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Bony–Brezis Theorem
In mathematics, the Bony–Brezis theorem, due to the French mathematicians Jean-Michel Bony and Haïm Brezis, gives necessary and sufficient conditions for a closed subset of a manifold to be invariant under the flow defined by a vector field, namely at each point of the closed set the vector field must have non-positive inner product with any exterior normal vector to the set. A vector is an ''exterior normal'' at a point of the closed set if there is a real-valued continuously differentiable function maximized locally at the point with that vector as its derivative at the point. If the closed subset is a smooth submanifold with boundary, the condition states that the vector field should not point outside the subset at boundary points. The generalization to non-smooth subsets is important in the theory of partial differential equations. The theorem had in fact been previously discovered by Mitio Nagumo in 1942 and is also known as the Nagumo theorem. Statement Let ''F'' be clos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Michel Bony
Jean-Michel Bony (born 1 February 1942 in Paris) is a French mathematician, specializing in mathematical analysis. He is known for his work on microlocal analysis and pseudodifferential operators. Education and career Bony completed his undergraduate and graduate studies at the École Normale Supérieure, where he received his Ph.D in 1972 with thesis advisor Gustave Choquet. Bony became a professor at the University of Paris-Sud and is now a professor at the École Polytechnique. His doctoral students include Jean-Yves Chemin. Research Bony's research deals with microlocal analysis, partial differential equations and potential theory. In 1981 he published important results on paradifferential operators, extending the theory of pseudifferential operators published by Ronald Coifman and Yves Meyer in 1979. Bony applied his theory to the propagation of singularities in solutions of semilinear wave equations. Recognition In 1980, Bony received the Prix Paul Doistau–Émile Blutet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haïm Brezis
Haïm Brezis (born 1 June 1944) is a French mathematician, who mainly works in functional analysis and partial differential equations. Biography Born in Riom-ès-Montagnes, Cantal, France. Brezis is the son of a Romanian immigrant father, who came to France in the 1930s, and a Jewish mother who fled from the Netherlands. His wife, Michal Govrin, a native Israeli, works as a novelist, poet, and theater director. Brezis received his Ph.D. from the University of Paris in 1972 under the supervision of Gustave Choquet. He is currently a professor at the Pierre and Marie Curie University and a visiting distinguished professor at Rutgers University. He is a member of the Academia Europaea (1988) and a foreign associate of the United States National Academy of Sciences (2003). In 2012 he became a fellow of the American Mathematical Society. He holds honorary doctorates from several universities including National Technical University of Athens. Brezis is listed as an ISI highly cited res ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necessary And Sufficient
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of is guaranteed by the truth of (equivalently, it is impossible to have without ). Similarly, is sufficient for , because being true always implies that is true, but not being true does not always imply that is not true. In general, a necessary condition is one that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition. The assertion that a statement is a "necessary ''and'' sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false. In ordinary English (also natural language) "necessary" and "sufficient" indicate relations betw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set. The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, and Anosov flows. Flows may also be defined for systems of random variables and stochastic processes, and occur in the study of ergodic dynamical systems. The most celebrated of these is perhaps the Bernoulli flow. Formal definition A flow on a set is a group action of the additive group of real numbers on . More explicitl ... [...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] |
Mitio Nagumo
Mitio (Michio) Nagumo ( ja, 南雲 道夫; May 7, 1905 – February 6, 1995) was a Japanese mathematician, who specialized in the theory of differential equations. He gave the first necessary and sufficient condition for positive invariance of closed sets under the flow induced by ordinary differential equations ( Nagumo/Bony-Brezis theorem). Biography Mitio Nagumo graduated from the Department of Mathematics at the Imperial University of Tokyo in March 1928. In March 1931 he was appointed Lecturer in the Faculty of Technology at the Imperial University of Kyushu. In February 1932 he left Japan for an academic visit to Göttingen, where he remained for two years. Upon his return from Göttingen in March 1934, he was appointed Lecturer in the Department of Mathematics at the Imperial University of Osaka, and was promoted to Associate Professor in September that year, becoming Professor in the Faculty of Science in March 1936. In March 1937 Nagumo received a Doctor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipschitz Continuous
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the ''Lipschitz constant'' of the function (or '' modulus of uniform continuity''). For instance, every function that has bounded first derivatives is Lipschitz continuous. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. We have the following chain of strict inclus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. Name Integral curves are known by various other names, depending on the nature and interpretation of the differential equation or vector field. In physics, integral curves for an electric field or magnetic field are known as field lines, and integral curves for the velocity field of a fluid are known as streamlines. In dynamical systems, the integral curves for a differential equation that governs a system are referred to as trajectories or orbits. Definition Suppose that F is a static vector field, that is, a vector-valued function with Cartesian coordinates (''F''1,''F''2,...,''F''''n''), and that x(''t'') is a parametric curve with Cartesian coordinates (''x''1(''t''),''x''2(''t''),...,''x''''n''(''t'')). Then x(''t'') is an integral curve of F if it is a solution of the autonomous system of ordinary differential equations, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One-sided Derivatives
In calculus, a branch of mathematics, the notions of one-sided differentiability and semi-differentiability of a real-valued function ''f'' of a real variable are weaker than differentiability. Specifically, the function ''f'' is said to be right differentiable at a point ''a'' if, roughly speaking, a derivative can be defined as the function's argument ''x'' moves to ''a'' from the right, and left differentiable at ''a'' if the derivative can be defined as ''x'' moves to ''a'' from the left. One-dimensional case In mathematics, a left derivative and a right derivative are derivatives (rates of change of a function) defined for movement in one direction only (left or right; that is, to lower or higher values) by the argument of a function. Definitions Let ''f'' denote a real-valued function defined on a subset ''I'' of the real numbers. If is a limit point of and the one-sided limit :\partial_+f(a):=\lim_\frac exists as a real number, then ''f'' is called right differe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
