In Fall 2005 and Winter 2006, this seminar will meet at 3pm on Wednesdays commencing October 5, 2005 in AP&M 6218. Commencing Spring 2006, the seminar will meet at 4pm on Wednesdays in AP&M 7218.

**
DESCRIPTION: **
This will be an interdisciplinary reading seminar commencing in Fall 2005.
The seminar will meet once per week on a day and time to be arranged.
Faculty, postdocs, visiting scholars and graduate
students will take turns presenting research papers
on stochastic systems arising in applications.
Areas of application include
biology, operations management, neuroscience and finance.

Students wishing to enroll in this seminar for credit should enroll for Math 288, Lecture B -- Stochastic Systems. One unit of credit involves attendance and two units involves giving a presentation. Please address all enquiries concerning this seminar to Professor Williams at williams@math.ucsd.edu

** SOME SAMPLE PAPERS **

MATLAB and C++ code, as well as pdf file of the paper are available by clicking here.

For a paper related to this talk, click here. Another related work can be found by clicking here.

For slides from Professor Bitmead's talk click here for quicktime, here for pdf, and here for powerpoint.

For a related paper, click here.

For a related paper by Mandelbaum, Massey and Reiman, click here. For another related paper by T. Kurtz, in Stochastic Processes and their Applications, 6 (1978), 223-240, click here.

Erel Levine, Center for Theoretical Biological Physics, Postdoc, UCSD.

M. R. Evans and T. Hanney, Nonequilibrium statistical mechanics of the zero-range process and related models, J. Phys. A: Math. Gen. 38 R195-R240 (2005).

E. Levine, D. Mukamel, G. M. Schutz, Zero-Range Process with Open Boundaries, Journal of Statistical Physics 120, 759 - 778 (2005)

A. G. Angel, M. R. Evans, E. Levine, and D. Mukamel, Critical phase in nonconserving zero-range processes and rewiring networks, Phys. Rev. E 72, 046132 (2005).

Amber Puha, California State University, visiting UCSD.

Queues with many servers. For the paper on which this talk is based, click here.

Peter Rowat, Institute for Neural Computation, UCSD.

The stochastic Hodgkin-Huxley equations.

Abstract: The deterministic Hodgkin-Huxley equations describe the generation of the action potential, or voltage "spike", in squid giant axon. These equations are fundamental to theoretical neuroscience. The inter-spike interval (ISI), which carries information in the nervous system, has intrinsic jitter due to channel noise, when the applied current is constant. For some ranges of the current this jitter is very large (CV>1). I have investigated a previously unnoticed anomaly in the distribution of ISIs. When ISIs are converted to instantaneous frequencies, the associated histogram is bimodal. The underlying mechanism arises from the interaction of channel noise with a sub-critical Hopf bifurcation in the noise-free dynamics. This mechanism and associated statistics of ISIs will be described. A difficulty with defining the biological notion of "threshold" will also be mentioned.

Dr. Shirin Handjani, Center for Communications Research.

A Particle System on the Tree with Unusual Asymptotic Behavior. Click here for an abstract.

No seminar this week due to Information theory and applications conference at UCSD, February 6-10, 2006.

R. J. Williams, Mathematics Department, Fluid approximation for an Internet congestion control model with fair bandwidth sharing.

Sumit Bhardwaj, graduate student, ECE.

Maxweight Scheduling in a Generalized Switch: State Space Collapse and Workload Minimization in Heavy Traffic. The paper on which this talk is based is by Alexander L. Stolyar, published in the Annals of Applied Probability 2004, Vol. 14, pp. 1-53. For a copy of this paper, click here.

Professor Jason Schweinsberg, Mathematics Department, UCSD.

Title: Mutation patterns in populations with large family sizes.

Abstract: Suppose we take a sample of size $n$ from a population and follow the ancestral lines backwards in time until the most recent common ancestor. Under standard assumptions, this process can be approximated by Kingman's coalescent, in which two lineages merge at rate one. Mutations that have occurred since the time of the most recent common ancestor will lead to segregating sites, which are positions in the DNA at which not all individuals in the sample are the same. If we denote by $M_k$ the number of mutations that affect $k$ individuals in the sample, then the sequence $M_1, \dots, M_{n-1}$ is called the site frequency spectrum. We will explain how the site frequency spectrum would be affected if some individuals have large numbers of offspring, so that the coalescent process that describes the genealogy of the population can have multiple mergers of ancestral lines. This model may be realistic for certain marine species. This is joint work with Julien Berestycki and Nathanael Berestycki.

Professor Tara Javidi, Department of Electrical and Computer Engineering, UCSD

Cooperative and Non-cooperative Resource Sharing: Delay Perspective

Abstract: From multi-description/multi-path routing to content distribution in P2P networks to community networking, many forms of resource sharing have, recently, been proposed to improve the network performance. From the perspective of any one user and when ignoring the interaction among users, all such schemes reduce to various forms of providing parallelism. In this talk, we argue that focusing on parallelism is by no means sufficient. When considering more users in the system, these strategies provide forms of statistical multiplexing advantage, while possibly increasing the network load via increased redundancy, overhead increase, and even contention inefficiency.

In this talk, we illustrate the issue of resource sharing in the above context via a multi-queue multi-server problem. Even though such model might not be perfectly realistic, it does capture some of the above trade-offs. We use this model to provide analytical results in a special case of homogeneous users and servers. Furthermore, we prove the robustness of a certain locally optimal strategy to non-cooperation in a Nash equilibrium/strategy context.

Professor Emanuel Todorov, Cognitive Science, UCSD

Stochastic optimal control models of biological movement.

Professor Yoav Freund, Computer Science and Electrical Engineering, UCSD

Title: Boosting and Brownian motion

Abstract: The computational task that lies in the core of many machine learning problems is the minimization of a cost function called the training error. This problem is frequently solved by local search algorithms such as gradient descent. The training error can usually be expressed as a sum over many terms, each corresponding to the loss of the model on a single training example. We show that the iteratively minimizing a cost function of this form by local search is closely related to the following game: Imagine you are a shepherd in charge of a large herd of sheep and your goal is to concentrate the sheep in a particular small area by nightfall. Your influence on the sheep movements is represented by vectors which define the direction in which you want each sheep to move. The lengths of the vectors correspond to the fraction of your "energy" that you spend on moving the particular sheep, and these lengths sum to one. The sheep then have to respond by moving in a way that has a slight correlation with the influence direction on average. We characterize the min/max solution to this game and show that by taking the appropriate small-step/continuous-time limit, this solution can be characterized by a stochastic differential equation. By solving this differential equation we re-derive some known boosting as well as design some new ones with desirable properties.

Professor J. F. Delmas, visiting UCSD.

Title: Limit theorems for bifurcating Markov chains and application to the detection of cellular aging (following Julien Guyon (CERMICS-ENPC)).

For a copy of the paper on which the talk is based, click here.

Abstract: A general method is given to study dependent data in a binary tree, where an individual in one generation gives rise to two different offspring, one ot type 0 and one of type 1 in the next generation. For any specific characteristic of these individuals, we assume that the characteristic is stochastic and depends on its ancestor only through its mother's characteristic. The dependency structure may be described by a transition probability P(x, dydz) which gives the probability that the pair of daughters' characteristic is around (y,z) given that the mother's is x, y (resp. z) corresponding to the characteristic of the offspring of type 0 (resp. 1). This defines a bifurcating Markov chain. A strong law a large numbers and central limit theorem are derived for such a model. The results are then used to detect cellular aging of E. Coli.

Professor Massimo Franceschetti, ECE, UCSD

Title: Random Networks for Communications.

Abstract: The theory of random graphs is a useful mathematical tool to describe many real world systems. Recently, the mathematical and engineering communities have shown a renewed interest in the geometric version of these models. The nodes are geometrically distributed at random, and pairs of nodes are connected by edges, whose presence depends on the random positions of the nodes in the plane. One emerging application is in the field of wireless communications, where radio transmitting stations communicate by radiating electromagnetic waves.

In this talk, first I review several models of random networks for communication that are directly related to continuum percolation. Then, I show some recent results on connectivity of dependent percolation models of interference limited networks. Finally, I introduce the Gupta-Kumar concept of throughput scaling, and argue how this can be obtained as a natural consequence of the RSW theorem in percolation theory.

Kristin Jehring, Graduate student, Mathematics Department, UCSD.

On genetic networks and queueing theory.

Paper on which this talk is based: Arazi, Ben-Jacob, Yechiali, Bridging genetic networks and queueing theory, Physica A, 332 (2004), 585-616.

Dr. Ben Raphael, Postdoc, Computer Science and Engineering, UCSD.

Title: Models of Tumorigenesis.

Professor Jun Liu (Rady School of Management)

Title: Long-Lived Private Information in a Continuous Time Economy: Portfolio Choice, Optimal Consumption, and Utility Gain

Abstract: We study the consumption-investment problem of an agent with a constant relative risk aversion (CRRA) preference function, who possesses private information about the future prospects of a stock. We examine the value of the information to the agent by comparing the utility equivalent with and without the information of the agent. The value of private information to the agent depends linearly on the wealth of agents and decreases with both the propensity to intermediate consumption and risk aversion. Agents with low coefficients of relative risk aversion value private information more highly. Consistent with the empirical literature, the optimal portfolio holdings of informed agents are correlated with expected returns on the risky asset. Highly risk averse informed agents consume a greater fraction of their wealth when they are informed than when they are uninformed, but the opposite is true of agents with low degrees of risk aversion.

Last updated September 28, 2005.