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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. Early life and career 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, Ad ... [...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 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. The Loewner semigroup generalizes the no ... [...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. Geometric ... [...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 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 ... [...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. Although it is commonly discussed on matrices (as a finite-dimensional case), the Loewner order is also well-defined on operators (an infinite-dimensional case) in the analogous way. 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 proper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pao Ming Pu
Pao Ming Pu (; August 1910 – February 22, 1988), was a Chinese 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 ''Frank''. Pu returned to mainland China in February 1951. (Katz '07) suggests that Pu may have been forc ... [...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 f ... [...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 for 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 normalization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adriano Garsia
Adriano Mario Garsia (20 August 1928 – 6 October 2024) was a Tunisian-born Italian American mathematician who worked in analysis, combinatorics, representation theory, and algebraic geometry. He was a student of Charles Loewner and 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. Life 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 gr ... [...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. Education and career 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 Vien ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagonal Lattice
The hexagonal lattice (sometimes called triangular lattice) is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° and are of equal lengths, : , \mathbf a_1, = , \mathbf a_2, = a. The reciprocal lattice of the hexagonal lattice is a hexagonal lattice in reciprocal space with orientation changed by 90° and primitive lattice vectors of length : g=\frac. Honeycomb point set The honeycomb point set is a special case of the hexagonal lattice with a two-atom basis. The centers of the hexagons of a honeycomb form a hexagonal lattice, and the honeycomb point set can be seen as the union of two offset hexagonal lattices. In nature, carbon atoms of the two-dimensional material graphene are arranged in a honeycomb point set. Crystal classes The ''hexagonal lattice'' class names, Schönflies notation, Hermann-Mauguin notation, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brown University
Brown University is a Private university, private Ivy League research university in Providence, Rhode Island, United States. It is the List of colonial colleges, seventh-oldest institution of higher education in the US, founded in 1764 as the ''College in the English Colony of Rhode Island and Providence Plantations''. One of nine colonial colleges chartered before the American Revolution, it was the first US college to codify that admission and instruction of students was to be equal regardless of the religious affiliation of students. The university is home to the oldest applied mathematics program in the country and oldest engineering program in the Ivy League. It was one of the early doctoral-granting institutions in the U.S., adding masters and doctoral studies in 1887. In 1969, it adopted its Open Curriculum (Brown University), Open Curriculum after student lobbying, which eliminated mandatory Curriculum#Core curriculum, general education distribution requirements. In 197 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
