Friday, December 9, 2011

Einstein on pure mathematics

"Pure mathematics is, in its way, the poetry of logical ideas
(Albert Einstein)

Tuesday, December 6, 2011

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; for the abstract go to the Archive of Speakers and Talks - 2011).

Saturday, September 10, 2011

Tuesday, August 30, 2011

Oamenii de lângă noi

Cotidianul LUMINA - 17 August 2011
Oamenii de lângă noi
de Monica Patriche
În memoria Prof. Univ. Dr. Constantin Tudor
Un model de modestie şi de viaţă trăită din plin, împlinită prin faptul că nu a îngropat talantul primit, ci dimpotrivă. Numai gânduri bune pentru cel care aşteaptă din parte-ne ceva pozitiv. Şi-mi aduc aminte cum povestea părintele Galeriu că la înmormântarea părintelui Stăniloae vorbea cu acesta ca şi când ar fi fost în viaţă, cu convingerea că este auzit. Aceasta este, cu siguranţă, ceea ce e bine să facem şi noi.

Simulating the galaxy formation


http://youtu.be/VQBzdcFkB7w
ERIS: World's first realistic simulation of the formation of the Milky Way

Wednesday, August 24, 2011

Tuesday, August 23, 2011

Solution to B1074 - in FQ

The solution to FQ-B1074, provided by the ONU-Solve problem group was published in the "Elementary Problems and Solutions" section of the August 2011 Issue of the Fibonacci Quarterly.

Friday, August 19, 2011

"One, Two, Three / Absolutely Elementary Mathematics" by David Berlinski

From a review that I wrote for "One, Two, Three / Absolutely Elementary Mathematics" by David Berlinski, that I recently wrote (and submitted):
Personally, I believe this is a particularly nice way to look at the “deep field” of mathematics: that’s a whole lot of treasure stuff out there (actually more abundant than the one in the physical world, as the author notices), enough for the unfolding, in complete Cantorian freedom, of the process of self-discovery of every human person. That may apply for the “deep field” of the arts and, why not, for that of physical/cosmological reality itself. Ultimately, such a reading of the mathematical experience in a phenomenological key may prove to be truly beneficial.

Image source (Hubble Deep Field (full mosaic) released by NASA on January 15, 1996)

Wednesday, August 17, 2011

Fermat 410

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.
(Pierre de Fermat; source)

Fermat's Fallibility (from mathpages.com)
Mac Tutor biography
Pierre de Fermat's birthday celebrated in Google Doodle (The Telegraph - UK)

Thursday, July 21, 2011

Lecture notes on finite fields (III)

Applications of finite fields (3/3) - Dickson polynomials and discrete logarithms. Older lecture notes from a notebook which I don't want to lose (unfortunately it is in danger of falling apart...)

Monday, July 18, 2011

Saturday, July 16, 2011

Florin & Mihai Caragiu: "Matematică şi Poezie – Confluenţe"


Publicat în revista „Sinapsa”, Nr. 8/2011, pp. 65-75 (foto: Diana Popescu)

În gândirea grecească, pe lângă ordine şi cosmos, logosul a fost o altă noţiune, cu dublu statut, obiectiv şi subiectiv. Logosul există atât în om, cât şi în lucruri. Cosmosul şi ordinea lui se opun unui aşa-numit haos iniţial şi sunt inseparabile de ideea de inteligibilitate, de logos. Pitagora, cel dintâi, a numit cuprinderea tuturor lucrurilor kosmos, din pricina rânduielii ce domnea în cuprinsul lor. Cosmosul era universul ordonat, altfel spus, normat. O altă proprietate fundamentală a kosmosului pitagoreic, pe lângă caracterul lui ordonat, este aceea a caracterului finit. Grecii, s-a spus, se temeau de infinit. Ei asociau valoarea binelui cu finitul şi, implicit, cu cognoscibilul. Pentru ei, infinitul era ceva de ordin iraţional, dăunător eticii. De altfel, esenţa învăţăturii pitagoreice e conceptul de număr în explicarea lumii. (Desigur, pe acea vreme, conceptul de număr era foarte primitiv, legat de ilustrări practice, cu pietricele sau din puncte.) Pitagora e ultimul punct nodal al sistemului încă nedivizat al imaginii lumii mitice, în centrul învăţăturilor sale aflându-se, într-un tot unitar, muzica, matematica, astronomia şi ritul. Există o mistică a numerelor pitagoreice care şi-a atins apogeul în şcoala neopitagoreicilor, în aşa-numitele teologii aritmetice ş. a. Se stabileau corelaţii între numere şi alte concepte matematice, numere şi muzică, numere şi morală, numere şi corpul uman, numere şi elementele universului.
E interesant de urmărit întâlnirea dintre poezie şi matematică în acest punct de convergenţă generală care este mitul, unde funcţiile spiritului nu sunt încă diferenţiate în discipline autonome, şi unde gnoseologicul şi ontologicul, cunoaşterea şi existenţa se suprapun. De altfel, suprapunerea dintre existenţă şi cunoaştere este o temă fundamentală a gândirii antice, şi care, de altfel, a cunoscut revalorificări interesante în modernitate. Parmenide afirma, de exemplu, că „acelaşi lucru este a cunoaşte şi a fi”. Cu alte cuvinte, există o identitate de natură între intelect şi restul existenţei, care face posibilă cunoaşterea. Ajungem, astfel, la ideea importantă de reflexivitate a gândirii, la ceea ce s-a numit, în legătură cu gândirea greacă şi nu numai, critica gândirii, cunoaşterea cunoaşterii.
În dialogul „Timaios”, Platon se referea la cele cinci poliedre regulate, spunând că sunt figuri cosmice la intersecţia dintre raţional şi iraţional, şi corespund treptelor universului său. Teoria lui Platon corespunde în mare măsură principiului economiei, în sensul folosirii unui număr minim de elemente pentru redarea diversităţii fenomenelor naturii, ca şi dezideratelor raţiunii, regularităţii şi ordinii de care e pătrunsă gândirea greacă. Existenţa ascultă de acelaşi logos ca şi intelectul [1].

Thursday, June 30, 2011

Gheorghe Vrânceanu (1900-1979)


Gheorghe Vrânceanu (June 30, 1900, Valea Hogei, Lipova, Bacău County – April 27, 1979, Bucharest)

Gheorghe Vrânceanu at MGP

Mac Tutor Biography

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)

Wednesday, June 22, 2011

Monday, June 20, 2011

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

Sunday, June 19, 2011

Wednesday, May 25, 2011

Monday, May 23, 2011

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.

Thursday, May 19, 2011

A Forgetful Number



A Forgetful Number - a poem by Vasko Popa (who "was the first [poet] in post-World War II Yugoslavia to break with the Socialist Realism") in the volume Secondary Heaven (Collected Poems), "Yawn of Yawns" (1968).

Image source (Riemann zeta function ζ(s) in the complex plane).

Solution to FQ-B1078

Solution to FQ-B1078 (Fibonacci Quarterly 48 (2010), no. 4, 367) submited on May 9, 2011 by the ONU-SOLVE problem group (faculty advisor - Mihai Caragiu).

Solution to AMM 11537

Solution to AMM 11537 (The American Mathematical Monthly 117 (2010), no. 10, p. 929), written by Mihai Caragiu and submitted on April 21, 2011.

Solution to AMM 11536

Solution to AMM 11536 (The American Mathematical Monthly 117 (9), November 2010, p. 835) written by Mihai Caragiu and submitted on March 30, 2011.

Solution to AMM 11527

Solution to AMM 11527 (The American Mathematical Monthly 117 (2010), no. 8, 742) written by Mihai Caragiu and submitted on February 28, 2011.

Solution to FQ-B1074

Solution to FQ-B1074 (Fibonacci Quarterly 48 (2010), no. 3, p. 278) submited on February 10, 2011 by the ONU-SOLVE problem group (faculty advisor Mihai Caragiu)

A classroom snippet

A linear algebra classroom snippet: symmetric matrices... with nice numbers. A grab bag of symmetric matrices

Thursday, May 5, 2011

An MGPF Path Towards a Fixed Point

MGPF - multidimensional greatest prime factor sequences
The MGPF conjecture: all MGPF sequences are ultimately periodic.
An-MGPF-Path-Towards-a-Fixed-Point

Wednesday, April 20, 2011

"An Euler-Fibonacci Sequence" (forthcoming)

Forthcoming (Far East Journal of Mathematical Sciences) - An Euler-Fibonacci Sequence
by Mihai Caragiu and Ashley Risch
(some raw data - here)

Wednesday, April 6, 2011

The OEIS entry on the greatest prime factor


From the Online Encyclopedia of Integer Sequences:
A006530 - Largest prime dividing n (with a(1)=1) includes a reference to the 2010 paper,
G. Back and M. Caragiu,
The greatest prime factor and recurrent sequences,
Fib. Q., 48 (2010), 358-362.

Photo © Mihai Caragiu

Tuesday, March 15, 2011

Does Mathematical Beauty Pose Problems for Naturalism?

Does Mathematical Beauty Pose Problems for Naturalism?
Russell W. Howell
Prof. of Mathematics, Westmont College - Santa Barbara, California
July 28, 2005
The process of pure mathematical thought and the engagement with ever-surprising, profound new statements and their proofs (fulfillable mathematical intentions) reveals, alongside with a never-ending, other-worldly depth of pure mathematical discovery, the true meaning of Cantor's words,
'the essence of mathematics lies in its freedom'
Through that participative process, the mind acquires a sense of distinctiveness and autonomy of mathematics with respect to the physical world/processes. The (contingent) physical world(s)/universe(s)? No need for such hypotheses (turning the tables on Laplace, here). No need for a 'working' hypothesis of a physical world out there in the depth of pure mathematical exploration. This is not good news for naturalism.The same case for profound freedom can be made by poets and artists.  (MC)

Tuesday, February 22, 2011

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

Wednesday, February 16, 2011

Hubert Dreyfus' criticism of AI

Dreyfus's critique of artificial intelligence (AI) concerns what he considers to be the four primary assumptions of AI research. The first two assumptions he criticizes are what he calls the "biological" and "psychological" assumptions. The biological assumption is that the brain is analogous to computer hardware and the mind is analogous to computer software. The psychological assumption is that the mind works by performing discrete computations (in the form of algorithmic rules) on discrete representations or symbols.

Dreyfus claims that the plausibility of the psychological assumption rests on two others: the epistemological and ontological assumptions. The epistemological assumption is that all activity (either by animate or inanimate objects) can be formalised (mathematically) in the form of predictive rules or laws. The ontological assumption is that reality consists entirely of a set of mutually independent, atomic (indivisible) facts. It's because of the epistemological assumption that workers in the field argue that intelligence is the same as formal rule-following, and it's because of the ontological one that they argue that human knowledge consists entirely of internal representations of reality.

On the basis of these two assumptions, workers in the field claim that cognition is the manipulation of internal symbols by internal rules, and that, therefore, human behaviour is, to a large extent, context free (see contextualism). Therefore a truly scientific psychology is possible, which will detail the 'internal' rules of the human mind, in the same way the laws of physics detail the 'external' laws of the physical world. But it is this key assumption that Dreyfus denies. In other words, he argues that we cannot now (and never will) be able to understand our own behavior in the same way as we understand objects in, for example, physics or chemistry: that is, by considering ourselves as things whose behaviour can be predicted via 'objective', context free scientific laws. According to Dreyfus, a context free psychology is a contradiction in terms.

Dreyfus's arguments against this position are taken from the phenomenological and hermeneutical tradition (especially the work of Martin Heidegger). Heidegger argued that, contrary to the cognitivist views on which AI is based, our being is in fact highly context bound, which is why the two context-free assumptions are false. Dreyfus doesn't deny that we can choose to see human (or any) activity as being 'law governed', in the same way that we can choose to see reality as consisting of indivisible atomic facts...if we wish. But it is a huge leap from that to state that because we want to or can see things in this way that it is therefore an objective fact that they are the case. In fact, Dreyfus argues that they are not (necessarily) the case, and that, therefore, any research program that assumes they are will quickly run into profound theoretical and practical problems. Therefore the current efforts of workers in the field are doomed to failure.

Source: Hubert Dreyfus - Wikipedia - http://en.wikipedia.org/wiki/Hubert_Dreyfus

Hubert Dreyfus on Husserl and Heidegger
Section 1 - http://www.youtube.com/watch?v=aaGk6S1qhz0


Section 2 - http://www.youtube.com/watch?v=ylKnb6WtYqU
Section 3 - http://www.youtube.com/watch?v=LgUDaml7ZJY
Section 4 - http://www.youtube.com/watch?v=QzAqfzWJTq4
Section 5 - http://www.youtube.com/watch?v=VfsKTSM5Sns

Sunday, February 6, 2011

Argument from Reason

“That grand myth which I asked you to admire a few minutes ago is not for me a hostile novelty breaking in on my traditional beliefs. On the contrary, that cosmology is what I started from. Deepening distrust and final abandonment of it long preceded my conversion to Christianity. Long before I believed Theology to be true I had already decided that the popular scientific picture at any rate was false. One absolutely central inconsistency ruins it; it is the one we touched on a fortnight ago. The whole picture professes to depend on inferences from observed facts. Unless inference is valid, the whole picture disappears. Unless we can be sure that reality in the remotest nebula or the remotest part obeys the thought--laws of the human scientist here and now in his laboratory-in other words, unless Reason is an absolute--all is in ruins. Yet those who ask me to believe this world picture also ask me to believe that Reason is simply the unforeseen and unintended by-product of mindless matter at one stage of its endless and aimless becoming. Here is flat contradiction. They ask me at the same moment to accept a conclusion and to discredit the only testimony on which that conclusion can be based. The difficulty is to me a fatal one; and the fact that when you put it to many scientists, far from having an answer, they seem not even to understand what the difficulty is, assures me that I have not found a mare's nest but detected a radical disease in their whole mode of thought from the very beginning. The man who has once understood the situation is compelled henceforth to regard the scientific cosmology as being, in principle, a myth; though no doubt a great many true particulars have been worked into it.”
(C. S. Lewis - as cited in Is Theology Poetry - GregorianStorage, 1/07/2011)
Image source: http://en.wikipedia.org/wiki/File:Hubble_-_infant_galaxy.jpg
Argument from Reason - Wikipedia.
C. S. Lewis' The cardinal difficulty of naturalism

Monday, January 24, 2011

Constantin Brâncuşi - Mathematics Interferences


An interesting paper:
BRANCUSI AND MATHEMATICS INTERFERENCES
by SAMOILĂ Gheorghe Ştefan,
Aplimat – Journal of Applied Mathematics, Vol. 2 (2009), No. 1, 227-234.

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

Tuesday, January 11, 2011

Kierkegaard on truth

The truth is a trap: you can not get it without it getting you; you cannot get the truth by capturing it, only by its capturing you. (Søren Kierkegaard)
This 'being captured by the truth' is echoed, in a sense, in mathematical research (and in science as well). It's as if a certain picture of the world (visible/physical or invisible/mathematical) progressively unfolds inside us, acquiring in the process a 'pointing beyond', iconic feature, that lifts ourselves.