Rudi Mathematici

x4-8180x3+25090190x2-34200948100x+17481136677369=0 Rudi Mathematici Gennaio

1 1 M (1803) Guglielmo LIBRI Carucci dalla Somaja APMO 1989 [1] (1878) Agner Krarup ERLANG (1894) Satyendranath BOSE K (1912) Boris GNEDENKO Siano x1 , x2 , , xn numeri reali 2 G (1822) Rudolf Julius Emmanuel CLAUSIUS n (1905) Lev Genrichovich SHNIRELMAN positivi e sia S = x . (1938) Anatoly SAMOILENKO å i i=1 3 V (1917) Yuri Alexeievich MITROPOLSHY 4 S (1643) Isaac NEWTON Provare che e`: 5 D (1838) Marie Ennemond Camille JORDAN n n S i (1871) Federigo ENRIQUES (1+ x ) £ (1871) Gino FANO Õ i å i! 2 6 L (1807) Jozeph Mitza PETZVAL i=1 i=0 (1841) Rudolf STURM 7 M (1871) Felix Edouard Justin Emile BOREL Dizionario di Matematica (1907) Raymond Edward Alan Christopher PALEY 8 M (1888) Richard COURANT Chiaramente: Non ho nessuna voglia (1924) Paul Moritz COHN di scrivere tutti i passaggi. (1942) Stephen William HAWKING 9 G (1864) Vladimir Adreievich STELKOV Prima Legge della Matematica 10 V (1875) Issai SCHUR Applicata: tutte le serie infinite (1905) Ruth MOUFANG convergono al loro primo termine. 11 S (1545) Guidobaldo DEL MONTE (1707) Vincenzo RICCATI (1734) Achille Pierre Dionis DU SEJOUR A mathematician's reputation rests 12 D (1906) Kurt August HIRSCH on the number of bad proofs he has 3 13 L (1864) Wilhelm Karl Werner Otto Fritz Franz WIEN given. (1876) Luther Pfahler EISENHART (1876) Erhard SCHMIDT Abram BESICOVITCH 14 M (1902) Alfred TARSKI Probabilities must be regarded as 15 M (1704) Johann CASTILLON analogous to the measurements of (1717) Mattew STEWART (1850) Sofia Vasilievna KOVALEVSKAJA physical magnitudes; that is to say, 16 G (1801) Thomas KLAUSEN they can never be known exactly, but 17 V (1847) Nikolay Egorovich ZUKOWSKY only within certain approximation. (1858) Gabriel KOENIGS 18 S (1856) Luigi BIANCHI Emile BOREL (1880) Paul EHRENFEST I have no certainties, at most 19 D (1813) Rudolf Friedrich Alfred CLEBSCH (1879) Guido FUBINI probabilities. (1908) Aleksandr Gennadievich KUROS Renato CACCIOPPOLI 4 20 L (1775) Andre` Marie AMPERE (1895) Gabor SZEGO What I tell you three times is true. (1904) Renato CACCIOPPOLI 21 M (1846) Pieter Hendrik SCHOUTE Charles DODGSON (1915) Yuri Vladimirovich LINNIK 22 M (1592) The proof of the Hilbert Basis (1908) Lev Davidovich LANDAU Theorem is not : it is 23 G (1840) Ernst ABBE (1862) David HILBERT theology. 24 V (1891) Abram Samoilovitch BESICOVITCH Camille JORDAN (1914) Vladimir Petrovich POTAPOV 25 S (1627) Robert BOYLE Probabilities must be regarded as (1736) Joseph-Louis LAGRANGE analogous to the measurement of (1843) Karl Herman Amandus SCHWARTZ physical magnitudes: they can never 26 D (1799) Benoit Paul Emile CLAPEYRON be known exactly, but only within 5 27 L (1832) Charles Lutwidge DODGSON certain approximation. 28 M (1701) Charles Marie de LA CONDAMINE (1892) Carlo Emilio BONFERRONI Emile BOREL 29 M (1817) William FERREL (1888) Sidney CHAPMAN God not only plays dice. He also 30 G (1619) sometimes throws the dice where they 31 V (1715) Giovanni Francesco FAGNANO dei Toschi cannot be seen. (1841) Samuel LOYD (1896) Sofia Alexandrovna JANOWSKAJA Stephen HAWKING

www.rudimathematici.com Rudi Mathematici Febbraio

5 1 S (1900) John Charles BURKILL APMO 1989 [2] 2 D (1522) Lodovico FERRARI Provare che l'equazione 6 3 L (1893) Gaston Maurice JULIA 2 2 2 2 4 M (1905) Eric Cristopher ZEEMAN 6 * (6a + 3b + c ) = 5n 5 M (1757) Jean Marie Constant DUHAMEL non ha soluzioni intere tranne 6 G (1612) Antoine ARNAULD (1695) Nicolaus (II) BERNOULLI a = b = c = n = 0 7 V (1877) Godfried Harold HARDY (1883) Eric Temple BELL Dizionario di Matematica 8 S (1700) Daniel BERNOULLI (1875) Francis Ysidro EDGEWORTH Banale: Se devo spiegarvi come si fa 9 D (1775) Farkas Wolfgang BOLYAI questo, avete sbagliato aula. (1907) Harod Scott MacDonald COXETER 7 10 L (1747) Aida YASUAKI Ci sono due gruppi di persone al 11 M (1800) William Henry Fox TALBOT (1839) Josiah Willard GIBBS mondo: quelli che credono il mondo (1915) Richard Wesley HAMMING possa essere diviso in due gruppi di 12 M (1914) Hanna CAEMMERER NEUMANN persone e gli altri. 13 G (1805) Johann Peter Gustav Lejeune DIRICHLET Connaitre, decouvrir, communiquer. 14 V (1468) Johann WERNER (1849) Hermann HANKEL Telle est la destinée d'un savant (1896) Edward Artur MILNE François ARAGO 15 S (1564) (1861) Alfred North WHITEHEAD Common sense is not really so (1946) Douglas HOFSTADTER common 16 D (1822) Francis GALTON (1853) Georgorio RICCI-CURBASTRO Antoine ARNAULD (1903) Beniamino SEGRE 8 17 L (1890) Sir Ronald Aymler FISHER "Obvious" is the most dangerous word (1891) Adolf Abraham Halevi FRAENKEL in mathematics. 18 M (1404) Leon Battista ALBERTI Eric Temple BELL 19 M (1473) Nicolaus COPERNICUS 20 G (1844) Ludwig BOLTZMANN ...it would be better for the true 21 V (1591) Girard DESARGUES if there were no (1915) Evgenni Michailovitch LIFSHITZ mathematicians on hearth. 22 S (1903) Frank Plumpton RAMSEY Daniel BERNOULLI 23 D (1583) Jean-Baptiste MORIN (1951) Shigefumi MORI ...an incorrect theory, even if it cannot 9 24 L (1871) Felix BERNSTEIN be inhibited bay any contradiction 25 M (1827) Henry WATSON that would refute it, is none the less 26 M (1786) Dominique Francois Jean ARAGO incorrect, just as a criminal policy is 27 G (1881) Luitzen Egbertus Jan BROUWER none the less criminal even if it 28 V (1735) Alexandre Theophile VANDERMONDE cannot be inhibited by any court that (1860) Herman HOLLERITH would curb it. Jan BROUWER Mathemata mathematici scribuntur Nicolaus COPERNICUS

www.rudimathematici.com Rudi Mathematici Marzo

9 1 S (1611) John PELL APMO 1989 [3] 2 D (1836) Julius WEINGARTEN Siano A1, A2, A3 tre punti sul piano e 10 3 L (1838) George William HILL (1845) sia, per notazione, A4=A1, A5=A2. Per 4 M (1822) Jules Antoine LISSAJUS n=1, 2, e 3 supponiamo che Bn sia il 5 M (1512) Gerardus MERCATOR punto medio di AnAn+1, e che Cn sia il (1759) Benjamin GOMPERTZ (1817) Angelo GENOCCHI punto medio di AnBn. Supponiamo 6 G (1866) Ettore BORTOLOTTI che AnCn+1 e BnCn+2 si incontrino in 7 V (1792) William HERSCHEL Dn, e che AnBn+1 si incontrino in En. (1824) Delfino CODAZZI Calcolare il rapporto tra l'area del (1851) George CHRYSTAL 8 S triangolo D1D2D3 e l'area del 9 D (1818) Ferdinand JOACHIMSTHAL (1900) Howard Hathaway AIKEN triangolo E1E2E3. 11 10 L (1864) William Fogg OSGOOD Dizionario di Matematica 11 M (1811) Urbain Jean Joseph LE VERRIER (1853) Salvatore PINCHERLE Si puo` facilmente dimostrare che: 12 M (1685) George BERKELEY Servono non piu` di quattro ore per (1824) Gustav Robert KIRKHHOFF (1859) Ernesto CESARO dimostrarlo. 13 G (1861) Jules Joseph DRACH (1957) Rudy D'ALEMBERT Teorema: tutti i numeri sono noiosi. 14 V (1864) Jozef KURSCHAK (1879) Albert EINSTEIN Dimostrazione (per assurdo). 15 S (1860) Walter Frank Raphael WELDON Supponiamo x sia il primo numero (1868) Grace CHISOLM YOUNG non noioso. Chi se ne frega? 16 D (1750) Caroline HERSCHEL (1789) Georg Simon OHM Mathematics is the most beautiful (1846) Magnus Gosta MITTAG-LEFFLER 12 17 L (1876) Ernest Benjamin ESCLANGON and the most powerful creation of the (1897) Charles FOX human spirit. Mathematics is as old 18 M (1640) Philippe de LA HIRE (1690) Christian GOLDBACH as Man. (1796) Jacob STEINER Stefan BANACH 19 M (1862) Adolf KNESER (1910) Jacob WOLFOWITZ In mathematics the art of proposing a 20 G (1840) Franz MERTENS question must be held on higher value (1884) Philip FRANCK (1938) Sergi Petrovich NOVIKOV than solving it. 21 V (1768) Jean Baptiste Joseph FOURIER (1884) George David BIRKHOFF Georg CANTOR 22 S (1917) Irving KAPLANSKY When writing about transcendental 23 D (1754) Georg Freiherr von VEGA issues, be transcendentally clear. (1882) Emmy Amalie NOETHER (1897) John Lighton SYNGE Rene` DESCARTES 13 24 L (1809) Joseph LIOUVILLE (1948) Sun-Yung (Alice) CHANG The search for truth is more 25 M (1538) Christopher CLAUSIUS important than its possession. 26 M (1848) Konstantin ADREEV Albert EINSTEIN (1913) Paul ERDOS 27 G (1857) Karl PEARSON Property is a nuisance. 28 V (1749) Pierre Simon de LAPLACE Paul ERDOS 29 S (1825) Francesco FAA` DI BRUNO Don't worry about people stealing (1873) Tullio LEVI-CIVITA (1896) Wilhelm ACKERMAN your ideas. If your ideas are any good, 30 D (1892) Stefan BANACH you'll have to ram them down 14 31 L (1596) Rene` DESCARTES people's throat. Howard AIKEN is the noblest branch of physics. William OSGOOD

www.rudimathematici.com Rudi Mathematici Aprile

14 1 M (1640) Georg MOHR (1776) Marie-Sophie GERMAIN APMO 1989 [4] (1895) Alexander Craig AITKEN Sia S un insieme formato da m 2 M (1934) Paul Joseph COHEN coppie (a,b) di interi positivi con la 3 G (1835) John Howard Van AMRINGE (1892) Hans RADEMACHER proprieta` che 1 £ a < b £ n . (1900) Albert Edward INGHAM (1909) Stanislaw Marcin ULAM Mostrare che esistono almeno (1971) Alice RIDDLE n 2 4 V (1809) Benjamin PEIRCE (1842) Francois Edouard Anatole LUCAS m - (1949) Shing-Tung YAU 4m * 4 5 S (1588) Thomas HOBBES 3n (1607) Honore` FABRI (1622) Vincenzo VIVIANI triple(a,b,c) tali che (a,b), (a,c) e (1869) Sergi Alexeievich CHAPLYGIN (b,c) appartengono a S. 6 D 15 7 L (1768) Francais Joseph FRANCAIS Dizionario di Matematica 8 M (1903) Marshall Harvey STONE Verificate per vostro conto: Questa e` 9 M (1791) George PEACOCK la parte noiosa della dimostrazione. (1816) Charles Eugene DELAUNAY (1919) John Presper HECKERT E` provato che la celebrazione dei 10 G (1857) Henry Ernest DUDENEY compleanni e` salutare. Le 11 V (1953) Andrew John WILES statistiche mostrano che chi celebra 12 S (1794) Germinal Pierre DANDELIN (1852) Carl Louis Ferdinand Von LINDEMANN piu` compleanni diventa piu` (1903) Jan TINBERGEN vecchio. 13 D (1728) Paolo FRISI (1813) Duncan Farquharson GREGORY How wonderful that we have met (1879) Francesco SEVERI with a paradox. Now we have some 16 14 L (1629) Christiaan HUYGENS hope of making progress. 15 M (1452) Leonardo da VINCI (1548) Pietro Antonio CATALDI Niels BOHR (1707) Leonhard EULER (1809) Herman Gunther GRASSMANN The notion of a set is too vague for 16 M (1682) John HADLEY the continuum hypothesis to have a (1823) Ferdinand Gotthold Max EISENSTEIN positive or negative answer 17 G (1798) Etienne BOBILLIER (1853) Arthur Moritz SCHONFLIES Paul COHEN 18 V (1907) Lars Valerian AHLFORS (1918) Hsien Chung WANG Any good idea can be stated in fifty (1949) Charles Luois FEFFERMAN words or less. 19 S (1880) Evgeny Evgenievich SLUTSKY (1883) Richard VIN MISES Stanislaw ULAM (1901) Kiyoshi OKA (1905) Charles EHRESMANN Mathematicians have tried in vain to 20 D (1839) Francesco SIACCI this day to discover some order in the 17 21 L (1652) Michel ROLLE sequence of prime numbers, and we (1774) Jean Baptiste BIOT (1875) Teiji TAKAGI have reason to believe that it is a 22 M (1811) Otto Ludwig HESSE mystery into which the human mind (1887) Harald August BOHR will never penetrate. 23 M (1858) Max Karl Ernst Ludwig PLANCK Leonhard EULER 24 G (1863) Giovanni VAILATI 25 V (1849) Felix Christian KLEIN If anybody says he can think about (1900) Wolfgang PAULI quantum problems without getting (1903) Andrei Nicolayevich KOLMOGOROV giddy, that only shows he has not 26 S (1889) Ludwig Josef Johan WITTENGSTEIN understood the first thing about 27 D (1755) Marc-Antoine PARSEVAL des Chenes them. 18 28 L (1906) Kurt GODEL 29 M (1854) Jules Henri POINCARE` Max PLANCK 30 M (1777) Johann Carl Friedrich GAUSS (1916) Claude Elwood SHANNON

www.rudimathematici.com Rudi Mathematici Maggio

18 1 G (1825) Johann Jacob BALMER APMO 1989 [4] 2 V (1860) D`Arcy Wentworth THOMPSON (1905) Kazimierz ZARANKIEWITZ Determinare tutte le funzioni f da R 3 S (1842) Otto STOLZ in R per cui: (1860) Vito VOLTERRA 4 D (1845) William Kingdon CLIFFORD 1. f(x) e` strettamente crescente 19 5 L (1833) Lazarus Emmanuel FUCHS 2. f(x)+g(x)=2x per qualsiasi x (1897) Francesco Giacomo TRICOMI reale. 6 M (1872) Willem DE SITTER (1906) Andre` VEIL dove g(x) e` tale che f(g(x))=x e 7 M (1926) Alexis Claude CLAIRAUT (1854) Giuseppe VERONESE g(f(x))=x per qualsiasi x reale. (1881) Ebenezer CUNNINGHAM (1896) Pavel Sergieievich ALEXANDROV Dizionario di Matematica 8 G (1859) JOhan Ludwig William Valdemar JENSEN Suggerimento: Il modo piu` 9 V (1746) Gaspard MONGE (1876) Gilbert Ames BLISS complicato (dei vari possibili) per 10 S (1788) Augustin Jean FRESNEL dimostrare il teorema (1847) William Karl Joseph KILLING (1958) Piotr Rizierovich SILVERBRAHMS Le statistiche sono come i bikini. 11 D (1918) Richard Phillips FEYNMAN Quello che mostrano e` molto 20 12 L (1845) Pierre Rene`Jean Baptiste Henry BROCARD interessante, ma quello che (1902) Frank YATES nascondono e` cruciale. 13 M (1750) Lorenzo MASCHERONI 14 M (1832) Rudolf Otto Sigismund LIPSCHITZ (1863) John Charles FIELDS You may always depend on it that 15 G (1939) Brian HARTLEY which cannot be translated into good English and sound common 16 V (1718) Maria Gaetana AGNESI (1821) Pafnuti Lvovi CHEBYSHEV sense is bad algebra 17 S William CLIFFORD 18 D (1850) Oliver HEAVISIDE (1892) Bertrand Arthur William RUSSELL And since geometry is the right 21 19 L (1919) Georgii Dimitirievich SUVOROV foundation of all painting, I have 20 M (1861) Henry Seely WHITE decided to teach its rudiments and 21 M (1471) Albrecht DURER principles to all youngsters eager for (1792) Gustave Gaspard de CORIOLIS art... 22 G (1865) Alfred Cardew DIXON 23 V (1914) Lipa BERS Albrecht DURER 24 S Where did we get Schrodinger 25 D (1838) Karl Mikailovich PETERSON equation from? It's not possible to 22 26 L (1667) Abraham DE MOIVRE derive it from anything you know. It (1896) Yuri Dimitrievich SOKOLOV came out of the mind of Schrodinger. 27 M (1862) John Edward CAMPBELL Richard FEYNMAN 28 M (1676) Jacopo Francesco RICCATI (1710) Johann (II) BERNOULLI Nature is not embarrassed by 29 G (1882) Harry BATEMAN difficulties of analysis 30 V (1814) Eugene Charles CATALAN Augustin FRESNEL 31 S (1926) John KEMENY

www.rudimathematici.com Rudi Mathematici Giugno

22 1 D (1796) Sadi Leonard Nicolas CARNOT (1851) Edward Bailey ELLIOTT APMO 1990 [1] (1899) Edward Charles TITCHMARSH Nel triangolo ABC, siano D, E, F i 23 2 L (1895) Tibor RADO` punti medi di BC, AC, AB 3 M (1659) David GREGORY rispettivamente e sia G il baricentro 4 M (1809) John Henry PRATT del triangolo. 5 G (1814) Pierre LAurent WANTZEL (1819) John Couch ADAMS Per ogni valore dell'angolo BAC, 6 V (1436) Johann Muller REGIOMONTANUS quanti triangoli non simili esistono (1857) Aleksandr Michailovitch LYAPUNOV (1906) Max ZORN per cui AEGF e` un quadrilatero 7 S (1863) Edward Burr VAN VLECK ciclico? 8 D (1625) Giovanni Domenico CASSINI (1858) Charlotte Angas SCOTT Dizionario di Matematica (1860) Alicia Boole STOTT Forza bruta: Quattro casi speciali, tre 24 9 L (1885) John Edensor LITTLEWOOD soluzioni per enumerazione, due 10 M (940) Mohammad ABU`L WAFA Al-Buzjani (1887) Vladimir Ivanovich SMIRNOV applicazioni kilometriche 11 M (1937) David Bryant MUMFORD dell'induzione. (1888) Zygmunt JANYSZEWSKI 12 G Qual'e` la domanda odiata dalla 13 V (1831) (1876) William Sealey GOSSET (Student) distribuzione di Cauchy? (1928) John Forbes NASH Hai un momento? 14 S (1736) Charles Augustin de COULOMB (1856) Andrei Andreyevich MARKOV (1903) Alonzo CHURCH The sciences are like a beautiful river, 15 D (1640) Bernard LAMY of which the course is easy to follow (1894) Nikolai Gregorievich CHEBOTARYOV when it has acquired a certain 25 16 L (1915) John Wilder TUKEY regularity; but if one wants to go back 17 M (1898) Maurits Cornelius ESCHER to the source, one will find it nowhere, 18 M (1858) Andrew Russell FORSYTH (1884) Charles Ernest WEATHERBURN because it is everywhere; it is spread 19 G (1623) Blaise PASCAL so much [as to be] over all the surface (1902) Wallace John ECKERT of the earth; it is the same if one want 20 V (1873) Alfred LOEWY to go back to the origin of the sciences, 21 S (1781) Simeon Denis POISSON (1828) Giuseppe BRUNO one will find only obscurity, vague 22 D (1860) Mario PIERI ideas, vicious circles; and one loses (1864) Hermann MINKOWSKY oneself in the primitive ideas. (1910) Konrad ZUSE 26 23 L (1912) Alan Mathison TURING Sadi CARNOT 24 M (1880) Oswald VEBLEN It is easier to square the circle than to 25 M (1908) William Van Orman QUINE get round a mathematician 26 G (1824) William THOMPSON, Lord Kelvin (1918) Yudell Leo LUKE 27 V (1806) Augustus DE MORGAN Algebra goes to the heart of the matter 28 S (1875) Henri Leon LEBESGUE at it ignores the casual nature of 29 D (1888) Aleksandr Aleksandrovich FRIEDMANN particular cases. 27 30 L (1791) Felix SAVART Edward TITCHMARSH A good mathematical joke is better mathematics than a dozen of mediocre papers. John LITTLEWOOD

www.rudimathematici.com Rudi Mathematici Luglio

27 1 M (1643) Gottfried Wilhelm von LEIBNITZ (1788) Jean Victor PONCELET APMO 1990 [2] 2 M (1820) William John Racquorn RANKINE (1852) Siano a1, a2,...,an numeri reali 3 G (1807) Ernest Jean Philippe Fauque de JONQUIERE positivi, e sia sk la somma dei (1897) Jesse DOUGLAS prodotti di a1, a2,...,an presi k alla 4 V (1906) Daniel Edwin RUTHERFORD (1917) Michail Samuilovich LIVSIC volta. Mostrare che, per k=1, 2,...,n-1: 2 5 S æ nö 6 D (1849) Alfred Bray KEMPE L Sk S n-k ³ ç ÷ a1a2 an 28 7 L (1816) Johann Rudolf WOLF èk ø (1906) William FELLER (1922) Vladimir Aleksandrovich MARCHENKO Dizionario di Matematica 8 M (1760) Christian KRAMP 9 M (1845) George Howard DARWIN Dimostrazione elegante: Non richiede 10 G (1862) Roger COTES conoscenze specialistiche e occupa (1868) Oliver Dimon KELLOGG meno di dieci righe 11 V (1857) Sir Joseph LARMOR (1890) Giacomo ALBANESE Discutere con uno statistico e` come 12 S (1875) Ernest Sigismund FISCHER (1895) Richard BUCKMINSTER FULLER lottare con un maiale. Dopo alcune 13 D (1527) John DEE ore, vi accorgete che lui si sta (1741) Karl Friedrich HINDENBURG divertendo. 29 14 L 15 M (1865) Wilhelm WIRTINGER All possible definitions of probability (1906) Adolph Andrej Pavlovich YUSHKEVICH fall short of the actual practice. 16 M (1678) Jakob HERMANN (1903) Irmgard FLUGGE-LOTZ William FELLER 17 G (1831) Victor Mayer Amedee` MANNHEIM (1837) Wilhelm LEXIS I am a passenger of the spaceship 18 V (1013) Hermann von REICHENAU Hearth. (1635) Robert HOOKE (1853) Hendrich Antoon LORENTZ Richard BUCKMINSTER FULLER 19 S (1768) Francois Joseph SERVOIS The imaginary number is a fine and 20 D wonderful resource of the human 30 21 L (1620) Jean PICARD (1848) Emil WEYR spirit, almost an amphibian between (1849) Robert Simpson WOODWARD being and not being 22 M (1784) Friedrich Wilhelm BESSEL Gottfriewd LEIBNIZ 23 M (1775) Etienne Louis MALUS (1854) Ivan SLEZYNSKY A quantity that is increased or 24 G (1851) Friedrich Herman SCHOTTKY decreased of an infinitely small (1871) Paul EPSTEIN (1923) Christine Mary HAMILL quantity is neither increased nor 25 V (1808) Johann Benedict LISTING decreased 26 S (1903) Kurt MAHLER Johann BERNOULLI 27 D (1667) Johann BERNOULLI (1801) George Biddel AIRY ...The science of Nature has already (1848) Lorand Baron von EOTVOS been too long made only a work of the (1871) Ernst Friedrich Ferdinand ZERMELO brain and the fancy. It is now high 31 28 L (1954) Gerd FALTINGS time that it should return to the 29 M plainness and soundness of 30 M observations on material and obvious 31 G (1704) Gabriel CRAMER things. (1712) Johann Samuel KOENIG Robert HOOKE

www.rudimathematici.com Rudi Mathematici Agosto

31 1 V (1861) Ivar Otto BENDIXSON (1881) Otto TOEPLITZ APMO 1990 [3] 2 S (1856) Ferdinand RUDIO (1902) Mina Spiegel REES Consideriamo tutti i triangoli ABC 3 D (1914) Mark KAC aventi una base fissa AB e la cui 32 4 L (1805) Sir William Rowan HAMILTON altezza da C sia una costante h. (1838) John VENN Per quali di questi triangoli il (1802) 5 M prodotto delle altezze e` massimo? 6 M (1638) Nicolas MALEBRANCHE (1741) John WILSON Dizionario di Matematica 7 G (1868) Ladislaus Josephowitsch BORTKIEWITZ Similmente: Almeno una riga di 8 V (1902) Paul Adrien Maurice DIRAC questa dimostrazione e` uguale ad 9 S (1537) Francesco BAROZZI (Franciscus Barocius) una riga della dimostrazione 10 D (1602) Gilles Personne de ROBERVAL precedente 33 11 L (1730) Charles BOSSUT (1842) Enrico D`OVIDIO La lotteria e` una tassa sulle persone 12 M (1882) Jules Antoine RICHARD (1887) Erwin Rudolf Josef Alexander SCHRODINGER che non capiscono la statistica. 13 M (1625) Erasmus BARTHOLIN (1819) George Gabriel STOKES The divergent series are the invention (1861) Cesare BURALI-FORTI of the devil. 14 G (1530) Giovanni Battista BENEDETTI (1842) Jean Gaston DARBOUX Niels ABEL (1865) Guido CASTELNUOVO (1866) Charles Gustave Nicolas de la VALLEE` POUSSIN Two seemingly incompatible 15 V (1863) Aleksei Nikolaevich KRYLOV conceptions can each represent an (1892) Louis Pierre Victor duc de BROGLIE (1901) Petr Sergeevich NOVIKOV aspect of the truth. They may serve in 16 S (12773) Louis Beniamin FRANCOEUR turn to represent the facts without (1821) Arthur CAYLEY ever entering into direct conflict. 17 D (1601) Pierre de FERMAT Louis DE BROGLIE 34 18 L (1685) Brook TAYLOR 19 M (1646) John FLAMSTEED As for everything else, so for a (1739) Georg Simon KLUGEL mathematical theory: beauty can be 20 M (1710) Thomas SIMPSON (1863) Corrado SEGRE perceived but not explained (1882) Waclav SIERPINSKI Arthur CAYLEY 21 G (1789) Augustin Louis CAUCHY This result is too beautiful to be false: 22 V (1647) Denis PAPIN it is more important to have beauty in 23 S (1683) Giovanni POLENI (1829) Moritz Benedikt CANTOR one's equations than to have them fit 24 D (1561) Bartholomeo PITISCUS experiment. (1942) Karen Keskulla UHLENBECK 35 25 L (1561) Philip van LANSBERGE Paul DIRAC (1844) Thomas MUIR 26 M (1728) Johann Heinrich LAMBERT And perhaps, posterity will thank me (1875) Giuseppe VITALI for having shown it that the ancients 27 M (1858) Giuseppe PEANO did not know everything. 28 G (1796) Irenee Jules BIENAYME` Pierre de FERMAT 29 V (1904) Leonard ROTH There are surely worse things than 30 S (1856) Carle David Tolme` RUNGE (1906) Olga TAUSSKY-TODD being wrong, and being dull and 31 D (1821) Hermann Ludwig Ferdinand von HELMHOLTZ pedantic are surely among them. Mark KAC

www.rudimathematici.com Rudi Mathematici Settembre

36 1 L (1659) Joseph SAURIN (1835) William Stankey JEVONS APMO 1990 [5] 2 M (1878) Mauriche Rene` FRECHET (1923) Rene` THOM Mostrare che per qualsiasi intero 3 M (1814) James Joseph SYLVESTER n ³ 6 , esiste un esagono convesso (1884) Solomon LEFSCHETZ (1908) Lev Semenovich PONTRYAGIN che puo` essere diviso in esattamente 4 G (1809) Luigi Federico MENABREA n triangoli congruenti. 5 V (1667) Giovanni Girolamo SACCHERI (1725) Jean Etienne MONTUCLA Dizionario di Matematica 6 S (1859) Boris Jakovlevich BUKREEV Dimostrazione in due righe: Vi scrivo (1863) Dimitri Aleksandrovich GRAVE solo la conclusione, meno ne sapete, 7 D (1707) George Louis Leclerc comte de BUFFON (1955) Efim ZELMANOV meno domande farete. 37 8 L (1584) Gregorius SAINT-VINCENT (1588) Marin MERSENNE Uno statistico e` un'esperto in grado 9 M (1860) Frank MORLEY di partire da una premessa azzardata 10 M (1839) Charles Sanders PEIRCE e arrivare ad una conclusione 11 G (1623) Stefano degli ANGELI evidente. (1877) sir James Hopwood JEANS 12 V (1891) Antoine Andre` Louis REYNAUD The attempt to apply rational (1900) Haskell Brooks CURRY arithmetic to a problem in geometry 13 S (1873) Constantin CARATHEODORY (1885) Wilhelm Johann Eugen BLASCHKE resulted in the first crisis in the 14 D (1858) Henry Burchard FINE (1891) Ivan Matveevich VINOGRADOV history of mathematics. The two 38 15 L (973) Abu Arrayhan Muhammad ibn Ahmad AL`BIRUNI relatively simple problems (the (1886) Paul Pierre LEVY determination of the diagonal of a 16 M (1494) Francisco MAUROLICO (1736) Johann Nikolaus TETENS square and that of the circumference 17 M (1743) Marie Jean Antoine Nicolas de Caritat de CONDORCET of a circle) revealed the existence of (1826) Georg Friedrich Bernhard RIEMANN new mathematical beings for which 18 G (1752) Adrien Marie LEGENDRE no place could be found in the 19 V (1749) Jean Baptiste DELAMBRE rational domain. 20 S (1842) Alexander Wilhelm von BRILL (1861) Frank Nelson COLE David DANTZIG 21 D (1899) Juliusz Pawel SCHAUDER [de Prony's Tables] will not serve in 39 22 L (1765) Paolo RUFFINI (1769) Louis PUISSANT the usual cases, but only in the (1803) Jaques Charles Francois STURM extraordinary cases. 23 M (1768) William WALLACE (1900) David van DANTZIG Jean DELAMBRE 24 M (1501) Girolamo CARDANO I believe that proving is not a natural (1625) Johan DE WITT (1801) Michail Vasilevich OSTROGRADSKI activity for mathematicians. 25 G (1819) George SALMON Rene' THOM (1888) Stefan MAZURKIEWICZ 26 V (1688) Willem Jakob `s GRAVESANDE The early study of Euclid make me a (1854) Percy Alexander MACMAHON (1891) Hans REICHENBACH hater of geometry. 27 S (1855) Paul Emile APPEL James SYLVESTER (1876) Earle Raymond HEDRICK (1919) James Hardy WILKINSON If it's just turning the crank it's 28 D (1698) Pierre Louis Moreau de MAUPERTUIS algebra, but if it's got an idea on it (1761) Ferdinand Francois Desire` Budan de BOISLAURENT (1873) Julian Lowell COOLIDGE it's topology. 40 29 L (1561) Adriaan van ROOMEN (1812) Adolph GOPEL Solomon LEFSCHETZ 30 M (1775) Robert ADRAIN For the intrinsic evidence of his (1829) Joseph WOLSTENHOLME (1883) Ernst HELLINGER creation, the Great Architect of the Universe now begins to appear as a pure mathematician. James JEANS

www.rudimathematici.com Rudi Mathematici Ottobre

40 1 M (1671) (1898) Bela KEREKJARTO` APMO 1991 [1] 2 G (1825) John James WALKER (1908) Arthur ERDELYI Sia G il baricentro del triangolo ABC 3 V (1944) Pierre Rene` DELIGNE e M sia il punto medio di BC. Sia X 4 S (1759) Louis Francois Antoine ARBOGAST su AB e Y su AC in modo tale che i (1797) Jerome SAVARY punti X, Y, e G siano collineari e XY e 5 D (1732) Nevil MASKELYNE (1781) Bernhard Placidus Johann Nepomuk BOLZANO BC siano parallele. Supponiamo che (1861) Thomas Little HEATH XC e GB si intersechino in Q e YB e 41 6 L (1552) Matteo RICCI GC si intersechino in P. Mostrare (1831) Julius Wilhelm Richard DEDEKIND (1908) Sergei Lvovich SOBOLEV che il triangolo MPQ e` simile al 7 M (1885) Niels BOHR triangolo ABC. 8 M (1908) Hans Arnold HEILBRONN Dizionario di Matematica 9 G (1581) Claude Gaspard BACHET de Meziriac (1704) Johann Andrea von SEGNER In breve: Sta finendo la mia ora, (1873) Karl SCHWARTZSCHILD quindi parlero` e scrivero` piu` veloce. 10 V (1861) Heinrich Friedrich Karl Ludwig BURKHARDT 11 S (1675) Samuel CLARKE Uno statistico ha qualcosa da dire (1777) Barnabe` BRISSON (1885) Alfred HAAR con i numeri, mentre un politico ha (1910) Cahit ARF da dire qualcosa sui numeri (1860) Elmer SPERRY 12 D 30 31 42 13 L (1890) Georg FEIGL 2 (2 -1) is the greatest perfect (1893) Kurt Werner Friedrich REIDEMEISTER (1932) John Griggs THOMSON number that will ever be discovered, 14 M (1687) Robert SIMSON for, as they are merely curious (1801) Joseph Antoine Ferdinand PLATEAU without being useful, it is not likely (1868) Alessandro PADOA 15 M (1608) that any person will attempt to find a (1735) Jesse RAMSDEN number beyond it. (1776) Peter BARLOW 16 G (1879) Philip Edward Bertrand JOURDAIN Peter BARLOW 17 V (1759) Jacob (II) BERNOULLI As professor in the Polytechnic School (1888) Paul Isaac BERNAYS in Zurich I found myself for the first (1741) John WILSON 18 S time obliged to lecture upon the 19 D (1903) Jean Frederic Auguste DELSARTE (1910) Subrahmanyan CHANDRASEKHAR elements of the differential 43 20 L (1632) Sir Cristopher WREN and felt more keenly than ever before (1863) William Henry YOUNG (1865) Aleksandr Petrovich KOTELNIKOV the lack of a really scientific 21 M (1677) Nicolaus (I) BERNOULLI foundation of the arithmetic. (1823) Enrico BETTI (1855) Giovan Battista GUCCIA Richard DEDEKIND (1893) William LEonard FERRAR Prediction is very difficult, especially 22 M (1587) Joachim JUNGIUS (1895) Rolf Herman NEVANLINNA about the future. (1907) Sarvadaman CHOWLA Niels BOHR 23 G (1865) Piers BOHL 24 V (1804) Wilhelm Eduard WEBER Newton is, of course, the greatest of (1873) Edmund Taylor WITTAKER all Cambridge professors; he also 25 S (1811) Evariste GALOIS happens to be the greatest disaster 26 D (1849) Ferdinand Georg FROBENIUS (1857) Charles Max MASON that ever befell not merely Cambridge (1911) Shiing-Shen CHERN mathematics in particular, but 44 27 L (1678) Pierre Remond de MONTMORT British mathematical science as a (1856) Ernest William HOBSON whole. 28 M (1804) Pierre Francois VERHULST 29 M (1925) Klaus ROTH Leonard ROTH 30 G (1906) Andrej Nikolaevich TIKHONOV A mathematician who is not also 31 V (1815) Karl Theodor Wilhelm WEIERSTRASS something of a poet will never be a perfect mathematician Karl WEIERSTRASS

www.rudimathematici.com Rudi Mathematici Novembre

44 1 S (1535) Giambattista DELLA PORTA APMO 1991 [3] 2 D (1815) George BOOLE Siano a1, a2,..., an e b1, b2,..., bn 45 3 L (1867) Martin Wilhelm KUTTA (1878) Arthur Byron COBLE numeri reali positivi per cui 4 M (1744) Johann (III) BERNOULLI n n (1865) Pierre Simon GIRARD a = b . 5 M (1848) James Whitbread Lee GLAISHER å i å i (1930) John Frank ADAMS i =1 i =1 6 G (1781) Giovanni Antonio Amedeo PLANA Mostrare che: 7 V (1660) Thomas Fantet DE LAGNY (1799) Karl Heinrich GRAFFE n (1898) Raphael SALEM a n 2 å i 8 S (1656) Edmond HALLEY ai i =1 (1846) Eugenio BERTINI ³ (1848) Fredrich Ludwig Gottlob FREGE å i =1 ai + bi 2 (1854) Johannes Robert RYDBERG (1869) Felix HAUSDORFF Dizionario di Matematica 9 D (1847) Carlo Alberto CASTIGLIANO (1885) Theodor Franz Eduard KALUZA Procedendo formalmente: Manipolare (1885) Hermann Klaus Hugo WEYL (1906) Jaroslav Borisovich LOPATYNSKY i simboli secondo le regole senza (1922) Imre LAKATOS avere la piu` pallida idea di cosa 46 10 L (1829) Helwin Bruno CHRISTOFFEL significhi. 11 M (1904) John Henry Constantine WHITEHEAD 12 M (1825) Michail Egorovich VASHCHENKO-ZAKHARCHENKO La statistica e` come il lampione per (1842) John William STRUTT Lord RAYLEIGH l'ubriaco: serve come supporto, non (1927) Yutaka TANIYAMA come illuminazione. 13 G (1876) Ernest Julius WILKZYNSKY (1878) Max Wilhelm DEHN Of the many forms of false culture, a 14 V (1845) Ulisse DINI premature converse with abstractions 15 S (1688) Louis Bertrand CASTEL (1793) Michel CHASLES is perhaps the most likely to prove (1794) Franz Adolph TAURINUS fatal to the growth of a masculine 16 D (1835) Eugenio BELTRAMI vigour of intellect. 47 17 L (1597) Henry GELLIBRAND (1717) Jean Le Rond D'ALEMBERT George BOOLE (1790) August Ferdinand MOBIUS Algebra is generous: she often gives 18 M (1872) Giovanni Enrico Eugenio VACCA (1927) Jon Leslie BRITTON more than is asked for 19 M (1894) Heinz HOPF (1900) Michail Alekseevich LAVRENTEV Jean D'ALEMBERT (1901) Nina Karlovna BARI Mathematics is the only instructional 20 G (1889) Edwin Powell HUBBLE (1924) Benoit MANDELBROT material that can be presented in an 21 V (1867) Dimitri SINTSOV entirely undogmatic way 22 S (1803) Giusto BELLAVITIS (1840) Emile Michel Hyacinte LEMOINE Max DEHN 23 D (1616) John WALLIS A can hardly meet with (1820) Issac TODHUNTER anything more undesirable than to 48 24 L (1549) Duncan MacLaren Young SOMERVILLE (1909) Gerhard GENTZEN have the foundations give way just as 25 M (1873) Claude Louis MATHIEU the work is finished. I was put in this (1841) Fredrich Wilhelm Karl Ernst SCHRODER position by a letter from Mr. Bertrand 26 M (1894) Norbert WIENER (1946) Enrico BOMBIERI Russell when the work was nearly 27 G (1867) Arthur Lee DIXON through in press. 28 V (1898) John WISHART Gottlob FREGE 29 S (1803) Christian Andreas DOPPLER (1849) Horace LAMB A modern mathematical proof is not (1879) Nikolay Mitrofanovich KRYLOV very different from a modern 30 D (1549) Sir Henry SAVILE machine: the simple fundamental principles are hidden under a of technical details. Hermann WEYL

www.rudimathematici.com Rudi Mathematici Dicembre

49 1 L (1792) Nikolay Yvanovich LOBACHEVSKY APMO 1991 [5] 2 M (1831) Paul David Gustav DU BOIS-RAYMOND (1901) George Frederick James TEMPLE Sono dati due cerchi tangenti tra di 3 M (1903) Sidney GOLDSTEIN loro e un punto P sulla loro tangente (1924) John BACKUS comune perpendicolare alla retta per 4 G (1795) Thomas CARLYLE i centri. Costruite con riga e 5 V (1868) Arnold Johannes Wilhelm SOMMERFELD (1901) Werner Karl HEISENBERG compasso tutti i cerchi tangenti ai 6 S (1682) Giulio Carlo FAGNANO dei Toschi due cerchi dati e passanti per P. 7 D (1647) Giovanni CEVA (1823) Leopold KRONECKER Dizionario di Matematica (1830) Antonio Luigi Gaudenzio Giuseppe CREMONA 50 8 L (1508) Regnier GEMMA FRISIUS Sorvoliamo sulla dimostrazione: (1865) Jaques Salomon HADAMARD Credetemi, e` vero (1919) Julia Bowman ROBINSON 9 M (1883) Nikolai Nikolaievich LUZIN La Vita, La Matematica e Tutto (1906) Grace Brewster MURRAY HOPPER (1917) Sergei Vasilovich FOMIN Quanto 10 M (1804) Karl Gustav Jacob JACOBI (1815) Augusta Ada KING Countess of LOVELACE · La vita e` complessa: ha 11 G (1882) Max BORN componenti reali e immaginarie 12 V (1832) Peter Ludwig Mejdell SYLOW 13 S (1724) Franz Ulrich Theodosius AEPINUS · Per un Matematico, la vita (1887) George POLYA reale e` un caso particolare. 14 D (1546) Tycho BRAHE 51 15 L (1802) Janos BOLYAI Errors using inadequate data are 16 M (1804) Wiktor Yakovievich BUNYAKOWSKY much less than those using no data at 17 M (1706) Gabrielle Emile Le Tonnelier de Breteuil du CHATELET all. (1835) Felice CASORATI (1842) Marius Sophus LIE Charles BABBAGE (1900) Dame Mary Lucy CARTWRIGHT Out of nothing I have created a 18 G (1917) Roger LYNDON strange new universe. 19 V (1783) Charles Julien BRIANCHON (1854) Marcel Louis BRILLOUIN Janos BOLYAI 20 S (1494) Oronce FINE (1648) Tommaso CEVA I am now convinced that theoretical (1875) Francesco Paolo CANTELLI physics is actual philosophy. 21 D (1878) Jan LUKASIEVIKZ (1932) John Robert RINGROSE Max BORN 52 22 L (1824) Francesco BRIOSCHI (1859) Otto Ludwig HOLDER It is a mathematical fact that the (1877) Tommaso BOGGIO casting of this pebble from my hand (1887) Srinivasa Aiyangar RAMANUJAN alters the centre of gravity of the 23 M (1872) Georgii Yurii PFEIFFER universe 24 M (1822) Charles HERMITE (1868) Emmanuel LASKER Thomas CARLYLE 25 G (1642) Isaac NEWTON (1900) Antoni ZYGMUND Shuffling is the only thing that 26 V (1780) Mary Fairfax Greig SOMERVILLE Nature cannot undo. (1791) Charles BABBAGE 27 S (1571) Johannes KEPLER Arthur EDDINGTON (1654) Jacob (Jacques) BERNOULLI Number theorists are like lotus eaters: 28 D (1808) Athanase Louis Victoire DUPRE` (1882) Arthur Stanley EDDINGTON having tasted this food they can never (1903) John von NEUMANN give it up. 53 29 L (1856) Thomas Jan STIELTJES Leopold KRONECKER 30 M (1897) Stanislaw SAKS 31 M (1872) Volodymyr LEVIYTSKY Logic merely sanctions the conquests (1896) Carl Ludwig SIEGEL of intuition. (1952) Vaughan Frederick Randall JONES Jaques HADAMARD If there is a problem you can't solve, then there is an easier problem you can't solve: find it. George POLYA

www.rudimathematici.com