Department of Mathematics

 Spring 2016

We will meet on Tuesdays at 4:00 pm.
Here is a tentative list of speakers (to be extended): Jayadev S. Athreya (University of Washington), Yevgeniy Kovchegov, Amber Puha (California State University, San Marcos)
Registration information: Mth 607, Sec 003 - CRN 54451



Title and abstract

Tuesday, April 5, 3:30 pm
GILK 115
Jayadev S. Athreya
University of Washington

"The Erdos-Szusz-Turan distribution"

Abstract. Dirichlet's theorem on diophantine approximation states that for any irrational number a, there are infinitely many rationals p/q so that q|aq-p| < 1. Erdos-Szusz-Turan asked the question of the probability that this estimate could be improved to q|aq-p|< A, with q in a fixed range [N, cN], and the behavior of this probability as N grows. We answer their question and provide a wide-ranging generalization to the setting of equivariant point processes. This is joint work with Anish Ghosh.
Tuesday, May 3, 4:00 pm
GILK 115
Yevgeniy Kovchegov
Oregon State University

"Horton-Strahler ordering and Tokunaga indexing in stochastic processes"

Abstract. We introduce a class of stochastic processes that we call hierarchical branching processes. By construction, the processes satisfy the Tokunaga, and hence Horton, self-similarity constraints. Taking the limit of averaged stochastic dynamics, we obtain the deterministic system of differential equations that describe the temporal dynamics of a Tokunaga branching system. In particular, we study the averaged tree width function to establish a phase transition in the Tokunaga dynamics that separates fading and explosive branching. We then describe a class of critical hierarchical branching processes (that happen at the phase transition boundary) that includes as a special case the celebrated critical Galton-Watson branching process. We illustrate efficiency of the critical hierarchical branching processes in describing diverse observed dendritic structures, and discuss the related critical phenomena from the point of view of respective applications.
Joint work with Ilya Zaliapin (University of Nevada Reno).
Tuesday, May 24, 4:00 pm
GILK 115
Amber Puha
California State University, San Marcos

"Diffusion Limits for Shortest Remaining Processing Time Queues under Nonstandard Spatial Scaling"

Abstract. In a shortest remaining processing time (SRPT) queue, the job that requires the least amount of processing time is preemptively served first. One effect of this is that the queue length is small in comparison to the total amount of work in the system (measured in units of processing time). In the case of processing time distributions with unbounded support, the queue length is so small that the sequence of queue length processes associated with a sequence of SRPT queues, rescaled with standard functional central limit theorem scaling and satisfying standard heavy traffic conditions, converge in distribution to the process that is identically equal to zero. This happens despite the fact that in this same regime the rescaled workload processes converge to a non-degenerate reflected Brownian motion. In particular, the queue length process is of smaller order magnitude than the workload process. In the case of processing time distributions that satisfy a rapid variation condition, we implement an alternative, unconventional spatial scaling that leads to a non-trivial limit for the queue length process. This result quantifies this order of magnitude difference between queue length and workload processes. We illustrate this result for Weibull processing time distributions.

Past probability seminars: Fall 2005, Winter 2006, Spring 2006, Fall 2006, Winter 2007, Spring 2007, Fall 2007, Winter 2008, Spring 2008, Fall 2008, Winter 2009, Spring 2009, Fall 2009, Winter 2010, Spring 2010, Fall 2010, Winter 2011, Spring 2011, Fall 2011, Winter 2012, Spring 2012, Fall 2012, Winter 2013, Spring 2013, Fall 2013, Winter 2014, Spring 2014, Fall 2014, Winter 2015, Spring 2015, Fall 2015, Winter 2016