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Maximum Principle
In the mathematical fields of differential equations and geometric analysis, the maximum principle is one of the most useful and best known tools of study. Solutions of a differential inequality in a domain ''D'' satisfy the maximum principle if they achieve their maxima at the boundary of ''D''. The maximum principle enables one to obtain information about solutions of differential equations without any explicit knowledge of the solutions themselves. In particular, the maximum principle is a useful tool in the numerical approximation of solutions of ordinary and partial differential equations and in the determination of bounds for the errors in such approximations. In a simple two-dimensional case, consider a function of two variables such that :\frac+\frac=0. The weak maximum principle, in this setting, says that for any open precompact subset of the domain of , the maximum of on the closure of is achieved on the boundary of . The strong maximum principle says that, unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pontryagin's Maximum Principle
Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It states that it is necessary for any optimal control along with the optimal state trajectory to solve the so-called Hamiltonian system, which is a two-point boundary value problem, plus a maximum condition of the control Hamiltonian. These necessary conditions become sufficient under certain convexity conditions on the objective and constraint functions. The maximum principle was formulated in 1956 by the Russian mathematician Lev Pontryagin and his students, and its initial application was to the maximization of the terminal speed of a rocket. The result was derived using ideas from the classical calculus of variations. After a slight perturbation of the optimal control, one considers the first-order term of a Taylor expansion with respect to the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Maximum Principle
The Hopf maximum principle is a maximum principle in the theory of second order elliptic partial differential equations and has been described as the "classic and bedrock result" of that theory. Generalizing the maximum principle for harmonic functions which was already known to Gauss in 1839, Eberhard Hopf proved in 1927 that if a function satisfies a second order partial differential inequality of a certain kind in a domain of R''n'' and attains a maximum in the domain then the function is constant. The simple idea behind Hopf's proof, the comparison technique he introduced for this purpose, has led to an enormous range of important applications and generalizations. Mathematical formulation Let ''u'' = ''u''(''x''), ''x'' = (''x''1, ..., ''x''''n'') be a ''C''2 function which satisfies the differential inequality : Lu = \sum_ a_(x)\frac + \sum_i b_i(x)\frac \geq 0 in an open domain (connected open subset of R''n'') Ω, where the symmetric matrix ''a''''ij'' = ''a''''ji ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Functions
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, \frac + \frac + \cdots + \frac = 0 everywhere on . This is usually written as \nabla^2 f = 0 or \Delta f = 0 Etymology of the term "harmonic" The descriptor "harmonic" in the name "harmonic function" originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as "harmonics." Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit ''n''-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and, over time, "harmo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nina Uraltseva
Nina Nikolaevna Uraltseva (born 1934, ) is a Russian mathematician, a professor of mathematics and head of the department of mathematical physics at Saint Petersburg State University, and the editor-in-chief of the ''Proceedings of the St. Petersburg Mathematical Society''. Her specialty is the study of nonlinear partial differential equations.Faculty profile (in Russian), Saint Petersburg State University, retrieved 2015-06-08.. Nina Uraltseva was born on 24 May 1934 in Leningrad, USSR (currently St. Petersburg, Russia), to parents Nikolai Fedorovich Uraltsev, (an engineer) and Lidiya Ivanovna Zmanovskaya (a school physics teacher). She received a diploma in physics from Saint Petersburg State University (then known as Leningrad State University) in 1956. She earned a Ph.D. in 1960 from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Olga Ladyzhenskaya
Olga Aleksandrovna Ladyzhenskaya ( rus, Ольга Александровна Ладыженская, links=no, p=ˈolʲɡə ɐlʲɪˈksandrəvnə ɫɐˈdɨʐɨnskəɪ̯ə, a=Ru-Olga Aleksandrovna Ladyzhenskaya.wav; 7 March 1922 – 12 January 2004) was a Russian mathematician who worked on partial differential equations, fluid dynamics, and the finite-difference method for the Navier–Stokes equations. She received the Lomonosov Gold Medal in 2002. She authored more than two hundred scientific publications, including six monographs. Biography Ladyzhenskaya was born and grew up in the small town of Kologriv, the daughter of a mathematics teacher who is credited with her early inspiration and love of mathematics. The artist Gennady Ladyzhensky was her grandfather's brother, also born in this town. In 1937 her father, Aleksandr Ivanovich Ladýzhenski, was arrested by the NKVD and executed as an "enemy of the people". Ladyzhenskaya completed high school in 1939, unlike her ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lawrence C
Lawrence may refer to: Education Colleges and universities * Lawrence Technological University, a university in Southfield, Michigan, United States * Lawrence University, a liberal arts university in Appleton, Wisconsin, United States Preparatory & high schools * Lawrence Academy at Groton, a preparatory school in Groton, Massachusetts, United States * Lawrence College, Ghora Gali, a high school in Pakistan * Lawrence School, Lovedale, a high school in India * The Lawrence School, Sanawar, a high school in India Research laboratories * Lawrence Berkeley National Laboratory, United States * Lawrence Livermore National Laboratory, United States People * Lawrence (given name), including a list of people with the name * Lawrence (surname), including a list of people with the name * Lawrence (band), an American soul-pop group * Lawrence (judge royal) (died after 1180), Hungarian nobleman, Judge royal 1164–1172 * Lawrence (musician), Lawrence Hayward (born 1961), British musician * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eberhard Hopf
Eberhard Frederich Ferdinand Hopf (April 4, 1902 in Salzburg, Austria-Hungary – July 24, 1983 in Bloomington, Indiana, USA) was a German mathematician and astronomer, one of the founding fathers of ergodic theory and a pioneer of bifurcation theory who also made significant contributions to the subjects of partial differential equations and integral equations, fluid dynamics, and differential geometry. The Hopf maximum principle is an early result of his (1927) that is one of the most important techniques in the theory of elliptic partial differential equations. Biography Hopf was born in Salzburg, Austria-Hungary, but his scientific career was divided between Germany and the United States. He received his Ph.D. in mathematics in 1926 and his ''Habilitation'' in mathematical astronomy from the University of Berlin in 1929. In 1971, Hopf was the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louis Nirenberg
Louis Nirenberg (February 28, 1925 – January 26, 2020) was a Canadian-American mathematician, considered one of the most outstanding Mathematical analysis, mathematicians of the 20th century. Nearly all of his work was in the field of partial differential equations. Many of his contributions are now regarded as fundamental to the field, such as his maximum principle, strong maximum principle for second-order parabolic partial differential equations and the Newlander–Nirenberg theorem in complex geometry. He is regarded as a foundational figure in the field of geometric analysis, with many of his works being closely related to the study of complex analysis and differential geometry. Biography Nirenberg was born in Hamilton, Ontario to Ukrainian Jewish immigrants. He attended Baron Byng High School and McGill University, completing his Bachelor of Science, BS in both mathematics and physics in 1945. Through a summer job at the National Research Council (Canada), National R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eugenio Calabi
Eugenio Calabi (May 11, 1923 – September 25, 2023) was an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, specializing in differential geometry, partial differential equations and their applications. Early life and education Calabi was born in Milan, Italy on May 11, 1923, into a Jewish family. His sister was the journalist Tullia Zevi Calabi. In 1938, the family left Italy because of the racial laws, and in 1939 arrived in the United States. In the fall of 1939, aged only 16, Calabi enrolled at the Massachusetts Institute of Technology, studying chemical engineering. His studies were interrupted when he was drafted in the US military in 1943 and served during World War II. Upon his discharge in 1946, Calabi was able to finish his bachelor's degree under the G.I. Bill, and was a Putnam Fellow. He received a master's degree in mathematics from the University of Illinois Urbana-Champaign in 1947 and his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theorem
In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Modulus Principle
In mathematics, the maximum modulus principle in complex analysis states that if f is a holomorphic function, then the modulus , f, cannot exhibit a strict maximum that is strictly within the domain of f. In other words, either f is locally a constant function, or, for any point z_0 inside the domain of f there exist other points arbitrarily close to z_0 at which , f, takes larger values. Formal statement Let f be a holomorphic function on some bounded and connected open subset D of the complex plane \mathbb and taking complex values. If z_0 is a point in D such that :, f(z_0), \ge , f(z), for all z in some neighborhood of z_0, then f is constant on D. This statement can be viewed as a special case of the open mapping theorem, which states that a nonconstant holomorphic function maps open sets to open sets: If , f, attains a local maximum at z, then the image of a sufficiently small open neighborhood of z cannot be open, so f is constant. Related statement Suppose th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary (topology)
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the Closure (topology), closure of not belonging to the Interior (topology), interior of . An element of the boundary of is called a boundary point of . The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include \operatorname(S), \operatorname(S), and \partial S. Some authors (for example Willard, in ''General Topology'') use the term frontier instead of boundary in an attempt to avoid confusion with a Manifold#Manifold with boundary, different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, ''Metric Spaces'' by E. T. Copson uses the term boundary to refer to Felix Hausdorff, Hausdorff's border, which is defined as the intersection ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
