|
Clebsch
Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Humboldt University of Berlin, Berlin. He subsequently taught in Berlin and University of Karlsruhe, Karlsruhe. His collaboration with Paul Gordan in University of Giessen, Giessen led to the introduction of Clebsch–Gordan coefficients for spherical harmonics, which are now widely used in quantum mechanics. Together with Carl Neumann at University of Göttingen, Göttingen, he founded the mathematical research journal ''Mathematische Annalen'' in 1868. In 1883 Adhémar Jean Claude Barré de Saint-Venant, Saint-Venant translated Clebsch's work on Elasticity (physics), elasticity into French and published it as ''Théorie de l'élasticité des Corps Solides''. Books by A. Clebsch Vorlesungen über Geometrie(Teubner, Leipzig, 1876-1891) edited by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clebsch Graph
In the mathematical field of graph theory, the Clebsch graph is either of two complementary graphs on 16 vertices, a 5-regular graph with 40 edges and a 10-regular graph with 80 edges. The 80-edge graph is the dimension-5 halved cube graph; it was called the Clebsch graph name by Seidel (1968) because of its relation to the configuration of 16 lines on the quartic surface discovered in 1868 by the German mathematician Alfred Clebsch. The 40-edge variant is the dimension-5 folded cube graph; it is also known as the Greenwood–Gleason graph after the work of , who used it to evaluate the Ramsey number ''R''(3,3,3) = 17.. Construction The dimension-5 folded cube graph (the 5-regular Clebsch graph) may be constructed by adding edges between opposite pairs of vertices in a 4-dimensional hypercube graph. (In an ''n''-dimensional hypercube, a pair of vertices are ''opposite'' if the shortest path between them has ''n'' edges.) Alternatively, it can be formed from a 5-dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clebsch–Gordan Coefficients
In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations (i.e., a reducible representation into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly). The name derives from the German mathematicians Alfred Clebsch and Paul Gordan, who encountered an equivalent problem in invariant theory. From a vector calculus perspective, the CG coefficients associated with the SO(3) group can be defined simply in terms of integrals of products of spherical harmonics and their complex conjugates. The addition of spins in qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clebsch Surface
In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by and , all of whose 27 exceptional lines can be defined over the real numbers. The term Klein's icosahedral surface can refer to either this surface or its blowup at the 10 Eckardt points. Definition The Clebsch surface is the set of points (''x''0:''x''1:''x''2:''x''3:''x''4) of P4 satisfying the equations :x_0 + x_1 + x_2 + x_3 + x_4 = 0, :x_0^3 + x_1^3 + x_2^3 + x_3^3 + x_4^3 = 0. Eliminating ''x''0 shows that it is also isomorphic to the surface :x_1^3 + x_2^3 + x_3^3 + x_4^3 = (x_1 + x_2 + x_3 + x_4)^3 in P3. Properties The symmetry group of the Clebsch surface is the symmetric group ''S''5 of order 120, acting by permutations of the coordinates (in ''P''4). Up to isomorphism, the Clebsch surface is the only cubic surface with this automorphism group. The 27 exceptional lines are: * The 15 images (under ''S''5) of the line of points o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clebsch Representation
In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field \boldsymbol(\boldsymbol) is: \boldsymbol = \boldsymbol \varphi + \psi\, \boldsymbol \chi, where the scalar fields \varphi(\boldsymbol), \psi(\boldsymbol) and \chi(\boldsymbol) are known as Clebsch potentials or Monge potentials, named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and \boldsymbol is the gradient operator. Background In fluid dynamics and plasma physics, the Clebsch representation provides a means to overcome the difficulties to describe an inviscid flow with non-zero vorticity – in the Eulerian reference frame – using Lagrangian mechanics and Hamiltonian mechanics. At the critical point of such functionals the result is the Euler equations, a set of equations describing the fluid flow. Note that the mentioned difficulties do not arise when describing the flow through a variational principle in the Lagrangian reference frame. In case of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Spherical harmonics originate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre–Clebsch Condition
__NOTOC__ In the calculus of variations the Legendre–Clebsch condition is a second-order condition which a solution of the Euler–Lagrange equation must satisfy in order to be a minimum. For the problem of minimizing : \int_^ L(t,x,x')\, dt . \, the condition is :L_(t,x(t),x'(t)) \ge 0, \, \forall t \in ,b/math> Generalized Legendre–Clebsch In optimal control, the situation is more complicated because of the possibility of a singular solution. The generalized Legendre–Clebsch condition, also known as convexity, is a sufficient condition for local optimality such that when the linear sensitivity of the Hamiltonian to changes in u is zero, i.e., : \frac = 0 The Hessian of the Hamiltonian is positive definite along the trajectory of the solution: : \frac > 0 In words, the generalized LC condition guarantees that over a singular arc, the Hamiltonian is minimized. See also * Bang–bang control In control theory, a bang–bang controller (2 step or on–off controlle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever. His contributions include the development of modern logic in the ''Begriffsschrift'' and work in the foundations of mathematics. His book the ''Foundations of Arithmetic'' is the seminal text of the logicist project, and is cited by Michael Dummett as where to pinpoint the linguistic turn. His philosophical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Gordan
__NOTOC__ Paul Albert Gordan (27 April 1837 – 21 December 1912) was a Jewish-German mathematician, a student of Carl Jacobi at the University of Königsberg before obtaining his PhD at the University of Breslau (1862),. and a professor at the University of Erlangen-Nuremberg. He was born in Breslau, Germany (now Wrocław, Poland), and died in Erlangen, Germany. He was known as "the king of invariant theory"... His most famous result is that the ring of invariants of binary forms of fixed degree is finitely generated. Clebsch–Gordan coefficients are named after him and Alfred Clebsch. Gordan also served as the thesis advisor for Emmy Noether. A famous quote attributed to Gordan about David Hilbert's proof of Hilbert's basis theorem, a result which vastly generalized his result on invariants, is "This is not mathematics; this is theology."Hermann Weyl, ''David Hilbert. 1862-1943'', Obituary Notices of Fellows of the Royal Society (1944). The proof in question was the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagram Map
In mathematics, the pentagram map is a discrete dynamical system on the moduli space of polygons in the projective plane. The pentagram map takes a given polygon, finds the intersections of the shortest diagonals of the polygon, and constructs a new polygon from these intersections. Richard Schwartz (mathematician), Richard Schwartz introduced the pentagram map for a general polygon in a 1992 paper though it seems that the special case, in which the map is defined for pentagons only, goes back to an 1871 paper of Alfred Clebsch and a 1945 paper of Theodore Motzkin. The pentagram map is similar in spirit to the constructions underlying Desargues' theorem and Poncelet's porism. It echoes the rationale and construction underlying a conjecture of Branko Grünbaum concerning the diagonals of a polygon. Definition of the map Basic construction Suppose that the vertex (geometry), vertices of the polygon P are given by P_1,P_3,P_5,\ldots The image of ''P'' under the pentagram map is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Von Brill
Alexander Wilhelm von Brill (20 September 1842 – 18 June 1935) was a German mathematician. Born in Darmstadt, Hesse, Brill was educated at the University of Giessen, where he earned his doctorate under supervision of Alfred Clebsch. He held a chair at the University of Tübingen, where Max Planck was among his students. In 1933, he joined the National Socialist Teachers League as one of the first members from Tübingen. The London Science Museum contains sliceform objects prepared by Brill and Felix Kleinbr> Selected publications''Vorlesungen über ebene algebraische Kurven und Funktionen.'' 1925.*''Vorlesungen über allgemeine Mechanik.'' 1928. *''Vorlesungen zur Einführung in die Mechanik raumerfüllender Massen.'' 1909. *''Graphische Darstellungen aus der reinen und angewandten Mathematik.'' 1894. *with Max Noether''Über algebraische Funktionen und ihre Anwendung in der Geometrie.'' Mitt. Göttinger Akad.1873 and their article with the same name in the Mathematischen Annal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperboloid Model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperboloid in (''n''+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and ''m''-planes are represented by the intersections of (''m''+1)-planes passing through the origin in Minkowski space with ''S''+ or by wedge products of ''m'' vectors. Hyperbolic space is embedded isometrically in Minkowski space; that is, the hyperbolic distance function is inherited from Minkowski space, analogous to the way spherical distance is inherited from Euclidean distance when the ''n''-sphere is embedded in (''n''+1)-dimensional Euclidean space. Other models of hyperbolic space can be thought of as map projections of ''S''+: the Beltrami–Klein model is the projection of ''S''+ through the origin onto a plane perpendic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Giessen
University of Giessen, official name Justus Liebig University Giessen (german: Justus-Liebig-Universität Gießen), is a large public research university in Giessen, Hesse, Germany. It is named after its most famous faculty member, Justus von Liebig, the founder of modern agricultural chemistry and inventor of artificial fertiliser. It covers the areas of arts/humanities, business, dentistry, economics, law, medicine, science, social sciences, and veterinary medicine. Its university hospital, which has two sites, Giessen and Marburg (the latter of which is the teaching hospital of the University of Marburg), is the only private university hospital in Germany. History The University of Giessen is among the oldest institutions of higher educations in the German-speaking world. It was founded in 1607 as a Lutheran university in the city of Giessen in Hesse-Darmstadt because the all-Hessian ''Landesuniversität'' (the nearby University of Marburg (''Philipps-Universität Marburg'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
