SUMMARY / RELATED TOPICS

Óbuda

Óbuda was a city in Hungary, merged with Buda and Pest on 17 November 1873. The name means Old Buda in Hungarian; the name in Croatian and Serbian for this city is Stari Budim, but the local Croat minority calls it Obuda. The island next to this part of the city today hosts the Sziget Festival, a huge music and cultural festival. Óbuda's centre is Fő tér, connected to a small square with a sculpture of people waiting for the rain to stop. It is accessible by HÉV. Settlements dating from the stone age have been found in Óbuda; the Romans built the capital of Pannonia province here. Hungarians arrived after 900 and it served as an important settlement of major tribal leaders kings; the site was the location of ecclesiastical foundations. Béla IV of Hungary built a new capital after the 1241-42 Mongol invasion in Buda, somewhat south of Óbuda. In the fourteenth century, Óbuda featured a cloister of the Poor Clares; the obscured historical remains of Óbuda, allied with the role it played in nineteenth-century poetry, has resulted it being subject to various historical disputes.

The Jewish Elementary School in Óbuda was victim of the Holocaust. On 13 June 2012, a commemorative plaque to the former teachers and students was affixed to the wall of the building erected on the site of the school. Quote: I will give them an everlasting name, that shall not be cut off. Károly Bebosculptor József Manes Österreicherphysician Pál Harrer – the first and only mayor of Óbuda Egon Orowanphysicist and metallurgist Aquincum Museum, small museum displays jewels, metal tools, wall paintings relating to the lives of ancient Romans living in Aquincum; the museum's outdoor site contains remnants of the town, including courtyards, baths, a market place, large columns, a stone sarcophagus. Roman ruins elsewhere in Óbuda include baths that served the Roman legionnaires stationed in Aquincum, the Hercules Villa, two amphitheatres, the Aquincum Civil Amphitheater and the larger Aquincum Military Amphitheatre. Kassák Museum, a branch museum of the Petőfi Literary Museum. Kerületi TUE, football team 33 FC, football team Official website A Walk through Old Buda

Bézout's theorem

Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves which do not share a common component. The theorem states that the number of common points of two such curves is at most equal to the product of their degrees, equality holds if one counts points at infinity and points with complex coordinates, if each point is counted with its intersection multiplicity, it is named after Étienne Bézout. Bézout's theorem refers to the generalization to higher dimensions: Let there be n homogeneous polynomials in n+1 variables, of degrees d 1, …, d n, that define n hypersurfaces in the projective space of dimension n. If the number of intersection points of the hypersurfaces is finite over an algebraic closure of the ground field this number is d 1 ⋯ d n, if the points are counted with their multiplicity; as in the case of two variables, in the case of affine hypersurfaces, when not counting multiplicities nor non-real points, this theorem provides only an upper bound of the number of points, reached.

This is referred to as Bézout's bound. Bézout's theorem is fundamental in computer algebra and effective algebraic geometry, by showing that most problems have a computational complexity, at least exponential in the number of variables, it follows that in these areas, the best complexity that may be hoped for will occur in algorithms that have a complexity, polynomial in Bézout's bound. Suppose that X and Y are two plane projective curves defined over a field F that do not have a common component; the total number of intersection points of X and Y with coordinates in an algebraically closed field E which contains F, counted with their multiplicities, is equal to the product of the degrees of X and Y. The generalization in higher dimension may be stated as: Let n projective hypersurfaces be given in a projective space of dimension n over an algebraically closed field, which are defined by n homogeneous polynomials in n + 1 variables, of degrees d 1, …, d n. Either the number of intersection points is infinite, or the number of intersection points, counted with multiplicity, is equal to the product d 1 ⋯ d n.

If the hypersurfaces are irreducible and in relative general position there are d 1 ⋯ d n intersection points, all with multiplicity 1. There are various proofs of this theorem. In particular, it may be deduced by applying iteratively the following generalization: if V is a projective algebraic set of dimension δ and degree d 1, H is a hypersurface of degree d 2, that does not contain any irreducible component of V the intersection of V and H has dimension δ − 1 and degree d 1 d 2. For a proof using the Hilbert series see Hilbert series and Hilbert polynomial#Degree of a projective variety and Bézout's theorem. Bézout's theorem has been further generalized as the so-called multi-homogeneous Bézout theorem. Bezout's theorem was stated by Isaac Newton in his proof of lemma 28 of volume 1 of his Principia in 1687, where he claims that two curves have a number of intersection points given by the product of their degrees; the theorem was published in 1779 in Étienne Bézout's Théorie générale des équations algébriques.

Bézout, who did not have at his disposal modern algebraic notation for equations in several variables, gave a proof based on manipulations with cumbersome algebraic expressions. From the modern point of view, Bézout's treatment was rather heuristic, since he did not formulate the precise conditions for the theorem to hold; this led to a sentiment, expressed by certain authors, that his proof was neither correct nor the first proof to be given. The most delicate part of Bézout's theorem and its generalization to the case of k algebraic hypersurfaces in k-dimensional projective space is the procedure of assigning the proper intersection multiplicities. If P is a common point of two plane algebraic curves X and Y, a non-singular point of both of them and, the tangent lines to X and Y at P are distinct the intersection multiplicity is one; this corresponds to the case of "transversal intersection". If the curves X and Y have a common tangent at P the multiplicity is at least two. See intersection number for the definition in general.

Two distinct non-parallel lines always meet in one point. Two parallel lines intersect at a unique point. To see how this works algebraically, in projective space, the lines x + 2 y = 3 and x + 2 y = 5

Prix Saint-Roman

The Prix Saint-Roman was a Group 3 flat horse race in France open to two-year-old thoroughbreds. For much of its history it was run at Longchamp over a distance of 1,800 metres, it was scheduled to take place each year in late September or early October. During the 1890s and early 1900s, the event was a 3,000-metre race for older horses, it was staged on the same day as the Prix du Conseil Municipal. The Prix Saint-Roman was restricted to two-year-olds and cut to 1,800 metres in 1907, it became part of the Prix de l'Arc de Triomphe meeting in 1920. The present system of race grading was introduced in 1971, the Prix Saint-Roman was classed at Group 3 level; the race was moved to the week before the Prix de l'Arc de Triomphe in 1989. It was transferred to Évry in 1991, switched to November in 1994, it was relocated to Saint-Cloud and shortened to 1,600 metres in 1997. The Prix Saint-Roman was closed to colts and geldings in 1998, it continued as a fillies' race until 2000. It was replaced the following year by a Group 3 race at Maisons-Laffitte.

* The 1910 race was a dead-heat and has joint winners. List of French flat horse races France Galop / Racing Post: 1979, 1980, 1981, 1982, 1983, 1984, 1985, 1986, 1987, 1988 1989, 1990, 1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998 1999, 2000