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Implicit function

In mathematics, an implicit equation is a relation of the form R = 0, where R is a function of several variables. For example, the implicit equation of the unit circle is x 2 + y 2 − 1 = 0. An implicit function is a function, defined implicitly by an implicit equation, by associating one of the variables with the others. Thus, an implicit function for y in the context of the unit circle is defined implicitly by x 2 + f 2 − 1 = 0; this implicit equation defines f as a function of x only if − 1 ≤ x ≤ 1 and one considers only non-negative values for the values of the function. The implicit function theorem provides conditions under which some kinds of relations define an implicit function, namely relations defined as the indicator function of the zero set of some continuously differentiable multivariate function. A common type of implicit function is an inverse function. Not all functions have a unique inverse function. If g is a function of x that has a unique inverse the inverse function of g, called g−1, is the unique function giving a solution of the equation y = g for x in terms of y.

This solution can be written as x = g − 1. Defining g − 1 as the inverse of g is an implicit definition. For some functions g, g − 1 can be written out explicitly as a closed-form expression — for instance, if g = 2 x − 1 g − 1 = 1 2. However, this is not possible, or only by introducing a new notation. Intuitively, an inverse function is obtained from g by interchanging the roles of the dependent and independent variables. Example The product log is an implicit function giving the solution for x of the equation y − xex = 0. An algebraic function is a function that satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable x gives a solution for y of an equation a n y n + a n − 1 y n − 1 + ⋯ + a 0 = 0 where the coefficients ai are polynomial functions of x; this algebraic function can be written as the right side of the solution equation y = f. Written like this, f is a multi-valued implicit function. Algebraic functions play an important role in algebraic geometry.

A simple example of an algebraic function is given by the left side of the unit circle equation: x 2 + y 2 − 1 = 0. Solving for y gives an explicit solution: y = ± 1 − x 2, but without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation as y = f, where f is the multi-valued implicit function. While explicit solutions can be found for equations that are quadratic and quartic in y, the same is not in general true for quintic and higher degree equations, such as y 5 + 2 y 4 − 7 y 3 + 3 y 2 − 6 y − x = 0. One can still refer to the implicit solution y = f involving the multi-valued implicit function f. Not every equation R = 0 implies a graph of a single-valued function, the circle equation being one prominent example. Another example is an implicit function given by x − C = 0 where C is a cubic polynomial having a "hump" in its graph. Thus, for an implicit function to be a true function it might be necessary to use just part of the graph.

An implicit function can sometimes be defined as a true function only after "zooming in" on some part of the x-axis and "cutting away" some unwanted function branches. An equation expressing y as an implic

Tikhvin Cemetery

Tikhvin Cemetery is a historic cemetery in the centre of Saint Petersburg. It is part of the Alexander Nevsky Lavra, is one of four cemeteries in the complex. Since 1932 it has been part of the State Museum of Urban Sculpture, which refers to it as the Necropolis of the Masters of Art. Opened in 1823 after the monastery's first cemetery, the Lazarevskoe, had become overcrowded, the cemetery was called the "New Lazarevsky", it acquired its name after the building of its cemetery church, consecrated to the icon of the Tikhvin Mother of God. It soon became a popular and prestigious burial ground; the first literary figure, Nikolay Karamzin, was buried in the cemetery in 1826, followed in 1833 by Nikolay Gnedich, an associate of Alexander Pushkin's. Several other friends of Pushkin were buried in the cemetery. Significant interments were those of Mikhail Glinka in 1857, Fyodor Dostoevsky in 1881, Modest Mussorgsky and Alexander Borodin in the 1880s, Pyotr Ilyich Tchaikovsky in 1891. During the Soviet period the cemetery was earmarked for development into a museum necropolis, envisaged as a landscaped park, with strategically placed memorials to important figures of Russian history.

With several notable artists having been buried in the cemetery, it was decided to designate it as the "Necropolis of the Masters of Art". During the 1930s many important Russian composers, sculptors and poets were exhumed from their original resting places across the city, brought, with or without their monuments, to be reburied in the Tikhvin cemetery. At the same time the monuments of those figures deemed not in keeping with the artistic theme of the cemetery were removed or destroyed. Several more burials of important artists took place during the Soviet period, as the cemetery established a role as a kind of national pantheon. Today the cemetery operates as a museum necropolis under the auspices of the State Museum of Urban Sculpture; the cemetery is located close to Alexander Nevsky Square, to the right of the pathway leading from the Gate Church to the River Monastyrka. This land had been occupied with ornamental and vegetable gardens; the first cemetery in the monastery, the Lazarevskoe Cemetery, had been established in 1717, by the early nineteenth century was becoming overcrowded.

In March 1823 the monastery authorities proposed the creation of a new burial ground opposite the St. Petersburg Theological Consistory; the new cemetery called the "New Lazarevsky", was established in the eastern part of the plot of land, between the pathway to the monastery, the consistory building, enclosed with a wooden fence. Over time it expanded to the west, into the areas occupied by monastic gardens, in the 1870s it was enclosed with a stone wall; the brothers D. M. and N. M. Polezhaev, wealthy merchants, funded the construction of a cemetery church, laid down on 26 September 1869 and built to the design of architect N. P. Grebyonki; the church was consecrated on 2 February 1873 in the name of the icon of the Tikhvin Mother of God, which from about 1876 became the common name of the cemetery. Two icons, one of Saint Dimitry of Rostov, one of Saint Mary of Egypt, were painted by Pavel Pleshanov for the church. In 1825 the church and cemetery were visited by Emperor Alexander I, prior to his journey to Taganrog.

The burial vault of Polezhaev family was in the crypt of the church, in 1901 the church underwent renovations. In 1918 archpriest Peter Skipetrov of the Gate Church, shot and killed during an early attempt by the Bolsheviks to requisition the monastery on 19 January 1918, was buried under the church's altar; the church was closed in 1931 and between 1935 and 1937 it was converted into a post office, with the destruction of its facades and interiors. With the establishment of the State Museum of Urban Sculpture, the building housed its scientific department, now houses an exhibition hall as part of the museum; the rate of burials in the Old and New Lazarevskoe cemeteries was about equal during the early years of the latter's existence, though by the 1830s the New Lazarevskoe Cemetery became more popular. Burials took place in the eastern part of the cemetery, in 1825 the holy fool monk Patermufy was buried there. In 1826 the writer Nikolay Karamzin was buried in the cemetery, followed in 1833 by Nikolay Gnedich, a contemporary of Pushkin's.

Gnedich's funeral on 6 February 1833 was attended by many prominent literary figures, including Pushkin, Ivan Krylov, Pyotr Vyazemsky, Pyotr Pletnyov, Fyodor Tolstoy and Alexey Olenin. With the exception of Pushkin, all would be buried in the Lavra's cemeteries. In 1844 another contemporary poet of Pushkin's, Yevgeny Baratynsky, was buried in the cemetery; the cemetery became a prestigious burial ground for those of many areas of society. The wealthy merchant A. I. Kosikovsky was buried under a monumental sarcophagus on a high pedestal surmounted by a canopy on eight fluted columns. Opposite it stood a grand monument to the statesman Pavel Demidov, which has since been lost. In 1857 the remains of the composer Mikhail Glinka were returned from Berlin and buried in the cemetery, with a grand monument erected two years to the design of architect I. I. Gornostayev, with sculptures by Nikolai Laveretsky. On 1 February 1881 the author Fyodor Dostoevsky was buried in the cemetery, with a large monument.

During the 1880s composers Modest Mussorgsky and Alexander Borodin were buried in the northern part of the grounds, with Pyotr Ilyich Tchaikovsky

Greatest Hits: From the Beginning (The Miracles album)

Greatest Hits from the Beginning is a compilation double LP by The Miracles released in 1965. This was the first double album released by the Motown Record Corporation, it covers most of the group's hits from their pre-1965 albums, such as "Shop Around", "Who's Lovin’ You", "You've Really Got A Hold On Me" and "Mickey's Monkey", as well as the non-album singles from 1964: "I Like It Like That" and "That's What Love Is Made Of". The album was a success, it was the first Miracles album to chart on the Billboard R&B Album chart, where it was an bigger success, peaking at #2. The compilation includes the group's early non-Motown singles in addition to their releases on Motown's Tamla label. "Got A Job" and "I Cry" were issued on End in 1958, while "Bad Girl" and "I Need a Change" were issued on Chess Records in 1959. "Bad Girl" was reissued by Motown the same year. The B-sides from all four non-Motown singles are included. However, this is not a complete collection of the group's pre-1960 singles; the single, "It" and its B-side, "Don't Say Bye Bye", issued on Argo and credited to Ron and Bill were omitted.

Other early charting singles missing are "Broken Hearted", "Ain't It Baby", "Mighty Good Lovin'", " The Man in You", "Come On Do The Jerk", "Everybody's Gotta Pay Some Dues". This album has since gone out of print. "Got a Job" "I Cry" "Mama Done Told Me" " Money" "Bad Girl" "I Love Your Baby" "I Need a Change" "All I Want Is You" " Depend on Me" "Who's Lovin' You" "That's What Love Is Made Of" "Mickey's Monkey" "I Gotta Dance to Keep From Crying" "You've Really Got a Hold on Me" "I Like It Like That" "A Love She Can Count On" "Shop Around" "Way Over There" "I've Been Good to You" "Would I Love You" "I'll Try Something New" "What's So Good About Goodbye" Smokey Robinson - lead vocals Ronnie White - backing vocals Bobby Rogers - backing vocals Warren "Pete" Moore - backing vocals, lead vocals and vocal arrangements Claudette Robinson - lead vocals, backing vocals Marv Tarplin - guitar The Funk Brothers: instrumentation Berry Gordy, Jr.: producer, album executive producer William "Smokey" Robinson, Jr.: producer, album executive producer Brian Holland, Lamont Dozier: producer

Dade Christian School

Dade Christian School is a private Christian school serving located in Miami, Florida. It is a ministry of New Testament Baptist Church. DCS has a 2018-2019 enrollment of 152, it was founded as a segregation academy in response to the court ordered desegregation of Miami-Dade public schools. New Testament Baptist Church was founded in 1954, Dade Christian School in 1961 by Pastor Al Janney. Al Janney founded Dade Christian School in reaction to court decisions removing mandatory prayer from public schools and as a segregation academy. Al Janney pastored New Testament Baptist Church until 1976, he founded the Florida Association of Christian Colleges and Schools, the American Association of Christian Schools, the Baptist University of America. The next pastor E. G. Robertson pastored until 1986 and oversaw Dade Christian when it was named a Blue Ribbon School in 1984. In 1973 a lawsuit was brought against Dade Christian School by an African-American couple named in the court documents as Mr. and Mrs. Johnny Brown, Jr.

At the time, Dade Christian was an all-white school. The Browns sought injunctive and monetary relief against the school for not allowing their two daughters to attend; the couple had been handed a card that said the policy of the school was "one of nonintegration" and had been asked to leave. The school claimed in their defense that it was against their religious belief to have a desegregated school because of their belief objecting to interracial marriages; the school lost, leaving Brown's attorney to comment that the last quasi-legal segregation had been eliminated. The Browns still wanted their children to attend the school; when Dade Christian School appealed the ruling in Brown v. Dade Christian School, Inc. in 1977, The United States Court of Appeals for the Fifth Circuit ruled in favor of the plaintiff in what was considered an open question left by the Supreme Court at the time. The court failed to produce a majority opinion, deciding to limit themselves to the specifics of the case.

They concluded that if it was a part of the school's religious beliefs, it was a minor one and thus outweighed by anti-discrimination rules. This brought up questions regarding the free exercise of religion if courts can determine what a religion holds. In 1984, the school forbade students from attending The Jackson 5 concert at the Orange Bowl. School officials sent a letter to parents justifying this decision and stating that "rock music is associated with dancing, the drug scene and other unacceptable behavior." At the 2006 Fellowship of Christian Cheerleaders National Competition in Orlando, the junior high team won first place with a rendition of Grease. The elementary stunt group finished first. New Testament Baptist Church is an independent Baptist church with two locations: Dade Christian School and The Master's Academy; the President and Senior Pastor was Dino J. Pedrone before he left to be the full-time president of Davis College on November 22, 2009. Former Miami Dolphins player Don McNeal is the Children's Ministry overseer.

Victoria Jackson, actress DCS segregation ruling: article from the Harvard Law Review

Bill Risley

William Charles Risley, is a former professional pitcher in Major League Baseball from 1992–1998 for the Montreal Expos, Toronto Blue Jays, Seattle Mariners. Risley graduated from Marist High School in Chicago and attended Truman College before being drafted by the Cincinnati Reds in 1987, he was traded to Montreal in 1991. He would be a relief pitcher throughout his MLB career, yet In the second game of a doubleheader at Dodger Stadium on July 8, 1992, Risley's big-league debut turned out to be his only time as a starter. Pressed into duty by manager Felipe Alou when the Expos needed to play two doubleheaders against the Los Angeles Dodgers in three days, Risley was the winning pitcher, going five innings and allowing the Dodgers just four hits, with relievers Bill Sampen and Mel Rojas closing out a 4-1 victory; that turned out to be Risley's only appearance at all for Montreal during the 1992 season. And he would be used just twice out of the bullpen in 1993, pitching in 41 games in the minors, before Montreal placed him on waivers.

Risley was claimed by Seattle. A career highlight came in 1995, when Risley pitched in the first four games of the American League division playoffs against the New York Yankees, earning a save in Game 4. Ace starter Randy Johnson's three innings of relief in an 11-inning victory in Game 5 sent Seattle on to the 1995 American League Championship Series, where Risley saw action again against the Cleveland Indians. Seattle did not get to the World Series that year. Risley was traded that December to the Toronto Blue Jays, where he spent the last three seasons of his career. Career statistics and player information from Baseball-Reference, or Baseball-Reference