Markov Chain Monte Carlo for Automated Tracking

Markov Chain Monte Carlo for Automated Tracking

of Genealogy in Microscopy Videos

Kathleen Champion

Abstract

Cell biologists use fluorescence time-lapse microscopy to follow the dynamics of proteins in organelles in time and space. Variation in timing during the cell division process can be studied in multinucleate cells by following individual nuclei through time to generate nuclear pedigrees. To undertake a quantitative analysis of mitosis timing, nuclei should be tracked through time over many frames of a time-lapse data set. This is challenging because the images have a low time resolution as well as a low signal to noise ratio, making both tracking and object identification challenging. While methods have been developed for tracking the movement of particles, there are few which have successfully incorporated mitosis to track dividing nuclei. In this paper, we treat the tracking problem as a high-dimensional statistical inference with noisy data and use Markov chain Monte Carlo to sample the posterior distribution. We present cases in 1D and 2D. We also introduce an algorithm for fitting 3D locations given multiple focal planes.

“Smoothed” Character Sums

Kamil Adamczewski

Abstract

The topic of ''smoothed'' character sums is very recent in mathematics. ''Smoothed'' character sum was defined in 2010 in a paper by Levin, Pomerance and Soundararajan and was further explored by Treviño in his PhD dissertation.

This thesis, which is a blend of number theory, abstract algebra and complex analysis, will explore and show how to construct Dirichlet character functions and both regular and “smoothed” character sums.

A Dirichlet character is a function from integers to complex numbers which has three properties: is periodic, completely multiplicative, and takes non-zero values if and only if the argument and the period are relatively prime.

In order to explore the upper bound of ''smoothed'' Dirichlet character sums, the mathematicians extended the widely-explored regular character sum and made it a weighted character sum by multiplying each term by a factor.

In this thesis, I will provide numerical evidence to how good the upper bound given by Levin, Pomerance and Soundararajan is. The importance of such data is evident in the case of regular character sums where the numerical evidence is in favor of the upper bound which assumes the generalized Riemann hypothesis.

The Kronecker Product of Two Monomial Symmetric Functions

Katherine Roddy

Abstract

This thesis investigates the problem of computing the Kronecker product of two monomial symmetric functions. Our main result is a formula that can be easily programmed to compute this product for any two monomial symmetric functions in commuting variables. The main tool in obtaining this formula is the algebra of symmetric functions in non-commuting variables. This algebra has basis elements that are indexed by set partitions. Hence, the formulas we obtain are in terms of the M¨obius function on the lattice of set partitions. A combinatorial interpretation for the coefficients of the Kronecker product of two monomial symmetric functions was given by Remmel and Whitehead in [1] in terms of an object called a primitive bi-brick cycle. In this thesis, we provide a formula for enumerating these objects and give alternate proofs for several of Remmel and Whitehead’s results. In particular, we can characterize the sign of the coefficients as well as when the coefficient is zero. Further, we give formulas for transition matrices between the monomial symmetric functions and the basis of power sum symmetric functions in commuting variables. These change of basis matrices are in terms of the M¨obius function on the lattice of set partitions. By using these transition matrices, we are able to produce another formula for the coefficients of the Kronecker product of two monomial symmetric functions. Finally, we give several explicit formulas for special cases of the product.

[1] J.B. Remmel and T. Whitehead, Transition matrices and kronecker product expansions of symmetric functions, Linear Multilinear Algebra 40 (1996), 337–352.