Charles Loewner

Charles Loewner
Charles Loewner in '63
Born (1893-05-29)29 May 1893
Lány, Bohemia
Died 8 January 1968(1968-01-08) (aged 74)
Stanford, California
Nationality American
Alma mater Karl-Ferdinands-Universität
Scientific career
Fields Mathematics
Institutions Stanford University
Syracuse University
University of Prague
Doctoral advisor Georg Alexander Pick
Doctoral students Lipman Bers
William J. Firey
Adriano Garsia
Roger Horn
Pao Ming Pu

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.[1][2]

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. Pu.

Loewner's torus inequality

In 1949 Loewner proved his torus inequality, to the effect that every metric on the 2-torus satisfies the optimal inequality

where sys is its systole. 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 .

Loewner matrix theorem

The Loewner matrix (in linear algebra) is a square matrix or, more specifically, a linear operator (of real functions) associated with 2 input parameters consisting of (1) a real continuously differentiable function on a subinterval of the real numbers and (2) an -dimensional vector with elements chosen from the subinterval; the 2 input parameters are assigned an output parameter consisting of an matrix.[3]

Let be a real-valued function that is continuously differentiable on the open interval .

For any define the divided difference of at as

if
                  , if .

Given , the Loewner matrix associated with for is defined as the matrix whose -entry is .

In his fundamental 1934 paper, Loewner proved that for each positive integer , is -monotone on if and only if is positive semidefinite for any choice of .[3][4][5] Most significantly, using this equivalence, he proved that is -monotone on for all if and only if is real analytic with an analytic continuation to the upper half plane that has a positive imaginary part on the upper plane.

Book by Loewner

  • Loewner, C.: Theory of continuous groups. Notes by H. Flanders and M. Protter. Mathematicians of Our Time 1, The MIT Press, Cambridge, Mass.—London, 1971.
    • Dover reprint. 2008.

See also

References

  • Berger, Marcel: À l'ombre de Loewner. (French) Ann. Sci. École Norm. Sup. (4) 5 (1972), 241–260.
  • Loewner, Charles; Nirenberg, Louis: Partial differential equations invariant under conformal or projective transformations. Contributions to analysis (a collection of papers dedicated to Lipman Bers), pp. 245–272. Academic Press, New York, 1974.
  1. Loewner Biography
  2. 2.2 Charles Loewner
  3. 1 2 Hiai, Fumio; Sano, Takashi (2012). "Loewner matrices of matrix convex and monotone functions". Journal of the Mathematical Society of Japan. 54 (2): 343–364. arXiv:1007.2478. doi:10.2969/jmsj/06420343.
  4. Löwner, Karl (1934). "Über monotone Matrixfunktionen". Mathematische Zeitschrift. 38 (1): 177–216. doi:10.1007/BF01170633.
  5. Loewner, Charles (1950). "Some classes of functions defined by difference or differential inequalities". Bull. Amer. Math. Soc. 56: 308–319. doi:10.1090/S0002-9904-1950-09405-1.
  • Stanford memorial resolution
  • Charles Loewner at the Mathematics Genealogy Project
  • O'Connor, John J.; Robertson, Edmund F., "Charles Loewner", MacTutor History of Mathematics archive, University of St Andrews .
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.