Menander was a Greek dramatist and the best-known representative of Athenian New Comedy. He took the prize at the Lenaia festival eight times, his record at the City Dionysia is unknown but may well have been spectacular. One of the most popular writers of antiquity, his work was lost during the Middle Ages and is known in modernity in fragmentary form, much of, discovered in the 20th century. Only one play, has survived entirely. Menander was the son of well-to-do parents, he derived his taste for comic drama from his uncle Alexis. He was the friend and pupil of Theophrastus, was on intimate terms with the Athenian dictator Demetrius of Phalerum, he enjoyed the patronage of Ptolemy Soter, the son of Lagus, who invited him to his court. But Menander, preferring the independence of his villa in the Piraeus and the company of his mistress Glycera, refused. According to the note of a scholiast on the Ibis of Ovid, he drowned while bathing, his countrymen honored him with a tomb on the road leading to Athens, where it was seen by Pausanias.

Numerous supposed busts of him survive, including a well-known statue in the Vatican thought to represent Gaius Marius. His rival in dramatic art was Philemon. Menander, believed himself to be the better dramatist, according to Aulus Gellius, used to ask Philemon: "Don't you feel ashamed whenever you gain a victory over me?" According to Caecilius of Calacte Menander was accused of plagiarism, as his The Superstitious Man was taken from The Augur of Antiphanes, but reworkings and variations on a theme of this sort were commonplace and so the charge is a complicated one. How long complete copies of his plays survived is unclear, although 23 of them, with commentary by Michael Psellus, were said to still have been available in Constantinople in the 11th century, he is praised by Plutarch and Quintilian, who accepted the tradition that he was the author of the speeches published under the name of the Attic orator Charisius. An admirer and imitator of Euripides, Menander resembles him in his keen observation of practical life, his analysis of the emotions, his fondness for moral maxims, many of which became proverbial: "The property of friends is common," "Whom the gods love die young," "Evil communications corrupt good manners".

These maxims were afterwards collected, with additions from other sources, were edited as Menander's One-Verse Maxims, a kind of moral textbook for the use of schools. The single surviving speech from his early play Drunkenness is an attack on the politician Callimedon, in the manner of Aristophanes, whose bawdy style was adopted in many of his plays. Menander found many Roman imitators. Eunuchus, Heauton Timorumenos and Adelphi of Terence were avowedly taken from Menander, but some of them appear to be adaptations and combinations of more than one play, thus in the Andria were combined Menander's The Woman from Andros and The Woman from Perinthos, in the Eunuchus, The Eunuch and The Flatterer, while the Adelphi was compiled from Menander and from Diphilus. The original of Terence's Hecyra is supposed to be, not by Menander, but Apollodorus of Carystus; the Bacchides and Stichus of Plautus were based upon Menander's The Double Deceiver and Brotherly-Loving Men, but the Poenulus does not seem to be from The Carthaginian, nor the Mostellaria from The Apparition, in spite of the similarity of titles.

Caecilius Statius, Luscius Lanuvinus and Atilius imitated Menander. He was further credited with the authorship of some epigrams of doubtful authenticity. Most of Menander's work did not survive the Middle Ages, except as short fragments. Federico da Montefeltro's library at Urbino reputedly had "tutte le opere", a complete works, but its existence has been questioned and there are no traces after Cesare Borgia's capture of the city and the transfer of the library to the Vatican; until the end of the 19th century, all, known of Menander were fragments quoted by other authors and collected by Augustus Meineke and Theodor Kock, Comicorum Atticorum Fragmenta. These consist of some 1650 verses or parts of verses, in addition to a considerable number of words quoted from Menander by ancient lexicographers; this situation changed abruptly in 1907, with the discovery of the Cairo Codex, which contained large parts of the Samia. A fragment of 115 lines of the Sikyonioi had been found in the papier mache of a mummy case in 1906.

In 1959, the Bodmer papyrus was published containing Dyskolos, more of the Samia, half of the Aspis. In the late 1960s, more of the Sikyonioi was found as filling for two more mummy cases. Other papyrus fragments continue to be published. In 2003, a palimpsest manuscript, in Syriac writing of the 9th century, was found where the reused parchment comes from a expensive 4th-century Greek manuscript of work

Darboux integral

In real analysis, a branch of mathematics, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, the values of the two integrals, if they exist, are equal; the definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral. Introductory textbooks on calculus and real analysis develop Riemann integration using the Darboux integral, rather than the true Riemann integral. Moreover, the definition is extended to defining Riemann-Stieltjes integration. Darboux integrals are named after Gaston Darboux; the definition of the Darboux integral considers upper and lower integrals, which exist for any bounded real-valued function f on the interval. The Darboux integral exists if and only if lower integrals are equal; the upper and lower integrals are in turn the infimum and supremum of upper and lower sums which over- and underestimate the "area under the curve."

In particular, for a given partition of the interval of integration, the upper and lower sums add together the areas of rectangular slices whose heights are the supremum and infimum of f in each subinterval of the partition. These ideas are made precise below: A partition of an interval is a finite sequence of values xi such that a = x 0 < x 1 < ⋯ < x n = b. Each interval is called a subinterval of the partition. Let ƒ:→ℝ be a bounded function, let P = be a partition of. Let M i = sup x ∈ m i = inf x ∈ f; the upper Darboux sum of ƒ with respect to P is P = ∑ i = 1 n M i. The lower Darboux sum of ƒ with respect to P is P = ∑ i = 1 n m i; the lower and upper Darboux sums are called the lower and upper sums. The upper Darboux integral of ƒ is U f = inf; the lower Darboux integral of ƒ is L f = sup. In some literature an integral symbol with an underline and overline represent the lower and upper Darboux integrals respectively. L f ≡ ∫ a b _ f d x U f ≡ ∫ a b ¯ f d x And like Darboux sums they are sometimes called the lower and upper integrals.

If Uƒ = Lƒ we call the common value the Darboux Integral. We say that ƒ is Darboux-integrable or integrable and set ∫ a b f d t = U f = L f

RCA Records

RCA Records is an American record label owned by Sony Music, a subsidiary of Sony Corporation of America. It is one of Sony Music's four flagship labels, alongside RCA's former long-time rival Columbia Records; the label has released multiple genres of music, including pop, rock, hip hop, electronic, R&B, blues and country. Its name is derived from the initials of its defunct parent company, the Radio Corporation of America, it was acquired by Bertelsmann in 1986, making it a part of Bertelsmann Music Group. It is the second-oldest record label in American history, after sister label Columbia Records; as of September 2019, artists signed to RCA Records include Britney Spears, Dave Matthews Band, G-Eazy, A$AP Rocky, Christina Aguilera, BROCKHAMPTON, Chris Brown, Miley Cyrus, Foo Fighters, Craig David, Becky G, Childish Gambino, Buddy Guy, Martin Garrix, H. E. R, Enrique Iglesias, Kesha, Alicia Keys, Lykke Li, Pentatonix, Mark Ronson, Shakira, SZA, Bryson Tiller, Justin Timberlake, TOOL, Walk the Moon, ZAYN and ATEEZ.

In 1929, the Radio Corporation of America purchased the Victor Talking Machine Company the world's largest manufacturer of phonographs and phonograph records. The company became RCA Victor. In absorbing Victor, RCA acquired the New World rights to the famous Nipper/"His Master's Voice" trademark. In 1931, RCA Victor's British affiliate the Gramophone Company merged with the Columbia Graphophone Company to form EMI; this gave RCA head David Sarnoff a seat on the EMI board. In September 1931, RCA Victor introduced the first 33⅓ rpm records sold to the public, calling them "Program Transcriptions"; these used a shallower and more spaced implementation of the large "standard groove" found on contemporary 78 rpm records, rather than the "microgroove" used for post-World War II 33⅓ rpm "LP" records. The format was a commercial failure because the new Victrolas with two-speed turntables designed to play these records were exorbitantly priced, the least expensive model retailing for $395.00 in the depths of the Great Depression.

The format was abandoned by 1933, two-speed turntables were no longer offered, but some Program Transcriptions lingered in the Victor record catalog until the end of the 1930s. During the early part of the Depression, RCA Victor made a number of attempts to create a successful cheap label to compete with the "dime store labels"; the first was the short-lived "Timely Tunes" label in 1931 sold at Montgomery Ward. Bluebird Records was created in 1932 as a sub-label of RCA Victor, it was an 8-inch record with a dark blue label, alongside an 8-inch Electradisk label. Neither were a success. In 1933, RCA Victor reintroduced Electradisk as a standard 10-inch label. Another cheap label, was produced; the same musical couplings were issued on all three labels and the Bluebird label still survives today, eight decades after Electradisk and Sunrise were discontinued. RCA Victor produced records for Montgomery Ward label during the 1930s. Besides manufacturing records for themselves, RCA Victor operated RCA Custom, the leading record manufacturer for independent record labels.

RCA Custom pressed many record compilations for The Reader's Digest Association. RCA sold its interest in EMI in 1935, but EMI continued to distribute RCA Victor recordings in the UK and its territories on the HMV label until the late 1950s. RCA manufactured and distributed HMV recordings on the RCA Victor and custom HMV labels in North America. During World War II, ties between RCA Victor and its Japanese affiliate JVC were severed. JVC's record company is known today as Victor Entertainment and still retains the Nipper/His Master's Voice trademark for use in Japan. From 1942 to 1944, RCA Victor was impacted by the American Federation of Musicians recording ban. All union musicians in the US and Canada were forbidden from making recordings during the period. One of the few exceptions was the eventual release of recorded radio broadcast performances from the NBC Symphony Orchestra conducted by Arturo Toscanini. However, RCA Victor lost the Philadelphia Orchestra during this period. Ormandy and the Philadelphia Orchestra would not return to RCA until 1968.

In the spring of 1946, "RCA Victor" replaced "Victor" on labels for shellac 78 rpm singles. In 1949, RCA Victor introduced the 7-inch 45 rpm micro-grooved vinylite record, marketed as the "45"; the new format, under development for several years, was intended to replace 78 rpm discs. By the time RCA Victor belatedly unveiled it, the 45 was now competing with the 10-inch and 12-inch 33⅓ rpm microgroove vinyl "LP" discs introduced by arch-rival Columbia Records in the early summer of 1948. In heavy promotion, RCA Victor sold compact, inexpensive add-on and stand-alone units that played the 45 rpm format exclusively. At first, RCA Victor's 45s wer