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As with good EPPS tradition, I will open this talk by discussing my life, showing family pictures and trying to be a little bit funny. I'll then move onto show pictures of colleagues that have been edited using Photoshop, or actually maybe I WILL NOT DO THAT since it is not so funny. I'll then move onto the real fun subject: Interactive Demonstrations with MATHEMATICA.

MATHEMATICA has a nice feature (since version 6) that allows to create simple interactive calculations: Manipulate[f[x],{x,0,1}] opens a window that displays the results of f[x], with a slider for x in the range [0,1] and re-evaluates f[x] in "real time" based on the position of the slider (which the user moves). This simple (almost trivial) feature allows one to create simple (and not so simple) demonstrations of mathematical phenomena which often give a-lot of insight. It is actually also a good research tool - I'll show a case or two where it helped me.

I'll then present some of the content of an undergraduate course that I taught in Israel: http://stat.haifa.ac.il/~yonin/interactive_demos_course_winter_09/int_demos.html  The purpose of that course was 3 fold: 1) To teach the students the basics of MATHEMATICA 2) To strengthen understanding (and give some intuition) for basic concepts of probability, statistics and queuing theory 3) To try to use the students for creating demonstrations for the so-called "Queuing Science Exploratorium": http://stat.haifa.ac.il/~yonin/qsm/main.html , a project I've been trying to build for hosting interactive demonstrations of queuing theory on the web

We show that in the stationary M/G/1 queue, if the service time distribution is IFR , then (a) The distribution of the number of customers in the system is also IFR (DFR), (b) The conditional distribution of the remaining service time given the number of customers in the system is also IFR  and (c) The conditional distribution of the remaining service time given the number of customers in the system, is stochastically decreasing with the number of customers in the system. In the DFR case, items a and b are change to DFR as well, while in item c, “decreasing” should be changed to “increasing”.

I will a give a talk about my current research topic the next day. So I want to take the opportunity to talk about something that (hopefully) most of you find interesting. I want to give a not too technical insight into the mathematical fundamental and idea of finance mathematics. 

In this talk I define invasion percolation cluster and state some recent results about its geometry.

We propose a probabilistic method for detection of signals and reconstruction of images in presence of random noise. The method uses results from percolation and random graph theories. We are able to detect and estimate not only regular images, but also weak signals, as well as fine structures such as curves. We describe a randomized algorithm that detects objects in noisy images very quickly. The algorithm works substantially faster than, say, wavelets-based algorithms. Moreover, a slightly modified version of this algorithm also produces reasonable estimates of images. Consistency and computational complexity of our algorithms will be discussed. It is expected that the algorithms are quick enough to be used in real-time systems. Some funny pictures will be also shown.

Alexander Ledovskikh has been graduated in National Technical University (Kiev Polytechnical Institute, Kiev, Ukraine), and then he has got his PhD in Institute For Sorption and Problems of Endoecology National Academy of Science of Ukraine, Kiev. After he has been appointed as a post-doc in TU/e his research activity is devoted to modelling of hydrogen storage in hydride-forming materials. Based on first principles chemical reaction kinetics and statistical thermodynamics, the model is able to describe the complex processes occurring in hydrogen storage systems, including phase transitions. A complete set of equations, governing pressure-composition isotherms and equilibrium potential in both solid-solution and two-phase coexistence regions has been obtained. The combination of thermodynamic and formal chemical kinetic descriptions is a good advantage of proposed model. The model has been tested on various hydride-forming materials. Good agreement between experimental and theoretical results has been found in all cases. The research results of Alexander have been published in famous scientific journals.

We will show how our transient interpretations of Little’s law and the ASTA (Arrivals See Time Averages) property can be used to provide new insight into the transient behavior of queueing systems that behave in a preemptive manner.  In particular, we will first show how our transient interpretations of Little’s law can be used to derive nice probabilistic expressions for all moments of $Q(t)$, which represents the number of customers at time $t$ in an M/G/1 queue that behaves in a preemptive Last-Come-First-Served manner (can be preemptive-resume, preemptive-repeat, or any random mixture of these disciplines), for any initial condition $Q(0)$.  As a consequence, by properly rescaling time and space and taking appropriate limits, these moments can also be used to derive analogous moment expressions for a regulated Brownian motion.  Next, we will show how the main ideas behind ASTA can be used to derive the Laplace-Stieltjes transform of the probability mass function (pmf) of $Q(t)$, for preemptive systems with state-dependent Poisson arrivals and services, and multiple deletions.  An interesting special case of our preemptive-queueing model is a birth-death process, and we will show how to use our results to quickly rediscover a probabilistic interpretation of the pmf of Q(t) for the M/M/1 model, which was previously derived in a work of Abate, Kijima and Whitt through the use of different methods.

Last updated 28/01/10

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