Graphing the Mandelbrot Set

Posted by Jason Polak on 13. June 2013 · Write a comment · Categories: analysis, elementary · Tags: , ,

A class of fractals known as Mandelbrot sets, named after Benoit Mandelbrot, have pervaded popular culture and are now controlling us. Well, perhaps not quite, but have you ever wondered how they are drawn? Here is an approximation of one:

From now on, Mandelbrot set will refer to the following set: for any complex number $c$, consider the function $f:\mathbb{C}\to\mathbb{C}$ defined by $f_c(z) = z^2 + c$. We define the Mandelbrot set to be the set of complex numbers $c\in\mathbb{C}$ such that the sequence of numbers $f_c(0), f_c(f_c(0)),f_c(f_c(f_c(0))),\dots$ is bounded.
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60 Linear (Matrix) Algebra Questions

Posted by Jason Polak on 02. April 2013 · Write a comment · Categories: elementary · Tags: , , ,

I sometimes am a teaching assistant for MATH 133 at McGill, and introductory linear algebra course that covers linear systems, diagonalisation, geometry in two and three dimensional Euclidean space, and the vector space $\mathbb{R}^n$, and I've collected a few theoretical questions here that I like to use in the hope that they may be useful to people studying this kind of material. I made up all of these questions, although obviously many of them in form are the same as elsewhere. Some of the questions are a bit unusual and curious, and none of them need special tricks to solve, just an understanding of the concepts in the course. They focus mostly on understanding the theory, and there are very few straight computational-type questions here.

Note. None of these questions are official in the sense that I do not write the final exams. The exact syllabus of the course should always be taken to be the class material and official course material on the course website. These are more for extra practice, but do cover the material of the course fairly closely.
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One Fair Coin and Three Choices

Posted by Jason Polak on 29. January 2013 · 2 comments · Categories: elementary, math · Tags: , ,

A few nights ago as I was drifting off to sleep I thought of the following puzzle: suppose you go out for ice cream and there are three flavours to choose from: passionfruit, coconut, and squid ink. You like all three equally, but can only choose one, and so you decide you want to make the choice randomly and with equal probability to each.

However, the only device you have to generate random numbers is a fair coin. So, how you do use your fair coin to choose between the three options of ice cream?

Of course, you can only use coin flips to make your choice. For instance, cutting the coin into three equal pieces, putting them in a bag to create a new stochastic process does not count.

Nonnegative Sums of Rows and Columns

Posted by Jason Polak on 11. December 2011 · Write a comment · Categories: elementary · Tags: , ,

For any $n\times n$ matrix $A$ with real entries, is it possible to make the sum of each row and each column nonnegative just by multiplying rows and columns by $-1$? In other words, you are allowed to multiply any row or column by $-1$ and repeat a finite number of times.

My fellow office mate Kirill, who also has a math blog, gave me this problem a few weeks ago and I thought about it for a few minutes here and there. The solution is in the fourth paragraph, so if you'd like to think about it yourself stop here before you get close.
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