Coda How to be a mathematician It takes a long time to learn to live-by the time you learn your time is gone. I spent most of a lifetime trying to be a mathematician-and what did I learn? What does it take to be one? I think I know the answer: you have to be born right, you must continually strive to become perfect, you must love mathematics more than anything else, you must work at it hard and without stop, and you must never give up. Born right? Yes. To be a scholar of mathematics you must be born with talent, insight, concentration, taste, luck, drive, and the ability to visualize and guess. For teaching you must in addition understand what kinds of obstacles learners are likely to place before themselves, and you must have sympathy for your audience, dedicated selflessness, verbal ability, clear style, and expository skill. To be able, finally, to pull your weight in the profession with the essential clerical and administrative jobs, you must be responsible, conscientious, careful, and organized-it helps if you also have some qualities of leadership and charisma. You can't be perfect, but if you don't try, you won't be good enough. To be a mathematician you must love mathematics more than family, religion, money, comfort, pleasure, glory. I do not mean that you must love it to the exclusion of family, religion, and the rest, and I do not mean that if you do love it, you'll never have any doubts, you'll never be discouraged, you'll never be ready to chuck it all and take up gardening instead. Doubts and discouragements are part of life. Great mathematicians have doubts and get discouraged, but usually they can't stop doing mathematics anyway, and, when they do, they miss it very deeply. 400 CODA 401 " Mathematician" is, to be sure, an undefined term, and it is possible that some people caned mathematicians nowadays (or ever) do not (or did not) love mathematics all that much. A spouse unsympathetic to mathematics demands equal time, a guilty parental conscience causes you to play catch with your boy Saturday afternoon instead of beating your head against the brick wall of that elusive problem-family, and religion, and money, comfort, pleasure, glory, and other calls of life, deep or trivial, exist for all of us to varying degrees, and I am not saying that mathematicians always ignore all of them. I am not saying that the love of mathematics is more important than the love of other things. What I am saying is that to the extent that one's loves can be ordered, the greatest love of a mathematician (the way I would like to use the term) is mathematics. I have known many mathematicians, great and small, and I feel sure that what I am saying is true about them. To mention some famous names: I'd be very surprised if Marston Morse, and Andre Weil, and Hermann Weyl, and Oscar Zariski didn't agree with me. Mind you, I am not recommending or insisting that you love mathematics. I am not issuing an order: "If you want to be a mathematician, start loving mathematics forthwith" - that would be absurd. What I am saying is that the love of mathematics is a hypothesis without which the conclusion doesn't follow. If you want to be a mathematician, look into your soul and ask yourself how much you want to be one. If the wish isn't very deep and very great, if it is not, in fact, maximal, if you have another desire that takes precedence, or even more than one, then you should not try to be a mathematician. The "should" is not a moral one; it is a pragmatic one. I think that you would probably not succeed in your attempt, and, in any event, you would probably feel frustrated and unhappy. As for working hard, I got my first hint of what that means when Carmichael told me how long it took him to prepare a fifty-minute invited address. Fifty hours, he said; an hour of work for each minute of the final presentation. When, many years later, six of us wrote our "history" paper ("American mathematics from 1940... "), I calculated that my share of the work took about 150 hours; I shudder to think how many man-hours the whole group put in. A few of my hours went toward preparing the lecture (as opposed to the paper). I talked it, the whole thing, out loud, and then, I talked it again, the whole thing, into a dictaphone. Then I listened to it, from beginning to end, six timesthree times for spots that needed polishing (and which I polished before the next time), and three more times to get the timing right (and, in particular, to get a feel for the timing of each part). Once all that was behind me, and I had prepared the transparencies, I talked the whole thing through one final rehearsal time (by myself-no audience). That's work. Archimedes taught us that a small quantity added to itself often enough becomes a large quantity (or, in proverbial terms, that every 402 HI. SEl\'IOR little bit helps). When it comes 10 accomplishing the bulk of the world's work, and, in particular, the work of a mathematician, whether it is proving a theorem, writing a book, teaching a course, chairing a department, or editing a journal, I claim credit for the formulation of the converse: Archimedes's way is the only way to get something done. Do a small bit, steadily every day, with no exception, with no holiday. As an example, I mention the first edition of my Hilbert Space Problem Book, which had 199 problems. I wrote most of the first draft during my Miami year, and I forced myself, compulsively, to write a problem a day. That doesn't mean that it took 199 days to write the whole book ~the total came to about three times that many. As for "never give up", that doesn't need explanation, and I have been trying to illustrate it with anecdotal evidence all along, but here, just for fun, is a pertinent little story. Somewhere around 1980 I was invited to give a talk to a "general" audience, and, having given it, I wrote it up and submitted it for publication in The Mathematics Teacher. In due course I received reports from a couple of referees; they read, in part, as follows ... The author apparently feels she/he is illustrating the thrill and power of abstraction. While his/her examples contain the potential for doing that, I don't feel that the presentation creates that effect. ... Much of the problem with the paper is that of a meandering style. The progression of ideas is not c1ear.... The mathematical topics focused on are of only moderate interest." The paper was firmly rejected. I didn't give up~ I just shrugged my shoulders, and submitted the same article, word for word, to what was then called The Two-Year College Mathematics Journal. It was accepted and printed, and, a year after that, it received the P61ya award from the Association. All these prescriptions and descriptions about how to be a mathematician arose, inevitably, from my own attempts to become one. Nobody can tell you what mathematicians should do, and I am not completely sure I know what in fact they do all I can really say is what I did. How close did I come'? What is my total mathematical contribution? The first answers that occur to me are a small but pretty proof (the monotone class theorem), a few reasonable theorems (mainly in my ergodic papers" Approximation theories ... " and .• In general a measurepreserving theorem is mixing "), and a good idea in logic (polyadic algebras). One thing that I am pretty good at is asking questions. Given a mathematical problem that I understand the statement of, one whose history I know something about and that I spent some time studying, one for which I am moderately up to date with the standard techniques ...... given those circumstances, I have some talent for identifying and formulating the central questions. If I look hard at a question for a month, try to answer it, and fail, then I am confident that it is not CODA 403 trivial. I am confident that a question like that is fit for mathematicians much better than I; when one of them solves it, he is bound to feel at least a small tingle of pride and pleasure. (Examples: the power inequality, and the Weyl-von Neumann theorem for normal operators.) Related to the questions I have asked are the concepts I discovered and introduced, notably subnormal and quasitriangular operators, and, possibly, capacity in Banach algebras. Important theories have grown out of such concepts; I think it's fair 10 call them contributions. I wrote a few good surveys and some pretty good books. Perhaps the best are Finite-Dimensional Vector Spaces and A Hilbert Space Problem Book-but then my vote on such matters is probably the one that weighs the least. My most nearly immortal contributions are an abbreviation and a typographical symbol. I invented "iff", for "if and only if" - but I could never believe that I was really its first inventor. I am quite prepared to believe that it existed before me, but I don't know that it did, and my invention (re-invention'1) of it is what spread it through the mathematical world. The symbol is definitely not my invention-it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like D, and is used to indicate an end, usually the end of a proof. It is most frequently called the "tombstone ", but at least one generous author referred to it as the "halmos". That's it; that's a life, that's a career. I was, in I think decreasing order of quality, a writer, an editor, a teacher, and a research mathematician. What's next? The writing of this book took a lot out of me. It took a year and a half of time and energy, during which I paid no attention to research. It was a deliberate gamble. 1 wanted to write this book, I wasn't at all sure that I could do it the way I dreamt, I wasn't sure I could tell the audience I had in mind what I wanted to say. If it turns out that I succeeded, I'll be happy; if not, I'll be sad. But, in either case, I'm not ready to crawl into a hole and pull it in after me. I'd like to write some more mathematics, to teach some more, and, who knows, even to prove a theorem. I'm going to try, that's for sure. I thought, I taught, I wrote, and I talked mathematics for fifty years, and I am glad I did. I wanted to be a mathematician. I still do. Index of Photographs Ambrose, W. 48 Anosov, D. V. 302 Bishop, E. 162 Brown, H. A. 364 Collingwood, E. 292 Conway, J. B. 364 Day, M. M. 231 de Branges, L. 326 Doob, J. L. 52 Eberlein, W. F. 242 Eilenberg, S. 248 Everett, H. S. 133 Feller, W. 94 Fillmore, P. A. 364 Fomin, S. V. 306 Furniss, W. T. 345 Gamkrelidze, R. V. 295 Gelfand, L M. 306 Halmos, P. R. 1931 21 1947 242 1949 247 1970 368 1980 386 Indiana University, functional analysts 364 Institute for Advanced Study, Fuld Hall common room 242 Kaplan, S. 242 Kirillov, A. A. 308 Kolmogorov, A. N. 309 Krein, M. G. 313 Lefschetz, S., letter from 87 Leibler, R. A. 242 ios, J. 316 Mac Lane, S. 248 Martin, W. T. 109 Massera, J. 186 Mautner, F. I. 242 McKnight, J. D. 331 McLaughlin, J. E. 332 Milman, D. 386 Minakshisundaram, S. 242 Mishchenko, E. F. 296 Mookini, E. 344 Moore, R. L. 256 Naimark, M. A. 310 Nakata, J. 357 Room. T. G. 281 Rubin, H. 242 Samelson, H. 102 Sarason, D. E. 276 Segal, I. E. 242 Shields, A. L. 72 Silber, J. R. 346 Smithies, F. 389 Spanier, E. H. 242 Stampfli, J. G. 364 Steenrod, N. E. 391 Stone, M. H. 145 von Neumann, J. 92 Wallen, L. J. 347 Williams, J. P. 364 Zorn, M. A. 362 Index academic titles, call me mister 27 advice, how to give 267, 269 advice to students, don't stay put 52 Alexander, J. W. and von Neumann 88 demotion 89 lectures 88 Alexandrov, P. S., German gentleman 310 algebra from Hazlett 45 from Shaw 32 gymnasium 14 high school, college, etc. 17,32 algebraic geometry at Illinois 36 from Emch 49 algebraic logic climax 211 flowers 212 representation theorem 212 all work and no play 53 almost periodic functions and Jessen 138 almost universal forms versus Ramanujan 42 excitement of progress 57 Ambrose, W. algebraic geometry exam 49 at Alabama 81 blase graduate student 50 first meeting? 29 music 29-30 pea pool 61 pins Doob down 69 politics 54 the light dawns 48 "American Mathematics from 1940 to the Day Before Yesterday" 397 AMG-B, AMG-C, etc. 114 AMS Board of Trustees 226 committee service 221 Council 226 Council voting procedure 226, 227 first meeting 78 functions of 226 in-group 224 analysis class report 47 conversion to 56 from Steimley 32 analytic geometry a revelation 24 mystery of rotation 25 solid 28 Ando, T., at Oberwolfach 386 Annals of Mathematics Studies 96 Anosov, D. V., Russian companion 301 anxiety attack 359 applied mathematician consulting 116 reformulation 117 Archimedes at Syracuse 108 converse of theorem 402 reasons better 205 area of ignorance 274 Argentina library problems 194 arithmetic in Hungarian II arithmetic races 12 Aronszajn-Smith, invariant subspaces 320 407 408 associate professor promotion 160 associate, vestigial rank 105 ASTP super course 111 Atiyah-Singer index theorem 398 Atomic Energy Commission fellowships 217 authors suggest referees 383 automathography vii axiology, the value of philosophy 39 Azumaya examination 271 bachelor's thesis with Harry Levy 33 Baer, R. assistants 62 at Illinois 51 integral closure 62 Bagley, R., with McKnight 330 Banach, S. in Fundamenta 65 book appears 49 book an addiction 78 book out of date 231 fixed point theorem 325 banquet in Russia 311-312 Barankin and Blakers on FDVS 95,96 Bartky talks about chairmanship 361 basic concepts of algebraic logic 390 basic minimum mathematics 60 Bateman, H., posthumous notes 294 Bear, H. S., Hawaii chairman 362 beer personal taste vii from cranberries 23 Berens, H., linguistic control 387 Berg, I. D., same as Cantor set theorem 325 Bergman, S., English 172 Berkson, E., helps found Wabash seminar 373 Berlitz for Spanish 170 Bernoulli numbers in topology 72 Bernstein, J., plays trumpet 242 Bernstein-Robinson, invariant subs paces 204, 320 Bessel's inequality, computational proof 4 Beth in Daigneault's thesis 214 Bieberbach conjecture solved 329 Bing, R. H., student of Moore 255 Birkhoff, G. D. conjectures ergodic case is general 112 ergodic theorem 68 prolific publisher 41 birthday problem 103 Artin and Cantor 103 Bishop, E. homework, a new theorem 137 thesis, first and second 161 INDEX Blackwell, D. brilliant freshman 45 Doob's student 80 blitz in chess 85 Blyth, T. S., at St. Andrews 388 BMC, Dundee, 1965 290 Boas, R. P. changes colors 381 Gaelic review 292 quashes review 120 Bocher's book is a mess 40 Bochner, S., as chairman 353 Bombieri, E., distinguished lecturer in Bloomington 268 Bonsall, F. F., introduction 291 book editor, how to be 228 book production takes long 144 book review editor, how to be 375-376 book reviewers where to find them 376 with a conscience 377 without a conscience 378 book reviews on all of pure mathematics 223 what are they 375 books read, only two 64 Boolean algebra free 206 as purified picture of mathematics 245 Bourbaki structure 47 article about 238 in Uruguay 185 Brahana, H. R. metabelian groupie 56 oral exam 60 teaches algebra 40 teaches calculus 27 Bram, J., sits in the back 161 bribe offered 44 Britain in 1958 243 brothers study medicine 19 Brown, A. A., helpful in Princeton 85 Brown, H. A. Brown-Halmos-Shields on Cesaro 319 Brown-Halmos on Toeplitz 319 turnabout jobs 161,362 undetermined Il 160 Brown, S., newsbreak 323 Budapest in the 1920's 5 revisited 284 the second time 315 Bulletin of the AMS, editorial committee 372 Butzer, P., at Oberwolfach 385 calculus a la Moore 258-259 not for freshmen 26 four step rule 26 INDEX Cambridge congress Lefschetz 163 Schwartz and Hadamard 163 panel on measure theory 164 Campbell-Baker-Hausdorff formula 3 Campbell, Sam, offers loan 83 Cantor set beautiful or nasty 63 in teaching 272 Cantor's proof not paradox 33 Caratheodory, a good book 58, 127 Carmichael, R. D. elliptic functions 56 fifty hours for fifty minutes 401 oral exam 60 teaches differential equations 61 teaches number theory 40, 41-42 The Dean 42 Theory of Groups, read and proofread 65 time for research 322 weak students 57 Castillo, H., Ph.D. exam 171 Cauchy integral formula, less beautiful than group 3 Cauchy's theorem is "in" 49 Cayley just a rumor 35 CBMS regional conferences 369 Cesaro is normal 320 chairman consult colleagues 351 how to select 349 length of service 352 foster contacts among faculty 354 pay attention to details 353 psychotherapist 361 "realist" 355 chairmanship nibbles 239 offers 343 secondary job 356 chairmen good and bad 350 three greats 353 Chambana daylight time 20 change fields often 82, 156 change notation to understand it 69 Charlottesville meeting of the AMS 79 Chebyshev, P. L., what did he do? 65 Chelsea Publishing Co., in fur shop 159 chemical engineering career 18 chemistry equations balance 19 laboratory 23 chess by correspondence 38 Chevalley, C. at Princeton 90 in Dundee 290, 293 on Council 58 Chicago college faculty 131 division faculty 130, I3! farewell conference 253, 254 fired from 249 glamorous 50's 200 master's exams 145, 146 Math J in the college 131 mathematical advisors 332 outstanding students 152-155 permanent visitors 138 Ph.D. students 159-162 salaries 1959 249 students in the 1950' s 202 The College 122 top curriculum 144 Christmas meetings of the AMS 107 circle squarer, 1T = 4 335 circus in Moscow 299 citizen identification card 91 classical mechanics, proud title 101 Clebsch, R. F. A., in algebraic geometry 49 Coble, A. B. beginning negotiation 95 by the short hairs 104 chairman 43 Ph.D. supervisor 43 salary 43 Coburn, father and son 274 Cockcroft, W. R., dean 294 cocky. definition 17 Cohen, P., accomplishment 203, 204 Cohn, P., at St. Andrews 388 collaboration Ambrose and Kakutani 99 Ewing 397 Gustafson 397 Leibler 95, 97 Lumer and Schaffer 192 marriage principle 98 Moolgavkar 397 Savage 151 Sunder 371 Vaughan 78 von Neumann 100-101 Wheeler 397 Ziemer 397 college at Chicago 131 Collingwood, E. offer of a drink 291 Sir 292 colloquium speaker, how to be 268 colloquium talks have a bad name 71 committees are inefficient 372 committees of one 372 bad ones 373 common room at Chicago 129 in Fine Hall 85 not Commons 85 communication, art versus science 10 communist by allegation 218 commutators, Brown-Pearcy 191 competition in research 323 409 410 complete logic is large 207 complex numbers not known 17 computability, how define 244 computer in Uruguay 184 conceptual versus computational 3 congress Cambridge 162 Edinburgh 243 consistency proofs, fascinating and admirable 203 consistent logic is small 207 constructive mathematics, a religion 161 continuity, definition, biggest single step 205 continuous functions of Hermitian operators 383 contributions of a life in mathematics 402-403 Coolidge versus Veblen and Young 33 counterexamples are good examples 63 covering course material 260-262 Cramer's rule, not much fun 203 cranks and Wigner 334 crank papers submitted 382 Crathorne, A. R. colloquium chairman 106 shaves moustache 37 0° = 7 135 culture American 16 early acquired 30 cylindric algebras, Tarski et al. 210 Daigneault, A., student via Church 214 Day, M. M., writes Ergebnisse book 231 de Branges-Rovnyak Fillmore's gap 328 invariant subspaces claimed 326 many readers 327 press release 328 Research Announcement 327 revised reprint 328 Dedekind cuts are useless 204 Dedekind on Gauss 61 deduction theorem via generated ideals 207 de la Vallee Poussin. real variable wisdom 56 demanding high school students 335 democracy in the AMS? 226 departmental recommendations via Jessie 358 department meetings at Quadrangle Club 130 Descartes' great victory 25 desk calculators in primeval days 37 Diamond Head Circle poolside seminar 362 INDEX Dickson's Moderrz Algebraic Theories, badly written 41 dictionaries 394, 395 American Heritage and Webster II 10 Dieudonne, J. committee member 392 recreational mathematics 325 dilation theory solves problems 157 dirty tricks in teaching 271 distinguished professor of the wrong sex 367 distinguished visitor, how to be 268 Dogpatch is Oak Ridge 114 Doob. J. L. arrives at Illinois 50 book as text 75 dedication like a monument 370 exploits ergodic theorem 68 first students 80 intellectual honesty or elegance 69 lecturer and teacher 51-52 martingale next to reprints 370 no research course 51 oral exam 60 suggests thesis direction 74 teaches topology 51, 61 thesis supervisor 61, 76 understands Oppenheimer 240 Douglas, R. G. in Oberwolfach 384 reads de Branges-Rovnyak 327 Dowker, C. H., von Neumann's assistant 93 draft mathematicians 105 Dresden, A., at Swarthmore 122 drinking during prohibition 23 driving in Scotland 291 duality, an instance of "same" 325 Dunford-Schwartz not yet in existence 231 Dushnik. B., teaches Savage 136 Dutch is funny 59 Dyer, E., Proceedings editor 224 Eckhart Hall, neo-gothic 128 Edinburgh application, Scotch joke 25 Edinburgh congress, department meeting 244 editor advertisement and advisor 228 judge 232 editorial work, part time, full time 222 Eells. J., at St. Andrews 388 Eilenberg, S. thesis students 273 who is? 123 Einstein and Straus 141 Eisenbud, father and son 90 Eisenhart, father and son 85 elliptic functions, dreadful stuff 56 INDEX Emch, A. oral exam 60 teaches algebraic geometry 49 Encyclopedia Britannica improves writing 395 English in Hungary, sometimes 287 entropy in ergodic theory 236 Erdelyi, A. in Dundee 290 in St. Andrews 388 on non-standard analysis 294 Erdos, P. dimension of rational points 91 helps with measure theory 92 nojob 90 Erdos is not Erdos 292 Ergebnisse, how to edit 230 ergodic course at Chicago 137 ergodic theorem, von Neumann and Birkhoff 68 ergodic theory, book and recent progress 235 Ernest, J., job nibble 370 Ethiopians in Budapest 289 Euclidean geometry in high school 18 advanced 34 Everett, H. S., brings Singer to Chicago 133 Euler's proof in Latin 58 Everitt, W. N. at Dundee 290 free dinner 291 Ewing, J. H. likes et al. 399 topological informant 365 examples, typical and degenerate 62 excommunicated research announcement 329 executive committee at Michigan doesn't rock 351 existential quantification is an operator 211 expository papers and surveys 390 faculty at University of Montevideo 180 engineering 182, 183 humanities and sciences 180, 181 false exercise replaced 143 false theorems for use on tests 146 father emigrates 14 lends money 83 offers college choice 18 teaches clock 10 FBI security checks, always necessary? 217 FDVS almost rejected 96 411 second edition rewarding? 238 sold by Savage 150 fee versus honorarium 267 Feinstein, A., changes his mind 146 Feller, W., writes severe review 94 fellowship applicants a pitiful lot 335 Fermat's last theorem, often proved 335 Fibonacci sequence, rabbit story 18 fiction as lies 8 Fiji on way to Sydney 282 filling in manuscripts 76 Fillmore, P., finds gap 328 Fine Hall Platonic 84 teas 85 finishing a book six times 143 finite simple groups versus Janko 282 fired from Chicago 249 first professors at Institute 84 Fisher, R. A., not a mathematician 65 Foia~, C. and Krein 314 at Gelfand seminar 307 Fomin, S. V. as translator 305 house, when painted? 306 Ford, L., father and son 112 foreign accent bane of existence 16 how to say "what" 16 not eliminated 106 formal power series and convergence 45 Forteza meets ship 173 at Instituto 185 "Foundations of Probability" corrected by Robbins 113, 114 exposition or survey 390 Fourier, learns something new 205 Fraenkel complains about review 223 free groups, like propositional calculus 206 French exam, free translation 38 Friday evening discussion 54 Friedman, M., friend of Savage 149 Fubini, G., at Institute 89 Fuchs, L., Budapest host 285,286 Fundamenta, great journal 65 Furniss, W. T., Hawaii dean 345 Galler, B., inherited from Stone 161,213 Galois, just a rumor 35 Galuten, A., becomes a publisher 159 gambling systems can't win 78, 79 game theory is born 100 Gamkrelidze and Mishchenko opera 300 salad bar 296 412 INDEX Garding, L. English 172 Riesz's assistant 138 Gardner, M., columns 237 Gaul's three parts 31 Gauss as examiner 61 Gelfand, I. M. always late 306 representation via Stone 245 three ring seminar 307 Gellert Hotel, no soap 286 general analysis at Chicago 122 generic means nothing in particular 49 geometry via Graustein 32 Gergen wants no refugees 80 German course in college 22 learn at two 9 reading mathematics 58 Gillman, L., 1976 meeting at San Antonio 397 "Glimpse into Hilbert Space" 390 Go is deep 86 goddam foreigner 15 Godel, K. accomplishment 203, 204 at Institute 90 completeness theorem 212 computability 244 no stories 141 proof 212 Goldman, 0., elderly at 21 142 Gomory, R., in Sydney 281 Gonzalez Dominguez, A., interview 169 Gottschalk teaches philosophy 33-34 grades on languages 136 Graduate Club at Illinois 66 Granville, Smith, and Longley the infamous 26 the light dawns 48 Green's theorem, a foggy mystery 32 Grundlehren is not Ergebnisse 230 Guggenheim fellowship before Morrow 40 finally 140 GUM for shopping 299 Gustafson, W. H., algebraic therapist 365 gymnasium curriculum 16 recitations 13 scary 12 Hadamard, J., dangerous? 163 Hadwin, D., at TCU 369 Hahn, H., among best sources 56, 127 halmos, tombstone 403 Halperin, I. S., square deal petition 216 Hamilton, N. T. flunks Latin 142 genius manque 333 Hamlet in Spanish 198 handwriting Americanized 7 Harary, F., at St. Andrews 388 Hardy and Littlewood on collaboration 98 Hardy Pure Mathematics versus Titchmarsh 47 Harrison, D., calls Sammy Sammy 278 Harrison, K., as student 368 Hartman-Wintner versus Toeplitz 319 Hasse on "higher algebra" 229 Hatcher, A. G. in the wrong picture 367 on theses 279 hats off to the ladies 30 I Hausdorff, F. beautiful and astonishing 57 just a rumor 35 not at Illinois 48 Hawaii chairmanship offer 343 glamorous visitors 360, 361 hires 20 assistant professors 360 midnight in January 282 offers to Shields and Wallen 347 Hay, G. E. makes Michigan offer 249 teaches Michigan procedures 254 Hazlett teaches algebra 45 healthy, becoming 240 Heisenberg uncertainty principle 191 Heller, A., takes three jobs 360 Hellinger, E. a refugee 157 on multiplicity 157 Helson, H., on shifts 254 Hermitage gigantic in Leningrad 300 white elephant 301 Hewitt, E., house, Porsche, and secretary 245 high school civics 17 geometry 18 history 17 in Chicago 15 Hilbert on Ph.D. supervision 272 Hilbert space and logic 211 book, first as lecture notes 158 book, too short 159 not at Illinois 49 problem book, one a day 402 history of mathematics in Monthly 381 Hobson frustrating 65 Hochschild, G., learns logic 215 Hoersch, V., teaches 4 x 4 matrices 28 honorarium versus fee 267 Hopf, E., invariant measure 236 Hotelling's notes on statistics 65 Howie, J., at St. Andrews 388 413 INDEX Huff, G., spoilsport 43 Hughes, D., wangles Michigan offer 249 Hungarian customs inspectors 285 in King's Cross 284 misinformation 318 vocabulary 16 Hungarians en masse 285 Hungary coffee during lecture 289 "Comrade Halmos" 289 democracy 316 English spoken 287 gallows humor 317 grumbles about Russian 287 how to reconfirm 288 interview 288 lecture in English 289 lighter after the war 286 middle class standard of living 287 New York Times is bad manners 287 polylogue 288 research institute versus university 288 scary 284 service is good and bad 317 Hutchins, M., invents the college 122, 123 Hyde Park, a segregated island 128 iff, who invented? 403 Iliev, L. G., at Oberwolfach 386 Illinois Bachelor of Science 35 first registration 20 freshman curriculum 22 going home to 105 second registration 24 statistics 31, 36 Illinois Institute of Technology, minus T 157 Indiana functional analysis seminar 363-365 goes down 363 invitation 362 law school and music school 365 less than Michigan 366 Ph.D. students, private seminar 367 indiscrete spaces 340 Infantozzi at Instituto 185 infinite regress, what's wrong with it? 213 "Innovation in Mathematics", a catastrophe 238 Institute Ambrose fellowship 83 first six professors 84 housing project 142, 240 in 1947 140, 141 in 1957 241 "insignificant" people office 141, 243 onerous administration Instituto de Matematica building 184 computer 184 founded by Laguardia library 185 people 185 integrally closed subrings f tt; 90 89 184 62 is not what you think 271 interpretation is homomorphism 208 interview about gambling 79 invariant measure problem 236 invariant subs paces Aronszajn-Smith 320 Bernstein-Robinson 204, 320 de Branges-Rovnyak 326 Italian, learned in a day 82 Izumi, S.-I., non-theorem 156 Jacobs, K., photographs 385 Janko, Z., versus Hilbert space 282 Jerison, M., helps found Wabash seminar 374 Jessen, B. English 172 lectures on almost periodic functions 138 Jessie Nakata makes recommendations 358 the perfect secretary 357 job hunting after the war 121 for no jobs 79 job pattern in U.S. 329-330 Jones, A. at Instituto 186 in Brazil 188 jUdging students is easy 147 Julia's Theorie Quantique 79 Kakutani, S., on Haar measure 98 Kalmar, L., lectures in German 287 Kaplansky, I. algebra and Mac Lane 248 at St. Andrews 384 teaches measure theory 145 Katok, A. B., in Sinai's seminar 313 Kazarinoff ruble exchange 304 Kelley, J. L. the name is Kelley 370 topological terminology 339 Ketchum, P., teaches analysis 47 Khintchine, A. is difficult 65 writes postcard 67 414 Kibbey, D. Doob's student 80 Real Variable Club 66 Kirillov, A., translates 308 Kleene, S. C., at Institute 89 Klein, F., Elementary Mathematics as culture 66 Kline, J. R., Secretary of AMS 79 Knopp, K., Theory of Functions translated 56, 65 knowing students is important 134-135 know versus teach 133 Kolmogorov, A. N. and discomfort 308-309 Grundbegriffe der Wahrscheinlichkeitsrechnung 49, 65 the new testament 65 Kolmogorov-Sinai, solve conjugacy problem 235 Krein, M. G. says yes 313 and Foia§ 314 mailing reprints 314 Krein-Milman theorem 386 Kremlin theater for food 300 Kriegspiel, three board chess 85 Kriete-Trutt versus Cesaro 320 Laguardia, R. arranges Punta del Este 192 at Cambridge congress 169 English 172 founds Instituto 184 meets ship 173 not on council 182 on time? 197 teaches engineers 183 Uruguay invitation 167 versus deans 183 versus Massera 187 Landau, E. organization 233 steep number theory 57 versus Dedekind 61 Lane, E. P. doesn't know Eilenberg 123 languages for mathematics 10 French and Fortran 9, 10 German, Latin, Hebrew 9 graded 136 Greek and Sanskirt 10 Larkin. P., libraries and bifocals 294 Lasker, E., plays Go 91 Lat is u.s.c., pleasant 384 law courses, here and there 18 law school at Indiana 365 Lawrence, T., at Rad. Lab. 115 INDEX Lax, P. edits de Branges-Rovnyak 327 on shifts 254 learning English in Hungary 14 to read 6 to write 7 Le Blanc. L., via measure theory 214 lectures have a bad name 71 Lefschetz. S. handwriting 86, 87 how to describe 395 left leaning affects social life 54 Leibler, R. A., plays pool 61 Leningrad more beautiful 299 train 303 Levy and Roth needs amplifying 65 Levy, H., teaches geometry 33 Levy, P., incomprehensible 65,81 Liapounov's theorem a la Kai Rander Buch 156 a la Lindenstrauss 156 library as laboratory 355 problems in Argentina 194 technique from Harry Levy 33 Lieber, T. C. Mits books 59 Lima da Silva Dias, Candido, tries to help 169 Lindemann and Littlewood 292 Linderholm. c., ergodic student and author 221 Lindstrum, A., Real Variable Club 66 linear algebra by Moore method 259 for two 106 from von Neumann 41 love-hate 40 not in analytic geometry 25 sold by Savage 150 linguistics is not philology 5 Liouville numbers in teaching 272 Littlewood, J. E. and Lindemann 293 zero-infinity law 121 log cabin 271 log, not In 271 logarithmic law of A. Weil 123,349 logic and Hilbert space 211 books to learn from 206 course in Chicago 215 differs from mathematics 203 examines symbols 205 formal and symbolic 202 from Kubitz 31 mathematics dictionary 207 Ph.D. students 212 "recessional" 216 INDEX revolutionized 202 summer institute 215 syllogistics 31 Lomnicki and Ulam mistake 68 Lomonosov newsbreak 323 Lorenzen, P., invitation 242 Los, J., linguistics courage 315 Lumer, G. at Oberwolfach 386 in Belgium 188 odd mathematics 187 swap problems 189 Lytle teaches Euclidean geometry 34 MacCluer, C., calls me mister 28 Mac Lane, S, chairman 248 Maharam review 120 Maker, P., Real Variable Club 66 Mandelbrot, B., and fractals 377 Markov, A. A., what did he do? 65 marriage problem, with Vaughan 78 Martin, R. S. as teacher 66 comes to Illinois 51 on Tarski 66 Martin, W. T. chairman at Syracuse 108 first appointments 108 leaves Syracuse 121 with Netzorg 46 martingale to Doob 370 Massera, J. helps found Instituto 184 in parliament 188 leave of absence 187 mathematics and politics 186 master key for gentlemen 130 master's degree, consolation prize 42 master's exam in philosophy 39 Mate, A., large cardinals 316 mathematical daydreams 30 mathematical machines, theory of 244 Mathematical Reviews, how it works 120 mathematicians are peripatetic 167 mathematics as an art 391 made difficult 222 not deductive 321 Maxwell "explained" 238 Mayer, W., half a homology 88 McCarthy hysteria 219 McKnight, J. D., at Miami 330 McLaughlin, J. E. on de Branges 328 shares linear algebra vice 333 measure spaces, all the same 100 Measure Theory begins 128 false exercise 143 415 how written 143 last word 142 measure theory panel at Cambridge congress 164 meetings of the AMS 106-107 memos from the chairman 361 Metallurgical Laboratory 114 method of successive approximations 325 Michal, A. D., peripatetic seminar 373 Michael, E., door buzzer and Saturday lunch 154 Michigan almost Chicago 366 Executive Committee 351 mathematical advisors 333 Ph.D. students 278 terms of offer 250 Miles, H., teaches algebra 28 Miller, G. A., millionaire 56 Miller, Blichfeldt, and Dickson Linear Groups 56 Miller, J. B., sends Ken Harrison 368 Milman, D., says they're crazy 387 minutes per week per class 269 mislead or lie 113 monotone class theorem, a proof 402 Monthly articles versus notes 382 editorial committee 373 editors 380 sneered at 379 Mookini, E. Hawaii chairman 343 negotiations 345 Moore, E. H., dictum on generalization 191 Moore, R. L. and Lumer 187 "Challenge in the Classroom" 257 colloquium book 255 don't read, think 257 method and covering material 261 student interview 257 Texas, Socratic, discovery 257 versus empty set 255 Morgenstern, 0., and von Neumann 100 Morley triangle 34 Morrison, P., erudite 237 Morrow, G., villain and hero 38-40 Morse, M., loves math. 401 Moscow mathematicians 295 smells 299 visitors 314 musical taste Haydn vii Brahms and Bach 29 416 INDEX Nachbin, L chair candidacy 178 editor of Summa 157 trying to arrange Brazil visit 169 Nagy, Sz. B. at Oberwolfach 385 honorarium check 132 not in Edinburgh 244 Naimark, M. A., charming host 310 Naive Set Theory, on Hewitt's bed, mistranslated 246 Nakata, J., perfect secretary 357 napkins, stochastic at Oberwolfach 385 negative three, no-minus three, yes 271 Nelson, E., as Chicago student 154 Netzorg, D. explains linear algebra 46 formal power series 45 good Illinois Ph.D. 80 Newton, I., and the binomial theorem 31 New York Times in Budapest 287 Nikodym, 0., on Radon-Nikodym 79 Nikolskii, S. M., banquet 311 non-standard analysis for mathematicians 204 Noriund, N. E., prolific publisher 41 Norse mythology from Flom 34 Northrop, E., at the College 122 Note about notes 229 number theory from Carmichael 40 Oberwolfach isolated institute 384 stochastic napkins 385 Olmsted, J., helpful in Princeton 85 ontogeny recapitulates phylogeny 270 operator methods in classical mechanics 101 Oppenheimer, J. R. as director 240 in Hungarian 317 optional skipping 75 ordering of employers 165 Oregon State, first job 83 Orloff, D., first student 159 Ornstein, D. and non-invariant measure 237 in courses 202 Parrott, S., breaks mug 278 passions, two are too many 53 passport application withdrawn 218 granted 219 patriotism in Hungary 8 Peano, G. algebra 212 arithmetic 215 axioms 203 pencil sharpening 322 peripatetic seminar 373 permanent visitors at Chicago 138 personal tastes: dark beer, long walks, TM vii Ph.D. candidates private seminar 279 versus advisor 274 Ph.D., do something else 278 Ph.D. rating system 66 Ph.D. supervision brainstorming 276 competition 274 Hilbert 272 hoarding 274 regular meetings 276 weaning students 277 Ph.D. theses zigzag 275 philology is not linguistics 5 philosophy as major? 24 courses 38 of mathematics in Monthly? 381 photograph sources 381 physics as spectator sport 14 in high school 15 Pitcher, E. on voting procedure 227 Pitt, H. R., in Nottingham 217 Plessner, A. I., on mUltiplicity 158 Poincare, H. conjecture and Smale 398 just a rumor 35 pointless spaces 120 polyadic algebras are mathematics 210 mirror mathematics 213 P6lya's dictum 77 P6lya-Szego and Savage 151 polynomially compact operators 320 pommy is English 281 Pontrjagin, L. S. discovers duality 210 English 312 Topological Groups, a thriller 93, 200 positive approximation and Dieudonne 324 Potapova, V. P., on shifts 254 prelims on basic knowledge 60 prepare for classes 135 Princeton matrix course 95 song 86 probability course bad at Illinois 34 is measure theory 325 Proceedings, AMS division of labor 223 managing editor 224 INDEX Proceedings, NAS, in mathematics library 106 proceedings of conferences are bad 369 professor promotion 160 Prohorov, Y. V., almost not met 311 prompt, plain, brief 334 proof, definition of 205 propositional calculus, what is it? 206 function 208 logic, computational lecture 39 pro's aim for perfection 265 are scholars 268 use medium 266 pseudo-metric, not semi- 339 Punta del Este conference Establier arrangements 192, 193 going there 193 invitations 193 lectures. monitored 194 note taking 194 Pythagorean theorem classified 115 Quadrangle Club, department meetings 130 quantification is cylindrification 210 quasitriangular, invariant subspaces? 320 questionnaires make bad theses 335 questions, open-ended 146 Radiation Laboratory invents radar 115 Radjavi-Rosenthal, operator te.rminology 340 Radon-Nikodjim theorem and sufficient statistics 151 Rainich, Y., thesis director 275 Ramsey's theorem enters logic 203 random alms in the real world 118 Rankin, R. A., reviews in Gaelic 292 R as a grade 153 Rasiowa and Sikorksi, algebraic logic 210 rating of mathematicians 305 rational points in Hilbert space 91 reading helps teaching 8 reading a paper 39 Real Variable Club at lIlinois 66 real variables at Indiana 366 "Recent Progress in Ergodic Theory" 235 recommendation how to write 334 Institute applicant 337 not the whole truth 337 Ph.D. program 341 publisher 341 Singer 338 should compare 337 recreational mathematics, Dieudonne 325 417 recruitment committee constituency 352 recursive number theory, brilliant and ugly 212 reductive algebra, not Hermitian 340 referee Boolean and prompt 121 how and why 119 referees, two at a time 383 Regellosigkeit 74 regional conferences 369 Reichenbach, H., mystic nonsense 68 rejected papers 321 reprint collection a good thing 67, 68 research by the clock 324 in bed 324 in ergodic theory 112 needs tranquility 118 teachers must do it 322 via special cases 324 via writing 323 why do it? 322 Revesz, P., Budapest host 285 reviews how and why 119 of books 223 of papers 120 Rey Pastor interview 169 rhetoric at Illinois 23 Riesz. F. and M. 138 Riesz, M., English 172 Riesz representation theorem, typical special case 324 Rimoldi, H. Chicago reminiscences 180 working conditions 18l Robbins, H. corrects "Foundations of Probability" 113, 114 threatens measure theory 127 Robinson, A. complains about review 223 sends preprint 204 Rogel's Thesaurus, how to use 395 Rohlin, V. A. and Plessner, on multiplicity 158 as host 303 disagreement 112 roommates Stanley Fry 20 AI Burton et al. 29 Room, T. G., and family 280 Rosenberg, A., editor of Monthly 399 Rosenthal, A., Landau's roommate 60 Rosenthal, P., finishes thesis 278 Rovnyak, J. defective merchandise 329 with de Branges 326 royalties, authors' and editors' 239 Rudin, M. E., student of Moore 256 Ruibal, Uruguay computer 184 418 INDEX rules, learn them 224 Rumanian school 320 Russell and Whitehead logic is attractive 202 Principia Mathematica 31 Russia automobiles vandalized 296 bugged room? 297 camera 295 domestic passports 299 doors to hotels 299 hotel service 296 royalty accounts 303 service in dining room 297-298 street maps 298 tipping 297 Russian exchange agreement 290 in Hungary 287 is not funny 59 Saaty, T. suspicious of . 'Glimpse into Hilbert Space" 390 St. Andrews colloquium. mathematical holiday 387-390 St. Isaac's Cathedral is beautiful 30 I Saks, S. not enough 127 the old testament 65 "same" mathematics 325 Samelson, H. arrives 102 birthday scam 103 Sammlung Goschen for straphangers 229 Sammy is Sammy 278 Santa Barbara invitation 370 Sarason, D. E. dinner guest 277 introvert, lecturer 276 writing thesis 276 Savage, L. J. joint course 75 learns from Dushnik 136 linear algebra 150 nystagmus 149 pemmican 151 P6lya-Szegii 151 sells FDYS 150 sufficient statistics 151 Wallis 152 SCFAS or Wabash west 375 Schaffer, J. J. impressive 187 in Pittsburgh 188 swap problems 189 Schreiber, M., thesis surprise 202, 222 Schiffer, M., as committee member 392 Schwartz. L., dangerous') 163 Scientific American. can't enjoy 237 scotophilia 243 second editions, don't do them 238,239 secretaries as officialdom 253 help and hindrance 223 Seifert-Threlfall, good book 58 semantic completeness is semisimplicity 207 semesters and quarters 269 seminar talks are not different n serendipity in library 65 service cop-out 220 departmental. professional, university 221 for failed scholars 221 to the profession 220 sesquicentennial seminar at Indiana 363 Severi, F .. in algebraic geometry 49 Shabat, B. Y., pays royalties 304 Shangri La is Los Alamos 114 Shaw, J. B., courtly gentleman 32 Sherman. S .. in Fine Hall 93 Shields, A. L., two-man seminar 73 shifts, paper on 254 shift transformations and entropy 236 Shilov, G. E., not met 312 Shirokov, F. Y .. not met 312 shortest letter 337 should I be a mathematicianry 148 Silber, J. blunt wholeheartedness 346 Texas negotiation 344, 345 simple topological spaces 340 simplify in research 77 Sinai, Y. G. at circus 299 seminar in English 312 Singer, L M" and Scheidy Everett 132133 Smale. S. and the Poincare conjecture 398 at Copacabana 93 Smithies, F. and N. at SI. Andrews 388 southern belles 256 Spanier. E., editor 327 Spanish at parties 172 Berlitz, Hugo, etc. 170 lecturing in 172 readings 171 through pictures 170 Spanish tutors. Castillo and Chiaraviglio 171 special cases, source of research 324 spectral inclusion theorems 191, 275 spectra paper. the weakest 95 spouse. believer') 70 419 INDEX Springer-Verlag editorship 228 fellow editors 232 square roots with Lumer and Schaffer 191 stable homotopy groups and Bernoulli numbers 72 stage fright, always 44 Stallings, J. R., and the Poincare conjecture 398 statistics departments as they grew 36 from Czuber 65 Steenrod, N. and his committee 39 j and thesis advice 274 death 393 stochastic processes 236 Steimley teaches higher analysis 32, 40 Steinhaus, H. in Fundamenta 65 on probability 68 Stokes' theorem, a foggy mystery 32 Stone. M. H. as chairman 123 book appears 49 chapter on mUltiplicity 69 four theorems 353 graduate curriculum 144 invites Measure Theory 127 promotion news 160 representation via Gelfand 245 resented at Chicago 249 thesis topics 274 Straus, E .. and Einstein 141 Strunk and White, Steenrod's hope 392 study, how to 69 Sturm, G. C. F., who was he~ 60 subnormal operators here to stay 157 sutlicient statistics with Savage 151 Summer Institute 1957 215 Sunder, V. S. at Indiana 370 at Santa Barbara 371 versus Korotkov and Bukhvalov 371 versus sexist language 371 sun worship 164 super course ASTP 110 surveys and expository papers 390 Swift, W" host to Wabash seminar 374 Sydney 1964 280 faculty 282 lecture notes lost 283 syllabi, good things 262, 263 syllogisms are obvious 202 symbolic and formal logic 202 syntactic concepts in logic 207 Syracuse colleagues 108, 109 grades 109 Master's candidates III standards 109 students III teaching load 112 Szabados, J., at Oberwolfach 386 Tamarkin, J. D., exam 271 Tarski, A. cylindric algebras 210 in Fundamenta 65 on truth 66 secrets 210 teach versus know 133 teaching assignments at Chicago 130 teaching begin with interesting facts 272 by challenge 271 do research 322 don '( lecture 273 freshman algebra 44 give advice 269 know the students 134 prepare 135 television course, expensive folly 359 thesis as roller coaster 76 not lifetime contract 214 "they" say what you said 70 Thompson, J. G., square root of the shift 202 Thrall, R, M. as graduate student 41 at Rad. Lab. 115 Tihonov, A. N. at banquet 311 theorem in marriage problem 78 Timoney, R., on Good Friday 113 tired liberal 55 Titchmarsh, E. C. double course 272 Theory of Functions versus Hardy 47 Toeplitz matrices, defined 319 tombstone, who invented0 403 topological groups course and students 200 exam 201 Tomier, E., mystic nonsense 68 topologies for measure-preserving transformations 112 town versus gown 366 Townsend's Real Variables is "good" 56 train to Leningrad, baffling 303 tranquil Tuesday for research Wednesday 118 translations into Russian 305 trigonometry how to prove identities 24 too much 24 420 INDEX trigonometry (cont.) vicarious 11 Tljitzinsky, W. J. Baire functions and Borel sets 58 thesis director 274 Truesdell, C. A., review of error 120 truth tables are homomorphisms 207 Trutt, D., in de Branges coterie 326 Tsar Pushka in Kremlin 307 Tukey, J. W., general topology and Zorn 36 Turing machines define computability 244 tutoring first experience 14 logic 31 two-person seminars 72 two referees at a time 383 typing 37 Ulam, S. Cambridge congress 164 measure theory 127 product spaces 68 unbiased estimation 121 undecidable propositions, fascinating and admirable 203 undetermined 1\ 160 unity of mathematical thought 211 University of Miami, Jimmy McKnight 330 University of Washington, visit 245 unreasonable effectiveness of mathematics 63 Uruguay arrival 173 automobile horns 198 banquet 197 bus doesn't stop 198 cold 177 computer 184 decision to go 168-170 Hamlet 198 heavenly haven 170 joint appointments 180 lecture audience 190 Martinis 176 meals 174 pensiones 173 probability course 190 red tape 183 research 191 schizophrenia 189 shabby 199 spying 196 Stone's comments 196 strikes 189 student control 183 teaching 188 teas 175 timing (hora Uruguaya) U.S. Embassy 195 vacation 183 weather 177 197 Vajda, S., at St. Andrews 388 valueless measures 120 van der Waerden, good book 58 Van Nostrand and Introduction to Hilbert Space 159 Mathematical Studies 229 undergraduate series 228 variable a pronoun 209 different sort 215 what is it? 208 Veblen and Young versus Coolidge 33 Veblen, O. lunches 243 operates 86 vector, what it is not 153 vector space over C 62, 64 Villegas, C. in Canada 188 nojob 182 Vinogradov, I. M., pontificates 303 von Mises, R. dicta about Kollektivs 65 mystic nonsense 68 Regel\osigkeit 74 von Neumann, J. guess on conjugacy problem 236 invariant measures 98 lectures, high-powered 92 the writing of the legend 393 Wabash seminar history 373 support it 266 Wald, A., a mathematics student 36 walking beginning 241 near Ann Arbor, and the record 331 Wallen, L. J. comes to Hawaii 347 in ergodic theory 202 Wallis, A., writes about Savage 152 Wang, T., shows off 47 war and the draft 105 exciting but remote 84 Weierstrass, K., just a rumor 35 Weil,A. at faculty meetings 130 L' integration. .. 200 logarithmic law 123, 349 loves math 401 monothetic groups 103 INDEX Ph.D. students 333 wants Grothendieck 130 WeIch, F. on Moore 255 Real Variable Club 66 West, T. T., and haitch 293 Weyl, H. dictates book 143 interview 89 loves math 401 Weyl-von Neumann theorem 403 white elephant in Hermitage 301 Whittaker and Watson, unlike Townsend 56 Wielandt, H., on commutators 191 Wiener, N. as teacher, 134 insecure 107 wife of forty years vii Wigner, E. and cranks 334 on applied mathematics 63 Wilder, R. L. at 80 74 student of Moore 255 Wilf, H., computer science editor 380 Wilhelmina versus vectors 152 Wilks, S. S., a mathematics student 36 Williams, Tennessee, in Hungarian 317 Wintner, A. on commutators 191 421 Spektraltheorie sweetens life 82 words Artin yes 5 Courant no 5 Lefschetz no 5 versus Bessel's inequality 4 versus numbers 3 Zariski yes 5 Wos, L., waves his hand 155 writing as thinking 8 wrong solutions to trivial problems 120 yellow series, at least two 230 Zaanen, A., at Oberwolfach 386 Zariski, 0., loves math 401 Zeeman, E. C., and the Poincare conjecture 398 Zermelo-Fraenkel axioms give set theory 213 zero-infinity law 121 Zorn, M. A. asks questions 363 exam 271 in German 387 lemma 36 Good Friday 114 Zygmund, A., at faculty meetings 130

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