1.
Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy

2.
Lev Landau
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Lev Davidovich Landau was a Soviet physicist who made fundamental contributions to many areas of theoretical physics. Landaus father was an engineer with the oil industry and his mother was a doctor. He learned to differentiate at age 12 and to integrate at age 13, Landau graduated in 1920 at age 13 from gymnasium. His parents considered him too young to attend university, so for a year he attended the Baku Economical Technical School. In 1922, at age 14, he matriculated at the Baku State University, subsequently, he ceased studying chemistry, but remained interested in the field throughout his life. In 1924, he moved to the centre of Soviet physics at the time. In Leningrad, he first made the acquaintance of theoretical physics and dedicated himself fully to its study, Landau subsequently enrolled for post-graduate studies at the Leningrad Physico-Technical Institute where he eventually received a doctorate in Physical and Mathematical Sciences in 1934. By that time he was fluent in German and French and could communicate in English and he later improved his English and learned Danish. After brief stays in Göttingen and Leipzig, he went to Copenhagen on 8 April 1930 to work at the Niels Bohrs Institute for Theoretical Physics and he stayed there till 3 May of the same year. After the visit, Landau always considered himself a pupil of Niels Bohr, after his stay in Copenhagen, he visited Cambridge, where he worked with P. A. M. Dirac, Copenhagen, and Zurich, where he worked with Wolfgang Pauli. From Zurich Landau went back to Copenhagen for the third time, apart from his theoretical accomplishments, Landau was the principal founder of a great tradition of theoretical physics in Kharkov, Soviet Union, sometimes referred to as the Landau school. During the Great Purge, Landau was investigated within the UPTI Affair in Kharkov, Landau developed a famous comprehensive exam called the Theoretical Minimum which students were expected to pass before admission to the school. The exam covered all aspects of physics, and between 1934 and 1961 only 43 candidates passed, but those who did later became quite notable theoretical physicists. In 1932, he computed the Chandrashekhar limit, however, he did not apply it to white dwarf stars, Landau was the head of the Theoretical Division at the Institute for Physical Problems from 1937 until 1962. After his release Landau discovered how to explain Kapitsas superfluidity using sound waves, or phonons, Landau led a team of mathematicians supporting Soviet atomic and hydrogen bomb development. Landau calculated the dynamics of the first Soviet thermonuclear bomb, including predicting the yield, for this work he received the Stalin Prize in 1949 and 1953, and was awarded the title Hero of Socialist Labour in 1954. His students included Lev Pitaevskii, Alexei Abrikosov, Evgeny Lifshitz, Lev Gorkov, Isaak Khalatnikov, Roald Sagdeev and Isaak Pomeranchuk. He received the 1962 Nobel Prize in Physics for his development of a theory of superfluidity that accounts for the properties of liquid helium II at a temperature below 2.17 K

3.
Eberhard Hopf
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The Hopf maximum principle is an early result of his that is one of the most important techniques in the theory of elliptic partial differential equations. Eberhard Hopf was born in Salzburg, Austria-Hungary, but his career was divided between Germany and the United States. He received his Ph. D. in Mathematics in 1926, in 1930 he received a fellowship from the Rockefeller Foundation to study classical mechanics with George Birkhoff at Harvard, but his appointment was at the Harvard College Observatory. In late 1931, with the help of Norbert Wiener, Hopf joined the Department of Mathematics of the Massachusetts Institute of Technology, while at MIT, Hopf did much of his work on ergodic theory. In Cambridge Hopf worked on mathematical and astronomical subjects. His book Mathematical problems of radiative equilibrium first appeared in 1934 and was reprinted in 1964, another important contribution from this period is the theory of Wiener-Hopf equations, which he developed in collaboration with Norbert Wiener. By 1960, a version of these equations was being extensively used in electrical engineering and geophysics. During this time, Hopf gained a reputation for his ability of illuminating the most complex subjects for his colleagues, because of this talent, many discoveries and proofs of other mathematicians became easier to understand after they had been described by Hopf. In 1936 Hopf received and accepted an offer of a professorship from the University of Leipzig. Hopf, with his wife Ilse and their infant daughter Barbara, returned to Germany, the book Ergodentheorie, most of which was written when Hopf was still at the Massachusetts Institute of Technology, was published in 1937. In that book, containing only 81 pages, Hopf presented a precise, in 1939 Hopf established ergodicity of the geodesic flow on compact manifolds of constant negative curvature. In 1940 Hopf was on the list of the lecturers to the International Congress of Mathematicians to be held in Cambridge. Because of the start of World War II, however, the Congress was cancelled, in 1942 Hopf was drafted to work in the German Aeronautical Institute. In 1944, one year before the end of World War II, on 22 February 1949 Hopf became a US citizen and joined Indiana University at Bloomington as a Professor of Mathematics. In 1962 he was made Research Professor of Mathematics, staying in position until his death. Hopf was never forgiven by many people for his moving to Germany in 1936, as a result, most of his work in ergodic theory and topology was neglected or even attributed to others in the years following the end of World War II. An example of this was the expulsion of Hopfs name from the version of the Wiener–Hopf equations. In 1971 Hopf was the American Mathematical Society Gibbs Lecturer, in 1981 he received the Leroy P. Steele Prize from the American Mathematical Society for seminal contributions to research

4.
Fluid dynamics
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In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids. It has several subdisciplines, including aerodynamics and hydrodynamics, before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, the foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum, and conservation of energy. These are based on mechanics and are modified in quantum mechanics. They are expressed using the Reynolds transport theorem, in addition to the above, fluids are assumed to obey the continuum assumption. Fluids are composed of molecules that collide with one another and solid objects, however, the continuum assumption assumes that fluids are continuous, rather than discrete. The fact that the fluid is made up of molecules is ignored. The unsimplified equations do not have a general solution, so they are primarily of use in Computational Fluid Dynamics. The equations can be simplified in a number of ways, all of which make them easier to solve, some of the simplifications allow some simple fluid dynamics problems to be solved in closed form. Three conservation laws are used to solve fluid dynamics problems, the conservation laws may be applied to a region of the flow called a control volume. A control volume is a volume in space through which fluid is assumed to flow. The integral formulations of the laws are used to describe the change of mass, momentum. Mass continuity, The rate of change of fluid mass inside a control volume must be equal to the net rate of flow into the volume. Mass flow into the system is accounted as positive, and since the vector to the surface is opposite the sense of flow into the system the term is negated. The first term on the right is the net rate at which momentum is convected into the volume, the second term on the right is the force due to pressure on the volumes surfaces. The first two terms on the right are negated since momentum entering the system is accounted as positive, the third term on the right is the net acceleration of the mass within the volume due to any body forces. Surface forces, such as forces, are represented by F surf. The following is the form of the momentum conservation equation

5.
Turbulence
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Turbulence or turbulent flow is a flow regime in fluid dynamics characterized by chaotic changes in pressure and flow velocity. It is in contrast to a flow regime, which occurs when a fluid flows in parallel layers. Turbulence is caused by kinetic energy in parts of a fluid flow. For this reason turbulence is easier to create in low viscosity fluids, in general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases. This would increase the energy needed to pump fluid through a pipe, however this effect can also be exploited by such as aerodynamic spoilers on aircraft, which deliberately spoil the laminar flow to increase drag and reduce lift. The onset of turbulence can be predicted by a constant called the Reynolds number. However, turbulence has long resisted detailed physical analysis, and the interactions within turbulence creates a complex situation. Richard Feynman has described turbulence as the most important unsolved problem of classical physics, smoke rising from a cigarette is mostly turbulent flow. However, for the first few centimeters the flow is laminar, the smoke plume becomes turbulent as its Reynolds number increases, due to its flow velocity and characteristic length increasing. If the golf ball were smooth, the boundary layer flow over the front of the sphere would be laminar at typical conditions. However, the layer would separate early, as the pressure gradient switched from favorable to unfavorable. To prevent this happening, the surface is dimpled to perturb the boundary layer. This results in higher skin friction, but moves the point of boundary layer separation further along, resulting in form drag. The flow conditions in industrial equipment and machines. The external flow over all kind of such as cars, airplanes, ships. The motions of matter in stellar atmospheres, a jet exhausting from a nozzle into a quiescent fluid. As the flow emerges into this external fluid, shear layers originating at the lips of the nozzle are created and these layers separate the fast moving jet from the external fluid, and at a certain critical Reynolds number they become unstable and break down to turbulence. Biologically generated turbulence resulting from swimming animals affects ocean mixing, snow fences work by inducing turbulence in the wind, forcing it to drop much of its snow load near the fence

6.
Fourier transform
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The Fourier transform decomposes a function of time into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies of its constituent notes. The Fourier transform is called the frequency domain representation of the original signal, the term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, linear operations performed in one domain have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the domain corresponds to multiplication by the frequency. Also, convolution in the domain corresponds to ordinary multiplication in the frequency domain. Concretely, this means that any linear time-invariant system, such as a filter applied to a signal, after performing the desired operations, transformation of the result can be made back to the time domain. Functions that are localized in the domain have Fourier transforms that are spread out across the frequency domain and vice versa. The Fourier transform of a Gaussian function is another Gaussian function, Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform can also be generalized to functions of variables on Euclidean space. In general, functions to which Fourier methods are applicable are complex-valued, the latter is routinely employed to handle periodic functions. The fast Fourier transform is an algorithm for computing the DFT, the Fourier transform of the function f is traditionally denoted by adding a circumflex, f ^. There are several conventions for defining the Fourier transform of an integrable function f, ℝ → ℂ. Here we will use the definition, f ^ = ∫ − ∞ ∞ f e −2 π i x ξ d x. When the independent variable x represents time, the transform variable ξ represents frequency. Under suitable conditions, f is determined by f ^ via the inverse transform, f = ∫ − ∞ ∞ f ^ e 2 π i ξ x d ξ, the functions f and f ^ often are referred to as a Fourier integral pair or Fourier transform pair. For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ, see Other conventions, the Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum. Many other characterizations of the Fourier transform exist, for example, one uses the Stone–von Neumann theorem, the Fourier transform is the unique unitary intertwiner for the symplectic and Euclidean Schrödinger representations of the Heisenberg group. In 1822, Joseph Fourier showed that some functions could be written as an sum of harmonics

7.
Andrey Kolmogorov
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Andrey Kolmogorov was born in Tambov, about 500 kilometers south-southeast of Moscow, in 1903. Kolmogorova, died giving birth to him, Andrey was raised by two of his aunts in Tunoshna at the estate of his grandfather, a well-to-do nobleman. Little is known about Andreys father and he was supposedly named Nikolai Matveevich Kataev and had been an agronomist. Nikolai had been exiled from St. Petersburg to the Yaroslavl province after his participation in the movement against the czars. He disappeared in 1919 and he was presumed to have killed in the Russian Civil War. Andrey Kolmogorov was educated in his aunt Veras village school, and his earliest literary efforts, Andrey was the editor of the mathematical section of this journal. In 1910, his aunt adopted him, and they moved to Moscow, later that same year, Kolmogorov began to study at the Moscow State University and at the same time Mendeleev Moscow Institute of Chemistry and Technology. Kolmogorov writes about this time, I arrived at Moscow University with a knowledge of mathematics. I knew in particular the beginning of set theory, I studied many questions in articles in the Encyclopedia of Brockhaus and Efron, filling out for myself what was presented too concisely in these articles. Kolmogorov gained a reputation for his wide-ranging erudition, during the same period, Kolmogorov worked out and proved several results in set theory and in the theory of Fourier series. In 1922, Kolmogorov gained international recognition for constructing a Fourier series that diverges almost everywhere, around this time, he decided to devote his life to mathematics. In 1925, Kolmogorov graduated from the Moscow State University and began to study under the supervision of Nikolai Luzin, Kolmogorov became interested in probability theory. In 1929, Kolmogorov earned his Doctor of Philosophy degree, from Moscow State University, in 1930, Kolmogorov went on his first long trip abroad, traveling to Göttingen and Munich, and then to Paris. He had various contacts in Göttingen. His pioneering work, About the Analytical Methods of Probability Theory, was published in 1931, also in 1931, he became a professor at the Moscow State University. In 1935, Kolmogorov became the first chairman of the department of probability theory at the Moscow State University, around the same years Kolmogorov contributed to the field of ecology and generalized the Lotka–Volterra model of predator-prey systems. In 1936, Kolmogorov and Alexandrov were involved in the persecution of their common teacher Nikolai Luzin, in the so-called Luzin affair. In a 1938 paper, Kolmogorov established the basic theorems for smoothing and predicting stationary stochastic processes—a paper that had military applications during the Cold War