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Paul Cohen

For other people named Paul Cohen, see Paul Cohen. Not to be confused with Paul Cohn. Paul Joseph Cohen was an American mathematician, he is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was awarded a Fields Medal. Cohen was born in Long Branch, New Jersey, into a Jewish family that had immigrated to the United States from what is now Poland, he graduated at age 16, from Stuyvesant High School in New York City. Cohen next studied at the Brooklyn College from 1950 to 1953, but he left without earning his bachelor's degree when he learned that he could start his graduate studies at the University of Chicago with just two years of college. At Chicago, Cohen completed his master's degree in mathematics in 1954 and his Doctor of Philosophy degree in 1958, under supervision of Antoni Zygmund; the title of his doctoral thesis was Topics in the Theory of Uniqueness of Trigonometrical Series. In 1957, before the award of his doctorate, Cohen was appointed as an Instructor in Mathematics at the University of Rochester for a year.

He spent the academic year 1958–59 at the Massachusetts Institute of Technology before spending 1959–61 as a fellow at the Institute for Advanced Study at Princeton. These were years. In Factorization in group algebras he showed that any integrable function on a locally compact group is the convolution of two such functions, solving a problem posed by Walter Rudin. In On a conjecture of Littlewood and idempotent measures Cohen made a significant breakthrough in solving the Littlewood Conjecture. On June 2, 1995 Cohen received an honorary doctorate from the Faculty of Science and Technology at Uppsala University, Sweden Cohen is noted for developing a mathematical technique called forcing, which he used to prove that neither the continuum hypothesis nor the axiom of choice can be proved from the standard Zermelo–Fraenkel axioms of set theory. In conjunction with the earlier work of Gödel, this showed that both of these statements are logically independent of the ZF axioms: these statements can be neither proved nor disproved from these axioms.

In this sense, the continuum hypothesis is undecidable, it is the most known example of a natural statement, independent from the standard ZF axioms of set theory. For his result on the continuum hypothesis, Cohen won the Fields Medal in mathematics in 1966, the National Medal of Science in 1967; the Fields Medal that Cohen won continues to be the only Fields Medal to be awarded for a work in mathematical logic, as of 2018. Apart from his work in set theory, Cohen made many valuable contributions to analysis, he was awarded the Bôcher Memorial Prize in mathematical analysis in 1964 for his paper "On a conjecture by Littlewood and idempotent measures", lends his name to the Cohen–Hewitt factorization theorem. Cohen was a full professor of mathematics at Stanford University, he was an Invited Speaker at the ICM in 1962 in 1966 in Moscow. Angus MacIntyre of the Queen Mary University of London stated about Cohen: "He was dauntingly clever, one would have had to be naive or exceptionally altruistic to put one's'hardest problem' to the Paul I knew in the'60s."

He went on to compare Cohen to Kurt Gödel, saying: "Nothing more dramatic than their work has happened in the history of the subject." Gödel himself wrote a letter to Cohen in 1963, a draft of which stated, "Let me repeat that it is a delight to read your proof of the ind of the cont hyp. I think that in all essential respects you have given the best possible proof & this does not happen frequently. Reading your proof had a pleasant effect on me as seeing a good play." While studying the continuum hypothesis, Cohen is quoted as saying in 1985 that he had "had the feeling that people thought the problem was hopeless, since there was no new way of constructing models of set theory. Indeed, they thought you had to be crazy to think about the problem.""A point of view which the author feels may come to be accepted is that CH is false. The main reason one accepts the axiom of infinity is that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe.

With the higher axioms of infinity. Now ℵ 1 is the cardinality of the set of countable ordinals, this is a special and the simplest way of generating a higher cardinal; the set C is, in contrast, generated by a new and more powerful principle, namely the power set axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the replacement axiom can reach C, thus C is ℵ ω, ℵ a, where a = ℵ ω, etc.. This point of view regards C as an rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. Generations will see the problem more and express themselves more eloquently."An "enduring and powerful product" of Cohen's work on the continuum hypothesis, one, used by "countless mathematicians" is k

Gibberulus gibberulus

Gibberulus gibberulus, common name the humpbacked conch, is a species of sea snail, a marine gastropod mollusk in the family Strombidae, the true conchs. There are two subspecies: Gibberulus gibberulus albus Gibberulus gibberulus gibberulus: represented as Gibberulus gibberulus Gibberulus gibberulus gibbosus The adult shell size varies between 30 mm and 70 mm; the smooth shell is gibbous. The spire is varicose; the body whorl is grooved at the base. The columella is smooth; the interior of the aperture is radiately striate. The shell is hieroglyphically marked with yellowish brown and white; the markings are arranged in a few or numerous interrupted revolving bands. The aperture is tinged violaceous, dark purplish brown; this species occurs in the Red Sea and in the Indian Ocean off Aldabra, Kenya, Mauritius, the Seychelles and Tanzania. Linnaeus, C. 1758. Systemae naturae per regna tria naturae, secundum classes, genera, cum characteribus, synonymis, locis.v. Holmiae: Laurentii Salvii 824 pp. Mörch, O.

A. L. 1852. Catalogus Conchyliorum quae reliquit d. Alphonso d'Aguirra & Gadea Comes de Yoldi, regis daniae cubiculariorum princeps, ordinis dannebrogici in prima classe & ordinis caroli Tertii eques. Part 1. Cephalophora. Copenhagen: Hafinae Vol. 1 170 pp. Frauenfeld, G. R. von 1869. Beitraege zur Fauna der Nicobaren. Verhandlungen der Zoologisch-Botanischen Gesellschaft in Wien XIX: 853-900 Jousseaume, F. 1888. Description des Mollusques Recueillis par M. Le Dr. Faurot dans la Mer Rouge et le Golfe D'Aden. Mémoires de la Société Zoologique de France 1: 166-223 Cotton, B. C. 1953. No. 3. Strombidae. Adelaide: Royal Society of South Australia, Malacological Section 4 pp. 1 pl. Abbott, R. T. 1960. The genus Strombus in the Indo-Pacific. Indo-Pacific Mollusca 1: 33-146 Walls, J. G.. Conchs and harps. A survey of the molluscan families Strombidae and Harpidae. T. F. H. Publications Ltd, Hong Kong. Wilson, B. 1993. Australian Marine Shells. Prosobranch Gastropods. Kallaroo, Western Australia: Odyssey Publishing Vol. 1 408 pp. "Gibberulus gibberulus gibberulus". Retrieved 24 March 2011

John Michael Rysbrack

Johannes Michel or John Michael Rysbrack, original name Jan Michiel Rijsbrack, was an 18th-century Flemish sculptor, who spent most of his career in England. His birth-year is sometimes given as 1693 or 1684. Rysbrack was born on 24 June 1694 in the son of the landscape painter Pieter Rijsbraeck, his older brother Pieter Andreas Rijsbrack was a landscape and still life painter. He studied drawings by Italian masters, before settling in London in 1720. In London, Rysbrack established himself as the leading sculptor, a position he was to retain until the mid-1740s, remaining one of the top three sculptors in Britain until shortly before his death, he produced vivid portraits and monuments of lively baroque composition establishing himself as a sought-after sculptor. He executed busts and funerary monuments of many of the most prominent men of his day, including the monument to Isaac Newton in Westminster Abbey, a statue of Marlborough, busts of Walpole and Pope. Dr Cox Macro commissioned him to make a bust of Flemish painter Peter Tillemans on his death in 1734.

In 1733 he carved a magnificent marble portrait bust of George Hamilton, 1st Earl of Orkney in the guise of a Roman centurion. Orkney was a distinguished general serving under the Duke of Marlborough. Orkney had taken the surrender of the French at the Battle of Blenheim in 1704, he took part in numerous subsequent battles during the War of the Spanish Succession. One of Rysbrack's greatest works, the bust of Lord Orkney is on display at the Victoria and Albert Museum, London, its special character owes something to a bond between the sculptor and Lord Orkney, one that had its origins nearly 30 years earlier and was no doubt enhanced in their conversation when Lord Orkney sat to the sculptor for the modelling of the bust. Rysbrack would have been aware of Lord Orkney’s heroism during the various campaigns in the Low Countries during the War of the Spanish Succession, not least the Battle of Ramillies on 23 May 1706 after which Orkney led the pursuit of the defeated French forces. Following the battle and pursuit, city after city – including Brussels and Antwerp – capitulated to Marlborough's forces.

In Antwerp, Rysbrack’s home city, to which Orkney was dispatched by Marlborough with re-enforcements for Major-General Cadogan, the Spanish Governor was in no mood to offer a token resistance, constrained the French part of the garrison to join him in surrendering the city on 6 June. The arrival in Antwerp of Marlborough’s victorious forces, led by Orkney is an event that Rysbrack a boy aged 12, would have witnessed; these events must have given sculptor and Lord Orkney a great deal of opportunity for shared reminiscences during the sittings for the bust. In St Michael and All Angels Church, there is another splendid monument by Rysbrack and dated 1754; the 2nd and the 3rd Duke of Beaufort are depicted in Roman costume, one standing, the other seated on the sarcophagus and holding a medallion. Decorative, asymmetrical drapery hangs down over the sarcophagus. Rysbrack cast the bronze equestrian statue of William III in Queen Square, Bristol in 1733, a monument to Edward Colston in All Saints, Bristol.

Rysbrack died in Vere Street, Westminster, in 1770. Portraits of Rysbrack "Model for the tomb of Sir Isaac Newton". Victoria and Albert Museum. Retrieved 19 November 2007. John Michael Rysbrack at Find a Grave

Grumman G-118

The Grumman G-118 was a design for an all-weather missile-armed interceptor aircraft for use on US Navy aircraft carriers. Conceived as an uprated F11F Tiger, it soon evolved into a larger and more powerful project. Although two prototypes were ordered in 1955, development was cancelled the same year in favor of the F4H Phantom II before any examples were built. Grumman's next carrier fighter would be the F-14 Tomcat, ordered in 1968; the Grumman Design 118 was a two-seat, twin-engined, rocket augmented, carrier-based all-weather supersonic fighter aircraft. It had a 45° swept wing, a "T-tail" empennage, two small folding ventral fins, a landing gear of tricycle configuration. For ejection, the tandem crew were ejected downwards, it featured a boundary layer control system to improve low speed handling. The G-118 was to be powered by two J79-GE-3 engines, with accommodations for the more powerful J79-GE-207 engines each producing 18,000 lbf of afterburning thrust. Similar to the contemporary Vought XF8U-3 Crusader III, it was designed with an additional throttleable liquid-fueled rocket engine using a mixture of JP-4 fuel and hydrogen peroxide oxidizer which produced 5,000 lbf of thrust.

Armament stores would have been under the fuselage in two semi-recessed hardpoints for the AIM-7 Sparrow air-to-air missile and an internal weapons bay for an additional AIM-7 or three AIM-9 Sidewinder missiles. Data from and Standard Aircraft CharacteristicsGeneral characteristics Crew: 2 Length: 57 ft 7 in Wingspan: 43 ft 11.69 in Height: 14 ft 10 in Wing area: 595 sq ft Empty weight: 26,355 lb Gross weight: 37,366 lb Max takeoff weight: 51,216 lb Powerplant: 2 × J79-GE-3 after-burning turbojet engines, 10,000 lbf thrust each dry, 15,600 lbf with afterburner Powerplant: 1 × throttleable rocket engine, 5,000 lbf thrustPerformance Maximum speed: Mach 2 Range: 1,352 nmi Service ceiling: 60,000 ft Wing loading: 62.8 lb/sq ft Armament Rockets: 2 x Ding Dong unguided nuclear rockets or Missiles: 3 x AIM-7 Sparrow III missiles or 2 x AIM-7 Sparrow and 3 x AIM-9 Sidewinder Related development F-11 TigerAircraft of comparable role and era F-4 Phantom II F8U-3 Crusader III Related lists List of fighter aircraft List of military aircraft of the United States Notes BibliographyAngelucci, Enzo.

The American Fighter from 1917 to the present. New York: Orion Books. Buttler, Tony. American Secret Projects: Fighters & Interceptors 1945-1978. Hinckley, England, UK: Midland Publishing. ISBN 978-1-85780-264-1; the Illustrated Encyclopedia of Aircraft. London: Aerospace Publishing. Taylor, Michael J. H.. Jane's Encyclopedia of Aviation. London: Studio Editions. World Aircraft Information Files. London: Bright Star Publishing

Alan Ramsbottom

Alan Ramsbottom was a professional racing cyclist from Clayton-le-Moors, who twice rode the Tour de France. Ramsbottom was a talented amateur in Britain in the late 1950s, he decided to race abroad when he wasn't selected for the Olympic Games of 1960 and moved to Troyes, France in 1961, after seeing an advertisement by the local club, UV Aube, for British riders. The club was run by the French national team manager. Ramsbottom was inspired by meeting Britain's leading professional rider, Brian Robinson, at a cycling club dinner in Blackburn, he won the second stage of the Tour de l'Avenir in France in 1961 turned professional and rode for the Pelforth-Sauvage team for 1962 and 1963. The team was advertised as riding Lejeune bicycles but Ramsbottom's was the Harry Quinn he had ridden as an amateur, sprayed in Lejeune colours, he came 45th in the Tour de France in 1962 as a first-year professional. And 16th in 1963, he finished eighth in Liège–Bastogne–Liège in 1963. and 11th in the Flèche Wallonne in both 1963 and 1965.

Bidot told Ramsbottom to think more of himself, to attack more and to force Pelforth to raise his salary. He won the Tour de Haute-Loire in France in 1964, he came fourth in Nice-Genoa, third in the Boucles Rouquevairoises. Ramsbottom planned to ride the Tour de France again that year but Pelforth dropped him from its team because of what Ramsbottom said was a misunderstanding between him and the manager, Maurice de Muer: I was building up to be in top form for the Tour de France; the year before, I had been 16th and am confident that I would have been in the first 10 if I hadn't been ordered to wait for Henry Anglade on the Forclaz, when he was beyond help. During the Dauphiné Libéré, which finished on 6 June just 16 days before the Tour started, Maurice de Muer said he wanted me to ride in the Tour of Luxembourg from 12–15 June. I reminded him of my heavy early season, my anxiety to do well in the Tour de France. I said. Did he mind if I missed the Luxembourg tour? He agreed, said it would be all right.

In the Dauphiné, my wife rang to say there was a letter from our team manager saying I had been picked for the Tour of Luxembourg. Assuming this letter to have been written before my talk with de Muer and that it had no more significance, I told her not to bother to reply and went off to the Isle of Man and finished fourth and returned to the Continent to find big stories in the papers saying I had failed to turn up at Luxembourg and cost Anglade the race, they wouldn't listen to my argument and I was chucked out of the Tour team. In 1964 he moved from Troyes to Belgium to join Tom Simpson in the Peugeot team But after that, he said, nothing went right.". He fell while training with another British professional, Vin Denson, caught his hand between cobbles and broke an arm, he said: If I had my chance on the Continent over again, I wouldn't change a thing until May 1964, when I moved to Belgium. Things never went right after that. Troyes - and most districts of France - give any rider a chance to shine.

Belgium - Flanders, that is, where the majority of races are held - suits only one type of rider: the strong, fearless man, prepared to rake risks and barge his way through gaps, but I am not one of them. Living in Ghent gave him the chance to ride more criteriums, the round-the-houses races where professionals in the 1960s made much of their money, he came third at Meerbeke in 1964 and third in London, at Crystal Palace, where he, runner-up Seamus Elliott and winner Tom Simpson lapped a field of domestic professionals. He didn't get in Peugeot's team for the Tour de France in 1965, he came second in a criterium at Wortegem, Belgium and at Zele and third at Aartrijke in 1965. In 1966 he returned to Britain because a glut of unemployed professionals on the Continent pushed wages lower than he thought necessary for a married man with two children, he moved to Great Harwood and went back to his former trade as a sewing machine mechanic, working in textile factories. At weekends he rode in 1966 and 1967 for Viking Cycles.

In 1965, Ramsbottom rode the Grand Prix des Gentilhommes at Lille with the journalist Jock Wadley. The race paired current riders with former racers or leisure riders in a two-man time-trial. Wadley called him:...a man of few words who seems lost in thought. He seems unaffected by the Tour and all that it entails, doesn't seem to care much what happens shrugging his shoulders when asked what his plans were. Whereas at the end of a tough day in the saddle his former team-mate Henry Anglade grabs a microphone and talks, Alan swallows a bottle of Perrier and pedals off to the hotel. Ramsbottom said the pain of his disappointment on the Continent "lessened over the years"


Cavtat is a village in the Dubrovnik-Neretva County of Croatia. It is on the Adriatic Sea coast 15 kilometres south of Dubrovnik and is the centre of the Konavle municipality; the original city was founded by the Greeks in the 6th century BC under the name of Epidaurus. The surrounding area was inhabited by the Illyrians; the town changed its name to Epidaurum when it came under Roman rule in 228 BC. Justinian I the Emperor of the Byzantine Empire sent his fleet to Cavtat during the Gothic War and occupied the town; the city was destroyed by the Avars and Slavs in the 7th century. Refugees from Epidaurum fled to the nearby island, Laus which over time evolved into the city of Dubrovnik; the town was re-established in the Middle Ages. After a short while it came under the control of the Republic of Ragusa; the modern Croatian name for the city reveals its link with Dubrovnik. Cavtat is derived from Civitas Vetus. Today, Cavtat is a popular tourist destination with many hotels and private households that rent rooms and apartments.

The seafront is filled with restaurants. A ferry boat connects the town to neighbouring Dubrovnik. There are many private luxury ships and yachts along the strand; the town cemetery on the hill contains a mausoleum belonging to the Račić family and decorated by the sculptor Ivan Meštrović. In year 2004 Cavtat got the title European Competition for Towns and Villages in Blooms for the well-tended green areas and flower arrangements on the beach promenade; the Epidaurus Festival of Music has been held annually in Cavtat since 2007. Vlaho Bukovac, painter Tino Pattiera, opera singer Luko Zore and Slavist Frano Supilo, politician Baltazar Bogišić, law historian and ethnologist Niko Koprivica, politician Dinko Zlatarić, poet and translator Raimondo Cunich, humanist Ljudevit Vulićević, Serbian writer and patriot Cavtat is twinned with: Bochnia, Poland Watsonville, California, USA Croatia Dubrovnik Dalmatia Republic of Ragusa Epidaurus Notes Cavtat Info, Villa Vidak: Cavtat Information, Up to date news from Cavtat and vicinity