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Liouville Equations
: ''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).'' : ''For Liouville's equation in quantum mechanics, see Von Neumann equation.'' : ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelfand equation.'' In differential geometry, Liouville's equation, named after Joseph Liouville, is the nonlinear partial differential equation satisfied by the conformal factor of a metric on a surface of constant Gaussian curvature : :\Delta_0\log f = -K f^2, where is the flat Laplace operator :\Delta_0 = \frac +\frac = 4 \frac \frac. Liouville's equation appears in the study of isothermal coordinates in differential geometry: the independent variables are the coordinates, while can be described as the conformal factor with respect to the flat metric. Occasionally it is the square that is referred to as the conformal factor, instead of itself. Liouville's equation was also taken as an example by David Hilbert in the formulation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville's Theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical mechanics, statistical and Hamiltonian mechanics. It asserts that ''the phase space, phase-space distribution function is constant along the Trajectory, trajectories of the system''—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability. Liouville's theorem applies to conservative systems, that is, systems in which the effects of friction are absent or can be ignored. The general mathematical formulation for such systems is the measure-preserving dynamical system. Liouville's theorem applies when there are degrees of freedom that can be interpreted as positions and momenta; not all measure-preserving dynamical systems have these, but Hamiltonian systems do. The general se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace–Beltrami Operator
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami. For any twice-differentiable real-valued function ''f'' defined on Euclidean space R''n'', the Laplace operator (also known as the ''Laplacian'') takes ''f'' to the divergence of its gradient vector field, which is the sum of the ''n'' pure second derivatives of ''f'' with respect to each vector of an orthonormal basis for R''n''. Like the Laplacian, the Laplace–Beltrami operator is defined as the divergence of the gradient, and is a linear operator taking functions into functions. The operator can be extended to operate on tensors as the divergence of the covariant derivative. Alternatively, the operator can be generalized to operate on differential forms using the divergence and exterior d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *'' Memoirs of the American Mathematical Society'' *'' Notices of the American Mathematical Society'' *'' Proceedings of the Ame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mary Frances Winston Newson
Mary Frances Winston Newson (August 7, 1869 December 5, 1959) was an American mathematician. She became the first American woman to receive a PhD in mathematics from a European university, namely the University of Göttingen in Germany.Grinstein and Campbell, p. 161 She was also the first person to translate Hilbert's problems into English. Early life Mary Newson was born Mary Frances Winston in Forreston, Illinois, the name Newson being the name of the husband she married. She was always known as May by her friends and family. Her parents were Thomas Winston, a country doctor, and Caroline Eliza Mumford. Thomas Winston had been born in Wales but had come to the United States at the age of two years with his parents. Caroline had been a teacher before her marriage, teaching French, art and mathematics. Mary was one of her parents' seven surviving children. She was taught at home by her mother, who had taught herself Latin and Greek so that she could prepare her children for a univ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nachrichten Von Der Königlichen Gesellschaft Der Wissenschaften Zu Göttingen, Mathematisch-Physikalische Klasse
''Nachrichten'' ('News') was a Volga German communist newspaper, published between 1918 and 1941.Geschichte der Wolgadeutschen"НАХРИХТЕН"/ref> ''Nachrichten'' was the organ of the Communist Party in the Volga German Autonomous Soviet Socialist Republic. The newspaper was founded under the name ''Vorwärts'' ('Forward') by the German Commissariat in Saratov in March 1918. ''Vorwärts'' was the first Bolshevik newspaper directed towards the Volga German colonists. It carried the by-line 'Organ of the Socialists in the German Volga territory' (''Organ der Sozialisten des deutschen Wolgagebiets'').Geschichte der Wolgadeutschen. Schiller, Franz P., Literatur zur Geschichte und Volkskunde der deutschen Kolonien in der Sowjetunion für die Jahre 1764 – 1926' The name was changed to ''Nachrichten'' in June 1918, as the Bolsheviks wanted to avoid any association with the SPD organ ''Vorwärts''.Heitman, Sidney. Germans from Russia in Colorado'. Fort Collins, Colo: Western Social ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Studies In Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats. List of books *1 ''The General Topology of Dynamical Systems'', Ethan Akin (1993, ) *2 ''Combinatorial Rigidity'', Jack Graver, Brigitte Servatius, Herman Servatius (1993, ) *3 ''An Introduction to Gröbner Bases'', William W. Adams, Philippe Loustaunau (1994, ) *4 ''The Integrals of Lebesgue, Denjoy, Perron, and Henstock'', Russell A. Gordon (1994, ) *5 ''Algebraic Curves and Riemann Surfaces'', Rick Miranda (1995, ) *6 ''Lectures on Quantum Groups'', Jens Carsten Jantzen (1996, ) *7 ''Algebraic Number Fields'', Gerald J. Janusz (1996, 2nd ed., ) *8 ''Discovering Modern Set Theory. I: The Basics'', Winfried Just, Martin Weese (1996, ) *9 ''An Invitation to Arithmetic Geometry'', Dino Lorenzini (1996, ) *10 ''Representations of Finite and Compact Groups'', Barry Simon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville Field Theory
In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation. Liouville theory is defined for all complex values of the central charge c of its Virasoro symmetry algebra, but it is unitary only if :c\in(1,+\infty), and its classical limit is : c\to +\infty. Although it is an interacting theory with a continuous spectrum, Liouville theory has been solved. In particular, its three-point function on the sphere has been determined analytically. Introduction Liouville theory describes the dynamics of a field \varphi called the Liouville field, which is defined on a two-dimensional space. This field is not a free field due to the presence of an exponential potential : V(\varphi) = e^\ , where the parameter b is called the coupling constant. In a free field theory, the energy eigenvectors e^ are linearly independent, and the momentum \alpha is conserved ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lobachevski Plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The hyperbolic plane is a plane where every point is a saddle point. Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane. The hyperboloid model of hyperbolic geometry provides a representation of events one temporal unit into the future in Minkowski space, the basis of special relativity. Each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Plane
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of each point (mathematics), point. It is an affine space, which includes in particular the concept of parallel lines. It has also measurement, metrical properties induced by a Euclidean distance, distance, which allows to define circles, and angle, angle measurement. A Euclidean plane with a chosen Cartesian coordinate system is called a ''Cartesian plane''. The set \mathbb^2 of the ordered pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called ''the'' Euclidean plane or ''standard Euclidean plane'', since every Euclidean plane is isomorphic to it. History Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the center (geometry), ''center'' of the sphere, and the distance is the sphere's ''radius''. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental surface in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is spherical Earth, often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pole (complex Analysis)
In complex analysis (a branch of mathematics), a pole is a certain type of singularity (mathematics), singularity of a complex-valued function of a complex number, complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity). Technically, a point is a pole of a function if it is a zero of a function, zero of the function and is holomorphic function, holomorphic (i.e. complex differentiable) in some neighbourhood (mathematics), neighbourhood of . A function is meromorphic function, meromorphic in an open set if for every point of there is a neighborhood of in which at least one of and is holomorphic. If is meromorphic in , then a zero of is a pole of , and a pole of is a zero of . This induces a duality between ''zeros'' and ''poles'', that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then the sum of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meromorphic Function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are ''poles'' of the function. The term comes from the Greek ''meros'' ( μέρος), meaning "part". Every meromorphic function on ''D'' can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on ''D'': any pole must coincide with a zero of the denominator. Heuristic description Intuitively, a meromorphic function is a ratio of two well-behaved (holomorphic) functions. Such a function will still be well-behaved, except possibly at the points where the denominator of the fraction is zero. If the denominator has a zero at ''z'' and the numerator does not, then the value of the function will approach infinity; if both parts have a zero at ''z'', then one must compare the multiplicity of these zeros. From an algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
