| 3/27/2013 | 4/3/12 (Next Week): | |||||
| Lorenz
Attractors, Chaotic Systems and Modeling Cancer |
Outreach at the Tobin Middle School |
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| Presenters:
Shayna Jackson and Matthew Shakespeare Abstract: Chaotic systems have been a difficult area of study. While they are deterministic in nature (not stochastic), their trajectories are complicated and oftentimes have a fractal dimension. Because of the complex nature of the system there are few single-valued invariants that can be used to describe them; namely Lyapunov exponents (which describe the divergence/convergence of the trajectories) and a few other notions of dimensions. One such notion of dimensions is Omega-Complexity. This is a novel notion of associating a number with a fractal trajectory that gives a quantitative measure of something qualitative about its behavior, that is, is there a dominant component or direction in which the fractal develops. We have written various Mathematica programs to test the properties of Omega-Complexity on 2D systems (Henon) and 3D systems (Lorenz) along with a few variations of Henon time delay embedding up to the 4th dimension. We have found experimentally that Omega-Complexity will vary with the number of points but appears to stabilize as the number of points increases. Also the Omega Complexity for x time delay coordinates is equal to the Omega Complexity for y time delay coordinates. We are now using these programs to test multiple systems and determine if any version of multi-dimensional visibility will detect Omega-Complexity of the underling system. Cancer is a widespread disease that emerges in a variety of ways, and its count is only growing with more and more subcategories entering into study. Each type of cancer requires its own approach and because of that cancer becomes harder and harder to cure. Many institutions have backed researchers and published documentation of the great strides in understanding cancer and its growth, and this key process of any research has been done over and over again, producing a variety of results from numerous perspectives to study as many forms of cancer as possible, with more still in development. With a great deal of the study work finished, the processes of application and interpretation become the playgrounds for breakthroughs in the field. Through the combination of fields, such as biology and chemistry, scientists have developed a wealth of knowledge for just what this disease is and how it works. By applying methodology from chaos theory to study the growth patterns of cancer a more accurate model for their growth patterns may be found. The aim is to develop a general model that can be fitted across many forms of cancer, and may even provide the doorway for new methods and approaches to be developed. |
Presenter:
Wentworth Math Club Abstract: During this Math Club meeting, we will be headed to the Tobin Middle School in Mission Hill to discuss visual cryptography and how to encode/decode some basic cyphers. |
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