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Hadwiger Conjecture (combinatorial Geometry)
In combinatorial geometry, the Hadwiger conjecture states that any convex body in ''n''-dimensional Euclidean space can be covered by 2''n'' or fewer smaller bodies homothetic with the original body, and that furthermore, the upper bound of 2''n'' is necessary if and only if the body is a parallelepiped. There also exists an equivalent formulation in terms of the number of floodlights needed to illuminate the body. The Hadwiger conjecture is named after Hugo Hadwiger, who included it on a list of unsolved problems in 1957; it was, however, previously studied by and independently, . Additionally, there is a different Hadwiger conjecture concerning graph coloring—and in some sources the geometric Hadwiger conjecture is also called the Levi–Hadwiger conjecture or the Hadwiger–Levi covering problem. The conjecture remains unsolved even in three dimensions, though the two dimensional case was resolved by . Formal statement Formally, the Hadwiger conjecture is: If ''K'' is any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadwiger Covering
Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss people, Swiss mathematician, known for his work in geometry, combinatorics, and cryptography. Biography Although born in Karlsruhe, Germany, Hadwiger grew up in Bern, Switzerland Bern (), or Berne (), ; ; ; . is the ''de facto'' Capital city, capital of Switzerland, referred to as the "federal city".; ; ; . According to the Swiss constitution, the Swiss Confederation intentionally has no "capital", but Bern has gov ..... He did his undergraduate studies at the University of Bern, where he majored in mathematics but also studied physics and actuarial science. He continued at Bern for his graduate studies, and received his Ph.D. in 1936 under the supervision of Willy Scherrer. He was for more than forty years a professor of mathematics at Bern. Mathematical concepts named after Hadwiger Hadwiger's theorem in integral geometry classifies the isometry-invariant val ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a -simplex is a -dimensional polytope that is the convex hull of its vertices. More formally, suppose the points u_0, \dots, u_k are affinely independent, which means that the vectors u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points C = \left\. A regular simplex is a simplex that is also a regular polytope. A regular -simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object. Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology. History Polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archiv Der Mathematik
'' Archiv der Mathematik'' is a peer-reviewed mathematics journal published by Springer, established in 1948. Abstracting and indexing The journal is abstracted and indexed in: Springer. 2022 * * * * According to the '' |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. The journal is devoted to shorter research articles. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The European Mathematical Society
'' Journal of the European Mathematical Society'' is a monthly peer-reviewed mathematical journal. Founded in 1999, the journal publishes articles on all areas of pure and applied mathematics. Most published articles are original research articles but the journal also publishes survey articles.Summary of the journal The journal has been published by until 2003. Since 2004, it is published by the . The first editor-in-chief was [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elemente Der Mathematik
''Elemente der Mathematik'' is a peer-reviewed scientific journal covering mathematics. It is published by the European Mathematical Society Publishing House on behalf of the Swiss Mathematical Society. It was established in 1946 by Louis Locher-Ernst, and transferred to the Swiss Mathematical Society in 1976. Rather than publishing research papers, it focuses on survey papers aimed at a broad audience. History The journal ''Elemente der Mathematik'' was founded in 1946 by Louis Locher-Ernst under the patronage of the Swiss Mathematical Society (SMG) to disseminate pedagogical and expository articles in mathematics and physics. Locher-Ernst outlined the scope and objectives—emphasising support for secondary and tertiary instruction—in a letter to the SMG president in August 1945 and at the autumn members' meeting in Fribourg later that year. Early editorial responsibilities were assumed by Locher-Ernst alongside Erwin Voellmy, Ernst Trost and Paul Buchner, while an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Mathematics Research Notices
The ''International Mathematics Research Notices'' is a peer-reviewed mathematics journal. Originally published by Duke University Press and Hindawi Publishing Corporation, it is now published by Oxford University Press. retrieved 2015-02-26. The Executive Editor is (). According to the '' |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borsuk's Conjecture
The Borsuk problem in geometry, for historical reasons incorrectly called Borsuk's conjecture, is a question in discrete geometry. It is named after Karol Borsuk. Problem In 1932, Karol Borsuk showed that an ordinary 3-dimensional ball in Euclidean space can be easily dissected into 4 solids, each of which has a smaller diameter than the ball, and generally -dimensional ball can be covered with compact sets of diameters smaller than the ball. At the same time he proved that subsets are not enough in general. The proof is based on the Borsuk–Ulam theorem. That led Borsuk to a general question: The question was answered in the positive in the following cases: * — which is the original result by Karol Borsuk (1932). * — shown by Julian Perkal (1947), and independently, 8 years later, by H. G. Eggleston (1955). A simple proof was found later by Branko Grünbaum and Aladár Heppes. * For all for smooth convex fields — shown by Hugo Hadwiger (1946). * For all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedrich Wilhelm Levi
Friedrich Wilhelm Daniel Levi (February 6, 1888 – January 1, 1966) was a German mathematician known for his work in abstract algebra, especially torsion-free abelian groups. He also worked in geometry, topology, set theory, and analysis. Early life and education Levi was born to Georg Levi and Emma Blum in Mulhouse in Alsace-Lorraine, then part of the German Empire. He received his Ph.D. in 1911 under Heinrich Martin Weber at the University of Strasbourg. Career Levi served his mandatory military service in the German Army in 1906–1907, and was called up again serving in the artillery during World War I, 1914–18. Awarded the Iron Cross, he was discharged as a lieutenant. In 1917, he married Barbara Fitting, with whom he eventually had three children ( Paul Levi, Charlotte, and Suzanne). He taught at the University of Leipzig from 1920 to 1935, when the Nazi government dismissed him because of his Jewish ancestry. Friedrich and Barbara moved to Calcutta, India. In 193 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zonotope
In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkowski sum of a set of line segments in three-dimensional space, or as a three-dimensional Projection (mathematics), projection of a hypercube. Zonohedra were originally defined and studied by Evgraf Stepanovich Fyodorov, E. S. Fedorove, a Russian Crystallography, crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope. Zonohedra that tile space The original motivation for studying zonohedra is that the Voronoi diagram of any Lattice (group), lattice forms a convex uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way can Honeycomb (geometry), tessellate 3-dimensional space and is called a primary parallelohedron. Each primary parallelohedron ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
