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Kantorovich
Leonid Vitalyevich Kantorovich ( rus, Леони́д Вита́льевич Канторо́вич, , p=lʲɪɐˈnʲit vʲɪˈtalʲjɪvʲɪtɕ kəntɐˈrovʲɪtɕ, a=Ru-Leonid_Vitaliyevich_Kantorovich.ogg; 19 January 19127 April 1986) was a Soviet mathematician and economist, known for his theory and development of techniques for the optimal allocation of resources. He is regarded as the founder of linear programming. He was the winner of the Stalin Prize in 1949 and the Nobel Memorial Prize in Economic Sciences in 1975. Biography Kantorovich was born on 19 January 1912, to a Russian Jewish family. His father was a doctor practicing in Saint Petersburg. In 1926, at the age of fourteen, he began his studies at Leningrad State University. He graduated from the Faculty of Mathematics and Mechanics in 1930, and began his graduate studies. In 1934, at the age of 22 years, he became a full professor. Later, Kantorovich worked for the Soviet government. He was given the task of optimiz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kantorovich Inequality
In mathematics, the Kantorovich inequality is a particular case of the Cauchy–Schwarz inequality, which is itself a generalization of the triangle inequality. The triangle inequality states that the length of two sides of any triangle, added together, will be equal to or greater than the length of the third side. In simplest terms, the Kantorovich inequality translates the basic idea of the triangle inequality into the terms and notational conventions of linear programming. (See vector space, inner product, and normed vector space for other examples of how the basic ideas inherent in the triangle inequality—line segment and distance—can be generalized into a broader context.) More formally, the Kantorovich inequality can be expressed this way: :Let :: p_i \geq 0,\quad 0 < a \leq x_i \leq b\texti=1, \dots ,n. :Let :Then :: |
Kantorovich Metric
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] |
Kantorovich Theorem
The Kantorovich theorem, or Newton–Kantorovich theorem, is a mathematical statement on the semi-local convergence of Newton's method. It was first stated by Leonid Kantorovich in 1948. It is similar to the form of the Banach fixed-point theorem, although it states existence and uniqueness of a zero rather than a fixed point. Newton's method constructs a sequence of points that under certain conditions will converge to a solution x of an equation f(x)=0 or a vector solution of a system of equation F(x)=0. The Kantorovich theorem gives conditions on the initial point of this sequence. If those conditions are satisfied then a solution exists close to the initial point and the sequence converges to that point. Assumptions Let X\subset\R^n be an open subset and F:X \subset \R^n \to\R^n a differentiable function with a Jacobian F^(\mathbf x) that is locally Lipschitz continuous (for instance if F is twice differentiable). That is, it is assumed that for any x \in X there is an ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gennadii Rubinstein
Gennadii Shlemovich Rubinstein (rus: Геннадий Шлемович Рубинштейн) was a Russian mathematician. His research focused on mathematical programming and operations research. His name is associated to the Kantorovich–Rubinstein metric which is commonly known as the Wasserstein distance used in optimal transport. Gennadii Rubinstein got his doctorate in St. Petersburg State University in 1956, under the supervision of Leonid V. Kantorovich. Alternate form of the first name: Gennady. Alternate forms of the last name: Rubinšteĭn, Rubinshtein. Biography Selected publications * * * * * * * See also * List of Russian mathematicians A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby union ... References External links * A web page about Gennadii Rubinstein's p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grigorii Mikhailovich Fichtenholz
Grigorii Mikhailovich Fichtenholz (or Fikhtengolts) (russian: Григо́рий Миха́йлович Фихтенго́льц) (June 8, 1888 in Odessa – June 26, 1959 in Leningrad) was a Soviet mathematician working on real analysis and functional analysis. Fichtenholz was one of the founders of the Leningrad school of real analysis. He also authored a three-volume textbook 'Differential and Integral Calculus'. The books cover mathematical analysis of function of one real variable, functions of many real variables and of complex functions. Due to depth and precision of presentation of material, these books are defined as classical position in mathematical analysis. Book was translated, among others, into German, Polish, Chinese, Vietnamese, and Persian however translation to English language has not been done still. Fichtenholz's books about analysis are widely used in Middle and Eastern European as well as Chinese universities due to its exceptionality of detailed and we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Smirnov (mathematician)
Vladimir Ivanovich Smirnov (russian: Влади́мир Ива́нович Смирно́в) (10 June 1887 – 11 February 1974) was a mathematician who made significant contributions in both pure and applied mathematics, and also in the history of mathematics. Smirnov worked on diverse areas of mathematics, such as complex functions and conjugate functions in Euclidean spaces. In the applied field his work includes the propagation of waves in elastic media with plane boundaries (with Sergei Sobolev) and the oscillations of elastic spheres. His pioneering approach to solving the initial-boundary value problem to the wave equation formed the basis of the spacetime triangle diagram (STTD) technique for wave motion developed by his follower Victor Borisov (also known as the Smirnov method of incomplete separation of variables). Smirnov was a Ph.D. student of Vladimir Steklov. Among his notable students were Sergei Sobolev, Solomon Mikhlin and Nobel prize winner Leonid Kantorovi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leningrad State University
Saint Petersburg State University (SPBU; russian: Санкт-Петербургский государственный университет) is a public research university in Saint Petersburg, Russia. Founded in 1724 by a decree of Peter the Great, the university from the beginning has had a focus on fundamental research in science, engineering and humanities. During the Soviet period, it was known as Leningrad State University (russian: Ленинградский государственный университет). It was renamed after Andrei Zhdanov in 1948 and was officially called "Leningrad State University, named after A. A. Zhdanov and decorated with the Order of Lenin and the Order of the Red Banner of Labour." Zhdanov's was removed in 1989 and Leningrad in the name was officially replaced with Saint Petersburg in 1992. It is made up of 24 specialized faculties (departments) and institutes, the Academic Gymnasium, the Medical College, the College of Physical Culture ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Svetlozar Rachev
Svetlozar (Zari) Todorov Rachev is a professor at Texas Tech University who works in the field of mathematical finance, probability theory, and statistics. He is known for his work in probability metrics, derivative pricing, financial risk modeling, and econometrics. In the practice of risk management, he is the originator of the methodology behind the flagship product of FinAnalytica. Life and work Rachev earned a MSc degree from the Faculty of Mathematics at Sofia University in 1974, a PhD degree from Lomonosov Moscow State University under the supervision of Vladimir Zolotarev in 1979, and a Dr Sci degree from Steklov Mathematical Institute in 1986 under the supervision of Leonid Kantorovich, a Nobel Prize winner in economic sciences, Andrey Kolmogorov and Yuri Prokhorov. Currently, he is Professor of Financial Mathematics at Texas Tech University. In mathematical finance, Rachev is known for his work on application of non-Gaussian models for risk assessment, option pricing, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saint Petersburg State University
Saint Petersburg State University (SPBU; russian: Санкт-Петербургский государственный университет) is a public research university in Saint Petersburg, Russia. Founded in 1724 by a decree of Peter the Great, the university from the beginning has had a focus on fundamental research in science, engineering and humanities. During the Soviet period, it was known as Leningrad State University (russian: Ленинградский государственный университет). It was renamed after Andrei Zhdanov in 1948 and was officially called "Leningrad State University, named after A. A. Zhdanov and decorated with the Order of Lenin and the Order of the Red Banner of Labour." Zhdanov's was removed in 1989 and Leningrad in the name was officially replaced with Saint Petersburg in 1992. It is made up of 24 specialized faculties (departments) and institutes, the Academic Gymnasium, the Medical College, the College of Physical Culture ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
