### Recent Posts

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##### Presentations and discussions / Re: Julia Set
« Last post by Andi on April 05, 2017, 12:56:26 AM »
Julia Sets:
zn+1 = c sin(zn)   zn+1 = c exp(zn)
zn+1 = c i cos(zn)   zn+1 = c zn (1 - zn)

A property of the the Julia Set is that if the domain of c is real numbers the the Julia Set it mirrored about the Real axis. If  c is a complex number with an imaginary component then then the symmetry is rotational at 180 degrees.
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##### Presentations and discussions / Brownian Motion
« Last post by Andi on April 05, 2017, 12:46:37 AM »
Properties of Brownian Motion

Brownian Motion is defined to be the observable random motion of particles in fluids. The motion is described analytically using the Wiener Process.

There are 4 properties describing this motion:

1. B(0) = 0
2. For all time intervals  t ≥ 0 the increments B(t) are independent random variables.
3.  for all t ≥ 0 and h > 0, the increments B(t + h) − B(t) are normally
distributed with expectation zero and variance h. (meaning there is no bias towards a certain direction for any variance h)
4. The function described by the Brownian Motion is almost always continuous.

References:

http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/McKnight.pdf
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##### Presentations and discussions / Brownian Motion
« Last post by leonoraboci on March 27, 2017, 11:23:51 PM »
Brownian Motion

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##### Presentations and discussions / Re: Julia Set
« Last post by leonoraboci on March 27, 2017, 11:19:20 PM »
Julia set with the parameter µ taken from the center of the circle on top of the cardioid.
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##### Presentations and discussions / Julia Set
« Last post by Thierry Serafin Nadeau on March 15, 2017, 06:07:38 PM »
Turns out that there are 4D analogues to the 2D Julia sets which use quaternions ($i^2 = j^2 = k^2 = ijk = -1$) instead of regular complex numbers. Seeing as they're 4D they can only be visualised as 3D slices of the whole set, which end up looking quite a bit different than the regular Julia sets depending on the chosen slice.

Here's a link to a video which passes through multiple 3D slices of a quaternion Julia set: https://www.youtube.com/watch?v=VkmqT6MQoDE
And bellow is an image of such a slice.
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##### Presentations and discussions / Julia Set
« Last post by Laura Campbell on March 14, 2017, 03:18:47 PM »
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##### Topics / MAT 475: Brownian Motion and the Heat Equation
« Last post by FangfeiLin on January 25, 2017, 06:36:38 PM »
Hi, my name is Fangfei. My project is 'Brownina Motion and the Heat Equation'.

Motivation:
Brownian motion is a diffusion process. PDEs of diffusion type(such as heat question) are useful tools of studying the diffusion process. Conversely, diffusion processes give insight into solutions of diffusion type PDE. The connection between the heat equation and brownian motion is remarkable. In my project, I will demonstrate the relationships between brownian motion and the heat equation and also introduce the heat equation from probabilistic view.

Also:
There is one sample presentation from last year called ‘Random walk and PDE’. The random walk in continuous time and space becomes brownian motion. If you are interested in this material, you may read that presentation first.

Hope you will enjoy it!
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##### Topics / Burgers' Equation
« Last post by Laura Campbell on January 24, 2017, 09:58:48 PM »
My presentation topic is Burgers' equation.  My plan is to discuss Burgers' inviscid equation, $u_t+uu_x=0$ and its solution, as well as its solution with particular initial conditions. Then considering the weak solutions to the equation and the Rankine-Hugoniot conditions. Finally, will go over Burgers' viscous equation, $u_t+uu_x= \epsilon u_{xx}$ and some examples of scenarios in which Burgers' equation is used - namely traffic flow and gas dynamics.
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##### Topics / Calculus of Variations (One Variable)
« Last post by leonoraboci on January 19, 2017, 09:51:14 PM »
My presentation topic is the Calculus of Variations (of one variable) with my teammate, Andi. We will be starting off the presentation with a very short overview of what we will be covering in our talk. We will be using graphs, proofs, and examples to demonstrate the main points of our topic, including some real world examples.
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##### Topics / My Presentation Topic
« Last post by Michael Chow on January 18, 2017, 07:07:46 PM »
Hi everyone,

I will be giving a presentation on a topic in Euclidean geometry -- Collinearity and Concurrency.

In my experience, students in Canada get $\varepsilon$ exposure to Euclidean geometry in both high school and university. The subject is often frowned upon and labeled as a "useless" subject, but I believe that any good mathematician, physicist, engineer, or architect should see an exposition of the subject to some degree. Not only is it an elementary foundation for whatever geometry one will need, but also it is a great source of beauty in math and great practice in creativity.

I will be assuming a very basic understanding of geometry, which includes results about angles, congruency and similarity of triangles and very basic geometry of the circle. (This presentation should be accessible even to middle school students with enough mathematical maturity.)

This presentation is concerned exactly with what I've said it's about, collinearity and concurrency, that is, when do three points lie on a line and when do three lines pass through a common point. These questions may sound simple and a bit mundane, but we will see many beautiful results, some of which discovered by the ancient greeks and some of which had elementary proofs discovered in the 20th century, that show collinearity and concurrency in some surprising configurations. Moreover, we will see a strange duality between the concepts of collinearity and concurrency (and therefore points and lines in a way).

We will start off with general results that give both \textit{necessary and sufficient} conditions for collinearity and concurrency in a given $\triangle ABC$. In fact, if points $D, E, F$ lie on $BC, CA, AB$ respectively, then the collinearity of $D, E, F$ and the concurrency of $AD, BE, CF$ depends only on the ratio $\frac{BD}{DC}\frac{CE}{EA}\frac{AF}{FB}$. We will quickly see some immediate corollaries of these powerful theorems. Then we will turn our attention to more special configurations and the related theorems, such as Euler's line, the Simson line, Pappus's theorem, and Desargues's Theorem to name a few. After developing some results on power of a point and radical axes, we will be able to prove the celebrated theorems of Pascal and Brianchon, which will naturally give rise to questions that belong in the realm of projective geometry, which we may touch on.

I hope this brief overview has peeked your interest and got you excited for this talk.

Mikey
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