to develop a theory of integrating functions on the manifold by means of the associated Lebesgue integral.


title: Metric tensor

text: A measure can be defined, by the Riesz representation theorem, by giving a positive linear functional on the space of compactly supported continuous functions on . More precisely, if is a manifold with a (pseudo-)Riemannian metric tensor , then there is a unique positive Borel measure such that for any coordinate chart ,


title: Metric tensor

text: for all supported in . Here is the determinant of the matrix formed by the components of the metric tensor in the coordinate chart. That is well-defined on functions supported in coordinate neighborhoods is justified by Jacobian change of variables. It extends to a unique positive linear functional on by means of a partition of unity.


title: Metric tensor

text: If is also oriented, then it is possible to define a natural volume form from the metric tensor. In a positively oriented coordinate system the volume form is represented as


title: Metric tensor

text: where the are the coordinate differentials and denotes the exterior product in the algebra of differential forms. The volume form also gives a way to integrate functions on the manifold, and this geometric integral agrees with the integral obtained by the canonical Borel measure.


title: Metric tensor

text: The most familiar example is that of elementary Euclidean geometry: the two-dimensional Euclidean metric tensor. In the usual coordinates, we can write


title: Metric tensor

text: In general, in a Cartesian coordinate system on a Euclidean space, the partial derivatives are orthonormal with respect to the Euclidean metric. Thus the metric tensor is the Kronecker delta δ in this coordinate system. The metric tensor with respect to arbitrary (possibly curvilinear) coordinates is given by


title: Metric tensor

text: The unit sphere in comes equipped with a natural metric induced from the ambient Euclidean metric, through the process explained in the induced metric section. In standard spherical coordinates , with the colatitude, the angle measured from the -axis, and the angle from the -axis in the -plane, the metric takes the form


title: Metric tensor

text: For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual length formula. For a timelike curve, the length formula gives the proper time along the curve.


title: Metric tensor

text: The Schwarzschild metric describes the spacetime around a spherically symmetric body, such as a planet, or a black hole. With coordinates


title: Metric tensor

text: where (inside the matrix) is the gravitational constant and represents the total mass-energy content of the central object.


title: Kern County, California

text: Kern County is a county located in the U.S. state of California. As of the 2020 census, the population was 909,235. Its county seat is Bakersfield.


title: Kern County, California

text: Kern County comprises the Bakersfield, California, Metropolitan statistical area. The county spans the southern end of the Central Valley. Covering , it ranges west to the southern slope of the Coast Ranges, and east beyond the southern slope of the eastern Sierra Nevada into the Mojave Desert, at the city of Ridgecrest. Its northernmost city is Delano, and its southern reach extends to just beyond Frazier Park, and the northern extremity of the parallel Antelope Valley.


title: Kern County, California

text: The county's economy is heavily linked to agriculture and to petroleum extraction. There is also a strong aviation, space, and military presence, such as Edwards Air Force Base, the China Lake Naval Air Weapons Station, and the Mojave Air and Space Port.


title: Kern County, California

text: With a population that is 54.9% Hispanic as of 2020, Kern is California's third-most populous majority-Hispanic county and the sixth-largest nationwide.


title: Kern County, California

text: Native Americans lived in this region for hundreds of years: Chumash, tribes grouped together under the settler name Yokuts, and others.


title: Kern County, California

text: Spain claimed the area in 1769. Entering from Grapevine Canyon to the south in 1772, Commander Don Pedro Fages became the first European to set foot in the area.


title: Kern County, California

text: The Battle of San Emigdio took place in Kern County in March 1824. The Chumash Native Americans of Mission Santa Barbara rebelled against the Mexican government and its taking over mission property and ejecting the natives. The battle occurred in the canyon where San Emigdio Creek flows down San Emigdio Mountain and the Blue Ridge, south of Bakersfield near today's Highway 166. Mexican forces from Monterey were commanded by Carlos Carrillo and the conflict was a low-casualty encounter, with only four Native Americans being killed and no Mexicans. The surviving Native Americans were pacified and brought back to Santa Barbara in June 1824 after a pursuit and negotiation in which many were allowed to keep their arms for the return march over the mountains.


title: Kern County, California

text: In the beginning, what was to become Kern County was dominated by mining in the mountains and in the desert. In 1855 the California legislature attempted to form a county in the area by giving the southeastern territory of Tulare County on the west of the Sierra Nevada Mountains to Buena Vista County. However, it was never officially organized prior to 1859, when the enabling legislation expired. The south of Tulare County was later organized as Kern County in 1866, with additions from Los Angeles and San Bernardino Counties. Its first county seat was the mining town of Havilah, in the mountains east of Bakersfield and north of Tehachapi.


title: Kern County, California

text: Settlers considered the flat land of the valley inhospitable and impassable at the time due to swamps, lakes, tule reeds and diseases such as malaria. This changed when residents started draining land for farming and constructing canals, most dug by hired Chinese laborers. Within 10 years the valley surpassed the mining areas as the economic power of the county, and as a result the county seat was moved from Havilah to Bakersfield in 1874.


title: Kern County, California

text: In 1899, the discovery well of the Kern River Oil Field was dug by hand and soon the towns of Oil City, Oil Center and Oildale came into existence.


title: Kern County, California

text: The county derives its name from the Kern River, which was named for Edward Kern, cartographer for General John C. Frémont's 1845 expedition, which crossed Walker Pass. The Kern River was originally named Rio Bravo de San Felipe by Father Francisco Garcés when he explored the area in 1776.


title: Kern County, California

text: Throughout recorded history, severe earthquakes have struck Kern County, including the 1857 Fort Tejon earthquake.


title: Kern County, California

text: On July 21, 1952, an earthquake occurred with the epicenter about south of Bakersfield. It measured 7.3 on the moment magnitude scale and killed 12 people. In addition to the deaths, it was responsible for hundreds of injuries and more than $60 million in property damage. The main shock was felt over much of California and as far away as Phoenix, Arizona and Reno, Nevada. The earthquake occurred on the White Wolf Fault and was the strongest to occur in California since the 1906 San Francisco earthquake. Tehachapi suffered the greatest damage and loss of life from the earthquake, though its effects were widely felt throughout central and southern California. The event had a significant aftershock sequence that persisted into July and August with the strongest coming on August 22, an M5.8 event with a maximum perceived intensity of VIII (Severe) and resulted in two additional deaths and an additional $10 million in property damage. Repercussions of the sequence of earthquakes were still being felt in the heavily damaged downtown area of Bakersfield well into the 1990s as city leaders attempted to improve safety of the surviving non-reinforced masonry buildings.


title: Kern County, California

text: Following the event, a field survey was conducted along the fault zone with the goal of estimating the peak ground acceleration of the shock based on visually evaluating rock formations and other indicators. Ground disturbances that were created by the earthquakes were also surveyed, both in the valley and in the foothills, with both vertical and horizontal displacements present in the epicenter area. The motion records that were acquired from the event were significant, and a reconnaissance report was recognized for its coverage of the event, and its setting a standard for similar engineering or scientific papers.


title: Kern County, California

text: Between 1983 and 1986, several ritual sex ring child abuse cases occurred in Kern County, resulting in numerous long prison sentences, all of which were overturned—some of them decades later, because the prosecutors had coerced false testimonies from the purported child victims. The details of these false accusations are covered extensively in the 2008 documentary "Witch Hunt", narrated by Sean Penn.


title: Kern County, California

text: According to the United States Census Bureau, the county has a total area of , of which is land and (0.4%) is water. It is the third-largest county by area in California. The tallest peak in the county is Sawmill Mountain with an elevation of 8822 ft. Its area is nearly the size of the state of New Hampshire; it extends:


title: Kern County, California

text: Kern County suffers from severe air pollution. Particulates cause poor visibility, especially in the winter. Western Kern County lies in the San Joaquin Valley and the topography traps pollutants. Although the topography is not as unfavorable in eastern Kern County, eastern Kern County is a non-attainment area for particulates. Air pollution caused by particulates is "in the unhealthy range an average of 40 days a year, according to the American Lung Association's (ALA) 2018 State of the Air Report.


title: Kern County, California

text: Chaparral comprises a considerable portion of the natural area within Kern County; the species diversity within these chaparral habitats, however, is considerably less than in many other regions of California. Whitethorn is a prominent example of chaparral species on the rocky slopes of the Sierra Nevada as well as the Inner Coastal Ranges. California Buckeye is a notable tree found in both chaparral and forests and whose southern range terminates in Kern County.


title: Kern County, California

text: According to the 2020 United States Census, Kern County's population was 909,235. It was the eleventh-largest county by population in California. The center of population of California is located in Kern County, in the town of Buttonwillow .


title: Kern County, California

text: "Note: the US Census treats Hispanic/Latino as an ethnic category. This table excludes Latinos from the racial categories and assigns them to a separate category. Hispanics/Latinos can be of any race."


title: Kern County, California

text: The 2010 United States Census reported that Kern County had a population of 839,631. The racial makeup of Kern County was 499,766 (59.5%) White, 48,921 (5.8%) African American, 12,676 (1.5%) Native American, 34,846 (4.2%) Asian, 1,252 (0.1%) Pacific Islander, 204,314 (24.3%) from other races, and 37,856 (4.5%) from two or more races. Hispanic or Latino of any race were 413,033 persons (49.2%); 43.4% of Kern County residents are of Mexican heritage, 1.0% Salvadoran, 0.5% Colombian, and 0.4% Guatemalan.


title: Kern County, California

text: According to the 2000 United States Census of 2000, there were 661,645 people, 208,652 households, and 156,489 families residing in the county. The population density was 81 people per square mile (31/km). There were 231,564 housing units at an average density of 28 per square mile (11/km). The racial makeup of the county was 61.6% White, 6.0% Black or African American, 3.4% Asian, 1.5% Native American, 0.2% Pacific Islander, 23.2% from other races, and 4.1% from two or more races. 38.4% of the population were Hispanic or Latino of any race. 8.4% were of German, 7.2% American and 5.7% Irish ancestry, according to the census. 66.8% spoke English, 29.1% Spanish and 1.0% Tagalog as their first language.


title: Kern County, California

text: There were 208,652 households, out of which 42.2% had children under the age of 18 living with them, 54.6% were married couples living together, 14.5% had a female householder with no husband present, and 25.0% were non-families. 20.3% of all households were made up of individuals, and 7.8% had someone living alone who was 65 years of age or older. The average household size was 3.03 and the average family size was 3.50.


title: Kern County, California

text: In the county, the age distribution of the population shows 31.9% under the age of 18, 10.2% from 18 to 24, 29.8% from 25 to 44, 18.7% from 45 to 64, and 9.4% who were 65 years of age or older. The median age was 31 years. For every 100 females there were 105.3 males. For every 100 females age 18 and over, there were 105.3 males.


title: Kern County, California

text: The median income for a household in the county was $35,446, and the median income for a family was $39,403. Males had a median income of $38,097 versus $25,876 for females. The per capita income for the county was $15,760. About 16.8% of families and 20.8% of the population were below the poverty line, including 27.8% of those under age 18 and 10.5% of those age 65 or over.


title: Kern County, California

text: Kern County is associated with the Bakersfield sound. The Buck Owens Crystal Palace is located in Bakersfield.


title: Kern County, California

text: The United States Office of Management and Budget has designated Kern County as the Bakersfield, CA Metropolitan Statistical Area. The United States Census Bureau ranked the Bakersfield, CA Metropolitan Statistical Area as the 63rd most populous metropolitan statistical area and the 68th most populous primary statistical area of the United States as of July 1, 2012.


title: Kern County, California

text: Kern County is a California Constitution defined general law county and is governed by an elected Board of Supervisors. The Board consists of five members, elected by districts, who serve four-year staggered terms. The county government provides countywide services such as elections and voter registration, some law enforcement, jails, vital records, property records, tax collection, public health, and social services. In addition, the County serves as the local government for all unincorporated areas.


title: Kern County, California

text: The Kern County Sheriff provides court protection, jail administration, and coroner services for the entire county of approximately 900,000 in population. It provides patrol and detective services for the unincorporated areas of the county and by contract to certain municipalities. The main Sheriff's office and station is at Bakersfield. There are 15 sheriff substations for the widespread county.


title: Kern County, California

text: Municipal police departments in the county are: Bakersfield, population 384,000; Delano, 54,000; Ridgecrest, 29,000; Wasco (sheriff contract city), 28,000; Arvin, 21,000; Shafter, 20,000; McFarland, 15,000; California City, 14,671; Tehachapi, 13,000; Taft, 9,327; Maricopa (sheriff contract city), 1,200.


title: Kern County, California

text: Kern is a strongly Republican county in Presidential and congressional elections. The last Democratic candidate for President to win a majority in the county was Lyndon Johnson in 1964. Kern remains the only county in Southern California that consistently votes Republican in recent elections. However, Republican margins in the county have shrunk recently, with Donald Trump's 10.2% margin of victory in 2020 being the smallest since Gerald Ford's 6.7% majority in 1976.


title: Kern County, California

text: On November 4, 2008, Kern County voted 75.29% in favor of Proposition 8, which amended the California Constitution to ban same-sex marriages.


title: Kern County, California

text: Fire protection within the county is provided by the Kern County Fire Department. Law enforcement within the county is provided by the Kern County Sheriff's Department.


title: Kern County, California

text: The Kern County Fire Department (KCFD) is an agency that provides fire protection and emergency medical services for the county of Kern, California, USA. Over 625 permanent employees and 100 extra help employees protect an area which spans over . KCFD provides fire protection services for over 500,000 citizens living in the unincorporated areas of Kern County and the cities of Arvin, Delano, Maricopa, McFarland, Ridgecrest, Shafter, Taft, Tehachapi and Wasco. This agency is contracted to provide dispatch services for the California City Fire Department, Kern Ambulance based in Wasco, and Liberty Ambulance of Bakersfield. Over 546 uniformed firefighters are stationed in 46 fire stations throughout the county.


title: Kern County, California

text: The Kern County Sheriff's Department is the agency responsible for law enforcement within the county of Kern. The department provides law enforcement within the county, maintains the jails used by both the county and municipal cities, and provides search and rescue. The department contains over 1,200 sworn deputies and civilian employees. Its jurisdiction contains all of the unincorporated areas of Kern County, approximately . The department headquarters is located at 1350 Norris Road in Bakersfield. There are 15 additional substations located throughout the county. The metro patrol area is divided into four regions: north, south, east, and west.


title: Kern County, California

text: In 2009, the district attorney claimed "the highest per capita prison commitment rate of any major California county." Kern County contains multiple state and federal prisons, including two private prisons. As a result, the courts have been known to sentence a higher than average number of defendants to long prison sentences to help the local economy. The county is among the most prolific with the death penalty, assigning death penalty sentences in 26 cases since 1976. In 2015 Kern County policemen from all departments killed more people per capita than any other American county. Because of the very harsh local criminal justice system, Kern County has been dubbed "the most punitive authoritarian jurisdiction on the west coast" and "Oklahoma of the west". In 2015, it was revealed that the Kern County Sheriff's office engaged in a longstanding program of attempted cash payoffs to women who had accused deputies of sexual assault. In the same year, a civil lawsuit filed by a survivor of a sexual assault committed by Kern County Sheriff's deputy Gabriel Lopez was settled for $1 million.


title: Kern County, California

text: Kern County also holds the distinction of having the most deaths per capita in the US by police shooting per an article published in The Guardian on December 1, 2015. In 2015 to the date of publication of the article, there have been 13 deaths by police shootings in a county of less than 875,000 population, or 0.016 per thousand persons. By comparison, during the same period of time in New York City, a population 10 times the size with a police force more than 20 times the size, there were 9 such deaths.


title: Kern County, California

text: The following table includes the number of incidents reported and the rate per 1,000 persons for each type of offense.


title: Kern County, California

text: The county has a large agricultural base and is a significant producer of oil, natural gas, hydro-electric power, Biomass, solar power, and wind power. Kern is noted for minerals, including gold, borate, and kernite. The largest open pit mine in California, which mines borax, is at Boron. As of October 1, 2016, Kern County contains nearly 25% of California's in-state renewable energy production, including 1,785 MW of solar power and 3,310 MW of wind power. Kern County is home to the Tehachapi Energy Storage Project, which was commissioned in 2014.


title: Kern County, California

text: Department of Defense facilities include Edwards Air Force Base and China Lake Naval Air Weapons Station. As home to Edwards Air Force Base the Air Force's main flight test facility, Kern has been the site of many milestones, including the first supersonic flight and the first landing of the Space Shuttle. The base has brought prosperity to the railroad towns of Mojave and Rosamond. Kern County is also the home of the first inland spaceport in the United States, the Mojave Spaceport.


title: Kern County, California

text: This has long been one of the county's biggest industries. Between 2012 and 2013 the produced value of ag products increased 6%, to a total of $6.8 billion. Grape is 31%, almond is 17%, milk is 13%, citrus is 11%, cattle + calves and pistachio are both 7%, carrots are only 6% (but that's 80% of carrots for the entire United States), hay is 4%, and cotton and potatoes are both 2%, of that. This is one of the highest-producing locations in the United States for vegetables, and also for watermelons. Vegetables are estimated to total $320 million every year. There are about 1,938 farms, at an average size of (however 41% are smaller than ), being the primary employment of 63% of operators.


title: Kern County, California

text: Major producer of almonds with production greater than annually. That is third of all the counties, 16% of the state's production. (See also almond in California.)


title: Kern County, California

text: Pistachio is another important employer here. The Michailides & Avenot group finds severe boscalid resistance in isolates of "Alternaria alternata" pathogenic on pistachio here. They find extensive such resistance in a swathe from the center down into the central southern part of the state, but especially here. (See also Pistachio in California and boscalid in California.)


title: Kern County, California

text: The Glassy-Winged Sharpshooter ("Homalodisca vitripennis") is a major insect pest in this county, including in this county's citrus groves. (See also Glassy-Winged Sharpshooter in California.)


title: Kern County, California

text: , Kern is California's top oil-producing county, with 78% of the state's 56,653 active oil wells and 71% of oil production. The county produced 144.5 million barrels of oil in 2015, accounting for about 4% of overall U.S. oil production.


title: Kern County, California

text: Oil development began with the 1894 discovery of the Midway-Sunset Oil Field, now the third-largest in the United States, in the southwestern portion of Kern County near Maricopa. The 1899 discovery along the Kern River was a breakthrough in oil production. Oil was refined here even before the establishment of the county. The Buena Vista Petroleum Company was organized and incorporated in 1864. Soon thereafter a refinery was built that operated until April 1867 when work ceased because of high freight charges.


title: Kern County, California

text: The 1910 Lakeview Gusher was the largest recorded oil strike in U.S. history. The well spewed approximately nine million barrels for 18 months before workers finally were able to cap it.


title: Kern County, California

text: Other big oil fields in southwestern Kern County discovered early in the 20th century include the Buena Vista, the South Belridge and the Cymric fields. The latter is the fastest-growing field in California in terms of barrels produced per year. Later large fields include the Kern River Oil Field, the fifth-largest in the U.S., the adjacent Kern Front Oil Field, the Mount Poso Oil Field in the lower foothills of the Sierra north-northeast of Bakersfield and the Fruitvale Oil Field, which underlies much of the city of Bakersfield, along and north of the Kern River.


title: Kern County, California

text: On July 22, 2009, Occidental Petroleum announced it had discovered the equivalent of 150 million to 250 million barrels of oil in Kern County, which the company called the largest oil discovery in California in 35 years. The find added about 10 percent to California's known reserves. Occidental's Ray Irani said it is likely that more oil would be found in the areas outside the initial six wells that tapped the discovery. Occidental has not revealed the exact location of the find, two-thirds of which is natural gas. BNET, an industry web publication, said the find would add to the company's 708 million barrels of proven reserves in California.


title: Kern County, California

text: The county today contributes more than three-quarters of all the oil produced onshore in California. Some of the large oil fields in Kern County which are still active include:


title: Kern County, California

text: Outdoor recreational activities include horseback riding, water skiing (Lake Isabella, Lake Buena Vista, Lake Ming, and private ski ranches). Off-roading and other motorsports take place at Jawbone Canyon, California City, Randsburg, Willow Springs, Buttonwillow, Bakersfield Speedway, Famoso Raceway, and the half-mile Kern Raceway. Hunting, paintball, white-water rafting, kayaking (Kern River), snow skiing (Shirley Meadows and Mount Pinos), shooting ranges (5 Dogs Range), hiking, biking (trails, paths, and roads), camping and fishing are also part of the recreational culture.


title: Kern County, California

text: However, California City, Ridgecrest, and other areas in the Mojave Desert regions of eastern Kern County may instead receive Los Angeles stations.


title: Kern County, California

text: The 2015 Disney film "McFarland, USA", starring Kevin Costner, is based on the cross-country team in the city of McFarland, California, which is located in northern Kern County.


title: Ganglion

text: A ganglion is a group of neuron cell bodies in the peripheral nervous system. In the somatic nervous system this includes dorsal root ganglia and trigeminal ganglia among a few others. In the autonomic nervous system there are both sympathetic and parasympathetic ganglia which contain the cell bodies of postganglionic sympathetic and parasympathetic neurons respectively.


title: Ganglion

text: Ganglia are primarily made up of somata and dendritic structures which are bundled or connected. Ganglia often interconnect with other ganglia to form a complex system of ganglia known as a plexus. Ganglia provide relay points and intermediary connections between different neurological structures in the body, such as the peripheral and central nervous systems.