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Fun with Fractions

January 9, 2018 Leave a comment

This post is inspired by tweet from @CutTheKnotMath and his post.

 

He posed this puzzle for the New Year:

Consider the list: a_i, 1\le i \le n

Consider the product: \prod_{i=1}^n (a_i+1) -1

For n\ge 2, replacing randomly chosen a_i, a_j with a_i+a_j+ a_i a_j in the expression does not change its value as (a_i+1)(a_j+1)=(a_i+a_j+a_i a_j)+1. So  repeating till there is one number allows you solve the problem, i.e. process ends a unique number. For the particular sequence this is particularly pleasing: \prod ^n_{i=1}(1+1/i)-1=\prod^n_{i=1}\frac{i+1}{i} -1= n+1-1=n.

 

Simulating a small example with Mathematica:

 

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Categories: Mathematica, Mathematics

A Bookmark

August 6, 2017 Leave a comment

This post is just a bookmark for a “history of cardiac catheterization”.

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Fun Animation

July 10, 2017 Leave a comment

This is an adaptation of code from:here.

me

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Stretching and Folding

May 2, 2017 Leave a comment


This post explores the Henon map:
x_{n+1}=y_n+1-a x_n^2\\\\ y_{n+1}=b x_n

1000 iterations of a point
msehenon

Exploring the “fate” of a disc after 10 iterations:

msehenon

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Harmony on a Donut

April 29, 2017 Leave a comment

“Harmony” was my sloppy look at the Kuramoto model. Looking at 2 oscillators:
\dot{\theta_1}=\omega_1+ k\sin (\theta_2-\theta_1)\\\\ \dot{\theta_2}=\omega_2+ k\sin (\theta_1-\theta_2)

The following starts “knotted” 2:3 with k=0, i.e. no coupling and then increasing the coupling…another visualization of harmony.

msesynchr.gif

Visualizing as motion on a circle:

 
msekuramoto

Knots on a Donut (Doughnut)

April 28, 2017 Leave a comment

Professor Strogatz lecture on quasiperiodicity motivates this post. Just using the simple system:

 \dot{\theta_1}=\omega_1\\\\  \dot{\theta_2}=\omega_2
when \omega_1/\omega_2 is rational the trajectory is closed, when irrational quasiperiodic. In the following the first gif is \omega_1=2,\omega_2=3 and the second: \omega_1=\sqrt{2},\omega_2=2

msetrefolimsequasiperiod

 

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Relaxation Oscillator

April 20, 2017 Leave a comment

Professor Strogatz Cornell course video lectures on “Nonlinear Dynamics and Chaos” are now online. This post pays homage to lecture 10 on analysis of the  Van der Pol equaltion:

\ddot{x}+x+\mu \dot{x}(x^2-1)=0

In the following the case for \mu =2,\mu =10 and the examination of the \mu >>1. It nicely illustrates the two time scales referred to, the slow then fast parts of the oscillation.

The phase portrait is the x, y plane where y is the the Lienard transformed and rescaled version of the equation.

So the system is:
\dot{x}=\mu(y-(x^3/3-x))
\dot{y}=-x/\mu

msevdp

msevdp2

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