Thursday, December 30, 2010

An abandoned education method

I did not follow the topic lately, but I am pleasantly surprised to notice that Reform mathematics appears listed in wikipedia's "List of abandoned education methods" (as of November 2009). See also MathematicallyCorrect.com and wheresthemath.com. Illustrating the spirit of the abandoned method, the following "mathematical" problem appears in a fifth-grade “Everyday Math” textbook (source: Fuzzy math: A nationwide epidemic):
A. If math were a color, it would be –, because –.
B. If it were a food, it would be –, because –.
C. If it were weather, it would be –, because –.

Fibonacci modulo m

An useful applet that computes the period of the Fibonacci sequence modulo m for m < 10000000 can be found here (part of Marc Renault's thesis).

Wednesday, November 24, 2010

Proof - a phenomenological touch

A phenomenological perspective on proof theory: of a particular interest here is a view on the proof as "fulfillment of a mathematical intention". See Richard Tieszen, Phenomenology, Logic, and the Philosophy of Mathematics, CUP, 2005, Ch. 13 - "Proofs and fulfillable mathematical intentions" (a  book which also addresses Kurt Gödel's involvement with phenomenological thought - also see Giuseppina Ronzitti's review in Philosophia Mathematica 16(2),264-276, 2008).

Monday, November 1, 2010

New paper. GPF-Tribonacci sequences

Starting with 5, 13, 7, each subsequent term is the greatest prime factor of the sum of the previous three terms. More about this type of sequences - in a new Fibonacci Quarterly article by Greg Back and Mihai Caragiu ("The Greatest Prime Factor and Recurrent Sequences" - Fibonacci Quarterly 48 (2010), no. 4, 358–362) - abstract here. 

In the main result on GPF-Fibonacci sequences (Theorem 3) we prove that all GPF-Fibonacci sequences (that is, prime sequences in which each subsequent term is the greatest prime factor of sum of the previous two terms) that are non-constant eventually enter the same 4-cycle 7,3,5,2.

Monday, October 25, 2010

The "phi-bonacci" sequence - an update

More data on the "phi-bonacci" sequence introduced previously here, after computing the first 500 terms:
  • a plot of the sequence of quotients X(n+1)/X(n) for n = 1,2,...,499:


X(n) is a multiple of 4 for n between 13 and 500.
Here is the updated raw data (last previously calculated term marked in red):

Friday, October 22, 2010

GPF stability?...


A visual on the behavior of the same recurrence as before, only with a different initial condition (an 8-digit prime, picked at random)
L[1] = 11631013
L[N] = P(26390*L[N-1] + 1103)
The limit cycle is the same. The choice
L[1] = 7654237825827857857221111238572389123865443346789678979
leads, again to the same limit cycle. This raises an interesting question of "stability" (that is, assuming ultimate periodicity holds, are there finitely many - if not a single one - limit cycles?*) Note that in this particular case, the limit cycle is not unique: for example, the choice
L[1] = 2250957258971258907129712971234237484736596896123596812363
leads to a different limit cycle, of period 18:

Thursday, October 21, 2010

A recurrence with primes inspired by a formula of Ramanujan

Inspired by the linear component appearing in the numerators of the terms of the Ramanujan's formula for pi, I looked into the recurrent sequence of primes defined as follows:   
L[1] = 2
L[N] = P(26390*L[N-1] + 1103)
where P is the greatest prime factor function. The prime sequence (L[N]) turns out to be ultimately periodic, with period
(1459, 30011, 15529243, 409816723873, 292299009270529, 701251895877205583, 15696384675317604187, 451826639233, 109391789076697, 151939437564949207, 74396630251, 29303389139179, 26646743, 2111734381, 55728670315693, 70865191, 18516162293, 487831, 12873861193, 1132987, 1921687, 330167, 968123137, 2901559, 14505047, 2091738751, 675347, 5940802811, 57192517, 137210047703, 938359, 24763295113, 6250999, 2795997707, 6922448587, 30040009, 46632696389, 3290305727, 282788861, 24151449977, 212452254964711, 5606615008518724393, 4871183188935589, 733134419023, 19347417318018073, 29451120121, 218661017, 242650193, 36178184149, 16311737023, 6797093683, 179375302295473, 29858734728029, 1862817989297131, 4993054962517, 15723597697, 311287129201, 13424039, 6946282163, 252731, 741063577, 106787, 26293, 10356319, 23227, 612961633, 511873, 4640443, 346915841, 3511741099, 33101671, 3527213, 6883, 20719, 433, 57427, 1529263, 510851287, 4456663, 117611337673, 999421747, 33078701, 2389547, 2335560979, 4285298911, 113089038262393, 30741542832733, 1801335599663, 688945601088517, 4790671, 7790117, 769966999, 239280127, 8821951, 756227, 34513, 4289, 37729271, 914302537, 11594639093, 39465611, 11110373, 478571, 11549, 859, 22670113, 176369, 1551459671, 40453493, 6714262147, 14927447, 43770591937, 70138194257, 2603300909203, 51190093, 942712181, 8292724819231, 24011017, 168928483, 6954793553, 270216913, 16301459, 1158523, 30573423073, 21129377, 5023461803, 1979, 17408971, 70381, 1798021, 88289, 467017, 540149, 39929, 4231837, 223803967, 5906186690233, 155864266755249973, 144041128039, 604427630617, 8896176894581, 1577342687387, 17151245784979, 452621376265596913, 878711663, 19359229, 2829679, 12128509, 8627, 41177, 65371, 1663589, 4878012757, 207798167, 609310403137, 1409652983, 1377805267499, 526960594337677, 86463252140809, 8419797874523803, 15369684199, 10962323405749, 1278583667, 132525119, 129531033019, 1221680363, 7825861, 206524472893, 5450180839647373, 24708859707660913, 652066807685171495173, 1941559636106473627171, 193285868819, 100015962316363, 1111967795377, 29344830120000133, 81694832084891, 718642206240091531, 18964967822676015504193, 4407663528935561, 857965267921, 42742695533, 20327255503, 11413537717559, 250324962137, 1275058048793, 11216260635882791, 857911268779, 13341354380129, 39119815787956157, 209783083, 325657385969, 72031783, 2035949, 163601, 72953, 7247, 21249937, 560785838533, 1345376207171543, 1075893275977485481, 9099479420859497, 98635636088557, 320878911329, 2822664823324471, 110249906411, 14055531546799, 499608015161, 548195730743, 160630728871, 385367721355163, 32609594737, 1641473, 83465267, 122840243, 140682811, 161071, 4250664793, 153443, 449929097, 26904379, 116579, 405499, 7715299, 3579647, 33083, 2881391)
Here is a logarithmic plot of this sequence:


This special case illustrates a general conjecture on the ultimate periodicity of GPF sequences. For this, and related sequences and algebraic structures, see
  • Greg Back and Mihai Caragiu, The Greatest Prime Factor and Recurrent Sequences, Fibonacci Quarterly (accepted for publication);
  • Mihai Caragiu and Greg Back, The Greatest Prime Factor and Related Magmas, JP J.of Algebra, Number Theory and Appl. 15 (2), 127-136 (December 2009);
  • Mihai Caragiu and Lisa Scheckelhoff, The Greatest Prime Factor and Related Sequences, JP J.of Algebra, Number Theory and Appl. 6(2), 403-409 (2006);

Thursday, September 9, 2010

A "phi-bonacci" sequence and its consecutive quotients

A most interesting sequence:
"phi-bonacci" ?...

X(0)=0, X(1)=1
X(n)=phi (X(n-1)+X(n-2)+1
if n is at least 2, where phi is the Euler's totient function.

This ensures that X(n) is never greater than the 'regular' Fibonacci number F(n)
 
Plotted - the sequence of quotients X(n+1)/X(n) for n = 1,2,...,324

The raw list of the first 325 non-zero terms follows:

1, 1, 2, 2, 4, 6, 10, 16, 18, 24, 42, 66, 108, 120, 228, 348, 576, 720, 1296, 2016, 3312, 5256, 7200, 12456, 17860, 25200, 40256, 37368, 39600, 72900, 112500, 185400, 282204, 364800, 517600, 805392, 1133988, 1939380, 2788176, 4727556, 6819120, 11539840, 18324852, 28220080, 46471680, 70297856, 77663160, 98640672, 173595168, 256221952, 408844800, 613907760, 1020322800, 1598868000, 2614401972, 3650502240, 6204873360, 9219832128, 14163287040, 23375208496, 37533203556, 59869153008, 77921885248, 136242824256, 171331767600, 280988047872, 412648088320, 492483317760, 759235553856, 1248565926960, 1825274073460,

Tuesday, August 31, 2010

Stefan Banach (1892-1945)

"Good mathematicians see analogies. Great mathematicians see analogies between analogies"
Stefan Banach - quote reportedly provided by Stanisław Ulam.
Banach died on August 31, 1945 at age 53
Home Page of Stefan Banach
Mac Tutor biography
Stefan Banach at MGP

Monday, August 30, 2010

Twenty-Five Years with Nicolas Bourbaki, 1949–1973

Interesting article by Armand Borel: Twenty-Five Years with Nicolas Bourbaki, 1949–1973 
Notices of the AMS
March 1998, 373-380

Constantin Brâncuşi


"I have grinded the matter to find the continuous line. And when I realized I could not find it, I stopped, as if an unseen someone had seen me and slapped my hands."

Constantin Brâncuşi (1876-1957). In the picture: Brâncuşi in Paris (1922), photograph taken by Edward Steichen (source)

Thursday, August 26, 2010

Taming the greatest prime factor...


Logarithmic plot of the first 500 terms of the prime sequence recursively defined as L[k]=P(2010*L[k-1]+83), with P being the greatest prime factor function, and L[1]=2. The sequence is eventually periodic, with a period of 120. The largest term of the sequence (and of the limit cycle too) is precisely... 2617998905649748013.

Monday, August 16, 2010

Recent Progress in Additive Prime Number Theory


Terence Tao, 2006 Fields Medal Recipient - on Recent Progress in Additive Prime Number Theory
2009 Moursund Lectures, Day 1

Abstract: Additive prime number theory is the study of additive patterns in the primes. We survey some recent advances in this subject, including the results of Goldston, Pintz, and Yildirim on small gaps between primes, the results of Green and myself on arithmetic progressions in the primes, and the results of Bourgain, Gamburd, and Sarnak for detecting almost primes in orbits.

Eadem mutata resurgo

Commemorating Jacob Bernoulli...

Eadem mutata resurgo
"Though changed I shall rise the same"

Inscribed on Jacob Bernoulli's tombstone (he died on August 16, 1705 in Basel), this motto refers to the logarithmic (equiangular) spiral (N.B. the spiral that was actually imprinted on the tombstone is not equiangular). Through this self-similar object Jacob Bernoulli symbolically points to the ‘fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self ’ - as quoted in Mario Livio's book "The Golden Ratio..." (via here).

Jacob Bernoulli at the Mathematics Genealogy Project.


The Whirlpool Galaxy...

Friday, July 30, 2010

Structure and Randomness in the Prime Numbers (Terence Tao)

Terence Tao: Structure and Randomness in the Prime Numbers, UCLA
Slides: pdf, powerpoint

Lecture for a general audience: Terence Tao is UCLA's Collins Professor of Mathematics, and the first UCLA professor to win the prestigious Fields Medal.

Tuesday, July 27, 2010

Johann Bernoulli (1667 - 1748) anniversary

Johann Bernoulli (1667 - 1748) was born on July 27, 7^3 years ago, in Basel, Switzerland. He was Euler's "mathematical parent".

And here is the... "sophomore's dream" - neat identities due to Johann Bernoulli (1697):

Wednesday, July 21, 2010

A traffic flow simulation

This traffic flow educational project with Matlab features a gradually increasing car density starting from an initial value of 0.2. There are 250 cells. The update rule (describing the transition from time t to time t + 1): apply rule 184, after which randomly select a position - if occupied, nothing happens, while if empty, introduce a car at the selected place with probability 0.7. The image follows the first 500 time units. Notice the transition to a congested regime happening at some point (emerging shock waves). In the picture, free cells are blue, while cells occupied by "cars" are red.

Shockwave traffic jams


Traffic jams replicated on the test-track (courtesy of Mathematical Society of Traffic Flow, Japan).

Sunday, July 18, 2010

Memories


Just discovered online an old colloquium announcement from 1997 - a talk that I gave at the University of Montana, Missoula. Great city, amazing people, unforgettable nature, and a very nice campus.

Tuesday, July 13, 2010

On the ranges of discrete exponentials

Florin Caragiu and Mihai Caragiu, “On the ranges of discrete exponentials,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 42, pp. 2265-2268, 2004. doi:10.1155/S016117120431 2056

Reference link

On the Ranges of Discrete Exponentials

Sunday, July 11, 2010

A great book by J. W. S. Cassels

"An introduction to the geometry of numbers" was the main reference for my 1987 senior capstone thesis (Methods of Geometry of Numbers) at the University of Bucharest. A great book! I just found out that today is the anniversary of the author, J. W. S Cassels (born on July 11, 1922 in Durham, England). Happy birthday!!!

Friday, June 11, 2010

Nicolai Vasilievich Bugaev (1837-1903)

Nicolai Vasilievich Bugaev (14 Sept 1837 - 11 June 1903) was one of the founders and leading members of the Moscow Mathematical Society (vice president from 1886,  president from 1891). He earned his doctoral degree in 1866 under Weierstrass and Kummer (see Bugaev's MGP entry). Dmitri Egorov earned his doctoral degree under Bugaev, in 1901. Pavel Florenskii studied mathematics with Bugaev at Moscow University (he entered the university in 1899). Bugaev played a central role in the creation and development of the Moscow school of functions of a real variable [1]. He considered mathematics to be grounded in the theory of functions, built a theory of discontinuous functions ("arithmology") and developed a system of analogies between functions appearing in elementary number theory and functions appearing in analysis.On a philosophical level, notable is Bugaev's move from  positivism to a personalistic metaphysics [2].


REFERENCES
[1] Demidov, S. - N. V. Bougaiev et la création de l'école de Moscou de la théorie des fonctions d'une variable réelle, Boethius Texte Abh. Gesch. Exakt. Wissensch., XII, Steiner, Wiesbaden, 1985.
[2] Shaposhnikov, V. A. - The philosophical views of N. V. Bugaev and Russian culture from the end of the nineteenth century to the beginning of the twentieth century, Istor.-Mat. Issled. (2) No. 7(42) (2002), 62-91, 366-367.
[3] Wikipedia entries for Nikolai Bugaev - in Russian and in English.

Tuesday, June 8, 2010

Harold Davenport


Harold Davenport (30 Oct 1907 - 9 June 1969) - MacTutor biography.

Pictured: quadratic residues and non-residues in the field with 6889 elements.

Saturday, June 5, 2010

A Thermodynamic Classification of Real Numbers

I just found a very interesting paper presentation (JNT on YT - link) by Thomas Garrity (Williams College) - "A Thermodynamic Classification of Real Numbers":


On arXiv - A Thermodynamic Classification of Real Numbers.

Thursday, May 27, 2010

An Evening with Leonhard Euler

An Evening with Leonhard Euler (talk given by William Dunham, Professor of Mathematics at Muhlenberg College). Presented by Philoctetes Center.

Alain Connes: Fun with F1


Alain Connes' YouTube Channel.
Alain Connes, Caterina Consani and Matilde Marcolli. Fun with F_1, Journal of Number Theory 129 (2009), no. 6, 1532-1561.
Alain Connes and Caterina Consani. On the notion of geometry over $\F_1$ (arXiv:0809.2926).
Journal of Number Theory on YouTube.

Wednesday, May 5, 2010

Élie Cartan (1869-1951)



Élie Cartan had significant contributions in areas such as Lie theory, differential geometry, exterior differential forms, the theory of spinors (introduced by him in 1913), etc. Cartan died on May 6, 1951.

Dieudonné places Cartan right after Poincaré and Hilbert when it comes to the lasting influence in shaping modern mathematics. He was a speaker at the 1924, 1932 and 1936 International Mathematical Congresses. He lectured in Romania in 1931. The letters that he exchanged with Albert Einstein, Gheorghe Ţiţeica, Alexandru Pantazi and Gheorghe Vrânceanu, have been published (as mentioned in M. A. Akivis and B Rosenfeld - Élie Cartan (1869-1951), Providence R.I., 1993).

Thursday, April 29, 2010

Paul Montel (1876-1975)

Paul Antoine Aristide Montel, Honorary Member of the Romanian Academy, advisor of Henri Cartan, Jean Dieudonné, Miron Nicolescu, Tiberiu Popoviciu and others (see Montel's entry at MGP), was born on April 29, 1876...

Saturday, April 24, 2010

Mathematics Genealogy Project - Edmund Husserl


Right here. Ph.D. 1881 (Universität Wien, advisor Leo Königsberger).

Max Planck on consciousness

"I regard consciousness as fundamental. I regard matter as derivative from consciousness. We cannot get behind consciousness. Everything that we talk about, everything that we regard as existing, postulates consciousness." (Max Planck - born on April 23, 1858)

link to quotation source
link to top picture source/credits (grave of Max Planck in Göttingen)
link to  bottom picture source/credits (NGC7090)
Mac Tutor Biography
Mathematics Genealogy Project - Max Planck
Planck units

Thursday, April 22, 2010

Marius Dabija (13 ianuarie 1969-22 iunie 2003)

De pe blogul lui Florin:

Un prieten căruia îi păstrez o vie amintire este Marius Dabija. Minte strălucită, scormonitoare, imprevizibil in acţiuni, căutând mereu soluţia surpriză. În clasele a VII-a şi a VIII-a am lucrat împreună la matematică, pregătindu-ne pentru Olimpiade. Ca să variem, jucam şah până nu mai ştiam de noi. Era talentat şi la fotbal, tenis de masă etc.

În liceu ne-am văzut mai rar. Eu eram la liceul I. L. Caragiale, el, la Mihai Viteazul. L-a avut profesor pe Eugen Onofraş. În clasa a XI-a, Marius a luat locul I la Olimpiada de matematică, faza Naţională şi la Olimpiada Internaţională de Matematică. După ce a absolvit Facultatea de matematică, a plecat la doctorat în America, unde munca de cercetare i-a fost încununată de reuşită. A luat doctoratul şi a scos nişte articole remarcabile.

Ca elev şi student, era în stare să-şi conducă profesorii de la agonie la extaz şi invers. Născocea pe loc soluţii din cele mai diverse la câte o problemă, după care, lăudat fiind de profesor, care nu reuşea totuşi să urmărească deplin şirul argumentărilor, revenea şi arătând că greşise într-un loc ştergea totul, scoţând ca din joben o altă demonstraţie fulger. Asta se putea întâmpla de câteva ori la rând...

Avea de regulă o deosebită poftă de viaţă, umor, voioşie, neastâmpăr, o doză sensibilă de nonconformism, atras de situaţii-limită, uneori fiind, e drept chinuit de gânduri şi întrebări, incertitudini existenţiale, întorcând lucrurile pe toate părţile în căutarea unei soluţii, construind şi deconstruind la nesfârşit.

Mie îmi părea câteodată, în unele momente ale sale de graţie, că regăsesc profilul unui Mozart al matematicii. Am aflat cu durere în inimă vestea că în America a trecut pe neaşteptate la cele veşnice, în plină activitate creatoare, la numai 34 de ani. Dumnezeu să-l ierte şi să-l odihnească!

Articole Publicate:

Dabija, M. "Algebraic and Geometric Dynamics in Several Complex Variables". PhD thesis, University of Michigan, 2000. ps.gz
Bonifant, A. and Dabija, M. "Contractive Curves". International Journal of Mathematics and Mathematical Sciences, 30(4), 2002. ps.gz
Bonifant, A. and Dabija,M. "Self-maps of P2 with invariant elliptic curves". Contemporary Mathematics, 311, 2002. ps.gz
Coman,D.and Dabija, M. "On the Dynamics of Some Diffeomorphisms of C2 near parabolic fixed points". Houston Journal of Mathematics, 24(1), 1998. pdf

Articole Nepublicate:

Dabija, M. "Bötcher divisors", 2000. ps
Dabija,M. "Self-maps of projective bundles on projective spaces",2000. ps
Dabija, M."Self-maps of ruled surfaces", 2000. ps
Dabija,M.and Jonsson, M. "Self-maps of P2 with an invariant curve of curves", 2002.

Sursa.

Wednesday, March 24, 2010

Matematica – un tip specific de cunoaştere

Interesându-se de filosofia matematicii, studiul nostru îşi propune să se refere la tipul cu totul specific de cunoaştere pe care îl reprezintă matematica (prin cel puţin o parte a ei), disciplină care nu poate fi înglobată în totalitatea ei gândirii ştiinţifice. Opoziţiile care se fac sunt de obicei între cunoaşterea mitică, comună, artistică, ştiinţifică şi filosofică. În prezent, nu numai că fizicienii folosesc modelul matematic în locul celui mecanic, dominant în secolul al XIX-lea, dar, mai mult decât atât, se poate vorbi despre o dominaţie a metodei matematice printr-o fascinaţie pe care o exercită asupra oricărei inteligenţe. Einstein rămâne cel mai bun exemplu de folosire a matematicii ca model, cu referire la folosirea aparatului matematic denumit calcul tensorial, în cadrul teoriei relativităţii generalizate. S-a afirmat că matematica este oglindirea gândirii; obiectul propriu-zis al gândirii matematice îl constituie produse ale gândirii, creaţii autonome ale spiritului. Constatarea că metodele matematice se dovedesc utile în studierea fenomenelor fizice şi de asemenea observaţia că există un progres în acest sens al teoriilor fac plauzibilă, pe cât de discutabilă, idea unei naturi cvasimatematice a realităţii. De notorietate sunt cazurile când formule matematice menite iniţial să uşureze calculul au condus, în procesul căutărilor, la concluzii verificate ulterior pe cale experimentală. Este de consemnat şi de comentat o opinie venită din partea unor filosofi logicieni, că deşi s-a încercat să se prezinte ştiinţa actuală ca având nevoie de o epistemologie nouă, totuşi epoca noastră este, încă, una carteziană. Ea nu respinge de fapt ideile clare şi distincte, cum ar părea că o face (a se vedea Louis de Broglie), ci lărgeşte cadrul, admiţând în corpul ştiinţei şi idei numai pentru „compatibilitatea lor” cu cele deja admise în construcţia unei ştiinţe determinate. Prin aceasta, însă, raţionalismul lucidităţii carteziene nu este afectat în esenţa lui, ci dimpotrivă amplificat (Anton Dimitriu). S-a spus că matematica prezintă o imagine ideală a unor fenomene ascunse. Este ca o hartă ideală pentru o ţară încă neidentificată cu precizie, sau care, poate, nici nu există în realitate. Omul rescrie lumea din el însuşi, prin procedee matematice. „La sfârşit, scria Jacques Maritain, gândirea umană va apărea ca un fel de demiurg, care fabrică lumea cognoscibilă cu conceptele sale, şi nu realitatea va cere ştiinţei să fie adevărată, ci ştiinţa va cere realităţii de a fi ştiinţifică”.

Tuesday, March 9, 2010

" I have followed its roots"...

"My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things".  (Georg Cantor - quoted in Journey Through Genius by William Dunham)

Tuesday, February 16, 2010

Ph.D. mathematician and NFL champion

In some sense, the stunning 2010 Super Bowl XLIV victory of New Orleans Saints led by Drew Brees against the Peyton Manning's Indianapolis Colts (my favorite team) may be analogue to a similar event that happened in 1964. Then the NFL quarterback Frank Ryan led the Cleveland Browns to the 1964 NFL Championship title in a 27-0 victory against Johnny Unitas' Baltimore Colts. To this one might add the impressive 1966 season in which the Cleveland Browns' legend Ryan threw for 2974 yards and scored 29 touchdowns.

What is especially relevant for this particular blog is that Frank Ryan is also the recipient of a Ph.D. in Mathematics awarded by Rice University in 1965, with a most interesting thesis, "A Characterization of the Set of Asymptotic Values of a Function Holomorphic in the Unit Disc", and that among the references cited in the thesis are Luzin's "Leçons sur les ensembles analytiques et leurs applications", Sierpinski's "General Topology" (University of Toronto Press, 1952), and Stoilow's "Les propriétés topologiques des fonctions analytiques d'une variable", Ann. Inst. H. Poincaré, 2 (1932), 233–266. In 1966 Frank Ryan also published two fundamental papers on the set of asymptotic values of a function holomorphic in the unit disc in Duke Mathematical Journal (he also published in Pacific Journal of Mathematics, Mathematische Zeitschrift, Michigan Mathematical Journal, etc).

I will conclude with mentioning a recent mathematical event - the amazing, super-entertaining after-dinner talk "Resolved, that a Football is a Mathematical Object" delivered by Frank Ryan at the 2007 Ohio MAA Meeting held at Wittenberg (a talk which I will never forget).

Friday, February 5, 2010

Gheorghe Ţiţeica (1873-1939)

Gheorghe Ţiţeica (1873-1939) died on February 5, 1939. He had important contributions in Differential Geometry, and served as the president of the geometry section at the International Congress of Mathematicians in Toronto (1924), Zürich (1932), and Oslo (1936). Also see Ţiţeica's Mac Tutor Biography, a cool elementary geometry problem - Ţiţeica's "five-lei coin problem", and the historical survey Gheorghe Ţiţeica and the origins of affine differential geometry (Historia Mathematica, 36 (2), 161-170 (2009)).

Thursday, January 14, 2010

Kurt Gödel (1906-1978)


Kurt Gödel (b. April 28, 1906, Brno, Moravia – d. January 14, 1978, Princeton, New Jersey, USA) - gravestone in the Princeton cemetery.


Einstein, Dirac, Gödel, Selberg, Harish-Chandra in Princeton - Gödel on left @ 1:06.

Wednesday, January 6, 2010

Georg Cantor (1845-1918)

From Wikipedia, under Georg Cantor: Cantor's philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to posit and prove concepts apart from the realm of physical phenomena, as expressions within an internal reality. The only restrictions on this metaphysical system are that all mathematical concepts must be devoid of internal contradiction, and that they follow from existing definitions, axioms, and theorems. This belief is summarized in his famous assertion that "the essence of mathematics is its freedom."

Georg Cantor died on January 6, 1918.
Mac Tutor Biography
Georg Cantor: His Mathematics and Philosophy of the Infinite by Joseph Warren Dauben, and a review (pdf, Bull. AMS)