title: Babson College

text: The main residential campus of Babson College is and located in the "Babson Park" section of Wellesley, Massachusetts, just fifteen miles west of Boston. It is adjacent to the Franklin W. Olin College of Engineering. Undergraduate and graduate students have the opportunity to take advantage of campus amenities including the student center, the cafeteria, Horn Library, multiple centers and institutes, the Webster fitness center, the Weissman Foundry the arts center, and a new centennial park known as the Kerry Murphy Healey Park, home of the second-largest rotating globe in the world at 28 feet in diameter. Executive and Enterprise Education visitors have the opportunity to stay in guest rooms adjacent to the meeting center.


title: Babson College

text: The Babson Boston classroom and event space, located at 100 High Street, gives Babson a presence in downtown Boston and connects the resources of the college with the innovative companies, organizations and leaders in Boston's Financial District. Opened in 2016, this facility provides the opportunity to offer MBA courses in a location convenient to where entrepreneurial students live and work. Over the course of the academic year, there are opportunities at the Boston location for students to engage with Babson offices and resources, including graduate programs, the Graduate Center for Career Development, Graduate Admissions and the college's alumni network. From 2011 to 2016, Babson Boston campus was at 253 Summer Street in Boston's Innovation District.


title: Babson College

text: Babson has a San Francisco campus which includes a Blended Learning MBA program, an undergraduate semester experience and custom executive education.


title: Babson College

text: In July 2017, President Kerry Healey announced Babson's plans to expand to Miami, Florida, where the Blended Learning MBA, Master of Science in Business Analytics (MSBA) and Certificate in Advanced Management (CAM), three graduate programs are offered. Babson's new hub in Miami is located at 1200 Brickell Avenue.


title: Babson College

text: Babson College offers all undergraduates a Bachelor of Science degree. Students are also given the option to declare concentrations their junior and senior year from a broad range of subjects in various business and other fields. Programs are accredited by Association to Advance Collegiate Schools of Business (AACSB) and the college itself has been regionally accredited by the New England Association of Schools and Colleges (NEASC) since 1950.


title: Babson College

text: In rankings, Babson was rated first among all colleges and universities in the nation by "Money Magazine" in 2014. In 2015, the magazine ranked Babson #1 in 10 Great Colleges for Business Majors. Babson's MBA program was ranked 58th overall in the "Bloomberg Businessweek" 2014 rankings. Babson's undergraduate business program is ranked 26th overall in the Bloomberg Businessweek 2014 rankings. Babson's undergraduate Entrepreneurship program has been ranked number one for the past 17 years by "U.S. News & World Report." In their 2013–2014 salary report, Payscale.com ranked Babson College at number five of all U.S. colleges and universities, ahead of schools such as Stanford, University of Pennsylvania, Harvard, Dartmouth, Columbia and Yale. This ranking represents an average mid-career salary of $123,000 and average starting salary of $59,700. In 2012, "Bloomberg Businessweek" ranked Babson eleventh among U.S. schools based on return on investment. CNN money ranked Babson eighth in their 2016 "Colleges with the highest-paid grads" rankings.


title: Babson College

text: Babson College offers master's degrees through its F. W. Olin Graduate School of Business, including a One-Year MBA Program, a Two-Year MBA Program, a 42-month Evening MBA Program and a Blended Learning MBA Program with campuses located in Boston, San Francisco and Miami. It also offers a Master's of Science in Entrepreneurial Leadership (MSEL), Business Analytics (MSBA), Finance (MSF) and a Certificate of Advanced Management (CAM).


title: Babson College

text: There are several fraternities and sororities on campus: Chi Omega, Delta Tau Delta, Kappa Kappa Gamma, Sigma Kappa and Sigma Phi Epsilon. There are also two professional business fraternities on campus: Delta Sigma Pi and Alpha Kappa Psi. Babson College Radio was started in 1998.


title: Babson College

text: Babson offers a variety of special interest housing, such as CODE (Community of Developers and Entrepreneurs), theStudio (the housing associated with CREATE, the arts-based student club), E-Tower, ONE Tower: Origins of Necessary Equality, GIVE Tower, Healthy Living Tower and Women Giving Back Tower.


title: Babson College

text: Babson's teams are known as the "Beavers" and its colors are green and white. The school has 22 varsity sports teams, the majority of which compete in the New England Women's and Men's Athletic Conference (NEWMAC) of the NCAA Division III. One of Babson's flagship sports is baseball which has won 7 Conference Championships and been to 5 NCAA Tournaments, including the 2019 College World Series. Additionally, the men's soccer team has established a history of success with 3 NCAA National Championships, 27 NCAA tournaments wins and 12 conference championships. The men's and women's alpine ski teams compete in the United States Collegiate Ski and Snowboard Association (USCSA) and the men's lacrosse team competes in the Pilgrim League. Babson College's men's hockey team competes in the Eastern College Athletic Conference (ECAC) and has won (1) NCAA D3 National title, (1) ECAC D2 title, six ECAC East Championships, appearing in the championship game in 9 of the last 12 seasons . Babson College's golf team competes in the New England Collegiate Conference (NECC) and won the title in 2011 giving them an automatic bid to the NCAAs. They were led by senior captain Joe Young who won NECC golfer of the year in 2011. Babson United Rugby Club won Northeast region of NSCRO 7's in 2016. In 2017, the school will begin construction of the Babson Recreation and Athletics Center, a major new facility that will support varsity, intramural and recreational sports and many other activities by the fall of 2019. In March 2017, Babson's basketball team won the Division III National Championship.


title: Black Celebration

text: Black Celebration is the fifth studio album by English electronic music band Depeche Mode, released on 17 March 1986 by Mute Records. The album further cemented the darkening sound created by Alan Wilder, which the band later used for their subsequent albums "Music for the Masses", "Violator", and "Songs of Faith and Devotion".


title: Black Celebration

text: "Black Celebration" reached number four on the UK Albums Chart, and has been cited as one of the most influential albums of the 1980s. To promote the album, the band embarked on the Black Celebration Tour. Three years after its release, "Spin" ranked it at number 15 on its "25 Greatest Albums of All Time" list.


title: Black Celebration

text: Contemporaneous reviews for "Black Celebration" in the British press were mixed. "Melody Maker"s Steve Sutherland lambasted the album and wrote that Depeche Mode came off as "pussycats desperate to appear perverted as an escape from the superficiality of teen stardom" and "Sounds" published a similarly scathing review. While criticizing chief songwriter Martin Gore's "adolescent fragments of despair", Sean O'Hagan of "NME" nonetheless praised "Black Celebration"s "perfectly constructed jigsaw melodies" and concluded, "When the songs address topics other than the composer's state of mind – as on the evocative exploration of loneliness that is 'World Full of Nothing' – Depeche Mode sound like a lot more than just a high tech, low-life melodrama." Betty Page of "Record Mirror" felt that the band should be admired for their "refusal to follow anything but their own fashion" and "unswerving ability to come up with great, fresh melodies."


title: Black Celebration

text: "Black Celebration" has since been reappraised in retrospective reviews. In 2007, Rob Sheffield of "Rolling Stone" referred to the album as an "instant classic for the band's fans" that at the time of its release had seemingly been "utterly ignored by everybody else."


title: Black Celebration

text: Trent Reznor of Nine Inch Nails cited "Black Celebration", and its subsequent tour, as an influence and said it helped inspire him to write the album "Pretty Hate Machine" (1989).


title: Black Celebration

text: In 2007, "Black Celebration" was re-released with a bonus DVD. It was released on 20 March 2007 in the United States, on 26 March in the United Kingdom and on 2 April in the rest of Europe, as a part of the third wave of reissues (along with "Construction Time Again"). The first CD was remastered and (except in the United States) released as a CD/SACD hybrid. The bonus DVD includes the album's B-sides in addition to the singles and B-sides for "Shake the Disease" and "It's Called a Heart", two singles released in the interim between "Some Great Reward" and "Black Celebration". The reissue also includes several live versions of some of the songs from "Black Celebration". The album is released as originally intended and ends with "New Dress" (not "Black Day" or "But Not Tonight").


title: Black Celebration

text: As with the other reissues, the accompanying DVD includes a documentary film. The film's title, "The Songs Aren't Good Enough, There Aren't Any Singles and It'll Never Get Played on the Radio", is a quote from the film in which Martin Gore paraphrases Daniel Miller's reaction to the album's early demos. The documentary includes much detail about the making of the album, its singles and the ensuing tour. Other highlights include the band meeting the Cure and behind-the-scenes footage of several of the music videos.


title: Black Celebration

text: The remastered album was released on vinyl on 2 April 2007 in Europe and on 11 September 2007 in the United States.


title: Euler's constant

text: Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().


title: Euler's constant

text: It is defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by formula_1


title: Euler's constant

text: The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled "De Progressionibus harmonicis observationes" (Eneström Index 43). Euler used the notations and for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations and for the constant. The notation appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time perhaps because of the constant's connection to the gamma function. For example, the German mathematician Carl Anton Bretschneider used the notation in 1835 and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.


title: Euler's constant

text: Euler's constant appears, among other places, in the following (where '*' means that this entry contains an explicit equation):


title: Euler's constant

text: The number has not been proved algebraic or transcendental. In fact, it is not even known whether is irrational. Using a continued fraction analysis, Papanikolaou showed in 1997 that if is rational, its denominator must be greater than 10. The ubiquity of revealed by the large number of equations below makes the irrationality of a major open question in mathematics.


title: Euler's constant

text: However, some progress was made. Kurt Mahler showed in 1968 that the number formula_4 is transcendental (here, formula_5 and formula_6 are Bessel functions). In 2009 Alexander Aptekarev proved that at least one of Euler's constant and the Euler–Gompertz constant is irrational; Tanguy Rivoal proved in 2012 that at least one of them is transcendental. In 2010 M. Ram Murty and N. Saradha showed that at most one of the numbers of the form


title: Euler's constant

text: with and is algebraic; this family includes the special case . In 2013 M. Ram Murty and A. Zaytseva found a different family containing , which is based on sums of reciprocals of integers not divisible by a fixed list of primes, with the same property.


title: Euler's constant

text: is related to the digamma function , and hence the derivative of the gamma function , when both functions are evaluated at 1. Thus:


title: Euler's constant

text: can also be expressed as an infinite sum whose terms involve the Riemann zeta function evaluated at positive integers:


title: Euler's constant

text: The error term in the last equation is a rapidly decreasing function of . As a result, the formula is well-suited for efficient computation of the constant to high precision.


title: Euler's constant

text: This formula indicates that when taking any positive integer n and dividing it by each positive integer k less than n, the average fraction by which the quotient n/k falls short of the next integer tends to formula_17 (rather than 0.5) as n tends to infinity.


title: Euler's constant

text: Closely related to this is the rational zeta series expression. By taking separately the first few terms of the series above, one obtains an estimate for the classical series limit:


title: Euler's constant

text: where is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, . Expanding some of the terms in the Hurwitz zeta function gives:


title: Euler's constant

text: The integral on the second line of the equation stands for the Debye function value of +∞, which is .


title: Euler's constant

text: One can express using a special case of Hadjicostas's formula as a double integral with equivalent series:


title: Euler's constant

text: for any formula_30. However, the rate of convergence of this expansion depends significantly on formula_31. In particular, formula_32 exhibits much more rapid convergence than the conventional expansion formula_33. This is because


title: Euler's constant

text: Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below.


title: Euler's constant

text: where are the "Bernoulli polynomials of the second kind", which are defined by the generating function


title: Euler's constant

text: Alabdulmohsin derived closed-form expressions for the sums of errors of these approximations. He showed that (Theorem A.1):


title: Euler's constant

text: The constant is important in number theory. Some authors denote this quantity simply as . equals the following limit, where is the th prime number:


title: Euler's constant

text: This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow using hypergeometric functions.


title: Euler's constant

text: The continued fraction expansion of begins , which has no "apparent" pattern. The continued fraction is known to have at least 475,006 terms, and it has infinitely many terms if and only if is irrational.


title: Euler's constant

text: Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Mascheroni attempted to calculate the constant to 32 decimal places, but made errors in the 20th–22nd and 31st-32nd decimal places; starting from the 20th digit, he calculated ...1811209008239 when the correct value is ...0651209008240.


title: Next Dutch general election

text: The next Dutch general election to elect the members of the House of Representatives is scheduled for March 2025, but may be held at an earlier date if a snap election is called.


title: Next Dutch general election

text: Pursuant to articles C.1, C.2 and C.3 of the electoral law, elections for the House of Representatives take place every four years in March. The 150 members of the House of Representatives are elected by open list proportional representation. The number of seats per list is determined using the D'Hondt method. There is an official electoral threshold of 1/150th (0.67%) of votes to secure a seat. Voters have the option to cast a preferential vote. The seats won by a list are first allocated to the candidates who, in preferential votes, have received at least 25 percent of the number of votes needed for one seat (effectively 0.17% of the total votes), regardless of their placement on the electoral list. If multiple candidates from a list pass this threshold, their ordering is determined based on the number of votes received. Any remaining seats are allocated to candidates according to their placement on the electoral list.


title: Dave Sheridan (actor)

text: David Christopher Sheridan (born March 10, 1969) is an American actor, comedian, writer, producer, and musician. He played Special Officer Doofy in the comedy "Scary Movie" (2000) (a spoof of deputy sheriff Dewey Riley from "Scream").


title: Dave Sheridan (actor)

text: Sheridan was born in Newark, Delaware. He was a member of Tau Kappa Epsilon fraternity at William Paterson University in Wayne, New Jersey.


title: Dave Sheridan (actor)

text: He began his career as an intern at "Saturday Night Live" during the 1991-1992 season. From there, he entered Chicago's famed comedy troupe, The Second City, where he wrote, directed, and produced "Dave Sheridan's America", a multimedia stage show. He starred on "Buzzkill", a 1996 MTV reality series that featured three slacker buddies staging elaborate pranks and catching it all on video.


title: Dave Sheridan (actor)

text: Sheridan is well known for his role in "Scary Movie" (2000) as Officer Doofy. He portrayed the character Doug in the 2001 film "Ghost World". Sheridan also appeared in the 2002 music videos for "By the Way" (playing a crazy taxi driver) and "Universally Speaking" by Red Hot Chili Peppers. Sheridan toured with his band that formed in 2004, Van Stone.


title: Dave Sheridan (actor)

text: Sheridan hosted a series called "Smile...You're Under Arrest!" on the Fox Reality Channel in 2008–09. In 2014, he portrayed Sheriff Lincoln in the parody film "The Walking Deceased", inspired by "The Walking Dead".


title: Dave Sheridan (actor)

text: In 2016, Sheridan started co-hosting "The Dave and Creech Show" podcast with actor/podcaster CJ Creech.


title: Scott Campbell (tattoo artist)

text: Scott Campbell (born May 13, 1977) is a New York–based American artist and tattoo artist whose clients include Howard Stern, Sting, Robert Downey Jr., Courtney Love, Orlando Bloom, Josh Hartnett, and Marc Jacobs.


title: Scott Campbell (tattoo artist)

text: Campbell grew up in New Orleans, Louisiana, with a conservative upbringing in a Southern Baptist family, that sparked his search for something different. He graduated in 1995 from Kingwood High School. Early on, Campbell debated the idea of a middle-class life. At the University of Texas, he studied biochemistry and planned a career as a medical illustrator. Soon he dropped out.


title: Scott Campbell (tattoo artist)

text: In his 20s, Campbell moved to San Francisco and began working as a copy editor for Lawrence Ferlinghetti, at City Lights Bookstore. It was then that he began tattooing. He traveled Asia and Europe, where he tattooed for cash, then moved to Williamsburg in 2001.


title: Scott Campbell (tattoo artist)

text: When he moved to New York in 2005 he opened his own tattoo shop, called Saved Tattoo. It was here that Scott's career led to fame. His first celebrity client was Heath Ledger, who commissioned a small bird in flight on his left forearm.


title: Scott Campbell (tattoo artist)

text: As a visual artist, Campbell is best known for a series of reliefs he creates using sheets of uncirculated US currency. He exhibits regularly internationally; his first show sold out at OHWOW, a gallery in Miami, Florida (April 2009). The gallery opened a location in New York at which Campbell had a solo show in April 2010.


title: Scott Campbell (tattoo artist)

text: Louis Vuitton Creative Director Marc Jacobs collaborated with Scott Campbell, whereby Scott customized a number of fancy leather bags for a men's line for the 2011 S/S collection.


title: Scott Campbell (tattoo artist)

text: In November 2015, Campbell debuted 'Whole Glory' at Milk Studios in New York City. The project saw 23 participants getting tattooed by Campbell through a hole in a wall. The participants did not communicate at all with the artist prior to being tattooed and he tattooed a piece of his choice. Whole Glory was also presented in Los Angeles, in Downtown LA, Moscow at the Garage Museum of Contemporary Art, and most recently at Covent Garden in London, presented by Lazarides Gallery. 'Whole Glory' London also debuted a series of tattooed pig skins sewn together with Campbell's designs encased in stainless steel frames in liquid preservative.


title: Scott Campbell (tattoo artist)

text: Campbell met actress Lake Bell on the set of her HBO show "How to Make It in America" in 2011. They were married on June 1, 2013, in New Orleans. In late October 2014, her representative confirmed that Bell had given birth to their daughter, Nova. In May 2017, Bell gave birth to their second child, a son named Ozzi, which is short for Ozgood. They separated in 2020.


title: Tyler Breeze

text: Matthew Clement (born January 19, 1988) is a Canadian professional wrestler and professional wrestling trainer. He is currently signed to WWE under the ring name Tyler Breeze as a trainer at the WWE Performance Center.


title: Tyler Breeze

text: From 2010 to 2012, he performed under the name Mike Dalton in WWE's developmental territory Florida Championship Wrestling (FCW). While in FCW, he won the FCW Florida Heavyweight Championship and the FCW Florida Tag Team Championship (with Leakee). In 2013, he transitioned to NXT and came up with the Tyler Breeze character, a narcissistic "pretty boy" obsessed with taking selfies of himself, even during matches. In 2015, Clement was promoted to WWE's main roster, where he would later team up with Fandango to form a team known as "Breezango" and later on as "The Fashion Police" after the two began appearing in a number of comedic skits known as "The Fashion Files". In 2019, Breezango returned to NXT, with the duo winning the NXT Tag Team Championship in August 2020.


title: Tyler Breeze

text: In 2019, Clement opened a wrestling school called Flatbacks Wrestling in Apopka, Florida, with Shawn Spears.


title: Tyler Breeze

text: Matthew Clement was born on January 19, 1988 in Penticton, British Columbia. He trained under fellow professional wrestler Lance Storm at Storm's academy in Calgary.


title: Tyler Breeze

text: Clement made his debut for Power Zone Wrestling in 2007 under the ring name Mattias Wild. His first match was a victory over Rage O'Riley on April 25, 2007. In 2008, Clement began wrestling for NWA Extreme Canadian Championship Wrestling (ECCW). While wrestling for ECCW, Clement was trained by Lance Storm in Calgary, Alberta. On May 2, Clement won a match against Alex Plexis to qualify for the first round of the 2009 Pacific Cup; he then lost in the first round to Billy Suede on June 6, 2009. On September 5, 2009, he lost a tag team match with Dan Myers against the team of "The Icons" Jamie Diaz and Nick Price in a first round match for the ECCW Tag Team Championship. The teams then began a feud, with Price and Diaz coming out on top on most occasions. Clement then worked for Prairie Wrestling Alliance in 2010, returning to ECCW for a last match on September 11, 2010, which he lost to Billy Suede.


title: Tyler Breeze

text: After signing with World Wrestling Entertainment (WWE), Clement made his debut for WWE's developmental territory Florida Championship Wrestling (FCW) on December 2, 2010, at a live event as Matt Clement, where he defeated Big E Langston, Darren Young, Leakee and 2 others. The following month he was called Mike McGrath, which lasted 3 months, only to be changed again to Mike Dalton in April 2011. Dalton made his television debut on the July 17, 2011 episode of "FCW TV" in a loss to Alexander Rusev. He earned another win in FCW on November 17, 2011, against Peter Orlov. On December 15, Dalton won a battle royal to be named the number-one contender to the FCW Florida Heavyweight Championship. He challenged Leo Kruger, but lost the match. Dalton won the FCW Heavyweight Championship on February 2, 2012, when he defeated Kruger. His reign lasted 21 days and he lost the rematch and the title to Kruger.


title: Tyler Breeze

text: When WWE rebranded FCW into NXT Wrestling, Dalton debuted as an enhancement talent on the June 20, 2012, episode of rebooted "NXT" show, where he teamed with CJ Parker and lost to The Ascension (Kenneth Cameron and Conor O'Brian). Dalton got his first victory when he teamed with Jason Jordan and defeated the team of Hunico and Camacho on August 1 but lost in a rematch against the two a few weeks later.


title: Tyler Breeze

text: In the beginning of 2013, seeing that Dalton's character hadn't receive much of the fans' attention, WWE officials informed him that in order to remain with WWE he should get a new character. On the July 24 episode of "NXT", Clement re–debuted with his new heel gimmick, as Tyler Breeze, a narcissistic "pretty boy" obsessed with taking selfies, even during matches; and later was revealed that Clement himself came up with the character, with Triple H giving him the suggestion of using an iPhone to take pictures as a way to make the character more modern. Soon after, Breeze started a feud with CJ Parker, after repeatedly being "photobombed" by Parker. After the two were put in a tag team match against Enzo Amore and Colin Cassady, which they lost after Parker refused to tag in Breeze, a match between the two took place on the October 16 episode of "NXT", which Breeze won. Two weeks later, Parker defeated Breeze in a rematch, and ended their feud.


title: Tyler Breeze

text: In December, Breeze helped Bo Dallas retain the NXT Championship during a lumberjack match against Adrian Neville, which resulted a match between the two on the January 15, 2014, episode of "NXT", where Neville was victorious. Breeze retorted later that Neville was not good-looking, mocking his nickname. The next month, in a rematch, Neville once again defeated Breeze. On February 27, at NXT Arrival, Breeze was scheduled to face Xavier Woods, but both men were attacked by Alexander Rusev, rendering the match a no contest. Throughout March, Breeze continued his feud with Woods, by attacking him before a match with Rusev, before the two faced each other in a match, which Breeze won. On the May 8 episode of "NXT", Breeze participated in a 20-man battle royal, for a shot at the NXT Championship, and was involved in a three-way tie. As a result, Breeze faced the other two winners, Sami Zayn and Tyson Kidd in a triple–threat match on the following week, which Kidd won. On May 29, at NXT TakeOver, Breeze defeated Sami Zayn in another number one contender's match but in mid–June he suffered a broken finger, which kept him out of action for a month. Breeze returned on the July 24 episode of "NXT", defeating Mojo Rawley and received his title match against champion Adrian Neville on the August 14 episode of "NXT", but was unsuccessful after Kidd interfered and caused a disqualification. This led to a match between Breeze and Kidd on the following week, which Kidd won, when Breeze counted himself out. On September 11, Breeze, along with Zayn and Kidd, once again challenged unsuccessfully for the title at . Prior to this, Breeze made his main roster debut on the September 8 episode of "Raw", teaming with Kidd in a match against Neville and Zayn, to hype their title match. Throughout the next months, Breeze competed in various matches, but was not involved in any feud.


title: Tyler Breeze

text: On the December 18 episode of "NXT", a cell phone vignette from Breeze aired, claiming that he was taking a short break from NXT to model overseas.<ref name="WWE NXT Results - 12/18/14 (Neville vs. Owens)"></ref> He returned on the January 8, 2015 episode of "NXT", defeating the debuting Chad Gable. On the January 21 episode of "NXT", Breeze was defeated by Hideo Itami in a first round match of a tournament to determine the new number one contender for the NXT Championship. In a rematch at , on February 11, Itami again defeated Breeze. Breeze's rivalry with Itami continued and the two along with Finn Bálor were scheduled for a triple–threat number one contender's match at , on May 20, however, Itami was attacked in the parking lot, and the match was turned into a one on one between Breeze and Bálor, which Bálor won. During the tapings of "NXT", on July 16, William Regal scheduled Breeze to wrestle Jushin Thunder Liger at , on August 22. Breeze called himself the "Face of the Rising Sun" until Liger won. Breeze then entered the "Dusty Rhodes Tag Team Classic" tournament, where he was paired with Bull Dempsey. The two were eliminated from the tournament by Tommaso Ciampa and Johnny Gargano on the September 9 episode of "NXT" when Dempsey got pinned. This led to a match between the two on the September 23 episode of "NXT", which Breeze won. On October 7, at , Breeze competed in a match against Apollo Crews, which he would lose. It would mark his final match at NXT, as Breeze was called up to the main roster after the event by Triple H.


title: Tyler Breeze

text: On the October 22, 2015 episode of "SmackDown", Breeze was promoted to WWE's main roster, appearing during a Miz TV segment, after being introduced by Summer Rae, and attacked Dolph Ziggler. On the November 9 episode of "Raw", Breeze made his in–ring debut, losing to Dean Ambrose in a first round match from the WWE World Heavyweight Championship tournament. Throughout November, Breeze continued his feud with Ziggler, resulting in a match at Survivor Series, which Breeze won. However, Ziggler defeated Breeze on "Raw" and "SmackDown", in two rematches, to officially end their feud. After competing in various matches, Breeze and Summer Rae amicably went their separate ways on the December 31 episode of "SmackDown".


title: Tyler Breeze

text: Starting on the January 11, 2016 episode of "Raw", Breeze went on a winless streak for over 20 televised matches, including losing to the likes of Zack Ryder and Jack Swagger on "Main Event" or "Superstars". At Royal Rumble, Breeze was eliminated within two minutes in his first Royal Rumble match by Roman Reigns and AJ Styles. On the February 15 episode of "Raw", Breeze competed in a five way match against Dean Ambrose, Kevin Owens, Stardust and Dolph Ziggler for Ambrose's WWE Intercontinental Championship, where Breeze was pinned by Owens. At WrestleMania 32, Breeze competed in the André the Giant Memorial Trophy, where he was eliminated by Mark Henry.


title: Tyler Breeze

text: On the April 28 episode of "SmackDown", Breeze formed an alliance with R-Truth. On the May 2 episode of "Raw", Breeze defeated Goldust after interference by Truth. On the May 12 episode of "SmackDown", a tag team match pitting Breeze and Truth against Goldust and Fandango ended with a quirky result when Breeze and Fandango turned on their partners, who had refused to fight each other. Breeze aligned with Fandango, adopting the name "Breezango". The new team defeated The Golden Truth (Goldust and R-Truth) a week later on "Raw" but lost a rematch at the Money in the Bank, ending the feud. After being drafted to "SmackDown" brand as a result of the WWE draft, Breezango defeated The Usos on July 24 at Battleground. Breezango then entered a tournament which would determine the inaugural SmackDown Tag Team Champions, they were eliminated in the first round by American Alpha. On the November 8 episode of "SmackDown Live", Breezango defeated The Vaudevillains to qualify for Team SmackDown for the 10-on-10 Survivor Series Tag Team Elimination match at Survivor Series; they were the first team eliminated from the match, which was ultimately won by Team Raw. During this time, Breezango began to refer themselves as "Fashion Police". At WrestleMania 33, Breeze took part in the André the Giant Memorial Battle Royal, which he failed to win.


title: Tyler Breeze

text: On the April 25, 2017 episode of "SmackDown Live", Breezango defeated The Ascension in a Beat the Clock challenge match, becoming number one contenders for the SmackDown Tag Team Championship. During this time, Breezango transitioned into fan favorites in the process, and this would be Breeze's first time as a face on the main roster. At Backlash, Breezango unsuccessfully challenged The Usos for the SmackDown Tag Team Championship. Breezango lost to Kevin Owens and Sami Zayn at Survivor Series and to The Bludgeon Brothers at Clash of Champions. On March 11, 2018, at Fastlane, Breezango teamed with Tye Dillinger to defeat Mojo Rawley, Chad Gable and Shelton Benjamin in a six-man tag team match. At WrestleMania 34, Breeze competed in the André the Giant Battle Royal, but was unsuccessful. On April 16, Breezango were moved to Raw brand as part of Superstar Shake-up. On that night, they defeated Cesaro and Sheamus. On July 6, it was reported that Fandango had suffered a left labrum tear in his shoulder that would put him out of action for six months, which was why Breeze had been inactive.


title: Tyler Breeze

text: At the end of the year, Breeze and Fandango were sent to NXT. According to Dave Meltzer, WWE assigned them to the brand to work as experienced veterans to work with the younger talent and to help NXT's rating during the Wednesday Night War against the All Elite Wrestling's "Dynamite". Breeze would make his return on the December 12 episode of "NXT", as the surprise opponent for Ricochet's NXT North American Championship match in a losing effort. On the December 17 episode of "Raw", Breeze answered Dean Ambrose's open challenge for the Intercontinental Championship, which he lost. At WrestleMania 35, Breeze competed in the André the Giant Memorial Battle Royal, which was won by Braun Strowman. On April 25, 2019, Breeze started appearing at NXT live events. On the May 22 episode of "NXT", Breeze returned to NXT as a full-time performer, where he interrupted NXT North American Champion Velveteen Dream and took a selfie with him, but then attacked him. It was announced that Breeze would face Velveteen Dream for the NXT North American Championship at , where he lost.


title: Tyler Breeze

text: Fandango returned from injury on the July 31, 2019 episode of "NXT", saving Breeze from a post-match attack by The Forgotten Sons (Jaxson Ryker, Steve Cutler and Wesley Blake). Breezango began a feud with The Forgotten Sons, defeating them in tag team action as well as in a six-man tag action with Isaiah "Swerve" Scott as their partner. However, in December, Fandango was sidelined with another injury as he announced he had undergone Ulnar collateral ligament reconstruction (Tommy John surgery) with an expected recovery of six to twelve months. On the June 3 episode of "NXT", Breeze once again reunited with Fandango after his return from injury as they defeated Oney Lorcan and Danny Burch, and The Undisputed Era in a number one contender's match for the NXT Tag Team Championship. On the June 17 episode of "NXT", Breezango failed to win the titles from Imperium (Fabian Aichner and Marcel Barthel), after interference from both Lorcan and Burch and Indus Sher (Rinku and Saurav).


title: Tyler Breeze

text: At NXT TakeOver XXX, Breezango defeated Oney Lorcan and Danny Burch, and Legado del Fantasma (Joaquin Wilde and Raul Mendoza) in a triple threat tag team match to become the number one contenders for the NXT Tag Team Championship. On the August 26, 2020 episode of "NXT", Breezango defeated Imperium to win the titles, marking Breeze's first title in WWE. on the October 21 episode of "NXT", Breezango lost the titles to Danny Burch and Oney Lorcan ending their reign at 56 days. On the November 11 episode of "NXT", they failed to regain the titles from Lorcan and Burch in a rematch.


title: Tyler Breeze

text: In early 2021, Breezango participated in Dusty Rhodes Tag Team Classic, but were defeated by The Undisputed Era (Adam Cole and Roderick Strong) in the first round. On the June 15 episode of "NXT", they defeated Imperium in what would be their final match in WWE. On June 25, 2021, Breeze, along with Fandango, were released from their WWE contracts.


title: Tyler Breeze

text: While appearing as a guest on the October 5, 2022 episode of "The Bump", Breeze announced that he had returned to WWE as a trainer at the WWE Performance Center.


title: Tyler Breeze

text: In February 2016, Clement started his own YouTube series called "Shootin' the Breeze", where Clement portrays his Breeze character answering fan questions. The first five episodes were released on Clement's own YouTube channel, before being moved to WWE's channel in March.


title: Tyler Breeze

text: Breeze is a recurring guest on Xavier Woods' YouTube channel UpUpDownDown, where he goes by the nickname 'Prince Pretty'. In November 2019, Breeze defeated Alexa Bliss in a game of "Saturday Night Slam Masters" to win the UpUpDownDown Championship, immediately after Bliss had won the championship in the finals of a "Brawlhalla" tournament. It was around this time that Breeze and Woods would start season 2 of their “Battle of the Brands” series, and Woods would challenge Breeze to a match for the first pick. However due to Woods being injured he chose Cesaro as his replacement. During the match Cesaro would turn on Woods and join Breeze forming “LeftRightLeftRight” along with Adam Cole.


title: Tyler Breeze

text: As Tyler Breeze, he made his video game debut as a playable character in "WWE 2K16", and has since appeared in "WWE 2K17", "WWE 2K18", "WWE 2K19", "WWE 2K20"., and WWE 2K22


title: Tyler Breeze

text: With his WWE release in 2021, 'Prince Pretty' would leave the show as a character and co-host temporarily, but returned in 2022 to take on Austin Creed once again in the 'Battle of the Brands' via WWE 2k22.


title: Tyler Breeze

text: In September 2016, Clement married Ashley Miller, better known as Audrey Marie. They have since split up.


title: Tyler Breeze

text: In May 2019, Clement and Ronnie Arneill (best known as Shawn Spears) opened a wrestling school in Apopka, Florida, called Flatbacks Wrestling.


title: Measuring instrument

text: A measuring instrument is a device to measure a physical quantity. In the physical sciences, quality assurance, and engineering, measurement is the activity of obtaining and comparing physical quantities of real-world objects and events. Established standard objects and events are used as units, and the process of measurement gives a number relating the item under study and the referenced unit of measurement. Measuring instruments, and formal test methods which define the instrument's use, are the means by which these relations of numbers are obtained. All measuring instruments are subject to varying degrees of instrument error and measurement uncertainty.


title: Measuring instrument

text: These instruments may range from simple objects such as rulers and stopwatches to electron microscopes and particle accelerators. Virtual instrumentation is widely used in the development of modern measuring instruments.


title: Measuring instrument

text: In the past, a common time measuring instrument was the sundial. Today, the usual measuring instruments for time are clocks and watches. For highly accurate measurement of time an atomic clock is used.


title: Measuring instrument

text: A gas meter measures energy indirectly by recording the volume of gas used. This figure can then be converted to a measure of energy by multiplying it by the calorific value of the gas.


title: Measuring instrument

text: A physical system that exchanges energy may be described by the amount of energy exchanged per time-interval, also called power or flux of energy.


title: Measuring instrument

text: Action describes energy summed up over the time a process lasts (time integral over energy). Its dimension is the same as that of an angular momentum.


title: Measuring instrument

text: This includes basic quantities found in classical- and continuum mechanics; but strives to exclude temperature-related questions or quantities.


title: Measuring instrument

text: Electrical charges interact via a field. That field is called electric field.If the charge doesn't move. If the charge moves, thus realizing an electric current, especially in an electrically neutral conductor, that field is called magnetic.


title: Measuring instrument

text: Electricity can be given a quality — a potential. And electricity has a substance-like property, the electric charge.


title: Measuring instrument

text: Energy (or power) in elementary electrodynamics is calculated by multiplying the potential by the amount of charge (or current) found at that potential: potential times charge (or current). (See Classical electromagnetism and Covariant formulation of classical electromagnetism)


title: Measuring instrument

text: Temperature-related considerations dominate thermodynamics. There are two distinct thermal properties: A thermal potential — the temperature. For example: A glowing coal has a different thermal quality than a non-glowing one.


title: Measuring instrument

text: And a substance-like property, — the entropy; for example: One glowing coal won't heat a pot of water, but a hundred will.


title: Measuring instrument

text: Energy in thermodynamics is calculated by multiplying the thermal potential by the amount of entropy found at that potential: temperature times entropy.


title: Measuring instrument

text: See also Temperature measurement and . More technically related may be seen thermal analysis methods in materials science.


title: Measuring instrument

text: Calorimeters are called passive if gauged to measure emerging energy carried by entropy, for example from chemical reactions. Calorimeters are called active or heated if they heat the sample, or reformulated: if they are gauged to fill the sample with a defined amount of entropy.


title: Measuring instrument

text: Phase change calorimeter's energy value divided by absolute temperature give the entropy exchanged. Phase changes produce no entropy and therefore offer themselves as an entropy measurement concept. Thus entropy values occur indirectly by processing energy measurements at defined temperatures, without producing entropy.


title: Measuring instrument

text: The given sample is cooled down to (almost) absolute zero (for example by submerging the sample in liquid helium). At absolute zero temperature any sample is assumed to contain no entropy (see Third law of thermodynamics for further information). Then the following two active calorimeter types can be used to fill the sample with entropy until the desired temperature has been reached: (see also Thermodynamic databases for pure substances)


title: Measuring instrument

text: Processes transferring energy from a non-thermal carrier to heat as a carrier do produce entropy (Example: mechanical/electrical friction, established by Count Rumford).


title: Measuring instrument

text: Either the produced entropy or heat are measured (calorimetry) or the transferred energy of the non-thermal carrier may be measured.


title: Measuring instrument

text: Entropy lowering its temperature—without losing energy—produces entropy (Example: Heat conduction in an isolated rod; "thermal friction").


title: Measuring instrument

text: Concerning a given sample, a proportionality factor relating temperature change and energy carried by heat. If the sample is a gas, then this coefficient depends significantly on being measured at constant volume or at constant pressure. (The terminiology preference in the heading indicates that the classical use of heat bars it from having substance-like properties.)


title: Measuring instrument

text: The temperature coefficient of energy divided by a substance-like quantity (amount of substance, mass, volume) describing the sample. Usually calculated from measurements by a division or could be measured directly using a unit amount of that sample.


title: Measuring instrument

text: This includes mostly instruments which measure macroscopic properties of matter: In the fields of solid-state physics; in condensed matter physics which considers solids, liquids, and in-betweens exhibiting for example viscoelastic behavior. Furthermore, fluid mechanics, where liquids, gases, plasmas, and in-betweens like supercritical fluids are studied.


title: Measuring instrument

text: This refers to particle density of fluids and compact(ed) solids like crystals, in contrast to bulk density of grainy or porous solids.


title: Measuring instrument

text: This section and the following sections include instruments from the wide field of , materials science.


title: Measuring instrument

text: Phase conversions like changes of aggregate state, chemical reactions or nuclear reactions transmuting substances, from reactants into products, or diffusion through membranes have an overall energy balance. Especially at constant pressure and constant temperature, molar energy balances define the notion of a substance potential or chemical potential or molar Gibbs energy, which gives the energetic information about whether the process is possible or not - in a closed system.


title: Measuring instrument

text: Energy balances that include entropy consist of two parts: A balance that accounts for the changed entropy content of the substances, and another one that accounts for the energy freed or taken by that reaction itself, the Gibbs energy change. The sum of reaction energy and energy associated to the change of entropy content is also called enthalpy. Often the whole enthalpy is carried by entropy and thus measurable calorimetrically.


title: Measuring instrument

text: For standard conditions in chemical reactions either molar entropy content and molar Gibbs energy with respect to some chosen zero point are tabulated. Or molar entropy content and molar enthalpy with respect to some chosen zero are tabulated. (See Standard enthalpy change of formation and Standard molar entropy)


title: Measuring instrument

text: The substance potential of a redox reaction is usually determined electrochemically current-free using reversible cells.


title: Measuring instrument

text: Microphones in general, sometimes their sensitivity is increased by the reflection- and concentration principle realized in acoustic mirrors.


title: Measuring instrument

text: Ionizing radiation includes rays of "particles" as well as rays of "waves". Especially X-rays and gamma rays transfer enough energy in non-thermal, (single-) collision processes to separate electron(s) from an atom.


title: Measuring instrument

text: This could include chemical substances, rays of any kind, elementary particles, and quasiparticles. Many measurement devices outside this section may be used or at least become part of an identification process.


title: Measuring instrument

text: For identification and content concerning chemical substances, see also Analytical chemistry, List of chemical analysis methods, and List of materials analysis methods.


title: Measuring instrument

text: Photometry is the measurement of light in terms of its perceived brightness to the human eye. Photometric quantities derive from analogous radiometric quantities by weighting the contribution of each wavelength by a luminosity function that models the eye's spectral sensitivity. For the ranges of possible values, see the orders of magnitude in:


title: Measuring instrument

text: Synthetic Aperture Radar (SAR) instruments measure radar brightness, Radar Cross Section (RCS), which is a function of the reflectivity and moisture of imaged objects at wavelengths which are too long to be perceived by the human eye. Black pixels mean no reflectivity (e.g. water surfaces), white pixels mean high reflectivity (e.g. urban areas). Colored pixels can be obtained by combining three gray-scaled images which usually interpret the polarization of electromagnetic waves. The combination R-G-B = HH-HV-VV combines radar images of waves sent and received horizontally (HH), sent horizontally and received vertically (HV) and sent and received vertically (VV). The calibration of such instruments is done by imaging objects (calibration targets) whose radar brightness is known.


title: Measuring instrument

text: Some instruments, such as telescopes and sea navigation instruments, have had military applications for many centuries. However, the role of instruments in military affairs rose exponentially with the development of technology via applied science, which began in the mid-19th century and has continued through the present day. Military instruments as a class draw on most of the categories of instrument described throughout this article, such as navigation, astronomy, optics, and imaging, and the kinetics of moving objects. Common abstract themes that unite military instruments are seeing into the distance, seeing in the dark, knowing an object's geographic location, and knowing and controlling a moving object's path and destination. Special features of these instruments may include ease of use, speed, reliability, and accuracy.


title: Althaea officinalis

text: Althaea officinalis, the marsh mallow or marshmallow, is a species of flowering plant indigenous to Europe, Western Asia and North Africa, which is used in herbalism and as an ornamental plant. A confection made from the root since ancient Egyptian times evolved into today's marshmallow treat, but most modern marshmallow treats no longer contain any marsh-mallow root.


title: Althaea officinalis

text: This herbaceous perennial has stems which die down in the autumn, They typically grow , but can reach and put out only a few lateral branches. The leaves are shortly petioled, roundish, ovate-cordate, long, and about broad, entire or three to five lobed, irregularly toothed at the margin, and thick. They are soft and velvety on both sides, due to a dense covering of stellate hairs. The lilac-pink flowers are shaped like those of the common mallow, but are smaller and of a pale colour, and are either axillary, or in panicles, more often the latter.


title: Althaea officinalis

text: The stamens are united into a tube, the anthers, kidney-shaped and one-celled. The flowers are in bloom during August and September, and are followed, as in other species of this order, by the flat, round fruit which are popularly called "cheeses".


title: Althaea officinalis

text: The common mallow is frequently called "marsh mallow" in colloquial terms, but the true marsh mallow is distinguished from all the other mallows growing in Great Britain by the numerous divisions of the outer calyx (six to nine cleft), by the hoary down which thickly clothes the stems and foliage, and by the numerous panicles of blush-coloured flowers, paler than the common mallow. The roots are perennial, thick, long and tapering, very tough and pliant, whitish yellow outside, white and fibrous within.


title: Althaea officinalis

text: Chemical constituents include altheahexacosanyl lactone ("n"-hexacos-2-enyl-1,5-olide), 2β-hydroxycalamene (altheacalamene) and altheacoumarin glucoside (5,6-dihydroxycoumarin-5-dodecanoate-6β--glucopyranoside), along with the known phytoconstituents lauric acid, β-sitosterol and lanosterol.


title: Althaea officinalis

text: Marshmallows are used in gardening as ornamental plants. Like most herbaceous perennials they require a sunny or partially-shaded position in moist, well-drained soil.


title: Althaea officinalis

text: The leaves, flowers and the root of "A. officinalis" (marshmallow) have been used in traditional herbal medicine. This use is reflected in the name of the genus, which comes from the Greek (""), meaning "to heal."


title: Althaea officinalis

text: Marshmallow is traditionally used as relief for irritation of mucous membranes, including use as a gargle for mouth and throat ulcers and gastric ulcers. In Russia, the root syrup is sold without a prescription by pharmacies, with intent to treat respiratory and digestive ailments.


title: Althaea officinalis

text: Most of the mallows have been used as food, and are mentioned by early classic writers with this connection. Mallow was an edible vegetable among the Romans; a dish of marsh mallow was one of their delicacies. Prospero Alpini stated in 1592 that a plant of the mallow kind was eaten by the Egyptians. Many of the poorer inhabitants of Syria subsisted for weeks on herbs, of which marshmallow is one of the most common. When boiled first and fried with onions and butter, the roots are said to form a palatable dish, and in times of scarcity consequent upon the failure of the crops, this plant, which grows there in great abundance, is collected heavily as a foodstuff.


title: Althaea officinalis

text: The young leaves can be cooked. The flower buds can be pickled. The roots can be peeled, sliced, boiled and sweetened to make candy. Water used to boil any part of the plant can be used as an egg white substitute.


title: Althaea officinalis

text: The root extract (halawa extract) is sometimes used as flavoring in the making of "halva". The later French version of the recipe, called ' (or ' for short), included an egg white meringue and was often flavored with rose water. "" more closely resembles contemporary commercially available marshmallows, which no longer contain "Althaea officinalis".


title: Law of sines

text: In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,


title: Law of sines

text: where , and are the lengths of the sides of a triangle, and , and are the opposite angles (see figure 2), while is the radius of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes stated using the reciprocals;


title: Law of sines

text: The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data (called the "ambiguous case") and the technique gives two possible values for the enclosed angle.


title: Law of sines

text: The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines.


title: Law of sines

text: According to Ubiratàn D'Ambrosio and Helaine Selin, the spherical law of sines was discovered in the 10th century. It is variously attributed to Abu-Mahmud Khojandi, Abu al-Wafa' Buzjani, Nasir al-Din al-Tusi and Abu Nasr Mansur.


title: Law of sines

text: Ibn Muʿādh al-Jayyānī's "The book of unknown arcs of a sphere" in the 11th century contains the general law of sines. The plane law of sines was later stated in the 13th century by Nasīr al-Dīn al-Tūsī. In his "On the Sector Figure", he stated the law of sines for plane and spherical triangles, and provided proofs for this law.


title: Law of sines

text: According to Glen Van Brummelen, "The Law of Sines is really Regiomontanus's foundation for his solutions of right-angled triangles in Book IV, and these solutions are in turn the bases for his solutions of general triangles." Regiomontanus was a 15th-century German mathematician.


title: Law of sines

text: The area of any triangle can be written as one half of its base times its height. Selecting one side of the triangle as the base, the height of the triangle relative to that base is computed as the length of another side times the sine of the angle between the chosen side and the base. Thus depending on the selection of the base, the area of the triangle can be written as any of:


title: Law of sines

text: When using the law of sines to find a side of a triangle, an ambiguous case occurs when two separate triangles can be constructed from the data provided (i.e., there are two different possible solutions to the triangle). In the case shown below they are triangles and .


title: Law of sines

text: Given a general triangle, the following conditions would need to be fulfilled for the case to be ambiguous:


title: Law of sines

text: If all the above conditions are true, then each of angles and produces a valid triangle, meaning that both of the following are true:


title: Law of sines

text: From there we can find the corresponding and or and if required, where is the side bounded by vertices and and is bounded by and .


title: Law of sines

text: If the lengths of two sides of the triangle and are equal to , the third side has length , and the angles opposite the sides of lengths , , and are , , and respectively then


title: Law of sines

text: the common value of the three fractions is actually the diameter of the triangle's circumcircle. This result dates back to Ptolemy.


title: Law of sines

text: As shown in the figure, let there be a circle with inscribed formula_10 and another inscribed formula_11 that passes through the circle's center O. The formula_12 has a central angle of formula_13 and thus formula_14. Since formula_15 is a right triangle,


title: Law of sines

text: where formula_17 is the radius of the circumscribing circle of the triangle. Angles formula_18 and formula_19 have the same central angle thus they are the same: formula_20. Therefore,


title: Law of sines

text: The area of a triangle is given by formula_24, where formula_25 is the angle enclosed by the sides of lengths and . Substituting the sine law into this equation gives


title: Law of sines

text: The sine rule can also be used in deriving the following formula for the triangle's area: Denoting the semi-sum of the angles' sines as formula_30, we have


title: Law of sines

text: Suppose the radius of the sphere is 1. Let , , and be the lengths of the great-arcs that are the sides of the triangle. Because it is a unit sphere, , , and are the angles at the center of the sphere subtended by those arcs, in radians. Let , , and be the angles opposite those respective sides. These are dihedral angles between the planes of the three great circles.


title: Law of sines

text: Consider a unit sphere with three unit vectors , and drawn from the origin to the vertices of the triangle. Thus the angles , , and are the angles , , and , respectively. The arc subtends an angle of magnitude at the centre. Introduce a Cartesian basis with along the -axis and in the -plane making an angle with the -axis. The vector projects to in the -plane and the angle between and the -axis is . Therefore, the three vectors have components:


title: Law of sines

text: The scalar triple product, is the volume of the parallelepiped formed by the position vectors of the vertices of the spherical triangle , and . This volume is invariant to the specific coordinate system used to represent , and . The value of the scalar triple product is the determinant with , and as its rows. With the -axis along the square of this determinant is


title: Law of sines

text: Repeating this calculation with the -axis along gives , while with the -axis along it is . Equating these expressions and dividing throughout by gives


title: Law of sines

text: where is the volume of the parallelepiped formed by the position vector of the vertices of the spherical triangle. Consequently, the result follows.


title: Law of sines

text: It is easy to see how for small spherical triangles, when the radius of the sphere is much greater than the sides of the triangle, this formula becomes the planar formula at the limit, since


title: Law of sines

text: A purely algebraic proof can be constructed from the spherical law of cosines. From the identity formula_56 and the explicit expression for formula_57 from the spherical law of cosines


title: Law of sines

text: Since the right hand side is invariant under a cyclic permutation of formula_59 the spherical sine rule follows immediately.


title: Law of sines

text: The figure used in the Geometric proof above is used by and also provided in Banerjee (see Figure 3 in this paper) to derive the sine law using elementary linear algebra and projection matrices.


title: Law of sines

text: which is the analog of the formula in Euclidean geometry expressing the sine of an angle as the opposite side divided by the hypotenuse.


title: Law of sines

text: By substituting , , and , one obtains respectively the Euclidean, spherical, and hyperbolic cases of the law of sines described above.


title: Law of sines

text: Let indicate the circumference of a circle of radius in a space of constant curvature . Then . Therefore, the law of sines can also be expressed as:


title: Law of sines

text: For an -dimensional simplex (i.e., triangle (), tetrahedron (), pentatope (), etc.) in -dimensional Euclidean space, the absolute value of the polar sine () of the normal vectors of the facets that meet at a vertex, divided by the hyperarea of the facet opposite the vertex is independent of the choice of the vertex. Writing for the hypervolume of the -dimensional simplex and for the product of the hyperareas of its -dimensional facets, the common ratio is


title: Law of sines

text: For example, a tetrahedron has four triangular facets. The absolute value of the polar sine of the normal vectors to the three facets that share a vertex, divided by the area of the fourth facet will not depend upon the choice of the vertex:


title: Right to education

text: The right to education has been recognized as a human right in a number of international conventions, including the International Covenant on Economic, Social and Cultural Rights which recognizes a right to free, compulsory primary education for all, an obligation to develop secondary education accessible to all with the progressive introduction of free secondary education, as well as an obligation to develop equitable access to higher education, ideally by the progressive introduction of free higher education. In 2021, 171 states were parties to the Covenant.


title: Right to education

text: In 2019, an estimated 260 million children worldwide did not have access to school education, and social inequality was a major cause.


title: Right to education

text: The Human Rights Measurement Initiative measures the right to education for countries around the world, based on their level of income.


title: Right to education

text: The right to education is reflected in article 26 of the Universal Declaration of Human Rights, which states:


title: Right to education

text: "Everyone has the right to education. Education shall be free, at least in the elementary and fundamental stages. Elementary education shall be compulsory. Technical and professional education shall be made generally available and higher education shall be equally accessible to all on the basis of merit. Education shall be directed to the full development of the human personality and to the strengthening of respect for human rights and fundamental freedoms. It shall promote understanding, tolerance and friendship among all nations, racial or religious groups, and shall further the activities of the United Nations for the maintenance of peace. Parents have a prior right to choose the kind of education that shall be given to their children."


title: Right to education

text: The right to education has been reaffirmed in the 1960 UNESCO Convention against Discrimination in Education, the 1966 International Covenant on Economic, Social and Cultural Rights, the 1981 Convention on the Elimination of All Forms of Discrimination Against Women, the 1989 Convention on the Rights of the Child, and the 2006 Convention on the Rights of Persons with Disabilities.


title: Right to education

text: In Africa, both the 1981 the African Charter on Human and Peoples' Rights and the 1990 African Charter on the Rights and Welfare of the Child recognize the right to education.


title: Right to education

text: In [Europe], Article 2 of the first Protocol of 20 March 1952 to the European Convention on Human Rights states that the right to education is recognized as a human right and is understood to establish an entitlement to education. According to the International Covenant on Economic, Social and Cultural Rights, the right to education includes the right to free, compulsory primary education for all, an obligation to develop secondary education accessible to all in particular by the progressive introduction of free secondary education, as well as an obligation to develop equitable access to higher education in particular by the progressive introduction of free higher education. The right to education also includes a responsibility to provide basic education for individuals who have not completed primary education. In addition to these access to education provisions, the right to education encompasses also the obligation to eliminate discrimination at all levels of the educational system, to set minimum standards, and to improve quality. The European Court of Human Rights in Strasbourg has applied this norm for example in the Belgian linguistic case. Article 10 of the European Social Charter guarantees the right to vocational education.


title: Right to education

text: According to Indian constitution under 86th Amendment act 2002, There is right to free and compulsory education up to 6–14 years of age.


title: Right to education

text: It has been argued that "International law provides no effective protection of the right to pre-primary education." Just two global treaties explicitly reference education prior to primary school. The Convention on the Elimination of All Forms of Discrimination against Women requires states to ensure equality for girls "in pre-school." And in the Convention on the Protection of the Rights of All Migrant Workers and Members of Their Families, states agree that access to "public pre-school educational institutions" shall not be denied due to the parents’ or child's "irregular situation with respect to stay." Less explicitly, the Convention on the Rights of Persons with Disabilities requires that "States Parties shall ensure an inclusive education system at all levels."


title: Right to education

text: In 2022, Human Rights Watch adopted a policy calling on states to make at least one year of free and compulsory, inclusive, quality pre-primary education available and accessible for all children. In doing so they advocated making one year of pre-primary education to be included as part of the minimum core of the right to education. They further called on all states to adopt a detailed plan of action for the progressive implementation of further years of pre-primary education, within a reasonable number of years to be fixed in the plan.


title: Right to education

text: Education is the access to formal institutional instructions. Generally, international instruments use the term in this sense and the right to education, as protected by international human rights instruments, refers primarily to education in a narrow sense. The 1960 UNESCO Convention against Discrimination in Education defines education in Article 1(2) as: "all types and levels of education, (including such) access to education, the standard and quality of education, and the conditions under which it is given."


title: Right to education

text: In a wider sense education may describe "all activities by which a human group transmits to its descendants a body of knowledge and skills and a moral code which enable the group to subsist". In this sense education refers to the transmission to a subsequent generation of those skills needed to perform tasks of daily living, and further passing on the social, cultural, spiritual and philosophical values of the particular community. The wider meaning of education has been recognised in Article 1(a) of UNESCO's 1974 "Recommendation concerning Education for International Understanding, Co-operation and Peace and Education relating to Human Rights and Fundamental Freedoms".


title: Right to education

text: "the entire process of social life by means of which individuals and social groups learn to develop consciously within, and for the benefit of, the national and international communities, the whole of their personal capabilities, attitudes, aptitudes and knowledge."


title: Right to education

text: The European Court of Human Rights has defined education in a narrow sense as "teaching or instructions... in particular to the transmission of knowledge and to intellectual development" and in a wider sense as "the whole process whereby, in any society, adults endeavour to transmit their beliefs, culture and other values to the young."


title: Right to education

text: The fulfilment of the right to education can be assessed using the 4As framework, which asserts that for education to be a meaningful right it must be available, accessible, acceptable and adaptable. The 4As framework was developed by the former UN Special Rapporteur on the Right to Education, Katarina Tomasevski, but is not necessarily the standard used in every international human rights instrument and hence not a generic guide to how the right to education is treated under national law.


title: Right to education

text: The 4As framework proposes that governments, as the prime duty-bearers, have to respect, protect and fulfil the right to education by making education available, accessible, acceptable and adaptable. The framework also places duties on other stakeholders in the education process: the child, which as the privileged subject of the right to education has the duty to comply with compulsory education requirements, the parents as the ‘first educators’, and professional educators, namely teachers.


title: Right to education

text: A number of international NGOs and charities work to realise the right to education using a rights-based approach to development.


title: Right to education

text: In Europe, before the Enlightenment of the eighteenth and nineteenth century, education was the responsibility of parents and the church. With the French and American Revolution, education was established also as a public function. It was thought that the state, by assuming a more active role in the sphere of education, could help to make education available and accessible to all. Education had thus far been primarily available to the upper social classes and public education was perceived as a means of realising the egalitarian ideals underlining both revolutions.


title: Right to education

text: However, neither the American Declaration of Independence (1776) nor the French Declaration of the Rights of Man and of the Citizen (1789) protected the right to education, as the liberal concepts of human rights in the nineteenth century envisaged that parents retained the primary duty for providing education to their children. It was the states obligation to ensure that parents complied with this duty, and many states enacted legislation making school attendance compulsory. Furthermore, child labour laws were enacted to limit the number of hours per day children could be employed, to ensure children would attend school. States also became involved in the legal regulation of curricula and established minimum educational standards.


title: Right to education

text: In "On Liberty" John Stuart Mill wrote that an "education established and controlled by the State should only exist, if it exists at all, as one among many competing experiments, carried on for the purpose of example and stimulus to keep the others up to a certain standard of excellence." Liberal thinkers of the nineteenth century pointed to the dangers to too much state involvement in the sphere of education, but relied on state intervention to reduce the dominance of the church, and to protect the right to education of children against their own parents. In the latter half of the nineteenth century, educational rights were included in domestic bills of rights. The 1849 "Paulskirchenverfassung", the constitution of the German Empire, strongly influenced subsequent European constitutions and devoted Article 152 to 158 of its bill of rights to education. The constitution recognised education as a function of the state, independent of the church. Remarkable at the time, the constitution proclaimed the right to free education for the poor, but the constitution did not explicitly require the state to set up educational institutions. Instead the constitution protected the rights of citizens to found and operate schools and to provide home education. The constitution also provided for freedom of science and teaching, and it guaranteed the right of everybody to choose a vocation and train for it.


title: Right to education

text: The nineteenth century also saw the development of socialist theory, which held that the primary task of the state was to ensure the economic and social well-being of the community through government intervention and regulation. Socialist theory recognised that individuals had claims to basic welfare services against the state and education was viewed as one of these welfare entitlements. This was in contrast to liberal theory at the time, which regarded non-state actors as the prime providers of education. In 1917 the Mexican constitution was the first to guarantee free and secular education. Later, socialist ideals were enshrined in the 1936 Soviet Constitution, which recognize the right to education with a corresponding obligation of the state to provide such education. The constitution guaranteed free and compulsory education at all levels, a system of state scholarships and vocational training in state enterprises. Subsequently, the right to education featured strongly in the constitutions of socialist states. As a political goal, right to education was declared in F. D. Roosevelt's 1944 speech on the Second Bill of Rights.


title: Right to education

text: Education in all its forms (informal, non-formal, and formal) is crucial to ensure dignity of all individuals. The aims of education, as set out in the International human rights law (IHRL), are all directed to the realization of the individual’s rights and dignity. These include, among others, ensuring human dignity and the full and holistic development of the human personality; fostering physical and cognitive development; allowing for the acquisition of knowledge, skills, and talents; contributing to the realization of the full potential of the individual; enhancing self-esteem and increasing confidence; encouraging respect for human rights; shaping a person’s sense of identity and affiliation with others; enabling socialization and meaningful interaction with others; enabling a person to shape the world around them enables their participation in community life; contributing to a full and satisfying life within society; and empowering and allowing for the increased enjoyment of other human rights.


title: Right to education

text: Education is also transformative for the state and society. As one of the most important mechanisms by which social groups, in particular indigenous peoples and minorities are maintained from generation to generation, passing on language, culture, identity, values, and customs, education is also one of the key ways states can ensure their economic, social, political, and cultural interests.


title: Right to education

text: International law does not protect the right to pre-primary education and international documents generally omit references to education at this level. The Universal Declaration of Human Rights states that everyone has the right to education, hence the right applies to all individuals, although children are considered as the main beneficiaries.


title: Right to education

text: Both secondary and higher education shall be made accessible "by every appropriate means, and in particular by the progressive introduction of free education".


title: Right to education

text: The realization of the right to education on a national level may be achieved through compulsory education, or more specifically free compulsory primary education, as stated in both the Universal Declaration of Human Rights and the International Covenant on Economic, Social and Cultural Rights.


title: Right to education

text: The rights of all children from early childhood stem from the 1948 Universal Declaration of Human Rights. The declaration proclaimed in article 1: ‘All human beings are born free and equal in dignity and rights’. The declaration states that human rights begin at birth and that childhood is a period demanding special care and assistance [art. 25 (2)]. The 1959 Declaration of the Rights of the Child affirmed that: ‘mankind owes to the child the best it has to give’, including education. This was amplified by the International Covenant on Economic, Social and Cultural Rights of 1966 which states that: ‘education shall be directed to the full development of the human personality and the sense of its dignity, and shall strengthen the respect for human rights and fundamental freedoms. [art. 13 (1)]


title: Right to education

text: The World Declaration on Education for All (EFA) adopted in 1990 in Jomtien, Thailand, states in article 5 that: ‘Learning begins at birth [...] This calls for early childhood care and initial education.’ A decade later, the Dakar Framework for Action on EFA established six goals, the first of which was: ‘expanding and improving early childhood care and education, especially for the most vulnerable and disadvantaged children.’ Protection of children of all ages from exploitation and actions that would jeopardize their health, education and well-being has also been emphasized by the International Labour Organization in Conventions No. 138 on the Minimum Age of Employment (1973) and No. 182 on the Prohibition and Immediate Action for the Elimination of the Worst Forms of Child Labour (1999). The United Nations contributed to such endeavours by the Declaration of the Rights of the Child unanimously adopted by the General Assembly in 1959.


title: Right to education

text: There are various NGO's working towards the right to education. EClickKart is one of such platforms initiated by Rohit N Shetty which states that Education is the basic right and EClickKart is working towards it.


title: Right to education

text: The privatization of education can have a positive impact for some social groups, in the form of increased availability of learning opportunities, greater parental choice and a wider range of curricula. However, it can also have negative effects resulting from insufficient or inadequate monitoring and regulation by the public authorities (schools without licences, hiring of untrained teachers and absence of quality assurance), with potential risks for social cohesion and solidarity. Of particular concern: "Marginalised groups fail to enjoy the bulk of positive impacts and also bear the disproportionate burden of the negative impacts of privatisation." Furthermore, uncontrolled fees demanded by private providers could undermine universal access to education. More generally, this could have a negative impact on the enjoyment of the right to a good quality education and on the realization of equal educational opportunities.


title: Right to education

text: Supplemental private tutoring, or ‘shadow education’, which represents one specific dimension of the privatization of education, is also growing worldwide. Often a symptom of badly functioning school systems, private tutoring, much like other manifestations of private education, can have both positive and negative effects for learners and their teachers. On one hand, teaching can be tailored to the needs of slower learners and teachers can supplement their school salaries. On the other hand, fees for private tutoring may represent a sizeable share of household income, particularly among the poor, and can therefore create inequalities in learning opportunities. And the fact that some teachers may put more effort into private tutoring and neglect their regular duties can adversely affect the quality of teaching and learning at school. The growth of shadow education, the financial resources mobilized by individuals and families, and the concerns regarding possible teacher misconduct and corruption are leading some ministries of education to attempt to regulate the phenomenon.


title: Right to education

text: In the year 2019, an estimated 260 million children did not have access to school education on a global scale.


title: Right to education

text: In the 21st century, gender inequality is still an obstacle to universal access to education. Conservative attitudes towards the female gender role challenges women's and girls' fully exercise of their right to education.


title: Right to education

text: Out of 750 million illiterate adults in the world, two-thirds are estimated to be women. This is due to gender inequality, misogynistic violence, as well as, marriage and pregnancy, often associated with poverty and geographic isolation.


title: Right to education

text: In the second decade of 21st century, the advocacy for women's right to access education became a global movement through the activism of Malala Yousafzai, a Pakistani Nobel laureate.


title: Right to education

text: The COVID-19 pandemic affected over 90% of the world's students and was responsible for the rise of social inequality in the access to education. The global recession immediate to the pandemic projected drastic consequences on education funding, causing long-lasting effects on the equal right to education.


title: Right to education

text: Globally, during the pandemic, markers of gender, class, and ethnicity presented themselves as factors of vulnerability in the access to basic rights such as education and health.


title: Right to education

text: In despite of E-learning historical objective walks towards the democratization of education access, depending on its quality, it can be a difficulty in the achievement of this right. Students lacking cultural capital, family support, and material conditions (including access to quality electronic equipment and internet) have had their access to education hindered by this modality of education. The return to classroom teaching during the COVID-19 pandemic generated a conflict between the right to health and the right to an education. By returning to the school before the pandemic was fully under control, students were exposed to the SARS-CoV-2. Another aspect chained by the pandemic, that also relates to the right to health, is the damage to students' mental health.


title: Katrina Lenk

text: Lenk originated the role of Dina in the Broadway musical "The Band's Visit", a performance for which she won the 2018 Tony Award for Best Actress in a Musical. She also performed the lead role of Bobbie in the 2021 gender-swapped revival of Stephen Sondheim's musical "Company". Lenk's additional stage credits include roles in the Broadway productions of "", "Once", and "Indecent", as well as roles in regional theater productions.


title: Katrina Lenk

text: Lenk was born in Chicago, Illinois to a family of Eastern European descent. She attended Barrington High School in Barrington, Illinois. She graduated from the School of Music at Northwestern University in 1997, majoring in viola performance and studying voice and musical theatre.


title: Katrina Lenk

text: Lenk appeared as Yitzak in "Hedwig and the Angry Inch" at the Broadway Theatre, Chicago, in May 2001. She performed as Linda Lovelace in the musical "" in 2008 at the Hayworth Theatre, Los Angeles, about which the "L. A. Weekly" wrote: "As Linda, Katrina Lenk is sensational—she has a dozen nuanced smiles that range from innocent and shattered to grateful, in order to express whatever passes as kindness when, say, a male co-star (Josh Greene) promises to make their scene fun."


title: Katrina Lenk

text: She made her Broadway debut in "The Miracle Worker" in March 2010 as the understudy for the roles of Annie Sullivan and Kate Keller. She joined the cast of the Broadway musical "Once" in May 2013, in the role of Réza, the violinist.


title: Katrina Lenk

text: In 2015 she appeared in the Yale Repertory Theatre world premiere of "Indecent" by Paula Vogel. She then appeared in the 2016 Off-Broadway production at the Vineyard Theatre, followed by the Broadway production at the Cort Theatre in 2017. In her review for "Newsday", Linda Winer noted the "...dancing, singing actors as the earthy, sensual Katrina Lenk." For her performance she received the 2017 Theatre World Award Dorothy Loudon Award for Excellence. A performance of the play was filmed for television and broadcast on PBS in November 2017.


title: Katrina Lenk

text: Lenk appeared on Broadway as Dina in the 2017 musical "The Band's Visit", for which she won the 2018 Tony Award for Best Actress in a Musical. Craig Nakano of the "Los Angeles Times" wrote: "For Lenk, the role is another turning point. After studying music and theater at Northwestern University, Lenk devoted herself to theater in California... it’s her poignant, heart-aching turn as Dina that is poised to make her a star."


title: Katrina Lenk

text: Lenk starred in the lead role of Bobbie, opposite Patti LuPone as Joanne, in a gender-swapped revival of Stephen Sondheim's "Company", which started previews on Broadway at the Bernard B. Jacobs Theatre on March 2 and was scheduled to open on March 22, 2020, until delayed by the COVID-19 pandemic. The production resumed previews on November 15, 2021, for an opening date of December 9, 2021. Lenk also appeared in the final season of "Ozark", playing the role of Clare.


title: Love Island (2015 TV series, series 6)

text: The sixth series of "Love Island" began broadcasting on 12 January 2020 on ITV2. The series was announced on 24 July 2019 when it was confirmed that two series would air in 2020, and is also the first winter edition of the series. Unlike the summer edition, the luxury villa is located in Cape Town. This series is also the first to be hosted by Laura Whitmore, having taken over from Caroline Flack. The series is narrated by Iain Stirling.


title: Love Island (2015 TV series, series 6)

text: On 15 February 2020, following the death of former presenter Caroline Flack, the "Unseen Bits" episode due to air that day was not broadcast; but was made available on the ITV Hub. The regular episode did not air the following day either. The show returned on 17 February with a tribute to Flack. On the same day, ITV2 confirmed that the companion series "Love Island: Aftersun" which was set to air after that episode's main show would not be broadcast. A further tribute to Flack aired during the final on 23 February, which featured a compilation of her time throughout her five series as host.


title: Love Island (2015 TV series, series 6)

text: On 23 February 2020, the series was won by Finn Tapp and Paige Turley, having received 44.52% of the final vote. Siânnise Fudge and Luke Trotman finished as runners-up with 43.61% of the vote, making it the closest voting percentage between the final two couples ever with less than 1% between them.


title: Love Island (2015 TV series, series 6)

text: The first 10-second trailer for the series aired on 2 December 2019 featuring the tagline "Do one, winter". It was then followed by a full-length trailer that used the song Lose Control by Meduza, Becky Hill & Goodboys was released on 6 December 2019 which featured Caroline Flack. On 17 December 2019, Flack announced that she would be standing down as host following allegations of assault towards her boyfriend. On 20 December, it was announced that fellow TV presenter Laura Whitmore would be her replacement. She also took the role of presenting "Love Island: Aftersun", which for the first time aired every Monday after the main show rather than its usual Sunday slot.


title: Love Island (2015 TV series, series 6)

text: For the first winter edition of the show, the Islanders will stay at the Midden Cottage, located in the "ultra-wealthy" Constantia suburb of Cape Town.


title: Love Island (2015 TV series, series 6)

text: The Islanders for the sixth series were released on 6 January 2020, just six days before the launch. These include Eve and Jess Gale, who are the second set of twins to compete in the show following John and Tony Alberti in 2015. The series was won by Finn Tapp and Paige Turley.


title: Love Island (2015 TV series, series 6)

text: Official ratings are taken from BARB and include ITV2 +1. Catch-up service totals are added to the official ratings. Because the Saturday episodes are weekly catch-up episodes rather than nightly highlights, these are not included in the overall averages. The week 6 Sunday episode did not air due to the passing of former host Caroline Flack.


title: Love Island (2015 TV series, series 6)

text: Viewers called for contestant Ollie Williams to be axed after pictures were obtained by "The Sun" showing him posing next to animal corpses from endangered species in South Africa. A representative of Williams claimed he is a "conservationist and worked with an anti-poaching unit in Mozambique" involved in the culling of sick animals. However, according to "Metro" Williams is the director of a hunting business known as Cornish Sporting Agency. In since-deleted images on his social media, Williams is seen posing with dead animals alongside his clients. It led to a Change.org petition to remove Williams from the series.


title: Love Island (2015 TV series, series 6)

text: Regulator Ofcom received 272 complaints after the first airing, 231 of which regarding Williams inclusion on the show. The producers stated they had no intention of removing Williams, saying they "refuse to back down to the public" and that they believed it would "blow over".


title: Bisection method

text: In mathematics, the bisection method is a root-finding method that applies to any continuous function for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and then selecting the subinterval in which the function changes sign, and therefore must contain a root. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods. The method is also called the interval halving method, the binary search method, or the dichotomy method.


title: Bisection method

text: For polynomials, more elaborate methods exist for testing the existence of a root in an interval (Descartes' rule of signs, Sturm's theorem, Budan's theorem). They allow extending the bisection method into efficient algorithms for finding all real roots of a polynomial; see Real-root isolation.


title: Bisection method

text: The method is applicable for numerically solving the equation "f"("x") = 0 for the real variable "x", where "f" is a continuous function defined on an interval ["a", "b"] and where "f"("a") and "f"("b") have opposite signs. In this case "a" and "b" are said to bracket a root since, by the intermediate value theorem, the continuous function "f" must have at least one root in the interval ("a", "b").


title: Bisection method

text: At each step the method divides the interval in two parts/halves by computing the midpoint "c" = ("a"+"b") / 2 of the interval and the value of the function "f"("c") at that point. If "c" itself is a root then the process has succeeded and stops. Otherwise, there are now only two possibilities: either "f"("a") and "f"("c") have opposite signs and bracket a root, or "f"("c") and "f"("b") have opposite signs and bracket a root. The method selects the subinterval that is guaranteed to be a bracket as the new interval to be used in the next step. In this way an interval that contains a zero of "f" is reduced in width by 50% at each step. The process is continued until the interval is sufficiently small.


title: Bisection method

text: Explicitly, if "f"("c")=0 then "c" may be taken as the solution and the process stops. Otherwise, if "f"("a") and "f"("c") have opposite signs, then the method sets "c" as the new value for "b", and if "f"("b") and "f"("c") have opposite signs then the method sets "c" as the new "a". In both cases, the new "f"("a") and "f"("b") have opposite signs, so the method is applicable to this smaller interval.


title: Bisection method

text: The input for the method is a continuous function "f", an interval ["a", "b"], and the function values "f"("a") and "f"("b"). The function values are of opposite sign (there is at least one zero crossing within the interval). Each iteration performs these steps:


title: Bisection method

text: When implementing the method on a computer, there can be problems with finite precision, so there are often additional convergence tests or limits to the number of iterations. Although "f" is continuous, finite precision may preclude a function value ever being zero. For example, consider ; there is no floating-point value approximating that gives exactly zero. Additionally, the difference between "a" and "b" is limited by the floating point precision; i.e., as the difference between "a" and "b" decreases, at some point the midpoint of ["a", "b"] will be numerically identical to (within floating point precision of) either "a" or "b".


title: Bisection method

text: First, two numbers formula_2 and formula_3 have to be found such that formula_4 and formula_5 have opposite signs. For the above function, formula_6 and formula_7 satisfy this criterion, as


title: Bisection method

text: In the first iteration, the end points of the interval which brackets the root are formula_10 and formula_11, so the midpoint is


title: Bisection method

text: The function value at the midpoint is formula_13. Because formula_14 is negative, formula_6 is replaced with formula_16 for the next iteration to ensure that formula_17 and formula_18 have opposite signs. As this continues, the interval between formula_2 and formula_3 will become increasingly smaller, converging on the root of the function. See this happen in the table below.


title: Bisection method

text: After 13 iterations, it becomes apparent that there is a convergence to about 1.521: a root for the polynomial.


title: Bisection method

text: The method is guaranteed to converge to a root of "f" if "f" is a continuous function on the interval ["a", "b"] and "f"("a") and "f"("b") have opposite signs. The absolute error is halved at each step so the method converges linearly. Specifically, if "c" = is the midpoint of the initial interval, and "c" is the midpoint of the interval in the "n"th step, then the difference between "c" and a solution "c" is bounded by


title: Bisection method

text: This formula can be used to determine, in advance, an upper bound on the number of iterations that the bisection method needs to converge to a root to within a certain tolerance.


title: Bisection method

text: The number "n" of iterations needed to achieve a required tolerance ε (that is, an error guaranteed to be at most ε), is bounded by


title: Bisection method

text: where the initial bracket size formula_23 and the required bracket size formula_24 The main motivation to use the bisection method is that over the set of continuous functions, no other method can guarantee to produce an estimate c to the solution c that in the "worst case" has an formula_25 absolute error with less than n iterations. This is also true under several common assumptions on function f and the behaviour of the function in the neighbourhood of the root.


title: Bisection method

text: However, despite the bisection method being optimal with respect to worst case performance under absolute error criteria it is sub-optimal with respect to "average performance" under standard assumptions as well as "asymptotic performance". Popular alternatives to the bisection method, such as the secant method, Ridders' method or Brent's method (amongst others), typically perform better since they trade-off worst case performance to achieve higher orders of convergence to the root. And, a strict improvement to the bisection method can be achieved with a higher order of convergence without trading-off worst case performance with the ITP Method.


title: Bart Freundlich

text: Bartholomew Freundlich (born January 17, 1970) is an American film director, television director, screenwriter, and film producer.


title: Bart Freundlich

text: Freundlich was born and raised in Manhattan, the son of Debbie, a marketing consultant, and Larry Freundlich, a writer and publisher. Freundlich is Jewish on his father's side. He graduated with a double major in Cinema Studies and Film and Television Production from New York University's Tisch School of the Arts. His brother, Oliver Freundlich, is an architect.


title: Bart Freundlich

text: Before his first full-length film, "The Myth of Fingerprints" (which would receive awards at the Sundance Film Festival) in 1994, Freundlich had written and directed a short film titled "A Dog Race in Alaska" and a documentary titled "Hired Hands".


title: Bart Freundlich

text: Freundlich has been in a relationship with actress Julianne Moore since 1996, when they met on the set of "The Myth of Fingerprints" (1997).


title: Bart Freundlich

text: When casting Moore's character, Mia, Freundlich explains that he “was looking for someone who had a lot of complication, a lot of sadness under the surface, and portrayed very, very little of it in her face.” During filming, Freundlich recalls seeing Moore standing outside, exposed to the elements without a coat: "I came up to her and stood with my back to the wind. I didn’t want her to be cold, and I also didn’t want to put my arm around her, or smother her, because I didn’t know her that well...Even though I couldn’t have articulated it then, I understood that I could keep her warm up to a point. But then the rest was going to be for her to do. We connected in that moment on an unspoken level, where she knew I saw that flame in her, and understood it, and was willing not to suffocate it but to protect it.”


title: Bart Freundlich

text: Moore and Freundlich married on August 23, 2003. The couple have a son, Caleb (born December 4, 1997) and a daughter, Liv Helen (born April 11, 2002). Moore has appeared in all of Freundlich's feature films with the exception of "Catch That Kid" (2004), "The Rebound" (2009), and "Wolves" (2016).


title: Nadir

text: The nadir (, ; ) is the direction pointing directly "below" a particular location; that is, it is one of two vertical directions at a specified location, orthogonal to a horizontal flat surface.