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Gino Fano
Gino Fano (5 January 18718 November 1952) was an Italian mathematician, best known as the founder of finite geometry. He was born to a wealthy Jewish family in Mantua, in Italy and died in Verona, also in Italy. Fano made various contributions on projective and algebraic geometry. His work in the foundations of geometry predates the similar, but more popular, work of David Hilbert by about a decade. He was the father of physicist Ugo Fano and electrical engineer Robert Fano and uncle to physicist and mathematician Giulio Racah. Mathematical work Fano was an early writer in the area of finite projective spaces. In his article on proving the independence of his set of axioms for projective ''n''-space, among other things, he considered the consequences of having a fourth harmonic point be equal to its conjugate. This leads to a configuration of seven points and seven lines contained in a finite three-dimensional space with 15 points, 35 lines and 15 planes, in which each lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mantua
Mantua ( ; it, Mantova ; Lombard and la, Mantua) is a city and '' comune'' in Lombardy, Italy, and capital of the province of the same name. In 2016, Mantua was designated as the Italian Capital of Culture. In 2017, it was named as the European Capital of Gastronomy, included in the Eastern Lombardy District (together with the cities of Bergamo, Brescia, and Cremona). In 2008, Mantua's ''centro storico'' (old town) and Sabbioneta were declared by UNESCO to be a World Heritage Site. Mantua's historic power and influence under the Gonzaga family has made it one of the main artistic, cultural, and especially musical hubs of Northern Italy and the country as a whole. Having one of the most splendid courts of Europe of the fifteenth, sixteenth, and early seventeenth centuries. Mantua is noted for its significant role in the history of opera; the city is also known for its architectural treasures and artifacts, elegant palaces, and the medieval and Renaissance cityscape. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate causes of phenomena, and usually frame their understanding in mathematical terms. Physicists work across a wide range of research fields, spanning all length scales: from sub-atomic and particle physics, through biological physics, to cosmological length scales encompassing the universe as a whole. The field generally includes two types of physicists: experimental physicists who specialize in the observation of natural phenomena and the development and analysis of experiments, and theoretical physicists who specialize in mathematical modeling of physical systems to rationalize, explain and predict natural phenomena. Physicists can apply their knowledge towards solving practical problems or to developing new technologies (also known as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1952 Deaths
Year 195 ( CXCV) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Scrapula and Clemens (or, less frequently, year 948 ''Ab urbe condita''). The denomination 195 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Events By place Roman Empire * Emperor Septimius Severus has the Roman Senate deify the previous emperor Commodus, in an attempt to gain favor with the family of Marcus Aurelius. * King Vologases V and other eastern princes support the claims of Pescennius Niger. The Roman province of Mesopotamia rises in revolt with Parthian support. Severus marches to Mesopotamia to battle the Parthians. * The Roman province of Syria is divided and the role of Antioch is diminished. The Romans annexed the Syrian cities of Edessa and Nisibis. Severus re-establish his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1871 Births
Events January–March * January 3 – Franco-Prussian War – Battle of Bapaume: Prussians win a strategic victory. * January 18 – Proclamation of the German Empire: The member states of the North German Confederation and the south German states, aside from Austria, unite into a single nation state, known as the German Empire. The King of Prussia is declared the first German Emperor as Wilhelm I of Germany, in the Hall of Mirrors at the Palace of Versailles. Constitution of the German Confederation comes into effect. It abolishes all restrictions on Jewish marriage, choice of occupation, place of residence, and property ownership, but exclusion from government employment and discrimination in social relations remain in effect. * January 21 – Giuseppe Garibaldi's group of French and Italian volunteer troops, in support of the French Third Republic, win a battle against the Prussians in the Battle of Dijon. * February 8 – 1871 French legislative electi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography. The early history of group theory dates from the 19th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Group
In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups have been studied extensively in the period of 1925 to 1940. Alfréd Haar, Haar and André Weil, Weil (respectively in 1933 and 1940) showed that the Integral, integrals and Fourier series are special cases of a very wide class of topological groups. Topological groups, along with continuous group actions, are used to study continuous symmetry, symmetries, which have many applications, for example, Symmetry (physics), in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Synthetic Geometry
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compass and straightedge, to draw conclusions and solve problems. Only after the introduction of coordinate methods was there a reason to introduce the term "synthetic geometry" to distinguish this approach to geometry from other approaches. Other approaches to geometry are embodied in analytic and algebraic geometries, where one would use analysis and algebraic techniques to obtain geometric results. According to Felix Klein Synthetic geometry is that which studies figures as such, without recourse to formulae, whereas analytic geometry consistently makes use of such formulae as can be written down after the adoption of an appropriate system of coordinates. Geometry as presented by Euclid in the ''Elements'' is the quintessential example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be emplo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein's Encyclopedia
Felix Klein's ''Encyclopedia of Mathematical Sciences'' is a German mathematical encyclopedia published in six volumes from 1898 to 1933. Klein and Wilhelm Franz Meyer were organizers of the encyclopedia. Its full title in English is ''Encyclopedia of Mathematical Sciences Including Their Applications'', which is ''Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen'' (EMW). It is 20,000 pages in length (6 volumes, ''i.e. Bände'', published in 23 separate books, 1-1, 1-2, 2-1-1, 2-1-2, 2-2, 2-3-1, 2-3-2, 3-1-1, 3-1-2, 3-2-1, 3-2-2a, 3-2-2b, 3-3, 4-1, 4-2, 4-3, 4-4, 5-1, 5-2, 5-3, 6-1, 6-2-1, 6-2-2) and was published by B.G. Teubner Verlag, publisher of '' Mathematische Annalen''. Today, Göttinger Digitalisierungszentrum provides online access to all volumes, while archive.org hosts some particular parts. Overview Walther von Dyck acted as chairman of the commission to publish the encyclopedia. In 1904 he contributed a preparatory report on the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PG(3,2)
In finite geometry, PG(3,2) is the smallest three-dimensional projective space. It can be thought of as an extension of the Fano plane. It has 15 points, 35 lines, and 15 planes. It also has the following properties: * Each point is contained in 7 lines and 7 planes * Each line is contained in 3 planes and contains 3 points * Each plane contains 7 points and 7 lines * Each plane is isomorphic to the Fano plane * Every pair of distinct planes intersect in a line * A line and a plane not containing the line intersect in exactly one point Constructions Construction from ''K''6 Take a complete graph ''K''6. It has 15 edges, 15 perfect matchings and 20 triangles. Create a point for each of the 15 edges, and a line for each of the 20 triangles and 15 matchings. The incidence structure between each triangle or matching (line) to its three constituent edges (points), induces a PG(3,2). Construction from Fano planes Take a Fano plane and apply all 5040 permutations of its 7 points. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Harmonic Conjugate
In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction: :Given three collinear points , let be a point not lying on their join and let any line through meet at respectively. If and meet at , and meets at , then is called the harmonic conjugate of with respect to . The point does not depend on what point is taken initially, nor upon what line through is used to find and . This fact follows from Desargues theorem. In real projective geometry, harmonic conjugacy can also be defined in terms of the cross-ratio as . Cross-ratio criterion The four points are sometimes called a harmonic range (on the real projective line) as it is found that always divides the segment ''internally'' in the same proportion as divides ''externally''. That is: :, AC, :, BC, = , AD, :, DB, \, . If these segments are now endowed with the ordinary metric interpretation of real num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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