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Tikhonov's Theorem (dynamical Systems)
In applied mathematics, Tikhonov's theorem on dynamical systems is a result on stability of solutions of systems of differential equations. It has applications to chemical kinetics. The theorem is named after Andrey Nikolayevich Tikhonov Andrey Nikolayevich Tikhonov (russian: Андре́й Никола́евич Ти́хонов; October 17, 1906 – October 7, 1993) was a leading Soviet Russian mathematician and geophysicist known for important contributions to topology, .... Statement Consider this system of differential equations: : \begin \frac & = \mathbf(\mathbf,\mathbf,t), \\ \mu\frac & = \mathbf(\mathbf,\mathbf,t). \end Taking the limit as \mu\to 0, this becomes the "degenerate system": : \begin \frac & = \mathbf(\mathbf,\mathbf,t), \\ \mathbf & = \varphi(\mathbf,t), \end where the second equation is the solution of the algebraic equation : \mathbf(\mathbf,\mathbf,t) = 0. Note that there may be more than one such function \varphi . Tikhonov's the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chemical Kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in which a reaction occurs but in itself tells nothing about its rate. Chemical kinetics includes investigations of how experimental conditions influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition states, as well as the construction of mathematical models that also can describe the characteristics of a chemical reaction. History In 1864, Peter Waage and Cato Guldberg pioneered the development of chemical kinetics by formulating the law of mass action, which states that the speed of a chemical reaction is proportional to the quantity of the reacting substances.C.M. Guldberg and P. Waage,"Studies Concerning Affinity" ''Forhandlinger i Videnskabs-Selskabet i Christiania'' (1864), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enzyme Kinetics
Enzyme kinetics is the study of the rates of enzyme-catalysed chemical reactions. In enzyme kinetics, the reaction rate is measured and the effects of varying the conditions of the reaction are investigated. Studying an enzyme's kinetics in this way can reveal the catalytic mechanism of this enzyme, its role in metabolism, how its activity is controlled, and how a drug or a modifier (inhibitor or activator) might affect the rate. An enzyme (E) is typically a protein molecule that promotes a reaction of another molecule, its substrate (S). This binds to the active site of the enzyme to produce an enzyme-substrate complex ES, and is transformed into an enzyme-product complex EP and from there to product P, via a transition state ES*. The series of steps is known as the mechanism: : E + S ⇄ ES ⇄ ES* ⇄ EP ⇄ E + P This example assumes the simplest case of a reaction with one substrate and one product. Such cases exist: for example, a mutase such as phosphoglucomutase ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrey Nikolayevich Tikhonov
Andrey Nikolayevich Tikhonov (russian: Андре́й Никола́евич Ти́хонов; October 17, 1906 – October 7, 1993) was a leading Soviet Russian mathematician and geophysicist known for important contributions to topology, functional analysis, mathematical physics, and ill-posed problems. He was also one of the inventors of the magnetotellurics method in geophysics. Other transliterations of his surname include "Tychonoff", "Tychonov", "Tihonov", "Tichonov." Biography Born in Gzhatsk, he studied at the Moscow State University where he received a Ph.D. in 1927 under the direction of Pavel Sergeevich Alexandrov. In 1933 he was appointed as a professor at Moscow State University. He became a corresponding member of the USSR Academy of Sciences on 29 January 1939 and a full member of the USSR Academy of Sciences on 1 July 1966. Research work Tikhonov worked in a number of different fields in mathematics. He made important contributions to topology, functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perturbation Theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In perturbation theory, the solution is expressed as a power series in a small parameter The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of \varepsilon usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, usually by keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. Perturbation theory is used in a wide range of fields, and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
