Maryam Mirzakhani - first woman to be awarded the Fields Medal - ICM 2014 Seoul

Maryam Mirzakhani - Iranian mathematician, full Professor of Mathematics at Stanford, first woman to be awarded the Fields Medal, at The International Congress of Mathematicians 2014 (Seoul, August 13-21)

"for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces" - see presentation and video @  http://www.icm2014.org/en/awards/prizes/f4

Joint Statistical Meetings Proceedings - not peer-reviewed

"If you give a presentation at JSM, you may submit a corresponding paper to be published in the conference proceedings. Papers are not peer-reviewed in the same manner as for journals, but authors are encouraged to have others examine their paper before submission. The proceedings are published online around November. Authors retain the right to publish their research later in a peer-reviewed journal."

SOURCE: "What Happens at JSM Should Not Stay at JSM / How to get the most out of the Joint Statistical Meetings"  AMSTAT News - May 2014
http://magazine.amstat.org/wp-content/uploads/2014/02/AN_May2014.pdf

Eu si... Antonin Scalia

Pe o scara personal/academica, momentul cel mai interesant al anului care a trecut a survenit atunci cind am fost comparat (la o conferinta, deci in public) cu... Antonin Scalia (!). Intentia nu a fost magulitoare, iar motivul fiind inistenta mea pe cercetarea intreprinsa de profesori, (nici vorba de "publish or perish" la un 4 years college, insa macar "publish involving undergraduates" - pentru un 4 years college ar fi minunat). Insistenta mea a trezit reactii mixte: unii s-au bucurat ("it's about time!"), altii au subliniat ceva in genul "teaching is paramount" (so?... does this exclude engaging students in research?), altii au fost relativ ostili, negind orice rol special acordat publicatiilor cu sau fara studenti co-autori. In sfirsit, ma bucur ca macar aceasta atitudine i-a pus pe ginduri pe unii.

Nature's secrets

"Nature conceals her secrets because she is sublime, not because she is a trickster" 
Albert Einstein, in a letter to Oscar Veblen

Cosmic inflation: 'Spectacular' discovery hailed

17 March 2014
Cosmic inflation: 'Spectacular' discovery hailed
By Jonathan Amos Science correspondent, BBC News


NY Times - Space & Cosmos 
Detection of Waves in Space Buttresses Landmark Theory of Big Bang 
By DENNIS OVERBYE, MARCH 17, 2014

RAMANUJAN: Letters from an Indian Clerk

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Why the brain sees maths as beauty (BBC News)

Mathematics: Why the brain sees maths as beauty 
By James Gallagher
Health and science reporter, BBC News
12 February 2014

"What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty. This is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating." (P. A. M. Dirac)

'On Ducci Sequences with Primes' - Fibonacci Quarterly

http://www.fq.math.ca/Abstracts/52-1/caragiu.pdf
Mihai Caragiu, Alexandru Zaharescu, and Mohammad Zaki 
On Ducci Sequences with Primes 
Fibonacci Quart. 52 (2014), no. 1, 32-38.

An analogue of the Proth-Gilbreath conjecture (new paper)

O fenomenologie a numerelor prime - in gen fantasy cu demonstratii plus "experiment" (computer).

Far East Journal of Mathematical Sciences (FJMS)

Volume 81, Issue 1, Pages 1 - 12 (October 2013)
AN ANALOGUE OF THE PROTH-GILBREATH CONJECTURE 
Mihai Caragiu, Alexandru Zaharescu and Mohammad Zaki
Communicated by Juliusz Brzezinski

Miami University Fall Conference on Undergraduate Research in Mathematics

Abstract of my presentation:

More than twelve years ago, a talk on "Ducci games" delivered jointly by two Ohio Northern University students at the 2001 Ohio MAA Spring meeting initiated a fairly long streak of undergraduate research in the area of number theory at our school. Since then, Ohio Northern University students presented 40 talks and posters in the broad area of number theory at various mathematics meetings, and were co-authors of 11 research articles in number theory which appeared in peer-reviewed mathematics journals. We were especially pleased to see our research on "greatest prime factor sequences" (published in Fibonacci Quarterly in 2010) featured alongside other "noteworthy variations on the Fibonacci numbers" in the keynote talk at the 15th International Conference on Fibonacci Numbers held in Budapest (June 25-30, 2012), and cited in various other journals. In the light of the speaker's experience as an undergraduate research advisor, we will try to address some issues of interest regarding the impact of undergraduate research in the outside mathematical community. This "impact" may be viewed as a long-sought fulfillment or closure of the combined efforts of faculty and students engaged in undergraduate research, which ultimately takes a life of its own. We will explore open-ended difficult questions such as: What does it mean to make an impact? Are there specific strategies for smaller schools? Can presentations make an impact? What is the relationship between undergraduate research and faculty research? Is it harder for pure mathematics?

Alba Iulia 2013

My talk (abstract) at the 2013 RMS-AMS Special Session of Discrete Mathematics and Theoretical Computer Science in Alba Iulia:
Uniform distribution for a class of k-paradoxical oriented graphs 
By using estimates for incomplete character sums with polynomial arguments, we provide uniform distribution results for the dominating sets in a class of k-paradoxical regular oriented graphs, including the Paley tournaments. Moreover, we will explore a method of quasi-random tournament generation from fi nite sets of natural numbers, by using the greatest prime factor function.

Work on Stirling numbers honored with an Allendoerfer Award (2013)

Khristo N. Boyadzhiev (Ohio Northern University) - 2013 Carl B. Allendoerfer Award for the paper "Close Encounters with the Stirling Numbers of the Second Kind" - Mathematics Magazine, 85:4 (2012), pp. 252-266.
More at: http://www.maa.org/news/2013-maa-awards-recipients-announced

UNIFORM DISTRIBUTION FOR A CLASS OF k‑PARADOXICAL ORIENTED GRAPHS

Mihai Caragiu, Donald Pleshinger and Jonathan C. Schroeder, UNIFORM DISTRIBUTION FOR A CLASS OF k‑PARADOXICAL ORIENTED GRAPHS, JP Journal of Algebra, Number Theory and Applications, Volume 29, Issue 2, Pages 107-117 (June 2013)

A major breakthrough on primes: Yitang Zhang

Bounded gaps between primes - by Yitang Zhang - Ann. Math. (forthcoming)

Mandelbrot set, zoom, matlab

http://www.youtube.com/watch?v=EQGGnIyIoDM

A parametric curve with... Bach's Prelude in G major

http://youtu.be/IvNQbTYSON8

A Uniform Distribution Result for k-Paradoxical Directed Graphs

The research I am conducting with two students of mine was presented at the 2013 MAA Undergraduate Poster Session in San Diego.

The Legacy of Srinivasa Ramanujan

Srinivasa Ramanujan: Going Strong at 125 (Krishnaswami Alladi, Editor - Notices of the AMS - Part I and Part II).

The Legacy of Srinivasa Ramanujan-An International Conference, University of Delhi, India

Mysterious WWII encrypted message found in UK

UK spies unable to crack coded message from WWII carrier pigeon (CNN)

A nice picture

A nice "Paley digraph" picture - with outgoing edges from x to x+1, x+2, x+3, x+4, x+6, x+8, x+9, x+12, x+13, x+16 and x+18 (modulo 23) for x = 0,1,...,22.

October 23 - ONU Mathematics Seminar on a-T-menability

The Role of Research at Undergraduate Institutions

An excellent article by Robert Gavin, in "Academic Excellence - The role of research in the physical sciences at undergraduate institutions" (Michael P. Doyle, Editor - published in the year 2000 by Research Corporation - a foundation for the advancement of science). Even if the paper, which emphasizes the role of publishing in a research-based education, refers to physical sciences, the ideas in there are even better suitable for mathematical sciences, where there is not an excessive need for laboratories and equipment.
Straight to the point: "Publishing research articles, especially those done in collaboration with undergraduate students, should be expected, encouraged and supported both before and after the tenure decision".

GPF sequences in Rutgers' Experimental Mathematics Seminar

Apparently the sequences introduced in the Fibonacci Quarterly paper by Greg Back and myself have been discussed in the RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR (Neil J. A. Sloane's presentation was on February 17, 2011

GPF sequences - a forum discussion

A discussion of the paper "The Greatest Prime Factor and Related Sequences" (JP Journal of Algebra, Number Theory and Applications 6(2), 403-409 (2006), by Mihai Caragiu and Lisa Scheckelhoff), with neat pictures, can be found here (Mathematical Oddities Thread - The Something Awful Forums)

An Euler-Fibonacci Sequence

An Euler-Fibonacci Sequence
by Mihai Caragiu and Ashley Risch
Far East Journal of Mathematical Sciences, Volume 52, Issue 1, Pages 1 - 7 (May 2011)
abstract - here

Undergraduate research: what is that ?

The Council on Undergraduate Research defines it as follows:

``An inquiry or investigation conducted by an undergraduate student that makes an original intellectual or creative contribution to the discipline.''

The word "original" is very important. An original contribution to knowledge rules out works of a severely expository or textbook nature, results that follow immediately from previous work (as in... use the previously obtained A=B to "discover" that 2A=2B, or something like that), or trivial derivations in existent or made-up ad hoc formal systems. The original contribution to the discipline must also go through a rigorous, external, peer-review process. Ideally, a rigorous, solid peer-review is a process which does not accept works simply because they are formally correct, indeed it demonstrates a pattern of rejecting a significant percentage of logically correct but otherwise not interesting enough (as judged by the reviewers) works. Also, note that being "peer-reviewed" is not the same with "being made public/disseminated" (a confusion that is still circulating). A valuable original contribution will be able to generate 'participative waves', engaging others in the area. Thus, when it comes to goals and assessment, 'undergraduate research' is not (and shouldn't be, in my opinion) different from good old 'research'. So it is a serious matter, and competitive universities recognize that. I found interesting the following straight-to-the-point statement (due to Jim Coleman, vice chancellor for research and professor of biology at the University of Missouri) on the central place of undergraduate research in the life of a good university:

``There is nothing more central to the mission of a university than activities associated with discovery, creation, innovation and scholarship. So, I think that what defines a great university is the integration of these activities into the entire fabric of the undergraduate experience.''

Integrating the research/scholarship into the undergraduate life is a challenging enterprise. There are no clear recipes, since there are students and students. Each individual case is unique and interesting in itself. The faculty's essential asset is their own involvement and demonstrated proficiency in research. Indeed the undergraduate research is driven, after all, by faculty research. Or, if you want, faculty research is a necessary condition for undergraduate research. One may ask, is it also a sufficient condition? This is not true, mainly because the student is a person, not a machine or a notebook on which the faculty mentor writes a paper. In the end, note that the complexities of (undergraduate) research that even an otherwise well prepared academic (mentor) faces, ultimately point to persons (as in real persons, and not ``the idea of a person''), and their participative experience.

Ion Barbu: matematică, poezie şi arte (un citat concis...)

Matematicile pun în joc puteri sufleteşti care nu sunt cu mult diferite de cele solicitate de poezie şi de arte.
(Ion Barbu; sursa - citatepedia.ro)

music and computation

Gottfried Wilhelm Leibniz's view on music ('the hidden arithmetical exercise of a mind unconscious that is calculating') has (at least) two straightforward interpretations. The first one is essentially reductionist (a 'fallacy of the misplaced concreteness' according to Alfred North Whitehead) and tends to suggest that music is nothing but computation (albeit in the background/unconscious, in a less obvious way). The second interpretation of the music-calculating connection runs somehow in the opposite direction, and tends to suggest that there is more to computation than meets the eye, an ethereal/ineffable/musical/higher-order quality. At this point, one might try to revisit the spirit of some traditional Gödelian themes...
Image source: http://en.wikipedia.org/wiki/File:FortranCardPROJ039.agr.jpg

Fractals and partitions

A recent major breakthrough is announced here (Emory University's site, eScienceCommons). Looks like ultimately periodic sequences made the news. Also see here a recent article by Ken Ono (The Last Words of a Genius, Notices of the AMS Volume 57, Number 11, 1410-1419), and here a relevant abstract by John Webb (An improved “zoom rate” for the Folsom-Kent-Ono l-adic fractal behavior of partition values).

Image source.

A relevant video - Ken Ono talk (Emory University YT Channel):

http://youtu.be/aj4FozCSg8g
New Theories Reveal the Nature of Numbers

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.

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):

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:

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);

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,

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...

Beauty in Mathematics

Michael Atiyah - Beauty in Mathematics

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.

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):

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.

Lothar Collatz anniversary

Lothar Collatz, who proposed (in 1937) the celebrated "3n+1 problem", was born 100 years ago in Arnsberg, Germany.

Mac Tutor Biography

Lothar Collatz at MGP

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.

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).

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...

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

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.

J.S. Bach - Crab Canon on a Möbius Strip

(YT link)

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).

Fibonacci numbers and the extreme and mean ratio - some history

Ruth Tatlow's article The Use and Abuse of Fibonacci Numbers and the Golden Section in Musicology Today (Understanding Bach, 1, 69-85, 2006) besides being very interesting in itself as a documented criticism of the "Golden numberism [that] has thoroughly infected musicology", incidentally provides useful references for those interested in the history of the representation of the EMR ("extreme and mean ratio" or "golden section") as the limit of the sequence of quotients F_n+1/F_n of consecutive Fibonacci numbers [1]. One may wonder who noticed this first? Leonardo Pisano (c. 1170 – c. 1250, also known as Fibonacci)? Not even close!

Evidence that this fact was noticed as early as the beginning of the 16th century was discovered by Leonard Curchin and Roger Herz-Fischler (handwritten annotation in the 1509 Luca Pacioli's edition of Elements [2]). When it comes to published work which associates the EMR with the limit of the sequence of quotients of consecutive Fibonacci numbers, Johannes Kepler wrote [3] the following, in 1611, about "this proportion that the geometers of today call divine":

It is impossible to provide a perfect example in round numbers. However, the further we advance from the number one, the more perfect the example becomes. Let the smallest numbers be 1 and 1... Add them, and the sum will be 2; add to this the greater of the 1s, result 3; add 2 to this, and get 5; add 3, get 8; 5 to 8, 13; 8 to 13, 21. As 5 is to 8, so 8 is to 13 approximately, and as 8 to 13, so 13 is to 21, approximately.

However, as communicated in 1995 by Peter Schreiber [4], in a rare book by the German reckoning master Simon Jacob (d. 1564) one can find a remark that the sequence of quotients of consecutive Fibonacci numbers approaches the EMR.

NOTES

[1] The term "extreme and mean ratio" goes back to Euclid's "Elements". Luca Pacioli introduced the "divine proportion" term in 1509, while the term "golden section" was introduced in 1835 by Martin Ohm. The exact value for the EMR is (1+sqrt(5))/2. The phrase "Fibonacci sequence" was coined by Edouard Lucas (1842-1891).
[2] Leonard Curchin and Roger Herz-Fischler, "De quand date le premier rapprochement entre la suite de Fibonacci et la division en extrême et moyenne raison?" (French) ["When were the first parallels drawn between the Fibonacci sequence and the golden section?"], Centaurus 28 (1985), no. 2, 129-138.
[3] Kepler, Johannes. Vom sechseckigen Schnee. (German) [On hexagonal snowflakes]. Translated from the Latin and with an introduction and notes by Dorothea Goetz. Ostwald's Classics of the Exact Sciences, 273. Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987.
[4] Peter Schreiber, "A Supplement to J. Shallit’s Paper "Origins of the analysis of the Euclidean algorithm"", Historia Mathematica, Vol. 22, Issue 4, 422-424 (1995). http://poncelet.math.nthu.edu.tw/disk5/js/history/1033a.pdf

Bernhard Riemann

Georg Friedrich Bernhard Riemann (September 17, 1826 – July 20, 1866) - one of the greatest mathematicians of all time.

• Full Mac Tutor Biography
• Wikipedia entry
• The Mathematical Papers
• Mathematics Genealogy
• "On the Number of Primes Less Than a Given Magnitude" (pdf)

Dedekind has a deeply touching account of Riemann's final moments, his passing away being marked by the words of Lord's prayer. Riemann's tombstone (see here and here) in Biganzolo (Italy) refers to Romans 8:28 ("And we know that all things work together for good to them that love God, to them who are the called according to his purpose"):

Here rests in God
Georg Friedrich Bernhard Riemann
Professor in Göttingen
born in Breselenz, September 17th, 1826
died in Selasca, Juli 20th, 1866
---
Those, who love God, all things
must serve to its best manner.

Fibonacci modulo m

The general problem of the periods of the Fibonacci sequence modulo m is definitely non-trivial (with the case m = p - prime - playing a very important role). An important reference can be found here ("The Fibonacci Sequence Under Various Moduli" - M.Sc. Thesis by Marc Renault, 1996). Also see the article (PDF) "The Fibonacci sequence modulo p^2...".

An example that teachers use relatively often as a middle-school problem: "Find the period of the sequence of the last digits of the Fibonacci numbers"! That will correspond to the modulus m=10, the answer is 60, and the elements of the period are

0,1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6,1,7,8,5,3,8,1,9,0,
9,9,8,7,5,2,7,9,6,5,1,6,7,3,0,3,3,6,9,5,4,9,3,2 ,5,7,2,9,1

If the modulus m is 2011 (that is the 305-th prime), the period of the Fibonacci sequence modulo m is 2010.