 |
The City University of
New York Applied Mathematics Seminar |
 |
Department
of Mathematics Graduate Center of CUNY 365 Fifth Avenue New York, NY 10016 Fridays 4:00PM-5:00PM Room: 3209 Organizers: Faranak Pahlevani and Hüseyin Yüce |
| Spring 2008 |
| May 16:
Tobias Schaefer (The Graduate Center and
College of Staten Island of CUNY) Mini Lecture Series in Financial Mathematics PLEASE NOTE THE ROOM CHANGE: 9204 Lecture IV: Continuous time models, risk-neutral measures and deriving the Black-Scholes formula using the Girsanov Theorem In this last talk of the mini lecture series in Financial Mathematics, I will show how the Black-Scholes formula can be derived using the Girsanov Theorem. For this purpose, I will briefly review the basic concepts of Brownian motion and Ito calculus and then show how the Girsanov Theorem provides a change of measure to derive the Black-Scholes formula. If time permits, I will comment on the connections to partial differential equations and path integrals. |
| May 9:
Philip Ording (Medgar Evers College - CUNY)
"Irrational
thoughts should be followed absolutely and logically" — Sol LeWitt In the 1960s, as notions of artistic uniqueness, spontaneity, and skill were being questioned, artists began developing strategies that emphasized ideas of reproducibility, computation, and variation. Several artists began explicitly using mathematical systems in their practice. This talk will highlight the role mathematical thinking has played in this tradition by investigating the work of several influential artists, three of whom I have had the opportunity to work with as a mathematics consultant. The questions I am interested in addressing include: How do mathematical models function in an artistic context? And how might art possibly raise new questions for mathematics itself? |
| May 2:
Kamyar Malakuti (New Jersey Institute of
Technology) The Numerical Analysis of Singular Solutions to Partial Differential Equations Singularities often occur in solutions to partial differential equations; important examples include the formation of shock fronts in hyperbolic equations and self-focusing type blow up in nonlinear parabolic equations. Information about formation and structure of singularities can have significant theoretical importance. For example, the question of singularity formation for the 3D Euler equations of incompressible inviscid flow has important implications in turbulence, and has been an open problem for more than a century. In this talk, we present a new method for the numerical analysis of complex singularities in solutions to partial differential equations. In the method, we analyze the decay of Fourier coefficients using a numerical form fit to ascertain the nature of singularities in two and three-dimensional functions. Our results generalize a well-known method for the analysis of singularities in one-dimensional functions to higher dimensions. As an example, we apply this method to analyze the complex singularities for the 2D inviscid Burger equations. |
| Apr. 25: No
meeting this week. GC is closed |
| Apr. 18:
Christina Mouser (Medgar Evers College - CUNY) Maintaining Phase of the Crustacean Tri-Phasic Pyloric Rhythm We construct and analyze a model network of the pyloric rhythm of the crustacean stomatogastric ganglion consisting of an oscillator neuron that inhibits two reciprocally inhibitory follower neurons. We derive analytic expressions that determine the phase of firing of the follower neurons with respect to the oscillator. An important aspect of the model is the inclusion of synapses that exhibit short-term synaptic depression. We show that these types of synapses allow there to be a complicated relationship between the intrinsic properties of the neurons and the synapses between them in determining phase relationships. Our analysis reveals the circumstances and ranges of cycle periods under which these properties work in concert with or independently from one another. In particular, we show that phase maintenance over a range of oscillator periods can be enhanced through the interplay of the two follower neurons if the synapses between these neurons are depressing. Since our model represents the core of the oscillatory pyloric network, the results of our analysis can be compared to experimental data and used to make predications about the biological network. |
| Apr. 11:
Onur Baser (University of Michigan - Ann Arbor) Time: 3:00PM-4:00PM, Room: 3209 Estimation of censored medical cost
Health care inflation is a concern in many industrialized countries. One
response is search for cost effective therapies which requires proper
analysis of treatment cost. Common problem with medical cost data is
censoring and statistical properties of estimating medical cost from a
censored data is not well developed. In this paper, we applied the
inverse probability weighted least-squares method to predict censored
total medical cost. Since survival time and medical costs may be subject
to right censoring and therefore are not always observable, the ordinary
least-squares approach cannot be used to assess the effects of certain
explanatory variables. Inverse probability weighted least-squares
estimation provides consistent asymptotic normal coefficients with
easily computable standard errors. A test is derived to compare the
differences between the coefficients estimated by the ordinary
least-squares approach and the inverse probability weighted
least-squares estimation. A study on the medical cost of lung cancer is
used as an application of the method. Time: 4:00PM-5:00PM, Room: 3209 Mini Lecture Series in Financial Mathematics (CANCELED) Lecture V: Transition to Continuous-Time Finance Continuation of Lecture IV |
| Apr. 4:
Holly Carley (New York City College of
Technology-CUNY) The Strong-coupling limit for the ground state of a particle harmonic oscillator interaction We will discuss a toy model for the polaron which essentially is a particle coupled with a harmonic oscillator. I will focus on one or two forms for the coupling and how one can learn something about the bottom of the spectrum. |
| Mar. 28:
Jesenko Vukadinovic (The Graduate Center and
College of Staten Island - CUNY) Title & Abstract : TBA |
| Mar. 21:
No meeting
this week. GC is closed |
| Mar. 14:
Bernd Kawohl
(University of Cologne, Germany) Time: 4:30 PM - 5:30 PM Variational versus PDE-based Approaches in Mathematical Image Processing In mathematical image processing we are often presented with amazing examples of image enhancement algorithms. Yet, when applied to different noisy images, they can produce unwanted effects. The analysis of such algorithms lags behind their intuitive development. Two essentially different models have found wide recognition: a variational approach according to Mumford and Shah and an approach via nonlinear diffusion equations. One of these equations is nonparabolic and was suggested by Perona and Malik. In my talk I will point out a connection between these two seemingly unrelated approaches and explain some connections with total variation flow. |
| Mar. 14:
Enea Parini
(University of Cologne,
Germany) Time: 4:00 PM - 4:30 PM Some results about Cheeger sets: uniqueness and non-uniqueness Given a bounded domain $\Omega\subset R^n$, the Cheeger problem consists of finding a set $E\subset\overline{\Omega}$ minimizing the ratio perimeter-area among all subsets of $\overline{\Omega}$. Such a set is a \emph{Cheeger set} for $\Omega$. In spite of its geometrical formulation, this problem finds application in questions related to the first eigenvalue and eigenfunction of the $p$-Laplace operator; in particular, it is known that the first eigenfunctions of the $p$-Laplacian converge, as $p\to 1$, to a function such that almost every of its level sets is a Cheeger set for $\Omega$. In my talk I will present some results about Cheeger sets and discuss in particular uniqueness questions. |
|
Mar. 7: Olympia Hadjiliadis
(Brooklyn College - CUNY) Mini Lecture Series in Financial Mathematics Lecture IV: Transition to Continuous-Time Finance (CANCELED) We will begin by the introduction of Markov Processes in discrete-time and the use of the Markov property in the derivation of recursive formulae for the valuation of derivative securities. We will then proceed to introduce the most fundamental continuous-time process, the Brownian motion process. We will define the stochastic integral and provide Ito's formula. We will introduce the Radon-Nikodym derivative process first in discrete and then in continuous time and the notion of change of measure through the Cameron-Martin-Girsanov theorem. We will then introduce the notion of self-financing strategies and derive the Black-Scholes formula for pricing derivative securities as well as hedging them. |
|
Feb. 29: Olympia Hadjiliadis
(Brooklyn College - CUNY) Mini Lecture Series in Financial Mathematics Lecture III: The Fundamental Theorem of Asset Pricing We will begin by introducing the notion of a probability measure. We will then introduce special families of processes such as martingales, Markov processes and the Radon-Nikodym derivative process all in discrete time. We will proceed to show that the discounted price of any asset is a martingale under a special measure called the risk-neutral measure. We will thus introduce the fundamental theorem of asset pricing that states that if there is a unique risk-neutral measure then there is no arbitrage and use the risk neutral pricing formula to price assets. Moreover we will demonstrate how to use the Markov property to algorithmically price assets through recursive equations and discuss their continuous time analogues. Finally, we will demonstrate how we can change probability measures (from actual to the risk-neutral) through the Radon-Nikodym derivative process. |
|
Feb. 22: Tobias Schaefer (The Graduate Center and
College of Staten Island of CUNY) Mini Lecture Series in Financial Mathematics Lecture II: The Black-Scholes Formula for European Call Options In this second talk of the mini lecture series on Financial Mathematics I will consider the transition from a stock process on discrete binomial tree to a continuous process. For a particular case, the Black-Scholes model, we will be able to derive the Black-Scholes formula for European call options. |
| Feb. 15:
Cyrill Muratov (New Jersey Institute of
Technology) Self-induced stochastic resonance: How new non-random behaviors can arise from the action of noise It is usually assumed that when a dynamical system is subjected to small random perturbations, its behavior remains essentially unchanged, apart from noise appearing on top of the otherwise deterministic dynamics. That this is not always the case is dramatically demonstrated by the phenomenon of stochastic resonance, whereby a small but finite amount of noise produces coherent phase locking between an applied periodic signal and the system's dynamical response. Perhaps even more surprisingly, the addition of small noise may produce new coherent behaviors that are fundamentally absent in the dynamics of the noise-free system. In other words, noise can actually play a constructive role in creating dynamics that are essentially non-random. This talk will present an overview of one robust mechanism by which such dynamics emerge out of noise which we termed self-induced stochastic resonance (SISR). I will demonstrate SISR in action for a range of systems whose common dynamical feature is excitability. I will show that both extrinsic and intrinsic noise in an excitable system may result in the onset of quasi-deterministic limit cycle oscillations, while in an excitable media the same mechanism produces traveling wave pacemakers. I will argue that one needs to re-examine the role of small noise in modeling the dynamics of complex dynamical systems. |
| Feb. 8:
Tobias Schaefer (The Graduate Center and
College of Staten Island of CUNY) Mini Lecture Series in Financial Mathematics Lecture I: Option Pricing on Binomial Trees This lecture is the beginning of a series of approximately five lectures that intend to give a brief introduction to the mathematical theory of option pricing. In this first introductory lecture, I will first explain the basic types of options and show how options can be priced using arbitrage principles. Then I will discuss in detail the binomial branch model and the binomial tree model for pricing European call and put options. |
|
Feb. 1: Organizational Meeting |
| Fall 2007 |
Dec. 14: Paul G. Ranky (New Jersey Institute of Technology) Multi-Variable Optimization Challenges Described by N-Dimensional Continuously Differentiable, Dynamic Vector-Functions, in the Digital Product Design and Digital Manufacturing Domain This research seminar will highlight the complexity of Multi-Variable Optimization Challenges Described by N-Dimensional Continuously Differentiable, Dynamic Vector-Functions, in the Digital Product Design and Digital Manufacturing Domain, as well as show some promising directions to come up with realistic solutions. In simple terms, Digital Design and Digital Manufacturing means to design, simulate, manufacture and test first on the computer screen, in the digital domain, and then build physically only when everything is working fine, and optimized. In our rapidly changing global and competitive world, the key is to continuously learn, innovate, and develop new, efficient and effective products, processes and services, that are successful. Since our systems are increasingly complex, with Digital Design and Digital Manufacturing we can create products, that work well and satisfy customers first time. Global challenges are pushing our industry and society to continuously improve, to become leaner, more cost effective and efficient, higher quality, and innovative, therefore we need to research and develop new optimization methods and tools in the Digital Design and Digital Manufacturing domain. This is why the above described research topic is essentially important. About Paul G Ranky: http://www.cimwareukandusa.com/aboutpgr.htm Some of his publications at: http://www.cimwareukandusa.com |
Dec. 7: Majid Hosseini (New Paltz College-SUNY) Lower Bounds for the Spectral Gap of Convex Domains It is known that the difference (gap) between the first two Dirichlet eigenvalues of a convex domain which is symmetric with respect to both coordinate axes is at least as large as the gap of the smallest oriented rectangle containing that domain. I will show how to obtain explicit lower bounds for the difference of the gaps. |
Nov. 30: Dana Draghicescu (Hunter College-CUNY) Modeling Probability Distribution Functions and Quantiles of Non-stationary Processes Modeling probability distribution functions (pdf's) and quantiles for dependent data is an important topic in theoretical and practical statistics, with a wide area of applicability. A quantile is a value above which a key quantity will occur a certain percent of the time. For instance, high pollution levels may cause severe respiratory problems, and large precipitation amounts can damage the environment. Thus we might want to know, say, an ozone level that is so high it only occurs 1% of the time. While there exist a large body of research on modeling trends (mean functions), comparatively little is known about temporal variations of pdf's or of extreme quantiles for environmental processes. In this talk I will analyze non-parametric estimators of pdf's and quantiles in a wide class of stochastic processes, allowing for non-stationarity and/or non-Gaussianity, as well as processes with infinite variance. I will discuss asymptotic properties and data-driven algorithms for selection of smoothing parameters. The methodology will be illustrated on Monte Carlo simulations and on applications to precipitation and air pollution data. |
Nov. 23: No meeting this week. |
Nov. 16: Urmi Ghosh-Dastidar (New York City College of Technology-CUNY) Understanding Bird Flu Propagation Bird flu is first located in Hong Kong in 1997 when a three-year old boy died from a strange influenza virus. This virus spreads directly from bird to humans without any other intermediate medium. In late 2003 and early 2004, eight countries of Asia (Cambodia, China, Indonesia, Japan, Laos, South Korea, Thailand, and Vietnam) are affected by the outbreak of this strange influenza H5N1 virus. Millions of birds died from this disease during that time. In June 2004, new cases of influenza H5N1 occurred among poultry in several Asian countries (Cambodia, China [Tibet], Indonesia, Kazakhstan, Malaysia, Mongolia, Russia [Siberia], Thailand, and Vietnam). Outbreaks of this viral infection are also noticed in Turkey, Romania, Ukraine, China, Croatia, and Mongolia. By July 14, 2006, more than 230 people worldwide were diagnosed with this virus (reported by the World Health Organization, WHO). Several of them died from this viral attack. Since people were never affected by this virus strain before, this incident raised a substantial amount of concerns. Here, a predator-prey model is modified to incorporate an SI (susceptible-infectious-death) infection among bird population and an SIR (susceptible-infectious-recovered or death) infection among human population in an endemic environment. We assumed that both species could be infective; however, only birds transmit the disease to humans or within its own community by contacts. This is reasonable at this point since human-to-human transmission is not yet proven. Stability of the equilibrium points is studied by linearizing the system about the points. Numerical simulations show that irrespective of initial infected population, if the contact rate is high, the disease prevail. |
Nov. 9: Yevgeniy Milman (Hunter College-CUNY) A Buckling Problem for Graphene Sheets We develop a continuum model that describes the elastic bending of a graphene sheet interacting with a rigid substrate by van der Waals forces. Using this model, we study a buckling problem for a graphene sheet perpendicular to a substrate. After identifying a trivial branch, we combine analysis and computation to determine the stability and bifurcations of solutions along this branch. Also presented are the results of atomistic simulations. The simulations agree qualitatively with the predictions of our continuum model but also suggest the importance, for some problems, of developing a continuum description of the van der Waals interaction that incorporates information on atomic positions. The research is based on: Research Experience for Undergraduates (REU) program at the University of Akron. |
Nov. 2: Tobias Schaefer (The Graduate Center and College of Staten Island of CUNY) Propagation of ultra-short optical pulses in nonlinear and random media The basic model for pulse propagation in optical media is the cubic nonlinear Schroedinger equation (NLSE). In the regime of ultra-short pulses, however, the basic assumption made in the derivation of the NLSE from Maxwell's equations as a slowly varying amplitude approximation is not valid anymore. In the talk, I will give first a sketch of the derivation of the NLSE of Maxwell's equations and then discuss applications of the basic model in the context of fiber optics. Then, I will present a different approximation, the short-pulse equation (SPE) and discuss the validity as well as the mathematical properties. In the last part of the talk I will show how to extend the derivation of the SPE to the case of random variations of the media. |
Oct. 26: Marcello Lucia (College of Staten Island-CUNY) Symmetry and Uniqueness of Steady States for Some Chemotaxis System We consider a class of parameter-dependent, nonlocal elliptic boundary value problems, that appear in context of the steady state problem for some chemotaxis systems. If the appearing parameter is less than some critical value, we establish some uniqueness results for a class of solutions that exhibit some special symmetry. |
Oct. 19: No meeting this week. |
Oct. 12: Faranak Pahlevani (Medgar Evers College-CUNY) Parametric Sensitivity Analysis and Computations for Fluid Models In recent years, Parametric Sensitivity Analysis has become a very important tool in analyzing fluid behavior. In this presentation we introduce the sensitivity of the computed flow solution using Large Eddy Simulation (LES) models with respect to the variation of the selected cut-off length scale. We demonstrate the analysis by using the Sensitivity Equation Method and provide numerical assessments to illustrate applications of the parametric sensitivity computations. |
| Oct. 05: Hüseyin Yüce (New York City College of Technology-CUNY) On the lowest eigenvalues of biharmonic equations and Applications to Plates
Biharmonic equations have applications in fluid dynamics, buckling and
vibration of plates which has extensive applications in civil,
mechanical, aerospace, and material engineering as well as vibration of
piezoelectric and acoustic devices. The biharmonic eigenvalue problem
has a general analytical solution in a circular domain given by linear
combinations of the Bessel functions. However, the difficulty in finding
solutions arises when the domain is no longer circular. |
Sept. 28: Organizational Meeting |