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Eberhard Hopf
Eberhard Frederich Ferdinand Hopf (April 4, 1902 in Salzburg, Austria-Hungary – July 24, 1983 in Bloomington, Indiana, USA) was a German mathematician and astronomer, one of the founding fathers of ergodic theory and a pioneer of bifurcation theory who also made significant contributions to the subjects of partial differential equations and integral equations, fluid dynamics, and differential geometry. The Hopf maximum principle is an early result of his (1927) that is one of the most important techniques in the theory of elliptic partial differential equations. Biography Hopf was born in Salzburg, Austria-Hungary, but his scientific career was divided between Germany and the United States. He received his Ph.D. in mathematics in 1926 and his ''Habilitation'' in mathematical astronomy from the University of Berlin in 1929. In 1971, Hopf was the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Salzburg
Salzburg is the List of cities and towns in Austria, fourth-largest city in Austria. In 2020 its population was 156,852. The city lies on the Salzach, Salzach River, near the border with Germany and at the foot of the Austrian Alps, Alps mountains. The town occupies the site of the Roman settlement of ''Iuvavum''. Founded as an episcopal see in 696, it became a Prince-Archbishopric of Salzburg, seat of the archbishop in 798. Its main sources of income were salt extraction, trade, as well as gold mining. The Hohensalzburg Fortress, fortress of Hohensalzburg, one of the largest medieval fortresses in Europe, dates from the 11th century. In the 17th century, Salzburg became a centre of the Counter-Reformation, with monasteries and numerous Baroque churches built. Salzburg has an extensive cultural and educational history, being the birthplace of Wolfgang Amadeus Mozart and being home to three universities and a large student population. Today, along with Vienna and the Tyrol (st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Bifurcation
In the mathematics of dynamical systems and differential equations, a Hopf bifurcation is said to occur when varying a parameter of the system causes the set of solutions (trajectories) to change from being attracted to (or repelled by) a fixed point, and instead become attracted to (or repelled by) an oscillatory, periodic solution. The Hopf bifurcation is a two-dimensional analog of the pitchfork bifurcation. Many different kinds of systems exhibit Hopf bifurcations, from radio oscillators to railroad bogies. Trailers towed behind automobiles become infamously unstable if loaded incorrectly, or if designed with the wrong geometry. This offers a gut-sense intuitive example of a Hopf bifurcation in the ordinary world, where stable motion becomes unstable and oscillatory as a parameter is varied. The general theory of how the solution sets of dynamical systems change in response to changes of parameters is called bifurcation theory; the term ''bifurcation'' arises, as the set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bifurcation Theory
Bifurcation theory is the Mathematics, mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematics, mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems (described by Ordinary differential equation, ordinary, Delay differential equation, delay or Partial differential equation, partial differential equations) and discrete systems (described by maps). The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior. Bifurcation types It is useful to divide bifurcations into two principal classes: * Local bif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Theory
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Astronomer
An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galaxies – in either observational astronomy, observational (by analyzing the data) or theoretical astronomy. Examples of topics or fields astronomers study include planetary science, Sun, solar astronomy, the Star formation, origin or stellar evolution, evolution of stars, or the galaxy formation and evolution, formation of galaxies. A related but distinct subject is physical cosmology, which studies the Universe as a whole. Types Astronomers typically fall under either of two main types: observational astronomy, observational and theoretical astronomy, theoretical. Observational astronomers make direct observations of Astronomical object, celestial objects and analyze the data. In contrast, theoretical astronomers create and investigate Con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematical model, models, and mathematics#Calculus and analysis, change. History One of the earliest known mathematicians was Thales of Miletus (); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem. The number of known mathematicians grew when Pythagoras of Samos () established the Pythagorean school, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of International Congresses Of Mathematicians Plenary And Invited Speakers
This is a list of International Congresses of Mathematicians Plenary and Invited Speakers. Being invited to talk at an International Congress of Mathematicians has been called "the equivalent, in this community, of an induction to a hall of fame." The current list of Plenary and Invited Speakers presented here is based on the ICM's post-WW II terminology, in which the one-hour speakers in the morning sessions are called "Plenary Speakers" and the other speakers (in the afternoon sessions) whose talks are included in the ICM published proceedings are called "Invited Speakers". In the pre-WW II congresses the Plenary Speakers were called "Invited Speakers". By congress year 1897, Zürich *Jules Andrade *Léon Autonne *Émile Borel *Nikolai Bugaev *Francesco Brioschi *Hermann Brunn *Cesare Burali-Forti *Charles Jean de la Vallée Poussin *Gustaf Eneström *Federigo Enriques *Gino Fano *Zoel García de Galdeano *Francesco Gerbaldi *Paul Gordan *Jacques Hadamard *Adolf Hurwitz *Felix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Josiah Willard Gibbs Lectureship
The Josiah Willard Gibbs Lectureship (also called the Gibbs Lecture) of the American Mathematical Society is an annually awarded mathematical prize, named in honor of Josiah Willard Gibbs. The prize is intended not only for mathematicians, but also for physicists, chemists, biologists, physicians, and other scientists who have made important applications of mathematics. The purpose of the prize is to recognize outstanding achievement in applied mathematics and "to enable the public and the academic community to become aware of the contribution that mathematics is making to present-day thinking and to modern civilization." The prize winner gives a lecture, which is subsequently published in the 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-expert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener–Hopf Method
The Wiener–Hopf method is a mathematical technique widely used in applied mathematics. It was initially developed by Norbert Wiener and Eberhard Hopf as a method to solve systems of integral equations, but has found wider use in solving two-dimensional partial differential equations with mixed boundary conditions on the same boundary. In general, the method works by exploiting the complex-analytical properties of transformed functions. Typically, the standard Fourier transform is used, but examples exist using other transforms, such as the Mellin transform. In general, the governing equations and boundary conditions are transformed and these transforms are used to define a pair of complex functions (typically denoted with '+' and '−' subscripts) which are respectively analytic in the upper and lower halves of the complex plane, and have growth no faster than polynomials in these regions. These two functions will also coincide on some region of the complex plane, typically, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Landau–Hopf Theory Of Turbulence
In physics, the Landau–Hopf theory of turbulence, named for Lev Landau and Eberhard Hopf, was until the mid-1970s, the accepted theory of how a fluid flow becomes turbulent. It states that as a fluid flows faster, it develops more Fourier modes. At first, a few modes dominate, but under stronger conditions, it forces the modes to become power-law distributed as explained in Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Soviet ...'s theory of turbulence. References * * Turbulence {{fluiddynamics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cole–Hopf Transformation
The Cole–Hopf transformation is a change of variables that allows to transform a special kind of parabolic partial differential equations (PDEs) with a quadratic nonlinearity into a linear heat equation. In particular, it provides an explicit formula for fairly general solutions of the PDE in terms of the initial datum and the heat kernel. Consider the following PDE:u_ - a\Delta u + b\, \nabla u\, ^ = 0, \quad u(0,x) = g(x) where x\in \mathbb^, a,b are constants, \Delta is the Laplace operator, \nabla is the gradient, and \, \cdot\, is the Euclidean norm in \mathbb^. By assuming that w = \phi(u), where \phi(\cdot) is an unknown smooth function, we may calculate:w_ = \phi'(u)u_, \quad \Delta w = \phi'(u)\Delta u + \phi''(u)\, \nabla u\, ^ Which implies that:\begin w_ = \phi'(u)u_ &= \phi'(u)\left( a\Delta u - b\, \nabla u\, ^\right) \\ &= a\Delta w - (a\phi'' + b\phi')\, \nabla u\, ^ \\ &= a\Delta w \end if we constrain \phi to satisfy a\phi'' + b\phi' = 0. Then we may transform ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
