Indiana University Math Club Jekyll 2013-11-18T22:13:15-05:00 http://www.indiana.edu/~mathclub// Tim Zakian http://www.indiana.edu/~mathclub// tzakian@indiana.edu <![CDATA[Sensitive Dependence for Unimodal Maps on the Interval]]> http://www.indiana.edu/~mathclub//articles/Marlies-Gerber http://www.indiana.edu/~mathclub//articles/Marlies-Gerber 2013-11-20T00:00:00-05:00 2013-11-20T00:00:00-05:00 Tim Zakian http://www.indiana.edu/~mathclub/ tzakian@indiana.edu <!-- mathjax config similar to math.stackexchange --> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ jax: ["input/TeX", "output/HTML-CSS"], tex2jax: { inlineMath: [ ['$', '$'] ], displayMath: [ ['$$', '$$']], processEscapes: true, skipTags: ['script', 'noscript', 'style', 'textarea', 'pre', 'code'] }, messageStyle: "none", "HTML-CSS": { preferredFont: "TeX", availableFonts: ["STIX","TeX"] } }); </script> <script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML" type="text/javascript"></script> <p>A function $~f : [a,b] \to [0,1]$ is unimodal if $~f(a) = f(b) = 0$, and there is a point $~c$ in $~(a,b)$ such that $~f$ is strictly increasing on $~[a,c]$ and strictly decreasing on $~[c,b]$. If $~[a,b] = [0,1]$, then we can consider the composition of $~f$ with itself $~n$ times, and we will denote that by $~f^n$. We can think of $~f$ as a dynamical system that gives a rule for how states change after time $~n$. (If the initial state is $~x$, then after one unit of time, the new state is $~f(x)$, and after two units of time, it is $~f^2(x)=f(f(x))$, etc. ) One of the features of a chaotic dynamical system is that for a given initial condition $~x$, you can find another initial condition $~y$, arbitrarily close to $~x$, such that after a long period of time $~n$, $~f^n(x)$ and $~f^n(y)$ are far apart. This is called sensitive dependence on initial conditions. Intuitively, it means that the long-term future cannot be accurately predicted, because there will always be some errors in measuring the initial conditions. I will discuss an easily verifiable condition on the derivatives of $~f$ (in the case of a three times differentiable unimodal map $~f$) that will guarantee sensitive dependence on initial conditions. Math M211 is the only prerequisite for this talk.</p> <p><a href="http://www.indiana.edu/~mathclub//articles/Marlies-Gerber">Sensitive Dependence for Unimodal Maps on the Interval</a> was originally published by Tim Zakian at <a href="http://www.indiana.edu/~mathclub/">Indiana University Math Club</a> on November 20, 2013.</p> <![CDATA[Poncelet's Porism]]> http://www.indiana.edu/~mathclub//articles/Matt-Bainbridge http://www.indiana.edu/~mathclub//articles/Matt-Bainbridge 2013-10-02T00:00:00-04:00 2013-10-02T00:00:00-04:00 Tim Zakian http://www.indiana.edu/~mathclub/ tzakian@indiana.edu <p>In this talk, Prof. Matt Bainbridge will talk about the <em>Poncelet&#8217;s Porism</em>. More specifically, suppose E and F are ellipses in the plane with E inside F. Poncelet&#8217;s Porism says that if there is a single polygon which is inscribed in F and circumscribed around E, then there are infinitely many such polygons. In this talk, I&#8217;ll give a beautiful proof of this theorem using some basic (but deep) properties of elliptic curves.</p> <p><a href="http://www.indiana.edu/~mathclub//articles/Matt-Bainbridge">Poncelet's Porism</a> was originally published by Tim Zakian at <a href="http://www.indiana.edu/~mathclub/">Indiana University Math Club</a> on October 02, 2013.</p> <![CDATA[Why finding arithmetic progressions is hard]]> http://www.indiana.edu/~mathclub//articles/Ciprian-Demeter http://www.indiana.edu/~mathclub//articles/Ciprian-Demeter 2013-09-25T00:00:00-04:00 2013-09-25T00:00:00-04:00 Tim Zakian http://www.indiana.edu/~mathclub/ tzakian@indiana.edu <p>In this talk, Prof. Ciprian Demeter will talk about finding arithmetic progressions of prime numbers. More specifically, he briefly explore the progress of (mathematical) technology over the last 100 years or so that has led to the recent proof that the prime numbers contain arbitrarily long arithmetic progressions.</p> <p><a href="http://www.indiana.edu/~mathclub//articles/Ciprian-Demeter">Why finding arithmetic progressions is hard</a> was originally published by Tim Zakian at <a href="http://www.indiana.edu/~mathclub/">Indiana University Math Club</a> on September 25, 2013.</p> <![CDATA[Mathematics of CAT scan]]> http://www.indiana.edu/~mathclub//articles/Jiri-Dadok http://www.indiana.edu/~mathclub//articles/Jiri-Dadok 2013-09-18T00:00:00-04:00 2013-09-18T00:00:00-04:00 Tim Zakian http://www.indiana.edu/~mathclub/ tzakian@indiana.edu <p>In this talk, Prof. Jiri Dadok will talk about the Mathematics behind the CAT scan, as well as talking about the history leading up to the mathematics used. This will include a brief discussion of the Radon and Fourier transforms.</p> <p><a href="http://www.indiana.edu/~mathclub//articles/Jiri-Dadok">Mathematics of CAT scan</a> was originally published by Tim Zakian at <a href="http://www.indiana.edu/~mathclub/">Indiana University Math Club</a> on September 18, 2013.</p> <![CDATA[The Music of Triangles]]> http://www.indiana.edu/~mathclub//articles/Chris-Judge http://www.indiana.edu/~mathclub//articles/Chris-Judge 2013-09-11T00:00:00-04:00 2013-09-11T00:00:00-04:00 Tim Zakian http://www.indiana.edu/~mathclub/ tzakian@indiana.edu <p>In this talk, Prof. Chris Judge will talk about what sorts of sounds a vibrating triangle can make. He will also set out to say what its `pure tones&#8217; are and how the answers depend on the shape of the triangle. He will also discuss some partial answers to these <em>mathematical</em> questions.</p> <p><a href="http://www.indiana.edu/~mathclub//articles/Chris-Judge">The Music of Triangles</a> was originally published by Tim Zakian at <a href="http://www.indiana.edu/~mathclub/">Indiana University Math Club</a> on September 11, 2013.</p>