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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] |
Robert McCann (mathematician)
Robert John McCann is a Canadian mathematician, known for his work in transportation theory. He has worked as a professor at the University of Toronto since 1998, and as Canada Research Chair in Mathematics, Economics, and Physics since 2020. Life and work McCann was raised in Windsor, Ontario. He studied engineering and physics at Queen's University before graduating with a degree in math, and earned a PhD in mathematics from Princeton University in 1994. McCann was a Tamarkin Assistant Professor at Brown University from 1994, before joining the University of Toronto Department of Mathematics in the fall of 1998. He served as editor-in-chief of the ''Canadian Journal of Mathematics'' from 2007 to 2016, and again since 2022. He was an invited speaker at the International Congress of Mathematicians in Seoul in 2014. He was elected a Fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cédric Villani
Cédric Patrice Thierry Villani (; born 5 October 1973) is a French politician and mathematician working primarily on partial differential equations, Riemannian geometry and mathematical physics. He was awarded the Fields Medal in 2010, and he was the director of Sorbonne University's Institut Henri Poincaré from 2009 to 2017. As of September 2022, he is a professor at Institut des Hautes Études Scientifiques. Villani has given two lectures at the Royal Institution, the first titled 'Birth of a Theorem'. The English translation of his book ''Théorème vivant'' (''Living Theorem'') has the same title. In the book he describes the links between his research on kinetic theory and that of the mathematician Carlo Cercignani: Villani, in fact, proved the so-called Cercignani's conjecture. His second lecture at the Royal Institution is titled 'The Extraordinary Theorems of John Nash'. Villani was elected as the deputy for Essonne's 5th constituency in the National Assembly, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes: Y_t = \int_0^t H_s\,dX_s, where is a locally square-integrable process adapted to the filtration generated by , which is a Brownian motion or, more generally, a semimartingale. The result of the integration is then another stochastic process. Concretely, the integral from 0 to any particular is a random variable, defined as a limit of a certain sequence of random variables. The paths of Brownian motion fail to satisfy the requirements to be able to apply the standard techniques of calculus. So with the integrand a stochastic process, the Itô stochas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical potential. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, as in spinodal decomposition. Diffusion is a stochastic process due to the inherent randomness of the diffusing entity and can be used to model many real-life stochastic scenarios. Therefore, diffusion and the corresponding mathematical models are used in several fields beyond physics, such as statistics, probability theory, information theory, neural networks, finance, and marketing. The concept of diffusion is widely used in many fields, including physics (Molecular diffusion, particle diffusion), chemistry, biology, sociology, economics, statistics, data science, and finance (diffusion of people, ideas, data and price v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
