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Dimensional Reduction
Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero. In physics, a theory in ''D'' spacetime dimensions can be redefined in a lower number of dimensions ''d'', by taking all the fields to be independent of the location in the extra ''D'' − ''d'' dimensions. For example, consider a periodic compact dimension with period ''L''. Let ''x'' be the coordinate along this dimension. Any field \phi can be described as a sum of the following terms: :\phi_n(x) = A_n \cos \left( \frac\right) with ''A''''n'' a constant. According to quantum mechanics, such a term has momentum ''nh''/''L'' along ''x'', where ''h'' is Planck's constant. Therefore, as L goes to zero, the momentum goes to infinity, and so does the energy, unless ''n'' = 0. However ''n'' = 0 gives a field which is constant with respect to ''x''. So at this limit, and at finite energy, \phi will not depend on ''x''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimensionality Reduction
Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension. Working in high-dimensional spaces can be undesirable for many reasons; raw data are often sparse as a consequence of the curse of dimensionality, and analyzing the data is usually computationally intractable (hard to control or deal with). Dimensionality reduction is common in fields that deal with large numbers of observations and/or large numbers of variables, such as signal processing, speech recognition, neuroinformatics, and bioinformatics. Methods are commonly divided into linear and nonlinear approaches. Approaches can also be divided into feature selection and feature extraction. Dimensionality reduction can be used for noise reduction, data visualization, cluster analysis, or as an intermediat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent. In coordinate-free terms, the gradient of a function f(\bf) may be defined by: :df=\nabla f \cdot d\bf where ''df'' is the total infinitesimal change in ''f'' for an infinitesimal displacement d\bf, and is seen to be maximal when d\bf is in the direction of the gradi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vicinity of black holes or similar compact astrophysical objects, such as neutron stars. Three of the four fundamental forces of physics are described within the framework of quantum mechanics and quantum field theory. The current understanding of the fourth force, gravity, is based on Albert Einstein's general theory of relativity, which is formulated within the entirely different framework of classical physics. However, that description is incomplete: describing the gravitational field of a black hole in the general theory of relativity leads physical quantities, such as the spacetime curvature, to diverge at the center of the black hole. This signals the breakdown of the general theory of relativity and the need for a theory that goes b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as the Minimal Supersymmetric Standard Model. Supergravity is the gauge theory of local supersymmetry. Since the supersymmetry (SUSY) generators form together with the Poincaré algebra a superalgebra, called the super-Poincaré algebra, supersymmetry as a gauge theory makes gravity arise in a natural way. Gravitons Like any field theory of gravity, a supergravity theory contains a spin-2 field whose quantum is the graviton. Supersymmetry requires the graviton field to have a superpartner. This field has spin 3/2 and its quantum is the gravitino. The number of gravitino fields is equal to the number of supersymmetries. History Gauge supersymmetry The first theory of local supersymmetry was proposed by Dick Arnowitt and Pran Nath in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries the gravitational force. Thus, string theory is a theory of quantum gravity. String theory is a broad and varied subject that attempts to address a number of deep questions of fundamental physics. String theory has contributed a number of advances to mathematical physics, which have been applied to a variety of problems in black hole physics, early universe cosmology, nuclear physics, and conde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giorgio Parisi
Giorgio Parisi (born 4 August 1948) is an Italian theoretical physicist, whose research has focused on quantum field theory, statistical mechanics and complex systems. His best known contributions are the QCD evolution equations for parton densities, obtained with Guido Altarelli, known as the Altarelli–Parisi or DGLAP equations, the exact solution of the Sherrington–Kirkpatrick model of spin glasses, the Kardar–Parisi–Zhang equation describing dynamic scaling of growing interfaces, and the study of whirling flocks of birds. He was awarded the 2021 Nobel Prize in Physics jointly with Klaus Hasselmann and Syukuro Manabe for groundbreaking contributions to theory of complex systems, in particular "for the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales." Career Giorgio Parisi received his degree from the University of Rome La Sapienza in 1970 under the supervision of Nicola Cabibbo. He was a researcher at the L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersymmetric Theory Of Stochastic Dynamics
Supersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The main utility of the theory from the physical point of view is a rigorous theoretical explanation of the ubiquitous spontaneous long-range dynamical behavior that manifests itself across disciplines via such phenomena as 1/f, flicker, and crackling noises and the power-law statistics, or Zipf's law, of instantonic processes like earthquakes and neuroavalanches. From the mathematical point of view, STS is interesting because it bridges the two major parts of mathematical physics – the dynamical systems theory and topological field theories. Besides these and related disciplines such as algebraic topology and supersymmetric field theories, STS is also connected with the traditional theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shang-keng Ma
Shang-keng Ma (September 24, 1940, Chongqing, Sichuan, China – November 24, 1983, La Jolla, California, ) was a Chinese theoretical physicist, known for his work on the theory of critical phenomena and random systems. He is known as the co-author with Bertrand Halperin and Pierre Hohenberg of a 1972 paper that "generalized the renormalization group theory to dynamical critical phenomena." Ma is also known as the co-author with Yoseph Imry of a 1975 paper and with Amnon Aharony and Imry of a 1976 paper that established the foundation of the random field Ising model (RFIM) Biography He transferred in 1959 from the National Taiwan University to the University of California, Berkeley. There he graduated in 1962 with a bachelor's degree in science and in 1966 with a Ph.D. His Ph.D. thesis ''Correlations of Photons from a Thermal Source'' was supervised by Kenneth M. Watson. As a postdoc in 1966, Ma went to the University of California, San Diego (UCSD) to study with Keith Brueckner ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yoseph Imry
Yoseph Imry (Hebrew: יוסף אמרי; born 23 February 1939 – 29 May 2018) was an Israeli physicist. He was best known for taking part in the foundation of mesoscopic physics, a relatively new branch of condensed matter physics. It is concerned with how the behavior of systems whose size is in between micro- and macroscopic, crosses over between these two regimes. These systems can be handled and addressed by more or less usual macroscopic methods, but their behavior may still show quantum effects. Awards and honours In 1996, 2001 and 2016, Imry received the Rothschild Prize, Israel Prize and Wolf Prize in physics, respectively. Imry was the 1996 Lorentz Professor at Leiden University. He was a member of the European Academy of Sciences and Arts (Salzburg), the European Academy of Sciences, Sciences and Humanities (Paris), the National Academy of Sciences, the American Physical Society and the Israel Academy of Sciences and Humanities. See also *List of Israel Prize reci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amnon Aharony
Amnon Aharony (Hebrew: אמנון אהרוני; born: 7 January 1943) is an Israeli Professor (Emeritus) of Physics in the School of Physics and Astronomy at Tel Aviv University, Israel and in the Physics Department of Ben Gurion University of the Negev, Israel. After years of research on statistical physics (critical phenomena, random systems, fractals, percolation), his current research focuses on condensed matter theory, especially in mesoscopic physics and spintronics. He is a member of the Israel Academy of Sciences and Humanities, a Foreign Honorary Member of the American Academy of Arts and Sciences and of several other academies. He also received several prizes, including the Rothschild Prize in Physical Sciences, and the Gunnar Randers Research Prize, awarded every other year by the King of Norway. Early life and education Amnon Aharony was born in Jerusalem, and grew up in Netanya, Israel. He received his B.Sc. in Physics and Mathematics in 1964 from the Hebrew unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
