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Otto Calculus
The Otto calculus (also known as Otto's calculus) is a mathematical system for studying diffusion equations that views the space of probability measures as an infinite dimensional Riemannian manifold by interpreting the Wasserstein distance as if it was a Riemannian metric. It is named after Felix Otto, who developed it in the late 1990s and published it in a 2001 paper on the geometry of dissipative evolution equations. Otto acknowledges inspiration from earlier work by David Kinderlehrer and conversations with Robert McCann and Cédric Villani. See also * Itô calculus Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The cent ... References Diffusion Partial differential equations Riemannian manifolds {{Differential-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffusion Equation
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws of diffusion). In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The diffusion equation is a special case of the convection–diffusion equation when bulk velocity is zero. It is equivalent to the heat equation under some circumstances. Statement The equation is usually written as: \frac = \nabla \cdot \big D(\phi,\mathbf) \ \nabla\phi(\mathbf,t) \big where is the density of the diffusing material at location and time and is the collective diffusion coefficient for density at location ; and represents the vector differential operator del. If the diffusion coefficient depends on the density then the equatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint (mutually exclusive) events by the measure should be the sum of the probabilities of the events; for example, the value assigned to the outcome "1 or 2" in a throw of a dice should be the sum of the values assigned to the outcomes "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology. Definition The requirements for a set function \mu to be a probability measure on a σ-algebra are that: * \mu must return results in the unit interval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifold, manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport. Any smooth surface in three-dimensional Eucl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wasserstein Distance
In mathematics, the Wasserstein distance or Kantorovich–Rubinstein metric is a distance function defined between probability distributions on a given metric space M. It is named after Leonid Vaseršteĭn. Intuitively, if each distribution is viewed as a unit amount of earth (soil) piled on ''M'', the metric is the minimum "cost" of turning one pile into the other, which is assumed to be the amount of earth that needs to be moved times the mean distance it has to be moved. This problem was first formalised by Gaspard Monge in 1781. Because of this analogy, the metric is known in computer science as the earth mover's distance. The name "Wasserstein distance" was coined by R. L. Dobrushin in 1970, after learning of it in the work of Leonid Vaseršteĭn on Markov processes describing large systems of automata (Russian, 1969). However the metric was first defined by Leonid Kantorovich in ''The Mathematical Method of Production Planning and Organization'' (Russian original 1939) in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Metric
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifold, manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport. Any smooth surface in three-dimensional Eucl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Otto (mathematician)
Felix Otto (born 19 May 1966) is a German mathematician. Biography He studied mathematics at the University of Bonn, finishing his PhD thesis in 1993 under the supervision of Stephan Luckhaus. After postdoctoral studies at the Courant Institute of Mathematical Sciences of New York University and at Carnegie Mellon University, in 1997 he became a professor at the University of California, Santa Barbara. From 1999 to 2010 he was professor for applied mathematics at the University of Bonn, and currently serves as one of the directors of the Max Planck Institute for Mathematics in the Sciences, Leipzig. Work Otto specialises in materials science, including work on the theory of partial differential equations. He is known for his work on the Otto–Villani theorem and the invention of the Otto calculus. Honours In 2006, he received the Gottfried Wilhelm Leibniz Prize of the Deutsche Forschungsgemeinschaft, which is the highest honour awarded in German research. In 2009, he was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Kinderlehrer
David Samuel Kinderlehrer (October 23, 1941, Allentown, Pennsylvania) is an American mathematician, who works on partial differential equations and related mathematics applied to materials in biology and physics. Kinderlehrer received in 1963 his bachelor's degree from Massachusetts Institute of Technology, MIT and in 1968 his Ph.D. from the University of California, Berkeley under Hans Lewy with thesis ''Minimal surfaces whose boundaries contain spikes''. He became in 1968 an instructor and in 1975 a full professor at the University of Minnesota in Minneapolis. For the academic year 1971–1972 he was a visiting professor at the Scuola Normale Superiore di Pisa. In 2003 he became a professor at Carnegie Mellon University. He works on partial differential equations, minimal surfaces, and variational inequality, variational inequalities, with mathematical applications to the microstructure of biological materials, to solid state physics, and to materials science, including crystalli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
