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Cauchy–Schwarz inequality

In mathematics, the Cauchy–Schwarz inequality known as the Cauchy–Bunyakovsky–Schwarz inequality, is a useful inequality encountered in many different settings, such as linear algebra, probability theory, vector algebra and other areas. It is considered to be one of the most important inequalities in all of mathematics; the inequality for sums was published by Augustin-Louis Cauchy, while the corresponding inequality for integrals was first proved by Viktor Bunyakovsky. The integral inequality was rediscovered by Hermann Amandus Schwarz; the Cauchy–Schwarz inequality states that for all vectors u and v of an inner product space it is true that | ⟨ u, v ⟩ | 2 ≤ ⟨ u, u ⟩ ⋅ ⟨ v, v ⟩, where ⟨ ⋅, ⋅ ⟩ is the inner product. Examples of inner products include the complex dot product. Equivalently, by taking the square root of both sides, referring to the norms of the vectors, the inequality is written as | ⟨ u, v ⟩ | ≤ ‖ u ‖ ‖ v ‖. Moreover, the two sides are only if u and v are linearly dependent. If u 1, …, u n ∈ C and v 1, …, v n ∈ C, the inner product is the standard complex inner product the inequality may be restated more explicitly as follows: | u 1 v ¯ 1 + ⋯ + u n v ¯ n | 2 ≤ or | ∑ i = 1 n u i v ¯ i | 2 ≤ ∑ j = 1 n | u j | 2 ∑ k = 1 n | v k | 2.

Let u and v be arbitrary vectors in a vector space over F with an inner product, where F is the field of real or complex numbers. We prove the inequality | ⟨ u, v ⟩ | ≤ ‖ u ‖ ‖ v ‖ and that equality holds if and only if either u or v is a multiple of the other. If v = 0, it is clear that there is equality, in this case u and v are linearly dependent, regardless of u, so the theorem is true. If u = 0. One henceforth assumes. Let z = u − u v = u − ⟨ u, v ⟩ ⟨ v, v ⟩ v. Then, by linearity of the inner product in its first argument, one has ⟨ z, v ⟩ = ⟨ u − ⟨ u, v ⟩ ⟨ v, v ⟩ v, v ⟩ =

Ben Chacko

Ben Patrick Chacko is an English journalist, the editor of the Morning Star. After joining the newspaper in 2010, he became editor in 2015. Chacko was born in London, he was brought up in Cheltenham, educated at the local Pate's Grammar School, St John's College, where he studied Mandarin Chinese. His father, Francis Chacko, who came to Britain from India at the age of eight, is an actuary, while his Lancastrian mother Sarah is a software engineer who studied for a PhD at the University of Oxford, his brother is the tax barrister Thomas Chacko, of Pump Court Tax Chambers. Chacko credits his conversion to communism to a recommendation from his mother when he was a teenager that he abandon the Socialist Worker newspaper, published by the Socialist Workers Party, for something more genuinely "leftie", such as the Morning Star. Chacko found the Star "a real revelation". By the age of 15, Chacko was attending meetings of the Young Communist League, he edited Challenge, the journal of the Young Communist League and was a member of the student union council at Oxford.

Chacko was appointed editor of the Morning Star in May 2015, the youngest editor of the paper since its founding editor, William Rust. He joined the paper as a sub-editor in 2010, was subsequently deputy features editor, assistant editor and deputy editor before being appointed acting editor in July 2014. "The Star is the most precious and only voice we have in the daily media", said the Labour MP Jeremy Corbyn at the time of Chacko's appointment in May 2015. "I look forward to working with Ben in promoting socialism and progress". After leaving university, Chacko lived in China for a few years; the country, according to him, is evidence that "you can run a society without surrendering to the idea that the market is always right". Post-Soviet Russia, on the other hand, he describes as "a gangster capitalist state run by oligarchs". In summer 2015, he told Josh Glancy of The Sunday Times: "We need a revolution in politics to overturn the power of private ownership... That doesn't mean, it could be a revolution, enacted by people in parliament as a result of a mass movement for change"

942 Romilda

942 Romilda is a background asteroid 36 kilometers in diameter, located in the outer region of the asteroid belt. It was discovered by German astronomer Karl Reinmuth at the Heidelberg Observatory on 11 October 1920; the assumed C-type asteroid has a rotation period of 6.97 hours. It was named "Romilda", a common German female name unrelated to the discoverer's contemporaries, taken from the almanac Lahrer Hinkender Bote. Romilda is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements, it orbits the Sun in the outer asteroid belt at a distance of 2.6–3.7 AU once every 5 years and 8 months. Its orbit has an eccentricity of 0.17 and an inclination of 11° with respect to the ecliptic. On 11 October 1920, Romilda was discovered by Karl Reinmuth at the Heidelberg-Königstuhl State Observatory in southwest Germany. On the same night, German astronomer Arnold Schwassmann independently discovered the asteroid at the Bergedorf Observatory in Hamburg.

However, the Minor Planet Center only credits Reinmuth as official discoverer. The body's observation arc begins at Heidelberg Observatory on 21 October 1920, the night after its official discovery observation; this minor planet was named "Romilda", after a female name picked from the Lahrer Hinkender Bote, published in Lahr, southern Germany. A Hinkender Bote was a popular almanac in the alemannic-speaking region from the late 17th throughout the early 20th century; the calendar section contains feast and name days, the dates of important fairs and astronomical ephemerides. For 25 March, the calendar gives "Romilda" as the German analogue next to the catholic and protestant feast days; as with 22 other asteroids – starting with 913 Otila, ending with 1144 Oda – Reinmuth selected names from this calendar due to his many asteroid discoveries that he had trouble thinking of proper names. These names are not related to the discoverer's contemporaries. Lutz Schmadel, the author of the Dictionary of Minor Planet Names learned about Reinmuth's source of inspiration from private communications with Dutch astronomer Ingrid van Houten-Groeneveld, who worked as a young astronomer at Heidelberg.

Romilda is an assumed carbonaceous C-type asteroid. In December 2005, a rotational lightcurve of Romilda was obtained from photometric observations over seven nights by Walter Cooney at the Blackberry Observatory in Louisiana. Lightcurve analysis gave a rotation period of 6.965±0.003 hours with a brightness variation of 0.35±0.05 magnitude. In January 2006, Italian astronomers Roberto Crippa and Federico Manzini at the Sozzago Astronomical Station determined a nearly identical period of 6.9659±0.0004 hours with an amplitude of 0.26±0.01 magnitude. According to the surveys carried out by the Japanese Akari satellite and the NEOWISE mission of NASA's Wide-field Infrared Survey Explorer, Romilda measures and kilometers in diameter, its surface has an albedo of and, respectively; the Collaborative Asteroid Lightcurve Link assumes a standard albedo for a carbonaceous asteroid of 0.057 and calculates a diameter of 35.12 km based on an absolute magnitude of 11. Further published mean-diameters and albedos by he WISE team include, with corresponding albedos of, and.

Lightcurve Database Query, at www.minorplanet.info Dictionary of Minor Planet Names, Google books Asteroids and comets rotation curves, CdR – Geneva Observatory, Raoul Behrend Discovery Circumstances: Numbered Minor Planets - – Minor Planet Center 942 Romilda at AstDyS-2, Asteroids—Dynamic Site Ephemeris · Observation prediction · Orbital info · Proper elements · Observational info 942 Romilda at the JPL Small-Body Database Close approach · Discovery · Ephemeris · Orbit diagram · Orbital elements · Physical parameters

Canute Peterson House

The Canute Peterson House is a historic residence in Ephraim, United States. In 1978, it was listed on the National Register of Historic Places. Built in 1869 by Canute Peterson, an early Latter-day Saint leader in Sanpete County, it was designed by architect William H. Folsom. Richard Nibley, brother of Mormon scholar Hugh Nibley and restored the home in the 1960s. For a time it was a breakfast. However, in 2014, Cache Valley Bank purchased and incorporated the old home into the newly constructed bank building, it is open to the public for tours Monday-Friday 8:30 am -- 5:00 pm. National Register of Historic Places listings in Sanpete County, Utah Former official website Historic American Buildings Survey No. UT-64, "Canute Peterson House, 10 North Main Street, Sanpete County, UT", 4 photos, 8 measured drawings, 7 data pages

Werner-Schaus Ministry II

The second Werner-Schaus Ministry was the government of Luxembourg between 6 February 1969 and 15 June 1974. Throughout the ministry, the Deputy Prime Minister was Eugène Schaus, replacing Henry Cravatte, Deputy Prime Minister in the Werner-Cravatte Ministry, it was a coalition between the Christian Social People's Party, the Democratic Party. At the discussions for the 1969 budget, disagreements surfaced in the government coalition. While the LSAP favoured a salary increase for civil servants and a pensions increase, the CSV believed that with the economic situation characterised by weak growth, it was not possible to give way to expensive pay demands; the two party heads, Pierre Werner and Henry Cravatte, decided to hold early elections, which would otherwise have been held in 1969. As a result of the elections of 15 December 1968, the CSV lost one seat and the LSAP lost three, while the Democratic Party regained its position of 1959, returning 11 Deputies. Despite their parties' losses and Cravatte would have liked to continue the cooperation between CSV and LSAP.

However, the trade union wing of the LSAP opposed such a continuation. The DP thus took the LSAP's place as a coalition partner in the government, while the LSAP went into opposition. Two ministerial reshuffles occurred during this government, one on 5 July 1971 with the addition of two secretaries of State, the other on 19 September 1972 with the resignation of Madeleine Frieden, due to a scandal. After this second change, Camille Ney was promoted to minister and Jacques Santer joined the government as a secretary of State. In the period 1969-1974, Luxembourg continued to play an important role in Europe thanks to ideas on monetary integration developed by its Prime Minister. In January 1968, invited to a congress of the CDU in Germany, Pierre Werner had presented a five-point plan of for European Economic and Monetary Union. Provoking the interest of the other member states of the EEC, Werner was invited to elaborate his views before the conference of ministers of finance in Rotterdam in September 1968 at the European summit of The Hague in December 1969.

At this last meeting, the governments declared their intention to elaborate a plan for steps towards creating an economic and monetary union. Currency was going to be the motor for European construction. In March 1970, Pierre Werner was made the head of a study group. In its final report, the commission of experts gave priority to the coordination of economic policies, the necessity for common decision-making instances, the centralisation of monetary policy by the creation of a committee of central bank governors, the limitation in variation of exchange rates and the creation of a European fund to sustain exchange rates; the “Werner Plan” was well-received, but the difficulties of the dollar and the pound sterling, the oil crisis and stagflation put a halt on its realisation. However, the European Monetary System, which came into force in 1979, made use of several of its elements, such as the European Monetary Cooperation Fund and the snake in the tunnel. In 1972, after 50 years of existence, the Belgium-Luxembourg Economic Union was due to expire.

The two governments decided to renew it every 10 years. Every renewal was the occasion for new negotiations. At the renewal of 1972, the question on the sharing of excise rights on petrol and alcohol was at the centre of negotiations. While the treaty of 1921 had established population size as the criterion for dividing up income from excise, the Luxembourgish government demanded a different method of dividing it up, which took more account of the economic realities. A protocol signed on 27 October 1971 tried to satisfy Luxembourgish demands in establishing at regular intervals, a new distribution of excise income. During the late 1960s, the Grand Duchy's economy had been sluggish: its growth rate was less than that of its EEC partners; the weak growth was due to the monolithic structure of the Luxembourgish economy. In 1968, the Economic and Social Council had warned the government that “everything leads us to believe that the times of great prosperity are over and will only return temporarily and in exceptional circumstances.

Subsequently, one should not expect the steel industry to continue to be, as it has for many years, the essential pillar of technical and social progress of the country”. From 1969, the economic situation improved. Investments in the steel industry increased by 150% compared with 1968. Steel production increased to reach the record number of 6,448,351 tonnes in 1974. Pay in the steel industry increased by 27% from 1969 to 1973; the wave of prosperity dimmed the structural fears which had provoked the fall of the preceding government. However, the full use of production capacities and increased global demand created a runaway situation which risked accelerating the spiral of inflation. From on, government action aimed to counter the effects of economic overheating by strengthening price controls, by ordering credit institutions and pension funds to slow their credit policy, by temporarily authorising work on Saturdays in the construction sector, affected by the lack of manpower, by reducing the rate of VAT on a number of consumed items with a strong influence on the price index.

A strong increase in the budget surplus made it possible to make public savings and lead a countercyclical policy. The state's budget surplus fed into various investment funds, such as the roads fund or the crisis fund created by the law of 27 July 1938; the introduction of VAT on 1 January 1970 sparked fears of inflation. The replacement of the old tax on revenue, the principal fiscal resource in the domain of indirect taxes, with the system of valu

FC Silmash Kharkiv

FC Silmash Kharkiv was an association football club of the Ukrainian SSR and the Russian Empire, created in 1910 as Sade FC. The club played at its own stadiums. Champion of Kharkiv: 1913, 1917 It represented an engine-building Gelferikh Sade Factory that existed since 1875 in Kharkiv and was producing an agricultural equipment. In 1918 the factory was nationalized and in 1922 was renamed into Sickle and Mallet. Since the fall of the Soviet Union, the factory stayed as a state company. In 2005 it was announced bankrupt. In 1919 Sade FC was reformed into FC Serp i Molot Kharkiv. In the fall of 1936 the Sickle and Mallet Factory fielded another football team in the Soviet Group B championship called FC Silmash Kharkiv. In 1941 the club was temporarily merged with FC Dynamo Kharkiv to form FC Spartak Kharkiv. Dynamo Kharkiv was dissolved as part of Dynamo sports society representation that took place in 1940 and included at least such clubs as FC Dynamo Odessa and FC Dynamo Dnipropetrovsk. Another FC Spartak has participated in the Soviet Group A competitions, yet was dissolved upon relegation in 1938.

There is information that after the World War II, FC Silmash Kharkiv was revived in lower league competitions, yet its further fate is uncertain. Helferikh-Sade Kharkiv. UkrSoccerHistory.com