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Dynamic Equations on the Sierpinski Gasket

Dynamic Equations on the Sierpinski Gasket

Runner-up, DSWeb Tutorials Contest

In this tutorial, we cover the basics of solving dynamic equations on the Sierpinski Gasket through numerical techniques. The work is divided up into three large lessons; lesson 1 covers the basics of the construction of the Sierpinski Gasket, and the construction of the Laplacian. The Laplacian can be defined on the Sierpinski Gasket through an explicit construction of a harmonic calculus, originally due to Jun Kigami; we follow his basic approach, while trying to keep the material accessible to those new in the area. Lesson 2 covers a few numerical algorithms that help solve differential equations, including an overview of a finite element method. Lesson 3 delves more in-depth into the finite element method. It illustrates how to use the finite element method to generate important function, such as the Green's Function and eigenfunctions.

As the Sierpinski Gasket is not a domain that one normally works in, special care is taken to illustrate the algorithms and methods visually. We often also point out the similarities between various concepts on the Sierpinski Gasket, and those in Euclidean spaces.

Finally, we provide custom-written software for producing all the graphs in the tutorials; this general software can be used for further exploration and solving differential equations on the Sierpinski Gasket. The software utilizes freely available tools to allow one to produce detailed 3d graphs. Lesson 3 discusses using the software.

Brain Dynamics: The Mathematics of the Spike

Brain Dynamics: The Mathematics of the Spike

Every second, a spike happens more than 100 billion times in your brain. Spikes are sudden electrical impulses, shot through one brain cell on its way to the next. Spikes are the currency of information in the brain and they drive everything we think and do. There are two basic questions that brain science must answer: how are spikes formed and what do they mean? Unraveling how neurons spike is a crown jewel of twentieth century neuroscience and mathematics was central to this resolution. Resolving the second question, that is, understanding the neural code, will be a central focus of twenty-first century neuroscience and mathematics will surely contribute to this resolution as well. A critical challenge that is not yet fully answered is to understand how spikes emerge from tiny neurons, hundreds per pinhead. How is it that spikes are produced by a huge variety of cells that look wildly different? How can different patterns of spikes be turned on and off in a single cell in normal operation or through medicines? And since vastly different inputs produce the same spike, just how does a neuron decide when to spike?

These questions have gripped the scientific community ever since spikes were first seen more than 100 years ago. Hodgkin and Huxley, two physiologists, showed how mathematics could solve all of them at once, laying the groundwork for their 1963 Nobel Prize and for modern neuroscience. Today, mathematicians are still building on Hodgkin and Huxley's theory of the spike to forge ahead in brain science.

Follow the links to the right to see how nonlinear mathematics provides the framework that unlocked the secret mechanics of the neuron. You’ll discover how this mathematical framework is a nexus for modern neuroscience, and meet Hodgkin and Huxley, the scientists who discovered it – and won a Nobel Prize.

This tutorial appears on Why Do Math.

An Introduction to Rotation Theory

An Introduction to Rotation Theory

Prize winner, DSWeb Student Competition, 2007

This tutorial introduces one of the most fundamental dynamical systems by studying maps of the circle to itself. We are mainly going to investigate homeomorphisms of the circle. 

Homeomorphisms look easy at first sight, but this tutorial should convince you that this first impression is not quite correct. Indeed, we shall meet several surprising phenomena and discuss results, which demonstrate the complications arising already in 1-dimensional dynamics. 

Even in higher-dimensional problems it often turns out that we can reduce our system to a 1-dimensional setup, so that we are also trying to give major tools and results, which should be helpful even if you are not planning to stay in the 1-dimensional world. The theory associated with self-maps on the circle sometimes goes by the name "Rotation Theory" and as we shall see: This name is justified.
An Introduction to Coupled Oscillators: Exploring the Kuramoto Model

An Introduction to Coupled Oscillators: Exploring the Kuramoto Model

Prize winner, DSWeb Student Competition, 2007

This tutorial provides an introduction to the application and non-linear dynamics of globally coupled oscillator systems by considering the popular and well researched Kuramoto model. 
Vortices in Bose-Einstein Condensates: (Super)fluids with a twist

Vortices in Bose-Einstein Condensates: (Super)fluids with a twist

P.G. Kevrekidis, R. Carretero-González and D.J. Frantzeskakis showcase some recent experimental and theoretical work in the coldest temperatures in the universe involving vortices in the newest state of matter: the atomic Bose-Einstein condensates. The remarkable feature that these experiments and associated analysis illustrate is a new kind of ``classical mechanics'' for vortices, which revisits the integrability of the two-body system and opens up exciting extensions for N-body generalizations thereof.

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