tag:blogger.com,1999:blog-72625148356471367912017-04-17T00:28:43.614-04:00MATHEMATICAL TRIPSInvitations, in an educational vein, to problem solving, ideas and their history, biographical snapshots, visuals, classroom capsules, undergraduate research ideas, recreations, experiments, arts and poetry, philosophical and foundational matters and connections.Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comBlogger245125tag:blogger.com,1999:blog-7262514835647136791.post-61812053583052690172017-04-17T00:27:00.000-04:002017-04-17T00:28:43.638-04:00Forthcoming experimental/computational number theory book: Sequential Experiments with Primes (Springer 2017 - by Mihai Caragiu)<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-mZLNXT04hPk/WPRBzKXkwzI/AAAAAAAACg8/Fh5_vfaVyuo5kOv2-1jW9aDR-Uhq3I3bgCPcB/s1600/cover.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="640" src="https://4.bp.blogspot.com/-mZLNXT04hPk/WPRBzKXkwzI/AAAAAAAACg8/Fh5_vfaVyuo5kOv2-1jW9aDR-Uhq3I3bgCPcB/s640/cover.jpg" width="424" /></a></div><span class="highlightNode">Forthcoming book:</span><br /><span class="highlightNode">Sequential Experiments with Primes</span> © 2017<br /><br /><a href="http://www.springer.com/us/book/9783319567617" rel="nofollow" target="_blank">http://www.springer.com/us/book/9783319567617</a><br /><br />Presents experimental and computational number theory in a new and interesting context<br />Offers new mathematical ideas and problems involving prime numbers and related sequences<br />Perfect for graduate and advanced undergraduate students studying applied and computational number theory<br /><div class="text_exposed_show"><br />Due: July 26, 2017<br />ISBN 978-3-319-56762-4<br /><br />With a specific focus on the mathematical life in small undergraduate colleges, this book presents a variety of elementary number theory insights involving sequences largely built from prime numbers and contingent number-theoretic functions. Chapters include new mathematical ideas and open problems, some of which are proved in the text. Vector valued MGPF sequences, extensions of Conway’s Subprime Fibonacci sequences, and linear complexity of bit streams derived from GPF sequences are among the topics covered in this book. This book is perfect for the pure-mathematics-minded educator in a small undergraduate college as well as graduate students and advanced undergraduate students looking for a significant high-impact learning experience in mathematics.<br /><br />Mihai Caragiu is a professor of mathematics at Ohio Northern University. He is the recipient of the 2010 Outstanding Teaching Faculty Award for the Department of Mathematics and Statistics.</div>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-39625665019795853032016-11-02T00:03:00.002-04:002016-11-02T00:03:21.089-04:00The amazing Grothendieck's view on research...<blockquote class="tr_bq"><span style="font-size: large;"><i>I am not really doing research, just trying to cultivate myself.</i></span></blockquote><span style="font-size: large;">Alexander Grothendieck (1928–2014)</span>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-66513441760818322142016-03-27T17:57:00.000-04:002016-03-27T17:57:17.072-04:00An amazing "prime conspiracy" discovered by Lemke Oliver and Soundararajan<span style="font-size: large;">QUANTA MAGAZINE / Number Theory</span><br /><a href="https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/"><b><span style="font-size: large;">Mathematicians Discover Prime Conspiracy</span></b></a><br /><span style="font-size: large;"><i>A previously unnoticed property of prime numbers seems to violate a longstanding assumption about how they behave. </i></span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">Terrence Tao: Biases between consecutive primes </span><br /><span style="font-size: large;"><a href="https://terrytao.wordpress.com/2016/03/14/biases-between-consecutive-primes">https://terrytao.wordpress.com/2016/03/14/biases-between-consecutive-primes</a></span><br /><br /><span style="font-size: large;">Lemke Oliver and Soundararajan ArXiv article</span><br /><span style="font-size: large;"><a href="http://arxiv.org/abs/1603.03720">http://arxiv.org/abs/1603.03720</a> </span>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-52878515734116143182016-03-27T17:46:00.000-04:002016-03-27T17:46:41.943-04:00Richard Feynman: 'Greek' versus 'Babylonian' mathematics LIVE<iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/YaUlqXRPMmY" width="420"></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-71646216680125159802016-03-27T17:43:00.000-04:002016-03-27T17:47:43.301-04:00Mathematics as Hidden Reality - Edward Frenkel<blockquote class="tr_bq"><i>"There's a secret world out there. A hidden parallel universe of beauty and elegance, intricately intertwined with ours. And it's invisible to most of us."</i><br /><i><br /></i><i>Imagine that you had to take an art class in which they taught you only how to paint a fence or a wall, but never showed you the paintings of the great masters. Alas, this is how math is taught, and so for most of us it becomes the intellectual equivalent of watching paint dry. Edward Frenkel wants to open this secret world to all of us because it can teach us so much about the mysteries of the Universe. In this talk, he weaves the discovery of math with his personal journey, addressing the existential questions of finding out who we are; of truth, courage, and passion.</i><br /><i><br /></i><i>Edward Frenkel is a professor of mathematics at the University of California, Berkeley, which he joined in 1997 after being on the faculty at Harvard University. He is a member of the American Academy of Arts and Sciences, a Fellow of the American Mathematical Society, and the winner of the Hermann Weyl Prize in mathematical physics. Frenkel has authored three books and over eighty scholarly articles in academic journals, and he has lectured on his work around the world. His YouTube videos have garnered over 3 million views combined.</i><br /><i><br /></i><i>Frenkel’s latest book "Love and Math" was a New York Times bestseller and has been named one of the Best Books of 2013 by both Amazon and iBooks. It is being translated into 14 languages. Frenkel has also co-produced, co-directed and played the lead in the film "Rites of Love and Math" (2010).</i></blockquote><br /><a class="yt-uix-sessionlink " data-sessionlink="ei=WVP4VruZJtTH-AX5v7uABA" href="http://edwardfrenkel.com/" rel="nofollow" target="_blank">http://edwardfrenkel.com</a><br /><br /><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/YnqQ-BWMHrE" width="560"></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-30083607249289793282016-03-27T17:40:00.000-04:002016-03-27T17:40:19.829-04:00Edward Frenkel: Let's Stop Hating Math <iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/PcAZWgvFc1Q" width="560"></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-28139148688884402252016-03-11T15:58:00.002-05:002016-03-11T16:03:36.327-05:00Fields arranged by purity<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-76jwn2fnM84/VuMw6h-MIKI/AAAAAAAACfM/UoQ3LBkkwsIB-7bt_8JA83C2x1yv4h__w/s1600/12794576_1232996386730253_1014139511116104118_n.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-76jwn2fnM84/VuMw6h-MIKI/AAAAAAAACfM/UoQ3LBkkwsIB-7bt_8JA83C2x1yv4h__w/s1600/12794576_1232996386730253_1014139511116104118_n.jpg" /></a></div><br /><span style="font-size: large;">Permanent link: <a href="http://xkcd.com/435/">http://xkcd.com/435/</a></span>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-56346558544159409032016-02-27T21:10:00.003-05:002016-02-27T21:12:37.942-05:00primes as "experimental data"...<blockquote class="tr_bq"><i><span style="font-size: large;">"Although the prime numbers are rigidly determined, they somehow feel like experimental data." </span></i></blockquote>T. Gowers, "Mathematics: A Very Short Introduction (Oxford Univ. Press, 2002) p.118 (apud <a href="http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/surprising.htm">source</a>)Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-73724729771482561532016-01-10T00:04:00.002-05:002016-01-10T00:04:59.403-05:00Amazing Resonance Experiment<iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/wvJAgrUBF4w" width="560"></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-81773213929312051612016-01-08T19:11:00.000-05:002016-01-08T19:11:06.573-05:00Paul Vicol's website<b><span style="font-size: large;"><a href="http://www.paulvicol.com/">http://www.paulvicol.com/</a></span></b>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-76329960853075114172016-01-07T01:08:00.002-05:002016-01-07T01:15:45.264-05:00Tenure matters. Research by proxy - a big "no"<span style="font-size: large;">That's a potential problem with tenure-track faculty at smaller colleges. In order to get credit for scholarly activity, some of them encourage/coach students to present materials at conferences (mainly state or local). The materials are mostly elementary and well known (so that they would not count as a significant scholarly activity for any self-respecting faculty working towards his/her tenure), however having student presenters is strongly encouraged by the organizers of said conferences, and their contributed presentations are benevolently welcomed (and rightfully so). While it is OK (and arguably a formative experience) for students to present such materials, it is not OK for their faculty advisors to claim credit for scholarship (via 'undergraduate research') on the basis of having student advisees presenting at conferences (while they avoid having their names as coauthors alongside their students). That's a big "no". There is no such thing as "research by proxy". If they help students to give contributed talks on general topics, this is actually a credential for teaching effectiveness, not a scholarly credential. If they want scholarly credential they should "man up" and sign up as co-authors in the presentations with their advisees (and consequently face the "tune" of assessment after they claim scholarly credit for said presentations), or, even better - so that every doubt will be removed - publish with student co-authors (or without, for that matter). There is no undergraduate research without faculty research.</span>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-63085227719048628992015-12-29T21:37:00.001-05:002015-12-29T21:37:17.063-05:00FYI: Springer make many logic books freely downloadable<a href="http://www.logicmatters.net/2015/12/28/springer-make-many-logic-books-freely-downloadable/"><span style="font-size: large;">http://www.logicmatters.net/2015/12/28/springer-make-many-logic-books-freely-downloadable/</span></a>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-1637277929414137572015-12-28T00:44:00.000-05:002015-12-28T00:45:31.294-05:00Why mathematics is exceptional<span style="font-size: large;"><i>“Mathematical knowledge is unlike any other knowledge. While our perception of the physical world can always be distorted, our perception of mathematical truths can’t be. They are objective, persistent, necessary truths. A mathematical formula or theorem means the same thing to anyone anywhere – no matter what gender, religion, or skin color; it will mean the same thing to anyone a thousand years from now. And what’s also amazing is that we own all of them. No one can patent a mathematical formula, it’s ours to share. There is nothing in this world that is so deep and exquisite and yet so readily available to all. That such a reservoir of knowledge really exists is nearly unbelievable. It’s too precious to be given away to the “initiated few.” It belongs to all of us.”</i> ― Edward Frenkel (Love and Math: The Heart of Hidden Reality - via goodreads.com) </span>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-15133199062678582082015-12-22T01:38:00.001-05:002015-12-22T01:49:21.429-05:00Bridges - 2016 conference<iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/rexr77S3FbU" width="560"></iframe> <br /><br />Atentie pentru cei interesati: submission deadlines:<br /><blockquote class="tr_bq"><i>Regular Papers: 1 February 2016 </i><br /><i>Workshop Papers: 1 March 2016 </i><br /><i>Short Papers: 15 March 2016 </i><br /><i>Acceptance Notifications: 30 March 2016 </i><br /><i>Final Submissions: 30 April 2016</i></blockquote><a href="http://bridgesmathart.org/bridges-2016/">http://bridgesmathart.org/bridges-2016/</a>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-72130826112807947232015-12-22T01:13:00.002-05:002015-12-22T01:14:02.246-05:00bridges - mathematics and art<span style="font-size: large;">The Bridges Archive: an online repository of every paper ever published in the annual <a href="http://bridgesmathart.org/">Bridges Conference</a> on art and mathematics.</span>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-72854318312857979292015-12-11T00:12:00.001-05:002015-12-11T00:12:33.340-05:00The Heisenberg and Witt algebras<span style="font-size: large;"><a href="https://amathew.wordpress.com/2012/03/01/the-heisenberg-and-witt-algebras/"><b>Here</b></a> - a nice introduction by Akhil Mathew @ <a href="https://amathew.wordpress.com/">Climbing Mount Burbaki</a></span>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-87718922355672643442015-12-09T23:02:00.002-05:002015-12-09T23:04:21.355-05:00The attractiveness of primes for experimental mathematics...<blockquote class="tr_bq"><i><span style="font-size: large;">'Although the prime numbers are rigidly determined, they somehow feel like experimental data'</span></i></blockquote><span style="font-size: large;"><b>Timothy Gowers</b>. <b>Mathematics: A Very Short Introduction</b> (Oxford Univ. Press, 2002) p.118; as cited in <a href="http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/surprising.htm">"surprising connections between number theory and physics"</a></span>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-25793035567294087422015-11-30T22:54:00.000-05:002015-11-30T22:57:01.042-05:00All primes in terms of one: non-associative algebra and Google cloud computing<span style="font-size: large;">Under addition, the positive integers 1, 1+1, 1+1+1,.... form a cyclic (semigroup) structure generated by 1. We will explore a way of representing prime numbers in terms of the single generator g = 2, under a non-associative, non-commutative binary operation "o" on the set A of all primes, defined by setting x o y:=P(2x+y) where P is the greatest prime factor function. For example, 3 = 2 o 2, 5 = 2 o (2 o (2 o (2 o 2)))), etc. In a joint work with Paul A. Vicol from Simon Fraser University, we managed to verify that all primes up to 7259167 can be expressed as non-associative products involving the symbols 2, o, and parentheses ),(. This computational evidence points towards a cyclicity conjecture for the (magma) structure (A, o). Moreover, we searched for other similar non-associative algebraic structures on A (prime magmas) that might be cyclic, established a set of fairly restrictive necessary conditions for cyclicity, formulated a more general cyclicity conjecture for special prime magmas, and found computational evidence (after days of running Julia programs on a GCE platform) for the cyclicity of the structures in a representative set. (October 24, 2014 - MAA Ohio Fall Meeting)</span>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-68173558806189349652015-11-28T01:15:00.000-05:002015-11-28T01:15:06.705-05:00Love and Tensor Algebra - from "The Cyberiad" by Stanislaw Lem<blockquote class="tr_bq"><i>"Come, let us hasten to a higher plane<br />Where dyads tread the fairy fields of Venn,<br />Their indices bedecked from one to n<br />Commingled in an endless Markov chain!</i><br /><i><br /></i><i>Come, every frustrum longs to be a cone<br />And every vector dreams of matrices.<br />Hark to the gentle gradient of the breeze:<br />It whispers of a more ergodic zone.</i><br /><br /><i>In Riemann, Hilbert or in Banach space<br />Let superscripts and subscripts go their ways.<br />Our asymptotes no longer out of phase,<br />We shall encounter, counting, face to face.</i><br /><br /><i>I'll grant thee random access to my heart,<br />Thou'lt tell me all the constants of thy love;<br />And so we two shall all love's lemmas prove,<br />And in our bound partition never part.</i><br /><br /><i>For what did Cauchy know, or Christoffel,<br />Or Fourier, or any Bools or Euler,<br />Wielding their compasses, their pens and rulers,<br />Of thy supernal sinusoidal spell?</i><br /><br /><i>Cancel me not - for what then shall remain?<br />Abscissas some mantissas, modules, modes,<br />A root or two, a torus and a node:<br />The inverse of my verse, a null domain.</i><br /><br /><i>Ellipse of bliss, converge, O lips divine!<br />the product o four scalars is defines!<br />Cyberiad draws nigh, and the skew mind<br />Cuts capers like a happy haversine.</i><br /><br /><i>I see the eigenvalue in thine eye,<br />I hear the tender tensor in thy sigh.<br />Bernoulli would have been content to die,<br />Had he but known such a^2 cos 2 phi!"</i></blockquote>SOURCE: <a href="http://www.aleph.se/Trans/Cultural/Art/tensor.html">http://www.aleph.se/Trans/Cultural/Art/tensor.html</a>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-34292987816142924212015-10-04T22:57:00.001-04:002015-10-04T22:57:31.862-04:00The OEIS Movie. Experimental Mathematics<iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/LCWglXljevY" width="420"></iframe> <br /><br /><span style="font-size: x-large;"><b><a href="https://www.youtube.com/user/kgnang/">Experimental mathematics channel</a> - You Tube </b></span>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-72965641271356068602015-09-27T18:14:00.000-04:002015-09-27T18:14:38.384-04:00Erdős' discrepancy problem - recently solved<blockquote class="tr_bq"><span style="font-size: large;"><b><a href="http://phys.org/news/2015-09-blog-comment-collab-maths-problem.html"><i>Blog comment, collab help man attack old maths problem</i></a> </b></span><br /><span style="font-size: large;"><b>Phys.Org - September 26, 2015 by Nancy Owano </b></span></blockquote><br /><span style="font-size: large;"><b>The Erdos Discrepancy Problem - Terrence Tao - <a href="http://arxiv.org/abs/1509.05363">http://arxiv.org/abs/1509.05363</a></b></span>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-61386923566490273362015-09-07T22:09:00.002-04:002015-09-07T22:11:21.298-04:00The farthest known galaxy...<blockquote class="tr_bq"><span style="font-size: large;"><i>"the recently discovered EGS8p7 galaxy is a whopping 13.2 billion years old, making it both the farthest known galaxy to date and just 600 million years younger than the universe as we know it."</i> (cf. <b>The farthest known galaxy is 13.2 billion years old</b> - <a href="http://www.engadget.com/2015/09/07/scientists-spot-farthest-ever-galaxy/">http://www.engadget.com/2015/09/07/scientists-spot-farthest-ever-galaxy/</a> ) </span></blockquote>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-63545336310276803522015-08-30T09:14:00.001-04:002015-08-30T09:14:23.400-04:00Psychology - the 36 percent replication problem<blockquote class="tr_bq"><span style="font-size: large;"><i>"Despite this careful planning, less than half of the replication studies reproduced the original results. While 97 percent of the original studies produced results with a “statistically significant” p-value of 0.05 or less, only 36 percent of the replication studies did the same."</i> (cf. <a href="http://fivethirtyeight.com/datalab/psychology-is-starting-to-deal-with-its-replication-problem/">"Psychology Is Starting To Deal With Its Replication Problem"</a>, Christie Aschwanden in FiveThirtyEight - August 27, 2015)</span><i><br /></i></blockquote>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-35642806844036630142015-07-26T22:56:00.000-04:002015-07-26T22:59:59.761-04:00The Singular Mind of Terry Tao<a href="http://www.nytimes.com/2015/07/26/magazine/the-singular-mind-of-terry-tao.html?_r=0"><b><span style="font-size: large;">The Singular Mind of Terry Tao </span></b></a><br /><i><span style="font-size: large;">A prodigy grows up to become one of the greatest mathematicians in the world. </span></i><br /><span style="font-size: large;">New York Times</span><br /><span style="font-size: large;">By GARETH COOK, July 24, 2015 </span>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-85328334549728820742015-07-26T22:46:00.001-04:002015-07-26T22:47:05.945-04:00A beautiful apology of pure mathematics<blockquote class="tr_bq"><b><span style="font-size: large;">Why Do We Pay Pure Mathematicians? </span></b><br /><i><span style="font-size: large;">Or, the Many Uses of Uselessness </span></i><br /><span style="font-size: large;"><b><a href="http://mathwithbaddrawings.com/2015/02/24/why-do-we-pay-mathematicians/">http://mathwithbaddrawings.com/2015/02/24/why-do-we-pay-mathematicians/</a></b></span></blockquote>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.com