Lothar Collatz—On the occasion of his 75th birthday

Lothar Collatz-On the Occasion of his 75th Birthday L. Elmer

Fakultiit ftir Mathematik t’niversitiit Bielefeld Postfach 8640 4800 Bielefeld, West Gennuny K. P. Hadeler

Lehrstuhl ftir Biomuthemutik der Universitiit Tiibingen Auf der Morgenstelle 10 7400 Tiibingen, West Gemany

Submitted

by Hans Schneider

Lothar Collatz was born on July 6, 1910 in Amsberg, Westphalia. Because his father worked as a surveyor in government service, the family moved quite often. Lothar Collatz graduated from high school in Stettin in Pomerania (now Szeczin, Poland). Between 1928 and 1933 he studied at the universities of Gottingen, Berlin, Greifswald, and Munich. In Berlin, he attended the lectures of Issai Schur, together with Alfred and Richard Brauer, Walter Ledermann, and Helmut Wielant, who were to become well-known algebraists. At Gottingen he studied with Richard v. Mises, Richard Courant, and Edmund Landau. At that university Carl Runge had founded in about 1904 the first group in applied mathematics and numerical analysis in Germany, and Gottingen continued to offer an active program in these areas. Collatz soon came into contact with problems in numerical analysis and mathematical physics. It would seem that he himself felt attached to the work of Runge. Collatz earned his doctoral degree in Berlin 1935, and he obtained his habilitation in 1937 at the Technische Hochschule Karlsruhe, where he became a Privat-Dozent. It was here that he met his wife Martha, nee Togny. During the war, like many other mathematicians, Collatz worked for the aircraft industry. He was faced, in the era of mechanical calculators, with huge numerical problems from mechanical engineering and hydrodynamics. In 1943 he was offered a chair of mathematics at the Technische Hochschule Hannover. The following period brought about close cooperation with colleagues from engineering departments and a tremendous scientific output mainly in the fields of numerical analysis of differential equations and of LINEAR ALGEBRA G Elsevier

Science

52 Vanderbilt

AND ITS APPLICATIONS 68:1-B

(1985)

Publishing Co., Inc., 1985

Ave., New York, NY 10017

00243795/85/$3.30

2

L. ELSNER AND K. P. HADELER

eigenvalue problems, resulting in two important monographs, reprinted several editions [2, 41. In 1952 Collatz accepted a chair of mathematics at the University

in of

Hamburg, in a highly reputed department. He founded the Institute of Applied Mathematics at the university which became the home of several generations of young mathematicians. He also laid the foundation for one of the first university computer centers in Germany. He retired officially in 1978, but he is still teaching and actively participates in the scientific and social life of the Institute. Lother Collatz’s outstanding scientific work and his active role in the scientific community have been honored on many occasions. In spite of his strong emotional ties to southern Germany and the mountains he refused an offer in 1969 from the University of Stuttgart. He is currently a member of the Academia Leopoldina (Halle), the Academies of Bologna and Modena, honorary member of the Hamburgische Mathematische Gesellschaft, and of the Gesellschaft fur Angewandte Mathematik und Mechanik (GAMM). He received honorary doctoral degrees from the Universities of Sao Paulo, Dundee, Hannover, from Technische Hochschule Vienna, and from Brunel University. He has been and remains a member of the board of editors of several journals; he had been an associate editor of this journal at one time. Though in this brief account many outstanding features of his career must be omitted, mention must be made of Collatz’s early recognition of the importance of electronic computers, his interest in teaching applied mathematics in high schools, the many Oberwolfach Conferences organized by him, and his vigorous efforts to maintain contacts between mathematicians in Eastern Europe and the West. His strenuous hiking tours with students and fellow scientists, his hospitality to visitors and staff (shared by his wife and his daughter Gudrun), and his personal modesty, are memorable to his many friends and colleagues, whom he delights with Seasons Greetings, illustrated by himself, showing sights of last year’s travel. The most important aspect of Lothar Collatz’s work lies in the fields of numerical analysis and applied mathematics. His scientific career began early. He started with difference schemes for ordinary and partial differential equations. As early as 1934 he gave a talk on this topic at a GAMM-meeting (one of only 16 lectures). In 1938, at the GAMM meeting in Gottingen, he presented the numerical treatment of eigenvalue problems in a survey lecture, which was published in two long articles [13] and became the nucleus of the monograph [l]. In this way he came in contact with the then developing matrix theory. The problem of finding error bounds lead him to quotient theorems [21]. He investigated monotonicity of matrices ([41, 501) and the convergence of iterative methods for linear systems [32, 33, 481. In particular,

LOTHAR COLLATZ-ON

THE OCCASION OF HIS 75TH BIRTHDAY

3

two of his contributions [21,41] had a very large influence on the subsequent development of matrix theory. During his stay at the Technical Universities Karlsruhe and Hannover, while in close contact with engineers, he wrote a series of (often joint) papers on problems in mechanical and mathematical physics. For extended periods Lothar Collatz directed his interests towards other fields of mathematics, such as approximation theory, optimazition, combinatorics, and bifurcation theory. It appears that he often provided an important stimulus by introducing a basic idea or by writing a textbook. Others carried on his ideas, while he moved on to another field. The paper by Collatz and Quade [lo] on Abminderungsfaktoren, and his paper with Sinogowitz [56] on spectra of graphs, are particularly well known. As a student, Collatz showed a great interest in geometric and combinatorial problems and he still enjoys to pose and solve such problems. It seems that he was one of and probably the first inventor of the now famous 3n + 1 algorithm (Collatz-, Kakutani-, Hasse-, Syracuse-Algorithm) which has intrigued so many mathematicians (see e.g., the article of R. Guy, Am. Math. Monthly 90: 35-41 (1983)). Collatz had many students, among them G. Bertram, J. Schrijder, J. Albrecht, H. Bartsch, G. Gloistehn, H. Werner, G. Meinardus, H. J. Weinitschke, W. Wetterling, R. Nicolovius, H. Feldmann, W. Krabs, E. Bohl, K. P. Hadeler, L. Elsner, J. Werner, G. Opfer, F. Natterer, E. Bredendiek, F. Lempio, J. Spiess, B. Monien, B. Werner, several of whom have contributed to the theory of matrices, in such areas as monotonicity, iterative methods, Kantorovich inequality, linear programming, positive matrices, inverse, and nonlinear eigenvalue problems. BOOKS Eigenwertprobleme und ihre numerische Behandlung, Akademische Verlagsgesellschaft, Leipzig, 1945, 338 pp. Eigenwertprobleme mit technischen Anwendungen, Akademische Verlagsgesellschaft, Leipzig, 1949, 466 pp. 2nd Edition. 1962, 500 pp. Diffeentialgleichungen fir Ingenieure, Wissenschaftliche Verlagsanstalt K. G. Hannover, 1949, 156 pp. 2nd Edition: Teubner-Verlag, Stuttgart, 1960, 197 pp. 6th Edition, 1981, 287 PP. Numerische Behandlung eon LIiff~entialgleichungen, Springer-Verlag. 1951, 458 pp. 2nd Edition: 1955, 526 pp. 3rd Edition: The Numerical Treatment of Differential Equations, (Transl. P. G. Williams), Springer-Verlag, 1960, 568 pp., 2nd printing 1966. Funktionalonalysis und Numerische Mathematik, Springer-Verlag, Berlin, 1964, 371 pp. Functional Analysis and Numerical Analysis, (Transl. Hansjijrg oser), Academic Press, New York, 1966, 473 pp. (with W. Wetterling) Optimierungsaufgaben (Heidelberger Taschenbiicher, Bd 15), Springer-Verlag, 1966, 181 pp. 2nd ed. 1971, 222 pp.

4

L. ELSNER AND K. P. HADELER 8

Optimization

9

(with

J.

Braunschweig, 10

(with

(Transl.

Probbns

Alhrecht)

Part I: 1972,

W. Krabs)

aus

Springer-Verlag,

der Angewandten

141 pp., Part II: 1973,

Approximutionstheorie.

Teuhner-Verlag,

dungen,

6. Wadsack),

Aufguben

Stuttgart,

1975. Un-Text,

Muthematik,

Vieweg,

141 pp.

Tchebyscheffsche

Approximotiown

mit Anuw-

1973, 208 pp.

PAPERS 1

Fehlerahsch’tzung

2

Gleichungen

3

fiir Differenzenverfahren,

geometrischer

Eigenschwingungen 315-317

Ornamente,

gleichseitig

Z. Angew.

Verdgerneinerung

5

Differenzenverfahren

dreieckiger

Membran,

des Differenzenverfahrens, mit hdherer (1935).

6

@adratwurzelziehen

mit der Rechenmaschine,

7

Differenzenverfahren

hei

Anger.

16:239-247

10

M&h.

und Fehlerahsch&ung Differentialgleichungen,

Schranken

fiir ersten

(with W. quade)

I\4ec/t. 14:

Muth.

(with

Th.

Eigenwert,

Konvergenz

des

Gen&erte

18:186-194

Berechnung

16:.59%60 (19:36). Z.

hei Eigenwertaufga2: 189-215

Mothemntik

(1937).

(1937).

der reellen periodischen

Funktionen,

Si!;.

Bw

(1938).

homogener

Maschinen

wit

Zusatztlrel~rnassen,

Z.

(1938).

Differenzenverfahrens Math.

Deutsche

McJ~/I.

Differentialgleichungen,

fiir Differenzenverfahren

Zng. Arch. 8:325-331

Eigenfrequenzen

M&I.

(1934).

SCVI. Inst. ungew.

A~cJ(~~/I. Mrd~.

partieller

Phys. -Mat/~. Kl. 30:383-429

P&&l),

Math.

14:35%351

iM~~t/l.Mwh. math.

Z. Angeco.

Deutsche

Zur Interpolationstheorie

Wiss.

zialgleichungen, 13

Angew

(1936).

Konvergenzheweis

Angew. 12

Mech.

Srhr.

Anfangswertprohlemen

hen gewiihnlicher

Preup Akad. 11

Z.

Z. Angelo.

Approximation.

Clnio. Berlin 3:1-34

9

(1933). (1934).

(1934).

4

8

1:3:56-57

McJ~/~.MK~.

Math. ~Va/urw.Unterr.64:164-169

Z.

bei

3:2OC-212

von Eigenwerten,

Eigenwertproblemen

partieller

Differew

(1938).

Mat/r. Mcd~. 19:224&249,

Z. Angetc.

297-318

(1939). 14

Ginstiger

15

Hornersches

Wert

der Kopplungskonstanten,

Schema

hei

komplexen

Zng. Arch. Z.

Wurzeln,

l&269-282 Angerc.

(1939). ,M&h.

Mcth.

2023.5-236

(1940). 16

Schrittweise

17

Vergleich

der

Aswonom.

Nwhr.

Ntierungen

18

EinschlieBungssatz

19

(with

R.

271:116-120

LM~zt/l.Z. 46:692-708 Bucerius

von Inte~algleichungell,

Interpolationsverfahren Z. Angetu.

(with R. Zurmiihl)

van

mit

dem

(1940).

Ritzschen

Verfahren,

(1941).

fiir Eigenwerte

‘Zurmiihl)

ferentialgleichungen, 20

hei I~~tegralgleichungen,

Integralgleict~ut~gsmethode

Genauigkeit

Moth.

der

Me&

verschiedener

A4c~t/2,Z. d7:395-398

numerischen

22:42%55

Integration

(1941). von

Dif

(1942).

Integrationsverfahren,

Ing.

A&.

13:34&36

(1942). 21

EinschlieBungssatz

fiir charakteristische

Zahlen

van

Matrizen,

McJ~/~.Z. 48:

221-226

(1942). 22

Nat&&he

Schrittweite

hei numerischer

Integration,

Z. Angctu.

M&h.

Me&.

22:216-225

(1942). 23

Fehlerahsch%.tzung fiir Iterationsverfahren Math. Me&. 22:357-361 (1942).

24

Graphische

Liisung

gleichungen

2. Ordnung,

25

(with

R.

Zurmiihl)

VDI-Zeitschr. 26

Liisung

von

gewisser

Randwertprohlemen Z. Angetu.

Gritten

88:511-515

hei linearen

Math.

und Vertafeln

hei Me&.

Gleichurrgssyste~~le~l,

gewiihnlichen 23:237-239

empirischer

linearen

Z.

Angew.

Differential-

(1943).

Funktionen

mitt&

Differenzen,

(1944).

Differentialgieichungen

mit dem harmonischen

Analysator,

Bwicht

ii/xv

LOTHAR COLLATi-ON

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

THE OCCASION OF HIS 75TH BIRTHDAY

die Mathematiket-Tagung in Tiibingen, 60-61 (1946). Stabilit’at von Regehmgen mit Nachlaufzeit, 2. Angew. Math. Mech. 25/27:6X-63 (1947). Eigenwertaufgaben bei Iinearen IntegmDifferentialgleichungen, Z. Anger. Math. Mech. 25/27: 129- 130 (1947). Differenzenschemaverfahren filr gewijhnliche Differentialgleichungen n-ter Ordnung, Z. Angew. Math. Mech. 29:199-209 (1949). Graphische und numerische Verfahren: in FIAT-tie&m ofGerman Science 1939- 1946, Wiesbaden, 1948. Iterationsverfahren fiir komplexe NullsteUen algebraischer Gleichungen, Z. Angew. Math. Mech. 30:97-101 (1950). Konvergenzkriterien bei Iterationsverfahren fiir lineare Gleichungen, Math. Z. 53:149-161 (1950). Zur Herleitung van Konvergenzkriterien, Z. Anger. Math. Mech. 30:278-280 (1950). Das Mehrstellenverfahren bei Plattenaufgaben, Z. Angew. Math. Mech. 30:385-388 (1950). Neuere Forschungen iiber numerische Behandlung van Differentialgleichungen, Z. Angew. Math. Mech. 31:230-236 (1951). Zur Stabilitat des Differenzenverfahrens bei der Stabschwingungsgleichung, Z. Angew. Math. Mech. 31:392-393 (1951). SuIla maggiorazione dell’errore nel probleme de Dirichlet, Atte del IV. Congesso della Unione matem. Ital., Taormina, 25-31, 1951 Einschliessungss’atze fiir Iteration und Relaxation, Z. Angew. Math. Mech. 32:76-84 (1952). Numerische Bestimmung periodischer Liisungen bei nichtlinearen Schwingungen, Z. Angew. Math. Phys. 3:193-205 (1952). Zur 1. Randwertaufgabe bei elliptischen Differentialgleichungen, Z. Anger. Math. Mech. 32:202-211 (1952). Aufgaben monotoner Art, Arch. Math. 3:36&376 (1952). Einschliessungss’gtze fiir Prod&e von Eigenwerten, Wiss. Z. TH Dresden, 343-346 and 347-352 (1952/1953). (with H. Gortler) Rohrstromung mit schwachem DraU, Z. Anger. Math. Phys. 5:95-110 (1954). Fehlerabschatzungen zum Iterationsverfahren bei hnearen und nichtlinearen Randwertaufgaben, Z. Angew. Math. Mech. 33:116-127 (1953). Instabilitat bei Verf. der zentralen Differenzen fiir Differentialgleichungen 2. Ordnung, Z. Angew. Moth. Phys. 4:153-l% (1953). Funktion alytische Methoden in der praktischen Analysis, Z. Angew. Math. Phys. 4:327-357 (1953). Das vereinfachte Newtonsche Verfahren bei algebraischen und transzendenten Gleichungen, Z. Angew. Math. Mech. 34:70-71 (1954). Zur Fehlerabschatzung bei linearen Gleichungssystemen, Z. Angau Math. Me&. 34:p.72 (1954). Vereinfachtes Newtonsches Verfahren bei nichtlinearen Randwertaufgaben, Arch. Math. 5:233-240 (1954). iiber monotone Systeme Iinearer Ungleichungen, I. reine Angem. Math. 194:193-194 (1955). Numerische und graphische Methoden Artikel Handb. Phys., herausgegeben von S. Fhigge, 11:349-470 Springer, 1955. Fehlermassprinzipien in der praktischen Analysis, Proc. Znt. Math. Congress Amsterdam, 111:209-215 (1954).

6

L. ELSNER AND K. P. HADELER

53 54

Approximation

v. (1956).

Carl Runge

als Angewandter

4:1-10

wien

Zur Berechnung

<56

(with (with

J.

,5S

Albrecht)

61

hiiherer

2:66-75

Auswertung

(with

J.

Albrecht)

Ordnung

4:67-68

Abhnndl.

J.

Mech.

Hochschrde

(1957).

Math.

Sem. (‘Go.

Hamburg

Integrale,

Z.

Angrtc.

~Math. Labor

T&n.

Hochschulr

in BanachrLmen,

Arch.

Rutional

Me&.

Mech.

Schrtier)

mit unendlichem

Grundgebiet,

Z. Angetc.

(1958). fiir

Numerische

Mathernatik

Unterricht

Methoden

Math.

bei numerischer

Schulunterricht,

DIT

(1958/59).

Behandhmg

van Differentialgleichungen,

(1958).

38:264&267

EinschlieDen

inr

11:252-258

der L.i%ungen

van

Randwertaufgaben.

Numc+r

M&/I.

(1959).

Monotonic

65

Tschebyscheffsche

operators

24:70-78

and boundary Ann&erung

Monotone

Operatoren

67

Monotonic

Proceed.

Funktionen,

68

Funktionalanalysis

bei Anfangswertaufgaben,

u. Extremalprinzip.

Wien,

value problems,

mit rationalen

Madison ADA. Moth.

35-45

59,

(1959).

Scnrr. Uninic. Hun-

(1960).

66

schule

Mitteilungen

f. Gleichungen

Beispiele

Funktionalanalytische

64

70

Labor Techn.

mehrdimensionaler

part. Differentialgleichungen

Phys. IXa:llS-128

1:61-72

69

Math.

(1958).

Randwertaufgaben

burg

Graphen,

zum 60. Geburtstag,

Math.

(with

Moth.

Angew.

(1958).

NAerungsverf.

Z. Angew. 63

Z.

(1958).

~~ath~nlatisch-Nuturwiss~chaftliche 62

Mitteilungen Schi~fitechnik

endlicher

Numerische

38:1-15

5:71-73

Analysis

60

Spektren

Prof. Dr. A. Walther Wien

Mathematiker,

Vertiderl.,

(1957).

LMath. Me&.

58

mehreren

van Potentialfunktionen,

U. Sinogowitz)

21:63-77

van

(1957).

55

57

Funktionen

36:198-211

u. numerische

8:101-105

Mathematik,

Verf.,

Z. 74:150-160 Numer.

Mitteilungen

(196Q.

Mad.

Math.

3:99-106 Labor

(1961).

Techn.

Ho&

(1961).

Monotones

Verhalten

Numerique,

Mons, 61-88

Funktionenfolgen

M&It.

bei Newtonschen

bei

gewGhn1.

u. part.

Differentialgleichungen,

&loll.

[‘Atu~lyse

(1961).

mit 2 dreigliedrigen

Rekursionsformeln,

Monutschefle

Math. 66:193-202

(1962). 71

(with

H.

Werner)

tionalanalytische 72

Eigenvalue

73

Theoretische

74

Potenzbilder

75

Giinther

Anfangswertaufgaben

Beh.,

Problems,

Math.

Phys.

Differentialgleichungen

der numerischen

Mathematik,

in

funk-

(1962).

9:22-40

of EngineeringMechanics,

in Handbook

Grundlagen

gewBhn1.

Semesterbtichte

New York, 1962,

luhresb&cht

DW’

Chap.

18.

65:72-96

(1962). endlicher

Schulz

+,

Gruppen,

Z. Angau.

I. Reine Angelo. Math.

fiir Eigenwerte

Mech.

Math. 211:169-175

43:96

76

Einschliessungssatz

77

Angew. Math. Mech. 43:277-280 (1963). iiber einige neuere Entwicklungen bei Differenzenverfahren tieller Differentialgleichungen,

78

Inclusion

theorems

several variables, 79

Applications

80

(RaU, Ed.), Functional

81

(1965). tiler Nmnerische

Apl.

for the minimal

bei partiellen

Math. distance

in Proc. Approrimation

of functional

analysis

Mathematik,

Differentialgl.

10: 165-177

Tschebyscheff

mathematics, 11:133-148

fiir Randwertaufgaben

(Garabedian,

to error estimation,

Leopoldina

2. und 4. Ordnung,

Z. par-

(1965).

in rational

ofFunctions

Vol. 2, lS65, pp. 253-269. analysis as an aid in numerical

(1962).

(1963).

in Error Proc. (1965).

approximation

with

Ed.), 1965, pp. 43-,56. in cfigitul computation,

IFIP-Congress

1:145-153

LOTHAR 82 83 84

COLLATZ-ON

Hans Ehrmann

+,

Einige

abstrakte

1 nung),

Computing

Monotonicity

Z. Angew.

Begriffe

85 86

and related

Alwin Walther Monotonic

87 88

Inclusion

Mathematische Nichtlineare

van Laugwitz,

Approximationstheorie Applications

Math.

equations

(Greenspan,

Mech. 47:213-215 4.

problems,

in Proc.

(1967).

Ordnung,

Intern.

Ser.

Num.

und

3:113-l=

(1968).

Tschebyscheff-Approximation 413-421

Anwendungen,

bei

der M&em&k,

(1969).

und Anwendungen,

Inter. Ser. Num. Math.

to approximation

15:X-127

151-162,

Some applications

(1970).

problems, in Integer and nonlinear

(J. Abadie, Ed.), North Holland, 1970, pp. 285-308. bei Randwertaufgaben, Proc. Internat. Congr.

Warna,

mehreren

(1968).

iiberblicke

ofConstructioe

T-Approximation

Functions,

der HaIbord-

Ed.), 1966, pp. 65-87.

Colloque Besun~on 1966, 63-73

rationalen

of nonlinear optimization

programming,

95

der

Optimierungsaufgaben

92

94

differential

equations,

Differentialgleichungen

for matrix problems,

91

Einseitige

in nonlinear

Apl. Math. 13:137-146 (1968). Optimienmg, Bild der Wissenschoft,

herausgegeben

93

methods

Z. Anger.

Behandlung

Vertiderlichen,

89 90

(Anwendungen

(1967).

theorems

Numerische

Mathematik

(1966).

gewijhnlichen

Math. 7:171-178

Mech. 563 (1966).

ofnonlineardiff@xmtial

1898-1967,

bei

Moth.

in der numerischen

1:235-255

Numerical solution

7

THE OCCASION OF HIS 75TH BIRTHDAY

1970.

of functional analysis, particularly

Nonlinear Functional

Analysis and Applications,

l%chenzerlegungen

1.11, Math.

to nonlinear integral equations,

Madison,

l-43

Bild o’er Wissenschuft,

Kabinett,

Proc.

(1971). 404-412,

510-516

(1971). 96

Einseitig

Tschebyscheff-Approximation

hren der Math. 97

Approximationstheorie 16X-39

98

Approximation Finite

bei Differentialgleichungen:

Methoden

und Verfa-

(1971).

und Dualit% bei Optimierungsaufgaben,

Intern.

Ser. Num. Math.

(1972).

Dundee 99

Phys. 5:89-99

1972,

Conf.

Theory.

Springer Lecture Notes in Math. 280:16-31

by functions

of fewer

(1972).

und infinite Optimierung

variables,

mit einigen

Anwendungen

VI.

ord.

part.

diff.

IKM Weimnr,

Equ.,

148-157

(1972). 100

Hermitean

Numer. 101

Anwendungen aufgaben,

102

methods

Methods,

Ordnung

for initial value problems in partial differential

Dublin, 41-61

der

Intern.

Dualitat

und Abstand in der numerischen

Monotonicity March

104

Optimierungstheorie

Ser. Num. Math. 17:21-27

Mathematisch-physikalische 103

der

Mathematik,

in partial

Different

applications

equations,

Symp. Dundee, July 1974,

105

Bemerkungen

106

(with H. Giinther metrischen

auf nichtlineare

zur verketteten

F. Reutter zum 60. Ceburtstag,

differential

and nonconvex

Approximation,

an einfachen

equations,

Conference

Dundee

(1974).

optimisation,

especially

to differential

Springer Lecture Notes in Math. 448:112-125

and J. Sprekels) VergIeich

Methoden

Approximations-

(1973).

2056-71

Springer Lecture Notes in Math. 415:85-102 of convex

Proc. Sym.

(1973).

Semesterberichte

with discontinuities

1974,

equations,

(1972).

Intern. zwischen

Testbeispielen,

Ser. Num.

(1975).

Math. 26:41-45

Diskretierungsverfahren Z.

Angew.

M&h.

(1975). und para-

Mech.

56:1-11

(1976). 107

Einige

Beziehungen

zwischen Graphentheorie,

Num. Math. 29:27-56 108

(with E. Bredendiek)

Math. 30:147-174 109

Approximation

Geometric

und Kombinatorik,

Intern. Ser.

(1975). Simultanapproximation

bei Randwertaufgaben,

Intern.

Ser. Num.

(1976). of fixed points of contractive

and expanding

operators.

Proc. Conftience

8

L. ELSNER AND K. P. HADELER

Austin, Approx. Theory II, Austin:327-340 (1976) 110 Approximation methods for expanding operators, Proc. Conjiicnce Numer And. Springer Lecture Notes in Math. 506:149-59 (1976). 111 Bifurcation diagrams, Proc. Confmence Or&n. Part. Diff. Equ., Springer Lecture Notes in Math. 564:41-s (1976). 112 Numerical treatment of some singular boundary value problems, Colloyu. Mdz. So&et. Janos Bolyni, 22 Num. Meth., Keszthely, Hungary, 133-151 (1977). 113 Verzweigungsdiagmmme und Hypergraphen, Intern. Ser. Num. Math. 38%42 (1977). 114

Graphen

bei

362-46

(1977).

Ornamenten

und

Intern. ST. Mrn~. Math.

Verzweigungsdiagrammen,

of some singular boundary value problems, 115 Numerical treatment Springer Lecture Notes in Math. 636:41-50 (1978). 116 Spektren periodischer Graphen, Resultate Math. 1:42-53 (1978). 117

Typen

van Hypergraphen

Math. 46:37-65

innerhalb

Idee und Erfahrung

119

Lower

Infern.

Ser.

Nurrr.

120

Approxim.

Methods

boundaries,

Proc. 4, Symp.

aus der Sicht des Mathematikers,

and upper

lems, Durham,

Monotonicity

der Mathematik,

And.,

(1979).

118

121

und aui3erhalb

Proc. Nwnrr.

bounds

147-157

GAMM-Mitteilungen, 3-24 (1976). Proc. Symp. Free. Boundary Vduc Prob.

for free boundaries, (1979).

for

boundary

and free boundary

value

problems

with

unbounded

Math. (I. Mar&, Ed.), Plzen, CSSR,

Nun.

value problems,

Springer Lecture

domains 51-61

No’otesin

or

free

(1978).

Math. 773:33-45

(1980). 122

Numerical Congress

123

Iteration Bolyui

124

procedures

Anwendung Probleme

van der

Inclusion

with hidden

singularities,

I’roc.

(1986).

using only one step of iteration.

Monotoniesatzen

Proc. Colloqtricz mech

Sot.

Jwws

der

Liisurlgen

van

Gleichungen,

(Prod&ion

van Badeanziigen),

Tetm,

Z. (%cvr.

(1982).

of certain

by Numerical

EinschlieBung

(1981).

Optimierung

52-53

of solutions

zur

189-225

Math.

infiniten

Math.,

Equations

value problems

III, Austin: 311-329

(1986).

iiberblicke

Natorwiss. 126

of some boundary Tlieory

35:399-322

Jahrbuch 125

treatment on Appror.

types

of integral

(Chris Baker,

Methods

equations,

in Trecztment

G. F. Miller, Eds.) Academic

of Integral Press.

1982,

pp. 4777488. I27

Some numerical

Applications, 128

test problems

(Whiteman,

Anwachsende

Schwingungen

Verzogenmgsglied, 129

EinschlieBung gleichungen. Richard

(with H. Krisch,

H. Werner,

Elektronische

theorems

monotonicity,

1985

bei

einer

Math. 62:49-s Mathematiker,

Rechenanlagen

for solutions

of Finite Elements

Differentialgleichungen

Math. 62:37-47

and P. Janssen)

Springer Lecture

14 Junuary

einigen

Losungen

van Mises als numerischer

Mathematik,

Received

bei

Ser. Num.

in The Mathem.

and

1982, pp. 65-76.

Ser. Num.

periodischer

131

Inlcusion

Intern.

Intern.

130

132

with singularities,

Ed.),

mit

Klasse

van

Differenzen-Differential-

(1983). Z. Angew.

Math.

Der EinfluB

der Computer

25: 118-125

(1983).

of differential

Notes in Math.

multiplikativem

(1983).

equations 1632:93-191

Mcch. (1983). auf die Numerische

with aid of pointwise (1983).

or vector