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Birkhoff–Kellogg Invariant-direction Theorem
In functional analysis, the Birkhoff–Kellogg invariant-direction theorem, named after G. D. Birkhoff and O. D. Kellogg, is a generalization of the Brouwer fixed-point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Egbertus Jan Brouwer, L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a nonempty compactness, compact convex set to itself, the .... The theorem states that: Let ''U'' be a bounded open neighborhood of 0 in an infinite-dimensional normed linear space ''V'', and let ''F'':∂''U'' → ''V'' be a compact map satisfying , , ''F''(''x''), , ≥ α for some α > 0 for all ''x'' in ∂''U''. Then ''F'' has an invariant direction, ''i.e.'', there exist some ''x''o and some ''λ'' > 0 satisfying ''x''o = ''λF''(''x''o). The Birkhoff–Kellogg theorem and its generalizations by Schauder and Leray have applications to partial differential equations. References Theorems in functional ana ... [...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 (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. 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, for example, continuous function, continuous or unitary operator, unitary 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 v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George David Birkhoff
George David Birkhoff (March21, 1884November12, 1944) was one of the top American mathematicians of his generation. He made valuable contributions to the theory of differential equations, dynamical systems, the four-color problem, the three-body problem, and general relativity. Today, Birkhoff is best remembered for the ergodic theorem. The George D. Birkhoff House, his residence in Cambridge, Massachusetts, has been designated a National Historic Landmark. Early life He was born in Overisel Township, Michigan, the son of two Dutch immigrants, David Birkhoff, who arrived in the United States in 1870, and Jane Gertrude Droppers. Birkhoff's father worked as a physician in Chicago while he was a child. From 1896 to 1902, he would attend the Lewis Institute as a teenager. Career Birkhoff was part of a generation of American mathematicians who were the first to study entirely within the United States and not participate in academics within Europe. Following his time at the Lewis Insti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oliver Dimon Kellogg
Oliver Dimon Kellogg (10 July 1878 – 27 August 1932) was an American mathematician. His father, Day Otis Kellogg, was a professor of literature at the University of Kansas and editor of the American edition of the ''Encyclopædia Britannica''. In 1895 Oliver Kellogg began his undergraduate study at Princeton University, where he earned his master's degree in 1900. With a John S. Kennedy stipend he first studied at the Humboldt University of Berlin and then in 1901/1902 at Georg-August-Universität Göttingen. At Göttingen in 1902 he earned his PhD with a thesis ''Zur Theorie der Integralgleichungen und des Dirichlet'schen Prinzips'' under the direction of David Hilbert. After completing his thesis, Kellogg became an instructor at Princeton and from 1905 at the University of Missouri, where he became a professor in 1910. In World War I he was a scientific advisor at the Coast Guard Academy in New London, Connecticut, where he worked on submarine detection. Kellogg became a l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brouwer Fixed-point Theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Egbertus Jan Brouwer, L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a nonempty compactness, compact convex set to itself, there is a point x_0 such that f(x_0)=x_0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed Disk (mathematics), disk D to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset K of Euclidean space to itself. Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem, the invariance of dimension and the Borsuk–Ulam theorem. This gives it a place ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Juliusz Schauder
Juliusz Paweł Schauder (; 21 September 1899 – September 1943) was a Polish mathematician known for his work in functional analysis, partial differential equations and mathematical physics. Life and career Born on 21 September 1899 in Lwów to a lawyer father of Jewish descent, he was drafted into the Austro-Hungarian Army right after his graduation from school and saw action on the Italian front. He was captured and imprisoned in Italy. He entered the university in Lwów in 1919 and received his doctorate in 1923. He got no appointment at the university and continued his research while working as teacher at a secondary school. Due to his outstanding results, he obtained a scholarship in 1932 that allowed him to spend several years in Leipzig and, especially, Paris. In Paris he started a very successful collaboration with Jean Leray. Around 1935 Schauder obtained the position of a senior assistant in the University of Lwów. Schauder, along with Stanisław Mazur, was an I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Leray
Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. Life and career He was born in Chantenay-sur-Loire (today part of Nantes). He studied at École Normale Supérieure from 1926 to 1929. He received his Ph.D. in 1933. In 1934 Leray published an important paper that founded the study of weak solutions of the Navier–Stokes equations. In the same year, he and Juliusz Schauder discovered a topological invariant, now called the Leray–Schauder degree, which they applied to prove the existence of solutions for partial differential equations lacking uniqueness. From 1938 to 1939 he was professor at the University of Nancy. He did not join the Bourbaki group, although he was close with its founders. His main work in topology was carried out while he was a prisoner of war in a camp in Edelbach, Austria from 1940 to 1945. He concealed his expertise on differential equations, fearing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |