Who: Ashlie Singer
Date:Wednesday, May 2
Time: 5:30
Location: Applied Math Conference Room
Discussion:
I have been working on finding short pulse solutions to Maxwell's equations. These solutions describe the output of the fairly new ultrafast laser. Assuming that we have a short pulse solution leads to a simpler set of equations that describe the pulse. I incorporated the method of multiple scales, asymptotic expansion, Fourier analysis, and spectral methods(FFTs) in effort to find short pulse solutions to Maxwell's equations.
Who: Saverio Spagnolie and Mark Snyder
Date:Wednesday, October 11
Time: TBA
Location: Applied Math Conference Room
Discussion:
Mark Snyder's Research:
If a data set satisfies Benford's Law, then it's significant digits will be distributed according to a logarithmic density. Our work involves creating discrete one-dimensional dynamical systems that satisfy Benford's Law. We construct dynamical systems satisfying Benford's Law that are conjugate to other dynamical systems that have been studied intensely (like the logistic map). The dynamical systems we create inherit the measures of ergodicity from the systems they are conjugate to. We also generate dynamical systems that satisfy Benford's Law by solving the inverse Frobenius-Perron problem for a given density. We are able to calculate the Lyapunov exponents for these new dynamical systems. In addition to these exact results we use Monte Carlo methods to analyze other dynamical systems. Because of the many physical data sets satisfying Benford's Law (tax data, geophysical data, statistics from diffusion of gasses) we believe our work may have applications in mathematical modeling.
Saverio Spagnolie's Research:
We have studied recurrence in numerical studies of two Hamiltonian differential equations. We began our investigation by exploring the well known cubic non-linear Schrodinger equation (NLSE), iAt=Axx+A|A|^2. As a volume preserving equation, recurrence is predicted in numerical solutions to the NLSE by the Poincare' recurrence theorem; this was observed when the Hamiltonian structure was preserved numerically. The second equation investigated, iAt=Axx+A|A|, has been less studied than the NLSE, but similarities suggested comparable behavior between the two. In contrast to the aforementioned theorem, researchers from LANL have observed a tendency towards the formation and persistence of a large scale solitary wave in numerical studies of the equation. In our own studies of the equation, we have found that not all of the quantities preserved analytically by the equation are behaving in the same way numerically. Thus, we have yet to reach a point from which we might comment confidently about recurrence.
Who: Holly Lewis, Josh Wells, and Jeremy Horgan-Kobelski
Date: Wednesday, November 15
Time: 5 - 6:15 pm
Location: Applied Math Conference Room
Discussion:
Jeremy Horgan-Kobelski's research:
I will be discussing the Hansen-Patrick family of iterative root finding methods. This is a parameter dependent family of cubically convergent methods for finding the roots of a function f(z) = 0. I will be looking at this class of methods from a dynamical systems perspective and will present several numerical experiments performed on different classes of complex polynomials. Halley's method, one special case contained in the family of methods will be presented in detail for the polynomials z(z-1)(z-r). Halley's method is unique since it corresponds to the only finite parameter value of the Hansen-Patrick family for which the method is rational. Dynamics of Halley's method in the complex plane will be presented along with some analytical work.
Holly Lewis and Josh Wells' research:
We have been studying the set of fractal images called the Sierpinski relatives using linear algebra techniques. The 512 images in the set can be created using several techniques; the most common is the use of iterated function systems. However, for the purpose of linear algebraic analysis, we have chosen to implement (in Matlab) a variation of Peitgen's method, called the tiling algorithm. These images are generated for sizes that are powers of two, up to 256x256 pixels. During our analysis we have used several linear algebra decomposition methods, including the SVD. Related to the SVD is the "Eigenface" technique, which we have adapted. This method determines a basis for the set of images and is used to confirm that the number of unique images is 456. While exploring the singular value (SV) spectra for each image, we have found that for 56 of the images, the SVs can be created using an equation related to Pascal's triangle and the golden number, [1+sqrt(5)]/2. The non-standard Haar Wavelet Decomposition (HWD) has been used in conjunction with the SVD for the 56 images that are related to Pascal's triangle and the golden number. The averages and differences matrices, which result from the HWD, also have interesting SV properties. For example, their SVs can be created using another equation involving Pascal's triangle and the golden number. Currently, we are investigating the results when permuting the SVs of each image and then reconstructing using A_new = U*S_new*V'. Under this transformation, some of the images map to other images in the set, some of these map to fractal images that are not in the Sierpinski set, and others map to images that are unrecognizable. Of primary interest are the new fractal images, whose properties will be analyzed.
Keep checking for announcements about other Undergraduate Seminars!!
As with all SIAM functions, Food will be provided!!!