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Nevanlinna–Pick Interpolation
In complex analysis, given ''initial data'' consisting of n points \lambda_1, \ldots, \lambda_n in the complex unit disc \mathbb and ''target data'' consisting of n points z_1, \ldots, z_n in \mathbb, the Nevanlinna–Pick interpolation problem is to find a holomorphic function \varphi that interpolates the data, that is for all i \in \, :\varphi(\lambda_i) = z_i, subject to the constraint \left\vert \varphi(\lambda) \right\vert \le 1 for all \lambda \in \mathbb. Georg Pick and Rolf Nevanlinna solved the problem independently in 1916 and 1919 respectively, showing that an interpolating function exists if and only if a matrix defined in terms of the initial and target data is positive semi-definite. Background The Nevanlinna–Pick theorem represents an n-point generalization of the Schwarz lemma. The invariant form of the Schwarz lemma states that for a holomorphic function f:\mathbb\to\mathbb, for all \lambda_1, \lambda_2 \in \mathbb, : \left, \frac\ \leq \left, \frac\. Setting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear engineering, nuclear, aerospace engineering, aerospace, mechanical engineering, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is Analyticity of holomorphic functions, analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louis De Branges De Bourcia
Louis de Branges de Bourcia (born August 21, 1932) is a French-American mathematician. He is the Edward C. Elliott Distinguished Professor of Mathematics at Purdue University in West Lafayette, Indiana. He is best known for proving the long-standing Bieberbach conjecture in 1984, now called de Branges's theorem. He claims to have proved several important conjectures in mathematics, including the generalized Riemann hypothesis. Born to American parents who lived in Paris, de Branges moved to the US in 1941 with his mother and sisters. His native language is French. He did his undergraduate studies at the Massachusetts Institute of Technology (1949–53), and received a PhD in mathematics from Cornell University (1953–57). His advisors were Wolfgang Fuchs and then-future Purdue colleague Harry Pollard. He spent two years (1959–60) at the Institute for Advanced Study and another two (1961–62) at the Courant Institute of Mathematical Sciences. He was appointed to Purdue in 196 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Allen Tannenbaum
Allen Robert Tannenbaum (born January 25, 1953) is an American/Israeli applied mathematician and presently Distinguished Professor of Computer Science and Applied Mathematics & Statistics at the State University of New York at Stony Brook. He is also Visiting Investigator of Medical Physics at Memorial Sloan Kettering Cancer Center in New York City. He has held a number of other positions in the United States, Israel, and Canada including the Bunn Professorship of Electrical and Computer Engineering and Interim Chair, and Senior Scientist at the Comprehensive Cancer Center at the University of Alabama, Birmingham. He received his B.A. from Columbia University in 1973 and Ph.D. with thesis advisor Heisuke Hironaka at the Harvard University in 1976. Tannenbaum has done research in numerous areas including robust control, computer vision, and biomedical imaging, having almost 500 publications. He pioneered the field of robust control with the solution of the gain margin and phase ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robust Control
In control theory, robust control is an approach to controller design that explicitly deals with uncertainty. Robust control methods are designed to function properly provided that uncertain parameters or disturbances are found within some (typically compact) set. Robust methods aim to achieve robust performance and/or stability in the presence of bounded modelling errors. The early methods of Bode and others were fairly robust; the state-space methods invented in the 1960s and 1970s were sometimes found to lack robustness, prompting research to improve them. This was the start of the theory of robust control, which took shape in the 1980s and 1990s and is still active today. In contrast with an adaptive control policy, a robust control policy is static, rather than adapting to measurements of variations, the controller is designed to work assuming that certain variables will be unknown but bounded. (Section 1.5) In German; an English version is also available Criteria for robustn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley Transform
In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators . Real homography The Cayley transform is an automorphism of the real projective line that permutes the elements of in sequence. For example, it maps the positive real numbers to the interval ˆ’1, 1 Thus the Cayley transform is used to adapt Legendre polynomials for use with functions on the positive real numbers with Legendre rational functions. As a real homography, points are described with projective coordinates, and the mapping is : ,\ 1= \left frac ,\ 1\right\thicksim - 1, \ x + 1= ,\ 1begin1 & 1 \\ -1 & 1 \end . Complex homography On the Riemann sphere, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus. If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a ''toroid'', as in a square toroid. Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings. Eyeglass lenses that combine spherical and cylindrical correction are toric lenses. A torus should not be confused with a ''solid torus'', which is formed by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gábor Szegő
Gábor SzegÅ‘ () (January 20, 1895 – August 7, 1985) was a Hungarian-American mathematician. He was one of the foremost mathematical analysts of his generation and made fundamental contributions to the theory of orthogonal polynomials and Toeplitz matrices building on the work of his contemporary Otto Toeplitz. Life SzegÅ‘ was born in Kunhegyes, Austria-Hungary (today Hungary), into a Jewish family as the son of Adolf SzegÅ‘ and Hermina Neuman.Biography on the homepage of Kunhegyes (in Hungarian) He married the chemist Anna Elisabeth Neményi in 1919, with whom he had two children. In 1912 he started studies in at the |
Reproducing Kernel Hilbert Space
In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions f and g in the RKHS are close in norm, i.e., \, f-g\, is small, then f and g are also pointwise close, i.e., , f(x)-g(x), is small for all x. The converse does not need to be true. Informally, this can be shown by looking at the supremum norm: the sequence of functions \sin^n (x) converges pointwise, but do not converge uniformly i.e. do not converge with respect to the supremum norm (note that this is not a counterexample because the supremum norm does not arise from any inner product due to not satisfying the parallelogram law). It is not entirely straightforward to construct a Hilbert space of functions which is not an RKHS. Some examples, however, have been found. Note that ''L''2 spaces are not Hilbert spaces of functions (and hence not RKH ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy Space
In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In real analysis Hardy spaces are certain spaces of distributions on the real line, which are (in the sense of distributions) boundary values of the holomorphic functions of the complex Hardy spaces, and are related to the ''Lp'' spaces of functional analysis. For 1 ≤ ''p'' < ∞ these real Hardy spaces ''Hp'' are certain s of ''Lp'', while for ''p'' < 1 the ''Lp'' spaces have some undesirable properties, and the Hardy spaces are much better behaved. There are also higher-dimensional generalizations, consisting of certain holomorphic functions on |
Ciprian Foias
Ciprian Ilie Foiaș (20 July 1933 – 22 March 2020) was a Romanian-American mathematician. He was awarded the Norbert Wiener Prize in Applied Mathematics in 1995, for his contributions in operator theory. Education and career Born in Reșița, Romania, Foias studied mathematics at the University of Bucharest. He completed his dissertation in 1957, but was not allowed to defend his thesis by the Communist government until 1962. He received his doctorate in 1962 under supervision of Miron Nicolescu. Foias defected to France following his lecture at the International Congress of Mathematicians in 1978. He later emigrated to the United States. Foias taught at his alma mater (1966–1979), Paris-Sud 11 University (1979–1983), and Indiana University (1983 until retirement). Beginning in 2000, he was a teacher and researcher at Texas A&M University, where he was a Distinguished Professor. The Foias constant is named after him. Foias is listed as an ISI highly cited researcher. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Béla Szőkefalvi-Nagy
Béla Szőkefalvi-Nagy (29 July 1913, Kolozsvár – 21 December 1998, Szeged) was a Hungarian mathematician. His father, Gyula Szőkefalvi-Nagy was also a famed mathematician. Szőkefalvi-Nagy collaborated with Alfréd Haar and Frigyes Riesz, founders of the Szegedian school of mathematics. He contributed to the theory of Fourier series and approximation theory. His most important achievements were made in functional analysis, especially, in the theory of Hilbert space operators. He was editor-in-chief of the ''Zentralblatt für Mathematik'', the ''Acta Scientiarum Mathematicarum'', and the ''Analysis Mathematica''. He was awarded the Kossuth Prize in 1953, along with his co-author F. Riesz, for his book ''Leçons d'analyse fonctionnelle.'' He was awarded the Lomonosov Medal in 1979. The Béla Szőkefalvi-Nagy Medal honoring his memory is awarded yearly by Bolyai Institute. His books * Béla Szőkefalvi-Nagy: ''Spektraldarstellung linearer Transformationen des Hilbertschen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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