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Minkowski Plane
In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes (the others being Möbius plane and Laguerre plane). Classical real Minkowski plane Applying the pseudo-euclidean distance d(P_1,P_2) = (x'_1-x'_2)^2 - (y'_1-y'_2)^2 on two points P_i = (x'_i, y'_i) (instead of the euclidean distance) we get the geometry of ''hyperbolas'', because a pseudo-euclidean circle \ is a hyperbola with midpoint M. By a transformation of coordinates x_i = x'_i + y'_i, y_i = x'_i - y'_i, the pseudo-euclidean distance can be rewritten as d(P_1,P_2) = (x_1 - x_2) (y_1 - y_2). The hyperbolas then have asymptotes parallel to the non-primed coordinate axes. The following completion (see Möbius and Laguerre planes) ''homogenizes'' the geometry of hyperbolas: * the set of points: \mathcal P := \left(\R \cup \left\\right)^2 = \R^2 \cup \left(\left\ \times \R\right) \cup \left(\R \times \left\\right) \ \cup \left\ \ , \ \infty \notin \R, * the set of cycles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Minkowski
Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number theory, mathematical physics, and the theory of relativity. Minkowski is perhaps best known for his foundational work describing space and time as a four-dimensional space, now known as "Minkowski spacetime", which facilitated geometric interpretations of Albert Einstein's special theory of relativity (1905). Personal life and family Hermann Minkowski was born in the town of Aleksota, the Suwałki Governorate, the Kingdom of Poland, part of the Russian Empire, to Lewin Boruch Minkowski, a merchant who subsidized the building of the choral synagogue in Kovno, and Rachel Taubmann, both of Jewish descent. Hermann was a younger brother of the medical researcher Oskar (born 1858). In different sources Minkowski's nationality is variously giv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elsevier
Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell'', the ScienceDirect collection of electronic journals, '' Trends'', the '' Current Opinion'' series, the online citation database Scopus, the SciVal tool for measuring research performance, the ClinicalKey search engine for clinicians, and the ClinicalPath evidence-based cancer care service. Elsevier's products and services also include digital tools for data management, instruction, research analytics and assessment. Elsevier is part of the RELX Group (known until 2015 as Reed Elsevier), a publicly traded company. According to RELX reports, in 2021 Elsevier published more than 600,000 articles annually in over 2,700 journals; as of 2018 its archives contained over 17 million documents and 40,000 e-books, with over one billion annual downloads. Researchers have criticized Elsevier for its high profit marg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incidence (geometry)
In geometry, an incidence relation is a heterogeneous relation that captures the idea being expressed when phrases such as "a point ''lies on'' a line" or "a line is ''contained in'' a plane" are used. The most basic incidence relation is that between a point, , and a line, , sometimes denoted . If the pair is called a ''flag''. There are many expressions used in common language to describe incidence (for example, a line ''passes through'' a point, a point ''lies in'' a plane, etc.) but the term "incidence" is preferred because it does not have the additional connotations that these other terms have, and it can be used in a symmetric manner. Statements such as "line intersects line " are also statements about incidence relations, but in this case, it is because this is a shorthand way of saying that "there exists a point that is incident with both line and line ". When one type of object can be thought of as a set of the other type of object (''viz''., a plane is a set of points) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Francis Buekenhout
Francis Buekenhout (born 23 April 1937 in Ixelles near Brussels) is a Belgian mathematician who introduced Buekenhout geometries and the concept of quadratic sets. Career Buekenhout studied at the University of Brussels under Jacques Tits and Paul Libois. Together with his teacher Jacques Tits, he developed concepts with the diagram geometries, also called Buekenhout geometries or Buekenhout–Tits geometries. These largely disregard the concrete axiom systems of a projective or affine geometry and put these and many other incidence geometries into a common framework. He worked at the ULB from 1960 to 1969 as an assistant to Libois. He was then appointed as extraordinary professor 1969 to 1998, and as ordinary professor from 1977 until his retirement in 2002. He has been a member of the Académie Royale des Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walter Benz
Walter Benz (May 2, 1931 Lahnstein – January 13, 2017 Ratzeburg) was a German mathematician, an expert in geometry. Benz studied at the Johannes Gutenberg University of Mainz and received his doctoral degree in 1954, with Robert Furch as his advisor. After a position at the Johann Wolfgang Goethe University Frankfurt am Main, he served as a professor at Ruhr University Bochum, University of Waterloo, and University of Hamburg. Benz was honoured with the degree of a Dr. h.c. Based on his book ''Vorlesungen über Geometrie der Algebren'' (Springer 1973), certain geometric objects are called Benz planes. Inner product spaces over the real numbers provide the basis of a 2007 book by Benz: ''Classical Geometries in Modern Contexts''.W. Benz (2007) ''Classical Geometries in Modern Contexts: Geometry of Real Inner Product Spaces'', Birkhäuser, See also * List of University of Waterloo people The University of Waterloo, located in Waterloo, Ontario, Canada, is a comprehensive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Geometry
In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two dimensions, conformal geometry may refer either to the study of conformal transformations of what are called "flat spaces" (such as Euclidean spaces or spheres), or to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics that are defined up to scale. Study of the flat structures is sometimes termed Möbius geometry, and is a type of Klein geometry. Conformal manifolds A conformal manifold is a pseudo-Riemannian manifold equipped with an equivalence class of metric tensors, in which two metrics ''g'' and ''h'' are equivalent if and only if :h = \lambda^2 g , where ''λ'' is a real-valued smooth function defined on the manifold and is called the conformal factor. An equivalence cla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Set
In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space). Definition of a quadratic set Let \mathfrak P=(,,\in) be a projective space. A quadratic set is a non-empty subset of for which the following two conditions hold: :(QS1) Every line g of intersects in at most two points or is contained in . ::(g is called exterior to if , g\cap , =0, tangent to if either , g\cap , =1 or g\cap =g, and secant to if , g\cap , =2.) :(QS2) For any point P\in the union _P of all tangent lines through P is a hyperplane or the entire space . A quadratic set is called non-degenerate if for every point P\in , the set _P is a hyperplane. A Pappian projective space is a projective space in which Pappus's hexagon theorem holds. The following result, due to Francis Buekenhout, is an astonishing statement for finite projective ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in ''D'' + 1 variables; for example, in the case of conic sections. When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a ''degenerate quadric'' or a ''reducible quadric''. In coordinates , the general quadric is thus defined by the algebraic equationSilvio LevQuadricsin "Geometry Formulas and Facts", excerpted from 30th Edition of ''CRC Standard Mathematical Tables and Formulas'', CRC Press, from The Geometry Center at University of Minnesota : \sum_^ x_i Q_ x_j + \sum_^ P_i x_i + R = 0 which may be compactly written in vector and matrix notation as: : x Q x^\mathrm + P x^\mathrm + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stereographic Projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to the diameter through the point. It is a smooth function, smooth, bijection, bijective function (mathematics), function from the entire sphere except the center of projection to the entire plane. It maps circle of a sphere, circles on the sphere to generalised circle, circles or lines on the plane, and is conformal map, conformal, meaning that it preserves angles at which curves meet and thus Local property, locally approximately preserves similarity (geometry), shapes. It is neither isometry, isometric (distance preserving) nor Equiareal map, equiareal (area preserving). The stereographic projection gives a way to representation (mathematics), represent a sphere by a plane. The metric tensor, metric induced metric, induced by the inverse s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
