Seminars and Colloquia by Series

Mixed-mode and relaxation oscillations in slow-fast predator-prey problems

Series
CDSNS Colloquium
Time
Monday, March 30, 2020 - 11:15 for 1 hour (actually 50 minutes)
Location
Skyles 006
Speaker
Susmita SadhuGeorgia College & State University Milledgeville

The entry-exit function for the phenomenon of delay of stability loss (Pontryagin’s delay) arising in certain classes of slow-fast planar systems plays a key role in establishing existence of limit cycles that exhibit relaxation oscillations. In the first part of my talk, I will present an elementary approach to study the entry-exit function for a general class of slow-fast systems, and apply this function to a broad class of slow-fast planar systems to obtain existence, global uniqueness and asymptotic orbital stability of periodic solutions that exhibit relaxation oscillations. The obtained results will then be applied to some predator-prey models. This research was conducted in collaboration with Dr. Shangbing Ai. In the second part of my talk, I will present a slow-fast system comprising of three species where further complex oscillatory patterns such as mixed mode oscillations (MMOs) are observed. MMOs are concatenations of small amplitude oscillations and large amplitude oscillations which are of relaxation types. In a neighborhood of singular Hopf bifurcation, these types of oscillations occur as long lasting chaotic transients as the system approaches a periodic attractor. The transients could persist for thousands of generations, reflecting that dynamics on an ecological timescale can be completely different than asymptotic dynamics. The goal of the talk is to find conditions that will determine whether a trajectory exhibits another cycle of MMO dynamics before reaching its asymptotic state.

Counting extensions revisited

Series
Combinatorics Seminar
Time
Friday, March 27, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lutz WarnkeGeorgia Tech

We consider rooted subgraphs in random graphs, i.e., extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices. 
In 1989, Joel Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices.  
For the important strictly balanced case, Spencer also raised the fundamental question whether these conditions are necessary. 
We answer this question by a careful second moment argument, and discuss some intriguing problems that remain open. 

Iterated Jackknives, Two-Sided Variance Inequalities, and $\Phi$-Entropy

Series
Stochastics Seminar
Time
Thursday, March 26, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christian HoudréGeorgia Institute of Technology

In analogy with the Gaussian setting, we provide, at first, inequalities on the variance of a function of $n$ independent random variables generalizing results obtained for i.i.d. ones. In particular, we obtain various upper and lower bounds on this variance, via the iterated Jackknife statistics, which can be considered as generalizations of the Efron-Stein inequality. Relations with Hoeffding decomposition are then presented. Finally, the case of the $\Phi$-entropy is also considered.

Joint work with O. Bousquet from Google Brain Team.

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