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Martin Hairer
Sir Martin Hairer (born 14 November 1975) is an Austrian-British mathematician working in the field of stochastic analysis, in particular stochastic partial differential equations. He is Professor of Mathematics at EPFL (École Polytechnique Fédérale de Lausanne) and at Imperial College London. He previously held appointments at the University of Warwick and the Courant Institute of New York University. In 2014 he was awarded the Fields Medal, one of the highest honours a mathematician can achieve. In 2020 he won the 2021 Breakthrough Prize in Mathematics. Early life and education Hairer was born in Geneva, Switzerland. He attended the Collège Claparède Geneva where he received his high school diploma in 1994. He entered a school science competition with sound editing software that was developed into Amadeus, and later continued to maintain the software in addition to his academic work; it continued to be widely used . He then attended the University of Geneva, where he ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imperial College London
Imperial College London (legally Imperial College of Science, Technology and Medicine) is a public research university in London, United Kingdom. Its history began with Prince Albert, consort of Queen Victoria, who developed his vision for a cultural area that included the Royal Albert Hall, Victoria & Albert Museum, Natural History Museum and royal colleges. In 1907, Imperial College was established by a royal charter, which unified the Royal College of Science, Royal School of Mines, and City and Guilds of London Institute. In 1988, the Imperial College School of Medicine was formed by merging with St Mary's Hospital Medical School. In 2004, Queen Elizabeth II opened the Imperial College Business School. Imperial focuses exclusively on science, technology, medicine, and business. The main campus is located in South Kensington, and there is an innovation campus in White City. Facilities also include teaching hospitals throughout London, and with Imperial College Health ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, recognising excellence in science, supporting outstanding science, providing scientific advice for policy, education and public engagement and fostering international and global co-operation. Founded on 28 November 1660, it was granted a royal charter by King Charles II as The Royal Society and is the oldest continuously existing scientific academy in the world. The society is governed by its Council, which is chaired by the Society's President, according to a set of statutes and standing orders. The members of Council and the President are elected from and by its Fellows, the basic members of the society, who are themselves elected by existing Fellows. , there are about 1,700 fellows, allowed to use the postnominal title FRS (Fellow of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
King Faisal Prize
The King Faisal Prize ( ar, جائزة الملك فيصل, formerly King Faisal International Prize), is an annual award sponsored by King Faisal Foundation presented to "dedicated men and women whose contributions make a positive difference". The foundation awards prizes in five categories: Service to Islam; Islamic studies; the Arabic language and Arabic literature; science; and medicine. Three of the prizes are widely considered as the most prestigious awards in the Muslim world. The first King Faisal Prize was awarded to the Pakistani scholar Abul A'la Maududi in the year 1979 for his service to Islam. In 1981, Khalid of Saudi Arabia received the same award. In 1984, Fahd of Saudi Arabia was the recipient of the award. In 1986, this prize was co-awarded to Ahmed Deedat and French Roger Garaudy. Award process Designation of subjects Each year, the selection committees designate subjects in Islamic Studies, Arabic Literature, and Medicine. Selected topics in Islamic Studi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rough Path
In stochastic analysis, a rough path is a generalization of the notion of smooth path allowing to construct a robust solution theory for controlled differential equations driven by classically irregular signals, for example a Wiener process. The theory was developed in the 1990s by Terry Lyons (mathematician), Terry Lyons. Several accounts of the theory are available. Rough path theory is focused on capturing and making precise the interactions between highly oscillatory and non-linear systems. It builds upon the harmonic analysis of L.C. Young, the geometric algebra of K.T. Chen, the Lipschitz function theory of H. Whitney and core ideas of stochastic analysis. The concepts and the uniform estimates have widespread application in pure and applied Mathematics and beyond. It provides a toolbox to recover with relative ease many classical results in stochastic analysis (Wong-Zakai, Stroock-Varadhan support theorem, construction of stochastic flows, etc) without using specific probabil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transition Path Sampling
Transition path sampling (TPS) is a Rare Event Sampling method used in computer simulations of rare events: physical or chemical transitions of a system from one stable state to another that occur too rarely to be observed on a computer timescale. Examples include protein folding, chemical reactions and nucleation. Standard simulation tools such as molecular dynamics can generate the dynamical trajectories of all the atoms in the system. However, because of the gap in accessible time-scales between simulation and reality, even present supercomputers might require years of simulations to show an event that occurs once per microsecond without some kind of acceleration. Transition path ensemble TPS focuses on the most interesting part of the simulation, ''the transition''. For example, an initially unfolded protein will vibrate for a long time in an open-string configuration before undergoing a transition and fold on itself. The aim of the method is to reproduce precisely those fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Homogenization
In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients, such as : \nabla\cdot\left(A\left(\frac\right)\nabla u_\right) = f where \epsilon is a very small parameter and A\left(\vec y\right) is a 1-periodic coefficient: A\left(\vec y+\vec e_i\right)=A\left(\vec y\right), i=1,\dots, n. It turns out that the study of these equations is also of great importance in physics and engineering, since equations of this type govern the physics of inhomogeneous or heterogeneous materials. Of course, all matter is inhomogeneous at some scale, but frequently it is convenient to treat it as homogeneous. A good example is the continuum concept which is used in continuum mechanics. Under this assumption, materials such as fluids, solids, etc. can be treated as homogeneous materials and associated with these materials are material properties such as shear modulus, elastic moduli, etc. Frequently, inhomogeneous mate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiple-scale Analysis
In mathematics and physics, multiple-scale analysis (also called the method of multiple scales) comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large values of the independent variables. This is done by introducing fast-scale and slow-scale variables for an independent variable, and subsequently treating these variables, fast and slow, as if they are independent. In the solution process of the perturbation problem thereafter, the resulting additional freedom – introduced by the new independent variables – is used to remove (unwanted) secular terms. The latter puts constraints on the approximate solution, which are called solvability conditions. Mathematics research from about the 1980s proposes that coordinate transforms and invariant manifolds provide a sounder support for multiscale modelling (for example, see center manifold and slow manifold). Example: undamped Duffing equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Process
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probabilit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodicity
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity. Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motion of particles, that is, geodesics on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits return to the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Functions
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions. For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hörmander's Condition
In mathematics, Hörmander's condition is a property of vector fields that, if satisfied, has many useful consequences in the theory of partial and stochastic differential equations. The condition is named after the Swedish mathematician Lars Hörmander. Definition Given two ''C''1 vector fields ''V'' and ''W'' on ''d''-dimensional Euclidean space R''d'', let 'V'', ''W''denote their Lie bracket, another vector field defined by :, W(x) = \mathrm V(x) W(x) - \mathrm W(x) V(x), where D''V''(''x'') denotes the Fréchet derivative of ''V'' at ''x'' ∈ R''d'', which can be thought of as a matrix that is applied to the vector ''W''(''x''), and ''vice versa''. Let ''A''0, ''A''1, ... ''A''''n'' be vector fields on R''d''. They are said to satisfy Hörmander's condition if, for every point ''x'' ∈ R''d'', the vectors :\begin &A_ (x)~,\\ & _ (x), A_ (x),\\ & A_ (x), A_ (x) A_ (x)]~,\\ &\quad\vdots\quad \end \qquad 0 \leq j_, j_, \ldots, j_ \leq n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Swiss National Science Foundation
The Swiss National Science Foundation (SNSF, German: ''Schweizerischer Nationalfonds zur Förderung der wissenschaftlichen Forschung'', SNF; French: ''Fonds national suisse de la recherche scientifique'', FNS; Italian: ''Fondo nazionale svizzero per la ricerca scientifica'') is a science research support organisation mandated by the Swiss Federal Government. The Swiss National Science Foundation was established under private law by physicist and medical doctor Alexander von Muralt in 1952. Organisation The SNSF consists of three main bodies: Foundation Council, National Research Council and Administrative Offices. The Foundation Council is the highest authority and makes strategic decisions. The National Research Council is composed of distinguished researchers who mostly work at Swiss institutions of higher education. They assess research proposals submitted to the SNSF and make funding decisions. The National Research Council comprises up to 100 members and is subdivided in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
