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Igor Girsanov
Igor Vladimirovich Girsanov (russian: И́горь Влади́мирович Гирсанов; (10 September 1934 – 16 March 1967) was a Russian mathematician. He made major contributions to probability theory and its applications. Early life Igor Girsanov was born on 10 September 1934, in Turkestan (then Kazakh ASSR). He studied in Baku until his family moved to Moscow in 1950. While at school he was an active member of the Moscow State University maths club and won multiple Moscow mathematics olympiads. Education Between 1952 and 1960 Girsanov was an undergraduate and graduate student at Moscow State University. After his graduation he joined the faculty. In 1965 he became Head of the newly formed Probability and Statistics Laboratory at MSU. Academic work One can distinguish two periods in Girsanov's academic work. Prior to 1961, he worked as a member of a group of mathematicians united around E. B. Dynkin who were developing the theory of Markov processes. Hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turkistan (city)
Turkistan ( kz, Түркістан, ''Türkıstan'') is a city and the administrative center of Turkistan Region of Kazakhstan, near the Syr Darya river. It is situated north-west of Shymkent on the Trans-Aral Railway between Kyzylorda to the north and Tashkent to the south. Its population has increased within ten years from to Turkistan's most prominent historical and cultural asset is the Mausoleum of Khoja Ahmed Yasawi, a UNESCO World Heritage site. The city is served by Hazrat Sultan International Airport. In 2021, Turkistan was proclaimed by the Organization of Turkic States as "Spiritual Capital of the Turkic World". In the same year, Turkistan was named as one of the top ten tourist destinations in Kazakhstan. History Turkistan is one of Kazakhstan's historic cities with an archaeological record dating back to the 4th century. It became a commercial centre after the final demise of Otrar, the medieval city whose ruins lie near the Syr Darya to the southeast. Through ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feller Process
In probability theory relating to stochastic processes, a Feller process is a particular kind of Markov process. Definitions Let ''X'' be a locally compact Hausdorff space with a countable base. Let ''C''0(''X'') denote the space of all real-valued continuous functions on ''X'' that vanish at infinity, equipped with the sup-norm , , ''f'' , , . From analysis, we know that ''C''0(''X'') with the sup norm is a Banach space. A Feller semigroup on ''C''0(''X'') is a collection ''t'' ≥ 0 of positive linear maps from ''C''0(''X'') to itself such that * , , ''T''''t''''f'' , , ≤ , , ''f'' , , for all ''t'' ≥ 0 and ''f'' in ''C''0(''X''), i.e., it is a contraction (in the weak sense); * the semigroup property: ''T''''t'' + ''s'' = ''T''''t'' o''T''''s'' for all ''s'', ''t'' ≥ 0; * lim''t'' → 0, , ''T''''t''''f'' − ''f'' , , = 0 for every ''f'' in ''C''0('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1934 Births
Events January–February * January 1 – The International Telecommunication Union, a specialist agency of the League of Nations, is established. * January 15 – The 8.0 Nepal–Bihar earthquake strikes Nepal and Bihar with a maximum Mercalli intensity of XI (''Extreme''), killing an estimated 6,000–10,700 people. * January 26 – A 10-year German–Polish declaration of non-aggression is signed by Nazi Germany and the Second Polish Republic. * January 30 ** In Nazi Germany, the political power of federal states such as Prussia is substantially abolished, by the "Law on the Reconstruction of the Reich" (''Gesetz über den Neuaufbau des Reiches''). ** Franklin D. Roosevelt, President of the United States, signs the Gold Reserve Act: all gold held in the Federal Reserve is to be surrendered to the United States Department of the Treasury; immediately following, the President raises the statutory gold price from US$20.67 per ounce to $35. * February 6 – F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chemical Reactor
A chemical reactor is an enclosed volume in which a chemical reaction takes place. In chemical engineering, it is generally understood to be a process vessel used to carry out a chemical reaction, which is one of the classic unit operations in chemical process analysis. The design of a chemical reactor deals with multiple aspects of chemical engineering. Chemical engineers design reactors to maximize net present value for the given reaction. Designers ensure that the reaction proceeds with the highest efficiency towards the desired output product, producing the highest yield of product while requiring the least amount of money to purchase and operate. Normal operating expenses include energy input, energy removal, raw material costs, labor, etc. Energy changes can come in the form of heating or cooling, pumping to increase pressure, frictional pressure loss or agitation.Chemical reaction engineering is the branch of chemical engineering which deals with chemical reactors and their d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimal Control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory. Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Economics
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity. Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial or contentious subjects that would be impossible without mathematics. Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships asserted to clarify ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-invariant Measure
In mathematics, a quasi-invariant measure ''μ'' with respect to a transformation ''T'', from a measure space ''X'' to itself, is a measure which, roughly speaking, is multiplied by a numerical function of ''T''. An important class of examples occurs when ''X'' is a smooth manifold ''M'', ''T'' is a diffeomorphism of ''M'', and ''μ'' is any measure that locally is a measure with base the Lebesgue measure on Euclidean space. Then the effect of ''T'' on μ is locally expressible as multiplication by the Jacobian determinant of the derivative (pushforward) of ''T''. To express this idea more formally in measure theory terms, the idea is that the Radon–Nikodym derivative of the transformed measure μ′ with respect to ''μ'' should exist everywhere; or that the two measures should be equivalent (i.e. mutually absolutely continuous): :\mu' = T_ (\mu) \approx \mu. That means, in other words, that ''T'' preserves the concept of a set of measure zero. Considering the whole equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the flow of water in a pipe, the Brownian motion, random motion of particles in the air, and population dynamics, the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real number, real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a Set (mathematics), set, without the need of a Differentiability, smooth space-time structure defined on it. At any given time, a dynamical system has a State ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Differential Equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock prices or physical systems subject to thermal fluctuations. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process. However, other types of random behaviour are possible, such as jump processes. Random differential equations are conjugate to stochastic differential equations. Background Stochastic differential equations originated in the theory of Brownian motion, in the work of Albert Einstein and Smoluchowski. These early examples were linear stochastic differential equations, also called 'Langevin' equations after French physicist Langevin, describing the motion of a harmonic oscillator subject to a random force. The mathematical theory of stochasti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
