857:
386:
1804:
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,
1548:, and his lectures were reproduced in the form of duplicated notes. Loewner planned to write a detailed book on continuous groups based on these lecture notes, but the project was still in the formative stage at the time of his death."
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1591:
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586:
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683:
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1635:
1615:
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1508:
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1433:
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1208:
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657:
551:
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2053:
852:{\displaystyle f^{}(s,t)={\begin{cases}\displaystyle {\frac {f(s)-f(t)}{s-t}},&{\text{if }}s\neq t\\f'(s),&{\text{if }}s=t\end{cases}}}
1974:
237:
Karl
Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sigmund Löwner was a store owner.
2001:
1530:
is real analytic with an analytic continuation to the upper half plane that has a positive imaginary part on the upper plane. See
501:
1704:
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269:
241:
164:
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1348:
313:
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125:
117:
1762:
A reviewer said, "The reader is helped by illuminating examples and comments on relations with analysis and geometry."
477:
504:
with elements chosen from the subinterval; the 2 input parameters are assigned an output parameter consisting of an
2099:
1594:
1532:
109:
1354:
866:
257:
53:
1288:
933:
2058:
273:
395:. The boundary case of equality is attained if and only if the metric is flat and homothetic to the so-called
1781:
1015:
381:{\displaystyle \operatorname {sys} ^{2}\leq {\frac {2}{\sqrt {3}}}\operatorname {area} (\mathbb {T} ^{2}),}
1710:
249:
1696:
1128:
1671:
1644:
1556:"decided to revise and correct the original lecture notes and make them available in permanent form."
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176:
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99:
57:
1843:
Hiai, Fumio; Sano, Takashi (2012). "Loewner matrices of matrix convex and monotone functions".
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507:
252:
in the first highly nontrivial case of the third coefficient. The technique he introduced, the
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1970:
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113:
2016:
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218:
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1941:
1924:
1748:
1801:: À l'ombre de Loewner. (French) Ann. Sci. École Norm. Sup. (4) 5 (1972), 241–260.
316:, to the effect that every metric on the 2-torus satisfies the optimal inequality
1991:
289:
202:
186:
146:
1925:"Some classes of functions defined by difference or differential inequalities"
293:
198:
79:
1190:
In his fundamental 1934 paper, Loewner proved that for each positive integer
1866:
301:
1901:
476:
functions) associated with 2 input parameters consisting of (1) a real
17:
553:
be a real-valued function that is continuously differentiable on the
260:; it was used in the final solution of the Bieberbach conjecture by
248:. One of his central mathematical contributions is the proof of the
1857:
399:, i.e. torus whose group of deck transformations is precisely the
1990:
Loewner, Charles; Flanders, Harley; Protter, Murray H. (2008).
1415:. Most significantly, using this equivalence, he proved that
1641:(page 10). The distinction is made between an abstract group
845:
1759:
have equal left and right invariant densities (page 48).
480:
function on a subinterval of the real numbers and (2) an
1888:
2105:
Stanford
University Department of Mathematics faculty
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325:
1544:"During 1955 visit to Berkeley he gave a course on
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32:
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380:
217:(29 May 1893 – 8 January 1968) was an American
8:
1845:Journal of the Mathematical Society of Japan
1558:Charles Loewner: Theory of Continuous Groups
1408:{\displaystyle t_{1},\ldots ,t_{n}\in (a,b)}
920:{\displaystyle t_{1},\ldots ,t_{n}\in (a,b)}
1340:{\displaystyle L_{f}(t_{1},\ldots ,t_{n})}
985:{\displaystyle L_{f}(t_{1},\ldots ,t_{n})}
29:
1940:
1856:
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2054:MacTutor History of Mathematics Archive
1810:
256:, has had far-reaching implications in
1838:
1836:
403:spanned by the cube roots of unity in
1057:{\displaystyle (t_{1},\ldots ,t_{n})}
7:
2085:20th-century American mathematicians
1736:{\displaystyle J({\overset {u}{v}})}
240:Loewner received his Ph.D. from the
1703:. These linear transformations are
1677:
1650:
27:American mathematician (1893–1968)
25:
1180:{\displaystyle f^{}(t_{i},t_{j})}
1688:{\displaystyle {\mathfrak {g}},}
1661:{\displaystyle {\mathfrak {g}},}
307:
1942:10.1090/S0002-9904-1950-09405-1
1755:(page 46). Loewner proves that
264:in 1985. Loewner worked at the
2095:American mathematical analysts
1751:, which Loewner attributes to
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1:
1567:In Loewner's terminology, if
254:Loewner differential equation
244:in 1917 under supervision of
2038:Stanford memorial resolution
628:{\displaystyle s,t\in (a,b)}
418:{\displaystyle \mathbb {C} }
1967:Theory of Continuous Groups
478:continuously differentiable
312:In 1949 Loewner proved his
2121:
1533:Operator monotone function
308:Loewner's torus inequality
126:Loewner's torus inequality
110:Operator monotone function
1965:Loewner, Charles (1971).
1923:Loewner, Charles (1950).
1890:Mathematische Zeitschrift
1777:Schramm–Loewner evolution
1564:, and re-issued in 2008.
1083:{\displaystyle n\times n}
523:{\displaystyle n\times n}
445:or, more specifically, a
258:geometric function theory
208:
135:
2059:University of St Andrews
1560:(1971) was published by
274:University of Louisville
1782:Loop-erased random walk
288:. His students include
1737:
1697:linear transformations
1689:
1662:
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1611:
1587:
1586:{\displaystyle x\in S}
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1006:
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679:
653:
629:
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547:
524:
494:
470:
429:Loewner matrix theorem
419:
382:
1929:Bull. Amer. Math. Soc
1867:10.2969/jmsj/06420343
1772:Löwner-John ellipsoid
1738:
1690:
1668:and a realization of
1663:
1632:
1612:
1588:
1525:
1505:
1485:
1483:{\displaystyle (a,b)}
1451:
1430:
1410:
1349:positive semidefinite
1342:
1280:
1278:{\displaystyle (a,b)}
1246:
1225:
1205:
1182:
1120:
1118:{\displaystyle (i,j)}
1085:
1059:
1007:
987:
922:
854:
680:
654:
630:
583:
581:{\displaystyle (a,b)}
548:
525:
495:
471:
469:{\displaystyle C^{1}}
420:
383:
250:Bieberbach conjecture
233:Early life and career
130:Loewner–Heinz theorem
2090:Czech mathematicians
2045:Robertson, Edmund F.
1787:Systoles of surfaces
1743:(page 41). The term
1711:
1701:group representation
1672:
1645:
1621:
1601:
1571:
1514:
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1355:
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996:
934:
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560:
537:
508:
484:
453:
407:
323:
270:University of Prague
266:University of Berlin
242:University of Prague
177:Georg Alexander Pick
165:University of Prague
100:University of Prague
2043:O'Connor, John J.;
1829:2.2 Charles Loewner
678:{\displaystyle s,t}
286:Stanford University
282:Syracuse University
161:Syracuse University
157:Stanford University
1902:10.1007/BF01170633
1733:
1685:
1658:
1627:
1607:
1583:
1520:
1500:
1480:
1446:
1425:
1405:
1351:for any choice of
1337:
1275:
1241:
1220:
1200:
1177:
1115:
1080:
1064:is defined as the
1054:
1002:
982:
917:
849:
844:
785:
675:
649:
637:divided difference
625:
578:
543:
520:
490:
466:
415:
378:
284:and eventually at
2100:Jewish scientists
2049:"Charles Loewner"
1818:Loewner Biography
1745:invariant density
1728:
1630:{\displaystyle x}
1610:{\displaystyle S}
1554:Murray H. Protter
1546:continuous groups
1540:Continuous groups
1523:{\displaystyle f}
1503:{\displaystyle n}
1449:{\displaystyle n}
1428:{\displaystyle f}
1244:{\displaystyle n}
1223:{\displaystyle f}
1203:{\displaystyle n}
1005:{\displaystyle f}
831:
791:
780:
652:{\displaystyle f}
546:{\displaystyle f}
493:{\displaystyle n}
401:hexagonal lattice
397:equilateral torus
391:where sys is its
349:
348:
212:
211:
183:Doctoral students
137:Scientific career
114:Systolic geometry
16:(Redirected from
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2025:
2017:Deane Montgomery
2014:
2008:
2007:
1987:
1981:
1980:
1962:
1956:
1955:Preface, page ix
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1920:
1914:
1913:
1885:
1879:
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1815:
1747:is used for the
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1597:is performed on
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992:associated with
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945:
926:
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923:
918:
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314:torus inequality
278:Brown University
262:Louis de Branges
191:William J. Firey
172:Doctoral advisor
118:Loewner equation
71:
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30:
21:
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2119:
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2109:
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2042:
2034:
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1977:
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1917:
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1827:
1823:
1816:
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1768:
1709:
1708:
1670:
1669:
1643:
1642:
1619:
1618:
1599:
1598:
1569:
1568:
1550:Harley Flanders
1542:
1512:
1511:
1510:if and only if
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1491:
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1459:
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1437:
1417:
1416:
1377:
1358:
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1324:
1305:
1292:
1287:
1286:
1285:if and only if
1255:
1254:
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1212:
1211:
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1191:
1164:
1151:
1132:
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1126:
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937:
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870:
865:
864:
843:
842:
826:
806:
803:
802:
786:
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730:
695:
690:
689:
661:
660:
641:
640:
593:
592:
558:
557:
535:
534:
506:
505:
482:
481:
456:
451:
450:
447:linear operator
431:
405:
404:
360:
326:
321:
320:
310:
235:
221:. His name was
215:Charles Loewner
201:
197:
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128:
124:
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116:
112:
96:Alma mater
83:
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69:
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51:
45:
43:
35:
34:Charles Loewner
28:
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15:
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2032:External links
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2009:
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1982:
1976:0-262-06-041-8
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1935:(4): 308–319.
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439:linear algebra
435:Loewner matrix
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298:Adriano Garsia
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195:Adriano Garsia
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107:
106:Known for
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84:
74:
72:(aged 74)
68:8 January 1968
66:
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26:
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2013:
2010:
2005:
2003:9780486462929
1999:
1995:
1994:
1993:Dover reprint
1986:
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1978:
1972:
1968:
1961:
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1907:
1903:
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1854:
1850:
1846:
1839:
1837:
1833:
1830:
1825:
1822:
1819:
1814:
1811:
1803:
1800:
1797:
1796:
1792:
1788:
1785:
1783:
1780:
1778:
1775:
1773:
1770:
1769:
1765:
1763:
1760:
1758:
1754:
1750:
1746:
1725:
1722:
1714:
1706:
1702:
1699:that yield a
1698:
1682:
1655:
1640:
1624:
1604:
1596:
1580:
1577:
1574:
1565:
1563:
1562:The MIT Press
1559:
1555:
1551:
1547:
1539:
1537:
1535:
1534:
1517:
1497:
1474:
1471:
1468:
1457:
1443:
1422:
1399:
1396:
1393:
1387:
1382:
1378:
1374:
1371:
1368:
1363:
1359:
1350:
1329:
1325:
1321:
1318:
1315:
1310:
1306:
1297:
1293:
1269:
1266:
1263:
1252:
1238:
1217:
1197:
1188:
1169:
1165:
1161:
1156:
1152:
1140:
1133:
1109:
1106:
1103:
1092:
1077:
1074:
1071:
1046:
1042:
1038:
1035:
1032:
1027:
1023:
999:
974:
970:
966:
963:
960:
955:
951:
942:
938:
930:
911:
908:
905:
899:
894:
890:
886:
883:
880:
875:
871:
839:
836:
833:
823:
817:
810:
807:
799:
796:
793:
782:
776:
773:
770:
762:
756:
753:
747:
741:
731:
726:
720:
717:
714:
703:
696:
688:
687:
686:
672:
669:
666:
646:
638:
619:
616:
613:
607:
604:
601:
598:
589:
572:
569:
566:
556:
555:open interval
540:
531:
517:
514:
511:
503:
500:-dimensional
487:
479:
461:
457:
448:
444:
443:square matrix
440:
436:
428:
426:
402:
398:
394:
375:
367:
354:
351:
345:
341:
336:
331:
327:
319:
318:
317:
315:
305:
303:
299:
295:
291:
287:
283:
279:
275:
271:
267:
263:
259:
255:
251:
247:
243:
238:
232:
230:
228:
225:in Czech and
224:
220:
219:mathematician
216:
207:
204:
200:
196:
192:
188:
185:
181:
178:
175:
173:
169:
166:
162:
158:
155:
151:
148:
145:
141:
138:
134:
131:
127:
123:
122:Loewner order
119:
115:
111:
108:
104:
101:
98:
94:
90:
86:
81:
77:
67:
63:
59:
55:
42:
38:
31:
19:
2052:
2012:
1992:
1985:
1966:
1960:
1951:
1932:
1928:
1918:
1893:
1889:
1883:
1848:
1844:
1824:
1813:
1761:
1749:Haar measure
1744:
1695:in terms of
1638:
1637:is called a
1595:group action
1566:
1557:
1543:
1531:
1189:
928:
862:
636:
590:
532:
434:
432:
396:
390:
311:
239:
236:
226:
223:Karel Löwner
222:
214:
213:
153:Institutions
136:
70:(1968-01-08)
2080:1968 deaths
2075:1893 births
635:define the
290:Lipman Bers
229:in German.
227:Karl Löwner
203:Pao Ming Pu
187:Lipman Bers
147:Mathematics
88:Nationality
50:29 May 1893
2069:Categories
1793:References
1125:-entry is
294:Roger Horn
246:Georg Pick
199:Roger Horn
80:California
46:1893-05-29
1910:121439134
1875:117532480
1858:1007.2478
1705:Jacobians
1578:∈
1456:-monotone
1388:∈
1372:…
1319:…
1251:-monotone
1075:×
1036:…
964:…
900:∈
884:…
797:≠
774:−
754:−
608:∈
515:×
449:(of real
355:
337:≤
1766:See also
1707:denoted
1639:quantity
1490:for all
830:if
811:′
790:if
591:For any
530:matrix.
302:P. M. Pu
91:American
76:Stanford
2023:0315038
1617:, then
441:) is a
393:systole
58:Bohemia
18:Loewner
2000:
1973:
1908:
1873:
1593:and a
1093:whose
1091:matrix
927:, the
863:Given
502:vector
300:, and
143:Fields
82:, U.S.
1906:S2CID
1871:S2CID
1853:arXiv
1805:1974.
1998:ISBN
1971:ISBN
1552:and
1012:for
533:Let
437:(in
433:The
352:area
65:Died
54:Lány
40:Born
1937:doi
1898:doi
1863:doi
1458:on
1435:is
1347:is
1253:on
1230:is
685:as
659:at
639:of
328:sys
2071::
2057:,
2051:,
2047:,
2020:MR
1996:.
1969:.
1933:56
1931:.
1927:.
1904:.
1894:38
1892:.
1869:.
1861:.
1849:54
1847:.
1835:^
1536:.
1210:,
1187:.
588:.
425:.
304:.
296:,
292:,
280:,
276:,
272:,
268:,
78:,
56:,
2006:.
1979:.
1945:.
1939::
1912:.
1900::
1877:.
1865::
1855::
1731:)
1726:u
1723:v
1718:(
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1683:,
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1478:)
1475:b
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1469:a
1466:(
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1423:f
1403:)
1400:b
1397:,
1394:a
1391:(
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1303:(
1298:f
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1273:)
1270:b
1267:,
1264:a
1261:(
1239:n
1218:f
1198:n
1175:)
1170:j
1166:t
1162:,
1157:i
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1149:(
1144:]
1141:1
1138:[
1134:f
1113:)
1110:j
1107:,
1104:i
1101:(
1078:n
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1028:1
1024:t
1020:(
1000:f
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971:t
967:,
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948:(
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915:)
912:b
909:,
906:a
903:(
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891:t
887:,
881:,
876:1
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859:.
840:t
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834:s
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818:s
815:(
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800:t
794:s
783:,
777:t
771:s
766:)
763:t
760:(
757:f
751:)
748:s
745:(
742:f
732:{
727:=
724:)
721:t
718:,
715:s
712:(
707:]
704:1
701:[
697:f
673:t
670:,
667:s
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620:b
617:,
614:a
611:(
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602:,
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576:)
573:b
570:,
567:a
564:(
541:f
518:n
512:n
488:n
462:1
458:C
412:C
376:,
373:)
368:2
363:T
358:(
346:3
342:2
332:2
48:)
44:(
20:)
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