tag:blogger.com,1999:blog-73791109607960141702017-12-10T22:21:18.298-05:00Clueless FundatmaA random walk through a subset of things I care about. Science, math, computing, higher education, open source software, economics, food etc.Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.comBlogger716125tag:blogger.com,1999:blog-7379110960796014170.post-22128876508627036192017-12-07T12:05:00.000-05:002017-12-07T12:05:50.978-05:00More is Different<div dir="ltr" style="text-align: left;" trbidi="on">Last week, I read a nearly 50 year old essay by P. W. Anderson (h/t fermatslibrary) entitled "More is Different" (<a href="http://fermatslibrary.com/s/more-is-different">pdf</a>). It is a fascinating opinion piece.<br /><ul style="text-align: left;"><li>"Quantitative differences become qualitative ones" - Marx</li><li>Psychology is not applied biology, nor is biology applied chemistry.</li></ul><div>This other essay on the "<a href="http://arthur.shumwaysmith.com/life/content/the_arrogance_of_physicists">arrogance of physicists</a>" speaks to a similar point:</div><blockquote class="tr_bq">But training and experience in physics gives you a very powerful toolbox of techniques, intuitions and approaches to solving problems that molds your outlook and attitude toward the rest of the world. Other fields of science or engineering are limited in their scope. Mathematics is powerful and immense in logical scope, but in the end it is all tautology, as I tease my mathematician friends, with no implied or even desired connection to the real world. Physics is the application of mathematics to reality and the 20th century proved its remarkable effectiveness in understanding that world, from the behavior of the tiniest particles to the limits of the entire cosmos. Chemistry generally confines itself to the world of atoms and molecules, biology to life, wonderful in itself, but confined so far as we know to just this planet. The social sciences limit themselves still further, mainly to the behavior of us human beings - certainly a complex and highly interesting subject, but difficult to generalize from. Engineering also has a powerful collection of intuitions and formulas to apply to the real world, but those tend to be more specific individual rules, rather than the general and universal laws that physicists have found. </blockquote><blockquote class="tr_bq">Computer scientists and their practical real-world programming cousins are perhaps closest to physicists in justified confidence in the generality of their toolbox. Everything real can be viewed as computational, and there are some very general rules about information and logic that seep into the intuition of any good programmer. As physics is the application of mathematics to the real world of physical things, so programming is the application of mathematics to the world of information about things, and sometimes those two worlds even seem to be merging.</blockquote></div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/euR8bcTJEpA" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/12/more-is-different.htmltag:blogger.com,1999:blog-7379110960796014170.post-23590341854404443662017-11-26T19:14:00.004-05:002017-11-26T19:14:57.650-05:00Post-Thanksgiving Links<div dir="ltr" style="text-align: left;" trbidi="on">Some links to interesting scientific content:<br /><br />1. How Wikipedia Tackles Fringe Nonsense (<a href="http://theness.com/neurologicablog/index.php/how-wikipedia-tackles-fringe-nonsense/">neurologica</a>)<br /><br />2. Seven Academic-World Lies (<a href="https://www.linkedin.com/pulse/7-lies-academic-world-keeps-telling-you-mariana-cerdeira">Mariana Cerdeira</a>)<br /><br />3. Numerically Approximating Ghosts (<a href="https://www.johndcook.com/blog/2009/08/11/approximating-a-solution-that-doesnt-exist/">John D Cook</a>)<br /><br />4. An Archive of Projects Using Differential Equations (<a href="https://www.cengage.com/math/book_content/0495108243_zill/projects_archive/">Zill</a>)</div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/-qTNd0WkJJA" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/11/post-thanksgiving-links.htmltag:blogger.com,1999:blog-7379110960796014170.post-35404593113827032492017-11-14T08:09:00.001-05:002017-11-14T08:09:24.423-05:00History of PowerPoint<div dir="ltr" style="text-align: left;" trbidi="on">The history of MS Office is riveting.<br /><br />This <a href="https://spectrum.ieee.org/tech-history/cyberspace/the-improbable-origins-of-powerpoint">essay</a> in IEEE Spectrum recounts the "Improbable Origins of PowerPoint". I did not know that Xerox PARC had such a direct influence of on MS Office (including <a href="https://en.wikipedia.org/wiki/Microsoft_Word">MS Word</a>).<br /><br />Reading the essay, one gets a sense for how fluid the desktop computer landscape was between the advent of the Apple Lisa and Microsoft's bundling of Word, Excel, and PowerPoint.</div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/QnLfNXbLixU" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/11/history-of-powerpoint.htmltag:blogger.com,1999:blog-7379110960796014170.post-75537918885840305222017-11-13T14:50:00.000-05:002017-11-26T11:48:45.799-05:00Exporting Numpy Arrays and Matrices to LaTeX<div dir="ltr" style="text-align: left;" trbidi="on">Over the past couple of years, a lot of my "numerical experimentation" work has moved from Octave to python/numpy.<br /><br />I incorporate a lot of this work into my classes and presentations (made using beamer), and having a script to translate vectors and matrices to LaTeX format is handy.<br /><br />In the past, I <a href="http://sachinashanbhag.blogspot.com/2012/11/exporting-matrices-in-octavematlab-to.html">shared</a> a Matlab/Octave <a href="https://docs.google.com/open?id=0Bww3OZktvGQucmxnd1FJNElCVGc">script</a> which does this.<br /><br />Here is a python/numpy <a href="https://gist.github.com/shane5ul/ab47124f9c22796e369948122c80037b">script</a> which does something similar. The script<br /><br /><ul style="text-align: left;"><li>autodetects integers and floats</li><li>allows you to control the number of decimals for floats</li><li>allows you optionally render floats in scientific format</li><li>right-justify using the bmatrix* environment (good for -ve numbers)</li><li>suppress small values near zero (~ 1e-16)</li></ul><div><br /></div></div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/-M5ZixqEFLQ" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com4http://sachinashanbhag.blogspot.com/2017/11/exporting-numpy-arrays-and-matrices-to.htmltag:blogger.com,1999:blog-7379110960796014170.post-16451153819106103062017-11-06T17:27:00.000-05:002017-11-06T17:27:28.981-05:00Python: Orthogonal Polynomials and Generalized Gauss Quadrature<div dir="ltr" style="text-align: left;" trbidi="on">A new (to me) <a href="https://github.com/nschloe/orthopy">python library</a> for easily computing families of orthogonal polynomials.<br /><br />Getting standard (generalized) Gauss quadrature schemes is extremely simple. For example to get 13 nodes and weights for Gauss-Laguerre integration, correct up to 50 decimal places:<br /><br /><pre style="background-color: #f6f8fa; border-radius: 3px; box-sizing: border-box; color: #24292e; font-family: SFMono-Regular, Consolas, "Liberation Mono", Menlo, Courier, monospace; font-size: 13.6px; line-height: 1.45; overflow: auto; padding: 16px; word-break: normal; word-wrap: normal;">pts,wts <span class="pl-k" style="box-sizing: border-box; color: #d73a49;">=</span> orthopy.schemes.laguerre(<span class="pl-c1" style="box-sizing: border-box; color: #005cc5;">13</span>, <span class="pl-v" style="box-sizing: border-box; color: #e36209;">decimal_places</span><span class="pl-k" style="box-sizing: border-box; color: #d73a49;">=</span><span class="pl-c1" style="box-sizing: border-box; color: #005cc5;">50</span>)</pre><br />The numpy <a href="https://docs.scipy.org/doc/numpy/reference/routines.polynomials.package.html">Polynomial</a> package provides similar functionality:<br /><br /><span style="font-family: "courier new" , "courier" , monospace;">pts, wts = numpy.polynomial.laguerre.laggauss(13)</span><br /><div><span style="font-family: "courier new" , "courier" , monospace;"><br /></span></div><div><span style="font-family: inherit;">A nice feature (besides arbitrary precision) is that you can derive custom orthogonal polynomials and quadrature rules. All you need to provide is a weight function and domain of the polynomials. From the project webpage:</span></div><div><pre style="background-color: #f6f8fa; border-radius: 3px; box-sizing: border-box; color: #24292e; font-family: SFMono-Regular, Consolas, "Liberation Mono", Menlo, Courier, monospace; font-size: 13.6px; line-height: 1.45; overflow: auto; padding: 16px; word-break: normal; word-wrap: normal;"><span class="pl-k" style="box-sizing: border-box; color: #d73a49;">import</span> orthopy<br />moments <span class="pl-k" style="box-sizing: border-box; color: #d73a49;">=</span> orthopy.compute_moments(<span class="pl-k" style="box-sizing: border-box; color: #d73a49;">lambda</span> <span class="pl-smi" style="box-sizing: border-box;">x</span>: x<span class="pl-k" style="box-sizing: border-box; color: #d73a49;">**</span><span class="pl-c1" style="box-sizing: border-box; color: #005cc5;">2</span>, <span class="pl-k" style="box-sizing: border-box; color: #d73a49;">-</span><span class="pl-c1" style="box-sizing: border-box; color: #005cc5;">1</span>, <span class="pl-k" style="box-sizing: border-box; color: #d73a49;">+</span><span class="pl-c1" style="box-sizing: border-box; color: #005cc5;">1</span>, <span class="pl-c1" style="box-sizing: border-box; color: #005cc5;">20</span>)<br />alpha, beta <span class="pl-k" style="box-sizing: border-box; color: #d73a49;">=</span> orthopy.chebyshev(moments)<br />points, weights <span class="pl-k" style="box-sizing: border-box; color: #d73a49;">=</span> orthopy.schemes.custom(alpha, beta, <span class="pl-v" style="box-sizing: border-box; color: #e36209;">decimal_places</span><span class="pl-k" style="box-sizing: border-box; color: #d73a49;">=</span><span class="pl-c1" style="box-sizing: border-box; color: #005cc5;">30</span>)</pre></div><div>This generates a 10-point scheme for integrating functions over the interval [-1, 1], with weight function \(w(x) = x^2\).</div><div><span style="font-family: inherit;"><br /></span></div></div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/ZecSJXzGX-I" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/11/python-orthogonal-polynomials-and.htmltag:blogger.com,1999:blog-7379110960796014170.post-82962477701486582172017-10-27T12:21:00.000-04:002017-10-27T12:21:12.582-04:00Science Links<div dir="ltr" style="text-align: left;" trbidi="on">1. When the Revolution Came for Amy Cuddy (<a href="https://www.nytimes.com/2017/10/18/magazine/when-the-revolution-came-for-amy-cuddy.html">Susan Dominus in the NYT</a>)<br /><blockquote class="tr_bq">But since 2015, even as she continued to stride onstage and tell the audiences to face down their fears, Cuddy has been fighting her own anxieties, as fellow academics have subjected her research to exceptionally high levels of public scrutiny. She is far from alone in facing challenges to her work: Since 2011, a methodological reform movement has been rattling the field, raising the possibility that vast amounts of research, even entire subfields, might be unreliable. Up-and-coming social psychologists, armed with new statistical sophistication, picked up the cause of replications, openly questioning the work their colleagues conducted under a now-outdated set of assumptions. The culture in the field, once cordial and collaborative, became openly combative, as scientists adjusted to new norms of public critique while still struggling to adjust to new standards of evidence.</blockquote>2. When correlations don't imply causation, but something far more screwy! (<a href="https://www.theatlantic.com/business/archive/2012/05/when-correlation-is-not-causation-but-something-much-more-screwy/256918/">the Atlantic</a>)<br />2a. John D. Cook <a href="https://www.johndcook.com/blog/2017/09/10/negative-correlation-introduced-by-success/">follows up</a> with "negative correlations" induced by success.<br /><br />3. <a href="http://pathwaystoscience.org/Grad.aspx">STEM resources</a> for students from K-PhD, and beyond (<a href="http://pathwaystoscience.org/Library.aspx">PathwaysToScience</a>)<br /><br /></div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/qc_M0ir5-H0" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/10/science-links.htmltag:blogger.com,1999:blog-7379110960796014170.post-4611223909872094832017-10-24T12:05:00.000-04:002017-10-24T12:05:14.479-04:00Gauss and Ceres<div dir="ltr" style="text-align: left;" trbidi="on"><a href="https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss">Car Friedrich Gauss</a> was an intellectual colossus, whose work informed or revolutionized broad and seemingly unrelated swathes of science and math. In computational science, his name is attached to numerous methods for solving equations, integrating functions, and describing probabilities.<br /><br />Interestingly, perhaps two of his most enduring contributions - <a href="https://en.wikipedia.org/wiki/Gaussian_elimination">Gaussian elimination</a> to solve systems of linear equations, and normal or <a href="https://en.wikipedia.org/wiki/Normal_distribution">Gaussian distribution</a> are linked through the fascinating story of how <a href="https://www.schillerinstitute.org/fid_97-01/982_orbit_ceres.pdf">Gauss determined the orbit of Ceres</a> (great read!).<br /><br />While there is plenty of geometry involved, this example illustrates how multiple observations of the asteroid by astronomers, lead to an over-determined system of equations. Assuming that these measurements were tainted by normal or Gaussian error, Gauss built the resulting "normal equations" and solved for the orbit.<br /><br />When Ceres was lost to the glare of the sun, he was able to use these calculations to direct astronomers to the part of the sky where they should point their telescopes.</div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/cenJemefENk" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/10/gauss-and-ceres.htmltag:blogger.com,1999:blog-7379110960796014170.post-67392633546992557562017-10-21T12:04:00.004-04:002017-10-21T12:04:51.281-04:00Pascal's Wager<div dir="ltr" style="text-align: left;" trbidi="on">I enjoyed <a href="http://rationallyspeakingpodcast.org/show/rs-190-amanda-askell-on-pascals-wager-and-other-low-risks-wi.html">this recent conversation</a> between Julia Galef and Amanda Askell on the nuances of <a href="https://en.wikipedia.org/wiki/Pascal%27s_Wager">Pascal's wager</a>. According to wikipedia:<br /><blockquote class="tr_bq">Pascal argues that a rational person should live as though God exists and seek to believe in God. If God does actually exist, such a person will have only a finite loss (some pleasures, luxury, etc.), whereas they stand to receive infinite gains (as represented by eternity in Heaven) and avoid infinite losses (eternity in Hell).</blockquote>I always thought this was something of a tongue-in-cheek argument because "of course" the argument fails the smell test. However, if we take it seriously, we find that it resists simple attempts at tearing it down. This blog post ("<a href="http://rationallyspeakingpodcast.org/show/rs-190-amanda-askell-on-pascals-wager-and-other-low-risks-wi.html">Common objections to Pascal's wager</a>") outlines some of the rebuttals. It makes for interesting reading.<br /><br />One of the things from the podcast that stuck with me was a comment about whether belief in climate change maps neatly onto Pascal's wager. Simplistically, let C be the claim that climate change is true, and ~C be the opposite claim. Let A denote action (taken to avert C), and ~A denote inaction (business as usual).<br /><br />Then, we have the following four possibilities, A|C (action given climate change), A|~C, ~A|C, and ~A|~C.<br /><br /><b>A|C = mildly painful</b><br /><br />An analogy might be something like an appendectomy. There is a problem (inflamed appendix or climate change), and appropriate corrective action is applied (surgical removal, CO2 reduction).<br /><br /><b>A|~C = mildly painful</b><br /><br />An analogy would be unused insurance. You buy home insurance for a year, and nothing happens. You had to fork over premiums (which is mildly painful), but you accept that as reasonable risk against catastrophe.<br /><br /><b>~A|C = catastrophe</b><br /><br />Piggybacking on the previous analogy, here your house is in flames and you realize you skimped on fire insurance. The external "shock" is bad (climate change or house catching fire), but your "penny-wise but pound-foolish" behavior made a bad situation much much worse.<br /><br /><b>~A|~C = mildly pleasurable</b><br /><br />An analogy (which strikes close to home) might be skipping the annual dental checkup, and finding out nothing is wrong with your teeth. As someone once remarked to me, sometimes "pleasure is simply the absence of pain."<br /><br />Note that the catastrophic outcome 3 (~A|C), with its "infinities", crowds out the others.<br /><br />Hence, Pascal might argue that we should believe in both, God and climate change.</div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/1x5FzSLXcqE" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/10/pascals-wager.htmltag:blogger.com,1999:blog-7379110960796014170.post-37398639984647060462017-10-12T08:37:00.000-04:002017-10-12T08:40:05.360-04:00Introduce Concepts in Historical Order?<div dir="ltr" style="text-align: left;" trbidi="on">Let me confess: I have read very few scientific classics in the original.<br /><br />I haven't read the <a href="https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica">Principia</a>, the <a href="https://en.wikipedia.org/wiki/On_the_Origin_of_Species">Origin of Species</a>, or the <a href="https://en.wikipedia.org/wiki/Euclid%27s_Elements">Elements</a>.<br /><br />I had not even read Einstein's 1905 classic on <a href="https://en.wikipedia.org/wiki/Brownian_motion">Brownian motion</a>, until a few years ago, even though half of my research is directly or indirectly animated by it.<br /><br />Ever since I saw this amazing <a href="https://sachinashanbhag.blogspot.com/2017/09/complex-numbers-part-deux.html">series on complex numbers</a>, I have been wondering whether presenting the historical progression of ideas might be "better" than the standard textbook introduction. Here are some of my observations.<br /><br />The <b>historical approach</b> (HA) is inherently interesting, because it is about ideas and the <b>people</b> behind them. Stories of humans exploring and pushing boundaries, regardless of domain, are fascinating. These stories often have imperfect people grappling with new ideas, getting confused by their implications, arguing back and forth, improving, and gradually perfecting them over centuries. This happened with classical mechanics, evolution, complex numbers, quantum mechanics, etc.<br /><br />The <b>standard approach</b> (SA), on the other hand, steers away from messy pasts, leaps of intuition that came seemingly from nowhere, the entertaining bickering, and the trials and errors. It trims away the excess fat of distractions, consolidates different viewpoints, and presents a sanitized account of an idea. It is, without question, the quickest and cleanest way to learn a new concept. This is an extremely desirable feature in university courses, which have a mandate to "cover" a set of concepts, often in limited time.<br /><br />Perhaps, a good practical compromise is to start with an example rooted in the historical approach to motivate the topic, and transition to the standard textbook approach to teach the meat of the topic. It might be interesting to conclude once again with a historical perspective, perhaps mixed with a discussion of the current state of art and open questions.</div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/jLT6Hx6b_zI" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/10/introduce-concepts-in-historical-order.htmltag:blogger.com,1999:blog-7379110960796014170.post-8182794377619660842017-10-04T14:34:00.000-04:002017-10-04T14:34:07.364-04:00LaTeX: Figure Captions<div dir="ltr" style="text-align: left;" trbidi="on">A minimal working example.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-WWLXRbasKs8/WbLiSvJSjFI/AAAAAAAADoQ/lPt8x-WIJ0McAhTWLb3Hrh6dLmnICqYcgCLcBGAs/s1600/Screenshot%2Bfrom%2B2017-09-08%2B14-29-11.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="717" data-original-width="374" src="https://2.bp.blogspot.com/-WWLXRbasKs8/WbLiSvJSjFI/AAAAAAAADoQ/lPt8x-WIJ0McAhTWLb3Hrh6dLmnICqYcgCLcBGAs/s1600/Screenshot%2Bfrom%2B2017-09-08%2B14-29-11.png" /></a></div><br /><br /><script src="https://gist.github.com/shane5ul/3e0a7bd37c1871939ac837a949793d5a.js"></script> <br /><br /></div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/86XZGOs0BRU" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/10/latex-figure-captions.htmltag:blogger.com,1999:blog-7379110960796014170.post-29158010016493949802017-09-25T14:45:00.000-04:002017-09-25T14:45:10.376-04:00Prony Method<div dir="ltr" style="text-align: left;" trbidi="on">Given N equispaced data-points \(F_i = F(t = i \Delta t)\), where \(i = 0, 1, ..., N-1\), <a href="http://sachinashanbhag.blogspot.com/2017/08/exam-question-on-fitting-sums-of.html">the Prony method</a> can be used to fit a sum of m decaying exponenitals: \[F(t) = \sum_{i=1}^{m} a_i e^{b_i t}. \] The 2m unknowns are \(a_i\) and \(b_i\).<br /><div><br /></div><div>In the Prony method, the number of modes in the exponential (m) is pre-specified. There are other methods, which are more general.</div><br />Here is a python subprogram which implements the Prony method.<br /><script src="https://gist.github.com/shane5ul/8b360bb605baa9e29e5e6ede364f4d7d.js"></script> If you have arrays t and F, it can be called as:<br /><br /><span style="font-family: "courier new" , "courier" , monospace;">a_est, b_est = prony(t, F, m)</span></div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/phcwnJW_BZ0" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/09/prony-method.htmltag:blogger.com,1999:blog-7379110960796014170.post-18444359853313708372017-09-22T11:17:00.000-04:002017-09-22T11:17:07.568-04:00MCMC Samplers Visualization<div dir="ltr" style="text-align: left;" trbidi="on">A really nice <a href="https://chi-feng.github.io/mcmc-demo/app.html#HamiltonianMC,banana">interactive gallery of MCMC</a> samplers<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/--dfx5PsCiGw/Wb6RySOpg5I/AAAAAAAADpY/l_zZmiaD-vYjMpJVL2c4fbQbchqybYxhgCLcBGAs/s1600/Screenshot%2Bfrom%2B2017-09-17%2B11-16-13.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="802" data-original-width="1600" height="200" src="https://2.bp.blogspot.com/--dfx5PsCiGw/Wb6RySOpg5I/AAAAAAAADpY/l_zZmiaD-vYjMpJVL2c4fbQbchqybYxhgCLcBGAs/s400/Screenshot%2Bfrom%2B2017-09-17%2B11-16-13.png" width="400" /></a></div><br />You can choose different algorithms, and target distributions, change method parameters and observe the chain evolve.<br /><br />This might come in handy next semester, when I teach a Monte Carlo class.</div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/iC1ZS30g9pA" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/09/mcmc-samplers-visualization.htmltag:blogger.com,1999:blog-7379110960796014170.post-14982650959493721682017-09-19T11:14:00.000-04:002017-09-19T11:14:01.919-04:00Some Useful Math Links!<div dir="ltr" style="text-align: left;" trbidi="on">1. The history of the division symbol (obelus) is fascinating! <a href="https://divisbyzero.com/2017/09/15/the-division-symbol-goes-viral/">(DivisionByZero</a>)<br /><br />2. On the same blog: "<a href="https://divisbyzero.com/2008/09/22/what-is-the-difference-between-a-theorem-a-lemma-and-a-corollary/">What is the difference between a theorem, a lemma, and a corollary?</a>"<br /><br />3. The "<a href="http://physicsinsights.org/glue_function.html">glue function</a>"<br /><br />4. Free <a href="http://linear.axler.net/LinearAbridged.html">abridged Linear Algebra</a> book from Sheldon Axler.</div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/caNPCNmgDn4" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/09/some-useful-math-links.htmltag:blogger.com,1999:blog-7379110960796014170.post-75871480611555653592017-09-16T11:06:00.000-04:002017-09-16T11:06:11.608-04:00Implicit Bias Test<div dir="ltr" style="text-align: left;" trbidi="on">I thoroughly enjoyed <a href="http://rationallyspeakingpodcast.org/show/rs-192-jesse-singal-on-the-problems-with-implicit-bias-tests.html">this</a> Jesse Singal interview on Rationally Speaking on the problems with the "<a href="https://en.wikipedia.org/wiki/Implicit-association_test">implicit association test</a>" for diagnosing implicit bias.<br /><br />The following <a href="https://www.youtube.com/watch?v=n5Q5FQfXZag">Dateline</a> video shows how the test was sold to the public as scientifically robust.<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/n5Q5FQfXZag/0.jpg" src="https://www.youtube.com/embed/n5Q5FQfXZag?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><br />For fun, you can <a href="https://implicit.harvard.edu/implicit/takeatest.html">take the test yourself</a>.<br /><br />For the problems with the test, check out Jesse Singal's piece from earlier this year, "<a href="http://nymag.com/scienceofus/2017/01/psychologys-racism-measuring-tool-isnt-up-to-the-job.html">Psychology’s Favorite Tool for Measuring Racism Isn’t Up to the Job</a>". It is a thoughtful essay, that should be read in its entirety.<blockquote class="tr_bq">A pile of scholarly work, some of it published in top psychology journals and most of it ignored by the media, suggests that the IAT falls far short of the quality-control standards normally expected of psychological instruments. The IAT, this research suggests, is a noisy, unreliable measure that correlates far too weakly with any real-world outcomes to be used to predict individuals’ behavior — even the test’s creators have now admitted as such. The history of the test suggests it was released to the public and excitedly publicized long before it had been fully validated in the rigorous, careful way normally demanded by the field of psychology.</blockquote>Singal is careful to point out that just because IAT is flawed it doesn't imply that implicit bias doesn't exist. I liked an analogy he used in the podcast. If a thermometer is flawed, you can't use it to determine if a person has a fever. The person may or may not have a fever, but the thermometer should probably be tossed away. </div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/gupZzfUJcWE" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/09/implicit-bias-test.htmltag:blogger.com,1999:blog-7379110960796014170.post-78944696895034369942017-09-12T12:39:00.000-04:002017-09-12T12:39:09.633-04:00Euler and Graph Theory<div dir="ltr" style="text-align: left;" trbidi="on">I have been enjoying Marcus du Sautoy's fine <a href="http://www.bbc.co.uk/programmes/b00srz5b/episodes/downloads">podcast</a> of famous mathematicians for BBC4. <div><br /></div><div>Yesterday, I listened to the Leonhard Euler episode. While I always knew Euler was one of the top mathematicians of all time, his <a href="https://en.wikipedia.org/wiki/Leonhard_Euler#Contributions_to_mathematics_and_physics">contributions</a> are truly remarkable.<div><br /></div><div>The podcast talks about how he solved the seven bridges of Konigsberg problem by inventing graph theory, and proving its first theorem. I looked at that theorem as it applies to a "kids game" in a <a href="http://sachinashanbhag.blogspot.com/2016/07/sunday-afternoon-fun.html">previous blog</a>.</div></div></div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/xKLgwBKkUIw" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/09/euler-and-graph-theory.htmltag:blogger.com,1999:blog-7379110960796014170.post-24559383829480851902017-09-05T18:31:00.000-04:002017-09-05T18:31:06.080-04:00Teacher's Day 2017<div dir="ltr" style="text-align: left;" trbidi="on">I did not expect writing this post would be so bittersweet. Last <a href="http://sachinashanbhag.blogspot.com/2016/09/teachers-day.html">Teacher's Day</a>, I decided I would use the occasion to highlight specific teachers, who have had an outsized impact on me.<br /><br />Today, I am going to tell you about Kartic C. Khilar, or KCK as he was called at IIT Bombay. KCK was a central figure, and participant, as I navigated a period of multiple transitions.<br /><br />Interestingly, I first "met" KCK even before I met him. The year I took the Joint Entrance Exam (JEE) to apply for admission to IIT, he was the principal administrator. The only reason I remember is because he had a "killer" last name (so juvenile, I know!).<br /><br />Like 200,000 other rats, I studied relentlessly for two years. JEE is like academic Olympics. We trained like mental athletes: cardio, weights, pilates, the whole nine yards. Then, the starting gun went off, and we scampered. The first two thousand got in.<br /><br />Miraculously, I tumbled my way into IIT Bombay first, and then to the chemical engineering department. KCK was the head of the department, when my "batch" arrived.<br /><br />He taught us fluid mechanics and solid-fluid operations. He was a fantastic teacher - one of the best I've had. His lectures were crisp. He was always cheerful. And he cared about all his students - not just toppers.<br /><br />He had one striking attribute: no ego. No made up sense of self-importance, which is all the more remarkable given the power gap between teachers and students (especially in India). If you went to his office, he would listen, despite how busy he was, or how unimportant you were.<br /><br />A highlight of the undergrad program at IIT is the B. Tech project (BTP), which is the undergrad equivalent of a PhD dissertation. Again, due to a random set of circumstances, he ended up being my BTP mentor. Over the course of the last year and half at IIT our interaction deepened, if only because we met one-on-one on a weekly basis to discuss research.<br /><br />Research in the Fluid Mechanics lab was fun. I don't think I would have embarked on a research career, if I hadn't enjoyed this experience so much. This work on "colloid-facilitated contaminant transport" with KCK and his grad student at that time - Tushar Sen - would end up becoming my first peer-reviewed <a href="http://www.sciencedirect.com/science/article/pii/S0927775703005545">publication</a>.<br /><br />I ended up at the University of Michigan as a grad student, in no small part due to his kind word. Michigan was his alma mater too. He visited Ann Arbor twice, while I was there. Once, when I was a PhD student, and later just before I started my new academic job at Florida State. Each time I went to Bombay, I would meet him; usually over lunch or dinner.<br /><br />Throughout this period, he selflessly offered his mind for me to pick, and his ocean of experience for me to draw from. At several points during this journey, I abandoned hopes of an academic career. Each time, he listened without judgment, and quietly held a mirror to my desire for autonomy and passion for teaching. For better or for worse, he was instrumental in me ending up on the trajectory I am currently on.<br /><br />And I couldn't be more grateful! Sometimes you try to peek over the horizon, but you can't see what a taller person who has been to more places can (in my case, that is literally true too).<br /><br /><br />In 2009, I shut the door to my office and wept, when I learnt about his <a href="http://www.che.iitb.ac.in/online/story/obituary-prof-kartic-khilar-a-tribute-his-career-and-compassion">untimely passing</a>. He was 57, in great mental and physical shape, and I always expected him to be around forever.<br /><br />When I first encountered KCK in 1994, I knew him as an administrator. Later at IIT he became my chairman and teacher, before becoming my BTP supervisor.<br /><br />Somewhere along the way, he became a mentor, and a close friend; emails that started with "Dear Prof. Khilar" eventually started with "Dear Kartic".<br /><br />Today, even though I knew it would bounce, I nearly wrote (to his familiar email address), "Dear Kartic, you are sorely missed." </div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/gk3DA0q1xvY" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/09/teachers-day-2017.htmltag:blogger.com,1999:blog-7379110960796014170.post-84800244643808376002017-09-02T18:29:00.002-04:002017-09-02T18:29:50.342-04:00Complex Numbers: Part Deux<div dir="ltr" style="text-align: left;" trbidi="on">I was pointed to this excellent <a href="https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF">series on complex numbers from Welch labs</a>, following my last post on complex numbers. It in the <a href="https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw">3Blue1Brown</a> mold, with just the right dose of insight and animation. The complex number series starts with basic ideas, and ends with a discussion of Riemann surfaces.<br /><br />I also came across an interesting way of proving exp(ix) = cos x + i sin x (@fermatslibrary), which I feel compelled to share, since we are already talking about complex numbers.<br /><br />Let \(f(x) = e^{-ix} (\cos x + i \sin x)\).<br /><br />The derivative of this function is \[f'(x) = e^{-ix} (i\cos x - i \sin x) - i e^{-ix} (\cos x + i \sin x) = 0.\] Since \(f'(x) = 0\), the function is a constant.<br /><br />Also f(0) = 1, which implies f(x) = 1.<br /><br />Thus, \(e^{ix} = \cos x + i \sin x\).<br /><br /><b>PS</b>: One of my students told me last week about the new podcast (<a href="https://www.benbenandblue.com/">Ben, Ben, and Blue</a>) that Grant Sanderson (of 3Blue1Brown) hosts on math, computer science and education. It is delightful.</div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/b_lUtdfBtg0" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/09/complex-numbers-part-deux.htmltag:blogger.com,1999:blog-7379110960796014170.post-80034317707909862572017-08-29T20:27:00.000-04:002017-08-29T20:27:01.433-04:00Anomalous Diffusion<div dir="ltr" style="text-align: left;" trbidi="on">I've been taking a deep dive into the world of anomalous diffusion over the past month. It is a fascinating subject that integrates applications from a variety of different fields.<br /><br />For someone interested, I'd recommend the following resources:<br /><br />1. A Physics World feature on "Anomalous diffusion spreads its wings" (<a href="http://www.tau.ac.il/~klafter1/ar1.pdf">pdf</a> - currently not paywalled)<br /><br />2. A <a href="https://www.youtube.com/watch?v=ZKjQKWq02_4">YouTube video</a> on anomalous diffusion in crowded environments<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/ZKjQKWq02_4/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/ZKjQKWq02_4?feature=player_embedded" width="320"></iframe></div><br /><br />3. A gentle introduction/tutorial on <a href="https://arxiv.org/pdf/0805.0419.pdf">normal and anomalous diffusion</a>, which introduces the intuition and mechanics of fractional calculus<br /><br />4. A more academic <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.473.7695&rep=rep1&type=pdf">review</a> of anomalous diffusion and fractional dynamics (may be <a href="http://www.sciencedirect.com/science/article/pii/S0370157300000703">paywalled</a>)</div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/HWzmQrddno0" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/08/anomalous-diffusion.htmltag:blogger.com,1999:blog-7379110960796014170.post-22103573058764484342017-08-23T10:10:00.003-04:002017-08-23T10:10:50.136-04:00If $1 = 1 sec ...<div dir="ltr" style="text-align: left;" trbidi="on">If $1 were equal to 1 second, the median US household income per year of $50,000 would correspond to half a day.<br /><br />This helps puts millions, billions, and trillions into perspective.<br /><br />Roughly,<br /><ul style="text-align: left;"><li>$1 million = 2 weeks</li><li>$1 billion = 32 years</li><li>$1 trillion = 300 centuries (before recorded history)</li></ul>A trillion is a really large number! </div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/IHZqc2SN44E" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/08/if-1-1-sec.htmltag:blogger.com,1999:blog-7379110960796014170.post-69677843180920213992017-08-15T12:49:00.002-04:002017-08-15T12:49:40.166-04:00Diffusion: A Historical Perspective<div dir="ltr" style="text-align: left;" trbidi="on">The paper (<a href="https://pdfs.semanticscholar.org/24c0/5e77599f7ec211b9c0bbf326138607889415.pdf">pdf</a>) "One and a half century of diffusion: Fick, Einstein, Before and Beyond" by Jean Philibert traces the history of diffusion phenomena.<br /><br />It starts with <a href="https://en.wikipedia.org/wiki/Thomas_Graham_(chemist)">Thomas Graham</a> (of dialysis fame) who perhaps made the first systematic observations, which were integrated into phenomenological law by German physiologist <a href="https://en.wikipedia.org/wiki/Adolf_Eugen_Fick">Adolf Fick</a> in 1855, at the age of 26.<br /><br />Fick observed the analogy between mass diffusion and heat conduction (now considered obvious), and piggy-backed on Fourier's law of conduction (1822). The paper cites the opening lines of Fick's work:<br /><blockquote class="tr_bq">A few years ago, Graham published an extensive investigation on the diffusion of salts in water, in which he more especially compared the diffusibility of different salts. It appears to me a matter of regret, however, that in such an exceedingly valuable and extensive investigation, the development of a fundamental law, for the operation of diffusion in a single element of space, was neglected, and I have therefore endeavoured to supply this omission.</blockquote>Next, the paper talks about the contributions of W. C. Roberts-Austen (an assistant to Thomas Graham, and successor as Master of the Mint) to quantification of diffusion in solids.<br /><br />In 1905, Einstein integrated Robert Brown's observations of random zig-zag trajectories and Fick's phenomenological laws, with the crucial observation that it was the mean-squared displacement, and not the mean displacement that was related to diffusion.<br /><br />Following Einstein's paper, the experimental work of Perrin was responsible helping the world accept the link between the microscopic (MSD is proportional to diffusivity and time) and macroscopic worlds (flux is proportional to concentration gradient).<br /><br />It is always interesting to look at the chronological development of (now familiar) ideas. These uncontroversial ideas were once strongly wrestled with. It took centuries for scientists to come up with a comprehensive understanding, and to develop interesting applications based off of it.</div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/RlwSUOuAfIg" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com2http://sachinashanbhag.blogspot.com/2017/08/diffusion-historical-perspective.htmltag:blogger.com,1999:blog-7379110960796014170.post-67709878068357351832017-08-12T14:30:00.000-04:002017-08-12T14:31:00.383-04:00Exam Question on Fitting Sums of Exponentials to Data<div dir="ltr" style="text-align: left;" trbidi="on">I wrote the question below for our PhD qualifiers. It addresses a problem I have been thinking about for over a decade now - starting from my time as a graduate student: how to fit a sum of decaying exponentials to data?<br /><br />The question explores a method called the <a href="https://en.wikipedia.org/wiki/Prony%27s_method">Prony</a> method. Here is the question:<br /><br />A classical problem in data analysis involves fitting a sum of exponentials to a time series of uniformly sampled observations. Here, let us suppose we are given N observations \((t_i, f_i)\), where \(t_i = i \Delta t\) for \(i = 0, 1, ..., N-1\).<br /><br />We want to fit the data to a sum of two exponentials. The <b>model equation</b> is, \[\hat{f}(t) = a_1 e^{b_1 t} + a_2 e^{b_2 t}.\] The general nonlinear regression problem to determine \(\{a_j, b_j\}\) becomes difficult as the number of exponentials in the sum increases. A number of quasi-linear methods have been developed to address this. In the question, we will explore one of these methods, and determine the fitting parameters.<br /><div><br />(a) First, generate a synthetic dataset \((t_i, f_i)\) with true \(a_1^* = a_2^* = 1.0\), \(b_1^* = -2.0\), \(b_2^* = -0.2\). Use \(t_0 = 0\), \(\Delta t = 1\), and N = 20. Attach a plot of the synthetic dataset. Use this dataset for numerical calculations below.<br /><br />(b) If \(b_1\) and \(b_2\) are known, then we can determine \(a_1\) and \(a_2\) by linear least squares. Set \(u_1 = e^{b_1 \Delta t}\) and \(u_2 = e^{b_2 \Delta t}\). Recognize that \(e^{b_i t_j} = e^{b_i j \Delta t} = u_i^j\). Hence from the model eqn, we can get a <b>linear system</b>:<br />\begin{align}<br />f_0 & = a_1 u_1^0 + a_2 u_2^0 \nonumber\\<br />f_1 & = a_1 u_1^1 + a_2 u_2^1 \nonumber\\<br />\vdots & = \vdots \nonumber\\<br />f_{N-1} & = a_1 u_1^{N-1} + a_2 u_2^{N-1}<br />\end{align}<br />Write a program to determine \(a_1\) and \(a_2\), given the data, \(b_1\) and \(b_2\).<br /><br />(c) Consider the polynomial \(p(z)\), which has \(u_1\) and \(u_2\) as its roots, \(p(z) = (z-u_1)(z-u_2) = z^2 - d_1 z -d_2 = 0\). Express \(u_1\) and \(u_2\) in terms of \(d_1\) and \(d_2\).<br /><br />(d) Now we seek to take linear combinations equations in the linear system above with the goal of eliminating \(a_j\). For example, consider the first three equations. If we multiply the first eqn by \(d_2\), the next by \(d_1\), and the third by -1 and sum them up.<br />\begin{align*}<br />d_2 f_0 & = a_1 d_2 + a_2 d_2\\<br />d_1 f_1 & = a_1 u_1 d_1 + a_2 u_2 d_1 \\<br />-1 f_2 & = -a_1 u_1^2 - a_2 u_2^2.<br />\end{align*}<br />We get \(-F_2 +d_1 F_1 + d_2 F_0 = -a_1(u_1^2 - d_1 u_1 - d_2) -\) \( a_2(u_2^2 -d_1 u_2 - d_2) = 0\), since \(p(u_i) = 0\).</div><div><br />We can pick the next set of three equations, and repeat the process (multiply by \(d_2\), \(d_1\), and -1 before summing up). Show that we end up with the following linear system:<br />\[\begin{bmatrix} f_{1} & f_0 \\ f_2 & f_1 \\ <br />\vdots & \vdots \\<br />f_{N-2} & f_{N-3} \\<br />\end{bmatrix} \begin{bmatrix} d_1 \\ d_2 \end{bmatrix} = \begin{bmatrix} f_2 \\ f_{3} \\ \vdots \\ f_{N-1} \end{bmatrix}\]<br />Determine \(d_1\) and \(d_2\), and hence \(u_1\) and \(u_2\). From this, find the estimated \(b_1\) and \(b_2\).<br /><br />(e) Once you know \(b_1\) and \(b_2\) find \(a_1\) and \(a_2\) by linear least squares solution of linear system.<br /><br /></div></div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/tTNGdIInFrM" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/08/exam-question-on-fitting-sums-of.htmltag:blogger.com,1999:blog-7379110960796014170.post-10001900818720983522017-08-09T16:35:00.000-04:002017-08-09T23:05:02.175-04:00Complex Numbers<div dir="ltr" style="text-align: left;" trbidi="on">1. What really are <a href="http://robjlow.blogspot.com/2017/06/what-is-this-thing-called-i.html?m=1">complex numbers</a>?<br /><br />2. The joy of <a href="https://mathwithbaddrawings.com/2017/05/17/the-joy-of-slightly-fishy-proofs/">slightly fishy proofs</a>.<br /><br />3. This <a href="https://math.stackexchange.com/questions/4961/interesting-results-easily-achieved-using-complex-numbers">discussion</a> on MathOverflow</div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/DWrFECnrRo4" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com3http://sachinashanbhag.blogspot.com/2017/08/complex-numbers.htmltag:blogger.com,1999:blog-7379110960796014170.post-24432746365791598692017-08-02T15:52:00.000-04:002017-08-02T15:52:04.594-04:00NumPy and Matlab<div dir="ltr" style="text-align: left;" trbidi="on">This post bookmarks two sites that provide handy cheat sheets of numpy equivalents for Matlab/Octave commands.<br /><br />The ones for linear algebra are particularly handy, because that is one subdomain where Matlab's notation is more natural.<br /><br />1. Numpy for Matlab users (<a href="http://mathesaurus.sourceforge.net/matlab-numpy.html">Mathesaurus</a>)<br /><br />2. Cheatsheets for Numpy, Matlab, and Julia (<a href="https://cheatsheets.quantecon.org/">quantecon</a>)</div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/vcft0PJS-hs" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/08/numpy-and-matlab.htmltag:blogger.com,1999:blog-7379110960796014170.post-49677357244099734842017-07-28T21:19:00.000-04:002017-07-24T08:59:15.425-04:00Interesting Scaling Laws<div dir="ltr" style="text-align: left;" trbidi="on">I recently read Geoffrey West's book "<a href="https://www.amazon.com/Scale-Universal-Innovation-Sustainability-Organisms/dp/1594205582">Scale</a>", and thought it was really great. Here are some resources to prime you for the subject.<br /><br />1. <a href="https://www.ted.com/talks/geoffrey_west_the_surprising_math_of_cities_and_corporations">TED Talk</a><br /><br />2. Talk <a href="https://www.youtube.com/watch?v=GoHD1ROPiUc">@ Google</a><br /><br />3. Essay at the <a href="https://www.edge.org/conversation/geoffrey_west-why-cities-keep-growing-corporations-and-people-always-die-and-life-gets">Edge</a><br /><br />4. Essay on <a href="https://medium.com/sfi-30-foundations-frontiers/scaling-the-surprising-mathematics-of-life-and-civilization-49ee18640a8">Medium</a></div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/vZqgD0Jj11I" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/06/interesting-scaling-laws.htmltag:blogger.com,1999:blog-7379110960796014170.post-41031123725894969792017-07-25T10:44:00.000-04:002017-07-25T10:44:00.167-04:00Russell's paradox<div dir="ltr" style="text-align: left;" trbidi="on"><div class="tr_bq">I came across this interesting paradox on a recent podcast. According to <a href="https://en.wikipedia.org/wiki/Russell%27s_paradox">wikipedia</a>:</div><blockquote>According to naive set theory, any definable collection is a set. Let ''R'' be the set of <b>all</b> sets that are not members of themselves. If ''R'' is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox. </blockquote><blockquote>Symbolically:<br />\[\text{Let } R = \{ x \mid x \not \in x \} \text{, then } R \in R \iff R \not \in R\]</blockquote>There is a nice commentary on the paradox in <a href="https://www.scientificamerican.com/article/what-is-russells-paradox/">SciAm</a>, and a superb entry on the <a href="https://plato.stanford.edu/entries/russell-paradox/">Stanford Encyclopedia of Philosophy</a></div><img src="http://feeds.feedburner.com/~r/CluelessFundatma/~4/y5s1tE1Vc8o" height="1" width="1" alt=""/>Sachin Shanbhaghttps://plus.google.com/115150474038005608083noreply@blogger.com0http://sachinashanbhag.blogspot.com/2017/07/russells-paradox.html