Indiana University Math ClubJekyll2013-11-18T22:13:15-05:00http://www.indiana.edu/~mathclub//Tim Zakianhttp://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-Gerber2013-11-20T00:00:00-05:002013-11-20T00:00:00-05:00Tim Zakianhttp://www.indiana.edu/~mathclub/tzakian@indiana.edu<!-- mathjax config similar to math.stackexchange -->
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<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-Bainbridge2013-10-02T00:00:00-04:002013-10-02T00:00:00-04:00Tim Zakianhttp://www.indiana.edu/~mathclub/tzakian@indiana.edu<p>In this talk, Prof. Matt Bainbridge will talk about the <em>Poncelet’s
Porism</em>. More specifically, suppose E and F are ellipses in the plane with
E inside F. Poncelet’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’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-Demeter2013-09-25T00:00:00-04:002013-09-25T00:00:00-04:00Tim Zakianhttp://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-Dadok2013-09-18T00:00:00-04:002013-09-18T00:00:00-04:00Tim Zakianhttp://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-Judge2013-09-11T00:00:00-04:002013-09-11T00:00:00-04:00Tim Zakianhttp://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’ 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>