The Whittaker-Shannon-Kotel′nikov sampling theorem and the Paley-Wiener theorem state that an entire function of exponential type (band-limited function) can be reconstructed exactly from its values at discrete sampling points through the cardinal series. For functions that are analytic in a horizontal strip, the [...]]]>

The Whittaker-Shannon-Kotel′nikov sampling theorem and the Paley-Wiener theorem state that an entire function of exponential type (band-limited function) can be reconstructed exactly from its values at discrete sampling points through the cardinal series. For functions that are analytic in a horizontal strip, the cardinal series is still highly accurate, with the approximation error converging to zero exponentially in the length of the sampling interval. In option valuation applications in finance, the characteristic function of the log return process of an asset is naturally analytic. We explore such analyticity and propose Hilbert transform based methods for the valuation of European, American and exotic options and Monte Carlo simulation from such characteristic functions. Millions of option contracts are traded daily on Chicago Board Option Exchange and Chicago Mercantile Exchange. Most of exchange traded options are of American style. The valuation of an American option reduces to an optimal stopping problem. In this talk, we present the Hilbert transform approach for the valuation of American options.

Liming Feng is an associate professor in the Department of Industrial and Enterprise Systems Engineering at the University of Illinois at Urbana-Champaign. He obtained his Ph.D. in Industrial Engineering and Management Sciences from Northwestern University in 2006. His main research interests are in quantitative finance. He is interested in developing theory and efficient computational methods for solving various quantitative finance problems. He is affiliated with the Master of Science in Financial Engineering program at the University of Illinois at Urbana-Champaign.

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Renming Song

Department of Mathematics, University of Illinois

Abstract:

In this talk, I will give a survey of the recent results on spatial central limit theorems for a large class of supercritical branching Markov processes.

]]>Renming Song

Department of Mathematics, University of Illinois

Abstract:

In this talk, I will give a survey of the recent results on spatial

central limit theorems for a large class of supercritical branching

Markov processes.

The talks scheduled this fall will be:

September 10: Georgios Fellouris, Statistics

October [...]]]>

The talks scheduled this fall will be:

September 10: Georgios Fellouris, Statistics

October 8: Andrew Ferguson, MatSE

October 15: Renming Song, Mathematics

November 19: Liming Feng, ISE

November: Daniel Work, CEE

Naira Hovakimyan, MechSE, talking on Planning with monsieur Bezier We will introduce a heuristic planar trajectory-generation framework for multiple vehicles. Desired feasible trajectories are generated using Pythagorean Hodograph Bezier curves that satisfy the dynamic constraints of the vehicles, and guarantee spatial separation [...]]]>
**Grainger Library**, *room 335*. The last IMSE seminar of the year will have

*Naira Hovakimyan*, MechSE, talking on Planning with monsieur Bezier

We will introduce a heuristic planar trajectory-generation framework for multiple vehicles. Desired feasible trajectories are generated using Pythagorean Hodograph Bezier curves that satisfy the dynamic constraints of the vehicles, and guarantee spatial separation between the paths for safe operation. It is shown that the trajectory generation framework can be cast into a constrained optimization problem where a set of (sub)optimal desired trajectories are obtained by minimizing a cost function. To show the efficiency of the algorithm, a simulation example is given, where three fixed-wing Unmanned Aerial Vehicles are following and coordinating along feasible trajectories that are generated by the algorithm.

followed by
*Richard Laugesen*, Math, who, sticking with the French, speaks about

The Laplacian: conjectured spectral bounds

Eigenvalues of the Laplacian have intrigued physicists and mathematicians for hundreds of years. They represent physical quantities such as frequencies of drums and energy levels of quantum particles, while encoding geometric information such as area and perimeter.

I will discuss recent numerical work (not by me) that hints at a possible isoperimetric type inequality for high energies.

(This talk is accessible to anyone who has heard of the trigonometric eigenfunctions of the rectangle or the Bessel eigenfunctions of the disk.)

Alejandro Dominguez-Garcia, ECE, talks on Analysis of Power System Dynamics Subject to Stochastic Power Injections We propose a framework to study the impact of stochastic active/reactive power injections on power system dynamics with a focus on time scales involving electromechanical phenomena. In this framework the active/reactive power injections evolve according to [...]]]>
**Grainger Library**, *room 329*.

*Alejandro Dominguez-Garcia*, ECE, talks on Analysis of Power System Dynamics Subject to Stochastic Power Injections

We propose a framework to study the impact of stochastic active/reactive power injections on power system dynamics with a focus on time scales involving electromechanical phenomena. In this framework the active/reactive power injections evolve according to a continuous-time Markov chain (CTMC), while the power system dynamics are described by the standard differential algebraic equation (DAE) model. The DAE model is linearized around a nominal set of active/reactive power injections; and the combination of the linearized DAE model and the CTMC forms a stochastic process known as a Stochastic Hybrid System (SHS). The extended generator of the SHS yields a system of ordinary differential equations that governs the evolution of the power system dynamic and algebraic state moments. We illustrate the application of the framework through numerical case studies. We close by discussing several extensions to the SHS model that allow capturing additional phenomena of interest, but make

the analysis of the resulting SHS substantially harder.

followed by the analysis of the resulting SHS substantially harder.

This work is joint with Lee DeVille, Sairaj Dhople, and Jiangmeng Zhang.

Moment closure for stochastic hybrid systems in equilibrium

We consider the formalism of stochastic hybrid systems (SHS) in general, and some specific problems motivated by applications. We seek to uncover the evolution of the moments of the process when it is at equilibrium or approaching equilibrium. We present a family of such processes for which classical techniques are ineffective and present a few differing perspectives on how to attack such problems, using both maximum entropy and renewal process ideas. We show that even when every component of the system is linear, the dynamics can be quite complex, and some surprising challenges arise.

This work is joint with Sairaj Dhople, Alejandro Dominguez-Garcia, and Jiangmeng Zhang.

Steven Levinson, ECE, talks on A Geometric Interpretation and Proof of Baum's Algorithm for Estimation of the Parameters of a Probabilistic Function of a Markov Process,A probabilistic function of a finite state Markov process is a stochastic process, , that generates [...]]]>
**Grainger Library**, *room 335*.

*Steven Levinson*, ECE, talks on A Geometric Interpretation and Proof of Baum's Algorithm for Estimation of the Parameters of a Probabilistic Function of a Markov Process,

A probabilistic function of a finite state Markov process is a stochastic process, , that generates a sequence of discrete symbols, , that is neither stationary nor Markovian of any order. The likelihood function for H, , is the probability that generates given that is specified by the parameter . An important problem is to determine given . The classical solution is to find that for which takes its maximum value. This solution yields a consistent estimator of the true process from which was produced. Baum’s algorithm is a computationally efficient way of maximizing with respect to subject to a stochasticity constraint on , . In particular, Baum derives a transformation, such that if , then except if is a critical point of in which case is a fixed point of . Iterative application of to an initial value of the parameter will cause to grow to a local maximum and, asymptotically in the length of , the unique global maximum.

There are two standard proofs of this result, one based on the convexity of and another based on Hölder’s inequality. These proofs are long and rather opaque giving little insight into the remarkable properties of . We give a new, short, and intuitively appealing geometric interpretation and proof of Baum’s algorithm. This proof begins by noticing that has the form of a homogeneous polynomial in with positive coefficients and restricts to lie on a manifold defined as that part of the hyperplane bounded by the first orthant of the space . From these observations it follows that the line segment from to has a positive projection on the gradient of everywhere on the the segment. Thus is, as Baum shows, a growth transformation.

followed by There are two standard proofs of this result, one based on the convexity of and another based on Hölder’s inequality. These proofs are long and rather opaque giving little insight into the remarkable properties of . We give a new, short, and intuitively appealing geometric interpretation and proof of Baum’s algorithm. This proof begins by noticing that has the form of a homogeneous polynomial in with positive coefficients and restricts to lie on a manifold defined as that part of the hyperplane bounded by the first orthant of the space . From these observations it follows that the line segment from to has a positive projection on the gradient of everywhere on the the segment. Thus is, as Baum shows, a growth transformation.

Using various techniques from harmonic analysis and number theory we demonstrate the dependence of the qualitative behavior of the solution of NLS on the algebraic properties of time. To obtain the results we use some recent advances in low regularity theory of dispersive PDE. This is joint work with M. B. Erdogan.

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The basic mechanisms involved in wide variety of engineering and biological processes depend on properties of non integrable distributions define by collections of vector fields. It often happens that the most efficient trajectories define closed subriemannian geodesics. In this talk we will establish some of the more remarkable properties of such trajectories, with emphasis on some especially tractable situations in which the manifold admits the structure of a Riemann symmetric space with the subriemannian structure being “matched” to the structure of the symmetric space in an obvious way. We also show that the geodesic sphere associated with these problems can not be differentiable and discuss the relationship between this fact and feedback stabilization.

Seminar is in **Altgeld Hall 345**, on ** Mondays 1:30pm**.

The program will be comprised of four 2-hour tutorials and invited technical talks.

Details here.

]]>The program will be comprised of four 2-hour tutorials and invited technical talks.

Details here.

]]>Last talks of the semester are by Kay Kirkpatrick, Math, on Heisenberg and XY models of ferromagnets and superconductors, Beyond the Ising model of ferromagnetism, there are the more realistic and challenging XY model (spins in the circle) and the classical Heisenberg [...]]]>
**Grainger Library**, *room 335*.

Last talks of the semester are by

*Kay Kirkpatrick*, Math, on * Heisenberg and XY models of ferromagnets and superconductors,*

Beyond the Ising model of ferromagnetism, there are the more realistic and challenging XY model (spins in the circle) and the classical Heisenberg model (spins are in the 2-sphere). I’ll mention some recent work with Elizabeth Meckes on asymptotics of the total spin for the mean-field Heisenberg model, through and at a phase transition. I’ll also discuss work in progress with student Jack Weinstein on metastability and hysteresis in the XY model, with connections to superconductors.

We give an introduction to Banach space tools in around

RIP and indicate how one can reduce the number of

measurements by optimizing over certain family of Banach spaces.

The structure of hierarchical networks in biological and physical systems has long been characterized using the Horton-Strahler ordering scheme. The scheme assigns an integer order to each edge in the network based on the topology of branching such that the order increases from distal parts of the network (e.g., mountain streams or capillaries) to the “root” of the network (e.g., the river outlet or the aorta).

However, Horton-Strahler ordering cannot be applied to networks with loops because they they create a contradiction in the edge ordering in terms of which edge precedes another in the hierarchy. In this talk I will present a generalization of the Horton-Strahler order to weighted planar reticular networks, where weights are assumed to correlate with the importance of network edges, e.g., weights estimated from edge widths may correlate to flow capacity. The new method assigns hierarchical levels not only to edges of the network, but also to its loops, and classifies the edges into reticular edges, which are responsible for loop formation, and tree edges. I will show that the sensitivity of the hierarchical levels to weight perturbations can be analyzed in a rigorous way. I will also discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.

There will be two interlinked talks, by Marius Junge, Math, and Kiryung Lee, ECE on RIP off Compressed Sensing We give an introduction to Banach space tools in around RIP and indicate how one can reduce the number of measurements by optimizing [...]]]>
**Grainger Library**, *room 335*.

There will be two interlinked talks, by *Marius Junge*, Math, and *Kiryung Lee*, ECE on

* RIP off Compressed Sensing
*

We give an introduction to Banach space tools in around

RIP and indicate how one can reduce the number of

measurements by optimizing over certain family of Banach spaces.