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Loewner
Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German. Karl Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sigmund Löwner was a store owner. Loewner received his Ph.D. from the University of Prague in 1917 under supervision of Georg Pick. One of his central mathematical contributions is the proof of the Bieberbach conjecture in the first highly nontrivial case of the third coefficient. The technique he introduced, the Loewner differential equation, has had far-reaching implications in geometric function theory; it was used in the final solution of the Bieberbach conjecture by Louis de Branges in 1985. Loewner worked at the University of Berlin, University of Prague, University of Louisville, Brown University, Syracuse University and eventually at Stanford University. His students include Lipman Bers, Roger Horn, Adriano Garsia, and P. M. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loewner Differential Equation
In mathematics, the Loewner differential equation, or Loewner equation, is an ordinary differential equation discovered by Charles Loewner in 1923 in complex analysis and geometric function theory. Originally introduced for studying slit mappings (conformal mappings of the open disk onto the complex plane with a curve joining 0 to ∞ removed), Loewner's method was later developed in 1943 by the Russian mathematician Pavel Parfenevich Kufarev (1909–1968). Any family of domains in the complex plane that expands continuously in the sense of Carathéodory to the whole plane leads to a one parameter family of conformal mappings, called a Loewner chain, as well as a two parameter family of Holomorphic function, holomorphic univalent function, univalent self-mappings of the unit disk, called a Loewner semigroup. This semigroup corresponds to a time dependent holomorphic vector field on the disk given by a one parameter family of holomorphic functions on the disk with positive real part. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loewner's Torus Inequality
In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus. Statement In 1949 Charles Loewner proved that every metric on the 2-torus \mathbb T^2 satisfies the optimal inequality : \operatorname^2 \leq \frac \operatorname(\mathbb T^2), where "sys" is its systole, i.e. least length of a noncontractible loop. The constant appearing on the right hand side is the Hermite constant \gamma_2 in dimension 2, so that Loewner's torus inequality can be rewritten as : \operatorname^2 \leq \gamma_2\;\operatorname(\mathbb T^2). The inequality was first mentioned in the literature in . Case of equality The boundary case of equality is attained if and only if the metric is flat and homothetic to the so-called ''equilateral torus'', i.e. torus whose group of deck transformations is precisely the hexagonal lattice spanned by the cube roots of unity in \mathbb C. Alter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Systolic Geometry
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slower-paced Introduction to systolic geometry. The notion of systole The ''systole'' of a compact metric space ''X'' is a metric invariant of ''X'', defined to be the least length of a noncontractible loop in ''X'' (i.e. a loop that cannot be contracted to a point in the ambient space ''X''). In more technical language, we minimize length over free loops representing nontrivial conjugacy classes in the fundamental group of ''X''. When ''X'' is a graph, the invariant is usually referred to as the girth, ever since the 1947 article on girth by W. T. Tutte. Possibly inspired by Tutte's article, Loewner started thinking about systolic questions on surfaces in the la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pao Ming Pu
Pao Ming Pu (the form of his name he used in Western languages, although the Wade-Giles transliteration would be Pu Baoming; ; August 1910 – February 22, 1988), was a mathematician born in Jintang County, Sichuan, China.. He was a student of Charles Loewner and a pioneer of systolic geometry, having proved what is today called Pu's inequality for the real projective plane, following Loewner's proof of Loewner's torus inequality. He later worked in the area of fuzzy mathematics. He spent much of his career as professor and chairman of the department of mathematics at Sichuan University. Biography Pu received his Ph.D. at Syracuse University in 1950 under the supervision of Charles Loewner, resulting in the publication in 1952 of the seminal paper containing both Pu's inequality for the real projective plane and Loewner's torus inequality. .99 The listing at the Mathematics Genealogy Project indicates that his first name, according to Syracuse University records, was ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loewner Order
In mathematics, Loewner order is the partial order defined by the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/convex scalar functions to monotone and concave/convex Hermitian valued functions. These functions arise naturally in matrix and operator theory and have applications in many areas of physics and engineering. Definition Let ''A'' and ''B'' be two Hermitian matrices of order ''n''. We say that ''A ≥ B'' if ''A'' − ''B'' is positive semi-definite. Similarly, we say that ''A > B'' if ''A'' − ''B'' is positive definite. Properties When ''A'' and ''B'' are real scalars (i.e. ''n'' = 1), the Loewner order reduces to the usual ordering of R. Although some familiar properties of the usual order of R are also valid when ''n'' ≥ 2, several properties are no longer valid. For instance, the comparability of two matrices may no longer be valid. In fact, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipman Bers
Lipman Bers ( Latvian: ''Lipmans Berss''; May 22, 1914 – October 29, 1993) was a Latvian-American mathematician, born in Riga, who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups. He was also known for his work in human rights activism.. Biography Bers was born in Riga, then under the rule of the Russian Czars, and spent several years as a child in Saint Petersburg; his family returned to Riga in approximately 1919, by which time it was part of independent Latvia. In Riga, his mother was the principal of a Jewish elementary school, and his father became the principal of a Jewish high school, both of which Bers attended, with an interlude in Berlin while his mother, by then separated from his father, attended the Berlin Psychoanalytic Institute. After high school, Bers studied at the University of Zurich for a year, but had to return to Riga again because of the difficulty of transferring money from Latvia in the international fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Branges' Theorem
In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was posed by and finally proven by . The statement concerns the Taylor coefficients a_n of a univalent function, i.e. a one-to-one holomorphic function that maps the unit disk into the complex plane, normalized as is always possible so that a_0=0 and a_1=1. That is, we consider a function defined on the open unit disk which is holomorphic and injective ('' univalent'') with Taylor series of the form :f(z)=z+\sum_ a_n z^n. Such functions are called ''schlicht''. The theorem then states that : , a_n, \leq n \quad \textn\geq 2. The Koebe function (see below) is a function in which a_n=n for all n, and it is schlicht, so we cannot find a stricter limit on the absolute value of the nth coefficient. Schlicht functions The normalizations : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georg Alexander Pick
Georg Alexander Pick (10 August 1859 – 26 July 1942) was an Austrian Jewish mathematician who was murdered during The Holocaust. He was born in Vienna to Josefa Schleisinger and Adolf Josef Pick and died at Theresienstadt concentration camp. Today he is best known for Pick's theorem for determining the area of lattice polygons. He published it in an article in 1899; it was popularized when Hugo Dyonizy Steinhaus included it in the 1969 edition of Mathematical Snapshots. Pick studied at the University of Vienna and defended his Ph.D. in 1880 under Leo Königsberger and Emil Weyr. After receiving his doctorate he was appointed an assistant to Ernst Mach at the Charles-Ferdinand University in Prague. He became a lecturer there in 1881. He took a leave from the university in 1884 during which he worked with Felix Klein at the University of Leipzig. Other than that year, he remained in Prague until his retirement in 1927 at which time he returned to Vienna. Pick headed the committ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adriano Garsia
Adriano Mario Garsia (born 20 August 1928) is a Tunisian-born Italian American mathematician who works in analysis, combinatorics, representation theory, and algebraic geometry. He is a student of Charles Loewner and has published work on representation theory, symmetric functions, and algebraic combinatorics. He and Mark Haiman made the N!_conjecture. He is also the namesake of the Garsia–Wachs algorithm for optimal binary search trees, which he published with his student Michelle L. Wachs in 1977. Born to Italian Tunisians in Tunis on 20 August 1928, Garsia moved to Rome in 1946. , he had 36 students and at least 200 descendants, according to the data at the Mathematics Genealogy Project. He was on the faculty of the University of California, San Diego. He retired in 2013 after 57 years at UCSD as a founding member of the Mathematics Department. At his 90 Birthday Conference in 2019, it was notable that he was the oldest principal investigator of a grant from the Nati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roger Horn
Roger Alan Horn (born January 19, 1942) is an American mathematician specializing in matrix analysis. He was research professor of mathematics at the University of Utah. He is known for formulating the Bateman–Horn conjecture with Paul T. Bateman on the density of prime number values generated by systems of polynomials. His books ''Matrix Analysis'' and ''Topics in Matrix Analysis'', co-written with Charles R. Johnson, are standard texts in advanced linear algebra. Career Roger Horn graduated from Cornell University with high honors in mathematics in 1963, after which he completed his PhD at Stanford University in 1967. Horn was the founder and chair of the Department of Mathematical Sciences at Johns Hopkins University from 1972 to 1979. As chair, he held a series of short courses for a monograph series published by the Johns Hopkins Press. He invited Gene Golub and Charles Van Loan to write a monograph, which later became the seminal ''Matrix Computations'' text book. He late ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Function Theory
Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem. Topics in geometric function theory The following are some of the most important topics in geometric function theory: Conformal maps A conformal map is a function which preserves angles locally. In the most common case the function has a domain and range in the complex plane. More formally, a map, : f: U \rightarrow V\qquad with U,V \subset \mathbb^n is called conformal (or angle-preserving) at a point u_0 if it preserves oriented angles between curves through u_0 with respect to their orientation (i.e., not just the magnitude of the angle). Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. Quasiconformal maps In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louis De Branges
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] |
