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Crossing Number (graph Theory)
In graph theory, the crossing number of a graph is the lowest number of edge crossings of a plane drawing of the graph . For instance, a graph is planar if and only if its crossing number is zero. Determining the crossing number continues to be of great importance in graph drawing, as user studies have shown that drawing graphs with few crossings makes it easier for people to understand the drawing. The study of crossing numbers originated in Turán's brick factory problem, in which Pál Turán asked for a factory plan that minimized the number of crossings between tracks connecting brick kilns to storage sites. Mathematically, this problem can be formalized as asking for the crossing number of a complete bipartite graph. The same problem arose independently in sociology at approximately the same time, in connection with the construction of sociograms. Turán's conjectured formula for the crossing numbers of complete bipartite graphs remains unproven, as does an analogous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon (not self-intersecting). Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points. A strictly convex polygon is a convex polygon such that no line contains two of its edges. In a convex polygon, all interior angles are less than or equal to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees. Properties The following properties of a simple polygon are all equivalent to convexity: *Every internal angle is strictly less than 180 degrees. *Every point on every line segment between two points inside or on the boundary of the polygon remains inside or on the boundary. *The polygon is entirely contained ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
On-line Encyclopedia Of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009. Sloane is chairman of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. , it contains over 350,000 sequences, making it the largest database of its kind. Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. History Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics. The database was at first stored on punche ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard K
Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include "Richie", "Dick", "Dickon", " Dickie", " Rich", "Rick", " Rico", " Ricky", and more. Richard is a common English, German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Catalan "Ricard" and the Italian "Riccardo", among others (see comprehensive variant list below). People named Richard Multiple people with the same name * Richard Andersen (other) * Richard Anderson (other) * Richard Cartwright (other) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructivism (art)
Constructivism is an early twentieth-century art movement founded in 1915 by Vladimir Tatlin and Alexander Rodchenko. Abstract and austere, constructivist art aimed to reflect modern industrial society and urban space. The movement rejected decorative stylization in favor of the industrial assemblage of materials. Constructivists were in favour of art for propaganda and social purposes, and were associated with Soviet socialism, the Bolsheviks and the Russian avant-garde. Constructivist architecture and art had a great effect on modern art movements of the 20th century, influencing major trends such as the Bauhaus and De Stijl movements. Its influence was widespread, with major effects upon architecture, sculpture, graphic design, industrial design, theatre, film, dance, fashion and, to some extent, music. Beginnings Constructivism was a post-World War I development of Russian Futurism, and particularly of the 'counter reliefs' of Vladimir Tatlin, which had been exhibited i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Ernest
John Ernest (May 6, 1922 – July 21, 1994) was an American-born constructivist abstract artist. He was born in Philadelphia, in 1922. After living and working in Sweden and Paris from 1946 to 1951, he moved to London, England, where he lived and worked from 1951. As a mature student at Saint Martin's School of Art he came under the influence of Victor Pasmore and other proponents of constructivism. During the 1950s together with Anthony Hill, Kenneth Martin, Mary Martin, Stephen Gilbert and Gillian Wise he became a key member of the British constructivist (a.k.a. constructionist) art movement. Ernest created both reliefs and free standing constructions. Several of his works are held at Tate Britain, including the Moebius Strip sculpture. He designed both a tower and a large wall relief at the International Union of Architects congress, South Bank, London, 1961. The exhibition structure also housed works by several of the other British constructivists. Ernest had a lifelong fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anthony Hill (artist)
Anthony Hill (23 April 1930 – 13 October 2020) was an English artist, painter and relief-maker, originally a member of the post-World War II British art movement termed the Constructionist Group whose work was essentially in the international constructivist tradition. Biography His fellow members in this group were Victor Pasmore, Adrian Heath, John Ernest, Kenneth Martin, Mary Martin, Gillian Wise (artist) and Stephen Gilbert. He was born on 23 April 1930 in London, and studied at the St Martin's and the Central Schools of Art 1948–51. He began painting in the style of Dada and Surrealism in 1948 but quickly moved on to geometric abstract idioms. He made his first relief in 1954 and abandoned painting for relief-making in 1956. One feature of these reliefs has been the use of non-traditional materials such as industrial aluminium and Perspex. His first one-man show of reliefs was held at the Institute of Contemporary Arts in 1958. He has participated in exhibitions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper And Lower Bounds
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element of that is less than or equal to every element of . A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. Examples For example, is a lower bound for the set (as a subset of the integers or of the real numbers, etc.), and so is . On the other hand, is not a lower bound for since it is not smaller than every element in . The set has as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that . Every subset of the natural numbers has a l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kazimierz Zarankiewicz
Kazimierz Zarankiewicz (2 May 1902 – 5 September 1959) was a Polish mathematician and Professor at the Warsaw University of Technology who was interested primarily in topology and graph theory. Biography Zarankiewicz was born in Częstochowa. He studied at the University of Warsaw, together with Zygmunt Janiszewski, Stefan Mazurkiewicz, Wacław Sierpiński, Kazimierz Kuratowski, and Stanisław Saks. During World War II, Zarankiewicz took part in illegal teaching, forbidden by the German authorities, and eventually was sent to a concentration camp. He survived and became a teacher at Warsaw University of Technology. He visited universities in Tomsk, Harvard, London, and Vienna. He served as president of the Warsaw section of the Polish Mathematical Society and the International Astronautical Federation. He died in London, England. Research contributions Zarankiewicz wrote works on cut-points in connected spaces, on conformal mappings, on complex functions and number t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposing military alliances: the Allies and the Axis powers. World War II was a total war that directly involved more than 100 million personnel from more than 30 countries. The major participants in the war threw their entire economic, industrial, and scientific capabilities behind the war effort, blurring the distinction between civilian and military resources. Aircraft played a major role in the conflict, enabling the strategic bombing of population centres and deploying the only two nuclear weapons ever used in war. World War II was by far the deadliest conflict in human history; it resulted in 70 to 85 million fatalities, mostly among civilians. Tens of millions died due to genocides (including the Holocaust), starvation, massa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zarankiewicz K4 7
Kazimierz Zarankiewicz (2 May 1902 – 5 September 1959) was a Polish mathematician and Professor at the Warsaw University of Technology who was interested primarily in topology and graph theory. Biography Zarankiewicz was born in Częstochowa. He studied at the University of Warsaw, together with Zygmunt Janiszewski, Stefan Mazurkiewicz, Wacław Sierpiński, Kazimierz Kuratowski, and Stanisław Saks. During World War II, Zarankiewicz took part in illegal teaching, forbidden by the German authorities, and eventually was sent to a concentration camp. He survived and became a teacher at Warsaw University of Technology. He visited universities in Tomsk, Harvard, London, and Vienna. He served as president of the Warsaw section of the Polish Mathematical Society and the International Astronautical Federation. He died in London, England. Research contributions Zarankiewicz wrote works on cut-points in connected spaces, on conformal mappings, on complex functions and number t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transversality (mathematics)
In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology. It is defined by considering the linearizations of the intersecting spaces at the points of intersection. Definition Two submanifolds of a given finite-dimensional smooth manifold are said to intersect transversally if at every point of intersection, their separate tangent spaces at that point together generate the tangent space of the ambient manifold at that point. Manifolds that do not intersect are vacuously transverse. If the manifolds are of complementary dimension (i.e., their dimensions add up to the dimension of the ambient space), the condition means that the tangent space to the ambient manifold is the direct sum of the two smaller tangent spaces. If an intersection is transverse, then the intersection will be a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
